UNIVERSITEIT GENT
FACULTEIT ECONOMIE EN BEDRIJFSKUNDE
ACADEMIEJAAR 2011 – 2012
LINKING THE VEHICLE ROUTING
PROBLEM AND THE CREW
SCHEDULING PROBLEM: PAST,
PRESENT AND FUTURE RESEARCH
Masterproef voorgedragen tot het bekomen van de graad van
Master of Science in de Bedrijfseconomie
Jordy Batselier
onder leiding van
Prof. dr. Broos Maenhout
PERMISSION
Ondergetekende verklaart dat de inhoud van deze masterproef mag geraadpleegd en/of
gereproduceerd worden, mits bronvermelding.
Undersigned declares that the contents of this thesis may be consulted and/or reproduced,
provided acknowledgment.
Jordy Batselier
II
Acknowledgements
The thesis in front of you is written on behalf of Ghent University, the university at which I am
proud to be a student for almost 6 years now. In my first 5 years I successfully waded through
the Engineering program. Although specialized in the field of construction, I also chose to
broaden my purely technical horizon and signed up for a minor in Business Administration. It
was there that I first encountered and learned to greatly appreciate the area of Operations
Research. Also my interest in economics was strongly sparked, which was the impetus to begin
a complementary study in Business Economics. When looking for an interesting subject for my
new master’s thesis, I was immediately attracted by the topics presented by the department of
Operations Management. The eventual subject of this thesis was clearly my favorite given my
previously developed interest in transportation, and I therefore want to thank my promotor prof.
dr. Broos Maenhout very much for willing to adapt the original subject for me (since it was
actually intended for a two-year Business Engineering thesis) and give me the necessary
guidance. The writing of this thesis has also further contributed to my affection for Operations
Research and thus played a significant role for the endeavor of commencing a PhD at the
department of Operations Management.
I would also like to thank my parents and my girlfriend’s parents for ensuring the necessary
peace and quiet while writing my thesis during weekends, my friends for asking me about my
thesis so that the discussions that followed could lead to new insights on my part, and last but
certainly not least my girlfriend Lynn for taking care of me, supporting me and (sometimes)
putting up with me during the busy days of writing this thesis.
Jordy Batselier
Ghent, May 2012
III
Contents
Acknowledgements ................................................................................................................... III
Abbreviations ............................................................................................................................ VI
Definitions ................................................................................................................................ VII
List of figures ........................................................................................................................... VIII
List of tables ............................................................................................................................ VIII
0.
Introduction ......................................................................................................................... 1
1.
Past ..................................................................................................................................... 5
1.1.
Individual routing and scheduling problems .................................................................. 5
1.1.1.
The Vehicle Routing Problem (VRP) ..................................................................... 6
1.1.2.
The Vehicle Scheduling Problem (VSP) ................................................................ 8
1.1.2.1.
Single depot case........................................................................................... 8
1.1.2.2.
Multiple depot case ........................................................................................ 9
1.1.3.
The Crew Scheduling Problem (CSP) ................................................................... 9
1.2.
The planning process in transportation companies ..................................................... 11
1.3.
The scheduling problem ............................................................................................. 16
1.3.1.
Traditional approach ........................................................................................... 17
1.3.2.
Defining the Vehicle and Crew Scheduling Problem (VCSP) ............................... 17
1.3.2.1.
Public transport context ................................................................................ 18
1.3.2.2.
Distribution context ....................................................................................... 19
1.3.3.
1.4.
Earlier literature on the VCSP ............................................................................. 19
1.3.3.1.
Partial integration ......................................................................................... 20
1.3.3.2.
Complete integration .................................................................................... 21
Combining routing and scheduling: the Vehicle and Crew Routing and Scheduling
Problem (VCRSP) ................................................................................................................. 22
2.
Present.............................................................................................................................. 23
2.1.
Categorization of recent VCSP approaches ............................................................... 23
2.1.1.
Categorization according to practical problem ..................................................... 24
2.1.1.1.
Type of transportation problem..................................................................... 24
IV
2.1.1.2.
Mode of transportation ................................................................................. 25
2.1.1.3.
Number of depots......................................................................................... 25
2.1.1.4.
Objectives .................................................................................................... 25
2.1.1.5.
Size/practicality of the problem..................................................................... 26
2.1.1.6.
Degree of urbanization ................................................................................. 27
2.1.1.7.
Regularity of the timetable ............................................................................ 27
2.1.1.8.
Admission of changeovers ........................................................................... 28
2.1.2.
Categorization according to solution method ....................................................... 28
2.1.2.1.
Degree of integration .................................................................................... 28
2.1.2.2.
Degree of network segmentation.................................................................. 29
2.1.2.3.
Model ........................................................................................................... 29
2.1.2.4.
Algorithm ...................................................................................................... 30
2.1.2.5.
Dynamism of the solution approach ............................................................. 30
2.1.3.
Actual categorization ........................................................................................... 31
2.1.4.
Conclusions ........................................................................................................ 38
2.2.
Evolution of the VCRSP ............................................................................................. 49
2.2.1.
Overview of existing approaches ......................................................................... 50
2.2.2.
Conclusions ........................................................................................................ 52
3.
Future research ................................................................................................................. 57
4.
General conclusion............................................................................................................ 65
Bibliography .............................................................................................................................. IX
Appendix A .............................................................................................................................. A.1
Appendix B .............................................................................................................................. B.1
Appendix C............................................................................................................................. C.1
Appendix D............................................................................................................................. D.1
Appendix E .............................................................................................................................. E.1
Appendix F .............................................................................................................................. F.1
Nederlandse samenvatting ................................................................................................... SV.1
V
Abbreviations
We provide an overview of the abbreviations which will been used throughout the thesis. In
alphabetical order:
CBN
Connection-Based Network
CG
Column Generation
CSP
Crew Scheduling Problem
CVRP
Capacitated Vehicle Routing Problem
EA
Evolutionary Algorithm
GDA
Great Deluge Algorithm
GRASP
Greedy Randomized Adaptive Search Procedure
ICSP
Independent Crew Scheduling Problem
LNS
Large Neighborhood Search
MDVCSP
Multiple Depot Vehicle and Crew Scheduling Problem
MDVRP
Multiple Depot Vehicle Routing Problem
MDVSP
Multiple Depot Vehicle Scheduling Problem
MIP
Mixed-Integer Programming
PAP
Personnel Assignment Problem
RCSP
Resource Constrained Shortest Path
RFS
Road Feeder Service
RRT
Record-to-Record Travel
SA
Simulated Annealing
SDVCSP
Single Depot Vehicle and Crew Scheduling Problem
SDVRP
Single Depot Vehicle Routing Problem
SDVSP
Single Depot Vehicle Scheduling Problem
TS
Tabu Search
TSN
Time-Space Network
VCRSP
Vehicle and Crew Routing and Scheduling Problem
VCSP
Vehicle and Crew Scheduling Problem
VCSPTW
Vehicle and Crew Scheduling Problem with Time Windows
VRP
Vehicle Routing Problem
VRPPD
Vehicle Routing Problem with Pickup and Delivery
VRPTW
Vehicle Routing Problem with Time Windows
VSP
Vehicle Scheduling Problem
VI
Definitions
We provide an overview of the most important terms, also introduced by (Huisman, 2004) and
(Steinzen, 2007), which will been used throughout the thesis. In alphabetical order:
changeover
change of vehicle of a crew
compatible trips
trips that can be executed consecutively by the same vehicle
deadhead (trip)
vehicle movement without carrying passengers or products, such as
movements from a depot to the first trip and from the last trip to a depot or
idle times outside the depot
depot
maintenance and storage facility for vehicles that are not in use for some
time
duty
sequence of pieces of work that can be assigned to the same crew
piece (of work)
sequence of tasks without a break that can be performed without
interruption by a single crew staying with the same vehicle
relief point
location and time where a crew may change vehicles
task
part of a vehicle block between two consecutive relief points that can be
assigned to a crew
timetable
item that defines the set of trips that are used to carry passengers or
deliver products
trip
vehicle movement with passengers or products specified by start and end
locations and times
vehicle block
sequence of compatible trips and deadheads, starting and ending in the
depot, that can be executed by a single vehicle
vehicle type
a set of vehicles with the same capacity, speed and equipment
VII
List of figures
Figure 1:
Planning process in transportation companies
p. 12
Figure 2:
Planning process in transportation companies (extended)
p. 16
List of tables
Table 1:
Summarized categorization of recent VCSP approaches
p. 33-37
Table 2:
Two-dimensional classification of relevant VCRSP papers
p. 51
VIII
0. Introduction
Vehicle routing and crew scheduling are perhaps the most important problems faced in today’s
transport companies, since vehicles and especially crews are the main resources for providing
services to their customers. This thesis actually covers the whole of the planning process in
transportation, comprising both routing and scheduling, more particular the Vehicle Routing
Problem (VRP), the Vehicle Scheduling Problem (VSP) and the Crew Scheduling Problem
(CSP)1. Slightly bluntly, we could say that the VRP constructs routes so that a number of
customers can be serviced with a fleet of vehicles, while the VSP assigns vehicles to cover
these routes and the CSP allocates crews to operate the vehicles (and routes). For every one of
these planning problems the main objective is to minimize the total costs, taking into account
certain constraints. Foregoing definitions of course do not show the total scope and complexity
of the subjects, which we will try to clearly and completely describe throughout this thesis. A
very important aspect in this regard is the consideration of the possibilities for integration, within
the scheduling, so between VSP and CSP leading to an integrated Vehicle and Crew
Scheduling Problem (VCSP)2, as well as on a higher level, that is between routing and
scheduling and thus defining a Vehicle and Crew Routing and Scheduling Problem (VCRSP).
The title of this thesis, Linking the Vehicle Routing Problem and the Crew Scheduling Problem:
Past, Present and Future Research, does in fact reflect every one of these aspects of the
transportation planning process. The title talks about the linking of the VRP and the CSP,
without mentioning the VSP. Yet, this is not an incompleteness. Namely, when using a
traditional sequential scheduling approach, the VSP always has to be solved first before being
able to start with the CSP, which uses the results of the VSP. In other words, solving the CSP
actually implies solving the VSP, so that the title is indeed not incomplete. Because of this
incorporation of the VSP in the CSP, it is a logical proceeding to also discuss the VCSP.
Moreover, the linking of the VRP and the CSP can be understood as the integration between
routing and scheduling and thus announces the discussion of the VCRSP.
However, the VCRSP does not emerge in every type of transportation environment, but only in
a distribution context and therefore not in public transport. For a first general understanding,
public transport companies can be either bus, tram, metro, train or airline operators which are
concerned with transporting passengers, whereas distribution companies (e.g. freight transport,
1
These and other abbreviations will be used throughout the entire thesis, mainly with the aim of not
overcharging the text. We refer to the introductory section Abbreviations. The reader should also take a
good look at the other introductory section Definitions, because some of the terms described there will be
used in the initial part of the thesis, before the definition is given in the text.
2
Simultaneous Vehicle and Crew Scheduling Problem or just Vehicle and Crew Scheduling Problem are
synonyms for integrated Vehicle and Crew Scheduling Problem.
1
delivery services, mail distribution) deliver products to their customers. The distinction between
public transport problems and distribution problems will be an important aspect throughout the
entire thesis. To the best of our knowledge, we are the first to make clear use of this natural
division and thus also to indicate the impact it has on the general approach of a particular
problem. Other papers do mention the existence of both contexts, but do not explicitly point to
the implications for a specific problem situated in one or the other context.
Of course, the above is not the only contribution of this thesis. In the first place, the thesis
comprises a general overview of all planning problems which can arise in a transportation
company and discusses them individually and well structured, defining links between the
problems afterwards. This is different to other papers that nearly always focus on a single
problem like the VRP or the VCSP, which complicates the situating within the bigger picture of
the entire planning process. We do provide such a general situating. A well structured overview
does indeed appear useful, because it is not unthinkable for a not so experienced reader to no
longer see the wood for the trees when reading many publications all discussing specific but
somehow connected problems. Own experiences only confirm these findings. We therefore try
to provide some sort of guide for reading transportation planning literature. Moreover, through
the use of an overall view, interesting and still unidentified subject for future research could be
discovered. And even more importantly, it enables us to find a conclusive answer to the central
research question of this thesis: ‘Is it (always) advantageous to use an integrated approach for
planning problems in transportation?’ There are two aspects to this central research question,
being (1) the integration of vehicle and crew scheduling and (2) the integration of routing and
scheduling. In other words, we try to determine whether or not the VCSP and the VCRSP,
respectively, are beneficial compared to the traditional approaches. To the best of our
knowledge, this is one of the first papers – if not the first – to try to answer these questions in a
very general and objective way, considering multiple relevant publications all together which is
very different from conventional papers in which only the self-proposed method is evaluated.
Our goal is to find a more generalized and conclusive answer about the potential of integration
as a methodology.
Maybe most importantly, this thesis goes further than the mere listing of existing papers. It also
does something with it. To the best of our knowledge, it is the first work to introduce an
extensive categorization procedure for VCSP approaches. This provides a much needed orderly
overview for a quite highly dispersed research area facing a recent booming of publications.
The reader (e.g. a planner of a transport company) should thus be able to find the most relevant
approach for the particular scheduling problem he is dealing with, among all the available VCSP
publications. The proposed categorization is also of use for purely research oriented issues. It
can be the basis for identifying gaps within the research area, in that way providing a new
2
perspective on the opportunities and needs for future research. Although not yet extensively
elaborated, an impetus for categorizing VCRSP approaches is also given.
It is beyond the scope of this thesis to explicitly present mathematical formulas and thus fully
describing the proposed models and algorithms. For this we always refer to the concerning
papers. A clear but brief description of the most important models and algorithms will be
provided though. However, this will never be detailed enough to be able to make propositions
for the adaptation of specific models or algorithms. It is also not our objective to identify the
‘best’ method to tackle a certain problem. We confine ourselves to mentioning which methods
are ‘good’.
Another limitation concerns the maximum allowed volume of this thesis. Since we attempt to
give a broad overview of the planning process in transportation, not all aspects can be
extensively treated. We especially limit the literature review for non-integrated problems and
less recent integrated approaches, for obvious reasons. Moreover, some less essential parts of
methods and definitions will not be presented in the main body. Those are added as
appendices, mostly literally quoting from existing papers. The imposed time limitation (together
with the volume restriction) made it impossible to deeply delve into all possible aspects of the
planning problem. Nevertheless, the presented quotes are selected to perfectly complement3
the discussion held.
The organization of the thesis can be inferred from the title, for we introduce the chapters Past,
Present and Future research. For several reasons, the year 2004 was imposed as the boundary
between Past and Present. Every chapter is further subdivided into sections. At the end, we
summarize the most important elements and conclusions of this thesis in the closing chapter
General conclusion. We now briefly describe the content of the chapters composing the main
body.
In the chapter Past we first discuss the individual routing and scheduling problems, so the VRP,
the VSP and the CSP. We then situate these individual problems within the larger whole of the
planning process in transportation companies, also elaborating on the distinction between public
transport and distribution context. Afterwards, the scheduling problem is described, first the
traditional sequential approach (VSP first - CSP second) and then the integrated VCSP
approach, defined in both public transport and distribution context. Subsequently, the most
3
The term complement indicates that the thesis can be understood without the reading of the
appendices, which are thus not an inherent part of the thesis, however providing useful extra information
and insights.
3
important earlier (pre-2004) literature on the VCSP is briefly reviewed4. To close the chapter,
the concept of combining routing and scheduling, thus the VCRSP, is introduced.
The chapter Present starts with the most important part, being the categorization of recent
VCSP approaches. Of course, we first define the categorization criteria, which either relate to
the practical problem considered or the solution method used for it. Once this is done, we can
proceed to the actual categorization of the VCSP approaches. Next, conclusions are derived
from the presented categorization, including an answer to the first aspect of the central research
question. The second part of the chapter is dedicated to the VCRSP, beginning with an
overview of the existing approaches and providing an impetus for categorization (after the
image of that for the VCSP). Finally, conclusions are stated for the VCRSP, now comprising an
answer to the second aspect of the central research question.
In the chapter Future research, we orderly list and discuss the research topics deduced from
the conclusions for the VCSP and the VCRSP in the previous chapter. The relevance of the
topics is substantiated by literal recommendations for future research presented by other
authors.
Following the logical time scale, we start our exposition with the chapter Past.
4
When literature on a certain problem is presented in this thesis, this will always be done in a
chronological manner, so the evolution over time of the can be tracked. If for some reason we should
deviate from the chronological ordering, this will always be declared.
4
1. Past
In order to place the following discussion into the right perspective, let us start with a definition
of what is meant by the term past. Simply stated, the past comprises all the papers that were
published before 2004. These papers and the methods they discuss are thus regarded as
somewhat older or less recent. Now why did we choose the year 2004 (more precisely January
1st of this year) as the transition point between past and present? There are several reasons for
this.
As will become clear in section 1.3.3, only a few publications in the pre-2004 operations
research literature address a simultaneous approach to vehicle and crew scheduling, which is a
very important topic in this thesis and therefore makes a logical criterion for dividing our time
line. Moreover, none of those publications makes a comparison between simultaneous and
sequential scheduling. Hence, they do not provide any indication of the benefit of a
simultaneous approach. This means that the central research question of this thesis (‘Is it
(always) advantageous to use an integrated approach for planning problems in transportation?’)
was not considered until 2004 ((Huisman, 2004) was the first). And most importantly, the
quantity (and quality) of publications on the VCSP increased rapidly since 2004. Of course, we
do not only discuss the integration of vehicle and crew scheduling in this thesis, the integration
of the routing aspect and the scheduling aspect is also considered. Papers on the latter topic
are however even more scarce and the relevant ones date from 2006 (Hollis et al., 2006) at the
earliest. Therefore, the choice of 2004 as transition point was deemed logical.
This chapter is organized as follows. In section 1.1 we first discuss the individual routing and
scheduling problems, so the VRP, the VSP and the CSP. Then, in section 1.2, we situate these
individual problems within the larger whole of the planning process in transportation companies,
also elaborating on the distinction between public transport and distribution context. In section
1.3, the scheduling problem is described, first the traditional sequential approach (VSP first CSP second) and then the integrated VCSP approach, defined in both public transport and
distribution context. We also briefly review the most important earlier (pre-2004) literature on the
VCSP. Finally, the concept of combining routing and scheduling, thus the VCRSP, is introduced
in section 1.4.
1.1.
Individual routing and scheduling problems
In this section, we will give a short description of the Vehicle Routing Problem (VRP), the
Vehicle Scheduling Problem (VSP) and the Crew Scheduling Problem (CSP) as separate
5
problems. Slightly bluntly, we could say that the VRP constructs routes so that a number of
customers can be serviced with a fleet of vehicles, while the VSP assigns vehicles to cover
these routes and the CSP allocates crews to operate the vehicles (and routes). For every one of
these planning problems the main objective is to minimize the total costs, considering certain
constraints. A more accurate description follows in the next subsections, for which the papers of
(Toth & Vigo, 2002), (Huisman, 2004) and (Steinzen, 2007) were a great inspiration.
In the following section, we describe the three individual problems, also citing some milestone
papers. Partially against the structure of this thesis (we are now in the chapter Past), we also
cite a few publications from 2004 and later. We do this for the benefit of the completeness of the
discussion and because in the chapter Present, the individual problems (VRP, VSP and CSP)
will not be re-examined.
1.1.1. The Vehicle Routing Problem (VRP)
The goal of the Vehicle Routing Problem (VRP) is to determine the optimal set of routes to be
performed by a fleet of vehicles (operated by drivers5) in order to service a given set of
customers. (Toth & Vigo, 2002) Because of the diversity that exists within the subject of VRP,
this section mainly consists of an overview of the most important types of VRP's. We perform
this discussion in the distribution or product delivery context, because this is the context in
which the VRP will nearly always occur, unlike in public transport. This will be explained later on
in section 1.2.
The standard version of the VRP deals with the distribution of a single product from a central
depot to a collection of geographically dispersed customers with a deterministic demand,
within a single period6 planning horizon. We also call this the standard Single Depot Vehicle
Routing Problem (SDVRP).
A set of possible vehicle routes for the supply of the customers is designed, taking into
account certain constraints. These constraints usually comprise the limited capacity of a vehicle
and the customer demand. A VRP that considers the first constraint, is sometimes specified as
a Capacitated Vehicle Routing Problem (CVRP). A minimization of the total transport costs is
always envisaged. It is implicitly assumed that the supply capacity of the central depot is
5
In this thesis, driver is a general term that not only represent the obvious bus, train and truck drivers, but
also pilots, etc. Even stronger, the term driver may also comprise other possible crew members that do
not actually drive the vehicle, like conductors (for trains) and flight attendants (for airlines). In that case,
driver can in fact be seen as a synonym for crew. Nevertheless, the term driver is often intuitively used
(just like here) because most of the time concepts are initially presented for the most widely discussed
mode of transport, namely road traffic (where generally there is only one crew member – the driver – per
vehicle).
6
This means that a particular time frame, usually a period of one day, is respected as a planning horizon.
6
sufficient to meet the needs of all customers and that each customer can be reached from the
central depot by a roundtrip on the same day. In this way, the standard VRP will eventually
strive for the determination of routes which can be ridden by a single driver on a particular day.
In the next three paragraphs, we discuss the main subdivisions and variations of the VRP.
When the capacity of the supplier is large enough, it may happen that several depots are
responsible for the supply of many customers. The problem, which is called Multiple Depot
Vehicle Routing Problem (MDVRP), then becomes more extensive as there is a choice from
which depot products have to be supplied. In practice, a commonly used technique consists of
assigning a certain number of customers to a depot, producing a cluster of customers. It thus
becomes sufficient to solve the VRP corresponding to the single depot case for each cluster
separately.
The distinction between single depot and multiple depot will also prove to be very important for
other planning problems discussed later on, because it is a distinction that really determines the
nature of the problem and therefore the solution methods used for it.
An extension of the VRP which has become more and more popular in recent years because of
its applicability to a wide area of real-world problems, is the VRPTW or Vehicle Routing Problem
with Time Windows. This problem can be described as a CVRP where each customer defines a
certain time period – the time window – in which the pickup or delivery of a product should take
place. Remark that the standard VRP does not consider these time windows. The ultimate goal
of the VRPTW is to determine routes that begin and end at a depot, serve a certain number of
customers en route within the specified time window and ensure that the capacity limitation of
the vehicles is not exceeded, all while keeping the routes as short as possible. Not surprisingly,
the VRPTW proves to be NP-hard.
Another variation of the standard problem is the Vehicle Routing Problem with Pickup and
Delivery (VRPPD) in which an amount of goods needs to be transported from certain pickup
locations to other delivery locations. The goal is again to find optimal routes for a fleet of
vehicles to visit the pickup and drop-off locations.
A very important aspect of the VRP is that almost all existing approaches do not distinguish
between a vehicle and its driver. A driver is thus identified with the vehicle he drives. This
means that the constraints imposed on drivers (work regulations, etc.) are imbedded in those
associated with the corresponding vehicles. (Toth & Vigo, 2002)
7
For more information about the VRP and its variants, we refer to (Toth & Vigo, 2002). More
recent evolutions in this area of research are discussed in (Golden et al., 2008).
1.1.2. The Vehicle Scheduling Problem (VSP)
Vehicle scheduling is the process of assigning trips to vehicles (or vehicles to trips if you will, in
any case there has to be a mutual assignment) so that the total vehicle costs are minimal.
Generally, it is assumed that start and end locations are fixed for all trips. When in a public
transport context, the same can be said for the start and end times of the trips. In a distribution
context however, we usually implement time windows so trip times would indeed not be fixed.
More information on this subject will follow in section 1.2. As stated in previous section, the
distinction between single depot and multiple depot determines the nature of a problem.
Therefore, we will approach the discussion of the Vehicle Scheduling Problem (VSP)
accordingly. Elements of (Huisman, 2004) and (Steinzen, 2007) were used as a basis and
complemented by additional considerations.
1.1.2.1.
Single depot case
In the case that there is only one depot, a homogeneous fleet and no time constraints, we are
dealing with the standard Single Depot Vehicle Scheduling Problem (SDVSP). The standard
problem can be defined as finding an assignment of trips to vehicles so that each trip is
assigned exactly once, each vehicle performs a feasible sequence of trips (two consecutively
assigned trips should be compatible7), each sequence starts and ends at the same depot, and
vehicle costs are minimized. The vehicle costs consist of a fixed component for every vehicle
(investment and maintenance costs) and variable operational costs for idle and travel time. In
most practical situations, companies try to minimize their fixed costs first and leave operational
cost minimization as a secondary objective. Usually, it is allowed that a vehicle returns to its
own depot between two trips if there is enough time to do this. (Huisman, 2004), (Steinzen,
2007)
The SDVSP is not only an interesting problem in itself, it also appears as a subproblem in much
more complicated problems like the Multiple Depot Vehicle Scheduling Problem (MDVSP)
7
“If two trips i and j are consecutively assigned to the same vehicle, the start time of trip j should be larger
than or equal to the end time of trip i plus the travel time from the end location of trip i to the start location
of trip j.” (Huisman, 2004, p. 24)
8
discussed in the next subsection and the integrated Vehicle and Crew Scheduling Problem
(VCSP) discussed in section 1.3.2. That is why there has been paid a lot of attention to the
subject. For a fairly recent survey on the SDVSP and its practical extensions we refer to (Bunte
& Kliewer, 2006).
1.1.2.2.
Multiple depot case
When there are multiple depots, we are of course dealing with a Multiple Depot Vehicle
Scheduling Problem (MDVSP). The transport company now operates its (homogeneous) fleet
out of several depots, where each vehicle is associated with a single depot. Again, every trip
has to be assigned to exactly one vehicle. The MDVSP can be extended by introducing multiple
vehicle types and by the constraint that some trips have to be serviced by vehicles from a
certain subset of depots. In some cases there are also other constraints concerning depot
capacity or route time, which can be imposed to make instances more realistic and thus more
challenging. The main objective remains of course the minimization of the vehicle costs, which
can be defined in the same way as in the single depot case. (Huisman, 2004), (Steinzen, 2007)
The number of depots proves to be an important aspect defining the complexity of a specific
problem instance. Whereas the SDVSP is solvable in polynomial time, (Bertossi et al., 1987)
showed that the MDVSP is NP-hard from the moment there are at least two depots. Because
the problem is always NP-hard, early works – dating back about 30 years – mainly focused on
heuristic algorithms. A fairly recent comparison of different heuristic approaches to the MDVSP
can be found in (Pepin et al., 2006). It took until 1989 with (Carpaneto et al., 1989) before exact
methods where used to solve the MDVSP. (Fischetti et al., 2001) categorize the exact solution
approaches into three basic types, based on the mathematical formulation used: singlecommodity flow, multicommodity flow or set partitioning. An explanation of these formulations
will be provided in section 2.1.2.3. (Huisman, 2004) and especially (Steinzen, 2007) provide an
elaborate presentation of the three types of exact methods. For a more extensive survey on the
specific models for the MDVSP the reader is again referred to (Bunte & Kliewer, 2006).
1.1.3. The Crew Scheduling Problem (CSP)
Again, elements of (Huisman, 2004) and (Steinzen, 2007) were used as a basis and
complemented by additional considerations. Firstly, we are inclined to state that the Crew
Scheduling Problem (CSP) is perhaps the most important planning problem in a transport
company since crew costs generally dominate vehicle costs (Bodin et al., 1983). It therefore has
9
received considerable attention in the operations research literature since the late eighties. The
problem deals with assigning tasks to crew duties so that each task is performed, each duty is
feasible with respect to a set of working rules and the total labor costs are minimized. The
working rules may comprise federal laws, safety regulations and (collective) in-house
agreements. Examples of working rules are minimum/maximum driving time, maximum spread
(length) of the duty, minimum break length and allowed start and end time of a duty. Of course,
in practical applications there can be many more of such rules. It is mainly because of this wide
variety of duty feasibility constraints that the CSP is more complex than the VSP, whereas in
essence both problems are quite congruous.
In the basic version of the problem, there are only working rules that are defined on an
individual duty, so no rules which require that a minimum/maximum number or percentage of
the duties has certain properties are taken into account8. Another basic assumption is that all
crews are equal since individual crew members are not considered. This assumption can be
extended by introducing multiple crew types. As already mentioned, the main objective is to
minimize the labor costs. In practice, this objective is often simplified to minimizing the total
number of duties so that only fixed crew costs are necessary to take into account. An extension
can consist of also considering minimization of the total working time, which implies the
introduction of an hourly rate for working time. On top of that, crew scheduling problems are
often subject to non-linear costs like overtime bonuses. (Fischetti et al., 1989) show that in
principle every CSP is NP-hard, even when only very basic constraints are considered.
(Huisman, 2004), (Steinzen, 2007)
The CSP is usually formulated as a set partitioning or set covering problem9. These are usually
solved with a column generation approach. Such an approach generally consists of the
following (consecutive) components: the mathematical formulation, the master problem, the
pricing problem (or column generation subproblem) and the construction of feasible (integer)
solutions. In such a model, the working rules in a duty have to be taken into account only in the
pricing problem. The master problem can be solved by LP relaxation10 or by Lagrangian
relaxation. The first approach generally proves to be the most popular. To get integer solutions,
different algorithms, exact or heuristical, can be used. Alternatively to column generation, the
8
“For instance, the percentage of split duties that have two pieces of work – one in the early morning and
another in the late afternoon with a long break in the middle – is often restricted.” (Steinzen, 2007, p. 7)
9
The formulation of the CSP as a set covering problem allows tasks to be over-covered, as opposed to
the set partitioning formulation. “In practice, this over-covering is often not acceptable, but solutions of this
model often contain little or no over-covers (since it is cheaper to assign only one driver to a task). The
main advantage of a set covering over a set partitioning formulation is that continuous and integer
solutions can be easier computed.” (Steinzen, 2007, p. 36)
10
LP relaxation is a synonym for linear programming relaxation or the shorter linear relaxation. All three
terms are used interchangeably throughout the thesis.
10
set of columns can be enumerated or generated heuristically. Furthermore, several
metaheuristics have also been proposed for solving the crew scheduling problem.
For an overview of the concrete CSP solution methods belonging to one of the above classes,
we again refer to (Huisman, 2004) and (Steinzen, 2007). A presentation of even earlier works
on crew scheduling can be found in (Carraresi & Gallo, 1984).
1.2.
The planning process in transportation companies
In this section, we will elaborate on the different phases of the planning process for
transportation problems. Now first, why is this planning process so important in the transport
sector, especially in public transport?
The last few decades, public transport experiences strong competition from private traffic.
Especially for environmental and health reasons, it is important to reverse this balance back
towards the side of public transport. However, given the difficulty of imposing restrictions or
penalties on private drivers, or discounts on the side of public transport, the designated tool for
obtaining this reversal is the quality of public transport. It soon becomes clear that a well
thought out planning is crucial in this view, as two opposing goals must be satisfied: delivering
an attractive service to clients and doing so with as little money as possible. This planning is
even more challenging when one realizes that the conflict between quality and cost gives rise to
a high degree of e.g. technical requirements (engineering). (Schellinck, 2009) Therefore, the
planning process is of the utmost importance.
We already described the VRP, VSP and CSP in previous sections, but how do they fit within
the greater whole of the planning process of a transport company, in public transport as well as
in the distribution context? From the last part of previous sentence, an important subdivision of
practical planning problems, which will be explained later in this section, already becomes
apparent. For the present, the intuitively obvious distinction between a public transport or mass
transit company (which can either be a bus, tram, metro, train or airline operator that transports
passengers) and a distribution or product delivery company (e.g. freight transport, delivery
services and mail distribution, which deliver goods to particular customers) is sufficient to be
able to comprehend the following paragraphs. Note that we include airlines in public transport,
while in some papers it is seen as a separate category (Hollis, 2011).
Before we answer the opening question of the preceding paragraph, let us clearly distinguish
between routing problems (VRP) and scheduling problems (VSP and CSP). If the customers
being serviced have no time restrictions and no precedence relations exist, then the problem is
11
a pure routing problem. If there is a specified time for the service to take place (so there is a
specified timetable), then a scheduling problem exists. This is the case for public transport
problems. Otherwise, we are dealing with a combined routing and scheduling problem. Think for
example of a distribution problem where the customers have to receive their order within a
given time window. This is not a pure routing problem, because the time windows do impose
certain time restrictions.
Now we discuss the situating of the VRP, VSP and CSP within the planning process of a public
transport company11. Later on, we point out the similarities and differences of this process in a
distribution company.
The entire planning process in a public transport company is very complex and for this reason
not (yet) computationally tractable as a whole. Traditionally, it is therefore divided into three
consecutive phases: a strategic, tactical, and operational one. Some phases are further split
into several subproblems that have to be sequentially solved. Figure 1 represents the complete
planning process. We now concisely describe each phase and the subproblems of which they
consist, loosely following (Steinzen, 2007).
Figure 1: Planning process in transportation companies
11
The exposition given here is based on the planning for a bus company, so for a road traffic mode of
transportation. The planning process for other sorts of public transport companies (train, airline, etc.) is
mostly similar, though there are some differences. In section 2.1.1.2 we elaborate on this issue.
12
The strategic phase comprises network design and route planning12. In a normal public
transport company, we typically deal with a planning horizon of several years. To start the
planning procedure, a so called origin-destination matrix is needed. This matrix actually
represents the demand data, that is the number of passengers that want to travel between any
two locations in the network on a particular time of the day. During network design we then
determine links of sufficient capacity within the network in such a way that the construction
costs are kept as low as possible.
The subsequent route planning problem then aims at satisfying the passenger demand by
selecting a set of line routes and their frequencies, given the transportation network designed in
the previous subproblem. Two conflicting goals can be identified, namely the maximization of
passenger comfort (measured e.g. by the total transit time or the number of direct connections)
and the minimization of line operating costs. From this discussion we can conclude that the
strategic phase, especially the route planning phase, largely corresponds to the VRP as
described in section 1.1.1. Also the customer-relatedness is an important binding aspect.
In the tactical planning phase a timetable is constructed. This timetabling problem is solved on a
seasonal basis (e.g. once in winter, once in summer). We presume that line routes, the traveling
times along them, their frequencies and any possible layover times at depots or other relief
points are known. The actual objective in this phase is to transpose the desired line frequencies
into a detailed timetable. Such a timetable defines the start and end locations and times of the
set of trips to be performed by the company. Notice that we incorporate the timetabling problem
in the tactical phase (following the idea of (Desaulniers & Hickman, 2006)) instead of seeing it
as a part of the operational phase as was traditionally done, like e.g. in (Huisman, 2004). Since
timetabling has a direct impact on both passengers (or more general customers) and staff, it
can be seen as an interface between the strategic and operational phase. As timetabling is an
interface, we do not consider it to be an inherent part of either the routing or the scheduling
problem.
The operational planning is concerned with the construction of vehicle and crew roundtrips with
the aim of minimizing total costs while considering a variety of operational constraints and work
regulations. One can notice that the operational phase is indeed more focused on the staff (or
crew) of the company than on the customers, as was more the case at the strategic level. Now
for the different subproblems. First, the Vehicle Scheduling Problem (VSP) consists of assigning
vehicles to trips, resulting in a schedule for each vehicle. Such a schedule is then split into
12
The terms line routes or lines are synonyms for routes, but where routes is a more general term, line
routes or lines are more appropriate in a public transport context (think of e.g. bus lines). Here we used
the more general term of route planning because we will also discuss a distribution context. Although, for
a discussion of public transport, this could more fittingly be called line planning.
13
several vehicle blocks, where a new vehicle block starts at each departure from a depot. Every
vehicle block is in itself a sequence of tasks. Each of these tasks then needs to be covered by a
crew duty (or just duty), which is the mission of the Crew Scheduling Problem (CSP). A duty
represents the workload on a daily basis of a not yet specified driver. Of course, a number of
work regulations (e.g. sufficient rest time) must be satisfied. From the short-term (daily) duties,
the crew rostering problem – which we do not discuss further in this thesis – constructs longterm (monthly) work schedules, logically called crew rosters, again considering multiple work
regulations. Unlike duties, crew rosters are assigned to specific drivers.
In a distribution company, the general structure of the planning process is more or less similar.
The natural sequence of the emerging problems remains the same: VRP, VSP and then CSP.
Only the definition of the problems is somewhat different. To indicate this, we will reconsider the
three phases in the planning process and point out the similarities and differences with respect
to the explanation mentioned for a public transport company.
In the distribution context the VRP does of course not consist of designing line routes (e.g. bus
lines) that remain the same for a long time and have the aim of picking up passengers at fixed
places. In fact, there is no such thing as ‘passengers’ here, it are the products ordered by the
customers that have to be transported, not the customers themselves. Those customers are not
always known well in advance, nor are their location or their demand (i.e. the demand data
constantly changes). Therefore, the routes will not be developed e.g. every few years, but much
more frequently, for example with a planning horizon of a few weeks or even less (think of a
delivery service that delivers products to customer’s home, within a week from ordering). For the
other aspects of strategic planning, we can state the following. Obviously, the goal of the
network design problem is still to determine the links of the network in such a way that
construction costs are minimized. Meanwhile, the concept of frequency of a chosen route is no
longer relevant, because the routes themselves may constantly change. The route planning
problem has similar objectives as in public transport: maximizing customer satisfaction and
minimizing operating costs of the routes. The objective of customer satisfaction (related to
service level) originates from the fact that customers like to get their ordered product on time,
which is actually more an operational concern than a strategic one.
In the timetabling phase, it is presumed that the routes and their frequencies are known. In the
product delivery context however, routes are not fixed and the concept of frequencies is not
even relevant. In this respect, we could state that the timetabling problem as such does not
arise in a distribution context. At some point though, mostly not long before the actual
14
execution, the routes will be laid out and an exact starting time for the tasks situated within their
time windows will be chosen, so actually a ‘timetable’ is indeed created. However, this process
is run through so many times – perhaps even every day – that the timetable will always be
different. Therefore, we say that timetables are not given in a distribution context. From above
discussion, we may certainly conclude that the tactical phase has a whole different nature for
distribution companies.
Then again, the operational phase in distribution companies is very similar to that in public
transport. Still, a difference worth mentioning is that in a distribution context, we also have to
consider the imposed time windows during the scheduling phases.
The division into public transport and distribution (or product delivery) problems can now be
made and is an important classification criterion for the practical planning problems – and for
the papers in which they are discussed – considered in this thesis. This classification criterion is
quite high-level, because it really suggests a different nature of the planning problem that has to
be solved. As already mentioned, public transport problems and distribution problems have
quite a few similarities, but then again there are some differences making the problems
intrinsically distinct.
We will clarify the most important difference through following reasoning. In public transport, the
strategic planning horizon is typically several years. So, once designed, the line routes remain
unaltered for a long period (e.g. at a particular bus stop, busses are generally expected to pick
up and drop off passengers at the exact same time every day). Moreover, the corresponding
timetable is adapted only to take seasonal effects into account (e.g. the alternation between
school days and vacations influence the frequencies of bus lines). This means that timetables
are mostly given for public transport problems. In other words, decisions about which lines to
operate and how frequently (i.e. the timetable), are input for the operational planning process in
public transport. They can be either determined by the marketing department of the company or
imposed by local, regional or national authorities. (Huisman, 2004) This last sentence denotes
that public transport companies are not always free to develop their own (optimal) routes, which
is generally not the case for distribution companies.
Therefore, it is not surprising that the vast majority of papers about planning in public transport
do not consider the strategic and tactical phase, but only the operational phase. This means
that in most practical public transport problems, only the scheduling aspect and not the routing
aspect is considered. Of course, for distribution problems this does not apply. There we have to
design new routes much more frequently, so the routing problem is a substantial part of the
planning problem that cannot be neglected. Obviously, the scheduling problem must also be
15
tackled in the distribution context. From this perception, we might be tempted to conclude that
distribution problems are more difficult than public transport problems because both the routing
and scheduling aspect have to be considered. This is however somewhat short-sighted,
because – as already mentioned in the beginning of this section – the delicate trade-off between
customer and cost is of great importance in public transport so more circumspection and
therefore more effort is required.
In the preceding description of the general planning process in public transport, we included the
strategic and tactical phases. While in practice, these phases are usually not considered.
Consequently, we may generally indicate public transport problems and distribution problems
as follows in Figure 2.
Figure 2: Planning process in transportation companies (extended)
1.3.
The scheduling problem
Although the routing problem precedes the scheduling problem in the planning process (see
Figure 2), we first discuss the latter. This because scheduling has to be performed for every
practical transportation problem, in public transport as well as in a distribution context. The
routing on the other hand is generally only considered in papers about distribution problems, so
16
this item is somewhat more specific. Moreover, the research and associated literature in the
field of scheduling is more extensive and will therefore cover the major part of this thesis.
In this section, we actually present the evolution of integration within the scheduling: from VSP
first - CSP second to VCSP.
The organization is as follows. We first describe the traditional sequential approach (VSP first CSP second) in section 1.3.1 and proceed to the integrated VCSP approach (section 1.3.2),
defining it in both a public transport and a distribution context. In the last section 1.3.3 we briefly
review the most important earlier (pre-2004) literature on the VCSP, introducing the division into
partial and complete integration approaches.
1.3.1. Traditional approach
The traditional sequential approach for vehicle and crew scheduling concerns the solving of first
the vehicle scheduling problem, and then the crew scheduling problem (in short: VSP first - CSP
second). Vehicle schedules are thus determined before crew schedules. In other words, we first
assign trips to vehicles and then schedule crews based on the resulting vehicle blocks. In fact,
the sequential method for solving a vehicle and crew scheduling problem basically comes down
to solving the CSP. Because the solution method for a standard CSP is based on the output of
the VSP, the VSP always has to be solved first, which corresponds to the definition of the
sequential approach. In Appendix A we present a detailed sequential method following these
principles and based on (Huisman, 2004). Of course, other sequential approaches do exist (e.g.
with incorporation of time windows when in a distribution context), but the presented one is quite
common and provides a basis for several other algorithms, so it will at least give a general
understanding of how a sequential method for vehicle and crew scheduling is performed
concretely. Looking into the method described in Appendix A may also be useful – though not
necessary – as an introduction to the integrated problem introduced in the next section since
integrated problems sometimes include traditional sequential vehicle and crew scheduling
problems as subproblems. Furthermore, the traditional approach is used as a point of
comparison to evaluate the (added) efficiency of integrated methods.
1.3.2. Defining the Vehicle and Crew Scheduling Problem (VCSP)
Although the traditional independent scheduling of vehicles and crews as described in previous
section was already seriously criticized in the early eighties by (Bodin et al., 1983), long after,
most of the algorithms published in literature still followed the sequential approach. Fortunately,
17
it became more and more clear that this was not the right track to follow and thus an evolution
towards an integrated approach was needed (an overview of the potential benefits of integration
is provided in (Freling et al., 1999)). We repeat that it is in fact so that personnel costs are
usually higher than vehicle costs and it would therefore certainly be useful to consider crew
scheduling simultaneously with vehicle scheduling. Another argument for integration follows
from the straightforward dependence between the two problems. Indeed, it should be
understood that the characteristics of the vehicle-related aspects will influence the resulting
personnel tasks, while an optimal personnel allocation will surely result in a different vehicle
use. It was therefore gradually accepted that an integrated approach of the VSP and the CSP
might well lead to significant cost reductions. Thus arose the integrated Vehicle and Crew
Scheduling Problem (VCSP).
The VCSP arises in both the public transport and the distribution context. In order to incorporate
domain specific concepts, we need different descriptions for both contexts. Therefore, we
successively and independently of one another discuss the definition of the VCSP, first for the
public transport context, and then for the distribution context. There will be overlap to some
extent, but we believe that the separate treatment of both problems will benefit the clarity. The
discussions are based on (Huisman, 2004) and (Steinzen, 2007), supplemented with additional
considerations.
1.3.2.1.
Public transport context
The VCSP in public transport is the following: “given a set of service requirements or trips within
a fixed planning horizon, it minimizes the total sum of vehicle and crew costs such that both the
vehicle and the crew schedule are feasible and mutually compatible. Each trip has fixed starting
and ending times (so there is a given set of timetabled trips) and can be assigned to a vehicle
and a crew member.” (Huisman, 2004, p. 78) In the multiple depot case, we must add ‘from a
certain set of depots’ to this last sentence. Of course, if every trip can only be assigned to a
vehicle and a crew from one specific depot, the multiple depot problem decomposes to a
number of single depot VCSP’s. Furthermore, the travelling times between all pairs of locations
are assumed to be known. The VCSP is NP-hard since (at least) the crew scheduling part is
NP-hard. When considering multiple depots, we know from section 1.1.2.2 that the vehicle
scheduling subproblem is also NP-hard, unlike in the single depot case. The remainder of the
definition is included in Appendix B, because it is not per se needed for the further reading of
this thesis (in fact, the presented terms in the introductory section Definitions are sufficient and
were actually already explained in the discussion of the operational phase of the planning
18
process in section 1.2), although of course very interesting and possibly providing some extra
insight into the subject. For example, it should become clear from the extended discussion that
a wide variety of practical scheduling problems exist which can be modeled in just as many
different ways. In this respect, a categorization of these practical problems and their solution
methods would prove very useful to get a well structured overall view on the topic of scheduling
in a transportation context. Therefore, this is one of the main goals of this thesis and will be
performed in the chapter Present. The exposition is strongly based on (Huisman, 2004), but
was slightly reorganized and supplemented in order to make it somewhat completer, mainly to
clarify the relation and transition between the single depot and multiple depot case.
1.3.2.2.
Distribution context
In public transport, the trips (and thus tasks) to be performed by a company are usually defined
by a given timetable and have fixed starting times. Therefore, the concept of time windows is
not relevant in that context. In the distribution context on the other hand, each task can be
characterized by a starting time window, within which the task can be commenced. These time
windows originate from the fact that usually there are time windows for deliveries, defined by an
earliest pickup time of the product at the depot and a latest drop-off time at the customer. This
implements that the routes themselves will have a time window within which they can be
operated, seeking to deliver all products on time. This practical application leads to the
distribution-specific Vehicle and Crew Scheduling Problem with Time Windows (VCSPTW). The
problem is that of creating mutually compatible vehicle and crew schedules that lead to a
minimum total cost, where vehicles and crews are based at multiple depots13 and have to cover
a given set of tasks with associated starting time windows. A detailed definition following (Hollis,
2011) is again added to Appendix C, mainly for the same reasons as mentioned in previous
subsection. This description of the VCSPTW considers not only multiple depots, but also
multiple vehicle and crew types, which is the most general case.
1.3.3. Earlier literature on the VCSP
Now we have defined the VCSP in previous section, we can proceed to an overview of the
associated literature from the past, so prior to 2004. We note that all the papers presented in
this section deal with the public transport problem. This because, to the best of our knowledge,
13
For the distribution problem we do not make the distinction between single and multiple depot anymore
and only discuss the more general latter case. We do this because we do not want overcharge the text,
especially since differences concerning the number of depots are very similar to those for public
transport.
19
there is no existing literature before 2004 that proposes a general treatment of the VCSP in the
distribution context, where timetables are not given in advance and time windows should be
implemented (i.e. the VCSPTW). Given the wider practical significance and the greater difficulty
of planning in public transport, this may not be surprising.
Although simultaneous vehicle and crew scheduling is was demonstrated to be of significant
practical interest by (Bodin et al., 1983), not many approaches of this kind have been proposed
in the literature before 2004. They mainly dealt with bus and driver scheduling and fall into the
category of either partial or complete integration. We use this last distinction to organize the
presentation of the relevant papers. Although some authors (e.g. (Hollis, 2011)) do not see
methods for partial integration as a VCSP (according to them, only methods for complete
integration – as defined in previous section – may carry that title), we also discuss the partial
methods because this outlines the evolution of the integration of vehicle and crew scheduling.
1.3.3.1.
Partial integration
In the eighties and early nineties, a few authors began to become involved in the VCSP.
However, it took some while before problems of considerable size and with multiple depots
could be handled with a fully integrated method. Therefore, earlier approaches – up to the late
nineties – were based on a heuristic or so called partial integration of vehicle and crew
scheduling. Similar to (Freling, 1997), we present two types of partial integration and mention
the most relevant papers using them:
-
a crew first - vehicle second approach:
performing crew scheduling while
including vehicle considerations, construction of a feasible vehicle schedule is
done afterwards
Most of the approaches of the first category are based on a heuristic procedure proposed by
(Ball et al., 1983). (Tosini & Vercellis, 1988), (Falkner & Ryan, 1992), and (Patrikalakis &
Xerocostas, 1992) all based their approaches on this paper.
-
a vehicle first - crew second approach:
performing vehicle scheduling while
including crew considerations, construction of a feasible crew schedule is done
afterwards
20
The two earliest heuristic approaches of the second type were proposed by (Scott, 1985) and
(Darby-Dowman et al., 1988). More recently, (Borndörfer et al., 2002) presented another
method belonging to this category.
(Haase et al., 2001) and (Freling et al., 2003) provide more extensive overviews of methods that
partially integrate vehicle and crew scheduling.
1.3.3.2.
Complete integration
Previous subsection indicates that only very few partially integrated approaches have been
suggested for the VCSP. Since the late nineties however, several researchers started to
develop different approaches introducing models and solution techniques based on
mathematical programming.
The first mathematical formulation for the single depot case was proposed by (Patrikalakis &
Xerocostas, 1992). However, their model is computationally intractable and therefore the
authors had to resort to a partially integrated solution method, which is why this paper was also
mentioned in previous subsection. (Freling et al., 1995) slightly altered this formulation and a
couple of years later, (Freling, 1997) proposed the first integrated treatment of vehicle and crew
scheduling concerning both model and solution method. His solution approach has inspired
multiple later publications (e.g. (Huisman, 2004)).
(Haase & Friberg, 1999) proposed another exact algorithm for the single depot VCSP. A few
years later, (Haase et al., 2001) introduced an interesting crew scheduling formulation for urban
mass transit with side constraints for the vehicles, more specifically a bus counter variable. In
the airline world, (Cordeau et al., 2001) and (Klabjan et al., 2002) used a similar approach to
integrate aircraft routing (which is equivalent to vehicle scheduling) and crew scheduling.
(Valouxis & Housos, 2002) described a VCSP that is actually a CSP since drivers are tied to
their vehicle. (Freling et al., 2003) considered the SDVCSP in an urban public transport context
with a homogeneous vehicle fleet and two crew types.
Note that all papers discussed above focus on the single depot problem. To the best of our
knowledge, only one pre-2004 paper deals with integration in the multiple depot case, and that
is the one written by (Gaffi & Nonato, 1999).
For more extensive discussions of the papers presented above, we can refer the reader to
(Huisman, 2004), (Steinzen, 2007) and (Hollis, 2011).
21
1.4.
Combining routing and scheduling: the Vehicle and Crew Routing
and Scheduling Problem (VCRSP)
The combination of the routing problem and the scheduling problem (the latter was discussed in
section 1.3) only occurs in the distribution context (e.g. freight distribution, postal organizations
and delivery services) and not in public transport, where generally line routes and timetables are
fixed, dictated for example by a local authority. It is so that, as could already be suspected from
the introduction of section 1.3.3, we could not find any relevant pre-2004 paper on this subject.
This means that for this section, we are limited to providing a brief description of the problem
and postpone further explanation to section 2.2 in the chapter Present.
We get straight to the point by providing a simplified description of the traditional planning
process in a distribution company. This process typically involves constructing a set of vehicle
routes, then producing vehicle schedules by assigning vehicles to the vehicle routes, and
building a set of crew schedules so that all vehicle routes are covered. (Hollis, 2011)
For the ease of notation and also to show a certain conformity with the VCSP, we will name a
distribution problem where routing and scheduling are combined, a (simultaneous or integrated)
Vehicle and Crew Routing and Scheduling Problem (VCRSP). We can provide a formal
definition of the VCRSP, which should in fact already be clear from the definition of a distribution
problem: the VCRSP concerns a planning problem where the required transports have no
given timetable (i.e. distribution context), and where a driver-vehicle combination is not
considered an inseparable unit. (Drexl et al., 2011) The latter characteristic stresses the
difference with a VRP, where a driver is nearly always identified with the vehicle he drives
(recall from section 1.1.1). As a consequence, a change in one route may have effects on the
feasibility of other routes in a VCRSP (this is called the interdependence problem), which is not
the case in a classic VRP where independence of routes is implicitly assumed. Thus we have
discovered an additional difficulty of the VCRSP. From this discussion, it should become
apparent that a VCRSP consists of some sort of merger between a VRP and a VCSP.
Now let us get a little ahead of things and already broach the subject of integration in the
distribution context. If in previous section about the scheduling problem, the integration within
the scheduling was discussed, then we are now dealing with an integration on a higher level,
namely between routing and scheduling (from a traditional routing first - scheduling after
approach to simultaneously routing and scheduling).
22
2. Present
Since all pre-2004 papers on planning in transportation were said to be part of the past, all
publications from 2004 up to now of course belong to the present.
The chapter starts with the most important part, being the categorization of recent VCSP
approaches. In section 2.1, we first define the categorization criteria, which either relate to the
practical problem considered or the solution method used for it. Once this is done, we proceed
to the actual categorization of the VCSP approaches. At the end of the section, conclusions are
derived from the presented categorization, including an answer to the first aspect of the central
research question. The second part of the chapter (section 2.2) is dedicated to the VCRSP,
beginning with an overview of the existing approaches and providing an impetus for
categorization (after the image of that for the VCSP). Finally, conclusions are stated for the
VCRSP, now comprising an answer to the second aspect of the central research question.
2.1.
Categorization of recent VCSP approaches
Above title already reveals the intent of this section, namely to categorize recent VCSP
approaches. Of course, we cannot consider all VCSP approaches introduced as from 2004, but
rather select the most important ones. A comprehensive categorization of these approaches
(and actually also of the corresponding papers) allows the reader to easily detect e.g. for which
practical problem a certain solution method has already been used (and was found to be
'good'), which solution methods are most commonly used for a particular practical problem,
which practical problems have been described most frequently, and which solution methods are
the most popular. From these observations, we can come to interesting conclusions that may
give rise to recommendations for future research (see also chapter Future research). A few
possibilities: a solution method that turned out to be very effective for a particular problem may
perhaps also be used for other practical problems, some practical problem that did not receive
much attention so far – inconsistent with its importance in real-life – can be identified, etc.
As could already be suspected from previous listings, we will classify the various VCSP
approaches according to two major aspects, namely (1) the practical problem that is considered
and (2) the way this problem is dealt with, so the solution method. Within these two overarching
categories we define multiple categorization criteria. We identify 8 criteria for the practical
problem, and 5 for the solution method. The criteria are presented in descending order of
23
importance (in terms of impact on the nature of the problem14 and its solution method). The
proposed criteria are obviously not exhaustive, but we believe to have identified some of the
most relevant ones. They are not too detailed, but still comprehensive enough so that all
approaches can be categorized sufficiently accurate.
This section is organized as follows. We start with the discussion of the categorization criteria
within both major categories, again under the pretext of ‘most important first’, so first for the
practical problem in section 2.1.1 and then for the solution method in section 2.1.2.
Categorization according to practical problem is more important because this actually defines
the nature of the problem, something that is in fact fixed and thus an inherent characteristic.
Whereas the nature of a given practical problem is unchangeable, the solution method for it can
still be chosen to some extent. Of course, the nature of the practical problem will partly
determine the solution method (kind of model, etc.) to be used. Therefore, we can state that the
categorization according to practical problem is higher level than that according to solution
method, which is intuitively logical. In section 2.1.3, we proceed to the actual categorization of
the recent VCSP approaches according to the proposed criteria. Finally, we draw conclusions
(links and trends) from this categorization, which we can use as a basis for making
recommendations for future research. This is included in section 2.1.4.
2.1.1. Categorization according to practical problem
The 8 criteria for categorization according to practical problem are (in descending order of
importance): type of transportation problem, mode of transportation, number of depots,
objectives, size/practicality of the problem, degree of urbanization, regularity of the timetable,
and admission of changeovers. For the sake of unity, literal definitions are extracted from one
and the same paper of (Steinzen, 2007).
2.1.1.1.
Type of transportation problem
The first and most high level
criterion is one that was already frequently mentioned and
discussed thoroughly in section 1.2, namely the subdivision in public transport problems and
distribution problems. We will not resume the entire discussion here, but rather refer to the
mentioned section.
14
W hen we are simply talking about a problem in this section, we obviously refer to a VCSP.
24
2.1.1.2.
Mode of transportation
We can identify three major modes of transportation: road traffic, railway and airline. Under
the title of railway we include trains, trams and metros. Clearly, this subdivision mainly concerns
public transport. Although one can imagine freight trains and cargo planes being used in the
distribution context, the vast majority of the distribution problems deal with transport by road.
The trucks, lorries and delivery vans used for this are categorized under road traffic, just like the
busses used in public transport.
The distinction in terms of mode of transportation is an important aspect regarding the nature of
– and therefore the solution method for – a scheduling problem, especially its complexity. We
therefore include an elaborate discussion of the differences between bus, railway and airline
planning (as was promised in section 1.2) in Appendix D. This discussion – inspired by
(Huisman, 2004) – is certainly interesting, but not necessary for the further reading of this
thesis.
2.1.1.3.
Number of depots
This criterion does not refer to the exact number of depots considered, but makes the distinction
between the single depot case and the multiple depot case. This distinction has been
discussed repeatedly in preceding sections and is therefore not repeated here. However, we do
recall that the number of depots, also the exact15 number, will have a significant impact on the
complexity of the problem.
2.1.1.4.
Objectives
Needless to say that for any scheduling problem, (1) cost reduction (minimization of costs) is
nearly always the main goal. However, there are also other objectives that can be pursued. We
will consider two important additional goals.
A first one is (2) the optimization of the service level, which concretely means that delays are to
be minimized. This objective is more relevant in a public transport context because timetables
are very strict here, whereas for distribution problems there is some tolerance due to the
15
By which we mean that, although two problems considering e.g. 2 and 4 depots both concern a multiple
depot case (and are thus the same according to the ‘number of depots’-criterion), the exact number of
depots does of course have an impact on complexity, that is, the problem considering 4 depots will be
more complex and thus harder to solve.
25
existence of time windows (but still, customers of a distribution company want to get their goods
on time so service level can also be relevant here).
A second additional objective is related to (3) the quality of crew schedules. We consider the
regularity of crew schedules as the defining quality aspect. A crew schedule is called regular if it
can be repeated many times. “Regularity is an important aspect for crew schedules since
regular solutions can improve operational reliability and reduce training costs. Furthermore,
regular solutions are less error-prone and crews often prefer to repeat itineraries.” (Steinzen,
2007, p. 151) Generally, regular crew schedules are much more difficult to obtain in a
distribution context than in a public transport context because of the ever changing timetables in
the former case. We do not consider the regularity of vehicle schedules as a relevant objective.
Indeed, vehicles are rather insensitive to the quality of their schedules (as opposed to crew
members).
Notice that objective (2) is aimed at enhancing the quality for customers, while (3) tries to
improve the quality for the crew (and the company).
2.1.1.5.
Size/practicality of the problem
It is difficult to unambiguously categorize the considered problem in terms of size (i.e. number of
trips, depots, vehicle types, etc.) as being small, medium-sized or large. Where do we set the
boundaries and how do we deal with the fact that these will shift over time? Because of these
limitations, it seems recommended to evaluate the size of a problem in terms of practicality. We
therefore classify the problems (and with them the papers in which they are discussed) as either
theoretical or application-based. In this thesis, the first class of theoretical problems deals
with randomly generated test instances. Mostly, the constraints considered in such a problem,
e.g. with respect to the feasibility of a duty, are kept rather simple. In the second class, the
proposed algorithms are applied to solve problems which arise from real-life applications and
thus contain many complex practical constraints.
We could state that theoretical problems generally correspond to rather small problems (i.e.
‘easy’ randomly generated test instances), whereas application-based problems correspond to
medium-sized and large problems (i.e. based on data from real-life transport companies). Of
course, one can think of a paper in which a proposed method is tested on randomly generated
instances as well as on real-life problems. In that case, the application-based classification
dominates the theoretical classification.
26
2.1.1.6.
Degree of urbanization
We immediately start with a definition: “Public transport scenarios can be categorized according
to the structure of the underlying transportation network. Urban service provides connections
within the city while ex-urban16 (regional) service connects the city with the suburbs and minor
towns in the region of the city.” (Steinzen, 2007, p. 152) Although one could notice a difference
in gradation, the terms suburban17 and ex-urban are assumed to be equivalent here. Ex-urban
services are thus defined as all services that are not strictly urban, which seems to be common
practice in transportation literature. Of course, many companies offer a mixture of both urban
and ex-urban services.
Similar general statements apply in the distribution context. Above explanation is of course
suitable for road traffic. For airlines, however, the (ex-)urban categorization is not very relevant,
because the distances traveled here far exceed the dimensions of cities. For rail traffic, we
could make the analogy by stating that trains operate within an ex-urban scenario while trams
and especially metros have a more urban character.
2.1.1.7.
Regularity of the timetable
A regular timetable is a timetable that remains the same for each day of the week, every week.
“In practice however, timetables may consist of many trips serviced every day and some
exceptions that do not repeat daily. In particular, service trips to schools, production facilities, or
public swimming baths are often subject to change, e.g., trips may be operated on every day
except Sunday or on Monday only.” (Steinzen, 2007, p.11) This results in irregular timetables.
“Notice that public transport companies face a similar situation whenever they change their
timetable, e.g., scheduled timetable changes in summer or winter. Typically, these changes
involve only a small portion of the complete timetable” (Steinzen, 2007, p. 12), as is usually the
case for irregular timetables.
We remark that regularity of the timetable should be seen in the context of presented regularity
of the timetable. For example, a certain transportation company can have an irregular timetable
in reality (e.g. different scenarios for weekdays and Sundays, i.e. more trips on weekdays), but
when this irregularity is not taken into account in the paper describing the real-life instances
(e.g. only the weekday scenario is mentioned, so the reader cannot know of the (non-)existence
16
Ex-urban is a synonym for extra-urban, but throughout this thesis we will keep using the term ex-urban.
Suburban settings have more relief opportunities and smaller distances between the depots than exurban settings.
17
27
of a Sunday scenario), then the timetable is classified as regular. When, for example, both
weekday and Sunday scenarios are presented in the paper, we deal with an irregular timetable.
The exposition above primarily relates to the public transport context. In the distribution context,
where the timetable is not given, there are many trips – usually even the vast majority – that do
not repeat daily. We can therefore state that, for distribution problems, we will nearly always
have to deal with (highly) irregular timetables.
2.1.1.8.
Admission of changeovers
A changeover is the change of vehicle of a driver whenever there is a relief point. This means
that, when changeovers are admitted, a crew is allowed to work on more than one vehicle
during a duty. We can deal separately with the situation where changeovers are and are not
allowed. The first category can be further divided, that is, the admission of changeovers can be
restricted or unrestricted. Restricted changeovers may imply assuming that “each crew is
assigned to a depot and may only conduct tasks on vehicles from this particular depot (which is
of course only a relevant criterion for multiple depot problems) and furthermore, assuming that a
driver may only change his vehicle during a break, i.e., between two pieces of work. In other
words, a restricted changeover is only allowed between vehicles from the same depot and the
driver must take a break after leaving his vehicle. […] However, when changeovers are
unrestricted, a driver may change between two vehicles of different depots whenever there is a
relief point (no matter if he takes a break or not).” (Steinzen, 2007, p. 110)
2.1.2. Categorization according to solution method
The 5 criteria for categorization according to solution method are (in descending order of
importance): degree of integration, degree of network segmentation, model, algorithm, and
dynamism of the solution approach.
2.1.2.1.
Degree of integration
For the categorization according to solution method, the most high level criterion is that
concerning the degree of integration between vehicle and crew scheduling. As already
mentioned in section 1.3.3, we can distinguish between methods for partial integration and
methods for complete integration. The former can be further subdivided into crew first 28
vehicle second and vehicle first - crew second approaches. For more information about this
subdivision, we refer back to section 1.3.3.1.
2.1.2.2.
Degree of network segmentation
To solve the scheduling problem, some transportation companies consider each route
separately (e.g. each bus line), while others consider a part of the network or even the whole
network at once.
We remark that this categorization criterion is only relevant when the proposed method is tested
on an application-based (real-life) problem. This because theoretical randomly generated
instances are almost always tailored in such a way that they can be solved as a whole.
2.1.2.3.
Model
A model defines the mathematical formulation of a practical problem. Generally, a model can be
classified
according
to
the
type
of
formulation
it
uses:
single-commodity
flow,
multicommodity flow, set partitioning or set covering. It is certainly possible that more than
one type of formulation is used in a model. We will give a very limited description of these
formulation principles. For a more detailed discussion we refer to e.g. (Toth and Vigo, 2002) and
(Huisman, 2004).
The single-commodity flow problem (also called minimum cost flow problem) consists of
“determining a least cost shipment of a commodity through a network that will satisfy the flow
demands at certain nodes from available supplies at other nodes. [...] The multicommodity flow
problem is an extension of the single-commodity flow problem, since instead of a single
commodity several commodities use the same underlying network. The different commodities
now have different origins and destinations.” (Huisman, 2004, pp. 10-11)
Given a set of elements (called the universe) and a number of sets whose union comprises the
universe, a set covering problem aims to identify the smallest number of sets whose union still
contains all elements in the universe. Note that one or more customers may be visited more
than once here. In the set partitioning problem this is not the case, because it is imposed that
each customer must be covered by exactly one of the selected routes.
The four above formulation types are the most common ones, though we mention another one
that is also used in rare cases, namely the set packing formulation. In the set packing problem,
29
given a certain universe and a family of subsets of that universe, the task is to find a
subfamily that uses the largest number of pairwise disjoint sets.
Still other types of formulations are possible, but are only used sporadically. If such a
formulation occurs in the actual categorization, we will give a brief description of it there.
2.1.2.4.
Algorithm
Basically, an algorithm defines the way of solving a practical problem that was converted into
mathematical terms forming a model (see previous subsection).
A large portion of the VCSP algorithms rely on column generation. Therefore, we present the
general structure of such an approach. We can divide a column generation-based algorithm into
following consecutive stages: the master problem, the column generation subproblem (or
pricing problem) and the construction of feasible solutions. We remark that these components
were already mentioned in section 1.1.3 about the CSP. For each of the three stages, multiple
solution techniques are possible. A general understanding of the column generation method,
together with the presentation of more specific techniques that can be used for the different
stages, will follow from the actual categorization in section 2.1.3.
When dealing with a VCSP approach based on column generation in the actual categorization,
we will always try to describe the specific techniques used for all three components in an orderly
fashion. Of course, other algorithms do exist that are not based on column generation (e.g.
metaheuristics). These will be described just as well, although the three stages applicable for
column generation will obviously not occur here.
2.1.2.5.
Dynamism of the solution approach
“Traditionally, a VCSP is solved only once, some time (this varies from a few days or weeks in
the distribution context to a few months in the public transport context) before the new timetable
starts, and it will not be changed for the whole period that the timetable is valid.” (Huisman,
2004, p. 111) Furthermore, travel times are assumed to be fixed. This is called the static
approach for solving a scheduling problem.
The basic idea of a dynamic approach is that schedules are constructed several times a day.
This means that we have to reschedule a few times during the day, reassigning vehicles and
crews to trips. Moreover, we can take into account different scenarios for future travel times.
30
(Huisman, 2004) In some papers (see (Barnhart & Laporte, 2007)) the dynamic approach to
scheduling is reflected in a real-time control phase (succeeding the three planning phases
defined in section 1.2) in which the whole process is evaluated, adjusted and maintained. As
one can imagine, a dynamic approach is much more labor intensive than a traditional static one.
2.1.3. Actual categorization
Having defined the relevant criteria, performing the actual categorization of the VCSP
approaches is now fairly straightforward. Yet, we should make some remarks about the
concrete form the categorization will take. First of all, each approach is logically denoted by the
paper in which it is presented. When multiple approaches are contained in one paper, a
particular approach will be indicated by the extra mentioning of the section in which it is
described (e.g. (Huisman, 2004), Ch. 3.3). We also notice that it is not uncommon that quite a
few papers actually present virtually the same approach. In that case, we will not categorize all
those papers individually, but only perform the categorization for the most relevant one (i.e. the
most recent or most comprehensive paper) and mention the other ones as ‘related papers’. In
order to provide an orderly overview, the discussion will be performed chronologically.
Of course, all criteria for practical problem and solution method will be evaluated. When the
categorization according to a certain criterion is not straightforwardly obvious, we will always
add a word of explanation. This is certainly useful when describing the algorithm, which is
usually not unambiguously definable in a few words, let alone one specific term (as opposed to
the model). Therefore, the algorithm will always be described somewhat more extensively.
Concerning the criterion size/practicality of a problem, we always name the specific company
from which real-life data is used when discussing an application-based approach (paper). To
provide some sort of numerical notion about the size of the treated – and thus treatable –
problems using a particular method, we also mention the maximum number of trips, depots and
vehicle types considered.
The discussion of an approach always ends with a conclusion mainly concerning the
performance of the proposed method (i.e. a summary of the computational results).
The most important remark is still to come. Although all three modes of transport – road traffic,
railway and airline – were mentioned in the corresponding criterion, we will mainly focus our
categorization on the road traffic case. This has multiple reasons.
First of all, it is quite simply so that most relevant publications on planning in transportation deal
with road traffic (mainly busses), partially because a larger benefit of integration can be
31
obtained here18. This also explains why we almost always based our discussion of a certain
problem or method on the road traffic case. Also the definitions of the categorization criteria are
primarily focused on road traffic. For example, the classification urban/ex-urban is very relevant
for road traffic, but not at all for airlines. However, for the railway case we can use the exact
same categorization criteria presented in previous sections 2.1.1 and 2.1.219. We will prove this
by categorizing an interesting approach for the VCSP in a railway context, leaving the
categorization criteria unaltered. Moreover, recent railway publications are far less numerous,
so many more relevant papers were not to be found anyway.
On the other hand, terms used in airline planning – and therefore in the categorization criteria
for airline problems – do not always entirely correspond to the terms used for road traffic (and
railway), although there certainly are some parallels to be discovered. For example, the
classification short-haul flights/long-haul flights can be seen as an equivalent for urban/ex-urban
and it was already mentioned that the process of aircraft routing is similar to vehicle scheduling
(see section 1.3.3.2). As regards solution methods, there are also some similarities to be
identified between airline and road traffic, e.g. the model introduced in (Sandhu & Klabjan,
2007) is very similar to the models used in (Hollis, 2011). The presented categorization
procedure for VCSP’s can thus fairly easily be extended to cope with airline problems.
Nevertheless, we do not explicitly do this ourselves, mainly because of the imposed limitations
in size of this thesis. Thus, an issue for future research is discovered.
The actual categorization is now performed for 18 of the most important VCSP approaches. For
the purpose of not overcharging the main body, the detailed categorization tables are included
in Appendix E. We immediately present the summarizing Table 1 here. First, a few comments
on this table. Related papers are mentioned (in italics) below the titles of the discussed
approaches. For the sake of clarity, the extra explanation of the criteria has been omitted.
Sometimes there will be two columns for one approach when listing maximum number of trips,
depots and vehicle types. This occurs when the three characteristics are maximum for different
instances. Most importantly, the algorithms are described only very briefly, using no more than a
few summarizing terms. Nevertheless, these terms should be sufficient to provide a clear view
on which type of algorithm was used (a column generation-based method, a metaheuristic,
etc.), so comparison on a general level is possible. The meaning of the abbreviations used in
the discussion of the algorithms can be found in the introductory section Abbreviations or in the
comprehensive categorization of the concerning approach in Appendix E.
18
This can be explained by the fact that the relative difference between crew and vehicle costs is higher
for road traffic than for the other modes of transportation. (Huisman, 2004)
19
Although the vehicle scheduling part of the algorithm is much more complicated for railways, but this
does not affect the categorization.
32
practical problem
transportation
problem type
mode of
transportation
number of depots
objectives
(Huisman, 2004),
Ch. 3.3
(Huisman, 2004),
Ch. 3.4
(Huisman, 2004),
Ch. 4.2-4.3
(Huisman et al.,
2005)
(Huisman, 2004),
Ch. 4.4
(Huisman et al.,
2005)
public transport
public transport
public transport
public transport
road traffic
road traffic
road traffic
road traffic
single depot
single depot
multiple depot
multiple depot
cost reduction
cost reduction
cost reduction
cost reduction
size/practicality of
application-based application-based application-based application-based
the problem
max trips
259
259
653
653
max depots
1
1
4 (avg 1.71)
4 (avg 1.71)
max vehicle types
1
1
1
1
degree of
urban
urban
ex-urban
ex-urban
urbanization
regularity of the
regular timetable regular timetable regular timetable regular timetable
timetable
admission of
restricted
restricted
restricted
no changeovers
changeovers
changeovers
changeovers
changeovers
solution method
degree of
complete
complete
complete
complete
integration
integration
integration
integration
integration
degree of network
each route
each route
part of the
part of the
segmentation
separately
separately
network
network
model
set partitioning
set covering
set partitioning
set partitioning
algorithm
dynamism of the
solution approach
Lagrangian
relaxation,
column
generation,
Lagrangian
heuristic
Lagrangian
relaxation,
column
generation,
Lagrangian
heuristic
Lagrangian
relaxation,
column
generation,
Lagrangian
heuristic
Lagrangian
relaxation,
column
generation,
Lagrangian
heuristic
static approach
static approach
static approach
static approach
Table 1 a: Summarized categorization of recent VCSP approaches
33
practical problem
transportation
problem type
mode of
transportation
number of depots
objectives
(Huisman, 2004),
Ch. 5.3
(Huisman &
Wagelmans,
2006)
(Borndörfer et
al., 2004)
(Rodrigues et al.,
2006)
(Steinzen, 2007),
Ch. 2.3-2.4
(Gintner, 2007)
public transport
public transport
public transport
public transport
road traffic
road traffic
road traffic
road traffic
multiple depot
multiple depot
single depot
multiple depot
cost reduction
and service level
cost reduction
cost reduction
cost reduction
size/practicality of
application-based application-based application-based application-based
the problem
max trips
304
1,414
634
395
653
max depots
4 (avg 1.71)
1
3
1
4
max vehicle types
1
3
5
1
1
degree of
ex-urban
urban/ex-urban
urban
ex-urban
urbanization
regularity of the
irregular
regular timetable
regular timetable regular timetable
timetable
timetable
admission of
restricted
restricted
restricted
no changeovers
changeovers
changeovers
changeovers
changeovers
solution method
degree of
complete
complete
complete
complete
integration
integration
integration
integration
integration
degree of network
part of the
whole network at
each route
part of the
segmentation
network
once
separately
network
model
multicommodity
multicommodity
set covering/
set partitioning
flow and set
flow and set
set packing
partitioning
partitioning
algorithm
Lagrangian
Lagrangian
Lagrangian
hybrid algorithm,
relaxation,
relaxation,
relaxation,
mathematical
column
column
column
programming
generation,
generation,
generation,
and (greedy)
Lagrangian
branch-andLagrangian
heuristic
heuristic
bound
heuristic
dynamism of the
dynamic
static approach
static approach
static approach
solution approach
approach
Table 1 b: Summarized categorization of recent VCSP approaches
34
practical problem
transportation
problem type
mode of
transportation
number of depots
(Steinzen, 2007),
Ch. 3.1-3.3
(Steinzen et al.,
2010)
(Steinzen, 2007),
Ch. 3.4
(Steinzen, 2007),
Ch. 4
(Steinzen, 2007),
Ch. 6
(Steinzen et al.,
2009)
public transport
public transport
public transport
public transport
road traffic
road traffic
road traffic
road traffic
multiple depot
multiple depot
multiple depot
single depot
cost reduction
cost reduction
and quality of
crew schedules
theoretical
application-based
200
4
1
433
1
1
ex-urban
ex-urban
objectives
cost reduction
cost reduction
size/practicality of
application-based application-based
the problem
max trips
653
653
max depots
4
4
max vehicle types
1
1
degree of
ex-urban
ex-urban
urbanization
regularity of the
regular timetable regular timetable
timetable
admission of
restricted
unrestricted
changeovers
changeovers
changeovers
solution method
degree of
complete
complete
integration
integration
integration
degree of network
segmentation
model
algorithm
dynamism of the
solution approach
part of the
network
multicommodity
flow and set
partitioning
Lagrangian
relaxation,
column
generation,
heuristic branchand-price
part of the
network
multicommodity
flow and set
partitioning
Lagrangian
relaxation,
column
generation,
heuristic branchand-price
static approach
static approach
regular timetable
restricted
changeovers
irregular
timetable
restricted
changeovers
complete
integration
partial
integration
(crew - vehicle)
n/a
n/a
multicommodity
flow and set
partitioning
set covering
hybrid algorithm,
mathematical
programming
and EA
Lagrangian
relaxation,
column
generation,
local and followon branching
static approach
static approach
Table 1 c: Summarized categorization of recent VCSP approaches
35
practical problem
transportation
problem type
mode of
transportation
number of depots
objectives
size/practicality of
the problem
max trips
max depots
max vehicle types
degree of
urbanization
regularity of the
timetable
admission of
changeovers
solution method
degree of
integration
degree of network
segmentation
model
(Kéri & Haase,
2007)
(Mesquita &
Paias, 2008)
(Mesquita et al.,
2006, 2011)
(Laurent & Hao,
2008)
(Bartodziej et al.,
2009)
public transport
public transport
public transport
distribution
road traffic
road traffic
road traffic
road traffic
single depot
multiple depot
single depot
single depot
cost reduction
and service level
cost reduction
cost reduction
cost reduction
theoretical
theoretical
133
1
1
400
4
1
249
1
2
urban
ex-urban
ex-urban
ex-urban
irregular
timetable
regular timetable
regular timetable
irregular
timetable
no changeovers
unrestricted
changeovers
no changeovers
no changeovers
complete
integration
complete
integration
complete
integration
n/a
n/a
n/a
complete
integration
whole network at
once
set partitioning
multicommodity
flow and mixed
set partitioning/
covering
constraint-based
set partitioning
application-based application-based
algorithm
dynamism of the
solution approach
1,400
1
n/a
779
1
27
linear relaxation,
column
generation,
round-up method
linear relaxation,
column
generation,
branch-andbound
metaheuristic,
GRASP:
constraint
programming
and local search
1) linear
relaxation and
column
generation
2) local searchbased
metaheuristics
(SA, GDA, RRT)
static approach
static approach
static approach
static approach
Table 1 d: Summarized categorization of recent VCSP approaches
36
practical problem
transportation
problem type
mode of
transportation
number of depots
objectives
(Sato et al.,
2009)
(Hollis, 2011),
Ch. 4
(Hollis et al.,
2006)
public transport
distribution
railway
road traffic
multiple depot
multiple depot
cost reduction
and service level
cost reduction
size/practicality of
application-based
the problem
max trips
786
max depots
n/a
max vehicle types
n/a
degree of
ex-urban
urbanization
regularity of the
regular timetable
timetable
admission of
restricted
changeovers
changeovers
solution method
degree of
complete
integration
integration
degree of network
each route
segmentation
separately
model
multicommodity
flow
algorithm
heuristic flow
modification,
local search
dynamism of the
solution approach
dynamic
approach
application-based
1,016
23
3
urban/ex-urban
irregular
timetable
restricted
changeovers
complete
integration
whole network at
once
set covering
linear relaxation,
column
generation,
branch-andbound
static approach
Table 1 e: Summarized categorization of recent VCSP approaches
37
2.1.4. Conclusions
In this section, we will discuss the trends that can be observed for the VCSP approaches. We
will also indicate some links between the presented criteria. Maybe even more important, we try
to discover gaps between and within the criteria (which in fact correspond to constraints, e.g. for
changeovers) so that we can identify interesting topics for future research that actually consist
of filling these gaps. Finally, we are now able to provide an answer to the first aspect of the
central research question: ‘Is it (always) advantageous to use an integrated approach for
scheduling problems in transportation?’
The order of the proposed criteria for our categorization of VCSP’s (type of transportation
problem - mode of transportation - number of depots - objectives - size/practicality of the
problem - degree of urbanization - regularity of the timetable - admission of changeovers degree of integration - degree of network segmentation - model - algorithm - dynamism of the
solution approach, supplemented by the conclusions for the presented approaches) will be used
as a guideline for the division into paragraphs of the upcoming exposition, though it is not strictly
followed.
It was already mentioned that the great majority of the papers consider a public transport VCSP,
as our categorization also confirms (only 2 of the 18 described approaches are situated in a
distribution context). However, we cannot draw too many conclusions from this regarding future
research (in the sense of ‘the expansion of distribution VCSP literature is needed’) because all
papers presented in next section 2.2 do consider the planning problem in the distribution
context, just not from a mere scheduling point of view (i.e. the development of a specific VCSP
approach) but rather integrating scheduling and routing in one general planning approach.
Therefore, suggestions for future research in the distribution context will be provided in next
section. It is also remarkable that the distribution papers are two of the most recent ones
described, the oldest dating from 2009. So indeed, planning in a distribution context is a very
recent research area.
Remember that we chose to only consider VCSP approaches for road traffic, supplemented by
one paper on railways. The exposition in this section has thus to be understood in the road
traffic context. An obvious working point that was already mentioned, concerns the adaptation of
the categorization procedure for airline problems and also its expansion in terms of number of
considered papers for the other modes of transport, especially railways.
38
Also notice that for railways, it appears from only one considered paper that difficulties like
multiple depots, a service level objective and allowed changeovers can be taken into account,
just as for road traffic.
A small majority of the approaches (11 out of 18) consider the more extensive case of multiple
depots. Sometimes authors consider an ‘easier’ single depot case because the addressed
practical problem quite simply exists of only one central depot (e.g. the RFS in (Bartodziej et al.,
2009) and the RET in (Huisman, 2004), Ch. 3.3 and Ch. 3.4), but mostly they do this to simplify
the introduction of a new solution approach (e.g. a hybrid algorithm in (Rodrigues et al., 2006), a
method that can deal with irregular timetables in (Steinzen, 2007), Ch. 6, a model that
incorporates trip shifting in (Kéri & Haase, 2007) and a novel metaheuristic in (Laurent & Hao,
2008)). An obvious subject for further research is of course to extend these solution approaches
to a multiple depot case.
Although not generally valid, one could detect a certain relation between number of depots and
degree of urbanization. In most cases, an urban context corresponds to a single depot and an
ex-urban context to multiple depots. This can be intuitively explained. In an urban context,
companies operate within the boundaries of a city, which means trips will never be of great
distance so that the vehicles should be able to return to a central depot located somewhere in
the city. In an ex-urban context on the other hand, customers are generally much more
geographically dispersed so that one single depot might not be able to service all customers
(within the imposed regulations), which means that several extra depots would be needed. As
already mentioned, this relation is not definite. It is of course possible that an urban transport
company chooses to operate from more than one depot, just as it may be that an ex-urban
transportation firm is nevertheless able to serve their customers from just one depot.
We can state that the service level objective, which aims at reducing delays for customers,
actually corresponds to applying a dynamic approach to the problem. Indeed, in both (Huisman,
2004), Ch. 5.3 and (Sato et al., 2009), increasing service level is an objective while a dynamic
approach is used to tackle the problem. The conformity of both criteria is in fact quite obvious.
When using a traditional static approach, there are no changes in the schedules for the whole
period that a timetable is valid. This means that when a delay occurs on a certain day, the next
trip to be performed by the concerning vehicle and/or driver may start late, possibly provoking
new delays which, in their turn, may cause a similar snowball effect. (Huisman, 2004) Since in a
dynamic approach we reschedule a few times per day, we can reassign vehicles and crews to
trips and thus may be able to prevent the delays at the start of a trip in many cases, improving
the service level for customers.
39
We remark that (Kéri & Haase, 2007) also have service level as an objective although they do
not use a dynamic approach. However, we already mentioned in the actual categorization of
Appendix E that although not explicitly performed in the paper, it should be logical (and possible
– regarding the computation time – for very small instances) to use the proposed approach in a
dynamic environment. We attribute the not carrying out of a dynamic approach to the very
limited size of the considered paper. Yet, it should certainly be possible to apply the proposed
method in a dynamic context, after some minor adaptations. This means the correspondence
between the service level objective and the dynamic solution approach is also affirmed by (Kéri
& Haase, 2007).
The other secondary objective – quality (regularity) of crew schedules – is also related to
another criteria, namely that of regularity of the timetable. This should not be surprising, since
crew schedule regularity is indeed only a relevant objective when the timetable is irregular
(when the timetable is regular, obviously the crew schedules will be too). If a ‘normal’ VCSP
approach is used for a problem where the timetable is irregular, regularity of the crew schedules
cannot be guaranteed. This can only be the case when the quality of crew schedules is explicitly
considered while developing the solution method. (Steinzen, 2007), Ch. 6 is the only approach
that does this.
However, we notice that there are more authors who deal with irregular timetables (in total 5 out
of 1820), but do not adapt their solution methods to take crew schedule regularity into account21.
In other words, these approaches ((Borndörfer et al., 2004), (Kéri and Haase, 2007), (Bartodziej
et al., 2009) and (Hollis, 2011), Ch. 4) will almost certainly produce irregular crew schedules,
which are undesirable for – obviously – the crew but also for the company. Future papers
should therefore pay more attention to the demand for more regular crew schedules when
dealing with an irregular timetable.
Although preventing delays from occurring should also certainly become a more investigated
subject in the future, we see that yet more approaches are concerned with service level (3) than
with quality of crew schedules (1). This is the same as saying that customers are generally
considered to be more important than employees. On top of that, a higher service level is
obtained by applying a dynamic approach, which implies that the schedules are only known just
before they should be executed, asking a more flexible way of working from the staff. Those
20
In our categorization, a clear majority of the papers consider regular timetables. However, it should be
noticed that since 2007, half of the approaches acknowledge the possible irregularity of timetables. We
therefore assume that future publications will increasingly work with irregular timetables.
21
This usually includes solving the problem for a multiple period (e.g. a week) instead of a single period
planning horizon (e.g. one day) (see section 1.1.1). One can imagine a transport company with different
scenarios for weekdays and Sundays, where planning for a single period would mean scheduling
weekdays and Sundays separately (in fact considering them as separate instances) so that the resulting
schedules would be unrelated and the entire weekly schedule would not be regular. This pursued
regularity is only possible when weekdays and Sundays are both considered in a multiple period planning
horizon.
40
crew members might very easily become frustrated by such an approach. Naturally customer
remains king, but it would not be a bad thing to tilt the attention a little more towards the
employees wellbeing.
The vast majority of VCSP papers apply the proposed approaches to real-life problems (15 of
the 18 papers are application-based). This practically oriented view can only be encouraged
and is certainly the path to be followed in the future as well. Furthermore, theoretical
approaches should also be tested on real-world instances to prove their practical relevance and
applicability.
From our survey, it appears that the largest instance tackled so far is of the size of 1,414 trips.
This is already quite large, although in practice there do exist transportation networks
comprising several thousands or even tens of thousands of trips22. Up to now, these instances
are not tractable as a whole, or at least no attempt has been made. It should be an objective to
keep pushing the size boundaries so even huge real-world problems can maximally benefit from
integrated approaches. So far, this seems not to have been a primary objective, since instances
have not really increased in size from 2004 to 2011 (the largest instance already dates back to
2004, considered in (Borndörfer et al., 2004)). The last statement is true when we consider the
number of trips as the absolute measure of size for a problem. However, we should also look at
the number of depots and vehicle types considered, because these characteristics are just as
well an aspect of the ‘size’ of an instance and actually increase the veracity of the problem (e.g.
a transport company that uses only one or two vehicle types is not always realistic). In this
respect, we do notice an evolution towards more extensive instances: (Bartodziej et al., 2009)
consider no less than 27 vehicle types while (Hollis, 2011), Ch. 4 introduces a staggering
number of 23 depots, also considering multiple vehicle types. Therefore, increasing the number
of depots and vehicle types does not seem to be a bottleneck for expanding the tractable size of
instances, the emphasis should be on the number of trips.
It is noteworthy that we consider number of vehicle types as a measure of size and thus
complexity of the problem, whereas number of crew types is not mentioned. This is because, in
almost all cases, only one single crew type is assumed (so placing a ‘1’ in every column for
number of crew types would not contribute a lot to the categorization). This is of course not very
realistic, since employees can differ significantly with respect to skills, but also rights and
responsibilities. For example, in an ex-urban distribution context, an older driver may have
earned the right to drive the shorter routes so that he does not have to spend a night away from
home, in contrast with junior employees. Seniority is only one criterion for defining different crew
types, one can easily think of many others. When dealing with irregular timetables, a distinction
22
Our figures seem to demonstrate that a whole network of a respectable transportation company
consists of at least a thousand trips.
41
between crew types can also be relevant for the objective of quality (regularity) of the crew
schedule. For example, senior employees are often spared from irregular working hours, while
starters should be willing to work overtime. Thus, a difference in importance of constructing a
regular crew schedule should be implemented, depending on crew type. It should be clear from
this exposition that the consideration of multiple crew types would be a meaningful addition for
future papers.
It is clear that most of the VCSP approaches are situated in an ex-urban context (12 out of 18,
plus 2 approaches that consider both urban and ex-urban problems). The main reason for this
observation can be found in (Huisman, 2004), stating “that it is generally expected that the
savings of using an integrated approach (VCSP) in an ex-urban setting are much more
significant than in an urban context, since there are much less opportunities to relief one driver
for another one. These reliefs are only allowed at depots and certain other specified locations,
which are much further away from each other than in the urban context. If first an optimal
vehicle schedule is constructed (i.e. traditional sequential approach), there can be vehicles
which do not pass a relief location for hours. Therefore, it is very well possible that there does
not exist a feasible crew schedule at all, or more probably, that the crew schedule will be very
inefficient.” (Huisman, 2004, p. 77) That is why an integrated approach is almost obligatory for
ex-urban scenarios and should thus be thoroughly investigated (more thoroughly than urban
scenarios).
In more than two thirds of the cases (13 out of 18), changeovers are allowed. This can be
explained by the fact that for larger problems (say of more than a couple of hundred trips)
allowing changeovers can save many drivers and vehicles, whereas an integrated approach
gives the same results for the cases with and without changeovers only for very small problems
((Huisman, 2004), Ch. 3.4). Of the scenarios where changeovers are allowed, only 2 consider
unrestricted changeovers. It is a relatively underexposed characteristic to not impose
restrictions to changeovers. Initial research showed that better and faster solutions can be
obtained for instances with more than 80 trips if changeovers are not restricted ((Steinzen,
2007), Ch. 3.4). Still, we reckon further research is needed so we can be more conclusive
regarding this aspect.
It could be suspected beforehand that almost all post-2004 approaches make use of complete
integration. In fact, only one approach still relies on partial integration ((Steinzen, 2007), Ch. 6).
The reason (Steinzen, 2007), Ch. 6 uses a partially integrated method – somewhat outdated
and less performant, but easier to implement than a fully integrated method we might state –
could be found in the fact that it is the only approach that tries to construct regular crew
42
schedules from an irregular timetable, which is an innovative and non-straightforward objective.
Since this specific problem focuses strongly on the wellbeing of the crew, it is not surprising that
the crew is scheduled first and independently of the vehicles. In the future though, the problem
should also be tackled using a completely integrated approach. Other future papers are
evidently also expected to rely on a full integration of vehicle and crew scheduling.
It was already mentioned that VCSP approaches which tackle the whole existing transportation
network at once consist of at least a thousand trips in our categorization. We detect 3 of these
approaches. We also identify 4 methods that solve each route separately and 6 that consider
part of the network. Obviously, these categorizations strongly depend on the specific real-life
problems that are considered in the paper (e.g. for a transportation company that operates
about 1,000 trips it will be easier to tackle the whole network at once than for a company that
operates 2,000 trips), but still they can give some insight of the scope of an approach. It should
be the eventual goal of every transport company to solve their whole network at once. In that
way, all possible elements (trips, depots, vehicle types, crew types, etc.) are taken into account
all together so that a maximum effect of integration can be obtained.
The set partitioning formulation is clearly the most popular one, it is used in 12 approaches, of
which half are pure set partitioning. In other ‘non-pure’ cases, set partitioning is combined with
another formulation type, mainly multicommodity flow. This multicommodity flow formulation is
mostly used only for the vehicle scheduling part of the model (while set partitioning is then used
for the crew scheduling part), which explains why only one approach employs a pure
multicommodity flow. Set covering seems to be quite a lot less popular than set partitioning.
Although, set covering formulations have the advantage that the number of variables, as the
solution space, is considerably reduced. With a view to solvability, it could thus be advisable to
base the model on a set covering formulation for larger and more complex future problems (high
number of trips, depots, vehicle types, crew types). The largest instance tackled so far
considering all measures of size combined (number of trips, depots and vehicle types) was
introduced by (Hollis, 2011), Ch. 4 and is indeed modeled using set covering, supporting the
previous argument. A more exotic formulation is the constraint-based one proposed by (Laurent
& Hao, 2008), which is specifically designed to provide a flexible basis for the introduced
metaheuristic (GRASP).
We do not elaborate too extensively on the various algorithms used, as we did not explicitly
present the mathematical formulas describing them fully (remember this was not within the
scope of this thesis). What we can observe, is that column generation is clearly the most
popular component of a solution approach. No less than 14 (out of 19, and not 18, because
43
(Bartodziej et al., 2009) propose two algorithms) solution methods are based on column
generation while in another two approaches column generation – which is a mathematical
programming method – is one of the two parts of a hybrid algorithm. This is not surprising, since
it is so that to solve a (relaxation of a) set partitioning or set covering problem – which are used
in almost every approach but a few – one has to resort to column generation (Huisman, 2004).
Indeed, the two rather unusual models – the constraint-based by (Laurent & Hao, 2008) and the
pure multicommodity flow by (Sato et al., 2009) – are not solved using column generation, but
with a novel metaheuristic (GRASP) including constraint programming and a heuristic flow
modification technique, respectively. Thus, non-mathematical programming methods are not so
well established yet. We can distinguish between heuristics and metaheuristics23. We identify
two methods that are more or less situated in the first category, namely the hybrid algorithm of
(Rodrigues et al., 2006) comprising a greedy heuristic and the heuristic flow modification
approach of (Sato et al., 2009). Also, three metaheuristics are proposed: a hybrid evolutionary
algorithm by (Steinzen, 2007), Ch. 4, a so-called GRASP by (Laurent & Hao, 2008) and some
local search-based metaheuristics (SA, GDA, RRT) by (Bartodziej et al., 2009). The hybrid
evolutionary algorithm of (Steinzen, 2007), Ch. 4 was actually the first attempt to apply a
metaheuristic to the VCSP. Since then, some other metaheuristic approaches have been
proposed, but still it remains an area for which further research is necessary. We should seek to
answer questions such as ‘Which particular metaheuristic performs best (for which particular
problem)? 24’ and ‘Can we make use of metaheuristics to solve even larger/more complex reallife problems (compared to mathematical programming25), and if so, how much larger/more
complex may those problems be?26’ in order to discover the true potential of metaheuristics.
Until now, the practical problems tackled with a metaheuristic (or heuristic or hybrid) approach
23
For a comprehensive overview of solution methods (comprising constraint programming and
mathematical programming approaches) we refer to (Ernst et al., 2004). A clear distinction between the
nature of (classical) heuristics and metaheuristics can be found in (Toth & Vigo, 2002). These authors
also state that metaheuristics outperform classical methods in terms of solution quality (and sometimes
even in terms of computing time), so there is little room left for significant improvement in the area of
classical heuristics and thus future focus should be on metaheuristics.
24
E.g. according to (Toth & Vigo, 2002) (they consider the VRP, not the VCSP), tabu search emerges as
the most effective approach. Procedures based on pure genetic algorithms (a subclass of evolutionary
algorithms) and on neural networks were clearly outperformed, while those based on simulated or
deterministic annealing and on ant systems were not quite competitive at the time (remember this was
about 10 years ago). Although it is stated that hybrid ant systems and genetic algorithms might, in the
future, be able to match the effectiveness of existing tabu search heuristics.
Also, (Bartodziej et al., 2009) conclude that the metaheuristics considered by them, “SA, GDA and RRT,
show a relatively similar convergence. […] Although, RRT has the fastest convergence at the beginning
and is only caught up if the running times are relatively large. […] On the other hand, SA and GDA are
slightly preferable when the first objective is to minimize the number of auxiliary vehicles.” (Bartodziej et
al., 2009, pp. 426-427) Finally, preference was given to RRT.
25
Generally, algorithms based on a mathematical programming approach will still achieve the lowest cost
solutions.
26
(Bartodziej et al., 2009) give a first indication of the truthfulness of this statement, since the largest
instances (> 695 trips and up to 1,400 trips) considered could not be solved by column generation, but
were indeed solvable using metaheuristics.
44
are overall not more complex (not only regarding the known measures of size, but also
considering e.g. the admission of changeovers) than the ones that are solved using column
generation. In fact, rather the opposite is true. Also, the capabilities of constraint programming
to solve highly constrained, thus very complex, transportation problems should be investigated.
A remarkable aspect concerning the algorithms based on column generation is that, although
linear relaxation was said to be more popular (see section 1.1.3), up to 2007 almost all
presented VCSP approaches used a Lagrangian relaxation to solve the master problem. This
could be explained by the fact that (Huisman, 2004) chose to relax the proposed formulation in
a Lagrangian way. Being a real landmark paper, several authors who published in the next few
years based their approach on that of (Huisman, 2004) (e.g. (Steinzen, 2007)). Since 2008,
however, LP relaxation is again most widely used, reclaiming its popularity.
Regarding the construction of feasible solutions (again for algorithms using column generation),
Lagrangian heuristics and branch-and-bound methods appear to be the preferred approaches.
Furthermore, the latter seem to have become the most popular nowadays. This evolution can
be explained by the fact that a higher quality of the algorithm (i.e. closing the gap with the lower
bound) can be obtained by using an exact (e.g. branch-and-price) method instead of
Lagrangian heuristics. Some other feasible solution construction methods have also been
proposed, as there is a heuristic branch-and-price method in (Steinzen, 2007), Ch. 3.1-3.3, a
method based on local and follow-on branching in (Steinzen, 2007), Ch. 6 and a round-up
method in (Kéri & Haase, 2007).
We already mentioned that the VCSP is solved dynamically in only 2 of the 18 approaches.
Now what is the reason that this topic has not yet received that much attention?
It was proved that for small instances with a single depot, a dynamic approach performs well to
reduce delays. However, computation times were still high for applying such an approach in
practice. On the other hand, for medium-sized instances with multiple depots – which is a much
more realistic situation – a traditional static approach with inclusion of buffer times performed
much better. ((Huisman, 2004), Ch. 5.3) This might indicate that some authors presume that the
idea of dynamically solving itself does not work so well. Nevertheless, it is recommended to
invest further research in speeding up the suggested algorithms. With faster computers and
better algorithms the dynamic approach could outperform the static one with buffer times for
larger problem instances as well. However, to the best of our knowledge, no existing paper has
yet explicitly showed that a dynamic approach may also be beneficial for larger instances,
compared to a traditional static approach with buffer times.
Previously, we related the dynamic approach to the service level objective which aims at
reducing delays for customers. This is certainly true in public transport, but does not completely
hold in a distribution context. For a distribution company, in fact, the prevention of delays is far
45
less pertinent because of the existence of time windows for deliveries, so that a certain delay
tolerance is already present. A dynamic approach in the product delivery context could instead
be understood as the possibility of including a new customer in a predefined route, during the
same day that particular customer places his order. A situation like this occurs in (Bartodziej et
al., 2009): “during operation of the fixed timetable, airlines may ask the trucking company for
additional transportation tasks on the spot.” (Bartodziej et al., 2009, p. 406) In such situations, a
dynamic scheduling approach would indeed be useful in order to rearrange the existing
schedule in such a way that it remains not far from optimal under the changed circumstances.
Such a dynamic approach in the distribution context we did however not yet encounter, but it
could be an interesting addition to the existing literature.
It is not surprising that virtually all authors present a general conclusion stating that their
approach can get good solutions (mostly also for specific real-world instances of considerable
size) within reasonable computation times. In fact, this is and should always be an obvious goal
when designing a new solution method.
More importantly, we now have enough information to solve the central research question, or at
least the first aspect of it, which regards the scheduling problem27: ‘Is it (always) advantageous
to use an integrated approach for scheduling problems in transportation?’
Actually, we should not make too much fuss about how to respond to this question, because the
answer is clearly and indisputably: ‘Yes’. We can give many literal quotes from the presented
VCSP approaches that evidence – and none that contradict – the positive answer, as we will do
here:

“The main conclusion is that we can save vehicles and/or crews by integrating the
vehicle and crew scheduling problem, which may lead to a big decrease in costs.”
(Huisman, 2004, p. 76), Ch. 3.3 and Ch. 3.4

“There are significant savings compared to the traditional sequential approach, where
first the vehicle scheduling and afterwards the crew scheduling problem is solved.”
(Huisman, 2004, p. 100), Ch. 4.2-4.3 and Ch. 4.4

“The solutions produced can be decidedly better in several respects at once than the
results of various types of sequential planning.” (Borndörfer et al., 2004, p. 20)
27
Recall that the second aspect of the central research question concerns the combined routing and
scheduling problem and will be considered in the next section.
46

“There is an efficiency gain compared to sequential planning. […] Even the number of
vehicles is always minimal, i.e., equals the number of vehicles when sequential planning
is performed, where vehicles are scheduled first (and therefore are optimal).” (Steinzen,
2007, pp. 116-117), Ch. 3.1-3.3

“The results show that there is an efficiency gain if vehicle and crew scheduling are
treated in an integrated way.” (Steinzen, 2007, p. 121), Ch. 3.4

“The approach discloses significant savings compared to the traditional sequential
approach without requiring a fully integrated solution method.” (Steinzen, 2007, p. 135),
Ch. 4

“First, one observes that the integrated approach always outperforms the sequential
one, or at least furnishes equivalent results. In particular, the savings in terms of number
of drivers are significant […]. The sequential approach provides a lower bound for the
number of vehicles that is always reached in the integrated solutions. […] Second, the
integrated approach is more powerful than the sequential one in the sense that the
sequential approach failed to solve a particular instance where the integrated approach
succeeded. […] These results show the dominance of the integrated approach over the
sequential one.” (Laurent & Hao, 2008, p. 474)
Notice that, except for (Laurent & Hao, 2008), none of the most recent VCSP papers (say dating
from the last 4 to 5 years) directly compare their approach with a traditional sequential one,
although they do make a comparison with other existing integrated approaches (see below).
This could be perceived as the existence of the general conception that, whatever the specific
problem is, an integrated VCSP approach will always prove to be more advantageous than a
sequential one, so that an explicit comparison is not useful anymore. Moreover, it is self-evident
that when a particular integrated method is more efficient than another earlier one – which has
been proven to outperform the sequential approach – that new method will of course also be
more beneficial than the sequential approach.
Given the amount of VCSP approaches we considered in this thesis – we did not just evaluate
the performance of one single proposed method relative to the traditional sequential approach –
we may assume to have the ‘right’ and the opportunity to state that it is always beneficial to use
an integrated approach for transportation scheduling problem, which can be seen as an
innovative contribution of this thesis.
47
Although the use of an integrated method is always recommended, it can prove more or less
beneficial depending on the specific conditions, reflected by the different criteria. For example,
we already mentioned in this section that it is generally expected that the savings of using an
integrated approach (VCSP) in an ex-urban setting are much more significant than in an urban
context. (Laurent & Hao, 2008) confirm this by stating that “the integrated approach is
indispensable especially when relief opportunities are rare” (Laurent & Hao, 2008, p. 475), so in
an ex-urban context. There are still other conditions that may influence the degree of profitability
of an integrated approach. We will list some of them here.

“The interpretation of the results depends on the ratio between the fixed vehicle and
crew costs. If fixed vehicle costs are much higher as compared to crew costs it becomes
less attractive to apply the integrated approach. On the other hand, if crew costs are
higher the integrated approach becomes more attractive.” (Huisman, 2004, p. 62), Ch.
3.3

“In the case that no changeovers are allowed, the benefit of integration may be very
significant because more vehicles and/or crews can then be saved.” (Huisman, 2004, p.
76), Ch. 3.4

“When we do allow changeovers, it is possible to reduce the total costs by allowing
changeovers more often, so by making the changeovers less restricted.” (Huisman,
2004, p. 76), Ch. 3.3
(Steinzen, 2007), Ch. 3.4 confirms this by stating that “it is worthwhile for planners in
practice to allow unrestricted changeovers since the additional flexibility results in
efficiency gains.” (Steinzen, 2007, p. 121)
The reader must not forget that an integrated approach is still advantageous over a sequential
one, even if the conditions are such that only a lower profitability can be obtained by integration.
Some authors do in fact explicitly compare their approaches to other existing methods. We will
not present these results here (interested readers are referred to the conclusions of the
comprehensive categorization in Appendix E), because it falls outside the scope of this thesis to
identify the ‘best’ available method at this time.
It should be mentioned that – besides the fact that not all authors directly compare their
approach to others – another important difficulty in the search for the ‘best’ VCSP method is the
absence of lower bounds in more than a few approaches. For instance, (Rodrigues et al., 2006)
do not compute lower bounds and, consequently, the quality of their solutions cannot be
48
assessed. Also for (Kéri & Haase, 2007) it should be noticed that the approach was not
compared with any other existing method from literature, nor were lower bounds calculated.
Just as for (Laurent & Hao, 2008), who themselves mention that “a more complete assessment
would compare the results with tight lower bounds, which are unfortunately unavailable yet.”
(Laurent & Hao, 2008, p. 474) Therefore, identifying the ‘best’ integrated scheduling method for
transportation problems is left to future papers.
2.2.
Evolution of the VCRSP
A Vehicle and Crew Routing and Scheduling Problem (VCRSP; see section 1.4) concerns the
combination of the routing problem and the scheduling problem, which only occurs in the
distribution context (e.g. freight distribution, postal organizations and delivery services). We do
not encounter VCRSP’s in a public transport context, because line routes (and generally also
timetables28) are fixed here, dictated for example by a local authority. In section 1.4, the subject
of VCRSP was not extensively discussed because no relevant pre-2004 papers were to be
found. The last few years though, some papers have been devoted to the planning process in a
distribution company. We will discuss these papers chronologically in order to give an overview
of the evolution in this area of research. More specifically, we examine to what extent and in
which way the routing aspect and the scheduling aspect are integrated. Recall that we are now
dealing with an integration that is higher-level than the one in the previous section, which was
limited to the integration within the scheduling problem (or in other words, the VCSP). Note that
the methods presented here will not be categorized in the same extensive manner as we did for
the VCSP methods in section 2.1, mainly because only few papers on the VCRSP exist so that
an extended categorization would not (yet) be very significant. Still, we can provide a
meaningful two-dimensional classification for the discussed VCRSP approaches.
This section is divided into following subsections. First (section 2.2.1) we provide an overview of
existing approaches for the planning problem in the distribution context (including a twodimensional classification) and afterwards (section 2.2.2) we draw conclusions based on this
overview, identifying gaps within the research area with the aim of detecting topics for future
research. Finally, we try to answer the central research question related to the VCRSP.
28
This was not the case in (Rodriguez et al., 2006) and (Kéri & Haase, 2007) discussed in the preceding
section 2.1, where the timetable was prepared during these approaches. This comes down to the
inclusion of the timetabling phase (which is sometimes seen as a part of the operational phase – see
section 1.2 – and therefore its occurrence in a scheduling problem is partially explained) in the solution
approach. Of course, once the timetables are defined (timetables apply for a long period in public
transport), we fall back to a standard VCSP.
49
2.2.1. Overview of existing approaches
In order to provide a starting base, we repeat the definition of the traditional sequential
approach for routing and scheduling. The traditional planning process for distribution problems
typically involves constructing a set of vehicle routes, then producing vehicle schedules by
assigning vehicles to the vehicle routes, and building a set of crew schedules so that all vehicle
routes are covered. (Hollis, 2011)
As was also indicated by (Drexl et al., 2011), most recent solution methods for distribution
problems are based on the recurring idea of decomposing the considered problem into several
stages and, in first stages, take some aspects into account which are needed for obtaining
feasible solutions in later stages (which actually comes down to partial integration). Yet,
different approaches in doing this are possible. As already mentioned in the introduction of this
section, we can identify two dimensions to categorize the different approaches. (Drexl et al.,
2011) provided us with the inspiration for this classification, but did not develop an actual
categorization themselves, nor did they make the connections we will present here.
A first dimension for categorization concerns the allowance of changeovers. On the one hand,
there are papers in which drivers can change vehicles only at a central depot. Obviously, this
comes down to a single depot case, more specific in an ex-urban context (where distances are
such that changeovers are often restricted to only take place at the depot(s)). On the other
hand, there are papers where changeovers are allowed in other relief points as well, possibly
but not necessarily depots. Normally, this comes down to a multiple depot case, although an
arrangement with a single central depot supplemented by several other relief points – that are
not depots – is also possible. The locational exibility of the second class adds an additional
degree of freedom. Hence, problems in the second class turn out to be significantly harder to
solve than those in the first class.
A second dimension regards the solution method used. Three approaches can be identified: (1)
first determining routes and assigning concrete vehicles and crew members afterwards, (2)
calculating routes for predetermined crew-vehicle pairs, and (3) performing a direct selection of
vehicles, crews and tasks. Remark that (2) actually takes the opposite way of (1), with respect
to the sequence of routing and scheduling. We point to the clear similarities of this classification
with the ‘degree of integration’-criterion for the VCSP approaches introduced in section 2.1.2.1.
(1) and (2) could both be seen as partial integration approaches, while (3) is more of a strive for
complete integration. When we make a further subdivision, the routing first - scheduling second
50
approaches of (1) are parallel to vehicle first - crew second for a VCSP, and the scheduling first
- routing second of (2) to crew first - vehicle second.
Notice that the two defined dimensions coincide perfectly with the two overarching
categorization aspects – practical problem and solution method – defined for the VCSP in
section 2.1. The first dimension deals with the admission of changeovers, while the number of
depots and degree of urbanization are also mentioned, all known criteria for the practical
problem. It was already explicitly mentioned that the second dimension regards to the solution
method used, more specifically the degree of integration. These observations prove that it
would certainly be possible and quite straightforward (due to the similarities) to design a
categorization for the VCRSP that is parallel and just as elaborate as the one introduced for the
VCSP in previous section. As already said, the existing literature on the VCRSP might not yet
be extensive enough to develop such an elaborate categorization at this time, but it certainly is
an interesting possibility for the future.
The relevant recent publications on distribution problems that we will consider here are, in
chronological order: (Hollis et al., 2006), (Xiang et al., 2006), (Laurent & Hao, 2007), (Zäpfel &
Bögl, 2008), (Kim et al., 2010), (Prescott-Gagnon et al., 2010) and (Drexl et al., 2011). This
selection is not arbitrarily, but corresponds to the VCRSP publications deemed most important
by the last mentioned authors. In Table 2, we categorize the papers according to the two
defined dimensions.
partial integration
changeovers only at
central depot
changeovers at
multiple relief points
complete
integration
routing - scheduling
scheduling - routing
(Xiang et al., 2006)
(Zäpfel & Bögl, 2008)
(Laurent & Hao, 2007)
(Prescott-Gagnon et al., 2010)
-
(Hollis et al., 2006)
(Drexl et al., 2011)
-
(Kim et al., 2010)
Table 2: Two-dimensional classification of relevant VCRSP papers
A brief chronological overview of above papers, describing the specific problems that are
considered and the way these problems are tackled, is included in Appendix F. Since (Drexl et
al., 2011) already presented such an overview, it would not be a contribution to do the exact
same thing ourselves. For the sake of completeness and to make verification of our presented
two-dimensional classification possible, we chose to include the (Drexl et al., 2011) overview in
Appendix F nonetheless. Also, the results from computational studies mentioned in the
51
concerning papers themselves were added. We will need those to draw our conclusions in next
section, however the relevant parts will also be cited there.
2.2.2. Conclusions
It is noteworthy that, although it are always vehicles and drivers that need to be scheduled in
the papers, the concrete application contexts are almost all different (limousine rental, mail
distribution, oil delivery, just to name a few). Consequently, we could say that a fairly wide
research area is already being covered, regarding the considered practical problems. It is
certainly a good thing that this kind of variation is propagated by the different papers, so
possible benefits of integration can be obtained for every particular distribution problem. On the
other hand, VCRSP methods that are initially designed for one specific application (e.g. oil
delivery) and perform well for it, should be adapted in order to make it more generally applicable
for other practical problems (e.g. mail distribution) as well. This defines a first recommendation
for further research.
Although the categorization of the papers in Table 2 is rather limited, some conclusions can be
derived from it. First of all, we see that most papers consider a setup where changeovers are
only allowed at the central depot. This single depot case is mostly assumed because it is
remarkably easier to solve than the scenario with admittance of changeovers at multiple relief
points, usually depots. Multiple depot instances are however very common in a distribution
context and are furthermore more comprehensive. That is why it is recommended for upcoming
papers to focus somewhat more on multiple depot problems, where changeovers are not
constricted to only take place at a single location.
When we take a look at the other dimension, it immediately stands out that virtually every
publication on the VCRSP so far uses a partially integrated approach. The routing first scheduling second approach proves to be the most popular one. This is not surprising, since it
corresponds best to the natural flow of the traditional planning process for distribution problems.
A gap is identified for the scheduling first - routing second methods that solve a problem where
changeovers are allowed at multiple relief points. This void should be filled, evaluating whether
such an approach is more or less beneficial than one based on routing first - scheduling second.
An advantage might indeed show, given that crew costs are mostly dominant (certainly for road
traffic) so that scheduling of crews becomes the most important aspect, meaning it would be
beneficial to construct the crew schedules as early on as possible in order to attain a better
approximation of optimality. However, a more important gap to be filled is that of VCRSP
52
approaches that pursue a complete manner of integration. Up to now, only one paper has made
such an attempt, namely (Kim et al., 2010). Although immediately considering the more difficult
case where changeovers are allowed at multiple relief points, the authors do not present any
relevant computational results and moreover do not develop any good lower bounds for the
problem. Consequently, the solution quality of the proposed algorithm cannot yet be evaluated.
Testing the approach of (Kim et al., 2010) on a practical instance is an idea for future research,
but more importantly, other approaches pursuing complete integration should be developed
(and applied, followed by an evaluation).
It does not take a lot of insight to detect that the approaches for the VCRSP clearly lag behind
those for VCSP, not only in number but also and foremost in terms of completeness of
integration. Whereas almost all recent VCSP papers present an approach based on complete
integration, for VCRSP papers this is actually a virtually non-existent research area. Of course,
distribution problems are more extensive since the routing of vehicles has to be performed as
well, on top of the scheduling of those vehicles and the crews. In section 1.2 we also mentioned
that public transport problems are generally seen as more important than distribution problems
because of the presence of certain specific difficulties (e.g. a more delicate trade-off between
customer and cost), which too could explain and to some extent ratify the lead of VCSP
approaches on VCRSP methods. Nonetheless, we feel it is time to dedicate some extra
attention and effort to the VCRSP, since distribution problems are all the same an import aspect
of today’s transport issues. (Hollis et al., 2006) actually provide a perfect indication of the
current subordination of integration between routing and scheduling (VCRSP), as they consider
a distribution problem, but with a focus on the introduction of a new VCSP method, only useful
for the scheduling part of course.
Although we managed to identify some topics for further research, the categorization in Table 2
is to limited to be able to recognize all the interesting future possibilities. An extension of this
categorization towards a one as presented for the VCSP approaches in section 2.1 was already
said to be possible, and would indeed prove very useful to identify more and clearer gaps in the
research area and therefore provide a base for more comprehensive recommendations for
future investigations. The design of such an extended categorization for VCRSP’s can thus in
itself be understood as a subject for future research.
To conclude, we try to answer the second aspect of the central research question29, concerning
the combined routing and scheduling problem: ‘Is it (always) advantageous to use an integrated
approach for combined routing and scheduling problems in transportation?’
29
The first aspect of the central research question concerned just the scheduling problem (VCSP) and
was already answered in section 2.1.4.
53
First of all, let us remark that providing a conclusive answer to this question is not so easy as for
the VCSP, merely because of the fact that far less papers are available on the VCRSP. Another
significant problem is that some authors do not even give any relevant prove of their VCRSP
approaches merely being ‘good’. For example, (Kim et al., 2010) present no computational
results whatsoever and admit that good lower bounds for the problem still need to be developed
to evaluate the solution quality of the proposed algorithm. They only show that their proposed
approach outperforms a simple greedy algorithm, which does not say much about the actual
solution quality. Also (Prescott-Gagnon et al., 2010), who do obtain computational results on
some instances derived from a real dataset, only mutually compare their own proposed
methods, without testing them to existing approaches or providing lower bounds. Again, this
does not really prove anything about the actual solution quality.
Most of the authors though, do more or less evidence that their approach is ‘good’. Some literal
quotes:

“The solution technique employed […] has been shown to find high quality solutions. […]
Moreover, Australia Post is currently using software developed based upon the ideas
presented in this paper to assist in the management of ongoing changes to the mail
distribution networks in major cities throughout Australia.” (Hollis et al., 2006, p. 149)

“The performance of the heuristic was evaluated by intensive computational tests on
some randomly generated instances, revealing that this method is capable of quickly
obtaining acceptable results. Moreover, the method is flexible to cope with many
practical constraints and does not contain any case-sensitive empirical parameter.”
(Xiang et al., 2006, pp. 1135-1136)

“Within a short time, the software supplies very good quality schedules in which the
major part of the trips is assigned, satisfying all constraints. […] The approach also
proves to be flexible. […] Moreover, the decision support system based on this research
is operating in the examined company and proves to be extremely useful.” (Laurent &
Hao, 2007, p. 557)

“Computer simulations have demonstrated that the constructed heuristic is suitable for
practice.” (Zäpfel & Bögl, 2008, p. 980) “All in all, the tabu search procedure can solve a
complex decision problem with a tremendous number of variables and constraints in an
efficient manner.” (Zäpfel & Bögl, 2008, p. 995)
54

“First and foremost, the presented algorithm is shown to be capable of solving large realworld instances and is able to achieve consistent and practically relevant results.” (Drexl
et al., 2011, p. 16)
Of course, the central research question does not ask whether an integrated approach is ‘good’,
but whether it is better than a traditional approach. Only two papers respond to this question, in
some sense, explicitly:

“The algorithms and solution techniques presented in the paper have been used by
network planners at Australia Post to demonstrate a potential transport network
operational cost saving of 10% compared to the actual practice for the 2003 Melbourne
metropolitan mail distribution network.” (Hollis et al., 2006, p. 149)

“Results obtained on real data sets show a significant improvement in terms of quality,
operational costs and elaboration time, compared with the actual practice in the
examined company.” (Laurent & Hao, 2007, p. 557)
The actual practice for both Australia Post (Hollis et al., 2006) and the examined company in
(Laurent & Hao, 2007) relied on the traditional planning process for distribution problems
described at the very beginning of section 2.2.1. So indeed, we can state that these two papers
prove that an integrated approach for the VCRSP is more advantageous than a traditional one.
In other words, it looks as if an integrated approach for routing and scheduling has indeed the
potential to be beneficial, but for the present, too little proof is provided to make this into a
general conclusion. Moreover, there is still the aspect of the integrated approach always being
advantageous. (Drexl et al., 2011) actually come to the conclusion that, for their given data set,
lorry/driver changes offer no savings potential, so that making a fixed lorry-driver assignment
would be the best choice. This in fact means that, for their specific problem, solving it as a
standard VRP is more beneficial than approaching it as a VCRSP. However, the authors stress
that these results are only valid for the considered business field and that the developed
algorithm may well lead to very different results with other data or in other application areas.
Anyhow, the statement of an integrated approach for combined routing and scheduling always
being more advantageous seems to have been invalidated.
Summarizing, there certainly is potential in the integration between routing and scheduling, at
first sight less pronounced than for the VCSP, but in any case more research in the area of
VCRSP is needed in order to evolve towards a universal conclusion. This further research can
thereby lead to new methods (those existing today are in fact only the first steps) which may
55
well ever produce better results than the traditional approach, so we could then wholeheartedly
answer ‘Yes’ to the central research question, just as for the VCSP.
56
3. Future research
As already mentioned, it is beyond the scope of this thesis to propose improvements for specific
existing solution methods, more particular for models and algorithms. We rather identify more
general and problem-inherent future research topics. Those were already deduced and
presented in sections 2.1.4 and 2.2.2 for the VCSP and the VCRSP, respectively. We will now
repeat the resulting recommendations in an orderly fashion (in order of increasing specificity, so
starting with the most general topic) and also mention – when available – some quotes for each
topic, literally extracted from the considered papers (on VCSP as well as on VCRSP) that
substantiate the relevance of the suggested recommendations for future research.
First for the VCSP:

Continue on the path of mainly application-based papers and test initially theoretical
approaches on real-world instances to prove their practical relevance and applicability.
-
“There can be quite some practical situations, which require more research.”
(Huisman, 2004, p. 149)
-
“The intention now is to submit both the presented algorithm and the interface to
intensive field tests at selected companies.” (Rodrigues et al., 2006, p. 861)

Future papers should all rely on a full integration of vehicle and crew scheduling.
-
“Computational results for the partially integrated (ex-urban) vehicle and crew
scheduling indicate that the regularity can be improved while maintaining cost
optimality30. However, we left a fully integrated consideration for future research.”
(Steinzen, 2007, p. 179)

It should be the eventual goal of every transport company to solve their whole network at
once so that a maximum effect of integration can be obtained.
30
This concerns the (Steinzen, 2007), Ch. 6 VCSP approach which was categorized in section 2.1.3.
Indeed, this approach had the objective of producing regular crew schedules and was the only approach
considered that relied on a partial integration.
57

Shift the attention a little from customer satisfaction (service level objective) towards
wellbeing of employees (quality of crew schedules objective), although customer
remains king.

Adapt the presented categorization procedure of section 2.1 for airline problems and
expand it in terms of number of considered papers for the other modes of transport,
especially railways.

Focus more on ex-urban (than on urban) scenarios, since an integrated approach is
almost obligatory in that setting.

Take into account the demand for more regular crew schedules when dealing with an
irregular timetable.
-
“We deem it worthwhile to include timetable considerations (i.e. irregular
timetables still leading to regular crew schedules) into the integrated treatment of
multiple-depot vehicle and crew scheduling.” (Steinzen, 2007, p. 180)
-
“We suggest to continue research on aspects related to the quality of work
conditions (i.e. regular crew schedules).” (Steinzen, 2007, p. 180)

Introduce a dynamic scheduling approach for the distribution context.
-
“A next step will be the integration of the system with the dispatching system, i.e.
to allow feasibility checking and the proposal of optimal insertions of ad-hoc trips
into RFS-plans31 in a dynamic environment.” (Bartodziej et al., 2009, pp. 428429)

Extend solution approaches designed for a single depot case to a multiple depot case,
especially when a single depot was assumed to simplify the introduction of a new
solution method.

Pay more attention to the development of dynamic scheduling approaches (to prevent
delays from occurring), including explicitly showing that a dynamic approach may also
31
Remember a RFS is indeed a distribution service.
58
be beneficial for larger instances, compared to a traditional static approach with buffer
times.
-
“We suggest to continue research on faster algorithms to solve the dynamic
integrated vehicle and crew scheduling problem, since the dynamic approach
was only able to solve small problem instances due to the large computation
time. […]Moreover, we suggest to test such a dynamic approach in a real-world
environment such that our assumptions can be checked and to see if such an
approach also works in practice.” (Huisman, 2004, p. 149)
-
“Introducing the ideas of dynamic approach in airline and railway environments
would be an interesting subject for future research.” (Huisman, 2004, p. 149)
-
“We suggest to continue research on aspects related to the quality of vehicle and
crew schedules such as robustness32.” (Steinzen, 2007, p. 180)
-
“Let us mention that the general solution approach shown in the paper is also
suitable for dynamic adjustment of schedules by local re-optimization.” (Laurent
& Hao, 2008, p. 475)
-
“For future work, the proposed solution method needs improving of the search
efficiency to handle larger disruptions.” (Sato et al., 2009, p. 150)

Keep pushing the size boundaries (especially number of trips, since increasing the
number of depots and vehicle types does not seem to be a bottleneck) of instances so
even huge real-world problems can maximally benefit from integrated approaches.
-
“Although some progress has been made over the past years, we are not aware
of an approach that could deal with several thousands or even tens of thousands
of trips. However, problem instances of such size with many depots are common
in big cities such as the German towns of Munich, Hamburg, or Berlin. Therefore,
we suggest to pursue further research on this topic.” (Steinzen, 2007, p. 180)

Consider multiple crew types instead of by default assuming only a single crew type.
32
A more robust solution is understood to be one where disruptions in the schedule (due to delays) are
less likely to be propagated into the future, causing delays of subsequent trips. This exactly corresponds
to the objective of dynamic scheduling.
59
-
“Our approach can easily be extended to the case with non-identical crews.”
(Steinzen, 2007, p. 62)

Further research is needed to be more conclusive regarding the possible advantage of
allowing unrestricted changeovers (compared to restricted changeovers).
-
“For future work, there are several conditions to be considered according to each
situation, which have not been covered by the proposed solution method. These
include rides that crews take without driving or conducting (corresponds to
making changeover less restricted).” (Sato et al., 2009, p. 150)
-
“An even more involved extension of our problem is to allow that a change of
driver/lorry at a relay station is performed without the driver taking a daily rest
before switching to another lorry (i.e. allowing unrestricted changeovers).” (Drexl
et al., 2011, p. 18)33

Identify the ‘best’ integrated scheduling method for transportation problems, which asks
for the computation of (tight) lower bounds and/or the comparison of the solutions
obtained by different approaches34.
-
“A more complete assessment would compare the results with tight lower
bounds, which are unfortunately unavailable yet.” (Laurent & Hao, 2008, p. 474)
-
“It is also important to design a mathematical solution method based on the
relaxation techniques. Because the mathematical method provides such a good
lower bound of the optimal solution, we are able to numerically evaluate the
capabilities of the proposed method.” (Sato et al., 2009, p. 150)

Base models on a set covering formulation to increase solvability of larger and more
complex future problems.
33
This paper actually considers a VCRSP, but the specific aspect of changeovers is clearly mainly – if not
only – related to the scheduling part of the problem, and thus to a VCSP.
34
Preferably calculated on the same computer, considering the exact same problem (or with only very
little variations, possibly needed to make a specific algorithm applicable for the problem).
60

Explore the capabilities of constraint programming to solve highly constrained, thus very
complex, transportation problems.

Further research in the area of metaheuristics35 is necessary, not only to identify which
particular metaheuristic performs best, but primarily to discover their true potential for
solving
larger/more
complex
real-life
problems
(compared
to
mathematical
programming).
-
“Further research in the field of metaheuristics will focus on how to partition the
trips assigned to a depot among vehicles and drivers with a local search
heuristic.” (Steinzen, 2007, p.135)
-
“We can evaluate whether a mathematical method has the potential to be more
suitable for the crew/vehicle rescheduling than the proposed heuristic
method.”(Sato et al., 2009, p. 150)
-
“Further research directions include the application of other metaheuristics.”
(Zäpfel & Bögl, 2008, p. 995)36
And for the VCRSP:

Dedicate some extra attention and effort to the whole of the VCRSP, since it is
underdeveloped compared to the VCSP while nonetheless, distribution problems are an
import aspect of today’s transport issues.

More general research on the VCRSP is needed for progressing towards a valid
conclusion whether or not an integrated approach is (always) more advantageous than a
traditional one. Authors should therefore explicitly compare their VCRSP method to the
traditional approach (or to other VCRSP methods).

Develop, apply and evaluate approaches pursuing complete integration.
35
For completeness, we could in fact also consider (classic) heuristics.
Although the authors actually consider a VCRSP, the metaheuristics are used for the scheduling part
so the mentioning of the paper is relevant here.
36
61
-
“Good lower bounds need to be developed to evaluate the solution quality of the
proposed algorithm37 and future algorithms, which will be devised.” (Kim et al.,
2010, p. 8430)

Test existing theoretical approaches on practical instances.

Design an extended categorization for VCRSP’s (Table 2 can provide a basis),
analogous to that for VCSP’s of section 2.1.

VCRSP methods that are initially designed for one specific application and perform well
for it, should be adapted in order to make it more generally applicable for other practical
problems as well.
-
“Although only the static dial-a-ride problem is solved in this paper, some
proposed techniques can be used to solve other problems.” (Xiang et al., 2006,
p. 1136)
-
“This paper deals with a real-world driver and vehicle scheduling problem in a
particular application context. Even if some aspects are specific, others are
general ones. In particular, the notion of simultaneous scheduling of drivers and
vehicles is quite general and relevant to many other scheduling applications.”
(Laurent & Hao, 2007, p. 557)
-
“The current problem can also be extended in many ways. These include
introducing time window constraints requiring that the service at each customer
starts within an associated time window.” (Kim et al., 2010, p. 8430)
-
“As a future research direction, one can consider extending the proposed
heuristics to treat a multiple product version of the problem where the vehicles
have several compartments of fixed sizes.” (Prescott-Gagnon et al., 2010, p. 14)
-
“The developed algorithm for simultaneous vehicle and crew routing and
scheduling may well lead to very different results with other data or in other
application areas, so that its further study is justified.” (Drexl et al., 2011, p. 18)
37
The algorithm proposed by (Kim et al., 2010) does indeed rely on complete integration.
62

Focus more on multiple depot problems, where changeovers are not constricted to only
take place at a single location.
-
“Further research directions include the extension of the solution concept to
multi-depot combined vehicle routing and personnel planning problems, which
arise when connected regions of a logistic service provider must be included in
this context.” (Zäpfel & Bögl, 2008, p. 995)
-
“A deeper study of problems where elementary objects may join and separate on
the fly at many different locations constitutes a challenging research area.” (Drexl
et al., 2011, p. 18)

Fill the void of scheduling first - routing second methods for solving a problem where
changeovers are allowed at multiple relief points, evaluating whether such an approach
is more or less beneficial than one based on routing first, scheduling second.
We already indicated that and explained why the VCRSP recommendations for future research
are quite a lot less comprehensive and concrete than those for VCSP. Now this also clearly
shows from the above listings. By adequately following the first and fifth recommendation for the
VCRSP, this gap can however quite easily be closed. Indeed, more relevant papers on the
VCRSP will ensure a broader base for coming to the right conclusions and an extensive
categorization of these papers may lead to the identification of many concrete research topics,
as was the case for the presented VCSP categorization.
Notice that almost half of the suggested research topics were not explicitly mentioned in the
considered papers. This certainly does not mean that these recommendations are irrelevant. In
fact, there are three equally important reasons which ratify their absence. A first one is that we
also included recommendations that are most probably already known and followed by the
collection of authors and were thus not explicitly mentioned by them (e.g. focus more on exurban (than on urban) scenarios, since an integrated approach is almost obligatory in that
setting). We however did indicate these topics for the sake of completeness. Secondly, some
subjects may be ‘new’ because we used a very broad and general view on the planning process
in transportation, whereas the vast majority of papers focuses on a rather specific problem. For
instance, research issues such as shifting the attention a little from customer satisfaction
towards wellbeing of employees can only be identified when looking at the bigger picture. And
third, a few topics were only revealed through the introduction of our own categorization
63
procedure, so will of course be presented for the very first time in this thesis. An example is the
recommendation for filling the void of scheduling first - routing second VCRSP methods for
solving a problem where changeovers are allowed at multiple relief points.
Furthermore, it is remarkable that many authors seem to acknowledge the possibility and
importance of dynamically approaching the scheduling problem. However, we observed that
only very few papers actually present such a dynamic approach. We have thus identified a high
need and a low presence, so consequently, dynamic scheduling should be treated as one of the
hottest amongst the topics. For the VCRSP, the generalization of methods specifically designed
for a particular problem, so that they can be applied for other problems as well, is the most
frequently mentioned research subject. This is of course not surprising, since it concerns a very
general recommendation that actually reflects the logical lifecycle of a newly developed solution
method.
We have to remark that not all research topics will be completely satisfiable at this point,
because quite a few of them entail an increase in problem complexity and thus a decrease in
tractability. These difficulties can only be overcome by faster computers and better algorithms.
The latter aspect is an obvious point of research (e.g. “the quality of the algorithms to solve the
integrated vehicle and crew scheduling problem can potentially be improved” (Huisman, 2004,
p. 149) and “we suggest to pursue further research on faster solution procedures for integrated
problems” (Steinzen, 2007, p. 180)) and can be directly influenced by the transportation
planning researchers themselves, whereas the former aspect cannot. It should therefore be
noted that the evolution of technology (i.e. computers) is a real factor concerning the tractability
of a specific transportation problem and thus forms an actual part of future research, although
planning researchers have no direct impact on it. Because researchers (i.e. the authors of the
considered papers) have no direct impact on it, the topic of improving computer performance
was not listed. Moreover it should never be the standard procedure to just wait until there is a
computer strong enough to solve a particular complex problem with an already existing
algorithm38. That is not scientific progress, developing better algorithms that can solve the
complex problem on an ordinary computer is.
38
Besides, some aspects of computation time are independent of the speed of the computer and thus
also apply for even the most powerful computers. E.g. the computation time for solving a problem
optimally can explode for a certain number of depots, if that number only slightly increases. (Huisman,
2004)
64
4. General conclusion
In this thesis, we discussed the routing and scheduling of vehicles and crews, perhaps the most
important problems faced in today’s transport companies. This thesis thus actually covered the
whole of the planning process in transportation.
Before starting with our actual exposition, we defined the year 2004 (more precisely January 1 st
of this year) as the transition point between past and present, primarily because the central
research question of this thesis, ‘Is it (always) advantageous to use an integrated approach for
planning problems in transportation?’, was not considered until then.
For the past, we provided the definitions for the individual routing and scheduling problems –
the Vehicle Routing Problem (VRP), the Vehicle Scheduling Problem (VSP) and the Crew
Scheduling Problem (CSP). In short, we could say that the VRP constructs routes so that a
number of customers can be serviced with a fleet of vehicles, while the VSP assigns vehicles to
cover these routes and the CSP allocates crews to operate the vehicles (and routes). The main
objective is always to minimize total costs, taking into account certain constraints.
We then situated these individual problems within the larger whole of the planning process in
transportation companies, which is divided into a strategic, tactical, and operational phase. The
routing (VRP) is mainly a strategic aspect, whereas the scheduling (VSP and CSP) is part of the
operational phase. We also elaborated on the distinction between public transport (transporting
passengers) and distribution context (delivering goods to customers), which was – just like the
distinction between single and multiple depot problems – an important guideline throughout the
thesis. In public transport, routes and timetables are mostly given (e.g. by a local authority) and
remain unchanged for a long period of time, which is not the case for distribution companies.
This led to the understanding that the combination of routing and scheduling is generally a
concern only for those distribution companies, while in public transport the focus lies mainly on
the mere scheduling of vehicles and crews. Planning in a distribution context thus comprises
more aspects, but is certainly not intrinsically more difficult or more important than the public
transport counterpart, where there is a much more delicate trade-off between customer and
cost.
The scheduling problem was described, first the traditional sequential approach (which basically
seemed to come down to solving the CSP, because this always incorporates the solution of the
VSP first) and then the integrated Vehicle and Crew Scheduling Problem or VCSP (including a
brief pre-2004 literature review), defined in both public transport and distribution context to point
out similarities and differences. An critical aspect is the possibility of introducing time windows in
a distribution context, which is never done for public transport, where trips have to be punctual.
65
If the VCSP is about integration within the scheduling, the problem referred to as Vehicle and
Crew Routing and Scheduling Problem (VCRSP) represents the strive for integration between
routing and scheduling, and therefore only appears in a distribution context. It was proven that a
VCRSP can be seen as some kind of merger between a VRP and a VCSP.
The VCSP papers from the present were categorized according to a new procedure presented
in this thesis. Of course, we first defined the categorization criteria, which either related to the
practical problem considered or the solution method used for it. In descending order of
importance, we identified 8 criteria of the first kind (type of transportation problem, mode of
transportation, number of depots, objectives, size/practicality of the problem, degree of
urbanization, regularity of the timetable, and admission of changeovers) and 5 of the second
(degree of integration, degree of network segmentation, model, algorithm, and dynamism of the
solution approach).
The actual categorization was then performed for 18 of the most important VCSP approaches,
along with the mentioning of a dozen other papers strongly related to them. We chose to only
consider VCSP approaches for road traffic (the most discussed mode of transportation),
supplemented by one paper on railways, but no airline problems. Extension of the
categorization procedure for airlines and also for railways is thus an obvious working point.
Some links between the proposed criteria were discovered and explained:
·
urban scenarios mostly correspond to single depot cases and ex-urban to multiple
depots,
·
the service level objective (reducing delays) can be achieved by using a dynamic
scheduling approach, and
·
the quality (regularity) of crew schedules is only a relevant objective when the timetable
is irregular.
Also several interesting trends were observed for the VCSP resulting from the categorization:
·
the vast majority of authors apply the proposed approaches to real-life problems,
·
the largest instance tackled so far is of the size of 1,414 trips and these sizes have not
really increased in recent years whereas the number of depots and vehicle types
considered did,
·
clearly more ex-urban scenarios are considered,
·
virtually all approaches make use of complete integration (as opposed to partial
integration),
·
a set partitioning formulation is still the most popular one, and
66
·
solution methods based on column generation remain the most commonly used with an
evolution towards linear relaxation (as opposed to Lagrangian relaxation) to solve the
master problem and branch-and-bound methods (as opposed to Lagrangian heuristics)
to obtain feasible solutions.
Most importantly, we showed that we can clearly and indisputably answer the VCSP-related
part of the central research question, ‘Is it (always) advantageous to use an integrated approach
for scheduling problems in transportation?’ with a sound ‘Yes’.
For the description of recent developments of the VCRSP, we provided an overview of the
existing approaches (7 relevant papers) and presented a first step towards an extensive
categorization similar to that for the VCSP. A two-dimensional classification was introduced with
dimensions corresponding to admission of changeovers and degree of integration, thus again
identifying the two overarching categorization aspects of practical problem and solution method.
The existing literature on the VCRSP is perhaps not yet extensive enough for an elaborate
categorization to be of use at this time, but it is certainly an interesting possibility for the future.
A couple of meaningful observations were made, namely that the concrete application contexts
of the VCRSP are very varied (limousine rental, mail distribution, oil delivery, just to name a few)
and that virtually every publication on the VCRSP so far uses a partially integrated approach
(notice the contrast with the VCSP). The VCRSP-related part of the central research question,
‘Is it (always) advantageous to use an integrated approach for combined routing and scheduling
problems in transportation?’, could not be as decisively answered as the VCSP-related part. We
cannot just respond ‘Yes’, but nuance it by stating that there certainly is potential in the
integration between routing and scheduling, at first sight less pronounced than for the VCSP,
but in any case more research in the area of VCRSP is needed in order to evolve towards a
universal conclusion. This further research can thereby lead to new methods which may well
ever produce better results than the traditional approach, so we could then wholeheartedly
answer ‘Yes’ to the central research question, just as for the VCSP.
It was beyond the scope of the thesis to propose improvements for specific existing solution
methods, more particular for models and algorithms. We rather identified more general and
problem-inherent future research topics. These were primarily discovered by identifying gaps
within the VCSP categorization and the VCRSP two-dimensional classification and then making
recommendations for filling those gaps. When available, we also mentioned some quotes for
each topic, literally extracted from the considered papers in order to substantiate the relevance
of the topics. Some of the most important recommendations for the VCSP are:
67
·
strive for the solution of the whole network at once for every transport company,
·
shift the attention a little from customer satisfaction (service level objective) towards
wellbeing of employees (quality of crew schedules objective),
·
take into account the demand for more regular crew schedules when dealing with an
irregular timetable,
·
keep pushing the size boundaries of instances,
·
consider multiple crew types instead of by default assuming only a single crew type,
·
investigate the possible advantage of allowing unrestricted changeovers (compared to
restricted changeovers),
·
further research in the area of metaheuristics and constraint programming, and
·
pay more attention to the development of dynamic scheduling approaches (and more
specifically introduce such an approach for the distribution context).
It is remarkable that many authors seem to acknowledge the importance of the last mentioned
topic. However, we observed that only very few papers actually present such a dynamic
approach. We have thus identified a high need and a low presence, so consequently, dynamic
scheduling should be treated as one of the hottest amongst the topics.
We also came to the conclusion that it might be needed to dedicate some extra attention and
effort to the whole of the VCRSP, since it is underdeveloped compared to the VCSP while
nonetheless, distribution problems are an import aspect of today’s transport issues. Other
significant VCRSP recommendations are:
·
develop and evaluate approaches pursuing complete integration,
·
test existing theoretical approaches on practical instances,
·
focus more on multiple depot problems where changeovers are not constricted to only
take place at a single location, and
·
adapt VCRSP methods that are initially designed for one specific application in order to
make them more generally applicable for other practical problems as well.
The last topic is the most frequently mentioned research subject in other papers. This is of
course not surprising, since it concerns a very general recommendation that actually reflects the
logical lifecycle of a newly developed solution method.
68
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XIV
Appendix A
“We solve the traditional vehicle and crew scheduling problem, or in other words the CSP, in
two steps. First we have to solve the vehicle scheduling problem (either the SDVSP or the
MDVSP dependent of the number of depots), generate all feasible pieces and (optional) all
feasible duties first. Then we select the optimal duties by solving the corresponding set covering
model with a Lagrangian heuristic. In the remainder of this discussion we will assume that there
is only one depot. However, the algorithm can be straightforwardly generalized to the multiple
depot case. The solution of the SDVSP gives a set of vehicle blocks on which we can define the
relief points. This results in a set of tasks from which we can easily enumerate all feasible
pieces, since a piece is a feasible sequence of consecutive tasks on the same vehicle block
only restricted by its duration.
Next, we generate all feasible duties. Since duties often consist, in practice, of at most three
pieces, we can do this by enumerating all possible combinations of pieces and check if such a
combination is feasible. One may use a duty generation network, as proposed by (Freling,
1997), to generate duties consisting of more than three pieces. Finally, we select the duties with
the Lagrangian heuristic, for which a global description is given here:
Step 0: Initialization
Generate a set of pieces such that each task can be covered by at least one piece. The initial
set of columns consists of these pieces.
Step 1: Computation of dual multipliers
Solve a Lagrangian dual problem with the current set of columns. This yields a lower bound for
the current set of columns.
Step 2: Selection of additional columns
Select columns from the previously generated duties with negative reduced cost. If no such
columns exist, meaning that the lower bound computed in Step 1 is a lower bound for the
overall problem (or another termination criterion is satisfied), go to Step 3; else, return to Step 1.
Step 3: Construction of feasible solution
Use all the columns selected in Step 0 and Step 2 to construct a feasible solution. We compute
a feasible solution by solving a set covering problem in which we consider all the columns which
have been selected along the way. We can either do this exactly, using a general or specialized
integer programming solver, or heuristically.” (Huisman, 2004, pp. 32-33)
For a more profound description of this method, we of course refer to (Huisman, 2004).
A.1
Appendix B
In general, we have a feasible vehicle schedule “if each trip is assigned to a vehicle, and each
vehicle performs a feasible sequence of trips, where a sequence of trips is feasible if it is
feasible for a vehicle to execute each pair of consecutive trips in the sequence. In the multiple
depot case, we must also make sure that each trip is assigned to a vehicle from a depot that is
allowed to drive this trip. The vehicle costs consist of a fixed component for every vehicle and
variable costs for idle and travel time. It is allowed that a vehicle returns to a depot between two
trips if there is enough time to do so.
From a vehicle schedule it follows which trips have to be performed by the same vehicle and
this defines so-called vehicle blocks. The blocks are subdivided at relief points, defined by
location and time, where and when a change of driver may occur and drivers can enjoy their
break. A task is defined by two consecutive relief points and represents the minimum portion of
work that can be assigned to a crew. We distinguish between two types of tasks: trip tasks
corresponding to (parts of) trips, and deadhead tasks of course corresponding to deadheading.
A deadhead is a period that a vehicle is moving to or from the depot, or a period between two
trips that a vehicle is outside of the depot (possibly moving without passengers). All trip tasks
need to be covered by a crew, while the covering of deadhead tasks depends on the vehicle
schedules and determines the compatibility between vehicle and crew schedules. In particular,
each deadhead task needs to be assigned to a crew if and only if its corresponding deadhead is
assigned to a vehicle. Note that more than one trip task may correspond to a single trip,
depending on the relief points along that trip. Similarly, more than one deadhead task may
correspond to a single deadhead.
The tasks then have to be assigned to crew members. The tasks that are assigned to the same
crew member define a crew duty. Together the duties constitute a crew schedule. We only
consider the basic crew scheduling problem as discussed in section 1.1.3, where a schedule is
feasible if each task is assigned to one duty, and each duty is a sequence of tasks that can be
performed by a single crew, both from a physical and a legal point of view. In particular, each
duty must satisfy several complicating constraints corresponding to work load regulations for
crews. Typical examples of such constraints are maximum working time without a break,
minimum break duration, maximum total working time, and maximum duration. These
constraints can differ between different types of duties, e.g. early, split and late duties. The cost
of a duty is usually a combination of fixed costs such as wages, and variable costs such as
overtime payment. Finally, we define a piece (of work) as a sequence of tasks on one vehicle
block without a break that can be performed by a single crew member without interruption.
B.1
We also need to make some assumptions that will determine the modeling of the problem (i.e.
the constraints in the model) and thus the complexity and veracity of the approach. An
assumption that is almost always made (in the single depot case as well as in the multiple depot
case), is that the cost function for the VCSP is the summation of the vehicle and crew
scheduling cost functions, where the primary vehicle and crew scheduling objectives are to
minimize the number of vehicles and crews, respectively. An additional term can be added to
the cost function which corresponds to variable vehicle costs (e.g. distance travelled) and
variable crew costs (e.g. overtime payment). Another common assumption is that the feasibility
of a piece only depends on its duration, which is limited by a minimum and maximum piece
length that follows from legal regulations. For the single depot case, it is also frequently
assumed that all vehicles are available at all time and that they are identical. This because with
these characteristics, the underlying vehicle scheduling problem is polynomially and thus rapidly
solvable. The assumptions cited above hold in a lot of real-world applications arising from public
transport scheduling.” (Huisman, 2004, pp. 38-39) Although, one can easily think of a situation
where this will no longer be the case, for example in a transportation company that disposes of
more than one vehicle type.
For the MDVCSP, it is often assumed that “each vehicle and each crew member has its/his own
depot, which respectively means that a vehicle starts and ends in the same depot (while the
number of vehicles used per depot is unlimited) and that a duty of a single crew member has
only tasks on vehicles from that depot (however, it is not necessary that every duty starts and
ends in this depot).” (Huisman, 2004, p. 79) Of course, these assumptions impose restrictions
on the associated models and algorithms. “Especially, the second assumption that all crew have
their own depot and are only allowed to perform tasks on vehicles from their own depot, is a
crucial one. If this assumption is not valid, the crew scheduling problem cannot be solved
independently for each depot anymore. Moreover, in the integrated models and algorithms the
constraints linking the vehicles and the crews for the different depots will not be independent
anymore. Such a situation can occur in practice, where one can think of a situation where the
driver first has a few tasks on a vehicle from his own depot and then on a vehicle from another
depot, where he will be relieved at the end of his duty on a location close to his own depot.
However, it is unlikely that such a situation occurs frequently, since not all trips are allowed to
be assigned to each depot and if it is allowed, it is often possible to change vehicles between
the depots.” (Huisman, 2004, p. 100)
Other possible assumptions are that “there is continuous attendance, i.e. there is always a
driver present if the vehicle is outside the depot (however, vehicle attendance at the depot is not
necessary) and that changeovers, which is the change of vehicle of a driver during his break,
are allowed. These last two assumptions imply that if a driver has no changeover, i.e. before
B.2
and after the break he drives the same vehicle, there should be another driver on this vehicle
during the break of the former driver, since there is otherwise nobody attending this vehicle.”
(Huisman, 2004, pp. 79-80) Again, one can easily imagine a transport company where
changeovers are not allowed, which means that the last assumption has to be altered and thus
the associated model (i.e. the constraints) will be different.
B.3
Appendix C
“A task is characterized by a start and end location, a duration, a set of vehicle types that can
be used to cover it (usually dictated by a minimum capacity requirement on the task, however
there are often operational restrictions that further reduce the set of compatible vehicle types)
and a time window, bounded by an earliest and latest starting time, within which it must be
commenced. A vehicle schedule is a set of tasks connected by empty repositioning or
deadhead trips as required, for the same vehicle type that starts and ends at the same depot
and usually spans less than one 24 hour period. A crew schedule or duty describes the tasks
undertaken by one driver for one shift. It is a collection of contiguous activities whose
composition is governed by the tasks it covers, the vehicle schedules these tasks are assigned
to and a set of regulations that usually vary for different types of drivers. A duty must start and
end at the same depot. It begins with a sign on activity followed by a set of drive activities
describing the tasks and deadheads covered, with an optional intervening break activity which
must occur at one of a predefined set of break locations, usually the depots, not necessarily the
home depot. A duty ends with a sign off activity. Deadhead trips may be needed in a duty for
many reasons: to connect different tasks covered by the duty, to travel from(to) the depot
to(from) the start(end) location of the first(last) task covered by the duty or to return to a depot to
incorporate a break activity. Note that a driver must always drive a vehicle when deadheading.
This is a characteristic that is unique to the distribution domain. In the public transport context
drivers are usually allowed to move within the network on their own (by walking, catching a bus
or plane, etc.). A special set of activities are associated with any vehicle changes (which usually
take place at a depot) that occur in a duty. Before a vehicle may be driven as part of a duty,
time is given to the driver to prepare the vehicle and before a driver finishes using a vehicle,
time is given to return it. Thus, from a vehicle view point, a duty can be thought of as being
composed of what are commonly called pieces of work. A piece of work is defined as a
contiguous set of activities within the same duty, bounded by prepare and return activities,
within which the same physical vehicle is being used. Note that we can choose whether or not
drivers are only allowed to drive vehicles based at their home depot. A prescribed set of
regulations exists for each duty type which govern the legality of a duty. This set of regulations
includes a maximum duration, the maximum duration before and after a break activity and
specific durations for sign on, prepare, break, return and sign off activities. A set of vehicle
movements for all vehicle types between allowable locations are used to construct the
deadheads that appear in duties. This set of vehicle movements obeys triangle inequalities for
cost, distance and duration. The operational cost of a duty is a non decreasing function of the
C.1
duration of the duty. It includes the drivers salary, which is proportional to the time worked, and
the running costs associated with the vehicles used during the duty, which is the sum of the
costs of the individual vehicle movements performed during the duty when covering tasks and
deadheading. There is also a fixed cost associated with each vehicle and duty type.” (Hollis,
2011, pp. 41-42)
C.2
Appendix D
The following discussion is situated in a public transport context, although similar statements
can be made for product delivery. Since we based our description of the planning process in
section 1.2 on that for a bus company, we will now compare the known bus planning process
with that for a railway and an airline operator, consecutively.
So we start with the differences between railway planning and the planning process in a bus
company. First of all, “timetabling in railway planning is much more complicated than in bus
transport, since the infrastructure of a railway system has to be taken into account. For
example, between each pair of trains on the same track there should be a minimum headway
due to safety regulations. Furthermore, differences in speed should be taken into account: a fast
Intercity service cannot start just after a slow train in the same direction. Similar remarks can be
made about the railway stations, where for instance cross-platforms connections between
certain trains are preferred. In a bus system all those complicating problems do not occur.
The vehicle scheduling problem is also slightly different, since different train units can be
combined to form one larger train. This is not possible with buses. The order of the units in a
train is another complicating factor. Furthermore, other problems that do not exist in the bus
context, like shunting at railway stations should be taken into account. Finally, the movement of
empty trains is often restricted, which makes the problem in this aspect somewhat easier than
bus scheduling, since there are much less possible solutions.
The crew scheduling and rostering problem are very similar. There is only one major difference:
in general, a train needs more crew members (driver and guards) than a bus (only a driver).
However, this does not make the problem much more complicated, since drivers and guards
can either be scheduled independently or they are scheduled as teams consisting of a driver
and one or two guards.
From these differences we can conclude that, in general, railway planning is more complicated
than bus planning.
Now for the differences between bus and airline planning, which are generally much smaller.
For instance, since most airlines have a hub-and-spoke network, the timetabling step can be
compared with the one in regional bus transport where connections should be ensured on the
hubs. However, the number of slots (capacity) at the airports is limited for each airline.
Vehicle scheduling is divided into two steps for most airlines: fleet assignment and aircraft
routing. In the fleet assignment problem, types of planes are assigned to each flight, while in the
D.1
aircraft routing problem the actual schedules are determined. The first problem has a planning
horizon of a season and the second one of a week. This is the major difference with bus
scheduling where the planning horizon is one day. On the other side, the trips in airline planning
are much longer than in bus scheduling.
Crew scheduling (and rostering) differ in the number of crew members that has to be scheduled.
However, pilots, copilots, cabin personnel can be scheduled independently of each other, in
general. A more important difference is that the crew related processes are mostly divided into
three steps: duty scheduling, crew pairing and crew rostering. A pairing is defined here as a
sequence of duties without a (long) rest period. There are also some differences from a
computational point of view. Since the trips are much longer, the number of trips per duty is
much smaller than for bus driver scheduling. Furthermore, crew is much more restricted to
certain types of airplanes. Therefore, large problems can be split up easily.
If we compare the differences above, it seems that airline and bus planning have the same
degree of difficulty. This in contradiction with railway planning, which is much more complicated
than bus and airline planning.” (Huisman, 2004, pp. 6-7)
D.2
Appendix E
E.1
(Huisman, 2004), Ch. 3.3
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
public transport
road traffic
single depot
cost reduction
application-based
RET, the mass transit company in Rotterdam
maximum of 259 trips (1 depot, 1 vehicle type)
urban
RET operates within the city of Rotterdam
regular timetable
restricted changeovers
a changeover is defined as the change of vehicle of a
driver during his break
solution method
degree of integration
degree of network segmentation
complete integration
each route separately
at the RET, the whole planning process is solved line-byline, so the data corresponds to bus and driver
scheduling of individual bus lines
model
set partitioning
The mathematical formulation proposed for the VCSP is a combination of the quasi-assignment
formulation for the vehicle scheduling problem, and the set partitioning formulation for crew
scheduling, similar to (Freling et al., 2003).
algorithm
The author uses the relaxation of the proposed model, where all set partitioning type of constraints are
first replaced by set covering constraints, which are subsequently relaxed in a Lagrangian way. Then the
remaining Lagrangian subproblem can be solved by pricing out the duty variables (related to the crew)
and solving a SDVSP for the trip variables (related to the vehicles). We can use the auction algorithm of
(Freling et al., 2001) for solving the SDVSP. The author proposes a two phase procedure for the column
generation pricing problem: in the first phase, a piece generation network is used to generate a set of
pieces of work which serve as input for the second phase where duties are generated. At the end a
feasible crew schedule is computed given the (feasible) vehicle schedule which resulted from solving the
last Lagrangian subproblem. This is done by solving the CSP.
dynamism of the solution approach
static approach
conclusion
The results reported show that we can get good solutions within reasonable computation times on a
personal computer. It was not expected that it would be useful to integrate under every circumstance.
When changeovers are allowed in the test problems, one driver may be saved by considering an
integrated approach. Of course, the saving of one driver is considerable when taking into account that
the author investigated instances for one bus line each. The interpretation of the computational results
depends on the ratio between the fixed vehicle and crew costs. If fixed vehicle costs are much higher as
E.2
compared to crew costs it becomes less attractive to apply the integrated approach. On the other hand,
if crew costs are higher the integrated approach becomes more attractive. The author also applied the
integrated approach to vehicle and crew scheduling to the exact RET case. He handled complicating
constraints, which have not been considered in the literature before. He also showed the results for two
individual bus lines, including the RET bus line with the largest number of trips. For these lines the
integrated problem could be solved in a reasonable amount of time, where the gap between the lower
bound and the best feasible solution was less than 10% in all cases. The main conclusion is that we can
save vehicles and/or crews by integrating the vehicle and crew scheduling problem, which may lead to a
big decrease in costs. Another important result is that sometimes it is indeed possible to reduce the total
costs by allowing changeovers more often (so by making the changeovers more unrestricted).
E.3
(Huisman, 2004), Ch. 3.4
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
single depot
cost reduction
application-based
RET, the mass transit company in Rotterdam
maximum of 259 trips (1 depot, 1 vehicle type)
urban
RET operates within the city of Rotterdam
regular timetable
no changeovers
complete integration
each route separately
at the RET, the whole planning process is solved line-byline, so the data corresponds to bus and driver
scheduling of individual bus lines
model
set covering
The VCSP without changeovers can be modeled in a straightforward way as a set covering problem due
to the introduction of combined duties. A combined duty is defined as a feasible vehicle duty and one or
more corresponding feasible crew duties.
algorithm
The algorithm here is almost similar to the one given in (Huisman, 2004), Ch. 3.3. Since the model here
is a pure set covering model, Lagrangian relaxations can be obtained in a similar way as for a traditional
CSP. The column generation pricing problem needs an additional procedure with respect to (Huisman,
2004), Ch. 3.3 in order to generate combined duties using previously generated crew duties as input.
This results in a three phase procedure. The other difference with (Huisman, 2004), Ch. 3.3 is that we
obtain a feasible solution in a similar way as for CSP by solving a set covering problem with the columns
generated during the lower bound phase. So the difference is that we do not compute a feasible vehicle
schedule first, since combined duties already define a vehicle schedule in itself.
dynamism of the solution approach
static approach
conclusion
The conclusion is the same as the one for (Huisman, 2004), Ch. 3.3, except for this addition. Based on
the results we can conclude that the benefit of integration may be significant when changeovers are not
allowed. In practice, it often occurs that either changeovers are not possible due to long distances or
changeovers are not allowed for legal or technical reasons. Especially, in the case that (almost) no
changeovers are allowed, integration is very attractive because more vehicles and/or crews can saved
here.
E.4
(Huisman, 2004), Ch. 4.2-4.3
related papers: (Huisman et al., 2005)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
public transport
road traffic
multiple depot
cost reduction
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 653 trips (4 depots39, 1 vehicle type)
ex-urban
regular timetable
restricted changeovers
definition of a changeover: the change of vehicle of a
driver during his break
solution method
degree of integration
degree of network segmentation
complete integration
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. They consider 8 different
problem instances for which the number of trips varies
between 194 and 653 trips (< 1,104).
model
set partitioning
It is an extension of the model in (Huisman, 2004), Ch. 3.3 to the multiple depot setting.
algorithm
The algorithm that is proposed to solve the presented model, is a combination of column generation
and Lagrangian relaxation and is an extension of the algorithm proposed in (Huisman, 2004), Ch.
3.3.The main part of the algorithm is used to compute a lower bound and therefore the author uses a
column generation algorithm. He solves the master problem with Lagrangian relaxation. Furthermore,
he generates the duties in the column generation subproblem (pricing problem), using the two phase
procedure of (Huisman, 2004), Ch. 3.3. Since there is no interaction between the different depots in the
column generation subproblem, we can solve them separately for every depot. An extra phase
comprising the deletion of columns is introduced, compared to the single depot case in (Huisman, 2004),
Ch. 3.3. Finally, feasible solutions are computed using a Lagrangian heuristic.
dynamism of the solution approach
static approach
conclusion
The results reported indicate that medium-sized problem instances with multiple depots can be solved
by using an integrated approach for the vehicle and crew scheduling problem. Furthermore, there are
significant savings compared to the traditional sequential approach, where first the vehicle scheduling
and afterwards the crew scheduling problem is solved.
39
However, not all trips were allowed to be driven by a vehicle from every depot, so the average number
of depots a trip can be operated from was actually only 1.71 (<4).
E.5
(Huisman, 2004), Ch. 4.4
related papers: (Huisman et al., 2005)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
public transport
road traffic
multiple depot
cost reduction
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 653 trips (4 depots40, 1 vehicle type)
ex-urban
regular timetable
restricted changeovers
definition of a changeover: the change of vehicle of a
driver during his break
solution method
degree of integration
degree of network segmentation
complete integration
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. They consider 8 different
problem instances for which the number of trips varies
between 194 and 653 trips (< 1,104).
model
set partitioning
The author proposes a mathematical formulation which has only variables related to crew duties. The
vehicle schedule can be obtained implicitly from the crew schedule. This formulation can be derived
from the one presented in (Huisman, 2004), Ch. 4.2-4.3, but is also equivalent to the formulation of
(Haase et al., 2001) in the case of a single depot. The formulation is characterized by the adding of an
extra decision variable to count the number of vehicles.
algorithm
The algorithm is again a Lagrangian heuristic based on column generation, similar to (Huisman, 2004),
Ch. 4.2-4.3.The author proposes an algorithm that consists of two phases. In the first phase, a lower
bound is computed using the proposed model by again combining column generation and Lagrangian
relaxation. The author uses the columns generated during the first phase in the second one to find a
feasible vehicle schedule and a corresponding crew schedule. The second phase is similar to the
construction of feasible solutions described in (Huisman, 2004), Ch. 4.2-4.3. The important differences
with (Huisman, 2004), Ch. 4.2-4.3 are thus in computing the lower bound, where the author uses a
different model.
dynamism of the solution approach
static approach
conclusion
Same conclusions apply as for (Huisman, 2004), Ch. 4.2-4.3. However, the lower bounds obtained by this
algorithm are rarely stronger than the bounds obtained by the one in (Huisman, 2004), Ch. 4.2-4.3 and
40
However, not all trips were allowed to be driven by a vehicle from every depot, so the average number
of depots a trip can be operated from was actually only 1.71 (<4).
E.6
regularly weaker. If the solutions of the (Huisman, 2004), Ch. 4.2-4.3 algorithm are compared with the
ones of the algorithm here, it is difficult to conclude which one is better, since sometimes the first one
gives the best solution and sometimes the second one. However, for larger random problem instances,
the (Huisman, 2004), Ch. 4.2-4.3 algorithm performs better.
E.7
(Huisman, 2004), Ch. 5.3
related papers: (Huisman & Wagelmans, 2006)
practical problem
type of transportation problem
public transport
mode of transportation
road traffic
number of depots
multiple depot
objectives
cost reduction and service level
size/practicality of the problem
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 304 trips (4 depots41, 1 vehicle type)
degree of urbanization
ex-urban
regularity of the timetable
regular timetable
admission of changeovers
restricted changeovers
changeovers only occur during the break
solution method
degree of integration
complete integration
degree of network segmentation
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. The considered problems
consist of 164 and 304 trips (< 1,104).
model
set partitioning
The mathematical formulation is based on that of (Huisman, 2004), Ch. 3.3. The notation with respect
to the vehicle scheduling part of the formulation is completely similar to (Huisman et al., 2001). An
important feature is that different scenarios for the travel times can be introduced. Hereby, the author
assumes that one of the scenarios is the main scenario and that the variables related to the crew
scheduling part of the problem are defined on this scenario.
algorithm
In the case of multiple-depots, all algorithms use the cluster-reschedule heuristic, so that multiple depot
problems can be solved as several single-depot problems (the actual algorithm thus solely focuses on
the single depot case). The algorithm solves a sequence of integrated vehicle and crew scheduling
problems and is based on that in (Huisman, 2004), Ch. 3.3. The generation of columns is exactly the
same since another vehicle scheduling problem does not influence this. The main difference lies in the
approach for the crew-related constraints which are first replaced by set covering constraints and
subsequently relaxed in a Lagrangian way. At the end a feasible crew schedule is computed given the
(feasible) vehicle schedule for the main scenario which resulted from solving the last Lagrangian
subproblem.
dynamism of the solution approach
dynamic approach
conclusion
The extension of the dynamic vehicle scheduling problem of (Huisman et al., 2001) to the situation
where crews are also considered did not always give the results which was expected beforehand. For
the small instance with a single depot the dynamic approach performs well. However, computation
41
However, not all trips were allowed to be driven by a vehicle from every depot, so the average number
of depots a trip can be operated from was actually only 1.71 (<4).
E.8
times are still high for applying such an approach in practice. On the other hand for the medium-sized
instance with multiple depots, the traditional static approach with buffer times performed much better.
The first reason is that this approach used the cluster-reschedule heuristic, i.e. all trips are assigned
beforehand to a certain depot. In other words, the overall result is dependent on the chosen
assignment. Another reason could be that the computation times allowed to solve the approach
dynamically were set too small, although by extending these times the results did not improve so much.
The final mentioned reason is that the idea of dynamically solving itself does not work so well. Since the
dynamic approach worked well for the small instance where the data set was not divided into several
smaller ones, and since extending the computation times did not lead to significant improvement, we
can conclude that the way the problem is split up, is the bottleneck. Therefore, the author recommends
to invest further research in speeding up the suggested algorithms. With faster computers and better
algorithms the dynamic approach should outperform the static one with buffer times for larger problem
instances as well. Finally, the author makes several remarks about the practical applicability of such a
dynamic approach. First of all, it will be difficult to test the assumptions that were made in a practical
environment. For instance, how can one measure if the travel times of the trips are really independent
of the actual chosen schedule? Secondly, it is important how drivers (but also planners and managers)
react on such a way of working. It is very easy for them to frustrate such an approach. Therefore, the
author concludes the discussion with the fact that there is still a long way to go before such an
approach can be used in practice.
E.9
(Borndörfer et al., 2004)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
multiple depot
cost reduction
application-based
the Regensburger Verkehrsbetriebe GmbH (RVB), a
medium sized public transportation company in
Germany and the Regionalverkehrsbetrieb Kurhessen
(RKH), a regional carrier in the middle of Germany
maximum of 1,414 trips (1 depot42, 3 vehicle types)
urban/ex-urban
the integrated scheduling method is applied to halfregional, half-urban instances and other mainly regional
instances
irregular timetable
the RVB operation has a different scenario for Sundays
and for workdays
restricted changeovers
complete integration
whole network at once
the authors consider instances that contain the entire
RVB operation
model
multicommodity flow and set partitioning
The model consists of a multicommodity flow model for vehicle scheduling and a set partitioning model
for duty scheduling on timetabled trips. These two models are joined by a set of coupling constraints for
the deadhead trips.
algorithm
The authors use a Lagrangian relaxation approach to solve the proposed model. Relaxing the coupling
constraints results in a Lagrangian master problem, a vehicle scheduling and a duty scheduling problem.
They use the method of (Löbel, 1997) to solve the vehicle scheduling problem, column generation to
solve the duty scheduling problem and an inexact adaptation of the proximal bundle method to solve
the Lagrangian master problem, producing dual and additional primal information as opposed to a
subgradient algorithm. After computing a lower bound, the bundle core is called repeatedly in a branchand-bound type procedure (backtracking procedure) to produce integer solutions.
dynamism of the solution approach
static approach
conclusion
The authors show that it is possible to tackle large-scale, complex, real-world integrated vehicle and
duty scheduling problems. The largest and most complex instance up to now has that been attacked
with integrated scheduling techniques is solved in about 125 hours. Furthermore, they compare their
42
For another instance, 3 depots were considered with 634 trips and 5 vehicle types.
E.10
approach with (Huisman, 2004), Ch. 4.2-4.3 on the same set of artificial instances. Using the same
assumptions, their approach clearly outperforms Huisman's method and solves instances with up to 400
trips and 2 (4) depots in 3.3 (12) hours. Also, the solutions produced can be decidedly better in several
respects at once than the results of various types of sequential planning.
E.11
(Rodrigues et al., 2006)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
public transport
road traffic
single depot
cost reduction
application-based
three companies that operate in the large metropolitan
regions of São Paulo and São Bernardo do Campo, Brazil
maximum of 395 trips (1 depot, 1 vehicle type)
urban
regular timetable
passenger demand on a typical day is considered
no changeovers
a crew is designated to a single vehicle for the entire
duration of its daily work schedule
solution method
degree of integration
degree of network segmentation
complete integration
each route separately
7 different lines are considered
model
set covering/set packing
The techniques used are based on integer programming models.
algorithm
A hybrid strategy combining mathematical programming models and heuristics is proposed. The former
produce good feasible solutions, while the latter improve the quality of the final solutions. The
algorithm has four phases: a preliminary schedule generator, a vehicle block generator, a final schedule
generator, and a heuristics that adjusts trip departure times. In the first phase, a bipartite graph is
constructed in order to obtain a set of primary start times. In the next phase, the primary start times are
used to generate vehicle blocks. In the third phase, the vehicle blocks are used in a classical packing or
covering model in order to construct the schedule. After once or twice through the cycle involving the
first three phases, the fourth and final phase starts. Here, a simple greedy heuristic adjusts the trip
departure times that are still not adequately well spaced by the end of the cycle.
dynamism of the solution approach
static approach
conclusion
This hybrid strategy was able to produce quite adequate solutions, in a fraction of the time that experts
take to construct manual solutions. Also, the operational cost of the proposed method showed
considerable gains over the manual scheduling approach.
E.12
(Steinzen, 2007), Ch. 2.3-2.4
related papers: (Gintner, 2007)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
multiple depot
cost reduction
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 653 trips (4 depots43, 1 vehicle type)
ex-urban
regular timetable
restricted changeovers
complete integration
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. They consider 8 different
problem instances for which the number of trips varies
between 194 and 653 trips (< 1,104).
model
multicommodity flow and set partitioning
The formulation is introduced by (Gintner, 2007) and combines a multicommodity network flow
formulation for vehicle scheduling with a set partitioning formulation for crew scheduling. The
underlying vehicle scheduling network is structured as a time-space network44.
algorithm
The solution method is a combination of column generation and Lagrangian relaxation and has been
inspired by (Huisman, 2004), Ch. 4.2-4.3. More precisely, column generation is used to compute a lower
bound where Lagrangian relaxation is applied to solve the master problem. Instead of solving the
restricted master problem with the simplex method to optimality, a subgradient method is used to solve
the Lagrangian dual approximately. Then, the two phase pricing procedure for the column generation
pricing problem proposed in (Huisman, 2004), Ch. 3.3 is used. The final step of the solution method aims
at finding a pair of feasible and compatible vehicle and crew schedules with a Lagrangian heuristic.
dynamism of the solution approach
static approach
conclusion
This method was not tested in itself, because it was improved within the same paper, namely in
(Steinzen, 2007), Ch. 3.1-3.3. Therefore, we refer to the conclusion for that approach.
43
Here a different setting (compared to (Huisman, 2004), Ch. 4.2-4.3) is considered, where every trip
may be serviced from every depot. Obviously, this makes the problems more difficult since the solution
space is expanded.
44
Multicommodity network flow formulations for multiple depot vehicle scheduling problems can be
classified by the underlying network structure. In a connection-based network (CBN), each feasible
connection between two trips corresponds to an explicit arc in the network while in a time-space network
(TSN) only connections between groups of compatible trips are considered. A time-space network
approach reduces the number of connection arcs dramatically if the number of start and end locations is
small compared to the number of trips. (Steinzen, 2007)
E.13
(Steinzen, 2007), Ch. 3.1-3.3
related papers: (Steinzen et al., 2010)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
multiple depot
cost reduction
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 653 trips (4 depots45, 1 vehicle type)
ex-urban
regular timetable
restricted changeovers
complete integration
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. They consider 8 different
problem instances for which the number of trips varies
between 194 and 653 trips (< 1,104).
model
multicommodity flow and set partitioning
The model is the same as in (Steinzen, 2007), Ch. 2.3-2.4.
algorithm
The solution approach is based on Lagrangian relaxation in combination with column generation. Again,
column generation is used to compute a lower bound where Lagrangian relaxation is applied to solve
the master problem. The two phase pricing procedure for the column generation pricing problem of
(Huisman, 2004), Ch. 3.3 is used, but now with a novel time-space network46 (instead of a classic
connetion-based network) for the duty generation phase. Also, a new dynamic programming approach
(including labeling approaches) to the resource constrained shortest path problems that appear in the
duty generation phase is proposed. The performance of the standard version of the method is then
considerably improved by using preprocessing (both generic and problem-specific) and further
acceleration techniques (multiple pricing, restricted networks, state space reduction and label pruning).
The author discusses three methods to compute integer solutions: a Lagrangian heuristic (sequential
approach), a branch-and-bound approach with novel branching schemes (branching on variables and
branching on follow-ons), and a novel heuristic branch-and-price algorithm (fix-and-optimize) which
enhances the method of (Huisman, 2004), Ch. 4.2-4.3 by regenerating columns in the integer phase and
applying depth-first (heuristic) branching in combination with different fixing strategies (fixing service
trips/follow-ons to depots). The latter approach proved to be the best and was therefore used.
dynamism of the solution approach
static approach
45
Here a different setting (compared to (Huisman, 2004), Ch. 4.2-4.3) is considered, where every trip
may be serviced from every depot. Obviously, this makes the problems more difficult since the solution
space is expanded.
46
The author also presented a similar aggregated time-space network, but since this network resulted in
rather poor solutions it was immediately discarded in the paper, as in this thesis.
E.14
conclusion
Notice that this setting is different to the results published in (Huisman, 2004), Ch. 4.2-4.3 and (Huisman
et al., 2005) since the solution space is expanded. As a consequence, the results cannot be directly
compared. The results show that real-world instances with up to 653 trips and 4 depots can be solved.
Furthermore, there is an efficiency gain compared to sequential planning. It can be observed that the
computational time does not always increase with the problem size. The author concludes that his
algorithm performs better if the density of the columns is small. The author reports a strong impact of
column density on the computational burden of a column generation algorithm for a multiple depot
vehicle scheduling problem. Finally, the number of vehicles is always minimal for the presented
approach, i.e., equals the number of vehicles when sequential planning is performed. The approach was
also applied to randomly generated data instances. Similar to the results on real-world problem
instances, the total number of vehicles and drivers can be remarkably reduced if integrated planning is
performed. It is shown that our approach clearly outperforms all other approaches from literature
((Gintner et al., 2006), (Borndörfer et al., 2004) and (Huisman et al., 2005)) in terms of solution quality
and solution time. Furthermore, the author has so far tackled the largest instances with 4 or more
depots. Also, approaches based on models like (Steinzen, 2007), Ch. 2.3-2.4 (also used by (Gintner et al.,
2006)) are beneficial compared to the classic connection-based model of (Huisman, 2004), Ch. 4.2-4.3.
The presented approach requires between 29% and 73% of the computational time of (Borndörfer et al.,
2004). Furthermore, the results indicate that the approach is the overall fastest known method for
integrated vehicle and crew scheduling problems under the stated assumptions. The results indicate
that the proposed approach can efficiently cover duty types with many pieces of work and complex
feasibility rules.
E.15
(Steinzen, 2007), Ch. 3.4
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
multiple depot
cost reduction
application-based
Conexxion, the largest bus company in the Netherlands
maximum of 653 trips (4 depots47, 1 vehicle type)
ex-urban
regular timetable
unrestricted changeovers
complete integration
part of the network
The total set consists of 1,104 trips and 4 depots in the
area between Rotterdam, Utrecht and Dordrecht, three
large cities in the Netherlands. We consider 8 different
problem instances for which the number of trips varies
between 194 and 653 trips (< 1,104).
model
multicommodity flow and set partitioning
The model is very similar to that of (Steinzen, 2007), Ch. 2.3-2.4. The main difference is that, if we allow
unrestricted changeovers, drivers may use vehicles from all depots. So that the duty variables and
linking constraints, which are separated by depot in (Steinzen, 2007), Ch. 2.3-2.4, do not need to be
separated anymore here.
algorithm
Similar to the solution approach in (Steinzen, 2007), Ch. 2.3-2.4 column generation in combination with
Lagrangian relaxation is applied. Basically, the same constraints are relaxed in a Lagrangian way and
again the same two phase pricing procedure is used as proposed in (Huisman, 2004), Ch. 3.3. However,
we no longer have a separate pricing problem for each depot since in the proposed model the duty
variables are not separated by depot. Consequently, the author sets up a single piece generation
network, which is an acyclic directed time-space network. Feasible solutions are found by using a
heuristic branch-and-price approach similar to (Steinzen, 2007), Ch. 3.1-3.3.
dynamism of the solution approach
static approach
conclusion
Basically, the results show that there is an efficiency gain if vehicle and crew scheduling are treated in
an integrated way. Similar to the restricted case, most of the time is spent in the integer phase. The
author concludes that the model with unrestricted changeovers is computationally more attractive than
that with restricted changeovers. Furthermore, he believes it is worthwhile for planners in practice to
allow unrestricted changeovers since the additional exibility results in efficiency gains. The results show
47
Here a different setting (compared to (Huisman, 2004), Ch. 4.2-4.3) is considered, where every trip
may be serviced from every depot. Obviously, this makes the problems more difficult since the solution
space is expanded.
E.16
that the presented approach outperforms the method of (Mesquita & Paias, 2008) in terms of
computational time and/or solution quality. Furthermore, instances with 640 trips and 4 depots are
solved that have not been tackled before. Finally, the author mentions that a valid lower bound can be
computed with his method while this is not possible for the method of (Mesquita & Paias, 2008) (since
they heuristically define the set of tasks).
E.17
(Steinzen, 2007), Ch. 4
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
public transport
road traffic
multiple depot
cost reduction
theoretical
randomly generated instances48
maximum of 200 trips (4 depots, 1 vehicle type)
ex-urban
regular timetable
restricted changeovers
complete integration
although fully integrated fitness evaluation is only
performed for the first few iterations (after that, a
traditional sequential approach is used), it can be used
throughout the entire evaluation process as well
degree of network segmentation
n/a
model
multicommodity flow and set partitioning
The model is initially the same as (Steinzen, 2007), Ch. 2.3-2.4, but it is decomposed into different
subproblems.
algorithm
The author presents a novel hybrid evolutionary algorithm that combines mathematical programming
techniques with an evolutionary algorithm49. The algorithm is based on a problem decomposition that
first assigns trips to depots (providing a trip-depot vector) and thus reduces the multiple-depot
integrated problem to several integrated problems with a single depot. The EA is used to find a good
trip-depot assignment where the fitness of a chromosome (individual) is evaluated using mathematical
programming techniques. In particular, the author uses column generation in combination with
Lagrangian relaxation. The computation of the fitness of the individuals can be done in three different
ways: sequential, partially integrated, or fully integrated. When we use a fully integrated evaluation,
then the model of (Steinzen, 2007), Ch. 2.3-2.4, for a given trip-depot assignment, reduces to a
minimum cost flow problem in combination with a set partitioning problem. Then column generation is
used in combination with Lagrangian relaxation in a similar way as in (Steinzen, 2007), Ch. 2.3-2.4.
dynamism of the solution approach
static approach
conclusion
The results reported in the previous section indicate that medium-sized problem instances with multiple
48
Proposed by (Huisman, 2004).
“An Evolutionary Algorithm (EA) simulates evolutionary processes in nature by creating an initial
population of individuals and applying genetic operators in each generation/reproduction. Each individual
is represented by a string or chromosome and corresponds to a possible solution to the (combinatorial)
optimization problem. The fitness of an individual represents the value of the objective function.
Furthermore, individuals with a high fitness get the opportunity to reproduce among each other by
exchanging genetic information.” (Steinzen, 2007, p. 125)
49
E.18
depots can be solved by using the proposed evolutionary algorithm. Furthermore, the approach
discloses significant savings compared to the traditional sequential approach without requiring a fully
integrated solution method. Although the presented algorithm performs worse than the best known
integrated algorithm, it proved to be competitive with other integrated approaches from literature
especially for medium-sized instances.
E.19
(Steinzen, 2007), Ch. 6
related papers: (Steinzen et al., 2009)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
public transport
road traffic
single depot
cost reduction and quality of crew schedules
application-based
not mentioned which particular real-world problem
maximum of 433 trips (1 depot, 1 vehicle type)
ex-urban
irregular timetable
restricted changeovers
partial integration (crew first - vehicle second)
the Independent Crew Scheduling Problem (ICSP) is
solved first and, then, the vehicle rotations from the
crew scheduling solution are put together such that the
vehicle schedule is feasible
degree of network segmentation
n/a
not mentioned which particular real-world problem is
considered, so not applicable
model
set covering
The ICSP can be formulated as set covering problem.
algorithm
First, the ICSP is solved using a column generation algorithm in combination with Lagrangian relaxation.
The author solves the corresponding Lagrangian dual with a subgradient algorithm to obtain
approximate dual values. The column generation pricing problem corresponds to a resource constrained
shortest path problem and is solved with a dynamic programming algorithm. The columns generated in
the column generation phase serve as input to the second phase where an appropriate integer solution
is sought. The author suggests a method for the second phase that takes the trade-off between costs
and regularity into account (so the regularity objective is taken into account in the feasible solution
construction phase). The basic idea of this method is to systematically search an optimal solution
among all optimal solutions that is as similar as possible to a given reference solution. In particular,
local branching cuts are used to select suitable solution subspaces and explore these subspaces with an
adapted version of follow-on branching. Once the ICSP is solved, the vehicle rotations from the crew
scheduling solution are put together such that the vehicle schedule is feasible.
dynamism of the solution approach
static approach
conclusion
A computational study that involved randomly generated and real-life data showed the applicability of
the proposed techniques. The author concludes that local branching effectively improves the regularity
while a carefully chosen follow-on branching scheme is well suited to improve solution quality and time.
The combination of both methods leads to improved solutions in terms of both cost and regularity
compared to a traditional approach with a default branch-and-bound approach.
E.20
(Kéri & Haase, 2007)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
public transport
road traffic
single depot
cost reduction and service level
loss of service quality can be avoided: the waiting times
of passengers does not change if the trips of the lines
whose connection are important are in the same flexible
groups
theoretical
two smaller artificial test instances
maximum of 133 trips (1 depot, 1 vehicle type)
urban
irregular timetable
a flexible timetable where starting times of the trips can
be shifted, will not be regular (not the same for each
day)
no changeovers
admission of changeovers
solution method
degree of integration
complete integration
degree of network segmentation
n/a
model
set partitioning
The authors extend the model introduced in (Haase et al., 2001) to incorporate what is called trip
shifting, where each trip is assigned an allowable set of shifted starting times.
algorithm
A heuristic method is proposed, nevertheless based on column generation. The linear relaxation of the
proposed model represents the main problem of the column generation process. In each iteration of the
process this relaxed problem is solved to optimality with the actual columns. Solving the subproblems is
equivalent to finding a resource constrained shortest path in each driver network. The authors use an
adapted dynamic programming algorithm (labeling algorithm) to solve this. After finding the optimal
solution of the linear relaxation of the proposed model, a round-up method is used to achieve an integer
solution.
dynamism of the solution approach
static approach
although not explicitly performed in the paper, it should
be logical (and possible – regarding the computation
time – for very small instances) to use the proposed
approach in a dynamic environment
conclusion
The authors have run the test first without using flexible timetable (but still using the same algorithm),
then with using it. The new approach yields a much better solution regarding the number of buses, and
the crew cost. However it requires much more time to solve the problem.
E.21
(Mesquita & Paias, 2008)
related papers: (Mesquita et al., 2006) and (Mesquita et al., 2011)50
practical problem
type of transportation problem
public transport
mode of transportation
road traffic
number of depots
multiple depot
objectives
cost reduction
size/practicality of the problem
theoretical
randomly generated data problems51
maximum of 400 trips (4 depots, 1 vehicle type)
degree of urbanization
ex-urban
not explicitly mentioned, but inferred from the test
instances used
regularity of the timetable
regular timetable
admission of changeovers
unrestricted changeovers
A break can occur during a changeover (but does not
have to). After taking his break (or not), the driver may
pick the same or another vehicle from any depot.
solution method
degree of integration
complete integration
degree of network segmentation
n/a
model
multicommodity flow and mixed set
partitioning/covering
The VCSP is described by an integer linear programming formulation combining a multicommodity
network flow model for the vehicle scheduling with a mixed set partitioning/covering model for the
crew scheduling.
algorithm
The authors propose an algorithm that starts with a pre-processing phase, based on the optimal
solution of the vehicle scheduling problem without requiring that vehicles return to the source depot, to
define the set of tasks and to obtain an initial set of duties. In a second phase, they solve the linear
programming relaxation of the models using a column generation scheme. The columns corresponding
to the duties can be seen as paths in an adequate network and are generated as needed by solving
shortest path problems with resource constraints (pricing problem). The pricing problem is solved by a
heuristic procedure using dynamic programming and a reduced state space where states are associated
to crew duties and the stages to tasks. Whenever the resulting solution is not integer, branch-andbound techniques are used over a subset of feasible duties for the crews.
dynamism of the solution approach
static approach
conclusion
Regarding results in (Huisman et al., 2005) and (Borndörfer et al., 2004) for the same test instances, the
presented approach led to a smaller number of crews although, in some cases, a greater number of
vehicles. However, better values for the sum of vehicles and crews were obtained, so a better general
50
Although (Mesquita et al., 2011) also comprises crew rostering within the operational planning phase,
in addition to vehicle and crew scheduling.
51
Proposed by (Huisman, 2004).
E.22
quality of the solution. The authors think that an important improvement over the existing methods is
the time consumed by the proposed algorithm to obtain these results. They cannot make a direct
comparison, since different computers have been used by the different authors. However, they state
that when the size of the problem increases, the time spent by the presented algorithm becomes
significantly smaller than the time spent by the algorithm proposed in (Borndörfer et al., 2004). From a
transportation company point of view, it is an important feature of an algorithm to produce quick and
‘good’ solutions. Moreover, the resulting solutions have few over-covers and are similar to partitiontype solutions and this makes them easier to implement in a real situation. In conclusion, the proposed
method seems to be a promising tool for dealing with large instances of the integrated VCSP.
E.23
(Laurent & Hao, 2008)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
solution method
degree of integration
degree of network segmentation
public transport
road traffic
single depot
cost reduction
application-based
7 non-specified real-world instances
maximum of 249 trips (1 depot, 2 vehicle types)
ex-urban
regular timetable
no changeovers
complete integration
n/a
it is not mentioned which real-life network was
considered, although the maximum size of the instances
gives away that the network was not tackled as a whole
model
constraint-based
The authors introduce an original formulation relying on a constraint satisfaction and optimization
model. This constraint-based formulation offers a natural modeling of the initial problem and provides a
flexible basis to implement various metaheuristics.
algorithm
They present the first application of a metaheuristic, Greedy Randomized Adaptive Search Procedure
(GRASP), to the VCSP. Within the GRASP algorithm, constraint programming techniques are first used to
build initial solutions. Improvements of these solutions are achieved with a local search algorithm which
embeds a powerful ‘ejection chain’ neighborhood exploration mechanism.
dynamism of the solution approach
static approach
conclusion
First, one observes that the integrated approach always outperforms the sequential one, or at least
furnishes equivalent results. In particular, the savings in terms of number of drivers are significant. The
sequential approach provides a lower bound for the number of vehicles that is always reached in the
integrated solutions. Across the 7 instances, the results are also quite stable with very small standard
deviations. Second, the integrated approach is more powerful than the sequential one in the sense that
the sequential approach failed to solve a particular instance where the integrated approach succeeded.
Moreover, in some cases, the integrated approach is indispensable, especially when relief opportunities
are rare. These results show the dominance of the integrated approach over the sequential one.
E.24
(Bartodziej et al., 2009)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
distribution
road traffic
more specific a Road Feeder Service52
single depot
trips starting and ending at the central hub
cost reduction
application-based
a major German RFS-carrier
maximum of about 1,400 trips (1 depot, number of
vehicle types not mentioned53)
ex-urban
very large distances need to be traveled
irregular timetable
during operation of the fixed timetable, airlines will
eventually ask the trucking company for additional
transportation tasks on the spot, so-called ad-hocs
no changeovers
Drivers stick with their truck. Trips with two drivers are
possible, but a relief in this context does not correspond
to a changeover as defined, because the relieved driver
does not switch to another vehicle but stays in the same
truck. In fact, the two drivers could be seen as an entity.
solution method
degree of integration
degree of network segmentation
complete integration
whole network at once
The largest instance, representing a set of lines in a
bidding round, has to be evaluated on the strategic
planning level. Since design of the network is included
here, of course a whole network is considered.
model
set partitioning
The problem can be represented by a set partitioning type of formulation.
algorithm
two different algorithms were presented:
1) The authors describe an algorithm for obtaining near-optimal solutions for the proposed model by
solving an LP relaxation via column generation and by using the generated columns to construct a
feasible integer solution. In their implementation they use two complex compatibility graphs to
represent all feasible round trips (columns). These graphs are constructed in a first pre-processing
phase.
52
A Road Feeder Service (RFS) concerns the part of the air cargo transport process that is done over
ground by a trucking company.
53
Another instance considers no less than 27 vehicle types for 779 trips.
E.25
2) The authors also describe several local search-based metaheuristics for solving the proposed model.
They combine complex and problem specific operations for the successive improvement of an initial
solution from two different classes: block operations (which remove one or several blocks from the
round trips of the current solution, combine the associated trips to new blocks and finally reassign each
new block to a round trip/vehicle observing all constraints) and trip operations (which remove and
reinsert single trips from/to the current solution, respectively). In each iteration of the local search
improvement phase a type of neighborhood (two where defined) is selected randomly and applied to
the current solution. Then, depending on the specific metaheuristic criterion (Simulated Annealing (SA),
Great Deluge Algorithm (GDA) or Record-to-Record Travel (RRT)) the modified solution is accepted or
rejected.
dynamism of the solution approach
static approach
conclusion
On the small examples (< 167 trips), the solution of the LP relaxation was optimal in most of the cases
and the gap to the optimal integer solution was very small for the remaining instances. Yet, it was not
possible to apply CG to the set of large instances (> 695 trips). As can be seen from the results the
limiting factor is not computational time but too high in-core memory requirements. In order to solve
the large instances, one must resort to the proposed metaheuristics. SA, GDA and RRT show a relatively
similar convergence. The mean percentage deviation from the near-optimal integer CG-solutions (for
the small instances) is for all metaheuristics smaller than 3% after 3 min running time. The authors state
that RRT has the fastest convergence at the beginning and is only caught up if the running times are
relatively large. This is probably due to the fact that the RRT-control is directly connected to the quality
of the solution. On the other hand they report that when the first objective is to minimize the number of
auxiliary vehicles SA and GDA are slightly preferable. Finally, RRT was applied to the largest instances. It
took about 3 min to construct an initial feasible solution. Then, within 10 s only the initial solution could
be improved by 24% and at the end of running time an improvement of 33% was gained. After all, the
model and the heuristics have shown to be appropriate to be implemented in a decision support system
for RFS-planning.
E.26
(Sato et al., 2009)
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
public transport
railway
multiple depot
cost reduction and service level
prevent delays from escalating is an objective (i.e. equal
to the service level objective)
application-based
a not explicitly named Japanese railway line
maximum of 786 trips (exact number of depots or
vehicle types was not mentioned)
ex-urban
railway trains operating between multiple cities are
considered (no metros or trams)
regular timetable
the line has about 200 vehicles and approximately 800
trains that are operated on the line everyday
restricted changeovers
there are certain resources (drivers, conductors) which
cannot be allocated to certain trips for some reason,
such as vehicle type
solution method
degree of integration
degree of network segmentation
complete integration
each route separately
the Japanese railway lines form a huge and complicated
traffic network because of their interconnectedness, a
single railway line is considered
model
multicommodity flow
A 0-1 integer programming formulation based on a network flow is proposed.
algorithm
The authors propose the following two-phase solution approach. In the first phase a feasible solution is
generated by using a partial exchange (is the exchange of a part of a schedule with another one,
performed when some flows have become infeasible because of transport disruption and timetable
changes) as a heuristic flow modification to make a feasible schedule. After setting this solution as an
initial solution, the solution method searches for alternatives by a local search in the second phase.
Local search is a kind of generate-and-test method in which a neighborhood of the temporal solution is
generated at each iteration step. This gives a set of solutions similar to the temporal solution, and the
solution with the best evaluation value is selected as the improved temporal solution.
dynamism of the solution approach
dynamic approach
a method for rescheduling is of course dynamic
conclusion
The proposed formulation is able to represent the differences between the new and original schedules,
E.27
which is a significant criterion for the rescheduling problem, though taking these differences into
account is difficult for other related formulations based on the set partitioning/covering models.
Computational results of real-world vehicle rescheduling data from the railway line indicated that the
proposed method generated a feasible solution within a practical amount of time, and on the basis of a
two-phase solution approach, the proposed method improved the evaluation values of the solution. The
authors believe that their network-oriented modeling and solution approach is promising for developing
a practical computer system for the rescheduling problem that would effectively support train recovery
operations under strict time limitations.
E.28
(Hollis, 2011), Ch. 4
related papers: (Hollis et al., 2006)54
practical problem
type of transportation problem
mode of transportation
number of depots
objectives
size/practicality of the problem
degree of urbanization
regularity of the timetable
admission of changeovers
distribution
road traffic
multiple depot
cost reduction
application-based
a range of postal and courier organizations, primarily
Australia Post
maximum of 1,016 trips (23 depots, 3 vehicle types55)
urban/ex-urban
both metropolitan (urban) and national level (ex-urban)
distribution networks are considered
irregular timetable
with a cyclical nature; the network is usually repeated
(with minor operational variations) each weekday
restricted changeovers
drivers are only allowed to drive vehicles based at their
home depot56
solution method
degree of integration
degree of network segmentation
complete integration
whole network at once
the largest instances tackled can be seen as a whole
network
model
set covering
The formulation used is based on set covering for duties, with an embedded circulation for vehicles.
algorithm
The author uses restricted enumeration followed by column generation to solve the linear relaxation of
the proposed formulation. The enumeration heuristic is used to quickly provide a good set of initial
columns. Column generation, incorporating connection fixing, is then used to solve the linear relaxation.
Connection fixing, or branching on follow-ons, during column generation (this combination is sometimes
referred to as price-and-branch57) is a heuristic fixing rule which involves identifying specific pairs of
tasks and forcing them to appear consecutively in a driver duty. A dynamic programming algorithm is
proposed to solve the associated column generation subproblem, generating specifically timed driver
54
Although (Hollis et al., 2006) mainly describes the combination of routing and scheduling, the
scheduling problem in itself is also discussed.
55
Maximum of 8 vehicle types in another instance with less trips.
56
The computational study performed in (Hollis et al., 2006) investigated the effect of relaxing this
restriction (i.e. relaxation to the case of unrestricted changeovers) for Australia Post product delivery
networks showing that, on average, it resulted in an improvement of only 0.08%. Clearly this negligible
improvement does not outweigh: the loss of operational simplicity, the exponential increase in the number
of variables required when this restriction is relaxed, and a potential increase in algorithmic complexity for
the column generation pricing problem. (Hollis et al., 2006)
57
Price-and-branch is not to be confused with branch-and-price where column generation is incorporated
within branch-and-bound to prove optimality.
E.29
duties. This algorithm is capable of: efficiently handling starting time windows for tasks, modeling
generic driver duty regulations when the tasks covered have starting time windows (where task starting
times can slide freely within their associated starting time window), and incorporating the added
complexity of a concurrent vehicle scheduling problem. Finally, an integer solution is found from the set
of duties produced by the enumeration and column generation process using branch-and-bound.
dynamism of the solution approach
static approach
conclusion
The results show that, on average, solutions where tasks can slide freely within their associated starting
time windows improve upon those where task starting times are fixed (i.e. equivalent to a public
transport context) by 7%. Furthermore, the former approach results in an average improvement of 2%
and is over six times faster than the time window discretisation approach (used in (Hollis et al., 2006)).
Very high quality solutions (within 0.05% on average of the best solution found) can be generated on
average three times faster, by employing only an intelligent subset of the domination criteria (used for
the elimination of labels in the dynamic programming algorithm).
E.30
Appendix F
(Hollis et al., 2006) “describe a simultaneous vehicle and crew routing and scheduling
application for urban letter mail distribution at Australia Post. They are the first to consider a
problem with multiple depots, where vehicles and drivers may be stationed and interchanged.
The authors use a two-stage approach. In the first stage, they determine ‘abstract’ vehicle
routes by solving a pickup-and-delivery problem with time windows, multiple depots, a
heterogeneous fleet of vehicles as well as several working time restrictions for drivers. A pathbased mixed-integer programming (MIP) model is presented and solved by heuristic column
generation. In the second stage, concrete vehicle and crew schedules are determined taking an
integrated vehicle and crew scheduling approach. This is again done by solving an MIP with
heuristic column generation. In the second-stage MIP, the tasks to be performed correspond to
the vehicle routes computed in the first stage.” (Drexl et al., 2011, p. 4)
“The authors examine the affect of different types of vehicle routing solutions on the vehicle and
crew scheduling solution, comparing the different levels of integration (with respect to the set of
allowable depots for a vehicle schedule covering a particular vehicle route) that are possible
with the new vehicle and crew scheduling algorithm.” (Hollis et al., 2006, p. 133) “The solution
technique employed […] has been shown to find high quality solutions. […] On average,
simultaneous solutions improve upon those found using the sequential method by 8.31%. […]
The algorithms and solution techniques presented in the paper have been used by network
planners at Australia Post to demonstrate a potential transport network operational cost saving
of 10% for the 2003 Melbourne metropolitan mail distribution network. Australia Post is currently
using software developed based upon the ideas presented in this paper to assist in the
management of ongoing changes to the mail distribution networks in major cities throughout
Australia.” (Hollis et al., 2006, p. 149)
(Xiang et al., 2006) “describe a static dial-a-ride problem which involves the scheduling of
heterogeneous vehicles and a group of drivers with different qualifications. The solution
procedure is a heuristic composed of a construction phase to obtain an initial solution, an
improvement phase, and an intensification phase to fine-tune the solution. The important aspect
of the procedure is that, initially, ‘abstract’ routes with a fixed schedule are determined, and only
in the last stage, concrete vehicles and drivers are assigned to the routes.” (Drexl et al., 2011,
p. 4)
“The performance of the heuristic was evaluated by intensive computational tests on some
randomly generated instances, […] revealing that this method is capable of quickly obtaining
F.1
acceptable results. […] More specific, small gaps to the lower bounds from the column
generation method were obtained in very short time for instances with no more than 200
requests. Although the result is not sensitive to the initial solution, the computational time can
be greatly reduced if some effort is spent to construct a good initial solution. With this good
initial solution, larger instances up to 2,000 requests were solved in less than 10 hours on a
popular personal computer.” (Xiang et al., 2006, p. 1117) “Moreover, the method is flexible to
cope with many practical constraints and does not contain any case-sensitive empirical
parameter.” (Xiang et al., 2006, p. 1136)
(Laurent & Hao, 2007) “consider the problem of simultaneously scheduling vehicles and drivers
for a limousine rental company. The required transports are pickup-and-delivery trips with given
time windows. The authors use a two-stage solution approach which aims to find a feasible
crew and vehicle schedule by assigning a driver-limousine pair to each trip. First, an initial
feasible solution is constructed by means of a greedy heuristic similar to the well-known best-fitdecreasing strategy for the bin packing problem, using constraint programming techniques for
domain reduction. Second, an improvement procedure based on local search embedded in a
simulated annealing metaheuristic is performed.” (Drexl et al., 2011, p. 3)
“Results obtained on real data sets show a significant improvement in terms of quality,
operational costs and elaboration time, compared with the actual practice in the examined
company. Within a short time, the software supplies very good quality schedules in which the
major part of the trips is assigned, satisfying all constraints. The approach also proves to be
flexible. It unifies the treatment of the static and dynamic parts of this problem in a single
framework. The decision support system based on this research is operating in the company
and proves to be extremely useful. […] Although this paper deals with a real-world driver and
vehicle scheduling problem in a particular application context where some aspects are specific,
others are general ones. Therefore, the proposed method can be relevant to many other
scheduling applications. For instance, the presented constraint-based model and some solution
techniques of this work have been successfully transposed to the domain of transportation by
bus in rural areas (see (Laurent & Hao, 2008) in section 2.1.3 and Appendix E).” (Laurent &
Hao, 2007, p. 557)
(Zäpfel & Bögl, 2008) “consider an application of local letter mail distribution. Pickup routes and
delivery routes (but no combined pickup-and-delivery routes) have to be planned within a
planning horizon of one week. In pickup routes, outbound shipments are transported from local
post offices to a letter mail distribution centre. Conversely, in delivery routes, shipments are
transported from the distribution centre to post offices. Schedules are planned for both drivers
and vehicles, taking into account European Union social legislation. The problem is solved
F.2
heuristically, by decomposing it into a Generalized VRP with Time Windows (GVRPTW) and a
Personnel Assignment Problem (PAP). First, a feasible solution to the GVRPTW is computed
[…]. Then, the PAP tries to find a feasible driver assignment for the GVRPTW solution. The
assignment is achieved by creating a table with all feasible combinations of drivers and routes.
Each table entry represents the costs resulting from the driver performing the route. A complete
personal assignment is computed, using three different strategies, among them a greedy and a
random procedure. After that, an improvement procedure embedded into a metaheuristic
follows.” (Drexl et al., 2011, p. 4)
“Computer simulations have demonstrated that the constructed heuristic is suitable for practice.”
(Zäpfel & Bögl, 2008, p. 980) “It is shown that especially an embedded Tabu Search procedure
is very competitive for solving real problems as considered in the paper. […] All in all, this Tabu
Search procedure can solve a complex decision problem with a tremendous number of
variables and constraints in an efficient manner.” (Zäpfel & Bögl, 2008, p. 995)
(Kim et al., 2010) “study a combined vehicle routing and staff scheduling problem where a
certain number of tasks has to be fulfilled in a fixed sequence at customers. Among the tasks,
an end-to-start relationship is assumed. In order to fulfill the tasks, different teams of workers
are available. Each team is qualified to perform one specific type of task. The teams cannot
move by themselves; instead, a set of vehicles is used to transport the teams. There is no fixed
assignment of a vehicle to a team, and each vehicle may carry at most one team at a time. The
authors develop an astonishingly simple procedure, in which the vehicles, the teams, and the
next tasks for each customer are stored in three lists, along with the relevant information on
times and locations. In each iteration, a triplet (vehicle, team, task) is selected from the lists,
using a best-fit criterion. Then, the lists are updated to reflect the situation resulting when the
selected vehicle transports the selected team to the location of the selected task.” (Drexl et al.,
2011, p. 4)
No relevant computational results were presented. “Good lower bounds for the problem still
need to be developed to evaluate the solution quality of the proposed algorithm. […] Although, it
is shown that the proposed approach significantly outperforms a simple greedy algorithm […]
and that the solution quality is not too sensitive to the random numbers that it uses.” (Kim et al.,
2010, pp. 8429-8430)
(Prescott-Gagnon et al., 2010) “study the problem of planning oil deliveries to customers by
lorry. To solve the problem, the authors develop three metaheuristics, a Tabu Search (TS)
algorithm, a Large Neighborhood Search (LNS) heuristic based on this TS algorithm and
another LNS heuristic based on a Column Generation (CG) heuristic which uses the TS
algorithm to generate columns. They use a greedy construction heuristic which sequentially
F.3
builds up routes for driver-lorry pairs by inserting the temporally closest customer. In the
destruction phase of the LNS algorithm, different heuristics […] are used to determine the
vertices to be removed from the routes of the current solution. The reconstruction phase applies
either TS or CG. Among other move types, the TS procedure employs a driver switch move
which tries to switch a pair of drivers, that is, have one driver drive the other driver's route and
vice versa.” (Drexl et al., 2011, pp. 3-4)
“Computational results obtained on instances derived from a real dataset indicate that the LNS
methods outperform the TS heuristic. Furthermore, the LNS method based on CG tends to
produce better quality results than the TS-based LNS heuristic, especially when sufficient
computational time is available. However, the LNS-TS heuristic can be considered as a good
alternative for a company that does not want to invest into a commercial linear programming
solver that is required for the LNS-CG method.” (Prescott-Gagnon et al., 2010, p. 14)
(Drexl et al., 2011) “study a simultaneous vehicle and crew routing and scheduling problem
arising in long-distance road transport: pickup-and-delivery requests have to be fulfilled over a
multi-period planning horizon by a heterogeneous fleet of lorries and drivers. They allow
lorry/driver changes at geographically dispersed relay stations. […] European driver rules are
considered completely and correctly for a planning horizon spanning one week. […] The
solution approach is based on a heuristic decomposition of the overall problem into two stages,
similar to the approach of (Hollis et al., 2006). […] In the first stage, routes for concrete lorries
(as opposed to the abstract routes in (Hollis et al., 2006)) are determined, taking into account
some driver rules. In the second stage, routes for drivers are computed, based on the lorry
routes from the first stage and taking into account the remaining driver rules to ensure
feasibility. […] Both stages can be solved with essentially the same rather simple and
straightforward algorithm. This algorithm primarily relies on an appropriate network
representation of the problem that is then solved using a large neighborhood search heuristic.”
(Drexl et al., 2011, p. 1-6)
“Extensive computational experiments have been performed with real-world data provided by a
major freight forwarder. […] First and foremost, the presented algorithm is shown to be capable
of solving large real-world instances and is able to achieve consistent and practically relevant
results. […] However, for the given data set, lorry/driver changes offer no savings potential, so a
fixed lorry-driver assignment seems to be the right set-up. These results, though, are only valid
for the considered business field. The developed algorithm may well lead to very different
results with other data or in other application areas.” (Drexl et al., 2011, p. 16-18)
F.4
Nederlandse samenvatting
In deze thesis behandelden we het routeren en plannen van voertuigen en personeel, enkele
van de belangrijkste uitdagingen in hedendaagse transportbedrijven. Er werd dus getracht om
het volledige planningsproces in de transportsector te beslaan.
Vooraleer van start te gaan met de eigenlijke uiteenzetting, kozen we het jaar 2004 (meer
precies 1 januari 2004) als het overgangspunt tussen verleden en heden. Deze keuze volgde
vooral uit het feit dat de centrale onderzoeksvraag van deze thesis, ‘Levert het gebruik van een
geïntegreerde benadering voor planningsproblemen in de transportsector (altijd) voordeel op?’,
niet beschouwd werd vóór 2004.
Voor de beschrijving van het verleden definieerden we eerst de individuele routering- en
planningsproblemen,
zijnde
het
voertuigrouteringprobleem
(VRP),
het
voertuigplanningsprobleem (VSP) en het personeelsplanningsprobleem (CSP). Kortweg kunnen
we stellen dat het VRP routes opbouwt zodat een aantal klanten bediend kunnen worden met
een voertuigvloot, terwijl het VSP voertuigen aanduidt om die routes af te leggen en het CSP
personeel toewijst aan de voertuigen (en de routes). De hoofddoelstelling is steeds het
minimaliseren van de totale kosten, rekening houdend met bepaalde randvoorwaarden.
Daarna situeerden we deze individuele problemen binnen het grotere geheel van het
planningsproces in transportbedrijven. Dit planningsproces is opgedeeld in een strategische,
een tactische en een operationele fase. Het routeren is een hoofdzakelijk strategisch aspect,
daar waar het plannen (VSP en CSP) eerder deel uitmaakt van de operationele fase. We
gingen ook verder in op het onderscheid tussen openbaar vervoer (transporteren van
passagiers) en distributiesector (producten leveren aan klanten), net zoals het onderscheid
tussen problemen met een enkel depot en met meerdere depots een belangrijke leidraad
doorheen de thesis. In het openbaar vervoer zijn routes en dienstregelingen bijna steeds
gegeven (bv. opgelegd door een lokale overheid) en blijven deze ongewijzigd gedurende lange
periodes, wat niet het geval is in de distributiesector. Dit leidde tot het inzicht dat de combinatie
van routering en planning over het algemeen een bezorgdheid is voor dit laatste type
transportbedrijven, terwijl voor openbaar vervoer de focus vooral ligt op de planning van
voertuigen en personeel. Het totale planningsproces in de distributiesector omvat dus meer
aspecten, maar is intrinsiek zeker niet moeilijker of belangrijker dan zijn tegenhanger voor
openbaar vervoer, waar de afweging tussen klanten en kosten veel delicater is.
Het planningsprobleem werd beschreven, beginnend met de traditionele sequentiële aanpak
(die in wezen neerkomt op het maken van de personeelsplanning, omdat dit altijd het eerst
SV.1
opstellen van de voertuigplanning met zich meebrengt) en dan overgaand naar de
geïntegreerde voertuig- en personeelsplanning (VCSP), inclusief een beknopt overzicht van de
literatuur van vóór 2004. Het VCSP werd gedefinieerd in beide contexten – openbaar vervoer
en distributie – om zo de gelijkenissen en verschillen aan te kunnen duiden. Een belangrijk
aspect is de mogelijkheid om rekening te houden met tijdsvensters in een distributiecontext, wat
niet van toepassing is in openbaar vervoer waar klanten een exact tijdstip van vertrek en
aankomst verwachten.
Een hoger niveau van integratie wordt nagestreefd wanneer we de routering en de planning
simultaan proberen aan te pakken, we spreken dan over het voertuig- en personeels- routeringen planningsprobleem (VCRSP). Aangezien het routeringaspect ook deel uitmaakt van dit
probleem, treedt het enkel op in de distributiesector. Het werd aangetoond dat een VCRSP
gezien kan worden als een soort fusie van een VRP en een VCSP.
De VCSP papers uit het heden werden gecategoriseerd aan de hand van een nieuwe
procedure voorgesteld in deze thesis. Uiteraard dienden we eerst de categorisatiecriteria te
definiëren – die ofwel betrekking hebben op het beschouwde praktisch probleem, ofwel op de
oplossingsmethode die ervoor gebruikt wordt. In volgorde van dalende belangrijkheid,
identificeerden we 8 criteria van de eerste soort (type transportprobleem, transportmiddel,
aantal depots, doelstellingen, grootte/bruikbaarheid van het probleem, verstedelijkingsgraad,
regelmatigheid van de dienstregeling en toelating van overstappen) en 5 van de tweede (graad
van integratie, graad van netwerksegmentatie, model, algoritme en dynamisme van de
oplossingsmethode).
De eigenlijke categorisatie werd dan uitgevoerd voor 18 van de belangrijkste VCSP
benaderingen, samen met de vermelding van een tiental papers die er sterk mee
samenhangen. We kozen ervoor om enkel VCSP benaderingen voor wegverkeer – de meest
besproken vervoerswijze – te beschouwen, aangevuld met één paper over spoorverkeer, maar
geen luchtvaart. Uitbreiding van de categorisatieprocedure voor luchtverkeer en ook voor
spoorverkeer is dus een evident werkpunt. Enkele verbanden tussen de voorgestelde criteria
werden ontdekt en toegelicht:
·
bij stedelijke scenario’s hebben we meestal te maken met een enkel depot en bij
regionale scenario’s met meerdere depots,
·
de doelstelling van dienstverleningsniveau (het beperken van vertragingen) kan
verwezenlijkt worden met behulp van een dynamische planningsaanpak, en
·
de kwaliteit (regelmatigheid) van personeelsschema’s is enkel een relevante doelstelling
als de dienstregeling onregelmatig is.
SV.2
Ook werden een aantal interessante trends geobserveerd voor het VCSP, als resultaat van de
categorisatie:
·
de overgrote meerderheid van auteurs passen de door hen voorgestelde methodes toe
op bestaande levensechte problemen,
·
de grootste problemen tot nu toe beschouwd zijn van een grootte van 1414 ritten en
deze groottes zijn de laatste jaren niet echt toegenomen terwijl dit voor het aantal depots
en voertuigtypes wel het geval was,
·
er worden duidelijk meer regionale scenario’s beschouwd,
·
zo goed als alle methodes maken gebruik van complete integratie (tegenover partiële
integratie), en
·
een set partitioning formulering is nog steeds het populairst en oplossingsmethodes
gebaseerd op kolomgeneratie blijven het vaakst gebruikt waarbij een evolutie kan
worden ontdekt in de richting van lineaire relaxatie (tegenover Lagrange relaxatie) voor
de oplossing van het hoofdprobleem en branch-and-bound methodes (tegenover
Lagrange heuristieken) om bruikbare oplossingen te bekomen.
Uiteindelijk toonden we ook aan dat men een duidelijk en onbetwistbaar antwoord kan geven op
het deel van de centrale onderzoeksvraag dat gerelateerd is aan het VCSP: ‘Levert het gebruik
van een geïntegreerde benadering voor het voertuig- en personeelsplanningsprobleem (altijd)
voordeel op?’. Dit antwoord is een volmondige ‘Ja’.
Voor de beschrijving van de recente ontwikkelingen van het VCRSP verstrekten we een
overzicht van de bestaande benaderingen (7 relevante papers) en zetten we een eerste stap
naar een uitgebreide categorisatie gelijkaardig aan die voor het VCSP. Een tweedimensionale
classificatie werd geïntroduceerd waarbij de dimensies overeenstemmen met de toelating van
overstappen en de graad van integratie, waaruit opnieuw de twee overkoepelende
categorisatie-aspecten praktisch probleem en oplossingsmethode blijken. Ook al is de
bestaande literatuur over het VCRSP misschien nog niet uitgebreid genoeg opdat een
doorgedreven categorisatie van nut zou zijn op dit moment, toch is het zeker een interessante
opportuniteit voor de toekomst.
Er werden een paar betekenisvolle observaties gedaan omtrent het VCRSP, namelijk dat de
concrete
toepassingsgebieden
zeer
gevarieerd
zijn
(limousineverhuur,
postbedeling,
oliedistributie, om er maar enkele te noemen) en dat nagenoeg elke publicatie tot dusver een
partieel geïntegreerde aanpak gebruikt (let op het contrast met het VCSP). Het aan het VCRSP
gerelateerde deel van de centrale onderzoeksvraag, ‘Levert het gebruik van een geïntegreerde
benadering voor het voertuig- en personeels- routering- en planningsprobleem (altijd) voordeel
SV.3
op?’, kan niet zo vastberaden beantwoord worden als het deel dat betrekking heeft tot het
VCSP. We kunnen niet onomwonden ‘Ja’ antwoorden, maar nuanceren dit door te stellen dat er
zeker potentieel zit in het integreren van routering en planning, op het eerste zicht minder dan
voor het VCSP, maar in elk geval is er meer onderzoek nodig omtrent het VCRSP zodat we tot
een universele conclusie kunnen komen. Dit verdere onderzoek kan daarenboven leiden tot
nieuwe methodes die mogelijk wel steeds betere resultaten opleveren dan de traditionele
benadering, zodat we dan wel volmondig ‘Ja’ kunnen antwoorden op de centrale
onderzoeksvraag, net zoals bij het VCSP.
Het viel buiten het opzet van deze thesis om verbeteringen voor te stellen voor specifieke
bestaande oplossingsmethodes, meer bepaald voor modellen en algoritmes. Liever
identificeerden we meer algemene en probleeminherente onderwerpen voor toekomstig
onderzoek. Deze werden in de eerste plaats ontdekt door het identificeren van leemtes binnen
de VCSP categorisatie en de tweedimensionale VCRSP classificatie en dan aanbevelingen te
doen voor het opvullen van die leemtes. Wanneer voorhanden, vermeldden we ook enkele
citaten betreffende elk onderzoeksonderwerp, letterlijk onttrokken aan de beschouwde papers
teneinde de relevantie van de onderwerpen te staven. Enkele van de belangrijkste
aanbevelingen voor het VCSP zijn:
·
streven naar het oplossen van het gehele netwerk ineens voor elk transportbedrijf,
·
een
kleine
aandachtsverschuiving
van
klanttevredenheid
(doelstelling
van
dienstverleningsniveau) naar welzijn van werknemers (doelstelling van kwaliteitsvol
personeelsschema),
·
rekening houden met de wens naar meer regelmatige personeelsschema’s wanneer
men te maken heeft met een onregelmatige dienstregeling,
·
de grenzen betreffende de grootte van de beschouwde problemen blijven verleggen,
·
meerdere personeelstypes beschouwen in plaats van standaard slechts één enkel
personeelstype te onderstellen,
·
het mogelijke voordeel van het toelaten van onbeperkt overstappen van crew
onderzoeken (tegenover beperkte overstappen),
·
verder onderzoek uitvoeren in het gebied van metaheuristieken en constraint
programming, en
·
meer aandacht besteden aan de ontwikkeling van dynamische planningsmethodes (en
meer specifiek, een dergelijke methode introduceren in de distributiecontext).
Het is opmerkelijk dat vele auteurs het belang van het laatst vernoemde onderzoeksonderwerk
lijken te erkennen, hoewel we vaststelden dat slechts zeer weinig papers ook daadwerkelijk
SV.4
zo’n dynamische aanpak hanteren. We identificeerden hier dus een hoge nood en een lage
beschikbaarheid, met als gevolg dat dynamische planning zeker gezien mag worden als een
heet hangijzer.
We kwamen ook tot de conclusie dat het nodig zou kunnen blijken om wat extra aandacht te
laten uitgaan naar het VCRSP in zijn geheel, aangezien het onderontwikkeld is ten opzichte van
het VCSP, terwijl distributieproblemen niettemin een essentieel deel uitmaken van de
hedendaagse transportactiviteiten. Andere significante aanbevelingen voor het VCRSP zijn:
·
ontwikkelen en evalueren van methodes die complete integratie nastreven,
·
testen van bestaande theoretische methodes op praktische problemen,
·
meer focussen op problemen met meerdere depots waar mogelijke overstaplocaties niet
beperkt zijn tot één bepaalde plaats, en
·
VCRSP methodes aanpassen die initieel ontworpen waren voor een specifiek
toepassingsgebied zodat ze meer algemeen toepasbaar worden en ook andere
praktische problemen kunnen behandelen.
Dit laatste onderzoeksonderwerp werd ook het meest aangehaald in andere papers. Wat
uiteraard niet verbazend is, aangezien het een zeer algemene aanbeveling betreft die in feite de
logische levenscyclus van een nieuw ontwikkelde oplossingsmethode weerspiegelt.
SV.5