Applied Intelligence 20, 179–194, 2004
c 2004 Kluwer Academic Publishers. Manufactured in The United States.
CLSS: An Intelligent Crane Lorry Scheduling System
HON WAI CHUN AND REBECCA Y. M. WONG
Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR
andy.chun@cityu.edu.hk
csymwong@cityu.edu.hk
Abstract. Companies that provide crane-lorry services are faced with the daily need to perform vehicle and driver
allocation and scheduling. Many companies still do this manually due to the lack of suitable technologies. This
manual approach is both time consuming and inaccurate and most probably will not lead to an optimized plan that
can reduce operational costs. In this paper, we describe the design of a system called “Crane Lorry Scheduling
System” (CLSS) that we have developed for the largest crane lorry company in Hong Kong. A crane lorry company
is a company that provides lorries with different types of mounted crane equipment and drivers to service different
types of moving and lifting jobs. CLSS is a Web-based application that streamlines communication with customers,
subcontractors and employees/lorry drivers. We modeled the lorry-assignment problem as a constraint-satisfaction
problem (CSP) algorithm, which we call the “Crane Lorry Optimizing Engine” (CLOE). CLOE was designed to
be easily customizable to match the needs and requirements of different crane lorry companies. We experimented
with two versions of CLOE, regular CLOE that finds “best” solutions and X-CLOE that finds “optimal” solutions.
Results from our tests show that CLOE is faster and generates better quality plans than the manual approach.
Keywords: constraint satisfaction problem, CSP, constraint programming, vehicle scheduling, optimization
1.
Introduction
The main purpose of our research is to investigate how
constraint-satisfaction techniques might be applied to
solve a crane-lorry scheduling problem. The main research issues included the modeling of the problem as a
constraint-satisfaction problem (CSP), the representation of domain knowledge as CSP constraints, the generation of “feasible,” “best” and “optimal” solutions,
and the satisfaction of potentially contradicting objectives. This research was performed using data from the
largest crane-lorry company in Hong Kong. Besides
this company, there are roughly 30 other crane-lorry
companies and hundreds of self-employed drivers that
operate their own crane-lorry vehicles. On a daily basis, crane-lorry companies assign vehicles and drivers
to service customer orders. Practically all companies
here currently perform this manually. A human planner must have a rich knowledge of the problem domain
as well as a good memory of historical patterns. In this
paper, we describe how we encapsulated the knowl-
edge of these human planners and modeled the problem as a Constraint Satisfaction Problem (CSP); the
knowledge is represented as CSP constraints and search
heuristics.
Background on Crane-Lorry Companies
A crane-lorry combines the mobility of a truck with the
lifting ability of a crane, which is usually permanently
mounted on the truck. These vehicles are especially
popular here due to the large number of major capital
projects, the narrow and winding roads, and the small
operating spaces on many construction sites. There can
be a wide variety of crane to lorry combinations. The
exact combination to be dispatched depends greatly
on the job requirements, such as the size and weight
of the payload, the placement of the payload and the
geography of the worksite.
Cranes can be installed either at the front (behind the
cab), center or rear of a truck. Crane trucks with crane
180
Figure 1.
Chun and Wong
Side and top view of truck with crane mounted in front.
mounted in the front can carry larger payloads, but
are usually considered less flexible or versatile because
the cab of the truck blocks part of the crane’s path of
movement (see Fig. 1). One the other hand, crane trucks
with the crane mounted at the rear (R-type lorry) are
considered more versatile as crane movement can cover
a much wider area (see Fig. 2). Rear-mounted cranes
are, however, limited by the size, weight and length of
the payload that they can carry.
In addition to the positioning of the crane, the size
of the crane also affects versatility and capacity. Larger
payloads require larger cranes, which in turn require
larger trucks, for example construction material or
even shipping containers. The lifting capacities of the
cranes may range from a few to 60 or more tons. In
Hong Kong, all cranes are assumed to have a 2-meter
outreach.
Since there are many different types of cranes, each
with different features and operational procedures that
can be substantially different from each other, crane
Figure 2.
Side and top view of truck with crane mounted in rear.
operators must be adequately trained on the particular
equipment he/she is operating. In most cases, the crane
operator is also the lorry driver. For safety reasons,
crane-lorry companies usually keep the crane-lorry to
driver/operator assignment fixed for a period of time to
minimize chances of operational errors. This assignment consistency gives the operator time to get familiar with the crane-lorry equipment and improves safety
and operational efficiency. This assignment is usually
only changed when a new crane-lorry is purchased or
an old one is decommissioned; or when there is change
in staffing or if the regular driver is not available. Moreover, although drivers have a set number of “days off”
each month, their working days are not fixed. That is,
they are assumed available whenever a customer order
is assigned to them. Since the driver assignment schedule changes on an ad hoc basis, our algorithm does
not generate the driver schedule. Instead, it assumes
this schedule is a manually maintained table within the
system.
CLSS: Crane Lorry Scheduling System
The Manual Scheduling Process
The scheduling and allocation of crane-lorries to customer orders is the most critical daily task. The schedule must first of all be accurate—assigning the “right”
vehicle-equipment combination to satisfy each customer order. It must also be optimizing—assigning the
crane-lorry with just the right level of capabilities. If
not, some jobs might need to be subcontracted out due
to lack of suitably equipped vehicles. This not only
means lost revenue, but even worse, lost customers that
might not return. Another factor affecting revenue is
the operational cost of assigning a suitable lorry with
necessary capability to fit the customer’s budget requirement so as to avoid assigning an over-equipped
lorry which will cost more than the customer is willing
to pay and unduly increases the operational cost to the
company. Another consideration is the traveling distance to the customer location; all factors being equal,
the crane-lorry closest to the customer will be assigned
to reduce operational costs.
The priority of the customer and his/her preferences
are the most important factors to be considered in generating the daily schedule. Other considerations include
the features and capabilities of the lorry and its mounted
crane, driver experience, past order history, and geographical location. To automate the entire scheduling
process, our CLOE algorithms encapsulate these considerations as CSP constraints and search heuristics.
Search heuristics are used to improve scheduling performance and quality. Quality is measured in terms
of how well the schedule matches customer preferences, maximizes lorry-crane utilization and reduces
cost. Since each crane-lorry company may have different business objectives and operational constraints,
CLOE was designed to be easily customizable to fit the
needs of different companies.
Most of the service orders arrive by the end of each
business day. At that time, the human planner will begin to construct the schedule for the following day. The
planner must complete the schedule as soon as possible in order to determine which orders, if any, must
be subcontracted, and then to contact subcontractors to
reserve the required capacity. Finding suitable subcontractors that have the appropriate equipment available
and confirming the subcontract orders can be a very
time consuming process. The sooner a schedule is produced, the better the chance of finding a suitable subcontractor. As time passes, it becomes more and more
difficult to find suitable crane-lorries, as other compa-
181
nies will have contracted them already. This implies
that lorries will have to be leased from a more expensive company or a lesser quality or larger lorry must be
used to satisfy customer needs.
When the scheduling process is performed manually,
there is always a chance of mistakes because of the large
amount of data and large number of constraints and
criteria that must all be considered simultaneously and
under tight time constraints. Therefore, one of the key
objectives in the design of our algorithm is to reduce the
scheduling time while improving the quality of schedules. Under the same circumstances and conditions,
a human planner can take up to an hour or more, depending on experience, to complete a schedule whereas
CLOE can produce a schedule within seconds. It takes
several months for a new planner to fully understand all
the scheduling criteria and constraints in order to produce an error-free schedule; even the most experienced
planner may make mistakes and cannot guarantee an
optimal schedule can be produced. There is no way a
human planner can thoroughly consider driver experience or historical data when performing scheduling.
Only the most experienced planner can keep all the
historical details in his/her head when constructing a
schedule. Since all relevant constraints and all historical data are considered by CLOE, the system guarantees no mistakes will ever be made and the quality of
schedule is best if not optimal.
The measurement of how “good” a schedule is may
differ from company to company. Our design allows
companies to dynamically adjust CLOE to employ the
heuristics that best reflect the company’s business objectives. For example, the primary objectives of the
company we did this research on are to minimize operational cost and at the same time maximize customer
satisfaction. Obviously, the second factor is very important in any service-oriented business because customer satisfaction is a critical part of business referrals
and repeat business, both of which directly affect the
company’s bottom line.
In the following sections, the architecture of CLSS
and the detail of the CLOE algorithms will be
presented.
2.
Architecture
The software architecture (see Fig. 3) can be divided
into four key modules according to the key business
processes—Sales Module, Scheduling Module, Subcontracting Module, and Dispatching Module. Figure 3
182
Figure 3.
Chun and Wong
Software architecture.
outlines the general architecture and operation of each
module.
Sales module responsible for maintaining customer and
order information. It receives customer orders, handles customer queries and provides order tracking.
Customer requests and orders come in throughout
the day. The sales staff will process customers’ enquiries and provide product information, such as
which types of lorries and cranes are available, and
their respective capabilities and differences. After
an order is finalized, it is passed to the “Scheduling
Module” that assigns a suitable crane-lorry to each
order.
Scheduling module responsible for resource allocation and contains the CLOE algorithm. Scheduling is performed at a predetermined time each day.
Allocation is done according to customer needs,
driver experience, and other business and operational rules and considerations. In addition, the
“Scheduling Module” will decide whether subcontractors are needed and which orders are to be
contracted out. If there is an excess supply of
crane-lorries, the planner will also determine which
lorries are on “stand-by” for possible non-reserved
orders. The scheduling process produces the following results: (1) which crane-lorries are still available, (2) which orders need to be subcontracted
out, and (3) which crane-lorry is assigned to which
order. The scheduling results are then passed to
the Sales Module, the Subcontracting Module, and
the Dispatching Module respectively for further
processing.
Dispatching module responsible for dispatching orders
to drivers and for receiving confirmation from the
drivers on their job status. Dispatching is done using mobile wireless devices. In the evening, after
the Scheduling Module has completed the vehicle
assignment, the Dispatching Module dispatches the
next day’s jobs to the drivers. As jobs are mainly local, they are usually completed within the same day
or overnight.
Subcontracting module subcontracts out any excess orders it cannot handle to other companies. From the
Scheduling Module, it receives a daily list of orders that cannot be fulfilled with internal company
resources. The company then tries to subcontract
these orders out to business partners. There is usually a pre-negotiated pricelist for different types of
services. Consequently, all the subcontractors need
to do is simply to confirm whether they have the
capacity to handle the excess order that has been
passed to them. If not, the order is sent to the next
subcontractor in the pool. The key purpose behind
subcontracting is to minimize revenue loss while
maximizing quality of service. This means keeping the more important or larger jobs, especially
those that offer long-term work, within the company
itself.
In this paper, we will focus mainly on the Scheduling Module that contains the CLOE algorithms. In this
CLSS: Crane Lorry Scheduling System
following Section, we will describe the knowledge encoded in and used by CLOE for scheduling.
3.
Business Rules and Constraints
The business logic, business objectives and operational
constraints will impact how a scheduling algorithm is
designed. The first piece of information used during
scheduling is the details of the customer order. Each
order is represented by a list of attributes, such as:
• Date and time vehicle is needed
• Duration of job—half day, whole day or overnight
• Required crane-lorry capacity, including the size and
type of lorry
• Location of job
• Lorry/driver preference if any
• Contact information, etc.
During the scheduling process, a human scheduler
will not only need to consider these basic attributes, but
also how to maximize customer satisfaction while minimizing costs. As these factors closely affect each another, it is important to achieve a balance between them.
Customer Satisfaction
The meaning of customer satisfaction can be different
for different companies. CLOE is designed to be customizable and allows measures to be easily adjusted
to suit the needs of different companies. For this company, customer satisfaction is defined as a combination
of customer preference, driver performance and driver
experience. For instance, a customer may prefer using a particular equipment or driver that it has utilized
before. Assigning a driver with more experience can
also increase customer satisfaction. Driver experience
is defined as the number of times a driver has been
assigned to a crane-lorry combination for a particular type of job before. The driver’s job performance
is generated from historical statistics on customer’s
after-service feedback. The net effect this will have on
CLOE is that it will provide some regularity and consistency in vehicle-driver assignments over time, which
this company has found to be highly desirable among
its customers. An additional benefit is that drivers will
be more familiar with their customers and their operations, which in many cases improves job satisfaction
for the drivers.
183
Operational Cost
Two key factors directly affecting the operational cost
are the resource utilization and transportation costs.
Maximizing resource utilization ensures resources
are not underused. This also increases the likelihood
that the company can satisfy more orders itself and
minimize the need for subcontracting. Even if subcontracting is needed, the company should try to satisfy the
larger orders and only subcontract out smaller, lowermargin orders to other companies. Besides subcontracting, assigning a crane-lorry with just the right capability
is another way to reduce operational costs, because the
size of the lorry is directly proportional to the operational cost of maintaining and operating it.
Allocating crane-lorries that are closer to the customer sites will reduce transportation costs. This is of
course assuming that the lorries have the same capability and degree of potential customer satisfaction as
defined previously.
Rules and Objectives
The following is a list of the more important business
knowledge extracted during business analysis. They are
implemented either as CSP constraints (indicated by a
“C” in the identifier), demons (“D”) or search heuristics
(“H”). Implementation details will be described in later
sections of this paper.
• [C1] No overlap constraint: each lorry should only
be assigned to one order at a time.
• [C2] Fulfill workload constraint: the lorry assigned
to a job must be adequately equipped for the job.
• [D1] Minimize number of subcontract orders constraint: when subcontracting, minimize the number
of orders given to subcontractors.
• [D2] Minimize subcontract order size constraint:
when subcontracting, subcontract orders that require
smaller-sized lorries to reduce operational costs
(considered by the “Subcontracting Module”).
• [H1] Maximize driver experience and customer satisfaction: the most experienced available lorry-driver
should be assigned to the most important customer
and if the customer has a special preference for a particular lorry, try to satisfy the customer’s preference
whenever possible.
• [H2] Maximize utilization: if it is necessary to assign
an over-equipped lorry in order to fill all remaining customer orders, the lorry with the closest level
184
Chun and Wong
of capacity should be assigned without frustrating
another order. In another words, the total tonnage
required to carry out all customer requests should be
minimized.
• [H3] Minimize travel distance: where possible, assign the closest available lorry to the worksite.
Underutilization Guidelines
As we mentioned before, the customer may explicitly
request a particular equipment/truck or driver when ordering. In some cases, this will cause the equipment to
be underutilized and will lower its overall utilization
rate and increase operational cost. On the other hand,
fulfilling customer requests will improve customer satisfaction. In order to achieve a balance between lowering operational costs and improving customer satisfaction, the company has these “Underutilization
Guidelines:”
• Underutilization must not increase the number of
subcontract orders nor the order size
• Underutilization must not cause another order to be
unsatisfiable
Furthermore, the “importance” or priority of the customer will also be taken into consideration before a decision is made. In any case, the size-utilization1 of a
crane-lorry should not be lower than 40% due to operational considerations; larger crane-lorry with less
carrying capacity actually operates less efficiently.
4.
Related Research
There are many different areas of optimization and
scheduling-related research on vehicles. For example,
there is the vehicle routing problem (VRP), vehicle
scheduling problem (VSP), automated guided vehicles
(AGV), etc. However, there are only a few research are
related to what we call the vehicle assignment problem (VAP), which this paper is about. Vehicle scheduling, in general, have played an important role in operations research (OR) ever since works from Dantzig
and Ramser [3] and Clarke and Wright [4] have been
published.
More recently, researchers have been experimenting
with a wider variety of techniques, including mathematical/linear programming (LP) [34, 37], constraint
programming (CP) [5], genetic algorithms (GA) [6] as
well as swamp/ant algorithms (AA) [38, 39] in solving
these types of scheduling problems. LP represents the
vehicle scheduling problem as a set of linear equations
that can be solved using mathematical programming
tools like CPLEX [36]. CP represents these problems in
terms of variables and constraints and uses arc consistency techniques to improve search efficiency. GA, on
the other hand, represents a vehicle schedule as a chromosome and uses survival-of-the-fitness techniques to
iteratively evolve a solution. Similar to GA and borrowing from ants in nature, AA also iteratively evolves
a solution. AA uses pheromone trail laying techniques
to record past evaluation/heuristic function values as
artificial pheromone.
The vehicle routing problem (VRP) is basically a
type of traveling salesman problem (TSP) where we
need to schedule vehicles, instead of salesmen, to visit
a set of customers to deliver and/or pickup goods.
Amongst other things, there are constraints on the vehicles, customers, drivers, goods and time. The objective
is to minimize cost, which is usually measured in terms
of the number of vehicles and/or distance traveled and
time spent.
Shaw [7] used CP techniques combined with local
searches in a way similar to the shuffling technique
used in job-shop scheduling to solve the vehicle routing problem. On the other hand, Ochi et al. [8], used
parallel GA combined with a scattered search and a
decomposition-into-petal procedure to solve a similar
problem. Interestingly, Barnier and Brisset [9] offer another approach to solve VRP by combining both CP and
GA techniques. This combined approach was shown to
perform well even for large search spaces. CP is used to
compute feasible solutions in a subspace of the search
space, while GA is used on the space formed by the
subspaces to perform the optimization.
There are also different variations to the vehicle routing problem. For example, Frizzell and Giffin [10] use
a heuristic approach to solve an extended VRP with
split delivery, allowing delivery to a customer to be
split between two or more vehicles, and a timeframe
when delivery must be made. Shang and Cuff [11] also
used a heuristic algorithm to solve pickup and delivery
problems within a Health Maintenance Organization
(HMO). In this problem, vehicles are used to transport
patient records, equipment and supplies. The objective
was to minimize vehicle expense, tardiness, and travel
time. Their research showed that their heuristic algorithm performed better at meeting these objectives than
the manual approach.
CLSS: Crane Lorry Scheduling System
Since a wide variety of approaches have been tested
on the vehicle routing problem, formal systems are being developed to help formalize and compare these approaches. For example, Desrochers et al. [12] outlined
a formal system that can be used to define models and
algorithms for vehicle routing and scheduling.
Related to the vehicle routing problem is the vehiclescheduling problem (VSP) [13]. In simple terms, VSP
is the problem of assigning vehicles to timetabled trips
such that each trip is performed by one vehicle, a set
of constraints is satisfied, and a cost function is minimized. The objectives usually include minimizing the
number of vehicles and different operational costs, such
as gas, labor, and deadheading trips. For example, bus
scheduling is a typical vehicle-scheduling problem.
Many real-life factors must be considered in solving
VSP, such as the number of depots, travel time at different locations and times of the day, labor regulations
that govern work hours of drivers, the types of vehicles
available, refueling needs, etc. For example, Park and
Song [14, 15] used heuristic algorithms combined with
a simple estimation model for time varying travel times
to solve VSP.
A different set of problems related with vehicles and
their scheduling is the problem of automated guided vehicles (AGV). AGV are used in many different environments such as factories, container terminals, and warehouses to transport goods and materials from one location to another. For example, Shawik [16] described research work in a flexible manufacturing system (FMS)
and its related AGV scheduling problem using a heuristic approach. Ulusoy et al. [17] solves a similar problem for scheduling AGV within a FMS, by using
genetic algorithms. Shawik, Ulusoy et al., combined
manufacturing machine scheduling with AGV scheduling. Other researchers focused just on AGV scheduling. For example, Wu [18] worked on a restricted
AGV scheduling problem using problem decomposition combined with a local search. Huang and Hallam
[19] used a totally different approach of distributed AI
(DAI) and negotiation among software agents to perform conflict resolution within the AGV scheduling
problem.
The work performed in the crane-lorry allocation
is related to these research areas but with some differences. We call this the vehicle assignment problem
(VAP). It is a specialized combination of the vehicle
routing problem (VRP) and vehicle-scheduling problem (VSP). In our crane-lorry assignment problem, we
need to satisfy pickup and delivery jobs by assigning
185
vehicle-driver pairs to each job. Unlike assumptions
made by many VRPs and VSPs, our vehicles are not
homogeneous and may contain different types of equipment. The routing is simplified, as the vehicles are
highly specialized and unique and can only perform
certain types of jobs. Hence, the vehicles usually perform only one or two jobs a day. The key problem to
be solved in VAP is not to compute the most efficient
route or schedule, but to find the most appropriate assignment of the equipment to each jobs while maximizing utilization and customer satisfaction. The “quality”
of the assignment plays an important role as this can
directly impact customer satisfaction. Hence VAP can
be modeled as a resource assignment problem.
Gill [20], in his SunRay V system, solved a related
VAP for freight-forwarders also using CP techniques.
Sunray V’s scheduling algorithm determines how many
jobs the company can handle, which jobs to subcontract, and assigns trucks or towheads to different jobs.
The key objective is to minimize subcontracted jobs,
meet all job deadlines and minimize inter-job idle time.
However, in the case of Sunray V, the towheads are
generally homogeneous, unlike the crane-lorries in this
problem. On the other hand, it is not necessary to schedule the jobs as the customer dictates when the lorry is
needed. For the present problem, customer satisfaction
and compatibility of driver’s experience with the job
assignment is more important than minimizing travel
time.
Before actual scheduling, the SunRay V algorithm
identifies the best pairings of tasks to minimize the
number of paid-trips before the actual scheduling. The
paid-trip also captures the transport and non-transport
related execution times and associated auxiliary trips.
There are also other preferences and considerations that
must be taken into account in the actual connection
selection procedure. The SunRay V system divides the
scheduling into two parts. The first part produces a
predictive schedule and identifies the most appropriate
trip for sub-contracting if resources are insufficient.
In the second part, the schedule produces an online
schedule, which assigns each towhead with one trip
after the executing or completing the current trip.
Table 1 shows the differences between the
SunRay V container-trucking problem and our cranelorry scheduling problem.
Due to the high variation in the execution time and
connection-time between each trip, the SunRay V requires an online scheduling module and must divide
the schedule into parts. In addition, it pairs tasks before
186
Chun and Wong
Table 1.
Differences between the SunRay V container trucking and the crane-lorry scheduling problems.
SunRay V scheduling
Crane-lorry scheduling
The scheduling module is divided
into two consecutive parts.
Produces the schedule and makes the sub-contracting
decision simultaneously.
Many factors affect the duration of the job and
the connection-time between jobs.
The customer specifies the required time (usually half a day,
whole day or mid-night) when making the order.
A Towhead may be assigned more than one trip
a day and the assignment depends on the
currently executing or completing trips.
Each lorry is assigned one order per day (if it is a full day order). If two half-day
orders suit the same lorry it will be assigned more than one order. Usually a
lorry will not be assigned more than two orders on the same day.
There is a high variability in job execution time,
response time and communication time with the
customer, which can cause frequent rescheduling.
Customer preference and driver working experience are
important considerations during the scheduling process and are the
primary factors determining the schedule quality.
Online schedule
Offline schedule
the actual scheduling in order to minimize the number
of trips. In the crane-lorry scheduling problem, fewer
time constraints need to be considered than in SunRay
V, and the most important consideration is whether a
cost effective schedule can be produced while satisfying customer preference.
Another related research is Duncan’s Schedule-IT
[21] application. Schedule-IT also uses a CP algorithm
to allocate drivers and vehicles for the delivery of customer orders. Like SunRay V and CLOE, excess work
is subcontracted out to other companies. There are a
number of constraints concerning the capabilities of
the resources (vehicles and drivers) and limitations on
working hours. Schedule-IT uses a driver-driven algorithm that schedules all the jobs of one driver before moving on to the next. In this way, the scheduler partitions the problem into sub-problems. In the
case at hand, all the orders are solved in one global
set. To meet the objective of maximizing the degree
of match between driver experience and the work order, the global approach was determined to yield better
solutions.
Table 2 shows the differences between the ScheduleIT and the crane-lorry scheduling system.
Table 2. Differences between the Schedule-IT and the crane-lorry
scheduling problems.
Schedule-IT
Completes one driver’s schedule
after another, sequentially.
The search space is split into
separable sub-problems.
A driver may be assigned
different vehicles.
JSolver scheduling module
All orders are solved in
one global set.
The search space is not split
into sub-problems.
Each driver is responsible for
one lorry; usually, no
changes are made.
5.
Model and Implementation
Our CLOE scheduling algorithms were implemented
using a Java constraint-programming language called
JSolver [1, 2]. JSolver extends the object-oriented
programming paradigm of Java with constraint-based
declarative programming. The basic underlying mechanisms include an arc consistency checking algorithm,
constraint propagation, and backtracking. It allows the
development of scheduling and resource management
systems in Java and deployment of these systems over
the Web or within an Enterprise Java Bean (EJB) application server. JSolver has been successfully used to
solve different types of engineering problems as well
as e-trade negotiation problems [22].
We modeled the crane-lorry assignment problem as a
constraint-satisfaction problem (CSP) [23–27]. In general, any scheduling or resource allocation problems
can be formulated as a constraint-satisfaction problem
(CSP), which involves the assignment of values to variables, subjected to a set of constraints. We have successfully used CSP algorithms for other transportationrelated resource allocation problem before [28–30].
CSP can be defined as consisting of a finite
set of n variables v1 , v2 , . . . , vn , a set of domains d1 , d2 , . . . , dn , and a set of constraint relations
c1 , c2 , . . . , cm . Each di defines a finite set of values
(or solutions) that variable vi may be assigned. A
constraint c j specifies the consistent or inconsistent
choices among variables and is defined as a subset
of the Cartesian product: c j ⊆ d1 × d2 × · · · × dn .
The goal of a CSP algorithm is to find one tuple from
d1 × d2 × · · · × dn such that n assignments of values to variables satisfy all constraints simultaneously.
CSP algorithms increase the efficiency of the search
CLSS: Crane Lorry Scheduling System
by looking ahead, and actively using the constraints to
prune the search space, thus minimizing backtracking.
When the lorry-assignment problem is formulated
as a CSP, each variable or unknown represents the assignment of a lorry to an order. The domain of each
variable, that is, the possible orders a lorry may be assigned to, will initially contain the set of all orders for
the next day. The CSP constraints are restrictions on
how these lorries can be assigned based on business
and operational rules and objectives.
The CLOE Algorithm
The CLOE algorithm is a recursive search algorithm
that iteratively assigns a value to a variable. We experimented with two models—a lorry-centric model
that used variables to represent crane-lorries, and an
order-centric model that used variables to represent
customer orders. In this paper, we only documented the
lorry-centric model as their performances were similar. For the lorry-centric model, at each recursion level,
CLOE selects the next “best” unassigned crane-lorry
and either assigns the “best” order to it or marks it
as “unassignable,” while satisfying all the predefined
constraints. A crane-lorry is “unassignable” if it is
under-equipped to handle any of the outstanding orders. On the other hand, assignment can still be made if
the crane-lorry is over-equipped. Over-equipped cranelorries should only be assigned, however, if there are
no other options left, in order to avoid under-utilization
or, even worse, making another order unfulfillable due
to a lack of suitable equipment.
We use search heuristics to define what “best” unassigned crane-lorry means, as well as what “best”
order to assign means. These heuristics guide the
search as well as improve the quality of the resulting schedule. Further use of an arc-consistency algorithm is made to perform constraint-propagation and
domain reduction in order to further improve search
performance.
The following is the pseudo-code for the CLOE algorithm using notations similar to those used by Tsang
[23]:
[Z ] finite set of variables (the set of all unassigned
crane-lorries)
[D] finite set of all variable domains (domain contains
all the orders for the day)
[C] finite set of constraints (restrictions on how lorries
can be assigned)
187
PROCEDURE CLOE (Z, D, C)
BEGIN
Solution ← NIL;
D ← Propagate (Z, D, C, 0);
IF (no domain in D is empty)
THEN solution ← Search (Z, 0, D , C);
END
The CLOE procedure first calls the Propagate procedure to perform domain reduction on posted constraints. (The last parameter to Propagate is the variable binding that triggered this propagation. This is
null for the initial constraint posting.) This pre-search
arc-consistency check will improve performance by removing all “obvious” infeasible choices from the domains of the variables. Infeasible choices are defined
through constraints contained in the set of constraints
“C” that governs how lorries can be assigned. This set
contains instances for several types of constraints. At
this pre-search stage of the algorithm, the constraint
(see Section 4) that has the largest impact is:
• [C2] Fulfill Workload Constraint: the lorry assigned
to job must be adequately equipped for the job
There is one “Fulfill Workload Constraint” per variable. The effect of posting each “Fulfill Workload Constraint” is that the domain of that variable will be guaranteed to contain only orders it can fulfill. This is
done through domain reduction and is performed by
the Propagate procedure prior to calling the Search.
All orders that the lorry is not equipped to fulfill are
removed from its domain. If the problem is “overconstrained,” some of the resulting domains will be
empty. When this happens, there is no solution and the
algorithm exits. Otherwise, after the initial search space
has been reduced, CLOE simply initializes the set of
committed labels, i.e., the partial solution set, (second
parameter to the Search procedure) to nil and starts the
actual search.
The Search procedure is the recursive algorithm that
performs the actual search and is defined below. The
following notations are used:
x: an unassigned variable (an unassigned crane-lorry)
Dx: the domain of variable x (the set of all orders that
the lorry can fulfill)
x, v: assignment of value v to variable x (the assignment of an order to a lorry)
U : set of unlabelled variables (set of unassigned cranelorries)
188
Chun and Wong
CL: set of committed labels (set of assigned cranelorries)
C: finite set of constraints (restrictions on how lorries
can be assigned)
We first evaluate the number and size of the order
to be subcontracted. After an order is assigned to a
lorry, we can check whether the remaining order and
unassigned lorry will result in increase the number and
PROCEDURE Search (U, CL, D, C)
BEGIN
fail ← FALSE;
IF (U = {}) THEN RETURN (CL)
//all lorries have been assigned orders
ELSE BEGIN
x ← SelectLorry (U );
//find next “best” lorry to schedule
REPEAT
v ← SelectOrder (Dx);
//find next “best” order to assign
Dx ← Delete (v, Dx);
//delete selected order from Dx
IF (AddValueDemon = pass)
//if CL + {x, v} violate no constraints
THEN BEGIN
D ← Propagate (U − {x}, D, C, x, v); // constraint propagation
If (no domain in D is empty)
THEN BEGIN
// recursively search
result ← Search (U − {x}, D, CL + {x, v}, D , C);
IF (result ! = NIL) THEN RETURN (result);
END
ELSE fail ← TRUE;
// domain in D is empty
END
ELSE fail ← TRUE;
// new compound label violates constraints
UNTIL (Dx = {} / fail ! = TRUE);
RETURN (NIL);
END
END
The main tasks for Search are: (1) to select the next
best unassigned lorry using the SelectLorry procedure;
(2) assign the next best order with SeletOrder; (3)
after a value is assigned to the variable the AddValueDemon will execute to check whether the new compound label violates any constraint; and finally (4) to
Propagate constraints. SelectLorry defines the heuristic for selecting the next “best unassigned crane-lorry.”
SelectOrder defines the heuristic for selecting the next
“best order to assign.” AddValueDemon is executed
just after a value is assigned to a variable in order to
check those relational constraints. The following relational constraints defined in Section 4 are implemented
as demons and processed in AddValueDemon:
• [D1] Minimize Number of Subcontract Order Constraint: minimize the number of orders given to subcontractors
• [D2] Minimize Subcontract Order Size Constraint:
when subcontracting, subcontract orders that require
smaller-sized lorries to reduce operational costs
size of order that need to be subcontracted to an outside company. If the constraint has been violated, the
algorithm will backtrack to a previous choice point.
Propagate is the constraint propagation algorithm
that performs domain reduction. It is a type of lookahead strategy [23, 31] to detect constraint violations.
The key constraints (see Section 4) that are propagated
during the search are instances of:
• [C1] No Overlap Constraint: each lorry should only
be assigned to one order at a time
There is one “No Overlap Constraint” per variable,
which represents a lorry-to-order assignment. The effect of posting each “No Overlap Constraint” is that
whenever an order has been assigned a lorry, all the
domains of the remaining unassigned lorries will have
their domains reduced by removing this order, as the
order will no longer be a valid choice.
Whenever any domain gets reduced to an empty set
or when a legal lorry-order assignment cannot be found,
CLSS: Crane Lorry Scheduling System
the Search algorithm returns NIL. This forces the algorithm to backtrack to a previous level and continue
the search at the previous choice points.
The Search Heuristics
The above describes how CLOE implements “hard”
constraints of the crane-lorry assignment problem.
Hard constraints are requirements that must never be
violated. With just hard constraints, CLOE will be able
to generate “feasible” solutions. CLOE uses search
heuristics to further improve the quality of the search
to produce “best” solutions. Best solutions are those
that try to satisfy other “soft” requirements such as
customer preferences. In CLOE, soft constraints are
implemented in the search heuristics SelectLorry and
SelectOrder.
189
received from the customer on how “satisfied” they
were with the service provided.
We found that these heuristics performed equally
well if schedule quality was defined in terms of “costs”
only. However, if we consider “customer satisfaction”,
the Experience Heuristic performed significantly better
[32]. CLOE implements these heuristics as a weighted
sum with the weights adjustable for different companies.
The SelectCrane Procedure
The SelectCrane procedure defines the heuristic for
what the “best” order to assign to a crane-lorry. The
heuristic used by CLOE is:
The SelectLorry Procedure
• Maximize Utilization Heuristic: Always select the
largest orders a lorry can handle and from these the
order that it has most experience in handling.
The SelectLorry procedure defines the heuristic for
what the next “best” unassigned crane-lorry means.
Many different domain-specific heuristics [32] were
experimented with in order to find the one that
produces the best improvement in the quality of
the generated schedule. These include the Versatility Heuristic, the Size Heuristic and the Experience
Heuristic:
The definition of “largeness” may vary from company to company and usually means the orders that
have the most stringent requirements on lorry capabilities. This may be in terms of maximum gross vehicle
weight, maximum capacity, size of the lorry, or combinations of these factors. In the company we performed
this work on, we define “largeness” as a weighted sum
of all these factors.
• Versatile Heuristic: By selecting the most “versatile” lorry first and assigning the best order
to it, we try to maximize customer satisfaction
by assigning the “most preferred” equipment first
and also maximize equipment utilization. “Versatility” is defined in terms of the normalized maximum ton-meters of the crane, normalized maximum gross vehicle weight of the lorry, normalized payload area of the vehicle, mounting location of the crane and company-specific weighting
factors.
• Size Heuristic: The Size Heuristic is a simplified
form of the Versatility Heuristic and focuses just on
the workload handling capability of the lorry. “Size”
is defined to be the normalized maximum gross vehicle weight (GVW) of the lorry.
• Experience Heuristic: This heuristic is based on the
historical statistics of the crane-lorry-driver combination in performing similar tasks (potential jobs in
the variable domain) in the past and the feedback
The Objective Function
An extended version of the CLOE algorithm, called
X-CLOE, was also implemented to experiment with
generating solutions that optimized an objective function. X-CLOE is similar to CLOE with the addition of
“branch and bound” optimization. Multi-criterion decision modeling is used to formulate this objective function, TF, which returns a number that reflects the overall “goodness” of the schedule. This goodness measure
includes driver experience, customer satisfaction, size
utilization and the travel distance. The definition of this
function may differ from company to company and depends on their respective business strategies.
TF is the total sum of the goodness measure, Fcxj ,
for each assignment in the schedule. For this company,
Fcxj is defined as:
Fcxj = (w1 ∗ lc ∗ Pcxj ) + (w2 ∗ Scj ) + (w3 ∗ Dcj ) (1)
190
Chun and Wong
where
Fcxj : the “goodness” measure of assigning crane-lorry
c and driver x to job j
lc : the “importance” level or priority of a customer
Pcxj : customer feedback historical statistics on performance of crane-lorry-driver for job j; range: 0..100,
Pcxj = 100 if customer highly recommends the combination and Pcxj = 0 if no preference.
Scj : size utilization if crane-lorry c is assigned to job
j; range: 0..100, Scj = 100 if resource c will become
fully utilized.
Dcj : the normalized travel distance between home location of crane-lorry c to location of job j; range
0..100, Dcj = 0 if the crane-lorry is right next to the
worksite.
wi : weighting factors that vary from company to company. Higher values for w1 increases customer satisfaction, while w2 and w3 improves utilization and
reduces costs.
After each solution, a new TF value is computed
and compared with the previous using a “branch and
bound” technique [33]. This way, X-CLOE incrementally finds the next best solution that optimizes TF.
Depending on the number of orders, the search space
can be quite large. Instead of waiting of an “optimal”
solution, X-CLOE also implements a timer mechanism
to return the best “near optimal” solution so far up to
the pre-set time limit. For all the X-CLOE experiments
in this paper, we set the timer to return a near optimal
solution to 20 seconds.
The CLOE algorithms can be customized for different crane-lorry companies simply by adjusting the
weights used in Eq. (1) to reflect a company’s unique
business objectives and operational needs. Alternatively, a totally different objective function with a different formula can be “plugged in” to replace the default one used by CLOE.
6.
Test Results
The following are results from our tests on the CLOE
and X-CLOE algorithms using seven sets of actual data
reflecting typical orders received by the project company. Historical data, used to determine “experience,”
was compiled using the actual schedule from the prior
two weeks as well as customer feedback on driver performance.
The results from testing these two algorithms were
then compared according to two key criteria: “cost”
and “quality”. “Cost” is measured in terms of distance
traveled and the total tonnage required to fulfill all customer requests. These criteria represent the primary operational costs of the company. “Quality” is measured
in terms of customer satisfaction and this translates to
a historical preference-level.
The Cost Factor
Operational cost is one of the factors we used to compare CLOE with the manual approach. Operational cost
is directly proportional to the total tonnage and the traveling distance. The “total tonnage” used is measured in
terms of the number and type of vehicles that were
allocated. Table 3 shows the number and types of vehicles used in the manual schedule. The vehicle type is
important, as larger vehicles require more resources to
operate. For simplicity, the vehicle type is represented
as the maximum lifting capacity (payload weight) of
the mounted crane. Crane outreach is normalized to
2 meters. For example, “25T” indicates that a vehicle
has a crane that can lift a payload of 25 tons with an
outreach of 2 meters.
The “total tonnage” is pretty much dictated by the
customer orders and room for improvement is limited.
We used this measure purely to verify the accuracy of
the CLOE algorithms. As Table 4 and Chart 1 illustrate,
both CLOE and X-CLOE performed equally well, if
not better than, the manual approach in terms of “total
tonnage.”
CLOE and X-CLOE performed better than the manual schedule in Test 5. After comparing results of the
manual and CLOE generated schedules for Test 5, the
analysis indicated that the manual schedule assigned a
larger crane-lorry to one of the orders, which resulted in
the increase in total tonnage used. This is one example
of how CLOE can prevent human errors in allocation.
Table 3. Number of vehicles used for each set of test data (manual
approach).
Manual
Test 1
Test 2
Test 3
Test 4
Test 5
Test 6
Test 7
12T
0
0
0
0
0
0
0
15T
0
1
2
2
2
2
0
17T
1
3
4
2
2
4
4
20T
0
3
3
1
3
3
3
25T
2
5
6
3
6
5
6
35T
2
2
2
1
0
2
1
55T
0
0
0
1
0
1
0
CLSS: Crane Lorry Scheduling System
Chart 1.
Percentage reduction in terms of “total tonnage” on schedule generated by CLOE and X-CLOE for each of the test case.
“Travelling distance” is another way to measure cost.
Table 5 shows the total distance between the parking location of the lorry and the job location. The percentage
cost reduction from CLOE is shown in Chart 2.
Chart 2 shows that the CLOE performed better on
Test 3, and equally well on Tests 1, 2 and 4 compared
with the manual schedule. However, CLOE did slightly
Table 4.
tonnage.
191
Percentage reduction in operation cost in terms of total
Test 1 Test 2 Test 3 Test 4 Test 5 Test6 Test 7
Manual
137
321
378
249
294
408
313
CLOE
137
321
378
249
281
408
313
X-CLOE
137
321
378
249
284
408
313
%Reduction—
CLOE
0
0
0
0
4.42
0
0
%Reduction—
X-CLOE
0
0
0
0
3.4
0
0
Table 5. The total distance and the corresponding percentage reduction in each test case.
Distance
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
Test 7
Manual
185
430
465
340
365
645
445
CLOE
185
430
440
340
385
655
495
X-CLOE
185
395
440
340
370
650
445
%Improve—
CLOE
0
0
5.38
0 −5.48 −1.55
%Improve—
X-CLOE
0
8.14
5.38
0 −1.37 −0.78
−11.24
0
worse on Tests 5, 6 and 7. Compared with manual
schedules, X-CLOE performed better on Tests 2 and
3, and equally well on Tests 1, 4 and 7. Although XCLOE performed slightly poorer, around 1%, on Tests
5 and 6, it did obtained an overall 11.3% cost reduction
in terms of distance. In all our test cases, the X-CLOE
performed better than the CLOE in reducing the overall
distance traveled.
The Quality Factor
The second key criterion in our evaluation of the CLOE
scheduling algorithms is whether they can generate
“quality” assignments, which is measured in terms of
degree of “customer satisfaction.” This is measured by
consolidating historical statistical data and customer
feedback and reflects driver experience and customer
preference. This “preference-level” measure is the parameter Pcxj defined in our objective function (Eq. (1)
in the previous section). Table 6 shows the Pcxj values
for the CLOE algorithms compared with the manual
approach. Chart 3 shows the percentage improvements.
As Chart 3 shows, CLOE algorithms performed
better than manual approach, except for Tests 3 and
5. After comparing results between manually and
CLOE-generated schedules for Test 3, we found that
the manual schedule incorrectly allocated a highly
over-equipped crane-lorry to one of the orders (a 25T
crane-lorry was assigned to an order requiring an 8T
crane-lorry) resulting in only 32% size utilization.
Underutilization improves the preference score by
increases cost. CLOE ensures utilization is always
192
Chun and Wong
Chart 2.
Percentage cost reduction in terms of distance between the location of the crane-lorry and the worksite.
Chart 3.
Percentage improvement in quality (in terms of preference level) for each of the algorithm compared with manual approach.
maximized. Overall, CLOE obtained 8% improvement
in customer satisfaction. In all our test cases, X-CLOE
performed better than CLOE and obtained an overall
12.6% improvement.
Table 6. Total preference-level and the percentage improvement
for each of algorithm compared with manual approach.
Preference
Test 1 Test 2 Test 3 Test 4 Test 5
Test 6 Test 7
Manual
130
510
536.8
480
CLOE
130
510
530
480
454.94 464.57 420
450
480
450
X-CLOE
130
510
530.4
480
470.4
480
450
%Improve
(CLOE)
0
0
−1.19
0
−1.086
3.32
7.14
%Improve
(X-CLOE)
0
0
−1.19
0
3.31
3.32
7.14
By combining the percentage improvements in cost
and quality factors, see Table 7, X-CLOE performs
better than the manual approach and regular CLOE in
most of the test cases.
On the surface, these percentage improvements
may seem small. However, there are numerous other
significant benefits from CLOE that are hard to
quantify. For example, long-term business benefits
Table 7.
Overall % improvement of cost and quality factors.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6 Test 7
%Improvement—
CLOE
%Improvement—
X-CLOE
0
0
0
4.19
0
8.14 4.19
0
−2.146 1.77 −4.1
5.34
2.54 7.14
CLSS: Crane Lorry Scheduling System
from better overall customer satisfaction cannot be
underestimated. There is also the operational benefit
for having a precious 1 to 2 hour lead time over
competitors in acquiring necessary vehicles from
business partners—time saved by using CLOE. This
lead time can make a big difference in being able to
satisfy all customer orders as well as getting a better
price from subcontractors. Expertise in crane-lorry
scheduling requires years of experience in the same
company. Unfortunately, this experience does not
map well to another company as each company’s
equipment and team, as well as business requirements
and objectives can be very different. With increasing
consolidation in this industry, companies must rely
on automated systems such as CLSS and CLOE as
human expertise does not scale well for this problem.
7.
Conclusion
The quality of service provided and the degree of
customer satisfaction are important differentiators for
service-oriented companies such as crane-lorry leasing
companies. Schedules for operations must be created
with these crucial business objectives in mind. In this
paper, we presented a CLOE scheduling algorithm that
achieves these objectives by capturing domain knowledge in the crane-lorry leasing industry and representing them as constraints and heuristics in a CSP algorithm. The X-CLOE algorithm then extends the original
CLOE with optimizing branch-and-bound to incrementally search for better solutions. Overview, the CLOE
algorithms produce more cost effective schedules than
manual approaches. Our tests show that by considering
the experience of the crane-lorry-driver combination,
traveling distance and the size utilization, the CLOE
algorithms were able to generate schedules that were
equal or better than those produced by human planners
and in greatly reduced time—within seconds. Although
this type of constraint-based scheduling is rarely used
in the vehicle leasing industry, we hope that this work
can excite interest in this area by demonstrating the
business values for improving customer satisfaction
and lowering operational costs.
Acknowledgment
The work described in this paper was substantially supported by a grant from the Research Grants Council of
the Hong Kong Special Administrative Region, China
(Project No. 9040517, CityU 1109/00E). This work
193
was also partially supported by a grant from City University of Hong Kong (Project No. 7001286).
Note
1. Size utilization = size of the order ∗ 100/size of the lorry. Example: the size utilization for a 35T lorry to carry out a 25T order is
equal to 71%.
References
1. H.W. Chun, “Constraint programming in Java with JSolver,” in
PACLP 99, London, April 1999.
2. http://www.ilog.com
3. G.B. Dantzig and J.H. Ramser, “The truck dispatching problem,”
Management Science, vol. 6, pp. 80–91, 1959.
4. G. Clarke and J.W. Wright, “Scheduling of vehicles from a central depot to a number of delivery points,” Operations Research,
vol. 12, pp. 568–581, 1964.
5. J.-F. Puget, “Object-oriented constraint programming for transportation problems,” in ILOG Solver Collected Papers, ILOG
A, France, 1994.
6. J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1976.
7. P. Shaw, “Using constraint programming and local search methods to solve vehicle routing problems,” in CP’98, edited by M.
Maher and J.-F. Puget, vol. LNCS 1520, pp. 417–431, 1998.
8. L.S. Ochi, D.S. Vianna, L.M.A. Drummond, and A.O. Victor, “A
parallel evolutionary algorithm for the vehicle routing problem
with heterogeneous fleet,” Future Generation Computer Systems, vol. 14, pp. 285–292, 1998.
9. N. Barnier and P. Brisset, “Optimization by hybridization of a
genetic algorithm with constraint satisfaction techniques,” Evolutionary Computation Proceedings, pp. 645–649, 1998.
10. P.W. Frizzell and J.W. Giffin, “The split delivery vehicle scheduling problem with time windows and grid network distances,”
Computers Operations Research, vol. 22, no. 6, pp. 655–667,
1995.
11. J.S. Shang and C.K. Cuff, “Multicriteria pickup and delivery
problem with transfer opportunity,” Computers & Industrial Engineering, vol. 30, no. 4, pp. 631–645, 1996.
12. M. Desrochers, C.V. Jones, J.K. Lenstra, M.W.P. Savelsbergh,
and L. Stougie, “Towards a model and algorithm management
system for vehicle routing and scheduling problems,” Decision
Support Systems, vol. 25, pp. 109–133, 1999.
13. F. Baita, R. Pesenti, W. Ukovich, and D. Favaretto, “A comparison of different solution approaches to the vehicle scheduling
problem in a practical case,” Computers & Operations Research,
vol. 27, pp. 1249–1269, 2000.
14. Y.B. Park and S.H. Song, “Vehicle scheduling problems with
time-varying speed,” Computers & Industrial Engineering,
vol. 33, nos. 3/4, pp. 853–856, 1997.
15. Y.B. Park, “A solution of the bicriteria vehicle scheduling problem with time and area-dependent travel speeds,” Computers &
Industrial Engineering, vol. 38, pp. 173–187, 2000.
16. T. Sawik, “A multilevel machine and vehicle scheduling in a flexible manufacturing system,” Mathematical Computer Modeling,
vol. 23, no. 7, pp. 45–57, 1996.
194
Chun and Wong
17. G. Ulusoy, F.S.-Serifoglu, and U. Bilge, “A genetic algorithm
approach to the simultaneous scheduling of machines and automated guided vehicles,” Computers Operations Research,
vol. 24, no. 4, pp. 335–352, 1997.
18. X.B. Wu, “The application of analytic process of resource in an
AGV scheduling,” Computers & Industrial Engineering, vol. 35,
no. 1/2, pp. 169–172, 1998.
19. X. Huang and J. Hallam, “Spring-based negotiation for conflict
resolution in AGV scheduling,” IEEE International Conference
on Systems, Man and Cybernetics, vol. 1, pp. 789–794, 1995.
20. A.J. Gill, “SUNRAY V—An intelligent container trucking
management system,” in Proceedings of the First ILOG
SOLVER and ILOG SCHEDULE International User’s Meeting,
1995.
21. Tim Duncan, “Schedule-IT: An intelligent vehicle scheduling,”
in Proceedings of the First ILOG SOLVER and ILOG SCHEDULE International User’s Meeting, 1995.
22. http://www.cmis.csiro.au/aai/
23. E. Tsang, Foundations of Constraint Satisfaction, Department of
Computer Science, University of Essex, Colchester, Essex, U.K.
24. J. Cohen, Constraint Logic Programming, Communications of
the ACM, vol. 33, no. 7, pp. 52–68, 1990.
25. G.L. Steele Jr., The Definition and Implementation of a Computer Programming Language Based on Constraints, Ph.D.
Thesis, MIT, 1980.
26. V. Kumar, “Algorithms for constraint satisfaction problems: A
survey,” Al Magazine, vol. 13, no. 1, pp. 32–44, 1992.
27. P. Van Hentenryck, Constraint Satisfaction in Logic Programming, MIT Press, 1989.
28. H.W. Chun, “Scheduling as a multi-dimensional placement
problem,” Engineering Applications of Artificial Intelligence,
vol. 9, no. 3, pp. 261–274, 1996.
29. H.W. Chun and R.W.T. Mak, “Intelligent resource simulation for an airport check-in counter allocation system,”
IEEE Transactions on Systems, Man, and Cybernetics—Part
C: Applications and Reviews, vol. 29, no. 3, pp. 325–335,
1999.
30. H.W. Chun, S. Chan, F. Tsang, and D. Yeung, “Stand allocation
system (SAS)—A constraint-based system developed with
software components,” Al Magazine, vol. 21, no. 4, pp. 63–74,
2000.
31. D. Frost and R. Dechter, “Look-ahead value ordering for constraint satisfaction problems,” in Proceeding of the Fourteenth
International Joint Conference on Artificial Intelligence, 1995,
vol. 1.
32. R.Y.M. Wong and A.H.W. Chun, “An intelligent crane-lorry
scheduling system using constraint satisfaction,” in Proceedings
of 4th World Multiconference on Systemics, Cybernetics and
Informatics (SCI2000), Orlando, Florida, July 23–26, 2000.
33. P.H. Winston, Artificial Intelligence, 3rd edition. AddisonWesley Pub. Co., 1992.
34. H. Paul Williams, Model Building in Mathematical Programming, 4th edition, John Wiley & Sons, 1999.
35. K. Marriott and P.J. Stuckey, Programming with Constraints:
An Introduction, MIT Press, 1998.
36. CPLEX, http://www.ilog.com/products/cplex/
37. W.L. Winston, et al., Introduction to Mathematical Programming: Applications and Algorithms, Wadsworth Publishing
Company, 1997.
38. J. Kennedy, et al., Swarm Intelligence, Morgan Kaufmann
Publishers, 2001.
39. E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence:
From Natural to Artificial Systems, Oxford University Press,
1999.