EUROPEAN
JOURNAL
OF OPERATIONAL
RESEARCH
ELSEVIER
European Journal of Operational Research 83 (1995) 182-199
Theory and Methodology
Properties of expected costs and performance measures
in stochastic models of scheduled transport
Malachy Carey a,*, Andrzej Kwiecifiski a,b
a Faculty of Business and Management, The University of Ulster, Northern Ireland, BT37 OQB, UK
b Mathematical Institute, University of Wroctaw, pl. Grunwaldzki 2 / 4, 41-386 Wroctaw, Poland
Received February 1993; revised July 1993
Abstract
In scheduled (timetabled) transport systems (for busses, trains, etc.) it is desirable at the planning stage to know
what effect proposed or planned changes in the schedule may have on expected costs, expected lateness, and other
measures of cost or reliability. We consider such effects here, taking account of the random deviations of actual
times (or arrivals, departures, etc.) from the corresponding scheduled times. We also take account of various forms
of interdependence (knock-on effects) between the timings (arrivals, departures, connections, lateness, etc.) of
different transport units. We formulate a stochastic model of such a complex transport system. (For generality, the
underlying deterministic version of the model is consistent with versions of various existing deterministic transport
models). We show that expected costs, and various measures of reliability, behave well (are convex) with respect to
any changes in the schedule. We derive this convexity, (a) without assuming any particular functional form for the
probability distributions of any of the random variables (trip times, wait times, etc.), (b) assuming very general
operating rules, (c) assuming a quite general transport network. These convexity properties assist transport planners
and managers in predicting the effects of schedule changes. They also ensure that various search algorithms can be
used to find improved or optimal schedules.
Keywords: Transportation; Optimization; Reliability; Networks
1. Introduction
T w o m a j o r objectives in o p e r a t i n g s c h e d u l e d t r a n s p o r t systems a r e to i n c r e a s e r e l i a b i l i t y a n d d e c r e a s e
costs o f t h e service. T o d o this efficiently it w o u l d b e v e r y useful to k n o w h o w e x p e c t e d costs a n d
r e l i a b i l i t y w o u l d v a r y as w e v a r y t h e p a r a m e t e r s o f t h e system, in p a r t i c u l a r as w e v a r y t h e scheduled
arrival a n d d e p a r t u r e times, w a i t times, h e a d w a y s , c o n n e c t i o n times, etc. A b e t t e r u n d e r s t a n d i n g o f
* This research was supported by Science and Engineering Research Council grant and GR/H/50432. It had earlier support from a
British Rail/Fellowship of Engineering Senior Research Fellowship which Dr Carey held at Oxford University.
* Corresponding author.
Elsevier Science B.V.
SSDI 0377-2217(93)E0248-V
M. Carey, A. KwieciFlski/ European Journal of Operational Research 83 (1995) 182-199
183
properties of expected costs and reliability (such as monotonicity and convexity) can give insight into the
behaviour of the system, can be used to guide both managerial behaviour and numerical algorithms in
improving or optimizing the system, and allows us to apply more accurate or faster converging
algorithms. We are interested here in whether these properties are implied by the general structure of
the underlying system.
For specificity in the discussion, we will mainly refer to high frequency passenger train services on a
rail network, of the type common in Britain and Europe. However, the results also apply to other forms
of scheduled transport. We chose this form of transport since it is very complex and interdependent in its
operating rules, so that other forms of transport are then covered as special cases. Complex rules arise
for rail because minimum headways are required between trains using the same section of track, or the
same station platform, etc., and trains usually cannot (or cannot easily) pass on the same line or section
of track. We assume that the sequence order of trains using a particular piece or line between station is
usually specified (see comment in Step 2, Section 3.8), and that this prespecified order can be different
for the same trains on successive section of track. Similar rules apply to other forms of scheduled
transport than rail, though they may be less strictly enforced for other forms. Later in the paper (Section
4 and 5) we relax the somewhat strict requirement of fixed prespecified order of trains by introducing
probabilistic operating rules, that is, operating rules which allow the order of trains (or other aspects of
system's operation) to vary accordingly to independent randomly changing circumstances.
Most previous discussions of probabilistic models of timetabling scheduled services consider only a
single link or a single line consisting several of links (e.g. PoweU and Sheffi, 1983; Chen and Harker,
1990; Carey, 1994; Carey and Kwiecifiski, 1992). Here we instead consider a more general network with
multiple connecting services. Because of the complexity of such a stochastic network model we take the
sequence order in which the services (trains, etc.) use links, stations, etc. as fixed, or at least temporarily
fixed (with the exception of probabilistic operating rules discussed in Section 5). In this case, the results
apply to adjusting or optimizing a network of services after the basic time order of the services has been
set. However, the results are also useful if one wishes to evaluate and compare many different possible
sequence orders for trains, etc. For each of these sequence orders the results in this paper apply and can
be used in computing cost and reliability measures for each sequence order considered. One can then
choose the best of these sequence orders. Such an evaluation .scheme could be introduced into existing
deterministic algorithms for generating schedules or timetables.
Transport networks consist of a set of nodes (e.g., railway stations) and a set of links (rail lines)
connecting pairs of nodes. The planned movement of trains in the network is described by fixed
timetable or scheduled _T of arrival and departure times, and hence scheduled trip times and wait times.
The actual movement of trains in the network is described by the actual times t of these events (arrivals,
departures, etc.), and perhaps certain other events on the train journeys. These actual times are random
variables which usually differ from the planned times. The actual time t i of an event is determined by: (i)
the actual times t of preceding events; (ii) the scheduled times _T fixed at the planning stage; (iii)
random variables associated with the current stage of the journey, e.g., random variation in trip times on
links, in time taken for passenger boarding alighting, etc.; (iv) rules or relationships describing operation
of the system - for example, trains must traverse links in a prespecified order, or a train may not be
permitted to depart from a station before its scheduled departure time. We argue in Section 3 that the
rules in (iv) can be stated as functions ~-~i so that the actual time of each new event i can be stated
recursively as
actual time ti
of new event i]
f[ actual times 1
='-'~iIlt
ofprevious/,
-
k
events
]
[Scheduled] [ r a n d o m ]/
[ times _T J' [variables J / '
1
(1)
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M. Carey, A. Kwiecifiski /European Journal of Operational Research 83 (1995) 182-199
Measures of reliability of the system, or cost of operating the system, can be defined as functions of
the actual or observed (random) times of arrivals, departures, and other successive time events in the
system. We show that, under commonly satisfied assumptions about the system, the expected values (and
other characteristics) of the cost and reliability measures behave well (are convex) with respect to
changes in the scheduled times of events (arrivals, departures, etc.) in the system. The required
assumptions are satisfied even in fairly complex models of rail networks.
We use the standard definition of a convex function. For the theory of convex functions see for
example Roberts and Varberg (1973) and for their use in optimization problems see for example Bazaraa
et al. (1993).
Definition 1.1. A function f : ~" ~ R k is said to be convex if and only if f ( a x + / 3 y ) < a f ( x ) + / 3 f ( y ) ,
coordinatewise, for all x, y ~ ~ " and a , / 3 >_ 0, ( a +/3) -- 1.
Section 3 contains a brief discussion of functional recursive rules in railway models. Some more
complicated rules describing 'knock-on' effect (interaction of trains travelling over the same link in a
short period of time) are presented in Section 3.5. Proposition 3.1 in Section 3.8 states that, given the set
of rules discussed in Sections 3.1-3.7, expected costs are convex with respect to parameters of the
system. That is, expected costs are convex with respect to any changes in scheduled arrival times,
departure times, and the various types of wait times, connection times, headways, etc. The proof,
however, is given in Sections 2 for a slightly more general model, and the result may therefore also be
used in contexts other than transport networks. Section 4 contains a short summary of some techniques
recently developed for the purpose of investigating convexity properties of stochastic systems, see e.g.,
Shaked and Shanthikumar (1988), Shanthikumar and Yao (1991). We give a simple application of this so
called sample path approach in Section 5 where we discuss operating rules assuming different forms
accordingly to independent randomly changing circumstances. In fact, the convexity results in Sections 2
and 3 are a special, simple case of those techniques. Finally, the Appendix gives a simple example
illustrating the usefulness of convexity in solving service planning problems.
2. General recursive model and mathematical preliminaries
In this section we formalize the simple concept of recursively defined random events outlined in
Section 1 (see (1)).
Let _T~ ~ nr be a vector of n T deterministic parameters or decision variables in the underlying system,
say _T= ( T I , . . . , TnT). Let z_= ( z l , . . . , 7nr) be a vector of random variables in the system, say Z : 12 ~ ~nr
where 12 is a probability space. We assume that z_ does not depend on _T (in the functional sense, that is,
the probability distribution of _z does not depend on _T). The random variables zl . . . . , "mr do not have to
be stochastically independent, though such an assumption is often realistic and independence of some
kind is postulated in many transport models (see e.g., Cern3~ and Va~iEek, 1977, Barnett, 1978, Marguier
and Ceder, 1984, and even further simplifications concerning particular distributions of random variations in Hall, 1985).
Now we recursively define a sequence of state variables t = (to, tl, t 2 . . . . ) as functions .9~i of
parameters _T and random variables _z, thus
t i =~i(_T, _z, t o . . . . , t i _ l ) ,
(2)
M. Carey, A. Kwiecitlski~European Journal of Operational Research 83 (1995) 182-199
185
where ~ i : ( ~ nT X ~ nr X ~i)...+ ~, for i > 0. This corresponds to (1). For convenience we denote ~ =
( ~ 0 , ~1, ~ 2 , - - . ). In Section 3 we will interpret (T, r_, ~ ) as a model of an underlying transport system
and random variables t i, i > 0, as time events occurring in the system. However, a number of other
interpretations are possible.
The recursive rule ~ may explicitly involve only some coordinates of vectors _T, r_ and ( t o , . . . , ti_l). It
is, however, more general (and still formally correct) to include all of them in (2). Similarly, it is possible
that there is only a finite number of recursive relationships but it is convenient not to impose such a
restriction.
The information in the model is contained entirely in three elements: T, _r and ~ . The state variables
ti, i > 0, describe the evolution of the underlying process. Note from (2) that t~ are variables defined on
the space O × Enr rather than /2 alone. To emphasize (when necessary) the dependence of ti on the
vector of parameters we shall write t~(_T). Since ti(_T) has also another argument, from the probability
space £2 (this argument, often denoted as ~o E g2, is traditionally not explicitly shown, e.g., random
variable X rather than X(o~)), the notation we have just introduced does not mean that t~ is fully
determined by the value of _T. It only indicates that the probability law of t i depends on, and is
determined by _T. For brevity the vector of random variables (t o. . . . . ti_ 1) will be denoted by _ti. 1 or
_ti_ 1(£).
In order to avoid expanding the notation in stating the following proposition, we use the same symbols
(_T, ~, _t~_i) to denote elements of the model and to denote arguments of ~-~i, when describing properties
of function ~9~. The context clearly defines the meaning of the notation.
Proposition 2.1. Consider a model consisting of T, v_ and ~ as described above. Let ti, i > 0, be the
sequence of random events defined by (2). Suppose that for each i >_0, the function t i = ~ i ( T , r, ti_ i) is
convex with respect to ( T_, ti_ 1) (with v fixed) and nondecreasing with respect to t_i_1 ( coordinatewise, with
T and • fixed).
Then for each i and for any nondecreasing convex cost function c : E i+i ~ ~ the expected cost function
~ ( T ) = E[c(ti(T))] is convex with respect to the vector of parameters T_.
Remark 2.2. The statement of Proposition 2.1 may at first seem trivial, since it asserts convexity of
certain cost function assuming first that other cost functions (and functional rules defining the model)
are convex. It is true that the proposition reflects fairly basic properties of convex functions (see the
proof). On the other hand, it is worth pointing out that the convexities of the functions c (assumption)
and ~ (assertion) are of quite different nature. The function c is a cost function defined o n the space of
random state variables _ti and characterizes costs arising in the system given a particular (random) value
of these variables, regardless of the underlying values of the 'control' parameters _T. The function ~ is
defined on the space of parameters _T and reflects expected costs in the system given the set of
parameters, regardless any particular outcome of the random state variables _tt.
Proof of Proposition 2.1. In order to prove the proposition we first prove a stronger property of ti(_T). We
show that for a fixed unexplicit argument ¢o ~ O (or, equivalently, for fixed ~) the state vector _ti(_T) is
convex w.r.t. _T. The proof is by induction, the induction assumption being that _ti_a(_T) is convex, for
given eo ~ J2 (or r), that is
_ti_l(O/T 1 Jr-/3_T2) _~ o/ti_ I(T1) --}-/3_ti_l(r2) ,
(3)
where _T1, _T2 ~ R nr, a , / 3 ~ (0, 1), ( a +/3) --- 1, and i > 1. This is, in fact, convexity of _ti_ l(_Z). Note that
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M. Carey,A. Kwiecifiski/ European Journal of OperationalResearch 83 (1995) 182-199
for i = 1 this assumption is true, since t o is not a function of any other state variable ti, and we assumed
that all functions ~ i (hence ~ 0 ) are convex w.r.t.T. Then we have
ti(aT_l +/3T2) = * i ( a T 1 +/3T2, l , t_i-l(aT_l +~T_2))
*i( T1 + T2, z,
+
-~<°f~'~i(~rl, Z, _ti-l(~rl)) -[-/~i(~r2, Z,-ti-l(~r2))
(from (3) and monot, of ~ i in .ti_l)
(from convexity of ~'i)
= otti(rl) q-/~ti(T2).
This proves that ti(T
- ) (hence also _ti(_T)) is convex w.r.t. _T, the second argument o9 ~ g2 being fixed. This
extends easily to c(.ti(_T1)): a discussion similar to the above applies, or equivalently one can use the
well-known fact that any nondecreasing convex function f of a convex function g is itself convex, see
e.g., Roberts and Varberg (1973). Thus,
c(ti(o~T1 Jr- ~T2) ) _~ ac(ti(T1) ) --}-/~c(ti(Z2) ).
(4)
The same inequality holds with the expected values of both sides of (4) taken, which proves the
proposition. []
The above proposition easily extends to cost functions of the form c(_t, _T) = c(t_i(Z), T ) convex in
(_t, _T) and nondecreasing in t. These arise when the parameters _T affect costs not only via the state
variables _ti, but also more directly (see Section 3.7 for examples). However, we chose t o prove the
proposition in its present, notationally slightly simpler form.
The following two corollaries follow easily from Proposition 2.1.
Corollary 2.3. Suppose that for i > O, /z(/1)(_T)=E[ti(T)] is the expected or mean value of the random
variable ti(T),_ and the assumptions of Proposition 2.1 hold. Then ~i'O~(T)_ is convex w.r.t. _T.
Corollary 2.4. Suppose that for i > 0, /x!k)(_T)= E[(ti(T))k], where k >_ 1, is the k-th moment the random
variable ti(T_), and that the assumptions of Proposition 2.1 hoM. If additionally the random variable ti(T
- ) is
nonnegative, then ix~)(T) is convex w.r.t. T_.
(The nonnegativity of ti(T
- ) is needed to ensure that (ti(Z)) k= c(ti(T_)) is nondecreasing and convex as
required in the Proposition.)
There are two equivalent sets of parameters or decision variables convenient in investigating transport
problems. One is the set of scheduled departure and arrival times, the other is the set of scheduled trip
times on links and wait times at stops. These can be obtained from each other by simple addition/
subtraction, that is, by linear operation. The following simple property of convex functions ensures that
these two ways of defining a timetable or schedule are also equivalent in terms of convexity properties of
the above model.
Lemma 2.5. Let f(T_) be a convex function of T_, and let _T= qff_S), where ~o is linear. Then the function
f(qffS)) is convex with respect to S.
3. Functional recursive rules in railway systems
In this section we set out a stochastic operating or planning model for a rail network. We assume that
the number of trains to run and the sequence order in which they arrive, depart, meet, etc., is already
M. Carey, A. Kwieci~ski/ European Journal of Operational Research 83 (1995) 182-199
187
given. This does not mean that the latter has already been decided, since the present model could be
used to evaluate and compare several different plans or proposals for the number of trains and their
sequence order.
To ensure relevance and usefulness of the model set out here, it is constructed so that the 'underlying'
deterministic model (a set of operating rules) is consistent with continuous (nondiscrete) versions of
well-known or standard train timetabling models (e.g., Jovanovic and Harker, 1991, Carey and Lockwood, 1992, Petersen et al., 1986, and surveys in Assad, 1980a,b). Similarly, the model is also consistent
with nondiscrete deterministic models for other forms of scheduled transport. By 'consistent with' we
mean that if the random variables in the model all have zero variance then the resulting model includes
existing deterministic models as special cases.
To define the stochastic operating model we focus on typical activities of trains at or near passenger
stations, since these are the key activities for passenger train services. Activities at other nodes of the
network, such as track crossings, track sidings, depots, marshalling yards, etc., can be discussed similarly.
Also, the model can be restated or extended to the activities of busses, airlines, or other scheduled
services.
Following the notation from the previous section we use sub- and superscripted capital letters T to
denote the scheduled parameters influencing the movement of the train (the timetable times of arrival,
departure, etc.). R a n d o m variables not dependent on the system parameters are denoted by Greek
letters (e.g. % X) with subscripts and superscripts if necessary. Lowercase letters (e.g. t, l) denote state
variables such as the actual departure times, delays, etc.
The numerical indices used in Section 2 to discriminate between the consecutive variables are for
convenience replaced here by more informative superscripts corresponding to the meaning of the
variables (e.g., t d, the actual departure time; I a, the actual delay of arrival; T a, the scheduled arrival
time; and so on). When it is necessary to distinguish time events taking place at different stations, say at
stations A and B, we use additional indices A and B (e.g., t~, lg and so on).
Sections 3.1-3.6 contain recursive definitions describing typical behaviour of trains in a rail network.
Section 3.7 gives examples of cost functions associated with parameters or state variables of the model.
These cost functions correspond to (components of) the functions c and ~ in Proposition 2.1 or in
comments following the proposition. In Section 3.8 we discuss in detail the correspondence of the rail
model stated here to the more general setting in Section 2.
3.1. The 'ready to arrive' time
Suppose that the train departs from station A towards station B at time t d. Then its 'ready to arrive'
time t~ a at station B is
t~a=t~+(TAB+'rAl3)
(5)
~'AB being the random journey time (first equation) or the random deviation from the scheduled journey
time TAB (second equation). The actual arrival time t a may be different from the 'ready to arrive' time
t rta for the reasons stated in Section 3.2. If this is the only train on the link or if the headway separating
it from the previous train is sufficiently large than the probability distribution of the link trip time (~'AB or
TAB + ~-~a3)is independent of the time at which preceding train(s) enter or exit the link. A more complex
situation, involving train interaction ('knock-on' effects) within the link, is described in Section 3.5.
trta=tAa+~'AB
,
B
or
3.2. The arrival time
Even if the train is ready to enter the station (at time t~ta), a random time X is needed for entering the
station, hence the simplest possible relationship between the arrival time t a and t rta is
t a = trta + X "
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M. Carey,A. Kwiec#iski~European Journalof OperationalResearch 83 (1995) 182-199
However, there may also be other time events which must take place before the train can arrive, for
instance, departure of another train occupying the platform at which the arriving train should stop. Let
t(1),..., t (m) denote such events and let "r0),..., Y (m) denote the corresponding random delays between
occurrence of these events and the arrival of the train. The latter random variables may include delays
caused by the signalling system and times taken by preceding trains to exit from the station. Thus
ta = max{ trta + X, t(1) + 'r(1), ... , t(m)
+
(6)
7(m)} •
3.3. The "ready to depart' time
When the train arrives at the station several tasks (i = 1 , . . . , m) must be performed before it is ready
to depart. Suppose that task i starts at time t TASK/and takes random time ~.TASKi to complete. Then we
have
t ~td = max(t TAsKI + r TASK1. . . .
, t TASKm q-
(7)
q'TASKm} .
The start time t TAsKi of the i-th task may be related in various ways to random or scheduled times of
other events, for instance:
t a, or
max{ta, Ta},
tTASK/=
or
max(ta, T d - A T } ,
or
max{tTASKil q- 7"TASKi~,..., tTASKik+ "rTASK~k} for some i l , . . . , i k < i,
(8)
or
[time event related to another train].
The five options in (8) embrace a wide range of transport operating practice, and can be described briefly
as follows.
(i) The i-th task starts immediately after train's arrival, e.g., passengers alighting, and some technical
services, may start immediately after arrival.
(ii) The i-th task starts after the scheduled arrival time or actual arrival time, whichever is greater.
E.g., even if the train arrives earlier than scheduled, passengers boarding (or alighting) may not be
permitted until the scheduled arrival time.
(iii) The i-th task starts after the actual arrival, but not earlier than AT before the scheduled
departure time T d. For instance it might be the operating policy that passenger boarding does not
start until such a time, especially when the alighting and boarding take place at separate points
(which is often the case in bus operations).
(iv) The i-th task starts only after some other tasks are completed, e.g., passengers may not be allowed
to board (say task 3) until alighting has been completed (task 1) and the train has been checked
and serviced (task 2).
(v) The i-th task starts after a time event related to another train's journey, e.g., the train must wait
for passengers from another train and must allow them sufficient time for changing train.
3.4. The departure time
The simplest case is when nothing obstructs the train and it can depart as soon as it is ready to depart,
but not earlier than its scheduled departure time, that is:
t d = max{t rtd, Td}.
See Section 3.5 for more complex situations.
(9)
M. Carey, A. Kwiecitlski/European Journal of Operational Research 83 (1995) 182-199
189
3.5. T w o aspect signalling m o d e l o f trains interaction
The following signalling model, along with more complex three aspect signalling model and simple
stochastic approximations to the trip times obtained from these models, were discussed in Carey and
Kwiecifiski (1992).
In order to control the movement of trains within the link A -+ B, and for safety reasons, the link is
divided into a number of sections or signal blocks. The sections of the link are numbered from 1 to d,
section 1 being the closest to station A. There are k trains travelling from station A to B (trains 1 . . . . , k).
Their numbers indicate the order of departing from station A, i.e., train 1 departs first. Let t rtd and T/d
denote respectively the 'ready to depart' time and the scheduled departure time of the i-th train,
i=l,...,k.
To maintain headways between trains only one train is allowed in each section at a time. This is
enforced by red and green signals as follows. Trains cannot pass a red signal. When a train enters a
section, the signal at the beginning of the section turns from green to red and reverts to green only when
the train passes into the next section. Thus if another train arrives a t a n occupied section it has to wait
until the preceding train has exited before it can enter. In particular, a train cannot depart from station
A if there is another train in the first section of the link. There is also a signal at the end of the last
section, at the entrance to the station B, prohibiting a train from entering the station if a designated
platform is not available.
The actual trip time of the i-th train on the j-th section, i = 1 , . . . , k, j = 1 , . . . , d, is a random variable;
let us denote it by rij. This random trip time does not include any waiting time within the j-th section
before the signal separating it from the ( j + 1)st section. It reflects only factors such as the condition,
topography and length of the section, the train's technical characteristics, and driver behaviour. Denote
by t.~~., j , i = 1, • . . , k, j = O, • . ., d, the actual (observed) time when the i-th train completes the j-th section
and enters the ( j + 1)st one (or arrives at the signal at the entrance to the station B if j = d). Hence t i,0
c
is the actual departure time from station A. The time t.t,J
~. includes the waiting time at the end Of the j-th
section. Note that t i,d
c is in fact the 'ready to arrive' time of the i-th train at the station B.
The section entering times of the 1st train are simply equal to its actual departure time (that is, its
'ready to depart' time or scheduled departure time whichever is greater) increased by successive section
trip times 'rl,j,
tCl,o --- max{t( td , Td},
(lOa)
J
t~,i = max{t[ td, TO} + E ~'1,.,
J = 1 , . . . , d.
(10b)
v=l
The times t~.
for = 2,..
l,J
tCi,o -- maxltytat t
, ~di ,
"'
k , satisfy the following recursion:
t/c- 1,1},
t~=,.j max{t/cj-1 + ri,j,
i--1,1+1}'
t c
t ~,a
c = t ~,a_~
¢
+ r~,d.
(11a)
J = 1, .., d - 1,
(llb)
(llc)
Let ~-~ be the free running trip time of train i, that is, the time it would need to complete its journey in
the same circumstances if it was the only (or the first) train on the link,
d
rfi r = ~ ri,,,
i = 1 ..... k.
(12)
M. Carey, A. Kwiecifiski /European Journal of Operational Research 83 (1995) 182-199
190
T h e 'ready to arrive' time t rta = t i,d
c given by the set of recursions (10)-(11) can be well approximated
by a simple formula. For the 2nd train,
t~ta = max{t~ td + "/'2
fr, t( td + T1fr q - S } ,
(13)
where s is an adjustment constant related to properties of the trains and the link. The approximation can
be also applied in the case of m o r e complex trains interaction models, see Carey and Kwiecifiski (1992)
for detailed discussion.
3,6. The lateness of arrival~departure
Lateness is the difference between the actual and scheduled a r r i v a l / d e p a r t u r e time when this is
positive. Thus the lateness of arrivals is
l a = max{0, t a -- Ta},
w h e r e T a and t a denote respectively the scheduled and actual arrival time of the train at the given
station. Using the notation max{0, u} = (u)+:
I a = ( t a -- Ta) +.
(14)
Similarly
l ° = (t ° - T °) +,
(15)
l d being the lateness of departure.
3. 7. Cost and reliability functions
To compare alternative timetables or sets of timetable parameters, various cost or reliability functions
may be used. These functions are associated with the decision variables fixed at the planning stage (the
timetable times) a n d / o r the state variables (the actual times of events). In the general case the expected
cost of a given schedule may be defined as
~(_T) = E [ c ( _ T , _t(_T))],
(16)
where c is a convex cost function nondecreasing in t. This cost can be divided into several m o r e specific
costs or benefits, for example as follows (e.g., see Carey, 1994).
Fixed costs of the schedule. Some of the costs are directly dependent on the scheduled times, e.g., costs
of fuel usage, crew payments, and changes in the traffic volume (via demand elasticities). In the simplest
model, these costs, e.g., for a particular link AB, can be assumed to be proportional to the scheduled trip
time on the link, say
,(TL
= a x
TAd),
or
= a X TAB,
where a is a unit cost, Td and Tt] are the departure and arrival times, or alternatively TAB is the trip
time.
Timetable times vs. desirable timetable times. It is usual to put bounds on some or all of the timetable
p a r a m e t e r s in the planning process. A certain value within these bounds may be thought particularly
desirable and any deviation from this value perceived as causing an additional cost. For instance, it may
be required that trains on a given route depart at roughly constant intervals, say between 8 and 13
minutes past the hour with the preferred value being 10 minutes. Let T o be such a preferred time (e.g.
desirable departure time) and T be the time actually included in the schedule. Fig. 1 shows a typical cost
function associated with these variables. This function is convex w.r.t.T.
M. Care'y, A. KwieciFtski /European Journal of Operational Research 83 (1995) 182-199
191
Cost
i
I
Lower
bound
Time T
Upper
bound
Preferred
time T p
m
Fig. 1. Cost of deviation from the desirable timetable.
Costs of being late or early. For transport users, and operators, there are usually costs associated with
arriving or departing later than scheduled. T h e r e may also be benefits associated with arriving earlier
than scheduled. The combined cost of lateness or earliness can be stated as
V3(T ) = E [ c 3 ( t -
r)],
where T is the scheduled arrival or departure time and t is the corresponding actual time. The benefit
per minute of earliness is usually less than the cost per minute of lateness, as illustrated in Fig. 2 (e.g. see
Black et al., 1984). This ensures that, if c3(t - T) is piecewise linear as in Fig. 2, then it is convex in
(t, T ) and nondecreasing in t. More generally, c3(.) may be nonlinear. In particular, the cost of lateness
usually increases at a constant or increasing rate with lateness. This again ensures that c 3 ( t - T) is
convex in (t, T ) and nondecreasing in t.
If there is no benefit for earliness then the cost of lateness/earliness reduces to ~ 4 ( T ) = E [ c 4 ( ( t T)+)]. A special case of this is expected lateness E[(t - T)+], which is also a measure of reliability of the
service and may be an object of optimization on its own. E.g., a transport company might wish to improve
reliability by adding say 20 minutes of additional slack time to the timetable of a given service. An
optimization process could answer the question, how exactly to split the slack time between the links (see
the Appendix for an example).
3.8. Properties of the model
We wish to apply Proposition 2.1 to derive certain properties of the model in Sections 3.1-3.7 above
(see Proposition 3.1 and corollaries below). However, to do that we must first show that the model
/I
Cost
c3(t -- T )
/
./"
/
./
/
/
/
/
/
/
/ J
f
Earliness
J
(o,o)
Lateness
(t - Ti
J
Fig. 2. Cost or benefit from arriving/departing l a t e r / e a r l i e r than scheduled.
192
M. Carey,A. Kwieci~ski/European Journal of Operational Research 83 (1995) 182-199
satisfies the assumptions of Proposition 2.1. We show this in Steps 1-3 below. These steps themselves,
and in particular the lengthy Step 2, provide useful insights into the properties of the model. The model
consists of Eqs. (5)-(9), definitions (14)-(15) with optional replacement of (5) by recursions such as
(10)-(11) or approximation (12)-(13), and the cost (16).
Step 1. First we define the variables _T, _z and t from Section 2 in terms of the rail network variables of
Sections 3.1-3.6. Thus, as we have already indicated at the beginning of this section, the vector of
scheduled parameters T from Section 2 corresponds to the scheduled departure, arrival and trip times
TAd, T~ and TAB for all trains and stations. The vector of random variations r consists of random
variables denoted throughout Sections 3.1-3.5 by Greek letters X and r, again for all trains and links
(stations). The vector _t of state variables consists of the variables on the left hand sides of Eqs. (5)-(15),
for all trains and links.
Step 2. Second, we show that each of the Eqs. (5)-(15) can be written in the form
ti = J t i ( T , z, to . . . . , ti-1).
Each of the Eqs. (5)-(15) has an element of t on the 1.h.s. and has one or more elements of _T or _r on
the r.h.s. The remaining elements of _T and z_ can be thought of as appearing implicitly in ~9~i(-) with zero
coefficients.
To complete Step 2 it remains only to show that the Eqs. (5)-(15) can be placed in a sequence such
that:
(A1) A n y elements oft_ on the r.h.s, o f an equation have already appeared on the l.h.s, o f earlier equations
in the sequence.
If we consider only one train, this sequence is evident form the titles of these equations, namely:
'ready to arrive' time, arrival time, 'ready to depart' time, departure time. Thus the sequence is (in
order): (5)-(9), (14)-(15), with optional replacement of (5) by recursions (10)-(11) or approximations
(12)-(13). This sequence has to be repeated for each train for all stations/stops which the train
encounters. For any one train the sequence order of times needed to satisfy (A1) is obtained by simply
repeating the above order in turn for each successive station/stop traversed by the train from its origin
to its destination.
If the actual times (arrival, departure, etc.) of each train were independent of the actual times of all
other trains, then we could satisfy assumption (A1) by simply treating each train as representing an
entirely separate system, or alternatively constructing a timing sequence for each train as above, then
simply appending these sequences to each other. However, the times of different trains are not
independent. For example, a train travelling through 10 stations, 1 to 10, may not be allowed to enter say
station 6 until another train travelling in the opposite direction from station 10 to station 1 has exited
from station 6 (this would be enforced by Eq. (6) above).
How can we be sure that, for multiple trains, the event times can be arranged in sequence order so as
to satisfy requirement (A1) above? The answer is as follows. The Eqs. (5)-(13) are decision rules or
operating rules which describe and govern the operation and running of the rail system (Eqs. (14)-(15)
are simply definitional). Some of these rules simply state physical logic (e.g., a train arrival time must
precede its departure time), others are part of the train plan or timetable, and others are written or
unwritten operating rules. If these rules are to be operational or executable, then the values of all
variables on the right hand side of each rule (from Eqs. (5)-(13)) must be known before the rule can be
applied (i.e. its 1.h.s. can be calculated). We can assume that train operators do not use operating rules
which would require certainty regarding the times, t i of random events which have not yet occurred.
M. Carey,A. Kwiecitlski/ European Journal of OperationalResearch 83 (1995) 182-199
193
Even if such rules were drawn up at the planning stage they would have to be adjusted by operators on
the day so as to be able to make decisions and make the system function.
We therefore assume that the set of rules (5)-(13) are in fact 'executable' or 'operational', which we
can define as follows:
(A2) In applying the operating rules (5)-(13) to all trains at all stations~stops any time dependencies
between train times are restricted to a set which are operationally executable, i.e. they depend only on
times already known from the execution o f previous decision rules.
This assumption is equivalent to (A1), hence ensures that Eqs. (5)-(15), when applied to all trains at all
stations/stops, can be rewritten in the required sequential or recursive form.
Note that the sequence order defined above is not unique, since the timings of many trains are
unrelated or only very indirectly related to each other and are not restricted to a particular order. In
general, the Eqs. (5)-(13) impose a definite order only on trains at the same station, or on the same train
at different stations. However, even for trains using the same station the order of arrivals, or departures,
may be fixed (via Eqs. (5)-(13)) for only some pairs of trains.
In practice, the usual way in which train operators seek to achieve the train order requirements (hence
(A1) and (A2)) is at the planning stage. In the train timetable or schedule all events, such as arrivals or
departures, are assigned scheduled times _T. This automatically implies a planned sequence order for all
arrivals and departures, etc. of all trains at all stations. This order should also generally immediately
satisfy (A2) and (A1) above. Of course, in practice, on the day, the actual times _t are subject to random
variation _z. These variations in times t may yield various actual train orders which will violate many
(unessential) aspects of the scheduled order but, as explained in the previous paragraph, train operators,
and Eqs. (5)-(13), are concerned only with enforcing a fixed order for only some parts of the schedule.
For example, a departing passenger train may have to wait until a preceding connecting train has
arrived with its passengers. Also, all trains required to use the same physical track between stations; or
the same platform at a station, may have their sequence order fixed, so as to try to prevent trains out of
sequence from blocking and delaying other trains. However, if there is a choice of multiple tracks
between stations, or of platforms at a station, then the train order may not be prespecified (via Eqs,
(5)-(13)). Furthermore, if we are considering say an airline system, or especially if we are considering a
bus system, then the above sequence order restrictions are much less likely to be required by the
operator.
Step 3. The final step in showing that the system (5)-(15), for all trains and all stations/stops, satisfies
the assumptions of Proposition 2.1, is to show that the r.h.s, of Eqs. (5)-(15) are, (a) convex in whichever
elements of _T a n d / o r _t are present, and (b) nondecreasing in whichever elements of t are present. It is
easy to confirm that both (a) and (b) hold. Recall that the function max{x, y , . . . } is convex and
(coordinatewise) nondecreasing.
The next proposition follows immediately from Proposition 2.1 and Steps 1-3 above.
Proposition 3.1. Suppose that the costs associated with operating the service are given by a function
c(t_, T ) = c(t_(T_), T_) convex in (t_, T_) and nondecreasing in t_, e.g. as in Section 3.7, where t and T are
respectively vectors o f random state variables (times) and scheduled times, defined in Step 1 above and in
Eqs. (5)-(15). Then the expected cost ~(T_) = E[c(t_(T_), _T)] is convex w.r.t. T_.
We point out again that the convexity of the functions c and ~ respectively are of quite different
nature, as explained in Remark 2.2, and therefore the assertion of the above proposition does not simply
repeat its assumption, as it might at first seem.
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M. Carey,A. Kwieci~ski/European Journal of OperationalResearch 83 (1995) 182,199
Lateness of departure and arrival (or other scheduled events), as defined in Section 3.6, are special
cases of state variables. As a nonnegative random variable, lateness satisfies the additional assumption of
Corollary 2.4, hence:
Corollary 3.2. Under the assumptions of Proposition 3.1 the expected value as well as higher moments of the
lateness of arrival (departure) l, defined by/z(k)(T) = E[(I(T_))k, is convex w.r.t. T_.
The strict assumption that if a sequence order of trains is specified it must be adhered to is slightly
relaxed in Section 5. Section 4 summarizes the techniques used in Section 5.
4. Stochastic convexity and related notions
The convexity results in Propositions 2.1 and 3.1 are a special case of a stronger property of 'sample
path' convexity of families of random vectors indexed by a vector of parameters _T. The 'sample path'
property holds not in terms of expectations, but rather in terms of the random variables themselves for
specific values of the parameters. More specifically, for a given random vector X ( T ) not only E[X(_T)] is
convex w.r.t. _T, but also for all T_I, T_2 ~ R nT and a,/3 ~ (0, 1) such that a +/3 = 1
_X(a_Tx +/3_7"2) -<a_X(_Ta) +/3_X(-T2)
a.s.
(17)
Here 'a.s.' means that the inequality holds almost surely or with probability one (see the Proof of
Proposition 2.1).
Inequality (17) is a very strong version of 'sample path' convexity. An approach based on a less strong
property of a collection of random variables parameterized by a univariate parameter was developed in
Shaked and Shanthikumar (1988). Their technique, based on the notion of sample path stochastic
convexity, was extended to more general random elements and other convexity-like properties, e.g., in
Chang (1990), Chang et al. (1991), and Meester and Shanthikumar (1990). The original idea of sample
path stochastic convexity from Shaked and Shanthikumar (1988) was generalized to a situation with a
multivariate parameter space in Shanthikumar and Yao (1991). We recall here the notion of the strong
stochastic convexity (SSCX) from that paper, as we shall use it in Section 5.
Let X ( T ) be a family of random vectors parameterized by the vector T ~ Enr.
Definition 4.1. The family X ( T ) , T ~ R nr, satisfies strong stochastic convexity (SSCX) if for any _T1,
_T2 ~ ~nr and a,/3 ~ (0, 1), (~ +/3)-= 1, there exists a probability space with three random vectors X(_T1),
_X(T2) and X ( a T x + ~3Tz), such that _X(~) st _X(_T,:)for i = 1, 2 and ) ( ( a T l +/3T_2) st X ( a T t +/3T2), and
X(a_T x +fl_T2) <a£(_T~) +/3_X(T2)
a.s.
(18)
The symbol ' s t , in the above definition means 'equal in distribution' (or 'stochastically equal'), i.e., the
two random variables or vectors have the same probability distributions.
Note that the property defined in Definition 4.1 is weaker than (17), though similar to it. Clearly, (17)
implies SSCX. But in contrast, the random vector ~ for which (18) holds, though stochastically equal to
X, is constructed on some probability space separately for each pair of parameters T1, T2. The 'original'
random vector X does not have to satisfy any 'sample path' inequalities. In particular, (17) need not
necessarily hold.
195
M. Carey, A. Kwiecitlski/ European Journal of Operational Research 83 (1995) 182-199
Remark 4.2. Suppose that X(_T) satisfies SSCX. Let c:Rk---> R be a convex and (coordinatewise)
nondecreasing function. T h e n (18) implies
C(8( O~T1"4-~T)) -~<O/C(8(T1)) -4-flC(~(T2))
a.s.
(19)
The inequality remains true when the expectations of the both sides are taken. This proves that
~(T) = E[c(X(T))] is convex w.r.t. T_, and this property is known as stochastic convexity (SCX),
R e m a r k 4.3. SSCX satisfies numerous closure properties: see Shanthikumar and Yao (1991). E.g., convex
and increasing functions of a (parameterized) random vector satisfying SSCX also satisfies SSCX, this
follows from (19). Hence the statement of Proposition 2.1 can be rephrased as follows: Given the
assumptions about the functions ~ i , the state variables (vectors) _ti(T) satisfy SSCX (and hence SCX).
The property we shall use in Section 5 is preservation of SSCX in random mixtures. The lemma below
is related to T h e o r e m 2.10 in Shanthikumar and Yao (1991).
Lemma 4.4. Let X_(T) and Y_(T_) be families of random vectors parameterized by T_. For a given T_,let
Z_(T)
[ X_(T)
Y_( T_)
with probability p,
with probability 1 - p,
(20)
where the random mechanism p is independent of (X_(T_), Y_(T_)). If (X(T_), Y_(T_)) satisfies SSCX, then so
does Z(T_ ).
This lemma can be easily generalized to a mixture of more than two families of random vectors. For
simplicity, we chose to state the lemma in its present form.
The assumption that the switching mechanism is independent of (X(T), _Y(_T)) can be easily formalized by introducing a family of zero-one random variables, say e(_T), such that P{e(T_) = 1} --p and e(_T)
is independent of ( X ( T ) , _Y(_T)). Of course, definition (20) should be then adjusted in the obvious way.
This assumption can be slightly weakened, e.g. by postulating that ( X ( T ) , Y(_T), e(T_)) satisfies SSCX,
but not totally relaxed as counterexamples can be easily constructed.
Note also the joint strong stochastic convexity of _X and _Y in L e m m a 4.4, that is, SSCX is satisfied by
(X(_T), _Y(_T)), the concatenation of X(_T) and Y(T), and not by X and _Y alone.
5. Altering the prespecified order of trains
In this section we relax the assumption that the prespecified order of trains must be always adhered
to. The order of trains in the model is enforced by operating rules ~ i , for i = 1, 2 . . . . . Altering the
ordering of trains means changing one or more of the rules. Similarly, introducing occasional additional
trains or cancelling existing ones, adjusting the timetable to allow for engineering works, altering the
timetable on weekends or holidays, etc., all require changes in the operating rules.
Consider the time t i of an event, for some i > 1. Let event t / b e defined by a set of operating rules
rather than a specific unique rule. Assume for simplicity, as in Lemma 4.4, that the set of rules consists
of exactly two rules. That is, instead of having a unique recursive rule ~g such that tg =oq2'g(_T,!, _tg-1), we
have
(./o)(£,
z , _t,_0,
ti = ~
~,~/a)(_T, _z, _ti_l),
or
196
M. Carey, A. Kwiecifiski / European Journal of Operational Research 83 (1995) 182-199
where ~/~1) and ~ ( 2 ) are two alternative recursive rules applied accordingly to the randomly changing
circumstances.
As an illustration, consider the following example. Two sections of track intersect. There is a train on
each track, one of them (say train a) having higher priority then the other (train/3). If the arrivals of the
trains at the intersection coincide, train/3 gives way to train a and is therefore delayed. We consider the
effect this situation has on train/3. Let:
t~x = Time when train/3 arrives at the intersection.
7 x = Time needed by train/3 to pass through intersection.
t px = Time when train a completes passing the intersection.
Then the time t~x when train fl completes passing the intersection can be defined as
t px + r x
t~x = ~ t~x + ~.x
if the trains meet,
if they do not.
Such dual recursive rules can be included into the basic model outlined in Sections 2 and 3 (see also
Remark 4.3) if the assumptions of the following proposition are met. (The proposition is in fact a
corollary from Lemma 4.4.)
Proposition 5.1. Suppose that ~ 1 ) and ~ i ~2) satisfy the convexity and monotonicity properties listed in
Proposition 2.1. Let
t/(T)
[~/~1)(
T,_
Z, _ti-l(_T))
~q~2)(_T, Z, .ti-l(_T))
with probability p,
with probability 1 , p ,
where the random mechanism p is independent of ti_l(_T). I f _ti_l(T) satisfies SSCX, then so does
_ti(T ) = (t i_ !(T), ti(T)).
The above proposition in its present form has somewhat limited applications, due to its two main
assumptions. These assumptions are that the underlying probabilistic operating rule, indicating which
formula applies, is (a) stochastically independent of the current value of the state variables _ti_ 1 and is (b)
governed by the probability p independent of the chosen value of the parameters _T. Thus the above
probabilistic operating rule does not explicitly take into account, for example, opportunities to change
train priorities on the day depending say on which train arrives first. Such variability can only be
implemented as an independent random occurrence with probability based on frequency of the
corresponding train despatching decision. Because of this, any cost and reliability measures computed
using the above rule are likely to be conservative - they will overestimate costs and underestimate
reliability characteristics.
Assumptions of this or similar nature are, however, common to many transport models. For example,
Chen and Harker (1990) develop a model of a single-track rail line with sidings to allow trains to meet
and pass. Train i is delayed if it is directed into a siding to let train j pass. However, the probability that
train i suffers a delay when it meets train j (Po in their paper) is assumed to be independent of the
actual times of the trains, or when or where they actually meet.
In some cases assumptions (a) and (b) may more closely approximate operating behaviour. For
example, the first assumption holds approximately if random variations of the state variables _ti_ 1 are
relatively small, that is, when the service is comparatively reliable. The second assumption holds
approximately if the ranges of the parameters involved are small. In that case Proposition 5.1 is
applicable at least in tuning existing timetables obtained from less flexible models.
M. Carey,A. Kwieci~ski/ EuropeanJournalof OperationalResearch83 (1995) 182-199
197
Appendix. Optimizing a simple numerical example
In this section we give a simple example of the usefulness of convexity in analyzing timetable design
for scheduled transport services. Convexity enables one to employ search algorithms or methods which
are faster, m o r e accurate or just simpler than would otherwise be possible, and to m a k e statements about
the nature (uniqueness, optimality, etc.) of the results. To illustrate this we consider a simplified
timetable design problem having several decision variables. Convexity ensures that we can find the global
solution by a simple search method. It also ensures that when we impose or vary a constraint in the
problem the optimal solution behaves in simple predictable way.
Consider a train travelling through a sequence of stops, starting from stop 0 and ending at stop n (in
our example n = 4). Let t i and T/, for i = 1 , . . . , n, be respectively the train's actual (random) and
scheduled trip times on link i (from stop i - 1 to i). For simplicity, assume that the scheduled (and
minimum required) waiting times at stops are zero, so that the scheduled arrival and departure times are
the same. T h a t is, the train leaves each stop as soon as it arrives, unless this is before its scheduled
a r r i v a l / d e p a r t u r e time. T h e scheduled a r r i v a l / d e p a r t u r e time at stop i is thus T 1 + ... + T/.
W e are concerned with lateness of arrival of the train at each stop. (This is the same as lateness of
departure as we assume scheduled arrival and departure times are the same and there is no required
wait time.) Let l / b e the lateness at stop i. Assuming that the train departs from stop 0 on time, we have
10=0,
l i = ( l i _ l - t - t i - T i ) +, i = l , . . . , n .
Recall that we write ( x ) ÷ for max{0, x}.
Let the scheduled trip times T/ consists of two components: a fixed minimum journey time k i and
'slack' or 'recovery' time 8i, that is T / = k / + 8i. If we choose large values for these scheduled slack times
(8i's), this allows time for trains to get back on schedule if they h a p p e n to be running late. However, as a
policy constraint, we assume that the sum of the scheduled slack times (~,inlsi) in the timetable should
not exceed a certain constant A. W e wish to choose the values of these scheduled times so as to minimize
an expected lateness function of the form
~(~1,'"
"'~n)= ~ aiE[li],
(A.1)
i=1
where ai's are some (nonnegative) weights corresponding, for example, to the average n u m b e r of
passengers alighting at each stop.
Table A.1
Optimum slack times and objective function ~1, in minutes
Total slack
A
10
12
14
16
18
20
22
Optimal slack times
Value of obj. function
;~1
~2
~3
~4
minimum
' naive'
2.75
3.07
3.45
3.87
4.31
4.78
5.26
3.85
4.18
4.54
4.92
5.31
5.71
6.13
3.22
3.72
4.19
4.65
5.11
5.56
6.01
0.15
1.01
1.80
2.55
3.26
3.93
4.58
9.08
7.30
5.84
4.66
3.72
2.97
2.38
9.90(-8.9%)
7.81(-6.9%)
6.16(-5.4%)
4.86(-4.3%)
3.84(-3.4%)
3.~(-2.6%)
2.43(-2.1%)
M. Carey,A. Kwieci~ski~European Journal of OperationalResearch 83 (1995) 182-199
198
Table A.2
Optimum slack times and objective function ~'2, in minutes
Total slack
Optimal slack times
A
81
82
83
10
12
14
16
18
20
22
1.69
2.01
2.38
2.80
3.25
3.73
4.23
3.06
3.41
3.78
4.17
4.56
4.96
5.37
3.03
3.46
3.88
4.29
4.70
5.11
5.53
Value of obj. function
84
2.19
3.09
3.94
4.72
5.47
6.17
6.85
minimum
16.98
13.16
10.16
7.84
6.07
4.71
3.69
'naive'
18.43 ( - 8.5%)
14.47 ( - 9.9%)
11.37 (-11.9%)
8.98 ( - 14.5%)
7.15 ( - 17.8%)
5.75 ( - 21.9%)
4.68 ( - 26.8%)
Let the probability distributions of the actual link trip times be shifted negative exponential, and be
the same for each link, that is,
P{ti <x}
if x < k ,
= ]0
1
e -x(x-k)
ifx>k,
where k = k 1 = ... = k 4 is the minimum trip time common to all links. The expected lateness at stops
E[I i] can now be calculated by easy, though lengthy, integration. Thus,
E[/I] = (l/A) e -~1,
E[12] = ( 1 / A ) ( e -x82 + (1 + 82) e-A(~l+~2)).
and so on. We assume k = 15 minutes and h = 0.35. This gives an expected link trip time of just under 18
minutes, with a standard deviation of about 3 minutes.
Proposition 2.1 ensures that the cost/reliability function, (A.1) subject to (EinlBi < A), is convex w.r.t.
(81 . . . . . 8n). A local minimum of (A.1) subject to (En=18i < A) is therefore also a global minimum. H e n c e
we can easily find the global minimum by applying one of the known fast algorithms for finding a local
minimum of a multivariate function. For instance, we used the ~ i n d Mi n i mu m function of Mathematica
(see Wolfram, 1988, Ch. 3.9.7).
Table A.1 summarizes the results assuming an objective function of the form
~71(81, 82, 83, 84) =E[ll]-4-E[12]-[-f[13]-lt-E[/4].
The table also contains values of the objective function corresponding to a commonly used 'nffive' way of
assigning slack time, by splitting it equally between links (81 = . . . 8 4 ~ ¼A). The percentage figures in the
last column show how far the 'nffive' solution is from the optimum.
The cost function ~ l corresponds to the situation when all stops are of equal importance. Suppose
instead that, say, roughly half of all passengers alight at the last stop, and the other half are distributed
equally between stops 1, 2 and 3. This is reflected in the following objective function:
~2(81, 82, 83, 84) = E [ l l ] --[-E [ / 2 ] + E [ 1 3 ] -~-3E[14].
The optimum slack times for this function are given in Table A.2. This time we d e f i n e the 'naive'
solution a s 84 = 71 A a n d 3 1 = 8 2 = 8 3 = ~A1
Acknowledgements
The authors wish to thank British Rail, T h e Fellowship of Engineering, and the Science and
Engineering Research Council for their support and cooperation.
M. Carey, A. Kwieci~ski/ European Journal of Operational Research 83 (1995) 182-199
199
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