- Email: [email protected]
makeup model
‘hnspn Rex-B Vol. 14B. pp. 101-114 Pergamon Press Ltd., 1980. Printed in Great Britain
MODELLING
OF RAIL NETWORKS:
ROUTING/MAKEUP
ARJANC College
of Business
and Management,
TOWARD
A
MODEL
A. ASSAD
University
of Maryland,
College
Park,
Maryland,
U.S.A.
Abstract-Freight flow management in rail systems involves multicommodity flows on a network complicated by node activities (queueing and classification of cars at marshalling yards). Routing in these systems should account for technology requirements of motive power and traction as well as resource allocation (cars to blocks, blocks to trains). In this paper, we propose a hierarchial taxonomy of modelling issues and describe a class of models dealing with car routing and train makeup from the viewpoint of network flows and combinatorial optimization. We compare our mode1 with two previous rail network models and discuss possibilities for algorithmic development.
1. INTRODUCTION
Railroads in the United States have been facing fierce competition in the area of freight transportation since 1940. The steadily falling market share in intercity freight transport and studies indicating poor utilization of available resources point to a need for rational planning methods for rail operations (TRB, 1975). Various models already exist for various subsystems of the rail transport system (Assad, 1977). The simulation approach dominates the literature in this area. For planning purposes, however, optimization models provide meaningful information on resource allocation in the presence of trade-offs. Motivated by the success of mathematical programming methodology in other transportation studies (see Florian, 1976a), this paper proposes an optimization model of routing and makeup activities in a rail network. The plan of this paper is as follows: in Section 2 we describe a number of issues in rail operating policies and the decisions they pose to the manager. Following Anthony (1965), we suggest a hierarchical view of the decision-making process. We formulate a general model for train routing and makeup in Section 3 using the terminology of network flows. Section 4 discusses various forms of the general model and explores its capabilities and limitations. Section 5 provides a brief review of two existing optimization models for rail networks and contrasts these efforts to our proposed model. Finally, we conclude by delineating further directions for the application and implementation of the model in this paper. 2. ISSUES
IN
RAIL
PLANNING
AND
THE
HIERARCHICAL
APPROACH
Broadly speaking we may view rail operating policies as a sequence of decisions striving to meet demand by a suitable allocation of resources and facilities available to the railroad (which we may view as the supplier of services). On the demand side, we assume that data is available in the form of the traffic volume to be moved between a given origin-destination pair. In practice, the demand requirements may be more complicated. The shipper might specify a maximum allowable delivery time or specific constraints on routing. On the supply side, the specified set of resources available to the railroad determines the feasible train routes, allowable train itineraries, crew and motive power availabilities, and yard facilities. The operating policies determine an assignment of the resources to each class of traffic (determined by its origin and destination as well as possibly traffic type). Operating policies may be roughly divided into line and yard policies. The former determine the routing of each traffic class on the physical rail 101
102
ARJANG A. ASSAD
network as well as the assignment of traffic to carriers. Yard policies specify the operations performed on different classes of traffic in the yards they visit. At each yard the incoming traffic is regrouped according to final destination. The classification of cars into these destination-oriented blocks is called the grouping or blocking policy. The blocks of traffic are next placed on classification or departure tracks where they wait for outbound trains. Each outbound train has a “take-list” specifying the blocks of traffic it may pick up at a given yard. The decision as to which blocks of traffic a given train may carry is called the makeup policy. Clearly the makeup decision interacts highly with both yard classification and network routing policies. We may therefore regard it as an important linking factor between line and yard policies. In this paper we attempt to model this interaction. The decisions of interest to rail management vary substantially in scope, time horizon, investment requirements, and the level of decision-making. We wish to preface our model by a hierarchical framework for the planning process of rail freight transport. We follow Anthony (1965) and the literature of hierarchical planning in production systems (Armstrong and Hax, 1974) in identifying three levels of decisions-namely strategic, tactical, and operational. We regard the following categorization as a tentative aid allowing us to position our model at a suitable level of aggregation. Strategic decisions The decisions involve resource acquisition over long time horizons and typically require major capital investments. Due to the pervasive and long-lasting impact of strategic decisions on the future of the system, top-level management is usually directly involved with their resolution. Prime examples of such decisions in the context of rail systems include : (a) Network design and improvement. Track abandonment. (b) Location of yards and major classification facilities within large yards. (c) Highly aggregate routing decisions. Long-term planning of train services. The network improvement model of LeBlanc (1967) is one example of a strategic problem studied in the rail literature. The problem of choosing a set of feasible routings over a long planning horizon may fall into the category of strategic decisions if highly aggregate measures of traffic demand are used. The railroad may also want to estimate the costs and impact of providing regular service on a given set of routes and use such aggregate routing models to evaluate its supply capabilities or use them as inputs into larger models dealing with trip distribution or modal split decisions. Tactical decisions These decisions have medium-term planning horizons and focus on effective allocation of existing resources rather than major acquisitions. The planning horizon and level of aggregation in a model addressing tactical decisions must allow it to take account of broad changes in system parameters and data (such as seasonalities in the traffic volumes and imbalances resulting from lack of uniformity in the geographical pattern of shipments) without having to incorporate day-to-day changes in the data-base. Some examples of tactical decisions in rail systems follow: (a) Train selection and traffic routing: What trains should run and what should the required frequency of each train be to accommodate traffic demand? (b) Train makeup: What groups (or blocks) of traffic should a train be allowed to carry (its take-list) at a given yard of its itinerary? (c) Yard classification policy: Into what groups or blocks should the incoming traffic to the yard be consolidated? (d) Allocation of classification work among yards: What is the total amount of classification work performed in the system? How should this workload be distributed among the various yards to account for the fact that they might have different technological capabilities?
Modelling of rail networks: toward a routing/makeup
model
(e) Train lengths: What train lengths should we consider economically Is it better to have shorter more frequent trains?
103
attractive?
These issues have long been at the heart of planning rail operations. Due to the complicated interactions among the preceding issues, no one model can hope to fully capture all aspects of tactical planning in rail transport. We envisage a tactical planning model as being solved on a monthly or quarterly basis to respond only to major, stable changes in traffic patterns. Resetting makeup and classification policies on a daily basis may result in confusion for yard personnel. Operational
decisions
Such decisions deal with day-to-day activities in a fairly detailed and dynamic environment. Correspondingly, only lower levels of management (say, yardmasters) are directly concerned with operational issues. Some examples are: (a) Train timetables: Determining arrival/departure times of each train at any intermediate station of its itinerary. (b) Track scheduling and priority policy: Assigning trains to tracks if track capacity is limited. Planning for meets and overtakes according to a priority scheme. (c) Engine scheduling: Planning for daily distribution of motive power units over a specified set of train schedules. (d) Empty car distribution: Distributing empty cars over the rail network to meet demand and rectify imbalances due to uneven freight movement. (e) Yard receiving and dispatching policies: Determining a priority scheme for processing the queue of trains incoming to a classification yard. Setting rules for departure times of outbound trains. (f) Line-haul and yard maintenance operations: Scheduling maintenance and inspection for cars, engines, tracks, and yard facilities. The element of timing is crucial to most of the decisions in this list. The aim of computerized information systems for railroads is to record the position of cars in real-time or on an hourly basis. Thus models for empty-car and engine distribution may well be resolved on a daily basis. We note that the distinction between tactical and operational issues may be blurred at times. For example, a more aggregate model guiding empty-car distribution on a zonal basis could be viewed as a tactical problem. Similarly the derivation of train timetables from train schedules could assume tactical dimensions and generally does so in the context of passenger trains. The main advantage of a hierarchical approach is to avoid the pitfall of dealing with all the decisions outlined above simultaneously through a monolithic model. Even if computer and algorithmic capabilities permitted the solution of a large-scale detailed rail model (which is presently not the case), this approach would still be inappropriate since it would not be responsive to managerial needs at each level of the organization. Thus we do not try to integrate service and route selection problems for trains with delay timetabling and empty-car distribution problems into the same model, since the two classes of problems relate to different levels of the managerial hierarchy. 3. A GENERAL
RAIL
NETWORK
MODEL
In this section we formulate a general model aimed at (i) modelling the interaction between routing decisions and yard activities, and (ii) capturing the economies associated with consolidating blocks of traffic into a single train. Since most delays in freight delivery occur in yards, an effective policy for reducing yard congestion and delays is to schedule bypass trains that deliver blocks of traffic directly from origin to destination with no intermediate reclassification. The savings associated with this option, however, have to warrant the additional costs of allocating one or more direct trains to that traffic. Our model is constructed to address this tradeoff aiming to strike a rational balance between customer service and rail operating costs. Consider a directed graph
ARIANG A. ASSAD
104
GP = (N, AP) representing the physical rail network comprised of a set of nodes (yards), N, and a set of arcs AP (referring to physical track sections between yards). We assume demand data is available in terms of required flows I P4between origin-destination or OD pair (p, 4). These requirements may be expressed in terms of number of cars travelling from origin p to destination q or in some form of equivalent tonnage. We endow the physical network GP with a route structure that specifies the set of feasible routes on which trains may be scheduled. Each route has a unique origin i and destination j, i and j being yards in the rail network, as well as a physical path (i.e. a series of links in AP) from i to j which is completely specified. In the rail literature such a route is occasionally referred to as a train. It corresponds to the itinerary of a unique train from i to j. A number of trains (in the sense of a string of cars provided with locomotives) may be run on each train route. Thus we distinguish between a train route (which specifies the itinerary through the physical network) and train frequency (which specifies the number of actual trains dispatched along that route). It is useful to think of train routes as a bus number (line) which has a specified itinerary. Obviously buses with the same number may be run on a given itinerary with any desired frequency, however, all of these share the same routing and stop-schedule.
Fig. 1. The network
of train
routes-G.
A train service from i to j, denoted [i, j], maintains its identity throughout its itinerary. It is therefore made up at yard i and broken up for reclassification at its destination j. No intermediate yards perform any classification activities on the train. In this model we shall ignore the option of stopping at an intermediate yard to set off or pick up. We make this assumption merely to simplify our exposition. A further refinement of our model may incorporate such stops explicitly with only minor modifications. Train service [i, j] therefore “bypasses” all intermediate yards between i and j and traffic groups that make “connections” travel on more than one train route. The train routes may be represented as a network by adding route arcs between nodes i and j for each feasible train service. Figure 1 shows the route network G for a physical network GP of a line of 4 yards. For example, traffic for OD pair (1,4) may be placed on train [l, 31, be classified at yard 3, and then be transferred to train [l, 41. In general the route network for physical network of n yards along a line will have n(n - 1)/2 arcs although, in practice, other managerial considerations (say inspection requirements or maximum lengths for itineraries) may rule out some of these routes. We denote the route network by G = (N, A). For each OD pair (p, q) we define the decision variables xv = number of cars (or tons of freight) travelling from p to 4 on route [i, j] yij = number of trains provided
on route [i, j].
We may view the variables xc? as arc flows on the route network G. Since the flows corresponding to different OD pairs should be distinguished from one another, we are dealing with a multicommodity flow on G (see Assad, 1978). For each arc (i, j) of G, we also specify the number of cars C(ijthat may be assigned to train [i, j]. Alternatively, if we interpret yij as the number of motive units provided on route [i, j], GLij will specify the allowable car/engine ratio on that route. Let Ii and Oi be the set of incoming and outgoing trains (respectively) at yard i, that is: Ii = {kl(k,i)EA},
Oi = {kl(i, k)EA}.
Modelling of rail networks: toward a routing/makeup
model
105
Let ~j denote the take-list of train [i,j], that is, the set of all OD pairs whose traffic may be assigned to that train. In general, this set is specified by the network configuration. Indeed, let us call node j reachable from node i if there is a path going from i to j in G. Then ~j = {(p, q) 1i is reachable from p, and q is reachable from j}. For the simple case of a line network of n consecutive yards where and
N = {l,...,n}
A={(i,j)ll
we have
Ii=
{k/l 5 k < i},Oi=
{k(i<
k<
ql
n}.
n}
and IT;i={p,q)llS
i
PI
If we assume that the route structure imposed on GP is such that G is an acyclic network, the reachability relation induces a partial order < on the node set N, that is we write i c j whenever j is reachable from node i. The flow balance and motive power constraints are: r*q -rpq 0
CXPJP - &,xfT= iS0i
ifi=p ifi=q otherwise
(1)
for all nodes i used OD pairs (p, q)
all x0 2 O.yij 2 0 and integral for all (i,j)E A.
(3)
In (2) xii denotes the total flow of traffic assigned to train service [i,j]. Note that for fixed values of yij the constraint setis that of a capacitated multicommodity flow problem. Our approach is to maintain this feasible set (which is shared with many other transportation models involving flow of goods and carriers) and incorporate complications into the objective function. The objective function will contain two types of costs: (i) main-haul or over-the-road costs and (ii) yard costs. For a given train service [i, j], let Xii be a vector with components xpj for all Cp,q) in the take list Tj listed, say, according to lexicographic ordering of pq. This vector will fully describe the makeup (or composition) of the train [i, j], i.e. how many cars of each traffic class (p, q) is placed on the train. At a given yard j we also form a supervector Xj containing the vectors X, for all trains [i, j] incoming to yard j (i.e. for all k E lj)e This vector contains the composition of all traffic brought into yard j. As an example for a yard j of the line network. 4
= (X,j,X,j,
.e*,Xj-l,j)
and Xii = (x!j,,
,
.
.
.
,
Xij”,
xif;‘,
.
.
.
,
x$, . . . , x#, . . . , xi;).
We consider two types of costs: train costs relating to running a train [i, j] over its route, with a specified load assumed to be expressible as a cost function $i,{Xijy yij); yard costs at a typical yard j with an incoming traffic Xj? given by the functional form 4JXj). Then a formal model may be set up as follows: (PI: min 27 =
C (LJ3E.4
subject to eqns (l)-(3).
$ijlxij3 Yij) + j& 4j@j>
(4)
ARIANG A. ASSAD
106
We shall refer to this model as the routing/make-up model (P). We now proceed to specify possible functional forms for train and yard costs. Train costs may include: (i) Crew costs-A crew should be engaged over the entire length of the train route. (ii) Fuel costs-These will depend on the train weight (total tonnage) and length (number of cars) as well as on the train’s speed and the geographical terrain of its route. It is reasonable to take fuel costs as being proportional to train weight over a given link. (iii) Costs of motive power-We attach costs to providing and running an engine over each link. (iv) Over-the-road delay costs-The delay incurred on the main-haul legs, due to travel-time, congestion, meets and passes, and so forth, may be attached a dollar value to reflect the value of capital tied up in the system pipelines and, more importantly, the shipper’s devaluation rate. Yard costs could include: (i) Inspection and classification costs-These are attached to the operations involved in inspecting and breaking up incoming trains for reclassification into outbound groups. They may reflect the use of yard equipment (yard-engine-hours) or labour resources (inspection crews). (ii) Yard delay costs-We may associate time (devaluation) costs to the total delay suffered by cars in the yard due to queueing effects and waiting times of various yard operations: receiving, classification, outbound inspection and assembly, accumulation and connection delays. Naturally in a yard with fixed resources (crew and equipment) component of yard costs is formed by delays incurred by different classes of traffic at that yard. One simple functional form for train costs is: ~ij(Xij,
yij) =
C~j'Yij + Cfj' Xij
(5)
where cFj = cost of providing a unit of motive power for train [i, j] cfj = hauling cost per car on the route of [i, j]
and xij, we recall, is the total load on train [i,j] as defined in eqn (2). Thus we have variable costs corresponding to the number of cars on the train (and track parameters for the train route) as well as engine costs which exhibit a discrete character peculiar to rail systems. Later, we shall point out why we believe modelling this discreteness in the cost function is important. For the moment, note that for fixed values of x~JJ (and hence xij), given that the cost function is strictly increasing in yij, the values of the yij variables may be deduced from eqn (2) to be yij = [CrGl . Xij] + where [ .]’ denotes rounding up to the nearest integer. Consequently the graph of $ij as a function of xij exhibits a stepwise nature as shown in Fig. 2. The magnitude of the jumps is ci’j and slope of the linear sections equals cfj We now turn to the yard cost function +j In the simplest case, 4j depends only on the total throughout of yard j, that is 4jGj>
= fj
( > C xij
.
ialj
To allow for congestion effects, we may take fj to be a convex increasing function. Moreover the network structure of the problem will be preserved if the node for yard j is split into two nodes joined by a “throughput arc” with flow c Xij and arc cost islj
Modelling
of rail networks:
toward
a routing/makeup
model
!
107
xiI 2Qij
aij
Fig. 2. The train cost function.
A(
2.G).
For
a tactical model, however, we consider this expression to be too crude. It
does ‘not distinguish the delaying effects of one class of traffic on others. Following Thomet (1971a, b) we introduce a delay function at a given yard j of the form Wj + Vj’Xij
where Wj is the fixed delay for processing a train coming into the yard and Vj is the variable delay (in units of time per car). Such a delay is incurred for any train [i,j] which is classified and processed at yard j. To account for the effect of train composition in the processing costs, we should note that trains composed purely of cars for yard j need not be classified at that yard. The traffic on such trains will be delivered to local industrial sidings and will thus exit the main rail network. We note that the total number of cars destined for a yard j (to stay) is a constant determined by traffic requirements and hence should not affect the optimization. That is 1 idj
x$ = 1 rpi = constant
(PsJ?STij
(8)
P
where the last summation is over all origins p that have traffic destined for yard j (i.e. all p to which node j is accessible). Let us define a variable 8ij corresponding to the classification status of train [i, j] as follows:
=
variables may
1 0
train [i, otherwise.
should be
defined in
at yard
of arc
0,
=
by the
1 Xv
6 [
(P.4)
1
(9)
where the summation runs over all pairs (p, q) such that (p, q) E Tj and j < q. 6(x) is the delta function that equals 0 for x = 0 and 1 for x > 0, commonly used in fixed charge problems. Equation (9) simply says that the train [i,j] should be classified at yard j whenever it includes any traffic travelling beyond yard j. Then the delay at yard j for processing trains [i,j] is d zij
(Wj +
=
VjXij)eij
(10)
We could translate this delay term to money costs by attaching a time cost dij to each unit of time delay. This should reflect in dollar terms the undesirability of delaying cars at yard j. The yard cost function may then be written as @j(Xj) = C dijOij(Wj + UjXij). idj
TR-B Vol. 14, Nos I-2-H
(11)
ARIANG A. ASSAD
108
We wish, however, to point out a complication in transferring delay times into cost terms: the devaluation costs dij should logically depend on the train composition. As in inventory holding costs, dij usually reflects the total dollar value of freight that undergoes delay. If the average devaluation rate of traffic corresponding to OD pair (p, 4) is c$‘Idollars/car/day, then the train devaluation rate would be dij =
C
C$"XPj4
(12)
(P4)ETi j
Since dij depends on the train composition, measuring delay in terms of car-days, rather than just days, leads to a quadratic objective function. 4. CAPABILITIES
AND
LIMITATIONS
OF
THE
NETWORK
MODEL
In this section we shall relate the general model presented above to the tactical issues listed in Section 2. Rail systems are distinguished from many other transportation modes by their inherent ability to move a large number of shipments as a single unit. We feel an adequate model of rail freight transportation should take explicit account of such economies associated with train length and the corresponding blocking policy of consolidating diverse classes of traffic into large groups in which members share some leg of their itinerary. The train length economies operate in two directions: on the one hand they set lower limits on the total traffic assigned to a given train so that it would be profitable to run. On the other, they allow us some flexibility of “stretching” the train capacity by assigning additional engines so that the train could accommodate more traffic. This latter feature is absent from the competing trucking mode, for example. On the negative side of the economies suggested by large shipments may well be negated by the necessary intermediate sorting and grouping operations on the cars which tend to increase operating costs delays while lowering customer service and equipment utilization. Our model should address both the economies and diseconomies mentioned above. When economies of scale are present, it is imperative that a model aiming to incorporate them be on the same hierarchical level as where such economies are realized. This consideration will serve as a guide for choosing a suitable level of aggregation and planning horizon for the model. For example, in a strategic model for routing traffic on a rail network we may measure the flow variables xc9 in units of cars/year corresponding to annual requirements rpq. If we choose aggregate costs functions $ij(xij) for the routes [i,j] and ~j(Uj) for yard costs as in eqn (6), then, since we expect the variables yij to assume large values, we may relax the integrality requirement on yij’s and use the relation
The program (P) then becomes a convex network minimization problem. While such a model is useful for aggregate planning purposes, it abstracts away from the issue of assigning traffic to trains and therefore is unable to address economies of the type occurring in the following example. Example. Consider three nodes (yards) i, j, and k of a rail network with requirements as shown in Fig. 3: Suppose moreover that we have a car/engine ratio of 40 for trains travelling over the routes [i, j], [i, k], and [k, j]. Sending cars directly from each origin to destination requires a total of 7 engines: one over the route (i, j) and 3 over each of the routes (i, k) and (k, j). However, diverting the traffic from i to j through yard k changes the loads on links (i, k) and (k, j) from 100 to 120. Then one engine is saved over the route (i, j) and, moreover, the other engines will be fully utilized (to capacity). There is, of course, a corresponding processing and classification cost that cars of (i,,j) will incur at yard k, but the balance may well be in favor of traffic diversion.
Modelling
of rail networks:
toward
a routing/makeup
model
109
20
j loo
too
v Fig. 3. Twenty
cars from
k
i to j; 100 cars from i to k; 100 cars from k toj.
The above example shows that the integrality issue in the provision of motive power units is a key factor in two questions that have long plagued tail operations: that of train length (short vs long trains) and of high equipment utilization. Maintaining the discreteness in our cost function allows for a more rational evaluation of the economies of shipping in train-loads. We see the economies are realized on the level of loading (and blocking) decisions for cars. We now list some of the issues that a successful solution of the routing/makeup model of Section 3 will clarify. (a) Service selection and frequency-The variables yij determine the choice of trains to provide service. If yij = 0 no trains will be sent along the route [i,j]. For yij = 1 only one train will be run. For larger values of Yij longer (or more frequent) trains will travel on the service [i,j]. (b) Train makeup-The optimal values of the flow variables xpj” will specify the optimal train composition on a given route [i, j]. In our model these variables will also provide guidelines for the blocking policy. (c) Yard grouping policy-At each yard the incoming traffic may be reclassified according to outbound destinations. The number of groups formed at the yard can be derived from the total number of outbound services operative at optimality. We may think of each classification track as containing all the traffic belonging to the takelist of a given outbound train. (d) Yard workload-The total workload of each classification yard may be obtained from the total yard delay as expressed by the terms rt in eqn (10). The yard parameters Wj and Uj should be calibrated for each yard and will differ from one yard to another according to yard type (flat or hump yard) and the technological profile of the yard. The model strives for an efficient allocation of classification work among yards. For example, classification activity may be diverted from a small flat yard to a large automatic hump yard where it may be performed much more quickly. (e) Train length-The train length variable xii will be influenced by the discrete marginal costs ct attached to yi? The discreteness of these variables will specify a number of “regimes” of possible train lengths with different costs to the railroad. We thus see that the routing/makeup mode, if solved, will address a number of the tactical issues raised in Section 2 and consequently has the potential of being a valuable planning tool. We shall close this section with some technical observations concerning our model for routing and makeup. As it stands, the model involves continuous flow variables xv, integer variables Yij, and rather complicated cost terms in the objective function as specified by eqn (11). Let us start by noting how this formulation may be transformed to a network flow model with arc costs involving set-ups. In the process we shall also eliminate the 0,l variables 8, which specify the classification status of the train [i,j]. Consider a given yard j. A train [i,j] will be classified at j if it carries freight destined for yards beyond yard j (i.e. for some nodes k > j). On the other hand, a train composed entirely of traffic due for yard j will not require reclassification at that yard. This situation can be partially captured by splitting the node for yard j into two nodes j’ and j” as shown in Fig. 4. The node j’ acts as a sink for all traffic staying at yard j (all traffic classes with
110
ARJANG A. ASSAD
Fig. 4. Network
representation
of processing
delay.
final destination j) while node j” acts as a source of outbound traffic from yard j. We may decompose the traffic volume of a given train [i, j] as follows Xij
=
1
xf9
=
=
p
zixff+jg_q=uj + uj+ p-zi
Pl¶
where uj and uj+ stand for the first and second summations, respectively. We now attach the classification and processing costs of yard j to this arc (j’, j”) by specifying an arc cost function of the form 4j,jfs(x)=
L,+v x E:cr 0” 1
j
Then pure trains, in the sense of uj+ = 0 will incur no processing delays at yard j. An alternative to capturing cost functions of eqns (lo)-(12) is to seek simpler cost functions such as cbj(Y)=
4’
iz,Yij J
that captures some measure of both train length and frequency for all incoming trains at yard j. Of course, ideally, one may wish to calculate processing delays for different trains separately. Since Wj refers to processing a single train, a better choice for delays due to fixed set-ups is
Finally we wish to note that our general model can be considerably enriched by the addition of rather simple constraints. Maximum train lengths (or frequencies according to the interpretation of YijS) could be enforced by adding the constraints yij < yij, for a given upper bound yij, on a route [i, j]. These constraints could arise from limitations of departure tracks or crew availability. Other researchers-Bodin et al. (1977Finclude explicit constraints on the blocking strategy at each yard to reflect the railroad’s policy. We may also address the allocation of motive power by specifying engine availabilities at each yard at the beginning of the planning horizon and impose flow constraints on variables yij. Naturally, the inclusion of motive power consideration depends on whether or not the railroad faces fleet size bottlenecks. At any rate, the approach at this point should be aggregate: we do not recommend the incorporation of detailed engine scheduling, as in Florian et al. (1976b), at this stage of the modelling process. 5. OTHER
RAIL
NETWORK
MODELS:
REVIEW
AND
COMPARISON
This section contains a brief review of two major attempts in providing optimization models of rail systems and compare them with our own. We begin by introducing the work of Thomet (1971a, b) that has stimulated our formulation. The basic component of Thomet’s algorithm is a cancellation procedure that cancels direct shipments in favor of a series of connections. Suppose that a direct
Modelling
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model
111
train going from i to j is cancelled and that its traffic is diverted upon a sequence of connecting routes from i to j. Set [k, 11 denote one such intermediate train. This train travels from yard k to yard 1 (both on the route from i to j) with no intermediate classification. The impact of this cancellation on the total costs is as follows: there are benefits measured by a reduction in train-miles and a decrease in motive power requirements (recall Fig. 3) and the elimination of other costs of running a direct train between i and j. The major increase in costs is due to additional classification and processing costs at intermediate yards: an intermediate train [k, fl must be regrouped at yard I and all traffic going beyond yard 1 (say to j) should be reclassified. The diversion may thus change the classification status t&r of that train and incur delay costs as given by eqn (11). We may therefore compute y&-the net cost of diverting traffic from train [i, j] to train [k, II-for each intermediate train [k, lJ. The best sequence of connecting trains, on to which the [i, j] traffic may be shifted, can be found by solving a shortest route problem as follows: Consider the physical route of train [i, j] through the network Gp We may represent this itinerary by a sequence of yards i, = i, i2, iJ,. . . i, = j. If m 2 3, (i) construct another network G(i, j) form (k, 1) = (i,, i,) for 1 I r < s I (ii) compute the costs # for all such (iii) find Tithe length of the shortest running [i, j] directly the savings #j
=
;lij
_
on the node set {iI,. . . , i,} with arcs of the
m; arcs, and route from i to j on G(i, j). If 1” is the cost of due to cancellation are defined as -ij Y
.
For example, in the network of Fig. 1, suppose train [l, 41 is cancelled and that the shortest path with respect to arc costs y:> is (1,2), (2,4) with length y14 = y:; + y$ Then the traffic can be diverted upon the two trains [1,2] and [2,4] resulting in a savings of A’4 -yi4. Thomet’s algorithm may now be described as follows: initiate the algorithm by assigning direct trains to all OD pairs with nonzero requirements [all (p, 4) with rpq > 01. This is called the Minimum Transit Time Policy (MTT) and involves no classification work (initially all ep4 = 0). At a given cancellation step, compute the savings sii of all trains [i, j] as described above and cancel the train with maximum savings. Transfer the traffic of this train to the intermediate trains and update train parameters accordingly. To find the minimum cost policy, this strategy is continued (one cancellation at a time) until no positive savings can be found, that is until sij < 0 for all remaining trains [i, j]. Thomet’s approach is similar to drop heuristics for plant location Feldman et al. (1966). One starts with all trains and successively cancels trains according to a myopic criterion of savings in total costs. We may immediately point out several limitations of Thomet’s approach : (a) As Thomet himself realizes this sequential cancellation procedure does not guarantee optimality upon its termination. Rather it provides a local optimum in the sense that there will be no “one-move” (cancellation of a single train) which could further reduce costs. (b) In the algorithm described above the physical routing of traffic is never changed. All traffic follows the shortest route between its origin and destination on the physical graph Gp What changes is the loading pattern of the traffic on the intermediate trains on the shortest route itinerary. However, as the small example of Fig. 3 indicates, it may be advantageous to divert traffic away from its shortest route. In practice rail freight is known to be routed through such circuitous paths, occasionally on the basis of a reduction in the railroad operating costs. This diversion may become necessary if flow on certain routes is limited by capacity constraints. This limitation of routing alternatives to shortest paths may result in further suboptimality in Thomet’s model. (c) The diversion of traffic from one train to another is always performed in bulk form, that is, the entire traffic load of a train is shifted. One may easily envisage cases where
112
ARJANG A. ASSAD
only partial diversion of traffic is necessary. For example, consider the situation of Fig. 3 with the demand from i to j changed from 20 to 60. Then we may retain 40 cars on the train [i, j] and only divert the additional 20 to the route going through yard k. This possibility is allowed by our routing/makeup model (P) but not by Thomet where a typical variable sv is limited to one of two values-0 or rP4. Indeed we may recast our planning model into Thomet’s form by the following device. Let zpj”be a new variable defined as zfJ/rpq. If we then substitute the quantity rpqzpiSfor xPpin eqns (lH4) of (P) we realize that for each (p, 4) the variables zpi”obey flow conservation equations for a flow value of 1. Restricting the variables to be integral will now mimic Thomet’s procedure. Another major rail optimization model developed at Queen’s University at Kingston is the Railcar Network model (Petersen and Fullerton, 1975). The study evolved over 5 years into a comprehensive rail planning model involving over-the-road and yard activities. The object to the model is to route freight on the rail network to meet demand at minimal total delay (in car-hours). The approach is to derive delay functions for component rail operations as a function of the flow of freight handled by each operation. If all the delay functions are convex, the routing problems may be viewed as a minimum cost network problem with convex costs for which a number of algorithms are available (see Sections V and VI of Assad, 1978). Algorithmically, the model is thus identical to the work in traffic assignment-Florian (1976abwhere the delay functions (also called service functions in road traffic assignment literature) reflect over-the-road congestion delays. The derivation of the delay function, however, is very different as it is based on average waiting times derived from queueing theory. The main components of delay are as follows: (i) inbound inspection time for incoming trains, (ii) classification time, (iii) train assembly time, i.e. cars waiting to be assembled into an outbound train, (iv) outbound inspection time, and (v) over-the-road time measuring link traversal times in the presence of congestion, meets and passes, etc. Factors (i)-(iv) refer to yard delays and are attached to intrayard arcs whose flows reflect yard throughout. Inbound and outbound inspection delays are taken to be constant while classification and assembly delays are derived from queueing formulas that yield average delay as a convex function of yard flows. Over-the-road delays are expressed analytically by considering priority rules for track usage. As mentioned before, we regard the railcar model as a major contribution. We wish, however, to bring out some basic differences between their work and our approach in this report: Our model makes a basic distinction between the flow of cars and the flow of trains on a given service [i,j]. The effect of running a train is determined by integer variables Yij, in contradistinction to the continuous flow variables Xij for the cars. The railcar model, instead, derives train flows from car flows by using an average train length of 1 cars/train. Thus a flow of xij cars over a route automatically results in Xi$ trains. Moreover the number of trains affects only the delay time, no attention is paid to the cost of scheduling an additional train. In Section 4 we discussed how these costs may directly influence routing and makeup decisions, possibly resulting in train cancellations if such costs are high. Our model attempts to capture some of the economies of the routing process more directly to allow us to decide, for example, whether or not a certain train should run or a particular traffic class should be incorporated into a train’s makeup. The same distinction arises with respect to classification costs. A component of our classification costs refers to the train-via the fixed charge term ~jeij in eqn (lob-and not merely to yard throughout. In fact, this term attempts a first approximation to the effect of train composition on yard costs. While the railcar model chooses a purely stochastic approach to evaluating waiting times, we have opted for using a delay term for which the effect of train cancellation or composition is more apparent. Finally we regard some of the delay terms in the railcar model as being highly dynamic in nature and thus not particularly suitable for static waiting time analysis. In particular, the assembly delay term will
Modelling
of rail networks:
toward
a routing/makeup
model
113
depend on the yard dispatching policy. Such policies, which belong to the operational level, should be studied separately incorporating the time element into direct account. To conclude this section we wish to position the models described above in the hierarchial framework of Section 2. We believe the railcar model relates the strategic level of decision-making. It specifies the configuration of traffic flow on the rail network using aggregate measures of yard and over-the-road delays. Our model, however, belongs to the tactical level: our direct concern with traffic routing and train scheduling reflects a desire to have firmer control on the choice of trains to run and their composition. Indeed one may be well advised to use the railcar model and our model sequentially. The former will provide an aggregate picture of traffic flows and supply a “base-level” of yard and line activities. Thereupon we may pass on to the routing/makeup model to evaluate certain more detailed decisions (for example, the use of direct versus local trains, and long vs short trains).
6. SPECIALIZATIONS
We now consider certain algorithmic developments.
OF
specializations
THE
MODEL
AND
CONCLUSIONS
of the model (P) with a view towards
possible
(a) Suppose we have a set of operative services and regard each train route without capacity restrictions. This corresponds to a scenario where a sufficient number of trains operate on each route so that we may assign as much traffic as we wish to that route. Mathematically, C(ij can be assigned a large value so that the constraints in eqn (2) are no longer binding. Thus only routing and yard costs will affect the flow pattern. The result is a traffic assignment problem with convex costs on the network of feasible routes. If the cost functions are simplified even further, say by taking vi = 0 in eqn (7) then the model yields a shortest path problem that will specify the optimal sequence of train connections for each class of OD traffic. Indeed, the model proposed by Truskolaski (1968) seems to be very close to this problem although he does not use shortest paths. (b) We may now refine the above specialization by attaching fixed costs to service provided on a given route [i,f still maintaining, however, the assumption of unlimited capacity. The yijs are then cl variables indicating if a service [i,j] is operative or not. This results in an uncapacitated network design problem. Once the Yij variables are fixed, the problem reduces to shortest path or traffic equilibrium problems as in part (a). Additional constraints limiting the total operating budget on the number of services used may be added to the model. (In the latter case we have a p-median problem.) In the context of rail networks, this approach is reminiscent of the paper by Pierick and Weigand (1976). For recent algorithmic work on the network design problem, the reader may consult Dionne and Florian (1977), Magnanti and Wong (1977). (c) The next level of refinement, where the capacity limitations are included explicitly, results in the routing/makeup models of Sections 3 and 4. Again, for a given assignment of yijs, we obtain capacitated multicommodity flow problems. We propose this last model for algorithmic development as it is typical of a variety of “loading problems” in transportation studies that seek to determine the optimal flows of traffic and carriers simultaneously. Richardson (1976) has used Benders Decomposition on an airline routing problem with a structure similar to our model (P). This work may serve as a useful point of departure. Another promising area of investigation involves developing efficient heuristic techniques for the class of models exemplified by (P). We have briefly discussed one such heuristic, due to Thomet. Other strategies may be proposed and possibly evaluated analytically along the lines of Cornuejols et al. (1977). In this paper we have proposed one approach to integrate routing and makeup activities on a rail network. We took a hierarchical viewpoint of the models concerning traffic routing and suggested that our model is appropriate to the tactical level. This
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model may interact with other strategic (or more aggregate) models as in (a) and (b) above. Our basic goal has been to provide a framework for modelling rail networks. Naturally, the ultimate test of our proposed model will involve computational experience with a realistic data-base from some railroad. We envisage such an undertaking in the future and hope to announce the results in follow-up publications. Since rail networks are likely to be large (20-40 yards and some 200-300 train routes), the computational difficulties in solving the model (P) have to be overcome. Here, we may try to use techniques such as Benders Decomposition or Lagrangean Relaxation to attack the general mixed integer program. Alternatively, we may concentrate on simplified versions of the model, say, scheduling on a line network with special assumptions on yard and line costs. The insights provided by these simpler problems may serve as a basis for developing efficient heuristics for the general problem. Progress along either one of these dimensions will substantially increase the promise of optimization-based models in aiding the planning process for rail systems. Acknowledgements-The author wishes to thank Professors R. W. Simpson and T. L. Magnanti of MIT for Pheir careful readings of the manuscript. This work was supported, in part, by the U.S. Department of Transportation under contract DOT-TSC-1058 (TARP). REFERENCES Allman W. P. (1966) A network simulation approach to the railroad freight train scheduling and car sorting problem. Unpublished Ph.D. dissertation, Northwestern University. Anthony R. N. (1965) Planning and Control Systems: A Framework for Analysis. Harvard Graduate School of Business Administration, Boston. Armstrong R. J. and Hax A. C. (1974) A hierarchial approach for a naval tender job shop design. MIT Oper. Res. Center Technical Report No. 101. Assad A. A. (1977) Analytical models in rail transportation. MIT Oper. Res. Center, Working Paper OR 066-77. Assad A. A. (1978) Multicommodity network flows: a survey. Networks 8, 37-91. Cornuejoils G., Fischer M. and Nemhauser G. L. (1977) Location of bank accounts to optimize float: an analytical study of exact and approximate algorithms. Man. Sci. 23, 789-810. Dionne R. and Florian M. (1977) Exact and approximate algorithms for optimal network design. University of Montreal, Centre de Recherche sur les Transports, Publication 41. Feldman E. F., Lehrer A. and Ray T. L. (1966) Warehouse location under continuous economies of scale. Man. Sci. 2, 670-684. Florian M. (1976a) (Ed), Traffic Equilibrium Methods. Lecture Notes in Economics and Mathematical Systems, Vol. 118. Springer-Verlag, New York. Florian M. (1976b) The engine scheduling problem in a railway network. INFOR. 14, 121-138. Leblanc L. J. (1976) Global solutions for a nonconvex nonconcave rail network model. Man. Sci. 23, 131-139. Magnanti T. L. and Wong R. T. (1977) Accelerating Benders Decomposition for network design. Discussion Paper, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain, Belgium. Petersen E. R. and Fullerton H. V. (1975) The railcar network model. Queen’s University at Kingston, CIGGT Report Number 75-l 1. Pierick K. and Weigand K. D. (1976) Methodical formulations for optimization of railway long-distance passenger transport. Rail Int. 7, 328-334. Richardson R. (1976) An optimization approach to routing aircraft. Trans. Sci. 10, 52-71. Thomet M. A. (1971a) A combinatorial-search approach to the freight scheduling problem. Unpublished Ph.D. dissertation, Department of Electrical Engineering, Carnegie-Mellon University. Thomet M. A. (1971b) A user-oriented freight railroad operating policy. IEEE Trans. on Systems, Man and Cyber. SMC-1, 349-356. Truskolaski A. (1968) Application of digital computer to the optimization of freight train formation plans. Bull. Int. Railway Cony. Assoc. 5, 211-226. TRB (1975) Railroad research study background paper, Transportation Research Board, Richard B. Cross, Oxford, Indiana.
MODELLING
OF RAIL NETWORKS:
ROUTING/MAKEUP
ARJANC College
of Business
and Management,
TOWARD
A
MODEL
A. ASSAD
University
of Maryland,
College
Park,
Maryland,
U.S.A.
Abstract-Freight flow management in rail systems involves multicommodity flows on a network complicated by node activities (queueing and classification of cars at marshalling yards). Routing in these systems should account for technology requirements of motive power and traction as well as resource allocation (cars to blocks, blocks to trains). In this paper, we propose a hierarchial taxonomy of modelling issues and describe a class of models dealing with car routing and train makeup from the viewpoint of network flows and combinatorial optimization. We compare our mode1 with two previous rail network models and discuss possibilities for algorithmic development.
1. INTRODUCTION
Railroads in the United States have been facing fierce competition in the area of freight transportation since 1940. The steadily falling market share in intercity freight transport and studies indicating poor utilization of available resources point to a need for rational planning methods for rail operations (TRB, 1975). Various models already exist for various subsystems of the rail transport system (Assad, 1977). The simulation approach dominates the literature in this area. For planning purposes, however, optimization models provide meaningful information on resource allocation in the presence of trade-offs. Motivated by the success of mathematical programming methodology in other transportation studies (see Florian, 1976a), this paper proposes an optimization model of routing and makeup activities in a rail network. The plan of this paper is as follows: in Section 2 we describe a number of issues in rail operating policies and the decisions they pose to the manager. Following Anthony (1965), we suggest a hierarchical view of the decision-making process. We formulate a general model for train routing and makeup in Section 3 using the terminology of network flows. Section 4 discusses various forms of the general model and explores its capabilities and limitations. Section 5 provides a brief review of two existing optimization models for rail networks and contrasts these efforts to our proposed model. Finally, we conclude by delineating further directions for the application and implementation of the model in this paper. 2. ISSUES
IN
RAIL
PLANNING
AND
THE
HIERARCHICAL
APPROACH
Broadly speaking we may view rail operating policies as a sequence of decisions striving to meet demand by a suitable allocation of resources and facilities available to the railroad (which we may view as the supplier of services). On the demand side, we assume that data is available in the form of the traffic volume to be moved between a given origin-destination pair. In practice, the demand requirements may be more complicated. The shipper might specify a maximum allowable delivery time or specific constraints on routing. On the supply side, the specified set of resources available to the railroad determines the feasible train routes, allowable train itineraries, crew and motive power availabilities, and yard facilities. The operating policies determine an assignment of the resources to each class of traffic (determined by its origin and destination as well as possibly traffic type). Operating policies may be roughly divided into line and yard policies. The former determine the routing of each traffic class on the physical rail 101
102
ARJANG A. ASSAD
network as well as the assignment of traffic to carriers. Yard policies specify the operations performed on different classes of traffic in the yards they visit. At each yard the incoming traffic is regrouped according to final destination. The classification of cars into these destination-oriented blocks is called the grouping or blocking policy. The blocks of traffic are next placed on classification or departure tracks where they wait for outbound trains. Each outbound train has a “take-list” specifying the blocks of traffic it may pick up at a given yard. The decision as to which blocks of traffic a given train may carry is called the makeup policy. Clearly the makeup decision interacts highly with both yard classification and network routing policies. We may therefore regard it as an important linking factor between line and yard policies. In this paper we attempt to model this interaction. The decisions of interest to rail management vary substantially in scope, time horizon, investment requirements, and the level of decision-making. We wish to preface our model by a hierarchical framework for the planning process of rail freight transport. We follow Anthony (1965) and the literature of hierarchical planning in production systems (Armstrong and Hax, 1974) in identifying three levels of decisions-namely strategic, tactical, and operational. We regard the following categorization as a tentative aid allowing us to position our model at a suitable level of aggregation. Strategic decisions The decisions involve resource acquisition over long time horizons and typically require major capital investments. Due to the pervasive and long-lasting impact of strategic decisions on the future of the system, top-level management is usually directly involved with their resolution. Prime examples of such decisions in the context of rail systems include : (a) Network design and improvement. Track abandonment. (b) Location of yards and major classification facilities within large yards. (c) Highly aggregate routing decisions. Long-term planning of train services. The network improvement model of LeBlanc (1967) is one example of a strategic problem studied in the rail literature. The problem of choosing a set of feasible routings over a long planning horizon may fall into the category of strategic decisions if highly aggregate measures of traffic demand are used. The railroad may also want to estimate the costs and impact of providing regular service on a given set of routes and use such aggregate routing models to evaluate its supply capabilities or use them as inputs into larger models dealing with trip distribution or modal split decisions. Tactical decisions These decisions have medium-term planning horizons and focus on effective allocation of existing resources rather than major acquisitions. The planning horizon and level of aggregation in a model addressing tactical decisions must allow it to take account of broad changes in system parameters and data (such as seasonalities in the traffic volumes and imbalances resulting from lack of uniformity in the geographical pattern of shipments) without having to incorporate day-to-day changes in the data-base. Some examples of tactical decisions in rail systems follow: (a) Train selection and traffic routing: What trains should run and what should the required frequency of each train be to accommodate traffic demand? (b) Train makeup: What groups (or blocks) of traffic should a train be allowed to carry (its take-list) at a given yard of its itinerary? (c) Yard classification policy: Into what groups or blocks should the incoming traffic to the yard be consolidated? (d) Allocation of classification work among yards: What is the total amount of classification work performed in the system? How should this workload be distributed among the various yards to account for the fact that they might have different technological capabilities?
Modelling of rail networks: toward a routing/makeup
model
(e) Train lengths: What train lengths should we consider economically Is it better to have shorter more frequent trains?
103
attractive?
These issues have long been at the heart of planning rail operations. Due to the complicated interactions among the preceding issues, no one model can hope to fully capture all aspects of tactical planning in rail transport. We envisage a tactical planning model as being solved on a monthly or quarterly basis to respond only to major, stable changes in traffic patterns. Resetting makeup and classification policies on a daily basis may result in confusion for yard personnel. Operational
decisions
Such decisions deal with day-to-day activities in a fairly detailed and dynamic environment. Correspondingly, only lower levels of management (say, yardmasters) are directly concerned with operational issues. Some examples are: (a) Train timetables: Determining arrival/departure times of each train at any intermediate station of its itinerary. (b) Track scheduling and priority policy: Assigning trains to tracks if track capacity is limited. Planning for meets and overtakes according to a priority scheme. (c) Engine scheduling: Planning for daily distribution of motive power units over a specified set of train schedules. (d) Empty car distribution: Distributing empty cars over the rail network to meet demand and rectify imbalances due to uneven freight movement. (e) Yard receiving and dispatching policies: Determining a priority scheme for processing the queue of trains incoming to a classification yard. Setting rules for departure times of outbound trains. (f) Line-haul and yard maintenance operations: Scheduling maintenance and inspection for cars, engines, tracks, and yard facilities. The element of timing is crucial to most of the decisions in this list. The aim of computerized information systems for railroads is to record the position of cars in real-time or on an hourly basis. Thus models for empty-car and engine distribution may well be resolved on a daily basis. We note that the distinction between tactical and operational issues may be blurred at times. For example, a more aggregate model guiding empty-car distribution on a zonal basis could be viewed as a tactical problem. Similarly the derivation of train timetables from train schedules could assume tactical dimensions and generally does so in the context of passenger trains. The main advantage of a hierarchical approach is to avoid the pitfall of dealing with all the decisions outlined above simultaneously through a monolithic model. Even if computer and algorithmic capabilities permitted the solution of a large-scale detailed rail model (which is presently not the case), this approach would still be inappropriate since it would not be responsive to managerial needs at each level of the organization. Thus we do not try to integrate service and route selection problems for trains with delay timetabling and empty-car distribution problems into the same model, since the two classes of problems relate to different levels of the managerial hierarchy. 3. A GENERAL
RAIL
NETWORK
MODEL
In this section we formulate a general model aimed at (i) modelling the interaction between routing decisions and yard activities, and (ii) capturing the economies associated with consolidating blocks of traffic into a single train. Since most delays in freight delivery occur in yards, an effective policy for reducing yard congestion and delays is to schedule bypass trains that deliver blocks of traffic directly from origin to destination with no intermediate reclassification. The savings associated with this option, however, have to warrant the additional costs of allocating one or more direct trains to that traffic. Our model is constructed to address this tradeoff aiming to strike a rational balance between customer service and rail operating costs. Consider a directed graph
ARIANG A. ASSAD
104
GP = (N, AP) representing the physical rail network comprised of a set of nodes (yards), N, and a set of arcs AP (referring to physical track sections between yards). We assume demand data is available in terms of required flows I P4between origin-destination or OD pair (p, 4). These requirements may be expressed in terms of number of cars travelling from origin p to destination q or in some form of equivalent tonnage. We endow the physical network GP with a route structure that specifies the set of feasible routes on which trains may be scheduled. Each route has a unique origin i and destination j, i and j being yards in the rail network, as well as a physical path (i.e. a series of links in AP) from i to j which is completely specified. In the rail literature such a route is occasionally referred to as a train. It corresponds to the itinerary of a unique train from i to j. A number of trains (in the sense of a string of cars provided with locomotives) may be run on each train route. Thus we distinguish between a train route (which specifies the itinerary through the physical network) and train frequency (which specifies the number of actual trains dispatched along that route). It is useful to think of train routes as a bus number (line) which has a specified itinerary. Obviously buses with the same number may be run on a given itinerary with any desired frequency, however, all of these share the same routing and stop-schedule.
Fig. 1. The network
of train
routes-G.
A train service from i to j, denoted [i, j], maintains its identity throughout its itinerary. It is therefore made up at yard i and broken up for reclassification at its destination j. No intermediate yards perform any classification activities on the train. In this model we shall ignore the option of stopping at an intermediate yard to set off or pick up. We make this assumption merely to simplify our exposition. A further refinement of our model may incorporate such stops explicitly with only minor modifications. Train service [i, j] therefore “bypasses” all intermediate yards between i and j and traffic groups that make “connections” travel on more than one train route. The train routes may be represented as a network by adding route arcs between nodes i and j for each feasible train service. Figure 1 shows the route network G for a physical network GP of a line of 4 yards. For example, traffic for OD pair (1,4) may be placed on train [l, 31, be classified at yard 3, and then be transferred to train [l, 41. In general the route network for physical network of n yards along a line will have n(n - 1)/2 arcs although, in practice, other managerial considerations (say inspection requirements or maximum lengths for itineraries) may rule out some of these routes. We denote the route network by G = (N, A). For each OD pair (p, q) we define the decision variables xv = number of cars (or tons of freight) travelling from p to 4 on route [i, j] yij = number of trains provided
on route [i, j].
We may view the variables xc? as arc flows on the route network G. Since the flows corresponding to different OD pairs should be distinguished from one another, we are dealing with a multicommodity flow on G (see Assad, 1978). For each arc (i, j) of G, we also specify the number of cars C(ijthat may be assigned to train [i, j]. Alternatively, if we interpret yij as the number of motive units provided on route [i, j], GLij will specify the allowable car/engine ratio on that route. Let Ii and Oi be the set of incoming and outgoing trains (respectively) at yard i, that is: Ii = {kl(k,i)EA},
Oi = {kl(i, k)EA}.
Modelling of rail networks: toward a routing/makeup
model
105
Let ~j denote the take-list of train [i,j], that is, the set of all OD pairs whose traffic may be assigned to that train. In general, this set is specified by the network configuration. Indeed, let us call node j reachable from node i if there is a path going from i to j in G. Then ~j = {(p, q) 1i is reachable from p, and q is reachable from j}. For the simple case of a line network of n consecutive yards where and
N = {l,...,n}
A={(i,j)ll
we have
Ii=
{k/l 5 k < i},Oi=
{k(i<
k<
ql
n}.
n}
and IT;i={p,q)llS
i
PI
If we assume that the route structure imposed on GP is such that G is an acyclic network, the reachability relation induces a partial order < on the node set N, that is we write i c j whenever j is reachable from node i. The flow balance and motive power constraints are: r*q -rpq 0
CXPJP - &,xfT= iS0i
ifi=p ifi=q otherwise
(1)
for all nodes i used OD pairs (p, q)
all x0 2 O.yij 2 0 and integral for all (i,j)E A.
(3)
In (2) xii denotes the total flow of traffic assigned to train service [i,j]. Note that for fixed values of yij the constraint setis that of a capacitated multicommodity flow problem. Our approach is to maintain this feasible set (which is shared with many other transportation models involving flow of goods and carriers) and incorporate complications into the objective function. The objective function will contain two types of costs: (i) main-haul or over-the-road costs and (ii) yard costs. For a given train service [i, j], let Xii be a vector with components xpj for all Cp,q) in the take list Tj listed, say, according to lexicographic ordering of pq. This vector will fully describe the makeup (or composition) of the train [i, j], i.e. how many cars of each traffic class (p, q) is placed on the train. At a given yard j we also form a supervector Xj containing the vectors X, for all trains [i, j] incoming to yard j (i.e. for all k E lj)e This vector contains the composition of all traffic brought into yard j. As an example for a yard j of the line network. 4
= (X,j,X,j,
.e*,Xj-l,j)
and Xii = (x!j,,
,
.
.
.
,
Xij”,
xif;‘,
.
.
.
,
x$, . . . , x#, . . . , xi;).
We consider two types of costs: train costs relating to running a train [i, j] over its route, with a specified load assumed to be expressible as a cost function $i,{Xijy yij); yard costs at a typical yard j with an incoming traffic Xj? given by the functional form 4JXj). Then a formal model may be set up as follows: (PI: min 27 =
C (LJ3E.4
subject to eqns (l)-(3).
$ijlxij3 Yij) + j& 4j@j>
(4)
ARIANG A. ASSAD
106
We shall refer to this model as the routing/make-up model (P). We now proceed to specify possible functional forms for train and yard costs. Train costs may include: (i) Crew costs-A crew should be engaged over the entire length of the train route. (ii) Fuel costs-These will depend on the train weight (total tonnage) and length (number of cars) as well as on the train’s speed and the geographical terrain of its route. It is reasonable to take fuel costs as being proportional to train weight over a given link. (iii) Costs of motive power-We attach costs to providing and running an engine over each link. (iv) Over-the-road delay costs-The delay incurred on the main-haul legs, due to travel-time, congestion, meets and passes, and so forth, may be attached a dollar value to reflect the value of capital tied up in the system pipelines and, more importantly, the shipper’s devaluation rate. Yard costs could include: (i) Inspection and classification costs-These are attached to the operations involved in inspecting and breaking up incoming trains for reclassification into outbound groups. They may reflect the use of yard equipment (yard-engine-hours) or labour resources (inspection crews). (ii) Yard delay costs-We may associate time (devaluation) costs to the total delay suffered by cars in the yard due to queueing effects and waiting times of various yard operations: receiving, classification, outbound inspection and assembly, accumulation and connection delays. Naturally in a yard with fixed resources (crew and equipment) component of yard costs is formed by delays incurred by different classes of traffic at that yard. One simple functional form for train costs is: ~ij(Xij,
yij) =
C~j'Yij + Cfj' Xij
(5)
where cFj = cost of providing a unit of motive power for train [i, j] cfj = hauling cost per car on the route of [i, j]
and xij, we recall, is the total load on train [i,j] as defined in eqn (2). Thus we have variable costs corresponding to the number of cars on the train (and track parameters for the train route) as well as engine costs which exhibit a discrete character peculiar to rail systems. Later, we shall point out why we believe modelling this discreteness in the cost function is important. For the moment, note that for fixed values of x~JJ (and hence xij), given that the cost function is strictly increasing in yij, the values of the yij variables may be deduced from eqn (2) to be yij = [CrGl . Xij] + where [ .]’ denotes rounding up to the nearest integer. Consequently the graph of $ij as a function of xij exhibits a stepwise nature as shown in Fig. 2. The magnitude of the jumps is ci’j and slope of the linear sections equals cfj We now turn to the yard cost function +j In the simplest case, 4j depends only on the total throughout of yard j, that is 4jGj>
= fj
( > C xij
.
ialj
To allow for congestion effects, we may take fj to be a convex increasing function. Moreover the network structure of the problem will be preserved if the node for yard j is split into two nodes joined by a “throughput arc” with flow c Xij and arc cost islj
Modelling
of rail networks:
toward
a routing/makeup
model
!
107
xiI 2Qij
aij
Fig. 2. The train cost function.
A(
2.G).
For
a tactical model, however, we consider this expression to be too crude. It
does ‘not distinguish the delaying effects of one class of traffic on others. Following Thomet (1971a, b) we introduce a delay function at a given yard j of the form Wj + Vj’Xij
where Wj is the fixed delay for processing a train coming into the yard and Vj is the variable delay (in units of time per car). Such a delay is incurred for any train [i,j] which is classified and processed at yard j. To account for the effect of train composition in the processing costs, we should note that trains composed purely of cars for yard j need not be classified at that yard. The traffic on such trains will be delivered to local industrial sidings and will thus exit the main rail network. We note that the total number of cars destined for a yard j (to stay) is a constant determined by traffic requirements and hence should not affect the optimization. That is 1 idj
x$ = 1 rpi = constant
(PsJ?STij
(8)
P
where the last summation is over all origins p that have traffic destined for yard j (i.e. all p to which node j is accessible). Let us define a variable 8ij corresponding to the classification status of train [i, j] as follows:
=
variables may
1 0
train [i, otherwise.
should be
defined in
at yard
of arc
0,
=
by the
1 Xv
6 [
(P.4)
1
(9)
where the summation runs over all pairs (p, q) such that (p, q) E Tj and j < q. 6(x) is the delta function that equals 0 for x = 0 and 1 for x > 0, commonly used in fixed charge problems. Equation (9) simply says that the train [i,j] should be classified at yard j whenever it includes any traffic travelling beyond yard j. Then the delay at yard j for processing trains [i,j] is d zij
(Wj +
=
VjXij)eij
(10)
We could translate this delay term to money costs by attaching a time cost dij to each unit of time delay. This should reflect in dollar terms the undesirability of delaying cars at yard j. The yard cost function may then be written as @j(Xj) = C dijOij(Wj + UjXij). idj
TR-B Vol. 14, Nos I-2-H
(11)
ARIANG A. ASSAD
108
We wish, however, to point out a complication in transferring delay times into cost terms: the devaluation costs dij should logically depend on the train composition. As in inventory holding costs, dij usually reflects the total dollar value of freight that undergoes delay. If the average devaluation rate of traffic corresponding to OD pair (p, 4) is c$‘Idollars/car/day, then the train devaluation rate would be dij =
C
C$"XPj4
(12)
(P4)ETi j
Since dij depends on the train composition, measuring delay in terms of car-days, rather than just days, leads to a quadratic objective function. 4. CAPABILITIES
AND
LIMITATIONS
OF
THE
NETWORK
MODEL
In this section we shall relate the general model presented above to the tactical issues listed in Section 2. Rail systems are distinguished from many other transportation modes by their inherent ability to move a large number of shipments as a single unit. We feel an adequate model of rail freight transportation should take explicit account of such economies associated with train length and the corresponding blocking policy of consolidating diverse classes of traffic into large groups in which members share some leg of their itinerary. The train length economies operate in two directions: on the one hand they set lower limits on the total traffic assigned to a given train so that it would be profitable to run. On the other, they allow us some flexibility of “stretching” the train capacity by assigning additional engines so that the train could accommodate more traffic. This latter feature is absent from the competing trucking mode, for example. On the negative side of the economies suggested by large shipments may well be negated by the necessary intermediate sorting and grouping operations on the cars which tend to increase operating costs delays while lowering customer service and equipment utilization. Our model should address both the economies and diseconomies mentioned above. When economies of scale are present, it is imperative that a model aiming to incorporate them be on the same hierarchical level as where such economies are realized. This consideration will serve as a guide for choosing a suitable level of aggregation and planning horizon for the model. For example, in a strategic model for routing traffic on a rail network we may measure the flow variables xc9 in units of cars/year corresponding to annual requirements rpq. If we choose aggregate costs functions $ij(xij) for the routes [i,j] and ~j(Uj) for yard costs as in eqn (6), then, since we expect the variables yij to assume large values, we may relax the integrality requirement on yij’s and use the relation
The program (P) then becomes a convex network minimization problem. While such a model is useful for aggregate planning purposes, it abstracts away from the issue of assigning traffic to trains and therefore is unable to address economies of the type occurring in the following example. Example. Consider three nodes (yards) i, j, and k of a rail network with requirements as shown in Fig. 3: Suppose moreover that we have a car/engine ratio of 40 for trains travelling over the routes [i, j], [i, k], and [k, j]. Sending cars directly from each origin to destination requires a total of 7 engines: one over the route (i, j) and 3 over each of the routes (i, k) and (k, j). However, diverting the traffic from i to j through yard k changes the loads on links (i, k) and (k, j) from 100 to 120. Then one engine is saved over the route (i, j) and, moreover, the other engines will be fully utilized (to capacity). There is, of course, a corresponding processing and classification cost that cars of (i,,j) will incur at yard k, but the balance may well be in favor of traffic diversion.
Modelling
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109
20
j loo
too
v Fig. 3. Twenty
cars from
k
i to j; 100 cars from i to k; 100 cars from k toj.
The above example shows that the integrality issue in the provision of motive power units is a key factor in two questions that have long plagued tail operations: that of train length (short vs long trains) and of high equipment utilization. Maintaining the discreteness in our cost function allows for a more rational evaluation of the economies of shipping in train-loads. We see the economies are realized on the level of loading (and blocking) decisions for cars. We now list some of the issues that a successful solution of the routing/makeup model of Section 3 will clarify. (a) Service selection and frequency-The variables yij determine the choice of trains to provide service. If yij = 0 no trains will be sent along the route [i,j]. For yij = 1 only one train will be run. For larger values of Yij longer (or more frequent) trains will travel on the service [i,j]. (b) Train makeup-The optimal values of the flow variables xpj” will specify the optimal train composition on a given route [i, j]. In our model these variables will also provide guidelines for the blocking policy. (c) Yard grouping policy-At each yard the incoming traffic may be reclassified according to outbound destinations. The number of groups formed at the yard can be derived from the total number of outbound services operative at optimality. We may think of each classification track as containing all the traffic belonging to the takelist of a given outbound train. (d) Yard workload-The total workload of each classification yard may be obtained from the total yard delay as expressed by the terms rt in eqn (10). The yard parameters Wj and Uj should be calibrated for each yard and will differ from one yard to another according to yard type (flat or hump yard) and the technological profile of the yard. The model strives for an efficient allocation of classification work among yards. For example, classification activity may be diverted from a small flat yard to a large automatic hump yard where it may be performed much more quickly. (e) Train length-The train length variable xii will be influenced by the discrete marginal costs ct attached to yi? The discreteness of these variables will specify a number of “regimes” of possible train lengths with different costs to the railroad. We thus see that the routing/makeup mode, if solved, will address a number of the tactical issues raised in Section 2 and consequently has the potential of being a valuable planning tool. We shall close this section with some technical observations concerning our model for routing and makeup. As it stands, the model involves continuous flow variables xv, integer variables Yij, and rather complicated cost terms in the objective function as specified by eqn (11). Let us start by noting how this formulation may be transformed to a network flow model with arc costs involving set-ups. In the process we shall also eliminate the 0,l variables 8, which specify the classification status of the train [i,j]. Consider a given yard j. A train [i,j] will be classified at j if it carries freight destined for yards beyond yard j (i.e. for some nodes k > j). On the other hand, a train composed entirely of traffic due for yard j will not require reclassification at that yard. This situation can be partially captured by splitting the node for yard j into two nodes j’ and j” as shown in Fig. 4. The node j’ acts as a sink for all traffic staying at yard j (all traffic classes with
110
ARJANG A. ASSAD
Fig. 4. Network
representation
of processing
delay.
final destination j) while node j” acts as a source of outbound traffic from yard j. We may decompose the traffic volume of a given train [i, j] as follows Xij
=
1
xf9
=
=
p
zixff+jg_q=uj + uj+ p-zi
Pl¶
where uj and uj+ stand for the first and second summations, respectively. We now attach the classification and processing costs of yard j to this arc (j’, j”) by specifying an arc cost function of the form 4j,jfs(x)=
L,+v x E:cr 0” 1
j
Then pure trains, in the sense of uj+ = 0 will incur no processing delays at yard j. An alternative to capturing cost functions of eqns (lo)-(12) is to seek simpler cost functions such as cbj(Y)=
4’
iz,Yij J
that captures some measure of both train length and frequency for all incoming trains at yard j. Of course, ideally, one may wish to calculate processing delays for different trains separately. Since Wj refers to processing a single train, a better choice for delays due to fixed set-ups is
Finally we wish to note that our general model can be considerably enriched by the addition of rather simple constraints. Maximum train lengths (or frequencies according to the interpretation of YijS) could be enforced by adding the constraints yij < yij, for a given upper bound yij, on a route [i, j]. These constraints could arise from limitations of departure tracks or crew availability. Other researchers-Bodin et al. (1977Finclude explicit constraints on the blocking strategy at each yard to reflect the railroad’s policy. We may also address the allocation of motive power by specifying engine availabilities at each yard at the beginning of the planning horizon and impose flow constraints on variables yij. Naturally, the inclusion of motive power consideration depends on whether or not the railroad faces fleet size bottlenecks. At any rate, the approach at this point should be aggregate: we do not recommend the incorporation of detailed engine scheduling, as in Florian et al. (1976b), at this stage of the modelling process. 5. OTHER
RAIL
NETWORK
MODELS:
REVIEW
AND
COMPARISON
This section contains a brief review of two major attempts in providing optimization models of rail systems and compare them with our own. We begin by introducing the work of Thomet (1971a, b) that has stimulated our formulation. The basic component of Thomet’s algorithm is a cancellation procedure that cancels direct shipments in favor of a series of connections. Suppose that a direct
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train going from i to j is cancelled and that its traffic is diverted upon a sequence of connecting routes from i to j. Set [k, 11 denote one such intermediate train. This train travels from yard k to yard 1 (both on the route from i to j) with no intermediate classification. The impact of this cancellation on the total costs is as follows: there are benefits measured by a reduction in train-miles and a decrease in motive power requirements (recall Fig. 3) and the elimination of other costs of running a direct train between i and j. The major increase in costs is due to additional classification and processing costs at intermediate yards: an intermediate train [k, fl must be regrouped at yard I and all traffic going beyond yard 1 (say to j) should be reclassified. The diversion may thus change the classification status t&r of that train and incur delay costs as given by eqn (11). We may therefore compute y&-the net cost of diverting traffic from train [i, j] to train [k, II-for each intermediate train [k, lJ. The best sequence of connecting trains, on to which the [i, j] traffic may be shifted, can be found by solving a shortest route problem as follows: Consider the physical route of train [i, j] through the network Gp We may represent this itinerary by a sequence of yards i, = i, i2, iJ,. . . i, = j. If m 2 3, (i) construct another network G(i, j) form (k, 1) = (i,, i,) for 1 I r < s I (ii) compute the costs # for all such (iii) find Tithe length of the shortest running [i, j] directly the savings #j
=
;lij
_
on the node set {iI,. . . , i,} with arcs of the
m; arcs, and route from i to j on G(i, j). If 1” is the cost of due to cancellation are defined as -ij Y
.
For example, in the network of Fig. 1, suppose train [l, 41 is cancelled and that the shortest path with respect to arc costs y:> is (1,2), (2,4) with length y14 = y:; + y$ Then the traffic can be diverted upon the two trains [1,2] and [2,4] resulting in a savings of A’4 -yi4. Thomet’s algorithm may now be described as follows: initiate the algorithm by assigning direct trains to all OD pairs with nonzero requirements [all (p, 4) with rpq > 01. This is called the Minimum Transit Time Policy (MTT) and involves no classification work (initially all ep4 = 0). At a given cancellation step, compute the savings sii of all trains [i, j] as described above and cancel the train with maximum savings. Transfer the traffic of this train to the intermediate trains and update train parameters accordingly. To find the minimum cost policy, this strategy is continued (one cancellation at a time) until no positive savings can be found, that is until sij < 0 for all remaining trains [i, j]. Thomet’s approach is similar to drop heuristics for plant location Feldman et al. (1966). One starts with all trains and successively cancels trains according to a myopic criterion of savings in total costs. We may immediately point out several limitations of Thomet’s approach : (a) As Thomet himself realizes this sequential cancellation procedure does not guarantee optimality upon its termination. Rather it provides a local optimum in the sense that there will be no “one-move” (cancellation of a single train) which could further reduce costs. (b) In the algorithm described above the physical routing of traffic is never changed. All traffic follows the shortest route between its origin and destination on the physical graph Gp What changes is the loading pattern of the traffic on the intermediate trains on the shortest route itinerary. However, as the small example of Fig. 3 indicates, it may be advantageous to divert traffic away from its shortest route. In practice rail freight is known to be routed through such circuitous paths, occasionally on the basis of a reduction in the railroad operating costs. This diversion may become necessary if flow on certain routes is limited by capacity constraints. This limitation of routing alternatives to shortest paths may result in further suboptimality in Thomet’s model. (c) The diversion of traffic from one train to another is always performed in bulk form, that is, the entire traffic load of a train is shifted. One may easily envisage cases where
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ARJANG A. ASSAD
only partial diversion of traffic is necessary. For example, consider the situation of Fig. 3 with the demand from i to j changed from 20 to 60. Then we may retain 40 cars on the train [i, j] and only divert the additional 20 to the route going through yard k. This possibility is allowed by our routing/makeup model (P) but not by Thomet where a typical variable sv is limited to one of two values-0 or rP4. Indeed we may recast our planning model into Thomet’s form by the following device. Let zpj”be a new variable defined as zfJ/rpq. If we then substitute the quantity rpqzpiSfor xPpin eqns (lH4) of (P) we realize that for each (p, 4) the variables zpi”obey flow conservation equations for a flow value of 1. Restricting the variables to be integral will now mimic Thomet’s procedure. Another major rail optimization model developed at Queen’s University at Kingston is the Railcar Network model (Petersen and Fullerton, 1975). The study evolved over 5 years into a comprehensive rail planning model involving over-the-road and yard activities. The object to the model is to route freight on the rail network to meet demand at minimal total delay (in car-hours). The approach is to derive delay functions for component rail operations as a function of the flow of freight handled by each operation. If all the delay functions are convex, the routing problems may be viewed as a minimum cost network problem with convex costs for which a number of algorithms are available (see Sections V and VI of Assad, 1978). Algorithmically, the model is thus identical to the work in traffic assignment-Florian (1976abwhere the delay functions (also called service functions in road traffic assignment literature) reflect over-the-road congestion delays. The derivation of the delay function, however, is very different as it is based on average waiting times derived from queueing theory. The main components of delay are as follows: (i) inbound inspection time for incoming trains, (ii) classification time, (iii) train assembly time, i.e. cars waiting to be assembled into an outbound train, (iv) outbound inspection time, and (v) over-the-road time measuring link traversal times in the presence of congestion, meets and passes, etc. Factors (i)-(iv) refer to yard delays and are attached to intrayard arcs whose flows reflect yard throughout. Inbound and outbound inspection delays are taken to be constant while classification and assembly delays are derived from queueing formulas that yield average delay as a convex function of yard flows. Over-the-road delays are expressed analytically by considering priority rules for track usage. As mentioned before, we regard the railcar model as a major contribution. We wish, however, to bring out some basic differences between their work and our approach in this report: Our model makes a basic distinction between the flow of cars and the flow of trains on a given service [i,j]. The effect of running a train is determined by integer variables Yij, in contradistinction to the continuous flow variables Xij for the cars. The railcar model, instead, derives train flows from car flows by using an average train length of 1 cars/train. Thus a flow of xij cars over a route automatically results in Xi$ trains. Moreover the number of trains affects only the delay time, no attention is paid to the cost of scheduling an additional train. In Section 4 we discussed how these costs may directly influence routing and makeup decisions, possibly resulting in train cancellations if such costs are high. Our model attempts to capture some of the economies of the routing process more directly to allow us to decide, for example, whether or not a certain train should run or a particular traffic class should be incorporated into a train’s makeup. The same distinction arises with respect to classification costs. A component of our classification costs refers to the train-via the fixed charge term ~jeij in eqn (lob-and not merely to yard throughout. In fact, this term attempts a first approximation to the effect of train composition on yard costs. While the railcar model chooses a purely stochastic approach to evaluating waiting times, we have opted for using a delay term for which the effect of train cancellation or composition is more apparent. Finally we regard some of the delay terms in the railcar model as being highly dynamic in nature and thus not particularly suitable for static waiting time analysis. In particular, the assembly delay term will
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depend on the yard dispatching policy. Such policies, which belong to the operational level, should be studied separately incorporating the time element into direct account. To conclude this section we wish to position the models described above in the hierarchial framework of Section 2. We believe the railcar model relates the strategic level of decision-making. It specifies the configuration of traffic flow on the rail network using aggregate measures of yard and over-the-road delays. Our model, however, belongs to the tactical level: our direct concern with traffic routing and train scheduling reflects a desire to have firmer control on the choice of trains to run and their composition. Indeed one may be well advised to use the railcar model and our model sequentially. The former will provide an aggregate picture of traffic flows and supply a “base-level” of yard and line activities. Thereupon we may pass on to the routing/makeup model to evaluate certain more detailed decisions (for example, the use of direct versus local trains, and long vs short trains).
6. SPECIALIZATIONS
We now consider certain algorithmic developments.
OF
specializations
THE
MODEL
AND
CONCLUSIONS
of the model (P) with a view towards
possible
(a) Suppose we have a set of operative services and regard each train route without capacity restrictions. This corresponds to a scenario where a sufficient number of trains operate on each route so that we may assign as much traffic as we wish to that route. Mathematically, C(ij can be assigned a large value so that the constraints in eqn (2) are no longer binding. Thus only routing and yard costs will affect the flow pattern. The result is a traffic assignment problem with convex costs on the network of feasible routes. If the cost functions are simplified even further, say by taking vi = 0 in eqn (7) then the model yields a shortest path problem that will specify the optimal sequence of train connections for each class of OD traffic. Indeed, the model proposed by Truskolaski (1968) seems to be very close to this problem although he does not use shortest paths. (b) We may now refine the above specialization by attaching fixed costs to service provided on a given route [i,f still maintaining, however, the assumption of unlimited capacity. The yijs are then cl variables indicating if a service [i,j] is operative or not. This results in an uncapacitated network design problem. Once the Yij variables are fixed, the problem reduces to shortest path or traffic equilibrium problems as in part (a). Additional constraints limiting the total operating budget on the number of services used may be added to the model. (In the latter case we have a p-median problem.) In the context of rail networks, this approach is reminiscent of the paper by Pierick and Weigand (1976). For recent algorithmic work on the network design problem, the reader may consult Dionne and Florian (1977), Magnanti and Wong (1977). (c) The next level of refinement, where the capacity limitations are included explicitly, results in the routing/makeup models of Sections 3 and 4. Again, for a given assignment of yijs, we obtain capacitated multicommodity flow problems. We propose this last model for algorithmic development as it is typical of a variety of “loading problems” in transportation studies that seek to determine the optimal flows of traffic and carriers simultaneously. Richardson (1976) has used Benders Decomposition on an airline routing problem with a structure similar to our model (P). This work may serve as a useful point of departure. Another promising area of investigation involves developing efficient heuristic techniques for the class of models exemplified by (P). We have briefly discussed one such heuristic, due to Thomet. Other strategies may be proposed and possibly evaluated analytically along the lines of Cornuejols et al. (1977). In this paper we have proposed one approach to integrate routing and makeup activities on a rail network. We took a hierarchical viewpoint of the models concerning traffic routing and suggested that our model is appropriate to the tactical level. This
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ARJANG A. ASSAD
model may interact with other strategic (or more aggregate) models as in (a) and (b) above. Our basic goal has been to provide a framework for modelling rail networks. Naturally, the ultimate test of our proposed model will involve computational experience with a realistic data-base from some railroad. We envisage such an undertaking in the future and hope to announce the results in follow-up publications. Since rail networks are likely to be large (20-40 yards and some 200-300 train routes), the computational difficulties in solving the model (P) have to be overcome. Here, we may try to use techniques such as Benders Decomposition or Lagrangean Relaxation to attack the general mixed integer program. Alternatively, we may concentrate on simplified versions of the model, say, scheduling on a line network with special assumptions on yard and line costs. The insights provided by these simpler problems may serve as a basis for developing efficient heuristics for the general problem. Progress along either one of these dimensions will substantially increase the promise of optimization-based models in aiding the planning process for rail systems. Acknowledgements-The author wishes to thank Professors R. W. Simpson and T. L. Magnanti of MIT for Pheir careful readings of the manuscript. This work was supported, in part, by the U.S. Department of Transportation under contract DOT-TSC-1058 (TARP). REFERENCES Allman W. P. (1966) A network simulation approach to the railroad freight train scheduling and car sorting problem. Unpublished Ph.D. dissertation, Northwestern University. Anthony R. N. (1965) Planning and Control Systems: A Framework for Analysis. Harvard Graduate School of Business Administration, Boston. Armstrong R. J. and Hax A. C. (1974) A hierarchial approach for a naval tender job shop design. MIT Oper. Res. Center Technical Report No. 101. Assad A. A. (1977) Analytical models in rail transportation. MIT Oper. Res. Center, Working Paper OR 066-77. Assad A. A. (1978) Multicommodity network flows: a survey. Networks 8, 37-91. Cornuejoils G., Fischer M. and Nemhauser G. L. (1977) Location of bank accounts to optimize float: an analytical study of exact and approximate algorithms. Man. Sci. 23, 789-810. Dionne R. and Florian M. (1977) Exact and approximate algorithms for optimal network design. University of Montreal, Centre de Recherche sur les Transports, Publication 41. Feldman E. F., Lehrer A. and Ray T. L. (1966) Warehouse location under continuous economies of scale. Man. Sci. 2, 670-684. Florian M. (1976a) (Ed), Traffic Equilibrium Methods. Lecture Notes in Economics and Mathematical Systems, Vol. 118. Springer-Verlag, New York. Florian M. (1976b) The engine scheduling problem in a railway network. INFOR. 14, 121-138. Leblanc L. J. (1976) Global solutions for a nonconvex nonconcave rail network model. Man. Sci. 23, 131-139. Magnanti T. L. and Wong R. T. (1977) Accelerating Benders Decomposition for network design. Discussion Paper, Center for Operations Research and Econometrics (CORE), Catholic University of Louvain, Belgium. Petersen E. R. and Fullerton H. V. (1975) The railcar network model. Queen’s University at Kingston, CIGGT Report Number 75-l 1. Pierick K. and Weigand K. D. (1976) Methodical formulations for optimization of railway long-distance passenger transport. Rail Int. 7, 328-334. Richardson R. (1976) An optimization approach to routing aircraft. Trans. Sci. 10, 52-71. Thomet M. A. (1971a) A combinatorial-search approach to the freight scheduling problem. Unpublished Ph.D. dissertation, Department of Electrical Engineering, Carnegie-Mellon University. Thomet M. A. (1971b) A user-oriented freight railroad operating policy. IEEE Trans. on Systems, Man and Cyber. SMC-1, 349-356. Truskolaski A. (1968) Application of digital computer to the optimization of freight train formation plans. Bull. Int. Railway Cony. Assoc. 5, 211-226. TRB (1975) Railroad research study background paper, Transportation Research Board, Richard B. Cross, Oxford, Indiana.