scheduling direct and indirect trains and containers in an...

Scheduling Direct and Indirect Trains andContainers in an Intermodal Setting

ALEXANDRA M. NEWMAN

Division of Economics and Business, Colorado School of Mines, Golden, Colorado 80401

CANDACE ARAI YANO

Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720

The focus of our research is on rail transportation of intermodal containers. We address theproblem of determining day-of-week schedules for both direct and indirect (via a hub) trainsand allocating containers to these trains for the rail (linehaul) portion of the intermodal trip.The goal is to minimize operating costs, including a fixed charge for each train, variabletransportation and handling costs for each container and yard storage costs, while meetingon-time delivery requirements. We formulate the problem as an integer program and develop anovel decomposition procedure to find near-optimal solutions. We also develop a method toprovide relatively tight bounds on the objective function values. Finally, we compare oursolutions to those obtained with heuristics designed to mimic current operations, and show thata savings of between 5 and 20% can be gained from using our solution procedure.

Intermodal transportation consists of combiningmodes, usually ship, truck, or rail to transportfreight. The focus of our research is on rail transpor-tation of intermodal containers for the long-haulportion of their journey. For distances over 500miles, train transportation is more efficient thantruck transportation, and results in savings in oper-ating costs and labor. Because rail transport divertssome freight traffic from the roads, congestion andwear and tear on highways is partially alleviated(MCKENZIE, 1989). Despite recent advances in theefficiency of intermodal operations, difficulties re-main. Some obstacles the railroads face result frominadequate infrastructure including a shortage oftrack, and the lack of a fully operational, continuoustranscontinental railroad. Other difficulties arise, inpart, due to basic management and information lim-itations, which lead to poor train routes and sched-ules and inadequate priority rules for sending ship-ments. Because of the extra delay incurred and theincreased potential for mishandled containers at in-termodal terminals, it is important that time andcost considerations be taken seriously for inter-modal transportation to compete effectively withlong-haul trucking.

To improve the scheduling and coordination oftrains, we address the problem of how to scheduledirect and indirect trains and which containers tosend on each train for the rail (linehaul) portion ofthe intermodal trip to minimize operating costswhile meeting on-time delivery requirements. Inter-modal rail operations differ from conventional railoperations in several important respects. First, be-cause of the high cost of container handling equip-ment, intermodal networks have relatively few,widely spaced terminals. For example, the IllinoisCentral Railroad has only nine intermodal termi-nals, a few of which are small. Networks with morethan about two dozen major intermodal terminalsare uncommon. With such a structure, economies ofscale can be realized not only in container handling,but also in train movements from terminal to termi-nal. Transport from the customer to the nearestintermodal terminal is handled by truck or by re-gional or feeder railroads. Second, because of thedistances between intermodal terminals, a typicalcontainer makes few stops and is transferred be-tween trains only a few times on its journey. Thiseliminates the need to consider blocks, i.e., groups ofrailcars that travel as a unit for one or more seg-

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Transportation Science, © 2000 INFORMS 0041-1655 /00 /3403-0256 $05.00Vol. 34, No. 3, August 2000 pp. 256–270, 1526-5447 electronic ISSN

ments of their journey (to reduce train reassemblytime at rail yards), which are essential in conven-tional rail scheduling and routing decisions. Finally,shorter delivery leadtimes are promised for inter-modal freight, and, consequently, there is a greaterneed to schedule trains to achieve desired levels ofcustomer service. Under conventional operations,some freight may wait while enough railcars accu-mulate to form a block. The first two factors reducethe number of decisions required for intermodalfreight versus conventional freight, but the thirdfactor dramatically increases the importance of care-ful train scheduling and routing decisions.

Most of the research on train scheduling and con-tainer routing uses average demand rates and hasthe goal of determining steady-state train frequen-cies and container allocations. KEATON (1989) con-sidered direct and indirect train scheduling, therouting of railcars, and their grouping (or blocking)on a train. CRAINIC and ROUSSEAU (1986) developeda general framework for multimode freight trans-portation including the design of the network (e.g.,which modes to use and what frequency of service toprovide) and the traffic routing scheme through thisnetwork. MARIN and SALMERON (1996) investigatedthe problem of determining an optimal train sched-ule for a rail network, and the optimal assignment ofrail cars to these trains such that each train carriescars of a single service class. They used simulatedannealing and tabu search to solve the problem. Fora recent survey on train scheduling and relatedproblems, see CORDEAU, TOTH, and VIGO (1998).

The literature on intermodal transportation isgrowing. MORLOK and SPASOVIC (1994) developed amodel to reduce drayage costs without affecting thetimeliness of pickups and deliveries. DIAL (1994)sought to minimize trailer-on-flatcar costs incurredby United Parcel Service by choosing whether toship freight with a trailer owned by United ParcelService, or to lease one from the railroad. BARNHARTand RATLIFF (1993) developed a model that sought tominimize transportation costs for a set of trailermovements by truck and/or rail, considering the pos-sibility of pairing trailers from different sources onthe same flatcar.

The research most similar to ours is by NOZICKand MORLOK (1997) and GORMAN (1998a,b). Nozickand Morlok took the train schedule as given andaddressed freight movement as well as equipmentand locomotive repositioning decisions. Their objec-tive was to minimize the cost of on-time deliverysubject to constraints on equipment availability.The primary differences between Gorman’s modeland ours are that he considered yard and rail linecapacity in an aggregate way, but focused on the

special case of a single origin and single destinationwith multiple routes between them. He used a tabu-enhanced genetic search to arrive at solutionswithin 10% of the optimum for this case. He thenapplied his procedure to a problem with multipleinterdependent origins and destinations. The solu-tion provided significant improvements in cost andcustomer service over current policies, but the solu-tion was not evaluated in an absolute sense.

To the best of our knowledge, this problem ofsimultaneously determining direct and indirecttrain-scheduling and container-routing decisions formultiple interdependent origins and destinationsusing a formal optimization approach has not beenaddressed in the literature. Four aspects of our prob-lem that make it challenging are: (i) more than onetrain may be sent on each segment each day, (ii)both direct and indirect trains may be scheduled,(iii) containers arrive dynamically during the deci-sion horizon, and (iv) customer orders have distinctdue dates.

The remainder of the paper is organized as fol-lows. In the following section, we present a problemstatement and formulation. In Section 2, we intro-duce a new decomposition approach, which is basedon a partitioning of the underlying network. In Sec-tion 3, we show how this decomposition approachcan be modified to handle larger problem instancesmore effectively. In Section 4, we develop valid ine-qualities that allow us to obtain tighter lowerbounds on the solutions. In Section 5, we discusssimple heuristics designed to mimic current prac-tice. In Section 6, we present numerical results. Asummary and directions for future research appearin Section 7.

1. PROBLEM STATEMENT AND FORMULATION

OUR RESEARCH WAS motivated by the train-schedul-ing and container-routing problem that we observedat the intermodal division of a major railroad. Therailroad has several intermodal terminals on thewest coast of the United States, a single major hubin the west-central part of the United States, and afew other intermodal terminals east of the Missis-sippi River. The flow of traffic eastbound is greaterthan it is westbound, as it is for most U.S. railroads.

Much of the eastbound rail transport capacity isdedicated to moving sea cargo for major interna-tional shipping lines, for which the transoceanictransit time is fairly predictable (approximately oneweek). Remaining train capacity is utilized to ser-vice smaller customers within a few hours’ drive ofthe railroad’s intermodal terminals. These custom-ers typically use intermodal retailers to coordinate

257INTERMODAL TRAIN AND CONTAINER SCHEDULING /

the truck and rail movements for their goods. Inter-modal retailers often reserve space on trains in ad-vance, and then sell this space to their customers.These reservations also contribute to the predict-ability of demand for the railroad. Overall, demandexhibits weekly patterns due to freighter schedules,and seasonal patterns due to factors such as tradi-tional cycles in retail demand, and agricultural andmanufacturing production.

The railroad offers several speeds (or levels) ofservice and charges a premium for faster (promised)delivery. Trains may be sent directly from an originintermodal terminal to the destination terminalwithout stopping at a hub, providing the fastestavailable service. Alternately, trains carrying con-tainers bound for several destinations may be sentto a hub, where containers are consolidated by des-tination onto outbound trains. This consolidationactivity may cause a few days’ delay for transferringcontainers or repositioning rail cars between trains.Further delays also occur when inbound and out-bound schedules are not coordinated.

Each train has a limited capacity, where the ca-pacity is expressed in terms of number of containersin our model. We assume that containers are homo-geneous in terms of their use of train capacity, whichdepends upon the power of the locomotives and theterrain over which the train must travel. Typically,decisions regarding locomotive capacity for eachtransportation segment are determined in advanceon the basis of demand forecasts. We assume thatthe capacities of all trains on a given segment arethe same, which reflects the situation in our moti-vating application.

Yard storage space for containers waiting to beshipped, awaiting a transfer at the hub, or waitingto be picked up, is limited at all terminals. As thenumber of containers in storage increases, contain-ers are stacked higher and more densely. This in-creases the time required to retrieve a container andplaces a further burden on material-handling equip-ment, which may already be a bottleneck. Our modeldoes not constrain the number of containers that canbe stored at a yard, but we do assess a cost forcontainer storage (discussed below) to deter unnec-essary container inventory.

From our observations of intermodal terminal op-erations, the train schedules and container routingdecisions do not appear to be affected strongly, if atall, by what speed of delivery has been promised, orwhat rate has been charged to the customer. Thismotivated us to investigate how to schedule trainsand route containers to achieve on-time delivery atminimum cost.

We address a short-term, finite-horizon, discrete-

time scheduling problem for the linehaul portion ofthe intermodal trip. Given container demands dif-ferentiated by origin, destination, arrival date atorigin, and due date, the objective is to determine atrain schedule (for both direct and indirect trains)and container-shipment plan to minimize the totalcost while meeting on-time delivery requirementsand adhering to train capacity restrictions. We dis-cuss the cost elements in more detail below.

The costs incurred by the rail company for trans-portation on a segment consist of both a fixed-chargecomponent, or fixed cost, for each train and a vari-able (per container) component. The fixed cost con-sists primarily of operators’ wages and the opportu-nity cost of locomotive use. We assume that eachtrain on a specific segment incurs the same fixedcost.

The variable cost per container consists of threemain components: (i) transportation costs, such asfuel, oil, and track maintenance; (ii) handling costsincurred for moving containers on and off the railcars, or for repositioning the cars at an intermediateterminal; and (iii) yard storage costs associated withholding containers in inventory at the origin or at anintermediate terminal. We assume that the variabletransportation costs are constant over time and thatthey depend only on the route. The assumption ofconstant transportation costs over time is quite rea-sonable over a short horizon (e.g., a week or two),and the assumption that costs depend only on therail segment is consistent with our assumption re-garding homogeneity of containers in terms of theiruse of transport capacity.

Handling costs for moving containers or reposi-tioning rail cars depend more heavily on the equip-ment used for such operations than they do upon thecontainer itself, or its origin or destination. Thus,the assumption of constant handling costs for agiven terminal is quite mild. Inventory costs consistprimarily of yard storage costs and the opportunitycost of having a container unavailable for use, be-cause the opportunity cost of capital for in-transitgoods is borne either by the shipper or by the con-signee. Yard storage costs in our model are assumedto be equal for all containers for all locations andtime periods. We assume that customers will acceptdelivery upon arrival at the destination, so no inven-tory is held at the destinations. Generalizations toconsider other linear cost structures and delivery-acceptance rules are straightforward.

We assume that hub delay and transit times aredeterministic, constant across time, and that bothtransit times and delays at the hub are expressed asan integral number of time periods, where a timeperiod is typically one day. In practice, transit times

258 / A. M. NEWMAN AND C. A. YANO

are rarely predictable, but because time is expressedin days, not hours or minutes, there is implicit slackin the schedule. As with airlines and other transpor-tation providers, additional slack in both scheduledtransit times and scheduled hub delays may be builtinto the schedule to help ensure on-time delivery.For instance, one major railroad adjusts scheduleswith the goal of ensuring a 12-hour delivery windowfor a container with a 4-day transit time.

Finally, we assume there is no limit on the num-ber of trains that can be sent each day, although inreality, locomotive and crew availability may be lim-ited with respect to location and time. Our modelcan be generalized to handle limits on the number oftrains, provided train availability is adequate to en-sure on-time delivery, which we enforce as a hardconstraint. In practice, terminal operators mightchoose to delay the shipment of some containers toavoid sending a train with a small number of con-tainers. Our model can be generalized to allow tardydeliveries and associated penalties.

To summarize, our problem is to choose train sched-ules and container routes for each day over a shorthorizon to achieve on-time delivery at minimum cost,where the total cost consists of a fixed charge per train,a variable transportation cost per container, both ofwhich are dependent on the rail segment, handlingcosts per container dependent upon the location, andinventory holding costs for containers held at any ter-minal prior to their arrival at the destination.

Note that our model is intended to aid in establish-ing schedules in the “bottleneck” direction and doesnot address locomotive repositioning. Empty containerrepositioning can be handled as part of the demand.The notation for our model is described below.

Subscripts:

i �index of originsj �index of hubsk �index of destinationst �index of periods in the time horizon, t � 1,

2, . . . , Tl �index of level of service, i.e., the due date of

the container at the destination

Parameters:

�ik �direct transportation time between origin iand destination k

�ij �transportation time between origin i and hubj

�jk �transportation time between hub j and desti-nation k

�j �delay time incurred from passing throughhub j

C �capacity of a train (number of containers)

h �holding cost of a container ($/container/day)cik

a �variable unit cost of transporting a containerdirectly from i to k

cijke �variable unit cost of transporting a container

from i via j to kSik

ao �fixed cost of running a train directly betweenorigin i and destination k

Sijeo �fixed cost of running a train between origin i

and hub jSjk

h �fixed cost of running a train between hub jand destination k

gio �cost of placing a container on the train at

origin igj

h �cost of rearranging a container at hub jgk

d �cost of removing a container from the train atdestination k

biktl �the number of containers that arrive at i onday t bound for k, due at time l

Decision variables:

Iiktlo �container inventory held at i at time t, which

is due at k by time lIijktl

h �container inventory from i, held at j at time t,due at k by time l

Iiktld �container inventory from i due by time l

which is held at k at time txiktl

ao �number of containers shipped directly from ito k at time t, due by time l

xijktleo �number of containers shipped from i to j at

time t, due at k by time lxijktl

h �number of containers from i, shipped at timet from j to k, due by time l

ziktao �the number of trains sent directly from origin

i to destination k at time tzijt

eo �number of trains sent from origin i to hub j attime t

zjkth �number of trains sent from hub j to destina-

tion k at time t

The formulation follows.

�P�

min Z � �iktl

hIiktlo � �

ijktl

hIijktlh � �

ijkl

�w�t��ij

t��ij��j

hxijkwleo

� �iktl

cika xiktl

ao � �ijktl

cijke xijktl

eo

� �iktl

gio xiktl

ao ��ijktl

gio xijktl

eo � �ijktl

gjh xijktl

h

� �iktl

gkd xiktl

ao ��ijktl

gkd xijktl

h � �ikt

Sikao zikt

ao��ijt

Sijeo zijt

eo

� �jkt

Sjkh zjkt

h


subject to

biktl � Iik�t�1�lo � Iiktl

o � xiktlao � �

j

xijktleo @i, k, t, l

(1)

Iijk�t�1�lh � xijk�t��ij��j�l

eo � Iijktlh � xijktl

h

@i, j, k, l, t � t � 1 � � ij � � j (2)

Iik�t�1�ld � xik�t��ik�l

ao � �j

xijk�t��jk�lh � Iiktl

d � biktl

@i, k, l, t � t � 1 � � ik (3)

�l

xiktlao � Czikt

ao @i, k, t (4)

�kl

xijktleo � Czijt

eo @i, j, t (5)

�il

xijktlh � Czjkt

h @j, k, t (6)

All variables restricted to be nonnegativeand integer.

where Iiktlo is set equal to 0 unless t � 1 and l � t �

�ik; Iijktlh is set equal to 0 unless t � 1 � �ij � �j and

l � t � �jk; and Iiktld is set equal to 0 unless t � 1 �

�ik and l � t. Note also that xijktleo � 0 if l � t �

�ij � �jk � �j and xijktlh � 0 if l � t � �jk. These

constraints ensure that appropriate inventory vari-ables are initialized to zero; others are constrainedto be non-negative. Similarly, indirect containershipments are constrained to be zero if due datesnecessitate a direct routing.

In our analysis, we assume only one hub, althoughthe formulation is written for the more general casein which there may be multiple hubs and each con-tainer may pass through at most one hub. We alsoassume that direct travel time between an originand a destination is strictly less than the total tran-sit and delay time for a container shipped indirectly,i.e., �ik � �ij � �jk � �j.

The objective function consists of inventory-hold-ing costs at the origin and at the hub; transportationcosts for directly and indirectly shipped goods; han-dling costs at the origin for both direct and indirectshipments, handling costs at the hub for indirectshipments, and handling costs at the destination fordirect and indirect shipments; and finally, fixedcosts at the origin for direct trains and trains boundfor a hub, and fixed costs at the hub for indirecttrains.

Constraints 1, 2, and 3 represent conservation offlow of containers at the origins, hubs, and destina-tions, respectively. Although we assume that no in-

ventory is held at the destination, the variables Iiktld

are used for bookkeeping purposes to ensure thatdemand is satisfied. To properly reflect that contain-ers are not actually held, we set the correspondinginventory-holding cost coefficients to zero. Con-straints 4 require that, for all origins, destinations,and time periods, the number of containers sent ondirect trains must not exceed the total capacity ofthe trains operating on that segment. Likewise, con-straints 5 and 6 ensure that train capacity is notexceeded on trains bound for the hub and trainsleaving the hub, respectively. Finally, nonnegativityand integrality constraints are imposed on all deci-sion variables.

A typical problem has thousands of general inte-ger variables, and the nature of the tradeoffs con-tributes further to the difficulty of the problem. Di-rect trains are both faster and less expensive thanindirect trains, but they service only one destina-tion. Thus, one faces difficult choices such aswhether to send containers on a relatively full indi-rect train today, or, alternately, to send them withinthe next few days on a direct train that may beunderutilized. Indeed, even when the train scheduleis fixed, the best priority scheme for assigning con-tainers to trains is not evident. In particular, it maynot be optimal to ship containers bound for the samedestination in earliest due-date order.

Our problem has the structure of a multiple-fixed-cost, multicommodity network flow problem. Figure1 depicts such a network for two origins, two desti-nations, and two time periods, assuming transittimes and hub delays are zero for simplicity. Inaddition to a source and a sink, the network containsthree sets of nodes, with each node representing alocation–time period pair, where the location may bean origin, hub, or destination. An arc links twonodes representing different locations if a containermay travel between the locations beginning andending exactly at the time periods associated withthe two nodes. Additional arcs from a location in onetime period to the same location in the subsequenttime period permit inventory flows from period toperiod. The network has a multicommodity struc-ture because flows on the same arc may differ withrespect to their origin, destination, and/or due date,depending upon the specific arc in question. Differ-ent variable costs may be associated with each com-modity on each arc. A commodity is assigned aninfinite cost if its due date prohibits it from travelingon a particular arc. Upper bounds on the total flowon each arc depend on the decisions regarding thenumber of trains between the relevant locations in agiven time period, and a fixed charge is assessed foreach train. Multicommodity (uncapacitated) net-


work flow problems with a fixed-charge structureare known to be NP-complete (GAREY and JOHNSON,1979), and our problem is complicated further by themultiple fixed costs. Although algorithms have beendeveloped for fixed-charge networks, none can pro-vide near-optimal solutions efficiently for our prob-lem. The problem may also be formulated as a singlecommodity network flow with bundle constraintswhose capacities depend upon the train decisions,but the problem is no less difficult to solve in thisform (see AHUJA, MAGNANTI, and ORLIN, 1993).

We attempted to develop solution procedures us-ing Lagrangian relaxation and Benders’ decomposi-tion. Relaxing the train capacity constraints usingLagrange multipliers led to poor results because themultipliers could not accurately capture the step-function nature of the costs associated with the trainvariables. In our application of Benders’ decomposi-tion, the train decisions are included in the masterproblem, and the container flow variables appear inthe subproblem. Although this decomposition ap-peared to be the most natural one, the dual priceinformation from the subproblem is insufficient toaid in selecting better train decisions because of thelarge fixed charge associated with each train. Forfurther details on Lagrangian relaxation and Bend-ers’ decomposition, see NEMHAUSER and WOLSEY(1988), and for further discussion of the applicationof these techniques to our problem, see NEWMAN(1998). Because of the difficulty of adapting tradi-tional techniques to our problem, we developed anew decomposition that takes advantage of the

physical structure of the system and the underlyingnetwork flow problems.

2. NEW DECOMPOSITION TECHNIQUE

OUR DECOMPOSITION APPROACH is motivated, inpart, by the observation that, if the optimal patternof container arrivals at the hub (by origin, destina-tion, arrival date at origin, and due date) wereknown, we could infer which containers were to besent on direct trains. Moreover, for any given pat-tern of container arrivals at the hub, the problemdecomposes into three sets of subproblems:

P0: scheduling direct trains and containers for eachorigin–destination pair;

P1: scheduling trains and containers into the hubfrom each origin; and

P2: scheduling trains and containers from the hubto each destination.

Each of these subproblems has a single origin anda single terminus (the hub or a destination), and,although they remain network flow problems withmultiple fixed charges on each arc, they can besolved optimally in polynomial time under the as-sumptions of our model. See YANO and NEWMAN(1998) for details. For convenience, let us refer to theobjective of P0 for origin i and destination k asZP0

(i, k), the objective of P1 for origin i as ZP1(i) and

the objective of P2 for destination k as ZP2(k).

Let us define B(i, k, t, t, l) as the number ofcontainers that arrive at origin i in period t, arrive

Fig. 1. Multi-commodity network depiction of our problem.


at the hub at period t, and are due at destination kin time period l. Then, assuming instantaneoustravel time for simplicity, the matrix, D, of arrivalsat the origin that must be shipped on direct trains isdefined by the values

Diktl � biktl �t�t

l

B�i, k, t, t, l � @i, k, t, l,

and the aggregate number of containers arriving atthe hub at time t, bound for destination k and due attime l can be represented as

�i, t

B�i, k, t, t, l � @k, t, l.

With these definitions, problem (P) can be re-stated as

minB

� min �i, k

�ZP0�i, k� �D�

� min �i

�ZP1�i� �B� � min �k

�ZP2�k� �B�� .

The matrix B is constrained by the pattern ofarrivals at the origin and the on-time delivery con-straints, and directly influences the matrix D. Theproblem of finding the optimal matrix B is difficult,even without considering the on-time delivery con-straints. Many different B matrices may lead tosimilar solutions for the individual subproblems be-cause each non-urgent container may take one ofseveral routes with similar, or even identical, costs.Moreover, because the total cost is the sum of thecosts of many subproblems, there may be many dif-ferent B matrices that lead to similar overall costs.For example, one solution in which a given originsends many containers indirectly and another sendsmany containers directly may have a similar cost toone in which the allocation of direct and indirect ship-ments is reversed. Our strategy is based on the con-jecture that finding “good” B matrices should providethe foundation for identifying a near-optimal solution.

Rather than searching for good B matrices directly,we solve a problem of the following form, which weterm the “origin scheduling problem” to determine thedirect and indirect train schedules and related con-tainer flows outbound from each origin i:

min �k

ZP0�i, k� � ZP1�i�, (7)

where, in ZP1(i), we assume that for each train in-

bound to the hub from origin i, there is a trainoutbound from the hub whose fixed cost is the de-

mand-weighted average of the costs of trains fromthe hub to the various destinations. Also, the han-dling cost per container is the sum of the handlingcost at the origin and at the hub. Thus, rather thanusing fixed and handling costs that reflect only thefirst transportation segment, we use adjusted coststhat reflect estimates for the entire route:

Sijeo � Sij

eo � Sjh gij

o � gio � gj

h ,

where Sjh is the fixed cost at hub j, obtained by using

an average or a weighted average of Sjkh across des-

tinations.Our motivation for making these cost adjustments

is to incorporate the first-order effects of sendingtrains from the origin to the hub on the costs thatare incurred after the train reaches the hub. In otherwords, the cost adjustment is an estimate of the“cost to go” outbound from the hub. Although thenumber of trains into and out of the hub may not beexactly equal within a short time horizon, in practi-cal applications, these values are fairly well bal-anced. If the train capacities are well utilized in-bound to the hub, on-time delivery requirementsmake it difficult to hold containers at the hub forlong enough to achieve significant additional consol-idation outbound from the hub. Here, we are implic-itly assuming that the capacities of trains inboundto and outbound from the hub are the same. If traincapacities vary by segment, then appropriate adjust-ments can be made in the cost-to-go estimates.

More formally, the origin scheduling problem foreach origin i is

min �ktl

hIiktlo � �

ktl

cika xiktl

ao � �jktl

cijke xijktl

eo

��ktl

gio xiktl

ao � �jktl

gijo xijktl

eo � �kt

Sikao zikt

ao

� �jt

Sijeo zijt

eo

subject to constraints 1, 4, 5, and nonnegativity andintegrality constraints on all variables.

Having solved the above problem for each origin i,we solve P2, the “hub scheduling problem,” for eachdestination k given container flows into the hub, i.e.,xijktl

eo from the origin scheduling problem. Letting cjke

denote the transportation cost between hub j anddestination k, the problem for destination k is

min �ijtl

hIijktlh � �

ijtl

� gjh � gk

d� xijktlh � �

ijtl

cjke xijktl

h

� �jt

Sjkh zjkt

h ,


subject to constraints 2, 6, and nonnegativity andintegrality constraints on all variables. More com-plex methods based on the same general strategyappear in NEWMAN and YANO (2000).

Observe that this solution strategy allows us tosolve each origin scheduling problem independently,and to solve an independent hub scheduling problemfor each destination. By simultaneously consideringdirect and indirect shipments from each origin inconstraint 7 and approximating the cost to go for theindirect shipments, we are able to find solutions that

reflect tradeoffs related to the type (direct or indi-rect), number, and timing of trains to service goodsarriving at the same origin. We expect there will besome loss of optimality from our approximation ofthe cost to go, but we trade this off against the lossof optimality from suboptimal solutions to the orig-inal, monolithic problem. We explore these issuesfurther in Section 6.

Figure 2 depicts the network representation of theorigin scheduling problem for a single origin, a hub,and two destinations for a given train schedule, and

Fig. 2. Network depiction of the origin scheduling problem for a given train schedule.

Fig. 3. Network depiction of the hub scheduling problem for a given train schedule.


Figure 3 depicts the hub scheduling problem for asingle hub–destination pair for a given train sched-ule. The examples have two time periods, two duedates, and both travel times and hub delays areassumed to be zero. Upper and lower bounds oncontainer flows are given in parentheses, and arccosts are given in brackets.

It is important to note that, with the train sched-ules fixed, both the origin and hub scheduling prob-lems can be represented as single-commodity net-work flow problems. In contrast, the originalproblem with a fixed train schedule remains a multi-commodity network flow problem. Thus, our solu-tion strategy takes advantage of not only the directbenefits of the decomposition in creating smallersubproblems, but also the indirect benefits due tothe structure of the resulting subproblems. In par-ticular, the fact that the embedded networks aresingle-commodity network flow problems allows usto relax the integrality constraints on the containerflows when solving the origin and hub schedulingproblems. In the next section, we show how ourmethodology can be modified to provide good solu-tions for larger problems.

3. PREPROCESSING METHOD TO DETERMINEDIRECT TRAINS FOR MANY DESTINATIONS

OUR METHOD TO OBTAIN solutions for problems witha larger number of destinations relies on a heuristicpreprocessing step to set the values of direct trainvariables in the origin scheduling subproblems. Ourrationale for heuristically setting the direct trainvariables (to reduce the size of the remaining prob-lem) is that these decisions depend primarily uponthe demand between a single origin and a singledestination, and are only indirectly affected by whenand how containers are sent to other destinations.Moreover, the primary indirect effect can be cap-tured largely in the flows of containers sent via thehub from the designated origin to all other destina-tions. Our preprocessing method is motivated bythese observations.

For each origin, the preprocessing step proceedsas follows. We construct K different subproblems,where K is the number of destinations. In the kthsubproblem, k � 1, . . . , K, we partition the set ofdestinations into two groups: a single destination, k,and the remaining K � 1 destinations, which weaggregate into a “super-destination.” Demands areaggregated across destinations within the super-destinations, taking into account differences intravel times. (In effect, demands with the same lat-est departure dates are grouped together.) Weightedaverage fixed and variable costs are assessed for the

direct and indirect routes between the origin and theaggregated destination.

This problem is now treated as an origin schedul-ing problem with two destinations. Direct and indi-rect train schedules for both the single (kth) and theaggregated destination are derived, along with thecorresponding container routing schemes, but onlythe direct train schedule for the kth destination isretained. Therefore, at the end of this preprocessingstep, for each origin, we have established directtrain schedules for all K destinations. Having setthe direct train variables for all origin–destinationpairs in the preprocessing step, we solve the originscheduling problems to determine indirect trainschedules and all container flows.

This procedure generally will not provide an opti-mal solution to the original origin scheduling prob-lem because the direct train schedules are deter-mined without full consideration of the details of theindirect train schedules. However, recall that theorigin scheduling problem is an approximation initself. From the viewpoint of solving the originalproblem, it would appear that there is greater loss ofoptimality from decoupling the origins to create theorigin scheduling problems (and from the inability ofcommercial software to find an optimal solution tothe original origin scheduling problem) than there isfrom the use of this aggregation procedure in solvingthe individual origin scheduling problems. More-over, because this preprocessing step entails collaps-ing a many-destination problem into a two-destina-tion problem, it can be used even when the numberof destinations is large.

In the next section, we develop valid inequalitiesto obtain tight lower bounds for the original model.

4. LOWER BOUNDS

TO EVALUATE THE performance of our procedure, weuse lower bounds as one type of benchmark. Thelower bounds provided by commercial software arebased on linear programming relaxations, which arepoor because they ignore the fixed-charge nature ofthe train costs. These bounds can be improved con-siderably by adding valid inequalities (cuts) to themonolithic problem. These cuts are used to tightenthe lower bound on the original problem (but do notnecessarily yield an improved integer solution).

The cuts pertain to the minimum number of trainsrequired to service certain subsets of the demand.Because of the substitutability of trains across timeto service non-urgent containers, it is difficult toobtain tight lower bounds on the number of trains onany segment in any time period. We can, however,obtain fairly tight lower bounds on the following:


i. the total number of trains outbound from eachorigin to each destination during the horizon(single origin–single destination constraint type1, or sosd1);

ii. the total number of trains outbound from eachorigin to all destinations (collectively) during thehorizon (single origin–all destinations, or soad);

iii. the total number of trains inbound to each des-tination from each origin during the horizon(single origin–single destination constraint type2, or sosd2); and

iv. the total number of trains from all origins in-bound to each destination during the horizon (allorigins–single destination, or aosd).

Let Mik, Mi, and Mk be the lower bounds on thenumber of trains sent during the horizon associatedwith the origin–destination pair (i, k), with an ori-gin i, and with a destination k, respectively. Thefour types of cuts are stated algebraically as:

�t�1

T

ziktao � �

t�1

T �j

zijteo � Mik @i, k, (sosd1)

�t�1

T �k

ziktao � �

t�1

T �j

zijteo � Mi @i, (soad)

�t�1

T

ziktao � �

t�1

T �j

zjkth � Mik @i, k, (sosd2)

�t�1

T �i

ziktao � �

t�1

T �j

zjkth � Mk @k. (aosd)

To obtain values of Mik, Mi, and Mk for all i andk, we introduce the concept of a “supertrain,” a fic-titious train type with the advantages of both directand indirect trains. A supertrain emanating fromthe origin can deliver shipments to any destination,giving it the geographical consolidation advantage ofindirect trains. However, like direct trains, it incursneither the cost nor time delays associated withpassing through a hub. Travel time to any destina-tion is the direct travel time for the origin–destina-tion pair. To obtain lower bounds on the number oftrains into each destination (sosd2 and aosd), wereverse the network and reindex time appropriately.

Before solving each supertrain problem, we firstdetermine whether there are any demands that ne-cessitate direct trains. We set the corresponding di-rect train variables to 1 (or to a value large enoughto accommodate these demands, if more than onetrain is needed for an origin–destination pair). Wethen solve the problem of finding a supertrain sched-

ule and container flows on both the supertrains andpre-set direct trains that minimize the total numberof supertrains outbound from the origin while satis-fying on-time delivery requirements. To establishthe value for Mi, we solve the supertrain problem fora single origin and all destinations. To establish thevalue for Mk, we solve the supertrain problem on areversed version of the network and consider allorigins and a single destination. To establish thevalue for Mik, we solve the supertrain problem forjust a single origin–destination pair on either theoriginal or the reversed network.

We now demonstrate that the optimal objectivevalue for the supertrain problem (which has theobjective of minimizing the number of trains) is alower bound on the optimal number of trains for thecost-minimization (i.e., the original) problem. Notethat the assumptions in the supertrain problem leadto differences in the constraints, not just in theobjective function. Therefore, for completeness, wedemonstrate this result formally.

We consider the train schedule component of theoptimal solution to the cost minimization problem,say {zikt

ao�, zijteo�, zjkt

h� }. From this schedule, we showhow to construct a feasible solution to the supertrainproblem with the same number of trains. Becausethe optimal solution to the supertrain problem has anequal or fewer number of trains than any feasiblesolution to that problem, the optimal solution to thesupertrain problem provides a valid lower bound onthe number of trains in the cost minimization problem.

We now show how to construct a feasible solutionto the supertrain problem from {zikt


h� }. Letzit

o be the number of supertrains departing fromorigin i at time t and zkt

d be the number of super-trains arriving at destination k at time t. Thesesupertrains will substitute for both direct and indi-rect trains in the original minimum-cost solution.First, substitute a supertrain for each direct trainoutbound from origin i in the minimum cost solutionand retain the container assignments. The super-train has the same transit time as a direct train, andthus satisfies on-time requirements for the contain-ers assigned to that train. Next, substitute a super-train for each indirect train outbound from origin iin the minimum cost solution, again retaining thecontainer assignments. The supertrain also has thesame transit time as a direct train for any destina-tion, and thus also satisfies on-time requirementsfor the containers on that train.

We can now set zito � k zikt

ao� � j zijteo�. The analysis

for trains inbound to destination k parallels the anal-ysis above, and we can set zkt

d � i ziktao� � j zjkt

h�. Thus,we have constructed a feasible solution to the super-train problem from {zikt


h�} that has the same


number of trains as in the minimum cost solution.Consequently, the minimum number of trains in thesupertrain problem provides a lower bound on thenumber of trains in the minimum cost problem.

5. SIMPLE HEURISTICS

WE DEVISE TWO heuristics that are designed tomimic current operating policies. These simple heu-ristics provide additional benchmarks and allow usto estimate the potential savings from using ourprocedure. At the intermodal operation that moti-vated our research, most, if not all, containers aresent via a hub, which leads to some late deliveries.Because we require on-time delivery in our model,these two simple heuristics include the provision fordirect shipments when required to satisfy on-timedelivery requirements.

In both heuristics, all containers are sent out as soonas possible after they arrive at the origin or at the hub.Containers requiring expedited service are sent ondirect trains. Non-urgent containers are assigned todirect trains (which have lower transportation andhandling costs) to the extent space is available. Then,all remaining non-urgent containers are assigned toindirect trains. Thus, the heuristics need only specifythe number of trains of each type.

The main difference between the two heuristics liesin the conditions for sending direct trains, beyond theminimum required to service expedited containers. InHeuristic 1, direct trains also are sent when there areenough containers to fill a train, even if direct serviceis not required. We also allow up to one additionaldirect train with non-urgent containers for each orig-in–destination pair, provided it is at least full, 0 � � 1. In practice, is a parameter determined bymanagement, taking into account the tradeoff be-tween the opportunity cost of operating a less-than-fulldirect train and the additional costs for containers sentthrough the hub. In Heuristic 2, only necessary directtrains are sent; no trains that contain exclusively non-urgent containers are sent directly. The first heuristicattempts to minimize costs by sending as many con-tainers as possible, or as practicable, on direct trains(which have lower costs). The second heuristic foregoesthis opportunity, emphasizing instead the opportunityfor consolidation on trains traveling into and out of thehub.

We now describe Heuristic 1 in more detail. Foreach origin–destination pair, compute the numberof direct trains:

Set ziktao � �

l�t��ik

t��ij��jk��j�1

biktl

C � y @i, k, t,

where y represents the smallest number of addi-tional direct trains to be sent other than those car-rying containers requiring expedited service, and isthe smallest nonnegative integer such that

� �l�t��ij��jk��j

biktl � C � ziktao �

l�t��ik


biktl� � �

� � C � � y 1� � C.

The corresponding number of containers that aresent directly is given as

xiktlao � biktl @i, k, t, t � � ik � l � t � � ij � � jk � � j ,

xiktlao � biktl @i, k, t, l � t � � ij � � jk � � j ,

where biktl � biktl @i, k, t, l � t � �ij � �jk � �jcontainers not requiring expedited service are cho-sen for direct shipment because either they fill adirect train (with or without expedited containers),or they constitute at most one train not carrying anyexpedited containers, but which is at least full.

Then, for each origin and time period, compute thenumber of indirect trains needed to service the re-maining containers not sent on a direct train:

Set zijteo �

� �k� �

l�t��ij��jk��j

biktl

� C � ziktao �

l�t��ik


biktl� � � �

C @i, j, t.

For the case of a single hub, the correspondingnumber of containers sent indirectly to the hub isgiven as

xijktleo � biktl biktl

@i, k, t, l � t � � ij � � jk � � j , and j � 1.

Compute the number of containers from each or-igin i ready to depart the hub for destination k attime t � �ij � �j as

�l

xijk�t��ij��j�lh

� � �l�t��ij��jk��j

biktl � C � ziktao �

l�t��ik


biktl� � �

@i, k, t, and j � 1.


Finally, for each destination and time period, com-pute the number of trains needed to accommodatethe number of containers ready to depart from thehub:

Set zjk�t��ij��j�h �

�i

�l�t��ij��jk��j

xijk�t��ij��j�lh

C @k, t, and j � 1.

Heuristic 2 can be described as follows: Send adirect train between an origin and a destinationwhen necessitated by the due dates of containers.After the containers requiring urgent service havebeen given priority, fill the remaining space withnon-urgent containers in any order (which we justifybelow). Send as many indirect trains from the originas necessary each day to ship all remaining contain-ers (i.e., those not sent directly). Send indirect trainsfrom the hub to each destination each day to ship thecontainers arriving at the hub.

Any allocation of remaining non-urgent contain-ers to indirect trains is feasible and has the samecost. Feasibility follows because the containers canbe sent on either train type. Total variable costs areequal because of our cost structure. Total fixed costis equal because the number of daily direct trainsdepends only on pre-determined demand, and thenumber of daily indirect trains is independent of theslack in the non-urgent containers’ schedules.

6. NUMERICAL RESULTS

WE GENERATED 30 problems with one hub, three tosix origins and destinations, and with different con-tainer-demand patterns and cost structures. Wesummarize problem characteristics in Tables I andII. All problems have eight time periods in whichcontainers become available at the various origins.All trains have a capacity of 200 containers. Con-tainer demand was generated for each origin–desti-nation–arrival time–due date combination with a

probability of 0.55 of being randomly generated froma discrete uniform distribution between 10 and 65,and a probability of 0.45 of being 0. Scenarios inwhich less expedited service is demanded are gen-erated as described above, except that, for each orig-in–destination–arrival time–due date combinationsuch that the shipment necessitated transport viadirect train (i.e., t � �ik � l � �ij � �jk � �j),demands that were originally positive are indepen-dently set to zero with probability 0.4–0.5.

Table II gives ranges of values for container ar-rival rates and the cost parameters. Industry datasuggest that fixed and variable transportation costsfor shipping a full train are approximately equal. Weset the fixed charge associated with each train to beproportional to the distance, based on our observa-tion that train-operator labor constitutes the major-ity of this cost. Handling costs per container arebased on the hourly wage of yard operators and theapproximate time needed to load, unload, or reposi-tion a container. The yard storage cost per containerper day is assigned a small value that provides in-centive to ship earlier rather than later, all elsebeing equal. For the first heuristic, we set � 0.65for all problems.

The problems were solved on a Sun SparcStation20 with 128 megabytes of RAM using CPLEX 6.0.We obtained solutions and lower bounds for themonolithic problem by using CPLEX (with a timelimit of 9000 seconds), with and without the validinequalities discussed in Section 4. We also obtainedsolutions using our decomposition procedure and thetwo simple heuristics. For all executions of theCPLEX software, we use depth-first search, andstrong branching, i.e., the branching variable is se-lected whose resolution is most likely to yield thegreatest improvement in the objective functionvalue. Additionally, we had CPLEX implement itsbuilt-in heuristic to find integer solutions. This com-bination of rules provided the best results for our setof problems. We also implemented a priority branch-ing scheme, but it did not lead to significant perfor-mance improvement.

TABLE ITest Problem Characteristics

Problems

Number ofOrigins–HubsDestinations

Relative ProportionExpedited Service

Demanded

1–5 3–1–3 �20%6–10 3–1–3 �10%

11–13 3–1–4 �20%14–15 3–1–4 �10%16–18 4–1–3 �20%19–20 4–1–3 �10%21–25 6–1–6 �20%26–30 6–1–6 �10%

TABLE IIParameters for Test Problem Instances

ParameterRange Used inTest Problems

Container arrival rate per day 0–65Fixed cost at origin (direct train) ($/train) 11000–15000Fixed cost at origin (indirect train) ($/train) 5900–8500Fixed cost at hub ($/train) 6300–8500Transportation cost ($/container) 40–100Handling cost (all locations) ($/container) 1–2Inventory holding cost ($/container/day) 1.5–2


Although the valid inequalities were quite effec-tive for improving the lower bounds (as we discuss inmore detail later), we found them to be ineffectivefor improving the best integer solution found prior tothe time limit or for reducing overall computationaleffort. One reason is that the cuts specify bounds onthe total number of trains outbound from an originor inbound to a destination, and thus also on thesum of fixed costs associated with the trains. Withsuch cuts, we are able to get much larger lowerbounds (on costs) at all levels of the search tree, andthus, also, a tighter lower bound upon termination.In contrast, such aggregate constraints provide littleguidance in the search process. Meanwhile, modestcomputational effort must be expended to computethe right-hand-side values for the valid inequalities,and the introduction of the valid inequalities slowsthe overall execution of CPLEX in solving the prob-lem, often resulting in inferior solutions. For thisreason, in what follows, we report solutions andCPU times for the monolithic problem without validinequalities.

Results appear in Table III. All results are re-

ported as the ratio of the relevant objective value tothat of the objective value from the decompositionprocedure. As shown in the second and third col-umns, the simple heuristics yield solutions that areabout 12% more costly than those found using ourdecomposition technique. The results demonstratethat significant cost savings can be realized from ourproposed procedures over systematic but simplerheuristics. Results in the fourth column show thatour decomposition procedure yields an average im-provement of approximately 1% versus the best in-teger solution obtainable from a straightforward im-plementation of CPLEX with a 9000-second timelimitation. In 6 of the 30 problems, the objectivevalue from our decomposition approach is 1–3%above that of the straightforward CPLEX solution,but, as we will show later, this small loss of solutionquality comes with a significant reduction in CPUtime. Moreover, some or all of this loss can be re-gained by using refinements of our basic approachthat require very little computing effort (see New-man and Yano, 2000).

The fifth and sixth columns report the ratio of thedecentralized objective value to lower bounds, wherethe lower bounds are obtained by solving the mono-lithic problem with and without the valid inequali-ties (discussed in Section 4), respectively. It is evi-dent that the addition of the valid inequalitiesimproves the bounds substantially. Furthermore,the tighter bounds demonstrate that our decompo-sition procedure performs well in an absolute sense.On average, the solutions from the decompositionprocedure are within 6.6% of the correspondinglower bounds, which is remarkable considering thestrong role of the fixed costs in our problem and thefact that the bounds are (still) based on linear pro-gramming relaxations. The bounds are tighter ininstances for which more expedited service is re-quired because many direct train variables must beset to 1 (or more). This, in turn, makes the validinequalities effectively tighter. By contrast, thestraightforward application of CPLEX yields boundsthat average 18% less than the corresponding objec-tive function values from the decomposition proce-dure.

Table IV contains CPU times that we report intwo different ways. The third column reports thesum of the CPU times for all origin scheduling sub-problems and the hub scheduling subproblems. Thefourth column reports the elapsed time that wouldbe required if the origin scheduling problems couldbe solved in parallel, and, subsequently, the hubscheduling problems could be solved in parallel. Thesolution times from the monolithic problem are re-ported as the time at which the best integer solution

TABLE IIIObjective Values and Lower Bounds as a Ratio of the Objective

from the Decomposition Procedure

ProblemHeuristic

1Heuristic

2

CPLEXBest

IntegerSolution

OurLowerBound

CPLEXLowerBound

1 1.12 1.12 1.02 0.98 0.802 1.12 1.12 1.00 0.98 0.803 1.14 1.14 1.01 0.96 0.854 1.17 1.17 1.02 0.99 0.815 1.15 1.15 1.00 0.97 0.866 1.14 1.15 1.01 0.93 0.857 1.14 1.15 1.00 0.90 0.818 1.11 1.12 1.03 0.92 0.869 1.19 1.19 1.03 0.92 0.85

10 1.19 1.18 1.03 0.94 0.8411 1.09 1.09 1.01 0.94 0.8212 1.13 1.13 1.01 0.97 0.8013 1.15 1.16 1.01 0.96 0.8414 1.16 1.16 1.04 0.91 0.8215 1.15 1.16 1.01 0.90 0.8316 1.10 1.10 1.01 0.96 0.7917 1.18 1.18 1.01 0.97 0.8118 1.11 1.12 1.00 0.96 0.8419 1.19 1.19 1.03 0.93 0.8220 1.22 1.24 1.02 0.93 0.8421 1.05 1.05 0.99 0.91 0.7922 1.06 1.06 0.98 0.92 0.7923 1.08 1.08 0.99 0.93 0.8124 1.08 1.08 1.00 0.94 0.8025 1.06 1.06 1.00 0.93 0.7926 1.07 1.05 0.99 0.90 0.8327 1.08 1.06 1.00 0.90 0.8228 1.07 1.06 1.00 0.89 0.8129 1.08 1.07 0.97 0.87 0.8330 1.08 1.08 0.99 0.90 0.83


is identified within the preset time limit of 9000seconds. In all cases, the 9000-second time limit isreached without a verified optimal solution. For vir-tually all of the problems, the CPU time for thedecomposition procedure is a small fraction of thatrequired to first identify the best integer solutionfound within the 9000-second time limit, and thelatter times could be achieved in practice only if onewere clairvoyant about the best time to terminatethe search.

The majority of the CPU time is associated withsolving the preprocessing step required for thelarger origin scheduling subproblems (i.e., thosewith six destinations in our data sets). This prepro-cessing step is executed (optimally) in about oneminute or less, on average, for our problems, but, asthe problem size grows, a greater number of theseproblems must be solved. Although the CPU timesfor serial processing increase with the number oforigins and destinations, parallel run times remain

very modest. The computing effort is dominated bythe origin scheduling problems; the solution timesfor the hub scheduling problems are negligible inalmost all instances. In general, problems with moreexpedited service are solved more quickly, as certaindirect-train variables can be fixed (or are set inadvance in the CPLEX pre-solve operation).

7. CONCLUSIONS

WE HAVE ADDRESSED the problem of simultaneouslydetermining train scheduling and container routingdecisions in a rail intermodal setting. We have de-veloped a decomposition procedure that takes ad-vantage of the embedded network structure, andyields near-optimal solutions in less than one-thirdthe time of commercial optimization software. Wehave also developed methods for obtaining tightlower bounds using valid inequalities. The entiredecomposition procedure yields solutions with objec-tive function values about 12% lower, on average,than those obtained with the simple heuristics de-scribed in Section 5. Managerially, our procedurehas the advantage of allowing decisions to be madelocally at each terminal while achieving, on average,better performance than the best solutions obtainedfrom commercial software for the monolithic prob-lem.

For problems with three or four origins and des-tinations, we are able to obtain optimal solutions forthe origin scheduling subproblems in our decompo-sition procedure. It is difficult to obtain (verified)optimal solutions to these subproblems (which mustconsider all destinations simultaneously) when thenumber of destinations grows larger. To deal withsuch situations, we have developed a variation of ourdecomposition method that relies on a preprocessingstep to set certain direct-train variables heuristi-cally before the remainder of the problem is solved.For a network with a single hub, this preprocessingstep is effective in reducing the size of the remainingorigin scheduling subproblems, allowing us to solvethem to optimality (for problems of the size consid-ered in our experiments). The method for construct-ing valid inequalities described in Section 4 providesmuch tighter lower bounds than those provided by astraightforward implementation of CPLEX anddemonstrates that our decomposition procedureyields solutions within 6.6% of the optimum, on av-erage.

Future research is needed to capture other prac-tical aspects of the problem. We mentioned earlierthat, in practice, some containers may be shippedlate to avoid sending a train with only a few contain-ers. Our model can be generalized to allow tardy

TABLE IVCPU Time Performance

Problem

Number ofOrigins–HubsDestinations

Serial Run Time(Decentralized

Approach)(sec.)

Parallel Run Time(Decentralized

Approach)(sec.)

CentralizedApproach

(sec.)*

1 3–1–3 † † 1825‡

2 3–1–3 † † 9000‡

3 3–1–3 † † 6802‡

4 3–1–3 † † 6051‡

5 3–1–3 † † 4605‡

6 3–1–3 † † 7892‡

7 3–1–3 142 142 8622‡

8 3–1–3 103 103 5662‡

9 3–1–3 † † 192‡

10 3–1–3 † † 5008‡

11 3–1–4 250 250 3824‡

12 3–1–4 97 97 1266‡

13 3–1–4 † † 534‡

14 3–1–4 517 290 2258‡

15 3–1–4 2112 1800 4700‡

16 4–1–3 † † 3008‡

17 4–1–3 † † 683‡

18 4–1–3 † † 497‡

19 4–1–3 † † 2910‡

20 4–1–3 † † 2800‡

21 6–1–6 972 221 1295‡

22 6–1–6 158 69 6469‡

23 6–1–6 114 69 1223‡

24 6–1–6 736 139 5870‡

25 6–1–6 810 113 1185‡

26 6–1–6 2492 355 2642‡

27 6–1–6 1019 226 6383‡

28 6–1–6 912 896 1000‡

29 6–1–6 536 207 5498‡

30 6–1–6 670 111 8708‡

*Indicates time at which best integer solution is first identi-fied.

†Indicates CPU time is less than five seconds.‡Time limit of 9000 seconds is reached.


deliveries (with tardiness penalties). Other general-izations include incorporation of hub capacity forcontainer handling, and networks with multiplehubs. There may be parallel hubs, with each ship-ment passing through (at most) one hub, or serialhubs, with shipments passing through one or morehubs en route to their destinations.

In the case of parallel hubs, the origin schedulingproblem is similar to that for the single hub case; thedecisions outbound from the origin include directtrain schedules and corresponding container ship-ments to each destination, and indirect train sched-ules and corresponding container movements toeach of several hubs. Estimates of the cost to go canbe made in much the same way as for a single-hubproblem. Once the origin scheduling problems aresolved, a hub scheduling problem is solved for eachhub–destination pair, because these problems canbe decoupled once the arrivals to the hubs areknown.

For scenarios involving two or more hubs in se-ries, intermediate hub scheduling subproblemsmust be solved sequentially starting from the ori-gins and moving toward the destinations. Becausethere may be several rail segments on a path be-tween an origin and destination, accurately estimat-ing the cost to go may be more difficult. Preliminarycomputational studies yield good results even with-out highly accurate cost-to-go estimates. Methodsfor obtaining valid inequalities for the single hubcase can be extended in a straightforward mannerfor multiple (parallel or serial) hub scenarios.

We believe that the decomposition concepts pro-posed here provide an opportunity for considerablecost reduction while maintaining the advantages oflocal decision-making. The benefits derive from look-ing ahead, not only with respect to downstreamfreight flows, but also with respect to forecastedfuture demands and opportunities to consolidateshipments across time.

ACKNOWLEDGMENTS

THIS WORK HAS BEEN supported in part by NationalScience Foundation Grant GER/HRD 93-96288 tothe University of California, Berkeley, and by fund-ing from the United States Department of Transpor-tation and the California Department of Transpor-tation, awarded by the University of CaliforniaTransportation Center. We also appreciate com-ments of the referees and Editor on an earlier ver-sion of this paper.

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(Received: October 1998; revisions received: June 1999, October1999; accepted: October 1999)


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