Transportation Research Part B 46 (2012) 100–119

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Transportation Research Part B

journal homepage: www.elsevier .com/ locate / t rb

Integrated operations planning and revenue management for railfreight transportation

Benoit Crevier a,b, Jean-François Cordeau a,⇑, Gilles Savard b

a Canada Research Chair in Logistics and Transportation and CIRRELT, HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7b Département de mathématiques et de génie industriel, École Polytechnique de Montréal, C.P. 6079, succursale Centre-Ville, Montréal, Canada H3C 3A7

a r t i c l e i n f o

Article history:Received 15 October 2010Received in revised form 3 September 2011Accepted 4 September 2011

Keywords:Rail freight transportationRevenue managementOperations planningPricingBilevel optimization

0191-2615/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.trb.2011.09.002

⇑ Corresponding author. Fax: +1 514 340 6834.E-mail address: jean-francois.cordeau@hec.ca (J.

a b s t r a c t

In the rail industry, profit maximization relies heavily on the integration of logistics activ-ities with an improved management of revenues. The operational policies chosen by thecarrier have an important impact on the network yield and thus on global profitability. Thispaper bridges the gap between railroad operations planning and revenue management. Wepropose a new bilevel mathematical formulation which encompasses pricing decisions andnetwork planning policies such as car blocking and routing as well as train make-up andscheduling. An exact solution approach based on a mixed integer formulation adapted tothe problem structure is presented, and computational results are reported on randomlygenerated instances.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

For many years, service companies have realized that a better management of their logistics operations can improve theirstrategic position on the market. The revenue aspect of day-to-day activities is of greater importance nowadays since manyindustries have noticed that profit maximization relies heavily on the integration of logistics activities with an improvedmanagement of revenues. That is why revenue management is becoming more and more vital to many industries such astelecommunications, hospitality and transportation (Talluri and van Ryzin, 2005).

In this paper, we analyze rail freight transportation which is an important economic sector. In 2007, the major NorthAmerican rail freight companies generated approximately 67.4 billion $US in revenues (Association of American Railroads,2009). Over time, competition has evolved to the point where market and other transportation modes have a predominantimpact over the policies of the rail industry. In the United States, for example, the Staggers Rail Act of 1980 gave more free-dom to the rail companies while competition acquired a greater regulation role over the tariffs set for the transportation offreight. As a response to this pressure from the market, rail carriers were forced to review their business management pro-cesses and pricing policies in order to be competitive (Wilson and Burton, 2003).

With the evolution of modeling and optimization techniques, many industries have updated their operations planning totake advantage of these innovations by establishing a better management of their activities. However, rail operations aremade up of complex interrelated policies such as car blocking, train scheduling, make-up and routing, yard management,locomotive assignment, empty car distribution and crew scheduling. These are the main reasons why rail transportationhas always been confronted to operations management tools which did not encompass all the realism of day-to-day

. All rights reserved.

-F. Cordeau).

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 101

operations. Part of the complexity also follows from the fact that railroads operate on an existing physical network thatimposes constraints of its own.

Service companies are also trying to include revenue management in their decision support systems. The success of thisintegration in the airline industry has encouraged others to follow in the same footsteps. The rail industry is characterized bycustomers with distinct attributes and perception of the proposed services (for example, the sensitivity to tariffs or servicereliability). Furthermore, rail transportation has more flexibility over capacity utilization and traffic speed through the linksof the network than other transportation modes. Moreover, a majority of companies are switching from tonnage-based toscheduled operations. This philosophy increases the reliability of the service from a customer’s perspective but imposesnew constraints on the carrier. These are market conditions, according to Talluri and van Ryzin (2005), where the use of rev-enue management is likely to be beneficial. The analysis of an integrated approach combining operations and revenue man-agement is therefore an interesting perspective of research in an attempt to study the connections between networkplanning and pricing policies.

The integration of revenue management and rail freight operations planning has not been considered very frequently inthe literature even though a growing interest by rail freight carriers for research on this topic can be observed. In this paper,we introduce a new bilevel mathematical formulation which combines both pricing decisions and network planning policies.Two pricing policies and their impact on the formulation are analyzed as well as resulting properties and valid inequalitieswhich are used to strengthen the mathematical model. An exact solution approach is proposed, and computational resultsare reported on randomly generated instances.

The remainder of the paper is organized as follows. Section 2 presents a description of the main aspects of rail operationsand revenue management. In Section 3, we provide a definition of the problem and introduce a path-based mathematicalformulation using specific networks which are also presented. Section 4 describes the reformulation of the problem as amixed integer mathematical program. Section 5 studies the impact of the pricing policies on the formulation. Resulting prop-erties and valid inequalities are also addressed. Section 6 shows how we generated the instances on which the solutionmethodology was tested. Finally, computational experiments are presented in Section 7, and this is followed by theconclusions.

2. Background

The complex operations of rail transportation have been, for a long time, limiting the implementation of state-of-the-artmethodologies and techniques designed to offer efficient decision support systems. Nevertheless, recent applications haveshown the potential gains that can result from the use of analysis tools such as mathematical optimization. In this perspec-tive, this paper proposes a model that bridges the gap between operations planning and revenue management decisionswhich are made, most of the time, independently. However, crew scheduling and management will not be treated here,but the interested reader is referred to the paper of Ernst et al. (2004) as well as those of Caprara et al. (1997, 1998, 1999).

2.1. Rail operations management

Rail carriers face numerous operational decisions. As demand is shipped locally to yards, blocking and car routing plansmust be designed. The blocking plan problem is concerned with the determination of the set of blocks that will be assembledat each yard. A block is a group of cars that are moved together, on one or more trains, from a common origin to a commondestination. Cars in a block may pass at intermediate yards but they will not be reclassified until the block has reached itsdestination. This destination might represent the end point of the itinerary of some cars or an intermediate stop in the routeof others that will be assigned to subsequent blocks. These activities must be performed and therefore planned at the dif-ferent yards according to the classification capacity. Yard management is consequently a central problem encountered inthe rail industry. Decisions are also made towards train routing, make-up and scheduling. The make-up policies specify theassignment of blocks to trains. These operating decisions will form the trip plan of each car which defines the itinerary thatwill be followed from origin to destination. The locomotive assignment problem specifies the management of locomotives inorder to satisfy pulling power requirements and repositioning for future demands. At destination, cars are released andempty car management decisions are made to redistribute cars through the network to efficiently respond to later requests.The size and complexity of the problems mentioned above prevent the solution of a totally integrated model. As a result, theglobal problem is usually decomposed and subproblems are tackled sequentially. Obviously, such an approach leads to asuboptimal operating plan. The integration of some of these problems has therefore been studied. The industry also triesto come up with decision support tools aimed at improving the global management of activities. Huntley et al. (1995),Ferreira (1997) and Ireland et al. (2004) have analyzed the obstacles and possible gains of implementing optimization toolsin the day-to-day activities of a rail carrier.

Rail operations have been studied extensively in the operations research literature and the main modeling and method-ological contributions have been surveyed by Assad (1980, 1981), Haghani (1987), Cordeau et al. (1998) and Newman et al.(2002). More recently, operational research methodologies in rail passenger transportation have been explored by Huismanet al. (2005). If we look more closely at integrated rail operations management contributions, a tactical level formulationcombining train service scheduling, car transit, blocking and make-up policies as well as yard classification work allocation

102 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

was introduced by Crainic et al. (1984) who proposed a heuristic method to minimize operating costs and delays. This ap-proach was later generalized to trucking and intermodal container transportation by Crainic and Rousseau (1986). Haghani(1989) has studied the movements of empty and loaded cars and trains as well as make-up policies. The problem was for-mulated as a time-space multi-commodity network problem and a heuristic approach was devised. Keaton (1989, 1992) hasconsidered train connections, service frequency, blocking policies and car movements. A Lagrangian relaxation approach wasfirst developed by Keaton (1989) and an extension of these results was proposed by Keaton (1992). An intermodal operationsplanning model was presented by Nozick and Morlok (1997). A heuristic methodology was proposed where the authors ana-lyze the linear relaxation solution of an integer programming model based on a time-space network. Train scheduling andtraffic assignment was studied by Gorman (1998a,b). An integer program, based on a 1-week horizon, is presented. Themethodology relies on a hybrid genetic and tabu search algorithm.

2.2. Rail revenue management

The revenue management aspects of railroad planning are of great importance today because of the increasingly compet-itive environment. Armstrong and Meissner (2010) provide a good overview of railway revenue management for both freightand passenger transportation. As mentioned previously, the rail industry relies on heterogeneous characteristics definingcustomer commodities and expectations towards the offered service. An appropriate segmentation can therefore be estab-lished accordingly. Pricing and capacity management policies can be elaborated to exploit that segmentation and a potentialcontrol or regulation of the demand can thus be obtained. As a consequence, carriers may possibly smooth demand, reduceglobal network congestion and make better use of assets. For most companies, access to a reliable transportation system rep-resents a key element in the selection of a carrier. In turn, the carrier aims at increasing service reliability by a more efficientmanagement of the operations coupled with adequate demand management policies in order to maximize profits. Revenuemanagement tries to tackle these problems. Service differentiation through market segmentation is a well known marketingapproach to provide customers with appropriate service according to their characteristics and sensitivity to service attri-butes. Rail service reliability analyses and the positive impact of a traffic segmentation in service or priority classes werepresented, among others, by Kwon (1994) and Kraft (1995).

Research in revenue management addresses problems such as overbooking, capacity management or pricing, notably inthe airline industry. McGill and van Ryzin (1999) have presented an overview of the main contributions. In the quest tomerge revenue management and operations planning, service companies are more and more interested in integrated capac-ity management and pricing decision support tools. Kimes (1989), Weatherford and Bodily (1992) and Talluri and van Ryzin(2005) have described favorable conditions for the practice of revenue management. Strasser (1996) and Kraft et al. (2000)have analyzed for their part the potential impacts of revenue management in rail transportation. However, very few publi-cations are concerned with such applications. In freight transportation in general, we note the papers of Powell et al. (1988)in the trucking industry, Maragos (1994) in maritime transportation as well as Kasilingam (1996) and Sobie (2000) in theairline industry. In rail transportation, revenue management has been mostly applied in the context of passenger transpor-tation. Kraft et al. (2000) as well as Johnston (2006) exposed the characteristics of policies designed by Amtrak to control seatand cabin sales according to booking class and market structure. Ben-Khedher et al. (1998) have described the implemen-tation of a strategic scheduling and planning system and a tactical capacity management system for high-speed train ser-vices at the Société Nationale des Chemins de Fer (SNCF) in France. Models were presented by Ciancimino et al. (1999)for setting booking limits on every origin–destination pair and maximizing expected profit. More recently, Côté and Riss(2006) and Riss et al. (2008) have described the development, at SNCF and Thalys, of tools designed to analyze optimalpricing policies, the marketing of products offered and the dynamic management of seat inventories. The competitiveenvironment is also considered.

In rail freight transportation, capacity management was considered by Harker and Hong (1994). A track pricing problemis presented for which different markets compete for accessing their ideal track utilization schedule. Kraft (1998, 2002)developed a methodology, based in part on a bid price approach, to jointly analyze the service offer and car movementsto maximize profits. For their part, authors have dealt with some aspects of revenue management in intermodal transpor-tation. Yan et al. (1995) have studied a problem where the main objective was the evaluation of the opportunity cost of de-mands and the corresponding pricing policies. A system of booking and revenue management was presented by Campbell(1996) who extended the results of Belobaba (1987, 1989) to create a methodology capable of analyzing the postponementof low priority loads to favor the movements of those with higher priority. Finally, Gorman (2001, 2005) studied the pricingpolicies for intermodal operations at the Burlington Northern and Santa Fe (BNSF) Railway. The integration of pricing andoperations management was studied by Li and Tayur (2005) in the intermodal transportation of trailers on flatcars. Usinga density function on each market, the relation between price and demand is evaluated and a revenue function is inferred.

Integrating pricing within the overall decision process has become a trend in the service industry. A recently proposedapproach to price optimization while considering customer reaction to pricing and other operational policies is based on bi-level mathematical programming which is used to formulate hierarchical decision making. It can be represented by a systemwhere an economic agent, called the leader, considers the reaction of a second agent, the follower, in its decision process.Colson et al. (2005) have proposed a study of the properties and main solution methodologies presented in the literaturefor linear, linear-quadratic or nonlinear programs. Bilevel programming was also treated by Dempe (2002).

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 103

A generic model that integrates pricing decision imposed by the leader on a set of goods or services and the reaction of theusers according to those policies was developed by Labbé et al. (1998a). An application in a multi-commodity transportationcontext was described. Labbé et al. (1998b) presented an approach inspired by the algorithm proposed by Gendreau et al.(1996) for the solution of linear bilevel programs. A similar methodology was exposed by Brotcorne et al. (2000) for pricingproblems on a single commodity transportation network. Heuristics were developed as well as improvement strategies. Mul-ti-commodity contexts were studied by Brotcorne et al. (2001). A combined network design and pricing model was pre-sented by Brotcorne et al. (2008). Côté et al. (2003) proposed a static and deterministic bilevel model for the pricing andallocation of capacity in the airline industry. A market segmentation based on service attributes such as price, transit dura-tion or service quality was devised. Finally, Castelli et al. (2004) have presented a bilevel model representing the behavior oftwo agents in a transportation network.

3. Integrated pricing and capacity management

In this section we present the main aspects involved in the decision process of railroad freight carriers. In a first step wepropose a representation of the relations between the most important decisions through a conceptual framework similar tothe one presented by Bartodziej et al. (2007) for air cargo transportation. Next we expose the ideas needed to formalize theseinteractions in a compound yield and pricing management model formulated by a path-based bilevel mathematical program.

3.1. Conceptual framework for railroad freight transportation

As mentioned earlier, most railroads are nowadays making business in a scheduled operations environment as opposed totraditional tonnage-based activities. Therefore, carriers need to design, at a tactical level, train schedules to meet expecteddemand. At an operational level, customer requests must then be assigned to the available transportation capacity. Trans-portation requests are commonly defined as a certain commodity for which specific attributes are known, such as the de-mand, which can be measured in number of cars to move or in an equivalent volume measure unit. Every request needsto be transported from a given origin to a given destination (OD) through one or several classification yards and is character-ized by a release time at which the commodity becomes available for transportation by the carrier and a due time represent-ing a deadline for the load to arrive at destination. Furthermore, requests can be segmented according to perception of thecustomer towards, for instance, transit duration or service quality (equipment type preferences, for example). The itinerariesfollow from the schedule and define the admissible paths for a commodity to be transported from origin to destination. Themovement of cars on an itinerary is limited by constraints such as block and locomotive pulling power capacities which areinduced by tactical decisions.

The problem addressed here is thus to simultaneously analyze capacity management and pricing in the relationship be-tween each OD and its corresponding set of admissible itineraries. These itineraries will be obtained through enumeration offeasible paths in networks that will be described in the next section.

3.2. Network representation

Let E be the set of equipment types in the network and K the set of commodities or requests for which a service demandhas been expressed. To each request k 2 K is assigned an origin–destination pair (o(k),d(k)) as well as a set of compatibleequipment types Ek. For each equipment type e 2 E, we define a multi-commodity space-time network Ge = (Ne,Ae) whereNe represents the vertex set, and Ae is the arc set of the graph where products circulating in the network are the commoditiesk such that e 2 Ek. Let B represent the set of blocks defined by the rail company. Therefore, Be � Ae describes the set of blocksfor which equipment e is available. This network is based on what authors like Crainic et al. (1984) and Fernández et al.(2004) call train services. Here we consider known services which specify scheduled trains with established frequenciesand predefined itineraries. Moreover, we assume that a train is characterized by a movement between two successive clas-sification yards. Therefore no stop is made between departure and arrival. Thus the capacity level that will be consideredalready takes into account the physical network structure of the segment the train will travel. The following representation,similar to others such as the one proposed by Kwon (1994) for instance, assumes that the blocking and make-up policies areknown. Fig. 1 presents the structure of a blocking network for a specific equipment type on a 1-day period.

3.2.1. Set Ne

Set Ne is composed of four types of vertices. The first two allow the identification of the departures and arrivals defined bythe scheduled trains. The others respectively designate the end of the block disassembly period for a specific train, and theend of the time window devoted to block assembly or, in other words, the latest time after which the assembly of cars intoblocks and train make-up would be impossible considering the scheduled departure time of the train.

3.2.2. Set Ae

Set Ae is also composed of four types of elements. The first two types of arcs represent the blocks assembly operations andthe make-up of the trains as well as the reverse operations at destination. Horizontal arcs indicate the storage of cars at each

end of the disassembly of blocks

end of time window for the assembly of blocks

...

Day 3

...

...

...

Day 2...

...

...

...

...

disassembly

assembly

storagemovements of blocks

A

E

D

C

B

Classification yard 1

Classification yard 2

Classification yard 3

Day 1

Legend: departurearrival

Fig. 1. Blocking network representation.

104 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

yard up to their departure time on a later block or up to their delivery to the consignee. Finally, arcs representing the move-ments of blocks on trains are set according to make-up policies. For instance, arc A in Fig. 1 specifies the movement of a blockwith yard 3 as origin and 1 as destination. This block will not be reclassified at intermediate yards along its path. Since block-ing and make-up policies are known beforehand, trains on which block A will travel are pre-established. One can observethat the block will travel from yard 3 to yard 2 where it will join block B for a final transit, on the same train, to yard 1 whichrepresents the destination of both blocks. Fig. 2 gives an example of train movements which are compatible with the definedblocks. Let {X;(y,z)} represent block X having yard y as origin and yard z as destination. In addition to the transit of blocks Aand B defined earlier, movements of blocks C, D and E are also presented. Therefore, blocks C and D move simultaneouslybetween yards 1 and 2 and, following removal of block C reaching its destination and the addition of E, a final transit is madeto yard 3.

3.3. Mathematical modeling: path-based formulation

We introduce a new formulation based on the bilevel mathematical programming paradigm. The rail carrier, identified asleader and denoted l in what follows, determines at the first level a pricing policy for the transit of commodities in the net-work under its control. The carrier considers the reaction of all customers, identified as the follower, in its decision making.The pricing scheme of the competition, denoted c, is also taken into account by the leader. The following formulation relieson feasible paths in the space-time networks developed previously. In order to establish the mathematical structure of theproposed model, we need to define in greater detail our notation. We therefore define the following sets:

Ile;k: set of itineraries offered by the leader for equipment e and request k;

Ilk: set of itineraries offered by the leader for request k, i.e. Il

k ¼ [e2EkIle;k;

Il: set of itineraries offered by the leader;Ick: set of itineraries offered by the competition for request k; an element i 2 Ic

k can represent an itinerary from trucking,maritime or competing rail companies between o(k) and d(k);

...

Day 3

...

...

...

Day 2...

...

...

...

...

Day 1

{A;(3,1)}

{D;(1,3)} & {E;(2,3)}

{C;(1,2)} & {D;(1,3)}{A;(3,1)} & {B;(2,1)}

Classification yard 1

Classification yard 2

Classification yard 3

Legend: departure

arrival

movements of trains

Fig. 2. Train movements representation.

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 105

H: set of trains;Ha: set of trains on which block a is assigned;Be

i;k: set of blocks making up itinerary i of request k in network Ge;Ea: set of equipments offered on block a;P: set of periods defined in the planning horizon (the duration of one period can be a day, for example);R: set of classification yards;Sr,p: set of itineraries composed of at least one assembly or disassembly arc of classification yard r 2 R during period p 2 P.Itineraries having an arc representing a block stopping at classification yard r at period p are also included in the set sinceeven though no disassembly or reclassification is made, track capacity is needed therefore limiting the global capacity ofthe yard,

as well as the exogenous parameters:

dk: demand of request k in number of cars;caph: capacity of train h in number of cars. This capacity is normally based on total pulling power of the locomotivesassigned to the train;capa: capacity of block a in number of cars. This capacity is often limited by the length of the classification tracks;cape,a: allowed capacity for equipment type e on block a in number of cars;capr: volume handling capacity of classification yard r during a period, in number of cars.

We also define the following variables:

tli: tariff set by the leader for moving one car on itinerary i. For the competition, the tariff tc

i is considered as known;tl: tariff vector of the leader;f li and f c

i : flow on itinerary i for the leader and the competition, respectively;fl and fc: flow vectors of the leader and competition,

106 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

and a characterization of each itinerary as well as the sensitivity of customers to service quality, as proposed by Marcotte andSavard (2002) and Côté et al. (2003) for airline passenger transportation:

Di: duration of itinerary i;Qi: service quality of itinerary i;ak: monetary equivalent of one unit of duration for the customer associated with request k;be,k: monetary equivalent to the loss of one unit of service quality by utilizing equipment type e for the customer asso-ciated with request k;cl

iða; bÞ and cci ða; bÞ: unit cost of the flow on itinerary i of the leader and the competition respectively. For itinerary i of

the leader defined in network Ge, this cost can be expressed, for request k, as cliða; bÞ ¼ tl

i þ qi;kða; bÞ where qi,k(a,b) =akDi + be,kQi is the part of the cost related to the service perception. For the competition, the cost iscc

i ða; bÞ ¼ tci þ akDi þ bkQi.

The developed model is deterministic and the itinerary generation phase allows us to evaluate the duration Di of eachpath i. This information is used, for instance, in the computation of the perceived costs just described. Furthermore, it canbe used to eliminate from a customer’s set of admissible itineraries those for which the duration exceeds a specific due timeestablished between the carrier and the shipper.

In the proposed formulation, as in actual railroad operations, the shipper does not choose the itineraries that will be usedto move its freight. However, every customer (or class of customers) will have its specific perception of the service offer. Forinstance, some of them will put more importance on transit duration than others and may be willing to pay a premium toaccess faster itineraries. In order to take this aspect into account in the combined analysis of freight assignment and pricingwe consider a linear combination of the tariff with some attributes of the shipper. This allows the carrier to characterize theperceived disutility of itineraries for each request and evaluate, in a planning phase, a proper freight assignment and corre-sponding pricing policy.

We can model the combined pricing and network capacity management problem (PCM) as follows:

ðPCMÞ maxtl ;f l ;f c

Xk2K

Xe2Ek

Xi2Il

e;k

tlif

li

s:t: tl P 0; ð1Þ

minf l ;f c

Xk2K

Xe2Ek

Xi2Il

e;k

cliða;bÞf l

i þXk2K

Xi2Ic

k

cci ða;bÞf c

i

s:t:Xe2Ek

Xi2Il

e;k

f li þ

Xi2Ic

k

f ci ¼ dk 8k 2 K; ðkkÞ ð2Þ

Xkje2Ek

Xija2Be

i;k

f li 6 cape;a 8a 2 B; e 2 Ea; ðge;aÞ ð3Þ

Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li 6 capa 8a 2 B; ðpaÞ ð4Þ

Xajh2Ha

Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li 6 caph 8h 2 H; ðchÞ ð5Þ

Xi2Sr;p

f li 6 capr 8r 2 R; p 2 P; ðxr;pÞ ð6Þ

f l; f c P 0; integer; ð7Þ

where k, g, p, c and x represent dual variable vectors associated with the different constraint sets.In this model, constraints (2) guarantee that the demand of each request is carried from origin to destination. Con-

straints (3) allow a maximal capacity for transporting a specific equipment type on a block. Constraints (4) limit equip-ment transit on blocks. Constraints (5) ensure the pulling capacities are respected. Car handling capacity ofclassification tracks at every yard is imposed by (6). Finally, (1) and (7) define constraints on decision variables of the firstand second levels.

From the presented formulation we can observe that network capacity resulting from scheduled operations planning isconsidered to have been determined beforehand. It is thus fixed and an unsold unit will be lost at the departure time of thecorresponding trains. In this OD-based model, similar to others proposed in the airline industry such as those of Marcotteand Savard (2002) and Côté et al. (2003), the capacity of each OD is induced by that set on the underlying routing segments.The objective is thus to manage the available capacity in order to satisfy demand while maximizing revenue. Furthermore,the predetermined operations planning decisions will incur most of the operating costs (train schedule, blocking and

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 107

make-up decisions, assignment of blocks to yard tracks, etc.). However, a variable cost, hi, could be considered to representthe handling cost of a car on itinerary i. We could then replace the first level objective function by:

maxtl ;f l ;f c

Xk2K

Xe2Ek

Xi2Il

e;k

tli � hi

� �f li

This is an extension to the described mathematical formulation that would not make the problem harder to solve.

4. Path-based model reformulation

In this section, we describe the methodology devised to solve the PCM. We present an exact approach to tackle this inte-grated yield and pricing management problem which is based on a reformulation of the bilevel model as a single-level mixedinteger mathematical program.

Bilevel models such as the formulation presented in the previous section have been studied by many authors. Labbé et al.(1998a,b), Brotcorne et al. (2000, 2001) and Marcotte and Savard (2002) have proposed solution methodologies based on thesubstitution of the second level mathematical program by its primal–dual optimality conditions. We present a similar meth-odology for the PCM.

First of all, the integrality of the commodity flow decision variables can be relaxed since, as authors like Marín and Salm-erón (1996) mention, the number of cars considered is large and the flexibility over network capacity allows some error mar-gin. Therefore, the fractional aspect of the commodity flow will not have a major impact on the optimal solution.

Next, since we do not have access to information about the networks of the competing carriers we consider that theircapacities are sufficient to meet the total demand for all commodities. Faced with the choice of an alternative to the itiner-aries of the leader, each request will thus select the competition itinerary with the least perceived cost. Therefore, for k 2 Klet f c

k ¼ f ci� and cc

kða; bÞ ¼ cci� ða; bÞ where i� ¼ arg mini2Ic

kcc

i ða; bÞ. Obviously, in order to have a positive flow on itinerary i 2 Ilk,

we must have that cliða; bÞ 6 cc

kða; bÞ.In the proposed model, constraints (2) impose that the follower’s demand must be satisfied by the leader or the compe-

tition. Thus, as pointed out in the last paragraph, the follower will consider the leader’s service offer as long as its perceivedcost falls below or is equal to the best offer on the market. Indeed, we suppose an optimistic optimization where for twoequivalent follower’s solutions, the one favoring the leader will be chosen. Moreover, some demand can also be lost to com-petition because of network capacity limitations. Therefore the optimization process aims at setting the leader’s tariffs, whileconsidering the pricing of competing carriers, to induce as much flow as capacity allows in order to maximize revenue.

We will now replace the second level program by its primal-dual optimality conditions. The dual mathematical programof the second level problem of the PCM can be stated as:

maxk;g;p;c;x

Xk2K

dkkk þXa2B

Xe2Ea

cape;age;a þXa2B

capapa þXh2H

caphch þXr2R

Xp2P

caprxr;p

s:t: kk þXa2Be

i;k

ðge;a þ paÞ þXa2Be

i;k

Xh2Ha

ch þXr2R

Xp2P

yir;pxr;p 6 cl

iða;bÞ 8k 2 K; e 2 Ek; i 2 Ile;k ð8Þ

kk 6 cckða;bÞ 8k 2 K ð9Þ

g 6 0 ð10Þp 6 0 ð11Þc 6 0 ð12Þx 6 0 ð13Þ

where parameter yir;p is equal to 1 when i 2 Sr,p and to 0 otherwise. The optimality conditions of the second level program

comprise primal feasibility (constraints (2)–(7)), dual feasibility (constraints (8)–(13)) as well as the following complemen-tarity constraints:

cliða;bÞ � kk þ

Xa2Be

i;k

ðge;a þ paÞ þXa2Be

i;k

Xh2Ha

ch þXr2R

Xp2P

yir;pxr;p

0@

1A

24

35f l

i ¼ 0 8k 2 K; e 2 Ek; i 2 Ile;k ð14Þ

cckða;bÞ � kk

� �f ck ¼ 0 8k 2 K ð15Þ

kk dk �Xe2Ek

Xi2Il

e;k

f li þ f c

k

0B@

1CA ¼ 0 8k 2 K ð16Þ

ge;a cape;a �X

kje2Ek

Xija2Be

i;k

f li

0@

1A ¼ 0 8a 2 B; e 2 Ea ð17Þ

108 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

pa capa �Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li

0@

1A ¼ 0 8a 2 B ð18Þ

ch caph �X

ajh2Ha

Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li

0@

1A ¼ 0 8h 2 H ð19Þ

xr;p capr �Xi2Sr;p

f li

0@

1A ¼ 0 8r 2 R; p 2 P: ð20Þ

We note that constraints (16) are always satisfied because of demand constraints (2). The proposed modifications transformthe bilevel program into a single level formulation:

XXX

ðPCM1Þ max

tl ;f l ;f c ;k;g;p;c;x k2K e2Ek i2Ile;k

tlif

li

s:t: tl P 0

Second level primal feasibility: constraints (2)–(7).Second level dual feasibility: constraints (8)–(13).

Second level complementarity: constraints (14), (15), (17)–(20).

The bilinear aspect of the leader’s objective requires its reformulation based on the strong duality theorem applied to thefollower’s problem. The theorem ensures that, at optimality, the primal and dual objectives of the follower will be equal:

Xk2K

Xe2Ek

Xi2Il

e;k

tlif

li þ

Xk2K

Xe2Ek

Xi2Il

e;k

qi;kða;bÞf li þ

Xk2K

cckða;bÞf c

k ¼Xk2K

dkkk þXa2B

Xe2Ea

cape;age;a

þXa2B

capapa þXh2H

caphch þXr2R

Xp2P

caprxr;p;

and thus:

Xk2K

Xe2Ek

Xi2Il

e;k

tlif

li ¼

Xk2K

dkkk þXa2B

Xe2Ea

cape;age;a þXa2B

capapa þXh2H

caphch þXr2R

Xp2P

caprxr;p

�Xk2K

Xe2Ek

Xi2Il

e;k

qi;kða; bÞf li �

Xk2K

cckða;bÞf c

k :

5. Pricing policies

In this section we discuss two pricing policies that carriers can put forward. We will consider cases where tariffs are seteither at an itinerary or at a request level. Appropriate pricing will take into account different criteria. Notably, the type offreight, origin and destination of the transit and the type of equipment used. The itineraries appearing in the formulationpresented in the previous section are characterized by those attributes. The preferred pricing policy will therefore need tobe dependent of these elements. It is then relevant to consider the impact on the revenue of interactions between tariffs.Thereby a pricing policy called disjoint will be studied. This allows the carrier to identify a revenue arising from a highlydisaggregated pricing strategy since each itinerary is assigned its own tariff. In a second analysis, the imposition of equalityconstraints between the tariffs for the itineraries of the same request will be considered. In this case, the prescribed tariff fora given request relies on an aggregation of information regarding the path in the network followed by the itineraries or theequipment used since a single tariff must be determined. Other pricing policies, which are not addressed here, might con-sider requests aggregation according to their origin–destination pair or freight transported in order to identify similar tariffsfor requests having common characteristics.

5.1. Disjoint pricing of the itineraries

In a first pricing approach, we consider that distinct tariffs can be assigned to every itinerary. The model presented earlierillustrates this context.

5.1.1. Moving constraints to the first levelThe difficulty of solving the proposed model comes mainly from the nonlinear complementarity constraints and their lin-

earization which would introduce a large number of binary variables. It is therefore essential to analyze the structure of themodel in order to identify the potential movements of constraints from the second to the first level. A similar study is pre-sented by Brotcorne et al. (2008). The authors show that, for a certain class of bilinear bilevel programs, the movement of

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 109

constraints from the second to the first level is valid if the corresponding dual variables take the value zero at optimality. Thischaracteristic can be verified for the capacity constraints of the proposed model. We will only study constraints (3), since asimilar reasoning can be made for constraints (4)–(6). We proceed by contradiction.

Proposition 1. For every optimal solution to PCM1 with a disjoint pricing policy of the itineraries, g⁄ = 0, p⁄ = 0, c⁄ = 0, x⁄ = 0.

Proof. Let ((tl)⁄, (fl)⁄, (fc)⁄,k⁄,g⁄,p⁄,c⁄,x⁄) be an optimal solution of the problem. Suppose that for this optimal solution g⁄– 0which implies that there is at least one dual variable such that g�e;a < 0. Set the vector g0 = 0, define (tl)0 where the tariff

assigned to itinerary i 2 Ile;k is ðtl

iÞ0 ¼ ðtl

i� �

Pa2Be

i;kg�e;a, and consider the solution ((tl)0, (fl)⁄, (fc)⁄,k⁄,g0,p⁄,c⁄,x⁄). This solution

is feasible since constraints (2)–(7) are necessarily satisfied because the flows are unchanged with respect to those of theoptimal solution. The same applies to constraints (9)–(13). For constraints (8), by hypothesis we have that:

k�k þXa2Be

i;k

g�e;a þ p�a� �

þXa2Be

i;k

Xh2Ha

c�h þXr2R

Xp2P

yir;px

�r;p 6 cl

iða; bÞ�

and thus:

k�k þXa2Be

i;k

g�e;a þ p�a� �

þXa2Be

i;k

Xh2Ha

c�h þXr2R

Xp2P

yir;px

�r;p 6 ðtl

iÞ� þ qi;kða;bÞ

k�k þXa2Be

i;k

p�a þXa2Be

i;k

Xh2Ha

c�h þXr2R

Xp2P

yir;px

�r;p 6 tl

i

� �� �Xa2Be

i;k

g�e;a

0@

1Aþ qi;kða;bÞ

k�k þXa2Be

i;k

g0e;a þ p�a� �

þXa2Be

i;k

Xh2Ha

c�h þXr2R

Xp2P

yir;pxr;p 6 cl

iða;bÞ0

The solution is therefore feasible for these constraints. Regarding complementarity constraints, only (14) and (17) need to beconsidered since the others only depend on the optimal vectors of the proposed solution. An equivalent analysis to the onefor constraints (8) can be made for complementarity constraints (14). Moreover, since g0 = 0, constraints (17) will be satis-fied. The only constraints left to verify are the first level non-negativity of the tariffs. By definition one has g⁄ 6 0 and byhypothesis there is at least one dual variable for which g�e;a < 0. Thus one can deduce that there is at least one itineraryi 2 Il

e;k such thatP

a2Bei;kg�e;a < 0 and consequently 0 6 ðtl

iÞ� < ðtl

iÞ0. We can conclude that the proposed solution is feasible

and generates a greater revenue since

Xk2K

Xe2Ek

Xi2Il

e;k

ðtliÞ�f l

i <Xk2K

Xe2Ek

Xi2Il

e;k

ðtliÞ0f l

i

This contradicts the optimal nature of ((tl)⁄, (fl)⁄, (fc)⁄,k⁄,g⁄,p⁄,c⁄,x⁄) and shows that g⁄ = 0. An identical conclusion can be ob-tained for constraints (4)–(6) and so p⁄ = 0, c⁄ = 0, x⁄ = 0 for every optimal solution. h

Moving capacity constraints to the first level is therefore admissible. The impact of these observations on the primal-dualformulation PCM1 with disjoint pricing is the simplification of constraints (8) and (14) as well as the elimination of con-straints (10)–(13) and constraints (17)–(20).

We can also show that, for every request, the optimal dual multiplier of the demand satisfaction constraint will always beequal to the least cost itinerary offered by the competition.

Proposition 2. Under an optimal pricing policy, k�k ¼ cckða; bÞ for each request k 2 K.

Proof. The proof is similar to that of Proposition 1. We proceed by contradiction. Let the vector ((tl)⁄, (fl)⁄, (fc)⁄,k⁄,g⁄,p⁄,c⁄,x⁄)be an optimal solution for the problem. Suppose that for this solution there exists k 2 K such that k�k – cc

kða; bÞwhich implies,

by constraints (9), that for this request k�k � cckða; bÞ < 0. Let k0 be such that k0k ¼ cc

kða; bÞ as well as (tl)0 for which the tariffassigned to itinerary i 2 Il

e;k is defined by tli

� �0 ¼ tli

� �� � k�k � cckða; bÞ

� �and consider solution ((tl)0, (fl)⁄, (fc)⁄,k0,g⁄,p⁄,c⁄,x⁄). By

a reasoning similar to Proposition 1 we can demonstrate that the proposed solution is feasible and produces a greater rev-enue than ((tl)⁄, (fl)⁄, (fc)⁄,k⁄,g⁄,p⁄,c⁄,x⁄), which contradicts its optimality and shows that k�k ¼ cc

kða; bÞ; 8k 2 K. h

Furthermore, since tariffs are disjoint, we can show that the tariff of an itinerary of the leader will always be equal to thetariff margin between this itinerary and the best itinerary offered by the competition for that request.

Corollary 1. For every itinerary i 2 Ilk; t

li ¼ cc

kða; bÞ � qi;kða; bÞ.

Proof. We know that cliða; bÞ 6 cc

kða; bÞ. From constraints (8), Propositions 1 and 2 we have cckða; bÞ 6 cl

iða; bÞ. Therefore,

cliða;bÞ ¼ cc

kða; bÞ ð21Þ

tli ¼ cc

kða; bÞ � qi;kða;bÞ � ð22Þ

110 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

Thus, as a consequence of Propositions 1, 2 and Eq. (21), constraints (8) and (9) as well as complementarity constraints(14) and (15) are satisfied.

Using the results of Propositions 1 and 2 we can simplify the linearized form of the leader’s objective function:

Xk2K

Xe2Ek

Xi2Il

e;k

tlif

li ¼

Xk2K

cckða;bÞ dk � f c

k

� ��Xk2K

Xe2Ek

Xi2Il

e;k

qi;kða;bÞf li :

The reformulation of PCM1 with disjoint pricing of the itineraries can now be described by:

maxf l ;f c

Xk2K

cckða;bÞ dk � f c

k

� ��Xk2K

Xe2Ek

Xi2Il

e;k

qi;kða;bÞf li

s:t:Xe2Ek

Xi2Il

e;k

f li þ f c

k ¼ dk 8k 2 K ð23Þ

Xkje2Ek

Xija2Be

i;k

f li 6 cape;a 8a 2 B; e 2 Ea ð24Þ

Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li 6 capa 8a 2 B ð25Þ

Xajh2Ha

Xe2Ea

Xkje2Ek

Xija2Be

i;k

f li 6 caph 8h 2 H ð26Þ

Xi2Sr;p

f li 6 capr 8r 2 R; p 2 P ð27Þ

f l; f c P 0; ð28Þ

which corresponds to the solution of a multi-commodity network flow problem.

5.2. Common pricing on the itineraries of a request

Consider the context where a common tariff must be assigned to the itineraries of a request. The leader has to select aunique tariff for the shipment of each car of a specific commodity, regardless of the itinerary on which the freight of the fol-lower will be assigned. The resulting model is more complex since moving constraints from the second to the first level is notpossible in this setting. All complementarity constraints must be considered. To model this pricing strategy, the only mod-ification needed is a small change to the definition of the perceived cost. For an itinerary i offered by the leader, the cost isnow cl

iða; bÞ ¼ tlk þ akDi þ be;kQi. Thus, the tariff now depends on request k, and not only on the considered itinerary. Further-

more, it can be shown that Proposition 2 still holds in this context.For this pricing strategy, the linearized form of the objective function of the leader is:

Xk2K

Xe2Ek

Xi2Il

e;k

tlkf l

i ¼Xk2K

cckða;bÞ dk � f c

k

� �þXa2B

Xe2Ea

cape;age;a þXa2B

capapa þXh2H

caphch

þXr2R

Xp2P

caprxr;p �Xk2K

Xe2Ek

Xi2Il

e;k

qi;kða; bÞf li :

The linearization of the complementarity constraints of the second level reformulation can be expressed as follows. In orderto simplify the presentation, only constraints (14) and (17) will be shown. Again constraints (15) and (16) are always satis-fied because, respectively, of Proposition 2 and of demand constraints (2). We have the following conditions where the xvariables are introduced to ensure that at least one term of the corresponding complementarity constraint is equal to zero:

ð14Þ ()

cliða;bÞ � cc

kða;bÞ �P

a2Bei;k

ðge;a þ paÞ �P

a2Bei;k

Ph2Ha

ch �Pr2R

Pp2P

yir;pxr;p 6 M1

i x1i

f li 6 M2

i x2i

x1i þ x2

i 6 1x1

i ; x2i 2 f0;1g

8>>>>>><>>>>>>:

ð29Þ

ð17Þ ()

cape;a �P

kje2Ek

Pija2Be

i;k

f li 6 M1

e;ax1e;a

�ge;a 6 M2e;ax2

e;a

x1e;a þ x2

e;a 6 1

x1e;a; x

2e;a 2 f0;1g

8>>>>>><>>>>>>:

ð30Þ

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 111

Constant parameters M represent valid upper bounds on the left-hand side of the corresponding constraints. The mixed inte-ger reformulation of PCM1 with a common pricing policy can be expressed as:

X � � XX X X XX XXX

max

tl ;f l ;f c ;g;p;c;x k2K

cckða;bÞ dk� f c

k þa2B e2Ea

cape;age;aþa2B

capapaþh2H

caphchþr2R p2P

caprxr;p�k2K e2Ek i2Il

e;k

qi;kða;bÞf li

s:t: tl P0

Second level primal feasibility: constraints (2)–(7).Second level dual feasibility: constraints (8)–(13).

Linearized form of second level complementarity constraints 14 and (17)–(20).

5.2.1. Valid inequalitiesWe now propose a set of valid inequalities to strengthen the previously proposed formulation. These cuts connect flow

variables with their corresponding tariffs.

Proposition 3. Let k 2 K and qi� ;kða; bÞ ¼min qi;kða; bÞji 2 Ilk

n o: The following inequalities are valid:

tlk þ ðqi;kða;bÞ � qi� ;kða;bÞÞx2

i 6 ðcckða; bÞ � qi� ;kða;bÞÞ 8i 2 Il

k

Proof. First, we know that the cost of every itinerary is bounded by the least cost itinerary of the competition,

cliða;bÞ 6 cc

kða;bÞ;8i 2 Ilk

Suppose Ilk is composed of n itineraries {i1, i2, . . . , in} for which, without loss of generality, qi1 ;k

ða; bÞ 6 qi2 ;kða; bÞ 6 � � � 6

qin ;kða; bÞ. Therefore qi� ;kða; bÞ ¼ qi1 ;kða; bÞ. Thus,

cli1ða; bÞ 6 cc

kða; bÞtl

k þ qi1 ;kða; bÞ 6 cc

kða;bÞtl

k 6 cckða; bÞ � qi1 ;k

ða;bÞ

which represents an upper bound on the tariff of the itineraries for request k. However, if in the optimal solution the flow onij – i1 is positive, where ij is such that qij ;k

ða; bÞ ¼max qi;kða; bÞji 2 Ilk; f

li > 0

n o, the bound on the tariff will become

tlk 6 cc

kða;bÞ � qij ;kða; bÞ

thus reducing the upper bound by cckða; bÞ � qi1 ;k

ða; bÞ� �

� cckða; bÞ � qij ;k

ða; bÞ� �

¼ qij ;kða; bÞ � qi1 ;k

ða; bÞ. Finally, the flow on ijcan be positive only if x2

ij¼ 1 and the bound reduction will be enforced in these circumstances. Thus we can conclude that

tlk 6 ðcc

kða;bÞ � qi1 ;kða;bÞÞ � ðqij ;k

ða;bÞ � qi1 ;kða;bÞÞx2

ij

tlk þ ðqij ;k

ða; bÞ � qi1 ;kða; bÞÞx2

ij6 ðcc

kða; bÞ � qi1 ;kða; bÞÞ

or

tlk þ ðqij ;k

ða;bÞ � qi� ;kða;bÞÞx2ij6 ðcc

kða;bÞ � qi� ;kða;bÞÞ

This relation can be defined for the set of itineraries of request k. h

The impact of the properties and valid inequalities presented in this section will be assessed in Section 7.

6. Generation of instances

This section presents the tool used to generate the family of instances on which the solution approach will be tested. Theproposed generator allows the construction of blocking networks. This requires the user to define the following parameters:

� jPj: number of periods defining the planning horizon;� jKj: number of requests;� jRj: number of yards;� jEj: number of equipment types available;� jBj: number of blocks;� jHj: number of train schedules.

Subsequently, train schedules are constructed and the network of train movements is established. For each train timeta-ble, the index of a yard representing the origin and another the destination are generated. The itineraries of trains between

112 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

the different origin–destination pairs are then enumerated. Following the determination of an origin and destination for eachblock, the blocking network is defined by the assignment, to every block, of a compatible train route. To each block is as-signed a number of equipments that will be valid on it. Then, the admissible block itineraries are enumerated for each sub-network resulting from a given equipment. Thus, according to the characteristics of each request k, the valid itineraries willbe added to the set Il

k. Finally a route of the competition is created for every request and represents the best service offer onthe market. It should be noted that unlike authors such as Bodin et al. (1980), Crainic et al. (1984), Newton et al. (1998) orAhuja et al. (2007), we do not consider the problem of constructing a blocking plan. As mentioned above, we assume thatblocking policies are designed beforehand.

In order for the instances to have a structure as close as possible to most North American railway networks, we focus onthe creation of demand corridors. To this end, we define an attraction factor for each yard in order to produce regions of highdemand density. Among the most dense centers, a number of yards will be selected and labeled as hubs. These will be located

more strategically. For the subsequently generated instances, we have max 1; jRj4

j kn ohubs. The other yards, considered as

satellites, will then be scattered throughout the network. Algorithm 1 shows the location procedure. The networks of move-ments of trains and blocks will then be generated so that the transportation capacity between the different yards is repre-sentative of the density of demand that transits there. The schedule of each train is obtained by generating a random startingtime and fixing the arrival time as the time required to travel between the two yards. To this time is added or subtracted acertain duration obtained by random sampling within a predefined interval to create some disturbance of the transit time.The travel time between two yards is derived from the Euclidean distance between their locations. This distance is then mul-tiplied by the average speed of a freight train which is about 32.19 km per hour (or 20 miles per hour). When generating theroute of the competition associated with a request k, the average duration Dk of the itineraries of the leader for this request isassessed and the chosen duration is selected randomly in the interval ½0:75Dk; 1:25Dk�.

Algorithm 1. Location of classification yards

1. Assignment of attraction factors:– Set i :¼ 1.while i 6 jRj do

– Let u be a random number in the interval 12jKjjRj

l m;2 jKj

jRj

l mh iaccording to a discrete uniform distribution. The attraction

factor (AF) of yard i is defined by AFi ¼ ujKj.

– Set i:¼i + 1.

2. Select the max 1; jRj4

j kn oyards with the highest attraction factors, deciding randomly in case of equality, and create

the hub set Rhub.3. Location of hubs:foreach r 2 Rhub do

if no hub has been located thenGenerate the coordinates in the [�50, 50]2 domain.

else(a) Generate the coordinates in the [�50, 50]2 domain.(b) Evaluate d, the average of the distances to the other hubs already located.(c) Let u be a random number in the [0,1] interval according to a continuous uniform distribution. If u P 1 � e�/d

(where / = 0.005), go back to (a).4. For each yard not in Rhub, generate the coordinates in the [�100,100]2 domain.

In regard to demand, we consider three classes of requests linked to the type of products moved through the network.Thus, we characterize the following classes:

– Class 1:(a) low-value products (e.g. coal, grain products);(b) central market of rail transportation;(c) low competition from other modes of transportation.

– Class 2:(a) intermediate value products (e.g. forest products);(b) second market segment in terms of importance;(c) strong competition.

– Class 3:(a) high-value products (e.g. motor vehicles);(b) least exploited market segment mainly due to high inventory and handling costs;(c) strong competition.

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 113

Moreover, according to the latest statistics obtained from The Railway Association of Canada (RAC), we can estimatethat, in Canada, Class 1 represents approximately 50–60% of the goods carried, 20–30% come from Class 2, and 10–20%from Class 3. We therefore distribute the requests according to these proportions. Algorithm 2 presents the request assign-ment to yards. A similar procedure is used with regard to train schedules and the creation of blocks. These will not bedescribed here.

Algorithm 2. Request assignment

– Set j :¼ 1.while j 6 jKj do

– Let iO be a random number in the [1, jRj] interval according to a discrete uniform distribution.– Let u be a random number in the [0,1] interval according to a continuous uniform distribution.if u < AFiO

then– Assign iO as the origin of the request j.– Let iD be a random number in the [1, jRj]niO according to a discrete uniform distribution.– Let v be a random number in the [0,1] interval according to a continuous uniform distribution.while v P AFiD

do– Regenerate iD.– Regenerate v.

– Assign iD as the destination of request j.– Set j :¼ j + 1.

As for the different capacities, some of them can be established using statistical data. For example, data collected by theRAC revealed that during the last 20 years, the number of cars per train is, on average, around 70. Thus, we establish thewindow of possible values for the capacity of a train as [60,80]. A similar analysis is performed for the other types of capacityso that the resulting instances are realistic. Finally, the windows of values for the various parameters related to the percep-tion of service are also generated to realistically portray the classes of request described above.

Tables 1 and 2 present the characteristics of the different instances and Table 3 the parameters of the demand classes. InTable 2, the instances keep most of the network features of the instances presented in Table 1 but are densified. In regard toTable 3, recall that the parameter a is the factor associated with the conversion, in monetary value, of one unit of durationand the same applies for b for the loss of a unit of quality of service. We thus set, for each parameter, an interval in which thevalue assigned to it will be selected. Note that the superscript indicates, once again, the leader or the competition. The rangesfor Qc were chosen while considering the interval for Ql defined by [50,100] in order to maintain consistency with the char-acterization of the request classes described earlier.

Finally, tariffs of the competition are generated to allow the viability of the leader’s itineraries. An itinerary i 2 Ilk for

which qi;kða; bÞ > cckða; bÞ must obviously be rejected. We therefore consider the itinerary i 2 Il

k such that

qi;k ¼max qi;kða; bÞji 2 Ilk

n o: Following the assignment of the perceived cost qc

k of the itinerary of the competition, let

� ¼ qi;k � qck. Therefore, when � 6 0, tc

k is randomly selected in the [50,150] interval, otherwise the value is chosen in[� + 50,� + 150].

7. Computational results

The models presented here have been solved using the IBM Ilog Cplex solver. However, it appears that Cplex sometimeshas difficulty in determining an initial solution, even if some are easily identifiable. Consider for example the solution forwhich the flow is only affected to the itineraries of the competition, generating a revenue of zero. Nevertheless, it is possibleto provide Cplex with a solution of better quality. Moreover, starting the solution process from a fixed solution in spite ofmodifications, including adding valid cuts, provides a comparison point from which we can more easily carry out the resultanalysis. We now propose a greedy procedure to achieve this goal. Algorithm 3 presents the different steps.

Table 1Instances Inst1.

jKj jRj jPj jBj jEj jHj

pr01 50 3 3 20 3 15pr02 150 5 5 40 3 25pr03 250 7 7 60 3 35pr04 350 9 3 80 6 45pr05 450 11 5 100 6 55pr06 550 13 7 150 6 65

114 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

Another procedure generating an initial solution to be compared to the one obtained by the greedy algorithm is based onsolving an inverse optimization problem. As authors such as Ahuja and Orlin (2001) mention, an inverse optimization prob-lem consists in inferring the values of the model parameters (cost coefficients, right-hand side vector, and the constraint ma-trix), given the values of observable parameters (optimal decision variables). The problem considered here arises from theflow generated when the tariffs are set to zero. Given the possible degeneracy of the mathematical program of the secondlevel, we successively solve the second level problem, starting with tariffs at zero, and use the optimal flows in order to gen-erate the inverse optimization problem and then identify the compatible tariffs which maximize the revenue. The procedureis repeated with the newly obtained tariffs until the revenue is stationary.

When solving the second level problem for a fixed tariff policy, an optimistic optimization is performed. The objectivefunction of the follower’s problem is modified to achieve this goal by considering the following alternative second levelobjective:

minf l ;f c

Xk2K

Xe2Ek

Xi2Il

e;k

ðð1� �Þtlk þ qi;kða; bÞÞf l

i þXk2K

Xi2Ic

k

cci ða; bÞf c

i ;

where � takes a small value. Thus, for two equivalent follower’s solutions, the one favoring the leader will be chosen. The twoinitial solutions are then compared and the one that yields the highest revenue is selected and given to Cplex as an initialsolution.

Algorithm 3. Greedy procedure for the construction of an initial solution

– Let G ¼ [e2EGe be the multigraph representing all block networks.– Let K :¼ K .while K–; do

– For all k 2 K and i 2 Ilk, let /i be the maximal flow on itinerary i represented by the minimum capacity of the path in

G.– Let Ui = min{dk,/i}, where dk is the demand associated with itinerary i.– Determine the itinerary �i generating the maximum marginal revenue:

�i ¼ arg maxk2K;i2Il

k

cckða; bÞ � qi;kða;bÞ

� �Ui

n o;

and let �k be the request associated with �i.if U�i > 0 then

– Assign the flow U�i to itinerary f l�i.

– Assign the residual demand, d�k �U�i, to the itinerary of the competition f c�k:

– Update the residual capacities of G by reducing the various capacities of �i by U�i.

– Set K :¼ K n �k.else

The network is saturated (assuming that 8k 2 K; i 2 Ilk; c

ckða; bÞ > qi;kða; bÞ) and therefore:

foreach k 2 K do– Assign demand dk to the itinerary of the competition f c

k :

– Set K :¼ K n k.

Following preliminary tests, and due to the limited number of inequalities generated, it seems unnecessary to use toolssuch as Cplex’s user cuts where valid cuts from a pool are dynamically added to the model. Furthermore, we will see thatinequalities have a major impact on the integrality gap at the root node of the branch-and-bound tree when these are im-posed at the beginning of the solution process. Other tests were also made to assess the impact, at different nodes of thebranching tree, of the identification of a feasible solution based on the tariffs obtained from the linear relaxation at the nodeconsidered. However, these tests have not shown, on average, that it is desirable to impose this procedure.

In the results reported in Tables 4 and 6, the column identified by GAPr indicates the integrality gap at the root node of thebranch-and-bound tree. This gap is obtained after dividing, by the objective value of the initial solution, the difference be-tween the objective value of the problem’s relaxation and the objective value of the initial solution. Similarly, the columnGAPf gives the final integrality gap. In the CPU column, the computing time in seconds is presented. Note that a limit of threehours of computing time is imposed. Thus, when the value of GAPf is positive, the time limit has been reached and the LimTlabel appears in the CPU column. Note that all the algorithmic procedures were coded in C++ and the Cplex 10.0 ConcertLibrary was used. Finally, the numerical tests were performed on an AMD Opteron 250 (2.4 GHz) computer.

Table 2Instances Inst2.

jKj jRj jPj jBj jEj jHj

pr07 75 3 3 20 4 15pr08 225 5 5 40 4 25pr09 375 7 7 60 4 35pr10 525 9 3 80 7 45pr11 675 11 5 100 7 55pr12 825 13 7 150 7 65

Table 3Parameters of the classes of request.

Class a bc bl Qc

1 0.1 0.6 1 11 1 11 100 1502 0.6 1.1 11 21 11 21 50 1003 1.1 1.6 21 31 21 31 10 50

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 115

Table 5 shows the gains from adding the inequalities from Proposition 3 when solving instances Inst1. In particular, thegains on the root node integrality gap indicate, on average, a reduction of about 28.63%. This gain is around 86% with respectto computation time. Moreover, optimality is now reached for all instances. These results clearly demonstrate the impact ofthe proposed cuts. A similar analysis is presented in Table 7 for instances Inst2 for which the average gain at the root node is18.92% and where three more instances have been solved to optimality. In addition, when the time limit is reached for pr11and pr12 the integrality gap is reduced by 56.76% and 71.77%, respectively, when the cuts are considered. Finally, a compu-tation time reduction of about 27.8% is observed.

Obviously, the use of an exact algorithm prevents, as it is often the case, the solution of very large instances. However, theapproach can be used to tackle problems where parts of the network are analyzed. That is why this paper focuses on thecreation of instances with demand corridors. In most railroads one can identify such subnetwork structures. Furthermore,these corridors often are the backbone of the carrier’s network. Therefore, carefully studying them could provide valuableinsights.

We have also analyzed the effect of the proposed properties and inequalities on achieving, for the leader, the optimal rev-enue or the best revenue identified by the approach presented. To this end, we estimate the gain of revenue when inequal-ities are imposed. We will assume as the reference point the CPU time needed by the algorithm to reach the optimal solutionor best obtained solution when the valid cuts are present. We then compare this revenue to the one identified by the exactmethod without inequalities after the same computation time. Tables 8 and 9 therefore present a measure, for instancesInst1 and Inst2, of the gain of revenue generated by the addition of the valid cuts.

We note that the average gain for the two sets of instances is 0.29%. We must however note that Cplex performs very wellwith regard to the identification of good solutions. Nevertheless it often requires an important computation time to establishthe optimality proof. This is where the addition of the valid inequalities offers the greatest impact. Despite this observation,the gain of revenue may be substantial.

In the last part of our computational experiments, we have compared the two pricing policies presented in Section 5. Thedisjoint pricing being a relaxation of the common policy, the expected revenue of the former will obviously be higher thanthe latter. In order to enforce the common policy, one could propose to evaluate, for each request, the average tariff obtainedfrom the disjoint policy and impose the resulting common rate to the follower. For request k such that

Pi2Il

kf li > 0, let tl

k bethe described average tariff:

Table 4Results

pr01pr02pr03pr04pr05pr06

tlk ¼

Pi2Il

kf li tl

iPi2Il

kf li

for Inst1.

Cplex Cplex + Proposition 3

GAPr (%) GAPf (%) CPU (s) GAPr (%) GAPf (%) CPU (s)

47.96 0.00 744.53 33.44 0.00 143.5423.06 0.00 2246.04 16.74 0.00 202.4828.40 0.00 7673.66 19.49 0.00 217.7324.54 1.21 LimT 18.54 0.00 1957.4323.18 0.00 688.14 16.92 0.00 180.2520.98 0.94 LimT 14.42 0.00 923.31

Table 5Gain analysis for Inst1.

GAPr (%) GAPf (%) CPU (%)

pr01 30.28 0.00 80.78pr02 27.41 0.00 90.99pr03 31.37 0.00 97.16pr04 24.45 100.00 81.88pr05 27.01 0.00 73.81pr06 31.27 100.00 91.45

Average 28.63 86.00

Table 6Results for Inst2.

Cplex Cplex + Proposition 3

GAPr (%) GAPf (%) CPU (s) GAPr (%) GAPf (%) CPU (s)

pr07 54.40 0.00 1107.22 42.92 0.00 224.27pr08 54.69 2.21 LimT 44.34 0.00 8614.70pr09 34.33 0.68 LimT 27.16 0.00 10004.90pr10 23.72 0.63 LimT 19.69 0.00 4376.24pr11 32.88 2.22 LimT 27.05 0.96 LimTpr12 29.00 2.09 LimT 23.81 0.59 LimT

Table 7Gain analysis for Inst2.

GAPr (%) GAPf (%) CPU (%)

pr07 21.10 0.00 79.74pr08 18.92 100.00 20.23pr09 20.89 100.00 7.36pr10 16.99 100.00 59.48pr11 17.73 56.76 0.00pr12 17.90 71.77 0.00

Average 18.92 27.80

Table 8Gain of revenue analysis for Inst1.

GAP (%)

pr01 2.25pr02 0.03pr03 0.08pr04 0.34pr05 0.00pr06 0.00

Average 0.45

Table 9Gain of revenue analysis for Inst2.

GAP (%)

pr07 0.35pr08 0.00pr09 0.10pr10 0.00pr11 0.14pr12 0.15

Average 0.12

116 B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119

Table 10Pricing policies comparison.

RG (%) RRRL (%)

pr01 33.18 �53.44pr02 10.03 �34.91pr03 14.32 �50.29pr04 9.48 �33.48pr05 11.53 �35.50pr06 11.49 �37.37pr07 24.23 �37.80pr08 18.61 �55.04pr09 11.04 �44.65pr10 7.99 �28.77pr11 10.41 �35.89pr12 9.42 �47.11

Average 14.31 �41.19

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 117

IfP

i2Ilkf li ¼ 0, set tl

k to a value greater than cckða; bÞ to prevent the use of these itineraries. The rational reaction to this tariff

vector ðtlÞ will induce a lower global revenue than the one generated from the common policy. Table 10 illustrates theseobservations. The column identified by RG provides the revenue gain of the disjoint policy over the common pricing policy.The RRRL column gives the rational reaction revenue loss when the average tariffs tl

k of the disjoint policy are imposed in thesecond level model. The value represents the loss percentage when the revenue corresponding to the second level reaction iscompared with the common pricing optimal revenue. From the results we notice that a disjoint pricing policy will provide,on average, 14.31% more revenue. Yet, when a common pricing policy is enforced, considering tl is clearly suboptimal. Herewe get, on average, a revenue 41.19% lower than what could be expected.

8. Conclusions

We have introduced a new model integrating operations planning and revenue management for a rail freight carrier. Wehave seen that there has been very few contributions on this topic by the scientific community, despite the undeniable prac-tical interest for this field of study. This is mainly due to the complexity of operations related to rail transportation, and thusto the integration of these operations with pricing optimization. By the analysis of two pricing policies, we have highlightedthe main model properties and shown the usefulness of some valid inequalities through their significant effect on both thebranch-and-bound tree’s root node integrality gap and overall CPU time. Future work will concentrate on the development ofother families of valid inequalities as well as heuristic approaches. Moreover, the formulation developed describes the objec-tive of the follower as a linear combination of different attributes. The analysis of the second level reaction based on choicemodels could represent an interesting extension which would increase the realism of the proposed formulations. These, likethe logit model (see for example Talluri and van Ryzin, 2005), introduce some probabilistic criteria in the selection process ofan alternative (an itinerary/tariff combination for example) by a user, resulting in a more accurate representation of itsbehavior.

Acknowledgements

Thanks are due to Vee Kachroo of the Canadian National Railway for fruitful discussions and for providing guidance ontest data generation. This work was supported by the Natural Sciences and Engineering Research Council of Canada andthe Fonds québécois de la recherche sur la nature et les technologies. This support is gratefully acknowledged. We are alsograteful to three anonymous referees for their valuable comments.

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