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Transportation Research Part B journal homepage: www.elsevier.com/locate/trb

Integrated operations planning and revenue management for rail freight transportation Benoit Crevier a,b, Jean-François Cordeau a,⇑, Gilles Savard b a b

Canada Research Chair in Logistics and Transportation and CIRRELT, HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Canada H3T 2A7 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 2010 Received in revised form 3 September 2011 Accepted 4 September 2011

Keywords: Rail freight transportation Revenue management Operations planning Pricing Bilevel optimization

a b s t r a c t In the rail industry, proﬁt maximization relies heavily on the integration of logistics activities with an improved management of revenues. The operational policies chosen by the carrier have an important impact on the network yield and thus on global proﬁtability. This paper bridges the gap between railroad operations planning and revenue management. We propose a new bilevel mathematical formulation which encompasses pricing decisions and network planning policies such as car blocking and routing as well as train make-up and scheduling. An exact solution approach based on a mixed integer formulation adapted to the problem structure is presented, and computational results are reported on randomly generated instances. Ó 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 their strategic position on the market. The revenue aspect of day-to-day activities is of greater importance nowadays since many industries have noticed that proﬁt maximization relies heavily on the integration of logistics activities with an improved management of revenues. That is why revenue management is becoming more and more vital to many industries such as telecommunications, 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 North American 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 predominant impact over the policies of the rail industry. In the United States, for example, the Staggers Rail Act of 1980 gave more freedom to the rail companies while competition acquired a greater regulation role over the tariffs set for the transportation of freight. As a response to this pressure from the market, rail carriers were forced to review their business management processes 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 to take advantage of these innovations by establishing a better management of their activities. However, rail operations are made 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 transportation has always been confronted to operations management tools which did not encompass all the realism of day-to-day

⇑ Corresponding author. Fax: +1 514 340 6834. E-mail address: [email protected] (J.-F. Cordeau). 0191-2615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.trb.2011.09.002

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operations. Part of the complexity also follows from the fact that railroads operate on an existing physical network that imposes constraints of its own. Service companies are also trying to include revenue management in their decision support systems. The success of this integration in the airline industry has encouraged others to follow in the same footsteps. The rail industry is characterized by customers with distinct attributes and perception of the proposed services (for example, the sensitivity to tariffs or service reliability). Furthermore, rail transportation has more ﬂexibility over capacity utilization and trafﬁc speed through the links of the network than other transportation modes. Moreover, a majority of companies are switching from tonnage-based to scheduled operations. This philosophy increases the reliability of the service from a customer’s perspective but imposes new constraints on the carrier. These are market conditions, according to Talluri and van Ryzin (2005), where the use of revenue management is likely to be beneﬁcial. The analysis of an integrated approach combining operations and revenue management is therefore an interesting perspective of research in an attempt to study the connections between network planning and pricing policies. The integration of revenue management and rail freight operations planning has not been considered very frequently in the 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 inequalities which are used to strengthen the mathematical model. An exact solution approach is proposed, and computational results are 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 operations and revenue management. In Section 3, we provide a deﬁnition of the problem and introduce a path-based mathematical formulation using speciﬁc networks which are also presented. Section 4 describes the reformulation of the problem as a mixed integer mathematical program. Section 5 studies the impact of the pricing policies on the formulation. Resulting properties and valid inequalities are also addressed. Section 6 shows how we generated the instances on which the solution methodology was tested. Finally, computational experiments are presented in Section 7, and this is followed by the conclusions. 2. Background The complex operations of rail transportation have been, for a long time, limiting the implementation of state-of-the-art methodologies and techniques designed to offer efﬁcient decision support systems. Nevertheless, recent applications have shown the potential gains that can result from the use of analysis tools such as mathematical optimization. In this perspective, this paper proposes a model that bridges the gap between operations planning and revenue management decisions which 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 plans must be designed. The blocking plan problem is concerned with the determination of the set of blocks that will be assembled at each yard. A block is a group of cars that are moved together, on one or more trains, from a common origin to a common destination. Cars in a block may pass at intermediate yards but they will not be reclassiﬁed until the block has reached its destination. This destination might represent the end point of the itinerary of some cars or an intermediate stop in the route of others that will be assigned to subsequent blocks. These activities must be performed and therefore planned at the different yards according to the classiﬁcation capacity. Yard management is consequently a central problem encountered in the rail industry. Decisions are also made towards train routing, make-up and scheduling. The make-up policies specify the assignment of blocks to trains. These operating decisions will form the trip plan of each car which deﬁnes the itinerary that will be followed from origin to destination. The locomotive assignment problem speciﬁes the management of locomotives in order to satisfy pulling power requirements and repositioning for future demands. At destination, cars are released and empty 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, the global problem is usually decomposed and subproblems are tackled sequentially. Obviously, such an approach leads to a suboptimal operating plan. The integration of some of these problems has therefore been studied. The industry also tries to 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 tools in 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 methodological 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 Huisman et al. (2005). If we look more closely at integrated rail operations management contributions, a tactical level formulation combining train service scheduling, car transit, blocking and make-up policies as well as yard classiﬁcation work allocation

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was introduced by Crainic et al. (1984) who proposed a heuristic method to minimize operating costs and delays. This approach 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 formulated as a time-space multi-commodity network problem and a heuristic approach was devised. Keaton (1989, 1992) has considered train connections, service frequency, blocking policies and car movements. A Lagrangian relaxation approach was ﬁrst developed by Keaton (1989) and an extension of these results was proposed by Keaton (1992). An intermodal operations planning model was presented by Nozick and Morlok (1997). A heuristic methodology was proposed where the authors analyze the linear relaxation solution of an integer programming model based on a time-space network. Train scheduling and trafﬁc assignment was studied by Gorman (1998a,b). An integer program, based on a 1-week horizon, is presented. The methodology 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 competitive environment. Armstrong and Meissner (2010) provide a good overview of railway revenue management for both freight and passenger transportation. As mentioned previously, the rail industry relies on heterogeneous characteristics deﬁning customer commodities and expectations towards the offered service. An appropriate segmentation can therefore be established accordingly. Pricing and capacity management policies can be elaborated to exploit that segmentation and a potential control or regulation of the demand can thus be obtained. As a consequence, carriers may possibly smooth demand, reduce global network congestion and make better use of assets. For most companies, access to a reliable transportation system represents a key element in the selection of a carrier. In turn, the carrier aims at increasing service reliability by a more efﬁcient management of the operations coupled with adequate demand management policies in order to maximize proﬁts. Revenue management tries to tackle these problems. Service differentiation through market segmentation is a well known marketing approach to provide customers with appropriate service according to their characteristics and sensitivity to service attributes. Rail service reliability analyses and the positive impact of a trafﬁc segmentation in service or priority classes were presented, among others, by Kwon (1994) and Kraft (1995). Research in revenue management addresses problems such as overbooking, capacity management or pricing, notably in the airline industry. McGill and van Ryzin (1999) have presented an overview of the main contributions. In the quest to merge revenue management and operations planning, service companies are more and more interested in integrated capacity 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 publications 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 the airline industry. In rail transportation, revenue management has been mostly applied in the context of passenger transportation. Kraft et al. (2000) as well as Johnston (2006) exposed the characteristics of policies designed by Amtrak to control seat and cabin sales according to booking class and market structure. Ben-Khedher et al. (1998) have described the implementation of a strategic scheduling and planning system and a tactical capacity management system for high-speed train services 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 optimal pricing policies, the marketing of products offered and the dynamic management of seat inventories. The competitive environment is also considered. In rail freight transportation, capacity management was considered by Harker and Hong (1994). A track pricing problem is 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 movements to maximize proﬁts. For their part, authors have dealt with some aspects of revenue management in intermodal transportation. Yan et al. (1995) have studied a problem where the main objective was the evaluation of the opportunity cost of demands 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 postponement of low priority loads to favor the movements of those with higher priority. Finally, Gorman (2001, 2005) studied the pricing policies for intermodal operations at the Burlington Northern and Santa Fe (BNSF) Railway. The integration of pricing and operations management was studied by Li and Tayur (2005) in the intermodal transportation of trailers on ﬂatcars. Using a 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 proposed approach to price optimization while considering customer reaction to pricing and other operational policies is based on bilevel mathematical programming which is used to formulate hierarchical decision making. It can be represented by a system where 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 literature for linear, linear-quadratic or nonlinear programs. Bilevel programming was also treated by Dempe (2002).

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A generic model that integrates pricing decision imposed by the leader on a set of goods or services and the reaction of the users according to those policies was developed by Labbé et al. (1998a). An application in a multi-commodity transportation context 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 pricing problems on a single commodity transportation network. Heuristics were developed as well as improvement strategies. Multi-commodity contexts were studied by Brotcorne et al. (2001). A combined network design and pricing model was presented by Brotcorne et al. (2008). Côté et al. (2003) proposed a static and deterministic bilevel model for the pricing and allocation of capacity in the airline industry. A market segmentation based on service attributes such as price, transit duration or service quality was devised. Finally, Castelli et al. (2004) have presented a bilevel model representing the behavior of two 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 we propose a representation of the relations between the most important decisions through a conceptual framework similar to the one presented by Bartodziej et al. (2007) for air cargo transportation. Next we expose the ideas needed to formalize these interactions 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 to traditional tonnage-based activities. Therefore, carriers need to design, at a tactical level, train schedules to meet expected demand. At an operational level, customer requests must then be assigned to the available transportation capacity. Transportation requests are commonly deﬁned as a certain commodity for which speciﬁc attributes are known, such as the demand, which can be measured in number of cars to move or in an equivalent volume measure unit. Every request needs to be transported from a given origin to a given destination (OD) through one or several classiﬁcation yards and is characterized by a release time at which the commodity becomes available for transportation by the carrier and a due time representing a deadline for the load to arrive at destination. Furthermore, requests can be segmented according to perception of the customer towards, for instance, transit duration or service quality (equipment type preferences, for example). The itineraries follow from the schedule and deﬁne the admissible paths for a commodity to be transported from origin to destination. The movement of cars on an itinerary is limited by constraints such as block and locomotive pulling power capacities which are induced by tactical decisions. The problem addressed here is thus to simultaneously analyze capacity management and pricing in the relationship between each OD and its corresponding set of admissible itineraries. These itineraries will be obtained through enumeration of feasible 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 demand has been expressed. To each request k 2 K is assigned an origin–destination pair (o(k), d(k)) as well as a set of compatible equipment types Ek. For each equipment type e 2 E, we deﬁne a multi-commodity space-time network Ge = (Ne, Ae) where Ne represents the vertex set, and Ae is the arc set of the graph where products circulating in the network are the commodities k such that e 2 Ek. Let B represent the set of blocks deﬁned by the rail company. Therefore, Be Ae describes the set of blocks for 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 frequencies and predeﬁned itineraries. Moreover, we assume that a train is characterized by a movement between two successive classiﬁcation yards. Therefore no stop is made between departure and arrival. Thus the capacity level that will be considered already 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 are known. 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 by the scheduled trains. The others respectively designate the end of the block disassembly period for a speciﬁc train, and the end of the time window devoted to block assembly or, in other words, the latest time after which the assembly of cars into blocks 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 and the make-up of the trains as well as the reverse operations at destination. Horizontal arcs indicate the storage of cars at each

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...

Day 1

...

Day 2

Classification yard 1

...

Day 3

... B

...

C A Classification yard 2

...

...

D E

Classification yard 3

...

...

Legend:

departure arrival

end of the disassembly of blocks end of time window for the assembly of blocks

movements of blocks storage

assembly disassembly

Fig. 1. Blocking network representation.

yard up to their departure time on a later block or up to their delivery to the consignee. Finally, arcs representing the movements of blocks on trains are set according to make-up policies. For instance, arc A in Fig. 1 speciﬁes the movement of a block with yard 3 as origin and 1 as destination. This block will not be reclassiﬁed at intermediate yards along its path. Since blocking and make-up policies are known beforehand, trains on which block A will travel are pre-established. One can observe that 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 which represents the destination of both blocks. Fig. 2 gives an example of train movements which are compatible with the deﬁned blocks. Let {X;(y, z)} represent block X having yard y as origin and yard z as destination. In addition to the transit of blocks A and B deﬁned earlier, movements of blocks C, D and E are also presented. Therefore, blocks C and D move simultaneously between yards 1 and 2 and, following removal of block C reaching its destination and the addition of E, a ﬁnal transit is made to 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 as leader and denoted l in what follows, determines at the ﬁrst level a pricing policy for the transit of commodities in the network 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 relies on feasible paths in the space-time networks developed previously. In order to establish the mathematical structure of the proposed 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. Ilk ¼ [e2Ek Ile;k ; Il: set of itineraries offered by the leader; Ick : set of itineraries offered by the competition for request k; an element i 2 Ick can represent an itinerary from trucking, maritime or competing rail companies between o(k) and d(k);

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...

Day 1

...

Day 2

Classification yard 1

...

Day 3

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

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

Classification yard 2

...

...

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

Classification yard 3

...

...

Legend:

departure

movements of trains

arrival Fig. 2. Train movements representation.

H: set of trains; Ha: set of trains on which block a is assigned; Bei;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 since even though no disassembly or reclassiﬁcation is made, track capacity is needed therefore limiting the global capacity of the 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 locomotives assigned 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 t ci is considered as known; tl: tariff vector of the leader; fil and fic : ﬂow on itinerary i for the leader and the competition, respectively; fl and fc: ﬂow vectors of the leader and competition,

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and a characterization of each itinerary as well as the sensitivity of customers to service quality, as proposed by Marcotte and Savard (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 associated with request k; cli ða; 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 cli ða; bÞ ¼ t li þ qi;k ða; 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 is cci ða; bÞ ¼ t ci þ ak Di þ bk Q i . The developed model is deterministic and the itinerary generation phase allows us to evaluate the duration Di of each path i. This information is used, for instance, in the computation of the perceived costs just described. Furthermore, it can be used to eliminate from a customer’s set of admissible itineraries those for which the duration exceeds a speciﬁc due time established 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 used to move its freight. However, every customer (or class of customers) will have its speciﬁc perception of the service offer. For instance, some of them will put more importance on transit duration than others and may be willing to pay a premium to access faster itineraries. In order to take this aspect into account in the combined analysis of freight assignment and pricing we consider a linear combination of the tariff with some attributes of the shipper. This allows the carrier to characterize the perceived disutility of itineraries for each request and evaluate, in a planning phase, a proper freight assignment and corresponding pricing policy. We can model the combined pricing and network capacity management problem (PCM) as follows:

ðPCMÞ

max tl ;f l ;f c

s:t: min f l ;f c

s:t:

XXX

t li fil

k2K e2Ek i2Il e;k

t l P 0; XXX

ð1Þ cli ða; bÞfil þ

fil þ

X

fic ¼ dk

8k 2 K; ðkk Þ

ð2Þ

i2Ick

e2Ek i2Il e;k

X X

cci ða; bÞfic

k2K i2Ick

k2K e2Ek i2Il e;k

XX

XX

fil 6 cape;a

8a 2 B; e 2 Ea ; ðge;a Þ

ð3Þ

kje2Ek ija2Bei;k

X X X

fil 6 capa

8a 2 B; ðpa Þ

ð4Þ

e2Ea kje2Ek ija2Bei;k

X X X X

fil 6 caph

8h 2 H; ðch Þ

ð5Þ

ajh2Ha e2Ea kje2Ek ija2Bei;k

X

fil 6 capr

8r 2 R; p 2 P; ðxr;p Þ

ð6Þ

i2Sr;p

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. Constraints (3) allow a maximal capacity for transporting a speciﬁc equipment type on a block. Constraints (4) limit equipment transit on blocks. Constraints (5) ensure the pulling capacities are respected. Car handling capacity of classiﬁcation tracks at every yard is imposed by (6). Finally, (1) and (7) deﬁne constraints on decision variables of the ﬁrst and second levels. From the presented formulation we can observe that network capacity resulting from scheduled operations planning is considered to have been determined beforehand. It is thus ﬁxed and an unsold unit will be lost at the departure time of the corresponding trains. In this OD-based model, similar to others proposed in the airline industry such as those of Marcotte and 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

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107

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

max

X X X

t l ;f l ;f c

t li hi fil

k2K e2Ek i2Il e;k

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 integrated yield and pricing management problem which is based on a reformulation of the bilevel model as a single-level mixed integer 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 the substitution of the second level mathematical program by its primal–dual optimality conditions. We present a similar methodology for the PCM. First of all, the integrality of the commodity ﬂow decision variables can be relaxed since, as authors like Marín and Salmerón (1996) mention, the number of cars considered is large and the ﬂexibility over network capacity allows some error margin. 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 their capacities are sufﬁcient to meet the total demand for all commodities. Faced with the choice of an alternative to the itineraries of the leader, each request will thus select the competition itinerary with the least perceived cost. Therefore, for k 2 K let fkc ¼ fic and cck ða; bÞ ¼ cci ða; bÞ where i ¼ arg mini2Ick cci ða; bÞ. Obviously, in order to have a positive ﬂow on itinerary i 2 Ilk , l c we must have that ci ða; bÞ 6 ck ða; bÞ. In the proposed model, constraints (2) impose that the follower’s demand must be satisﬁed by the leader or the competition. Thus, as pointed out in the last paragraph, the follower will consider the leader’s service offer as long as its perceived cost falls below or is equal to the best offer on the market. Indeed, we suppose an optimistic optimization where for two equivalent follower’s solutions, the one favoring the leader will be chosen. Moreover, some demand can also be lost to competition because of network capacity limitations. Therefore the optimization process aims at setting the leader’s tariffs, while considering 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 program of the second level problem of the PCM can be stated as:

max

k;g;p;c;x

s:t:

X

dk kk þ

k2K

kk þ

XX

cape;a ge;a þ

a2B e2Ea

X

X

capa pa þ

a2B

ðge;a þ pa Þ þ

a2Bei;k

X X

caph ch þ

XX

XX

capr xr;p

r2R p2P

h2H

ch þ

a2Bei;k h2Ha

X

yir;p xr;p 6 cli ða; bÞ 8k 2 K; e 2 Ek ; i 2 Ile;k

ð8Þ

r2R p2P

kk 6 cck ða; bÞ 8k 2 K

ð9Þ

g60 p60 c60 x60

ð10Þ ð11Þ ð12Þ ð13Þ

yir;p

where parameter 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 complementarity constraints:

2

0

4cl ða; bÞ @kk þ i

X

ðge;a þ pa Þ þ

a2Bei;k

X X

ch þ

a2Bei;k h2Ha

cck ða; bÞ kk fkc ¼ 0 8k 2 K 0

B kk @dk

XX e2Ek i2Il e;k

0

ge;a @cape;a

XX

13

yir;p

xr;p A5fil ¼ 0 8k 2 K; e 2 Ek ; i 2 Ile;k

ð14Þ

r2R p2P

ð15Þ

1 C fil þ fkc A ¼ 0 8k 2 K

X X kje2Ek ija2Bei;k

ð16Þ

1 fil A

¼ 0 8a 2 B; e 2 Ea

ð17Þ

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0

X X X

pa @capa

1 fil A

¼ 0 8a 2 B

ð18Þ

e2Ea kje2Ek ija2Bei;k

0

X X X X

ch @caph

1 fil A

¼ 0 8h 2 H

ð19Þ

ajh2Ha e2Ea kje2Ek ija2Bei;k

0

X

xr;p @capr

1 fil A

¼ 0 8r 2 R; p 2 P:

ð20Þ

i2Sr;p

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

ðPCM1 Þ

XXX

max

t l ;f l ;f c ;k;g;p;c;x

t li fil

k2K e2Ek i2Il e;k

s:t: t l 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 the follower’s problem. The theorem ensures that, at optimality, the primal and dual objectives of the follower will be equal:

XXX

tli fil þ

k2K e2Ek i2Il e;k

þ

X

XXX

qi;k ða; bÞfil þ

k2K e2Ek i2Il e;k

capa pa þ

a2B

X

caph ch þ

X

cck ða; bÞfkc ¼

X

k2K

XX

dk kk þ

XX

cape;a ge;a

a2B e2Ea

k2K

capr xr;p ;

r2R p2P

h2H

and thus:

XXX

t li fil ¼

k2K e2Ek i2Il e;k

X k2K

5. Pricing policies

dk kk þ

XX

cape;a ge;a þ

X

a

a2B e2E

XXX k2K e2Ek i2Il e;k

qi;k ða; bÞfil

capa pa þ

a2B

X

X h2H

caph ch þ

XX

capr xr;p

r2R p2P

cck ða; bÞfkc :

k2K

In this section we discuss two pricing policies that carriers can put forward. We will consider cases where tariffs are set either at an itinerary or at a request level. Appropriate pricing will take into account different criteria. Notably, the type of freight, origin and destination of the transit and the type of equipment used. The itineraries appearing in the formulation presented in the previous section are characterized by those attributes. The preferred pricing policy will therefore need to be 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 highly disaggregated pricing strategy since each itinerary is assigned its own tariff. In a second analysis, the imposition of equality constraints between the tariffs for the itineraries of the same request will be considered. In this case, the prescribed tariff for a given request relies on an aggregation of information regarding the path in the network followed by the itineraries or the equipment used since a single tariff must be determined. Other pricing policies, which are not addressed here, might consider requests aggregation according to their origin–destination pair or freight transported in order to identify similar tariffs for 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 earlier illustrates this context. 5.1.1. Moving constraints to the ﬁrst level The difﬁculty of solving the proposed model comes mainly from the nonlinear complementarity constraints and their linearization which would introduce a large number of binary variables. It is therefore essential to analyze the structure of the model in order to identify the potential movements of constraints from the second to the ﬁrst level. A similar study is presented by Brotcorne et al. (2008). The authors show that, for a certain class of bilinear bilevel programs, the movement of

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constraints from the second to the ﬁrst level is valid if the corresponding dual variables take the value zero at optimality. This characteristic can be veriﬁed for the capacity constraints of the proposed model. We will only study constraints (3), since a similar 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⁄ – 0 which implies that there is at least one dual variable such that ge;a < 0. Set the vector g0 = 0, deﬁne (tl)0 where the tariff P assigned to itinerary i 2 Ile;k is ðt li Þ0 ¼ ðt li Þ a2Be ge;a , and consider the solution ((tl)0 , (fl)⁄, (fc)⁄, k⁄, g0 , p⁄, c⁄, x⁄). This solution i;k

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

X

kk þ

ge;a þ pa þ

a2Bei;k

and thus:

kk þ

a2Bei;k

X

ge;a þ pa þ

a2Bei;k

kk

þ

X

þ

ch þ

a

p þ

X

X X

X X

a2Bei;k h2Ha 0 e;a

a

g þp

a2Bei;k

þ

h

c þ

XX

yir;p xr;p 6 cli ða; bÞ

r2R p2P

h2Ha

ch þ

a2Bei;k h2Ha

a2Bei;k

kk

X X

yir;p xr;p 6 ðt li Þ þ qi;k ða; bÞ

r2R p2P

XX

yir;p

r2R p2P

X X

XX

h

c þ

a2Bei;k h2Ha

x

r;p

0 1 l X @ 6 ti ge;a A þ qi;k ða; bÞ

XX

a2Bei;k

yir;p

xr;p 6 cli ða; bÞ0

r2R p2P

The solution is therefore feasible for these constraints. Regarding complementarity constraints, only (14) and (17) need to be considered since the others only depend on the optimal vectors of the proposed solution. An equivalent analysis to the one for constraints (8) can be made for complementarity constraints (14). Moreover, since g0 = 0, constraints (17) will be satisﬁed. The only constraints left to verify are the ﬁrst level non-negativity of the tariffs. By deﬁnition one has g⁄ 6 0 and by hypothesis there is at least one dual variable for which ge;a < 0. Thus one can deduce that there is at least one itinerary ^i 2 Il such that P e g < 0 and consequently 0 6 ðtl Þ < ðtl Þ0 . We can conclude that the proposed solution is feasible ^i ^i e;a a2B^ e;k i;k and generates a greater revenue since

XXX

ðt li Þ fil <

k2K e2Ek i2Il e;k

XXX

ðt li Þ0 fil

k2K e2Ek i2Il e;k

This contradicts the optimal nature of ((tl)⁄, (fl)⁄, (fc)⁄, k⁄, g⁄, p⁄, c⁄, x⁄) and shows that g⁄ = 0. An identical conclusion can be obtained 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-dual formulation PCM1 with disjoint pricing is the simpliﬁcation of constraints (8) and (14) as well as the elimination of constraints (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 be equal to the least cost itinerary offered by the competition. Proposition 2. Under an optimal pricing policy, kk ¼ cck ða; 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⁄) ^ 2 K such that k^ – cc ða; bÞ which implies, be an optimal solution for the problem. Suppose that for this solution there exists k ^ k k 0 c 0 by constraints (9), that for this request kk^ ck^ ða; bÞ < 0. Let k be such that kk ¼ cck ða; bÞ as well as (tl)0 for which the tariff 0 assigned to itinerary i 2 Ile;k is deﬁned by t li ¼ t li kk cck ða; 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 revenue than ((tl)⁄, (fl)⁄, (fc)⁄, k⁄, g⁄, p⁄, c⁄, x⁄), which contradicts its optimality and shows that kk ¼ cck ða; 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 the tariff margin between this itinerary and the best itinerary offered by the competition for that request. Corollary 1. For every itinerary i 2 Ilk ; tli ¼ cck ða; bÞ qi;k ða; bÞ. Proof. We know that cli ða; bÞ 6 cck ða; bÞ. From constraints (8), Propositions 1 and 2 we have cck ða; bÞ 6 cli ða; bÞ. Therefore,

cli ða; bÞ ¼ cck ða; bÞ

ð21Þ

tli

ð22Þ

¼

cck ð

a; bÞ qi;k ða; bÞ

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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:

XXX

t li fil ¼

k2K e2Ek i2Il e;k

X

XXX cck ða; bÞ dk fkc qi;k ða; bÞfil : k2K e2Ek i2Il e;k

k2K

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

max f l ;f c

s:t:

X

XXX cck ða; bÞ dk fkc qi;k ða; bÞfil k2K e2Ek i2Il e;k

k2K

XX

fil þ fkc ¼ dk

8k 2 K

ð23Þ

e2Ek i2Il e;k

X X

fil 6 cape;a

8a 2 B; e 2 Ea

ð24Þ

kje2Ek ija2Bei;k

X X X e2E

fil 6 capa

8a 2 B

ð25Þ

kje2Ek ija2Bei;k

a

X X X X

fil 6 caph

8h 2 H

ð26Þ

ajh2Ha e2Ea kje2Ek ija2Bei;k

X

fil 6 capr

8r 2 R; p 2 P

ð27Þ

i2Sr;p

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 a unique tariff for the shipment of each car of a speciﬁc commodity, regardless of the itinerary on which the freight of the follower will be assigned. The resulting model is more complex since moving constraints from the second to the ﬁrst level is not possible in this setting. All complementarity constraints must be considered. To model this pricing strategy, the only modiﬁcation needed is a small change to the deﬁnition of the perceived cost. For an itinerary i offered by the leader, the cost is now cli ða; bÞ ¼ t lk þ ak Di þ be;k Q i . Thus, the tariff now depends on request k, and not only on the considered itinerary. Furthermore, 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:

XXX k2K e2Ek i2Il e;k

t lk fil ¼

X

X X XX cck ða; bÞ dk fkc þ cape;a ge;a þ capa pa þ caph ch a2B e2Ea

k2K

þ

XX r2R p2P

capr xr;p

XXX

a2B

h2H

; bÞfil :

qi;k ða

k2K e2Ek i2Il e;k

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

8 P P P PP i > ðge;a þ pa Þ ch yr;p xr;p 6 M 1i x1i cl ða; bÞ cck ða; bÞ > > i e e h2H > r2R p2P a2Bi;k a2Bi;k a > > < 2 ð14Þ () fil 6 M i x2i > > 1 2 > xi þ xi 6 1 > > > : 1 2 xi ; xi 2 f0; 1g

ð29Þ

8 P P l fi 6 M 1e;a x1e;a cape;a > > > kje2Ek ija2Bei;k > > > < 2 2 ð17Þ () ge;a 6 Me;a xe;a > > 1 2 > xe;a þ xe;a 6 1 > > > : 1 2 xe;a ; xe;a 2 f0; 1g

ð30Þ

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

max

t l ;f l ;f c ;g;p;c;x

X

X X XX XXX XX cck ða;bÞ dk fkc þ cape;a ge;a þ capa pa þ caph ch þ capr xr;p qi;k ða;bÞfil

k2K

a2B e2Ea

a2B

h2H

r2R p2P

k2K e2Ek i2Il e;k

s:t: tl P 0 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 inequalities We now propose a set of valid inequalities to strengthen the previously proposed formulation. These cuts connect ﬂow variables with their corresponding tariffs. n o Proposition 3. Let k 2 K and qi ;k ða; bÞ ¼ min qi;k ða; bÞji 2 Ilk : The following inequalities are valid:

tlk þ ðqi;k ða; bÞ qi ;k ða; bÞÞx2i 6 ðcck ða; bÞ qi ;k ða; bÞÞ 8i 2 Ilk Proof. First, we know that the cost of every itinerary is bounded by the least cost itinerary of the competition,

cli ða; bÞ 6 cck ða; 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 ;k ða; bÞ 6 6 qin ;k ða; bÞ. Therefore qi ;k ða; bÞ ¼ qi1 ;k ða; bÞ. Thus,

cli1 ða; bÞ 6 cck ða; bÞ t lk þ qi1 ;k ða; bÞ 6 cck ða; bÞ t lk 6 cck ða; bÞ qi1 ;k ða; bÞ which represents an upper bound on the tariff of the itineraries for request k.oHowever, if in the optimal solution the ﬂow on n ij – i1 is positive, where ij is such that qij ;k ða; bÞ ¼ max qi;k ða; bÞji 2 Ilk ; fil > 0 , the bound on the tariff will become

tlk 6 cck ða; bÞ qij ;k ða; bÞ

thus reducing the upper bound by cck ða; bÞ qi1 ;k ða; bÞ cck ða; bÞ qij ;k ða; bÞ ¼ qij ;k ða; bÞ qi1 ;k ða; bÞ. Finally, the ﬂow on ij can be positive only if x2ij ¼ 1 and the bound reduction will be enforced in these circumstances. Thus we can conclude that

t lk 6 ðcck ða; bÞ qi1 ;k ða; bÞÞ ðqij ;k ða; bÞ qi1 ;k ða; bÞÞx2ij t lk þ ðqij ;k ða; bÞ qi1 ;k ða; bÞÞx2ij 6 ðcck ða; bÞ qi1 ;k ða; bÞÞ or

tlk þ ðqij ;k ða; bÞ qi ;k ða; bÞÞx2ij 6 ðcck ða; bÞ qi ;k ða; 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. The proposed 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 timetable, the index of a yard representing the origin and another the destination are generated. The itineraries of trains between

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the different origin–destination pairs are then enumerated. Following the determination of an origin and destination for each block, the blocking network is deﬁned by the assignment, to every block, of a compatible train route. To each block is assigned a number of equipments that will be valid on it. Then, the admissible block itineraries are enumerated for each subnetwork resulting from a given equipment. Thus, according to the characteristics of each request k, the valid itineraries will be added to the set Ilk . Finally a route of the competition is created for every request and represents the best service offer on the market. It should be noted that unlike authors such as Bodin et al. (1980), Crainic et al. (1984), Newton et al. (1998) or Ahuja et al. (2007), we do not consider the problem of constructing a blocking plan. As mentioned above, we assume that blocking 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 on the creation of demand corridors. To this end, we deﬁne an attraction factor for each yard in order to produce regions of high demand density. Among the most dense centers, a number of yards will be selected and labeled as hubs. These will be located n j ko more strategically. For the subsequently generated instances, we have max 1; jRj hubs. The other yards, considered as 4 satellites, will then be scattered throughout the network. Algorithm 1 shows the location procedure. The networks of movements of trains and blocks will then be generated so that the transportation capacity between the different yards is representative of the density of demand that transits there. The schedule of each train is obtained by generating a random starting time and ﬁxing the arrival time as the time required to travel between the two yards. To this time is added or subtracted a certain 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 multiplied by the average speed of a freight train which is about 32.19 km per hour (or 20 miles per hour). When generating the route of the competition associated with a request k, the average duration Dk of the itineraries of the leader for this request is assessed 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

h l m 1 2

jKj jRj

l mi ; 2 jKj according to a discrete uniform distribution. The attraction jRj

u factor (AF) of yard i is deﬁned by AF i ¼ jKj .

– Set i:¼i + 1.

n j ko yards with the highest attraction factors, deciding randomly in case of equality, and create 2. Select the max 1; jRj 4 the hub set Rhub. 3. Location of hubs: foreach r 2 Rhub do if no hub has been located then Generate 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.

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Moreover, according to the latest statistics obtained from The Railway Association of Canada (RAC), we can estimate that, 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 assignment to yards. A similar procedure is used with regard to train schedules and the creation of blocks. These will not be described 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 < AF iO 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 AF iD 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 the RAC revealed that during the last 20 years, the number of cars per train is, on average, around 70. Thus, we establish the window of possible values for the capacity of a train as [60, 80]. A similar analysis is performed for the other types of capacity so that the resulting instances are realistic. Finally, the windows of values for the various parameters related to the perception 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. In Table 2, the instances keep most of the network features of the instances presented in Table 1 but are densiﬁed. In regard to Table 3, recall that the parameter a is the factor associated with the conversion, in monetary value, of one unit of duration and 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 the value assigned to it will be selected. Note that the superscript indicates, once again, the leader or the competition. The ranges for Qc were chosen while considering the interval for Ql deﬁned by [50, 100] in order to maintain consistency with the characterization 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;k ða; bÞ > cck ða; bÞ must obviously be rejected. We therefore consider the itinerary ^i 2 Ilk such that n o q^i;k ¼ max qi;k ða; bÞji 2 Ilk : Following the assignment of the perceived cost qck of the itinerary of the competition, let

¼ q^i;k qck . Therefore, [ + 50, + 150].

when

6 0,

t ck is randomly selected in the [50, 150] interval, otherwise the value is chosen in

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

Table 1 Instances Inst1.

pr01 pr02 pr03 pr04 pr05 pr06

jKj

jRj

jPj

jBj

jEj

jHj

50 150 250 350 450 550

3 5 7 9 11 13

3 5 7 3 5 7

20 40 60 80 100 150

3 3 3 6 6 6

15 25 35 45 55 65

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Another procedure generating an initial solution to be compared to the one obtained by the greedy algorithm is based on solving an inverse optimization problem. As authors such as Ahuja and Orlin (2001) mention, an inverse optimization problem consists in inferring the values of the model parameters (cost coefﬁcients, right-hand side vector, and the constraint matrix), given the values of observable parameters (optimal decision variables). The problem considered here arises from the ﬂow generated when the tariffs are set to zero. Given the possible degeneracy of the mathematical program of the second level, we successively solve the second level problem, starting with tariffs at zero, and use the optimal ﬂows in order to generate the inverse optimization problem and then identify the compatible tariffs which maximize the revenue. The procedure is 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 objective function of the follower’s problem is modiﬁed to achieve this goal by considering the following alternative second level objective:

min f l ;f c

XXX

ðð1 Þtlk þ qi;k ða; bÞÞfil þ

XX

k2K e2Ek i2Il e;k

cci ða; bÞfic ;

k2K i2Ick

where takes a small value. Thus, for two equivalent follower’s solutions, the one favoring the leader will be chosen. The two initial solutions are then compared and the one that yields the highest revenue is selected and given to Cplex as an initial solution. Algorithm 3. Greedy procedure for the construction of an initial solution – Let G ¼ [e2E Ge 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 max k2K;i2Ilk

n o cck ða; bÞ qi;k ða; bÞ Ui ;

be the request associated with i. and let k if Ui > 0 then – Assign the ﬂow Ui to itinerary fil . – Assign the residual demand, dk Ui , to the itinerary of the competition fkc : – Update the residual capacities of G by reducing the various capacities of i by Ui . – Set K :¼ K n k. else The network is saturated (assuming that 8k 2 K; i 2 Ilk ; cck ða; bÞ > qi;k ða; bÞ) and therefore: foreach k 2 K do – Assign demand dk to the itinerary of the competition fkc : – Set K :¼ K n k.

Following preliminary tests, and due to the limited number of inequalities generated, it seems unnecessary to use tools such as Cplex’s user cuts where valid cuts from a pool are dynamically added to the model. Furthermore, we will see that inequalities have a major impact on the integrality gap at the root node of the branch-and-bound tree when these are imposed at the beginning of the solution process. Other tests were also made to assess the impact, at different nodes of the branching tree, of the identiﬁcation of a feasible solution based on the tariffs obtained from the linear relaxation at the node considered. 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 the branch-and-bound tree. This gap is obtained after dividing, by the objective value of the initial solution, the difference between the objective value of the problem’s relaxation and the objective value of the initial solution. Similarly, the column GAPf gives the ﬁnal integrality gap. In the CPU column, the computing time in seconds is presented. Note that a limit of three hours of computing time is imposed. Thus, when the value of GAPf is positive, the time limit has been reached and the LimT label appears in the CPU column. Note that all the algorithmic procedures were coded in C++ and the Cplex 10.0 Concert Library was used. Finally, the numerical tests were performed on an AMD Opteron 250 (2.4 GHz) computer.

115

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 Table 2 Instances Inst2.

pr07 pr08 pr09 pr10 pr11 pr12

jKj

jRj

jPj

jBj

jEj

jHj

75 225 375 525 675 825

3 5 7 9 11 13

3 5 7 3 5 7

20 40 60 80 100 150

4 4 4 7 7 7

15 25 35 45 55 65

Table 3 Parameters of the classes of request. Class

a

1 2 3

0.1 0.6 1.1

bc 0.6 1.1 1.6

1 11 21

bl 11 21 31

Qc

1 11 21

11 21 31

100 50 10

150 100 50

Table 5 shows the gains from adding the inequalities from Proposition 3 when solving instances Inst1. In particular, the gains on the root node integrality gap indicate, on average, a reduction of about 28.63%. This gain is around 86% with respect to computation time. Moreover, optimality is now reached for all instances. These results clearly demonstrate the impact of the proposed cuts. A similar analysis is presented in Table 7 for instances Inst2 for which the average gain at the root node is 18.92% and where three more instances have been solved to optimality. In addition, when the time limit is reached for pr11 and pr12 the integrality gap is reduced by 56.76% and 71.77%, respectively, when the cuts are considered. Finally, a computation 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, the approach can be used to tackle problems where parts of the network are analyzed. That is why this paper focuses on the creation 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 valuable insights. We have also analyzed the effect of the proposed properties and inequalities on achieving, for the leader, the optimal revenue or the best revenue identiﬁed by the approach presented. To this end, we estimate the gain of revenue when inequalities are imposed. We will assume as the reference point the CPU time needed by the algorithm to reach the optimal solution or best obtained solution when the valid cuts are present. We then compare this revenue to the one identiﬁed by the exact method without inequalities after the same computation time. Tables 8 and 9 therefore present a measure, for instances Inst1 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 well with regard to the identiﬁcation of good solutions. Nevertheless it often requires an important computation time to establish the 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. The disjoint pricing being a relaxation of the common policy, the expected revenue of the former will obviously be higher than the latter. In order to enforce the common policy, one could propose to evaluate, for each request, the average tariff obtained P from the disjoint policy and impose the resulting common rate to the follower. For request k such that i2Il fil > 0, let t lk be k the described average tariff:

P tlk

l l i2Ilk fi t i

¼ P

l i2Ilk fi

Table 4 Results for Inst1. Cplex

pr01 pr02 pr03 pr04 pr05 pr06

Cplex + Proposition 3

GAPr (%)

GAPf (%)

CPU (s)

GAPr (%)

GAPf (%)

CPU (s)

47.96 23.06 28.40 24.54 23.18 20.98

0.00 0.00 0.00 1.21 0.00 0.94

744.53 2246.04 7673.66 LimT 688.14 LimT

33.44 16.74 19.49 18.54 16.92 14.42

0.00 0.00 0.00 0.00 0.00 0.00

143.54 202.48 217.73 1957.43 180.25 923.31

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B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 Table 5 Gain analysis for Inst1. GAPr (%)

GAPf (%)

CPU (%)

pr01 pr02 pr03 pr04 pr05 pr06

30.28 27.41 31.37 24.45 27.01 31.27

0.00 0.00 0.00 100.00 0.00 100.00

80.78 90.99 97.16 81.88 73.81 91.45

Average

28.63

86.00

Table 6 Results for Inst2. Cplex

pr07 pr08 pr09 pr10 pr11 pr12

Cplex + Proposition 3

GAPr (%)

GAPf (%)

CPU (s)

GAPr (%)

GAPf (%)

CPU (s)

54.40 54.69 34.33 23.72 32.88 29.00

0.00 2.21 0.68 0.63 2.22 2.09

1107.22 LimT LimT LimT LimT LimT

42.92 44.34 27.16 19.69 27.05 23.81

0.00 0.00 0.00 0.00 0.96 0.59

224.27 8614.70 10004.90 4376.24 LimT LimT

Table 7 Gain analysis for Inst2. GAPr (%)

GAPf (%)

CPU (%)

pr07 pr08 pr09 pr10 pr11 pr12

21.10 18.92 20.89 16.99 17.73 17.90

0.00 100.00 100.00 100.00 56.76 71.77

79.74 20.23 7.36 59.48 0.00 0.00

Average

18.92

27.80

Table 8 Gain of revenue analysis for Inst1. GAP (%) pr01 pr02 pr03 pr04 pr05 pr06

2.25 0.03 0.08 0.34 0.00 0.00

Average

0.45

Table 9 Gain of revenue analysis for Inst2. GAP (%) pr07 pr08 pr09 pr10 pr11 pr12

0.35 0.00 0.10 0.00 0.14 0.15

Average

0.12

117

B. Crevier et al. / Transportation Research Part B 46 (2012) 100–119 Table 10 Pricing policies comparison. RG (%)

RRRL (%)

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 pr11 pr12

33.18 10.03 14.32 9.48 11.53 11.49 24.23 18.61 11.04 7.99 10.41 9.42

53.44 34.91 50.29 33.48 35.50 37.37 37.80 55.04 44.65 28.77 35.89 47.11

Average

14.31

41.19

P If i2Il fil ¼ 0, set t lk to a value greater than cck ða; bÞ to prevent the use of these itineraries. The rational reaction to this tariff k vector ðtl Þ will induce a lower global revenue than the one generated from the common policy. Table 10 illustrates these observations. 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 t lk of the disjoint policy are imposed in the second level model. The value represents the loss percentage when the revenue corresponding to the second level reaction is compared 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. Here we 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. We have seen that there has been very few contributions on this topic by the scientiﬁc community, despite the undeniable practical interest for this ﬁeld of study. This is mainly due to the complexity of operations related to rail transportation, and thus to the integration of these operations with pricing optimization. By the analysis of two pricing policies, we have highlighted the main model properties and shown the usefulness of some valid inequalities through their signiﬁcant effect on both the branch-and-bound tree’s root node integrality gap and overall CPU time. Future work will concentrate on the development of other families of valid inequalities as well as heuristic approaches. Moreover, the formulation developed describes the objective of the follower as a linear combination of different attributes. The analysis of the second level reaction based on choice models could represent an interesting extension which would increase the realism of the proposed formulations. These, like the logit model (see for example Talluri and van Ryzin, 2005), introduce some probabilistic criteria in the selection process of an alternative (an itinerary/tariff combination for example) by a user, resulting in a more accurate representation of its behavior. Acknowledgements Thanks are due to Vee Kachroo of the Canadian National Railway for fruitful discussions and for providing guidance on test data generation. This work was supported by the Natural Sciences and Engineering Research Council of Canada and the Fonds québécois de la recherche sur la nature et les technologies. This support is gratefully acknowledged. We are also grateful to three anonymous referees for their valuable comments. References Ahuja, R.K., Orlin, J.B., 2001. Inverse optimization. Operations Research 49 (5), 771–783. Ahuja, R.K., Jha, K.C., Liu, J., 2007. Solving real-life railroad blocking problems. Interfaces 37 (5), 404–419. Armstrong, A., Meissner, J., 2010. Railway Revenue Management: Overview and Models. Working Paper, Lancaster University Management School, 2010.

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