Intermodal hinterland network design with multiple actors

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Intermodal hinterland network design with multiple actors

Yann Bouchery, Jan Fransoo

Beta Working Paper series 449

BETA publicatie WP 449 (working paper)

ISBN ISSN NUR

804

Eindhoven March 2014

Intermodal hinterland network design with multiple actors

Yann BOUCHERY*, Jan FRANSOO

Eindhoven University of Technology, School of Industrial Engineering P.O. Box 513, 5600 MB Eindhoven, the Netherlands

March 14, 2014

Abstract: Hinterland transportation has become increasingly critical for efficient global container

supply chain performance. This paper analyzes the implications of the common presence of multiple

actors in intermodal hinterland supply chains. We propose several models to analyze the objectives of the

different actors and develop structural properties of the actors’ behavior. Our results show that, in general,

the objectives of the different actors involved in the design of intermodal hinterland networks are not

aligned. We also show that the actors’ behavior should be taken into consideration in the design phase of

intermodal hinterland networks and that the impact of being overly optimistic when estimating these

actors’ behavior may be substantial. The proposed results provide a better understanding of the dynamics

of intermodal hinterland networks and can help achieve better coordination across the container supply

chain.

Keywords: hinterland supply chain, intermodal transportation, hub-and-spoke network design,

multiple actors

* Corresponding author: Yann Bouchery, y.bouchery@tue.nl, Tel: +31 4 02 47 43 88

1. Introduction Containerization has been the main technological revolution of the maritime industry in the past 30 years.

This innovation has shaped current global supply chains by substantially reducing transportation costs.

For example, the freight rate on a port-to-port basis between Shanghai and Rotterdam for a 40-foot

container is €0.21 per km (OECD/ITF, 2009). This freight rate implies that the maritime transportation

cost for 32-inch television screens from Asia to Europe is less than €3 per screen. As a result, global

container traffic has been growing at almost three times world gross domestic product growth since the

early 1990s (UN-ESCAP, 2005). This paper analyzes the efficiency of container transportation systems in

the hinterland supply chain. Although the distance covered by the container in the hinterland is typically

small, inland transportation costs are often substantial. For example, the freight rate for inland

transportation by truck from the port of Rotterdam typically ranges from €1.50 per km to €4 per km,

depending on the distance and weight (OECD/ITF, 2009); this is 7–19 times higher than the maritime

transportation freight rate. Hinterland activities also include container handling operations. Summing up

all the costs related to hinterland operations, Notteboom and Rodrigue (2005) estimate that the proportion

of inland costs relative to the total transportation costs of container shipping ranges from 40% to 80%.

Thus, improving the efficiency of the hinterland supply chain could provide substantial benefits from a

global supply chain perspective.

Hinterland operations provide a critical contribution to the global performance of the container supply

chain. As a result, all the actors involved in the global container supply chain have increased their

presence in hinterland operations. For example, APM Terminals, a subsidiary of Maersk line (the world’s

largest container liner company), operates 74 container terminals and 166 inland services worldwide.

Terminal operating companies are also becoming increasingly involved in hinterland operations (see, e.g.,

Veenstra et al., 2012). In addition, hinterland access is a key determinant of port competitiveness (Van

den Berg and De Langen, 2011). Containerization has indeed increased the competition between ports by

enlarging the proportion of contestable hinterland—that is, the hinterland region that can be efficiently

accessed from more than one port (Notteboom, 2010). The trend toward bigger ships also puts more

pressure on the hinterland networks. Thus, congestion is becoming a key issue for most port authorities

worldwide (OECD/ITF, 2013). Acknowledging this, port authorities are becoming increasingly active in

the hinterland. For example, the port of Barcelona has actively contributed to the development of the

inland terminal of Zaragoza to reduce pressure on the road transportation network. As a result, the share

of rail volume between Barcelona and Zaragoza has increased from 9% in 2007 to 52% in 2009. Van den

Berg and De Langen (2011) provide a detailed case study of the Zaragoza inland terminal, and Rodrigue

et al. (2010) offer examples of the actors involved in inland terminals.

2

This paper analyzes the implications of the presence of multiple actors in most hinterland supply

chains. Indeed, no single actor usually fulfills the role of supply chain leader in such hinterland supply

chains (Bontekoning et al., 2004), and thus the performance of the hinterland network depends on the

behavior of the different actors involved (De Langen and Chouly, 2004). This issue has been frequently

emphasized and discussed in the maritime economics literature (Notteboom, 2008; Roso et al., 2009; Van

Der Horst and De Langen, 2008). For example, Song and Panayides (2008) empirically show a positive

correlation between port integration in the hinterland supply chain and port competitiveness. However,

model-based research on hinterland network design with multiple actors is scarce. Fransoo and Lee

(2013) recently argued that research from the operations management community is lacking on container

transportation systems despite its critical role in current global supply chains. They identify key industry

problems and other worthwhile areas for further research. Among the proposed problems, the

coordination of container shipments across the container supply chain can be particularly challenging.

This issue is clearly related to the multiple-actors feature of such supply chains.

We propose several models to account for the objectives and behaviors of the different actors

involved in hinterland networks. Our main contributions are twofold. First, we analyze and characterize

the structural properties of the different settings considered. These properties enable us to propose

efficient algorithms to solve the problem in these different settings. Second, we apply the results to an

example based on the features of the hinterland network in the Netherlands and provide related insights.

We prove that, in general, the objectives of the different actors involved in the design of hinterland

networks are not aligned. Our results also show that the actors’ behavior should be taken into account in

the design phase of hinterland networks and that the impact of being too optimistic when estimating these

actors’ behavior can be substantial. The results serve as a basis for appropriately taking multiple actors

into account in hinterland network design problems.

We organized the rest of the paper as follows: Section 2 analyzes the existing literature. Section 3

focuses on the description of the model and on the mathematical formulation of the different settings

considered. Section 4 analyzes the structural properties of the different problem settings considered.

Section 5 details the most important insights of our research by focusing on an example based on the

features of the hinterland supply chain in the Netherlands. Finally, section 6 offers conclusions.

2. Literature review When focusing on hinterland networks, the major impact of containerization is the increasing role of

intermodal transportation. Intermodal transportation involves the transportation of the load from origin to

destination in the same transportation unit without handling of the goods themselves when changing

modes (Crainic and Kim, 2007). The shipping container is the most common transportation unit used in

3

intermodal hinterland networks, and most of these networks are organized in hub-and-spoke setting

(Crainic and Kim, 2007). The first articles focusing on hub-and-spoke network design problems can be

traced back to 1986 (O’Kelly, 1986a, 1986b). The literature on hub location has since expanded rapidly.

Alumur and Kara (2008), Campbell and O’Kelly (2012), and Farahani et al. (2013) all provide reviews.

Several classical formulations of the hub location problem appear in the literature (e.g., hub median, hub

center, hub covering), but the most commonly used model for hinterland transportation network design is

the p-hub median problem. This problem consists of locating a definite number of hubs and deciding how

to allocate a set of origin/destination nodes to these hubs to minimize the total transportation costs of the

system. The cost of transporting one unit of flow per unit distance is discounted on interhub arcs to

represent the economies of scale achieved by such consolidation systems. This feature creates an

incentive to route origin/destination flow through more than one hub because, though this increases the

total distance travelled, it may lead to an overall cost reduction. The basic p-hub median model (and many

of its extensions) assumes that economies of scale are somehow exogenous to the decisions made about

hub location and origin/destination allocation. A fixed discount factor is typically applied to account for

economies of scale on interhub arcs. This limitation was first addressed by O’Kelly and Bryan (1998),

who account for flow-dependent economies of scale on interhub arcs by considering strictly increasing

concave transportation cost functions. They prove that the optimal hub locations may differ greatly from

the results obtained without taking flow-dependent economies of scale into account. The p-hub median

problem is known to be NP-hard. However, a great amount of research has strived to find efficient ways

of solving the problem (either optimally or by using heuristic approaches), and large-scale problems can

now be solved efficiently. The p-hub median problem can be considered an extension of the p-median

problem for facility location (Hakimi, 1964, 1965), which takes interdependency between facilities into

account. As for the facility location research, many extensions of the basic model settings have been

proposed.

Bontekoning et al. (2004) present a review of the early research on intermodal transportation for

hinterland supply chains. Following the same trend as global container traffic, the literature on intermodal

hub-and-spoke network design has quickly expanded in the past decade (Alumur, Kara and Karasan,

2012; Alumur, Yaman and Kara, 2012; Arnold et al., 2004; Groothedde et al., 2005; Ishfaq and Sox,

2010, 2011, 2012; Jeong et al., 2007; Limbourg and Jourquin, 2009; Meng and Wang, 2011; Racunica

and Wynter, 2005; Sörensen and Vanovermeire, 2013; Zhang et al., 2013). These articles primarily

extend the classical p-hub median problem by taking classical features of intermodal container

transportation into account. The most commonly considered aspects are flow-dependent economies of

scale for transportation and transshipment activities, travel time, service constraints, and congestion in the

system. These studies usually propose a new model that incorporates some features proved to be of

4

practical importance. Then, they develop new solution techniques to solve the proposed problem and

focus on assessing the efficiency of the considered solution technique.

To our knowledge, only two recently published articles consider multiple actors in intermodal hub-

and-spoke network design problems. Meng and Wang (2011) formulate user equilibrium constraints to

take the behavior of intermodal operators into account. User equilibrium constraints are commonly used

in the multicommodity network design literature (see, e.g., Yamada et al., 2009). Sörensen and

Vanovermeire (2013) argue that, in general, the location and transportations costs typically included in

intermodal hub location problems are not paid by the same actors. Thus, they consider these two types of

costs separately and develop a bi-objective optimization model to identify the existing trade-offs between

both types of costs. These two articles are particularly noteworthy here because they attempt to contribute

to what is often acknowledged as the most challenging aspect of container supply chains. However, the

authors primarily focus on identifying effective procedures to solve their proposed model. Thus, the

impact of having multiple actors involved in such supply chains cannot be assessed from the proposed

results. In the current paper, we attempt to better understand the consequences of having multiple actors

involved in hinterland supply chains. By focusing on multiple model formulations, we highlight the

implications of considering several objectives for the location and allocation decisions addressed in hub-

and-spoke network design problems.

3. Model description

3.1 Context The model we propose describes some real settings of many contemporary hinterland supply chains. For

the sake of clarity, we present the model in terms of import flows from a single port to various

destinations. The problem could be reversed by considering export flows from various origins to a single

port. Our results hold in that case as well. We assume that the flows under consideration are

containerized. Because the dimensions of containers have been standardized (Agarwal and Ergun, 2008),

the proposed model takes only one type of container into account. We consider that the containers must

be delivered to a fixed set of destinations with deterministic demand as this is usually the case in the hub

location literature. When arriving at their destination, the containers are unloaded. We do not explicitly

take into account empty container management but consider this a parameter of the model. Two options

are available for delivering a container from origin to destination: direct shipment by truck and intermodal

transportation. In the latter case, we assume that the containers are loaded on a train from the port to an

inland terminal. When directly connected with a sea port, the inland terminal can be referred to as an

“extended gate” (Veenstra et al., 2012), a “dry port” (Roso et al., 2009), or an “inland port” (Rodrigue et

5

al., 2010). We focus on train transportation because this is the most developed intermodal transportation

solution worldwide. However, the model is also valid for road-barge and road-short sea intermodal

transportation systems. After arriving at the terminal, the containers are transshipped to trucks to reach

their final destination.

The objective of the model is to analyze how to locate these inland terminals while acknowledging

that several actors are involved in such a decision. Table 1 provides a list of the potential actors involved

in inland terminal location decisions along with some examples of their respective objectives. We focus

on three objective functions, or location rules, in the proposed model. First, as we mentioned previously,

the port authorities involved in such inland terminal projects are mainly interested in maximizing the

volume transported by rail from the port to the inland terminal, to reduce road traffic congestion in the

port surroundings and thus increase their competitiveness. This objective may also be followed by the

train operator company. We note that this objective is equivalent to maximizing terminal utilization in the

proposed model. Maximizing terminal utilization is of interest to the inland terminal operator. Local

authorities may also view this objective as essential because efficient terminal utilization would positively

affect the local economy; thus, the first location rule is terminal utilization, or TU. Second, from a public

authority perspective, the main objective may be to minimize the total number of kilometers traveled by

truck because intermodal transportation is often considered less polluting. This objective is equivalent to

maximizing modal shift; thus, the second location rule is modal shift, or MS. Third, the total cost

minimization objective traditionally used in the hub location literature is of critical importance for making

intermodal transportation a viable option because cost is one of the major criteria for shippers and freight

forwarders; thus, the third location rule is total cost, or TC. The objectives followed by the different actors

as well as their individual bargaining power may strongly differ from one specific inland terminal

situation to another. Each practical situation has its own set of characteristics based on past behaviors,

political issues, trust, and willingness to collaborate of each actor.

Actors Example of objective Proposed location rule

Terminal operators Maximize terminal utilization TU

Port authorities Reduce road traffic congestion in the port surrounding TU

Public authorities Minimize the distance traveled by truck MS

Local authorities Maximize terminal utilization TU

Rail operators Maximize train service utilization TU

Shippers Minimize transportation costs TC

Freight forwarders Minimize transportation costs TC

Table 1: Actors involved in inland terminal location decisions

6

Compared with the classical p-hub median problem we described in section 2, we consider a single

origin and a single inland terminal to locate. This problem setting enables us to evaluate the dynamics of

inland terminal locations with multiple actors and is in line with Campbell and O’Kelly (2012), who

argue that “new single hub model formulations continue to provide intriguing formulation issues” (p.

163).

Our model takes into account flow-dependent economies of scale for train transportation and

transshipment operations at the inland terminal. We model the train transportation cost per container and

the transshipment cost per container as general concave nondecreasing functions of the total number of

operated containers. This feature creates a cost interdependency among the allocation decisions made for

each destination. The destinations are considered individual companies, such as retailers, requesting a

transportation service in the model. In addition to the inland terminal location problem, the allocation

problem consists of deciding, for each destination, which share of the total flow should be allocated to the

inland terminal (i.e., shipped by intermodal transportation). Three allocation rules are considered. When

focusing on costs, flow-dependent economies of scale and a multiple-actors setting lead to some

noteworthy allocation issues. In the classical single-actor vision of hub location problems, O’Kelly and

Bryan (1998) point out that “some origin-destination pairs may be routed via a path that is not their least-

cost path because doing so will minimize total network travel cost” (p. 608). This statement may not hold

in a multiple-actor setting. The situation is similar to a classical problem in the traffic assignment

literature. Because of congestion, the solution that minimizes the total traveling time in the system is not

equivalent to the solution that minimizes the travel times of the individual users. This leads to two

extreme behaviors that Wardrop (1952) describes as user equilibrium (each user minimizes its own travel

time) versus system optimum (the total travel time of the system is minimized). This paper examines both

user equilibrium (UE) and system optimum (SO). For the UE allocation rule, we consider that each

destination allocates the demand flow between direct and intermodal shipment to minimize its own

transportation costs. For the SO allocation rule, the destinations take the total costs in the system into

account in making their allocation decisions. The third allocation rule considered is to maximize the

modal shift (MS). A destination may indeed be willing to reduce its environmental impact by favoring

intermodal transportation. This objective of maximizing the modal shift is equivalent to minimizing the

distance travelled by road. In real-life situations, we acknowledge that a mix of these strategies would be

encountered. Each destination may have its own positioning with respect to MS, UE, and SO rules.

Moreover, the freight forwarders and truck operators often deal with several destinations; thus, some

pooling effects are encountered. However, we focus on the three extreme scenarios—all UE, all SO, and

all MS—in our analysis to examine the impact of allocation decisions on the overall performance of the

hinterland supply chain.

7

The strong interrelationship between the location and the allocation subproblem is the key feature of

any hub location problem. In the setting considered herein, we can obtain the optimal hub location

through enumeration; thus, the most complex part is the allocation subproblem. In what follows, we

combine the three allocation and three location rules to assess the effect of the interactions among

different actors on optimal decisions. By mixing the considered allocation and location rules, we propose

and analyze nine formulations.

3.2 Model formulations

The hinterland supply chain under study consists of a single port (considered as the origin) with *ℵ∈N

destinations. A deterministic constant flow in must be shipped from origin to destination { }Nj ;...;1∈ .

Here, jn is expressed in number of containers, and only one type of container is available (40-foot

containers); thus, we assume that *ℵ∈jn for all { }Nj ;...;1∈ . At most, one inland terminal must be

located among *ℵ∈M candidate locations. Each potential terminal location is referred to as location

{ }Mi ;...;1∈ . In addition, we refer to the origin (i.e., the port) as location 0=i .

The truck transportation cost function considered is linear in the distance and in the number of

containers shipped. The truck transportation cost per container.km may be destination dependent to

account for the different rates proposed by different truck operators as well as for the different empty

container management practices developed by each destination.

Thus, we can express the cost of transporting one container from location { }Mi ;...;0∈ to destination

{ }Nj ;...;1∈ as follows:

jtruckjiji ZZ ,,1, δ= , (1)

where

ji,δ = the distance from location i to destination j (expressed in km) and

jtruckZ , = the truck transportation cost per container.km for destination j (expressed in

€/container.km).

While delivery takes place through intermodal transportation, train transportation is used from the

origin to the inland terminal. We define the cost of shipping one container by train from the origin to the

inland terminal { }Mi ;...;1∈ as follows:

)()( ,02,0 KZKZ trainii δ= , (2)

where

8

i,0δ = the distance from the origin to inland terminal j (expressed in km),

)(KZtrain = the flow-dependent train transportation cost per container.km (in €/container.km),

and *+ℜ∈K = the total number of containers shipped by train.

The train transportation cost function is linear in distance. However, flow-dependent economies of

scale are taken into account; thus, the train transportation cost per container depends on the total amount

of containers shipped by train K . We further assume that )(. KZK train is a concave nondecreasing

function. Thus, the train transportation cost per container is nonincreasing in the total number of

containers shipped by train, and the marginal cost of shipping an additional container is nonnegative and

nonincreasing in the amount shipped.

In addition to train and truck transportation costs, intermodal transportation implies additional

container handling operations at the inland terminal. We define the cost of transshipping one container at

terminal { }Mi ;...;1∈ is follows:

)()(2,1 KKZ ii γ= , (3)

where

)(Kiγ = the flow-dependent transshipment cost per container at terminal i (in €/container) and *+ℜ∈K = the total number of containers transshipped at terminal i .

Note that the total number of containers transshipped at terminal i is equal to the total number of

containers shipped by train. The transshipment cost depends on the location considered to account for the

difference in land and labor costs as well as for the difference in terminal layout, equipment, and size.

This cost also accounts for the difference in cost from transshipping to a truck or to a train at the origin, if

any. In addition, we define 0)(0 =Kγ , for all *+ℜ∈K . The transshipment cost per container depends

on the total amount of containers transshipped at terminal i . We further assume that )(. KK iγ is a

concave nondecreasing function. Thus, the transshipment cost per container is nonincreasing in the total

number of containers transshipped, and the marginal cost of transshipping an additional container is

nonnegative and nonincreasing in the amount transshipped.

The total cost of shipping one container from the origin to destination { }Nj ;...;1∈ by intermodal

transportation through inland terminal { }Mi ;...;1∈ is equal to 1,

2,12,0 )()( jiii ZKZKZ ++ . To simplify

the notation, we define the following:

)()()( 2,12,0

3,0 KZKZKZ iii += . (4)

9

Note that 0)(30,0 =KZ for all +ℜ∈K . As is usually the case in the hub location literature, we further

define 10 , ≤≤ jiX as the proportion of flow from origin to destination j being routed through terminal

i . Here, 1,0 =jX indicates that the entire flow from origin to destination j is delivered by direct

shipment, while 1, =jiX , where { }Mi ;...;1∈ , indicates that the entire flow from origin to destination j

is delivered by intermodal transportation using inland terminal i .

We define the following three following objective functions, depending on the location rule under

consideration:

( )∑∑= =

+M

i

N

jjiijij XKZZnMIN

0 1,

3,0

1, )( , (5)

∑∑= =

M

i

N

jjijij XZnMIN

0 1,

1, , and (6)

MAX K . (7)

Objective 5 corresponds to minimizing the total cost (TC location rule). Objective 6 corresponds to

minimizing the total truck transportation costs—that is, maximizing the modal shift (MS location rule).

Finally, objective 7 aims to maximize the number of containers shipped by train—that is, maximizing

terminal utilization (TU location rule).

For any of the three location rules, the following set of constraints must be considered:

∑=

≤M

iiy

11 , (8)

{ } { }Miyi ;...;1,1;0 ∈∀∈ , (9)

{ } { }MiNjyX iji ;...;1,;...;1,, ∈∀∈∀≤ , (10)

{ }NjXM

iji ;...;1,1

0, ∈∀=∑

=, (11)

{ } { }NjMiX ji ;...;1,;...;0,0, ∈∀∈∀≥ , and (12)

∑∑= =

=M

i

N

jjij XnK

1 1, . (13)

10

As constraints 9 show, iy are binary variables equal to 1 if inland terminal i is open and 0 otherwise.

Constraints 8 ensure that, at most, one inland terminal can be opened. Constraints 10 imply that flow can

be routed only through an open terminal. Constraints 11 ensure that the total amount of flow is shipped

from origin to destinations. Constraints 12 ensure that the proportions of flow routed are nonnegative.

Finally, constraint 13 is used to account for the number of containers routed by intermodal transportation.

When the location and the allocation rules under consideration are different, some additional

constraints must be considered. We formulate these additional constraints in section 4.1. Note that

objective function 5 under constraints 8–13 corresponds to the TC/SO problem—that is, the classical hub

location formulation with the objective of minimizing the total transportation costs in the network. The

MS/MS problem is represented by objective function 6 under constraints 8–13.

4. The allocation subproblem

4.1 Constraints formulation

In the allocation subproblem, we assume that the inland terminal { }Mi ;...;1∈ is open. For the MS

allocation rule, the objective of maximizing modal shift is equivalent to the objective of minimizing the

traveled distance by road. Thus, for each destination, the decision is made by comparing the distance from

the inland terminal and the distance from the port. If the destination j is located closer to the inland

terminal, the entire demand flow jn will be shipped by intermodal transportation. Otherwise, the entire

demand flow will be shipped directly by truck. This single routing condition enables simplification of the

analysis. Theorem 1 shows that this condition also holds for the UE and SO allocation rules.

Theorem 1: The single routing condition holds for each destination for the UE, SO, and MS allocation

rules.

Proofs appear in appendix A. As we explained in section 3.2, the chosen allocation rule may result in the

addition of some constraints to the general problem (as soon as the location and allocation rules are

different). For the MS allocation rule, the following sets of constraints need to be taken into

consideration:

{ }NjZZXy jijji

M

ii ;...;1,0)( 1

,1,0,

1∈∀≥−∑

=, and (14)

{ }NjZZXy jjij

M

ii ;...;1,0)( 1

,01,,0

1∈∀≥−∑

=. (15)

11

Constraints 14 imply that 0, =jiX if jji ,0, δδ > . Constraints 15 imply that 0,0 =jX if jij ,,0 δδ > .

Objective 5 under constraints 8–15 corresponds to the TC/MS problem, while objective 7 under

constraints 8–15 corresponds to the TU/MS problem.

The UE allocation rule results in the following sets of constraints:

{ }NjnXKZZZXy jjiijijji

M

ii ;...;1,0)))1((( ,

3,0

1,

1,0,

1∈∀≥−+−−∑

=, and (16)

{ }NjZnXKZZXy jjjiijij

M

ii ;...;1,0)))1((( 1

,0,3,0

1,,0

1∈∀≥−−++∑

=. (17)

Constraints 16 imply that 0, =jiX if it is cheaper for destination j to be shipped directly. Constraints

17 imply that 0,0 =jX if it is cheaper for destination j to be shipped through terminal i . The TC/UE,

MS/UE, and TU/UE problems consist of objectives 5, 6, and 7, respectively, under constraints 8–13 as

well as constraints 16–17.

Finally the SO allocation rule implies the following sets of constraints:

( ) ,0)1(()())1(( ,3,0

3,0,

3,0

1,

1,0,

1≥

−+−+−+−−∑

=jjiii

jjjiijijji

M

ii nXKZKZ

nKnXKZZZXy { }Nj ;...;1∈∀ ,

and (18)

( ) ,0)1(()())1(( ,3,0

3,0,

3,0

1,0

1,,0

1≥

−+−−−++−∑

=jjiii

jjjiijjij

M

ii nXKZKZ

nKnXKZZZXy { }Nj ;...;1∈∀ .

(19)

Compared with constraints 16, constraints 18 also take into account the train transportation cost

reduction incurred by the other destinations. These constraints imply that 0, =jiX if it is globally

cheaper to ship the demand flow from destination j directly. The same line of reasoning is applied to

formulate constraints 19. These imply that 0,0 =jX if it is globally cheaper to ship the demand flow

from destination j through terminal i .

As explained previously, examination of the problems under the MS allocation rule is simple because

the allocation decisions may be taken independently for each destination and because we restrict our

attention to the case in which, at most, one inland terminal can be used. The next two sections are devoted

to the UE and SO allocations rules. The allocation subproblem is complex for these two allocation rules

because the decisions made for each destination are interdependent.

12

4.2 User equilibrium allocation As Fisk (1984) points out, Wardrop’s first principle of user equilibrium is equivalent to the Nash

equilibrium principle in noncooperative game theory. This concept is characterized by the property that

neither player can unilaterally reduce transportation costs by changing its decision. In this section, we

show that several Nash equilibria may exist in some settings of the UE allocation rule. Then, we propose

an algorithm to identify all the existing Nash equilibria for a given problem, based on some structural

properties of the proposed model.

We refer to the N players noncooperative game corresponding to the UE allocation rule as the UE

allocation game. In this game, each destination is considered a player of the game. Each destination aims

to maximize its relative profit compared with direct shipment costs. Using theorem 1, we can assert that

each destination { }Nj ;...;1∈ has two options { }1;0, ∈jiX and tries to maximize its relative profit jP ,

which depends not only on player j ’s action jiX , but also on the others players’ actions { kiji XX ,, =−

{ } }jkNk ≠∈ ,;...;1 . Then, 0, =jiX corresponds to the case in which destination j chooses direct

shipment. By definition, 0),0( , =− jij XP , independent of the decisions made by the other players. When

1, =jiX , this corresponds to the case in which destination j decides on delivery using terminal i . Then,

),1( , jij XP − may be either negative or positive, depending on the other players’ decisions, due to flow-

dependent economies of scale for train transportation and transshipment operations. We consider that

)(. 3,0 KZK i is divided among the actors by following the proportional allocation rule—namely, a player

transporting jn out of K containers by intermodal transportation would be charged )(. 3,0 KZn ij . We

obtain the following:

( ))(),1( 3,0

1,

1,0, KZZZnXP ijijjjij −−=− ∑

=

=N

kkki nXK

1, . (20)

Note that in the special case in which 0),1( , =− jij XP , we assume that player j would choose direct

shipment. This decision may indeed be viewed as less risky because it is not affected by the other players’

decisions. Nash equilibrium is obtained when neither player can individually increase its profit by

changing the decision.

We define { }*,

*1,

* ;...; Niii XXX = as a Nash equilibrium if the following properties hold for all

{ }Nj ;...;1∈ and for all { }1;0, ∈jiX :

),(),( *,,

*,

*, jijijjijij XXPXXP −− ≥ . (21)

13

),0(),1(1 *,

*,

*, jijjijji XPXPX −− ><=>= . (22)

Nash (1950) proves that there is at least one Nash equilibrium under general assumptions by applying

the fixed point theorem. However, this equilibrium may be obtained by requiring at least one player to

choose a probability distribution over the set of potential actions to protect against other players’

reactions. This type of strategy is a mixed strategy, as opposed to a pure strategy. A pure strategies Nash

equilibrium (PSNE) is such that each player chooses an action for sure. This type of equilibrium may not

always exist (e.g., the matching pennies game). Theorem 2 proves that at least one PSNE exists for the

UE allocation game. This is because 0),0( , =− jij XP , independent of the other players’ decisions. Thus,

each player can be protected against all the other players’ actions while following a pure strategy.

Theorem 2: The UE allocation game leads to at least one PSNE.

In N players noncooperative games, several PSNEs may exist. Theorem 3 states that this is also the case

for the UE allocation game.

Theorem 3: The UE allocation game may lead to several PSNEs.

As we show in the proof (see appendix A), this result holds even when considering only two players. In

this case, the situation is encountered when neither of the players has enough volume to make intermodal

transportation profitable in a stand-alone setting and when the combination of both players’ volumes

makes intermodal transportation profitable for each. This example can be viewed as a stag hunt game in

which two PSNEs exist. The first one, when the two players choose direct shipment, is called risk

dominant. The second, when both players choose intermodal transportation, is called payoff dominant.

Several Nash equilibria may exist, which is of critical importance. To our knowledge, the only article

considering UE constraints in an intermodal hub location problem is that of Meng and Wang (2011).

However, the authors minimize the total cost under these UE constraints without acknowledging that the

proposed solution may not lead to a unique Nash equilibrium. Thus, in a situation with several PSNEs,

only the payoff-dominant Nash equilibrium is considered. We examine the impacts of this assumption

further in section 5.

The structural properties of the UE allocation game enable us to simplify the analysis and design an

algorithm to identify all the existing PSNEs, as we show in proposition 1 (resulting from theorem 4 and

corollary 1).

14

Theorem 4: Assume that there are 2≥L PSNEs in a given setting of the UE allocation game. For all

{ }Ll ;...;1∈ , let liU , be the set of all players choosing intermodal transportation under equilibrium l (

liU , may be empty). Then, it is possible to order the 2≥L PSNEs such that

{ }1;...;1,,1, −∈∀⊂+ LlUU lili .

Corollary 1: Assume that there are L PSNEs in a given setting of the UE allocation game. Then,

12

+

≤

NL .

In the two-player setting with two PSNEs considered previously, we find that ∅=2,iU corresponds to

the risk-dominant equilibrium and that { }2;11, =iU correspond to the payoff-dominant equilibrium. We

observe that theorem 4 holds for this example, as 1,2, ii UU ⊂ . Moreover, we have all the PSNEs for this

game by using corollary 1.

Algorithm UE

Step 1: For all { }Nj ;...;1∈ , compute 1,

1,0, jijji ZZ −=Λ .

Step 2: Rank all destination { }Nj ;...;1∈ from the largest to the smallest value of ji,Λ .

Step 3: Set 0=τ , NN =τ .

Step 4: Solve { }

>Λ= ∑

=∈

q

jjiqiNq

nZq1

3,0,;...;1

* max;0maxτ

τ .

Step 5: { }{ }*1, ;...;1 ττ qjNjU i ≤∈=+ .

Step 6: Solve { }

≤Λ= ∑

=∈

q

jjiqi

qqnZp

1

3,0,

;...;1

**

max;0maxτ

τ .

Step 7: If 0* =τp , then stop. Otherwise, go to Step 8.

Step 8: Set 1+= ττ , set { }*1;...;0 −= ττ pN , and go to Step 4.

Proposition 1: Algorithm UE enables the identification of all the PSNEs of the UE allocation game.

15

We use algorithm UE in section 5 in an example in which several PSNEs exist. We further analyze the

impacts of having more than one Nash equilibrium and derive several insights.

4.3 System optimum allocation In the SO allocation rule, the destinations take the total cost of the system into account in making their

allocation decision. Compared with the UE allocation rule, some destinations may decide to use the

inland terminal even if this choice increases their individual cost, as soon as the volume added for train

transportation and transshipment operations provides a stronger cost reduction for the other players using

intermodal transportation. Because the number of destinations is bounded, we can ensure that there is an

allocation minimizing the total cost in the system. However, this allocation may not be unique. We define

the SO allocation as the allocation minimizing the total cost of the system with the minimum number of

destinations using intermodal transportation. For the SO allocation rule, )(. 3,0 KZK i should be

considered globally. This cost must be compared with the sum of individual savings in truck

transportation cost when the containers are shipped through terminal i instead of being shipped directly.

The following algorithm (algorithm SO) can determine the system optimum allocation—that is, the set iS

of the destinations being shipped to using intermodal transportation.

Algorithm SO

Step 1: For all { }Nj ;...;1∈ , compute 1,

1,0, jijji ZZ −=Λ .

Step 2: Rank all destination { }Nj ;...;1∈ from the largest to the smallest value of ji,Λ .

Step 3: Solve { }

−Λ= ∑∑∑

===∈

q

jji

q

jj

q

jjij

NqnZnnq

1

3,0

11,

;...;1

* maxargmin .

Step 4: If 0***

1

3,0

11, >

−Λ ∑∑∑

===

q

jji

q

jj

q

jjij nZnn , then { }{ }*;...;1 qjNjSi ≤∈= . Otherwise, ∅=iS .

Proposition 2: Algorithm SO enables the identification of iS .

Note that

−Λ ∑∑∑

===

q

jji

q

jj

q

jjij nZnn

1

3,0

11, may not always be a monotonic function of q ; thus, local

optima may exist. A remaining issue is to verify that the SO allocation is stable in the sense of

16

cooperative game theory when ∅≠iS . Indeed, the SO allocation would be of interest only if there is an

allocation of

∑∑==

q

jji

q

jj nZn

1

3,0

1 such that no group of destinations has the incentive to act independently

by setting up another service. This problem involves proving the nonemptiness of the core for the

cooperative game ( )vSi ;∅≠ with the following characteristic function:

( )iSSv ⊆

−Λ>− ∑∑∑

∈∈∈ Sjji

Sjj

Sjjij nZnn 3

,0, and ( ) 0=∅v . (23)

Theorem 5: The cooperative game ( )vSi ; is convex and thus has a nonempty core.

Note that determining the set of allocations in the core of the game is outside the scope of this paper.

4.4 Performance comparisons of the allocation rules The results we presented in the previous sections may also help in comparing the performance of the

allocation rules with respect to the different location objectives considered. Theorem 6 shows that for any

potential terminal location, the allocation rule inducing the highest terminal utilization is MS, followed by

SO and finally the payoff-dominant PSNE. In addition to theorem 4, theorem 6 allows for a direct ranking

of the different allocation rules in terms of terminal utilization for a given terminal location.

Theorem 6: Let iM be the set of destinations using terminal i for the modal shift allocation rule; then,

for all { }Mi ;...;1∈ ,

iii MSU ⊆⊆1, .

Theorem 6 also implies that the SO allocation rule performs better than the UE allocation rule in terms of

modal shift. In terms of total cost, the situation is not as clear. Indeed, in most of the cases, the MS

allocation rule would perform better than the UE allocation because the marginal train transportation and

transshipment costs per container added are relatively low in most practical situations. However, an

exception is when the UE and SO allocation rules lead to not using the terminal. In this case, the MS

allocation rule leads to an increase in total cost when the distance from at least one destination to the

terminal is lower than the distance to the port.

17

5. Example and insights This section is based on an example representing features of the hinterland network in the Netherlands.

The main objective here is to explore the implications of having multiple actors involved in such a supply

chain. We apply the modeling developed in section 3, and the theoretical results of section 4 enable us to

quickly solve the problem. In this example, 25=N and 10=M . The locations of the destinations and

potential terminal locations appear in figure 1. The crosses represent the destinations, the dots represent

the potential terminal locations, and the square represents the location of the port. The terminal location

numbers also appear in figure 1. In this example, we calculate the distances by considering the Euclidean

norm.

Figure 1: The hinterland supply chin considered

For each of the 10 potential inland terminal locations, we determine all the PSNEs, the SO allocation, and

the MS allocation. In each case, we estimate the total cost, the modal shift, and the terminal utilization.

We calculate terminal utilization by dividing the actual number of containers transshipped at the terminal

by the total amount of potential containers (i.e., ∑=

N

jjnK

1/ ). We also consider the situation in which no

terminal has been opened. We calculate modal shift by comparing the total distance traveled by truck in

any situation with this latter case. Note that we assume that the containers are loaded with 20 tons of

cargo; thus, the number of ton.kilometers shifted from the road can be estimated. The results appear in

table 2.

7

4

2

9

8

6

10

1

5

3

18

Total Cost (€) Modal Shift

(ton.km)

Terminal

Utilization (%) Total Cost (€)

Modal Shift

(ton.km)

Terminal

Utilization (%)

DS 11 749.89 - 0% S5 10 130.62 64 840 54% U1,1 9 891.01 65 300 46% M5 10 150.42 65 980 61% U1,2 10 886.05 44 160 30% U6,1 11 749.89 - 0% U1,3 11 749.89 - 0% S6 11 749.89 - 0% S1 9 597.84 72 720 57% M6 12 168.69 48 360 52% M1 9 614.09 74 440 65% U7,1 11 749.89 - 0% U2,1 10 129.35 56 860 46% S7 11 524.04 20 900 57% U2,2 11 749.89 - 0% M7 11 585.01 21 480 70% S2 9 556.69 68 740 67% U8,1 9 701.18 56 260 52% M2 9 562.98 69 460 72% U8,2 11 749.89 - 0% U3,1 11 749.89 - 0% S8 9 463.53 62 080 67% S3 11 446.99 60 000 43% M8 9 489.35 62 240 72% M3 11 448.94 60 300 46% U9,1 11 749.89 - 0% U4,1 10 618.40 29 660 76% S9 10 478.83 49 580 74% U4,2 11 749.89 - 0% M9 10 492.97 49 820 78% S4 10 590.89 30 540 80% U10,1 11 749.89 - 0% M4 10 609.24 30 740 85% S10 10 715.00 66 320 59% U5,1 10 817.76 49 440 28% M10 10 715.00 66 320 59% U5,2 11 749.89 - 0%

Table 2: Overall results

We can derive several insights from these results. We first focus on analyzing the impact of considering

several objectives for the location decision. We perform this analysis by focusing on the SO allocation

rule because this is the most commonly used rule in the literature. (Note that the same type of analysis can

be performed with the other allocation rules we consider.) When considering the TC location rule, the

best solution is terminal 8. For the MS location rule, the best solution is terminal 1. Finally, for the TU

location rule, the best solution is terminal 4. This demonstrates that the objectives of the different actors

involved in the design of intermodal hinterland networks are not aligned.

Insight 1: In general, the objectives of the different actors involved in the design of intermodal hinterland

networks are not aligned.

From a practical point of view, insight 1 means that the result obtained using a terminal utilization

maximization objective or a modal shift maximization objective may not be viable in practice if it ends up

in a solution that is too costly. This implies that port authorities, public authorities, and local authorities

may be required to subsidize such train services and terminal operations to make intermodal

19

transportation viable from a cost perspective. This feature is in line with the results presented in the

maritime economics literature (Van den Berg and De Langen, 2011). In addition, the location obtained

using a cost minimization objective, as is usually the case in the literature, may not accurately represent

the location decision made in practice, as other objectives may play a role.

Continuing with such an analysis, we note that some solutions that are not optimal for any of the

location rules considered represent interesting trade-offs. For example, terminal 2 is the second-best

solution in terms of total cost, the second-best solution in terms of modal shift, and the third-best solution

in terms of terminal utilization. We can conclude that identifying such trade-offs may be of significant

interest in helping align the objectives of the different actors involved. One way to obtain such a solution

is to consider multiobjective optimization techniques.

Insight 2: Multiobjective optimization may yield solutions that align the interests of the different actors

involved in the design of intermodal hinterland networks.

To our knowledge, Sörensen and Vanovermeire's (2013) study is the only one that considers

multiobjective optimization techniques to take into account the multiple-actors feature of intermodal

hinterland networks. However, their first attempt to model the multiple-actors feature of intermodal

supply chains using multiobjective optimization focuses mainly on proposing and assessing a solution

procedure for the problem. Additional research is required to generate insights into and solutions for

practical decision making.

Comparing the best locations for the three location objectives considered, we note that the optimal

TU location is close to the port, the optimal TC location is at a midrange distance, and the optimal MS

solution is farther inland. We could explain such a result as follows: The optimal MS location is farther

inland than the optimal TC location because the train transportation and transshipment costs are not

included when optimizing modal shift. Thus, locating the terminal farther inland may increase the

distance traveled by train. From a cost perspective, reducing the distance traveled by train while slightly

increasing the distance traveled by truck is a better option. Although the inland terminal is closer to the

port, the number of destinations that are closer from the terminal than from the port is increasing, and thus

the train transportation cost decreases (as the distance decreases and the volume increases). As a result,

terminal utilization increases.

Insight 3: Inland terminals close to the port are better in terms of utilization, midrange terminals are

better from a total cost perspective, and distant inland terminals perform better for modal shift.

20

Insight 3 can be used to understand the dynamics of inland terminal location in a hinterland network. As

Roso et al. (2009) point out, all types of terminals may be encountered in practice. The comparative

advantages and drawbacks of such terminal locations can be assessed in terms of total cost, terminal

utilization, and modal shift. Nevertheless, insight 3 needs to be taken with caution because the interaction

and competition among several terminals are not taken into account in the proposed model and may play

a role.

We derive the second set of insights from analyzing the impacts of taking several allocation rules into

account. First, the optimal location may depend on the considered allocation rule. This situation could be

encountered for the TC location rule by excluding terminal 8 from the analysis (by considering that this

terminal may not be available). In this case, terminal 1 is the best location if the UE allocation rule is

considered. Conversely, terminal 2 is the best option for the SO and the MS allocation rules. We can

conclude that the behaviors of the actors should be taken into account in the design phase of intermodal

hinterland networks.

Insight 4: The optimal location depends on the considered allocation rule. Thus, the behavior of the

actors should be taken into account in the design phase of intermodal hinterland networks.

In practice, actors’ behaviors are often difficult to determine accurately when the system is running; thus,

determining these behaviors in the design phase may be challenging. Additional research is required to

address such difficulties. In accordance with theorem 6, we also note in this example that the SO

allocation rule used in most of the hub location literature results in an overly optimistic view of real

situations.

Insight 5: The system optimum allocation rule leads to overly optimistic results in terms of total cost,

modal shift, and terminal utilization.

The impacts of being overly optimistic can be substantial. In some situations, the system optimum

allocation may lead to a solution that seems appealing, while the only existing PSNE has no destinations

using the terminal. This situation is encountered for terminals 3, 7, 9, and 10. To assess what might occur

in such a situation, we exclude terminals 8, 2, 1, and 5 from the analysis. The total cost optimal location

for the SO allocation rule in such a scenario is terminal 9, leading to a total daily cost of €10 478.83 and

74% terminal utilization. Assume now that the network is designed according to this solution and that the

allocation rule followed by the actors is UE instead of SO. Then, no destinations would use the terminal.

Conversely, choosing terminal 4 with an expected cost of €10 590.89 and an expected terminal utilization

21

of 80% for the SO allocation rule would have been a much better choice because, in this case, the payoff-

dominant user equilibrium leads to a total cost of €10 618.40 and terminal utilization of 76%.

Insight 6: The impacts of being overly optimistic when estimating actors’ behavior in the design phase

can be substantial.

Insight 6 can help explain why some intermodal transportation projects are predicted to be effective in

theory while being very difficult to turn into profitable projects in practice. This insight is in accordance

with Rodrigue et al. (2010), who report that “both public and private actors have a tendency to

overestimate the benefits and traffic potential and underestimate the costs and externalities of inland port

projects” (p. 528). The proposed model can be used to help the different actors involved in inland

terminal projects to better assess the traffic potential and related benefits of such projects.

Finally, we discuss some insights derived from having several PSNEs. First, we note that the risk-

dominant Nash equilibrium involves having no destination shipped to from the inland terminal for all the

potential terminal considered.

Insight 7: In most of the cases, the risk-dominant Nash equilibrium involves not using the inland

terminal.

Indeed, no actor is powerful enough to make intermodal transportation profitable by acting individually

because economies of scale are of key importance for efficient intermodal transportation. This issue is not

taken into account in the hub location literature. It follows that the actors, even if they are competitors,

need to develop mutual trust to make intermodal transportation viable.

Insight 8: Mutual trust among the actors is a prerequisite for efficient intermodal hinterland

transportation.

The question of how to promote such mutual trust in practice is of great interest. For example, several

destinations may agree on guaranteed minimum volumes shipped by intermodal transportation before the

implementation of the inland terminal. Such practice is currently employed in the Netherlands, where port

authorities and governmental agencies act as a platform for such mutual trust agreements. Indeed,

11 traders in the region of Westland have signed an agreement to transport 10 000–15 000 containers per

year by barge from the port of Rotterdam to the container terminal of Hook of Holland (project Fresh

22

Corridor 7). We refer to the maritime economics literature focusing on hinterland supply chains for

related discussions (see, e.g., Van Der Horst and De Langen, 2008).

Finally, the example we have presented shows that more than two PSNEs may exist, as in the case of

terminal 1. Understanding the dynamic behind the users’ behavior to forecast which equilibrium is more

likely to occur in practice is of great importance. For example, formulating user equilibrium constraints

while considering a total cost minimization objective, as proposed by Meng and Wang (2011), may not

accurately represent the current situation and may be viewed as being overly optimistic in the

performance of the intermodal hinterland transportation system if the cost-dominant Nash equilibrium is

not chosen by the actors in the networks. Additional research is required to understand actors’ behavior in

such intermodal hinterland networks.

6. Conclusion In this paper, we analyzed the implications of having multiple actors involved in intermodal hinterland

supply chains. Our main research contribution is to compare the solutions obtained while considering the

different objectives and behaviors of the actors involved. This process enables us to assess the impact of

not accurately estimating actors’ behavior when designing an intermodal hinterland network. We found

new theoretical results pertaining to some structural properties of the actors’ behavior. These results make

it possible to design algorithms to solve the allocation subproblems optimally. Our results also show that

the literature modeling the multiple-actors feature of current intermodal hinterland supply chains provides

only a partial representation of the existing actors’ equilibria. In addition, we derive new insights from an

example representing features of the hinterland network in the Netherlands. We prove that, in general, the

objectives of the different actors involved in the design of intermodal hinterland networks are not aligned.

Our results also show that the actors’ behavior should be taken into account in the design phase of

intermodal hinterland networks and that the impact of being overly optimistic when estimating these

actors’ behavior can be substantial. Finally, we show that multiobjective optimization can yield solutions

that balance the conflicting objectives of the different actors in the network design phase and that such a

technique can help coordinate the container shipments across the container supply chain.

This research underscores the importance of taking into account the multiple-actors setting of

intermodal hinterland networks. The results may also generalize to the entire container supply chain, but

further research is necessary for this end. Several techniques can be adequately used to take this multiple-

actors setting into account. Among others, we show that multiobjective optimization and game theory are

of primary relevance. We hope that our results help pave the way for further research from the operations

management community on container transportation systems.

23

Acknowledgments The research was partly funded by Dinalog, the Dutch Institute for Advanced Logistics.

Appendix A

Proof of Theorem 1

We need to prove that { }1;0, ∈jiX for all { }Nj ;...;1∈ , for the three considered allocation rules.

MS allocation rule For all { }Nj ;...;1∈ , if 1

,01, jji ZZ < , then 1,,0, ==>< jijji Xδδ . Otherwise,

0,,0,1,0

1, ==>>=>> jijjijji XZZ δδ .

UE allocation rule By contradiction, assume that 01 , >> jiX for a given { }Nj ;...;1∈ . This implies that

1,0

3,0

1,

1,0,

3,0

1,, ))(())(( jjijijjjjiijijji ZnKZZnZnXKZZnX <+=><+ . Because 3

,0 iZ is nonincreasing in K ,

we obtain 1,0,

3,0

1, )))1((( jjjjiijij ZnnXKZZn <−++ , implying that 1, =jiX .

SO allocation rule By contradiction, assume that 01

1, >> jiX for a given { }Nj ;...;11 ∈ . Using the results of the UE

allocation rule, we can conclude that choosing 11, =jiX will reduce the transportation cost for

destination 1j without increasing the costs for the other destinations (the costs could even be reduced

because the number of containers shipped through the inland terminal is increasing). We can conclude

that 11, =jiX at optimality.

Proof of Theorem 2

We can construct a PSNE as follows: We begin by setting 0, =jiX { }Nj ;...;1∈∀ . If none of the

players can increase their profit by individually using intermodal transportation, then considering direct

shipment for all the players is a PSNE. Otherwise, there is a destination { }Nk ;...;1∈ such that

)0,0()0,1( ,, =≥= −− jijjik XPXP . Let 1, =kiX . If none of the remaining players can increase their

profit by individually using intermodal transportation, then considering direct shipment for all the players

except for player k is a PSNE. Otherwise, the same procedure can be repeated, and any player included

in the set of players using intermodal transportation will never have any incentive to change its decision

24

to direct shipment (because KKZ i )(3,0 is nonincreasing in K). Because { }N;...;1 is a finite set, the

proposed procedure necessarily converges; thus, a PSNE always exists for the UE allocation game.

Proof of Theorem 3

Consider the special case in which the UE allocation game is restricted to only two players. The payoff

matrix above can be used to model this game. Player 1 can decide to choose either top or bottom, while

player 2 can decide to choose either left or right. The payoffs received by each player appear in each of

the four cells representing possible outcomes of the game; the first value is received by player 1, and the

second is received by player 2. Because of the special feature of the game, the payoff associated with

direct shipment is equal to 0. In addition, BA ≥ and ba ≥ as a result of economies of scale. Consider

the case in which BA >> 0 and ba >> 0 . In this case, the corresponding UE allocation game is

similar to a stag hunt game in which two PSNEs exist.

Proof of Theorem 4 Theorem 4 derives from the notion that any player included in the set of players using intermodal

transportation in any given PSNE will never have an incentive to change its decision to direct shipment,

while increasing the number of players using intermodal transportation due to flow-dependent economies

of scale.

Proof of Corollary 1 Assume that there are L PSNEs in a given setting of the UE allocation game. Using theorem 4, we can

deduce that { }1;...;1),()( ,1, −∈∀≤+ LlUCardUCard lili . Because 1+lS and lS are two distinct Nash

equilibria, we also find that { }1;...;1),()( ,1, −∈∀<+ LlUCardUCard lili .

By contradiction, assume that there are { }1;...;1 −∈ Ll , such that )(1)( ,1, lili UCardUCard =++ .

Then, only one player is added to the set of players using intermodal transportation. This player’s profit is

strictly greater than zero in the case of intermodal transportation given the other players’ decisions; thus,

25

1+lU is not a Nash equilibrium. This proves that { }1;...;1 −∈∀ Ll , 2)()( 1 +≥ +ll UCardUCard ; thus,

12

+

≤

NL .

Proof of Proposition 1

Let 0=τ . Assume that Nq =*τ ; then,

>Λ ∑

=

N

jjiNi nZ

1

3,0, and { }Nq ;...;1∈∀ ,

>Λ ∑

=

N

jjiqi nZ

1

3,0, .

It follows that { }Nq ;...;1∈∀ , 0)1,1( , >=−qiq XP , and thus { }{ }NjU i ;...;11, ∈= is the payoff-

dominant PSNE. If Nq <*τ , then player N will choose direct shipment even when all the players are

using the terminal because 0)1,1( ,1

3,0, ≤==>

≤Λ −

=∑ NiN

N

jjiNi XPnZ ; thus, 0*

, =NiX for all the

existing PSNEs. By induction, it follows with the same argument that 0*, =jiX for all *

τqq > . Using the

argumentation developed in the case in which Nq =*τ , we prove that { }{ }*

1, ;...;1 τqjNjU i ≤∈= is the

payoff-dominant PSNE.

Assume that 0* =τp ; then, *τqq ≤∀ ,

>Λ ∑

=

N

jjiqi nZ

1

3,0, . Considering first that all the

destinations are shipped to directly, 0)0,1( ,1

1

1

3,01, >==>

≤Λ −

=∑ Nij

jii XPnZ ; thus, player 1 can

individually decide to use the inland terminal. By induction, it follows that all the destinations *τqq ≤ can

iteratively decide to use the inland terminal while being profitable. Thus, 1U is the only PSNE of the

given UE allocation game. Otherwise, 0** >> ττ pq because

>Λ ∑

=

*

*

1

3,0,

τ

τ

q

jjiqi nZ . Using the same

argument as in the case of 0* =τp , we can show that all the destinations q , such that **ττ qqp ≤< , can

iteratively decide to use the inland terminal while being profitable, and we conclude that

{ }{ }*2, ;...;1 τpjNjU i ≤∈⊂ . The procedure may then be iterated to find 2,iU if this exists. Using

corollary 1, algorithm UE necessarily stops, and all the PSNEs can be identified.

26

Proof of Proposition 2

Proving proposition 2 implies showing that ranking all destinations { }Nj ;...;1∈ from the largest to the

smallest value of ji,Λ enables the identification of iS . By contradiction, assume that the destinations are

not ranked according to their ji,Λ value, and let { }

−Λ= ∑∑∑

===∈

q

jji

q

jj

q

jjij

NqnZnnq

1

3,0

11,

;...;1

* maxargmin .

If *qp > such that *,, qipi Λ>Λ , then

+

+−Λ+Λ<

−Λ ∑∑∑∑∑∑

======p

q

jjip

q

jjpip

q

jjij

q

jji

q

jj

q

jjij nnZnnnnnZnn

******

1

3,0

1,

1,

1

3,0

11, because

)(. 3,0 KZK i is concave in K . Thus, an optimum is reached at *q when *,, qiqi Λ≥Λ for all *qq ≤ and

when *,, qiqi Λ<Λ for all *qq > , and ranking all destinations { }Nj ;...;1∈ from the largest to the

smallest value of ji,Λ enables the identification of iS .

Proof of Theorem 5

A cooperative ( )vSi ;∅≠ game is convex when v is supermodular; that is, iSVT ⊆∀ , ,

( ) ( ) ( ) ( )VvTvVTvVTv +≥∩+∪ . This property is equivalent to

{ }( ) ( ) { }( ) ( )VvjVvTvjTv −∪≤−∪ , iSVT ⊆⊆∀ \{ }j , iSj∈∀ .

iSVT ⊆⊆∀ \{ }j , iSj∈∀ :

{ }( ) { }( ) ( ) =+∪−−∪ TvjTvVvjVv )(

{ } { } { } { }03

,03,0

3,0

3,0 ≥

+

−

−

∑∑∑∑∑∑∑∑∈∈∪∈∪∈∈∈∪∈∪∈ Vk

kiVk

kjVk

kijVk

kVk

kiVk

kjTk

kijTk

k nZnnZnnZnnZn

because )(. 3,0 KZK i is nondecreasing concave in K . Proposition 8 follows from the fact that any convex

game has a nonempty core.

Proof of Theorem 6

We first prove that { }Mi ;1∈∀ , ii MS ⊆ . Note that { }{ }0;...;1 , ≤Λ∈= jii NjM because

( ) jjijjijji ZtruckZZ .,,01,

1,0, δδ −=−=Λ . If ∅=iS , the result holds. Assume that ∅≠iS , and let

27

iSa∈ . Then, *,, qiai Λ≥Λ , where *q is defined as in algorithm SO. Moreover, because

{ }

−Λ= ∑∑∑

===∈

q

jji

q

jj

q

jjij

NqnZnnq

1

3,0

11,

;...;1

* maxargmin , 0*, >Λ qi ; thus, 0, ≥Λ ai and iMa∈ .

To prove that ii SU ⊆1, , we argue that any destination that is individually better when using the

inland terminal contributes to global cost minimization because this decision has no negative effect on the

other destinations. By transitivity of the inclusion, we can conclude that iii MSU ⊆⊆1, .

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30

Working Papers Beta 2009 - 2014 nr. Year Title Author(s) 449 448 447 446 445 444 443 442 441 440 439 438

2014 2014 2014 2014 2014 2014 2014 2014 2013 2013 2013 2013

Intermodal hinterland network design with multiple actors The Share-a-Ride Problem: People and Parcels Sharing Taxis Stochastic inventory models for a single item at a single location Optimal and heuristic repairable stocking and expediting in a fluctuating demand environment Connecting inventory control and repair shop control: a differentiated control structure for repairable spare parts A survey on design and usage of Software Reference Architectures Extending and Adapting the Architecture Tradeoff Analysis Method for the Evaluation of Software Reference Architectures A multimodal network flow problem with product Quality preservation, transshipment, and asset management Integrating passenger and freight transportation: Model formulation and insights The Price of Payment Delay On Characterization of the Core of Lane Covering Games via Dual Solutions Destocking, the Bullwhip Effect, and the Credit Crisis: Empirical Modeling of Supply Chain Dynamics

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437 436 435 434 433 432 431 430 429 428 427

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Methodological support for business process Redesign in healthcare: a systematic literature review Dynamics and equilibria under incremental Horizontal differentiation on the Salop circle Analyzing Conformance to Clinical Protocols Involving Advanced Synchronizations Models for Ambulance Planning on the Strategic and the Tactical Level Mode Allocation and Scheduling of Inland Container Transportation: A Case-Study in the Netherlands Socially responsible transportation and lot sizing: Insights from multiobjective optimization Inventory routing for dynamic waste collection Simulation and Logistics Optimization of an Integrated Emergency Post Last Time Buy and Repair Decisions for Spare Parts A Review of Recent Research on Green Road Freight Transportation Typology of Repair Shops for Maintenance Spare Parts

Rob J.B. Vanwersch, Khurram Shahzad, Irene Vanderfeesten, Kris Vanhaecht, Paul Grefen, Liliane Pintelon, Jan Mendling, Geofridus G. Van Merode, Hajo A. Reijers B. Vermeulen, J.A. La Poutré, A.G. de Kok Hui Yan, Pieter Van Gorp, Uzay Kaymak, Xudong Lu, Richard Vdovjak, Hendriks H.M. Korsten, Huilong Duan J. Theresia van Essen, Johann L. Hurink, Stefan Nickel, Melanie Reuter Stefano Fazi, Tom Van Woensel, Jan C. Fransoo Yann Bouchery, Asma Ghaffari, Zied Jemai, Jan Fransoo Martijn Mes, Marco Schutten, Arturo Pérez Rivera N.J. Borgman, M.R.K. Mes, I.M.H. Vliegen, E.W. Hans S. Behfard, M.C. van der Heijden, A. Al Hanbali, W.H.M. Zijm Emrah Demir, Tolga Bektas, Gilbert Laporte M.A. Driessen, V.C.S. Wiers, G.J. van Houtum, W.D. Rustenburg

426 425 424 423 422 421 420 419 418 417 416 415

2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013

A value network development model and Implications for innovation and production network management Single Vehicle Routing with Stochastic Demands: Approximate Dynamic Programming Influence of Spillback Effect on Dynamic Shortest Path Problems with Travel-Time-Dependent Network Disruptions Dynamic Shortest Path Problem with Travel-Time-Dependent Stochastic Disruptions: Hybrid Approximate Dynamic Programming Algorithms with a Clustering Approach System-oriented inventory models for spare parts Lost Sales Inventory Models with Batch Ordering And Handling Costs Response speed and the bullwhip Anticipatory Routing of Police Helicopters Supply Chain Finance: research challenges ahead Improving the Performance of Sorter Systems By Scheduling Inbound Containers Regional logistics land allocation policies: Stimulating spatial concentration of logistics firms The development of measures of process harmonization

B. Vermeulen, A.G. de Kok C. Zhang, N.P. Dellaert, L. Zhao, T. Van Woensel, D. Sever Derya Sever, Nico Dellaert, Tom Van Woensel, Ton de Kok Derya Sever, Lei Zhao, Nico Dellaert, Tom Van Woensel, Ton de Kok R.J.I. Basten, G.J. van Houtum T. Van Woensel, N. Erkip, A. Curseu, J.C. Fransoo Maximiliano Udenio, Jan C. Fransoo, Eleni Vatamidou, Nico Dellaert Rick van Urk, Martijn R.K. Mes, Erwin W. Hans Kasper van der Vliet, Matthew J. Reindorp, Jan C. Fransoo S.W.A. Haneyah, J.M.J. Schutten, K. Fikse Frank P. van den Heuvel, Peter W. de Langen, Karel H. van Donselaar, Jan C. Fransoo Heidi L. Romero, Remco M. Dijkman, Paul W.P.J. Grefen, Arjan van Weele

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2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013 2013

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Paul Grefen, Egon Lüftenegger, Eric van der Linden, Caren Weisleder Duygu Tas, Nico Dellaert, Tom van Woensel, Ton de Kok Marco Comuzzi, Guus Jacobs, Paul Grefen A. Al Hanbali, E.M. Alvarez, M.C. van der van der Heijden Frank P. van den Heuvel, Karel H. van Donselaar, Rob A.C.M. Broekmeulen, Jan C. Fransoo, Peter W. de Langen Anna Franceschetti, Dorothée Honhon,Tom van Woensel, Tolga Bektas, GilbertLaporte. J.A. Larco, V. Wiers, J. Fransoo J. Theresia van Essen, Mark van Houdenhoven, Johann L. Hurink Pieter Van Gorp, Marco Comuzzi Kasper van der Vliet, Matthew J. Reindorp, Jan C. Fransoo Edgar E. Blanco, Jan C. Fransoo Duygu Tas, Ola Jabali, Tom van Woensel

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The Service Dominant Business Model: A Service Focused Conceptualization Relationship between freight accessibility and Logistics employment in US counties A Condition-Based Maintenance Policy for Multi-Component Systems with a High Maintenance Setup Cost A flexible iterative improvement heuristic to Support creation of feasible shift rosters in Self-rostering Scheduled Service Network Design with Synchronization and Transshipment Constraints For Intermodal Container Transportation Networks Destocking, the bullwhip effect, and the credit Crisis: empirical modeling of supply chain Dynamics Vehicle routing with restricted loading capacities Service differentiation through selective lateral transshipments A Generalized Simulation Model of an Integrated Emergency Post Business Process Technology and the Cloud: Defining a Business Process Cloud Platform Vehicle Routing with Soft Time Windows and Stochastic Travel Times: A Column Generation And Branch-and-Price Solution Approach

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2012 2012 2012 2012 2012 2012 2012 2012 2012 2012 2012

Improve OR-Schedule to Reduce Number of Required Beds How does development lead time affect performance over the ramp-up lifecycle? Evidence from the consumer electronics industry The Impact of Product Complexity on Ramp- Up Performance Co-location synergies: specialized versus diverse logistics concentration areas Proximity matters: Synergies through co-location of logistics establishments Spatial concentration and location dynamics in logistics:the case of a Dutch province FNet: An Index for Advanced Business Process Querying Defining Various Pathway Terms The Service Dominant Strategy Canvas: Defining and Visualizing a Service Dominant Strategy through the Traditional Strategic Lens A Stochastic Variable Size Bin Packing Problem With Time Constraints

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380 379 378 377 375 374 373 372 371 370 369 368 367

2012 2012 2012 2012 2012 2012 2012 2012 2012 2012 2011 2011 2011

Coordination and Analysis of Barge Container Hinterland Networks Proximity matters: Synergies through co-location of logistics establishments A literature review in process harmonization: a conceptual framework A Generic Material Flow Control Model for Two Different Industries Improving the performance of sorter systems by scheduling inbound containers Strategies for dynamic appointment making by container terminals MyPHRMachines: Lifelong Personal Health Records in the Cloud Service differentiation in spare parts supply through dedicated stocks Spare parts inventory pooling: how to share the benefits Condition based spare parts supply Using Simulation to Assess the Opportunities of Dynamic Waste Collection Aggregate overhaul and supply chain planning for rotables Operating Room Rescheduling

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366 365 364 363 362 361 360 359 358 357 356 355 354

2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 2011

Switching Transport Modes to Meet Voluntary Carbon Emission Targets On two-echelon inventory systems with Poisson demand and lost sales Minimizing the Waiting Time for Emergency Surgery Vehicle Routing Problem with Stochastic Travel Times Including Soft Time Windows and Service Costs A New Approximate Evaluation Method for Two-Echelon Inventory Systems with Emergency Shipments Approximating Multi-Objective Time-Dependent Optimization Problems Branch and Cut and Price for the Time Dependent Vehicle Routing Problem with Time Window Analysis of an Assemble-to-Order System with Different Review Periods Interval Availability Analysis of a Two-Echelon, Multi-Item System Carbon-Optimal and Carbon-Neutral Supply Chains Generic Planning and Control of Automated Material Handling Systems: Practical Requirements Versus Existing Theory Last time buy decisions for products sold under warranty Spatial concentration and location dynamics in logistics: the case of a Dutch provence

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353 352 351 350 349 348 347 346 345 344 343 342 341 339 338

2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 2011 2010 2010 2010 2010

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335 334 333 332 331 330 329 328 327 326 325 324 323

2010 2010 2010 2010 2010 2010 2010 2010 2010 2010 2010 2010 2010

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E. Lüftenegger, S. Angelov, E. van der Linden, P. Grefen

309 2010 Design of Complex Architectures Using a Three Dimension Approach: the CrossWork Case R. Seguel, P. Grefen, R. Eshuis

308 2010 Effect of carbon emission regulations on transport mode selection in supply chains

K.M.R. Hoen, T. Tan, J.C. Fransoo, G.J. van Houtum

307 2010 Interaction between intelligent agent strategies for real-time transportation planning

Martijn Mes, Matthieu van der Heijden, Peter Schuur

306 2010 Internal Slackening Scoring Methods Marco Slikker, Peter Borm, René van den Brink

305 2010 Vehicle Routing with Traffic Congestion and Drivers' Driving and Working Rules

A.L. Kok, E.W. Hans, J.M.J. Schutten, W.H.M. Zijm

304 2010 Practical extensions to the level of repair analysis R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

303 2010 Ocean Container Transport: An Underestimated and Critical Link in Global Supply Chain Performance

Jan C. Fransoo, Chung-Yee Lee

302 2010 Capacity reservation and utilization for a manufacturer with uncertain capacity and demand Y. Boulaksil; J.C. Fransoo; T. Tan

300 2009 Spare parts inventory pooling games F.J.P. Karsten; M. Slikker; G.J. van Houtum

299 2009 Capacity flexibility allocation in an outsourced supply chain with reservation Y. Boulaksil, M. Grunow, J.C. Fransoo

298

2010

An optimal approach for the joint problem of level of repair analysis and spare parts stocking

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

297 2009 Responding to the Lehman Wave: Sales Forecasting and Supply Management during the Credit Crisis

Robert Peels, Maximiliano Udenio, Jan C. Fransoo, Marcel Wolfs, Tom Hendrikx

296 2009 An exact approach for relating recovering surgical patient workload to the master surgical schedule

Peter T. Vanberkel, Richard J. Boucherie, Erwin W. Hans, Johann L. Hurink, Wineke A.M. van Lent, Wim H. van Harten

295

2009

An iterative method for the simultaneous optimization of repair decisions and spare parts stocks

R.J.I. Basten, M.C. van der Heijden, J.M.J. Schutten

294 2009 Fujaba hits the Wall(-e) Pieter van Gorp, Ruben Jubeh, Bernhard Grusie, Anne Keller

293 2009 Implementation of a Healthcare Process in Four Different Workflow Systems

R.S. Mans, W.M.P. van der Aalst, N.C. Russell, P.J.M. Bakker

292 2009 Business Process Model Repositories - Framework and Survey

Zhiqiang Yan, Remco Dijkman, Paul Grefen

291 2009 Efficient Optimization of the Dual-Index Policy Using Markov Chains

Joachim Arts, Marcel van Vuuren, Gudrun Kiesmuller

290 2009 Hierarchical Knowledge-Gradient for Sequential Sampling

Martijn R.K. Mes; Warren B. Powell; Peter I. Frazier

289 2009 Analyzing combined vehicle routing and break scheduling from a distributed decision making perspective

C.M. Meyer; A.L. Kok; H. Kopfer; J.M.J. Schutten

288 2009 Anticipation of lead time performance in Supply Chain Operations Planning

Michiel Jansen; Ton G. de Kok; Jan C. Fransoo

287 2009 Inventory Models with Lateral Transshipments: A Review

Colin Paterson; Gudrun Kiesmuller; Ruud Teunter; Kevin Glazebrook

286 2009 Efficiency evaluation for pooling resources in health care

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

285 2009 A Survey of Health Care Models that Encompass Multiple Departments

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

284 2009 Supporting Process Control in Business Collaborations

S. Angelov; K. Vidyasankar; J. Vonk; P. Grefen

283 2009 Inventory Control with Partial Batch Ordering O. Alp; W.T. Huh; T. Tan

282 2009 Translating Safe Petri Nets to Statecharts in a Structure-Preserving Way R. Eshuis

281 2009 The link between product data model and process model J.J.C.L. Vogelaar; H.A. Reijers

280 2009 Inventory planning for spare parts networks with delivery time requirements I.C. Reijnen; T. Tan; G.J. van Houtum

279 2009 Co-Evolution of Demand and Supply under Competition B. Vermeulen; A.G. de Kok

278 277

2010 2009

Toward Meso-level Product-Market Network Indices for Strategic Product Selection and (Re)Design Guidelines over the Product Life-Cycle An Efficient Method to Construct Minimal Protocol Adaptors

B. Vermeulen, A.G. de Kok R. Seguel, R. Eshuis, P. Grefen

276 2009 Coordinating Supply Chains: a Bilevel Programming Approach Ton G. de Kok, Gabriella Muratore

275 2009 Inventory redistribution for fashion products under demand parameter update G.P. Kiesmuller, S. Minner

274 2009 Comparing Markov chains: Combining aggregation and precedence relations applied to sets of states

A. Busic, I.M.H. Vliegen, A. Scheller-Wolf

273 2009 Separate tools or tool kits: an exploratory study of engineers' preferences

I.M.H. Vliegen, P.A.M. Kleingeld, G.J. van Houtum

272

2009

An Exact Solution Procedure for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering

Engin Topan, Z. Pelin Bayindir, Tarkan Tan

271 2009 Distributed Decision Making in Combined Vehicle Routing and Break Scheduling

C.M. Meyer, H. Kopfer, A.L. Kok, M. Schutten

270 2009 Dynamic Programming Algorithm for the Vehicle Routing Problem with Time Windows and EC Social Legislation

A.L. Kok, C.M. Meyer, H. Kopfer, J.M.J. Schutten

269 2009 Similarity of Business Process Models: Metics and Evaluation

Remco Dijkman, Marlon Dumas, Boudewijn van Dongen, Reina Kaarik, Jan Mendling

267 2009 Vehicle routing under time-dependent travel times: the impact of congestion avoidance A.L. Kok, E.W. Hans, J.M.J. Schutten

266 2009 Restricted dynamic programming: a flexible framework for solving realistic VRPs

J. Gromicho; J.J. van Hoorn; A.L. Kok; J.M.J. Schutten;

Working Papers published before 2009 see: http://beta.ieis.tue.nl