1 Introduction
Due to its reliability, sustainability and its economical competitiveness, Intermodal Transportation (IT) has gained a good reputation. In fact, despite its lack of flexibility in the transport chain, intermodal transportation operators strikes to respect time schedules. Furthermore, the intermodal transportation is gaining ground over road transportation due to large investments in infrastructure development. Moreover, several customers focus, today, on environmentally solutions in the transport industry. For all these reasons, intermodal transportation has attracted the attention of researchers and industry operators. The location of intermodal terminals is among the most challenging issues in the scientific literature. We study in this paper the Intermodal Terminal Location Problem in incomplete networks. The remainder of this article is organized as follows: we provide the state of the art in the section 2, problem description is given in the section 3, we propose several extensions of the original formulation in the section 4 and we conclude in the section 5.
2 State of the art
The scientific literature on Intermodal Location Problems is relatively recent, but the number of articles dealing with this subject is steadily increasing. Arnold et al. (2001)
modeled the intermodal transshipment centers location as a linear program and proposed several extensions of the basic version.
Artmann and Fischer (2003) studied the sustainability issues of the traffic shift from road to rail. Bontekoning et al. (2004) provided a review synthesis about intermodal transportation in the field of operations research. Limbourg and Jourquin (2009) proposed an iterative heuristic based on the pmedian problem and on the multimodal assignement problem. Sorensen et al. (2012) proved that the intermodal terminal location problem is NPhard and proposed efficient heuristics to solve it. Tsamboulas et al. (2007) developed a methodology for the policy measures assessment for modal shift to intermodal transportation. Lin and Lin (2014) proposed a simplified version of the Sorensen model and proposed two efficient mathheuristics to solve it. Oudani et al. (2014)solved the problem using a genetic algorithm and proposed a new intermodal cost evaluation.
Lin and Lin (2018) proposed a twostage matheuristic approach to solve the problem. They proposed a program reformulation of the problem and test it using randomly generated data set. Abbassi et al. (2018) proposed a biobjective model for transportation of agricultural products from Morocco to Europe and developed heuristics to solve it. Recently, Mostert et al. (2018) proposed a biobjective mathematical model minimizing the transportation and environmental costs objectives. Abbassi et al. (2019) studied the robust intermodal freight transport problem and proposed two solutions approaches for solving the problem. To the best of our knowledge, the current paper is the first to consider the terminals network incompleteness.3 Problem description
3.1 The incomplete version motivation
As in hub location problems, most studies in intermodal terminal location problems assume a complete interterminals network, that is, every terminal pair is interconnected. In the incomplete network studied here, we relax this assumption. In fact, assuming a complete network increase the total investment cost and connecting all terminals directly may also become unnecessary and expensive.
Fig. 1 shows a small incomplete intermodal terminal network with 3 terminals and 4 customers. The induced graph by terminals is not complete. For instance, there is no rail link between terminal and .
3.2 Mathematical formulation
Let be the following parameters:
set of customers.
set of potentials sites for intermodal terminals.
number of links between located terminals.
the amount of goods to be transported from the customer to customer .
the unit cost for intermodal transportation from customer to customer through the two terminal and .
the unit unimodal cost for routing goods from customer to customer .
the cost for location of the terminal .
the capacity of the terminal .
Let be the following decision variables:
if the interterminals link between and is established, 0 otherwise.
the amount of goods transported by road from the customer to customer
the amount of goods transported by road from the customer to customer through the two terminal and
The incomplete intermodal terminal location problem may be modeled as follows:
(1) 
Subject to :
(2) 
(3) 
(4) 
(5) 
(6) 
(7) 
(8) 
The objective function (1) minimizes the total cost for routing goods by road and by using the intermodal railroad transportation. The constraint (2) states that the sum of the amount routed directly from to and that through the terminals and is equal exactly the total goods to be transported form to . The constraint (3) guarantees the respect of the terminals capacities. Inequalities (4) and (5) state that an interterminal link is established if the two terminals are located. The equation (6) specify that if an interterminal link is established in the two directions. The equation (7) states the establishment of the given number of the interterminal links and the last equation (8) forbids the interterminal flows between closed terminals. The decision variable controls the interterminal links to open. Thus, if for some reason, a railway link is inconceivable between two terminals du to geographic, economic or environmental constraints, this link is prohibited by .
The model is a Mixed 01 Integer Program (MIP). If we denote the cardinal of and by the cardinal of , then the program has constraints and variables.
If the triangular inequality holds for units cost then . This proposition demonstrates that this constraint used in several mathematical formulations in the literature is an unnecessary constraint [10]. The intermodal unit cost is calcultaed as follows: where are respectively the cost between the customer and the terminal , the interterminal cost between and and the cost between the terminal and customer with . The coefficient is the discount parameter expressing the scale economy generated by using rail mode between the terminal and . For instance, this coefficient is assumed to be equal to 0.5 in the work of Sörensen et al. [10]. Since, , then Let be the cardinal of the potentials sites set. If then the problem is unfeasible. A complete graph (fully connected) with vertices has at most edges.
3.3 Exact solutions
The model is validated by implementation in CPLEX 12.6 solver. To report numerical results, we used the instances randomly generated by Sö rensen et al. Customers and potentials sites coordinates are randomly generated between and . The goods demands are generated from the interval . The investment cost in the interval and potentials sites capacities are drawn from . After that, cost is equal the euclidean distance between the customers and while . The exact solutions for some instances are reported in the table 1. Optimal solutions for larger instances (more than 90 customers and 40 potential location) are not found in 1 hour.
Instance  Cost  Time (s)  # terminals 
10C10L2TL  10,2  0,55  6 
10C10L4TL  9,62  1,09  6 
10C10L6TL  9,42  1,00  7 
10C10L8TL  9,36  0,69  8 
10C10L10TL  9,31  1,42  6 
10C10L12TL  9,29  1,06  9 
20C10L2TL 
52,8  2,95  3 
20C10L4TL  52,7  2,52  5 
20C10L6TL  52,7  1,92  5 
20C10L8TL  52,7  1,63  5 
20C10L10TL  52,7  1,64  5 
20C10L12TL  52,8  3,08  6 
40C10L2TL 
201,5  31,02  7 
40C10L4TL  201  22,55  8 
40C10L6TL  200,8  36,92  8 
40C10L8TL  200,7  18,45  8 
40C10L10TL  200,6  18,78  8 
40C10L12TL  200,6  13,83  8 
80C10L2TL 
778,1  338,77  3 
80C10L4TL  778,8  1349,20  4 
80C10L6TL  778,8  599,92  4 
80C10L8TL  779,8  1529,13  5 
80C10L10TL  779,8  891,28  5 
4 Extensions
4.1 Minimizing the number of interterminals links
Instead of minimizing the number of terminals to be located, we consider the problem of minimizing the number of links between a given number of terminals to be located. This problem can be modeled as follows:
(9) 
Subject to:
(10) 
We report in the table 2 optimal solutions of this version for some instances with different values of number of terminals.
Instance  Cost  Time (s)  # links 

10C10L2T  11,24  2,31  2 
10C10L4T  10,2  0,55  5 
10C10L6T  9,82  0,63  7 
10C10L8T  9,74  0,45  9 
10C10L10T  9,74  0,44  10 
20C20L4T  50,9  161,41  6 
20C20L8T  48,2  270,94  12 
20C20L12T  46,9  3442,39  17 
20C20L16T  *  *  * 
20C20L20T  *  *  * 
30C30L4T  117,2  1658,91  10** 
30C30L8T  *  *  * 
30C30L12T  *  *  * 
30C30L16T  *  *  * 
30C30L20T  *  *  * 
40C40L4T  *  *  * 
40C40L8T  *  *  * 
40C40L12T  *  *  * 
40C40L16T  *  *  * 
40C40L20T  *  *  * 
50C50L8T  *  *  * 
50C50L12T  *  *  * 
50C50L16T  *  *  * 
50C50L20T  *  *  * 
60C60L4T  *  *  * 
60C60L8T  *  *  * 
60C60L12T  *  *  * 
60C60L16T  *  *  * 
60C60L20T  *  *  * 
4.2 Minimizing the handling cost in terminals
This version aims to minimize the operational handling cost in terminals. We denote the handling cost in terminal to terminal (we consider this cost as asymetric i.e ). This problem may be formulated as follows:
(11) 
Subject to:
We report in the table 3 optimal solutions of this version for some instances with different values of number of interterminals links.
Instance  Cost  Time (s)  # links 

10C10L2TL  11,24  2,31  2 
10C10L4TL  10,2  0,55  5 
10C10L6TL  9,82  0,63  7 
10C10L8TL  9,74  0,45  9 
10C10L10TL  9,74  0,44  10 
20C20L4TL  50,9  161,41  6 
20C20L8TL  48,2  270,94  12 
20C20L12TL  46,9  3442,39  17 
40C40L4TL  *  *  * 
40C40L8TL  *  *  * 
40C40L12TL  *  *  * 
40C40L16TL  *  *  * 
40C40L20TL  *  *  * 
50C50L8TL  *  *  * 
50C50L12T  *  *  * 
50C50L16TL  *  *  * 
50C50L20TL  *  *  * 
60C60L4TL  *  *  * 
60C60L8TL  *  *  * 
60C60L12TL  *  *  * 
60C60L16TL  *  *  * 
60C60L20TL  *  *  * 
4.3 Intermodal terminal location problem
In this version both the number of the terminals to be located and the number of the interterminals links are given. This problem can be modeled as follows:
(12) 
Subject to:
We report in the table 4 optimal solutions of this version for some instances.
Instance  Cost  Time (s) 

10C10L2T2TL  6,24  2,31 
10C10L4T4TL  9,12  0,35 
10C10L6T6TL  9,82  0,53 
10C10L8T8TL  9,74  0,55 
10C10L10T10TL  9,74  0,47 
20C20L4T4TL  54,9  171,67 
20C20L8T8TL  45,2  250,65 
20C20L12T12TL  41,9  3456,65 
40C40L4T4TL  *  * 
40C40L8T8TL  *  * 
40C40L12T12TL  *  * 
40C40L16T16TL  *  * 
40C40L20T20TL  *  * 
50C50L8T8TL  *  * 
50C50L12T12TL  *  * 
50C50L16T16TL  *  * 
50C50L20T20TL  *  * 
60C60L4T4TL  *  * 
60C60L8T8TL  *  * 
60C60L12T12TL  *  * 
60C60L16T16TL  *  * 
60C60L20T20TL  *  * 
5 Conclusion
In this paper, we proposed a general version of the classical Intermodal Terminal Location Problem (ITLP) when the induced graph by located terminals is not necessarily complete. We formulate the problem as 01 linear program and proposed several extentions. We reported numerical results on data set instances given in the literature using CPLEX solver. As perspectives, we envision:

To develop efficient heuristics to solve larger instances

To combine the problem with routing problems

To study a hybrid hub intermodal terminal location problem considering incomplete networks
References
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