Robust optimisation of the intermodal freight transport problem: Modeling and solving with an efficient hybrid approach

Accepted Manuscript Title: Robust optimisation of the intermodal freight transport problem: modeling and solving with an efficient hybrid approach Authors: Abderrahman Abbassi, Ahmed El hilali Alaoui, Jaouad Boukachour PII: DOI: Reference:

S1877-7503(18)30665-3 https://doi.org/10.1016/j.jocs.2018.12.001 JOCS 951

To appear in: Received date: Revised date: Accepted date:

20 June 2018 3 December 2018 3 December 2018

Please cite this article as: Abbassi A, Alaoui AEh, Boukachour J, Robust optimisation of the intermodal freight transport problem: modeling and solving with an efficient hybrid approach, Journal of Computational Science (2018), https://doi.org/10.1016/j.jocs.2018.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Robust optimisation of the intermodal freight transport problem: modeling and solving with an efficient hybrid approach Abderrahman Abbassi 1,2*, Ahmed El hilali Alaoui 1 and Jaouad Boukachour 2 1Modeling

and scientific computing laboratory, Faculty of Sciences and Technologies, USMBA University, Morocco Mathematics Laboratory of Le Havre, UNIHAVRE, Normandie University, France *Corresponding author: [email protected]

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2Applied

Highlights

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 (1) Robust optimisation models for the intermodal terminal location problem (ITLP).  (2) A developed population based simulated annealing approach.  (3) An efficient hybrid approach based on a meta-heuristic and an exact method for deterministic and robust ITLP.  (4) Real case study for the ITLP with and without uncertainties.  (5) Comparative studies prove that the proposed approaches provide better results.

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Abstract

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In the present paper, we propose robust optimisation models for the intermodal freight transport problem for dealing with uncertainties in using costs, and the capacities of terminals and uncertainties in transportation costs. Moreover, we propose two solution approaches for efficiently solving the problem. The first one is a populationbased simulated annealing (PBSA) with heuristics. The second one is a hybrid solution approach combining the PBSA with an exact method. In addition to a theoretical study of randomly-generated instances, a real network is addressed for testing the performance of the proposed methods and models and analysing the impact of robustness.

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1. Introduction

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Keywords: Robust Optimisation; Intermodal Transport; Terminal Location; Hybrid Approach; Simulated Annealing

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Intermodal freight transportation plays a key role in supply chain logistics. Because of globalization and growth of international trade, the use of intermodal transport has been growing remarkably [1]. According to the European Commission of 2011 [2], shifting goods from road mode to other transportation modes is greatly required and 50 percent (50%) of freight should be dispatched via multimodal transport by 2050. Intermodal transport can make a considerable impact to facilitate the international transport of goods and has become an important component of the economy of every country. This importance makes it a process deserving a rigorous study and allows it to receive serious financial, political, contractual and economical support [3]. It has improved efficiency of the import and the export of freight between countries involved in international trade. The concept of intermodality arises when transport between suppliers and their customers occurs not only directly by one mode but also by the use of at least two different modes of transport [4]. Intermodal freight transport problems have applications in many areas such as importation and exportation problems, transportation of hazardous materials (Hazmat), and passenger’s international transport [5, 6, 7]. Our country for example, exports products to Tunisia directly by road mode. It also exports goods to France by intermodal links; so products are firstly transported by road mode (trucks) from the suppliers to a port of our country (terminal of origin area), then by maritime mode towards a European port (terminal of destination area), and finally by road mode to France. That scenario is an intermodal road/ship transportation.

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It combines advantages of both modes for reducing greenhouse gas emissions and shipping large quantities of goods for long haul transport [8]. When a decision about terminal location is also required, the problem is then an intermodal terminal location problem ITLP. Given a set of customers and a set of terminals, the goal of the ITLP is to serve all customers while reducing the total transportation cost which comprises the using cost of terminals and the distribution cost. In addition, each customer has its demand, while each terminal has a limited capacity and a using cost. Customer-customer links are unimodal and represent door-to-door services. In our case, they are undertaken by road mode. In addition, intermodal links can be used by combining two modes and using two terminals. When these services are used, goods are transported from an origin node to a terminal, then to another terminal of the destination area, then finally to the destination node. In our case, customer-terminal links are ensured by road mode, terminal-terminal by maritime mode and finally terminal-customer linkings by road mode. The ITLP assumptions are described in more detail in section 3. International transport companies seek to improve the procurement and distribution of products by managing infrastructure (ports) and resources (transportation modes) in order to minimise the total cost of transport. Indeed, for an effective strategy of distribution, the terminals should be located and the paths and the modes of transportation should be optimally partitioned. For this purpose, the transport costs, the capacities and the usage costs of terminals, as well as the demands of customers are often suggested by the decision-maker to be taken into consideration as input data for making the appropriate decisions on intermodal freight transport. In many papers of the literature, the input parameters of intermodal transport problems are assumed to be deterministic and known. In fact, this assumption is not realistic. Long term decisions such as intermodal terminal location brings some challenges such as lack of information about some parameters of the problem or poorly knowing their future values due to estimations, measurements or implementations. This trend may affect negatively the quality of the decisions. It is therefore essential to guarantee robustness of the solutions against uncertainties. The capacity parameter may differ due to many components. It increases for example due to investments or expansion works made by the terminal operator year by year to be able to manage the flow of more containers. It may also decrease because of the internal management problems within terminals or due to temporary public works which disrupts places or equipment in the terminal. Another example in the same vein, simple unexpected breakdowns in the refrigeration systems of ports brings a variation of capacities assigned by the terminal operator for the carriers of the perishable products which lose their quality and need refrigeration services. The annual report "Ports en chiffres'' published in 2016 by the Moroccan ministry of the transport and logistics confirms this trend. We remark in this report that the capacity of Casablanca port increased by +3 % during the period 2007-2009, while the capacity of Layonne port decreased by 48 % during the same period. The uncertainty in capacities is considered in the transportation problem studied by Boukani et al. [9]. According to Almur et al 2012 [10], the using cost may vary due to many factors, such as the price of raw material and the price of possessions. Moreover, the increasing number of containers, the large dimensions of vessels, the delays in loading and unloading, congestion and the problems of the internal management of terminals, all have as consequences some variations and perturbations, which may make the usage costs of terminals an uncertain parameter. Uncertainty in using cost is assumed by Alumur et al. [10] and Boukani et al. [9]. Transportation cost is extremely dependent on several factors such as the quality of services provided and the prices of oil which vary. Therefore, transportation companies and decision makers -when they make long term decisions- prefer solutions that are robust against any changes even if these changes are minor, infrequent or incidental. Uncertainty in transportation costs are considered in the transportation problems studied by [11, 12, 13, 14]. In order to highlight the overall aim of this article, three major contributions have been investigated. An efficient method is proposed combining a developed population-based simulated annealing approach and an exact approach for solving the problem, a method for dealing with some instances where the Cplex solver

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fails to find solutions; and a method for obtaining better solutions than metaheuristics. In addition, the problem is addressed not only in terms of a deterministic case, but also in a robust version; this is done by proposing optimisation models for obtaining solutions that are robust against uncertainty, and by studying the impact of uncertainties on the transport design and the choice of modes, paths, amounts and terminals. Moreover, a real network case is studied for applying the deterministic and the proposed robust models and solution approaches on a real case and not only in theoretical instances. The remainder of this paper is organised as follows: Section 2 provides a literature review of the intermodal freight transport problem both with and without uncertainties. Section 3 presents the general optimisation model of the deterministic intermodal terminal location problem. The proposed solution approach for solving that model is described in detail in Section 4. The robust optimisation models proposed are presented in Section 5 for dealing with uncertainties. In Section 6, a real case study and a numerical analysis are reported. Finally, Section 7 concludes the paper with some discussions and future works. 2. Literature review

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Freight intermodal transport was defined by Crainic and Kim [4] as the transport of goods from production sites to customers using loading units (containers) with a combination of different transportation modes (maritime, road, rail, air, etc.). It has become an attractive area of research and the number of articles dealing with that problem has remarkably risen in the last few decades. Many applications have been investigated, many models have been presented and different solution approaches have been proposed. But only a small number of papers have addressed the problem of uncertainties. This section provides a literature review of some related papers; beginning with Arnold et al. [15], where a mixed-integer variables model is proposed for giving the best ways to locate terminals and construct routes in an intermodal rail-road network. The model is then extended to many variants and modeled with various mathematical formulations. The intermodal terminal location problem is studied by Sörensen et al [16]. They used a Greedy Randomised Adaptive Search Procedure (GRASP) and attribute-based Hill Climber method (ABHC) with an improvement phase of local search. The solution quality of the first method is nearly equivalent to the second algorithm. Lin et al. [17] focused also on this version of intermodal transport and the studied mathematical formulation is modified by integration of simplified constraints, so as to enhance the model and improve the efficiency of solving the problem in terms of running time. In their case, the routing of goods between customers can be made via terminals or directly via door-to-door services. Racunica and Wynter [18] studied an intermodal transportation problem in terms of an incapacitated hub location problem in a rail network. The problem is modeled as a mixed integer variable model with a concave objective function describing transportation costs between hubs and destinations for minimising the total distribution cost. In the same way, the problem addressed by Ishfaq and Sox [19] is a multimodal version of the p-hub median. They proposed a mathematical model for a transportation problem of the railroad network. Given a set of cities and a set of hubs, the model involves locating a subset of hubs and building routes between cities, including pairs of hubs. The service time allowed between two cities was considered in the model; without neglecting the possible delays at the open hubs. A tabu search metaheuristic is presented and the computational results prove that it finds good solutions with a deviation of 0.71 from the optimal solution. Their objective was to minimise the sum of costs of location and transportation through the constructed intermodal links. The intermodal freight transport has also received the attention of Meng and Wang [20]; they addressed the problem in terms of the hub-and-spoke network of multiple stockholders and many types of containers. In addition to the constraints in managing the components of the network, the proposed model deals also with budget limit constraints and the capacities of physical and trans-shipment links with an aim of minimising the sum of operation and transportation costs. A hybrid genetic algorithm is adapted for solving the problem with a real Asian network consisting of three modes and two types of containers. The algorithm is based on the standard genetic algorithm with its usual steps, except the fitness evaluation step where a Frank-Wolfe method is used to solve a sub-problem. The proposed algorithm is compared with an exhaustive enumeration

method. The computational time and the quality of solutions prove its performance. In terms of a truck scheduling problem, Nossack and Pesch [21] studied the intermodal transport by considering a set of original depots, a set of terminals and a set of pick-up and delivery requests with a time window. The aim was to ensure the pre-routing and haulage transport of containers (including empty containers) by constructing arcs to be traversed by trucks in addition to the beginning times of activities in order to minimise the total operating time.

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The decisions wanted by the decision-makers, the nature of the products to be transported, the challenging constraints and the objectives, all influence the type of problem and its outcomes. Intermodal freight transport copes with many applications and is based on many real networks and realistic case studies. To name just a few, a flow based mathematical model is proposed by Bhattacharya et al. [5] for strategic transport planning of a distribution network of an Indian region. Their model requires as input data the inventory and holding capacities of terminals, train capacities, and loading, unloading and routing costs in order to determine the best start, end and transit times associated with each terminal. In addition to the profit and the economical objective, the environmental sustainability and the performance of ports have a considerable impact on intermodality [22, 23, 24]. The sustainable intermodal transport was the subject of Baykasoğlu and Subulan [25]. They proposed a multi-objective model for a real multi-period intermodal transport problem. It is applied in a road-sea-rail network in Turkey. The main idea was to find the best import and export quantities at each period. They looked for minimisation of transportation cost and satisfaction of customers by minimising the transit time in addition to an environmental objective by reducing 𝐶𝑂2 emissions in the network. The greenhouse gas emissions are also considered in the integrated production-distribution model proposed by Meisel et al. [26]. They studied transportation of chemical products of an international company to some European regions. The model proposed combines intermodal transportation with inventory and production management. The aim was to determine the output quantity of each production site, managing inventories at each period and routing goods through rail or road transport arcs in order to minimise the total cost of transport and the share of demand transported using direct road transport as an environmental objective. A heuristic approach and an exact method are used for solving the studied problem.

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Characteristics and perishability of products transported intermodaly are focused upon by Abbassi et al. [27], and they deal with multimodal transport of agricultural goods from production sites to final customers. They presented a bi-objective model for minimising the total cost of transport and minimising the overtime for delivering products as soon as possible. That approach allows respecting capacities of production sites and satisfying demands of customers within their time window. Moreover, they proposed realistic constraints for guaranteeing quality of goods by assuming that the transportation time in addition to delays must not exceed the lifetime of the product. A multi-objective version of the greedy randomised adaptive search algorithm and a hybrid non-dominated sorting genetic algorithm are proposed for solving the problem. The computational results and the performance metrics demonstrate the superiority of the second approach. The model is validated by a real case study of exportation of vegetables and fruit from Morocco to Europe. Note that the nonperishable products which don’t require temporal constraints can be addressed in the intermodal terminal location problem discussed in our paper. Briefly, for more highlights about intermodal freight distribution, a survey is detailed in [28, 29]. As we mentioned in the introduction, the uncertainty deserves to be studied and robust solutions are needed by decision makers because of rapid daily changes in an economically and contractually unstable world. Moreover, various parties are involved with the intermodal transport network, which means many possible sources of uncertainty. For simplifying and for esthetic and organisational reasons, we provide in Table 1 some related works classified according to the variants studied, types of modes used in the network, the objective functions, the solution approaches adopted, the type of benchmarking and, finally, if the uncertainty is taken into consideration or not.

Table 1 Some related papers Paper

Problem type

Type of modes

Objectives

Methods

Uncertainty type

Intermodal terminals location

Ra , Ro

c

H

-

[30]

Intermodal transportation planning

Us

c

H

Demands

[19]

Intermodal terminals location

Ra , Ro

c

MH

-

[31]

Intermodal terminals location

Ra , Ro, W

c

Us

-

[20]

Intermodal hub-and-spoke

Ra , Ro, W

c

MH

-

[10]

Hub location

Us

c

Cp

Set-up cost , Demands

[16]

Intermodal terminals location

Ra , Ro

c

[32]

Intermodal transportation planning

Ra , Ro

c,r

[33]

Intermodal transportation planning

Ra , A , W

c,t

[26]

Production–intermodal distribution

Ra , Ro

c,e

[17]

Intermodal terminals location

Ra , Ro

c

[34]

Intermodal transportation planning

Ra , Ro, W

c , t, e

[35]

Intermodal transportation planning

Ra , Ro, W

c,t

[36]

Intermodal transportation planning

Ra , Ro

[25]

Intermodal transportation planning

Ra , Ro, W

[37]

Hub location

Ro

[11]

Hub location

Us

[38]

Intermodal allocation

Ra , Ro, W

[39]

Intermodal transportation planning

[40]

Hub location

[41]

Hub location

[27]

Intermodal transportation planning Intermodal terminal location

MH

-

MH

-

H

-

B&C , H

-

H

-

Cp

Demands , Time

Cp

-

Demands

L

-

c

B&C

-

c

B&C

Demand, Cost

c

Cplex

-

Ra , Ro, W

c

B&B, H

-

Us

c

Cp

Flows

Us

t,c

MH

Time, Demand, Cost

Ro , W

c,o

MH

-

Ro , W

c

MH+Exact

Cost , Capacity , Using cost

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MH

c, t , e

D

This paper

mr

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[15]

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Ra: Rail – Ro : Road – W : Water – Us: Unspecified – C: cost – T: time – e: environmental - o: overtime – mr: maximum regretH: heuristics- MH: metaheuristic – B&B: branch and bound- B&C: branch and cut- L: lingo - Cp: cplex.

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The table shows a dominance of combined rail-road transport which may be due to geographic and network topology factors. The transportation cost seems to be the most targeted objective and the number of works dealing with uncertainties is still very few. In addition, metaheuristics and heuristics are the most used solution methods. The interpolation between exact approaches and metaheuristics are also considered for solving some related problems [42]. In our paper we are interested in such methods which are known as matheuristics.

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We describe some similarities and differences between the problems of this table. In hub location problems HLP, we select a subset of nodes to be hubs for ensuring goods distribution between non-hub nodes. The capacity constraints refer to hub and non-hub nodes [43, 44] or to arcs of the network [45]. However, in the ITLP networks, there is set of nodes (customers) having demands and another set of terminals which are concerned by the location decision and the capacity constraints [16]. Moreover, in HLP papers, a single mode is used for transporting goods and sometimes it is unspecified; this is shown in the table. In addition, flows are just distributed through hub nodes and there is no direct links between non-hub nodes [46, 47, 48]. Nevertheless, in the ITLP, flows can be transported through direct links. It also involves the use of at least two different transportation modes when the transport is ensured through terminals. Applications of HLPs include telecommunication, distribution systems in urban areas, computer network, postal delivery, rapid transit systems and express package [11]. However, ITLPs are applied to exportation and importation problems that require integration of different modes, hazardous materials transportation and massively expanded transport of goods. In fact, the intermodality is widely practical in the transport of passengers

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traveling using several modes [49, 50, 51, 52]. However, we are interested in this paper by the freight transport. The difference between HLPs and the ITLPs is revealed in terms of topology network, assumptions, models and applications. This difference is in their deterministic version; it will be also in the robust one. However, Production-intermodal distribution problems combine production process and intermodal transport in a unique supply network. It includes decisions about the choice of transportation modes and determining flows consolidated, in addition to making decisions about the amount of goods produced in each production site. A limited capacity and unit cost of production is associated with each production site. The objective is to obtain a good decision for production and distribution operations in order to minimize the total cost which comprises the cost of production and the cost of distribution [26]. Nevertheless, the intermodal transport planning problem ITP seems to be more general variant of intermodal freight transport. But the planning is revealed in this variant because it requires the scheduling of many operations or to address the intermodal transportation in a multi-period horizon [28], which is not the case in ITLP.

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We observe also from the table that uncertainties have been widely studied in hub location problems. Zetina et al. [11], Meraklı and Yaman [53] and, more recently, Correia et al. [40] provide different uncertainty aspects. However, much less work has been done on intermodal transport with uncertainties. One of these works is the paper by Puettmann and Stadtler [30] for an intermodal transport problem with stochastic demands. Demir et al. [34] studied a green intermodal transport problem with uncertain demands and transit times based on a set of scenarios. Meraklı and Yaman [53] considered uncertain demands with a polyhedral uncertainty set; and Fotuhi and Huynh [36] studied scenarios where an intermodal freight transport problem was addressed with a multi-period horizon and probabilistic robust model. In the next section, the deterministic version of the studied problem is presented. 3. Deterministic intermodal terminal location problem

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The intermodal terminal location problem can be defined as a transportation problem consisting of distributing goods between customers using at least two transportation modes and benefiting from services of intermediate terminals. In such a case, decision makers look for locating terminals to be used and determining the flows to be transported by unimodal and intermodal links in order to minimise the total distribution cost.

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A simple example is shown in Fig. 1. There are three customers c1 , c2 and c3 and their associated demands are shown in the left part of the same figure.

Fig. 1. Example of the intermodal terminal location network

Suppose that the capacity of each terminal among the four possible terminals is set at 100 transport units (one unit is one container). A possible solution is given in the right part of the figure where Terminal 2 is unused. Terminals 1, 3 and 4 are selected to be intermediate terminals for transporting a fraction of the demands of customer 2 arriving from other customers. Moreover, the capacities of the used terminals are respected. Other demands are satisfied unimodaly.

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Generally, we assume that the data of the problem are fixed and their values are known in advance by the decision-maker. For the deterministic version of the problem of intermodal transport with the location of terminals, we presume the formulation presented in [16]. It ensures the transport between a set 𝑁 of customers directly via one mode or intermodaly via the use of the intermediate terminals chosen among a set 𝐾 of potential terminals. A demand 𝑞𝑖𝑗 to be satisfied is associated with each pair of origin-destination customers. A limited capacity 𝐶𝑘 and a fixed using cost 𝐹𝑘 are associated with each terminal 𝑘 in 𝐾. While 𝑘𝑚 the costs 𝑐𝑖𝑗 and 𝑐𝑖𝑗 indicate, respectively, the cost of unimodal transport between the customers 𝑖 and 𝑗 and the cost of intermodal transport between 𝑖 and 𝑗 via terminals 𝑘 and 𝑚. Let us note 𝑦𝑘 a binary decision variable which indicates the strategic decision of location where 𝑦𝑘 equals 1 if the terminal 𝑘 is chosen and 0 otherwise. The formulation also needs two operational decision variables which indicate flows to be transported, 𝑤𝑖𝑗 represents the amount of goods to transport unimodaly from customer i to customer j and the

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𝑘𝑚 second variable 𝑥𝑖𝑗 is the quantity of goods to be transported intermodaly from 𝑖 to 𝑗 via terminals 𝑘 and 𝑚. The following assumptions are made for the problem.

The demand of each origin-destination pair can be ensured by direct and/or intermodal transport.

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Each intermodal link (i,k,m,j) is ensured by combining transportation modes road-maritime-road and a pair (k,m) of terminals.

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The capacity of each terminal is limited

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Each closed terminal (non-used) is not to be part of the intermodal distribution strategy.

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The deterministic mathematical model of the intermodal transport with the location of terminals is in the following:

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𝑀𝑖𝑛 ∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘 𝑖,𝑗∈ 𝑁 𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

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∑ 𝑥𝑖𝑗𝑘𝑚 + 𝑤𝑖𝑗 = 𝑞𝑖𝑗

(1)

𝑘∈𝐾

𝑖, 𝑗 ∈ 𝑁

(2)

𝑘∈𝐾

(3)

𝑘,𝑚∈𝐾

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∑ 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑥𝑖𝑗𝑚𝑘 ≤ 𝐶𝑘

𝑖,𝑗∈ 𝑁 𝑚∈𝐾

𝑖,𝑗∈𝑁 𝑚∈𝐾

𝑥𝑖𝑗𝑘𝑚 ≤ 𝑞𝑖𝑗 . 𝑦𝑘

𝑖, 𝑗 ∈ 𝑁

𝑘, 𝑚 ∈ 𝐾

(4)

𝑥𝑖𝑗𝑘𝑚 ≤ 𝑞𝑖𝑗 . 𝑦𝑚

𝑖, 𝑗 ∈ 𝑁

𝑘, 𝑚 ∈ 𝐾

(5)

𝑖, 𝑗 ∈ 𝑁

𝑘, 𝑚 ∈ 𝐾

(6)

𝑥𝑖𝑗𝑘𝑚 ≥ 0 , 𝑦𝑘 ∈ {0,1}

𝑤𝑖𝑗 ≥ 0

𝑘∈𝐾

(7)

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The objective function (1) minimises the total cost which comprises the usage costs of selected terminals in addition to the unimodal and the intermodal transportation costs. In constraints (2), the demand of each origin-destination pair is exactly the total amount of goods transported unimodally and intermodaly from the origin node to the destination node. Constraints (3) take into consideration the limited capacity of each terminal. Terminal capacity has to be sufficient for the sum of the incoming and the outgoing amounts managed by that terminal. Constraints (4) and (5) ensure that if a terminal is not selected, there is no possibility to be used for transporting any intermodal amount. In other words, no intermodal amount can be transported via a non-open terminal. The last constraints – (6) and (7) – define the decision variables of the problem. They respectively ensure that only the positive amounts can be transported and only two possible choices regarding each terminal, to be used or not. At first, a theoretical study is done on randomly generated data. For each instance, the number of potential terminals and the number of customers are fixed. Their coordinates are randomly chosen in the Euclidian space between (0, 0) and (5000, 5000). Capacities of terminals and quantities of goods demanded by customers are respectively drawn in a random manner within [0, 10000] and [0, 500]. The fixed usage cost of each terminal is randomly taken from the interval [0, 500]. The costs of direct transport between the 𝑘𝑚 customers are exactly the Euclidian distances: in other words, 𝑐𝑖𝑗 = 𝑑𝑖𝑗 ∀ 𝑖, 𝑗 ∈ 𝑁. However, the cost 𝑐𝑖𝑗 equals the sum of three parts, the distance between customer i and origin terminal k, the distance between the destination terminal m and customer j and, finally, the half of the distance between terminals k and m [17]. We generated different instances of various sizes to represent data of the problem studied. They are available in https://www.dropbox.com/s/5sz7i8nadc2cprz/instances.rar?dl=0 . These instances will be used to test the efficiency of the proposed methods presented in the next section.

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4. Proposed method for the deterministic ITLP

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For solving the deterministic intermodal terminal location problem presented in the previous section, we propose a solution approach based on a metaheuristic and an exact method. We proceed in two stages. In the first stage, we propose a simulated annealing algorithm based on the population, combined with a series of heuristics, the best solution is returned. This best solution is used as an initial solution for the second stage, where the problem is reduced and an exact method is used for more adjusting and improving of our solution. Simulating annealing (SA) is a metaheuristic method proposed for the first time in 1993 by Metropolis et al. [54] and has proven its efficiency in solving many optimisation problems [55, 56]. Generally, the SA constructs an initial solution and generates new solutions by moving iteratively to the neighbourhood at each iteration. If the objective function value of the new solution is better than the value of the current solution, then an updating step is made. Otherwise, the difference between the objective values of both solutions is

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calculated (∆𝐸) and the new solution is accepted with a tolerance probability of 𝑒𝑥𝑝 (−

∆𝐸 ) where 𝑇

𝑇 is the

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temperature at the current iteration. The step-by-step decreasing value of the temperature allows for iteratively passing from large to reduced probabilities of accepting bad solutions. Thus, the SA returns a good final solution. Instead of the mono-solution aspect of the simulated annealing algorithm, we introduce a population-based modification for guaranteeing a more diverse search for a solution. Moreover, we used five heuristics in order to create neighbourhoods. Suppose that the network example of Fig. 2 (a) is a referential solution which represents the current solution. For creating its neighbour, the following heuristics can be used. Path-Swap: We used this heuristic to generate new solutions with different paths. Two paths in the current solution are randomly chosen and their origin terminals are altered. A simple illustration of this SwapOrigin-Terminal can be shown in Fig. 2 (b) which represents a neighbour of the referential solution (a). For creating a neighbouring solution with modified paths, we can also use a Swap-Destination-Terminal procedure to permute the destination terminals of two different origin-destination links; thus, the current solution (a) becomes a different solution as in Fig. 2 (c). Moreover, a path can be modified by permuting its

terminals. As an example, the path 1-2-4-2 of the current solution becomes 1-4-2-2 in the new generated solution. These Swap-OD-Terminal operators can generate new solutions with different paths like in Fig. 2 (d).

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Mode-Swap: This heuristic is used for modifying the amounts of goods transported along an origindestination pair. Two customers are randomly chosen and a part of the amount transported unimodally is added to the amount transported intermodaly via some terminals, or vice versa. And this is, of course, while ensuring the conservation of the total flow and respecting the capacity constraints of the terminals involved. See Fig. 2 (e). Relocate: In this heuristic, a non-used terminal in the current solution is randomly selected and opened while an open one is selected to be closed in the new solution. The transport along the origin-destination pairs where the old terminal was used is ensured using the new open terminal. Relocate functions of the referential solution (a) generates the solution shown in Fig. 2 (f). Open-Additional: According to this heuristic, a closed terminal is randomly selected to be part of the list of open terminals and be either an origin or destination terminal of some origin-destination links. This is applied to Terminal 3 of Fig. 2 (g) in comparison to the referential solution.

A

CC

EP

TE

D

M

A

N

U

Eliminate: In this move, an open terminal in the current solution is randomly chosen to be closed once and for all. The total flow passing from that terminal is shared by other open terminals and its fixed usage cost is neglected. This is what happened to Terminal 2, as shown in the new solution of Fig. 2 (h), compared to the referential solution.

Fig. 2. Neighborhood procedure: (b, c, d) Path-Swap; (e) Mode-Swap; (f) Relocate; (g) Open-Additional; (h) Eliminate.

The detailed pseudo-code of the proposed population-based simulated annealing algorithm PBSA is illustrated in Algorithm 1.

Algorithm 1 : A Population-Based Simulated Annealing algorithm PBSA 1:

Generating an initial list of feasible solution 𝑆1 𝑆2 , … , 𝑆𝑛

2:

Calculate the value of each solution 𝑓(𝑆𝑖 )

3:

𝑇0 ← 𝑇 , 𝑆̅𝑖 = 𝑆𝑖

4:

While ( 𝑇 > 𝑇𝑓 ) do

5: 6:

𝜏1 ← 𝑟𝑎𝑛𝑑(0,1)

7:

If (𝜏1 < 0.5) then

Generate a new solution using “ Eliminate” heuristic

8: 9:

else

Generate a new solution using “ Open-Additional” heuristic

10 : End if

12 :

Update the current solution

13 :

𝜏2 ← 𝑟𝑎𝑛𝑑𝑖(3)

14 :

If (𝜏2 = 1) then

U

11 :

N

Generate a new solution using “ Swap-Origin-Terminal” heuristic

15 :

If (𝜏2 = 2) then

16 :

A

Generate a new solution using “Swap-Destination-Terminal” heuristic

17 :

M

If (𝜏2 = 3) then

18 :

Generate a new solution using “Swap-OD-Terminal” heuristic

19 : End if

21 :

Update the current solution

22 :

Using “Relocate” heuristic for modifying the current solution

23 :

Using “Mode-Swap” heuristic for modifying the current solution

TE

∆𝐸𝑖 = 𝑓(𝑆̅𝑖 ) − 𝑓(𝑠𝑖 ) If (∆𝐸𝑖 > 0) then

EP

25 :

D

20 :

24 :

𝑆̅𝑖 = 𝑆𝑖 and

26 : 27 :

else 𝜌 = 𝑟𝑎𝑛𝑑(0,1)

29 :

If (𝜌 > exp(

CC

28 :

∆𝐸𝑖 𝑇

)) then

𝑆̅𝑖 = 𝑆𝑖

30 :

A

SC RI PT

For (each solution in the current population) do

31 :

End if

32 :

End if

33 :

End for

34 :

𝜒𝐵𝑒𝑠𝑡 ← 𝑆𝑙 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑓(𝑆𝑙 ) = min{𝑓(𝑆𝑖 )} 𝑖

= min{𝑓(𝑆𝑖 )}

35 :

𝐹𝐵𝑒𝑠𝑡

36 :

𝑇 = 𝛼. 𝑇 , 𝛼 ∈ ]0,1]

𝑖

37 :

End while

38 :

Return and Archive 𝜒𝐵𝑒𝑠𝑡 𝑎𝑛𝑑 𝐹𝐵𝑒𝑠𝑡

𝑖,𝑗∈ 𝑁 ̃ 𝑘,𝑚∈ 𝐾

N

∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗

∑ 𝑥𝑖𝑗𝑘𝑚 + 𝑤𝑖𝑗 = 𝑞𝑖𝑗

𝑖, 𝑗 ∈ 𝑁

̃ 𝑘,𝑚∈ 𝐾

𝑥𝑖𝑗𝑘𝑚 ≥ 0 ,

𝑖,𝑗∈𝑁 ̃ 𝑚∈ 𝐾

𝑤𝑖𝑗 ≥ 0

𝑖, 𝑗 ∈ 𝑁

TE

𝑖,𝑗∈ 𝑁 ̃ 𝑚∈ 𝐾

D

∑ 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑥𝑖𝑗𝑚𝑘 ≤ 𝐶𝑘

A

𝑖,𝑗∈𝑁

M

𝑀𝑖𝑛

U

SC RI PT

The algorithm PBSA begins with the generation of a set of feasible solutions and evaluating them. While a stopping criterion is not satisfied, we use the series of heuristics to build neighbourhoods for each solution. The first neighbour is generated by the alternative using either the Eliminate or Open-Additional heuristic. A random generated parameter determines which one can be used. The second neighbour is generated using the heuristic path-swap and a randomly generated parameter determines the operator to be used for guaranteeing this modification using one of Swap-Origin-Terminal, Swap-Destination-Terminal or Swap-OD-Terminals. The third and the fourth neighbours are generated using Relocate and Swap-Mode heuristics, respectively. Then the best neighbour of the current 𝑖 𝑡ℎ solution is kept if (∆𝐸𝑖 ) > 0 or with a probabilistic tolerance, where ∆𝐸𝑖 is the difference between the objective value of the 𝑖 𝑡ℎ solution and its best neighbour. Before passing to the following iteration, the best solution of the population is compared with the best global solution χ Best to replace it in case of dominance. In addition, the temperature is reduced by a parameter of reduction. When the algorithm reaches the end, the best global solution is returned. The best solution obtained by the PBSA can be a good initial solution for the exact method. It gives a good allusion about the location decision obtained after several improvements and it guarantees good distribution in terms of the total cost. We can count on this quality solution to fix terminals to be open, and afterwards, we use an exact approach to adjust and improve the flows transported in the network by focusing on – and only on – the open terminals. This new problem is a reduced sub-problem which we solved by the exact method intlinprog of ̃ be the set of open terminals in the best solution obtained by the first phase: in other Matlab-2014. Indeed, let 𝐾 ̃ = {𝑘 ∈ 𝐾 , 𝑦𝑘 = 1}. Then the reduced problem can be presented as follows: words, 𝐾

̃ 𝑘∈ 𝐾

̃ 𝑘, 𝑚 ∈ 𝐾

(8)

(9)

(10) (11)

A

CC

EP

The constraints (9) and (10) have the same role as constraints (2) and (3) respectively when focusing only on the open terminals. Constraints (4) and (5) of the original problem are eliminated because of their redundancy with the constraints (9) if they are added to the reduced problem. The total usage cost of terminals is eliminated from the objective function because in this case it represents a constant and a fixed value which is the sum of the using costs of the open terminals recommended by the first phase. In brief, the hybrid approach we proposed is a population based simulated annealing algorithm combined with an exact method and designated PBSA-Exact. The main steps of the PBSA-Exact are given in the following Algorithm 2. Algorithm 2: PBSA-Exact approach

1:

∗ Using the best solution of PBSA as an initial solution 𝜒 ∗𝐵𝑒𝑠𝑡 ← 𝜒 𝐵𝑒𝑠𝑡 and 𝐹𝐵𝑒𝑠𝑡 ← 𝐹𝐵𝑒𝑠𝑡

2:

Constructing the reduced model of the problem

3:

Representing the reduced model as a standard Linear program LP

4:

Solving the LP problem using intlinprog of Matlab software

5:

∗ ∗ Return the best solution obtained 𝜒𝐵𝑒𝑠𝑡 and its value 𝐹𝐵𝑒𝑠𝑡

SC RI PT

Recall that the initial model comprises three decision variables 𝑦, 𝑥 and 𝑤. The first one having a strategic nature determines the location decision. The second and the third decision variables are operational and they determine the amount of goods transported unimodaly and intermodaly. The PBSA method gives a best solution (𝑥, 𝑤, 𝑦) for the initial problem. Then it is the turn of the exact method for improving that solution. The exact method trusts and keeps the decision 𝑦 of terminal location suggested by the PBSA approach and improves the decision variables 𝑥 and 𝑤 by solving the reduced model. In other words, the exact approach determines other values 𝑥’ and 𝑤’ for the operational decision variables so that the solution (𝑥’, 𝑤’, 𝑦) is better than (𝑥, 𝑤, 𝑦). To the best of our knowledge, the proposed method PBSA-Exact which combines an exact approach and a population-based metaheuristic, is not presented or used in other related works of the literature. This is an interesting method, and its validity is shown by the tests performed and presented in Table 2. So, 38 instances of different sizes were randomly generated and tested with Cplex12.5, the standard SA, the proposed populationbased simulated annealing PBSA and the proposed hybrid method PBSA-Exact. The experiments are conducted on a personal computer HP core i3, 2.2 GHz with 4 GB of RAM. Table 2 Obtained results using Cplex, simulated annealing SA, PBSA and PBSA-Exact. Objective function value (× 𝟏𝟎𝟕 ) Cplex M1 M2 M3

Clients

10

5

5

1.49

1.52

1.49

1.49

1.34

1.26

1.26 5.81

CPU time M1 M2

M3

0.13

0.11

0.69

0.69

0.10

1.49

1.49

0.10

0.13

1.88

1.88

Cplex

5

10

0.15

15

10

5

5.71

6.13

5.88

20

10

10

5.89

5.98

5.95

5.91

0.25

0.28

5.05

5.05

1.09

1.06

1.05

8.10

0.23

4.35

4.35

27.49

27.29

0.20

0.56

1.22

1.22

20

25

20

5

27.12

30

10

20

5.09

30

20

10

26.36

35

5

30

1.36

35

30

5

59.11

5.66

A

5

27.38 5.19

5.15

0.60

0.76

17.70

17.71

27.10

26.96

26.73

0.60

1.33

3.31

3.31

1.46

1.37

1.36

0.50

0.45

9.03

9.03

59.98

M

25

1.05

N

15

1.26

U

Nodes

Number Terminals

59.45

2.20

0.82

2.73

2.73

5.93

5.12

4.99

1.30

1.70

36.29

36.30

11.50

11.50

10

30

40

20

20

25.65

27.39

27.00

26.41

3.30

4.18

40

30

10

57.22

58.83

58.35

58.12

3.70

2.15

7.19

7.19

40

1.11

1.26

1.12

1.12

0.70

0.67

15.72

15.73

5

108.20

108.77

108.54

108.40

1.00

1.52

3.61

3.86

5

45

40

50 50

40

4.35

5.74

5.05

4.77

2.40

3.42

8.72

8.72

40

10

104.81

107.65

107.18

106.17

5.40

5.06

11.41

11.44

5

50

0.88

1.09

0.89

0.88

0.90

1.08

24.70

24.70

5

164.22

165.99

165.79

165.67

1.70

2.70

5.08

5.09

165.24

165.12

164.27

10.30

8.65

13.34

13.35

10

CC

55

EP

45

55

TE

40

D

59.78

4.77

50 50

10

50

20

30

N

25.90

25.23

24.44

N

7.71

19.81

20.65

50

30

20

N

61.18

60.47

60.01

N

9.77

16.75

16.76

5.51

5.13

4.93

N

5.11

9.04

9.05

25.61

24.77

23.68

N

12.26

29.46

36.50

A

60

161.99

60

10

50

N

60

20

40

N

60

30

30

N

58.60

57.48

57.40

N

18.87

34.36

34.38

107.19

106.84

105.21

N

16.31

31.43

31.46

25.58

24.31

22.68

N

18.46

30.84

30.86

60

40

20

N

70

20

50

N

70

30

40

N

57.30

60.05

56.79

N

31.18

45.98

46.00

106.00

104.95

102.78

N

29.83

59.24

59.27

70

40

30

N

70

50

20

N

164.40

163.99

162.65

N

28.33

42.71

42.73

80

30

50

N

56.45

59.30

55.29

N

45.77

71.60

71.63

80

40

40

N

108.06

107.14

104.98

N

46.17

118.19

118.25

80

50

30

N

163.92

163.31

161.32

N

59.35

131.86

131.91

90

40

50

N

103.78

103.20

100.60

N

100.30

181.78

181.86

163.61

162.86

160.58

N

89.87

231.26

231.34

90

50

40

N

100

50

50

N

161.31

160.57

157.28

N

141.35

348.20

348.33

150

50

100

N

159.07

157.61

150.50

N

432.09

1060.04

1060.55

50

N

651.32

650.71

646.73

N

464.99

1110.86

1111.38

-

78.70

78.38

77.21

-

41.94

98.91

99.16

150

100 Average values

M 3: the PBSA

SC RI PT

N: a memory problem / M 1: Simulated annealing (SA) / M 2: population based simulated annealing (PBSA) / combined with exact approach (PBSA-Exact).

It should be noted that the bold is used in the table so as to indicate the best objective function value obtained for each instance.

CC

EP

TE

D

M

A

N

U

The first, the second and the third columns of Table 2 denote, respectively, the number of nodes in the network, the number of terminals and the number of clients in each instance. The fourth to the seventh columns represent, respectively, the results obtained by using Cplex, simulated annealing, population-based SA and the hybrid solution approach PBSA-Exact. However, the final row is for the average values. The obtained results indicate that Cplex does not give solutions for large and some medium instances while all instances are solved by the standard simulated annealing and the two other proposed methods. A comparison is made in terms of minimum value of the objective function. Results show also that there is a remarkable difference between the objective function value when using SA and the PBSA, which proves the advantage and the importance of developing population-based simulated annealing to get very good solutions much more effective than standard simulated annealing solutions. The column that represents the PBSA-exact method shows the effectiveness of the population-based simulated annealing algorithm combined with an exact approach. For small and some medium instances, the results of PBSA-exact are very close to those of Cplex if we speak in terms of deviations between both results. The PBSAexact method also shows superiority over SA and PBSA meta-heuristics for small, medium and large instances. This comparison gives us several important interpretations. The metaheuristic method (PBSA) if it is followed by an exact method (which gives PBSA-exact) can be very efficient and gives very good solutions for problems of different sizes in comparison with a method based on a meta-heuristic alone. Reciprocally, if the exact approach is firstly preceded by a meta-heuristic method, it can find a better starting solution and after reducing the problem, this combined method effectively solves all our instances of different sizes, including problems where the exact method alone (such as Cplex) failed to solve in a reasonable time or due to lack of memory. However, the two proposed methods are slightly more time-consuming for large instances. The following proposition proves why the PBSA-Exact approach has such advantage.

A

Proposition: The solution obtained by the PBSA-Exact approach for the reduced problem is a feasible solution for the initial problem. It is also better than the solution obtained by the PBSA approach. Proof: We note by 𝑃 the initial model of the ITLP and 𝑋 its associated set of feasible solutions. The objective function of this problem is: 𝑘𝑚 𝑘𝑚 𝑓(𝑧) = ∑ 𝑐𝑖𝑗 𝑥𝑖𝑗 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘 𝑖,𝑗∈ 𝑁 𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

𝑘∈𝐾

where 𝑧 = (𝑥, 𝑤, 𝑦) ∈ 𝑋, 𝑥 and 𝑤 are the intermodal and the unimodal transportation decision variables, while 𝑦 is the location decision variable.

Let 𝑃̃ be the reduced problem of 𝑃 and 𝑋̃ its associated set of feasible solutions. We define the objective function of this reduced problem by: 𝑔(𝑧) = ∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 𝑖,𝑗∈ 𝑁 ̃ 𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

̃ is the set of open terminals. Where 𝑧̃ = (𝑥, 𝑤) ∈ 𝑋̃ ⊆ 𝑋 and 𝐾

SC RI PT

Assume that 𝑧 = (𝑥, 𝑤, 𝑦) ∈ 𝑋 is the best solution obtained by the metaheuristic approach PBSA for the initial problem 𝑃. Consequently, 𝑓(𝑧) ≤ 𝑓(𝑧 ′ ) ∀ 𝑧 ′ ∈ 𝒱(𝑧) ∩ 𝑋 where 𝒱(𝑧) represents the set of neighbor solutions of 𝑧. When the initial problem is solved by the PBSA method, its solution 𝑧 = (𝑥, 𝑤, 𝑦) is returned. We keep the location decision y and we consider 𝑧̃ = (𝑥, 𝑤) as an initial solution for 𝑃̃. Assume that z̃ ∗ is the optimal solution obtained by the exact approach for the reduced problem 𝑃̃. Then 𝑔(𝑧̃ ∗ ) ≤ 𝑔(𝑧 ′′ ) ∀ 𝑧 ′′ ∈ 𝒱(𝑧̃ ∗ ) ∩ 𝑋̃. Because 𝑧̃ is an initial solution of 𝑃̃ we have also 𝑔(𝑧̃ ∗ ) ≤ 𝑔(𝑧̃ ) . Finally, the cost of the location decision part is included. This implies 𝑔(𝑧̃ ∗ ) + ∑ 𝐹𝑘 𝑦𝑘 ≤ 𝑔(𝑧 ′′ ) + ∑ 𝐹𝑘 𝑦𝑘 ∀ 𝑧 ′′ ∈ 𝒱(𝑧̃ ∗ ) ∩ 𝑋̃ 𝑘∈𝐾

𝑔(𝑧̃ ∗ ) + ∑ 𝐹𝑘 𝑦𝑘 ≤ 𝑔(𝑧̃ ) + ∑ 𝐹𝑘 𝑦𝑘 𝑘∈𝐾

∗

U

𝑘∈𝐾

and

𝑘∈𝐾

We know that 𝑧 = (𝑧̃ , 𝑦 ). We merge also the solution 𝑧̃ with the decision 𝑦 kept and we note 𝑧 = (𝑧̃ ∗ , 𝑦). 𝑓(𝑧 ∗) ≤ 𝑓(𝑧 ′′ )

∀ 𝑧 ′′ ∈ 𝒱(𝑧 ∗ ) ∩ 𝑋̃ ⊆ 𝒱(𝑧 ∗) ∩ 𝑋

and

N

So,

∗

𝑓(𝑧 ∗) ≤ 𝑓(𝑧)

M

A

This means that the best solution obtained by the PBSA-Exact approach for the reduced problem is a best feasible solution for the initial problem and it is also better than the solution found by the metaheuristic approach PBSA. The efficiency of the proposed hybrid approach PBSA-Exact is proven mathematically and the results of table 2 adhere to the proposition.

TE

D

So far, we have proposed a population-based simulated annealing and a hybrid approach to solving the intermodal terminal location problem which we considered in a deterministic version where all data are fixed and nominal. In the next section, we propose some robust optimisation models for the ITLP.

5. Robust optimisation models for the ITLP

A

CC

EP

In this section, we study the problem of intermodal transport under uncertainty. At first, we consider that the uncertainties affect the using cost of terminals. Then we propose a robust optimisation RO model with uncertainty on the capacities of terminals. Finally we talk about uncertainty on transportation costs. The main objective is to find robust solutions for the intermodal freight transport problem which are feasible and possible to be accredited no matter what realisation the problem data may take. These sources of perturbations that make data uncertain may be of different natures and their laws of probability are unknown and difficult to describe. To deal with uncertainties in the intermodal transport problem with terminal location, we apply the robust optimisation approach. We focus on the minimax regret formulations which are useful criteria in robust optimisation. They consist of considering some discrete scenarios, determining the objective function value for each one, and minimising the worst case [57]. We present in the following their principle. Generally, let (𝑃) be an optimization problem where 𝑢 is the decision variable, 𝑋 is the set of feasible solutions and 𝑓 is the objective function. 𝑀𝑖𝑛 𝑓(𝑢)

(P):

𝑠𝑡. 𝑢∈𝑋

Let 𝑆 be the set of possible scenarios for an uncertain parameter. We denote by 𝑋𝑠 and 𝑓𝑠 the set of feasible solutions for scenario s and the value of the corresponding objective function respectively. A solution 𝑢 can be a feasible robust solution if it is feasible for all scenarios; because we do not know which one will occur. This means 𝑢 ∈ 𝑋𝑠 ∀𝑠 ∈ 𝑆 [9]. A feasible robust solution 𝑢 is a best robust solution if it guarantees feasibility whatever the scenario occurred and minimizes the maximum regret value; in other words, finding a best robust solution requires solving the following problem [9]: 𝑀𝑖𝑛 𝑚𝑎𝑥 𝑅𝑠 𝑢∈𝑋

𝑠∈𝑆

𝑢 ∈ 𝑋𝑠

SC RI PT

𝑠𝑡. ∀𝑠 ∈𝑆

N

U

Where 𝑅𝑠 is the regret value defined by 𝑅𝑠 = 𝑓𝑠 (𝑢) − 𝑓𝑠∗ and 𝑓𝑠∗ is the value of the best solution found under scenario s. This is useful for problems having an uncertainty in the constraints or in the objective function; since a problem with uncertainty in objective function can be reformulated as a problem with certain objective when the function is integrated in the constraints. In our paper the set 𝑆 of scenarios refers to the set of uncertain using cost scenarios 𝑆𝑈𝐶 , uncertain capacities scenarios 𝑆𝐶 and uncertain transportation cost scenarios 𝑆𝑇𝐶 , for the three robust optimization models respectively. In the following subsections we propose the three cases of uncertainty.

A

5.1. Robust optimisation model with terminal using cost uncertainty

M

We present the robust optimisation model for the intermodal terminal location problem with uncertain using costs of terminals. We calculate the value of the objective function for each scenario and minimise the worst case function in the robust optimisation model. The following parameters are required for the new robust formulation: Set of scenarios for the uncertain using costs of terminals

𝐹𝑘𝑠 𝑓𝑠∗

:

The using cost of terminal 𝑘 under scenario s

:

The optimal (the best) objective function value of scenario s

TE

𝑅𝑓𝑠 :

D

𝑆𝑈𝐶 :

Regret value associated with the optimal (best) solution for scenario s

EP

The mathematical formulation of the ITLP that corresponds to the scenario s consists of minimising the objective function (12) and taking into consideration the constraints (2) - (7).

CC

𝑀𝑖𝑛 ∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘𝑠 𝑦𝑘 𝑖,𝑗∈ 𝑁 𝑘,𝑚∈ 𝐾

(12)

𝑘∈𝐾

(2) − (7)

A

s.t.

𝑖,𝑗∈𝑁

We look for a best robust solution. However, we do not know exactly which scenario will happen. Thus, the minmax regret model for the intermodal terminal location problem with uncertain using costs of the terminals is proposed as follows: 𝑀𝑖𝑛 max 𝑅𝑓𝑠

(13)

𝑠 ∈𝑆𝑈𝐶

s.t.

(2) − (7)

𝑅𝑓𝑠 =

∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘𝑠 𝑦𝑘 𝑖,𝑗∈ 𝑁 (𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

𝑘∈𝐾

− 𝑓𝑠∗ )

𝑠 ∈ 𝑆𝑈𝐶

(14)

5.2. Robust optimisation model with terminal capacity uncertainty

Similarly, we present the robust optimisation model for ITLP with uncertain capacities of terminals. We determine the function value for each scenario and minimise the worst-case function in the robust optimisation model. The additional parameters required for the new robust formulation are: Set of scenarios for the uncertain capacities of terminals

𝐶𝑘𝑠 𝑓𝑠∗

:

The capacity of terminal k under scenario s

:

The optimal (the best) objective function value of the scenario s

𝑅𝑓𝑠 :

SC RI PT

𝑆𝑐 :

Regret value associated with the optimal (best) solution for the scenario s

For a scenario s, the mathematical formulation of the ITLP consists of minimising the objective function (15) and satisfying the constraints (2) and (4) - (7). 𝑀𝑖𝑛 ∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘 𝑖,𝑗∈ 𝑁 𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

(15)

𝑘∈𝐾

∑ 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑥𝑖𝑗𝑚𝑘 ≤ 𝐶𝑘𝑠

N

𝑘 ∈ 𝐾 , 𝑠 ∈ 𝑆𝐶

𝑖,𝑗∈𝑁 𝑚∈𝐾

(16)

A

𝑖,𝑗∈ 𝑁 𝑚∈𝐾

U

(2) 𝑎𝑛𝑑 (4) − (7)

s.t.

M

For finding the best robust solution for the intermodal terminal location problem with uncertain using costs of terminals, and with the assumption that we do not know exactly which scenario will occur, the min-max regret model is proposed as follows:

D

𝑀𝑖𝑛 max 𝑅𝑓𝑠 𝑠 ∈𝑆𝑐

(17)

∑ 𝑐𝑖𝑗𝑘𝑚 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘

EP

𝑅𝑓𝑠 =

TE

s.t. (2) , (4) − (7)𝑎𝑛𝑑 (16)

𝑖,𝑗∈ 𝑁 (𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

𝑘∈𝐾

− 𝑓𝑠∗

𝑠 ∈ 𝑆𝑐

(18)

)

CC

5.3. Robust optimisation model with transportation cost uncertainty

A

As outlined previously for the other uncertain parameters, we present the robust optimisation model for the intermodal terminal location problem with uncertain transportation costs. The value of the objective function for each scenario is determined. We minimise the worst case function in the robust optimisation model. That formulation requires defining the following parameters: 𝑆𝑇𝐶

:

Set of scenarios for the uncertain transportation costs

𝑐𝑖𝑗 (𝑠) :

The unimodal transportation cost associated with the origin-destination link (𝑖, 𝑗) under scenario 𝑠

𝑘𝑚 𝑐𝑖𝑗 (𝑠):

𝑓𝑠∗

:

The intermodal transportation cost under scenario 𝑠 between the origin-destination nodes (𝑖, 𝑗) and via terminals 𝑘 and 𝑚 The optimal (the best) objective function value of the scenario s

𝑅𝑓𝑠

:

Regret value associated with the optimal (best) solution for the scenario s

Regarding this case of uncertain transportation costs, the mathematical formulation of the ITLP that corresponds to the scenario s consists of minimising the objective function (19) and taking into consideration the constraints (2) - (7). 𝑘𝑚 𝑘𝑚 (𝑠) 𝑥𝑖𝑗 𝑀𝑖𝑛 ∑ 𝑐𝑖𝑗 + ∑ 𝑐𝑖𝑗 (𝑠) 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘 𝑖,𝑗∈ 𝑁 𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

(19)

𝑘∈𝐾

(2) − (7)

s.t.

𝑀𝑖𝑛 max 𝑅𝑓𝑠

(20)

𝑠 ∈𝑆𝑇𝐶

(2) − (7)

𝑅𝑓𝑠 =

∑ 𝑐𝑖𝑗𝑘𝑚 (𝑠) 𝑥𝑖𝑗𝑘𝑚 + ∑ 𝑐𝑖𝑗 (𝑠) 𝑤𝑖𝑗 + ∑ 𝐹𝑘 𝑦𝑘 𝑖,𝑗∈ 𝑁 (𝑘,𝑚∈ 𝐾

𝑖,𝑗∈𝑁

𝑘∈𝐾

− 𝑓𝑠∗ )

𝑠 ∈ 𝑆𝑇𝐶

(21)

U

s.t.

SC RI PT

We do not know exactly which scenario will happen in this case of uncertainty. Thus, we introduce a min-max regret model for the intermodal terminal location problem with uncertain transportation costs; it can be presented as follows:

M

A

N

The three proposed robust optimisation models with respective functions (13), (17) and (20) are different. The first one considers the uncertainty in the using cost of terminals and scenarios are defined in the set 𝑆𝑈𝐶 of using costs 𝑠 ∈ 𝑆𝑈𝐶 . While the second and the third models deal with uncertainty in the capacities of terminals and uncertainty in the transportation cost, they are based on the sets of scenarios 𝑆𝐶 and 𝑆𝑇𝐶 respectively ( 𝑠 ∈ 𝑆𝑐 and 𝑠 ∈ 𝑆𝑇𝐶 ). The deterministic model and the three proposed robust optimisation models will be adapted to a real case network and solved with all approaches presented previously.

CC

EP

TE

D

When using the Cplex solver, robust models can be solved after linearisation by integration of a unique new variable greater than or equal to 𝑅𝑓𝑠 for all s. When using the proposed solution approaches, the robust optimization models are solved in the same manner and following the same steps as in the deterministic problem. But at first, in each robust model, we determine the solution value of scenarios, and then we look for the best solution of the Min-Max model. For example, in the case of uncertainty in capacities of terminals, we define the set 𝑆𝑐 of scenarios for capacities. For each scenario we apply the solution approaches for solving the model having the objective function (15), and we obtain the best solutions of scenarios and their associated regret values. Then, we use again the solution approach for solving the robust optimization model having the function objective (17) to determine the best robust solution. We proceed in the same manner for other robust optimization models. In the following section, we adapt the deterministic model and the proposed robust models and methods to a real case study.

A

6. Real case study and computational analysis

We apply our models and the proposed solution approach in a real worldwide intermodal transport network. There are 40 nodes corresponding to 20 customers and 20 seaport terminals. Fig. 3 shows the locations and the associated number of these nodes. The terminal-terminal transport is ensured by maritime mode and its transportation cost is calculated by multiplying the associated distances in terms of miles by a unit maritime transport cost. Customer-terminal, terminal-customer and customer-customer links are traversed by road mode and their transportation costs are calculated based on a unit cost. The using costs and the capacities of terminals are taken from the websites of the concerned seaports. The instance is completed by generating demands of customers in different levels: low, medium and high. Discussions and computational analysis are introduced in four subsections.

SC RI PT U

A

6.1. The impact of uncertainty on best solutions

N

Fig. 3. The real network studied: 20 customers and 20 terminals.

TE

D

M

The deterministic and robust optimisation models are implemented in Cplex and also solved by the simulated annealing SA; the developed population-based simulated annealing PBSA and the proposed hybrid method PBSA-Exact. We compared values of solutions of the robust optimization models with the deterministic model. Note that for each robust optimisation model, we consider five scenarios and the parameters of each scenario are randomly taken from an interval [ 𝑎̅ , 𝑎̅ + 𝑎̂ ] where 𝑎̅ is a nominal value and 𝑎̂ is the maximum variation. We compare the performance of the algorithms in deterministic and robust optimisation models. Table 3 summarises these values obtained by each method for each problem type. Table 3 Objective function values of the deterministic and the robust solutions obtained

Small

SA (× 𝟏𝟎𝟓 )

PBSA (× 𝟏𝟎𝟓 )

PBSA-Exact (× 𝟏𝟎𝟓 )

Deterministic

100.18

109.09

107.64

105.86

8.91

7.45

5.68

Uncertain using cost

101.07

110.74

110.69

109.40

9.67

9.62

8.33

Uncertain capacity

100.18

111.65

109.90

107.94

11.47

9.72

7.75

N

166.86

166.86

166.86

-

-

-

Deterministic

1482.05

1613.59

1598.54

1564.12

131.54

116.48

82.07

Uncertain using cost

1483.18

1641.04

1609.15

1570.79

157.86

125.97

87.60

Uncertain capacity

1482.40

1633.60

1614.13

1564.90

151.20

131.73

82.50

N

2492.67

2492.67

2453.49

-

-

-

Deterministic

15186.78 16256.68 16119.45

15795.69

1069.89

932.67

608.90

Uncertain using cost

15188.01 16738.01 16231.85

15881.89

1550

1043.83

693.88

Uncertain capacity

15162.99 16427.33 16265.80

15867.24

1264.34

1102.81

704.25

A

EP

Uncertain transportation cost

Medium

Uncertain transportation cost

Large

Dev 1 (× Dev 2 (× Dev 3 (× 𝟏𝟎𝟓 ) 𝟏𝟎𝟓 ) 𝟏𝟎𝟓 )

Cplex (× 𝟏𝟎𝟓 )

Model type

CC

Instance type

Uncertain transportation cost

N 24863.67 24863.67 24863.67 N: a memory problem, SA: Simulated annealing, PBSA: population based simulated annealing, PBSA-Exact: the PBSA combined with exact approach, Dev 1,2 and 3 are respectively deviations of SA, PBSA and PBSA-Exact from Cplex optimality.

SC RI PT

The first column of Table 3 represents demand levels. The second column represents type of the model, columns three to six are for the objective values of the best solutions obtained respectively by using Cplex, the standard SA, the developed PBSA, and the proposed hybrid PBSA-Exact. Cplex successfully solved the deterministic and robust problems, except in the case of transportation cost uncertainty, where it failed to find any solution, while the used meta-heuristics could solve all models. The results show that the population-based method PBSA gave better results than SA. However, the proposed hybrid method PBSA-Exact has shown superiority over the SA and PBSA methods in obtaining good solutions, in terms of results quality. This result is more easily shown by the deviations from the Cplex results in columns seven to nine. The proposed hybrid method PBSA-Exact shows lower deviations in comparison with the other metaheuristic methods. Note that the robust solutions for the model of the uncertain capacity of terminals have the least total cost compared to the solutions of other robust models. However, robust solutions of the uncertain transportation cost case have higher total costs among the solutions of the different robust models. And this trend can be noticed for all levels of demands. This consistency shows the impact of robustness on the total cost of the best solution of the intermodal freight transport problem. 6.2. The impact of uncertainty on the strategic decision

N

U

Robustness is valuable for strategic and operational decisions. Table 4 confirms this remark. It shows open terminals in each problem type with and without uncertainty. Table 4 The open terminals for the deterministic and the robust optimization problems Model

A

Instances type

1-2-5-6-8-9-11-12-13-15-16-18-19-20

Uncertain using cost Uncertain capacity

1-2-5-6-8-9-11-12-13-15-16-18-19-20

Deterministic

1-2-3- 4-5-6-8-9-10-11-12-13-14-15-16-17-18-19-20

TE

D

1-2-4-5-6-8-9-10-11-12-13-15-16-18-19-20

Uncertain using cost

1-2-5-6-8-9-10-11-12-13-14-15-16-17-18-19-20

Uncertain capacity

1-2-3-4-5-6- 8-9-10-11-12-13-14-15-16-17-18-19 20

Uncertain transportation cost

1-2- 4-5-6- 8-9-10-11-12-13-15-16-18-19-20

Deterministic

1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20

Uncertain using cost

1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20

Uncertain capacity

1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20

Uncertain transportation cost

1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20

A

CC

Large demands

1-2-5-6-8-11-12-13-15-16-18-19-20

Uncertain transportation cost

EP

Meduim demands

M

Deterministic Small demands

Selected terminals

Some terminal locations {1,2,5,6,8,11,12,13,15,16,18,19,20} stay common best choices for deterministic and robust optimisation models for capacity, using cost and transportation cost. In addition, the number of terminals selected increases as the demand level increases. Robustness causes terminal location changes. For example, for small demand levels, the terminals are located at nodes {1,2,5,6,8,9,11,12,13,15,16,18,19,20} in the deterministic case. On the other hand, for the uncertainty in using cost and uncertainty in capacity, the terminals are located at nodes {1,2,5,6,8,11,12,13,15,16,18,19,20} and {1,2,5,6,8,9,11,12,13,15,16,18,19,20}, respectively. While, for uncertain transportation costs, only four nodes are not located as terminals {3, 7, 14, 17}.

The robustness on the using cost may impose the selection of a fewer number of terminals, in comparison with the deterministic case, as seen for small- and medium-level demands. This decreasing requirement can also be noticed when the level of demand becomes low.

U

6.3. The impact of uncertainty on the operational decision

SC RI PT

Because of various results and many paths in the solutions, which can not be presented, we only report the results of the instance having medium level of demands and we describe only the paths in terms of connectivity. The connectivity of a terminal is defined by the number of paths where the terminal is connected. A terminal with higher connectivity is more important. In the deterministic case, because of ignoring the uncertainty, terminals 4, 5, 2 and 9 are with higher connectivity. The terminal 5 provides 48 intermodal links, while terminals 5, 2 and 9 provide 40, 36 and 30 intermodal links respectively. In the case of uncertainty on using costs, terminal 2 is in the first place, followed by terminal 12, 5 and 15. When the uncertainty affects the terminal capacity, terminals 3, 12,2,11 and 9 are the most connected in the network; they ensure large numbers of intermodal linking. Whereas, uncertainty on transportation cost results in higher connectivity for terminals 2, 12, 13 and 15. They ensure 52, 43, 37 and 28 intermodal linkings respectively. We observe also that terminals 2 and 12 have large connectivity in the three cases of uncertainty. In addition, there is a large difference between these cases and the deterministic situation. It may be concluded that there is a remarkable sensitivity of the strategic location decision to robustness.

CC

EP

TE

D

M

A

N

Regarding the operational decision, Fig. 4 shows the percentage of the total flow transported either unimodally (directly) or intermodaly (combination of two modes), according to the best obtained solutions for deterministic and robust models and with different levels of demands.

Fig. 4. Percentage of flows transported unimodaly and intermodaly.

A

The notation D represents the deterministic case, while UUC, UC and UTC are for uncertain using cost, uncertain capacity and uncertain transportation cost.

The distribution strategy, in cases of uncertainty, is totally different from the deterministic case and the total flows are not the same. Moreover, a slight decreasing of intermodality can be seen when the uncertainty affects parameters of terminals, such as their capacities or using costs. This trend proves the impact of robustness and the type of uncertainty on flows in particular, and on operational decisions in general. Nevertheless, the deterministic and the robust cases are all in agreement on the strength of intermodality in the studied network for satisfying a large percentage of freight demands compared to the unimodality regardless of the demand levels.

6.4. The impact of uncertainty of conservatism of robust solutions

D

M

A

N

U

SC RI PT

According to Gabrel and Murat [57], and Bertsimas and Sim [58], the conservatism of robust solutions is a very prominent issue in robustness. Indeed, the concept of conservatism refers to the difference between the objective function value of the best robust solution and the objective value of the nominal solution. It is implicitly the amount or the price to be paid for using the robust solution rather than the nominal one. Fig. 5 illustrates these deviations for each case.

TE

Fig. 5. Total cost difference between nominal and robust solutions.

EP

The notation UUC, UC and UTC are respectively for the cases of uncertain using cost, uncertain capacity and uncertain transportation cost.

A

CC

Fig. 5 shows not only that the PBSA-Exact hybrid approach is superior to the other methods, followed by the PBSA then the SA. But it also gives an idea on what type of uncertainty is more conservative. When uncertainty affects terminals capacity, the values of robust solutions have fewer deviations from the nominal solutions, followed by cases of uncertainty over the using cost of terminals. However, when uncertainty affects transportation costs, a huge gap is clearly visible and a consignable perturbation occurs. This result gives a clear indication about the impact of the type of robustness on the conservatism of the robust solutions and about the deviation of their total costs from the total cost of the nominal case that the decision-makers usually encounter and use each time.

7. Conclusion In this study, three robust models are proposed for the intermodal freight transport with terminals location for dealing with uncertainties and describing three sources; using costs of terminals, capacities of terminals and transportation costs. Moreover, instead of using the standard version of the simulated annealing algorithm for solving the deterministic and the robust problems, we developed it in a population-based method PBSA. Neighbors of solutions are generated with some suggested heuristics. Another method is also proposed. It is a hybrid approach based on the PBSA and an exact approach intlinprog of Matlab-2014.

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A comparative study was done on 38 randomly-generated instances and the obtained results show the efficiency of the hybrid method PBSA-Exact. A real network of 20 terminals and 20 customers was also considered for testing the models. The objective function value of solutions of the deterministic and the robust models show again the performance of the hybrid approach. There are several lessons learned. Developing a population based simulated annealing method can give better solutions than the usual mono-solution simulated annealing. Because in the population based method, we benefit from diversity, in addition to testing and evaluation of a large number of solutions. On the other hand, the efficiency of the hybrid method PBSA-Exact is also proved and justified by the conducted tests. Combining the metaheuristic PBSA with the exact method leads to benefit from advantages of both methods and it is efficient and preferred to the metaheuristic alone. Moreover, we underline the importance of taking into consideration the uncertainties in the freight intermodal transport problem with terminal location. When analysing the obtained solutions, we could see a clear contrast in the open terminals, the flows transported and the cost difference between nominal and robust solutions which proves the effect of uncertainties on terminal location and intermodal distribution decisions. Future work could investigate the intermodal transport of perishable products and dealing with other robust optimisation formulations and other uncertainty types, such as demands and temporal parameters. Those are real issues that are worthy of consideration and could change the whole distribution strategy and the network design.

U

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Author Biographies

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Abderrahman Abbassi Email: [email protected] Offices: - Faculty of Sciences and Technologies, B.P. 2202- Route d’Imouzzer, Fez, Morocco - Normandie University, UNIHAVRE, 76600 Le Havre, France

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Ahmed El hilali Alaoui

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Member of Modeling and Scientific Computing Laboratory of USMBA university in Morocco and Applied Mathematics Laboratory of Normandie university in France. His research areas of interest include transportation issues, multiobjective optimisation, metaheuristics, and robust optimisation.

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Email: [email protected] Office: Faculty of Sciences and Technologies, B.P. 2202- Route d’Imouzzer, Fez, Morocco

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Professor in Faculty of Sciences and Technologies (FST) and member of Modeling and Scientific Computing Laboratory of USMBA university in Morocco. He has supervised many Phd students working on optimisation, multimodal transport, vehicle routing problems and scheduling problems.

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Jaouad Boukachour Email: [email protected] Office: Normandie University, UNIHAVRE, 76600 Le Havre, France

Professor in University and Technology Institute (IUT), member of Applied Mathematics Laboratory of Normandie university in Le Havre-France and EURO Working Group on Vehicle Routing and Logistics Optimization. He has participated in several international collaborations and many industrial partnerships. His research areas of interest include multimodality, port logistics, transportation issues, traceability, risk management and scheduling problems.