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Top1. Introduction
Within the wide scope of logistics management, transportation plays a central role and is a crucial activity in the delivery of goods and services. The transport problem is one of the mainly essential combinatorial optimization problems that have taken the interest of several researchers. Huge research efforts have been devoted to the study of logistic problems and thousands of papers have been written on many variants of this problem such as Traveling Salesman Problem (TSP) (Toth et al., 2001), Vehicle Routing Problem (VRP) and supply chain management (SCM) (Pisinger et al., 2007).
The VRP (Golden et al., 2007) and (Toth et al., 2001) is one of the most studied combinatorial optimization problems and it consists of the optimal design of routes to be used by a fleet of vehicles to satisfy the demands of customers.
Many other related problems are associated with VRP such as the Heterogeneous Fleet Vehicle Routing Problem (HVRP), which differs from the classical VRP in that it deals with a heterogeneous fleet of vehicles having various capacities and both fixed and variable costs (Xu et al., 2014). Therefore, the HVRP involves designing a set of vehicle routes, each starting and ending at the depot, for a heterogeneous fleet of vehicles which services a set of customers with specific demands. Each customer is visited only once, and the total demand of a route should not exceed the loading capacity of the vehicle assigned to it. The routing cost of a vehicle is the sum of its fixed costs and variable costs incurred proportionately to the travel distance.
The main objective is to minimize the total of such routing costs. The number of available vehicles of each type is assumed to be unlimited and the total sum of the delays.
In the literature, three HVRP versions have been studied. The first one was introduced by (Golden et al., 1984), in which variable costs are uniformly given over all vehicle types with the number of available vehicles assumed to be unlimited for each type. This version is also called the Fleet Size and Mix VRP “FSM” (Ismail et al., 2008). This version that we are consider in this paper.
The second version considers the variable costs, depending on the vehicle type, which is ignored in the first version. This version is referred as HVRP (Gheysens, et al., 1986), the FSM with variable unit running costs (Taillard, 1999).
The third one, called the VRP with a heterogeneous fleet of vehicles (Toth et al., 2002), generalizes the second version by limiting the number of available vehicles of each type.
The remaining part of the paper is organized as follows: the next section introduces the notation used throughout the paper and describes the variants of heterogeneous VRPs studied in the literature then we introduce our problem: Multi-depot Fleet Size and Mix vehicle Routing Problem with time window. In section 3 we present the mathematical formulation of our own extension of the problem. Then, in the fourth part, we define our resolution method to satisfy our constraints, after that we implement our approach in MATLAB. The computational results for the benchmark instances are analyzed in Section 6. Finally, in section 7 draws the conclusion and future scope of the application.