Mathematical Problems in Engineering

Volume 2018, Article ID 2745437, 12 pages

https://doi.org/10.1155/2018/2745437

## Optimization of a Two-Echelon Location Lot-Sizing Routing Problem with Deterministic Demand

^{1}Department of Mathematics and Computing, MACS Laboratory, Faculty of Sciences, Hassan II University, Casablanca, Morocco^{2}Department of Applied Mathematics and Informatics, Ecole Normale Supérieure de l’Enseignement Technique, Mohamed V University, Rabat, Morocco

Correspondence should be addressed to Laila Kechmane; moc.liamg@alialenamhcek

Received 12 October 2017; Revised 13 April 2018; Accepted 20 May 2018; Published 14 June 2018

Academic Editor: Josefa Mula

Copyright © 2018 Laila Kechmane et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper aims to solve a multiperiod location lot-sizing routing problem with deterministic demand in a two-echelon network composed of a single factory, a set of potential depots, and a set of customers. Solving this problem involves making strategic decisions such as location of depots as well as operational and tactical decisions which include customers’ assignment to the open depots, vehicle routing organization, and inventory management. A mathematical model is presented to describe the problem and a genetic algorithm combined with a local search procedure is proposed to solve it and is tested over three sets of instances.

#### 1. Introduction

The design of a distribution network comprises three major problems: facility location, vehicle routing, and inventory control, these problems belong to different categories of decisions, and researchers classify the supply chain decisions into strategic, tactical, and operational: strategic level decisions are fixed for long periods time, which could be years, such as location of facilities, tactical decisions are made for medium periods of time such as inventory management, and operational level concerns short-term decisions which includes distribution decisions.

Recently, many researchers have studied the benefit of combining two decision levels into a single model; the inventory location problem (ILP) combines the facility location problem (FLP) and inventory management; many studies have been done on the different variants of ILP such as Daskin et al. [1], Shen [2], Ozsen et al. [3], Shen and Qi [4], Miranda and Garrido [5], and Melo et al. [6].

Many papers in the literature studied the location routing problem (LRP) which combines the facility location problem and the vehicle routing problem (VRP); the importance of tackling them simultaneously is studied by Salhi and Rand [7]. The LRP consists in determining the optimal number of facilities, their location, allocation of customers to the open facilities, and the vehicle routing organization. This problem has been studied by many researchers such as Prins et al. [8], Prins et al. [9], Duhamel et al. [10], Benlenger [11], Baldacci et al. [12], Ting et al. [13], Contardo et al. [14], and Kechmane et al. [15]. A recent survey on this problem is presented by Prodhon [16].

The inventory-routing problem (IRP) integrates inventory management and the vehicle routing problem, and the deterministic version of this problem has been studied by Archetti et al. [17], Oppen [18], Archetti et al. [19], Moin et al. [20], Coelho et al. [21], and Coelho and Laporte [22]. Adelman [23], Hvattum and Lokketangen [24], Hvattum et al. [25], and Mete and Zabinsky [26] studied the stochastic version of the problem. A survey on IRP is presented by Vidovic et al. [27]. Other papers in the literature integrate inventory, lot-sizing, and vehicle routing decisions such as Nananukul [28]; the author developed a reactive Tabu search procedure for solving a production-distribution supply chain and improved results using a path relinking strategy; Cetinkaya et al. [29] studied an integrated multiproduct inventory lot-sizing and vehicle routing problem and presented an iterative solution approach in which the problem is decomposed into inventory and routing subproblems; Chen and Sarker [30] established an integrated optimal model of inventory lot-sizing vehicle routing of multisupplier single-manufacturer with milk-run JIT delivery and solved it using an ant colony optimization algorithm; and Kechmane et al. [31] proposed a mathematical model to optimize transport cost and delivery lot-sizing in a single level logistic network and solved it using a MILP solver.

Few studies in the literature consider the strategic, operational, and tactical level decisions simultaneously, yet the vehicle routing organization depends on inventory control problem, which in turn depends on decisions taken in the location-allocation phase. The problem with the aim of facilities location, vehicle routing organization, and inventory management is known as location-inventory-routing problem (LIRP); inventory may be considered at plants, at final customers, or at both of them. Miranda and Garrido [32] studied the effect of ignoring inventory decisions while designing a supply chain network and the importance of managing inventories to reduce the total costs.

Liu and Lee [33] proposed a single product, multidepot location routing problem taking inventory control decisions into consideration where customers have stochastic demand and applied a two-phase heuristic method to solve the problem. Liu and Lin [34] divided the proposed problem by Liu and Lee [33] into a location-allocation problem and inventory-routing problem; they proposed a hybrid heuristic combining Tabu search with simulated annealing to solve the problem. Ambrossino and Scutella [35] proposed a linear programming model for the LIRP with deterministic demand where inventories must be managed at both plants and customers.

Shen and Qi [4] modified a LIRP model proposed by Daskin et al. [1] and used Lagrangian relaxation to solve subproblems. Ahmadi-Javid and Azad [36] proposed a stochastic model to simultaneously optimize location, allocation, inventory, and routing decisions; they proposed a hybrid algorithm of Tabu search and simulated annealing. Hiassat and Diabat [37] proposed a multiperiod model formulated as Mixed Integer Program for the LIRP with deterministic demand for perishable product. A multiphase heuristic algorithm based on simulated annealing and ant colony system (ACS) is proposed by Ahmadi-Javid and Seddighi [38] to solve a location-routing-inventory model with multisource distribution networks.

Guerrero et al. [39] proposed an algorithm for multiperiod LIRP with deterministic demand in a three-layer supply chain composed of a single supplier, potential depots, and customers; a hybrid heuristic and exact solution method are developed to solve the problem as well as an intensification phase to investigate the inventory and routing decision where the location and customers assignments are fixed and a dynamic lot-sizing problem is solved using a MIP solver. Nekooghadirli et al. [40] studied a biobjective location-routing-inventory model with multiperiod, multiproduct, stochastic demand and probabilistic travelling time among customers; their model aims to minimize the total cost and maximize the average time for delivering commodities to customers; and Zhang et al. [41] presented a hybrid metaheuristic combined with simulated annealing method to solve a LIRP in a two-echelon network composed of multiple capacitated depots and final customers.

Genetic algorithm (GA) is a stochastic optimization method proposed by J. Holland (1962) and then developed by D. Goldberg (1989); it is inspired by the natural evolution in genetics where a population of individuals each represented as a chromosome and representing a solution is subject to selection, crossover, and mutation operators at each generation to breed a new generation.

GA have been successfully applied to a variety of optimization problems including transport problems combining two decision levels; Abdelmaguid and Dessouky [42] developed a genetic algorithm approach to the integrated inventory distribution problem. A biased random key genetic algorithm approach is developed by Chan et al. [43] to solve the inventory-based multi-item lot-sizing problem. Rungreunganaun and Woarawichai [44] applied genetic algorithms to solve an inventory lot-sizing problem with supplier selection under storage space; the objective of their research is to calculate the optimal inventory lot-sizing suppliers and minimize the inventory cost. Prins et al. [8] proposed a memetic algorithm combined with population management, which is a technique developed by Sorensen and Sevaux [45], to solve a capacitated location routing problem; Park et al. [46] presented a GA for the vendor-managed inventory-routing problem with lost sales; Marinakis and Marinaki [47] presented a bilevel genetic algorithm and applied it to a real life location routing problem; a hybrid GA for the multiproduct multiperiod inventory-routing problem is presented by Moin et al. [20]; Liao et al. [48] used a nondominated sorting genetic algorithm (NSGA-II) to optimize a multiobjective location-inventory problem; and Kechmane et al. [49] developed a memetic algorithm to solve a capacitated location routing problem.

Recently, some genetic algorithms have been developed to solve problems combining location, inventory and routing decisions; Forouzanfar et al. [50] used a GA to optimize the total cost for a LIRP in a supply chain with risk pooling, a hybrid genetic-simulated annealing algorithm is proposed by Li et al. [51] for the LIRP considering returns under e-supply chain environment, and Hiassat et al. [52] developed a GA approach for a LIRP with perishable products.

This paper deals with a multiperiod Location lot-sizing routing problem (LLSRP) with deterministic demand that aims to design a two-echelon supply chain network consisting of potential depots supplied by a factory and facing deterministic nonconstant demands of customers. The objective is to determine the location of depots to open, to assign customers to the opened depots, to organize the vehicles routing, and to manage inventories at depots and customers by determining quantities to send to them during each period of the planning horizon so that the overall cost is minimized and customers’ demands are satisfied, such problem arises in many domains such as food distribution for supermarket chains, military operations, and pharmaceutical industry, to solve the LLSRP, and a hybrid genetic algorithm is developed. This paper is organized as follows: the description of the problem and a mathematical model are presented in Section 2, in Section 3 components of the proposed algorithm are presented in detail, computational results are given in Section 4, and a conclusion is given in the last section.

#### 2. Problem Description and Mathematical Formulation

Let be a complete, weighted, and directed graph, is the set of nodes in the graph where is the set of potential depots and is the set of customers. is the set of arcs linking the different nodes in the graph and with being the cost of the trip from node to node . is a discrete and finite planning horizon. Each depot has an opening cost and an ordering cost . Each node has a storage capacity . Each customer is assigned to a single depot. A homogenous unlimited fleet of vehicles is available. A vehicle might serve multiple customers during its tour but a customer is served by only one vehicle. Vehicles have a limited capacity and a cost of use . is the number of vehicles used from depot fleet during the period . Each vehicle ends its tour at the depot it starts from.

Before presenting the mathematical model for the LLSRP, let the decision variables be if the depot is opened, if the vehicle crosses the arc on period , and if the depot is replenished on period . if customer is affected to depot . is the delivery quantity received by depot on period . is the delivery quantity received by customer from depot in period via the vehicle . is the quantity needed by node in period t. is the available stock at node at the end of time period , represents the initial stock at node . is the cost of holding a unit from period to period at node .subject toThe objective function (1) sums the depots opening costs, the vehicles’ cost of use, the total routing costs, the ordering costs for every depot, and the holding costs at customers and depots over the planning horizon.

Constraints (2) define the depots’ need for every period. Inventory balance equations are presented by (3) and (4). Constraints (5) and (6) guarantee the satisfaction of customers and depots demand for every period. Equations (7) and (8) ensure that opened depots and customers storage capacities are respected. Equations (9) force the respect of vehicles capacity.

Constraints (10) ensure that each customer is assigned to a single depot and constraints (11) ensure that if a customer is assigned to a depot, the latter is opened. Constraints (12) force vehicles to leave every node they visit. Constraints (13) and (14) guarantee that each customer is visited by one vehicle at most per period. Constraints (15) ensure that each vehicle performs one route per period at the most.

Equations (16) are subtour elimination constraints, they guarantee the presence of a depot in every route. Constraint (17) links the vehicle routing variables and the assignment variables; it states that if a vehicle leaves a depot and visits a customer, then this customer must be assigned to the depot. Constraints (18) ensure that travels from a depot to another one are not allowed. Equations (19) provide the number of vehicles used from every depot fleet in every period. Finally (20)–(28) are standard integrality and nonnegativity constraints.

#### 3. A Hybrid Genetic Algorithm for the Location Lot-Sizing Routing Problem

In order to solve the Location lot-sizing routing problem with deterministic demand, a hybrid genetic algorithm is proposed, and the different components of the algorithm are presented in the sequel.

##### 3.1. Solution Representation

As mentioned earlier, in genetic algorithms, solutions are represented as chromosomes, this representation must be clearly defined; in the proposed algorithm, the encoding method for the location routing problem proposed by Prins et al. [8] is adopted; a chromosome consists of two parts; the first one relates to the depots; it indicates their status; if the gene corresponding to a depot is zero, it means that the depot is closed, else it contains the index of the first customer assigned to it. The second part of the chromosome indicates the set of customers assigned to each open depot. Figure 1 represents an example of a chromosome, the gene corresponding to depot 2 is zero, and this means that depot is closed. The genes corresponding to depots 1 and 3 contain, respectively, the values 1 and 7, this means that both of them are open, and the sets of customers assigned to depots 1 and 3 are, respectively and .