Mathematical Problems in Engineering

Volume 2015, Article ID 525980, 11 pages

http://dx.doi.org/10.1155/2015/525980

## An Endosymbiotic Evolutionary Algorithm for the Hub Location-Routing Problem

Hankuk University of Foreign Studies, Yongin 449791, Republic of Korea

Received 25 February 2015; Accepted 21 June 2015

Academic Editor: Emilio Insfran

Copyright © 2015 Ji Ung Sun. 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

We consider a capacitated hub location-routing problem (HLRP) which combines the hub location problem and multihub vehicle routing decisions. The HLRP not only determines the locations of the capacitated -hubs within a set of potential hubs but also deals with the routes of the vehicles to meet the demands of customers. This problem is formulated as a 0-1 mixed integer programming model with the objective of the minimum total cost including routing cost, fixed hub cost, and fixed vehicle cost. As the HLRP has impractically demanding for the large sized problems, we develop a solution method based on the endosymbiotic evolutionary algorithm (EEA) which solves hub location and vehicle routing problem simultaneously. The performance of the proposed algorithm is examined through a comparative study. The experimental results show that the proposed EEA can be a viable solution method for the supply chain network planning.

#### 1. Introduction

In supply chain management, the design of distribution networks is one of the most important problems because it offers a great potential to reduce costs and to improve service quality. An important aspect of designing a distribution network is the determination of the locations of facilities such as warehouses, depots, and distribution centers. Classical facility location models assume that each customer is served on a straight-and-back basis on a given route while computing distribution cost. This situation is true only if the demand of each customer is a full truckload. However, in many applications arising in practice, the demand of each customer may be less than a truckload such that multiple customers are served in a single route and distribution cost depends on the sequence of customers on the route [1]. In this case, to reflect accurately the distribution cost of routes within a location model, the location-routing problem (LRP) should be solved simultaneously.

The LRP deals with determining the location of facilities and the routes of the vehicles for serving the customers under some constraints such as facility and vehicle capacities and route length to satisfy demands of all customers and to minimize the total cost including routing costs, vehicle fixed costs, facility fixed, and operating costs. Location-routing models are especially necessary for systems where the time horizon for the facility location decisions is not too long, and location costs are comparable to the routing costs. For an extensive review and classification of LRP, see Nagy and Salhi [2].

This study is motivated by the observation made in the postal service. Hub and spoke structure is a well-known configuration implemented in postal systems. It provides economies of scale by consolidating the traffic flows at the hubs as opposed to connect directly each origin-destination (O-D) point. In the standard LRP, we assume a structure of facilities serving a number of customers, who are connected to their depot by means of vehicle tours. No routes connect facilities to each other. On the other hand, this study deals more complex network structure than the standard LRP. We consider a variant of the LRP known as the hub location-routing problem (HLRP). In the HLRP, flow of packages, mail, or passengers from different nodes are collected at hubs, transferred between hubs along hub links in order to economically consolidate flows on the route, and distributed to their destinations. In a typical hub and spoke network, the flow of each O-D pair consists of three components, in other words, collection, transfer, and distribution, respectively. For handling the HLRP it is convenient to distinguish between two types of decisions, that is, hub location problem (HLP) and multihub vehicle routing problem (MH-VRP). The HLP is concerned with determining which nodes in a network are designated as hubs and which nonhub nodes are allocated to each hub. In -hub median problem, the total number of hubs is fixed as and the locations of number of hubs are determined such that the sum of the flow costs is minimized. The HLP can be categorized into single allocation and multiple allocations hub problem. In the single allocation, each nonhub node must be allocated only one hub whereas, in the multiple allocations, each nonhub node may be allocated to more than one hub. Further, the HLP can be categorized as either capacitated or uncapacitated. The capacitated HLP considers capacity constraint on the amount of flow through hub. On the other hand, there are no capacity restrictions on the uncapacitated HLP. While the HLP that has received the most attention from researchers is the uncapacitated single allocation -hub median problem, this paper deals with the capacitated single allocation -hub median problem for the part of HLP. We prescribe that exactly hubs need to be chosen within a set of potential hub locations. Each candidate location of hub has capacity restriction and fixed cost for establishing it as hub. In addition, we determine the allocation of each customer node to exactly one hub among the determined number of hubs. The flow of each O-D customer pair should go through one hub and at most through two hubs. Once the HLP is solved, the HLRP reduces to the MH-VRP. A set of homogeneous vehicles with the same capacity are available. Each customer must be served by exactly one vehicle for pickup and delivery service, respectively. The loads of each route cannot exceed the vehicle capacity and the total demands of customers allocated to each hub also should not exceed the capacity of that hub. We determine the routes of vehicles so as to minimize the routing costs and fixed vehicle costs.

Kuby and Gray [3] consider the hub network design problem where they determine the least cost set of direct or stopover routes for the traffic from a given set of points to a predetermined hub. Aykin [4] addresses the HLRP in continuous solution space where the hubs can be located anywhere in the service region encountered in air cargo transportation. He presents an algorithm solving a series of shortest path problems and a multifacility location problem in an iterative manner. Even though these two studies consider routing possibilities, their approaches are restricted in parallel with their assumptions. Nagy and Salhi [5] introduce the many-to-many location-routing problem similar to the HLRP where several customers send goods to others and the locations of each hub are to be determined. In their study, a hierarchical heuristic solution framework based on the concept of nested methods is presented. Bruns et al. [6] consider a problem arising in the parcel delivery operations of a postal service. They determine the locations of delivery hubs and the allocation of customer area to delivery hubs. In the case of routing costs, they present a route length estimation formula. The problem studied by Wasner and [7] is closely related to the HLRP. However, all interhub flow must go through a central hub rather than allowing all hubs to be directly connected to each other. Çetiner et al. [8] consider the combined hubbing and routing problem in postal delivery systems and develop an iterative two-state solution procedure for the problem in order to produce a route-compatible hub configuration. Catanzaro et al. [9] investigate the partitioning-HLRP, a peculiar version of the HLP involving graph partitioning and routing features and mainly arising from the deployment of an internet routing protocol. The partitioning-HLRP consists of partitioning a given network into subnetworks, locating at least one hub in each subnetwork, and routing the traffic within the network at minimum cost.

The HLRP is very difficult to be solved since it is composed of two NP-hard problems, the HLP and the MH-VRP. Therefore, the relatively large number of papers devoted to develop the approximation methods and the heuristic approaches can be classified into three categories such as clustering-based, iterative, and hierarchical heuristics, respectively. Clustering-based methods begin by partitioning the customer set into clusters, one cluster per potential depot or one per vehicle route. Then, they locate a depot in each cluster and then solve a VRP for each cluster [10–13]. Iterative heuristics decompose the problem into its two subproblems. Then, the methods iteratively solve the subproblems, feeding information from one phase to the other [14–17]. Hierarchical heuristics considers the location problem as the main problem and routing as a subordinate problem where main algorithm is devoted to solving the location problem and refers in each step to a subroutine that solves the routing problem [18, 19]. To solve the problem, the HLRP is formulated as a 0-1 mixed integer programming model. The integer programming model can provide the optimal solutions for small sized problem instances, but it is not practical for problems of large size in a reasonable computational time. Therefore, a novel solution framework based on an endosymbiotic evolutionary algorithm (EEA) is proposed which solves the HLRP and the MHVRP simultaneously.

The remainder of this paper is organized as follows. In Section 2, a mathematical model of the HLRP is developed. Section 3 presents an EEA and the detailed descriptions. Section 4 provides the computational results. Finally, we present the conclusion and the discussion of some future research directions.

#### 2. Mathematical Model

The HLRP under study can be formulated as a 0-1 mixed integer programming model which is an extension of the many-to-many proposed by Nagy and Salhi [5]. The HLRP has some different characteristics with the many-to-many LRP of which exactly hubs have to be chosen within a set of potential hub locations and each candidate hub has a capacity restriction. In addition, the vehicle route of pickup and delivery purpose for each hub can be constructed separately as shown in Figure 1. Then the HLRP under study can be formulated mathematically using the following notations: : the set of nodes, , : the set of candidate hub locations, : the set of customer nodes, : the set of type of vehicle route = , : the fixed cost of establishing candidate hub node as a hub, , : the capacity of loads associated with candidate hub node , , : the distance between nodes and , , : the amount of flow between origin and destination , , : the number of hubs to be opened, : the transfer coefficient between two hub nodes, : the routing coefficient among the customer nodes, : the fixed vehicle operating cost, : the maximum capacity of each vehicle, : , if a route type can be constructed directly from node to node ; , : , if candidate hub is used as a hub; , : , if customer is assigned to hub ; , : , if the flow from origin to destination routed via hub and ; , : the amount of flow from hub to hub , : pickup load on vehicle just after having serviced customer , : delivery load on vehicle just before having serviced customer .