An electromagnetism-like metaheuristic for the uncapacitated multiple allocation p-hub median problem

Highlights

• An efficient EM approach for solving the UMApHMP is presented.

• EM approach is combined with scaling technique.

• Local search is implemented very efficiently.

• Experimental comparison with other methods indicates superiority of EM approach.

Abstract

This paper deals with the uncapacitated multiple allocation p-hub median problem (UMApHMP). An electromagnetism-like (EM) method is proposed for solving this NP-hard problem. Our new scaling technique, combined with the movement based on the attraction–repulsion mechanism, directs the EM towards promising search regions. Numerical results on a battery of benchmark instances known from the literature are reported. They show that the EM reaches all previously known optimal solutions, and gives excellent results on large-scale instances. The present approach is also extended to solve the capacitated version of the problem. As it was the case in the uncapacitated version, EM also reached all previously known optimal solutions.

Introduction

The problem of locating hub facilities is very popular in the last three decades since it has many practical applications in diverse areas such as telecommunications, transportation, and postal delivery systems. Here, we will mention only several recent applications: fuzzy p-hub center problem (Yang, Liu, & Yang, 2013), uncapacitated p-hub maximal covering problem (Hwang & Lee, 2012), uncapacitated hub median problem (Filipović, 2011) and hub location inventory model for bicycle sharing system design (Lin, Yang, & Chang, 2013). For comprehensive surveys on hub location problems, the reader is referred to Alumur and Kara, 2008, Campbell et al., 2002.

There are several types of hub network problems. Their general assumption is that the hubs are fully interconnected while the non-hub nodes must route all of their traffic indirectly via one or more hubs. In this paper, the uncapacitated multiple allocation p-hub median problem (UMApHMP) is considered primarily. The main properties of this problem are:

• The problem is “uncapacitated”, which means that there are no capacity limitations on the hubs or on the flow between arcs.

• “Multiple allocation” allows each non-hub node to be allocated to several or all hubs in such a way that the overall cost of satisfying the flow demand is minimized.

• The number of hubs must be exactly equal to p.

• Setup costs are ignored.

• It is assumed that there are known non-negative flows associated with each origin–destination pair of nodes.

For this problem, the objective function is the total flow cost, which can be computed as a sum of transportation costs between all pairs of nodes. Transportation cost between a pair of nodes depends on the distance between the nodes in the network, the amount of flow to be moved across these distances, and the type of link between the nodes (whether the flow is translated between hubs, collected from, or distributed to a non-hub node). The cost coefficients corresponding to these types of links are denoted as χ, α and δ, respectively. The capacitated multiple allocation p-hub median problem (CMApHMP) is another variant when the collection flows are bounded.

The UMApHMP is known to be NP-hard in a strong sense, except for the special cases (for example, sparse matrix of flows) that are solvable in polynomial time. If the set of hubs is fixed, then the resulting sub-problem can be polynomially solved using the shortest-path algorithm in O(n² · p) time. As the CMApHMP is the generalization of UMApHMP, it is also NP-hard. Its sub-problem is polynomially solvable when the hubs are fixed, but not by dynamic programming.

The UMApHMP was proposed by O’Kelly (1987). The problem was also formulated as a quadratic integer program with a non-convex objective function. The first mixed integer linear programming formulation (MILP), with a large number of variables and constraints, was given by Campbell (1994). Aykin (1995) described an enumeration and greedy interchange method for UMApHMP and its variants.

Ernst and Krishnamoorthy (1998) introduced MILP formulation, that required fewer variables and constraints than the formulation previously used in literature. They presented an exact method by explicit enumeration based on the shortest paths. The shortest path problems were solved by producing the lower bounds, used in a Branch-and-Bound (BnB) scheme to obtain exact solutions. Additionally, a heuristic algorithm for solving large-scale instances, also based on the shortest paths, were described. The authors presented computational results only for CAB (n ⩽ 25, p ⩽ 4) and smaller size AP (n ⩽ 50, p ⩽ 6) instances. Hybridization with BnB gave results on some larger AP instances (n = 100, p ⩽  5 and n = 200, p = 2, 3).

Preprocessing techniques and tightening constraints for existing MILP formulations were presented, with an appropriate heuristic and an exact Branch-and-Bound method (Boland, Krishnamoorthy, Ernst, & Ebery, 2004). Experimental results showed that the computational effort required to obtain optimal solutions on CAB (n ⩽ 25, p ⩽ 4) and smaller size AP instances (n ⩽ 50, p ⩽ 5) was effectively reduced.

Garcia, Landete, and Marin (2012) introduced an integer programming formulation with O(n²) variables. Based on this formulation, a branch-and-cut algorithm has been developed, which allows the solving of larger instances than those previously solved in literature, but it was still unable to solve large-scale UMApHMP instances to optimality.

Stanimirović (2008) proposed a genetic algorithm (GA) based on the binary representation. It used genetic operators designed to keep the feasibility of individuals in the population. The mutation operator with frozen bits was used to increase the diversity of the genetic material. The running time of the genetic algorithm was improved by caching technique. Computational results showed that the proposed method reached all previously known optimal solutions, and also gave results on large-scale AP instances (up to n = 200, p = 20) that were not considered in the literature before.

Another genetic algorithm with an integer representation was proposed by Milanović (2010). The applied encoding scheme provided the feasibility of individuals by default, thus enabling the usage of standard genetic operators. The experiments were carried out on the standard AP data sets with up to n = 200, p = 20. The proposed approach also reached all previously known optimal solutions, and achieved three new best-known solutions on large-scale AP instances.

Liu et al., 2012a, Liu et al., 2012b used hybrid approach of the genetic algorithm with the rough set data mining technique to reduce the range of hub choices. The authors presented computational results only for instance with n = 15 nodes.

Ma and Ting (2008) modified the solution construction rule of canonical ant colony optimization (ACO), which involves locations of p-hubs and allocations of non-hub nodes to the selected hubs. The computational results for three different combinations of allocation rules in ACO were presented, and the rules were implemented in either solution construction rule or in local search. The authors compared their results only with exact methods from (Ernst & Krishnamoorthy, 1998) on CAB (n ⩽ 25, p ⩽ 4) and smaller size AP instances (n ⩽ 50, p ⩽ 5; n = 100, p ⩽ 5 and n = 200, p = 2,3).