Integrating simplified swarm optimization with AHP for solving capacitated military logistic depot location problem
Introduction
Taiwan is a highly urbanized country surrounded by sea, thus, when homeland defense begins, urban warfare is unavoidable. For quickly establishing, deploying, and maintaining a distribution system to provide sufficient support to combat units, military logistics are incorporated into existing civilian buildings (facilities). During a war, combat units carry only the basic load for ease of combat; thus, each of their battalion level units (BLUs) has to construct a battalion-level depot (BLD) to supply its combat units using current buildings in its own combat area. Moreover, Regional Support Command requisitions civilian buildings for a number of region-level depots (RLDs), which are limited by a finite labor force, to serve all BLDs in its region and satisfy their demands.
The military logistic depot location problem (MLDLP) presented here can be briefly described with a network. As shown in Fig. 1, the network consists of two levels of depots: BLDs and RLDs, where only RLDs have limited capacity and all BLDs have specific demands from its combat units. Each BLD is served by only one RLD and only supplies its own combat units, thus, there are two sets of preselected alternative buildings corresponding to BLDs and RLDs, respectively, and those buildings have several attributes to be considered. Simultaneous decisions must be made regarding which buildings of both levels will be determined as depots. This model is hierarchical in nature and involves the facility location problem.
The facility location problem is a well-known combinatorial optimization method in operation research, which aims to select the best location for facilities from a given set of potential sites in a distribution system that can effectively and efficiently transport goods from plants to customers via depots. As it is a very critical and strategic problem for many areas, various facility location models have been investigated in literature [1], [2]. The main differences among these models consist of their variables regarding the different natures of the facilities, such as (1) single or multiple levels, (2) uncapacitated or capacitated, (3) single or multiple sourcing.
The multi-level model extends from the single level facility location problem, and present if facilities have been located simultaneously on several levels of the distribution system [2]. In the uncapacitated facility location problem, each facility is assumed to supply an infinite demand, and each customer receives all required demands from one facility. If its supply capacity is limited, the problem is referred to as a capacitated facility location problem. A special case of this problem, in which each customer receives its demand from exactly one facility, is called the single-source capacitated facility location problem [3], [4].
A particular facility location model, called the two-level, single-source, capacitated facility location problem (TSCFLP), is considered in this work. This model locates capacitated facilities in a manner that facilities in higher level can serve lower-level facilities subject to single source constraints. As facilities cannot be independently located at each level, there is a need to consider them as a hierarchical system [5]. This model is used in a variety of applications, such as telecommunications [6], distribution networks [7], [8], [9], remanufacturing networks [10], humanitarian relief logistics [11], and fiber-optic access networks [12]. For most of those works, the objective is to minimize the total cost; however, regarding military applications; it is indeed an intractable problem for commanders to plan a military logistics system during wartime, as cost is not the only factor affecting such decisions.
TSCFLP is a well-known NP-hard problem, since it generalizes the uncapacitated facility location problem which is NP-hard [13]. Existing approaches can be mainly divided into four categories: branch and bound [14], [15], Lagrangian relaxation [6], [11], [16], heuristic [3], [17], [18], [19], and evolutionary computation methods [20], [21], [22], [23]. Due to numerical difficulties and computational burdens, classical exact solution methods are only able to solve small-size problems. For large scale problems, most recent research has concentrated on the use of evolutionary computation methods that provide near-optimal solutions within a reasonable computation time [24], [25].
Over the past two decades, evolutionary computation methods have become important tools for solving the various combinatorial problems encountered in many practical settings. Among the different existing evolutionary computation methods, simplified swarm optimization (SSO) has become a very popular approach, as it identifies high-quality solutions for many problems [26], [27], [28], [29]. While SSO literature is very rich, none of the papers have addressed SSO to solve facility location problems.
The goal of this work is to investigate the application of facility location modeling techniques to the homeland defense problems faced by Taiwan’s army. The objective of MLDLP is to maximize the average utility of the requisitioned buildings, where the utility of a selected building depends on its attributes. In this pursuit, a two-stage method involving the analytic hierarchy process (AHP) and SSO is proposed to solve this problem. In the first stage, AHP is presented as a stand-alone methodology to estimate the relative weights of the attributes as the coefficients of the objective function [30]. In the second stage, SSO adopts a novel local search scheme based on AHP, called SSO, which is presented to deal with MLDLP. The performance of the proposed SSO model is evaluated via 36 randomly generated instances, which is compared with other evolutionary computation methods. Encouraging results are found in terms of the efficiency and effectiveness of the proposed method.
The outline of the paper is, as follows: Section 2 formulates the mathematical model for MLDLP. AHP and SSO are briefly described in Section 3. Section 4 discusses the procedure of the proposed two-stage method for the problem. The test problems are described and discuss the numerical results in Section 5, and Section 6 offers conclusions.
Section snippets
Preliminary material
The aim of MLDLP is to determine the optimal allocation of depots to establish an effective and efficient distribution network for maximizing the average utility of requisitioned buildings under practical constraints. Compared to business applications, cost (distance) is only one of the factors affecting decision-making in MLDLP, as military problems are critical and sensitive, thus, this work attempts to find the other way to construct the objective function for MLDLP. This section describes
Analytic hierarchy process (AHP)
The analytic hierarchy process (AHP), originally introduced by Saaty [33], is a well-known method for solving multi criteria decision-making problems [34]. When applying AHP on MLDLP, the hierarchy structure can be drawn as Fig. 2. As can be seen, it is a two levels AHP in which the first level (AHP-1) is used to determine the weight of each attribute while the second level (AHP-2) is applied for local search to improve the solution quality of SSO.
AHP-1 is described briefly, as follows:
Step 1.
Proposed method
The proposed method as shown in Fig. 3 has two stages using two methodologies for MLDLP. In the first stage, AHP-1 is applied independently to estimate the relative weights of attributes as the coefficients of the objective function for MLDLP. In the next step, a novel SSO with a local search based on AHP-2 is proposed to deal with MLDLP. There are many other alternatives of AHP in the literature that could also be used in our algorithm framework [39], [40], [41], [42]. Since our major
Experiment results and discussion
To evaluate the quality and performance of the proposed SSO for MLDLP, two experiments, EX 1 and EX 2, are carried out. The purpose of EX 1 is to observe the effect of Nahp and Nalt, which are the parameters of ALS, to find the best setting for both using the 9 designed treatments. In EX 2, SSO with the best setting is compared with existing algorithms, including discrete particle swarm optimization (PSO) [43], genetic algorithm (GA) [44], improved differential evolution algorithm (DE)
Conclusions
When the facility location problem is applied to plan a military logistics system, unlike most applications, cost is not the main factor affecting decision-making. Thus, this work proposes a novel model, MLDLP, which has the goal of determining the optimal allocation of two-level depots to establish an effective and efficient military distribution network for maximize the average utility of requisitioned buildings under practical constraints; moreover, it develops a two-stage method to solve
Acknowledgments
We are thankful to Lieutenant Guan-Jhong Syu for completing numerous questionnaires and this research was supported by the National Science Council of Taiwan , R.O.C under grant MOST 107-2221-E-606-009.
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