Integrating simplified swarm optimization with AHP for solving capacitated military logistic depot location problem

Applied Soft Computing Journal 78 (2019) 1–12

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

Integrating simplified swarm optimization with AHP for solving capacitated military logistic depot location problem Chyh-Ming Lai Institute of Resources Management and Decision Science, Management College, National Defense University, Taipei 112, Taiwan

highlights • • • •

Addressed a facility location problem in a military logistic system, named MLDLP. The objective of MLDLP is to maximize the average utility of requisitioned buildings. Proposed a novel local search scheme based on AHP to guide the proposed algorithm. Statistical results indicate that the proposed method is better than its competitors.

article

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Article history: Received 15 April 2018 Received in revised form 9 December 2018 Accepted 9 February 2019 Available online 14 February 2019 Keywords: Facility location problem Military logistic system Analytic hierarchy process Simplified swarm optimization

a b s t r a c t This work addresses a two-level facility location problem in a military logistic system, named the military logistic depot location problem (MLDLP). Unlike most previous research on facility location problems, the objective of MLDLP is to maximize the average utility of requisitioned buildings, and the utility of a selected building depends on its attributes. This work develops an integer programming model and provides a two-stage method for this problem. In the first stage, the analytic hierarchy process (AHP) is applied to estimate the relative weights of the attributes as the coefficients of the objective function. In the second stage, simplified swarm optimization (SSO) is combined with a novel local search scheme based on AHP, called SSO AHP , which is proposed to deal with the problem. SSOAHP is empirically verified on thirty-six randomly generated problems, and the corresponding results are reported comparing the results with those obtained by up-to-date evolutionary computation methods. © 2019 Elsevier B.V. All rights reserved.

1. 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 E-mail address: [email protected]. https://doi.org/10.1016/j.asoc.2019.02.016 1568-4946/© 2019 Elsevier B.V. All rights reserved.

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

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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, singlesource, 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 [7–9], 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 [3,17–19], and evolutionary computation methods [20–23]. 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 nearoptimal 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 [26–29]. 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 SSOAHP , which is presented to deal with MLDLP. The performance of the proposed SSOAHP 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. 2. 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 some preliminary materials, such as the introduction of attributes for an alternative building and the formulation of MLDLP. 2.1. Attributes of a potential building In the case of MLDLP, several quantitative attributes are considered, which can be used to measure the quality of a potential building and introduced as follow [31,32]: Available volume (capacity): Available volume (in cubic meters) refers to the capacity of a building that can be used to store supplies after the storage space in this building is planned. One of the shortcomings of the requisition of an existing building for a depot is that the capacity of the depot is limited by the architecture and cannot be expanded. Therefore, the selection of a building with enough capacity is beneficial to the storage of supplies and can increase the subsequent flexibility to increase the reserves. Level of main road: Whether the logistic supplies can be transported to the frontline as quickly as possible depends on the level of the main road adjacent to the depot. In Taiwan, roads are divided into 6 levels, including country roads, district highways, county highways, city highways, provincial highways, and national highways. The higher the level, the wider the road is, and the more connected by-paths. Thus, denser transportation networks will be formed. Structure of building: The capability of a building to resist bombing and protect its supplies depends on its structure and materials which has great effect on the location decision here. Taiwan’s common building material and structure is divided into 7 grades: iron sheet structures, brick structures with an iron roof, brick structures, masonry-concrete structures, reinforced concrete structures, steel structures, and steel reinforced concrete structures. Among them, steel reinforced concrete structures have the highest grade, while iron sheet structures have the lowest grade. Size of operation area: Operation area (in square meters) is an area where operational tasks occur, such as forklift operations for truck unloading and loading, and its size directly affects operational efficiency of a depot. Therefore, a larger operation area leads to better throughput. Distance: Distances (in kilometers) between two level depots or BLDs to its combat units are calculated by the Euclidean method. Obviously, a longer distance requires higher transportation costs, but more importantly, the risk of vehicles being exposed to air raids is also increased. As can be seen, the units used to measure various attributes are different; therefore, in order to avoid a meaningless summation, the units of the measurement of the attributes must be normalized. Not only that, if the scales of the attributes are different, then the normalization is necessary to transform all scales to a specific range, i.e., between 0 and 1, to avoid solutions biased toward those buildings that can achieve larger values. Thus, value yij for the ith attribute, as obtained by jth building, is normalized by Eq. (1), with the exception of the distance attribute, which uses Eq. (2). The normalized value y’ ij is bounded between [0, 1] and a higher value is better. y′ij = y′ij =

yij − min (yi )

(1)

max (yi ) − min (yi ) yij − min (yi ) min (yi ) − max (yi )

+1

(2)

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

2.2. MLDLP formulation To formulate the MLDLP, the following notations are used: Nblu Nbldi Nrld I Ji

The total number of BLUs The total number of potential BLDs for BLU i The total number of potential RLDs The set of BLUs, I = {1, 2, . . . , Nblu} The set of potential BLDs for BLUi , Ji = {bldij | 1 ≤ j ≤ Nbldi } The set of potential RLDs, K = {rldk | 1≤ k ≤ Nrld} The jth potential BLD for BLUi The kth potential RLD The weight of the mth attribute The capacity of bldij and rldk , respectively

K bldij rldk wm capij , capk lmr ij , lmr k lsbij , lsbk soaij , soak disij

The level of main road used by bldij and rldk , respectively The level for the structure of building used for bldij and rldk , respectively The size of the operation area for bldij and rldk , respectively The average distance from bldij to combat units which all belong to BLU i The distance from rldk to bldij add up disij The distance between rldk and rldl Demand for BLU i came from its combat units { 1, if a bldij is requisitioned to be a BLD for BLU i, ∀i ∈ I , j ∈ Ji , = { 0, otherwise 1, if a rldk is imposed to be a RLD, ∀k ∈ K , = 0, otherwise { 1, if a bldij is served by rldk , ∀i ∈ I , j ∈ J i , k ∈ K , = 0, otherwise Manpower required of rldk Maximum available manpower to operate all RLDs

disijk diskl demi xij yk zijk bk B

The objective function in Eq. (3) maximizes the average utility of all requisitioned buildings for BLDs and RLDs. Eq. (4) ensures that each BLU opens exactly one BLD. Eq. (5) ensures that each BLD is assigned to only one RLD. Eqs. (6) and (7) ensure that assignments are made only to requisitioned buildings for BLDs and RLDs. Eq. (8) ensures that the demand of BLU, as serviced by a certain RLD, cannot exceed its capacity. Eq. (9) indicates that the total labor force required to operate all RLDs must be less than or equal to the predefined allowable amount of B. Eq. (10) ensures that the distance between two RLDs must be greater than 1 kilometer for safety reasons. Finally, all variables used in the models are required to be binary. 3. Methodology 3.1. 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. Ask experts or decision-makers to decide the relative importance of the attributes by comparing each pair of attributes and ranking them on the Saaty rating scale [1, 9] to build pairwise comparison matrix C . Each element cij in C represents the relative importance of attribute i over j, which is limited in [1, 9]; the higher it is, the more important the ith attribute is in comparison with the jth attribute. The reciprocal characteristic induces the following constraints: cji =

1 cij

, ∀i ̸ = j ,

cii = 1, ∀i ∈ [1, n]

∑

∑

Maxmize f = i∈I j∈Ji ( ) × w1 capij + w2 lmrij + w3 lsbij + w4 soaij + w5 disij xij

+∑

1

∑

k∈K yk

(w1 capk + w2 lmrk + w3 lsbk + w4 soak ) yk ]

i∈I Ji w5 disijk zijk ∑ j∈∑

∑

+

[ k∈K

∑

i∈I

j∈Ji zijk

(3) subject to

∑

xij = 1, ∀i ∈ I

(4)

j∈Ji

∑∑

zijk = 1, ∀i ∈ I

(5)

j∈Ji k∈K

zijk ≤ xij , ∀i ∈ I , j ∈ Ji , k ∈ K

(6)

zijk ≤ yk , ∀i ∈ I , j ∈ Ji , k ∈ K ,

(7)

∑∑

(8)

i∈I

∑

demi zijk ≤ capk , ∀k ∈ K

j∈Ji

bk yk ≤ B

(12) (13)

If attribute i is absolutely more important than attribute j, then we have cij = 9. According to the above constraint of Eq. (12), attribute j must be absolutely less important than attribute i and cji = 1/9. If cij = 1, the two attributes are equally important.

We can then state the problem as: 1 Nbat

3

(9)

k∈K

diskl > 1, if yk and yl = 1, for k ̸ = l and ∀k, l ∈ K

(10)

xijk , yk , zijk ∈ {0, 1} , ∀i ∈ I , j ∈ Ji , k ∈ K

(11)

Step 2. The estimation of actual relative weights (denoted as w) could be obtained from Eq. (14), where λmax is the largest eigenvalue of the pairwise comparison matrix C , and w = [w1 , w2 , . . . , wn ]T is its right eigenvector. It must be that this eigenvector ∑noted n solution is normalized additively, i.e. i=1 wi = 1. C · w = λmax · w

(14)

Step3. Once λmax has been computed, the consistency index (CI) and the consistency ratio (CR) are used to verify the consistency of the comparison matrix, which is obtained using the following equations: CI = CR =

(λmax − n) (n − 1) CI RI

(15) (16)

Where RI is a random consistency index, which can be yielded from randomly generated reciprocal matrices. Saaty suggested that the consistency ratio (CR) should not exceed 0.1. The weights of all attributes obtained by AHP-1 are employed as the coefficients of the objective function in Eq. (3). AHP-2 as a

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C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

Fig. 1. Concept of MLDLP.

Fig. 2. Decomposition of the problem into a hierarchy process.

local search is used to select a RLD for a BLU in the second stage, and its steps are described briefly, as follows: Step 1. For a specific BLD, chose a certain number of alternative RLDs, say Nalt, with the corresponding Nalt ×n attribute value matrix A, in which each element aik is the attribute value of the RLD i with respect to the kth attribute, and weight vector w, as obtained from AHP-1. Step 2. Use Eq. (17) and (18), where max_ak is the maximum value of the kth column in A, calculate the Nalt × Nalt pairwise matrix V k , in which each element vkij represents the relative importance of RLD i over j with respect to the kth attribute. vkij is a value limited in [1, 10], as suggested by [35], where a higher value indicates that the more important RLD i is in comparison with RLD j. c , if aik ≥ ajk vijk = 1/c , otherwise ⏐ ⏐ ⏐ c = aik − sjk ⏐ × 9/max_ak + 1

Fig. 3. Flowchart of the proposed method.

{

(17) (18)

Step 3. Derive option performance matrix E = [eik ]Nalt ×n , as based on Eq. (19). eik =

1 Nalt

Nalt ∑

vijk

∑Nalt j=1

i=1

vijk

(19)

Step 4. According Eq. (20), calculate the priority vector ω = [ω1 , ω2 , . . . , ωNalt ]T in which ωi is the priority value of alternative RLD i, and those values are used to select a RLD for a BLD.

ω =E ·w

(20)

3.2. Simplified Swarm Optimization (SSO) The proposed SSOAHP is based on simplified swarm optimization (SSO). Before discussing the proposed method, the basic SSO is introduced in this sub-section. SSO is a novel and interesting population-based evolutionary algorithm, which was first proposed in 2009 by Yeh, for overcoming the drawbacks of particle swarm optimization (PSO) in solving discrete problems [36]. It has

been successfully adopted in a number of applications [26,27,29, 37,38]. The process of SSO is introduced, as follows. All candidate solutions in SSO are randomly generated within the search space, and updated according to its unique update mechanism, as shown in Eq. (21). Let Xit = (xti1 , xti2 , . . . , xtij , . . . , xtin ) be the ith solution in the population at iteration t; let Pit = (pi1 , pi2 , . . . , pij , . . . , pin ) be pBest, which is the best solution with the best fitness value in ith solution’s history; let G = (g1 , g2 , . . . , gj , . . . , gn ) be gBest, which is the best solution with the best fitness value among all solutions. Each variable xtij is updated successively to a value related to four sources: the gBest g j , its current pBest pij , its current value xtij−1 or a random feasible value x depending on a uniform random number ρ in [0, 1] as compared to three predetermined parameters: Cg , Cp, and Cw . Meaning a solution is updated to be a synthesis of four sources, and this method maintains the diversity of the population to enhance the capacity to escape from a local optimum.

xtij

=

⎧ g ⎪ ⎨pj

ij xtij−1

⎪ ⎩

x

if if if if

ρ ρ ρ ρ

∈ [0, Cg ) ∈ [Cg , Cp ) ∈ [Cp , Cw ) ∈ [Cw , 1]

(21)

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

5

Table 1 Consistency test for all individual pairwise comparison matrices. Expert i

1

2

3

4

5

6

7

8

9

10

λmax

5.0681 0.0170 0.0152

5.3537 0.0884 0.0790

5.4449 0.1112 0.0993

5.2955 0.0739 0.0660

5.4400 0.1100 0.0982

5.4088 0.1022 0.0912

5.1549 0.0387 0.0346

5.3972 0.0993 0.0887

5.4067 0.1017 0.0908

5.3989 0.0997 0.0891

CI CR

Table 2 Overall pairwise comparison matrix. Attribute

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 0.3243 1.1130 0.6667 2.5302

3.0837 1 2.9137 1.8295 4.6440

0.8985 0.3432 1 0.4066 1.3808

1.4998 0.5466 2.4595 1 3.0539

0.3952 0.2153 0.7242 0.3274 1

Weight

0.1864

0.0706

0.2392

0.1166

0.3872

λmax = 5.0405, CI = 0.0101, CR = 0.0090

4. 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 SSOAHP 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 [39–42]. Since our major purpose is to study the feasibility and efficiency of the proposed method. We only used the AHP in the experimental studies in this paper.

Table 3 The result for Example. RLD i

1

2

8

ω

0.6421 0.6421

0.2649 0.9070

0.0930 1

CPV

Nbldj ]. The variables xj = h, j = Nblu + 1, . . . , Nblu + 2, . . . , 2 × Nblu in the second half of the vector represent that hth RLD is open and services the (j – Nblu)th BLU. Thus, h is limited in [1, Nrld]. This representation guarantees that all solutions in SSOAHP satisfy the constraints listed in Eqs. (4)–(7). Fig. 1 depicts a military logistics system, which can be represented by X = (2, 1, 3, 1, 2; 1, 1, 4, 5, 5).

4.1. Assessment of weights for the factors using AHP Five attributes are selected for MLDLP, including capacity, level of main road, structure of the building, operation area, and distance. As described in Section 3.1, AHP-1 is used to calculate the weight of each attribute. After interviewing 10 local experts, individual pairwise comparison matrices are obtained as listed in Table A.1, and the corresponding consistency index CR values are all in [0.0152, 0.0993] as shown in Table 1, therefore all judgements are consistent. The overall pairwise comparison matrix between each attribute can be extracted by taking the geometric average of those pairwise comparison matrices, as listed in Table 2; and then, the weight for each attribute can be calculated according to Eq. (14) and as input to the next stage. Table 2 reveals CR = 0.0090 < 0.1, which shows that the comparison matrix is consistent. 4.2. The proposed SSOAHP for MLDLP After weighting the attributes, in the second stage, SSO is implemented to select an optimal allocation of depots for MLDLP. Some reported results show that SSO performs satisfactory global searches; however, such exploration ability may be harmful to the quality of the solution and its convergence speed [26,28]. Thus, this work proposes SSOAHP based on SSO and AHP, which is adopted as a local search algorithm, to further improve the disadvantage of SSO for solving MLDLP. 4.2.1. Solution representation In the initialization phase of SSOAHP , a population of initial solutions are randomly generated to initiate solution space exploration. For each solution, Xi = (xi,1 , xi,2 , . . . xi,Nblu ; xi,Nblu+1 , . . . , xi,j , . . . , xi,2×Nblu ), which is generated with a vector of 2×Nblu variables. The variables xj = k, j = 1, 2, . . . , Nblu in the first half of the vector represent that kth BLD is open for BLU j, and k is bounded in [1,

4.2.2. Fitness function The major difficulty in the applicability of SSOAHP to MLDLP is the lack of general methodology for handling constraints. One way of dealing with candidate solutions that violate the constraints is the use of a penalty function as listed in Eq. (23), which has the aim of decreasing the fitness of infeasible solutions to drive the population toward the feasible region. For a solution X, f (X ) is its objective value, which is calculated according to Eq. (3), dk (X ) is the total demand of BLUs serviced by RLD k corresponding to Eq. (8), b(X ) is the total labor force required to operate all RLDs corresponding to Eq. (9), and c(X ) is the total times to violate the constraint in Eq. (10). The fitness function used in SSOAHP is formulated as Eq. (22), where λ is the degree of penalization set to 103 , as determined by trial and error: F (X ) =

{

f (X ) if constraints in Eqs. (8) − (10) are all satisfied, f (X ) − λ · β (X ) otherwise. (22)

β (X ) =

∑

max [0, dk (X ) − capk ] + max [0, b (X ) − B] + c (X )

k∈K

(23) 4.2.3. AHP-based local search (ALS) This work proposes a novel local search strategy, called ALS, which is designed based on AHP-2 to improve solutions for SSO. ALS is triggered for each updated solution when condition ρ < Nahp is satisfied, in order to achieve a trade-off between global and local searches, where ρ is a uniform random number in [0, 1] and Nahp is a predefined parameter of ALS, as obtained based on the experiment in Section 5. Considering the computation costs and results of previous experiments, the local search is conducted once on a selected variable in the second half of a solution, i.e., xj ,

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C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

Fig. 4. Flowchart of the proposed SSOAHP .

Fig. 5. The main effects plot for EX 1.

j = Nblu + 1, . . . , Nblu + 2, . . . , 2 × Nblu. That is, AHP-2 is used to aid the selection of RLD for a BLU. The ALS procedure is given, as follows: Step 0. If a solution X = (x1 , x2 , . . . xNblu ; xNblu+1 , . . . , xj , . . . , x2×Nblu ) with fitness value F (X ) satisfies ρ < Nahp, then activate ALS; and let X ∗ = X and F (X ∗ ) = F (X ). Step 1. Randomly generate an integer in [Nblu + 1, 2 ×Nblu] for choosing a BLU, say h, and record the value of xh , say µ. Step 2. Randomly select a number of potential RLDs, say Nalt, which is determined in Section 5. Step 3. Apply AHP-2, as introduced in Section 3.1, to evaluate the priority values (ωk , k = 1, 2, . . . , Nalt) of the alternative RLDs, as based on their normalized attribute values. Step 4. Apply roulette-wheel selection using the priority values to select a RLD, say q.

Step 5. Let x∗j = q, if x∗j = µ, ∀j∈[Nblu + 1, 2 × Nblu] and calculate F (X ∗ ). Step 6. If F (X ∗ ) > F (X ), let X = X ∗ , F (X ) = F (X ∗ ); and update its pBest P and gBest G if necessary. Step 7. Halt. The ALs procedure in SSOAHP is demonstrated in the following example: Example, let Nblu = 5, Nbld = 5, Nrld = 10, Nalt = 3, a solution X = (4, 5, 4, 2, 1; 3, 7, 4, 3, 7) with its fitness value F (X ) = −134.6450 and suppose that ρ < Nahp is satisfied: Step 0. Activate ALS and let X ∗ = X and F (X ∗ ) = F (X ). Step 1. Randomly generate an integer in [6, 12], say h = 6, and let

µ = x6 = 3. Step 2. Randomly select 3 RLDs (Nalt = 3), i.e, RLD 1, 2 and 8.

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

7

Table 4 Value setting for the problem set. Parameter

Value

Type

Nblu

= (5, 20, 50, 70)

Integer

Nbld

= (5, 10 15) ⎧ (10, 15, 20), ⎪ ⎨ (30, 35, 40), = ⎪ ⎩(60, 65, 70), (80, 85, 90),

Nrld

if Nblu if Nblu if Nblu if Nblu

=5 = 20 = 50 = 70

Ncomi

The total number of combat unit belong to BLU i and limited in [2, 6]

rgei

The defense range of BLU i on the x-axis and set in [2 × Ncomi , 3 × Ncomi ]

Coordinate system

⎧ rgei ] , if ⎪ ⎨[st[i , sti +rge ] x ∈ combat unit of BLU i i i + 2 , 1].5 , if x ∈ BLD of BLU i x= N [ st∑ ⎪ ⎩ 0, Nblu i=1 rgei , if x ∈ RLD sti =

Float

{ 0, if i = 1 ∑i−1

j=1 rgej , otherwise ⎧ ⎨[3, 6] , if y ∈ combat unit y = [7, 10] , if y ∈ BLD ⎩[13, 16] , if y ∈ RLD

capij

N(15, 0.5)

capk

N(20, 5)

lmr ij , lmr k

[1, 6]

lsbij , lsbk

[1, 7]

soaij

[20, 50]

soak

[30, 60]

disij

Calculated based on the coordinate system

demi

The demand of combat unit n and demn ∼N(2, 0.2)

demi

=

bk

⎧3, ⎪ ⎪ ⎪3, ⎪ ⎪ ⎪ ⎨2, = 2, ⎪ ⎪ 2, ⎪ ⎪ ⎪ ⎪ ⎩1, 1,

B

= Nblu

Integer

∑Ncomi n=1

Float

demn

if capk > 30 if 25 < capk if 25 < capk if 20 < capk if 15 < capk if 15 < capk if capk ≤ 15

≤ 30 and rand() > 0.5 ≤ 30 and rand() ≤ 0.5 ≤ 25 ≤ 20 and rand() > 0.5 ≤ 20 and rand() ≤ 0.5

Step 3. Apply AHP-2 to a evaluate the priority values of selected RLDs, the result as listed in Table 3.

Table 5 The main-effects of different treatment combinations for Nahp and Nalt. Statistics

Step 4. Calculate cumulative priority value (CPV) of the priority values of selected RLDs, as shown in Table 3. Given a random number, say ρ = 0.7214 and compare with CPV. Because CPV1 = 0.4170 < ρ = 0.5214 < CPV2 = 0.9065, then RLD 2 (q = 2) is chosen for updating X. Step 5. Because x∗6 = x∗9 = µ = 3, let x∗6 = x∗9 = q = 2, that is X ∗ = (4, 5, 4, 2, 1; 2, 7, 4, 2, 7) and calculate F (X ∗ ) = 1.4246. Step 6. Because F (X ∗ ) > F (X ), let X = X ∗ = (4, 5, 4, 2, 1; 2, 7, 4, 2, 7), F (X ) = F (X ∗ ) = 1.4246; and update pBest and gBest, if necessary.

Nahp

Nalt

Average

10

20

30

F’ av g

0.1 0.4 0.7 Average

0.7109a 0.7743ab 0.7015 0.7289

0.6775 0.6865 0.7312ab 0.6984

0.5945 0.7210b 0.7181 0.6779

0.6610 0.7272 0.7169

T’ av g

0.1 0.4 0.7 Average

0.1166ab 0.1689a 0.2747a 0.1867

0.1482b 0.2978 0.3537 0.2665

0.1864b 0.3989 0.5507 0.3787

0.1504 0.2885 0.3930

F’ std

0.1 0.4 0.7 Average

0.0868a 0.0701ab 0.1226 0.0932

0.1034b 0.1230 0.1103a 0.1123

0.1066b 0.1114 0.1188 0.1123

0.0989 0.1015 0.1172

T’ std

0.1 0.4 0.7 Average

0.0842 0.0310ab 0.0989 0.0714

0.1031 0.0818 0.0439ab 0.0762

0.0812a 0.0638b 0.1281 0.0910

0.0895 0.0589 0.0903

Step 7. Halt. 4.2.4. The overall procedure of SSOAHP According to the descriptions in Sections 3.2 and 4.2, the flowchart of SSOAHP is depicted in Fig. 4, and the overall procedure is described, as follows:

Integer

a,b

The best value among the values of the same row and column, respectively.

SSOAHP PROCEDURE Step 0. Randomly generate Xi0 , calculate F (Xi0 ) using Eq. (22), let pBest P i = Xi , find gBest G among all solutions and let t = 1, for i = 1, 2, . . . , Nsol.

Step 3. Apply Eq. (21) to update Xit , calculate F (Xit ), let Pi = Xit , if

Step 1. Let i = 1.

Step 4. If ρ < Nahp, activate ALS to update Xit . Otherwise, go to Step

Step 2. If stopping criteria is met, halt. Otherwise, go to Step 3.

5.

F (Xit ) > F (Pi ) and G = Xit , if F (Xit ) > F (G).

8

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12 Table 6 The performance of each treatment combination on the four selected problems. Problem (Nblu, Nrld)

(5, 20) (20, 40) (50, 70) (70, 90) Average

Criteria

Nahp Nalt = 10

Fav g Tav g Fav g Tav g Fav g Tav g Fav g Tav g Fav g Tav g

Nalt = 20

Nalt = 30

0.1

0.4

0.7

0.1

0.4

0.7

0.1

0.4

0.7

1.656 0.836 1.631 9.741 1.617 70.734 1.633 159.802 1.634 60.278

1.658 0.797 1.631 9.973 1.620 72.528 1.635 159.736 1.636 60.759

1.656 0.959 1.631 10.405 1.620 73.278 1.628 165.278 1.634 62.480

1.655 0.808 1.630 9.998 1.618 71.609 1.631 159.658 1.634 60.518

1.656 0.842 1.629 10.623 1.618 74.388 1.634 164.909 1.634 62.691

1.658 0.945 1.630 10.730 1.621 75.356 1.631 166.659 1.635 63.423

1.655 0.770 1.628 10.406 1.616 71.905 1.627 161.239 1.632 61.080

1.655 0.863 1.631 10.789 1.620 76.141 1.632 170.653 1.634 64.611

1.656 1.167 1.631 11.831 1.620 78.416 1.629 170.950 1.634 65.591

Fig. 6. The box plots of the fitness values for GA, DE, SSO, and SSOAHP ..

Step 5. If i < Nsol, let i = i +1 and go to Step 2. Otherwise, let t = t + 1 and go to Step 1. 5. Experiment results and discussion To evaluate the quality and performance of the proposed SSOAHP 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, SSOAHP 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) [45], and the original SSO. In all experiments, two SSO-based methods: SSOAHP and SSO, adopt the same setting: Cg = 0.6, Cp = 0.85 and Cw = 0.99; for GA, the crossover rate and mutation rate are set to 0.7 and 0.3, respectively; for PSO, the maximum and minimum perturbation coefficients are set to 0.3 and 0, which are directly adopted from [43]; for DE, the maximum and minimum crossover rates are set to 0.9 and 0.1, which are directly adopted from [45]. For making a fair comparison of SSOAHP with these methods, the number required to calculate the fitness function (Ncfe) is adopted as the stopping criterion, and set to Ncfe = 2 × Nblu × Nbld × Nrld. All compared methods are coded in MATLAB language and executed on an intel Core i7 3.7GHz PC with 64GB memory. The runtime unit is in CPU seconds.

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

9

Table 8 Computational results of all compared algorithms when Nblu = 20.

Table 7 Computational results of all compared algorithms when Nblu = 5. (Nbld, Nrld)

Criteria

GA

PSO

IDE

SSO

SSOAHP

(Nbld, Nrld)

Criteria

GA

PSO

IDE

SSO

SSOAHP

(5, 10)

Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g

1.25521 1.36643 1.44616 0.04260 0.11615 1.46578 1.50244 1.53190 0.01722 0.11302 1.44886 1.55657 1.65283 0.04928 0.13646 1.46495 1.55377 1.62631 0.02949 0.13958 1.31679 1.46472 1.52042 0.04092 0.21146 1.58075 1.64292 1.69348 0.02815 0.27188 1.39389 1.48078 1.53173 0.03494 0.20521 1.49588 1.54374 1.57929 0.02165 0.30677 1.58368 1.62969 1.66134 0.02274 0.40938

1.24854 1.37778 1.55453 0.06364 0.08594 1.42065 1.45965 1.50528 0.02464 0.12708 1.46816 1.53784 1.60996 0.03454 0.15260 1.40737 1.50049 1.59046 0.04384 0.14948 0.00000 1.38035 1.49775 0.26326 0.21771 1.48807 1.54688 1.62346 0.03885 0.29219 1.36043 1.43050 1.51739 0.03082 0.21927 1.40756 1.45968 1.53078 0.02706 0.32292 1.47273 1.55767 1.63819 0.03770 0.43281

1.34673 1.41015 1.53497 0.03913 0.06927 1.50470 1.52566 1.53709 0.00710 0.10000 1.56709 1.62428 1.67160 0.03704 0.13646 1.53754 1.60771 1.65026 0.02787 0.13073 0.00000 1.45219 1.51309 0.27470 0.19010 1.62922 1.66924 1.69348 0.01882 0.25417 1.42656 1.50280 1.53210 0.02969 0.19323 1.49643 1.54445 1.57573 0.02055 0.28229 1.60464 1.63316 1.66174 0.01600 0.38073

0.00000 1.34732 1.46386 0.25740 0.07708 1.47850 1.50983 1.53709 0.01506 0.10625 1.43983 1.58434 1.67160 0.05864 0.14479 1.47320 1.59689 1.65026 0.04446 0.14063 0.00000 1.42172 1.52664 0.27221 0.20990 1.58683 1.66182 1.69350 0.03063 0.27552 1.41143 1.47975 1.53432 0.03535 0.21042 1.48550 1.55481 1.58348 0.03002 0.31094 1.57673 1.64755 1.66174 0.01899 0.41719

1.37696 1.46502 1.56890 0.05635 0.10156 1.51806 1.53422 1.53709 0.00425 0.11927 1.58447 1.66905 1.68311 0.02474 0.15833 1.59105 1.63376 1.65026 0.01531 0.15625 1.47011 1.50523 1.52664 0.01299 0.23698 1.63058 1.67977 1.69350 0.01942 0.31458 1.46761 1.51555 1.53432 0.01586 0.23385 1.55169 1.57775 1.58348 0.00900 0.34688 1.63503 1.65915 1.66174 0.00609 0.46979

(5, 30)

Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g

1.42388 1.46412 1.48948 0.01559 1.40781 1.44657 1.48179 1.54162 0.02041 1.63906 1.44841 1.50636 1.56698 0.02753 1.94375 1.50902 1.55104 1.59363 0.01819 2.75625 1.47741 1.50253 1.53127 0.01215 3.23802 1.44923 1.50851 1.57810 0.03051 3.70260 1.52302 1.55216 1.57708 0.01567 4.15000 1.41352 1.48321 1.55054 0.03071 4.87448 1.51900 1.54914 1.58112 0.01531 5.51667

1.34324 1.38492 1.44158 0.02081 1.78438 1.35833 1.39768 1.44318 0.01844 2.08854 0.00000 1.34798 1.44008 0.25593 2.45260 1.38220 1.42504 1.47159 0.01930 3.55313 1.39179 1.42957 1.46136 0.01461 4.14010 0.00000 1.32342 1.48152 0.36105 4.72656 0.00000 1.39058 1.49232 0.26391 5.33490 0.00000 1.33138 1.42764 0.25217 6.23021 1.35115 1.39670 1.43544 0.02195 7.07135

1.51308 1.52990 1.54607 0.00794 1.42708 1.51546 1.55130 1.56699 0.01110 1.70990 1.58535 1.60306 1.61824 0.00772 1.92083 1.58939 1.60872 1.62638 0.00892 2.82292 1.55902 1.57117 1.58068 0.00496 3.29115 1.62886 1.65179 1.66834 0.01058 3.76719 1.60957 1.62474 1.64338 0.00762 4.21042 1.54460 1.57645 1.60089 0.01551 4.94323 1.57293 1.59517 1.62521 0.01117 5.56615

1.49595 1.52126 1.54466 0.00974 1.42500 1.52501 1.54693 1.56673 0.00982 1.71250 1.55905 1.58997 1.61286 0.01197 1.89531 1.60923 1.62155 1.63683 0.00720 2.82448 1.53143 1.55630 1.57557 0.01098 3.30781 1.56268 1.62111 1.65517 0.02434 3.78281 1.61050 1.62632 1.64281 0.00808 4.23438 1.52936 1.57499 1.60995 0.02188 4.96354 1.61230 1.62817 1.63726 0.00652 5.59375

1.51637 1.54461 1.55883 0.01058 1.51615 1.56458 1.56978 1.57279 0.00179 1.79792 1.58711 1.61833 1.62938 0.00955 2.02031 1.60352 1.62748 1.64516 0.01151 2.98854 1.54061 1.57813 1.58632 0.01038 3.50104 1.60505 1.66190 1.67990 0.02332 4.00833 1.60452 1.63569 1.64925 0.01215 4.50156 1.52984 1.58673 1.61717 0.02715 5.28021 1.62640 1.63695 1.64080 0.00367 5.91823

Fav g Tav g

1.52678 0.21221

1.47232 0.22222

1.55218 0.19300

1.53378 0.21030

1.58216 0.23750

Fav g Tav g

1.51098 3.24763

1.38081 4.15353

1.59025 3.29543

1.58740 3.30440

1.60662 3.50359

(5, 15)

(5, 20)

(10, 10)

(10, 15)

(10, 20)

(15, 10)

(15, 15)

(15, 20)

Average

5.1. Problem set To test the performance of SSOAHP , a set of problems is randomly generated in terms of three key factors, which affect problem complexity, including Nblu, Nrld, and Nbldi . For simplify the generation of the problem, the number of potential BLDs of all BLU, i.e., Nbldi , is set to the same, i.e., Nbld. The values of Nblu are set to 5, 20, 50, and 70 against its own values of Nrld. For each pair of (Nblu, Nrld) are available for Nbld = 5, 10, and 15. Thus, 36 combinations of (Nblu, Nbld, Nrld) are constructed in the problem set. For each combination, the values of other parameters are generated randomly according to Table 4. 5.2. Results of EX 1 To efficiently select the best setting for the ALS parameters, a factorial experiment is conducted, which considers two factors: Nahp and Nalt, and both have three levels: 0.1, 0.4, and 0.7 for Nahp; and 10, 20, and 30 for Nalt, which are constructed by nine treatment combinations, and tested against four problems selected from the problem set, including (Nblu, Nrld) = (5, 20), (20, 40), (50, 70) and (70, 90) under fixed Nbld = 15. For each treatment,

(5, 35)

(5, 40)

(10, 30)

(10, 35)

(10, 40)

(15, 30)

(15, 35)

(15, 40)

Average

SSOAHP implements 10 independent runs with 30 population sizes on the selected problems. The fitness value Fij (X ) and CPU time Tij (X ), as obtained by SSOAHP in the jth run of the ith treatment, are normalized by Eq. (24), which have the same scale for easy observation. Yij (X ) = min (Yi (X )) Yij′ (X ) = (24) max (Yi (X )) − min (Yi (X )) The experimental results are summarized in Table 5, in which F’ av g and T’ av g represent the average of the normalized fitness value and CPU time over 10 runs, respectively; and the corresponding standard deviations are denoted by F’ std and T’ std . To assist in the practical interpretation of this experiment, Fig. 5 presents a plot of the two main effects. It shows that smallest and highest F’ av g are attained at Nahp = 0.1 and 0.4, respectively. Nalt has negative main effects on F’ av g ; meaning that increasing the value moves F’ av g downward. For T’ av g , both two factors have positive main effects; the reason is that ALS is less efficient in consuming Ncfe than the update mechanism of SSO. Thus, the more frequently ALS is conducted, the more time is required for SSOAHP . The performance of each treatment combination on the four selected problems are listed in Table 6, where the best values of all treatments are shown

10

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12 Table 10 Computational results of all compared algorithms when Nblu = 70.

Table 9 Computational results of all compared algorithms when Nblu = 50. (Nbld, Nrld)

Criteria

GA

PSO

IDE

SSO

SSOAHP

(Nbld, Nrld) Criteria GA

(5, 60)

Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g

0.00000 1.23144 1.41035 0.41811 12.18594 1.34129 1.39040 1.42581 0.02047 13.22083 1.33237 1.36267 1.39239 0.01566 14.23229 1.34450 1.41136 1.43832 0.02226 24.68333 1.41717 1.44544 1.47974 0.01560 26.40885 0.00000 1.11428 1.45065 0.56742 28.51042 1.40031 1.43318 1.46959 0.01839 36.99167 1.38175 1.42375 1.45496 0.01699 39.17969 1.43143 1.46099 1.48359 0.01317 42.94219

0.00000 0.04489 1.34666 0.24587 17.17969 0.00000 0.58426 1.37597 0.67965 18.63594 0.00000 0.35167 1.34265 0.59323 20.19948 0.00000 0.81791 1.38648 0.67932 34.93854 0.00000 0.60315 1.41729 0.70161 37.47865 0.00000 0.13699 1.38294 0.41801 40.42865 0.00000 0.63434 1.40893 0.68994 52.07344 0.00000 0.94659 1.39108 0.63048 55.90677 0.00000 0.91692 1.41031 0.65959 60.73281

0.00000 1.48469 1.55322 0.28074 13.12500 1.51927 1.55791 1.57285 0.01255 14.27188 1.49846 1.51632 1.52847 0.00728 15.31875 1.57476 1.58738 1.60053 0.00753 26.46615 1.59351 1.60557 1.62136 0.00633 28.14531 1.58160 1.60432 1.61714 0.00868 30.67656 1.57575 1.59584 1.61358 0.00868 39.13333 1.57037 1.58511 1.59728 0.00565 41.52240 1.59268 1.60628 1.61833 0.00599 45.50417

1.48375 1.50995 1.52394 0.00826 12.33750 1.51192 1.53207 1.54813 0.00820 13.50417 1.46254 1.48626 1.50405 0.00981 14.43438 1.54316 1.56870 1.58374 0.00871 25.12813 1.56039 1.57823 1.60049 0.00829 26.66771 1.54805 1.57710 1.59938 0.01135 29.05469 1.56278 1.58073 1.59193 0.00854 37.00052 1.56372 1.57958 1.59010 0.00710 39.67552 1.57770 1.59485 1.61058 0.00723 43.40938

0.00000 1.49416 1.55493 0.28225 12.69896 1.53645 1.56521 1.58386 0.00911 13.82500 1.50288 1.51898 1.53138 0.00754 14.75938 1.59183 1.60171 1.61552 0.00536 25.70833 1.60071 1.60825 1.61816 0.00375 27.32448 1.59911 1.61525 1.63066 0.00860 29.81406 1.59098 1.60804 1.62004 0.00784 37.88229 1.59403 1.60587 1.61512 0.00484 40.45521 1.61031 1.62110 1.62875 0.00444 44.53750

(5, 80)

Fav g Tav g

1.36372 26.48391

0.55964 37.50822

1.57149 28.24039

1.55639 26.80133

1.58206 27.44502

Average

(5, 65)

(5, 70)

(10, 60)

(10, 65)

(10, 70)

(15, 60)

(15, 65)

(15, 70)

Average

in bold. As can be seen, it has the same conclusion, that (Nahp, Nalt) = (0.4, 10) is the most effective combination at a moderate computation cost. Thus, SSOAHP is implemented with Nahp = 0.4 and Nalt = 10 in the EX 2. 5.3. Results of EX 2 The purpose of EX 2 is to compare the performance of the proposed method with its competitors: GA, PSO, DE, and SSO. All algorithms implement 30 runs with 30 population sizes for each artificial problem. All methods have the chance to converge to an infeasible solution, especially on large problems. If an infeasible solution is obtained by a method, the corresponding fitness value is amended as 0 to narrow the range of results and maintain comparability. The computational results, including the worst (Fworst ), the average (Fav g ), the best (Fbest ) fitness value, the standard deviation (Fstd ) of the obtained solutions and the average CPU times (Tav g ), when Nblu = 5, 20, 50, and 70, are reported in Tables 7–10, respectively. The best values of each problem are shown in bold, and the results lead to the following interpretation:

(5, 85)

(5, 90)

(10, 80)

(10, 85)

(10, 90)

(15, 80)

(15, 85)

(15, 90)

Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fworst Fav g Fbest Fstd Tav g Fav g Tav g

PSO

IDE

SSO

SSOAHP

1.33700 0.00000 1.51096 1.48632 1.54948 1.37607 0.09023 1.54296 1.50558 1.55629 1.41089 1.35512 1.55805 1.51617 1.56364 0.01775 0.34338 0.00843 0.00714 0.00471 29.58438 42.94531 32.23333 29.79323 29.91563 1.35678 0.00000 1.53211 1.49143 1.53288 1.38556 0.40330 1.53983 1.50288 1.54721 1.41891 1.36313 1.54929 1.51530 1.56225 0.01619 0.62663 0.00412 0.00632 0.00591 31.85156 45.67500 34.35833 32.28281 32.29583 0.00000 0.00000 1.54081 1.47199 1.54836 1.11961 0.04483 1.55189 1.49739 1.56150 1.38155 1.34485 1.56508 1.51851 1.56964 0.50984 0.24554 0.00587 0.01063 0.00558 33.36823 48.35313 36.24531 33.64635 34.06875 1.38121 0.00000 1.57843 1.54281 1.60641 1.40963 0.68890 1.58953 1.55744 1.61304 1.45005 1.40369 1.59576 1.57583 1.62092 0.01371 0.70076 0.00463 0.00675 0.00350 59.59323 85.70625 64.12135 59.85781 60.54948 1.41118 0.00000 1.57776 1.55404 1.60807 1.43178 0.18703 1.58914 1.56886 1.61678 1.45426 1.41693 1.60013 1.57958 1.62254 0.01233 0.48500 0.00529 0.00708 0.00350 64.56979 91.92188 69.17396 64.65885 65.29792 1.36974 0.00000 1.58636 1.54749 1.59594 1.40902 0.49905 1.60184 1.56492 1.61439 1.43065 1.38327 1.61219 1.57715 1.62398 0.01493 0.66711 0.00566 0.00714 0.00594 66.80417 97.15573 71.60521 67.10781 67.82604 1.37277 0.00000 1.57144 1.54040 1.60453 1.40712 0.09253 1.58540 1.55817 1.61060 1.43908 1.40781 1.59559 1.57074 1.61698 0.01729 0.35216 0.00541 0.00641 0.00272 89.26354 129.00469 94.96354 90.74583 91.10052 0.00000 0.00000 1.56079 1.52322 1.58214 1.28394 0.08956 1.57250 1.53654 1.58856 1.40256 1.35939 1.58125 1.55174 1.59382 0.34948 0.34085 0.00529 0.00732 0.00289 95.89115 141.10417 102.91458 98.37188 99.89740 1.34738 0.00000 1.59688 1.55888 1.62733 1.41166 0.74112 1.60930 1.57940 1.63672 1.45589 1.41165 1.62184 1.59882 1.64539 0.02192 0.70523 0.00547 0.00947 0.00463 102.17240 149.43229 110.28802 102.31979 103.27969 1.35938 63.67760

0.31517 92.36649

1.57582 68.43374

1.54124 64.30938

1.59390 64.91458

Table 11 The results of the statistical test. Method

Friedman’s test Ranks

GA PSO DE SSO SSOAHP

3.92 4.94 2.19 2.92 1.03

Statistic

132

Holm’s test p-value

p-value

APV

0.000000

0.000000 0.000000 0.001745 0.000000

0.000000 0.000000 0.001745 0.000001

1. PSO has the worst efficiency, as the update for each variable in a solution requires the calculations of three intermediate variables [43]. As a result, computation costs are higher than other methods. IDE is efficient only for smaller MLDLPs. The efficiency of SSOAHP is slightly lower than GA and SSO at first glance; however, the results show the trend that SSOAHP comes from behind. The reason might be that the inefficiency of the proposed ALS, as mentioned in Section 5.2, is mitigated when the size of the problem increases. 2. In addition, the quality of solutions obtained by PSO is worst. The reason might be that this discrete version of PSO is not suitable for the solution of MLDLP. IDE obtains higher quality

C.-M. Lai / Applied Soft Computing Journal 78 (2019) 1–12

11

Table A.1 10 individual pairwise comparison matrices. Expert 1

cap

lmr

lsb

soa

dis

Expert 2

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 0.33 0.50 0.25 2.00

3.00 1 2.00 0.50 4.00

2.00 0.50 1 0.33 3.00

4.00 2.00 3.00 1 5.00

0.50 0.25 0.33 0.20 1

cap lmr lsb soa dis

1 0.33 7.00 5.00 3.00

3.00 1 9.00 7.00 5.00

0.14 0.11 1 0.33 0.20

0.20 0.14 3.00 1 0.20

0.33 0.20 5.00 5.00 1

Expert 3

cap

lmr

lsb

soa

dis

Expert 4

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 0.17 4.00 0.25 0.33

6.00 1 7.00 4.00 5.00

0.25 0.14 1 0.20 0.20

4.00 0.25 5.00 1 3.00

3.00 0.20 5.00 0.33 1

cap lmr lsb soa dis

1 0.17 0.50 0.33 4.00

6.00 1 5.00 3.00 8.00

2.00 0.20 1 0.25 5.00

3.00 0.33 4.00 1 7.00

0.25 0.13 0.20 0.14 1

Expert 5

cap

lmr

lsb

soa

dis

Expert 6

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 2.00 1.00 3.00 6.00

0.50 1 1.00 0.50 3.00

1.00 1.00 1 0.50 1.00

0.33 2.00 2.00 1 2.00

0.17 0.33 1.00 0.50 1

cap lmr lsb soa dis

1 0.25 1.00 1.00 5.00

4.00 1 4.00 4.00 3.00

1.00 0.25 1 1.00 4.00

1.00 0.25 1.00 1 4.00

0.20 0.33 0.25 0.25 1

Expert 7

cap

lmr

lsb

soa

dis

Expert 8

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 0.50 5.00 2.00 7.00

2.00 1 7.00 5.00 9.00

0.20 0.14 1 0.33 3.00

0.50 0.20 3.00 1 5.00

0.14 0.11 0.33 0.20 1

cap lmr lsb soa dis

1 0.25 0.50 0.33 4.00

4.00 1 1.00 0.50 3.00

2.00 1.00 1 0.33 3.00

3.00 2.00 3.00 1 3.00

0.25 0.33 0.33 0.33 1

Expert 9

cap

lmr

lsb

soa

dis

Expert 10

cap

lmr

lsb

soa

dis

cap lmr lsb soa dis

1 0.33 0.17 0.33 8.00

3.00 1 0.50 2.00 6.00

6.00 2.00 1 2.00 7.00

3.00 0.50 0.50 1 7.00

0.13 0.17 0.14 0.14 1

cap lmr lsb soa dis

1 0.20 1.00 0.25 0.20

5.00 1 5.00 1.00 4.00

1.00 0.20 1 0.20 0.17

4.00 1.00 5.00 1 4.00

5.00 0.25 6.00 0.25 1

solutions than GA for all problems, and SSO for most problems, especially larger size problems. Compared to GA, PSO, IDE, and SSO, the SSOAHP method obtains the greatest scores in Fworst , Favg and Fbest for most problems. This empirically demonstrates that ALS, as proposed in this work, enhances the performance of SSOAHP to more effectively solve MLDLPs. The corresponding box plots as shown in Fig. 6 depict the full range of results, which helps summarize the complex results from EX 2, and provides much of the same information. 5.4. Nonparametric statistical test For further confirming whether or not SSOAHP offers a significant improvement over its competitors for MLDLP, statistical testing, as suggested by [46], is employed in this work. The significance level is set to α = 0.05 to determine whether the hypothesis is rejected in all the comparisons, while Fav g , as obtained by all methods for each problem, is adopted as the target value for statistical testing. Table 11 reveals the statistical test results. Friedman ranks indicate that SSOAHP is the best performing algorithm of the comparisons, with a rank of 1.4. The p-value = 0.000000, which suggests significant differences among all compared algorithms. Thus, Holm’s testing is conducted as a post hoc test to perform pairwise comparisons considering SSOAHP and its competitors. The results of Holm’s testing are listed in Table 11, in which the adjusted p-value (APV) is stricter than the p-value, as it considers the accumulated family errors [46,47]. Both p-values and APVs come to the same conclusion, that the proposed SSOAHP is statistically more effective than its competitors. 6. 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 MLDLP. AHP, which is independently employed in the first stage, uses pairwise comparison to calculate the weights of attributes, and then, applies these weights as the coefficients of the objective function in MLDLP. In the second stage, a novel SSOAHP , which is designed as a hybrid method based on SSO and AHP, is proposed to solve MLDLP. An extensive experimental study on 36 MLDLPs is conducted herein. The results indicate that the proposed ALS enhances the exploitation of SSOAHP to achieve better quality solutions than its competitors. Furthermore, statistical testing statistically confirms these findings. 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 1072221-E-606-009. Appendix See Table A.1. References [1] C.H. Aikens, Facility location models for distribution planning, European J. Oper. Res. 22 (1985) 263–279. [2] A. Klose, A. Drexl, Facility location models for distribution system design, European J. Oper. Res. 162 (2005) 4–29. [3] S. Tragantalerngsak, J. Holt, M. Ro, Lagrangian Heuristics for the two-echelon, single-source, capacitated facility location problem, European J. Oper. Res. 102 (1997) 611–625.

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