Artificial immune system and sheep flock algorithms ...

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Artificial immune system and sheep flock algorithms for two-stage fixedcharge transportation problem a

b

c

Devika Kannan , Kannan Govindan & Hamed Soleimani a

Department of Mechanical & Manufacturing Engineering, Aalborg University, Copenhagen, Denmark b

Department of Business and Economics, University of Southern Denmark, Odense, Denmark c

Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University (IAU), Qazvin, Iran Published online: 21 Mar 2014.

To cite this article: Devika Kannan, Kannan Govindan & Hamed Soleimani (2014) Artificial immune system and sheep flock algorithms for two-stage fixed-charge transportation problem, Optimization: A Journal of Mathematical Programming and Operations Research, 63:10, 1465-1479, DOI: 10.1080/02331934.2014.898148 To link to this article: http://dx.doi.org/10.1080/02331934.2014.898148

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Optimization, 2014 Vol. 63, No. 10, 1465–1479, http://dx.doi.org/10.1080/02331934.2014.898148

Artificial immune system and sheep flock algorithms for two-stage fixed-charge transportation problem

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Devika Kannana, Kannan Govindanb* and Hamed Soleimanic a Department of Mechanical & Manufacturing Engineering, Aalborg University, Copenhagen, Denmark; bDepartment of Business and Economics, University of Southern Denmark, Odense, Denmark; cFaculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University (IAU), Qazvin, Iran

(Received 13 March 2013; accepted 8 February 2014) In this paper, we cope with a two-stage distribution planning problem of supply chain regarding fixed charges. The focus of the paper is on developing efficient solution methodologies of the selected NP-hard problem. Based on computational limitations, common exact and approximation solution approaches are unable to solve real-world instances of such NP-hard problems in a reasonable time. These approaches involve cumbersome computational steps in real-size cases. In order to solve the mixed integer linear programming model, we develop an artificial immune system and a sheep flock algorithm to achieve better solutions in comparison to earlier approaches. The evaluations are set up based on two phases; first, comparing performances of two proposed algorithms with two previous studies with the same data (two previously proposed genetic algorithm and ant colony optimization methods) and second, evaluating two proposed algorithms in larger instances. Computational studies reveal that the proposed algorithms present acceptable performance by achieving solutions that are more robust in a proper time. Keywords: artificial immune system; distribution planning problem; fixed-charge transportation problem; genetic algorithm; sheep flock algorithm MSC code: 65K10; 90C08; 90C11

1. Introduction A supply chain in its classical form is a combination of processes to fulfil customer requests, and not only includes the manufacturers and suppliers, but also transporters, warehouses, retailers and customers themselves.[1] It stipulates that the main goal of a supply chain is fulfilling customers’ requests. In this regard, an efficient distribution system plays a vital role to cover customers’ demands. Cost efficiency is the most important issue in a distribution system when we know that the cost of distribution accounts for about 30% of the cost price of a product. Thus, in this paper, we address a two-stage distribution problem regarding fixed charges between sources and destinations. In a distribution-planning problem, we determine how many products should be

*Corresponding author. Email: [email protected] © 2014 Taylor & Francis

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transported from each plant to each distribution centre and from each distribution centre to each customer. Reviewing earlier research illustrates that although there are sufficient new developments in modelling of distribution planning problems (regarding different kinds of constraints), few works concentrate on practical solution methodologies. Meanwhile, the main concern of any research should be the practical applicability of the proposed model. Surely, due to their limitations, general solvers or analytical methods are unable to achieve the optimal solution(s) of practical instances for such NP-hard problems in a reasonable time. It should be noted that the fixed-charge distribution problem is an NPhard one.[2] Therefore, heuristic and meta-heuristic algorithms are extremely efficient in such cases and recently have been widely used.[3–6] There are some attempts in utilizing meta-heuristic algorithms like Jawahar and Balaji [7] and Gen et al. [8] in utilizing a genetic algorithm, and Yang and Liu [9] in benefiting from tabu search, but they are already not enough. They could not evaluate their algorithms in large-scale instances to analyse their practicability. In order to cover the mentioned gap, two elite meta-heuristic algorithms are considered: artificial immune system (AIS) and sheep flock algorithm (SFA). AIS and SFA are two of the well-known meta-heuristic algorithms, which are widely used in different fields of science. AIS is a robust, adaptive, autonomous algorithm, which is inspired by the biological immune system. The immune system consists of a multitude of cells and molecules, which interact in a variety of ways to detect and eliminate infectious agents (pathogens).[10] This well-behaved algorithm has been developed successfully in different fields of research; De Castro and Von Zuben [11] and Kodaz et al. [12] in medical applications and diagnose diseases, Coello and CruzCortes [13] and Omkar et al. [14] in multi-objective design optimizations, Basu [15] and Liao and Tsao [16], in power system optimization problems, etc. Such research helped us to develop this meta-heuristic algorithm in distribution planning problem. On the other hand, SFA is a well-known algorithm that simulates heredity of a sheep flock in a prairie. It is also an efficient algorithm to cope with large-scale problems.[17] SFA has been successfully developed in different kinds of problems, especially in various scheduling problems such as Anandaraman [18] in job shop scheduling and flow shop scheduling problems [19], in simultaneous scheduling of machines and two identical automated-guided vehicles in a flexible manufacturing system so as to minimize makespan and mean tardiness and also other fields.[20] Based on its well-behaved performance in large-scale problems, we decided to develop SFA for fixed-charge distribution planning problem as an NP-hard one. Finally, this paper tries to find some acceptable results for necessary questions regarding the solution methodologies in solving two-stage FCTP. Indeed, it is vital to know if we can improve the quality of solutions and time in the problem of two-stage FCTP. As such improvements can have a drastic effect on transportation costs and cost price of products,[7] finding better solutions is completely necessary. In order to develop efficient and robust solution methodologies for the two-stage fixed-charge transportation planning problem, we have presented two meta-heuristic algorithms: AIS and SFA. To the best of our knowledge, this is the first research which tries to develop these solution methodologies for two-stage fixed-charge distribution planning problem.

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The rest of the paper is organized as follows: Section 2 presents a complete literature review in distribution planning solution methodologies to clarify current gaps. The two-stage fixed-charge distribution planning model is explained in Section 3. Developed AIS and SFA solution methodologies and descriptions of their steps are discussed in Section 4. Section 5 is considered a complete computational analysis. In Section 6, the conclusion, the summary of the achieved results and some indications of future research are illustrated.

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2. Literature review Distribution planning problem is a vital issue in the cost efficiency of supply chains. Thus, we can find vast studies on this topic. In this paper, we focus on an advanced model of two-stage transportation problem, which considers fixed charges and continuous costs simultaneously, which is the most compatible model with real-world situations. This model is completely described by Jawahar and Balaji [7]. The two-stage transportation planning model contains plants, distributors and customers. As mentioned in the previous section, we focus on solution methodologies to achieve better results. We can categorize the solution methodologies of fixed-charge transportation problem in four classes: exact algorithms, approximation algorithms, heuristic algorithms and metaheuristic algorithms. Exact methods are the oldest and maybe the most complicated approaches to solving any kind of problems. However, based on their limitations and computational complexities, exact algorithms cannot handle even mid-size instances so they are often not practical in real-world situations. Steinberg [21], Gray [22] are two of the first papers to develop branch and bound methods. Definitely, the difficulty of such exact algorithms exponentially increases based on problem size augmentation. Branch and bound methods are the most popular approaches in exact algorithms like Kennington and Unger [23], McKeown [24], Cabot and Erenguc [25], Schaffer and O’Leary [26], Palekar et al. [27], Herer and Rosenblatt [28], and Bell et al. [29]. Clearly, some other exact methods are developed for fixed-charge transportation problem including constraint programming [30] and branch-and-cut algorithm.[31] The limitations of exact algorithms in finding the optimum points of real-world problems in a reasonable time lead researchers to inexact solutions such as approximate, heuristic and meta-heuristic algorithms. There are few attempts at developing approximation algorithms. The literature shows algorithms of this kind are almost as hard as exact algorithms if we insist on having an acceptance close-optimum approximation. The famous SWIFT method of Walker [32], which is based on simplex method, is one of the approximation algorithms. Based on the approximation of dual problem, Balakrishnan et al. [33] developed a well-behaved approximation technique. Finally, Kim and Pardalos [34] introduced the dynamic slopescaling procedure, which could achieve a not-so-good-approximate of the optimum. Regarding acceptable backgrounds and highly well-behaved performances in NPhard problems, heuristic and metaheuristic algorithms are vastly developed for FCTP. Table 1 presents a review on these algorithms. As is clear in Table 1, recent researchers also tried to develop the solution methodologies in two-stage FCTP. Seven well-known meta-heuristic algorithms are specified in Table 1 and the publications are categorized by the solution methodologies including genetic algorithm, ant colony, tabu search, heuristic algorithms, hybrid

Our research

Jo et al. [50] Jawahar and Balaji [7] Hajiaghaei et al. [51] Xie and Jia [35] DuraiRaj and Rajendran [3] Lotfi and Tavakkoli-Moghaddam [36] Vinay and Sridharan [37] Vinay et al. [4]

Wright and von Lanzenauer [45] Adlakha and Kowalski [46] Amiri [47] Adlakha et al. [48] Sun et al. [49] El-Sherbiny [5] Alhamali and El-Sherbiny [6]

Paper

Compared

*

* * * * * *

Genetic algorithm

Compared

*

Ant colony

Table 1. A review of heuristic and meta-heuristic algorithms for FCTP.

–

*

Tabu search

–

* * * *

Heuristic

–

PSOAIS

Hybrid

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– – – – – – –

– – – – – – –

*

– – – – – – –

– – – – – AIS

*

SFA

AIS

Linear

Linear Linear

Nonlinear Linear Nonlinear Nonlinear Linear Both

Linear Linear Linear Linear Linear Linear Linear

Linear

Two

Two Two

Single Two Single Single Two Single

Single Single Single Single Single Single Single

Stages

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algorithms, AIS and SFA. All of the mentioned references are related to FCTP but with different stages, which are segregated in Table 1 (one-stage and two-stage FCTP). The linear and non-linear approaches, which are used by researchers, are distinguished in Table 1 too. Recent publications prove the necessities of utilizing meta-heuristic algorithms. Xie and Jia [35], DuraiRaj and Rajendran [3], Lotfi and Tavakkoli-Moghaddam [36], and Vinay and Sridharan [37] applied and developed genetic algorithm. Besides, Vinay et al. [4] utilized ant colony to solve two-stage FCTP. Results of Table 1 reveal that the process of developing metaheuristics is continued by researchers. Analysing Table 1 precisely, we can find points that are more valuable: first, we can see the growing interest in meta-heuristic methodologies. This is because of their applicability in real-sized instances and their acceptable results. Indeed, exact solutions and general exact solvers are unable to solve NP-hard type of problems (such as two-stage FCTP) in a reasonable time so there should be some attempts to elevate and improve inexact methods such as meta-heuristic algorithms. Second, we can find the necessity of new methodologies that can elevate the current solutions, especially in two-stage transportation problem. The majority of papers deal with genetic algorithms and consider only single-stage problems. Yet different types of genetic algorithms perform well in achieving acceptable results. Definitely, we can elevate the acceptable performance of meta-heuristic algorithms and genetic algorithms by utilizing other well-behaved algorithms including AIS and SFA. In total, there are just three papers on two-stage transportation planning problems, which are based on genetic algorithms and ant colony optimizations. Finally, based on Table 1, we cannot find any developments of AIS or SF algorithms, which motivate us in this research. On the other hand, the gap in proposing new methodologies for two-stage transportation problem is revealed in the light of the above considerations in Table 1. Based on the above discussion, the necessities of new applicable methodologies that could solve the two-stage FCTP better than the current developed methods are clarified. Here, we try to cover this gap by developing two famous meta-heuristic algorithms: AIS and SFA. The performance evaluation study of the proposed algorithms is performed by comparing their results with the results of two newly presented genetic algorithm and ant colony method in Jawahar and Balaji [7], and Vinay et al. [4], respectively. Vinay et al. [4] tried to improve the achievements of Jawahar and Balaji [7] and we try to improve both. 3. Problem description As mentioned in the previous section, the model is inspired by Jawahar and Balaji [7]. The formulation is illustrated in this section. As discussed in this paper, the two-stage fixed-charge transportation model has been described for the two-echelon SC distribution, which has a linear objective function. Its main objective is to fix on the applied routes and the quantity of the shipments to be made on those routes to achieve as low a total cost of distribution as possible. The two-stage fixed-charge transportation problem is a kind of supply chain distribution problem. In the presented network, there are p plants, q distribution centres and r customers (retailers). Any of the plants can ship to any of the distribution centres at a transportation cost of Cij (per unit cost of shipment). Here we have a fixed charge too, so we should add a fixed charge of Fij which is specified for operating the route. Again,

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any of the distribution centres can ship to any of the customers at transportation cost of Cjk (per unit cost of shipment) plus a fixed charge of Fjk specified for operating the route. Besides, there are capacity constraints for plants and distribution centres. Each plant i ¼ 1; 2; 3; . . . p contains Si units of supply. Each distribution centre j ¼ 1; 2; 3; . . . q has Cj units of stock and each customer k ¼ 1; 2; 3; . . . r contains Dk units of demand. Figure 1, illustrates the regarded two-stage transportation problem. The mathematical formulation of the two-stage FCTP problem is as follows [7]: Minimize Z ¼

p X q X 

q X r   X   jk Xjk þ F jk Yjk Cij Xij þ Fij Yij þ C

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i¼1 j¼1

(1)

j¼1 k¼1

Subjected to: q X

Xjk ¼ Dk



8k; k ¼ 1, 2,3,. . .; r



(Demand constraint)

(2)

j¼1 q X

Xij  Si

ð8i; i ¼ 1; 2; 3;. . .; pÞ

(Supply capacity constraint)

(3)

j¼1

Xij  0 and Xjk  0 p X

Si 

i¼1 p X i¼1

Xij ¼

r X

Xjk

r X

(4)

Dk

(5)

(Flow conservation constraint)

(6)

k¼1

ð8j; j ¼ 1; 2; 3;. . .; qÞ

k¼1

 Yij ¼ Yjk ¼

Figure 1. Proposed two-stage FCTP.



1; if X ij [ 0 0; otherwise

(7)

1; if X jk [ 0 0; otherwise

(8)

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where Xij = Quantity of products transported from plant (i) to distributor (j). Cij = Costs of transportation between plant (i) and distributor (j). X jk = Quantity of products transported from distributor (j) to retailer (k).  jk = Costs of transportation between distributor (j) and retailer (k). C F ij = Fixed transportation costs between plant (i) and distributor (j).  jk = Fixed transportation costs between distributor (j) and retailer (k). F Si = Supply capacity at the plant (i). Cj = Capacity at the distributor (j). Dk = Demand at the customer (retailer) (k). In this paper, both the unbalanced and balanced capacity of the source are considered. This assumption means that supply is greater than or equal to the demanded quantities at the destination for both the single and two-stage FCTP. Even though FCTP is similar to the classical transportation problem, it is quite difficult to solve because of the discontinuity in the objective function due to the presence of fixed cost, which makes it unsolvable by the direct application of the transportation algorithm. 4. Solution methodology Over the past few decades, non-traditional optimization approaches were proposed by researchers for solving NP-hard problems in various fields(.[38–40] In this paper, two meta-heuristics (AIS and a sheep flock heredity algorithm) are proposed to solve twostage FCTP problems. The detailed procedure about both the algorithms is given in the next sub-sections. 4.1. Artificial immune system AISs are adaptive systems, which are inspired by the concept of theoretical immunology and observed immune functions, principles and models, which are typically applied as a problem-solving technique.[41] The AIS algorithm works on the two principles such as clonal selection and affinity maturation principle. The main aim of the clonal selection process is to identify the number of the best antibodies for undergoing the cloning process. The best antibodies are identified using affinity value or affinity function. In this work, the affinity function is defined by the following Equation (9). Affinity ðzÞ ¼

1 Total cos tðzÞ

(9)

Since the cloning is directly proportional to the affinity, the number of clones usually depends on the affinity values of the antibodies. For example in our case, the affinity function is the reciprocal of the objective function value, so the antibodies with lower affinity values are considered as the best antibodies. After the clonal selection, the next process is affinity maturation, which consists of two sub processes namely mutation and receptor editing. In mutation subprocess, the generated clones will undergo two types of mutation namely inverse mutation and pairwise mutation. This subprocess will ensure the betterment of the generated clone. Once the mutation subprocess is over, the next subprocess receptor editing is done by replacing the percentage of worst performing antibodies from the total population by the newly generated antibodies.[42]

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The procedure involved in the AIS is discussed below. For a more detailed description about AIS, refer Kannan [42], Prakash and Deskmuk [43], and [40]. The various steps involved in AIS are summarized as follows [14, 40, 42, 43]: (1) Generate the antibodies randomly equal to the population size. (2) Using the above Equation (9) determine the affinity value for all the antibodies. (3) Perform cloning process by selecting the antibodies whose affinity values are higher than the average affinity of all the antibodies. (4) After the cloning process, try to evaluate the cloned antibodies by finding their affinity values. If the cloned antibodies are better than the old antibodies, then replace the old low affinity antibodies with the newly cloned higher affinity antibodies. (5) Perform Inverse mutation and pairwise interchange mutation process. (6) Perform receptor editing by replacing the percentage of worst performing antibodies from the total population by the newly generated antibodies. (7) Check for the termination criterion, if it is not satisfied then repeat the above steps until the termination criterion is met. 4.2. Sheep flock heredity algorithm Sheep flock heredity algorithm (SFHA or SFA) is a new evolutionary computation method based on sheep flocks heredity. Like a genetic algorithm, SFHA also works with two kinds of operators: (1) sub-chromosome level genetic operator and (2) chromosome (global) level genetic operator. This hierarchical step is referred as multi-stage genetic operation [20]. The detailed steps involved in SFHA are summarized below [20, 24]. Step 1: Initialize the population Step 2: Stage 1: Select the parent Sub-chromosome level crossover Set sub-chromosome level crossover probability If population probability is less than or equal to Perform sub-chromosome level crossover Else retain the old sequences Sub-chromosome level mutation Set sub-chromosome mutation probability If population probability is less than or equal to Perform sub-chromosome level mutation Else retain the same sequences Step 3: Stage 2: Select two sequences from population Chromosome level crossover Set crossover probability If population probability is less than or equal to Perform chromosome level crossover Else retain the same sequences Chromosome level mutation Set mutation probability If population probability is less than or equal to

sub-chromosome level probability

sub-chromosome mutation probability

crossover probability

mutation probability

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Perform chromosome level mutation Else retain the same sequences End if the termination criterion is satisfied.

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5. Computational evaluations In this section, complete evaluation of the two proposed algorithms is discussed. At first, we evaluate them in comparison to current presented methodologies. Next, larger instances are considered and solved utilizing two developed algorithms. 5.1. Evaluation of four methodologies In order to evaluate the performance of the proposed algorithms in comparison to previous research, the AIS and SFA algorithms are coded using C++ programming language and the computational experiments are run on a 3 GHz Intel Pentium IV computer equipped with 4 GB of RAM. we use the same data and the same results of the proposed genetic algorithm of Jawahar and Balaji [7] and ant colony of Vinay et al. [4] to achieve fair evaluations. The AIS and SF parameters are selected based on some pilot experiments. The AIS parameters are as follows: 20 for population size, 0.3 for antibody elimination rate, 0.05 for mutation probability and 0.5 for elitist selection rate. The SF parameters are as follows: 10 for population size, 0.6 for sub-chromosome cross over probability, 0.03 for sub-chromosome mutation probability, 0.8 for crossover probability and 0.05 for mutation probability. The termination criterion is chosen as 500 iterations for both AIS and SF and the results are illustrated in Tables 2 and 3. Analysing the results of performances of four algorithms in Table 2, we can conclude the better achievements of the proposed algorithms in this paper. The results of proposed AIS and SFA are lower or equal than the other two previously developed algorithms in all of the 20 test problems. Indeed, based on the stability of their performances, they can completely dominate the GA of Jawahar and Balaji [7] and ACO of Vinay et al. [4]. It is mentioned that between these two proposed algorithms, SFA has also performed more favourably. SFA could dominate AIS in all 20 test problems. Regarding the average of the achievements in Table 3, both AIS and SFA could find lower average objective function values than the pre-presented GA and ACO. Considering the Table 3 analysis, we can conclude AIS performs 2.17% better than GA and 0.2% better than ACO. The basic reason behind the better performance of AIS is maturation process which eliminates the premature convergence which is one of the major limitation of GA. Meanwhile, SFA even presents lower values. It can dominate GA and ACO with improvements values of 2.41 and 0.44%, respectively. It can also improve the performance of AIS for about 0.24%. SFA achieve the faster rate of optimal solutions through chromosome level crossover and mutation process. Therefore, the acceptability of the performances of two proposed algorithms is proved. These comparisons can clearly highlight the capabilities of two proposed algorithms to cope with the NPhard problem of two-stage fixed-charge transportation problem. In terms of computational time, for the entire test problem, AIS took less than 90 s and SFA took less than 60 s to find the solution.

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Table 2. Total cost of transportations of proposed algorithms and previous algorithms. Proposed in this paper

Previously proposed

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Test problem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Problem size

GA of Jawahar and Balaji

ACO of Vinay et al.

AIS

SFA

2*2*3 2*2*4 2*2*5 2*2*6 2*2*7 2*3*3 2*3*4 2*3*6 2*3*8 2*4*8 2*5*6 3*2*4 3*2*5 3*3*4 3*3*5 3*3*6 3*3*7 3*3*7 3*4*6 4*3*5

112,600 237,750 180,450 165,650 162,490 59,500 32,150 69,670 264,680 85,200 94,565 47,140 178,950 57,100 152,800 132,890 106,615 302,350 83,500 118,450

112,600 237,750 180,450 165,650 162,490 59,500 32,150 69,045 258,730 80,900 80,865 47,140 178,950 57,100 152,800 132,890 105,715 281,730 77,250 118,450

112,600 237,750 180,450 165,650 162,490 59,500 32,150 67,380 258,730 77,400 80,865 47,140 178,950 57,100 152,800 132,890 105,715 281,730 77,250 118,450

112,600 237,750 180,450 165,650 162,490 59,500 32,150 67,380 258,730 77,400 80,865 47,140 175,350 57,100 152,800 132,890 104,115 281,100 76,900 118,450

Table 3. Summary of performance evaluations of algorithms. Criteria Average of 20 test problems Improvement percentage based on GA performances Improvement percentage based on ACO performances Improvement percentage based on AIS performances

GA

ACO

AIS

SFA

132,225 – – –

129,608 1.98% – –

129,349 2.17% 0.20% –

129,040 2.41% 0.44% 0.24%

5.2. Considering larger instances In the previous section, the performances of the proposed AIS and SFA are successfully evaluated in comparison to two previously elite developed algorithms for FCTP problem. They could both dominate previously proposed GA and ACO in the quality of solutions. In this section, 10 larger instances are considered in order to evaluate these proposed algorithms in real and larger size instances in terms of solutions and time. We can rarely see such kind of instances in previous research. The results are illustrated in Table 4 and Figure 2. It should be pointed out that the results illustrated in Table 4 are derived from the average of 10 runs per test problems for each methodology. Analysing the above illustration in Table 4 and Figure 2 confirms the previous rationales of comparing four algorithms through 20 test problems. We can derive from

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Table 4. Performance evaluations in large instances. Total cost of transportation

Test problem

Problem size

Objective valuea

Objective boundb

1 2 3 4 5 6 7 8 9 10 Average

5*6*7 5*6*8 6*7*9 6*7*10 7*8*11 7*8*12 8*9*13 8*9*14 9*10*15 9*10*16

138,679 141,270 160,283 177,084 190,172 211,132 223,154 241,278 256,874 274,084 201,401

138,679 141,270 158,046 172,664 186,246 205,245 226,548 231,269 248,645 269,235 197784.7

AIS

SFA

Diff: (AIS-SFA)/ AIS (%)

142,568 139,689 144,820 141,470 164,351 161,053 181,056 178,046 194,215 190,872 214,085 210,723 233,415 229,084 247,631 243,005 262,998 258,584 283,856 278,958 206899.5 203148.4

2.02 2.31 2.01 1.66 1.72 1.57 1.86 1.87 1.68 1.73 1.84

a

Upper bound solution (best known) obtained after interrupting the LINGO solver after 30 min. Lower bound solution obtained after interrupting the LINGO solver after 30 min.

b

282000 Objective Function Values

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Lingo

252000 222000 AIS 192000

SFA

162000 132000

1

2

3

4

5

6

7

8

9

10

Test Problem

Figure 2. Comparing solutions results of AIS and SFA.

the mentioned illustrations that in terms of time, both of the proposed algorithms could achieve their optimal points in just less than one minute in all instances. Such performances are completely well behaved and acceptable. On the other hand, in terms of quality of solutions, SFA could again prove its priority on AIS. The results show the differences of 1.84% on average between SFA and AIS. SFA also performed better in previous analysis of 20 test problems. Clearly, we can now conclude that SFA is the best meta-heuristic algorithm in terms of quality of solutions between these four discussed and proposed algorithms in mentioned two-stage fixed-charge transportation problem. On the other hand, in order to evaluate the performance of a general exact solver, LINGO is utilized and the lower and upper bound solutions (best known), obtained after interrupting the LINGO solver after 30 min, are recorded and presented in Table 4. As discussed, general exact solvers are unable to achieve the optimal solution of such

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NP-hard problems in a reasonable time, which is clearly proved by the results of Table 4. Results reveal that the proposed algorithms perform better than LINGO software. On average, AIS and SFA present 0.9 and 2.7% better results, respectively, in less than one minute, which can prove their acceptability in comparison to LINGO results.

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6. Conclusion and future research In this paper, we cope with a two-stage distributions planning of supply chain with a fixed charge. Two new methodologies are developed and evaluated for the selected NPhard problem; an AIS and a SFA. These algorithms are selected based on the literature and their capabilities regarding to FCTP. The proposed algorithms are developed and coded to solve two-stage FCTP. In order to evaluate proposed meta-heuristic algorithms, two previous studies are considered: Jawahar and Balaji [7] in utilizing a genetic algorithm and Vinay et al. [4] in developing an ant colony optimization. The same data analyses of mentioned papers are regarded here. Results prove that two proposed algorithms dominate the previous GA and ACO completely. The results show 2.17 and 2.41% improvement compared to pre-presented GA for the proposed AIS and SFA, respectively. Besides, 0.20 and 0.44% improvement are presented by AIS and SFA, respectively, in comparison to previous ACO presented in literature based on the same data. Generally, SFA perform better than AIS in all computational studies. On the other hand, all instances are run by LINGO software and then the lower and upper bound solutions, obtained after interrupting the LINGO solver after 30 min, are compared to proposed algorithms. Results reveal that the proposed algorithms perform better than LINGO software. On average, AIS and SFA present 0.9 and 2.7% better results, respectively, in less than one minute, which can prove their acceptability in comparison to LINGO results. In the second phase of computational analysis, some larger test problems are considered. Again, as we expected, SFA could prove its priority on AIS in the quality of solutions. Both of the proposed algorithms could achieve their optimal points just in less than one minute in all instances. The results show the differences of 1.84% on average between SFA and AIS. SFA also performed better in previous analysis of 20 test problems. Clearly, in the mentioned two-stage fixed-charge transportation problem, it can be concluded that between these four algorithms, SFA is the best one in the quality of solutions. There are some guides which can suggest future research directions. The next step could be researching the three-stage fixed-charge transportation problem. For instance, one can consider warehouses as another important entity. Different kinds of solution methodologies and model developments could be developed here. In the current twostage FCTP, the main categories of extensions can be regarded as stochastic or fuzzy parameters. This is a necessary field for development. Notes on contributors Devika Kannan is currently affiliated with Aalborg University, Copenhagen, Denmark. Her research interests include supply chain management, green supply chain management and sustainable supply chain management. She has published more than 10 papers in refereed international journals. Her publications have appeared in leading journals including EJOR, IJPE, IJPR, ANOR and JCP.

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Kannan Govindan is an associate professor of Operations and Supply Chain Management in the Department of Business and Economics, University of Southern Denmark, Odense M, Denmark. His research interests include logistics, supply chain management, green and sustainable supply chain management, reverse logistics and maritime logistics. He has published more than 80 papers in refereed international journals and more than 70 papers in conferences. He has published in leading journals including EJOR, Omega, C&OR, ANOR, IJPE, IJPR, JORS, TRE and JCP. He is an editor-in-chief of the International Journal of Advanced Operations Management and International Journal of Business Performance and Supply Chain Modelling. He is an editorial board member of the Resources Conservation and Recycling and Transportation Research Part E. Hamed Soleimani is an assistant professor of Industrial Engineering at the Islamic Azad University (IAU), Qazvin Branch, Iran. He has published papers in leading journals such as ESWA, EJOR, IJPR, IJAMT and ANOR. He is also a reviewer for COR, EJOR, IJOR and other journals. His fields of research are supply chain management, reverse logistics, stochastic optimization and metaheuristic algorithms.

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