a hybrid particle swarm algorithm with artificial immune learning...

Computers & Industrial Engineering 64 (2013) 610–620

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Computers & Industrial Engineering

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A hybrid particle swarm algorithm with artificial immune learning forsolving the fixed charge transportation problem q

0360-8352/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.cie.2012.12.001

q This manuscript was processed by Area Editor Iris F.A. Vis.⇑ Corresponding author. Current address: Operations Research Dept., Institute of

Statistical Studies and Research (ISSR), Cairo University, Egypt. Tel.: +201284665997; fax: +20 237482533.

E-mail addresses: [email protected], [email protected] (M.M. El-Sherbiny), [email protected] (R.M. Alhamali).

Mahmoud M. El-Sherbiny a,⇑, Rashid M. Alhamali b

a Operations Research Dept., Institute of Statistical Studies and Research (ISSR), Cairo University, Egyptb Dept. of Quantitative Analysis, College of Business Administration (CBA), King Saud University, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 August 2011Received in revised form 14 May 2012Accepted 2 December 2012Available online 20 December 2012

Keywords:Fixed charge transportationConvergenceParticle swarmGenetic algorithmArtificial immune system

Fixed Charge Transportation Problem (FCTP) is an NP-hard problem with many applications in both tra-ditional and modern industrial situations. This paper introduces a Hybrid Particle Swarm algorithm withartificial Immune Learning (HPSIL) for solving fixed FCTPs. In HPSIL algorithm a flexible particle (chromo-some) structure, decoding procedure and allocation procedure are used instead of a Prüfer number and aspanning tree that used with genetic algorithms. The proposed allocation procedure guarantees finding afeasible solution for each generated particle. The HPSIL algorithm can be used for solving both balancedand unbalanced FCTPs without introducing dummy supplier or dummy demand. With regard to solutionquality, the HPSIL algorithm can be considered as a viable alternative for solving FCTPs in addition to therecent algorithms.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The Fixed Charge Transportation Problem (FCTP) is consideredto be an NP-hard problem (Murty, 1968). Usually, a FCTP is formu-lated and solved as a mixed integer network programming prob-lem. Theoretically, the FCTP can be solved by any mixed integerprogramming; however, these methods are not employed becauseof their inefficient and expensive computations. Generally, themethods of solving FCTP are classified as either exact or heuristic.Exact methods for solving FCTP include the cutting plane method(Rousseau, 1973), the vertex ranking method (McKeown, 1975),and the branch-and-bound method (Palekar, Karwan, & Zionts,1990), amongst others. However, exact methods are not very use-ful when a problem reaches a certain level, because they do notmake the most use of the special network structure of the FCTP.Therefore, heuristic methods have been proposed, such as the adja-cent extreme point search method (Balinski, 1961; Sun et al.,1993), and the Lagrangian relaxation method (Wright et al.,1989, 1991). Although these methods are usually computationallyefficient, the major disadvantage of heuristic methods is the possi-

bility of terminating at a local optimum that is far distant from theglobal optimum. Recently, some meta-heuristic methods havebeen employed in solving FCTP, such as the Tabu search methodfor solving FCTPs (Sun, Aronson, & Mckeown, 1998), the hybrid Ge-netic Algorithm (GA) based on a spanning tree with Prüfer num-bers for solving bicriteria transportation problem (Gen, Ida, & Li,1998), and GAs based on a matrix permutation representation(Gottlieb, Julstrom, Rothlauf, & Raidl, 2001; Raidl & Julstrom,2003) which have improved the effective coding of the spanningtree method based on edge sets. The GA creates a sorted set ofedges to encode the spanning tree, which is more efficient com-pared to evolution strategies (ES) at a certain level (Su & Zhan,2006). Moreover, to improve solution quality a lot of evolutionaryalgorithms use a random procedure to generate a Prüfer numberwith sum of problem dimensions (the number of suppliers andcustomers) less two digits in range of unity and the sum of prob-lem dimensions. The generated Prüfer number, may not be trans-lated to a feasible spanning tree to represent the solution of theFCTP. In order to overcome this problem, Gen & Cheng (2000)developed a criterion for checking the feasibility of the Prüfer num-ber. Jo, Li, and Gen (2007) discovered that this technique fails togenerate the feasible solution when the difference between thenumber of suppliers and the number of customers is very largeand developed another feasibility criterion to check the feasibilityof the Prüfer number and then used a repairing procedure formaking it feasible. Jo et al. (2007) applied the spanning tree-basedgenetic algorithm which developed based on the solution structure

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for the linear transportation problem to solve the nonlinear FCTP.Comments based on the calculations of this work were presentedin Kannan el al. (2008). Actually, Kannan corrected (recalculated)the costs of the two examples presented in Jo et al. (2007). Sincethe repairing procedure may take long time to repair, Hajiaghaei,Keshteli, Molla-Alizadeh-Zavardehi, and Tavakkoli-Moghaddam(2010) addressed a nonlinear FCTP using a spanning tree based ge-netic algorithm and proposed a method to generate Prüfer numberat random which does not need a repairing procedure. Othman,Delavar, Behnam, and Lessanibahri (2011) used the fuzzy logic con-trollers to adopt the crossover parameters of the GA for solving FCTPand reached the local optimum remarkably faster. Molla-Alizadeh-Zavardehi et al. (2011) proposed an artificial immune algorithm(AIA) and a GA based on the spanning tree and Prüfer numberrepresentation for solving capacitated fixed-charge transportationproblem in a two-stage supply chain network and selecting the suit-able values of the algorithms’ parameters that gives the best perfor-mances of such algorithms. In addition, they investigated the impactof increasing the problem size on the performance. Based on thiswork comments on the mathematical model, the transportationgraph and the total cost for the example are presented by El-Sherb-iny (2012). Nevertheless, the quality of solutions attained largely de-pends on the randomness. On the other hand, the Particle SwarmOptimization (PSO) technique developed by Eberhart and Kennedy(1995) is a simple evolutionary algorithm that differs from otherevolutionary computation techniques in that it is motivated by thesimulation of social behavior. PSO has features of both GAs and ESs(Tandon, El-Mounayri, & Kishawy, 2002) and exhibits good perfor-mance in finding solutions of static optimization problems (Parso-poulos and Vrahatis, 2001). Several modifications to the originalPSO were developed to improve the solution quality, such as a com-bined PSO based on the previous global best and the current globalbest positions (El-Sherbiny, 2007), a modified algorithm for PSOwith a constriction coefficient (El-Sherbiny, 2009), and a particleswarm inspired optimization algorithm without the velocity equa-tion (El-Sherbiny, 2011).

To improve the quality of the solution, this paper introduces aHybrid Particle Swarm algorithm with artificial Immune Learning(HPSIL) for solving the FCTPs and studies the effect of various fac-tors on the performance thereof. In addition, the results of experi-mentally comparing its performance with that of the geneticalgorithms proposed in (Hajiaghaei et al., 2010; Othman et al.,2011) as well as the solution found by LINGO are presented.

The rest of the paper is organized as follows. In Section 2, theFCTP model is described, while in Section 3 some basic conceptsof PSO and Artificial Immune System (AIS) are presented. The pro-posed HPSIL algorithm is introduced in Section 4. The HPSIL perfor-mance study based on the experimental design and analysis arepresented in Section 5. In Section 6, numerical experiments withthe HPSIL algorithm are discussed. Finally, the conclusion and fu-ture work are reported in Section 7.

2. FCTP model

FCTP can be described as a distribution problem, with m suppli-ers (warehouses, factories or plants) and n customers (destinationsor demand points). Each of the m suppliers can ship to any of the ncustomers at a unit shipping cost cij (unit cost to ship from supplieri to customer j) plus a fixed cost fij, assumed for opening the route.Each supplier i = 1, 2, . . . , m has si units of supply and each cus-tomer j = 1, 2, . . . , n demands dj units. The objective is to determinewhich routes should be opened and the size of the shipment, sothat the total cost of satisfying the demand, given the supply con-straints, is minimized. The standard mathematical model of theFCTP can be expressed as follows:

Min z ¼Xm

i¼1

Xn

j¼1

ðcijxij þ fijyijÞ

s:t:Xm

i¼1

xij P dj for j ¼ 1; . . . ;n

ð1Þ

Xn

j¼1

xij P si for i ¼ 1; . . . ;m

xij P 0; 8i; j

yij ¼ 0 if xij ¼ 0

yij ¼ 1 if xij > 0

ð2Þ

where xij is the unknown quantity to be transported through theroute (i, j), that is, from supplier i to customer j.

3. Basic concepts of PSO and AIS

PSO has some advantages over other similar optimization tech-niques. It is not largely affected by the problem size, can convergeto the optimal solution in many problems where most analyticalmethods fail to converge (Valle, Venayagamoorthy, Mohagheghi,Hernandez, & Harley, 2008). PSO is more efficient in maintainingthe diversity of the swarm (Engelbrecht, 2006), since all the parti-cles use the information related to the most successful particle inorder to improve themselves and comparatively faster than someother optimization algorithm. Therefore, it can be effectively ap-plied to different optimization problems (Deb, Basanta, & Sarkar,2011).

On the other hand, AIS is self-organizing, learning capability,and not many system parameters are required, hence AIS offerspowerful and robust information processing capabilities for solvingcomplex optimization problems. The proposed algorithm tries touse the advantages of both PSO and AIS for solving FCTP. The fol-lowing subsections present the basic concepts related to bothPSO and AIS.

3.1. Particle swarm optimization

PSO is a population-based stochastic optimization technique,developed by Eberhart and Kennedy (1995). PSO simulates the so-cial behavior of organisms, such as birds in a flock or fish in aschool. This behavior can be described as an automatically andinteractively updating system. In PSO, each single candidate solu-tion can be considered to be a particle in the search space. Eachparticle makes use of its own memory as well as knowledge gainedby the swarm as a whole to find the best solution. All the particleshave fitness values which are evaluated by the objective functionof the FCTP. During movement, each particle adjusts its positionby changing its velocity according to its own experience and thatof a neighboring particle, thus making use of the best positionencountered by itself and its neighbor. Particles move throughthe problem space by following a current of optimum particles.The process is then iterated a fixed number of times or until a pre-determined minimum error is achieved (Kennedy, 2003).

Suppose the search space is L-dimensional and K particles formthe colony. The kth particle represents a L-dimensional vector ~pk

where k = 1, 2, . . . , K. This means that the kth particle is locatedat ~pk ¼ ðpk1; pk2; . . . ; pkLÞ 2 S of the search space (solution space).The position of each particle is a potential result (potential solu-tion) of the problem under study. We could calculate the particle’sfitness by putting its position into a designated objective function.The kth particle’s ‘‘flying’’ velocity at the tth iteration is denoted as~vkðtÞ, the local best position of the kth particle is denoted as ~hkðtÞand the global best position of the swarm is denoted as ~gðtÞ. Each

Fig. 1. Example of the proposed chromosome structure.

Fig. 2. Main architecture of the HPSIL.

612 M.M. El-Sherbiny, R.M. Alhamali / Computers & Industrial Engineering 64 (2013) 610–620

particle updates its position using (3) and (4) where b1 and b2 arepositive constants, called the acceleration constants, and r1, r2 e [0,1] are uniform random numbers.

~vkðt þ 1Þ ¼ ~vkðtÞ þ b1r1ð~hkðtÞ �~pkðtÞÞ þ b2r2ð~gðtÞ �~pkðtÞÞ ð3Þ~pkðt þ 1Þ ¼~pkðtÞ þ~vkðt þ 1Þ ð4Þ

The termination criterion for the iterations is determined bywhether the max generation or a designated value of the particlefitness is reached.

Compared to ~vkðtÞ and ~vkðt þ 1Þ is the summation of two com-ponents b1r1ð~hkðtÞ �~pkðtÞÞ and b2r2ð~gðtÞ �~pkðtÞÞ as shown in (3).The main aim is to guide each particle to fly towards its best posi-tion, with the latter component emphasizing the swarm learningwhereby a particle always learns from the best of the swarm andhas a tendency to fly toward the best. As a result, it has the poten-tial to compete for the best new particle.

3.2. Artificial immune system

A human immune system is made up of numerous B and T cells,which are constantly being produced in the bone marrow and thy-mus, respectively. The level of B cell simulation depends not onlyon the success of the match to the antigen, but also on how wellit matches with other B cells in the immune system. If the stimu-lation of B cells reaches a certain threshold, the B cell is trans-formed into a blast that begins to differentiate rapidly, producingclones that turn on a mutation mechanism that generates muta-tions in the gene coding for the antibody molecule, which is calledsomatic hypermutation. However, if the stimulation level falls be-low the threshold, the B cell does not replicate and ultimately dies.

Various special AISs have been developed to solve complexoptimization problems. One of these is aiNet (De Castro & Timmis,2002, 2002b), which was inspired by biological immune systems.Opt-aiNet (Timmis, Knight, Catro, & Hart, 2004), an application ofaiNet for function optimization, considers the optimized objectivefunction as an antigen and the candidate solutions as antibodies.The candidate antibodies evolve according to the matching degreeof fitness between the antibodies and antigen. The better thematch between them, the smaller is the mutation degree of thecandidate antibody, and vice versa.

The opt-aiNet algorithm consists of seven components: initiali-zation, affinity value, clone, mutation, selection, cell interaction,and recruitment. The mutation operation of candidate antibodiesis executed by the following equation:

newAB ¼ oldABþ r1b

e�affinity ð5Þ

where oldAB is the previous candidate antibody before mutation,newAB is the new antibody obtained after mutation, fitness is theobjective function value related to oldAB, b is a control parameterfor mutation, and r [0, 1] is a uniform random number. Affinity ismeasured based on the Euclidean distance between two antibodiesor between an antibody and an antigen (De Castro and Von Zuben,2001). The affinity measure results from an adapted distance func-tion, such that the affinity is maximum when the distance is mini-mum. It is clear that the larger the fitness value of the antibody, thesmaller is the level of mutation.

4. HPSIL algorithm

While most of the previous works are based on using the Prüfernumbers with a spanning tree, the proposed algorithm uses a newapproach considering the basic properties of the particle swarmoptimization and the classical methods for solving transportationproblems. In the HPSIL, a flexible chromosome structure with

length (m + n) is used (see Fig. 1), together with a decoding andan allocation procedures as illustrated in Figs. 3 and 4, respectively.Usage of such a chromosome structure allows more than one geneto have the same value. Thus, particle swarm optimization withinteger solutions can be used to solve the FCTP. In this section, ahybrid particle swarm algorithm with artificial immune learningis proposed to solve the FCTP. The new coding scheme and the cod-ing procedure are explained in Section 4.1. The allocation proce-dure to find a corresponding feasible solution for each generatedparticle is described in Section 4.2. In Section 4.3, six versions ofthe mutation equations for candidate antibodies used in hybrid im-mune networks with swarm learning to generate particles are pre-sented. The general steps of the HPSIL algorithm, as illustrated inFig. 2, are explained below:

Step 1: Set t = 0 and Generate an initial population of ~pkðtÞ par-ticles. Each particle in ~pkðtÞ has an order structure of (m + n)dimensional vector of integers. This order structure is used torepresent the solution space (particles) as represented in Fig. 1.Step 2: Apply the decoding procedure, as explained in Sec-tion 4.1, to find the suppliers sequence S and the customerssequence D based on population ~pkðtÞ resulting from step 1.Step 3: Apply the allocation procedure, as explained in Sec-tion 4.2, to find the corresponding transportation allocationbased on the suppliers sequence S and the costumers sequence

Fig. 3. An illustrative example of applying the decoding procedure.


D resulted from Step 2. The results of this procedure are thetransported quantities from each supplier S to each customer D.Step 4: Calculate the value of the objective function (fitnessvalue) of each particle in population ~pkðtÞ.Step 5: If the termination criterion is met, then the best solutionis reached for the FCTP. Otherwise, continue.Step 6: Update the population positions ~pkðt þ 1Þ by exe-cuting one of the selected mutation equations and go toStep 2.

4.1. The main schema

One of the most important issues when designing a PSO algo-rithm is concerning its solution (particle) representation. To con-struct a direct relationship between the problem domain and theHPSIL, the proposed coding scheme consists of a set of integernumbers in the interval [1, q], with the length of the schemeequal to m + n, where q = Max(m, n), m is the number ofsuppliers, and n is the number of customers. Therefore, the

length of each particle in the ~pkðtÞ is equal to the sum ofthe problem dimensions. i.e. the particle k is doneted as~pk ¼ ðpk1; pk2; . . . ; pkLÞ where L = m + n and the value of each gen-ome pkl [1, q], l = 1, 2, . . . , L. Fig. 1 depicts a sample particlewhich is used to code a 4 � 5 FCTP or any FCTP with m + n = 9.As shown in Fig. 1, the value of each genome pkl is between 1and q = 5. In fact, each particle shows a group of numbers be-tween 1 and q and each genome pkl points to the index of an ar-ray converted to a supplier or customer number. Moreover, itcan be noted that any number can be repeated and this repeti-tion makes the particle structure more flexible in applying anymutation equation. Thus based on the flexibility of the particlestructure, combined with the decoding procedure, any meta-heu-ristic techniques such as PSO, GA, Tabu search, and others can beapplied.

In this procedure, ~pk is an input particle (chromosome) thatmust be decoded to show two lists S and D, where the list Sand D represent the suppliers sequence and the customers se-quence respectively. As an example, Fig. 3 illustrates the results

Fig. 4. An illustrative example of applying the allocation procedure.


of applying the decoding procedure to the particle presented inFig. 1. The steps for the decoding procedure are presentedbelow:

Input: the particle ~pk .Output: the list S and the list D.Step 1: Create a collection list QS that includes the number ofsuppliers {1, 2, . . . , m}, and a collection list QD that includesthe number of customers {1, 2, . . . , n}.Step 2: Set i = 1.Step 3: Set j = Mod(pki, |QS|) + 1.Step 4: Add the item QS(j) to the list S.Step 5: Remove the item j from the collection list QS.Step 6: Increment i with 1.Step 7: Repeat the steps from 3 to 6 until i = n + 1.Step 8: Set j = Mod(pki, |QD|) + 1.Step 9: Add the item QD(j) to the list D.Step 10: Remove the item j from the collection list QD.Step 11: Increment i with 1.Step 12: Repeat the steps from 8 to 12 until i = m + n + 1.

where |QS| and |QD| are equal to the length of QS and QDrespectively.

4.2. Allocation procedure

This procedure allocates the transported units based on the listS and the list D resulted from the decoding procedure. In otherwords, this procedure finds a feasible solution for FCTP based onthe outputs of the decoding algorithm. The inputs of the allocationprocedure are the two lists S and D (the output of the decoding pro-cedure). Based on the two lists (S and D) the allocation procedureallocates Xij (feasible solution) units for FCTP. The steps for the allo-cation procedure are presented below:

Inputs: the lists S and D.Output: the feasible solution Xij.Step 1: Set k equal to 1.Step 2: Set i = S(1) and j = D(1).Step 3: If si = dj then {set xij = si, remove S(1), and remove D(1)}

If si > dj then {set xij = dj, set si = si � dj, and remove D(1)}.If si < dj then {set xij = si, set dj = dj � si, and remove S(1)}.

Step 4: Set Sol(k, 1) equal to i, Sol(k, 2) = j, and Sol(k,3)= xij whereSol is the solution array with (n + m � 1, 3) dimensions.Step 5: Update k = k + 1.Step 6: Repeat from step 2 to step 5 until jSj = 0 or jDj = 0.Step 7: Return the Sol array.


Fig. 4 represents an illustrative example of applying this algo-rithm. This procedure guarantees the validity of both constraints(1) and (2) in the mathematical model and guarantees the feasibil-ity of all the generated solutions. Step 6 of the procedures ensuresthat either the procedure continues until the length of supplier’sorder jSjor the length of customer’s order jDjbecomes zero, i.e.,the procedure terminates when either of the supplier or demandquantities are shipped and the surplus/slack quantity is automati-cally deemed to have been allocated to a dummy supplier/cus-tomer. Hence, this procedure applies to both balanced andunbalanced transportation problems without introducing a dum-my supplier or a dummy customer.

4.3. Mutation equations

While the mutation operation for candidate antibodies used inhybrid immune networks with swarm learning (Fu, Li, & Tan,2007) are applied as the mutation equations with PSO as in (5) in-stead of (3) and (4), the proposed mutation operation of the HPSILis defined by the following equation:

~pkðt þ 1Þ ¼~pkðtÞ þ c1r11b

e�Affinity þ c2r2ð~gðtÞ �~pkðtÞÞ ð6Þ

where~pkðt þ 1Þ and~pkðtÞ are the new and previous particles, respec-tively, ~gðtÞ is the position of the particle with the best global valuefunction, b is a control parameter for mutation, r1 and r2 [0, 1] areuniform random numbers, and c1 and c2 are the acceleration coeffi-cients. Affinity is measured based on the Euclidean distance be-tween two antibodies or between an antibody and an antigen (DeCastro and Von Zuben, 2001). Using this Euclidean distance as ameasure of affinity consumes more time than affinity functionbased on fitness value. One of the formulations based on makespanvalues of the schedules was presented by Costa, Vargas, Von Zuben,and França (2002) and also used by Orhan and Alper (2004). In theFCTP, particle~pk has a value function that refers to the affinity valueof that particle. Affinity value of each particle is calculated from thevalue function. Hence the affinity function for the HPSIL is definedas in (7) where the fitness of the particle ~pk is equal to the valueof its objective function.

Affinity ð~pkÞ ¼1

Fitnessð~pkÞð7Þ

The aim of (6) is to guide the particle to fly towards its best position,and the affinity value emphasizes swarm learning whereby the par-ticle always learns from the best of the swarm and has a tendencyto fly toward the best. As a result it has the potential to compete forthe best new particle. Carrying out the mutation guides the particleto fly toward its best position depending on several factors, includ-ing the affinity value, the position of the previous particle~hkðtÞ; andthe position of the global best particle~gðtÞ. Therefore, several candi-date modifications can be carried out on the structure of the muta-tion Eq. (6). The proposed modifications are explained in thefollowing paragraphs.

The first modification: The first term of the equation, which is theposition of the previous particle~pkðtÞ; is replaced with the best lo-cal particle position ~hkðtÞ.

The second modification: Since many affinity functions have beenused in the literature, three different modification to affinity func-tion (7) are proposed. The proposed affinity functions are asfollows.

The first affinity function is the function value (fitness) of the bestlocal particle position, and denoted as vð~hkðtÞÞ. This affinity func-tion is represented in the following equation:

Affinity1 ¼1

vð~hkðtÞÞð8Þ

The second affinity function is the difference between the value of thebest local particle vð~hkðtÞÞ and the best global particle vð~gðtÞÞ di-vided by the value of the best local particle vð~hkðtÞÞ as given inthe following equation:

Affinity2 ¼vð~hkðtÞÞ � vð~gðtÞÞ

vð~hkðtÞÞð9Þ

The third affinity function is the difference between the value of thebest local particle vð~hkðtÞÞ and the best global particle vð~gðtÞÞ di-vided by the value of the best global particle vð~gðtÞÞ as given in(10). Since the FCTP is a minimization problem, it can be observedthat vð~hkðtÞÞP vð~gðtÞÞ in all cases.

Affinity3 ¼vð~hkðtÞÞ � vð~gðtÞÞ

vð~gðtÞÞ ð10Þ

As a result of the three affinity functions and the two particle posi-tions (the current or previous), six versions of mutation Eq. (6) areconstructed. While in the original PSO algorithm the structure ofthe particle consists of real numbers, the FCTP requires a particlestructure with integer values to represent the number of supplierand number of demand. Hence, the PSO is modified to generatean integer value in the search space. Simply stated, in the HPSILalgorithm real numbers generated through PSO within the definedsearch space are truncated into integers using the Int [�] function,where Int [�] represents the truncation function. Hence, the recom-mended mutation equations used by the HPSIL algorithm to gener-ate a new population is one of (11)–(16).

~pkðt þ 1Þ ¼ Int½~pkðtÞ þ c1r11b

e� 1

vð~hk ðtÞÞ

� �þ c2r2ð~gðtÞ �~pkðtÞÞ� ð11Þ

~pkðt þ 1Þ ¼ Int½~pkðtÞ þ c1r11b

e� vð�hk ðtÞÞ�vð~gðtÞÞ

vð~hk ðtÞÞ

� �þ c2r2ð~gðtÞ �~pkðtÞÞ� ð12Þ

~pkðt þ 1Þ ¼ Int ~pkðtÞ þ c1r11b


vð~gðtÞÞ

� �þ c2r2ð~gðtÞ �~pkðtÞÞ

" #ð13Þ

~pkðt þ 1Þ ¼ Int ~hkðtÞ þ c1r11b

e� 1

vð~hk ðtÞÞ


" #ð14Þ



vð~hk ðtÞÞ


" #ð15Þ


evð�hk ðtÞÞ�vð~gðtÞÞ

vð~gðtÞÞ


" #ð16Þ

5. Performance study

The aim of this section is to define the factors that affect theperformance of the HPSIL algorithm and selecting the level of eachfactor which provide its best performance. In order to achieve thisaim, two measures considered are detailed. In addition the charac-teristics of the problems used for analyzing the performance arepresented. The analysis of experimental design presents the com-parison of results based on the main factor effects and interactioneffects for finding the best factor levels.

5.1. Experimental design

It can be observed that a number of factors affect the perfor-mance of the HPSIL algorithm. The factors include the particle po-sition (PP) used in the first part of swarm equation (best localparticle position or previous particle position), the population size(PSize), affinity functions (AFs), and the acceleration coefficients

Table 1Factors and their levels.

Factors Symbols Levels

Particle position PP Local best particlePrevious particle

Affinity function AF 1vð~hkðtÞÞvð~hkðtÞÞ�vð~gðtÞÞ

vð~hkðtÞÞ

vð~hkðtÞÞ�vð~gðtÞÞvð~gðtÞÞ

Population size PSize 406080

Acceleration coefficients (C1 and C2) 0.60, 0.400.75, 0.400.73, 0.370.70, 0.30

Table 2Characteristics of the FCT test problems.

Problemsize

Totalsupply

Rang of variable costs Rang of fixed costs

Lowerlimit

Upperlimit

Lowerlimit

Upperlimit

10 � 10 10,000 3 8 50 20010 � 20 15,000 3 8 100 40015 � 15 15,000 3 8 200 80010 � 30 15,000 3 8 400 160050 � 50 50,000 3 8 400 1600

Table 3Mean RPD responses.

Level PP AF PSize C1 and C2

1 0.0312 0.0267 0.0361 0.02882 0.0268 0.0310 0.0294 0.02863 0.0293 0.0215 0.02784 0.0309Delta 0.0044 0.0042 0.0146 0.0031Rank 2 3 1 4

Fig. 5. Main effects plot (fitted mean) for RPD at each level of the factors.


(C1 and C2). These factors and their levels are listed in Table 1. Thelevels considered for the PSize and (C1 and C2) are based on theemperical study using various combinations. It is observed thatthe values considered in Table 1 provides a good performance. Inorder to analyze the performance and solution quality of the HPSILalgorithm using the mutation Eqs. (11)–(16), the effects of suchmodifications have been analyzed.

In robust parameter design, the primary goal is to find factorsettings that minimize response variation, while adjusting (orkeeping) the process on target. After determining which factors af-fect variation, the settings for controllable factors are found thatwill either reduce the variation, make the HPSIL algorithm insensi-tive to changes in uncontrollable (noise) factors, or both. An algo-rithm designed with this goal will produce output that is moreconsistent. To achieve this goal, two measures are used in theexperiment. The first is the S/N ratio used by (Park, 1996), whichindicates the amount of variation present in the response variable,with the aim of maximizing the signal-to-noise ratio. This S/N ratiois calculated as follows:

S=N ratio ¼ �10log10ðobjective functionÞ2

The second measure considered is the Relative Percentage Devia-tion (RPD), which is used by Hajiaghaei et al. (2010), because thescales of the objective functions in each instance are different andcannot be compared directly. The RPD is calculated for each in-stance for making the comparisons on the same scale. RPD is calcu-lated as follows:

RPD ¼ Algsol �Minsol

Minsol� 100

where Algsol and Minsol are the obtained objective values for eachreplication of the trial in a given instance and the obtained bestsolution, respectively. After converting the objective values to RPDs,the mean RPD is calculated for each trial.

All the problem combinations in the experiment; that is,2 � 3 � 3 � 4 = 72 instances for each problem have been consid-ered. The reason for this is not to miss any instance that may affectthe analysis which may lead to an inappropriate conclusion. More-over, each problem is executed three times. Hence, a total of 216executions have been performed for each problem. In view of theexcessive time involved for each execution, five different problemslisted in Table 2 in addition to the two examples listed in the sec-tion dealing with numerical experiments are solved to evaluate theperformance of the HPSIL algorithm for solving the FCTPs.

5.2. Analysis of experimental design

The algorithm is coded in Visual Basic and executed on a PCwith a 1.83 GHz Intel Core 2 Duo processor and 4 GB of RAM.The factorial design and analysis are carried out using MINITAB Re-lease 14.1 statistical software. For the sake of fairness, the samenumber of iterations are set for all trials or combinations of oper-ator and parameter levels, that is, 1000 generations.

The mean RPD values of the trials are averaged at each level, aspresented in Table 3. These values are plotted in Fig. 5. The re-sponse tables show the average of each response characteristic(mean RPD) for each level of each factor. The table includes ranksbased on Delta statistics, which compare the relative magnitudeof effects. The ranks based on Delta values; rank 1 to the highestDelta value, rank 2 to the second highest, and so on are presentedin Table 3. The ranks indicate the relative importance of each factorto the response. It can be observed that the relative impact rank-ings on the performance of the proposed algorithm are the popula-tion size (PSize), the particle position (PP), the affinity function(AF), and the acceleration coefficients (C1 and C2) respectively. Thismeans the population size (PSize), followed by the particle position(PP) have the highest impact on the performance of the proposedalgorithm. Similarly the S/N ratios are averaged at each level andare presented in Table 4. Further, the mean S/N ratio results for

Table 4Mean S/N ratio responses.

Level PP AF PSize C1 and C2

1 �12.71 �12.70 �12.71 �12.702 �12.70 �12.71 �12.71 �12.703 �12.71 �12.69 �12.704 �12.71Delta 0.01 0.01 0.02 0.00Rank 2 3 1 4

Fig. 6. Main effects plot (fitted mean) for S/N ratio at each level of the factors.


each parameter level are shown in Fig. 6. As can be seen, the S/Nratios illustrate the best parameters for the factors, which confirmthe same results as the RPD values.

Referring to Figs. 5 and 6, we can observe that the performanceof the algorithm is gradually improving as the PSize is increased.Based on the slope of the plot, there seems to be significantimprovements in the performance at each level. Therefore, thehigher the PSize, the better is the performance of the algorithm.

Fig. 7. Main interaction plo

Also, the performance of the algorithm is better at level two ofthe PP, as compared to the first level, i.e., using the previous parti-cle position gives a better performance than using the local bestparticle position in the mutation equations presented in (11)–(16) of the proposed algorithm. Similarly, the first AF consideredat level 1 provides the best performance than the other two affinityfunctions, considered as levels 2 and 3 respectively. In other words,using the AF as in (8) will result in the algorithm performing betterthan using the affinity functions as in (9) and (10). In addition tothat, the performance of the algorithm differs marginally at thefour different levels of the (C1 and C2) considered. However, eventhough the relative best performance is achieved at the third level,which are at 0.73, 0.37 respectively, it seems that there is no signif-icant difference in the performance among the first three levels ofthe (C1 and C2) considered.

An interaction between the factors can magnify or diminish themain effects. Hence, evaluating the effect of interactions is impor-tant. The pairwise interaction among the factors affecting RPD andS/N ratio are plotted in Figs. 7 and 8, respectively. These plots showthe impact of changing the setting of one factor on another. Fromthe parts A–C of Fig. 7 it can be observed that the performance ofthe algorithm is uniformly better at level two of the particle posi-tion (PP) at all levels of the other three factors, viz., affinity function(AF), population size (PSize) and acceleration coefficients (C1 andC2). This conclusion can be arrived at since the interaction plotsof the PP with the other three factors result in lines which consis-tently result in better performance for level two of the PP. Simi-larly, Parts G–I of Fig. 7 show that the performance of thealgorithm is uniformly better for level three of the PSize. Thereseems to be considerable effect of interaction between the AF usedand the (C1 and C2). It can be observed from Parts D–F of Fig. 7 thatthe affinity function 1 provides a better solution for both the levelsof PPs, for higher values of PSize and when we use levels 1, 3 and 4of the (C1 and C2). However, the affinity functions 2 and 3 will pro-vide better solution for the second level of the acceleration coeffi-cients. The interactions presented in Parts J–L of Fig. 7 indicate thatthe second level of the PP yields a better performance for all levels

t (fitted mean) for RPD.

Fig. 8. Main interaction plot (fitted mean) for S/N ratio.

Table 5Unit variable cost in 4 � 5 problem.

Plants Customers

Shipping costs cij Fixed costs fij

1 2 3 4 5 1 2 3 4 5

1 8 4 3 5 8 60 88 95 76 972 3 6 4 8 5 51 72 65 87 763 8 4 5 3 4 67 89 99 89 1004 4 6 8 3 3 86 84 70 92 88


of the (C1 and C2). Further, the level 1 of the acceleration coeffi-cients yields equally good results for levels 1 and 3 of the affinityfunction. However, levels 2 and 3 of the (C1 and C2) provide goodresults for level 2 of the AF. Part L of the Fig. 7 confirms that level3 of the Psize provides uniformly better solutions for all levels ofthe (C1 and C2). The same results will be arrived at by interpretingthe interaction plots for S/N ratio presented in Fig. 8.

These results suggest that to obtain the minimum solutions forour FCTP, the setting for the particle position should be at the level2 and the population size at its third level which is the highest va-lue considered. Prior to considering the interactions, the analysissuggests that the affinity function at its first level, and the acceler-ation coefficients at their third level provide the best solution. Theanalysis of interaction effects also suggest that the acceleration

Table 6Unit variable cost in 5 � 10 problem.

Plants Customers

Shipping costs cij Fixed

1 2 3 4 5 6 7 8 9 10 1

1 8 4 3 5 2 1 3 5 2 6 1602 3 3 4 8 5 3 5 1 4 5 4513 7 4 5 3 4 2 4 3 7 3 1674 1 2 8 1 3 1 4 6 8 2 3865 4 5 6 3 3 4 2 1 2 1 156

coefficients at level 3 with affinity function at level 1 provide min-imum solution.

Based on the above analysis, the best setup of the HPSIL algo-rithm is setting: the population size at its third level which is equalto 80 particles or more, the particle position at the level 2, theaffinity function at its first level, and the acceleration coefficientsat their third level which are at 0.73, 0.37 respectively. Thesame conclusion can be arrived at using Table 4 and Fig. 6 for S/N ratio.

6. Numerical experiments

To evaluate the performance of the proposed algorithm, twoproblems of different sizes, previously addressed by Hajiaghaeiet al. (2010) and Othman et al. (2011) were solved, and comparedwith the solutions presented by the original researchers and withthe solution from LINGO. The sizes of the problems are 4 � 5 and5 � 10, respectively. The variable and fixed costs for the first prob-lem are given in Table 5, while those for the second problem aregiven in Table 6. The parameters used in the HPSIL algorithm forthese problems are optimally tuned parameters and operatorsbased on experimental results.

Regarding the first problem as illustrated in Table 5, the supplyand demand values from each plant (1–4) for each customer (1–5)are as follows: s1 = 57, s2 = 93, s3 = 50, s4 = 75, d1 = 88, d2 = 57,

costs fij

2 3 4 5 6 7 8 9 10

488 295 376 297 360 199 292 481 162172 265 487 176 260 280 300 354 201250 499 189 340 216 177 495 170 414184 370 292 188 206 340 205 465 273244 460 382 270 180 235 355 276 190

Table 7Transportation allocation matrix found LINGO for 4 � 5 problem.

D1 D2 D3 D4 D5

S1 38 19S2 88 5S3 19 31S4 42 33

Table 8Transportation allocation matrix found by Hajiaghaei et al. (2010) and by the HPSILfor 4 � 5 problem.

D1 D2 D3 D4 D5

S1 57S2 69 24S3 50S4 19 23 33

Table 9Transportation allocation matrix found by LINGO for 5 � 10 problem.

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

S1 51 106S2 79 90 124S3 15 88 47S4 225 71 215 64S5 10 167 133

Table 10Transportation allocation matrix found by Hajiaghaei et al. (2010) for 5 � 10 problem.

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

S1 90 67S2 150 124 19S3 5 88 57S4 225 210 63 77S5 273 37

Table 11Transportation allocation matrix found by the HPSIL for 5 � 10 problem.

D1 D2 D3 D4 D5 D6 D7 D8 D9 D10

S1 130 27S2 15 90 124 64S3 88 30 32S4 225 135 215S5 241 69


d3 = 24, d4 = 73, and d5 = 33. The local optimal solution obtainedusing the HPSIL algorithm is the same as the solution found byHajiaghaei et al. (2010), and Othman et al. (2011) that is, equalto 1484, while the local optimum obtained from LINGO is 1544.The transportation allocation matrixes for each algorithm areshown in Tables 7 and 8.

Regarding the second problem as illustrated in Table 6, the sup-ply and demand values from each plant (1–5) for each customer(1–10) are as follows: s1 = 157, s2 = 293, s3 = 150, s4 = 575,s5 = 310, d1 = 225, d2 = 150, d3 = 90, d4 = 215, d5 = 130, d6 = 88,d7 = 57, d8 = 124, d9 = 273, and d10 = 133. The local optimal solutionobtained for this problem using the proposed algorithm (HPSIL) is6296, while the solution found by Hajiaghaei et al. (2010) was6305, the local optimum found by Othman et al. (2011) was6406 and the local optimal solution obtained from LINGO was6719. The transportation allocation matrixes for each algorithmare presented in Tables 9–11.

Concerning the first problem, the local optimum solution foundby Hajiaghaei-Keshteli et al. (2010) seems to be a global optimumsolution of this problem, therefore the HPSIL algorithm cannot finda better solution. While, the solution found for the second problemby Hajiaghaei-Keshteli et al. (2010) is not an optimal solution, theHPSIL algorithm resulted in a better solution.

7. Conclusion

In this paper, a HPSIL for solving the FCTP has been proposed. Toinvestigate the influence of the parameters on the performance ofthe algorithm, experimental designs have been carried out. In theproposed HPSIL algorithm a flexible particle structure combinedwith decoding and allocation procedures are used instead of a Prü-fer number and a spanning tree used with a genetic algorithm. Oneof the major contributions is that the HPSIL generates the feasibil-ity of all generated solutions and can be used for solving both bal-anced and unbalanced FCTPs. Moreover, the chromosome (particle)structure combined with the decoding procedure can be used withany meta-heuristic techniques such as the Tabu search, geneticalgorithms, ant colonies and artificial immune systems, amongothers. The comparison of the HPSIL algorithm with the GAs pre-sented by Othman et al. (2011), Hajiaghaei et al. (2010), and LINGOshows that the HPSIL algorithm provides an equal or better solu-tion compared to the others. The performance of the HPSIL algo-rithm and the solution quality prove that HPSIL is highlycompetitive and can be considered as a viable alternative for solv-ing FCTPs.

Future work includes further experimentation with the param-eters for the HPSIL algorithm, such as studying the relation be-tween the population size and the problem dimension, analyzingthe effect of acceleration coefficients (C1 and C2) on the problemsolution. Further, the HPSIL algorithm will be tested on other reallife problems and investigate using of other meta-heuristic tech-niques combined with the proposed decoding and allocation pro-cedures for solving the FCTP.

Acknowledgement

The authors appreciate and acknowledge the comments of thereviewers, which greatly helped us in improving the contents ofthe paper. This paper is supported by the Research Center at theCollege of Business Administration and the Deanship of ScientificResearch at King Suad University, Riyadh.

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