research article a hybrid approach to solve a model of...

Research ArticleA Hybrid Approach to Solve a Model ofClosed-Loop Supply Chain

Nafiseh Tokhmehchi1 Ahmad Makui2 and Soheil Sadi-Nezhad1

1Department of Industrial Engineering Science and Research Branch Islamic Azad University Tehran Iran2Department of Industrial Engineering Iran University of Science and Technology Tehran Iran

Correspondence should be addressed to Nafiseh Tokhmehchi nafisetymailcom

Received 18 October 2014 Revised 11 January 2015 Accepted 2 February 2015

Academic Editor Lionel Amodeo

Copyright copy 2015 Nafiseh Tokhmehchi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper investigates a closed-loop supply chain network including plants demand centers as well as collection centers anddisposal centers In forward flow the products are directly sent to demand centers after being produced by plants but in thereverse flow reused products are returned to collection centers and after investigating are partly sent to disposal centers and theother part is resent to plants for remanufacturingThe proposed mathematical model is based on mixed-integer programming andhelps minimizing the total cost Total costs include the expenditure of establishing new centers producing new products cargotransport in the network and disposal The model aims to answer these two questions (1) What number and in which places theplants collection centers and disposal centers will be constructed (2)What amount of products will be flowing in each segment ofthe chain in order to minimize the total cost Four types of tuned metaheuristic algorithms were used which are hybrid forms ofgenetic andfirefly algorithms Finally an adequate number of instances are generated to analyse the behavior of proposed algorithmsComputational results reveal that iterative sequentialization hybrid provides better solution compared with the other approachesin large size

1 Introduction

Nowadays all countries are focusing on sustainable develop-ment which is only achieved by saving and optimal using ofthe limited and nonrenewable resources in that country Gov-ernments implement a vast range of activities to achieve suchpurpose including enacting rules and regulations to encour-age using environment-friendly raw materials in industrialcomplexes reduce the usage of fossil and oil energy andrecycle the products in both public and private sectors Sus-tainable supply chain management includes supersedingreusing recycling and defusing materials as well as improv-ing public wellbeing and reducing chain costs

When companies use a forward-reverse system (closed-loop) many environmental draw backs will be prevented forinstance energy waste high consumption and transporta-tion costs Supply chain networks of sustainable development

are considered suitable means for designing closed-loopproduction networks This method fits the procedures in aneconomic cycle which helps economic and environmentaldevelopment significantly

Forward activities include improving the design engi-neering production and supply marketing sales and after-sale services of a new product Reverse activities consist ofreverse logistics usage and disposal places testing classify-ing refreshing recycling marketing and reselling Econo-mic approaches governmental strategies and customer pres-sure are 3 main aspects of reverse logistics [1]

There is a key difference between reverse chain and wastemanagement Waste management means an efficient collec-tion procedure which does not reuse the products but inreverse-network the products are resurrected

A closed loop network contrives the forward and reverseactivities in a unique system In other words the integration

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 179102 18 pageshttpdxdoiorg1011552015179102

2 Mathematical Problems in Engineering

of forward and reverse supply chains forms closed-loop sup-ply chainThis integration can improve economic and enviro-nmental functions in most of the industries

In recent decades the pressure of world economy hasforced companies to introduce new core competencies Notonly are the companies encouraged by environmental rulesto recycle the used products but also they have learned theeconomic benefits of this approach hence developing reverselogistics and closed-loop networks have become a key focalpoint for researchers and companies

The idea of this research arises from a tire manufacturerthat aspires to supply the demands of their clients and reducetheir costs Knowing that they felt the need to design a closed-loop supply chain which includes locating and constructingnew plants for manufacturing and remanufacturing with twodifferent technologies collection centers and disposal cen-ters therefore this research tries to develop a mathematicalmodel for the industry in question Suitable solutions aresought for the same purpose

The remainder of the paper is organized as follows InSection 2 we offer a relevant literature on closed-loop supplychain network Section 3 is dedicated to define and mathe-matical formulation of the problem Comprehensive expla-nations of proposed solutions are discussed in Section 4Section 5 is dedicated to computational results including datagathering tuning the parameters and comparisons betweensolution approaches Finally in Section 6 conclusions areprovided and some future suggestions are given

2 Literature Review

There have been lots of studies about supply chain networkdesign In order to focus more precisely we divided ourreview in three parts including forward networks reversenetworks and closed-loop networks Additionally at the endof this section we will present 2 tables to provide some com-parison between different studies

Most of studies in supply chain design related to forwardnetwork such as [2ndash10] Sabri and Beamon [2] developed amixed integer linear programming (MILP) model with theaim tominimize cost andmaximize service level and flexibil-ity The mentioned model can be used for simultaneous stra-tegic and operational planning Syarif et al [3] considereda multistage logistic problem and formulated it with MILPmodel In order to be solved they utilized spanning tree-based genetic algorithm Jayaraman and Ross [4] focused onplanning and execution stages of a PLOT (production logis-tics outbound and transportation) system They also pre-sented anMILPmodel and solved it with a simulated anneal-ing methodology Miranda and Garrido [5] presented asimultaneous approach to incorporate inventory controldecisions in a distribution network design system In thisway they developed a mixed integer nonlinear programming(MINLP) model and solved it with the Lagrangian relaxationand the subgradient method Altiparmak et al [6] developeda MINLP model with some conflicting objectives consistingof cost service level and equity of the capacity utilizationratio For a solution they used a genetic algorithm approach

Amiri [7] developed a MILP model and provided a heuristicsolution procedure based on Lagrangian relaxation for a dis-tribution network Tsiakis and Papageorgiou [8] determinedthe optimal configuration of a production and distributionnetwork with respect to operational and financial limitationsIn this way they proposed aMILPmodel with the aim of costminimization and solved it with an exact method Pishvaee etal [9] considered a socially responsible supply chain networkdesign In addition to dealing with uncertain conditions in anetwork a robust possibilistic programming was developedPishvaee et al [10] developed a credibility-based fuzzy math-ematical programming model with the purpose to minimizetotal costs and the environmental impacts for a green logisticsdesign under uncertainty

In this part we focus on reverse logistic networks byreview papers [11ndash17] Govindan et al [11] classified 382papers by problem classes solution and modeling approa-ches Mentioned paper is generally divided into 4 partsincluding reverse logistics closed-loop sustainable and greenbut more focused on the first two parts Jayaraman et al [12]considered a reverse distribution network and developed anMILPmodel for it Apart from that they introduced a heuris-tic solution methodology Listes and Dekker [13] developed astochastic approach for product recovery network design andapplied it to a real case study on recycling sand from demo-lition waste in The Netherlands Aras et al [14] developeda mixed-integer nonlinear facility location-allocation modelunder pick-up strategywith capacitated vehiclesThe solutionapproach was based on a Tabu searchmethod Schweiger andSahamie [15] considered a combined continuous and dis-crete facility location problem containing home-recyclingtechniques as well as selling and disposing technologies Inaddition a hybrid Tabu search algorithm is used to solve itEskandarpour et al [16] proposed a biobjective MILP modelfor postsales reverse logistics network operated by a 3PLThey solved their model with a novel multistart variableneighborhood search with three new encodingndashdecodingmechanisms Roghanian and Pazhoheshfar [17] consideredmultiproduct multistage reverse logistics network understochastic environment and proposed a probabilistic mix int-eger linear model for it They also presented a priority basedgenetic algorithm to solve their mathematical model

Papers [18ndash28] belong to closed loop supply chain Fleis-chmann et al [18] presented a MILP facility location modeland used it to analyze the impact of product return flows onlogistics networks Ko and Evans [19] developed a MINLPmodel for a dynamic integrated forwardreverse logistics net-work and solved it by a genetic algorithm Pishvaee et al [20]tried to minimize the total costs andmaximize responsibilityThey applied mixed-integer programming with two objec-tives to find a set of nondominated solutions Pishvaee andTorabi [21] used stochastic approach in closed-loop supplychains with uncertainty conditions They aimed to preventsuboptimizing the proposed model This study combinedthe design decisions in both forward and reverse networksand also used fuzzy approach to solve the model Amin andZhang [22] considered a supply chain networkwith the aim tomaximize profit minimize the rate of defects and maximizeimportance of suppliers Their model included 2 phases In

Mathematical Problems in Engineering 3

the first phase the selection factor of suppliers is determinedin a reverse flow The second phase introduced a mixed-integer programming model This was the first attempt to simul-taneously select the suppliers and allocate the orders in aclosed-loop network Pishvaee and Razmi [23] applied amul-tiobjective fuzzy programming model to design an environ-ment-friendly supply chain Mentioned model had severalparameters of uncertainty and intended tominimize the costsand environmental effects

2013 and 2014 are remarkable years with regard to theresearches on closed-loop systems Numerous articles suchas [24ndash28] have been published in these years Amin andZhang [24] used amixed-integer linearmodel forminimizingthe total costs and at the same time considering environ-mental factors They applied Weighted Sum Method and 120576-constraint Method to develop the model They also tried toprobe the effect of demand uncertainty and return uncer-tainty in the network The mentioned article used stochasticprogramming technique for this purpose Ramezani et al[25] devised a multiobjective stochastic model to design aforward-reverse logistic chain in uncertainty environmentThis model incorporates 3 levels in forward flow and 2 levelsin reverse flow and attempted to maximize the interest resp-onsiveness to clients and quality in the logistic networkSoleimani et al [26] developed a mix integer linear model fora multiechelon multiproduct and multiperiod closed loopnetwork In their model plants warehouses and distributorscan stratify customer demand For a solution they utilizedCPLEX and genetic algorithm Faccio et al [27] introduceda linear programming model for a closed-loop problem withthe aim to minimize the total costs of this chain Parametricstudy and robust analysis of this model have been contrastedwith the classic forward-flow approach and social responsi-bility Devika et al [28] developed a mixed integer program-ming model in closed-loop supply chain based on triplebottom line approachThey also presented three novel hybridmetaheuristicmethods based on adapted imperialist compet-itive algorithms and variable neighborhood search to solve it

These mentioned researches indicate that metaheuristicalgorithms are becoming the most efficient approaches tosolve complex problems in supply chain network design

Table 1 is dedicated to coding system and Table 2 showsthe mentioned models in different studies

3 Problem Definition

The problem in this paper has been taken from a closed-loopsupply chain including plants collection centers demandcenters and disposal centers Plants are able to manufacturenew products and remanufacture the returned products Inforward flow the products are dispatched to demand cen-ters after being produced by factories In reverse flow thereturned products are returned to collection centers Collec-tion centers are in charge of the following

(i) collecting the returned products from demand centers

(ii) determining the quality of returned products byobserving and categorizing into two groups renewablenonrenewable

Candidate points for manufacturing andremanufacturing centers

collectioninspection centers

Fixed points for customer centersReverse flowForward flow

Candidate points for

Candidate points for disposal centers

Figure 1 Proposed closed-loop supply chain

(iii) sending renewable returned products to plants (forremanufacturing)

(iv) sending nonrenewable returned products to disposalcenters

The model tries to answer this question In which placesand what number of plants collection centers and disposalcenters will be established and what amount of productsmust go there to minimize the total cost Figure 1 provides apictorial representation of the closed-loop supply chain Inaddition to defining the problem precisely we consider thefollowing assumption and limitations

(i) All the returned products from demand centers arekept in collection centers The demand centers havefixed locations

(ii) The potential locations capacity of plants collectioncenters and disposal centers are already determined

(iii) The needs of every demand center can be met withthe help of several plants When founding a new plantdiverse manufacturing and remanufacturing technolo-gies are available The needs of demand centers will bemet completely

(iv) There is no cargo-flow between the same level facilities


Table 1 Coding of literature review

Type of network ModelingForward logistic F ContinuousReverse logistic R Continuous approximation CAClosed loop C Discrete

Layers of network Mixed integer linear programming MILPForward logistic stage Mixed integer nonlinear programming MINLPSupplier centers SC Stochastic mixed integer programming SMIPManufacturing centers MC Fuzzy mixed integer programming FMIPDistribution centers DC Probabilistic mixed integer linear programming PMILP

Reverse logistic stage Combined continuous and discreteRedistribution centers RDC Combined continuous and mixed integer programming CMIPRecycling centers RyC Objective of modelCollectioninspection centers CIC Cost CoDisposal centers DsC Profit PrRecovery centers RoC Quality QuDisassembly center DaC Environmental impact EIProcessing center PrC Social impact SIReturning center RC Flexibility FleRemanufacturing centers RMC Service level SL

Feature of model Tardiness TdProduct Defect rate DRSingle product SP Importance of external suppliers IESMultiproduct MP Capacity utilization CU

Period Solution methodologyMultiperiod MPr Exact (small sizes) ExSingle period SPr Lagrangian relaxation-based LR

Demand Metaheuristic MHDeterministic D Interactive fuzzy solution approach IFStochastic S 120576-constraint method ECFuzzy Fu Other heuristics HRandom Ra

Facility capacityCapacitated CaUncapacitated UCa

(v) A certain percent of products taken from demandcenters will be returned by collection centers later on

(vi) Every collection center is allowed to send nonrenew-able products to diverse disposal centers

31 Mathematical Model According to what is mentionedabove the mathematical model for this problem is mixed-integer programming The following indices variables andparameters are used to model the problem in hand

311 Indices and Sets

119894 index for potential location of plants 119894 = 1 119868

119897 index for potential location of collection-inspectioncenters 119897 = 1 119871

119899 index for technology types in plants 119899 = 1 119873

119904 index for potential location of disposal centers 119904 =

1 119878

119895 index for products 119895 = 1 119869

119896 index for fixed location of customers (demandcenters) 119896 = 1 119870

312 Variables

119906119894119896119895119899

quantity of product 119895 produced with technology119899 at plant 119894 for demand center 119896

V119896119897119895 quantity of returned product 119895 shipped from

demand center 119896 to collection-inspection center 119897

119908119897119894119895 quantity of returned product 119895 shipped from

collection-inspection center 119897 to plant 119894


Table 2 Review of previously published literature

Reference Type ofnetwork Layers of network Feature of model Modeling Solution

methodologyObjective of

model[2] F SC MCd DCd MP SPr D Ca MILP Ex Co Fle SL[3] F SC MCd DCd SP SPr D Ca MILP MH Co[4] F MC DCd MP SPr D Ca MILP MH Co[5] F MC DCd SP MPr S Ca MINLP LR Co[6] F SC MCd DCd SP SPr D Ca MINLP MH Co SL CU[7] F MCd DCd SP SPr D Ca MILP LR Co[8] F MCd DCd MP SPr D Ca MILP Ex Co[9] F MCd DCd SP SPr Fu Ca FMIP Ex Co SI[10] F MCd DCd SP SPr Fu Ca FMIP IF Co EI[12] R CICd RoCd SP SPr D Ca MILP H Co[13] R CICd RyCd MP SPr S Ca SMIP Ex Co[14] R CICd MP SPr D UCa MINLP MH Co[15] R SC MC CICd MP MPr D Ca CMIP MH Co[16] R MC DsC CIC RoCd MP SPr D Ca MILP MH Co Td[17] R MC DaCd PrCd RC RyC MP SPr Ra Ca PMILP MH Co[18] C CICd RoCd MCd DCd SP SPr D UCa MILP Ex Co[19] C RoC MC DCd MP MPr D Ca MINLP MH Co[20] C MCd RoCd CICd DsCd DCd SP SPr D Ca MILP MH Co SL[21] C MC DCd CICd RoCd RyCd SP MPr Fu Ca FMLP IF Co Td[22] C SCd RMCd DaCd MP SPr D Ca MILP Ex Pr DR IES[23] C MCd CICd RoC DsC SP SPr Fu Ca FMIP IF Co EI[24] C MCdRMCd CICd DsC MP SPr S Ca MILP EC Co EI[25] C SC MCd DCd CICd RMC DsCd MP SPr S Ca MILP EC Pr SL Qu[26] C SCd MCd DsCd DCd RDCd CICd MP MPr D Ca MILP MH Co[27] C MCd DsC DCd MP SPr D Ca MILP Ex Co

[28] C SC MCd DCd CICd RoCdRMCd RyCd DsCd SP SPr D Ca MILP MH Co EI SI

This paper C MCdRMCd CICd DsCd MP SPr D Ca MILP MH CodFacilities to be located in the network

119909119897119895119904 quantity of returned product 119895 shipped from col-

lection-inspection center 119897 to disposal center 119904

119910119894119899 1 if the plant center 119894 is to be established with tech-

nology 119899 0 otherwise

119911119897 1 if the collection-inspection center 119897 is to be estab-

lished 0 otherwise

119892119904 1 if the disposal center 119904 is to be established 0 other-

wise

313 Parameters

119898119894119895119899 unit cost of manufacturing product 119895 in plant 119894

with technology 119899

119905119896119895 unit cost of transportation product 119895 from plants

to customer centers (Per km)

119905119886119895 unit cost of transportation product 119895 from cus-

tomer centers to collection-inspection centers (Perkm)

119905119892119895 unit cost of transportation product 119895 from collec-

tion-inspection centers to plants (Per km)


tion-inspection centers to disposal centers (Per km)

119891119894119899 fixed cost of establishing plant 119894 with technology 119899

1198911015840

119897 fixed cost of establishing collection-inspection

center 119897

11989110158401015840

119904 fixed cost of establishing disposal center 119904

119886119895 unit saving cost of product 119895

119887119895119904 unit disposal cost of product 119895 at center 119904


119890119894119896 distance between plant 119894 and customer center 119896

ℎ119894119897 distance between collection-inspection center 119897 to

plant 119894

119900119897119904 distance between collection-inspection center 119897 to

disposal center 119904


customer center 119896

119901119896119895 demand of customer center 119896 for product 119895

119902119896119895 amount of returned product 119895 from customer

center 119896

119903119895 minimum disposal rate for product 119895

119888119894119895119899 maximum manufacturing capacity of plant 119894 for

product 119895 with technology 119899

1198881015840

119897119895 maximum capacity of collection-inspection center 119897

for product 119895

11988810158401015840

119904119895 maximum capacity of disposal center 119904 for product

119895

314 Proposed Mathematical Formulation The mathemati-cal model of this problem becomes

Min TC

= sum

119894

sum

119899

119891119894119899119910119894119899+sum

119897

1198911015840

119897119911119897+sum

119904

11989110158401015840

119904119892119904

+sum

119894

sum

119896

sum

119895

sum

119899

(119898119894119895119899

+ 119905119896119895119890119894119896) 119906119894119896119895119899

+sum

119896

sum

119897

sum

119895

119905119886119895119889119897119896V119896119897119895

+sum

119897

sum

119894

sum

119895

(minus119886119895+ 119905119892119895ℎ119894119897)119908119897119894119895

+sum

119904

sum

119897

sum

119895

(119887119895119904+ 119905119889119895119900119897119904) 119909119897119895119904

(1)

st sum

119894

sum

119899

119906119894119896119895119899

ge 119901119896119895 forall119896 119895 (2)

sum

119897

sum

119895

119908119897119894119895

+sum

119896

sum

119895

sum

119899

119906119894119896119895119899

le sum

119899

sum

119895

119888119894119895119899

119910119894119899

forall119894

(3)

sum

119897

V119896119897119895

le sum

119894

sum

119899

119906119894119896119895119899

forall119896 119895 (4)

119903119895sum

119896

V119896119897119895

le sum

119904

119909119897119895119904 forall119897 119895 (5)

sum

119896

sum

119895

V119896119897119895

le 119911119897sum

119895

1198881015840

119897119895 forall119897 (6)

sum

119896

V119896119897119895

= sum

119894

119908119897119894119895

+sum

119904

119909119897119895119904 forall119897 119895 (7)

sum

119897

sum

119895

119909119897119895119904

le 119892119904sum

119895

11988810158401015840

119895119904 forall119904 (8)

sum

119897

V119896119897119895

= 119902119896119895 forall119896 119895 (9)

sum

119899

119910119894119899

le 1 forall119894 (10)

119910119894119899 119911119897 119892119904isin 0 1 forall119894 119897 119899 119904 (11)

119906119894119896119895119899

V119896119897119895 119908119897119894119895 119909119897119895119904

ge 0 forall119894 119896 119897 119895 119899 119904 (12)

The objective is minimizing the total costs There are 7 typesof cost The first 3 are the constant expenses of establishingnew plants new collection centers and disposal centers The4th part includes the transportation andmanufacturing costsof producing new products The 5th expenditure will berecycling and transporting the returned products The 6thsegment is associated with the total costs of recycling andtransporting returned products from collection centers tofactories Finally the 7th part of target function represents thetransportation and disposal costs

Constraint (2) guarantees that the total number of manu-factured products for each demand center is equal to or great-er than their demand rate Constraint (3) shows the capacityof plants Constraint (4) shows that the flow rate in forwardmethod is higher than reverse flow Constraint (5) representsthe disposal amount Constraint (6) presents the limitedcapacity of collection centers Equation (7) indicates that theamount of returned products from customer centers is equalto the amount of returned products to plants and the amountof products sent to disposal centers from every collectioncenter for every product Constraint (8) illustrates the limitedcapacity of disposal centers Equation (9) refers to the retur-ned products Constraint (10) demonstrates that every plantif established enjoys only one type of technology In otherwords no combination of technologies will be accepted Thetwo final constraints in themodel focus on the decision varia-bles Three variables are associated with establishing and notestablishing the plants collection centers and disposal cen-ters respectively in their potential locations These variablesare binary Other decision variables in the last constraint aredefined as nonnegative

4 Solutions

For the optimization problemswith large scale and high com-plexity classic solutions will not be reliable The solutionsmust search the feasible space in amore sophisticated way Asmentioned in literature review nowadays many researchers


devise metaheuristic algorithms for complicated problems ofsupply chain

Before applying metaheuristic we must show that theclosed-loop problem is NP hard In this way we mentionedthe fact that the closed-loop problem is a capacitated loca-tion-allocation problem and also can be considered as amultiple-choice Knapsack problem for it is known to be NP-hard [3 29 30] So metaheuristic algorithm can be one of thebest candidate solvers for such problems The problem inquestion becomes more complicated by increasing the num-ber of facilities and product technologies This might makethe accurate and heuristic solutions inapplicable hencemetaheuristic algorithms become the focal point Amongthe metaheuristic solving techniques hybridization methodsplay an important role in optimizing the case

41 Genetic Algorithms Themain idea of genetic algorithmsarises from nature These are effective random searchingtechniques inmega-scale target areasThey search in differentdirections to find the optimum answer

Genetic algorithms usually generate several random solu-tions to the problem initially This set of solutions is calledldquoThe initial populationrdquo and each member of the set is titledldquoChromosomerdquo In order to find better chromosomes theoperators of genetic algorithm cross the chromosomes overeach other This cross-over leads to mutation so that newchromosomes are generatedThebestmembers are separatedThis is how the second generation is formed In other wordsthe operators of selecting crossing over and mutating everygeneration produce a new generation of chromosomes Thenew generation is different from the previous oneThe wholeprocess is repeated over and over for the next generationsuntil the ultimate solution is obtained The steps of thisalgorithm are explained below in detail respectively

(1) creating and apprising the initial population that isgenerated randomly

(2) selecting the parents with one of the selectionmethods(roulette wheel tournament and random)

(3) applying crossover operator on the selected chromo-somes as parents

(4) selecting the chromosomes for mutating

(5) applying the mutation operator

(6) creating a new population from the best members ofall the 3 main populations crossed-over mutated andinitial population

(7) appraising the new population

(8) restarting from step (2) if the ending conditions are notmet

411 The Solution Structure The first step to solve a problemwith metaheuristic algorithms is creating a structure fordifferent solutions This is called ldquocoding phaserdquo In low-complexity problems with a small number of variables

uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution

Figure 2 An illustration of chromosomal structure

the simplest method for presenting the solution structure isusing a string in problems Matrix view is sometimes appliedas well

With modern calculation tools such as Matlab Softwarestronger and more efficient structures have become availablefor presenting the solutions Data structure is one of themIn data structure every answer is associated with a structureThe structure will also have substructures based on thenumber of variables in the problem For the current problemevery answer is initially presented with one structure Thereare 7 key variables so every answer has 7 substructures whichare also divided into substructures according to the problemscale (number of indexes for manufacturers collection cen-ters demand centers disposal centers etc) The scheme ofone answer is shown in Figure 2

412 Selection For every member a pair of parents is select-ed The aim is to choose the best elements Even the weakestelement has the chance to be selectedThis prevents stoppingon local solutions Numerous selection operators have beenintroduced so far for example operators roulette wheel tour-nament and randomThis paper has coded all the 3methodsbut the vast range of tests proved that method roulette wheelis the best selection technique for our model

413 Crossover Genetic algorithms show the probability forcrossover operator which is between 0 and 1 This presentsthe probability of creating offspring This is the likelihood ofchromosomes being crossed over again The crossover ofevery pair of chromosomes will create an offspring which is anew member of the next generation This operator tries tofind suitable potential answers in the next generation

There are various cross-over methods such as methodspartial-mapped crossover (PMX) order crossover (OX)multipoint crossover and cycle crossover (CX) [29] Thecurrent study uses multipoint crossover technique which is adeveloped version of single-point crossover Two structuresare primarily considered Then several locations are selectedin the structure randomly The elements of these locationsare substituted with each other A scheme of this crossovertechnique comes in Figure 3


uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs

u998400ikjn u998400ikjn

998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s

Parent 1 Parent 2Offspring 1 Offspring 2

Figure 3 An illustration of crossover operations

414 Fitness Function Fitness function illustrates the qual-ification of each answer in every step of the algorithm Thisfunction is the objective function in most of optimizationproblems It must find the minimum or maximum amountexcept for penalty functions Penalty function was one of theessentials according to the conditions and nature of a prob-lem After applying genetic algorithms for chromosomes thenew solutions might not always conform to the conditions ofthe problem This is why penalty functions are used in thetarget function The penalties stick to the objective functionlike weights They do not let improper chromosomes repeatin the next generation or appear in the final solution

This paper creates the initial solutions with the techniquethat prevents generating any type of unfeasible answer Dur-ing cross-over and mutation steps if the changes in elementslead to such answers again a correction method heals theanswers towards the feasible space hence penalty functionwill not be necessary therefore the fitness function in thisstudy will be exactly the same as target function in (1) Thesuggested correction method is divided into 3 techniquesaccording to the constraints of thismodelThefirst one is ldquolessthan or equal to constraintsrdquo Assume constraint sum

119895119883119895119894

le

119877119894is applied and for certain reasons variable 119883

119895119894that is

generated in chromosome structure has violated feasibilityconditions (sum

119895119883119895119894≻ 119877119894) In such case we find the maximum

number in the structure of 119883119895119894and suppose it is equal to the

minimum number in the structure This is repeated over andover until we get back to feasibility

The second case is ldquogreater than or equal to constraintsrdquoAssume constraintsum

119895119883119895119894ge 119877119894exists and for certain reasons

the generated 119883119895119894created by the constraint is not feasible

(sum119895119883119895119894≺ 119877119894) In such case we find the minimum number in

the structure of119883119895119894and make it equal to the maximum num-

ber in the structureThis is repeated until we get back to feasi-bility

The 3rd case is ldquoequality constraintrdquo Assume the existenceof constraint sum

119895119883119895119894

= 119877119894 The generated variable 119883

119895119894does

not fit equality conditions (sum119895119883119895119894

= 119877119894) In such case again

we create a matrix of random numbers between 0 and 1 andcalled it 119883

1015840

119895119894 The new 119883

119895119894is obtained from the following

formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring

Figure 4 An illustration of mutation operations

It is noteworthy that all cases 1 2 and 3 can be generalized toseveral summations

415 Mutation Genetic algorithms have small and constantchange probability which is between 0 and 1 According tothis probability child chromosomes change randomly andcome across mutation There are various methods of mutat-ing such as methods inversion insertion and uniform

In the suggested chromosomesrsquo structure every variablesits in a fixed location therefore using mutation operatorsthat substitute the location of one variable with another leadsto losing the answer structure hence methods such as uni-form can be proper choices for mutating

The proposed mutation operator in the current studyselects one of the 7 variables randomly and causes somechange in it Figure 4 shows a scheme of mutating method

416 Finishing Conditions It is obvious that the designedalgorithm which searches through the answer space has tostop somewhere and somehow otherwise the search proce-dure will grow too long and sometimes does not come to anyresult at all

Several methods have been introduced and devised so farabout the finishing criteria of metaheuristic algorithms forinstance gaining an acceptable level in the answer and pass-ing through a certain number of generations in the algorithmand a particular number of loops without improvement inanswer

Some researchers use a time limit for ending the algo-rithmThey restrict the run time or the maximum time with-out gaining an improved answer Among all introduced cri-teria time-based techniques seem to be unreliable becausevarious computers have different processing speeds besidesevery year new models with higher speed and vaster featurescome to market


This paper regards the number of generations as finishingcriterion for the algorithm Every algorithm stops after cycl-ing through a certain number of generations

42 Firefly Algorithm Firefly algorithm was introduced firs-tly in 2007 by someone named Yang [31] This is one ofthe newest optimization algorithms whose idea arises fromnature and is based upon swarm intelligence In FF algo-rithm a few artificial fireflies are initially generated randomlywith a structure similar to the chromosomes in genetic algo-rithm

After generating and distributing the fireflies every fireflyradiates some light whose intensity represents the optimalitylevel of its location In other words the firefly with higherfitness ismore attractive and grabsmore attentionThe attrac-tiveness of fireflies is based on their brightness and light stren-gth hence the closer they are to the light source the moreattractive they are Additionally if a firefly is weak but close tothe light source again it can enjoy suitable attractivenessThelight intensity of every firefly is continuously compared withthe light strength of other fireflies and the difference of lightlevel stimulates the firefly tomoveThismeans if the firefly hashigher fitness than the contrasted firefly it moves to the newlocation which is an average between the location of higher-fitness firefly and the contrasted one

If the problem objective is minimization lighter fireflyis attracted towards the dimmer firefly In this algorithmeven the weakest firefly moves randomly inside the problemdomain with the aim of increasing the chance of findingglobal optimum

The decrease in light level is caused and determined bylight absorption factor and controlled within a certain scaleThe information exchange among fireflies is done via lightradiation towards each other Finally this whole group pro-cedure causes the fireflies to tend towards the more optimalpoint

421 The Procedure of FF The steps of this algorithm areshown below

(1) Generate the initial population of fireflies

(2) Calculate light intensity 119868 for every firefly (we canalways consider objective function of a firefly as lightintensity of it [31])

119868119909prop TC

119909 (14)

(3) For every firefly (as ldquo119894rdquo) and for every other firefly (asldquo119895rdquo) if the received light intensity of firefly ldquo119894rdquo is lessthan the received light of firefly ldquo119895rdquo (119868

119894≺ 119868119895) we pro-

duce new solution according to following formula

1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)

119883119894is the location of firefly 119894119883

119895is the location of firefly

119895 120572 is the mutation coefficient (it is between 0 and1 [31]) 1205730 is attraction coefficient base value (usuallyconsidered 1 [31]) 120576 is a random vector (usually

considered sign [randminus12]) and120575 is a light absorptioncoefficient (it typically varies from 001 to 100 [31])

In addition 119903 is the Cartesian distance to light sourceand calculated by the following equation [31]

119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

(4) Evaluate the new fireflies

(5) Determine the best detected answer

(6) Loop to step (3) if the ending conditions are not true

Again the structure of each firefly is similar to the chromo-some in Figure 2 Cost function is similarly calculated as well

43 Hybridization of Algorithms With the help of hybridiz-ing process two or more algorithms can be combined tocreate new methods which inherit part of the characteristicsof each creator algorithm and form a new algorithmwith newdifferent characteristics [32] The goal of algorithmsrsquo hybridi-zation is improving their characteristics or creating newcharacters that are not available in each algorithm aloneThatis why in the recent years more and more hybrid algorithmsare being born and applied

Among different hybrids combining a search methodwithin a genetic algorithm can upgrade the search perfor-mance and also it is a chance to hybrid optimization to attaintheir target [33]

The range of hybridization methods is enormous forexample sequentialization and parallel methods In currentstudy four hybridization methods have been devised andwith the help ofMatlab Software all of themwere coded besi-des a summary about the functionality of each of the 4hybridization methods is explained with the related chart

431 Algorithm PHGF The first suggested algorithm is asimple parallelization technique In this technique the cal-culations of genetic and firefly algorithms are performed in aparallel way with similar cycles and the final output is chosenfrom the best answers of both algorithms Figure 5 shows anillustration of simple parallelization hybridization

The steps of PHGF are shown below

(1) Randomly generate an initial population (119901119892)

(2) While (not stop condition)

(a) For ℎ = 1119867119892(119867119892is the number of individuals in

the population)(a1) Recombine 119901

119888percent of population to pro-

duce 1199011015840119892offspring and then evaluate them

(a2) Mutate 119901119898percent of population to produce

11990110158401015840

119892offspring and then evaluate them

(a3) Merge population offspring generated viarecombination and offspring generated frommutation in a unique set (119901

119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm

Geneticalgorithm Final solutionInputs

Figure 5 An illustration of simple parallelization hybridization

(a4) Truncate best members of merged set andgenerate newpopulation of size119901

119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while

(3) Save 119867119892members of best solutions obtain by GA in

finalga

(4) Randomly generate an initial population as artificialfireflies (119901

119891)


(b) For 119899 = 1119867119891(119867119891is the total number of fireflies)

(b1) For119898 = 1119867119891

If 119868119899lt 119868119898

Generate new solution and update it(119901119891) according to formulation in (15)

End ifEnd for119898

End for 119899

End while

(6) Save 119867119891members of best solutions obtain by FF in

finalff

(7) Merge best of GA and best of FF (finalga cup finalff ) rarr

finalℎ

(8) Rank the member of finalℎand select best of them as

PHGF solution

432 Algorithm EHGF The second suggested algorithm isbased on embedding technique In this technique one solu-tion is embedded in another According to the algorithmsused in this paper Genetic algorithm is considered the baseand firefly algorithm is only used for diversifying the answersIn other words firefly plays the part of mutating operatorin this suggested hybrid Figure 6 shows an illustration ofembedded hybridization

The steps of EHGF are shown below



Genetic algorithm

Firefly algorithm

Random solutions Final solution

Figure 6 An illustration of embedded hybridization

(a) for ℎ = 1119867 (119867 is the number of individuals)(a1) Recombine 119901


duce 1199011015840 offspring and then evaluate them(a2) Merge population and offspring generated

via recombination in a unique set (119901 cup 1199011015840)

and then sort them(a3) Truncate best members of merged set and

generate new population of size 119901 (119901 cup

1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898

Generate new solution and update itaccording to formulation in (15) toform 119901

119891

End ifEnd for119898

End for 119899(c) Mutate 119901

119898percent of population (119901

119891) to produce

11990110158401015840 offspring and then evaluate them

(d) Merge population generated via mutation andpopulation from firefly algorithm in a unique set(119901119891cup 11990110158401015840) and then sort them

(e) Truncate best members of merged set and gener-ate new population of size 119901 (119901

119891cup 11990110158401015840) rarr 119901

End while

(3) Select best of final generation (119901) as EHGF solution

433 Algorithm ISHGF In this technique the output ofgenetic algorithm is used as the input of firefly algorithm andthen the output of firefly algorithmwill be used as the input ofgenetic algorithm This will be done over and over until fin-ishing conditions come true Such new algorithm is obtainedfrom hybridization with iterative sequentialization methodand is called ISHGF

In ISHGF according to the sizes of GA and FF populationmembers 3 scenarios might occur The first scenario is119867

119892=

119867119891(119867119892is the number of individuals in the population and

119867119891is the total number of fireflies) In such case the answers

are sent continually fromGA to FF algorithm and from FF to


Geneticalgorithm

Fireflyalgorithm StopRandom

solutionsFinal solutionYes

No

Figure 7An illustration of iterative sequentialization hybridization

GA until gaining the desired answer The second scenario is119867119892gt 119867119891 In such case the best members of GA population

sit in the place of FF members and the remaining groupof members whose number is calculated by subtracting 119867

119892

from119867119891are thrown away Of course when the algorithm has

not reached the desired answer GA receives all the membersof FF and compensates its lacking members by generatingrandom answers The 3rd scenario is 119867

119892lt 119867119891 Hereon all

GAmembers go into FF algorithm and new random answersare generated as many as119867

119891minus119867119892 Of course in this scenario

again if the ending conditions do not arise the algorithmreturns to its beginning so GA selects only as many as 119867

119892

members among the best answers of 119867119891and throws the rest

away Figure 7 shows an illustration of iterative sequentializa-tion hybridization

The steps of ISHGF are shown below (119867119892gt 119867119891)








11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them(a4) Truncate best members of merged set and

generate newpopulation of size119901119892 (119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ(b) Set best members of final population in size 119867

119891

as artificial fireflies (119867119891is the total number of

fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm

Geneticalgorithm Final solution

Inputs 120573

1 minus 120573

Figure 8 An illustration of simple workload division hybridization

(d) Rank the solution update the best and also gener-ate (119867

119892minus 119867119891) members randomly and add them

to 119901119891(Added members + 119901

119891rarr 119901119892)

End while

(3) Select best of final generation as ISHGF solution

434 Algorithm WHGF The fourth suggested algorithm issimple workload division This technique is very similar tosimple parallelizationmeaning that every algorithmworks inparallel with and independent of other algorithms but in theend the collected answers appear in the final answer accord-ing to the determined share for every algorithm For examplefirefly algorithm has 60 and genetic algorithm has 40share in the ultimate answer Obviously the final solution isthe best answer among all the answers collected from bothparticipant algorithms in the hybrid process Figure 8 showsan illustration of simple workload division hybridization

The steps of WHGF are shown below








11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and


generate newpopulation of size119901119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga(4) Randomly generate an initial population as artificial

fireflies (119901119891)



(b) For 119899 = 1119867119891(119867119891is the total number of firefly)

(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff

(7) Merge 120573 best of GA and 1 minus 120573 best of FF(120573 (finalga) cup 1 minus 120573 (finalff )) rarr final

ℎ


WHGF solution

5 Computational Results

51 Data Gathering In order to investigate the performanceof suggested algorithms numerous tests are required Firstof all certain numbers of the parameters required in theproblem must be determined The performance of suggestedsolutions is to be appraised in various scales small mediumand large Knowing that direct input of numbers into theparameters and changing the numbers in every test willmakethe solution too complicated hence uniform statistical distri-bution was applied here Table 3 shows the values of requiredparameters for solving this problem

52 Tuning the Parameters For improving the performanceof suggested hybrid algorithms a particular process calledparameter tuning is required Parameter tuning keeps therequired parameters of the problem in the best possible rangeso that every parameter performs in the best possible wayNumerous techniques have been used so far for setting theparameters of metaheuristic algorithms including Trial AndError Full-Factorial Taguchi Response Surface and NeuralNetworks

This paper aims to set the parameters of all 4 suggestedalgorithms with Taguchi Method which was innovated byTaguchi in 1986 [34]The first step is finding and introducingthe influential parameters in algorithm Since the suggestedsolutions are all based on genetic and firefly algorithms theinfluential factors in performance of their hybrid also includethe parameters of those two

The parameters of genetic algorithm include mutationrate crossover rate number of iterations and number ofchromosomes in the population The selected parameters infirefly algorithm also include the number of artificial fireflieslight absorption coefficient mutation coefficient and themaximum number of iterations The parameter values aredefined in 3 levels small medium and largeThese values areshown with indexes 1 2 and 3 respectively

Table 3 Values of model parameters

Parameter Levels119898119894119895119899

simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)

Since the mathematical model of this problem is notsimilar to other studies the value of parameters cannot becompared with other researches therefore these levels havebeen determined with trial-and-error within their permittedrange Taguchi solution approach looks at the number offactors and the levels and determines the required number oftests accordingly Since this paper has selected 8 factors and3 levels 27 problems are to be solved Taguchi technique inthis problem has been performed by Minitab Software on aPCwith 3GBRAM Table 4 shows the considered parametersand their levels

For tuning the parameters the obtained values shouldbecome normalized so as to make them independent ofrange Numerous methods are available for making the para-meters dimensionless including methods fuzzy normaliza-tion and linear normalization This paper uses related per-centage deviation (RPD) technique with the following for-mula

RPD =

1003816100381610038161003816Algorihm Solution minus Best Solution100381610038161003816

1003816

Best Solutiontimes 100 (17)

Table 5 shows the results of 27 times problem solving withdifferent levels of factors and in RPD form

The ultimate aim of parameter tuning is adjusting each ofthe algorithms in their best possible position This is why asample problem is initially selected and solved with variouslevels of parameters and then the results are entered in acolumnThe best answer is the minimum value (because thisis a minimizing problem) According to RPD formula thebest answer turns into zero and the other answers are eval-uated according to their deviation from the best thereforesince every algorithm is evaluated separately every columnhas 1 zero The obtained numbers in every algorithm wereentered into Minitab separately The following graphs showthe results of suggested levels for parameters in TaguchiMethod Minitab Software introduces two graphs for everyalgorithm one associated with mean and the other one withSNR Every Mean diagram consists of 8 graphs indicating


Table 4 Levels of algorithm parameters

Parameter 1 2 3

Genetic algorithm

119862119875 crossover percentage 04 06 08

119872119875 mutation percentage 01 02 03

119868GA number of GA iterations 50 100 150119875119878 population size 50 70 100

Firefly algorithm

119873119865 number of fireflies 100 150 200

119871 light absorption coefficient 001 50 100119872119865 mutation coefficient 02 04 08

119868119865 number of FF iterations 50 100 150

Table 5 Values of proposed algorithms in parameter tuning

Problem number 119862119875

119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865

PHGF EHGF ISHGF WHGF1 1 1 1 1 1 1 1 1 523200 11200 140600 580002 2 2 2 2 1 1 1 1 712600 418600 168700 177003 3 3 3 3 1 1 1 1 140300 21800 144000 1878004 2 1 1 1 2 2 2 1 41300 118000 180400 4100005 3 2 2 2 2 2 2 1 129900 239200 381800 458006 1 3 3 3 2 2 2 1 31900 69100 127900 000007 3 1 1 1 3 3 3 1 24700 313700 72200 5393008 1 2 2 2 3 3 3 1 180600 00000 575700 1189009 2 3 3 3 3 3 3 1 318800 188100 411100 48230010 1 3 2 1 3 2 1 2 105400 37000 176400 1410011 2 1 3 2 3 2 1 2 934500 651600 624800 39330012 3 2 1 3 3 2 1 2 923300 127800 565400 17240013 2 3 2 1 1 3 2 2 109800 214500 431000 11410014 3 1 3 2 1 3 2 2 139500 147800 685200 12200015 1 2 1 3 1 3 2 2 177200 722900 394100 55930016 3 3 2 1 2 1 3 2 334700 238600 334600 16760017 1 1 3 2 2 1 3 2 193600 792100 416500 32160018 2 2 1 3 2 1 3 2 913600 663900 437200 23870019 1 2 3 1 2 3 1 3 287100 445700 537800 3050020 2 3 1 2 2 3 1 3 46800 06900 66500 6420021 3 1 2 3 2 3 1 3 238800 214600 820400 11890022 2 2 3 1 3 1 2 3 290600 723600 555700 74600023 3 3 1 2 3 1 2 3 821800 48300 516000 82540024 1 1 2 3 3 1 2 3 470000 815000 448400 46310025 3 2 3 1 1 2 3 3 198900 353300 00000 98070026 1 3 1 2 1 2 3 3 00000 70600 17000 18040027 2 1 2 3 1 2 3 3 788300 477000 376400 802100

parameter levels The best value in every level is the mini-mum for example in the diagram of crossover rate in simpleparallelization algorithm level 1 with the lowest value isconsidered the optimum value for that parameter

The other diagram is SNR which illustrates the optimumlevel of every parameter in another wayThe higher SNR levelis better Here is a caveat to keep in mind If the parametersare tuned properly the results of both Means and SNR dia-grams will be equal otherwise if the interpretation of every

parameter in average diagram is different from its optimumlevel in SNR diagram the tests must be repeated

Note that we have one type of population in the wholeEHGF algorithm therefore the number of fireflies does notdefine

Table 6 shows the tuned levels of parameters in everyalgorithm according to Figures 9 10 11 and 12

After entering the tuned value of every parameter thealgorithm will be used for solving 40 problems in small


Table 6 Tuned parameters

Algorithms 119862119875

119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865

PHGF 1 2 3 2 1 3 3 1EHGF 1 1 2 1 mdash 1 3 3ISHGF 1 3 2 1 1 1 3 1WHGF 1 1 3 2 2 2 3 1

321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns

Main effects plot for MeansData Means

LIGA NF MF IFCP MP PS

(a)

321 321 321 321 321 321 321 321

L

Main effects plot for SN ratiosData Means

minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio

sSignal-to-noise smaller is better

IGA NF MF IFCP MP PS

(b)

Figure 9 Main effects plots of Means (a) and SNR (b) for PHGF

321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321

Signal-to-noise smaller is better


Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)

Figure 10 Main effects plots of Means (a) and SNR (b) for EHGF

321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321

minus24minus25minus26minus27minus28minus29minus30minus31minus32minus33



Mea

n of

SN

ratio

s


(b)

Figure 11 Main effects plots of Means (a) and SNR (b) for ISHGF


50

40

30

20

10

321321321321321321321321

Mea

n of

Mea

nsMain effects plot for Means

Data MeansLIGA NF MF IFCP MP PS

(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)

Figure 12 Main effect plots for Mean (a) and SNR (b) of WHGF

PHGF EHGF ISHGF WHGF GA FF

15

20

25

30times105

Interval plot95 CI for the Mean

Small size

Figure 13 Comparison between algorithms in small size

medium and large scale Table 7 demonstrates the resultedvalues In addition we consider our base algorithms GA andFF to investigate effect of hybridization on the test problems

53 Comparisons The values obtained from all 6 solutionsare contrasted according to statistical comparison techniqueswith the aim to find the best solution For each problem theproposed algorithms are run 3 times and then the averagevalue is considered In order to determine the significant dif-ference between different values obtained by algorithmANOVAStatistical Test was devisedwith SPSS Software Notethat in order to compare more precisely each size consideredseparately Tables 8 9 and 10 and Figures 13 14 and 15 showthe comparisons between algorithms

As shown in Figure 13 and also from ANOVA Table 8(sig gt 005) there is no significant difference between algo-rithms in solving small size problems

Although it seems that there is a difference between algo-rithms inTable 9 (Sig lt 005) because of overlapping betweenplots in Figure 14 we cannot certainly report which one isbetter in solving medium size problems

Value obtained for sig lower than 005 is indicating thatthere is a significant difference between algorithms In addi-tion as shown in Figure 15 it is obvious that ISHGF achieveslower cost in large size than the other algorithms


Medium size


10121416182022242628times106

Figure 14 Comparison between algorithms in medium size



Large size

10

15

20

25

30

35

40times107

Figure 15 Comparison between algorithms in large size

6 Conclusion

This paper introduces a new model for reducing the costs ofclosed-loop supply chain The model includes several plantsthat are able tomanufacture and remanufacture various typesof products using diverse technologies In forward flow thedemanded products of customer centers were sent to themdirectly without any go-between In reverse flow the collec-tion centers are in charge of collecting and evaluating theproducts so as to decide about sending the products to plantsor disposal centers

The proposed mathematical model in this problem grewmore and more complicated by increasing the number offacilities products and technologies so that classic methodsbecame unreliable for solving the problem hence this paper


Table 7 Result of objective function in different sizes

Test number 119894 119899 119897 119896 119904 119895 PHGF EHGF ISHGF WHGF GA FFSmall size

1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641

Table 8 ANOVA of small size

Sum of squares df Mean square 119865 SigBetween groups 177311986411 5 354511986410

0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59


Table 9 ANOVA for medium size


2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83

Table 10 ANOVA for large size


4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95

tried to use 4 different types of hybrid algorithms based ongenetic and firefly algorithms

After tuning the optimum parameters of those two algo-rithms using Taguchi Approach inMinitab Software the par-ameters were used for solving 40 problems with small med-ium and large scale Finally ANOVA Statistical Test was dev-ised with SPSS Software and this shows that iterative sequen-tialization technique gave the best answers in large size

Closed-loop supply chain has considerable economicenvironmental and social effects therefore it has become afocal point in recent years Such model can consider socialand environmental factors which are influential in the net-work It can be formulated in multiobjective form

Increasing the number of levels and participants of thesupply chain design can help improve its compatibility withthe needs of that industry and society

Aside from that considering uncertainty in parametersand investigating the flexibility of model with regard togradual changes can also help make the model significantlymore robust

Using other metaheuristic methods and artificial neuralnetworks can also be considered among the future solutionideas

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] M T Melo S Nickel and F Saldanha-da-Gama ldquoFacilitylocation and supply chain managementmdasha reviewrdquo EuropeanJournal of Operational Research vol 196 no 2 pp 401ndash4122009

[2] E H Sabri and B M Beamon ldquoA multi-objective approach tosimultaneous strategic and operational planning in supply chaindesignrdquo Omega vol 28 no 5 pp 581ndash598 2000

[3] A Syarif Y S Yun and M Gen ldquoStudy on multi-stagelogistic chain network a spanning tree-based genetic algorithmapproachrdquo Computers and Industrial Engineering vol 43 no 1-2 pp 299ndash314 2002

[4] V Jayaraman andA Ross ldquoA simulated annealingmethodologyto distribution network design and managementrdquo European

Journal of Operational Research vol 144 no 3 pp 629ndash6452003

[5] P A Miranda and R A Garrido ldquoIncorporating inventorycontrol decisions into a strategic distribution network designmodel with stochastic demandrdquoTransportation Research Part ELogistics and Transportation Review vol 40 no 3 pp 183ndash2072004

[6] F Altiparmak M Gen L Lin and T Paksoy ldquoA genetic algo-rithm approach for multi-objective optimization of supplychain networksrdquo Computers and Industrial Engineering vol 51no 1 pp 196ndash215 2006

[7] A Amiri ldquoDesigning a distribution network in a supply chainsystem formulation and efficient solution procedurerdquoEuropeanJournal of Operational Research vol 171 no 2 pp 567ndash5762006

[8] P Tsiakis and L G Papageorgiou ldquoOptimal production allo-cation and distribution supply chain networksrdquo InternationalJournal of Production Economics vol 111 no 2 pp 468ndash4832008

[9] M S Pishvaee J Razmi and S A Torabi ldquoRobust possibilisticprogramming for socially responsible supply chain networkdesign a new approachrdquo Fuzzy Sets and Systems vol 206 pp1ndash20 2012

[10] M S Pishvaee S A Torabi and J Razmi ldquoCredibility-basedfuzzy mathematical programming model for green logisticsdesign under uncertaintyrdquo Computers and Industrial Engineer-ing vol 62 no 2 pp 624ndash632 2012

[11] K Govindan H Soleimani and D Kannan ldquoReverse logisticsand closed-loop supply chain a comprehensive review toexplore the futurerdquo European Journal of Operational Researchvol 240 no 3 pp 603ndash626 2015

[12] V Jayaraman R A Patterson and E Rolland ldquoThe design ofreverse distributionnetworksmodels and solution proceduresrdquoEuropean Journal of Operational Research vol 150 no 1 pp128ndash149 2003

[13] O Listes and R Dekker ldquoA stochastic approach to a casestudy for product recovery network designrdquo European Journalof Operational Research vol 160 no 1 pp 268ndash287 2005

[14] N Aras D Aksen and A G Tanugur ldquoLocating collectioncenters for incentive- dependent returns under a pick-up policywith capacitated vehiclesrdquo European Journal of OperationalResearch vol 191 no 3 pp 1223ndash1240 2008

[15] K Schweiger and R Sahamie ldquoA hybrid Tabu Search approachfor the design of a paper recycling networkrdquo TransportationResearch Part E Logistics and Transportation Review vol 50 no1 p 98 2013


[16] M Eskandarpour E Nikbakhsh and S H Zegordi ldquoVariableneighborhood search for the bi-objective post-sales networkdesign problem a fitness landscape analysis approachrdquo Com-puters amp Operations Research vol 52 part B pp 300ndash314 2014

[17] E Roghanian and P Pazhoheshfar ldquoAn optimization modelfor reverse logistics network under stochastic environment byusing genetic algorithmrdquo Journal of Manufacturing Systems vol33 no 3 pp 348ndash356 2014

[18] M Fleischmann P Beullens J M Bloemhof-Ruwaard and LN VanWassenhove ldquoThe impact of product recovery on logis-tics network designrdquo Production and Operations Managementvol 10 no 2 pp 156ndash173 2001

[19] H J Ko and G W Evans ldquoA genetic algorithm-based heuristicfor the dynamic integrated forwardreverse logistics networkfor 3PLsrdquo Computers amp Operations Research vol 34 no 2 pp346ndash366 2007

[20] M S Pishvaee R Z Farahani and W Dullaert ldquoA memeticalgorithm for bi-objective integrated forwardreverse logisticsnetwork designrdquo Computers and Operations Research vol 37no 6 pp 1100ndash1112 2010

[21] M S Pishvaee and S A Torabi ldquoA possibilistic programmingapproach for closed-loop supply chain network design underuncertaintyrdquo Fuzzy Sets and Systems vol 161 no 20 pp 2668ndash2683 2010

[22] S H Amin and G Zhang ldquoAn integrated model for closed-loop supply chain configuration and supplier selection multi-objective approachrdquo Expert Systems with Applications vol 39no 8 pp 6782ndash6791 2012

[23] M S Pishvaee and J Razmi ldquoEnvironmental supply chainnetwork design using multi-objective fuzzy mathematical pro-grammingrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol36 no 8 pp 3433ndash3446 2012

[24] S H Amin and G Zhang ldquoA multi-objective facility locationmodel for closed-loop supply chain network under uncertaindemand and returnrdquo Applied Mathematical Modelling vol 37no 6 pp 4165ndash4176 2013

[25] M Ramezani M Bashiri and R Tavakkoli-Moghaddam ldquoAnew multi-objective stochastic model for a forwardreverselogistic network design with responsiveness and quality levelrdquoApplied Mathematical Modelling vol 37 no 1-2 pp 328ndash3442013

[26] H Soleimani M Seyyed-Esfahani and M A Shirazi ldquoDesign-ing and planning a multi-echelon multi-period multi-productclosed-loop supply chain utilizing genetic algorithmrdquo Interna-tional Journal of Advanced Manufacturing Technology vol 68no 1ndash4 pp 917ndash931 2013

[27] M Faccio A Persona F Sgarbossa and G Zanin ldquoSustainableSC through the complete reprocessing of end-of-life productsbymanufacturers a traditional versus social responsibility com-pany perspectiverdquo European Journal of Operational Researchvol 233 no 2 pp 359ndash373 2014

[28] K Devika A Jafarian and V Nourbakhsh ldquoDesigning a sus-tainable closed-loop supply chain network based on triple bot-tom line approach a comparison of metaheuristics hybridiza-tion techniquesrdquo European Journal of Operational Research vol235 no 3 pp 594ndash615 2014

[29] M Gen and R Cheng Genetic Algorithms and EngineeringDesign John Wiley amp Sons New York NY USA 1997

[30] H-F Wang and H-W Hsu ldquoA closed-loop logistic modelwith a spanning-tree based genetic algorithmrdquo Computers andOperations Research vol 37 no 2 pp 376ndash389 2010

[31] X S Yang ldquoFirefly algorithm levy flights and global optimiza-tionrdquo in Research and Development in Intelligent Systems XXVIpp 209ndash218 Springer Berlin Germany 2010

[32] K Eshghi and M Kariminasab Combinatorial Optimization ampMetaheuristics Mehr Azin Press Tehran Iran 2012

[33] F G Lobo and D E Goldberg ldquoDecision making in a hybridgenetic algorithmrdquo inProceedings of the IEEE International Con-ference on Evolutionary Computation pp 121ndash125 IEEE PressIndianapolis Ind USA April 1997

[34] G Taguchi S Chowdhury and Y Wu Taguchirsquos Quality Engi-neering Handbook Wiley New York NY USA 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


of forward and reverse supply chains forms closed-loop sup-ply chainThis integration can improve economic and enviro-nmental functions in most of the industries

In recent decades the pressure of world economy hasforced companies to introduce new core competencies Notonly are the companies encouraged by environmental rulesto recycle the used products but also they have learned theeconomic benefits of this approach hence developing reverselogistics and closed-loop networks have become a key focalpoint for researchers and companies

The idea of this research arises from a tire manufacturerthat aspires to supply the demands of their clients and reducetheir costs Knowing that they felt the need to design a closed-loop supply chain which includes locating and constructingnew plants for manufacturing and remanufacturing with twodifferent technologies collection centers and disposal cen-ters therefore this research tries to develop a mathematicalmodel for the industry in question Suitable solutions aresought for the same purpose

The remainder of the paper is organized as follows InSection 2 we offer a relevant literature on closed-loop supplychain network Section 3 is dedicated to define and mathe-matical formulation of the problem Comprehensive expla-nations of proposed solutions are discussed in Section 4Section 5 is dedicated to computational results including datagathering tuning the parameters and comparisons betweensolution approaches Finally in Section 6 conclusions areprovided and some future suggestions are given

2 Literature Review

There have been lots of studies about supply chain networkdesign In order to focus more precisely we divided ourreview in three parts including forward networks reversenetworks and closed-loop networks Additionally at the endof this section we will present 2 tables to provide some com-parison between different studies

Most of studies in supply chain design related to forwardnetwork such as [2ndash10] Sabri and Beamon [2] developed amixed integer linear programming (MILP) model with theaim tominimize cost andmaximize service level and flexibil-ity The mentioned model can be used for simultaneous stra-tegic and operational planning Syarif et al [3] considereda multistage logistic problem and formulated it with MILPmodel In order to be solved they utilized spanning tree-based genetic algorithm Jayaraman and Ross [4] focused onplanning and execution stages of a PLOT (production logis-tics outbound and transportation) system They also pre-sented anMILPmodel and solved it with a simulated anneal-ing methodology Miranda and Garrido [5] presented asimultaneous approach to incorporate inventory controldecisions in a distribution network design system In thisway they developed a mixed integer nonlinear programming(MINLP) model and solved it with the Lagrangian relaxationand the subgradient method Altiparmak et al [6] developeda MINLP model with some conflicting objectives consistingof cost service level and equity of the capacity utilizationratio For a solution they used a genetic algorithm approach

Amiri [7] developed a MILP model and provided a heuristicsolution procedure based on Lagrangian relaxation for a dis-tribution network Tsiakis and Papageorgiou [8] determinedthe optimal configuration of a production and distributionnetwork with respect to operational and financial limitationsIn this way they proposed aMILPmodel with the aim of costminimization and solved it with an exact method Pishvaee etal [9] considered a socially responsible supply chain networkdesign In addition to dealing with uncertain conditions in anetwork a robust possibilistic programming was developedPishvaee et al [10] developed a credibility-based fuzzy math-ematical programming model with the purpose to minimizetotal costs and the environmental impacts for a green logisticsdesign under uncertainty

In this part we focus on reverse logistic networks byreview papers [11ndash17] Govindan et al [11] classified 382papers by problem classes solution and modeling approa-ches Mentioned paper is generally divided into 4 partsincluding reverse logistics closed-loop sustainable and greenbut more focused on the first two parts Jayaraman et al [12]considered a reverse distribution network and developed anMILPmodel for it Apart from that they introduced a heuris-tic solution methodology Listes and Dekker [13] developed astochastic approach for product recovery network design andapplied it to a real case study on recycling sand from demo-lition waste in The Netherlands Aras et al [14] developeda mixed-integer nonlinear facility location-allocation modelunder pick-up strategywith capacitated vehiclesThe solutionapproach was based on a Tabu searchmethod Schweiger andSahamie [15] considered a combined continuous and dis-crete facility location problem containing home-recyclingtechniques as well as selling and disposing technologies Inaddition a hybrid Tabu search algorithm is used to solve itEskandarpour et al [16] proposed a biobjective MILP modelfor postsales reverse logistics network operated by a 3PLThey solved their model with a novel multistart variableneighborhood search with three new encodingndashdecodingmechanisms Roghanian and Pazhoheshfar [17] consideredmultiproduct multistage reverse logistics network understochastic environment and proposed a probabilistic mix int-eger linear model for it They also presented a priority basedgenetic algorithm to solve their mathematical model

Papers [18ndash28] belong to closed loop supply chain Fleis-chmann et al [18] presented a MILP facility location modeland used it to analyze the impact of product return flows onlogistics networks Ko and Evans [19] developed a MINLPmodel for a dynamic integrated forwardreverse logistics net-work and solved it by a genetic algorithm Pishvaee et al [20]tried to minimize the total costs andmaximize responsibilityThey applied mixed-integer programming with two objec-tives to find a set of nondominated solutions Pishvaee andTorabi [21] used stochastic approach in closed-loop supplychains with uncertainty conditions They aimed to preventsuboptimizing the proposed model This study combinedthe design decisions in both forward and reverse networksand also used fuzzy approach to solve the model Amin andZhang [22] considered a supply chain networkwith the aim tomaximize profit minimize the rate of defects and maximizeimportance of suppliers Their model included 2 phases In








































1 119878



312 Variables

119906119894119896119895119899


















lished 0 otherwise


wise

313 Parameters












1198911015840


center 119897

11989110158401015840







plant 119894







center 119896




1198881015840


for product 119895

11988810158401015840


119895


Min TC

= sum

119894

sum

119899

119891119894119899119910119894119899+sum

119897

1198911015840

119897119911119897+sum

119904

11989110158401015840

119904119892119904

+sum

119894

sum

119896

sum

119895

sum

119899

(119898119894119895119899

+ 119905119896119895119890119894119896) 119906119894119896119895119899

+sum

119896

sum

119897

sum

119895

119905119886119895119889119897119896V119896119897119895

+sum

119897

sum

119894

sum

119895

(minus119886119895+ 119905119892119895ℎ119894119897)119908119897119894119895

+sum

119904

sum

119897

sum

119895

(119887119895119904+ 119905119889119895119900119897119904) 119909119897119895119904

(1)

st sum

119894

sum

119899

119906119894119896119895119899

ge 119901119896119895 forall119896 119895 (2)

sum

119897

sum

119895

119908119897119894119895

+sum

119896

sum

119895

sum

119899

119906119894119896119895119899

le sum

119899

sum

119895

119888119894119895119899

119910119894119899

forall119894

(3)

sum

119897

V119896119897119895

le sum

119894

sum

119899

119906119894119896119895119899

forall119896 119895 (4)

119903119895sum

119896

V119896119897119895

le sum

119904

119909119897119895119904 forall119897 119895 (5)

sum

119896

sum

119895

V119896119897119895

le 119911119897sum

119895

1198881015840

119897119895 forall119897 (6)

sum

119896

V119896119897119895

= sum

119894

119908119897119894119895

+sum

119904

119909119897119895119904 forall119897 119895 (7)

sum

119897

sum

119895

119909119897119895119904

le 119892119904sum

119895

11988810158401015840

119895119904 forall119904 (8)

sum

119897

V119896119897119895

= 119902119896119895 forall119896 119895 (9)

sum

119899

119910119894119899

le 1 forall119894 (10)

119910119894119899 119911119897 119892119904isin 0 1 forall119894 119897 119899 119904 (11)

119906119894119896119895119899

V119896119897119895 119908119897119894119895 119909119897119895119904

ge 0 forall119894 119896 119897 119895 119899 119904 (12)



4 Solutions
















uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution








uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































1 119878



312 Variables

119906119894119896119895119899


















lished 0 otherwise


wise

313 Parameters












1198911015840


center 119897

11989110158401015840







plant 119894







center 119896




1198881015840


for product 119895

11988810158401015840


119895


Min TC

= sum

119894

sum

119899

119891119894119899119910119894119899+sum

119897

1198911015840

119897119911119897+sum

119904

11989110158401015840

119904119892119904

+sum

119894

sum

119896

sum

119895

sum

119899

(119898119894119895119899

+ 119905119896119895119890119894119896) 119906119894119896119895119899

+sum

119896

sum

119897

sum

119895

119905119886119895119889119897119896V119896119897119895

+sum

119897

sum

119894

sum

119895

(minus119886119895+ 119905119892119895ℎ119894119897)119908119897119894119895

+sum

119904

sum

119897

sum

119895

(119887119895119904+ 119905119889119895119900119897119904) 119909119897119895119904

(1)

st sum

119894

sum

119899

119906119894119896119895119899

ge 119901119896119895 forall119896 119895 (2)

sum

119897

sum

119895

119908119897119894119895

+sum

119896

sum

119895

sum

119899

119906119894119896119895119899

le sum

119899

sum

119895

119888119894119895119899

119910119894119899

forall119894

(3)

sum

119897

V119896119897119895

le sum

119894

sum

119899

119906119894119896119895119899

forall119896 119895 (4)

119903119895sum

119896

V119896119897119895

le sum

119904

119909119897119895119904 forall119897 119895 (5)

sum

119896

sum

119895

V119896119897119895

le 119911119897sum

119895

1198881015840

119897119895 forall119897 (6)

sum

119896

V119896119897119895

= sum

119894

119908119897119894119895

+sum

119904

119909119897119895119904 forall119897 119895 (7)

sum

119897

sum

119895

119909119897119895119904

le 119892119904sum

119895

11988810158401015840

119895119904 forall119904 (8)

sum

119897

V119896119897119895

= 119902119896119895 forall119896 119895 (9)

sum

119899

119910119894119899

le 1 forall119894 (10)

119910119894119899 119911119897 119892119904isin 0 1 forall119894 119897 119899 119904 (11)

119906119894119896119895119899

V119896119897119895 119908119897119894119895 119909119897119895119904

ge 0 forall119894 119896 119897 119895 119899 119904 (12)



4 Solutions
















uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution








uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















lished 0 otherwise


wise

313 Parameters












1198911015840


center 119897

11989110158401015840







plant 119894







center 119896




1198881015840


for product 119895

11988810158401015840


119895


Min TC

= sum

119894

sum

119899

119891119894119899119910119894119899+sum

119897

1198911015840

119897119911119897+sum

119904

11989110158401015840

119904119892119904

+sum

119894

sum

119896

sum

119895

sum

119899

(119898119894119895119899

+ 119905119896119895119890119894119896) 119906119894119896119895119899

+sum

119896

sum

119897

sum

119895

119905119886119895119889119897119896V119896119897119895

+sum

119897

sum

119894

sum

119895

(minus119886119895+ 119905119892119895ℎ119894119897)119908119897119894119895

+sum

119904

sum

119897

sum

119895

(119887119895119904+ 119905119889119895119900119897119904) 119909119897119895119904

(1)

st sum

119894

sum

119899

119906119894119896119895119899

ge 119901119896119895 forall119896 119895 (2)

sum

119897

sum

119895

119908119897119894119895

+sum

119896

sum

119895

sum

119899

119906119894119896119895119899

le sum

119899

sum

119895

119888119894119895119899

119910119894119899

forall119894

(3)

sum

119897

V119896119897119895

le sum

119894

sum

119899

119906119894119896119895119899

forall119896 119895 (4)

119903119895sum

119896

V119896119897119895

le sum

119904

119909119897119895119904 forall119897 119895 (5)

sum

119896

sum

119895

V119896119897119895

le 119911119897sum

119895

1198881015840

119897119895 forall119897 (6)

sum

119896

V119896119897119895

= sum

119894

119908119897119894119895

+sum

119904

119909119897119895119904 forall119897 119895 (7)

sum

119897

sum

119895

119909119897119895119904

le 119892119904sum

119895

11988810158401015840

119895119904 forall119904 (8)

sum

119897

V119896119897119895

= 119902119896119895 forall119896 119895 (9)

sum

119899

119910119894119899

le 1 forall119894 (10)

119910119894119899 119911119897 119892119904isin 0 1 forall119894 119897 119899 119904 (11)

119906119894119896119895119899

V119896119897119895 119908119897119894119895 119909119897119895119904

ge 0 forall119894 119896 119897 119895 119899 119904 (12)



4 Solutions
















uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution








uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












plant 119894







center 119896




1198881015840


for product 119895

11988810158401015840


119895


Min TC

= sum

119894

sum

119899

119891119894119899119910119894119899+sum

119897

1198911015840

119897119911119897+sum

119904

11989110158401015840

119904119892119904

+sum

119894

sum

119896

sum

119895

sum

119899

(119898119894119895119899

+ 119905119896119895119890119894119896) 119906119894119896119895119899

+sum

119896

sum

119897

sum

119895

119905119886119895119889119897119896V119896119897119895

+sum

119897

sum

119894

sum

119895

(minus119886119895+ 119905119892119895ℎ119894119897)119908119897119894119895

+sum

119904

sum

119897

sum

119895

(119887119895119904+ 119905119889119895119900119897119904) 119909119897119895119904

(1)

st sum

119894

sum

119899

119906119894119896119895119899

ge 119901119896119895 forall119896 119895 (2)

sum

119897

sum

119895

119908119897119894119895

+sum

119896

sum

119895

sum

119899

119906119894119896119895119899

le sum

119899

sum

119895

119888119894119895119899

119910119894119899

forall119894

(3)

sum

119897

V119896119897119895

le sum

119894

sum

119899

119906119894119896119895119899

forall119896 119895 (4)

119903119895sum

119896

V119896119897119895

le sum

119904

119909119897119895119904 forall119897 119895 (5)

sum

119896

sum

119895

V119896119897119895

le 119911119897sum

119895

1198881015840

119897119895 forall119897 (6)

sum

119896

V119896119897119895

= sum

119894

119908119897119894119895

+sum

119904

119909119897119895119904 forall119897 119895 (7)

sum

119897

sum

119895

119909119897119895119904

le 119892119904sum

119895

11988810158401015840

119895119904 forall119904 (8)

sum

119897

V119896119897119895

= 119902119896119895 forall119896 119895 (9)

sum

119899

119910119894119899

le 1 forall119894 (10)

119910119894119899 119911119897 119892119904isin 0 1 forall119894 119897 119899 119904 (11)

119906119894119896119895119899

V119896119897119895 119908119897119894119895 119909119897119895119904

ge 0 forall119894 119896 119897 119895 119899 119904 (12)



4 Solutions
















uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution








uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























uikjn

klj

wlij

xlj

yin

zl

gs

A typical solution








uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










uikjn uikjn

klj klj

wlij wlij

xlj xlj

yin yin

zl zl

gs gs


998400klj 998400klj

w998400lij w998400

lij

x998400lj x998400lj

y998400in y998400in

z998400l z998400l

g998400s g998400s





119895119883119895119894

le


119895119894that is












119895119883119895119894


119895119894does




1015840

119895119894 The new 119883


formula

119883119895119894=

1198771198941198831015840

119895119894

sum1198951198831015840

119895119894

(13)

uikjn

klj

wlij

xlj

zl

gs

y998400in

uikjn

klj

wlij

xlj

yin

zl

gs

Parent Offspring


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119868119909prop TC

119909 (14)


119894≺ 119868119895) we pro-


1199091015840

119894= 119909119894+1205730119890minus120575119903

2(119909119895minus119909119894) + 120572120576 (15)






119903 =

10038171003817100381710038171003817119909119894minus119909119895

10038171003817100381710038171003817 (16)

















11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and

then sort them


Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Firefly algorithm




119892(119901119892cup1199011015840

119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while


finalga


119891)



(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


finalℎ


PHGF solution





Genetic algorithm

Firefly algorithm









1199011015840) rarr 119901

119892

End for ℎ(b) For 119899 = 1119867

(b1) For119898 = 1119867If 119868119899lt 119868119898


119891

End ifEnd for119898



119891) to produce




119891cup 11990110158401015840) rarr 119901

End while




119892=





Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Geneticalgorithm



No




119892



119892lt 119867119891 Hereon all




119892











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892


119891


fireflies)(c) For 119899 = 1119867

119891

(c1) For119898 = 1119867119891

If 119868119899lt 119868119898


119891

End ifEnd for119898

End for 119899

Firefly algorithm


Inputs 120573

1 minus 120573




to 119901119891(Added members + 119901

119891rarr 119901119892)

End while











11990110158401015840



119892cup 1199011015840

119892cup 11990110158401015840

119892) and



119892cup

11990110158401015840

119892) rarr 119901

119892

End for ℎ

End while



fireflies (119901119891)




(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(b1) For119898 = 1119867119891

If 119868119899lt 119868119898


End ifEnd for119898

End for 119899

End while


finalff


ℎ


WHGF solution








simUniform (100 150)119905119896119895119905119886119895119905119892119895119905119889119895

simUniform (04 07)119891119894119899

simUniform (50000 80000)1198911015840

119897simUniform (15000 25000)

11989110158401015840

119904simUniform (1000 5000)

119886119895

simUniform (20 50)119887119895119904

simUniform (1 3)119890119894119896ℎ119894119897119900119897119904119889119897119896

simUniform (50 1000)119901119896119895

simUniform (100 300)119902119896119895

simUniform (60 170)119903119895

simUniform (01 05)119888119894119895119899

simUniform (300 600)1198881015840

119897119895simUniform (50 150)

11988810158401015840

119904119895simUniform (50 100)



RPD =


1003816






Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Parameter 1 2 3

Genetic algorithm




Firefly algorithm






119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865











119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119872119875

119868GA 119875119878

119873119865

119871 119872119865

119868119865


321321321321321321321321

50

45

40

35

30

25

20

15

Mea

n of

Mea

ns



(a)

321 321 321 321 321 321 321 321

L


minus22

minus24

minus26

minus28

minus30

minus32

Mea

n of

SN

ratio



(b)


321 321 321 321 321 321 321

45

40

35

30

25

20

15

10

Mea

n of

Mea

ns


LIGA MF IFCP MP PS

(a)

321 321 321 321 321 321 321



Mea

n of

SN

ratio

s

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

LIGA MF IFCP MP PS

(b)


321321321321321321321321

Mea

n of

Mea

ns


45

40

35

30

25


(a)

321321321321321321321321




Mea

n of

SN

ratio

s


(b)



50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50

40

30

20

10

321321321321321321321321

Mea

n of

Mea



(a)

minus150

minus175

minus200

minus225

minus250

minus275

minus300

minus325

321 321321321321321321321



Mea

n of

SN

ratio

s


(b)



15

20

25

30times105


Small size








Medium size


10121416182022242628times106




Large size

10

15

20

25

30

35

40times107


6 Conclusion






1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 3 2 4 3 3 3 393576 393576 393576 393576 393576 3935762 4 3 5 3 4 5 698845 698845 698845 698845 698845 6988453 5 5 3 4 3 6 1067536 1064270 1064270 1065080 1068635 10670544 6 6 6 7 5 7 2158435 2148399 2148399 2201551 2399887 22025465 7 7 5 8 6 8 2858856 2814122 2748879 2857432 2987687 29766546 8 7 5 9 8 9 3636078 3607543 3606396 3767543 3896554 38655437 9 9 8 7 8 10 2158435 2158356 2148399 2201551 2376787 22597678 10 7 5 8 6 8 2858856 2814122 2748879 2857432 2976543 29055329 11 8 8 6 5 6 1742796 1684336 1678985 1701216 1865443 180121610 12 7 6 8 5 8 2737472 2719836 2700264 2735864 2865534 2798755

Medium size11 15 18 14 18 17 15 10996158 10979629 9774905 11995749 12924442 1195252512 16 14 15 16 15 12 8712242 8739928 8503092 8770480 10765551 1065544913 16 18 14 16 18 17 8156921 8006446 7509688 8175722 10678656 1065454814 17 13 17 12 17 13 7806508 7906765 7455443 7894554 11677166 1076615615 17 17 16 14 18 12 9793332 9845677 9325596 9776563 10976546 1061766516 18 19 15 15 12 14 9607417 9635377 9066544 9795875 11987761 1127676517 18 13 15 13 18 15 8977340 8764466 8053433 8657933 11876764 1248776518 20 25 24 23 22 25 20929984 20719064 13421243 21928467 29988761 2886655419 21 22 25 22 21 20 19332874 19664777 13766436 19147367 27745443 2687654720 22 20 24 23 22 25 20968413 21013803 13613456 22022560 30989866 2997754321 23 24 25 22 20 22 21257583 22043677 14075544 22290761 30445545 2999897722 24 22 24 24 24 23 24156236 24477472 14154453 25276767 31766544 3024331123 25 25 24 23 25 25 25132359 25632467 14879885 27698700 30775450 2913142524 27 28 25 22 26 21 23524389 20556578 14079875 23035363 28765433 27875432

Large size25 40 30 20 30 20 30 39350882 38687623 16568206 39137465 45243073 4489776526 50 35 25 35 30 35 53080191 54756375 35076364 58090324 61645455 5987645327 60 40 24 40 30 40 69001276 69013803 55613456 70022560 78865564 7765342228 70 45 25 45 35 45 86980117 85043677 60755440 89290761 95432121 9166553429 80 50 24 50 45 50 127136406 134477472 65554453 125276767 158654430 14854476830 90 55 24 55 50 55 129311201 125632467 71879885 127698767 175437681 17076644331 100 60 25 60 55 60 153461145 150556578 75479875 150353632 179866554 17868655432 110 65 36 65 60 65 200032401 212626371 91696575 200898322 308787651 29897663333 120 70 37 70 65 70 205869473 214363366 100689990 229642535 348776542 33676554334 130 75 38 75 60 75 235587893 233453673 115756433 245644424 358776523 35087564135 140 80 40 80 70 80 267022500 259080977 138777655 265543523 368776421 35797659236 160 90 41 90 80 90 336749110 326902525 147565442 344252522 408756455 40764432737 180 95 45 95 85 95 373113655 367699878 169676567 387668145 459896754 44908676538 190 95 45 100 80 95 392707538 382889301 181878668 437897798 480764533 47I86654339 200 100 50 100 90 100 410369288 402245224 199232433 405255598 489796875 47875453240 220 110 90 110 80 110 426265014 421533566 209631466 478326559 589886552 580875641



0031 1000Within groups 621811986413 54 115111986412

Total 6236E13 59




2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2998 0016Within groups 441311986415 78 565711986413

Total 5261E15 83



4137 0002Within groups 165911986418 90 184311986416

Total 2040E18 95









References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a hybrid approach to solve a model of...

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