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Applied Soft Computing 34 (2015) 655–677

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

uzzy adaptive genetic algorithm for multi-objective assembly linealancing problems

.H. Alavidoosta,∗, Mosahar Tarimoradia, M.H. Fazel Zarandia,b

Computational Intelligent Systems Laboratory, Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue,.O. Box 15875-4413, Tehran, IranKnowledge Intelligent Systems Laboratory, University of Toronto, Toronto, Canada

r t i c l e i n f o

rticle history:vailable online 10 June 2015

eywords:ssembly line balancingdaptive genetic algorithmvolutionary operatorsuzzy numbersuzzy controllerne-Fifth Success Rule

a b s t r a c t

This paper aims at multi-objective straight and U-shaped assembly line balancing problems with the fuzzytask processing times. The problems are referred to herein as f-SALBP and f-SULBP and the objectives thatare considered to be satisfied are: (a) minimizing the numbers of stations, (b) maximizing the fuzzy lineefficiency, (c) minimizing the fuzzy idleness percentage, and (d) minimizing the fuzzy smoothness index.In fact, the f-SALBP and f-SULBP are SALBP and SULBP generalization in fuzzy circumstance, respectively.Initially, the two problems are formulated and due to the uncertainty, variability and imprecision thatoften occurred in real-world production systems, the processing time of tasks are supposed as triangularfuzzy numbers. Then, to solve the problem, a hybrid multi-objective genetic algorithm is proposed. A One-Fifth Success Rule (OFSR) is deployed for the selection and mutation operators to improve the geneticalgorithm’s performance. The results in the genetic algorithm are being controlled in convergence and

diversity simultaneously by means of controlling the selective pressure (SP) and mutation rate. Likewise,a fuzzy controller to SP is employed for the OFSR toward a better implementation of the genetic algorithm.In addition, the Taguchi design of experiments is used for parameter control and calibration. Finally, thenumerical examples are presented to compare the performance of the proposed method with the existingones. The results show significantly better performance for the proposed algorithm.

© 2015 Elsevier B.V. All rights reserved.

. Introduction

The competitive market leads producers to promote their man-facturing systems by a more efficient and effective plan in a shorteriod of time. Thus, in the actual design of a manufacturing sys-em, programming an efficient assembly line continuously was anmportant and controversial issue in the past decades [1]. The man-facturing assembly line was introduced for the first time by Henryord in the early 1900s [2]. The assembly line balancing problemALBP) includes assigning the needed tasks for producing a prod-ct as series or batches to a set of stations, so that the objectiveunctions being optimized subject to the limitations [3]. From thisoint of view, the tasks sequence is the most important issue that

hould be considered in the assembly line development [4].

There are numerous reviews about ALBP in the literature, andhey are generally classified into two main types of Simple ALBP

∗ Corresponding author. Tel.: +98 9112116291; fax: +98 1135257759.E-mail addresses: mosatfa.alavidoost@aut.ac.ir (M.H. Alavidoost),

osahar@aut.ac.ir (M. Tarimoradi), zarandi@aut.ac.ir (M.H.F. Zarandi).

ttp://dx.doi.org/10.1016/j.asoc.2015.06.001568-4946/© 2015 Elsevier B.V. All rights reserved.

(SALBP) and Generalized ALBP (GALBP). GALBP versions have theextra features such as cost goals, station parallelization, mixed-model production, etc. in comparison with SALBPs [5]. From thegoal point of view, SALBP types are divided into SALBP-F, SALBP-1, SALBP-2, and SALBP-E. The SALBP-F is a feasibility problem fora given combination of time cycle and stations number. SALBP-1and SALBP-2 are the dual of each other, because the SALBP-1 goalis minimizing the station number for a given cycle time, while theSALBP-2 goal is minimizing the cycle time for a given stations num-ber. In the SALBP-E cycle, the time and stations number ought tobe minimized simultaneously so that efficiency can be maximized.In addition to the presented classification, the assembly lines canbe divided into two categories with respect to their layout, straightassembly lines and U-shaped assembly lines. The straight assem-bly line is considered as one of the most important traditional massproduction sections, and the U-shaped assembly lines are definedto reduce the costs and improve Just-In-Time (JIT) [6]. On the other

hand, they can be divided into single models and mixed modelswith respect to their types of products [3]. In the single model ofthe assembly line, only one product can be produced in the manu-facturing line, and others that can produce more than one product

6 Soft C

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pp[bapfÖebwpssmmmo[baabwtmtompYa[tndartostTsiotas

i

56 M.H. Alavidoost et al. / Applied

re called mixed model assembly lines. The SALB problem is a sin-le model for straight assembly line balancing and U-shaped layoutALB is called simple U-line balancing (SULB).

The ALBPs were proven to be NP-hard by Gutjahr andemhauser [7] and Ajenblit and Wainwright [8]. Therefore, accord-

ng to the difficulty of such problems, much effort has been made forhe development and expansion of heuristic methods such as theanked Positional Weighting Technique (RPWT), COMSOAL tech-ique [9], MALB technique [10], MUST technique [11], and LBHAethod [12], Critical Path Method (CPM) based approach [13], and

lso meta-heuristic methods such as genetic algorithm (GA) [8,14],imulated Annealing (SA) [15], Tabu Search (TS) [16,17], Parti-le Swarm Optimization (PSO) [18], and Ant Colony OptimizationACO) [19].

A multi-objective GA for solving the U-shaped assembly lineroblem was proposed by Hwang et al. [20], and they made a com-arison between straight and U-shaped assembly lines. Kim et al.21] rendered a mathematical model and GA for a two-sided assem-ly line. In Hwang and Katayama [22] work a multi-decision geneticpproach for solving mixed-model U-shaped lines has been pro-osed which was validated through a case study. A TS algorithmor solving the two-sided assembly line problem was prepared byzcan and Toklu [23] and the results were benchmarked by thexisting approaches. An adaptive GA for solving ALBP was offeredy Yu and Yin [24] in which their algorithm efficiency was provenith an example. In another noteworthy work, a hybrid GA wasroposed by Akpınar and Mirac Bayhan [25] and deployed forolving the ALB mixed model with parallel station and zoning con-traints. Kazemi et al. [26] proposed a two-stage GA for solvingixed-model U-shaped assembly lines. Nearchou [27] used a novelethod based on PSO for SALBP and compared it with the existingethod. Rabbani et al. [28] proposed a heuristic algorithm based

n GA for the mixed-model two-sided assembly line. Chang et al.29] focused on productivity in the printed circuit board assem-ly line and rendered a GA with external self-evolving multiplerchives in solving this problem. Chutima and Chimklai [30] used

PSO to solve the multi-objective two-sided mixed-model assem-ly line and showed that if their proposed algorithm was combinedith the local search, the quality of its solution set would be bet-

er. In another work, Purnomo et al. [31] offered a mathematicalodel for the two-sided assembly line and solved it with GA and

he iterative first-fit rule method, and finally compared the resultsf these methods. Manavizadeh et al. [32] proposed an SA for aixed model assembly U-line balancing type-I problem and com-

ared the algorithm results with the exact method. Hamzadayi andildiz [33] used an SA algorithm for the line balancing problemsnd modeled sequencing in U-shaped assembly lines. Dou et al.34] proposed a discrete PSO for solving SALBP-1 and comparedheir results with GA. Kalayci and Gupta [35] used a PSO with aeighborhood-based mutation operator for solving the sequence-ependent disassembly line balancing. Zha and Yu [36] proposed

hybrid ant colony algorithm for solving the U-line balancing andebalancing the problem and compared their algorithm results withhe existing methods. Among these meta-heuristic methods, mostf the studies were devoted to GA and these previous researchtudies have indicated that there must be sufficient motivationo use this popular algorithm for solving the emerging problem.o perform a controlled random search for identifying the optimalolution, an alternative traditional optimal technique was providedn the complex circumstances [37]. The focus of many researchersn GA and its popularity was the authors’ motivation to improvehe performance of this meta-heuristic through a modification as

part of the contribution of this paper and put it into practice toolve the mentioned controversial problem.

Numerous works have been reviewed that have solved ALBPsn crisp circumstances whilst actual world problems usually deal

omputing 34 (2015) 655–677

with uncertainty and vagueness. To represent uncertainty, fuzzynumbers can reflect the ambiguity of real data well. Considerableattention has been given to ALBPs in the literature, only some ofwhich have managed to solve such problems in the fuzzy envi-ronment. In other words, in comparison with crisp ALBPs, fewresearchers have focused on fuzzy ALBP so far [37,38]. Among thearticles focusing on solving the fuzzy ALBP by precise methods, fewresearches [39–42] are noticeable.

Studies in this area reveals those which have used heuristicand meta-heuristic methods for solving the ALBP in a fuzzy envi-ronment are rare. In the 90s Tsujimura et al. [43] and Gen et al.[44] initialized using fuzzy GA for this problem. With a typical GAprovided that the tasks processing time was presented in fuzzynumbers, they solved SALBP-1. While Brudaru and Valmar [45] pro-posed a combined GA with a Branch and Bound method to solveSALBP-1. Fonseca et al. [2] presented and modified the RankedPositional Weighting Technique and COMSOAL method with fuzzynumbers, and applied it to solve these sort of problems. Hop [46]proposed a heuristic method to solve a fuzzy mixed-model ALBP.Zhang et al. [47] prepared a heuristic method to solve SULBP withfuzzy numbers. Özbakır and Tapkan [48] presented a model for two-sided ALBP and solved this problem by Bees Algorithm. Zacharia andNearchou [49] also introduced a multi-objective GA to solve SALBP-2 with fuzzy numbers, in which they applied the weighted sum ofobjectives. Zacharia and Nearchou [50] presented a meta-heuristicalgorithm based on the genetic algorithm for solving SALBP-E.

As mentioned, since numerous researchers used GA and itspopularity, this paper tends to improve the performance of thisalgorithm through a modification. Likewise, it is noteworthy thatno research has considered and solved SULB-1 using meta-heuristicmethods in fuzzy circumstances. So this paper has considered theSALB-1 and SULB-1 in which a modified GA is presented with theOFSR that results in enhancing the performance. A fuzzy controllerfor better adaptation between GA and the OFSR has been renderedand also the parameters of the proposed algorithm have been cali-brated by the Taguchi design of experiments. Due to the uncertaintyin the real world, fuzzy numbers have been used to represent theassembly line cycle and processing time.

The rest of the paper is organized as follows: in Section 2, themain characteristics of SALBP and SULBP are represented. In Section3, the fuzzy arithmetic is provided as well as a number of criteria tosort the fuzzy numbers. To present the contribution of the geneticalgorithm, OFSR and also the procedure of genetic algorithm mod-ification with OFSR are presented in Section 4. In Section 5, theparameters of the proposed algorithm will initially be calibratedusing the Taguchi method, and then the proposed algorithm willbe examined by benchmarks and its results will be compared withthe existing methods. Finally, conclusions and some guidelines forfuture studies are provided in Section 6.

2. Problem formulation

This section represents the main characteristics of SALBP-1 andSULBP-1. As mentioned before, the assembly line is a series oflocations which are called stations, and a subset of tasks that areperformed and need to be processed for the production of a unit inthese locations [44]. For these problems, the available informationis as follows [51]:

• A given set of tasks J = {i|i = 1, 2, . . . n}.

• The set of tasks’ needed time which is shown as T = {ti|i =

1, 2, . . .n}.• Each task’s allocated time that will be presented as triangular

fuzzy number (TFN).

Soft Computing 34 (2015) 655–677 657

•

•

••

•

•••••••

sbsa⋃k

S

∑

csbttjd1eb

I

wSsStsa

t

c

I

Formulas (15)–(18) carry four basic operations on the closed

M.H. Alavidoost et al. / Applied

The set of precedence relations P = {(a, b|task a must be completedbefore taskb)}.Maximum allowed fuzzy cycle time (Cmax).

Symbols of this article are listed below:

ti: Fuzzy processing time that is represented by TFNs.Sk: Set of activities which are processed in station k Sk = {i|taski isprocessed at stationk} , ∀ k = 1, . . ., mt(Sk): Fuzzy time that every station needs to complete all therequired tasks.c: Assembly line’s fuzzy cycle time, i.e. max

k{t(Sk)}.

Cmax: Maximum allowed fuzzy cycle time.T: Total processing time.Ik: Fuzzy idle time for station Sk, (k = 1, . . ., m).E: Fuzzy balance efficiency.ID: Fuzzy idle percentage of assembly line.SX: Fuzzy smoothness index of assembly line.

In this problem, there are a number of stations which are pre-ented by a set as WS = {ws1, ws2, . . ., wsm}, and each task shoulde assigned only to one of these stations. In addition, the “J” sethould be allocated into stations, so that the limits of Eqs. (1)–(3)re satisfied [43,44]:

m

=1

Sk = J (1)

k

⋂k /= l

Sl = ∅ (2)

iεSk

ti ≤ Cmax k = 1, 2, . . ., m (3)

The first and second constraints guarantee that all tasks are allo-ated into the stations and each task will be allocated only to onetation. The third one ensures that each station cycle time will note greater than the maximum allowed fuzzy cycle time. In SALB,he jth work can be allocated to the kth station, only when its priorasks have been assigned to 1, 2,. . .,kth stations, whilst in SULB, theth task can be allocated to the kth station, only when all its pre-ecessor tasks and/or all its successor tasks have been allocated to, 2,. . .,kth stations [51]. Thus, in the tasks allocation, constraintquation (4) for SALB and constraint equation (5) for SULB shoulde met [51].

F(a, b) ∈ P, a ∈ Sk, b ∈ Sl, THEN k ≤ l, for all a; (4)

IF(a, b) ∈ P, a ∈ Sk, b ∈ Sl, THEN k ≤ l, for all a;

or, IF(b, c) ∈ P, b ∈ Sk, c ∈ Sr, THEN r ≤ k, for all c;(5)

Constraint (4) is defined for SALB and ensures its complianceith the predecessor constraints. Constraint (5) is also defined for

ULB, guaranteeing the compliance of at least one of the predeces-or or successor constraints. Besides the main goal of SALBP-1 andULBP-1, which is minimizing the number of stations, it is possibleo define other goals for comparing the solutions with the sametation numbers. According to the problem, the following resultsre as shown in Eqs. (6)–(12) [2,47]:

(Sk) =∑iεSk

ti, k = 1, . . ., m (6)

˜ = maxk

{t(Sk)} (7)

k = Cmax − t(Sk), k = 1, . . ., m (8)

Fig. 1. Triangular fuzzy number.

SX =

√√√√ m∑k=1

(Cmax − t(Sk))2

(9)

T =m∑k=1

t(Sk) (10)

E = T

m × c (11)

ID =

m∑k=1

(Cmax − t(Sk))

m × Cmax(12)

Eq. (6) calculates the fuzzy cycle time of each station and Eq. (7)calculates the fuzzy cycle time of the assembly line. Formula (8)calculates the fuzzy idle percentage of the assembly line. Formula(9) calculates the fuzzy smoothness index. By Eqs. (10) and (11),the fuzzy efficiency of assembly line, and by Eq. (12) the fuzzy idlepercentage of the assembly line could be calculated.

3. Fuzzy numbers

This section reviews the fuzzy arithmetic and a set of criteriato rank the fuzzy numbers. In this paper, as shown in Fig. 1, TFNsare used to present the processing time of the tasks. A TFN can becharacterized by three parameters A = (A1, A2, A3). Formula (13)represents a TFN membership function.

�A(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(x − A1

A2 − A1

), A1 ≤ x ≤ A2(

A3 − x

A3 − A2

), A2 ≤ x ≤ A3

0 O.W.

(13)

3.1. Fuzzy arithmetic

Basically, fuzzy arithmetic could be carried out based on aninterval arithmetic. In other words, fuzzy numbers should be trans-ferred from a singleton into an interval using �-cut and intervalarithmetic could be done. According to [52]. Suppose that the *donates four basic operations (+, − , · , &/), the interval arithmeticcould be defined by formula (14) as:

[d, e] ∗ [g, h] = {f ∗ u|d ≤ f ≤ e, g ≤ u ≤ h} (14)

intervals.

[d, e] + [g, h] = [d + g, e + h] (15)

[d, e] − [g, h] = [d − h, e − g] (16)

6 Soft Computing 34 (2015) 655–677

[

abaecs

A

A

A

0

3

mc

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58 M.H. Alavidoost et al. / Applied

d, e] · [g, h] = [min(d · g, d · h, e · g, e · h),

max(d · g, d · h, e · g, e · h)] (17)

[d, e][g, h]

=[min

(d

g,d

h,e

g,e

h

), max

(d

g,d

h,e

g,e

h

)]; 0 /∈ [g, h] (18)

Usually, using �-cut in computation leads to high complexitynd might produce non-linear results which increases the issuey itself. To overcome the trouble, some researchers have usedpproximation of the operation results to change it into a lin-ar form. Using such an approximation is a good method to faceomputational complexity, and preserves the fuzzy number shapeimultaneously. (Formulas (19)–(22)).

˜ + B = (A1 + B1, A2 + B2, A3 + B3) (19)

˜ − B = (A1 − B1, A2 − B2, A3 − B3) (20)

˜ × B = (min(A1 · B1, A1 · B3, A3 · B1, A3 · B3), A1 · B1,

max(A1 · B1, A1 · B3, A3 · B1, A3 · B3)) (21)

A

B=

(min

(A1

B1,A1

B3,A3

B1,A3

B3

),A1

B1, max

(A1

B1,A1

B3,A3

B1,A3

B3

));

˜ = (0, 0, 0) /∈ B (22)

.2. Ranking the fuzzy numbers

The operator ≤ is used for comparing two fuzzy numbers in for-ula (3) whilst for comparison and TFNs ranking, the following

riteria will be used for prioritization [53]:

Criterion 1: The number is greater which in terms of the threepoints, the weighted average (Beginning, Peak, End) is greater(F1 by Eq. (23)).Criterion 2: The number is greater which in the midpoint, isgreater (F2 by Eq. (24)).Criterion 3: The number is greater which in terms of distancebetween the beginning and end point is greater (F3 by Eq. (25)).

1 = A1 + 2 × A2 + A3

4(23)

2 = A2 (24)

3 = A3 − A1 (25)

For comparing TFNs, initially, criterion1 (Eq. (23)) is used, andf the first criterion cannot determine the major TFN, so criterion2Eq. (24)) is used, and so on.

. Solution procedure

To describe the procedure in this section, first the main steps in typical GA would be considered and after that, the adaptivity ofhe operators and also Diversity and Convergence Control for theroposed algorithm for SALBP and SULBP would be presented.

.1. Genetic algorithm

Genetic algorithm [54] is a popular meta-heuristic algorithm.he majority of GAs consists of the following steps:

Fig. 2. General diagram of genetic algorithm [54].

Procedure: Typical GABegin

Step 1. Determine population size (nPop), crossover rate (Pc), and themutation rate (Pm) (Pc and Pm are the ratio of childes which are generatedby the crossover and mutation operations, respectively).Step 2. Generate initial population, using random numbers.Step 3. Calculate fitness function for each chromosome.Step 4. In case of satisfying the stopping criteria, the algorithm stops,otherwise it goes to step 5.Step 5. Perform crossover (and mutation) on the Pc% (and Pm%) of thegeneration.Step 6. Create the new generation.Step 7. Repeat Step 3 and Step 4.

END

The GA’s general diagram is displayed in Fig. 2.If the GA operators are defined properly and adapted to the prob-

lem well, this algorithm could be efficient to solve the problem. So,first of all, the algorithm operators ought to be defined for the ALBP.

4.2. Chromosome encoding

There are different ways for chromosome encoding, but a simpleand popular method in ALBP, is strings of integers (see [37] for a sur-vey). Following the tradition, in this paper the integer permutationbetween [1, the number of tasks] is deployed. Such a permutationrepresents the priority of the tasks. Suppose an 8-task ALBP withits predecessor and successor relations is presented as Fig. 3. Thedigit in each circle is a task number and what is coped above is thefuzzy processing time in terms of a TFN. Fig. 4 exposes a case forgenotype of chromosome. In this example, the highest priority tobe assigned is for the task 6 (first gene), and it is followed by task 3(second gene), etc. (Fig. 4).

4.3. Chromosome decoding

Here, chromosome decoding induces tasks assignment into thestations. For chromosome decoding in ALBP, numerous methodsare primed [43,44,55,56]. Many of these methods use chromosomes

M.H. Alavidoost et al. / Applied Soft C

Fig. 3. Predecessor constraints graph for an 8-task ALBP.

tcwspnmmhcf

semdcctt(3shtbp

aber (Crossover Point) from [1, n − 1] interval, parents divide into

Fig. 4. An example for genotype of chromosome.

hat have not violated precedence constraints. These kinds ofhromosomes are called feasible sequence chromosomes. Theireakness will be evolved when they have got infeasible in

equence. The repairing mechanism to make them feasible is theath that GAs overcome to face this problem. But repairing doesot necessarily produce a unique sequence. This paper uses chro-osome decoding to get rid of the repairing mechanism. In thisethod, only a set of tasks that have no predecessor manages to

ave the chance to be assigned into the stations. Thus, precedenceonstraints are observed just in assigning and decoding. The stepsor chromosome decoding are as follows:

Procedure: Chromosome Decoding for ALBPBegin

Step 1. Set: k = 1, N =∅.Step 2. Define candidate task set (N) that are assignable into Sk .

For SALBP:N = {i|taski have no predecessor or its predecessor have been assignedformerly}

For SULBP:N = {i|taski have no predecessor/successor or its predecessor/successorhave been assigned formerly}Step 3. Determine the task of j in the set of N, which has captured thehighest priority for assigning (the priority is according to their order inchromosome).Step 4. IF

∑iεSk

(ti) + tj ≤ Cmax , THEN the task of j would be assigned into

Sk , otherwise a new station would be needed and k = k + 1, and also the taskof j would be assigned into a new station.Step 5. IF there is no task for assigning, go to step 2.Step 6. Return decoded chromosome.

END

With this method, a chromosome which is a permutation ofome numbers could be decoded. Decoding for the mentionedxample is though that for commence, this is task 1 (N = {1}) thatanaged to be assigned into the first station since it has no pre-

ecessor. After that, the N has to be updated and a couple ofandidates, the tasks 2 and 3 (N = {2,3}) are considered. Task 3 inomparison with task 2 has higher priority in the genotype, so it isask 3 that catches the chance to be assigned. The total processingime for task 1 and task 3 does not exceed the Cmax(10, 12, 14) ≤12, 15, 18), so there is no use in adding a new station and task

would be assigned into the first station (S1 = {1,3}). In the nexttep, tasks 2, 5, and 7 are assignable (N = {2,5,7}). Task 5 has theighest priority between them, and since assigning this task into

he first station leads to exceeding Cmax, thus a news station woulde needed (S2) and task 5 could be assigned into this station. Thisrocedure has to be continued until no task lasts unassigned in a

omputing 34 (2015) 655–677 659

proper station. Fig. 5 represents the chromosome decoding for thementioned example.

The decoding procedure for SULB is similar to what was pro-posed for SALB, but here in each step, those tasks are candidated tobe assigned which have no predecessors or successors. Fig. 6 repre-sents a part of this procedure for SULB in the mentioned example.

4.4. Fitness evaluation

Having decoded the solutions in hand, and subject to that themain goal to minimize the station numbers, the chromosome thathas the lower stations would be labeled with higher fitness. How-ever, just in case two are the same in station numbers, the nextgoals are maximizing E, minimizing ID, and minimizing SX (Formula(26)).⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩F1 ∝ 1/m

F2 ∝ Defuzzified{E}

F3 ∝ 1/Defuzzified{ID}

F3 ∝ 1/Defuzzified{SX}

(26)

where m donates the station numbers, F1, F2, F3 and F4 respectivelyare first, second, third, and fourth objective functions. To defuzzifythe E, ID, and SX the formula (23) comes in handy. Also, to deter-mine the fitness function for each chromosome, the weighted-sumof objectives by formula (27) is useful (fv is the fitness of the vthchromosome).

fv = {w1 × nomailzed(F1)} + {w2 × nomailzed(F2)}+ {w3 × nomailzed(F3)} + {w4 × nomailzed(F4)} (27)

The equilibrium between wi, should be though that the first goalimportance be more than the others, so let us establish it as w1 �w2 = w3 = w4.

4.5. Initial generation

The initial generation is produced randomly and using permu-tation of an integer between 1 and the number of the tasks.

4.6. Evolutionary operators

GA, with exploration in solution space and exploitation on thecaught solutions, primes optimum solution for the problem. Thegreater the focus on exploration, the weaker convergence ratethe algorithm would reach, and on the other hand, the morefocus on the exploitation, the more decrease in diversity it couldhave. Summing up, to meet a good solution in a feasible time, itneeds a tradeoff between them. GA deploys its main operators, i.e.crossover and mutation for exploration and exploitation, respec-tively. So, well-prepared operators lead to higher performance forthe algorithm.

4.6.1. CrossoverThere are different methods for this operator as single-

point crossover (SPX), double-point crossover (DPX), and uniformcrossover (UX).

. SPX: to implement this type of crossover, using a random num-

two parts. The offspring inherits the first part from one of themdirectly and the other half from the other parent. Note that to pre-vent elimination or repetition of the numbers, they inherit the

660 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Fig. 5. Decoding example for SALB problem.

Fig. 6. A case of chromosome decoding for the SULB mentioned example.

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 661

over:

b

c

4

mMM

b

Fig. 7. An example for cross

second part indirectly, i.e. they would own a numerical sequence(Fig. 7).

. DPX: to deploy this crossover, using two random numbers (Point1 and Point 2) from [1, n − 1] interval, the parents divide intothree parts. Here offspring inherit their first and third part fromone of the parents and the middle from another parent indirectlyi.e. numerical sequence (Fig. 7).

. UX: to use this one, a random array which is Boolean valuedand its length is equal to the number of the genes should beproduced. For each ˛i = 1 the offspring inherits the correspondinggene directly from one of the parents and in cases that ˛i = 0 thecorresponding gene would be inherited from the other parentindirectly i.e. by the numerical sequence (Fig. 7c).

.6.2. MutationThere are different methods for the mutation. Integer numerical

ethods, which are used in this paper are Swap Mutation (SWM),ultiple-Swap Mutation (MSM), Reverse Mutation (RM), and Shiftutation (ShM).

a. SWM: selection of two genes and change them (Fig. 8a).. MSM: this mutation is a generalized type of SWM so that the

SWM operators act on chromosomes several times (Fig. 8b). Foreach MSM, the number of times that SWM acts on chromosomescould be calculated using formula (28).

nSWM =⌈⌊CL

2

⌋× �

⌉(28)

where, nSWM is the number of SWM, CL donates chromosomelength, and � represents the mutation rate (between 0 and 1).Note that

⌊�⌋

and⌈�⌉

are floor and ceiling of the � respectively.Clearly, the bigger � tends to an increase in exploration and alsothe smaller � tends to an increase in algorithm exploitation.

c. RM: a gene would be randomly selected and using formula (29),

the place of the second gene would be determined and after thatthese genes would be reversed (Fig. 9a).

ML =⌈CL × �

⌉(29)

(a) SPX, (b) DPX, and (c) UX.

where ML represents the mutation length and its value changeswith � change.

d. ShM: selects two genes and transports the second gene into thefirst neighborhood (Fig. 9b).

4.7. Diversity and Convergence Control

A point is considered throughout the contribution of this paperis to propose a method to control diversity and convergence inGA so that it improves the GA performance. Formerly mentioned,one of the diversity control methods is using evolutionary opera-tors (i.e. crossover and mutation). In addition to these operators,the selection operator could to be effective in the diversity con-trol. This paper uses two methods for diversity control, mutationrate control (using OFSR and its implementation in integer num-bers) and SP control (with a formula for SP control and using OFSRand also a fuzzy controller, the diversity and convergence would becontrolled). Note that the mutation rate and SP control effect themutation and selection operator, respectively.

4.7.1. One-Fifth Success Rule (OFSR)OFSR is initially introduced by Rechenberg [57] for the Evolu-

tionary Strategies (ES) algorithm (it is a meta-heuristic method perse). Similar to GA, ESs use mutation and crossover of chromosomesfor evolution of the generations. Each chromosome is presentedas ( x1, x2, . . ., xn, �) in which xjs are the problem variablesand presented by real numbers, and � is the mutation step length.Rechenberg [57] had mathematically proven OFSR for the ESs witha chromosome and a child. This rule says that if the ratio of thenumber of successful mutations rather than the number of totalmutations is equal to one-fifth, then the convergence rate to theoptimum solution will be the maximum rate. For mutation in ES,

several methods are proposed. One of them is adding a randomnormal value to all genes as in formula (30).

x′j = xj + N(0, �) (30)

662 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Fig. 8. An example for mutation: (a) SWM and (b) MSM.

utatio

Og

•

•

•

cirdticsg(etta

4

mptO

•

Fig. 9. An example for m

The value of � is determined by the OFSR during the algorithm.FSR will follow three conditions to achieve the maximum conver-ence rate:

Mode 1: While the probability of success in past k-populations isequal to one-fifth, the mutation step length (�) will not change.Mode 2: While the probability of success in past k-populations ismore than one-fifth, the mutation step length (�) will increase.Mode 3: While the probability of success in past k-populations isless than one-fifth, the mutation step length (�) will decrease.

In the cited modes, mode 2 is used for preventing the prematureonvergence by creating diversity in the generations and mode 3s used for increasing the speed of convergence. In fact, this algo-ithm considers diversity and convergence, simultaneously. If theiversity of chromosomes is low, to escape the local optimum,he step length to search a wider area of the solution space isncreased here and if the convergence of chromosomes is low, foronverge chromosomes to reach an optimum solution, the searchpace is narrowed by reducing the step length. The general dia-ram of the OFSR is shown in Fig. 10. C is a constant, is equal to1/past k − populations), Ps is the probability of success in past k gen-rations, t is the iteration number, Xt presents the chromosomes athe time of t, F(Xt) donates to the chromosome fitness function athe time of t, �0 and � are the standard deviation from the first stepnd next steps, respectively.

.7.2. Controlling the mutation rateSince OFSR could be deployed for real variables, in this paper a

echanism is presented so that using OFSR for integer variables beossible. In this paper, mutation rate (�) control is used instead of

he mutation step length. So, according to the presented facts forFSR, four rules for controlling � come in handy:

Rule 1: IF Ps = 1/5, THEN � t+1 = � t.

n: (a) RM and (b) SHM.

• Rule 2: IF Ps > 1/5, THEN � t+1 = C × � t.• Rule 3: IF Ps < 1/5 AND (� t/C) < 1, THEN � t+1 = � t/C.• Rule 4: IF Ps < 1/5 AND (� t/C) > 1, THEN � t+1 = 1.

Note that, � and C are numbers lower than 1 and to increase the� value, we need its product with C (Rule 2). In addition, the � valuecould not be bigger than unit, so if the value of �/C is greater than1, the � value should be 1 (Rule 4). The Ps update method is thoughthat presented in the prior section.

4.7.3. SP controllerAs mentioned before, the OFSR considers diversity and con-

vergence simultaneously for a better search. Also in GA, for asimultaneous influence on diversity and convergence, one coulduse SP (the probability of selecting the best member of the popu-lation, divided into the average probabilities of selecting the othermembers of the population). In other words, by controlling the SP,diversity and convergence could be optimum simultaneously. To doso, the selection operator must be defined according to the fitnessand SP that has to be entered in the selective operator. In this paper,to make a link between fitness of each chromosome and its selec-tion probability, the Boltzmann method [58] is used as in Eq. (31)(SP is the selective pressure, fv is the fitness of the vth chromosome,and Pv is the probability of selecting the vth chromosome).

Pv ∝ eSP×fv (31)

With regard to Eq. (31), more fitness of the chromosome meansmore probability of selection of the chromosome. Summation ofall probabilities ought to be equal to one. Thus, the probability ofselecting each chromosome is divided by the summation of proba-

bilities (Eq. (32), N is the population size).

Pv = eSP×fv∑Nv=1e

SP×fv(32)

M.H. Alavidoost et al. / Applied Soft C

acahbt

W

Fig. 10. General diagram for OFSR.

In order to be able to successfully enter the OFSR in the prob-bilities, initially sort the existing population on the basis ofhromosome fitness. Then, by increasing or decreasing the SP, anttempt is made that the ratio of the total probability of the bottomalf of the population (the Weaker Half Probability (WHP)) shoulde equal to one-fifth. On the other hand, invoking the OFSR tendso the tuned SP (formula (33)).

HP =N∑

v=[N/2]+1

Pv = 15

(33)

Fig. 11. Fuzzy terms MFs: (a) two linear MF and a Triangular

omputing 34 (2015) 655–677 663

However, due to the continuous nature of the solution space,reaching the number of one-fifth is difficult (or even impossible).There are different ways to control a variable in a continuous space.The method that is deployed in this paper as a popular methodis fuzzy logic, presented by Zadeh [59]. The fuzzy logic helps usto model the knowledge in terms of fuzzy IF-THEN rules [60].The fuzzy set theory proposes a systemic approach by definingmembership functions (MFs) for computations related to linguisticinformation that improves the numerical experiments. As a resultof the meta-heuristics good performance and their importance insolving NP-hard problems, some researchers have used methods toimprove their performance using fuzzy logic, and to control thesealgorithms parameters are a case of their efforts [61,62]. In thispaper fuzzy terms and fuzzy rules are applied as the SP controller.The used fuzzy terms are defined as “Small”, “Good”, and “Big”.

If the WHP is small, the SP should be limited (Formula (34)) oth-erwise, it could be aroused (Formula (35)), (DoF = degree of firing).

IF WHP is Small; THEN SPt+1

= SPt ×{

1 +(WHP − 1

5

)× DoFSmall

}(34)

IF WHP is Big; THEN SPt+1

= SPt ×{

1 +(WHP − 1

5

)× DoFBig

}(35)

As seen in rules (34) and (35), more distance between WHPand one-fifth leads to more SP variation in the direction of one-fifth. Also, DoF helps for the lower fluctuation and SP convergence.In addition, the use of momentum can be useful to increase theconvergence celerity (Formula (36) and (37)).

IF WHP is Small; THEN SPt+1

= SPt ×{

1 +(WHP − 1

5

)× DoFSmall

}+ ˛(�SPt) (36)

IF WHP is Big; THEN SPt+1

= SPt ×{

1 +(WHP − 1

5

)× DoFBig

}+ ˛(�SPt) (37)

The � presents the momentum in formula (36) and (37). In fact,the variation of SP and its direction in iterations (�SPt) have an

effect on SP in the next iteration (SPt+1) that tends to convergencecelerity. Defined fuzzy rules have an effect on SP iteratively, whilstthe WHP satisfies the defined fuzzy term of good. Scilicet, WHP isapproximately being equal to the (1/5)(WHP ∼= (1/5)).

MF; (b) reverse sigmoid MF, sigmoid MF, Gaussian MF.

664 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Fig. 12. Selective pressure for two sets MFs through a hundred runs.

Table 1Parameters and their levels.

Class Parameter Symbol Level

Class A

Population size nPopA nPopA(1): 10, nPopA(2): 20, nPopA(3): 30Maximum number of iteration ItrA ItrA(1): 50, ItrA(2): 75, ItrA(3): 100Crossover rate ˇA ˇA(1) : 0.7, ˇA(2) : 0.8, ˇA(3) : 0.9Mutation rate �A �A(1) : 0.1, �A(2) : 0.15, �A(3) : 0.2

Class B

Population size nPopB nPopB(1): 30, nPopB(2): 50, nPopB(3): 70Maximum number of iteration ItrB ItrB(1): 150, ItrB(2): 200, ItrB(3): 250Crossover rate ˇB ˇB(1) : 0.7, ˇB(2) : 0.8, ˇB(3) : 0.9Mutation rate �B �B(1) : 0.1, �B(2) : 0.15, �B(3) : 0.2

Class C

Population size nPopC nPopC(1): 70, nPopC(2): 100, nPopC(3): 130Maximum number of iteration ItrC ItrC(1): 300, ItrC(2): 400, ItrC(3): 500Crossover rate ˇC ˇC(1) : 0.7, ˇC(2) : 0.8, ˇC(3) : 0.9

oataf

scot

Mutation rate

Each fuzzy term has a corresponding MF. In this paper two typesf MFs are deployed and their performances are considered. One is

set of MFs consisting of two linear MFs with a Triangular MF, andhe other one is a set that consists of a Gaussian MF, a sigmoid MF,nd a reverse sigmoid MF. Fig. 11 presents MFs for the mentioneduzzy terms.

Generally, for fuzzy controllers, sigmoid and Gaussian member-

hip functions (MFs) have smoother and better performances inomparison with linear and Triangular MFs. To compare these typesf MFs, their performances are simulated and the results show bet-er performance sigmoid and Gaussian MFs. Fig. 12 presents caught

Fig. 13. Mean of means and S/N ratio

�C �C(1) : 0.1, �C(2) : 0.15, �C(3) : 0.2

SP for the two types in 100 runs. As shown, despite the fact thatboth of them have controlled the SP in an interval of around 0.2,the controller with sigmoid and Gaussian MF shows much betterperformance (either in proximity with 0.2 and fluctuation in trend)than the other one.

5. Comparison

In this section, first of all, the parameter which controls themechanism would be considered and after that the proposed mod-ified GA would be benchmarked with the standard functions. Then,

for each parameter in class A.

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 665

Table 2Selected level of the parameters.

Parameter Symbol Class A Class B Class C

Population size nPopA nPopA(3): 30 nPopB(3): 70 nPopC(3): 130Maximum number of iteration ItrA ItrA(1): 50 ItrB(2): 200 ItrC(3): 500Crossover rate ˇA ˇA(2) : 0.8 ˇB(1) : 0.7 ˇC(3) : 0.9

�A(3

tar

5

retdw

ctild

Mutation rate �A

he proposed algorithm is examined with benchmarks of SALBP-1nd SULBP-1. Finally, the comparison between the proposed algo-ithm and the existing methods would be rendered.

.1. Problem parameters control using Taguchi method

There are various methods to calibrate the meta-heuristic algo-ithm parameters, some of which are full factorial design, i.e. theyxamine all possible combinations [63,64], which are intrinsicallyime and cost consuming. The Taguchi method [65] uses a specialesign of orthogonal arrays to study the entire parameters spaceith a small number of experiments.

The Taguchi method clusters the factors into two main groups:ontrollable and noise factors (uncontrollable). Since noise of fac-

ors are uncontrollable, their elimination is unpractical and almostmpossible, and the Taguchi method tries to reach the best control-able factors level from a robustness point of view. In addition toetermine the best factors level, Taguchi method establishes the

Fig. 14. Mean of means and S/N ratio

Fig. 15. Mean of means and S/N ratio

) : 0.2 �B(1) : 0.1 �C(2) : 0.15

relative importance of each factor with respect to its main impactson the objective function [66]. To analyze the experimental dataand find optimal factor combination, the Taguchi method uses acriterion entitled Signal-to-Noise (S/N) ratio which is expected tobe maximum.

The Taguchi method divides objective functions into threegroups:

The smaller the better: In case that approaching the objectivefunction value to zero is better, comes in handy. In this situation,S/N ratio would be calculated by formula (38), (e) determines thenumber of experiment, Obje is the objective function value in theeth experiment, and nExP is the number of parameters combinationwhich should be examined.

SN

ratio = −10log10

(∑e

Obj2enExp

)(38)

for each parameter in class B.

for each parameter in class C.

666 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Table 3Standard functions [68].

fb

Table 4Methods’ details.

Selection method Parent selection New generation

Roulette Wheel –Roulette Wheel

Roulette Wheel (fitnessproportion selection)

Roulette Wheel and Elitismselection

Tournament –Tournament

Tournament Tournament and Elitismselection

Random – RouletteWheel

Random selection Roulette Wheel and Elitismselection

Random – Tournament Random selection Tournament and Elitismselection

Random – Random Random selection Random selection andarchive

Random – OFSR withliner and TriangularMFs

Random selection Selection Base ScaledFitness Proportion (OFSR)and Elitism selection

Maxsol − Minsol

After transforming the objective values into RDIs, the RDI shouldbe calculated and changed into S/N ratio. To find the optimal factorcombination, both S/N ratio and RDI are needed to be maximized

Table 6Summarized reached solutions for SALBP-1.

%Optimalsolution

Maximum of%deviation

Average of%deviation

Problem class

100 0.00 0.00 Class A93.39 14.29 0.87 Class B65.83 14.29 2.53 Class C

86.41 14.29 1.13 Total

Table 7Summarized reached solutions for SULBP-1.

%Optimalsolution

Maximum of%deviation

Average of%deviation

Problem class

100 0.00 0.00 Class A73.54 16.67 2.93 Class B59.31 20.00 4.01 Class C

TS

The larger the better: In case that the upper value of objectiveunction is better, comes in handy. In this situation, S/N ratio woulde calculated by formula (39).

S[

( ¯Obj2/s2)

]

N

ratio = −10log10 nExp(39)

able 5elected benchmarks for each class (A–C).

Class A Class B

Benchmark name Benchmark size Benchmark name

Mertens 7 Roszieg

Bowman 8 Heskiaoff

Jaeschke 9 Buxey

Jackson 11 Sawyer

Mansoor 11 Lutz1

Mitchell 21 Gunther

Random – OFSR withSigmoid andGaussian MFs

Random selection Selection Base ScaledFitness Proportion (OFSR)and Elitism selection

Nominal is the best: In case that there is a specific target valuefor objective function, comes in handy. In this situation, S/N ratiowould be calculated by formula (40).

SN

ratio = −10log10

[∑e

(1/Obj2e )nExp

](40)

Note that since objective values in each instance are not co-dimensional, they should be transformed. For changing the indexesto a digit with no dimension, Related Deviation Index (RDI) [67] hasbeen used (Formula (41)).

RDI =∣∣Algsol − Bestsol

∣∣(41)

77.62 20.00 2.31 Total

Class C

Benchmark size Benchmark name Benchmark size

25 Kilbridge 4528 Hahn 5329 Warnecke 5830 Tonge 7032 Wee-Mag 7535 Arcus1 83

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 667

-260

-210

-160

-110

-60

-10

1 201 401 601 801 1001 1201 1401 1601 1801

Log

of

obje

ctiv

e

IterationRandom-OFSR Tri Random-OFSR Gaus Random-RouletteRandom -Tournament Roulette-R oulette To urna ment- Tour

Fig. 16. Algorithms results for De Jong’s function (sphere model).

-0.5

0

0.5

1

1.5

2

1 201 401 601 801 1001 1201 1401 1601 1801

Log

of

obje

ctiv

e

IterationRandom-OFSR Tri Random-OFS R Gaus Random-RouletteRandom-Tournament Roulette-Roulette Tournam ent- Tour

for Ro

rfmwnp

umtca

Fig. 17. Algorithms results

egardless of their objective function value. Summing up, a levelor parameters should be selected and in this level, S/N ratio has

aximum value while RDI has the minimum value in comparisonith the other levels, and just in case a level of these criteria areot satisfied simultaneously, another experiment for that specificarameter should be designed [55].

Controllable factors which are selected for this portion are pop-lation size, maximum number of iteration, crossover rate, andutation rate. The proposed algorithm has been examined with

hree types of benchmarks (benchmarks were classified into threelasses of A, B, and C according to their size). So, for every factor,ccording to the benchmark size, three levels were considered for

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

1 201 4

Log

of

obje

ctiv

e

IteraRandom -OFSR Tri Ra ndom Random -Tourna ment Roulette

Fig. 18. Algorithms results

senbrock’s valley function.

which each level value was obtained relying on the trial and error(Table 1).

The Minitab 17 was used for Taguchi method implementation.The Taguchi experiments for each three class of A, B, and C havebeen carried out separately. The S/N ratio and means criteria for A,B, and C classes are exposed in Figs. 13–15 respectively.

As it is clear in Fig. 13, for nPopA and �A the third levelshould be selected, because in comparison with the other lev-els, S/N ratio has maximum value and the means has minimum

value in this level. But for the determination of ItrA and ˇA, anextra experiment should be designed. The same analysis comes inhandy for Figs. 14 and 15. Using the exposed diagram and after

01 601 801

tion-OFSR G aus Ra ndom-R oulette-R oulette To urna ment- Tour

for Beale’s function.

668 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1 11 21 31 41 51 61 71

Log

of

obje

ctiv

e

IterationRandom -OFSR Tri Ra ndom -OFSR G aus Ra ndom-R oulette

oulett

s for G

cp

5

ril

Random -Tourna ment R

Fig. 19. Algorithms result

omplimentary experiments, the selected levels for parameters areresented as Table 2.

.2. Numerical results over evolutionary standard benchmarks

The proposed algorithm could be benchmarked through theendered algorithms using standard functions and this is alsonvoked for benchmarking the proposed modified GA with theisted standard function in Table 3 [68].

-34

-29

-24

-19

-14

-9

-4

1

1 201 401

Log

of

obje

ctiv

e

IterRandom -OFSR Tri Ra ndom Random -Tournament Roulette

Fig. 20. Algorithms results

-2

-1.5

-1

-0.5

0

0.5

1

1 201 401 601 801 1

Log

of

obje

ctiv

e

IterRandom -OFSR Tri Ra ndom Random -Tourna ment Roulette

Fig. 21. Algorithms results

e-Roulette To urna ment- Tour

oldstein–Price’s function.

The proposed GA improves the performance with respect to theselection operator. The preliminary tests on the SP controller showthat the best performance of the proposed GA is prepared when forselecting the individual of the new generation, the SP controlleris deployed and for parent selection (in evolutionary operators)

the Random selection is used. SP controller is compared with theother methods. There are several methods for the selection in whichRoulette Wheel, Tournament, and Random are compared with theSP controller (Table 4).

601 801 1001

ation-OFSR G aus Ra ndom-R oulette-Roulette To urnament- Tour

for Booth’s function.

001 1201 1401 1601 1801

ation-OFSR G aus Ra ndom-R oulette-Roulette To urna ment- Tour

for Bukin’s function.

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 669

-16

-14

-12

-10

-8

-6

-4

-2

0

1 21 41 61 81 101 121 141 161

Log

of

obje

ctiv

e

IterationRandom -OFSR Tri Ra ndom -OFSR G aus Ra ndom-R ouletteRandom-Tournament Roulette-Roulette Tournament-Tour

Fig. 22. Algorithms results for Ackley’s function.

-17

-15

-13

-11

-9

-7

-5

-3

-1

1 21 41 61 81 101 121 141 161 181 201 221 241 261 281 301 321 341 361 381

Log

of

obje

ctiv

e

IterationRandom -OFSR Tri Ra ndom -OFSR G aus Ra ndom-R ouletteRandom -Tournament Roulette-R oulette To urna ment- Tour

sults

b

vii

Fig. 23. Algorithms re

Figs. 16–24 present means in 10 runs of each method for eachenchmark.

As it can be observed in Figs. 16–24 from the quality point of

iew, the two proposed methods (SP controller with two sets MFs)n many of the benchmarks are converged into better results. Its only for Rosenbrock that the Tournament selection produced

-0.05

0.45

0.95

1.45

1.95

1 2 3 4 5 6 7 8 9 10

Log

of

obje

ctiv

e

IterRandom -OFSR Tri Ra ndom Random -Tournament Roulette

Fig. 24. Algorithms results

for Matyas’s function.

the better results after a couple of thousands of runs. From theconvergence rate point of view, the proposed methods are betterthan Random and Roulette Wheel but rather than Tournament it

is lower in some of its functions. As mentioned before, focusing onconvergence makes the convergence rate decreased and stoppingthe algorithm in the local solution and also focusing on diversity

11 12 13 14 15 16 17 18 19 20 21

ation-OFSR G aus Ra ndom-R oulette-Roulette To urna ment- Tour

for Easom’s function.

670 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Table 8Result of proposed algorithm for fuzzy SALBP-1.

Benchmark Optimal values fordeterministic SALBP-1

Result of proposed GA for fuzzy SALBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

Mertens

7 6 6 6 (5.80, 6.00, 6.20) (6.67, 19.44, 45.67) (0.76, 0.81, 0.85) (1.16, 3.87, 6.16)7 7 5 5 (6.80, 7.00, 7.20) (4.50, 17.14, 39.00) (0.79, 0.83, 0.87) (0.00, 3.16, 5.47)7 8 5 5 (6.80, 7.00, 7.20) (12.22, 27.50, 47.71) (0.79, 0.83, 0.87) (2.87, 5.20, 7.64)7 10 3 3 (9.80, 10.00, 10.20) (0.00, 3.33, 17.41) (0.92, 0.97, 1.00) (0.00, 1.00, 2.86)7 15 2 2 (14.70, 15.00, 15.30) (0.00, 3.33, 13.21) (0.92, 0.97, 1.00) (0.00, 1.00, 2.73)7 18 2 2 (14.70, 15.00, 15.30) (11.32, 19.44, 28.53) (0.92, 0.97, 1.00) (3.11, 5.00, 6.90)

Bowman 8 20 5 5 (16.80, 17.00, 17.20) (18.29, 25.00, 32.42) (0.86, 0.88, 0.90) (9.91, 12.21, 14.61)

Jaeschke

9 6 8 8 (5.80, 6.00, 6.20) (8.21, 22.92, 49.75) (0.73, 0.77, 0.82) (1.72, 4.80, 7.56)9 7 7 7 (6.80, 7.00, 7.20) (10.00, 24.49, 47.38) (0.72, 0.76, 0.80) (2.60, 5.29, 7.97)9 8 6 6 (7.80, 8.00, 8.20) (10.19, 22.92, 42.62) (0.73, 0.77, 0.81) (2.61, 5.20, 7.73)9 10 4 4 (9.70, 10.00, 10.30) (1.82, 7.50, 21.94) (0.88, 0.93, 0.98) (0.00, 2.24, 4.27)9 18 3 3 (16.60, 17.00, 17.40) (23.68, 31.48, 40.98) (0.69, 0.73, 0.76) (11.16, 12.69, 14.37)

Jackson

11 7 8 8 (6.70, 7.00, 7.30) (5.78, 17.86, 39.79) (0.77, 0.82, 0.88) (1.21, 4.47, 7.25)11 9 6 6 (8.70, 9.00, 9.30) (6.33, 14.81, 31.46) (0.80, 0.85, 0.90) (1.79, 4.47, 6.81)11 10 5 5 (9.70, 10.00, 10.30) (3.27, 8.00, 22.44) (0.87, 0.92, 0.97) (0.00, 3.16, 5.20)11 13 4 4 (11.70, 12.00, 12.30) (2.68, 11.54, 23.13) (0.91, 0.96, 1.00) (0.00, 3.16, 5.63)11 14 4 4 (11.70, 12.00, 12.30) (8.17, 17.86, 29.04) (0.91, 0.96, 1.00) (2.98, 5.29, 7.75)11 21 3 3 (15.60, 16.00, 16.40) (19.55, 26.98, 35.17) (0.91, 0.96, 1.00) (7.50, 9.85, 12.21)

Mansoor11 48 4 4 (47.40, 48.00, 48.60) (2.40, 3.65, 6.44) (0.95, 0.96, 0.98) (2.73, 5.00, 6.92)11 62 3 3 (61.60, 62.00, 62.40) (0.00, 0.54, 2.79) (0.98, 0.99, 1.00) (0.00, 1.00, 3.03)11 94 2 2 (92.70, 93.00, 93.30) (0.11, 1.60, 3.28) (0.99, 0.99, 1.00) (0.00, 2.24, 4.44)

Mitchell

21 14 8 8 (13.60, 14.00, 14.40) (0.67, 6.25, 16.44) (0.89, 0.94, 0.98) (0.00, 3.00, 6.26)21 15 8 8 (13.60, 14.00, 14.40) (4.30, 12.50, 22.41) (0.89, 0.94, 0.98) (2.15, 5.57, 9.02)21 21 5 5 (20.50, 21.00, 21.50) (0.00, 0.00, 7.10) (0.96, 1.00, 1.00) (0.00, 0.00, 3.18)21 26 5 5 (20.50, 21.00, 21.50) (13.26, 19.23, 25.68) (0.96, 1.00, 1.00) (8.01, 11.18, 14.36)21 35 3 3 (34.20, 35.00, 35.80) (0.00, 0.00, 5.00) (0.96, 1.00, 1.00) (0.00, 0.00, 2.95)21 39 3 3 (35.30, 36.00, 36.70) (5.75, 10.26, 15.00) (0.93, 0.97, 1.00) (4.25, 7.07, 9.96)

Roszieg

25 14 10 10 (13.50, 14.00, 14.50) (4.93, 10.71, 21.15) (0.84, 0.89, 0.94) (5.24, 7.81, 10.63)25 16 8 8 (15.60, 16.00, 16.40) (0.59, 2.34, 11.25) (0.93, 0.98, 1.00) (0.00, 2.24, 5.14)25 18 8 8 (15.60, 16.00, 16.40) (5.59, 13.19, 21.69) (0.93, 0.98, 1.00) (3.61, 7.00, 10.61)25 21 6 6 (20.50, 21.00, 21.50) (0.00, 0.79, 7.92) (0.95, 0.99, 1.00) (0.00, 1.00, 4.04)25 25 6 6 (20.50, 21.00, 21.50) (10.58, 16.67, 23.26) (0.95, 0.99, 1.00) (6.80, 10.25, 13.71)25 32 4 4 (31.40, 32.00, 32.60) (0.38, 2.34, 7.66) (0.94, 0.98, 1.00) (0.00, 2.24, 5.02)

Heskiaoff

28 138 8 8 (129.50, 130, 130.50) (6.22, 7.25, 8.28) (0.978, 0.985, 0.991) (25.19, 28.91, 32.67)28 205 5 6 (172.10, 173, 173.90) (15.95, 16.75, 17.55) (0.979, 0.987, 0.994) (0.00, 1.00, 4.05)28 216 5 5 (205.40, 206, 206.60) (4.44, 5.19, 5.93) (0.989, 0.994, 1.000) (21.79, 25.26, 28.74)28 256 4 4 (255.30, 256, 256.70) (0.00, 0.00, 0.67) (0.995, 1.000, 1.000) (0.00, 0.00, 3.40)28 324 4 4 (257.00, 258, 259.00) (20.40, 20.99, 21.58) (0.986, 0.992, 0.999) (132.7, 136.1, 139.5)28 342 3 3 (341.40, 342, 342.60) (0.00, 0.19, 0.76) (0.994, 0.998, 1.000) (0.00, 1.41, 4.69)

Buxey

29 27 13 13 (26.50, 27.00, 27.50) (4.26, 7.69, 12.69) (0.90, 0.92, 0.95) (6.10, 9.64, 13.27)29 30 12 12 (27.60, 28.00, 28.40) (5.67, 10.00, 14.63) (0.94, 0.96, 0.99) (7.21, 11.05, 15.14)29 33 11 11 (31.40, 32.00, 32.60) (6.93, 10.74, 15.03) (0.90, 0.92, 0.95) (9.78, 13.30, 17.07)29 36 10 10 (33.50, 34.00, 34.50) (6.24, 10.00, 13.97) (0.93, 0.95, 0.98) (9.32, 12.73, 16.45)29 41 8 8 (40.60, 41.00, 41.40) (0.00, 1.22, 4.66) (0.97, 0.99, 1.00) (0.00, 2.00, 5.49)29 47 7 8 (40.60, 41.00, 41.40) (10.70, 13.83, 17.09) (0.97, 0.99, 1.00) (14.79, 18.60, 22.43)29 54 7 7 (47.40, 48.00, 48.60) (11.45, 14.29, 17.22) (0.94, 0.96, 0.99) (17.64, 21.21, 24.84)

Sawyer

30 25 14 14 (24.50, 25.00, 25.50) (4.37, 7.43, 12.80) (0.899, 0.926, 0.953) (6.98, 10.30, 13.69)30 27 13 13 (25.60, 26.00, 26.40) (3.35, 7.69, 12.72) (0.935, 0.959, 0.983) (5.67, 9.00, 12.90)30 30 12 12 (27.60, 28.00, 28.40) (5.65, 10.00, 14.66) (0.942, 0.964, 0.987) (8.08, 11.66, 15.63)30 33 11 11 (31.40, 32.00, 32.60) (6.98, 10.74, 15.06) (0.895, 0.920, 0.947) (9.94, 13.45, 17.20)30 36 10 10 (33.40, 34.00, 34.60) (6.22, 10.00, 14.00) (0.928, 0.953, 0.979) (9.10, 12.57, 16.35)30 41 8 8 (40.60, 41.00, 41.40) (0.00, 1.22, 4.69) (0.969, 0.988, 1.000) (0.00, 2.00, 5.52)30 47 7 8 (40.60, 41.00, 41.40) (10.68, 13.83, 17.12) (0.969, 0.988, 1.000) (14.74, 18.60, 22.47)30 54 7 7 (47.30, 48.00, 48.70) (11.43, 14.29, 17.25) (0.942, 0.964, 0.988) (17.28, 20.93, 24.62)30 75 5 5 (64.30, 65.00, 65.70) (11.32, 13.60, 15.95) (0.977, 0.997, 1.000) (19.26, 22.83, 26.40)

Lutz1

32 1414 11 11 (1399.7, 1400, 1400.3) (8.99, 9.09, 9.19) (0.918, 0.9182, 0.919) (513.2, 516.7, 520.3)32 1572 10 10 (1525.6, 1526, 1526.4) (9.96, 10.05, 10.14) (0.926, 0.9266, 0.927) (557.4, 561.0, 564.7)32 1768 9 9 (1663.5, 1664, 1664.5) (11.05, 11.14, 11.22) (0.944, 0.9442, 0.945) (739.3, 742.4, 745.5)32 2020 8 8 (1859.4, 1860, 1860.6) (12.42, 12.50, 12.58) (0.950, 0.9503, 0.951) (764.2, 767.8, 771.4)32 2357 7 7 (2095.6, 2096, 2096.4) (14.23, 14.30, 14.37) (0.963, 0.9637, 0.964) (900.3, 904.0, 907.8)32 2828 6 6 (2487.4, 2488, 2488.6) (16.61, 16.67, 16.73) (0.947, 0.9472, 0.948) (1221, 1224.4, 1228)

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 671

Table 8 (Continued)

Benchmark Optimal values fordeterministic SALBP-1

Result of proposed GA for fuzzy SALBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

Gunther

35 41 14 14 (39.50, 40.00, 40.50) (12.67, 15.85, 19.38) (0.85, 0.86, 0.88) (39.63, 42.11, 44.90)35 44 12 12 (43.40, 44.00, 44.60) (6.28, 8.52, 11.72) (0.90, 0.91, 0.93) (16.76, 19.52, 22.58)35 49 11 11 (47.30, 48.00, 48.70) (7.71, 10.39, 13.35) (0.90, 0.91, 0.94) (16.17, 19.70, 23.46)35 54 9 10 (49.50, 50.00, 50.50) (7.91, 10.56, 13.30) (0.95, 0.97, 0.98) (15.32, 19.26, 23.32)35 61 9 9 (55.40, 56.00, 56.60) (9.59, 12.02, 14.54) (0.94, 0.96, 0.98) (22.03, 25.42, 29.01)35 69 8 8 (63.20, 64.00, 64.80) (10.27, 12.50, 14.80) (0.92, 0.94, 0.96) (25.98, 29.21, 32.62)35 81 7 7 (73.30, 74.00, 74.70) (12.80, 14.81, 16.88) (0.92, 0.93, 0.95) (37.44, 40.40, 43.52)

Kilbridge

45 56 10 10 (55.50, 56, 56.50) (0.25, 1.43, 4.09) (0.97, 0.99, 1.00) (0.00, 3.46, 7.56)45 57 10 10 (55.50, 56, 56.50) (0.90, 3.16, 5.80) (0.97, 0.99, 1.00) (2.05, 6.48, 10.75)45 62 9 10 (56.20, 57, 57.80) (8.49, 10.97, 13.52) (0.95, 0.97, 0.99) (26.31, 29.29, 32.66)45 69 8 9 (62.20, 63, 63.80) (8.81, 11.11, 13.48) (0.95, 0.97, 0.99) (23.27, 26.93, 30.82)45 79 7 8 (70.10, 71, 71.90) (10.55, 12.66, 14.82) (0.95, 0.97, 0.99) (25.21, 29.36, 33.59)45 92 6 6 (91.00, 92, 93.00) (0.00, 0.00, 1.92) (0.98, 1.00, 1.00) (25.31, 29.29, 35.66)45 110 6 6 (92.40, 93, 93.60) (14.64, 16.36, 18.12) (0.97, 0.99, 1.00) (40.97, 45.14, 49.34)45 111 5 5 (110.40, 111, 111.60) (0.00, 0.54, 2.27) (0.98, 0.99, 1.00) (0.00, 1.73, 5.81)45 138 4 4 (136.80, 138, 139.20) (0.00, 0.00, 1.55) (0.98, 1.00, 1.00) (0.00, 1.65, 4.78)45 184 3 3 (182.50, 184, 185.50) (0.00, 0.00, 1.37) (0.98, 1.00, 1.00) (0.00, 1.53, 3.81)

Hahn

53 2004 8 8 (1924.4, 1926, 1927.6) (12.42, 12.51, 12.60) (0.909, 0.9103, 0.911) (1026, 1028.8, 1032)53 2338 7 7 (2334.1, 2336, 2337.9) (14.22, 14.30, 14.38) (0.857, 0.8578, 0.859) (1476, 1478.5, 1481)53 2806 6 6 (2604.9, 2607, 2609.1) (16.62, 16.69, 16.76) (0.896, 0.8967, 0.898) (1365, 1368.7, 1372)53 3507 5 5 (3099.0, 3100, 3101.0) (19.95, 20.01, 20.08) (0.904, 0.9049, 0.906) (1890, 1893.2, 1897)53 4676 4 4 (4439.3, 4441, 4442.7) (24.96, 25.01, 25.07) (0.789, 0.7896, 0.790) (3159, 3161.3, 3164)

Warnecke

58 54 31 33 (52.60, 53.00, 53.40) (10.78, 13.13, 15.60) (0.875, 0.885, 0.895) (46.69, 51.75, 57.15)58 56 29 32 (55.70, 56.00, 56.30) (11.47, 13.62, 16.01) (0.856, 0.864, 0.872) (49.60, 54.64, 59.95)58 58 29 29 (57.70, 58.00, 58.30) (5.97, 7.97, 10.21) (0.91, 0.92, 0.93) (16.76, 19.52, 22.58)58 60 27 29 (58.60, 59.00, 59.40) (8.92, 11.03, 13.26) (0.895, 0.905, 0.914) (35.53, 40.94, 46.60)58 62 27 29 (59.80, 60.00, 60.20) (11.78, 13.90, 16.10) (0.883, 0.890, 0.896) (51.08, 56.21, 61.60)58 65 25 28 (63.60, 64.00, 64.40) (12.95, 14.95, 17.06) (0.855, 0.864, 0.873) (59.83, 64.65, 69.71)58 68 24 26 (66.60, 67.00, 67.40) (10.55, 12.44, 14.45) (0.880, 0.889, 0.897) (53.16, 57.58, 62.30)58 71 23 24 (69.60, 70.00, 70.40) (7.37, 9.15, 11.06) (0.913, 0.921, 0.930) (38.14, 42.45, 47.12)58 74 22 23 (72.60, 73.00, 73.40) (7.35, 9.05, 10.89) (0.914, 0.922, 0.931) (41.17, 45.12, 49.45)58 78 21 22 (76.70, 77.00, 77.30) (8.11, 9.79, 11.56) (0.907, 0.914, 0.921) (48.56, 52.35, 56.48)58 82 20 20 (80.50, 81.00, 81.50) (4.10, 5.61, 7.27) (0.95, 0.96, 0.97) (15.32, 19.26, 23.32)58 86 19 20 (84.50, 85.00, 85.50) (8.48, 10.00, 11.64) (0.902, 0.911, 0.919) (51.85, 55.57, 59.59)58 92 17 19 (87.60, 88.00, 88.40) (9.92, 11.44, 13.00) (0.918, 0.926, 0.934) (47.26, 52.10, 57.10)58 97 17 18 (93.50, 94.00, 94.50) (9.88, 11.34, 12.84) (0.907, 0.915, 0.923) (54.58, 58.77, 63.19)58 104 15 16 (101.5, 102.0, 102.5) (5.61, 6.97, 8.36) (0.940, 0.949, 0.957) (33.02, 37.04, 41.39)58 111 14 15 (108.5, 109.0, 109.5) (5.73, 7.03, 8.35) (0.939, 0.947, 0.955) (35.06, 39.00, 43.25)

Tonge

70 160 23 24 (159.60, 160, 160.40) (7.81, 8.59, 9.46) (0.910, 0.914, 0.918) (94.02, 98.22, 102.6)70 168 22 23 (165.30, 166, 166.70) (8.34, 9.16, 10.00) (0.914, 0.919, 0.925) (84.16, 89.10, 94.20)70 176 21 22 (174.50, 175, 175.50) (8.58, 9.35, 10.16) (0.907, 0.912, 0.916) (90.7, 95.53, 100.48)70 185 20 20 (184.50, 185, 185.50) (4.42, 5.14, 5.90) (0.944, 0.949, 0.953) 55.19, 59.19, 63.5070 195 19 20 (189.40, 190, 190.60) (9.26, 10.00, 10.75) (0.919, 0.924, 0.928) (99.4, 104.27, 109.3)70 207 18 18 (206.30, 207, 207.70) (5.23, 5.80, 6.50) (0.937, 0.942, 0.947) (70.98, 74.85, 78.88)70 220 17 18 (209.60, 210, 210.40) (10.68, 11.36, 12.05) (0.925, 0.929, 0.932) (120.97, 126, 131.1)70 234 16 16 (228.60, 229, 229.40) (5.61, 6.25, 6.89) (0.954, 0.958, 0.962) (71.17, 75.42, 79.86)70 251 14 15 (245.10, 246, 246.90) (6.16, 6.77, 7.39) (0.946, 0.951, 0.957) (77.20, 81.43, 85.83)70 270 14 14 (262.40, 263, 263.60) (6.56, 7.14, 7.73) (0.949, 0.953, 0.957) (87.96, 92.26, 96.70)70 293 13 13 (280.20, 281, 281.80) (7.30, 7.85, 8.40) (0.956, 0.961, 0.966) (89.86, 94.77, 99.74)70 320 11 12 (301, 302, 303) (8.07, 8.59, 9.12) (0.963, 0.969, 0.974) (95.75, 100.9, 106.1)70 364 10 10 (357.10, 358, 358.90) (3.10, 3.57, 4.05) (0.976, 0.980, 0.985) (39.47, 44.32, 49.29)70 410 9 9 (393.90, 395, 396.10) (4.43, 4.88, 5.32) (0.983, 0.987, 0.992) (55.42, 60.75, 66.08)70 468 8 8 (159.60, 160, 160.40) (5.84, 6.25, 6.66) (0.979, 0.984, 0.988) (81.25, 86.28, 91.34)70 527 7 7 (165.30, 166, 166.70) (4.46, 4.85, 5.24) (0.988, 0.993, 0.997) (62.88, 68.11, 73.35)

Wee-Mag

75 28 63 63 (26.80, 27.00, 27.20) (10.69, 15.02, 19.72) (0.87, 0.88, 0.89) (28.95, 36.76, 44.94)75 29 63 63 (27.60, 28.00, 28.40) (13.67, 17.95, 22.59) (0.83, 0.85, 0.87) (36.72, 44.70, 52.95)75 30 62 62 (29.70, 30.00, 30.30) (15.31, 19.41, 23.94) (0.79, 0.81, 0.82) (41.22, 49.28, 57.51)75 31 62 62 (29.70, 30.00, 30.30) (17.84, 22.01, 26.48) (0.79, 0.81, 0.82) (49.15, 57.25, 65.52)75 32 61 61 (31.70, 32.00, 32.30) (19.34, 23.21, 27.58) (0.76, 0.77, 0.78) (54.06, 62.15, 70.32)75 33 61 61 (31.70, 32.00, 32.30) (21.52, 25.53, 29.84) (0.76, 0.77, 0.78) (61.94, 70.01, 78.22)75 34 61 61 (31.70, 32.00, 32.30) (23.72, 27.72, 31.97) (0.76, 0.77, 0.78) (69.67, 77.80, 86.06)75 35 60 60 (34.60, 35.00, 35.40) (24.88, 28.62, 32.77) (0.70, 0.71, 0.73) (74.82, 82.86, 90.96)75 36 60 60 (34.60, 35.00, 35.40) (26.77, 30.60, 34.69) (0.70, 0.71, 0.73) (82.31, 90.37, 98.54)75 37 60 60 (34.70, 35.00, 35.30) (28.66, 32.48, 36.50) (0.70, 0.71, 0.72) (89.62, 97.78, 106.0)75 38 60 60 (34.60, 35.00, 35.40) (30.49, 34.25, 38.22) (0.70, 0.71, 0.73) (97.0, 105.3, 113.54)75 39 60 60 (34.70, 35.00, 35.30) (32.23, 35.94, 39.85) (0.70, 0.71, 0.72) (104.7, 112.9, 121.3)75 40 60 60 (34.80, 35.00, 35.20) (33.88, 37.54, 41.39) (0.71, 0.71, 0.72) (112.1, 120.4, 128.8)75 41 59 59 (40.50, 41.00, 41.50) (34.55, 38.03, 41.80) (0.61, 0.62, 0.63) (117.9, 126.0, 134.2)75 42 55 55 (41.70, 42.00, 42.30) (31.96, 35.11, 38.74) (0.64, 0.65, 0.66) (111.2, 118.8, 126.5)

672 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Table 8 (Continued)

Benchmark Optimal values fordeterministic SALBP-1

Result of proposed GA for fuzzy SALBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

75 43 50 50 (42.70, 43.00, 43.30) (27.54, 30.28, 33.74) (0.69, 0.70, 0.71) (101.0, 107.8, 114.7)75 47 33 35 (46.70, 47.00, 47.30) (6.90, 8.88, 11.71) (0.90, 0.91, 0.92) (40.44, 44.23, 48.48)75 49 32 33 (48.70, 49.00, 49.30) (5.24, 7.30, 10.01) (0.92, 0.93, 0.94) (28.41, 32.65, 37.44)75 50 32 33 (48.60, 49.00, 49.40) (6.65, 9.15, 11.84) (0.91, 0.93, 0.94) (30.62, 35.51, 40.99)75 52 31 32 (50.60, 51.00, 51.40) (7.45, 9.92, 12.53) (0.91, 0.92, 0.93) (28.15, 33.87, 39.99)75 54 31 31 (52.60, 53.00, 53.40) (8.06, 10.45, 12.99) (0.90, 0.91, 0.92) (31.80, 37.35, 43.27)75 56 30 30 (55.70, 56.00, 56.30) (8.75, 10.77, 13.24) (0.88, 0.89, 0.90) (35.62, 40.98, 46.55)

Arcus1

83 3786 21 22 (3690.9, 3691, 3691.1) (9.07, 9.11, 9.15) (0.9322, 0.9323, 0.9325) (2039.8, 2044.8, 2049.8)83 3985 20 21 (3857.5, 3858, 3858.5) (9.50, 9.53, 9.57) (0.9342, 0.9344, 0.9347) (1997.2, 2002.6, 2008.1)83 4206 19 19 (4177.4, 4178, 4178.6) (5.23, 5.26, 5.30) (0.9535, 0.9537, 0.9539) (1202.1, 1206.9, 1211.8)83 4454 18 19 (4241.6, 4242, 4242.4) (10.50, 10.54, 10.57) (0.9391, 0.9393, 0.9395) (2275.3, 2280.7, 2286.2)83 4732 17 17 (4678.6, 4679, 4679.4) (5.86, 5.89, 5.92) (0.9516, 0.9518, 0.9520) (1487.0, 1491.4, 1495.8)83 5048 16 16 (4954.3, 4955, 4955.7) (6.23, 6.27, 6.30) (0.9547, 0.9549, 0.9552) (1747.3, 1751.4, 1755.4)83 5408 15 15 (5281.5, 5282, 5282.5) (6.64, 6.67, 6.70) (0.9553, 0.9555, 0.9557) (1609.9, 1615.0, 1620.1)83 5824 14 14 (5565.5, 5566, 5566.5) (7.12, 7.15, 7.18) (0.9714, 0.9715, 0.9717) (1704.9, 1710.2, 1715.5)83 5853 14 14 (5599.3, 5600, 5600.7) (7.58, 7.61, 7.64) (0.9654, 0.9657, 0.9659) (1721.5, 1727.2, 1733.0)83 6309 13 13 (6042.2, 6043, 6043.8) (7.67, 7.69, 7.72) (0.9635, 0.9637, 0.9639) (1883.9, 1889.3, 1894.6)83 6842 12 12 (6508.9, 6510, 6511.1) (7.77, 7.79, 7.82) (0.9688, 0.9691, 0.9694) (2084.2, 2089.2, 2094.2)83 6883 12 12 (6561.5, 6562, 6562.5) (8.31, 8.34, 8.37) (0.9613, 0.9614, 0.9616) (2194.7, 2199.8, 2205.0)83 7571 11 11 (7090.7, 7091, 7091.3) (9.07, 9.09, 9.12) (0.9704, 0.9706, 0.9707) (2345.9, 2351.5, 2357.2)

24.2)

28.6)

1022

rTacsiw

tvGt

5

mamfitht

A

abbf

att

83 8412 10 10 (7921.8, 7923, 7983 8898 9 9 (8527.4, 8528, 8583 10,816 8 8 (10,220.6, 10222,

educes convergence rate and would be more time consuming.he tournament method rather than the proposed algorithm has

greater focus on convergence which tends to create a boost inonvergence rate and also makes cease the Tournament in localolution more possible. In other words, low focus on the diversitys the main cause of worst results for Tournament in comparison

ith the proposed algorithm.A comparison between the two proposed methods shows that

hey are almost the same for the quality of results, but for the con-ergence rate, the SP controller which works with sigmoid andaussian MFs has a better condition. The issue of root in this con-

roller type has been discussed before (Fig. 12).

.3. Numerical results over SALBP-1 and SULBP-1 benchmarks

In this section, the proposed algorithm will examine the bench-arks of SALBP-1 and SULBP-1. More details of these benchmarks

re reachable on Scholl [69] and http://alb.mansci.de. These bench-arks have been defined in crisp state, so for transferring to the

uzzy state, it is postulated that the input number in crisp states equal to the peak point (A2) in fuzzy state. For calculation ofhe beginning point (A1) and end point (A3) formula (42) comes inandy. In this paper, the values of and � for the tasks processingime are supposed to be 0.1 and for Cmax it is 1.

˜ = (A2 − �, A2, A2 + ) (42)

Selected benchmarks according to the number of tasks (n)re divided into A, B, and C. These groups consist of small-sizedenchmarks (n < 25), medium-sized benchmarks (25 ≤ n < 45), andig-sized benchmarks (n ≥ 45), respectively. Selected benchmarksor each class are shown in Table 5.

The proposed algorithm is implemented in MATLAB and run on computer with “Core 2 Duo 2.2, 2 GHz PC”. Because of the stochas-ic behavior in the meta-heuristic algorithm, the algorithm wasried out 10 runs by each benchmark and the best solutions are

(9.98, 10.00, 10.02) (0.9553, 0.9555, 0.9558) (2812.8, 2818.1, 2823.4)(5.44, 5.46, 5.49) (0.9862, 0.9864, 0.9866) (1474.2, 1479.9, 1485.6)

3.4) (12.49, 12.51, 12.53) (0.9256, 0.9258, 0.9260) (6124.0, 6126.9, 6129.8)

summarized in Tables 6 and 7. Output solutions are compared withoptimums by formula (43).

%Deviation ={

(x − x∗)x∗

}· 100 (43)

Tables 6 and 7 consist of the information below:

Average of %deviation: shows the average percent of deviations.%Optimal solution: exposes the optimum solution percentage.Maximum of %deviation: presents the maximum percent of devi-ations.

As it is cleared in Tables 6 and 7, the proposed algorithm tends tothe optimum results in all A class benchmarks which show the highperformance of this algorithm for this class. The bigger the solutionspace, the lower the solutions quality, which is clear in B and Cclasses. All in all, the proposed algorithm has good performance insolving Class B and is also good in Class C. SALB-1 rather than SULB-1 has better results compared to the algorithm, which stems fromthe smaller size of the solution space, although the algorithm hashigh performance in both SALB-1 and SULB-1 which is apparentlypresented in the last row of Tables 6 and 7. More results are exposedin details in Tables 8 and 9.

Run time dependence to the problem size is completely clear. Inother words, the greater the number of tasks in ALBP, the further thecomputer run-time. The average CPU time for running each bench-mark has been exposed in Fig. 25. As it is clear, process time variesbetween 3 and 360 s which is a proof for the algorithm valuableconvergence time per se.

5.4. Final comparison between proposed method and existingones

In this section, performance of the proposed algorithm is exam-

ined using the test problem Tsujimura et al. [43]. An exampleof the problem is solved by the proposed algorithm to illustratethe improvements and the results are compared with the existingmethods in this problem which are presented in Table 10. Likewise,

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 673

Table 9Result of proposed algorithm for fuzzy SULBP-1.

Benchmark Optimal values fordeterministic SULBP-1

Result of proposed GA for fuzzy SULBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

Mertens

7 6 6 6 (5.80, 6.00, 6.20) (6.67, 19.44, 45.67) (0.76, 0.81, 0.85) (1.16, 3.87, 6.16)7 7 5 5 (6.80, 7.00, 7.20) (4.50, 17.14, 39.00) (0.79, 0.83, 0.87) (0.00, 3.16, 5.47)7 8 5 5 (6.80, 7.00, 7.20) (12.22, 27.50, 47.71) (0.79, 0.83, 0.87) (2.87, 5.20, 7.64)7 10 3 3 (9.80, 10.00, 10.20) (0.00, 3.33, 17.41) (0.92, 0.97, 1.00) (0.00, 1.00, 2.86)7 15 2 2 (14.70, 15.00, 15.30) (0.00, 3.33, 13.21) (0.92, 0.97, 1.00) (0.00, 1.00, 2.73)7 18 2 2 (14.70, 15.00, 15.30) (11.32, 19.44, 28.53) (0.92, 0.97, 1.00) (3.11, 5.00, 6.90)

Bowman 8 20 5 5 (19.70, 20.00, 20.30) (3.21, 6.25, 12.89) (0.91, 0.94, 0.96) (1.06, 3.61, 5.49)

Jaeschke

9 6 8 8 (5.80, 6.00, 6.20) (8.21, 22.92, 49.75) (0.73, 0.77, 0.82) (1.72, 4.80, 7.56)9 7 7 7 (6.80, 7.00, 7.20) (11.61, 24.49, 47.38) (0.72, 0.76, 0.80) (2.60, 5.48, 8.08)9 8 6 6 (7.70, 8.00, 8.30) (12.22, 22.92, 42.62) (0.72, 0.77, 0.82) (2.92, 5.57, 7.95)9 10 4 4 (9.70, 10.00, 10.30) (0.00, 7.50, 21.94) (0.88, 0.93, 0.98) (0.00, 1.73, 4.03)9 18 3 3 (13.60, 14.00, 14.40) (22.98, 31.48, 40.98) (0.84, 0.88, 0.93) (8.65, 10.63, 12.70)

Jackson

11 7 7 7 (6.80, 7.00, 7.20) (0.00, 6.12, 26.43) (0.89, 0.94, 0.99) (0.00, 1.73, 4.38)11 9 6 6 (8.70, 9.00, 9.30) (6.00, 14.81, 31.46) (0.80, 0.85, 0.90) (1.22, 4.24, 6.68)11 10 5 5 (9.60, 10.00, 10.40) (3.45, 8.00, 22.44) (0.86, 0.92, 0.98) (0.00, 3.16, 5.15)11 13 4 4 (11.70, 12.00, 12.30) (2.50, 11.54, 23.13) (0.91, 0.96, 1.00) (0.00, 3.16, 5.65)11 14 4 4 (11.70, 12.00, 12.30) (8.17, 17.86, 29.04) (0.91, 0.96, 1.00) (3.05, 5.29, 7.73)11 21 3 3 (15.60, 16.00, 16.40) (19.55, 26.98, 35.17) (0.91, 0.96, 1.00) (7.64, 9.95, 12.28)

Mansoor11 48 4 4 (47.40, 48.00, 48.60) (2.40, 3.65, 6.44) (0.95, 0.96, 0.98) (2.73, 5.00, 6.92)11 62 3 3 (61.60, 62.00, 62.40) (0.00, 0.54, 2.79) (0.98, 0.99, 1.00) (0.00, 1.00, 3.03)11 94 2 2 (92.70, 93.00, 93.30) (0.11, 1.60, 3.28) (0.99, 0.99, 1.00) (0.00, 2.24, 4.44)

Mitchell

21 14 8 8 (13.50, 14.00, 14.50) (1.33, 6.25, 16.44) (0.89, 0.94, 0.99) (0.00, 3.32, 6.38)21 15 8 8 (13.50, 14.00, 14.50) (4.61, 12.50, 22.41) (0.89, 0.94, 0.99) (2.43, 5.74, 9.11)21 21 5 5 (20.50, 21.00, 21.50) (0.00, 0.00, 7.10) (0.96, 1.00, 1.00) (0.00, 0.00, 3.18)21 26 5 5 (20.50, 21.00, 21.50) (13.26, 19.23, 25.68) (0.96, 1.00, 1.00) (8.01, 11.18, 14.36)21 35 3 3 (34.30, 35.00, 35.70) (0.00, 0.00, 5.00) (0.96, 1.00, 1.00) (0.00, 0.00, 2.94)21 39 3 3 (34.30, 35.00, 35.70) (5.75, 10.26, 15.00) (0.96, 1.00, 1.00) (3.98, 6.93, 9.87)

Roszieg

25 14 9 9 (13.60, 14.00, 14.40) (0.00, 0.79, 10.68) (0.95, 0.99, 1.00) (0.00, 1.00, 4.26)25 16 8 8 (15.70, 16.00, 16.30) (0.00, 2.34, 11.25) (0.94, 0.98, 1.00) (0.00, 1.73, 5.02)25 18 7 7 (17.60, 18.00, 18.40) (0.00, 0.79, 8.82) (0.95, 0.99, 1.00) (0.00, 1.00, 4.09)25 21 6 6 (20.60, 21.00, 21.40) (0.00, 0.79, 7.92) (0.95, 0.99, 1.00) (0.00, 1.00, 4.07)25 25 5 5 (24.50, 25.00, 25.50) (0.00, 0.00, 6.25) (0.96, 1.00, 1.00) (0.00, 0.00, 3.35)25 32 4 4 (31.60, 32.00, 32.40) (0.00, 2.34, 7.66) (0.95, 0.98, 1.00) (0.00, 1.73, 4.88)

Heskiaoff

28 138 8 8 (130.30, 131, 131.70) (6.22, 7.25, 8.28) (0.969, 0.977, 0.985) (25.72, 29.36, 33.05)28 205 5 5 (204.20, 205, 205.80) (0.00, 0.10, 0.86) (0.992, 0.999, 1.00) (0.00, 1.00, 4.05)28 216 5 5 (204.20, 205, 205.80) (4.44, 5.19, 5.93) (0.992, 0.999, 1.00) (21.57, 25.06, 28.55)28 256 4 4 (255.30, 256, 256.70) (0.00, 0.00, 0.67) (0.995, 1.000, 1.00) (0.00, 0.00, 3.40)28 324 4 4 (256.30, 257, 257.70) (20.40, 20.99, 21.58) (0.991, 0.996, 1.00) (132.6, 136.0, 139.4)28 342 3 3 (341.50, 342, 342.50) (0.00, 0.19, 0.76) (0.994, 0.998, 1.00) (0.00, 1.41, 4.71)

Buxey

29 27 13 13 (26.50, 27.00, 27.50) (4.31, 7.69, 12.69) (0.90, 0.92, 0.95) (6.32, 9.85, 13.45)29 30 11 12 (28.60, 29.00, 29.40) (5.91, 10.00, 14.63) (0.91, 0.93, 0.95) (8.11, 11.75, 15.65)29 33 10 11 (30.50, 31.00, 31.50) (6.71, 10.74, 15.03) (0.93, 0.95, 0.97) (9.93, 13.30, 17.05)29 36 9 10 (33.40, 34.00, 34.60) (6.24, 10.00, 13.97) (0.93, 0.95, 0.98) (8.98, 12.49, 16.28)29 41 8 8 (40.40, 41.00, 41.60) (0.42, 1.22, 4.66) (0.96, 0.99, 1.00) (0.00, 2.83, 5.78)29 47 7 7 (46.60, 47.00, 47.40) (0.00, 1.52, 4.63) (0.97, 0.98, 1.00) (0.00, 2.24, 5.76)29 54 6 6 (53.40, 54.00, 54.60) (0.00, 0.00, 2.80) (0.98, 1.00, 1.00) (0.00, 0.24, 3.23)

Sawyer

30 25 14 14 (24.50, 25.00, 25.50) (4.73, 7.43, 12.80) (0.899, 0.926, 0.953) (6.98, 10.30, 13.69)30 27 13 13 (26.60, 27.00, 27.40) (4.01, 7.69, 12.72) (0.901, 0.923, 0.946) (5.67, 9.00, 12.90)30 30 11 12 (28.80, 29.00, 29.20) (5.81, 10.00, 14.66) (0.916, 0.931, 0.946) (8.08, 11.66, 15.63)30 33 10 11 (30.70, 31.00, 31.30) (6.68, 10.74, 15.06) (0.932, 0.950, 0.968) (9.94, 13.45, 17.20)30 36 9 10 (34.30, 35.00, 35.70) (6.76, 10.00, 14.00) (0.899, 0.926, 0.953) (9.10, 12.57, 16.35)30 41 8 8 (40.40, 41.00, 41.60) (0.00, 1.22, 4.69) (0.965, 0.988, 1.000) (0.00, 2.00, 5.52)30 47 7 7 (46.70, 47.00, 47.30) (0.48, 1.52, 4.66) (0.969, 0.985, 1.000) (0.00, 3.00, 5.52)30 54 6 7 (48.50, 49.00, 49.50) (11.43, 14.29, 17.25) (0.926, 0.945, 0.963) (17.28, 20.93, 24.62)30 75 5 5 (67.30, 68.00, 68.70) (11.32, 13.60, 15.95) (0.934, 0.953, 0.972) (19.26, 22.83, 26.40)

Lutz1

32 1414 11 11 (1399.5, 1400, 1400.5) (8.99, 9.09, 9.19) (0.918, 0.9182, 0.919) (540.1, 543.4, 546.8)32 1572 10 10 (1451.6, 1452, 1452.4) (9.96, 10.05, 10.14) (0.973, 0.9738, 0.974) (514.6, 518.5, 522.5)32 1768 9 9 (1603.6, 1604, 1604.4) (11.05, 11.14, 11.22) (0.979, 0.9795, 0.980) (603.6, 607.5, 611.5)32 2020 8 8 (1787.8, 1788, 1788.2) (12.42, 12.50, 12.58) (0.988, 0.9885, 0.989) (711.5, 715.5, 719.4)32 2357 7 7 (2057.5, 2058, 2058.5) (14.23, 14.30, 14.37) (0.981, 0.9815, 0.982) (895.3, 899.1, 902.9)32 2828 6 6 (2403.4, 2404, 2404.6) (16.61, 16.67, 16.73) (0.978, 0.9803, 0.981) (1155, 1158.7, 1162)

Gunther

35 41 12 13 (39.30, 40.00, 40.70) (6.47, 9.38, 12.79) (0.91, 0.93, 0.95) (16.12, 19.18, 22.66)35 44 12 12 (42.40, 43.00, 43.60) (5.65, 8.52, 11.72) (0.92, 0.94, 0.96) (11.73, 15.26, 19.13)35 49 10 11 (46.30, 47.00, 47.70) (7.55, 10.39, 13.35) (0.91, 0.93, 0.96) (15.31, 18.97, 22.90)35 54 9 10 (49.30, 50.00, 50.70) (7.91, 10.56, 13.30) (0.95, 0.97, 0.99) (15.11, 19.00, 23.04)35 61 8 9 (55.40, 56.00, 56.60) (9.59, 12.02, 14.54) (0.94, 0.96, 0.98) (19.00, 22.89, 26.87)35 69 7 8 (61.60, 62.00, 62.40) (10.27, 12.50, 14.80) (0.96, 0.97, 0.99) (20.85, 24.84, 28.86)35 81 6 6 (80.40, 81.00, 81.60) (0.00, 0.62, 2.60) (0.98, 0.99, 1.00) (0.00, 1.73, 5.27)

674 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Table 9 (Continued)

Benchmark Optimal values fordeterministic SULBP-1

Result of proposed GA for fuzzy SULBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

Kilbridge

45 56 10 10 (55.20, 56, 56.80) (0.21, 1.43, 4.09) (0.96, 0.99, 1.00) (0.00, 3.46, 7.49)45 57 10 10 (55.50, 56, 56.50) (0.90, 3.16, 5.80) (0.97, 0.99, 1.00) (1.87, 6.32, 10.74)45 62 9 9 (61.40, 62, 62.60) (0.14, 1.08, 3.55) (0.97, 0.99, 1.00) (0.00, 3.16, 6.92)45 69 8 9 (62.20, 63, 63.80) (8.81, 11.11, 13.48) (0.95, 0.97, 0.99) (23.53, 27.04, 30.81)45 79 7 8 (70.10, 71, 71.90) (10.55, 12.66, 14.82) (0.95, 0.97, 0.99) (25.23, 29.43, 33.70)45 92 6 6 (91.10, 92, 92.90) (0.00, 0.00, 1.92) (0.98, 1.00, 1.00) (25.31, 29.29, 35.66)45 110 6 6 (92.90, 94, 95.10) (14.64, 16.36, 18.12) (0.96, 0.98, 1.00) (44.25, 48.15, 52.11)45 111 5 5 (110.50, 111, 111.50) (0.00, 0.54, 2.27) (0.98, 0.99, 1.00) (0.00, 1.73, 5.84)45 138 4 4 (136.70, 138, 139.30) (0.00, 0.00, 1.55) (0.98, 1.00, 1.00) (0.00, 1.65, 5.78)45 184 3 3 (182.70, 184, 185.30) (0.00, 0.00, 1.37) (0.98, 1.00, 1.00) (0.00, 1.34, 3.66)

Hahn

53 2004 8 8 (1839.8, 1840, 1840.2) (12.42, 12.51, 12.60) (0.952, 0.9529, 0.953) (737.8, 742.6, 747.4)53 2338 7 7 (2040.6, 2042, 2043.4) (14.22, 14.30, 14.38) (0.980, 0.9813, 0.982) (883.9, 888.4, 893.0)53 2806 5 6 (2392.6, 2393, 2393.4) (16.62, 16.69, 16.76) (0.976, 0.9769, 0.977) (1145, 1149.5, 1154)53 3507 5 5 (2909.1, 2910, 2910.9) (19.95, 20.01, 20.08) (0.963, 0.9640, 0.965) (1609, 1613.5, 1618)53 4676 3 3 (4674.9, 4676, 4677.1) (0.00, 0.01, 0.07) (0.9992, 0.9999, 1.00) (0.00, 1.41, 6.21)

Warnecke

58 54 [30,31] 33 (53.70, 54.00, 54.30) (10.98, 13.13, 15.60) (0.86, 0.87, 0.88) (49.74, 54.63, 59.79)58 56 29 32 (54.80, 55.00, 55.20) (11.32, 13.62, 16.01) (0.87, 0.88, 0.89) (44.16, 49.76, 55.60)58 58 28 31 (56.60, 57.00, 57.40) (11.73, 13.90, 16.23) (0.87, 0.88, 0.89) (52.70, 57.55, 62.70)58 60 27 29 (58.60, 59, 59.40) (8.92, 11.03, 13.26) (0.90, 0.90, 0.91) (51.92, 56.99, 62.32)58 62 [26,27] 29 (61.60, 62, 62.40) (11.93, 13.90, 16.10) (0.85, 0.86, 0.87) (53.29, 58.24, 63.41)58 65 [24,25] 28 (62.70, 63, 63.30) (12.89, 14.95, 17.06) (0.87, 0.88, 0.89) (58.18, 63.15, 68.36)58 68 [23,24] 25 (67.60, 68, 68.40) (7.43, 8.94, 10.91) (0.90, 0.91, 0.92) (38.15, 42.40, 46.89)58 71 [22,23] 24 (69.60, 70, 70.40) (7.37, 9.15, 11.06) (0.91, 0.92, 0.93) (53.73, 58.27, 63.08)58 74 [21,22] 22 (73.60, 74.00, 74.40) (3.61, 4.91, 6.71) (0.94, 0.95, 0.96) (1.87, 6.32, 10.74)58 78 20 22 (76.50, 77, 77.50) (8.10, 9.79, 11.56) (0.90, 0.91, 0.92) (38.39, 43.10, 48.08)58 82 [19,20] 20 (81.60, 82, 82.40) (4.19, 5.61, 7.27) (0.94, 0.94, 0.95) (23.88, 27.96, 32.44)58 86 18 20 (84.60, 85, 85.40) (8.49, 10.00, 11.64) (0.90, 0.91, 0.92) (52.72, 56.41, 60.39)58 92 17 18 (91.60, 92, 92.40) (5.24, 6.52, 8.05) (0.93, 0.93, 0.94) (39.06, 42.36, 46.00)58 97 16 18 (92.60, 93, 93.40) (9.88, 11.34, 12.84) (0.92, 0.92, 0.93) (49.29, 54.00, 58.87)58 104 15 16 (101.50, 102, 102.50) (5.61, 6.97, 8.36) (0.94, 0.95, 0.96) (34.15, 38.05, 42.30)58 111 14 15 (108.40, 109, 109.60) (5.73, 7.03, 8.35) (0.94, 0.95, 0.96) (36.50, 40.26, 44.36)

Tonge

70 160 [22,23] 24 (0.921, 0.926, 0.930) (7.74, 8.59, 9.46) (0.921, 0.926, 0.930) (77.13, 82.24, 87.52)70 168 21 23 (0.910, 0.914, 0.917) (8.34, 9.16, 10.00) (0.910, 0.914, 0.917) (106.7, 110.5, 114.6)70 176 [20,21] 22 (0.907, 0.912, 0.917) (8.57, 9.35, 10.16) (0.907, 0.912, 0.917) (94.85, 99.62, 104.5)70 185 19 20 (0.944, 0.949, 0.953) (4.42, 5.14, 5.90) (0.944, 0.949, 0.953) (94.8, 99.67, 104.68)70 195 18 20 (0.919, 0.924, 0.928) (9.26, 10.00, 10.75) (0.919, 0.924, 0.928) (110.7, 115.1, 119.5)70 207 17 18 (0.937, 0.942, 0.947) (5.23, 5.80, 6.50) (0.937, 0.942, 0.947) (125.8, 130.1, 134.5)70 220 16 17 (0.934, 0.939, 0.943) (5.58, 6.15, 6.82) (0.934, 0.939, 0.943) (83.16, 86.88, 90.76)70 234 15 16 (0.959, 0.962, 0.966) (5.61, 6.25, 6.89) (0.959, 0.962, 0.966) (59.44, 64.61, 69.88)70 251 14 15 (0.943, 0.947, 0.952) (6.16, 6.77, 7.39) (0.943, 0.947, 0.952) (79.89, 84.05, 88.39)70 270 13 14 (0.949, 0.953, 0.957) (6.56, 7.14, 7.73) (0.949, 0.953, 0.957) (79.66, 84.29, 89.03)70 293 12 13 (0.956, 0.961, 0.966) (7.30, 7.85, 8.40) (0.956, 0.961, 0.966) (98.6, 103.16, 107.8)70 320 11 12 (0.955, 0.959, 0.963) (8.07, 8.59, 9.12) (0.955, 0.959, 0.963) (103.1, 107.9, 112.9)70 364 10 10 (0.973, 0.978, 0.982) (3.10, 3.57, 4.05) (0.973, 0.978, 0.982) (46.25, 50.32, 54.63)70 410 9 9 (0.978, 0.982, 0.987) (4.43, 4.88, 5.32) (0.978, 0.982, 0.987) (59.20, 64.27, 69.38)70 468 8 8 (0.981, 0.986, 0.991) (5.84, 6.25, 6.66) (0.981, 0.986, 0.991) (78.31, 83.55, 88.80)70 527 7 7 (0.987, 0.991, 0.995) (4.46, 4.85, 5.24) (0.987, 0.991, 0.995) (63.00, 68.26, 73.53)

Wee-Mag

75 28 63 63 (27.70, 28.00, 28.30) (11.07, 15.02, 19.72) (0.84, 0.85, 0.86) (32.24, 39.56, 47.22)75 29 63 63 (27.70, 28.00, 28.30) (13.66, 17.95, 22.59) (0.84, 0.85, 0.86) (36.69, 44.70, 52.97)75 30 62 62 (29.70, 30.00, 30.30) (15.31, 19.41, 23.94) (0.79, 0.81, 0.82) (41.28, 49.33, 57.54)75 31 62 62 (29.70, 30.00, 30.30) (17.85, 22.01, 26.48) (0.79, 0.81, 0.82) (49.19, 57.28, 65.55)75 32 61 61 (31.70, 32.00, 32.30) (19.34, 23.21, 27.58) (0.76, 0.77, 0.78) (54.62, 62.63, 70.74)75 33 61 61 (31.70, 32.00, 32.30) (21.50, 25.53, 29.84) (0.76, 0.77, 0.78) (61.69, 69.79, 78.02)75 34 61 61 (31.70, 32.00, 32.30) (23.72, 27.72, 31.97) (0.76, 0.77, 0.78) (69.13, 77.32, 85.63)75 35 60 60 (34.70, 35.00, 35.30) (24.83, 28.62, 32.77) (0.70, 0.71, 0.72) (75.50, 83.46, 91.51)75 36 60 60 (34.60, 35.00, 35.40) (26.77, 30.60, 34.69) (0.70, 0.71, 0.73) (82.33, 90.40, 98.58)75 37 60 60 (34.70, 35.00, 35.30) (28.66, 32.48, 36.50) (0.70, 0.71, 0.72) (89.05, 97.3, 105.54)75 38 60 60 (34.70, 35.00, 35.30) (30.49, 34.25, 38.22) (0.70, 0.71, 0.72) (96.63, 104.9, 113.2)75 39 60 60 (34.70, 35.00, 35.30) (32.23, 35.94, 39.85) (0.70, 0.71, 0.72) (104.5, 112.8, 121.1)75 40 60 60 (34.70, 35.00, 35.30) (33.88, 37.54, 41.39) (0.70, 0.71, 0.72) (111.7, 120.1, 128.5)75 41 59 59 (40.50, 41.00, 41.50) (34.55, 38.03, 41.80) (0.61, 0.62, 0.63) (117.6, 125.7, 133.9)75 42 55 55 (41.80, 42.00, 42.20) (31.90, 35.11, 38.74) (0.64, 0.65, 0.66) (111.4, 119.0, 126.7)75 43 50 50 (42.70, 43.00, 43.30) (27.58, 30.28, 33.74) (0.69, 0.70, 0.71) (101.0, 107.9, 114.7)75 47 [32,33] 35 (46.70, 47.00, 47.30) (6.90, 8.88, 11.71) (0.90, 0.91, 0.92) (40.44, 44.23, 48.48)75 49 [31,32] 33 (48.70, 49.00, 49.30) (5.07, 7.30, 10.01) (0.92, 0.93, 0.94) (29.39, 33.65, 38.32)75 50 [31,32] 32 (49.70, 50.00, 50.30) (3.95, 6.31, 8.96) (0.93, 0.94, 0.95) (22.24, 26.55, 31.71)75 52 31 31 (50.60, 51.00, 51.40) (7.48, 9.92, 12.53) (0.91, 0.92, 0.93) (18.36, 24.10, 30.08)75 54 31 31 (52.60, 53.00, 53.40) (8.08, 10.45, 12.99) (0.90, 0.91, 0.92) (30.00, 35.82, 41.94)75 56 30 30 (55.70, 56.00, 56.30) (8.75, 10.77, 13.24) (0.88, 0.89, 0.90) (35.84, 41.17, 46.70)

M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677 675

Table 9 (Continued)

Benchmark Optimal values fordeterministic SULBP-1

Result of proposed GA for fuzzy SULBP-1

Name Size Cmax m* or [LB*, UB*] m c ID E SXn

Arcus1

83 3786 21 22 (3690.6, 3691, 3691.4) (9.07, 9.11, 9.15) (0.9321, 0.9323, 0.9325) (1832.9, 1838.7, 1844.4)83 3985 20 21 (3872.7, 3873, 3873.3) (9.50, 9.53, 9.57) (0.9307, 0.9308, 0.9310) (2079.9, 2085.0, 2090.2)83 4206 19 19 (4191.4, 4192, 4192.6) (5.23, 5.26, 5.30) (0.9503, 0.9505, 0.9508) (1210.9, 1215.7, 1220.4)83 4454 18 18 (4437.5, 4438, 4438.5) (5.54, 5.57, 5.60) (0.9475, 0.9477, 0.9479) (1254.3, 1259.4, 1264.4)83 4732 17 17 (4663.8, 4664, 4664.2) (5.86, 5.89, 5.92) (0.9547, 0.9548, 0.9550) (1399.2, 1404.2, 1409.2)83 5048 16 16 (4989.5, 4990, 4990.5) (6.23, 6.27, 6.30) (0.9480, 0.9482, 0.9484) (1476.2, 1481.2, 1486.2)83 5408 15 15 (5237.4, 5238, 5238.6) (6.64, 6.67, 6.70) (0.9633, 0.9636, 0.9638) (1525.0, 1530.4, 1535.9)83 5824 14 14 (5618.4, 5619, 5619.6) (7.12, 7.15, 7.18) (0.9622, 0.9624, 0.9626) (1665.9, 1671.4, 1676.9)83 5853 13 14 (5585.3, 5586, 5586.7) (7.58, 7.61, 7.64) (0.9678, 0.9681, 0.9683) (1747.3, 1752.9, 1758.5)83 6309 13 13 (5992.3, 5993, 5993.7) (7.67, 7.69, 7.72) (0.9715, 0.9717, 0.9720) (1851.9, 1857.4, 1862.9)83 6842 12 12 (6516.4, 6517, 6517.6) (7.77, 7.79, 7.82) (0.9679, 0.9681, 0.9683) (1906.1, 1911.7, 1917.4)83 6883 12 12 (6522.5, 6523, 6523.5) (8.31, 8.34, 8.37) (0.9670, 0.9672, 0.9674) (2049.4, 2055.1, 2060.9)83 7571 11 11 (7058.4, 7059, 7059.6) (9.07, 9.09, 9.12) (0.9748, 0.9750, 0.9752) (2351.1, 2356.7, 2362.3)83 8412 10 10 (7911.9, 7913, 791483 8898 9 9 (8554, 8555, 8556)

83 10,816 7 8 (9924.2, 9925, 9925

Table 10Fuzzy processing time of tasks [43].

Task no. Taskprocessingtime

Task no. Taskprocessingtime

1 (5 6 8) 7 (15 16 18)2 (3 5 6) 8 (3 5 6)3 (7 8 9) 9 (5 7 8)4 (8 10 11) 10 (11 15 17)5 (5 7 8) 11 (9 10 11)6 (16 18 20) 12 (16 18 19)

Given Cmax = [49–51].

Table 11Result of existing method for test problem.

Algorithm K Allocated

Modified GA forSALB

1 1, 2, 4, 5,2 6, 7, 9

3 10, 11, 12Fuzzy efficiency = (0.7305, 0.969, 1)Fuzzy idle percentage = (3.9216, 16.6667, 34.0136)Fuzzy smoothness index = (3.7417, 14.5258, 29.1548)

Modified GA forSULB

1 1, 6, 12

2 2, 5, 8, 103 3, 4, 7, 9

Fuzzy efficiency = (0.7305, 0. 992, 1)Fuzzy idle percentage = (3.9216, 16.6667, 34.0136)Fuzzy smoothness index = (3.7417, 14.4568, 29.1890)1 1, 2, 4, 6

GA for SALB byTsujimura et al.[43]

2 5, 3, 8, 103 7, 9, 11

4 12

Fuzzy efficiency = (0.572, 0.8013, 1)Fuzzy idle percentage = (26.9608, 37.5, 51.53061)Fuzzy smoothness index = (33.7787, 40.7308, 51.9134

Fuzzy RPWT forSALB by Fonsecaet al. [2]

1 1, 4, 2, 3,2 7, 10, 6

3 9, 11, 12

Fuzzy efficiency = (0.624242, 0.85034, 1)Fuzzy idle percentage = (7.843137, 16.6667, 34.0136)Fuzzy smoothness index = (8.2462, 17.5214, 30.3645)

ModifiedCOMSOAL for SALBby Fonseca et al. [2]

1 1, 5, 4, 2,2 6, 3, 9, 8

3 10, 11, 12Fuzzy efficiency = (0.647799, 0.94697, 1)Fuzzy idle percentage = (5.228758, 16.6667, 34.0136)Fuzzy smoothness index = (3.1623, 15.1327, 29.1548)

.1) (9.98, 10.00, 10.02) (0.9565, 0.9567, 0.9570) (2905.8, 2910.9, 2916.1)(5.44, 5.46, 5.49) (0.9830, 0.9833, 0.9835) (1570.2, 1575.7, 1581.2)

.8) (12.49, 12.51, 12.53) (0.9533, 0.9535, 0.9537) (3978.0, 3983.6, 3989.1)

the predecessor and successor constraints of the test problem aredisplayed in Fig. 26.

The results from the proposed algorithm and the results fromother existing methods are shown in Table 11. The first and secondrows of Table 11 present the results from the proposed algorithmin this paper for SALB and SULB, respectively. The third row of thetable shows the results from the GA offered by Tsujimura et al. [43].The fourth and fifth lines also show the results of fuzzy RPWT andmodified COMSOAL proposed by Fonseca et al. [2]; finally, the last

line is dedicated to the results of a fuzzy heuristic algorithm for theSULB problem that is offered by Zhang et al. [47]. As it is observed,none of the existing methods, rather than the number of stations,

tasks t(Sk) Ik c

3, 8 (31, 41, 48) (1, 9, 20) (36, 43, 47)(36, 41, 46) (3, 9, 15)

(36, 43, 47) (2, 7, 15)

(37, 42, 47) (2, 8, 14) (37, 42, 47), 11 (31, 42, 48) (1, 8, 20)

(35, 41, 46) (3, 9, 16)

(32, 39, 45) (4, 11, 19) (32, 39, 45)

(26, 35, 40) (9, 15, 25)(29, 33, 37) (12, 17, 22)(16, 18, 19) (30, 32, 35)

)

5, 8 (31, 41, 48) (1, 9, 20) (11, 15, 21)(42, 49, 55) (0, 1, 9)(30, 35, 38) (11, 15, 21)

7 (36, 44, 51) (0, 6, 15) (6, 12, 20)(31, 38, 43) (6, 12, 20)

(36, 43, 47) (2, 7, 15)

676 M.H. Alavidoost et al. / Applied Soft Computing 34 (2015) 655–677

Table 11 (Continued)

Algorithm K Allocated tasks t(Sk) Ik c

HeuristicAlgorithm for SULBby Zhang et al. [47]

1 1, 4, 11, 12 (38, 44, 49) (0, 6, 13) (12, 17, 24)2 2, 3, 5, 7, 8, 9 (38, 48, 55) (0, 2, 13)3 6, 10 (27, 33, 37) (12, 17, 24)Fuzzy efficiency = (0.624242, 0.85034, 1)Fuzzy idle percentage = (7.843137, 16.6667, 34.0136)Fuzzy smoothness index = (8.1240, 18.1384, 30.2324)

Table 12Summarized result of existing method for SALBP-1.

Problem class Modified GA (%) GA by Tsujimuraet al. [43] (%)

Fuzzy RPWT by Fonsecaet al. [2] (%)

Modified COMSOAL byFonseca et al. [2] (%)

Average of%deviation

A 0.00 0.93 8.06 1.62B 0.87 3.28 3.30 4.30C 2.53 3.41 4.06 4.59Total 1.13 2.54 5.14 3.50

Maximum of%deviation

A 0.00 33.33 50.00 33.33B 14.29 25.00 16.67 25.00C 14.29 14.29 33.33 33.33Total 14.29 33.33 50.00 33.33

%Optimal solution

A 100 97.22 75.00 91.67B 93.39 75.66 72.22 64.81C 65.83 58.75 59.24 53.06Total 86.41 77.21 68.82 69.85

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90

CP

U T

ime

(sec

)

ibar

oS

aoia

Table 13Summarized result of existing method for SULBP-1.

Problem class Modified GA(%)

Heuristic for SULB byZhang et al. [47] (%)

Average of%deviation

A 0.00 10.67B 2.93 8.33C 4.01 6.30Total 2.31 8.44

Maximum of%deviation

A 0.00 50.00B 16.67 20.00C 20.00 33.33Total 20.00 50.00

%Optimal solution

A 100 63.89B 73.54 35.19C 59.31 46.60

Number of Tas ks

Fig. 25. CPU time vs problem size.

n terms of the line performance, by the idleness percentage, andy the smoothness index, are better than our proposed modifiedlgorithm. This shows the high performance of the proposed algo-ithm.

In addition to the detailed example, the proposed algorithm andther methods are examined with bench-marks of SALBP-1 andULBP-1 and the results are presented in Tables 12 and 13.

As presented in Tables 12 and 13, the results of the proposed

lgorithm are averagely better than other methods in “Averagef %Deviation”, “Maximum of %Deviation”, and “%Optimal Solution”ndexes each of which proves high performance for the proposedlgorithm.

Fig. 26. Predecessor and successor constraints graph [43].

Total 77.62 48.56

6. Concluding remarks and implications for future works

In this paper, the single model of straight and U-shaped assem-bly line balancing with fuzzy processing time (f-SALBP and asf-SULBP) have been considered. Consistent with the uncertainty,variability, and imprecision in actual systems, task processing timelike the problem input data is presented in terms of TFNs. Thegoals considered in this paper were minimizing the numbers ofstations, maximizing fuzzy line efficiency, minimizing fuzzy idle-ness percentage, and minimizing fuzzy smoothness index. After themathematical formulation of the problem in fuzzy state, a methodbased on OFSR for SP controller has been proposed, and then thealgorithm parameters were calibrated using the Taguchi method.Finally, the algorithm was examined on different benchmarks andthe experimental results were the proof of its powerful capabil-ity. However, it is limited by the assembly line balancing singlemodel, and it is hoped that in the future researchers manage tosolve more complex problems such as the mixed model of assem-bly line balancing using the proposed adaptive GA. In addition,

the fuzzy problem of SALB-1 and SULB-1 could be solved by othermeta-heuristic algorithms such as ACO and their results could becompared with the proposed hybrid fuzzy adaptive GA.

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Methodol. 60 (2014) 66–84.

M.H. Alavidoost et al. / Applied

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