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IFSJSP: A novel methodology for the Job-ShopScheduling Problem based on intuitionistic fuzzy setsXiaoge Zhang a , Yong Deng a b , Felix T.S. Chan c , Peida Xu d , Sankaran Mahadevan b & YongHu ea School of Computer and Information Science, Southwest University , Chongqing 400715 ,Chinab School of Engineering, Vanderbilt University , Nashville , TN 37235 , USAc Department of Industrial and Systems Engineering , The Hong Kong PolytechnicUniversity , Hung Hum , Kowloon , Hong Kongd School of Electronics and Information Technology, Shanghai Jiao Tong University , Shanghai200240 , Chinae Institute of Business Intelligence and Knowledge Discovery, Guangdong University ofForeign Studies , Guangzhou 510006 , ChinaPublished online: 09 Jun 2013.

To cite this article: Xiaoge Zhang , Yong Deng , Felix T.S. Chan , Peida Xu , Sankaran Mahadevan & Yong Hu (2013): IFSJSP:A novel methodology for the Job-Shop Scheduling Problem based on intuitionistic fuzzy sets, International Journal ofProduction Research, DOI:10.1080/00207543.2013.793425

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IFSJSP: A novel methodology for the Job-Shop Scheduling Problem based on intuitionisticfuzzy sets

Xiaoge Zhanga, Yong Denga,b*, Felix T.S. Chanc, Peida Xud, Sankaran Mahadevanb and Yong Hue

aSchool of Computer and Information Science, Southwest University, Chongqing 400715, China; bSchool of Engineering, VanderbiltUniversity, Nashville, TN 37235, USA; cDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University,Hung Hum, Kowloon, Hong Kong; dSchool of Electronics and Information Technology, Shanghai Jiao Tong University, Shanghai200240, China; eInstitute of Business Intelligence and Knowledge Discovery, Guangdong University of Foreign Studies, Guangzhou

510006, China

(Received 19 May 2012; final version received 31 March 2013)

The Job-Shop Scheduling Problem (JSP) is an important concern in advanced manufacturing systems. In realapplications, uncertainties exist practically everywhere in the JSP, ranging from engineering design to product manufac-turing, product operating conditions and maintenance. A variety of approaches have been proposed to handle the uncer-tain information. Among them, the Intuitionistic Fuzzy Sets (IFS) is a novel tool with the ability to handle vagueinformation and is widely used in many fields. This paper develops a method to address the JSP under an uncertainenvironment based on IFSs. Another contribution of this paper is to put forward a generalised (or extended) IFS to pro-cess the additive operation and to compare the operation between two IFSs. The methodology is illustrated using athree-step procedure. First, a transformation is constructed to convert the uncertain information in the JSP into the corre-sponding IFS. Secondly, a novel addition operation between two IFSs is proposed that is suitable for the JSP. Then anovel comparison operation on two IFSs is presented. Finally, a procedure is constructed using the chromosome of anoperation-based representation and a genetic algorithm. Two examples are used to demonstrate the efficiency of theproposed method. In addition, a comparison between the results of the proposed IFSJSP and other existing approachesdemonstrates that IFSJSP significantly outperforms other existing methods.

Keywords: intuitionistic fuzzy sets; job-shop scheduling problem; genetic algorithm; uncertain information

1. Introduction

The Job-Shop Scheduling Problem (JSP) has been known as a stubborn combinatorial optimisation problem ever sincethe 1950s (Graham 1966). In terms of computational complexity, JSP is an NP-hard problem in the strong sense(Lenstra, Rinnooy Kan, and Brucker 1977). As it characterises an important decision process in contemporary manufac-turing systems, much research has focused on the JSP. Due to its great complexity, recent research has focused on meta-heuristic approaches, such as genetic algorithms (GAs) (Sakawa and Kubota 2000; Prakash, Chan, and Deshmukh 2011;Zhang, Gao, and Shi 2011), tau search (TS) (Nowicki and Smutnicki 2005), particle swarm optimisation (PSO) (Lei2012a), ant colony optimisation (ACO) (Rossi and Dini 2007; Xing et al. 2010), and a differential evolution (DE)algorithm (Tasgetiren et al. 2006).

Most existing research is based upon the condition that all time parameters are known exactly. However, there aremany vaguely formulated relations and imprecisely quantified physical data values in real-world applications, and pre-cise details are simply not known. There could be uncertainty in a number of factors, such as equipment availability,processing times and costs. Thus, in these cases the solutions generated using deterministic models may not be veryrobust. Although the JSP has been investigated (Wong, Chan, and Chan 2009; Amiri et al. 2010), only a few studiestake into account the factor of uncertainty using fuzzy set theory or stochastic theory (Ip, Lau, and Chan 2003; Chanet al. 2008; Lei 2010b; Kumar and Chan 2011; Kumar et al. 2011; Lei and Guo 2011). Moreover, the probability distri-bution needs to be determined in stochastic theory, and the membership function needs to be set in fuzzy set theory. Alarge amount of data is required to build these functions, which is expensive or impossible in actual manufacturingsystems.

*Corresponding author. Email: ydeng@swu.edu.cn

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Since uncertainty is one of the features of real-world applications, many methods have been proposed to solve prob-lems under an uncertain environment, such as evidence theory (Deng et al. 2011a; Deng, Jiang, and Sadiq 2011b), inter-val theory (Ashtiani et al. 2009), and fuzzy set theory (He et al. 2006; Liu 2010; Deng et al. 2012; Liu et al. 2012). Ofthese, the intuitionistic fuzzy set is a novel tool with the ability to handle vague information and is widely used in manyfields such as pattern recognition (Khatibi and Montazer 2002; Deng et al. 2010) and decision making (Maji, Roy, andBiswas 2002; Liu and Wang 2007; Xu 2007a, Boran et al. 2009; Tan and Chen 2010; Wei 2010; Pei and Zheng 2012;Kang et al. 2012). In traditional fuzzy set (FS) theory, the membership function of a fuzzy set has only a single value.However, in practice, it cannot express the degree of ‘support’, ‘opposition’, and ‘uncertainty’ at the same time. As aresult, the traditional fuzzy set approach is not adequate when the environment becomes more complicated. The conceptof intuitionistic fuzzy sets can be viewed as an alternative approach to define a fuzzy set when available information isnot sufficient for the definition of a conventional fuzzy set. The concept of intuitionistic fuzzy sets is a generalisation ofthe concept of fuzzy sets that is well suited for dealing with vagueness (Atanassov 1999; 2000). Many studies havebeen published on decision making under an uncertain environment based on IFS theory (Liu and Wang 2007; Boranet al. 2009). It can be seen that intuitionistic fuzzy sets can be used to deal with uncertain information and handle anyactivity requiring human expertise and knowledge (Li 1999; De, Biswas and Roy 2001; Liu and Wang 2007; Xu2007c), which are inevitably imprecise or not totally reliable.

In the Job-Shop Scheduling Problem, many manufacturing parameters need to be optimised, such as minimising thetotal tardiness time (Lei and Guo 2011; Tasgetiren et al. 2011), the average completion time (Sitters 2010), the machineidle time (Khorshidian et al. 2011), the makespan (Haq et al. 2010; Low et al. 2010; Seo and Kim 2010; Yang andYang 2010; Wang and Wang 2012), etc. Of these parameters, the makespan is commonly used. Several studies havebeen reported on minimising the job makespan (Rajendran and Ziegler 2004; Allahverdi and Aydilek 2010; Chihaouiet al. 2011; Mokhtari, Abadi, and Cheraghalikhani 2011; Zheng et al. 2011). In this paper, we focus on minimising thejob makespan and applying intuitionistic fuzzy sets theory to the Job-Shop Scheduling Problem. A novel approach isproposed to solve the JSP with an uncertain processing time. To the best of our knowledge, this is the first time thatintuitionistic fuzzy sets have been applied to deal with the Job-Shop Scheduling Problem under an uncertain environ-ment.

Compared with stochastic theory, intuitionistic fuzzy sets have certain advantages. First, for intuitionistic fuzzy sets,it is simpler to determine the upper and lower bounds of a variable than to build the membership functions, whichrequires a lot of data in advance. Moreover, in real-world applications, we cannot obtain such data on a large scale aftera long period of production. Secondly, intuitionistic fuzzy sets provide more flexibility to handle the Job-Shop Schedul-ing Problem under an uncertain environment. When the environment changes, intuitionistic fuzzy sets are easy toamend. On the contrary, it is complicated when stochastic theory is used, which needs to re-build the membershipfunction. Intuitionistic fuzzy sets are more suitable for handling the uncertain information existing in the Job-ShopScheduling Problem compared with stochastic theory.

The remainder of the paper is organised as follows. Section 2 gives a review of the Job-Shop Scheduling Problem.Basic concepts of intuitionistic fuzzy sets and the problem are introduced in Section 3. Section 4 details the proposedmethod. A numerical example is illustrated to show the efficiency of the proposed method in Section 5. The conclusionsare summarised in Section 6.

2. Literature review of the Job-Shop Scheduling Problem

The scheduling problem consists of defining a schedule that can meet all timing and logical constraints of the tasksbeing scheduled, and, in general, it has been classified as NP-complete (Johnson 1982).

In the past decades, fuzzy scheduling and stochastic scheduling techniques have attracted much attention and severalmethods have been reported. For the fuzzy Job-Shop Scheduling Problem (FJSP), Sakawa and Kubota (2000) proposeda genetic algorithm (GA) to deal with the fuzzy processing time and fuzzy due date. Ip et al. (2002) proposed a ToolMonitoring System (TMS) to assign a machine schedule using fuzzy approaches based on the information from the toolsand part relationships. Niu, Jiao, and Gu (2008) presented a particle swarm optimisation with genetic operators (GPSO)to minimise the fuzzy makespan. Lei (2010a) presented a decomposition–integration genetic algorithm (DIGA), com-posed of a two-string representation, an effective decoding method and a main population, for the flexible Job-ShopScheduling Problem with fuzzy processing time. Rajkumar et al. (2011) proposed a greedy randomised adaptive searchprocedure (GRASP) to solve the flexible Job-Shop Scheduling Problem (FJSP) with limited resource constraints. Amiriet al. (2010) presented a variable neighbourhood search (VNS) algorithm. Various neighbourhood structures related toassignment and sequencing problems are used for generating neighbouring solutions in the process of the presentedalgorithm.

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For the stochastic Job-Shop Scheduling Problem, Tavakkoli-Moghaddam et al. (2005) proposed a hybrid methodbased on a neural network and simulated annealing (SA) that uses a neural network approach to generate an initial feasi-ble solution and a SA to improve the quality of the initial solution. Gu et al. (2010) proposed a novel competitive co-evolutionary quantum genetic algorithm (CCQGA) to minimise the expected value of the makespan. Azadeh, Negahban,and Moghaddam (2011) proposed a new algorithm based on computer simulation and artificial neural networks to selectthe optimal dispatching rule for each machine from a set of rules in order to minimise the makespan in the stochasticJob-Shop Scheduling Problem (SJSP).

However, for the stochastic theory, it is required to construct the membership function or the distribution of thestochastic variable under the environment of fuzzy and stochastic theories in advance. In actual manufacturing systems,it is expensive or impossible to produce the processing data for membership functions and probability distributions.Consequently, stochastic theory is not practical in many real-world applications due to the real-time requirement. Theintuitionistic fuzzy set (IFS) is a novel tool with the ability to handle vague information and has been applied in manyfields to describe uncertainty. Determining the IFSs of variables is simpler than building the membership function of thestochastic variable. It provides the possibility to model unknown information in a new way. However, to the authors’knowledge, there is a lack of studies that apply IFSs to the JSP. In this paper, taking advantage of intuitionistic fuzzysets having the ability to deal with problems under an uncertain environment, a novel methodology is proposed for theJob-Shop Scheduling Problem—IFSJSP.

In many cases, scheduling problems are classified as NP-hard. Therefore, applying a heuristic methodology to obtaina near-optimal solution in a relatively shorter period is more practical than mathematical programming. Of the variousheuristic approaches, GAs are recognised as being appropriate and efficient. Many researchers have applied GAs to theJSP. For example, Cheng, Gen, and Tsujimura (1996) gave a detailed tutorial survey of papers using GAs to solve theclassical Job-Shop Scheduling Problem (JSP) in their Part I survey. In Part II, they reviewed papers that use hybrid GAsto tackle the JSP. The method of Zhou, Cheung, and Leung (2009), who use a hybrid framework integrating a heuristicand a genetic algorithm (GA), is used for job-shop scheduling to minimise the weighted tardiness.

Furthermore, several researchers have proposed different kinds of adaptive GAs for particular problems to strategi-cally strengthen the genetic search in different phases of evolution (e.g., De Giovanni and Pezzella (2010) and Kuo andHan (2011)).

It can be seen that GAs are an efficient approach to the JSP. Therefore, it is meaningful to combine IFSs and GAsto develop a new approach to handle the JSP under an uncertain environment. The new approach has some advantagesover classical GAs. First, it does not increase the complexity of the classical GA because the IFS is expressed in a sim-ple way and its operations are not complicated (Atanassov 1986). Secondly, the IFS is flexible in expressing uncertaininformation existing in the JSP due to its having been widely used in real applications, such as decision making (Li2010; Wei, Zhao, and Lin 2010) and handling uncertain information (Xu 2007b). Compared with other methods dealingwith uncertain information, the IFS is simple and flexible. More importantly, it is simple to determine the upper andlower bounds of a variable than to build the membership functions to express changes in the variables (Lei 2012b).Finally, compared with the classical GA, it has the advantage of addressing the JSP under an uncertain environment,which the GA cannot do. As a consequence, in order to apply IFSs to the JSP, we combine the GA and the IFS todevelop a new genetic algorithm in which operation-based representation is applied to the JSP.

3. Basic theory

This section introduces the basic concepts of intuitionistic fuzzy sets and the Job-Shop Scheduling Problem.

3.1 Basic concepts of intuitionistic fuzzy sets

Definition 3.1: (Atanassov 1986) An intuitionistic fuzzy set in X is defined as

A ¼ fhx; lA(x); vA(x)ij x 2 Xg;

with the condition 0 � lA(x)þ vA(x) � 1 for all x 2 X .lA(x) and vA(x) denote, respectively, the degree of membership and the degree of non-membership of the element x

in set A. In addition, pA(x) denotes the degree of uncertainty towards x 2 X ,

pA(x) ¼ 1� lA(x)� tA(x); x 2 X :

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For example, for intuitionistic fuzzy sets (la; ta) ¼ (0:5; 0:3), it can be seen that la ¼ 0:5 and ta ¼ 0:3. This couldmean, for example, ‘‘A total of 10 persons vote. Of these, five vote for and three against. The other two abstain fromvoting’’.

The following expressions are defined by Atanassov (1986) for all A and B belonging to IFSs(X ):

(1) A � B if and only if lA(x) � lB(x) and vB(x) � vA(x) for all x 2 X ;(2) A ¼ B if and only if A � B and B � A;(3) Ac ¼ fhx; vA(x); lA(x)ij x 2 Xg.

3.2 Problem description of the IJSP

The classical Job-Shop Scheduling Problem can be stated as follows (Wang, Yin, and Wang 2009): there are m differentmachines and n different jobs that need to be scheduled. Every job consists of a finite set of operations and the opera-tion order on machines is pre-specified. Each operation is characterised by a required machine and a fixed processingtime. The following assumptions are made: (1) each machine is an entity; (2) no interruption is allowed; (3) there isonly one of each type of machine; (4) the technological constraints are known in advance and are immutable; and (5)there is no randomness, all the data are known and fixed. There are also several constraints on the jobs and machines:(1) a job does not visit the same machine twice; (2) there are no precedence constraints among the operations of differ-ent jobs; (3) each operation needs to be processed during an uninterrupted period of a given length on a given machine;(4) each machine can handle at most one operation at a time; and (5) neither the release times nor the due dates arespecified. The problem is to determine the operation sequence on the machines in order to minimise the makespan, i.e.the time required to complete all the jobs.

The Job-Shop Scheduling Problem with the makespan objective can be formulated as follows:

min max1� k�m

max1� k�m

fcikg� �

; (1)

s.t.

cik � tik ¼ M (1� aihk) � cih; i ¼ 1; 2; . . . ; n; h; k ¼ 1; 2; . . . ;m; (2)

cjk � cik ¼ M (1� xijk) � pjk ; i ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ;m; (3)

cik � 0; i ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ;m; (4)

xijk ¼ 0 or 1; i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ;m; (5)

where cjk is the completion time of job j on machine k, tjk is the processing time of job j on machine k, M is a largepositive number, and aihk is an indicator coefficient defined as

aihk ¼ 1; if processing on machine h precedes that on machine k for job i ;0; otherwise;

�

and xijk is an indicator variable defined as

xijk ¼ 1; if job i precedes job j on machine k ;0:

�

The objective is to minimise the makespan. Constraint (2) ensures that the processing sequence of operations foreach job corresponds to the prescribed order. Constraint (3) ensures that each machine can process only one job at atime.

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The Interval Job-Shop Scheduling Problem (IJSP) is the extended version of the JSP by introducing interval process-ing conditions, including interval processing times. The n� m IJSP is composed of n jobs Ji (i ¼ 1; 2; . . . ; n) and mmachines Mk (k ¼ 1; 2; . . . ;m). Operation Oij indicates the jth operation of Ji processed on a machine and its operationtime is represented as an interval number pij ¼ ½p

ij; pij�. The objective of the IJSP is to minimise the total processing

time.Other assumptions of the JSP remain suitable for the IJSP.

(1) Each machine can carry out at most one operation in each time unit.(2) Each operation has to be carried out on one machine in each time unit.(3) Once an operation starts, it will be finished without interruption.(4) Each operation can only start upon the completion of its preceding operation.

3.3 Genetic algorithm

The genetic algorithm (GA) (Holland 1975) is a heuristic search technique used to solve optimisation and search prob-lems. It is a simulation of the process of natural selection and biological evolution. The process of problem solvingusing a genetic algorithm is described in Figure 1. For more detailed information, see Holland (1975).

Before starting the genetic evolution, a genetic representation and a fitness function should be determined. Thegenetic representation, called the chromosome, is an expressing mode of the solution. The fitness function is defined tomeasure the quality of the represented solution. Once the genetic representation and the fitness function have beendefined, the evolution iterative solution process can start.

3.3.1 Initialisation

This is a process of initialising the population of solutions. In the first iteration, the initial population is formed by manyindividual solutions that are randomly generated. In other iterations, the population of solutions is generated accordingto the result of the last iteration.

Figure 1. Flowchart for the GA.

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3.3.2 Selection

In this step, a proportion of the existing population in the current generation is selected to breed a new generation.Individual solutions are selected through a fitness-based judgement. Those solutions that are fitter than others measuredby the fitness function are typically more likely to be selected. Typical selection methods include roulette wheel selec-tion and tournament selection.

3.3.3 Reproduction

Once the individual solutions to breed a new generation have been selected, the next generation population of solutionsis generated using genetic operators: crossover and mutation. Crossover is an operation that partly interchanges thestructure of two ‘parent’ individual solutions to generate new solutions. The mutation operation changes the structure ofan individual solution randomly with a low probability. The search ability of the genetic algorithm is improved rapidlyby crossover and mutation. Normally, the average fitness of the population will increase after the crossover and mutationoperations.

3.3.4 Termination

The above evolutionary process is repeated until a termination condition has been reached. Generally speaking, the ter-minating condition is either that a fixed number of generations has been reached or that the fitness of the population ofthe solutions does not increase, etc.

Due to its efficiency in stochastic optimisation, the GA has been widely use in many optimisation problems (Chanet al. 2004; Chan and Kumar 2009; Kumar et al. 2009; Toledo et al. 2009; Engin, Ceran, and Yilmaz 2011; Li et al.2011; Rahmani, Mousavi, and Kamali 2011; Subbaraj, Rengaraj, and Salivahanan 2011).

4. Proposed method

The proposed method takes advantages of intuitionistic fuzzy sets to model the uncertain information in the JSP. First, atransformation is constructed to convert the uncertain information in the JSP into corresponding intuitionistic fuzzy sets.Second, a new addition operation between two IFSs is proposed that is suitable for the JSP. Then a novel comparisonoperation of two IFSs is presented. Finally, a procedure is constructed using the chromosome of the operation-basedrepresentation and the genetic algorithm. A flow chart of the proposed method is shown in Figure 2.

4.1 Addition and comparison operations of intuitionistic fuzzy sets

Definition 4.1: Generalised intuitionistic fuzzy sets. A generalised intuitionistic fuzzy set is defined as

A ¼ fhx; lA(x); vA(x)ij x 2 Xg;

with the condition 0 � lA(x), 0 � vA(x).

Definition 4.2: Assume that there are two generalised intuitionistic fuzzy sets A ¼ fhx; lA(x); vA(x)ij x 2 Xg andB ¼ fhx; lB(x); vB(x)ij x 2 Xg, then the addition of A and B is defined as

Aþ B ¼ ½(lA(x)þ lB(x)); (vA(x)þ vB(x))�: (6)

Similarly, if there are n generalised intuitionistic fuzzy sets A1 ¼ fhx; l1(x); v1(x)ij x 2 Xg; A2 ¼fhx; l2(x); v2(x)ij x 2 Xg; . . . ;An ¼ fhx; ln(x); vn(x)ij x 2 Xg, then the addition of A1 to An is defined as

A1 þ A2 þ � � � þ An ¼ ½(l1(x)þ l2(x)þ � � � þ ln(x)); (v1(x)þ v2(x)þ � � � þ vn(x))�: (7)

Definition 4.3: Suppose there are two generalised intuitionistic fuzzy sets A ¼ fhx; lA(x); vA(x)ij x 2 Xg andB ¼ fhx; lB(x); vB(x)ij x 2 Xg, the possibility degree-based comparison of A and B is defined as

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P(A � B) ¼0; vA[vB;0:5; vA ¼ vB; lA ¼ lB;1; vA � vB:

8<: (8)

The possibility degree of intuitionistic fuzzy sets denotes the degree that one intuitionistic fuzzy set is larger than orsmaller than another. From Equation (8), the following features can be obtained:

(1) 0 � P(A � B) � 1;(2) if P(A � B) ¼ P(B � A), then A ¼ B.

If A � B, which means lA ¼ lB and vA ¼ vB, then P(A � B) ¼ P(B � A) ¼ 0:5. Therefore, we define A\pdB iffP(A � B)[0:5 or P(B � A)\0:5. Similarly, A[pd B iff P(A � B)\0:5 or P(B � A)[0:5.

4.2 Representation of the Job-Shop Scheduling Problem

Operation-based representation has been widely used in the JSP (Lei 2012a), and encodes a schedule as a sequence ofoperations and each gene stands for one operation. The representation can also be applied to the IJSP. For an n� mIJSP, the chromosome of the operation-based representation is an integer string (p1; p2; . . . ; pn�m) in which job Ji(i ¼ 1; 2; . . . ; n) occurs m times and each repetition (each gene) does not indicate a concrete operation of a job but refersto a unique operation that is context dependent. The initial chromosome of the JSP is randomly generated. Then the

Interval job schedulingproblems

Construct the intuitionistic fuzzy setcorresponding to IJSSP

Calculate the fitness value of each chrome

Analyze and obtain the result

Initialize the population

Termination Criterion is met or not

Yes

No

Process crossover operator

Get the chromosome with the best fitness value

Carry out mutation operator

Figure 2. Flow chart of the proposed method.

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chromosome is sequentially allocated a best available beginning time by adopting certain operations of intuitionisticfuzzy sets introduced in Section 2.

For example, for the 4� 3 IJSP shown in Table 1, Figure 3 displays two alternative chromosomes. There are threemachines and four jobs, with each job consisting of three operations and the operation order is predefined. In Figure 3(a), the first gene 3 means that J3’s first operation is processed. The reason for this is that the number 3 in thechromosome appears for the first time. Similarly, the next is J1’s first operation. The next is J2’s first operation. The nextnumber is 3 again. When the number 3 appears for the second time in the chromosome, it indicates J3’s second opera-tion. As this process continues, one candidate schedule for this IJSP problem can be obtained aso31; o11; o21; o32; o33; o41; o22; o12; o42; o13; o23; o43. In a similar way, a schedule corresponding to the chromosome inFigure 3(b) is obtained as o21; o22; o31; o23; o41; o32; o11; o33; o42; o12; o43; o13.

4.3 Raw scheduling of the IJSP

Again consider the 4� 3 IJSP shown in Table 1 composed of four jobs and three machines. Each job has threeoperations. The second column displays the processing time of each operation under an uncertain environment and thethird column illustrates the sequence of each job. First, the maximum processing time and the minimum processing timeof all operations are obtained as ½3; 7� and ½1; 3�, respectively. Then the addition of the upper value of the maximum pro-cessing time and the upper value of the minimum processing time is 3þ 7 ¼ 10. The following IFS is constructed. Foroperation o11, the processing time is ½1; 3�. Figure 4 shows the IFS corresponding to operation o11. For operation o11,l(x) ¼ 1=10 ¼ 0:1 and v(x) ¼ (10� 3)=10 ¼ 0:7. As a consequence, A ¼ fhx; 0:1; 0:7ijx 2 Xg. Other IFSs can beobtained similarly.

For the example shown in Table 1, the chromosome may be (1; 4; 4; 3; 4; 3; 2; 3; 1; 1; 2; 2). The corresponding list ofoperations is o11; o41; o42; o31; o43; o32; o21; o33; o12; o13; o22; o23. Then the raw schedule of the IJSP can be constructed asfollows.

Table 1. Illustration of the IJSP.

Job

Operation

1 2 3 1 2 3Interval processing time Processing routes

J1 [1,3] [3,5] [2,3] M1 M2 M3

J2 [2,4] [4,6] [3,5] M1 M3 M2

J3 [2,5] [3,7] [4,6] M3 M1 M2

J4 [5,6] [3,5] [2,4] M3 M2 M1

(a)

(b)

Figure 3. (a) Sample encoding of a chromosome. (b) Another sample encoding of a chromosome.

uncertainty

0 10

1 3 Av x

Figure 4. The corresponding IFS for operation o11.

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First, o11 is performed. Then the corresponding IFS11 ¼ ½0:1; 0:7� and the next is o41, which is processed on machineM3. Therefore, IFS31 ¼ ½0:5; 0:4�. For operation o42, its previous operation is o41, IFS21 ¼ ½0:5; 0:4� þ ½0:3; 0:5� ¼½0:8; 0:9�.

For operation o31, its operation machine is M3. However, o41 is performed on machine M3 before o31 is performed.As a consequence, IFS32 ¼ IFS31 þ ½0:2; 0:5�. Using Equation (6), it can be obtained that IFS32 ¼ ½0:70; 0:90�. For oper-ation o43, its previous operation is o42, whose IFS is ½0:80; 0:90� and its operation machine is M1, which is used byoperation o11 before operation o43. Thus, the IFSs between operation o42 and operation o11 need to be compared.According to Equation (8), P(½0:80; 0:90� � ½0:10; 0:70�) ¼ 0 and it can be obtained that operation o43 must be per-formed after operation o42. Hence, IFS12 ¼ ½0:8; 0:9� þ ½0:2; 0:6� ¼ ½1:0; 1:5�. The next operation is o32. Its previousoperation is o31 and its operation machine is M1, which is used by operation o43 before the operation itself. The IFSsbetween operation o31 and operation o43 should be compared. P(½0:5; 0:4� � ½1:0; 1:5�) ¼ 1. Therefore, operation o32 isprocessed after operation o43. IFS13 ¼ ½1:0; 1:5� þ ½0:3; 0:3� ¼ ½1:3; 1:8�.

For operation o21, its operation machine is M1. Similarly, it should be performed after operation o32 andIFS14 ¼ ½1:3; 1:8� þ ½0:2; 0:6� ¼ ½1:5; 2:4�. For operation o33, its previous operation is o32 and its operation machine isM2. Thus, the IFSs between operation o32 and operation o42 should be compared. P(½1:3; 1:8� � ½0:8; 0:9�) ¼ 0. As aresult, IFS22 ¼ ½1:3; 1:8� þ ½0:4; 0:4� ¼ ½1:7; 2:2�. Now its operation o12’s turn. Its previous operation is o11 and it isoperated on machine M2. Naturally, the IFSs between o33 and operation o11 should be compared.P(½1:7; 2:2� � ½0:1; 0:7�) ¼ 0. It can be obtained that operation o12 should be operated after operation o33. ThenIFS23 ¼ ½1:7; 2:2� þ ½0:3; 0:5� ¼ ½2:0; 2:7�. For operation o13, its previous operation is o12 and it is processed on machineM3. The IFSs between o12 and o31 should be compared. P(½2:0; 2:7� � ½0:7; 0:9�) ¼ 0. Operation o13 should be per-formed after operation o12. IFS33 ¼ ½2:0; 2:7� þ ½0:2; 0:7� ¼ ½2:2; 3:4�. For operation o22, it is operated on machine M3,which is used by operation o13 before the operation itself, and its previous operation is o21. The IFSs between o13 ando21 should be compared. P(½2:2; 3:4� � ½1:5; 2:4�) ¼ 0. Hence, operation o22 should be performed after operation o13,IFS34 ¼ ½2:2; 3:4� þ ½0:4; 0:4� ¼ ½2:6; 3:8�. For operation o23, its operation machine is M2 and its previous machine iso22. The IFSs between o12 and o22 need to be compared. P(½2:0; 2:7� � ½2:6; 3:8�) ¼ 1. Thus, o23 needs to be performedafter o22 is completed. IFS24 ¼ ½2:6; 3:8� þ ½0:3; 0:5� ¼ ½2:9; 4:3�. Figure 5 shows the raw schedule for the 4� 3 IJSP;the triangle under the line represents the beginning time of the operation and the triangle on the line indicates thecompletion time of each operation.

4.4 Genetic algorithm for the IJSP

Although the above procedures can generate raw schedules for the IJSP, the result of the raw scheduling is not optimal.A genetic algorithm (GA) is investigated to optimise the results in this section.

Figure 5. The raw schedule of the IJSP.

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4.4.1 Calculation of the fitness value

In this section, a GA is applied to build the schedule. The GA consists of four parts: calculation of the fitness value,selection, crossover and mutation, as shown in Figure 1. The calculation of the fitness value in the proposed method isas follows. As can be seen in Figure 5, the ultimate maximum processing time in machines M1, M2, and M3 is ½29; 47�,and the fitness value is f ¼ 1=(29þ 47) ¼ 0:0132.

4.4.2 Crossover operator

During crossover, a pair of chromosomes are randomly selected according to the roulette wheel selection approach, anda number of genes are randomly selected according to a predefined crossover rate. Figure 6 shows a sample crossover,in which the first four genes are selected to cross over. Sometimes, crossover may generate an invalid chromosome,such as redundant operations and inconsistent factors. For example, as can be seen in Figure 6, the numbers in dashedboxes display the redundant genes in Of1 and Of2. These redundant genes appear while the genes that should appeardo not appear. To avoid this, the redundant genes and the lost genes in the chromosome are found first. Then the lostgenes are used to replace the redundant genes in the chromosome. Finally, the correct chromosome after crossover isobtained as shown in Figure 6.

4.4.3 Mutation operator

During mutation, a pair of genes are randomly selected and swapped. As can be seen in Figure 7, genes 3 and 1 in thechromosome are selected to swap. After swapping, Of1 is obtained. The purpose of this mutation is to reschedule thescheduling priority of the jobs’ operations. For example, after swapping, the production priority for operation o12 isrescheduled to be processed before operation o32 as shown in Figure 7.

4.4.4 Optimisation results

The other parameters of the GA are a population scale of 20, a crossover probability of 0.6, a mutation probability of0.01, and a maximum generation of 100 for the 4� 3 IJSP.

After mutation, we check whether or not the termination criterion is met. If the termination criterion is met, thenexit the iteration loop and obtain the result. The objective of this section is to find the best schedule plan based on theprevious steps.

P1:

P2:

Of1:

Of2:

Of1:

Of2:

3 1 2 3 3 4 2 1 4 1 2 4

2 2 3 2 4 3 1 3 4 1 4 1

2 2 3 2 3 4 3 1 4 1 1 4

3 1 3 2 4 3 1 2 4 1 4 2

2 2 3 2 3 4 2 1 4 1 2 4

3 1 2 3 4 3 1 3 4 1 4 1

Step 1

Step 2

Figure 6. Sample crossover of the GA.

3 1 2 3 3 4 2 1 4 1 2 4

3 1 2 1 3 4 2 1 4 3 2 4

P1:

Of1:

Figure 7. Sample mutation of the GA.

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5. Numerical examples

In this section, examples are illustrated to demonstrate the efficiency of the proposed method. The computation resultsare also compared with the existing approaches. The parameter is first defined.

Definition 5.1: Suppose there are n computation results expressed by Ai ¼ (Li;Ui) (i ¼ 1; 2; . . . ; n), where Li and Ui

denote the lower and upper bounds of the ith result of the IJSP. Then the parameter called the average standard devia-tion (ASD) is defined as

ASDi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi

j¼1f½(Lj þ Uj)=2� � Ljg2i

s; j ¼ 1; 2 . . . ; i; i ¼ 1; 2 . . . ; n: (9)

The parameter reflects the average standard deviation of the computational results. In other words, it reflects theaverage fluctuations of all the solutions. We can determine whether the solutions converge to a stable result from thisparameter.

The following example illustrates how the IFS is more efficient than the traditional fuzzy set.

Example 5.2 Assume there are three distribution centre locations in a logistics company, A1, A2, and A3, that need tobe ranked. There are three attributes associated with each distribution centre: T1 for distance, T2 for capacity, and T3 forcost. The weight vector of the attributes is

x ¼ (x1;x2;x3)T ¼ (0:35; 0:25; 0:40):

The membership function for these three distribution centre locations is

T1 T2 T3

A1

A2

A3

0:75 0:80 0:400:60 0:68 0:750:80 0:46 0:65

0@

1A

Under these circumstances, the IFS degenerates into a traditional fuzzy set. For example, for T1 in group A1, the IFSfor this attribute can be expressed as (0:75; 0:25; 0). Using the weighted sum, the value of the degree of membership foreach group can be calculated as

lA1 ¼ 0:75� 0:35þ 0:80� 0:25þ 0:40� 0:40 ¼ 0:6225;

lA2 ¼ 0:60� 0:35þ 0:68� 0:25þ 0:75� 0:40 ¼ 0:6800;

lA3 ¼ 0:80� 0:35þ 0:46� 0:25þ 0:65� 0:40 ¼ 0:6550:

As can be seen from Figure 8, both the traditional fuzzy set and the IFS can obtain the same results. However, inmany cases, uncertainties exist practically everywhere from engineering design to product manufacturing, product life-time service condition and maintenance. As a consequence, for many real-world applications, the decision matrix fordifferent decision alternatives is fuzzy. For this example, suppose the fuzzy decision matrix is as follows:

T1 T2 T3

A1

A2

A3

(0:75; 0:10; 0:15) (0:80; 0:15; 0:05) (0:40; 0:45; 0:15)(0:60; 0:25; 0:15) (0:68; 0:20; 0:12) (0:75; 0:05; 0:20)(0:80; 0:20; 0:00) (0:46; 0:51; 0:03) (0:65; 0:30; 0:05)

0@

1A

The first element in each cell denotes the degree of membership function and the second element in each cell repre-sents the degree of non-membership function. The last element in each cell expresses the degree of uncertainty existing

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in the decision process. For this example, the traditional fuzzy set is not efficient because it cannot express the degreeof ‘support’, ‘opposition’, and ‘uncertainty’ at the same time. On the contrary, the IFS can deal with this situation effec-tively. Following the same steps, the IFS obtains the results shown in Figure 9. As can be seen, the IFS is more efficientin handling uncertain information when compared with the traditional fuzzy set.

Example 5.3 An example of a 5� 5 IJSP is shown in Table 2. The fourth column displays the actual processing timeof each operation. The other parameters of the GA are as follows: population scale 20, crossover possibility 0.6, muta-tion possibility 0.01, and maximum generations 100 for this problem. The program is run 25 times. Using the proposedmethod, Figure 10 shows the best result after each iteration. The top of each vertical bar in Figure 10 states the maxi-mum processing time of each iteration and the bottom of each vertical bar displays the minimum processing time. Themiddle of the vertical bar gives the average processing time. Figure 11 illustrates the fluctuation of the average standarddeviation when the iteration increases. Figure 11 can be obtained based on Equation (9). As can be seen from Figure 11,the fluctuations of the first 10 items are large. When the iteration increases, the result tends to be stable, which is shownin Figure 11 as an approximate straight line. As a consequence, it can be concluded that the proposed method canobtain an optimum schedule for this problem as most of the results converge to a stable value.

In addition, the computational results for the IFSJSP and the GA used by Lei (2012b) were processed. Table 3 dis-plays the parameters of the GA and IFSJSP. In Table 3, ‘Optimal solution’ indicates the solution whose fitness value isthe largest in the solution set. Figure 12 illustrates the computation results. As can be seen from Figure 12, the averagevalue of the IFSJSP is always lower than or equal to the value of the GA. Not only is the computational result for IFS-JSP better than that of the GA, but it needs fewer iterations and population scales.

A1 A2 A3Traditional Fuzzy Set 0.6225 0.68 0.655IFS 0.6225 0.68 0.655

0.590.6

0.610.620.630.640.650.660.670.680.69

Prio

rity

Wei

ghts

Traditional Fuzzy Set IFS

Figure 8. Priority weights for the three distribution centre locations.

0.5 1 1.5 2 2.5 3 3.5

0.450.5

0.550.6

0.650.7

0.750.8

0.850.9

0.95

Prio

rity

Wei

ghts

Decision Alternatives

A1

A2 A3

Figure 9. Priority weights for different decision alternatives under an uncertain environment.

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Table 2. A 5� 5 IJSP (units: minutes).

Job

Operation

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5Interval processing time Processing routes Actual processing time

J1 [2,4] [3,6] [2,5] [4,6] [1,3] M4 M3 M5 M2 M1 3 4 3 5 2J2 [2,4] [2,5] [2,5] [1,3] [4,5] M5 M4 M3 M1 M2 4 2 5 2 4J3 [2,5] [1,3] [3,5] [2,4] [3,6] M3 M5 M2 M1 M4 4 2 3 4 5J4 [1,3] [3,5] [1,5] [2,6] [2,5] M4 M5 M3 M2 M1 2 4 3 4 5J5 [2,4] [1,4] [1,4] [2,3] [3,5] M2 M3 M4 M5 M1 4 2 3 3 5

5 10 15 20 25

10

15

20

25

30

35

40

45

50

Iteration

Proc

essi

ng T

ime

Figure 10. Ultimate processing time of each iteration for the 5� 5 IJSP (units: minutes).

5 10 15 20 256

7

8

9

10

11

12

Iteration

Aver

age

stan

dard

dev

iatio

n (A

SD)

Figure 11. Average standard deviation for the 5� 5 IJSP.

Table 3. Parameter comparison between the IFSJSP and the GA for the 5� 5 IJSP (MG, maximum generation).

Approach MG Population scale Crossover probability Mutation probability Optimal solution Fitness value

GA 300 25 0.9 0.1 (17,36) 0.01885IFSJSP 100 20 0.6 0.01 (17,33) 0.02000

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Figure 13 shows one possible schedule using the actual processing time in Table 2. Rectangles of the same colourindicate operations belonging to the same job. The schedule can be obtained easily from Figure 13.

Example 5.4 Table 4 displays one example of the 10� 10 IJSP. The processing time and processing routes aredisplayed in the second and third columns, respectively. The parameters of the GA for this problem are as follows:population scale 40, crossover possibility 0.6, mutation possibility 0.01, and maximum generations 1000. Similarly, weran the program 25 times. Figure 14 shows the processing time for the 10� 10 IJSP. Figure 15 displays the averagefluctuation of the solutions according to Equation (9). As can be seen, the solution fluctuation is very subtle around thevalue 30. To some extent, this may can reflect convergence of the solution to a stable result. As the solutions havealmost no fluctuations, the plot of the fluctuation is approximately a straight line. As a result, it can be concluded thatthe proposed method can achieve a stable result, although the IJSP becomes more complex. As a consequence, the pro-posed method has the ability to handle IJSP with different levels of complexity.

In addition, we compared the basic parameters and computational results for the IFSJSP and the GA, shown inTable 5 and Figure 16, respectively. From Table 5, it can be seen that IFSJSP needs fewer population scales than theGA. As the IFSJSP’s population scale is less than that of the GA, its computational result is no better than the GA forthe 10� 10 IJSP. However, the IFSJSP can obtain approximate results, but does not consume as much in terms ofresources.

5 10 15 20 25

10

15

20

25

30

35

40

45

50

Proc

essi

ng T

ime

Iteration

GAIFSJSP

Figure 12. Comparison between the IFSJSP and the GA for the 5� 5 IJSP (units: minutes).

Figure 13. One possible schedule using the actual processing time for the 5� 5 IJSP (units: minutes).

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Table

4.A

10�10

IJSP(units:minutes).

Job

Operatio

n

12

34

56

78

910

12

34

56

78

910

Interval

processing

time

Processingroutes

J 1[3,5]

[10,13

][4,10]

[5,9]

[10,16

][10,13

][4,10]

[4,8]

[4,7]

[9,13]

M7

M9

M2

M5

M4

M1

M6

M3

M8

M10

J 2[10,15

][9,14]

[7,14]

[7,13]

[2,6]

[8,13]

[7,12]

[8,14]

[6,10]

[10,15

]M

10M

3M

5M

4M

8M

1M

2M

7M

6M

9

J 3[5,10]

[2,5]

[1,4]

[7,12]

[8,12]

[2,5]

[4,7]

[7,10]

[9,14]

[8,15]

M5

M7

M4

M3

M1

M8

M6

M2

M9

M10

J 4[5,10]

[4,8]

[7,14]

[3,8]

[4,6]

[8,12]

[2,4]

[2,5]

[6,11]

[6,11]

M5

M6

M4

M7

M8

M2

M1

M9

M3

M10

J 5[7,10]

[5,7]

[10,14

][2,4]

[9,13]

[5,9]

[5,8]

[1,4]

[3,6]

[2,6]

M2

M7

M4

M1

M3

M5

M9

M6

M8

M10

J 6[1,4]

[3,5]

[3,6]

[5,8]

[8,13]

[9,14]

[4,8]

[1,4]

[2,5]

[6,12]

M7

M3

M8

M2

M4

M9

M10

M1

M6

M5

J 7[10,14

][2,4]

[9,12]

[9,11]

[4,6]

[3,7]

[1,4]

[2,5]

[8,13]

[7,11]

M10

M8

M6

M3

M7

M2

M9

M4

M1

M5

J 8[7,15]

[9,15]

[5,9]

[8,13]

[6,9]

[8,10]

[9,13]

[1,4]

[8,14]

[6,12]

M6

M2

M8

M4

M7

M3

M9

M5

M10

M1

J 9[4,10]

[3,6]

[6,10]

[3,6]

[8,12]

[5,10]

[4,9]

[1,4]

[3,7]

[10,15

]M

7M

2M

8M

6M

9M

4M

10M

1M

3M

5

J 10

[1,3]

[8,13]

[7,9]

[6,12]

[9,15]

[7,15]

[10,18

][1,5]

[1,4]

[2,5]

M4

M1

M9

M10

M5

M2

M6

M3

M8

M7

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5 10 15 20 25

50

100

150

200

250

300

Proc

essi

ng T

ime

Iteration

Figure 14. Ultimate processing time of each iteration for the 10� 10 IJSP (units: minutes).

5 10 15 20 25

10

15

20

25

30

35

40

45

50

Aver

age

stan

dard

dev

iatio

n (A

SD)

Iteration

Figure 15. Average standard deviation for the 10� 10 IJSP.

Table 5. Parameter comparison between the IFSJSP and the GA for the 10� 10 IJSP (MG, maximum generation).

Approach MG Population scale Crossover probability Mutation probability Optimal solution Fitness value

GA 300 100 0.9 0.1 (88,150) 0.0042IFSJSP 300 20 0.6 0.01 (99,159) 0.0039

5 10 15 20 25

50

100

150

200

250

300

Proc

essi

ng T

ime

Iteration

GAIFSJSP

Figure 16. Comparison between the IFSJSP and the GA for the 10� 10 IJSP (units: minutes).

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6. Conclusions

The JSP under an uncertain environment has attracted much attention during the past decades. Taking advantage ofintuitionistic fuzzy sets (IFS) that have the ability to handle the uncertain information, a novel approach is proposed inthis paper to deal with uncertain processing times. Then we design an effective GA by calculation of the fitness value,crossover, mutation, etc., in which operation-based representation is used for decoding in the chromosome. Numericalexamples are illustrated to show the effectiveness of the proposed method.

Future research will include IJSP problems with actual processing constraints such as machine breakdowns andmaintenance under an uncertain environment, solving the IJSP with multi-objectives and dealing with additionalconstraints of the IJSP problem. In addition, we will also investigate new approaches to obtain high-quality solutions tothe problem, such as adopting new crossover and mutation operations, and using new mechanisms to improve the localsearch ability of the GA. Another important consideration is to implement the proposed approach in a parallelcomputing environment.

AcknowledgementsThe authors thank the anonymous reviewers for their valuable comments and suggestions which improved the paper. The workdescribed in this paper was partially supported by the facilities provided by The Hong Kong Polytechnic University. The work ispartially supported by the National Natural Science Foundation of China (grant Nos. 61174022 and 71271061), the ChongqingNatural Science Foundation (for Distinguished Young Scholars) (grant No. CSCT, 2010BA2003), the National High TechnologyResearch and Development Program of China (863 Program) (grant No. 2013AA013801), the Science and Technology PlanningProject of Guangdong Province, China (project No. 2010B010600034), the Southwest University Scientific & TechnologicalInnovation Fund for Postgraduates (grant No. ky2011011), and the Fundamental Research Funds for the Central Universities (grantNo. XDJK2013D010).

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