A fuzzy-mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates

ARTICLE IN PRESS Robotics and Computer-Integrated Manufacturing 25 (2009) 853–859

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

A fuzzy-mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates A.H. Gharehgozli, R. Tavakkoli-Moghaddam , N. Zaerpour Department of Industrial Engineering, College of Engineering, University of Tehran, P.O. Box: 11155-4563, Tehran, Iran

a r t i c l e in fo

abstract

Article history: Received 23 December 2006 Received in revised form 27 July 2008 Accepted 24 December 2008

This paper presents a new mixed-integer goal programming (MIGP) model for a parallel-machine scheduling problem with sequence-dependent setup times and release dates. Two objectives are considered in the model to minimize the total weighted flow time and the total weighted tardiness simultaneously. Due to the complexity of the above model and uncertainty involved in real-world scheduling problems, it is sometimes unrealistic or even impossible to acquire exact input data. Hence, we consider the parallel-machine scheduling problem with sequence-dependent set-up times under the hypothesis of fuzzy processing time’s knowledge and two fuzzy objectives as the MIGP model. In addition, a quite effective and applicable methodology for solving the above fuzzy model are presented. At the end, the effectiveness of the proposed model and the denoted methodology is demonstrated through some test problems. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Parallel machine scheduling Mixed-integer goal programming Fuzzy theory Sequence-dependent setup time Release date

1. Introduction Production scheduling consists of planning jobs that need to be performed in an orderly sequence of operations. It is a tool that optimizes the use of available resources. However, scheduling problems are very complex computationally and therefore it is so difficult to optimally solve in reasonable time due to its combinatorial nature. As a result, finding an optimal solution in reasonable time is not always possible [1]. Many researchers have assumed the parameters such as job processing times, release dates, and due dates to be deterministic [2]. However, in the realworld situation, these parameters are often encountered with uncertainties. Accordingly, there are basically two approaches to deal with the machine scheduling problems that encounter with uncertainties [3], including the stochastic probabilistic theory and possibility theory [4] or fuzzy set theory [5]. Obviously, one of the most important disadvantages and difficulties of using stochastic probabilistic theory is that it needs a great deal of knowledge about the statistical distribution of the uncertain parameter [6]. The fuzzy approach represents an alternative way to model imprecision and uncertainty, which is more efficient than the latter, especially when no historical information is available [7]. It was first introduced as a representation scheme and calculus for uncertain or vague notions. The fuzzy set theory provides a conceptual framework [8] which performs so efficiently in  Corresponding author. Tel.: +98 2182084183/+98 21 88021067; Mobile: +98 912 1580480; fax: +98 21 88013102. E-mail address: [email protected] (R. Tavakkoli-Moghaddam).

0736-5845/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2008.12.005

decreasing the scheduling problem computational complexity with respect to the same problem formulated by the probability theory. It should be noted that such an imprecision is due to the subjective and qualitative evaluations, rather than the effect of uncontrollable events. The use of the fuzzy sets theory in treating different scheduling problems has been so successful, particularly where judgment and intuition play an important role [9], such as customer demand [9], processing times [10], production due dates [11], or job precedence relations [12]. Prade et al. [13] published the earliest paper in fuzzy scheduling. Ishii and Tada [12] considered a singlemachine scheduling problem minimizing the maximum lateness of jobs with fuzzy precedence relations. Han et al. [14] considered single-machine scheduling problem with fuzzy due dates. Ishibuchi and Murata [15] presented a flow shop scheduling problem with fuzzy parameters, such as fuzzy due dates and fuzzy processing times. The fuzzy job shop scheduling problem was analyzed by Kuroda and Wang [10]. Konno and Ishii [16] discussed an open shop scheduling problem with fuzzy allowable time and fuzzy resource constraint. Itoh and Ishii [17] proposed a singlemachine scheduling model dealing with fuzzy processing times and due dates. Some new models for real-time task scheduling with fuzzy deadlines and processing times are presented by Litoiu and Tadei [18]. Tavakkoli-Moghaddam et al. [19] suggested a fuzzy mixed-integer goal programming (MIGP) model for singlemachine scheduling problem. In production scheduling, most researches are concerned with the minimization of single criterion. However, real-life production scheduling tends to consider multi-criteria simultaneously.

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Different problems in production scheduling have been treated using fuzzy multi-criteria decision making (MADM) methods. Among lots of examples, some of them are the most considerable ones such as: scheduling of single machine [12,20], job shop [21,22], and flow shop [15,23]. Although it is extremely important, little attention has been given to multi-criteria scheduling problems, especially in the case of multiple machines, which is because of the extreme complexity of these combinatorial problems [24]. A typical multi-objective optimization problem can be presented as follows: Z l ¼ min Z l ðxÞ

8l

(1)

s.t. ðAxÞi pbi xX0

8i

(2) (3)

where functions Zl are the objectives and X is the vector of variables. A feasible solution X* is efficient if and only if for all other feasible solutions X, Zl(X*)pZl(X) with at least one strict inequality. In the literature concerning multi-objective scheduling problems, five main approaches are distinguishable: (a) hierarchical approach, (b) utility approach, (c) goal programming (or satisfying approach), (d) simultaneous (or Pareto), and (e) interactive approaches. Each approach has its own strengths and weaknesses as described in the general literature on multi-objective optimization problems. Clearly depending on the aim of study or the context of the application treated, an essential approach is used. Chen and Bulfin [25] and Hoogeveen et al. [26] independently proved that only problems including flow time as primary criterion in a hierarchical optimization and problems minimizing flow time and maximizing tardiness can be solved in polynomial time. All other problems are either shown to be NP-hard one or remain open, as far as computational complexity is concerned. In the rest of the paper, we focus on a parallel-machine scheduling problem (FPMSP). In parallel processing, jobs are processed by one of several machines, allowing considerable reduction in makespan. The study of parallel machine problems is relevant from both the theoretical and the practical points of view [27]. From the practical point of view, it is important because we can find many examples of the use of parallel machines in the real world. From the theoretical point of view, it is a generalization of the single machine problem and a particular case of problems arising in flexible manufacturing systems. You may review the literature of parallel-machine scheduling problem in Refs. [26,28]. Chen and Powell [29] considered the scheduling problem n jobs on m identical, uniform, or unrelated machines with two particular objectives: (1) to minimize the total weighted completion time and (2) to minimize the weighted number of tardy jobs. They extended their work by proposing an approach for the problem of scheduling n jobs with an unrestrictive large, common due date on m identical parallel machines to minimize the total weighted earliness and tardiness [30]. Radhakrishnan et al. [31] addressed a parallel machine earliness/tardiness scheduling problem with varying process times and non-common due dates. They employed a simulated annealing algorithm in order to find the optimal sequence of jobs. The problem of scheduling a set of independent jobs with common due dates on identical parallel machines to minimize the total earliness–tardiness penalty of jobs is considered by Chen and Lee [32]. Chen and Powell [33] studied the minimization of the total weighted completion time of the jobs, when multiple job families should be scheduled on identical parallel machines with sequence-dependent or sequence-inde-

pendent setup times. Recently, some studies have provided interesting results on parallel machine scheduling problems, with the help of genetic and exact algorithms [34–37]. In classical parallel machine scheduling problems, all the parameters and variables are considered deterministic. However, since multiple sources of uncertainty and complex interrelationships at various levels between diverse entities exist in these kinds of problems, it is quite unreliable to set them as precise values. Some research [38] has modeled parallel machine problem by probability distribution that is usually predicted from historical data. However, whenever statistical data is unreliable or even unavailable, stochastic models may not be the best choice. Fuzzy set theory may provide an alternative approach for dealing with the uncertainty. Fuzzy set theory has been found extensive applications in various fields. A limited amount of the literature has been devoted to fuzzy parallel-machine scheduling problems. Peng and Liu [39] proposed a practical application. In addition to single-objective scheduling models, they considered the multi-objective FPMSPs and formulated as three-objective models, which not only minimize the fuzzy maximum tardiness, but also minimize the fuzzy maximum completion time (i.e., makespan) and the fuzzy maximum idleness. Motivated by the literature discussed above, this paper presents a parallel-machine scheduling problem with sequencedependent set-up times under the hypothesis of fuzzy processing time’s knowledge and two fuzzy objectives. The rest of the paper is organized as follows. Section 2 discusses about the multiobjective mixed-integer programming (MO-MIP) of parallel machines. Section 3 illustrates a quick review of the fuzzy theory principles. Section 4 describes about the fuzzy mixed-integer goal programming model and an apt procedure for solving it. Section 5 is devoted to numerical experiments conducted on the basis of generated random data that shows the effectiveness of the proposed approach. Section 6 is the remarking conclusion of this work.

2. Proposed multi-objective mixed-integer programming model In this paper, the problem can be formally formulated as follow: N jobs with sequence-dependent set-up times, different release dates, non-common due dates, and varying processing times are to be processed on M parallel machines. Let di be the due date of job i and pik be the processing time of job i on machine k. Let sij denote the set-up time for job j immediately following job i in the sequence. S0i is the set-up time for job i in sequence position 1. Let j be a large positive number. Moreover, let the real variables ti, ci, and ri denote the tardiness, completion, and release time of job i, respectively. In addition, we consider the following variables ( xkij ¼

1 0

( ykij

¼

if job j is immediately after job i in sequence on machine k; otherwise:

1

if job j is assigned to machine k;

0

otherwise:

The multi-objective mixed-integer programming model of the parallel machine scheduling problem can be written as follows: Z 1 ¼ min

N X i¼1

w1i t i

(4)

ARTICLE IN PRESS A.H. Gharehgozli et al. / Robotics and Computer-Integrated Manufacturing 25 (2009) 853–859

Z 2 ¼ min

N X

w2i ci

(5)

855

X a real number in the interval [0, 1]. Therefore, A˜ is completely characterized by the set Eq. (13):

i¼1

A ¼ fðx; mA ðxÞÞjx 2 Xg

s.t. ci þ jð1  xk0i ÞXs0i þ r i þ pki

8i; k

cj þ jð1  xkij ÞXmaxfr j ; ci þ sij g þ pkj N X

xk0i ¼ 1

(6) 8i; j; k;

iaj

(7)

(8)

8k

i¼1 N X

xkij ¼ ykj

8j; k

(9)

i¼0iaj N X

xkij pyki

8i; k

(10)

ykj ¼ 1

(11)

8j

k¼1

t i ¼ maxfo; ci  di g xkij ; ykj 2 f0; 1g ci ; t i X0

Definition 3-2. . The height of a fuzzy set is the largest membership grade attained by any element in that set. A fuzzy set A˜ in the universe of discourse X is called normal if and only if the height of A˜ is equal to 1. A fuzzy number is a fuzzy subset in the universe of discourse X that is both convex and normal. Definition 3-3. . A fuzzy set A˜ in the universe of discourse X is convex if and only if for every x1, x2 in X:

mA ðlx1 þ ð1  lÞx2 ÞXminðmA ðx1 Þ; mA ðx2 ÞÞ

(15)

where lA[0,1] and ‘‘min’’ denotes the minimum operator. Definition 3-4. The acut of fuzzy number A˜ is defined as Eq. (15): Aa ¼ fxjmA ðxi ÞXa and x 2 Xg

j¼1jai M X

(14)

8i; j;

8i

(12)

iaj

8i.

(13)

The objective functions (4) and (5) minimize the total weighted tardiness and total weighted flow time, respectively. Let wi1 and wi2 be weights of job i in each objective function, respectively. A dummy job 0 is introduced on each machine in order to take care of the setup time for the real job assigned to the first position in the sequence on each machine. Constraint (6) represents that the completion time of a real job assigned to the first position in the sequence on any machine should be greater than or equal to the sum of its initial set-up time and its processing time. Constraint (7) ensures that the completion time of a real job in a sequence on a machine will be at least equal to the sum of the completion time of the preceding job, the sequence-dependent set-up time, and the processing time of the present job. Constraint (8) guarantees that the dummy job 0 is positioned at the beginning of the sequence before all the real jobs on each machine. Constraint (9) depicts that if a real job is assigned to a machine, then it will be immediately preceded by one job (i.e., the real job in sequence position one will be preceded by the dummy job 0). Similarly, Constraint (10) portrays that if a real job is assigned to a machine then it can be succeeded by at most one job. The job in the last position of the sequence on a machine will not have a succeeding job. Constraints (9) and (10) together affirm that N real jobs are assigned over M machines. They also ensure that if job i immediately precedes job j on machine k, then both jobs i and j belong to machine k. Constraint (11) confirms that a real job is assigned to exactly one machine. Constraint (12) specifies the tardiness of each job.

3. A quick review of fuzzy set theory principles In this section, some basic definitions of fuzzy sets, fuzzy numbers, and linguistic variables are reviewed [40]. The following basic definitions and notations will be used throughout this paper until otherwise stated: Definition 3-1. . A fuzzy set A˜ in a universe of discourse X is characterized by a membership function mA(x), which is called the ˜ It associates with each element x in grade of membership of x in A.

(16)

where aA[0,1]. In other words, a is the minimum value of membership grade which is desirable. The symbol Aa represents a non-empty bounded interval contained in X, which can be denoted by Aa ¼ [Aal, Aau], Aal and Aau are the lower and upper bounds of the closed interval, ˜ if AalX0 and Aaup1 for all respectively. For a fuzzy number A, aA[0, 1], then A˜ is called a standard (normal) positive fuzzy number. Definition 3-5. A positive triangular fuzzy number A˜ can be defined as A˜ ¼ ða1 ; a2 ; a3 Þ, as shown in Fig. 1. A non-fuzzy number r ˜, can be expressed as (r, r, r, r). The acut of the fuzzy number A which can be denoted by Aa ¼ [Aal, Aau], is shown in Fig. 1. 4. Fuzzy mixed-integer goal programming A typical fuzzy multi-objective linear programming problem can be stated as follows: Z k ¼ min˜ Z k ðxÞ

(17)

8k

s.t. ðAxÞi pb˜ i

(18)

8i

xX0.

(19)

Consider that the aspiration level of the objective is defined as Z0. In addition, we assume bi and its corresponding tolerance Dbi

μA (x)

1

μA (x) =1, x = a2

x − a1 a2 − a1

a3 − x a3 − a2

, a1 ≤ x ≤ a2

, a2 ≤ x ≤ a3

α μA (x) = 0

μA (x) = 0, x ≥ a3

x ≤ a1

a1 Aαl

a3

a2 = (a2 − a1)α+a1

Aαu

x

= a3 −(a3 −a2)α

Fig. 1. A positive triangular fuzzy number, its membership function and acut.

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a 2 ½ 0 1 .

8i. The fuzzy objective and the fuzzy constraints are then considered without difference. Thus, the fuzzy mixed-integer goal programming model (F-MIGP) can be stated as follows:

xX0;

find x

(20)

4.1. Proposed F-MIGP model for parallel machine scheduling problem with sequence-dependent setup times and release dates

(21)

In the scheduling problems, the initial data such as job processing times, ready times, and due dates have been usually assumed to be deterministic. However, in the real-world situations, these parameters are often encountered with uncertainties. Thus, due to the complexity and uncertainty involved in real-world scheduling problems, it is sometimes unrealistic or even impossible to require exact data. A good scheduling model needs to tolerate vagueness or ambiguity. In this study, the model imprecision is represented by the fuzzy approach. A multi-objective programming model for a parallel-machine scheduling problem with sequence-dependent setup times under the hypothesis of fuzzy processing time’s knowledge and two fuzzy objectives is presented as follows:

s.t. Z k ðxÞpZ k ðAxÞi Xbi

8k

(22)

8i

xX0.

(23)

The fuzzy objective functions and the fuzzy constraints are defined by their corresponding membership functions. For simplicity, let us assume that the membership functions ml of the fuzzy objectives are non-increasing continuous linear functions, and the membership functions mi 8i, of the fuzzy constraints are non-increasing linear membership functions. Thus, the membership functions of constraints are as follows: 8 1 if Ax  bi > > > < ðAxÞi  bi if bi  Dbi pAx  bi 8i (24) mi ðxÞ ¼ 1 þ D bi > > > : 0 if Axpbi  Dbi To be more specific, let us describe the membership functions of the F-MIGP objectives as follows: 8 1 if Z k ðxÞ  Z lk > > > > u < Z  Z ðxÞ k k if Z lk pZ k ðxÞ  Z uk 8k (25) mZk ðxÞ ¼ u l > > Z k _Z k > > :0 u if Z ðxÞXZ k

k

where Zku and Zkl are the upper bound and lower bound of the kth objective function, respectively. The membership functions can be illustrated as Fig. 2. Zimmerman [41] then used Bellman and Zadeh’s max–min [42] operator to solve the equations. So the crisp mixed-integer goal programming model (C-MIGP) will be equivalent to:

Z˜ 1 ¼ min˜

N X

(34)

w1i t i

(35)

w2i ci

(36)

i¼1

Z˜ 2 ¼ min˜

N X i¼1

s.t. ci þ jð1  xk0i ÞXs0i þ r i þ p˜ ki

8i; k

cj þ jð1  xkij ÞXmaxfr j ; ci þ sij g þ p˜ kj

(37) 8i; j; k;

iaj

Constraints ð8Þ to ð13Þ.

(38) (39)

mk ðxÞXa 8k

(27)

mi ðxÞXa 8i

(28)

The fuzzy objective functions (35) and (36) imprecisely minimize the total weighted tardiness and total weighted flow time, respectively. Constraint (37) represents that the completion time of a real job assigned to the first position in the sequence on any machine should be greater than or equal to the sum of its initial setup time and its fuzzy processing time. Constraint (38) ensures that the completion time of a real job in sequence on a machine will be at least equal to the sum of the completion time of the preceding job, the sequence-dependent set-up time, and the fuzzy processing time of the present job. The other constraints are the same as MOMIP model. So the F-MIGP model will be equivalent to:

mk ðxÞ; mi ðxÞ 2 ½ 0 1  8k; i

(29)

max a

a 2 ½0 1

(30)

max a

(26)

s.t.

(40)

s.t. The above model can also be expressed as follows: max a

(31)

s.t. Z k ðxÞpZ uk

ðZ uk

a



Z lk Þ

ðAxÞi Xbi  Dbi ð1  aÞ

Z 1 ðxÞpZ u1  aðZ u1  Z l1 Þ

(41)

Z 2 ðxÞpZ u2  aðZ u2  Z l2 Þ

(42)

ci þ jð1  xk0i ÞXs0i þ pki þ r i  ð1  aÞDpk i

(32)

8k

(33)

8i

8i; k

(43)

cj þ jð1  xkij ÞXmaxfr j ; ci þ sij g þ pkj  ð1  aÞDpk j

8i; j; k;

μZ (x)

μi (x)

k

bi − Δb

i

bi

x

iaj

Constraints ð8Þ to ð13Þ;

Zkl

Zku

Z

Fig. 2. The membership functions of the fuzzy objective and fuzzy constraints.

(44)

a 2 ½ 0 1 .

(45)

4.2. The applicable methodology for solving the F-MIGP model We propose an apt methodology to solve the F-MIGP model for the parallel-machine scheduling problem with sequence-dependent

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setup times and fuzzy objectives. The proposed methodology is as follows: 4.2.1. Step 1. Establish the MO-MIP and F-MIGP models The proposed MO-MIP and F-MIGP models are established in a crisp and fuzzy environment, respectively. These models are used in the next steps in order to find the optimal sequence of jobs on parallel machines. The MIP model is employed to find the objective goals and the fuzzy one is utilized to find the optimal sequence. 4.2.2. Step 2. Establish the upper bounds and lower bounds of the fuzzy objectives One objective of the MIP model is taken at time and is solved subject to the MIP formulation constraints. Thus, the objective upper bound and lower bound are obtained.

857

up times for the real jobs assigned to the first position in the sequence on each machine. Tables 2 and 3 summarize the results of applying the proposed methodology in order to find the optimal solution of the F-MIGP model in the random generated test problem. Regarding the first and second steps of the suggested methodology mentioned in Section 4, first of all, the upper bounds and lower bounds of the objective functions should be obtained. This task is carried out by solving the MIP model for each objective. The results are depicted in Table 2. Then, using these values (upper bounds and lower bounds), the C-MIGP model developed by Zimmerman method should be solved. It is worthy noting that, Zimmerman method is among the first practical methods proposed to solve fuzzy

4.2.3. Step 3. Establish the membership functions for all objective functions Suitable membership functions for all objective functions are determined based on the MIP results of the individual objective functions. 4.2.4. Step 4. Establish the membership functions for all fuzzy constraints Suitable membership functions for all fuzzy constraints are decided considering known processing times. The corresponding tolerances are determined by experts. 4.2.5. Step 5. Establish the C-MIGP model The C-MIGP model is developed by Zimmerman’s method. Eventually, the model is solved and the optimal sequence, degree of achievement (a), and the optimal objective values are obtained. The above proposed methodology is shown in Fig. 3.

5. Numerical example In this section, we demonstrate the effectiveness of the F-MIGP technique for the parallel-machine scheduling problem with sequence-dependent setup times with assuming fuzzy processing times through some test problems. All test problems have been solved using Lingo 8.0 software. These tests have been done on a portable computer Intel(R) Pentium(R) M 1.86 GHz with 512 MB RAM and MS Windows XP professional Operating System. The initial inputs are produced as follows:

 The process times and release dates are created at random    

between ½ 1 10 ; The corresponding process time tolerances are assumed to be d*(pik) 8i,k, d 2 ½ 0 1 ; The weights of jobs in each objective function (wil, wi2a0) are created at random between ½ 1 6 ; The dependent setup times are created at random between ½ 2 4 . As commonly assumed in the literature [43], setup times satisfy the triangle inequality: sij+sjvXsiv, for all i,j,vAN; The corresponding due dates are also computed by di ¼ P k ð M k¼1 pi =MÞNð1  Rnd½ 0 1 Þ 8i as given in Ref. [44]. N is the number of jobs, M is the number of machines, and pi’s are the processing times.

First of all, we show the procedure of the proposed methodology for solving F-MIGP model on a random generated test problem. Table 1 provides the initial inputs of the problem. The setup times are similar for each machine, and s0j indicates the set-

Fig. 3. Proposed methodology to solve the F-MIGP for the parallel-machine scheduling problem with sequence-dependent set-up times and fuzzy objectives.

Table 1 Initial inputs of the intermediate-size test problem (d ¼ 0.4). Job

pi1

pi2

pi3

di

ri

wi 1

wi 2

s0j

1 2 3 4 5 6 7

5 1 9 3 5 6 7

3 8 9 6 9 6 7

1 5 3 8 2 8 3

6 2 17 10 3 20 12

3 4 7 8 4 4 7

3 5 5 2 4 2 3

5 4 2 3 2 2 4

4 3 3 3 3 4 2

Sij 2

3

0 63 6 6 63 6 62 6 6 63 6 6 44

2

2

3

4

3

4

0

3

2

3

3

3 2

0 3

4 0

4 3

4 3

2

4

3

0

3

3

4

4

3

0

3

4

3

3

4

4

37 7 7 37 7 47 7 7 37 7 7 25 0

Table 2 The output of the 2nd step of the methodology for the test problem. The obtained goals by solving the MIP model

Zkl

Zku

Total weighted tardiness (first objective) Total weighted flow time (second objective)

63 243

101 279

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programming models. The degree of goal achievement will be a ¼ 0.47 (Table 3). Fig. 4 illustrates the optimal sequence of jobs using fuzzy parameters and objectives. We also apply the proposed methodology to some other random generated test problems with different number of jobs and machines. The degrees of goal achievement (a) are illustrated in Table 4. A problem with 7 jobs and 3 machines is the ultimate problem which can be solved by a computer introduced above in reasonable amount of time using LINGO 8. Parallel machine scheduling problems are categorized as NP-hard problems even for two identical parallel machines [45]. We know that the number of jobs or machines increase, finding the exact solution becomes so hard and the problem cannot be solved in reasonable amount of time. Regarding what is discussed in Section 4, for each fuzzy problem three other problems should be solved which for 7 jobs and 3 machines, each takes about one hour. Experiments proved that the value of delta (d) may affect on the results of test problems. Table 5 shows that the degree of goal achievement generally grows as the delta becomes larger. It means that the solution is related to the processing time tolerances. As the value of delta increases, the value of goal achievement increases.

6. Conclusion It is well known that the optimal solution of single objective models can be quite different to those models consisting of multi objectives. In fact, the decision maker often wants to minimize the Table 3 The final output of the C-MIGP model for the test problem. The obtained solutions by solving the C-MIGP model

Values

Degree of goals achievement Optimal total weighted tardiness Optimal total weighted flow time

0.47 83.055 262.18

Table 5 Degree of goal achievement (a) through some test problems with different d. Number of jobs (n)

Alpha (a)

d

4 5 6 7

0.2

0.4

0.6

0.8

0.366197 0.325359 0.343008 0.272727

0.322289 0.33612 0.418879 0.487153

0.433824 0.404306 0.445755 0.414271

0.491548 0.484398 0.512031 0.451444

earliness/tardiness penalty or total flow time. Each of these objectives is valid from a general point of view. Since these objectives conflict with each other, a solution may perform well for one objective or it gives inferior results for others. For this reason, scheduling problems have a multi-objective nature. In this paper, we have proposed a new mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup times, release dates, and two objectives. In decision making situations, the high degree of fuzziness and uncertainties is included in the data set. The fuzzy set theory provides a framework for handling the uncertainties of this type. We have suggested a novel fuzzy mixed-integer goal programming model for a parallel-machine scheduling problem with sequence-dependent setup, release dates, and fuzzy process times as well as a suitable methodology for solving it. Additional research from this study may follow several directions. One trend may make an empirical study of manufacturing organizations, comparing the effectiveness of their current production planning with the theoretical policy, resulting from this model. As mentioned in the paper, because of the fact that the large-sized parallel-machine scheduling problem is NP-hard, another direction might be to employ metaheuristics algorithms to determine the solution of large-sized model.

Acknowledgements

Machine

J7

J5

3

2

J1

1

J6

J2 0

The authors would like to acknowledge the Iran National Science Foundation (INSF) for the financial support of this work. The authors would like to also thank the editor-in-chief as well as the anonymous reviewers for their helpful comments and suggestions, which greatly improved the presentation of this paper.

J3

References

J4 5

10 Time

15

20

Fig. 4. Optimal sequence of jobs in the random-generated test problem using fuzzy parameters and objectives.

Table 4 Degree of goal achievement (a) through some test problems (d ¼ 0.4). Number of machines (m)

1 2 3

Alpha (a) Number of jobs (n) 4

5

6

7

0.5 0.322289 0.470817

0.469565 0.33612 0.482353

0.362595 0.418879 0.324742

0.399796 0.487153 0.472222

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