Solving Constrained Flow-Shop Scheduling Problem through Multistage Fuzzy Binding Approach with Fuzzy Due Dates

This paper deals with constrained multistage machines ﬂow-shop (FS) scheduling model in which processing times, job weights, and break-down machine time are characterized by fuzzy numbers that are piecewise as well as quadratic in nature. Avoiding to convert the model into its crisp, the closed interval approximation for the piecewise quadratic fuzzy numbers is incorporated. The suggested method leads a noncrossing optimal sequence to the considered problem and minimizes the total elapsed time under fuzziness. The proposed approach helps the decision maker to search for applicable solution related to real-world problems and minimizes the total fuzzy elapsed time. A numerical example is provided for the illustration of the suggested methodology.


Introduction
Scheduling contains the sequence of jobs following the resource as well as time constraints, with a specific objective. e job scheduling and controlling through a production is a significant role in any industrial manufacturing unit. e FS scheduling model is the simple version where all jobs are operated on all the machines in order, is one of the recent issues in the field of production control, and is to determine the job sequence on the machines to minimize the makespan. e scheduling model usually consists of three components: time of transportation, job weight, and machine time for the break down.
Job scheduling problems, normally, occur such as programs for running on a sequence using some computer operators and to order the jobs for processing in a plant of manufacturing. Numerous researchers studied various FS scheduling problem and job scheduling problems and proposed algorithms in the crisp environment [1,2]. A new heuristic algorithm was introduced by Aggarwal et al. [3] for obtaining an optimal (near-optimal) sequence to bicriteria three-stage FS scheduling based on heuristic technique, which was further discussed by Patider et al. [4]. Abdullah and Abdolrazzagh-Nezhad [5] developed an algorithm for solving theatrical models for fuzzy job-shop scheduling. e FS scheduling model under the fuzzy processing time has been formulated by Ishibuchi et al. [6]. Afterwards, several researchers considered the machine sequence-dependent processing times. Ahonen and de Alvarenga [7] formulated and proposed a solution for the FS scheduling model, considering the recirculation and machine sequence-varying processing time. Qu et al. [8] proposed an algorithm to solve the no-wait FS scheduling problem based on the hormone modulation mechanism. Komaki et al. [9] introduced a consolidated survey of assembly FS models with their solution approach. Belabid et al. [10] proposed three methods for resolution of a permutation FS problem with independent setup time: mixed-integer LP model and two heuristics so as to minimize the maximum of job competition time.
In literature, authors, such as Zadeh [11] and Dubois and Prade [12], considered the FS problem with the consideration transportation cost. Hnaien et al. [13] presented the makespan minimization problem by describing the two-machine FS under a constraint related to availability of the first machine. A two-stage multiprocessor FS scheduling problem was considered under the deterioration of maintenance in a cleaner production [14]. Khatami and Zegordi [15] suggested the flexible maintenance time intervals.
Yang et al. [16] studied the FS scheduling of many production lines for precast production. Toumi et al. [17] presented the branch-and-bound technique for the solution of blocking FS scheduling problem under the assumption of makespan criterion. Yu et al. [18] presented the iterative method for batching and scheduling problem for the minimization of total job tardiness in two-stage hybrid FS. Shahvari and Logendran [19] presented a comparison of hybrid algorithm for a batch scheduling problem in hybrid FS under the assumption of learning effect. ey used a clustering-genetic algorithm-based technique.
A particular kind of FS problem is called the permutation FS scheduling problem, where the job processing order is the same for each subsequent step of the processing [20]. Over the years in literature, several authors studied the permutation FS problem. Damodaran et al. [21] proposed the particle swarm optimization procedure for solving the permutation FS. ey considered the scheduling batch processing machines in the model. Some multiobjective methods were also suggested by many researchers. Li and Ma [22] presented an artificial bee colony algorithm for multiobjective permutation FS problem with sequence varying with setup times. Chaouch et al. [23] presented a modified method of ant colony optimization algorithm to determine the optimal scheduling for the distributed job shop problem. Khalifa [24] analyzed the single-machine preparation issue in a fuzzy date setting.
Several researchers studied the fuzzy methods for solving the permutation FS problem, for instance, Tirkolaee et al. [25], Sioud and Gagne [26], and Kumar [27]. Tirkolaee et al. [25] studied a multitrip green capacitated arc routing problem with an application to urban services. ey used the hybrid genetic algorithm. Sioud and Gagne [26] proposed a special type solution method based on the enhanced migrating birds to permutation FS problem with the assumption of sequence-dependent setup times. Goli et al. [28] proposed a FS scheduling problem with outsourcing option on subcontractors. ey considered the just-in-time criteria in model formulation. Tirkolaee et al. [29] investigated the pollution-routing problem with cross-dock selection. ey used the Pareto-based algorithm to deal with the multiobjective optimization problem. Afterwards, Khalifa and Kumar [30] proposed the fuzzy solution approach to fully neutrosophic linear programming problem. ey also presented an application to stock portfolio selection. Very recently, Tirkolaee et al. [31] presented a FS scheduling problem with outsourcing option. ey used fuzzy programming and artificial fish swarm algorithm. Goli et al. [32] investigated a fuzzy production scheduling model. ey considered the automated guided vehicles as well as human factors.
In this paper, a novel method called multistage fuzzy binding for solving the problem under consideration in which jobs processing time, weights, and break-down machine are characterized as piecewise quadratic fuzzy numbers is proposed. Here, it is assumed that there is no power break up for dealing with break-down power as it has been assumed that the unit of production is still a small-scale one. e suggested method depends on the binding method applied by Pandian and Rajendran [33] which provides a noncrossing optimal sequence to the considered problem with the minimizing total fuzzy elapsed time.
e rest of the research work is organized as follows: the basic concept and arithmetic operations related to fuzzy numbers and their arithmetic operations are described in Section 2. Section 3 describes some of the assumptions and notations required in the proposed problem mathematical formulation. Section 4 formulates fuzzy constrained multistage FS scheduling problems. Section 5 proposes multistage fuzzy binding approach for obtaining a noncrossing optimal sequence. In Section 6, a numerical example to illustrate the methodology is introduced. Finally, some concluding remarks are reported in Section 7.

Preliminaries
is section introduces some of the basic concepts, and results related to fuzzy numbers, piecewise quadratic fuzzy numbers, and their arithmetic operations are recalled.
Definition 1 (see [34]). A piecewise quadratic fuzzy number (PQFN) is denoted by a PQ � (a 1 , a 2 , a 3 , a 4 , a 5 ), where a 1 ≤ a 2 ≤ a 3 ≤ a 4 ≤ a 5 are real numbers, and is defined by if its membership function μ a PQ as follows (as in Figure 1): e interval of confidence at level α for the PQFN is defined as follows: (2)

Advances in Fuzzy Systems
Definition 2 (see [34]). An interval approximation Definition 3 (see [35,36]). An interval on R is defined as where a L is the left limit and a R is the right limit of A.
Definition 4 (see [37]). e interval is also defined as where is the center and Definition 5 e associated ordinary numbers of PQFN corresponding to the closed interval approximation

Notation and Assumptions
3.1. Notation. e following notations can be used in the proposed FS scheduling problem.
S k : sequence resulted by applying Johnson's procedure, M ij : quadratic piecewise fuzzy processing time of the ith job on machine M j (i � 1, 2, . . . , n; j � 1, 2, . . . , m). P i : processes that require uninterrupted power supply and no break-down time are permitted. Q i : processes that require power supply and breakdown time are permitted. M i : processes that do not require power supply and may be continued during break-down time. F: fuzzy performance measure (i � 1, 2, . . . , n),

Assumptions.
In this FS scheduling problem, the following assumptions are made:  (ix) To feed a job on a second machine, it must be completed on the first machine. (x) Each job has m operations. (xi) Each job must be completed once it is started.

Problem Statement
e aim of the problem is to minimize the total piecewise quadratic fuzzy elapsed time that is to find the optimal sequence of the jobs. Assume that job i(i � 1, 2, . . . , n) is to be processed on machine j(j � 1, 2, . . . , m) in the existence of specified rental policy. Let M ij (i � 1, 2, . . . , n; j � 1, 2, . . . , m) be the processing time of job i on machine j characterized by PCF numbers, which may be classified into three categories: (1) e processes require uninterrupted power supply, and no break-down is permitted (say, P 1 , P 2 , . . .). In addition, let job i(i � 1, n) be assigned having fuzzy weights w i relative to the importance of performance in the sequence. e measure of the fuzzy performance is defined as where f i is the flow time of the i th job. Let the fuzzy breakdown approximate interval be [a, b]. Our aim is to determine the optimal sequence of jobs to minimize the total fuzzy elapsed time. e problem can be illustrated as in Table 1.
Assume that the considered problem satisfies one or both the following conditions:

Proposed Approach
e steps of the approach are as follows: Step 1: consider the piecewise quadratic fuzzy constrained multistage machines FS scheduling (PQFCMFSS) problem.
Step 2: convert the PQFCMFSS problem into the corresponding approximated closed-interval CMFSS problem.
Step 3: convert the CMFSS problem into a two-machine FS scheduling problem by introducing two fictitious machines H 1 and H 2 with  Step 4: applying the method introduced by Pandian and Rajendran [33] to obtain the optimal sequence. Step 6: identify the modified processing time on different jobs under categories P 1 , P 1 , . . ., and Q 1 , Q 2 , . . ..
Step 7: modify the fuzzy processing time after categorizing the jobs as follows: Step 8: determine the minimum total elapsed time and the weighted men-flow for the FS scheduling problem.  � (a 1 , a 2 , a 3 , a 4 , a 5 ).

Job
Machines with PQF processing times PCF weights of job

Numerical Example
In this section, we solve a numerical example to illustrate the suggested approach.
Step 3: convert the problem into two machines problem as in Table 4.
Step 4: using the binding method introduced by Pandian and Rajendran [33]; the modified processing times are as in Table 5. By applying Johnson's algorithm, the PQF constrained multistage machines FS scheduling problem is given by the following sequence: Hence, the PQF elapsed time is (105, 106, 107, 108, 109).
Based on Definition 6, Table 8 changes to Table 9 as follows.
It is obvious that the optimal sequence in fuzzy environment is Accordingly, Table 9 changes to Table 10 as follows.  � (a 1 , a 2 , a 3 , a 4 , a 5 ).

Conclusions
In this research article, a new approach, namely, multistage fuzzy binding method has applied for solving the PQF constrained multistage FS scheduling problems, where the processing times and the jobs weight are characterized by PQF numbers. e advantage of the approach is that there is no risk for the decision maker, it is more applicable for realworld problems, it is easy and simple for understanding, and it is an important tool to the managers who are dealing with the flow-job problems so as to provide a noncrossing optimal sequence. e main findings are particularly useful for a fuzzy FS scheduling problem, while the processing times and the jobs weight are fuzzy parameters. Some practical implications and managerial insights can be drawn from this proposed study, under fuzzy due dates. In industry and business sector, the decision maker can apply to schedule the flow-shop of the machines in the workshop under fuzzy due dates. is would optimize the usages of the machines and hence the revenue of the company. For future research, the proposed problem may be extended by considering the stochastic random variable, for the processing times as well as the jobs weight.

Data Availability
e data used to support the findings of this research are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest.