Minimizing the Machine Processing Time in a Flow Shop Scheduling Problem under Piecewise Quadratic Fuzzy Numbers

School of Mathematics and Physics, Bengbu University, Bengbu 233000, China Department of Operations Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt Department of Mathematics, College of Science and Arts, Qassim University, AlBadaya 51951, Saudi Arabia Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran 46818-53617, Iran Department of Applied Mathematics, Ayandegan Institute of Higher Education, Tonekabon, Iran


Introduction
Scheduling dilemma is concerned with determining the optimal or nearly optimal schedule under certain limitations. Numerous methods have been proposed by several researchers to solve this problem. Scheduling to achieve a speci c goal requires a variety of activities by spending time and budget. Flow shop is the most studied production setting in the literature on scheduling. In [1], one of the earliest results in ow shop scheduling is an algorithm for minimizing the completion time of all activities in a two or three-machine shop. Gupta [2] suggested a method for determining the best time to schedule a ow shop scheduling problem (FSSP) with a certain structure. e method [2] has important considerations and developed by several scholars; see [3− 7]. Narian and Bagga [9] investigated the problem of obtaining a sequence that provides the lowest possible cost of renting while minimizing the time spent. Schulz et al. [10] explored a mixture FSSP with varying discrete production speed levels. An upgraded multi-objective algorithm was used by Gheisarihe et al. [11] to solve the exible FSSP with sequence-based transportation time, a probable network, and setup time.
In the real-world applied scienti c problems, due to the complexity of di erent systems and the inaccuracy of data, classical methods cannot take into account inaccuracies in discussions. erefore, using tools such as fuzzy perspective [12] can be helpful in managing this important task (see also [13,14]). e theory of fuzzy sets and its applications in optimization were proposed by Zimmermann [15]. Kaufmann and Gupta [16] studied several fuzzy mathematical models with their applications to engineering and management sciences.
By using triangular fuzzy sets to describe work processing times, Petrovic and Song [17] studied the task sequence problem in a two-machine ow shop. Multi-product parallel multi-stage cell manufacturing organizations can apply Saracoglu and Suer's methodology [18] to create items on time. ey employed this methodology in the case study of a shoe manufacturing plant to produce products on time. Pang et al. [19] presented the FSSP and hybrid flow shop scheduling with the intention of determining the optimal scheduling approach for manufacturing facilities. Shao et al. [20] examined a distributed fuzzy blocking FSSP with processing times represented by fuzzy numbers, with the goal of minimizing the fuzzy makespan across all components. Recently, some papers are introduced to deal with real-world problems in fuzzy environments and their extensions (see [21][22][23][24][25][26]).
is study aims to investigate a particular n-job of scheduling with piecewise quadratic fuzzy number (PQFN). Given the total time elapsed, in which processing times are shown in PQFN, an innovative approach to sequencing tasks is proposed, which minimizes the cost of renting machines.

Research Gap and Motivation.
e following points may lead to motivation of the proposed study.
(1) e piecewise quadratic fuzzy number (PQFN) introduced by Jain [27] is an extended concept of fuzzy set. (2) In real-world scenarios, distinct parameters are further classified into disjoint sets having subparametric values. It presents the optimal selection with the help of suitable parameters. In decision making, the jury may endure some sort of tendency and proclivity while paying no attention to such parametric categorization during the decision. (7) e advantageous aspects of the proposed structure are discussed. e generalization of proposed structure is presented.

Paper Organization.
is paper is organized as follows. e next section introduces the preliminaries of PQFNs and some notations. A three-stage FSSP model is provided in Section 3. Section 4 provides an efficient method for determining the sequence of jobs that minimizes the cost of equipment rental. Section 5 gives a numerical example for illustration. Section 6 introduces a comparative study with the existing methods. Finally, the conclusions are drawn in Section 7.

Prerequisites
Here, we study some preliminaries that we need for the main sections (for more details, see [27]). 5 are real numbers, and its membership function μ W PQ is given by Figure 1 shows the graphical representation of a PQFN.
, we have the following properties: (6) e order relations: 2.1. Symbolization. Table 1 shows the symbols of our work.

Methodology
Before we discuss the issue formulation, let us define the rental cost.

Cost of Renting.
e machines are rented out if needed and returned if they are no longer needed. For example, the first machine is rented at the beginning of the work process, the second machine is rented when the first work is completed in the first machine, and so on.
Suppose that some tasks i, i � 1, n under the definite rental policy L are managed on three machines M j , j � 1, 2, 3. Let a PQ ij be the PQFPT of i-th task on j-th machine (see Table 2). Let S ij , i � 1, n; j � 1, 2, 3. Determine the related processing times with crisp number on devices M 1 , M 2 , and M 3 in such a way that either a j2 ≤ a i1 or a j2 ≤ a i3 ; ∀i, j. Our objective is to determine S k of the tasks that minimizes the cost of renting the equipment. e problem may be expressed mathematically as follows: Using the CIA of PQFN, model (10) may be reformulated as follows:

Proposed Algorithm
In this part, we show our strategy for minimizing the time and, consequently, the cost of renting a three-stage FSSP with PQF-based processing time while ignoring the makespan.
Step 1. Find the associated ordinary number for all tasks.
Step 2. If a j2 ≤ a i1 or a j2 ≤ a i3 ; ∀i, j, i.e., max a i1 ≥ min a j2 or max a i3 ≥ min a j2 ; ∀i, j, go to next step; otherwise, break. Step 3. Define dummy machines H 1 and H 2 , and their processing times H i 1 and H i 2 are as follows: Step 4. Use the existing algorithm [1] on H i and get S 1 .
Step 5. Put the 2 nd ,. . ., n th tasks of the S 1 in the first position and all other tasks of S 1 in the same order.
Step 6. For all possible sequences S k , k � 1, n, calculate: Step 7. Set min R(S k ) , k � 1, n as the optimal solution.

Numerical Example
Consider Table 5 as the problem. Now, we solve this problem by our model. At first, in Tables 6 and 7, we compute the related interval and crisp numbers for each PQFPT. en, using Step 3 of our algorithm, the processing times can be computed as shown in Table 8. Using Consumption time for M j that is necessary for S k CT(S k ) Whole completion time Corresponding normal time of the i-th task on M j R(S k ) Whole rental payment C Cost of renting Table 3: e problem with CIA matrix.
Tasks  Table 4: e problem with the corresponding crisp matrix form.

Comparative Study
In this section, the proposed approach is compared with some existing studies to illustrate the advantages of the proposed approach. e results for this analysis are summarized in Table 11. e symbol "↓" or "↑" shown in the table represents whether the associated feature satisfies or not.

Conclusions and Future Works
In this paper, the problem of minimizing the cost of renting machines for flow shop scheduling with a specific structure is investigated. An innovative approach to solve it is then proposed in which the processing times are fragmented as piecewise quadratic fuzzy numbers. e result shows that the proposed method has its advantage in flexible decision making corresponding to favorite priorities of alternatives. is study may be extended to additional fuzzy-like structures, such as interval-valued fuzzy set, Pythagorean fuzzy set, spherical fuzzy set, intuitionistic fuzzy set, picture fuzzy set, neutrosophic set, and so on, in future work.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.