On a Flow-Shop Scheduling Problem with Fuzzy Pentagonal Processing Time

Scheduling involves planning and arranging jobs across a coordinated set of events to satisfy the customer’s demands. In this article, we present a simple approach for the flow-shop (FS) scheduling problem under fuzzy environment in which processing time of jobs are represented by pentagonal fuzzy numbers. .is study is intended to reduce the rental cost of the machine in compliance with the rental policy. .e fuzzy FS scheduling problem is solved without converting the processing time into its equivalent crisp numbers using a robust ranking technique and a fuzzy arithmetic pentagonal fuzzy numbers. A numerical illustration indicates that the approach is workable, accurate, and relevant.


Introduction
Scheduling requires a variety of activities to accomplish a particular purpose with time and budget. e job schedule and management of its flows through a planning step are the most fundamental facets of any industrial production procedure. e management of a certain number of machines or facilities for a certain number of tasks or employment is one difficulty. e control of staff in any manner poses a largely unresolved obstacle to accomplish a certain objective. In planning a production or plan, a decision maker has difficulties encouraging prompt implementation and reducing demands such as flow times or completion periods. e scheduler's goal is to set starting times to achieve maximum performance subject to energy and technological constraints. e problem with flowshop preparation is that the simplistic version of which all staffs travel in the same sequence on all machinery is one of the most critical issues in production management. To minimize a machine's completion time, Argawal et al. [1] encountered three FS device scheduling problems. Vahedi-Nouri et al. [2] presented a more broad version of the FS strategy to minimize the average flow rate. In order to optimize the publication time, Ren et al. [3] explored the topic of FS programming. Laribi et al. [4] have introduced a mathematical model for two FS-limited machines that address FS time reduction problems where renewables are not constrained. Yazdani and Naderi [5] considered the scheduling problem with no-idle hybrid flow-shop and applied a mixed integer linear programming to formulate the problem. Qu et al. [6] designed a flower pollination algorithm based on the hormone modulation mechanism for nowait flow-shop scheduling problem, where the method uses available neighborhood search strategy based on dynamic selfadaptive variable work piece in the local search.
is would be made possible by a blueprint for several research fields such as computer analysis and organizational science. Decision-making in a fuzzy environment, developed by Bellman and Zadeh [7], has an improvement and a great help in the management decision problems. Fuzzy numerical data can be represented by means of fuzzy subsets of the real line, known as fuzzy numbers. Dubois and Prade [8] spread the expansion of algebraic operations into fuzzy numbers by a fuzzy hypothesis of real numbers. Panda and Pal developed the theory of pentagonal andantes fuzzy numbers (2016). Chakraborty et al. [9] extended the properties of the pentagonal fuzzy number to interval-value fuzzy and pentagonal fuzzy numbers. Many scientists [10,11] were involved in the problem of FS preparation. A method containing a variety of statistical and fuzzy flow-shop sets was introduced by Yao and Lin [12]. Hejari et al. [13] enhanced McCahon and Lee's [14] algorithm for work sequence with a fuzzy processing time (1990). For a smooth processing time, Ishibuchi and Lee established FS scheduling problem [15]. Temiz and Erol [16] also employed the fuzzy scheduling principle of the FS, updated the branch-and-bound of Ignall and Schrage [17], and reassigned the Ignall and Schrage algorithm to three FS frame problems. Gupta et al. [18] applied a heuristic algorithm for minimizing the rental cost of the machines for a specially structured three-stage flow-shop scheduling where triangular fuzzy numbers represent the processing times of the machine. Neira et al. [19] provided a general overview of the FS scheduling problem under uncertainties and its role in production logistics to draw up opportunities for further research. Komaki et al. [20] conducted a joint survey of their templates for assembly flow shops. Janaki and Ismail [21] explored the interval, modeling work using an efficient and heuristic make-up approach. Khalifa [2] analyzed the singlemachine preparation issue in a fizzy dates setting. Khalifa et al. [23] proposed a fuzzy binding approach for solving constrained flow-shop scheduling problem without converting the problem into its crisp.
In this paper, we consider the problem of flow-shop scheduling in a fuzzy environment where processing times are represented as pentagonal fuzzy numbers. We investigate a method to find the rental cost of the machines without converting the pentagonal fuzzy numbers into its classical number.
is study aims to minimize the rental costs of the machine as described in the rental policy. e remainder of the article is organized as follows. Section 2 addresses some preliminaries. Section 3 includes some fundamental assumptions and concepts that are central to the problem. Section 4 contains the pentagonal fuzzy number processing times' assertion of the FS programming problem. Section 5 offers an approach to the problem of fuzzy flow-shop scheduling. Section 6 includes a numerical illustration example. Section 7 presents a comparison between the solutions from the proposed approach with the others. Finally, concluding remarks are included in Section 8.

Preliminaries
In order to easily discuss the problem, it recalls basic rules and findings related to fuzzy numbers, pentagonal fuzzy numbers, and arithmetic operations of pentagonal fuzzy numbers and its ranking.
Definition 1 (see [21][22][23][24]). A fuzzy set A a set of real numbers R is called fuzzy if its membership function μ A (x): R ⟶ [0, 1] have the following properties: and the closure cl(Supp(A)) is compact set Definition 2 (see [8, 22, 23 to 9, 25, 26]). A fuzzy number A P � (a 1 , a 2 , a 3 , a 4 , a 5 ), a 1 ≤ a 2 ≤ a 3 ≤ a 4 ≤ a 5 on R is said to be a pentagonal fuzzy number if its membership function is Figure 1 shows the graphical representation of the pentagonal fuzzy number A P . 5 ) be two pentagonal fuzzy numbers and v ≠ 0. e arithmetic operations on A P and B P are 1 , a 2 , a 3 , a 4 , a 5 Definition 4 (see [25]). e ranking function of A P � (a 1 , a 2 , a 3 , a 4 , a 5 ) is defined as Definition 5. e order of relation between A P and B P based on the ranking function is defined as

Assumptions, Notations, and Rental Policy
In this flow-shop scheduling problem, the following assumptions are made.

Assumptions
(i) No job pre-emption is allowed (ii) Just one function can be performed by the machine (iii) All jobs are available at the beginning of schedule time (iv) e initialization times of the machines are negligible (v) All workers are processed throughout the deterministic phase (vii) Due dates are pentagonal fuzzy numbers (viii) e machine may be idle (viii) e production period is separate from the schedule (ix) e first machine must be completed to provide a second device with feeding (x) For each mission, M operations are needed (xi) Any job must be done as it has begun

Rental Policy.
e first machine will be taken on rent in the beginning of the processing of the jobs; the second machine will be taken on rent when the first job is completed on the first machine and transported to the second machine.
e third machine will be taken on rent when the first job is completed on the second machine and transported to the third machine and so on.

Problem Formulation
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 (A P ) ij (i � 1, 2, . . . , n; j � 1, 2, . . . , m) be the processing time of ith job on jth machine characterized by pentagonal fuzzy numbers. e problem can be formulated as which subjects to rental policy P.

Solution Method
In this section, the method to minimize the utilization time and then the rental cost of flow-shop scheduling problem under fuzzy environment is provided in the following steps.
Step 1 : convert the problem into a problem of two machines if one of the following conditions is reached:

Numerical Example
Consider 5 jobs and 3 machines' flow-shop scheduling problem having fuzzy processing times represented by pentagonal fuzzy numbers as in Table 1. e rental cost per unit time for machines M 1 , M 2 , and M 3 are 5, 3, and 4 units, respectively. Our aim is to minimize the total rental cost [28]. Table 1 illustrates the three machines scheduling problem with processing times characterized by pentagonal fuzzy numbers.
Hence, Table 3 provides the time in and out of the machines.
In addition, the times idle and rental costs are calculated in Table 4

Comparative Study
From Table 5, it is clear that the objective function value resulted from the Sathish and Ganesan algorithm is the same with the proposed approach, and the overall rental costs for the machines resulting from this strategy are clearly lower than those found by Gupta et al. [18].  1 , a 2 , a 3 , a 4 , a 5 ).

Job
Machine M 1 Machine M 2 Machine M 3

Concluding Remarks
In this paper, we provided a clear approach for the preparation of the flow-shop problem under fuzzy environment in which processing time of jobs are represented by pentagonal fuzzy numbers. e fuzzy FS scheduling problem was solved without converting the processing time into its equivalent crisp numbers using a robust ranking technique and a fuzzy arithmetic pentagonal fuzzy numbers. e strategy aims to reduce the rental costs of the machine in accordance with the leasing system. e advantage of this approach is more flexible where it allows the decision maker to choose the targets he/she is willing and useful for potential study in widespread fluctuations.

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

Conflicts of Interest
e authors declare no conflicts of interest.