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

Authors: Tingwei Zhou; Hamiden Abd El-Wahed Khalifa; Seyyed Esmaeil Najafi; S.A. Edalatpanah
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The piecewise quadratic fuzzy number (PQFN) can signify uncertain information that exists in scientific, technological, and engineering fields. Hence, it is a useful tool for describing information in scheduling problems. This study examines structured n-job flow shop scheduling with fuzzy piecewise quadratic processing times and three machines. Close interval approximation of PQFNs is also offered as one of the most effective approximate intervals. Furthermore, the leasing cost of equipment is minimized with the use of a fuzzy style and an inventive algorithm. To demonstrate how the proposed framework can be used, a numerical illustration is provided.


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## Body

## 1. 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 specific 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 flow 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 flow shop scheduling problem (FSSP) with a certain structure.The 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 flexible FSSP with sequence-based transportation time, a probable network, and setup time.In the real-world applied scientific problems, due to the complexity of different systems and the inaccuracy of data, classical methods cannot take into account inaccuracies in discussions. Therefore, using tools such as fuzzy perspective [12] can be helpful in managing this important task (see also [13, 14]).The 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 flow shop. Multi-product parallel multi-stage cell manufacturing organizations can apply Saracoglu and Suer’s methodology [18] to create items on time. They 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 [21–26]).This study aims to investigate a particularn-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.


### 1.1. Research Gap and Motivation

The following points may lead to motivation of the proposed study.(1)
The 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 sub-parametric 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.(3)
Inspired from the above literature, new notions of PQFN are conceptualized along with some elementary essential properties and generalized typical results. Moreover, decision-making algorithmic approaches are proposed.


### 1.2. Main Contributions and Advantages

The following are the main contributions of this proposed study:(1)
The existing relevant models are made adequate with the consideration of multi-argument approximate function through the development of the fuzzy set theory.(2)
The scenario where parameters are further partitioned into sub-parametric values in the form of sets is tackled by using PQFNs.(3)
Some fundamentals like elementary properties and arithmetic operations of PQFNs are characterized.(4)
Decision-making applications are discussed based on the proposal of PQFNS arithmetic operations.(5)
The results of the proposed similarity are compared with relevant existing models.(6)
The proposed structure is compared with relevant models under suitable evaluating indicators.(7)
The advantageous aspects of the proposed structure are discussed. The generalization of proposed structure is presented.


### 1.3. Paper Organization

This paper is organized as follows. The next section introduces the preliminaries of PQFNs and some notations. A three-stage FSSP model is provided in Section3. 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.


## 1.1. Research Gap and Motivation

The following points may lead to motivation of the proposed study.(1)
The 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 sub-parametric 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.(3)
Inspired from the above literature, new notions of PQFN are conceptualized along with some elementary essential properties and generalized typical results. Moreover, decision-making algorithmic approaches are proposed.


## 1.2. Main Contributions and Advantages

The following are the main contributions of this proposed study:(1)
The existing relevant models are made adequate with the consideration of multi-argument approximate function through the development of the fuzzy set theory.(2)
The scenario where parameters are further partitioned into sub-parametric values in the form of sets is tackled by using PQFNs.(3)
Some fundamentals like elementary properties and arithmetic operations of PQFNs are characterized.(4)
Decision-making applications are discussed based on the proposal of PQFNS arithmetic operations.(5)
The results of the proposed similarity are compared with relevant existing models.(6)
The proposed structure is compared with relevant models under suitable evaluating indicators.(7)
The advantageous aspects of the proposed structure are discussed. The generalization of proposed structure is presented.


## 1.3. Paper Organization

This paper is organized as follows. The next section introduces the preliminaries of PQFNs and some notations. A three-stage FSSP model is provided in Section3. 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.


## 2. Prerequisites

Here, we study some preliminaries that we need for the main sections (for more details, see [27]).Definition 1.
A PQFN is denoted byW˜PQ=w1,w2,w3,w4,w5, where w1≤w2≤w3≤w4≤w5 are real numbers, and its membership function μW˜PQ is given by(1)μW˜PQ=0,x<w1;121w2−w12x−w12,w1≤x≤w2;121w3−w22x−w22+1,w2≤x≤w3;121w4−w32x−w32+1,w3≤x≤w4;121w5−w42x−w42,w4≤x≤w5;0,x>w5.
Figure1 shows the graphical representation of a PQFN.Figure 1
Graphical representation of a piecewise quadratic fuzzy number (PQFN).Definition 2.
LetU˜PQ=u1,u2,u3,u4,u5 and V˜PQ=v1,v2,v3,v4,v5 be two PQFNs. Then, we have(i)
Addition:U˜PQ+V˜PQ=u1+v1,u2+v2,u3+v3,u4+v4,u5+v5.(ii)
Subtraction:U˜PQ−V˜PQ=u1−v5,u2−v4,u3−v3,u4−v2,u5−v1.(iii)
Scalar multiplication:kU˜PQ=ku1,ku2,ku3,ku4,ku5,k>0,ku5,ku4,ku3,ku2,ku1,k<0..Definition 3.
For the close interval approximation (CIA) of PQFN ofU=Uα−,Uα+, we call U^=Uα−+Uα+/2 as the associated real number of U.Definition 4.
ForU=Uα−,Uα+ and V=Vα−,Vα+, we have the following properties:(1)
Addition:U+V=Uα−+Vbα−,Uα++Vα+.(2)
Subtraction:U−V=Uα−−Vα+,Uα+−Vα−.(3)
Scalar multiplication:kU=kUα−,kUα+,k>0kUα+,kUα−,k<0.(4)
Multiplication:U×V, Uα+Vα−+Uα−Vα+/2,Uα−Vbα−+Uα+Vα+/2.(5)
Division:U÷V, 2Uα−/Vα−+Vα+,2Uα+/Vα−+Vα+,V>0,Vα−+Vα+≠02Uα+/Vα−+Vα+,2Uα−/Vα−+Vα+,V<0,Vα−+Vα+≠0.(6)
The order relations:(i)
U<˜V if Uα−≤Vα− and Uα+≤Vα+ or Uα−+Uα+≤Vα−+Vα+.(ii)
U is preferred to V if and only if Uα−≥Vα−,Uα+≥Vα+.


### 2.1. Symbolization

Table1 shows the symbols of our work.Table 1
List of symbols.
AbbreviationsDescriptionsSArrangement of jobs,i=1,n¯SkSequence obtained through the method [1],k=1,2,…,nMhMachineh,h=1,2,3MMinimum makespana˜lhPPQF processing time (PQFPT) for thel-th task on Mh,h=1,2,3aijPQClose interval estimate of the PQFPT of thei-th task in sequence Sk running on MjtijSkTime ofi-th task for Sk on MjUjSkConsumption time forMj that is necessary for SkCTSkWhole completion timeIijSkIdle timea^ijCorresponding normal time of thei-th task on MjRSkWhole rental paymentCCost of renting


## 2.1. Symbolization

Table1 shows the symbols of our work.Table 1
List of symbols.
AbbreviationsDescriptionsSArrangement of jobs,i=1,n¯SkSequence obtained through the method [1],k=1,2,…,nMhMachineh,h=1,2,3MMinimum makespana˜lhPPQF processing time (PQFPT) for thel-th task on Mh,h=1,2,3aijPQClose interval estimate of the PQFPT of thei-th task in sequence Sk running on MjtijSkTime ofi-th task for Sk on MjUjSkConsumption time forMj that is necessary for SkCTSkWhole completion timeIijSkIdle timea^ijCorresponding normal time of thei-th task on MjRSkWhole rental paymentCCost of renting


## 3. Methodology

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


### 3.1. Cost of Renting

The 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 tasksi,i=1,n¯ under the definite rental policy L are managed on three machines Mj,j=1,2,3. Let a˜ijPQ be the PQFPT of i-th task on j-th machine (see Table 2). Let Sij,i=1,n¯;j=1,2,3. Determine the related processing times with crisp number on devices M1,M2, and M3 in such a way that either a^j2≤a^i1 or a^j2≤a^i3;∀i,j. Our objective is to determine Sk of the tasks that minimizes the cost of renting the equipment.Table 2
Description of the problem with the PQFN matrix.
TasksM1M2M3ia˜i1PQa˜i2PQa˜i3PQ1a˜11PQa˜12PQa˜13PQ2a˜21PQa˜22PQa˜23PQ…………na˜n1PQa˜n2PQa˜n3PQThe problem may be expressed mathematically as follows:(2)minR˜PQSk=∑i=1na˜i1PQ×C1+U˜2PQSk×C2+U˜3PQSk×C3Subject to rental policy L.Using the CIA of PQFN, model (10) may be reformulated as follows:(3)minRα−Sk,Rα+Sk=∑i=1nai1α−,ai1α+×C1+U2α−Sk,U2α+Sk×C2+U3α−Sk,U3α+Sk×C3Subject to rental policy L.Table2 may be recreated in CIA of PQFN format as shown in Tables 3 and 4.Table 3
The problem with CIA matrix.
TasksM1M2M3iai1α−,ai1α+ai2α−,ai2α+ai3α−,ai3α+1a11α−,a11α+a12α−,a12α+a13α−,a13α+2a21α−,a21α+a22α−,a22α+a23α−,a23α+…………nan1α−,an1α+an2α−,an2α+an3α−,an3α+Table 4
The problem with the corresponding crisp matrix form.
TasksM1M2M3iai1α−+ai1α+/2ai2α−+ai2α+/2ai3α−+ai3α+/21a11α−+a11α+/2a12α−+a12α+/2a13α−+a13α+/22a21α−+a21α+/2a22α−+a22α+/2a23α−+a23α+/2…………nan1α−+an1α+/2an2α−+an2α+/2an3α−+an3α+/2


## 3.1. Cost of Renting

The 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 tasksi,i=1,n¯ under the definite rental policy L are managed on three machines Mj,j=1,2,3. Let a˜ijPQ be the PQFPT of i-th task on j-th machine (see Table 2). Let Sij,i=1,n¯;j=1,2,3. Determine the related processing times with crisp number on devices M1,M2, and M3 in such a way that either a^j2≤a^i1 or a^j2≤a^i3;∀i,j. Our objective is to determine Sk of the tasks that minimizes the cost of renting the equipment.Table 2
Description of the problem with the PQFN matrix.
TasksM1M2M3ia˜i1PQa˜i2PQa˜i3PQ1a˜11PQa˜12PQa˜13PQ2a˜21PQa˜22PQa˜23PQ…………na˜n1PQa˜n2PQa˜n3PQThe problem may be expressed mathematically as follows:(2)minR˜PQSk=∑i=1na˜i1PQ×C1+U˜2PQSk×C2+U˜3PQSk×C3Subject to rental policy L.Using the CIA of PQFN, model (10) may be reformulated as follows:(3)minRα−Sk,Rα+Sk=∑i=1nai1α−,ai1α+×C1+U2α−Sk,U2α+Sk×C2+U3α−Sk,U3α+Sk×C3Subject to rental policy L.Table2 may be recreated in CIA of PQFN format as shown in Tables 3 and 4.Table 3
The problem with CIA matrix.
TasksM1M2M3iai1α−,ai1α+ai2α−,ai2α+ai3α−,ai3α+1a11α−,a11α+a12α−,a12α+a13α−,a13α+2a21α−,a21α+a22α−,a22α+a23α−,a23α+…………nan1α−,an1α+an2α−,an2α+an3α−,an3α+Table 4
The problem with the corresponding crisp matrix form.
TasksM1M2M3iai1α−+ai1α+/2ai2α−+ai2α+/2ai3α−+ai3α+/21a11α−+a11α+/2a12α−+a12α+/2a13α−+a13α+/22a21α−+a21α+/2a22α−+a22α+/2a23α−+a23α+/2…………nan1α−+an1α+/2an2α−+an2α+/2an3α−+an3α+/2


## 4. 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.
Ifa^j2≤a^i1 or a^j2≤a^i3;∀i,j, i.e., maxa^i1≥mina^j2 or maxa^i3≥mina^j2;∀i,j, go to next step; otherwise, break.Step 3.
Define dummy machinesH1 and H2, and their processing times H1i and H2i are as follows: H1i=a^i1+a^i2,H2i=a^i2+a^i3;∀i.Step 4.
Use the existing algorithm [1] onHi and get S1.Step 5.
Put the 2nd,…, nth tasks of the S1 in the first position and all other tasks of S1 in the same order.Step 6.
For all possible sequencesSk,k=1,n¯, calculate: R^Sk=∑i=1na^ij×C1+U^2Sk×C2+U^3Sk×C3.Step 7.
SetminR^Sk,k=1,n¯ as the optimal solution.


## 5. Numerical Example

Consider Table5 as the problem. Now, we solve this problem by our model.Table 5
PQFN processing times for machines.
TasksM1M2M3ia˜i1PQa˜i2PQa˜i3PQ1(6,7,8,9,10)(5,6,7,8,10)(2,3,4,5,7)2(11,12,13,14,17)(4,5,6,7,9)(3,4,5,6,8)3(7,8,10,12,14)(3,4,5,6,9)(5,6,7,8,9)4(9,10,11,12,14)(3,5,6,7,9)(10,11,12,13,15)5(7,9,10,11,13)(2,5,6,8,10)(5,8,9,10,11)At first, in Tables6 and 7, we compute the related interval and crisp numbers for each PQFPT.Table 6
PQFN processing times with interval format.
TasksM1M2M3iai1PQai2PQai3PQ1[7,9][6,8][3,5]2[12,14][5,7][4,6]3[8,12][4,6][6,8]4[10,12][5,7][11,13]5[9,11][5,8][8,10]Table 7
PQFN processing times with crisp format.
TasksM1M2M3ia^i1a^i2a^i3187421365310574116125106.59Then, using Step 3 of our algorithm, the processing times can be computed as shown in Table8.Table 8
The related crisp numbers of the processing times.
TasksH1H211511219113151241718516.515.5Using procedure [1], S1:2-4-5-1-3.The subsequent viable sequences correspond to the minimal rental cost:S2:4−2−5−1−3;S3:1−2−4−5−3;S4:3−2−4−5−1.Tables9 and 10 illustrate the in-out flow for the sequence S1 in the PQFNs and CIA forms.Table 9
The in-out flow forS1 in the PQFNs.
TasksM1M2M3iIn- outIn- outIn- out2(11,12,13,14,17)(15,17,19,21,26)(18,21,24,27,34)4(19,22,24,26,31)(18,22,25,28,35)(28,32,36,39,49)5(26,31,34,37,44)(20,27,31,36,45)(33,40,45,49,60)1(32,38,42,46,54)(25,33,38,44,55)(35,43,49,54,67)3(39,46,52,58,68)(28,37,43,50,64)(40,49,56,62,76)Table 10
The in-out flow forS1 in CIA.
JobsMachineM1MachineM2MachineM32[12,14][17,21][21,27]4[22,26][22,28][32,39]5[31,37][27,36][40,49]1[38,46][33,44][43,54]3[46,58][37,50][49,62]For S1, we get the following.The completion time forS1 is CT˜PQS1=40,49,56,62,76,CTS1=49,62, and CT^S1=55.5.The consumption time for machineM2 is U˜2PQS1=11,23,30,38,53, U2S1=23,38, and U^2S1=30.5.The consumption time for machineM3 is U˜3PQS1=14,28,37,45,61, U3S1=28,45, and U^3S1=36.5.(4)R˜PQS1=∑i=15a˜i1×C1+U2S1×C2+U3S1×C3=572,726,831,935,1145,RS1=726,935andR^S1=830.5.Similarly, we have the following.ForS2:(5)CT˜PQS2=37,47,54,61,73,CTS2=47,61,andCT^S2=54,(6)U˜2PQS2=12,23,30,38,52,U2S2=23,38,andU^2S2=30.5,(7)U˜3PQS2=14,28,37,45,61,U3S2=28,45,andU^2S2=36.5,(8)R˜PQS2=∑i=15a˜i1PQ×C1+U˜2PQS2×C2+U˜3PQS2×C3=582,718,769,927,1131,RS2=718,927,andR^S1=822.5.ForS3, we have(9)CT˜PQS3=36,45,52,59,70,CTS3=45,59,andCT^S3=52,(10)U˜2PQS3=13,23,30,38,49,U2S3=23,38,andU^2S3=30.5,(11)U˜3PQS3=16,28,37,46,59,U3S3=28,46,andU^2S3=37,(12)R˜PQS3=∑i=15a˜i1PQ×C1+U˜2PQS3×C2+U˜3PQS3×C3=562,686,791,896,1075,RS3=686,896,andR^S3=791.ForS4, we have(13)CT˜PQS4=35,44,52,60,73,CTS4=44,60,andCT^S4=52,(14)U˜2PQS4=10,11,30,40,54,U2S4=11,40,andU^2S4=25.5,(15)U˜3PQS4=12,26,37,48,63,U3S4=26,48,andU^2S4=37,(16)R˜PQS4=∑i=15a˜i1PQ×C1+U˜2PQS4×C2+U˜3PQS4×C3=560,672,823,956,1161,RS4=672,956,andR^S4=814.ForS5, we have(17)R^S5=2066.6257.Thus,(18)R˜PQS3=∑i=15a˜i1PQ×C1+U˜2PQS3×C2+U˜3PQS3×C3=562,686,791,896,1075,(19)RS3=686,896.Therefore,S3:1−2−4−5−3 is the optimal sequence subject to the minimum rental cost, and R^S3=791 is the minimum rental cost irrespective of the total time passed.


## 6. Comparative Study

In this section, the proposed approach is compared with some existing studies to illustrate the advantages of the proposed approach. The results for this analysis are summarized in Table11. The symbol “↓” or “↑” shown in the table represents whether the associated feature satisfies or not.Table 11
Comparison of different researchers’ contributions.
AuthorProcessing timePiecewise quadratic fuzzy numbersClose approximate intervalMinimum rental costRuiz et al. [28]↓↓↓↑Liang et al. [29]↑↓↓↑Sanchez-Herrera et al. [30]↑↓↓↑Our proposed approach↑↑↑↑


## 7. 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. The result shows that the proposed method has its advantage in flexible decision making corresponding to favorite priorities of alternatives. This 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.



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