# A Novel Approach for Minimizing Processing Times of Three-Stage Flow Shop Scheduling Problems under Fuzziness

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## Abstract

**:**

## 1. Introduction

- (1)
- Introducing suitable terminologies and measures that consider the properties of a possible optimal scheduling;
- (2)
- Defining two methods for determining the best schedule, one based on the ordering of pentagonal fuzzy numbers and the other on PFN interval confidence;
- (3)
- Interacting the analyst with the DM to arrive to the optimal sequence.

- (1)
- To minimize the fuzzy professing time of the machines subject to the rental policy;
- (2)
- To study the inclusive study of pentagonal fuzzy numbers in the scheduling problem;
- (3)
- To specific the concept of the optimal scheduling for the scenario;
- (4)
- To validate the proposed study with the support of illustrative example.

## 2. Preliminaries

**Definition**

**1.**

- (1)
- ${\mathbf{\mu}}_{\tilde{\mathit{A}}}\left(\mathbf{x}\right)$is an upper semi—continuous membership function;
- (2)
- $\tilde{\mathit{A}}$is convex fuzzy set, i.e.,${\mathit{\mu}}_{\tilde{\mathit{A}}}\left(\mathit{w}\mathit{x}+\left(\mathit{1}-\mathit{w}\right)\mathit{y}\right)\ge \mathit{min}\left\{{\mathit{\mu}}_{\tilde{\mathit{A}}}\left(\mathit{x}\right),{\mathit{\mu}}_{\tilde{\mathit{A}}}\left(\mathit{y}\right)\right\}$for all $\mathit{x},\mathit{y}\in \mathbb{R};\mathbf{0}\le \mathit{w}\le \mathbf{1};$$\tilde{\mathit{A}}$is normal, i.e.,$\exists {\mathit{x}}_{\mathbf{0}}\in \mathbb{R}$for which${\mathit{\mu}}_{\tilde{\mathit{A}}}\left({\mathit{x}}_{\mathbf{0}}\right)=\mathbf{1};$
- (3)
- $Supp(\tilde{A})=\left\{x\in \mathbb{R}:{\mu}_{\tilde{A}}\left(x\right)0\right\}$is the support of$\tilde{A}$, and the closure$cl(Supp(\tilde{A}))$is compact set.

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

## 3. Notations and Assumptions

#### 3.1. Notations

#### 3.2. Assumptions

- i.
- Preemption was prohibited in all jobs. The machine only handled one work at a time.
- ii.
- All jobs were open at the beginning of schedule time. The duration of the production was independent of the schedule.
- iii.
- Each machine’s initial setup time was disregarded. Any machine could be unoccupied.
- iv.
- The deterministic phase was used to process each job. A task must be completed after it had been started.
- v.
- Before the second machine could handle the second work, the first job must have been completed in the first machine.
- vi.
- PFNs were used to represent the due dates.

## 4. Statement of the Problem

## 5. Solution Procedure

## 6. Numerical Example

**1st approach: Ranking method solution**

**2nd Approach named as Interval Confidence based method:**

- Make the optimization with the associated ordinary;
- Compute the true optimum for the intervals of confidence at the level $\alpha =0$;
- Compare the results with those obtained using the pentagonal fuzzy numbers. If the divergence is very small within acceptable limits, keep the results obtained by using the PFNs.

## 7. Comparative Study

- (a)
- The intermediate value of the pentagonal and the triangular fuzzy numbers was equal i.e., when the membership ${\mu}_{A}\left(x\right)=1,x=452$ for the total rental cost and $x=63$ for the total processing time;
- (b)
- The left and right fuzziness index values, defined in [24] were smaller than the values we obtained for the idle times of machines, total processing time and the rental cost;
- (c)

#### Advantages/Limitations of the Proposed Algorithm

- (1)
- The methodology does not involve a unified method because the DM’s vision, kind of fuzzy number, and $\alpha -$level set vary from one another, making it impossible to assign a united way for allocating the intriguing scenarios for the DM.
- (2)
- Many factors must be considered such as: (i) the possibility of formulating the problem, (ii) the possibility of formulating the problem and choosing the $\alpha -$ level set, and (iii) the capability of solving the problem’s selected scenarios and finding their exact optimal scheduling.

## 8. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphical Representation of a symmetric PFN [18].

T | $\mathit{i}$ | 1 | 2 | … | $\mathit{n}$ |
---|---|---|---|---|---|

Machine ${\mathcal{M}}_{1}$ | ${\tilde{\mathcal{A}}}_{i1}$ | ${\tilde{\mathcal{A}}}_{11}$ | ${\tilde{\mathcal{A}}}_{12}$ | … | ${\tilde{\mathcal{A}}}_{1n}$ |

Machine ${\mathcal{M}}_{2}$ | ${\tilde{\mathcal{A}}}_{i2}$ | ${\tilde{\mathcal{A}}}_{21}$ | ${\tilde{\mathcal{A}}}_{22}$ | … | ${\tilde{\mathcal{A}}}_{2n}$ |

Machine ${\mathcal{M}}_{3}$ | ${\tilde{\mathcal{A}}}_{i3}$ | ${\tilde{\mathcal{A}}}_{31}$ | ${\tilde{\mathcal{A}}}_{32}$ | … | ${\tilde{\mathcal{A}}}_{3n}$ |

Job | ${\mathcal{M}}_{1}$ | ${\mathcal{M}}_{2}$ | ${\mathcal{M}}_{3}$ |
---|---|---|---|

1 | (7, 7.5, 8, 8.5, 9) | (6, 6.5, 7, 7.5, 8) | (3, 3.5, 4, 4.5, 5) |

2 | (12, 12.5,13, 13.5, 14) | (5, 5.5, 6, 6.5, 7) | (4, 4.5, 5, 5.5, 6) |

3 | (8, 9, 10, 11, 12) | (4, 4.5, 5, 5.5, 6) | (6, 6.5, 7, 7.5, 8) |

4 | (10, 10.5, 11, 11.5, 12) | (5, 5.5, 6, 6.5, 7) | (11, 11.5, 12, 12.5, 13) |

5 | (9, 9.5, 10, 10.5, 11) | (5, 5.5, 6, 6.5, 7) | (8, 8.5, 9, 9.5, 10) |

Job | $\mathit{X}$ | $\mathit{Y}$ |
---|---|---|

1 | (13,14,15,16,17) | (9,10,11,12,13) |

2 | (17,18,19,20,21) | (9,10,11,12,13) |

3 | (12,13.5, 15,16.5,18) | (10,11,12,13,14) |

4 | (15,16,17,18,19) | (16,17,18,19,20) |

5 | (14,15,16,17,18) | (13,14,15,16,17) |

Machine 1 | Machine 2 | Machine 3 | ||||
---|---|---|---|---|---|---|

Job | Time in | Time out | Time in | Time out | Time in | Time out |

4 | (0, 0, 0, 0, 0) | (10, 10.5, 11, 11.5, 12) | (10, 10.5, 11, 11.5, 12) | (10, 10.5, 11, 11.5, 12) | (15, 16, 17, 18, 19) | (26, 27.5, 29, 30.5, 32) |

5 | (10, 10.5, 11, 11.5, 12) | (19, 20, 21, 22, 23) | (19, 20, 21, 22, 23) | (19, 20, 21, 22, 23) | (26, 27.5, 29, 30.5, 32) | (34, 36, 38, 40, 42) |

2 | (19, 20, 21, 22, 23) | (31, 32.5, 34, 35.5, 37) | (31, 32.5, 34, 35.5, 37) | (31, 32.5, 34, 35.5, 37) | (36, 38, 40, 42, 44) | (40, 42.5, 45, 47.5, 50) |

3 | (31, 32.5, 34, 35.5, 37) | (39, 41.5, 44, 46.5, 49) | (39, 41.5, 44, 46.5, 49) | (39, 41.5, 44, 46.5, 49) | (43, 46, 49, 52, 55) | (49, 52.5, 56, 59.5, 63) |

1 | (39, 41.5, 44, 46.5, 49) | (46, 49, 52, 55, 58) | (46, 49, 52, 55, 58) | (46, 49, 52, 55, 58) | (52, 55.5, 59, 62.5, 66) | (55, 59, 63, 67, 71) |

Jobs | ${\mathcal{M}}_{1}$ | ${\mathcal{M}}_{2}$ | ${\mathcal{M}}_{3}$ |
---|---|---|---|

1 | [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, 7] | [8, 10] |

Job | $\mathit{X}$ | $\mathit{Y}$ |
---|---|---|

1 | [13, 17] | [9, 13] |

2 | [17, 21] | [9, 13] |

3 | [12, 18] | [10, 14] |

4 | [15, 19] | [16, 20] |

5 | [14, 18] | [13, 17] |

Job | Machine-1 | Machine-2 | Machine-3 | |||
---|---|---|---|---|---|---|

Time in | Time out | Time in | Time out | Time in | Time out | |

4 | [0, 0] | [10, 12] | [10, 12] | [15, 19] | [15, 19] | (26,32) |

5 | [10, 12] | [19, 23] | [19, 23] | [24, 30] | [26, 32] | (34,42) |

2 | [19, 23] | [31, 37] | [31, 37] | [36, 44] | [36, 44] | [40, 50] |

3 | [31, 37] | [39, 49] | [39, 49] | [43, 55] | [43, 55] | [49, 63] |

1 | [39, 49] | [46, 58] | [46, 58] | [52, 66] | [52, 66] | [55, 71] |

Type of Fuzzy Number | Our Proposed Algorithm | Algorithm by Sathish and Ganesan [24] |
---|---|---|

Pentagonal fuzzy number | (55, 59, 63, 67, 71) | (61, 62, 63, 64, 65) |

Triangular fuzzy number | (59, 63, 67) | (61, 63, 65) |

Fuzziness index triangular fuzzy number | (63, 4, 4) | (63, 2, 2) |

Type of Fuzzy Number | Proposed Algorithm | Algorithm by Sathish and Ganesan [24] |
---|---|---|

Pentagonal fuzzy number | (229, 340.5, 452, 563.5, 675) | (444, 448, 452, 456, 460) |

Triangular fuzzy number | (340.5, 452, 563.5) | (444, 452, 460) |

Fuzziness index triangular fuzzy number | (452, 111.5, 111.5) | (452, 8, 8) |

Type of Fuzzy Number | ${\mathcal{I}}_{\mathit{j}}$ | Proposed Algorithm | Algorithm by Sathish and Ganesan [24] |
---|---|---|---|

Pentagonal fuzzy number | ${\mathcal{I}}_{1}$ | $\left(-3,4,11,18,25\right)$ | (9, 10, 11, 12, 13) |

${\mathcal{I}}_{2}$ | $\left(-13,2.5,18,33.5,49\right),$ | (16, 17, 18, 19, 20) | |

${\mathcal{I}}_{3}$ | $\left(-24,-7.5,9,25.5,42\right)$ | (7, 8, 9, 10, 11) | |

Triangular fuzzy number | ${\mathcal{I}}_{1}$ | $\left(4,11,18\right)$ | (9, 11, 13) |

${\mathcal{I}}_{2}$ | $\left(2.5,18,33.5\right),$ | (16, 18, 20) | |

${\mathcal{I}}_{3}$ | $\left(-7.5,9,25.5\right)$ | (7, 9, 11) | |

Fuzziness index triangular fuzzy number | ${\mathcal{I}}_{1}$ | (11, 7, 7) | (11, 2, 2) |

${\mathcal{I}}_{2}$ | (18, 15.5, 15.5) | (18, 2, 2) | |

${\mathcal{I}}_{3}$ | (9, 16.5, 16.5) | (9, 2, 2) |

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## Share and Cite

**MDPI and ACS Style**

Alburaikan, A.; Garg, H.; Khalifa, H.A.E.-W.
A Novel Approach for Minimizing Processing Times of Three-Stage Flow Shop Scheduling Problems under Fuzziness. *Symmetry* **2023**, *15*, 130.
https://doi.org/10.3390/sym15010130

**AMA Style**

Alburaikan A, Garg H, Khalifa HAE-W.
A Novel Approach for Minimizing Processing Times of Three-Stage Flow Shop Scheduling Problems under Fuzziness. *Symmetry*. 2023; 15(1):130.
https://doi.org/10.3390/sym15010130

**Chicago/Turabian Style**

Alburaikan, Alhanouf, Harish Garg, and Hamiden Abd El-Wahed Khalifa.
2023. "A Novel Approach for Minimizing Processing Times of Three-Stage Flow Shop Scheduling Problems under Fuzziness" *Symmetry* 15, no. 1: 130.
https://doi.org/10.3390/sym15010130