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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 34-43

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.34

© The Korean Institute of Intelligent Systems

Generalized Intuitionistic Fuzzy Flow Shop Scheduling Problem with Setup Time and Single Transport Facility

T. Yogashanthi1, Shakeela Sathish1, and K. Ganesan2

1Department of Mathematics, SRM Institute of Science and Technology, Ramapuram, Chennai, India
2Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chennai, India

Correspondence to :
Shakeela Sathish (shakeels@srmist.edu.in)

Received: April 2, 2022; Revised: October 24, 2022; Accepted: March 6, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Setup time is the amount of time required for a machine to adjust its settings or the preparation of a device at each stage to process and deliver a completed job. A novel approach for the n-job 2-machine generalized intuitionistic fuzzy flow shop scheduling problem, subject to the setup time, was proposed. When the machines are kept in different places, the transporting and return times of transport play a significant role in the production. Generalized triangular intuitionistic fuzzy numbers were considered to represent the processing, setup, transportation, and return times. This study aims to minimize the intuitionistic fuzzy total production time with less vagueness.

Keywords: Generalized intuitionistic fuzzy number, Flow shop scheduling, Left fuzziness index, Right fuzziness index, Euclidean distance

Conflict of Interest

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No potential conflict of interest relevant to this article was reported.

Biographies

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T. Yogashanthi received her Ph.D. degree in intuitionistic fuzzy sets from SRM Institute of Science and Technology. She is a faculty member in the Department of Mathematics at SRM Institute of Science and Technology, Ramapuram, Chennai, India. Her main research interests include fuzzy sets and intuitionistic fuzzy optimization.

E-mail: yogashanthi13@gmail.com

Shakeela Sathish is currently working as a professor in the Department of Mathematics at SRM Institute of Science and Technology, Ramapuram, Chennai, India. Her research area is focused on fuzzy optimization.

E-mail: shakeels@srmist.edu.in

K. Ganesan is currently a professor in the Department of Mathematics at SRM Institute of Science and Technology, Chennai, India. His research areas include fuzzy optimization, interval mathematics, operations research, and ordinary and partial differential equations. He has handled research projects funded by BRNS and visited countries like Iran and Japan, among others. Thus far, 17 candidates have obtained their Ph.D degrees under his guidance, and many are currently pursuing research under his direction.

E-mail: ganesank@srmist.edu.in

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 34-43

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.34

Copyright © The Korean Institute of Intelligent Systems.

Generalized Intuitionistic Fuzzy Flow Shop Scheduling Problem with Setup Time and Single Transport Facility

T. Yogashanthi1, Shakeela Sathish1, and K. Ganesan2

1Department of Mathematics, SRM Institute of Science and Technology, Ramapuram, Chennai, India
2Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chennai, India

Correspondence to:Shakeela Sathish (shakeels@srmist.edu.in)

Received: April 2, 2022; Revised: October 24, 2022; Accepted: March 6, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Setup time is the amount of time required for a machine to adjust its settings or the preparation of a device at each stage to process and deliver a completed job. A novel approach for the n-job 2-machine generalized intuitionistic fuzzy flow shop scheduling problem, subject to the setup time, was proposed. When the machines are kept in different places, the transporting and return times of transport play a significant role in the production. Generalized triangular intuitionistic fuzzy numbers were considered to represent the processing, setup, transportation, and return times. This study aims to minimize the intuitionistic fuzzy total production time with less vagueness.

Keywords: Generalized intuitionistic fuzzy number, Flow shop scheduling, Left fuzziness index, Right fuzziness index, Euclidean distance

Fig 1.

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Figure 1.

Generalized triangular intuitionistic fuzzy number.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 34-43https://doi.org/10.5391/IJFIS.2023.23.1.34

Table 1 . Conceptual model of the problem in matrix form.

JobiMachine M1t˜i,1→2Ir˜iIMachine M2
S˜i,1Ip˜i,1IS˜i,2Ip˜i,2I
1S˜1,1Ip˜1,1It˜1,1→2Ir˜1IS˜1,2Ip˜1,2I
2S˜2,1Ip˜2,1It˜2,1→2Ir˜2IS˜2,2Ip˜2,2I
3S˜3,1Ip˜3,1It˜3,1→2Ir˜3IS˜3,2Ip˜3,2I
4S˜4,1Ip˜4,1It˜4,1→2Ir˜4IS˜4,2Ip˜4,2I
–––––––
–––––––
nS˜n,1Ip˜n,1It˜n,1→2Ir˜nIS˜n,2Ip˜n,2I

Table 2 . Handling durations as GTIFNs.

JobiMachine M1t˜i,1→2Ir˜iIMachine M2
S˜i,1Ip˜i,1IS˜i,2Ip˜i,2I
1(〈2, 3, 4〉, 0.5, 0.5)(〈6, 7, 8〉, 0.5, 0.5)(〈4, 4, 4〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈2, 4, 6〉, 0.5, 0.5)(〈7, 8, 9〉, 0.5, 0.5)
2(〈1, 2, 3〉, 0.5, 0.5)(〈10, 11, 12〉, 0.5, 0.5)(〈8, 8, 8〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈2, 3, 4〉, 0.5, 0.5)(〈12, 13, 14〉, 0.5, 0.5)
3(〈2, 4, 6〉, 0.5, 0.5)(〈5, 7, 8〉, 0.5, 0.5)(〈10, 10, 10〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈1, 2, 3〉, 0.5, 0.5)(〈6, 7, 8〉, 0.5, 0.5)
4(〈1, 2, 3〉, 0.5, 0.5)(〈8, 10, 12〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈3, 4, 5〉, 0.5, 0.5)(〈9, 10, 11〉, 0.5, 0.5)
5(〈2, 3, 4〉, 0.5, 0.5)(〈7, 8, 9〉, 0.5, 0.5)(〈5, 5, 5〉, 0.5, 0.5)(〈3, 3, 3〉, 0.5, 0.5)(〈1, 2, 3〉, 0.5, 0.5)(〈4, 5, 6〉, 0.5, 0.5)

Table 3 . Parametric form of processing durations.

JobiMachine M1t˜i,1→2Ir˜iIMachine M2
S˜i,1Ip˜i,1IS˜i,2Ip˜i,2I
1(〈3, 1 − 2r, 1 − 2r〉, 〈3, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈7, 1 − 2r, 1 − 2r〉, 〈7, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈4, 0, 0〉, 〈4, 0, 0〉, 0.5, 0.5)(〈4, 0, 0〉, 〈4, 0, 0〉, 0.5, 0.5)(〈4, 2 − 4r, 2 − 4r〉, 〈4, −2 + 4r*, −2 + 4r*〉, 0.5, 0.5)(〈8, 1 − 2r, 1 − 2r〉, 〈8, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
2(〈2, 1 − 2r, 1 − 2r〉, 〈2, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈11, 1 − 2r, 1 − 2r〉, 〈11, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈8, 0, 0〉, 〈8, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈3, 1 − 2r, 1 − 2r〉, 〈3, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈13, 1 − 2r, 1 − 2r〉, 〈13, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
3(〈4, 2 − 4r, 2 − 4r〉, 〈4, −2 + 4r*, −2 + 4r*〉, 0.5, 0.5)(〈7.5, 2.5 − 4r, 0.5 − 2r〉, 〈7.5, −1.5 + 4r*, −1.5 + 2r*〉, 0.5, 0.5)(〈10, 0, 0〉, 〈10, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈2, 1 − 2r, 1 − 2r〉, 〈2, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈7, 1 − 2r, 1 − 2r〉, 〈7, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
4(〈2, 1 − 2r, 1 − 2r〉, 〈2, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈10, 2 − 4r, 2 − 4r〉, 〈10, −2 + 4r*, −2 + 4r*〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈4, 1 − 2r, 1 − 2r〉, 〈4, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈10, 1 − 2r, 1 − 2r〉, 〈10, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
5(〈3, 1 − 2r, 1 − 2r〉, 〈3, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈8, 1 − 2r, 1 − 2r〉, 〈8, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈5, 0, 0〉, 〈5, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈2, 1 − 2r, 1 − 2r〉, 〈2, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈5, 1 − 2r, 1 − 2r〉, 〈5, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)

Table 4 . Representation of two fictitious machines.

JobG˜iI=R˜i-1I+t˜i,1→2I+p˜i,1I-S˜i,2IH˜iI=R˜i-1I+t˜i,1→2I+p˜i,2I-S˜i,1I
1(〈7, 2 − 4r, 2 − 4r〉, 〈7, −2 + 4r*, −2 + 4r*〉, 0.5, 0.5)(〈9, 1 − 2r, 1 −2r〉, 〈9, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
2(〈16, 1 − 2r, 1 − 2r〉, 〈16, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)(〈19, 1 − 2r, 1 −2r〉, 〈19, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)
3(〈19, 2.5 − 4r, 1 − 2r〉, 〈19, −1.5 + 4r*, −1 + 2r*〉, 0.5, 0.5)(〈16.5, 2.5 − 4r, 2 − 4r〉, 〈16.5, −1.5 + 4r*, −2 + 4r*〉, 0.5, 0.5)
4(〈12, 2 − 4r, 2 − 4r〉, 〈12, −2 + 4r*, −2 + 4r*〉,0.5, 0.5)(〈14, 2 − 4r, 2 − 4r〉, 〈14, −2 + 4r*, −2 + 4r*〉,0.5, 0.5)
5(〈11, 1 − 2r, 1 − 2r〉, 〈11, −1 + 2r*, −1 + 2r*〉,0.5, 0.5)(〈7, 1 − 2r, 1 − 2r〉, 〈7, −1 + 2r*, −1 + 2r*〉, 0.5, 0.5)

Table 5 . In-out table.

JobMachine M1t˜i,1→2Ir˜iIy˜i-1Iz˜iIMachine M2
InOutInOut
1(〈3, 1−2r, 1−2r〉, 〈3,− 1+2r*, − 1+2r*〉, 0.5, 0.5)(〈10, 1−2r, 1−2r〉, 〈10,− 1+2r*, − 1+2r*〉, 0.5, 0.5)(〈4, 0, 0〉, 〈4, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)-(〈14, 1−2r, 1−2r〉, 〈14,− 1+2r*, − 1+2r*〉, 0.5, 0.5)(〈18, 2−4r, 2−4r〉, 〈18, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈26, 2−4r, 2−4r〉, 〈26, − 2+4r*, − 2+4r*〉, 0.5, 0.5)
4(〈12, 1−2r, 1−2r〉, 〈12,− 1+2r*, − 1+2r*〉, 0.5, 0.5)(〈22, 2−4r, 2−4r〉, 〈22,− 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈17, 1−2r, 1−2r〉, 〈17,− 1+2r*, − 1+2r*〉, 0.5, 0.5)(〈25, 2−4r, 2−4r〉, 〈25, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈30, 2−4r, 2−4r〉, 〈30, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈40, 2−4r, 2−4r〉, 〈40, − 2+4r*, − 2+4r*〉, 0.5, 0.5)
2(〈24, 2−4r, 2−4r〉, 〈24, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈35, 2−4r, 2−4r〉, 〈35, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈8, 0, 0〉, 〈8, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈28, 2−4r, 2−4r〉, 〈28, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈43, 2−4r, 2−4r〉, 〈43, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈46, 2−4r, 2−4r〉, 〈46, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈59, 2−4r, 2−4r〉, 〈59, − 2+4r*, − 2+4r*〉, 0.5, 0.5)
3(〈39, 2−4r, 2−4r〉, 〈39, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈46.5, 2.5−4r, 2−4r〉 〈46.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈10, 0, 0〉, 〈10, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈46, 2−4r, 2−4r〉, 〈46, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈56.5, 2.5−4r, 2−4r〉, 〈56.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈61, 2−4r, 2−4r〉, 〈61, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈68, 2−4r, 2−4r〉, 〈68, − 2+4r*, − 2+4r*〉, 0.5, 0.5)
5(〈 49.5, 2.5−4r, 2−4r 〉, 〈49.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈57.5, 2.5−4r, 2−4r 〉, 〈57.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈5, 0, 0〉, 〈5, 0, 0〉, 0.5, 0.5)(〈3, 0, 0〉, 〈3, 0, 0〉, 0.5, 0.5)(〈59.5, 2.5−4r, 2−4r〉, 〈59.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈62.5, 2.5−4r, 2−4r〉, 〈62.5, − 1.5+4r*, − 2+4r*〉, 0.5, 0.5)(〈70, 2−4r, 2−4r〉, 〈70, − 2+4r*, − 2+4r*〉, 0.5, 0.5)(〈75, 2−4r, 2−4r〉, 〈75, − 2+4r*, − 2+4r*〉, 0.5, 0.5)

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