# Development of Integrated Linear Programming Fuzzy-Rough MCDM Model for Production Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Methodology Steps

#### 2.2. Linear Programming

#### 2.3. IMF SWARA Method

#### 2.4. Development of a Novel Rough CRADIS Approach

## 3. Integration of Linear Programming and a Fuzzy-Rough MCDM Model for Production Optimization

#### 3.1. Special Case of Linear Programming Optimization

#### 3.2. Formation of MCDM Model—Defining Criteria and Alternatives

#### 3.3. Determining the Significance of Criteria Using the IMF SWARA Method

#### 3.4. Determining the Optimal Solution Using the Rough CRADIS Approach

_{1}and x

_{2}are defined, by including the values of x

_{1}and x

_{2}in each constraint, as e.g., for A1:

_{1}and x

_{2}are 2500 and 1000, respectively, and the utilization of the first group of machines M is not complete, but there are still 750 h available, while the second group of machines N is fully utilized. When it comes to meeting the requirements of the market with product A, it is fully satisfied, while there is a shortage of 2000 products B according to these parameters. The initial decision-making matrix represents equal low and upper numbers because it represents the difference in satisfying the set restrictions for each criterion separately which is essential for managers while having no influence on the application of the rough model.

## 4. Verification of the Developed Model and Discussion

#### 4.1. Sensitivity Analysis

#### 4.2. Comparative Analysis

#### 4.3. Limitations and Managerial Implications

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Linguistic Variable | Abbreviation | TFN Scale | ||
---|---|---|---|---|

Absolutely less significant | ALS | 1 | 1 | 1 |

Dominantly less significant | DLS | 1/2 | 2/3 | 1 |

Much less significant | MLS | 2/5 | 1/2 | 2/3 |

Really less significant | RLS | 1/3 | 2/5 | 1/2 |

Less significant | LS | 2/7 | 1/3 | 2/5 |

Moderately less significant | MDLS | 1/4 | 2/7 | 1/3 |

Weakly less significant | WLS | 2/9 | 1/4 | 2/7 |

Equally significant | ES | 0 | 0 | 0 |

x_{1} | X_{2} | S_{1} | S_{2} | S_{3} | S_{4} | Const. |
---|---|---|---|---|---|---|

0 | 0 | 1 | −3 | 3/2 | 0 | 750 |

0 | 1 | 0 | 2 | −2 | 0 | 1000 |

1 | 0 | 0 | 0 | 1 | 0 | 2500 |

0 | 0 | 0 | −2 | 2 | 1 | 2000 |

0 | 0 | 0 | −2000 | 0 | 0 | F − 6000000 |

0 | 0 | 2/3 | −2 | 1 | 0 | 500 |

0 | 1 | 0 | 2 | −2 | 0 | 1000 |

1 | 0 | 0 | 0 | 1 | 0 | 2500 |

0 | 0 | 0 | −2 | 2 | 1 | 2000 |

0 | 0 | 0 | −2000 | 0 | 0 | F − 6000000 |

0 | 0 | 2/3 | −2 | 1 | 0 | 500 |

0 | 1 | 4/3 | −2 | 0 | 0 | 2000 |

1 | 0 | −2/3 | 2 | 0 | 0 | 2000 |

0 | 0 | −4/3 | 2 | 0 | 1 | 1000 |

0 | 0 | 0 | −2000 | 0 | 0 | F − 6000000 |

t | x_{1} | x_{2} | |
---|---|---|---|

A_{1} | 1 | 2500 | 1000 |

A_{2} | 0.095 | 2475 | 1050 |

A_{3} | 0.090 | 2450 | 1100 |

A_{4} | 0.085 | 2425 | 1150 |

A_{5} | 0.080 | 2400 | 1200 |

A_{6} | 0.075 | 2375 | 1250 |

A_{7} | 0.070 | 2350 | 1300 |

A_{8} | 0.065 | 2325 | 1350 |

A_{9} | 0.060 | 2300 | 1400 |

A_{10} | 0.055 | 2275 | 1450 |

A_{11} | 0.050 | 2250 | 1500 |

A_{12} | 0.045 | 2225 | 1550 |

A_{13} | 0.040 | 2200 | 1600 |

A_{14} | 0.035 | 2175 | 1650 |

A_{15} | 0.030 | 2150 | 1700 |

A_{16} | 0.025 | 2125 | 1750 |

A_{17} | 0.020 | 2100 | 1800 |

A_{18} | 0.015 | 2075 | 1850 |

A_{19} | 0.010 | 2050 | 1900 |

A_{20} | 0.05 | 2025 | 1950 |

A_{21} | 0 | 2000 | 2000 |

$\overline{{\mathit{\wp}}_{\mathit{j}}}$ | $\overline{{\mathit{\Im}}_{\mathit{j}}}$ | $\overline{{\mathit{\aleph}}_{\mathit{j}}}$ | $\overline{{\mathit{w}}_{\mathit{j}}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|---|

C3 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.287 | 0.291 | 0.296 | |

C4 | ES | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.287 | 0.291 | 0.296 |

C1 | WLS | 1.222 | 1.250 | 1.286 | 0.778 | 0.800 | 0.818 | 0.223 | 0.233 | 0.242 |

C2 | WLS | 1.222 | 1.250 | 1.286 | 0.605 | 0.640 | 0.669 | 0.173 | 0.186 | 0.198 |

SUM | 3.383 | 3.440 | 3.488 |

C1 | C2 | C3 | C4 | |||||
---|---|---|---|---|---|---|---|---|

A1 | 750.00 | 750.00 | 1 | 1 | 1 | 1 | 2000 | 2000 |

A2 | 712.50 | 712.50 | 1 | 1 | 25 | 25 | 1950 | 1950 |

A3 | 675.00 | 675.00 | 1 | 1 | 50 | 50 | 1900 | 1900 |

A4 | 637.50 | 637.50 | 1 | 1 | 75 | 75 | 1850 | 1850 |

A5 | 600.00 | 600.00 | 1 | 1 | 100 | 100 | 1800 | 1800 |

A6 | 562.50 | 562.50 | 1 | 1 | 125 | 125 | 1750 | 1750 |

A7 | 525.00 | 525.00 | 1 | 1 | 150 | 150 | 1700 | 1700 |

A8 | 487.50 | 487.50 | 1 | 1 | 175 | 175 | 1650 | 1650 |

A9 | 450.00 | 450.00 | 1 | 1 | 200 | 200 | 1600 | 1600 |

A10 | 412.50 | 412.50 | 1 | 1 | 225 | 225 | 1550 | 1550 |

A11 | 375.00 | 375.00 | 1 | 1 | 250 | 250 | 1500 | 1500 |

A12 | 337.50 | 337.50 | 1 | 1 | 275 | 275 | 1450 | 1450 |

A13 | 300.00 | 300.00 | 1 | 1 | 300 | 300 | 1400 | 1400 |

A14 | 262.50 | 262.50 | 1 | 1 | 325 | 325 | 1350 | 1350 |

A15 | 225.00 | 225.00 | 1 | 1 | 350 | 350 | 1300 | 1300 |

A16 | 187.50 | 187.50 | 1 | 1 | 375 | 375 | 1250 | 1250 |

A17 | 150.00 | 150.00 | 1 | 1 | 400 | 400 | 1200 | 1200 |

A18 | 112.50 | 112.50 | 1 | 1 | 425 | 425 | 1150 | 1150 |

A19 | 75.00 | 75.00 | 1 | 1 | 450 | 450 | 1100 | 1100 |

A20 | 37.50 | 37.50 | 1 | 1 | 475 | 475 | 1050 | 1050 |

A21 | 1.00 | 1.00 | 1 | 1 | 500 | 500 | 1000 | 1000 |

${\mathit{\Re}}_{\mathit{i}}^{+}$ | ${\mathit{\Re}}_{\mathit{i}}^{-}$ | ${\mathit{\Re}}_{\mathit{o}}^{+}$ | ${\mathit{\Re}}_{\mathit{o}}^{-}$ | ${\mathit{\hslash}}_{\mathit{i}}^{+}$ | ${\mathit{\hslash}}_{\mathit{i}}^{-}$ | ${\mathit{\chi}}_{\mathit{i}}$ | Rank | |
---|---|---|---|---|---|---|---|---|

A1 | 0.9465 | 0.6685 | 0.6165 | 0.9985 | 0.6514 | 0.669542 | 0.660 | 1 |

A2 | 1.3343 | 0.2807 | 0.4621 | 0.281146 | 0.372 | 3 | ||

A3 | 1.3422 | 0.2729 | 0.4594 | 0.273271 | 0.366 | 7 | ||

A4 | 1.3446 | 0.2704 | 0.4585 | 0.270808 | 0.365 | 11 | ||

A5 | 1.3457 | 0.2693 | 0.4581 | 0.269709 | 0.364 | 14 | ||

A6 | 1.3463 | 0.2688 | 0.4580 | 0.269167 | 0.364 | 17 | ||

A7 | 1.3465 | 0.2685 | 0.4579 | 0.268913 | 0.363 | 19 | ||

A8 | 1.3466 | 0.2684 | 0.4578 | 0.268834 | 0.363 | 21 | ||

A9 | 1.3466 | 0.2685 | 0.4579 | 0.268876 | 0.363 | 20 | ||

A10 | 1.3464 | 0.2686 | 0.4579 | 0.269011 | 0.363 | 18 | ||

A11 | 1.3462 | 0.2688 | 0.4580 | 0.269222 | 0.364 | 16 | ||

A12 | 1.3459 | 0.2691 | 0.4581 | 0.269506 | 0.364 | 15 | ||

A13 | 1.3456 | 0.2695 | 0.4582 | 0.269862 | 0.364 | 13 | ||

A14 | 1.3451 | 0.2699 | 0.4583 | 0.270297 | 0.364 | 12 | ||

A15 | 1.3446 | 0.2704 | 0.4585 | 0.270829 | 0.365 | 10 | ||

A16 | 1.3439 | 0.2711 | 0.4587 | 0.271487 | 0.365 | 9 | ||

A17 | 1.3431 | 0.2719 | 0.4590 | 0.272337 | 0.366 | 8 | ||

A18 | 1.3419 | 0.2731 | 0.4594 | 0.273524 | 0.366 | 6 | ||

A19 | 1.3400 | 0.2751 | 0.4601 | 0.275485 | 0.368 | 5 | ||

A20 | 1.3351 | 0.2800 | 0.4618 | 0.280383 | 0.371 | 4 | ||

A21 | 1.0199 | 0.5951 | 0.6045 | 0.596012 | 0.600 | 2 |

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**MDPI and ACS Style**

Dordevic, M.; Tešić, R.; Todorović, S.; Jokić, M.; Das, D.K.; Stević, Ž.; Vrtagic, S.
Development of Integrated Linear Programming Fuzzy-Rough MCDM Model for Production Optimization. *Axioms* **2022**, *11*, 510.
https://doi.org/10.3390/axioms11100510

**AMA Style**

Dordevic M, Tešić R, Todorović S, Jokić M, Das DK, Stević Ž, Vrtagic S.
Development of Integrated Linear Programming Fuzzy-Rough MCDM Model for Production Optimization. *Axioms*. 2022; 11(10):510.
https://doi.org/10.3390/axioms11100510

**Chicago/Turabian Style**

Dordevic, Milan, Rade Tešić, Srdjan Todorović, Miloš Jokić, Dillip Kumar Das, Željko Stević, and Sabahudin Vrtagic.
2022. "Development of Integrated Linear Programming Fuzzy-Rough MCDM Model for Production Optimization" *Axioms* 11, no. 10: 510.
https://doi.org/10.3390/axioms11100510