# Bi-Level Multi-Objective Production Planning Problem with Multi-Choice Parameters: A Fuzzy Goal Programming Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Bi-level Programming

#### 2.2. Multi-choice Programming Problem

#### 2.3. Production Planning

## 3. Statement of the Model

- The model has multi-objectives, where we maximize profit, product liability, quality, and workers’ satisfaction in the industry.
- The multi-item production model is to be considered.
- One machine cannot perform more than one operation at a time.
- There is no shortage of materials in production.
- Demand should be only for final products.
- Machine and storage capacity cannot exceed the maximum level in any case.

**Nomenclature**

**Indices:**

**k**—Index for multi-choices, k = 1, 2,…, K**j**—Index for the manufactured product, j = 1, 2,…, J**i**—Index for machines i = 1,2,…,I**l**—Index for the level, l = 1,2**m**—Index for the objectives, m = 1,2,…,M

**Decision Variable:**

- ${x}_{j}$—Manufactured items

**Parameters:**

- ${P}_{j}$—Profit related to the product
- ${L}_{j}$—Liability of the product
- ${Q}_{j}$—Quality of the product
- ${W}_{j}$—Workers’ satisfaction
- ${m}_{j}$—Milling machine time on jth product
- $M$—The total available time of milling machine
- ${l}_{j}$—Lathe time on jth product
- $L$—The total available time of lathe
- ${g}_{j}$—Grinder time on jth product
- $G$—The total available time of grinder
- ${s}_{j}$—Jig saw time on jth product
- $S$—The total available time of jig saw
- ${d}_{j}$—Drill press time on jth product
- $D$—The total available time of drill press
- ${b}_{j}$—Band saw time on jth product
- $B$—The total available time of band saw

**I**is related to maximizing the profit:

**II**is related to maximizing product reliability:

**III**is related to maximizing the quality of the product:

**IV**is related to maximizing the workers’ satisfaction:

**I**is related to the milling of a product.

**II**is related to the lathe of a product.

**III**is related to the grinder of a product.

**IV**is related to the jigsaw of a product.

**V**is related to the drill press of a product.

**VI**is related to the band saw of a product.

## 4. General Formulation of Bi-Level Multi-Objective Programming Problem (BLMOPP)

_{1}and n

_{2}are the number of controllable variables in the first and second level, respectively, such that ${X}_{1}=({x}_{11},{x}_{12,\dots ,}{x}_{1{n}_{1}})$ and ${X}_{2}=({x}_{21},{x}_{22,\dots ,}{x}_{2{n}_{2}}).$ So, the BLMOPP of maximization/minimization type may be formulated as follows:

**[Ist level]**

**[IInd level]**

## 5. BLMOPP with Multi-Choices Interval-Type

**Case I.**In this case, we have considered a BLMOPP in which all cost coefficients ${C}_{lmj}^{k}(k=1,2,\dots K;l=1,2;m=1,2,\dots M;j=1,2,\dots ,J)$ are multi-choice interval type. Then the model is;

**Case II.**In this case, we have considered a BLMOPP where, ${a}_{ij}$ (i = 1 2,…,I; j = 1,2,…,J) are in the multi-choice interval types. After that, the model is rewritten as:

**Case III.**In this case, we have considered a BLMOPP where ${b}_{i}(i=1,2,\dots ,I)$ are in multi-choice interval type. The model is further rewritten as:

**Case IV.**In the final case, we consider a BLMOPP where all the parameters of the problem $({C}_{lmj},{a}_{ij},{b}_{i})$ are in the multi-choice interval type. The mathematical model of the problem given by:

## 6. Fuzzy goal Programming Formulation of BLMOPP with A Multi-Choice Interval Type

**Step 1:**Formulate the bi-level multi-level production planning problem with multi-choice interval type parameters.

**Step 2:**As explained in Section 5, convert the multi-choice interval type parameters into their equivalent deterministic form.

**Step 3:**Solve each objective function as a solitary objective problem using only one objective function at a time. The solutions thus obtained are considered to be ideal solutions for the objectives.

**Step 4:**Construct the payoff matrix using the ideal solution and then calculate the value of all other objective functions, respectively.

**Step 5:**Determine the best and worst solution of each objective functions from the payoff matrix.

**Step 6:**Construct the membership function for each objective as mention below:

**Step 7:**After getting the optimal solution from the first level, now we have to move to the second level by constructing the membership function for the decision variables and second level objection functions.

**Step 8:**In this approach, the over and under deviations are required to minimize the aspiration level of fuzzy goals. Therefore, considering the goals have the same priority in the achievement function, an equivalent fuzzy bi-level multi-objective linear goal programming model can be proposed as:

## 7. Numerical Illustration

**[Ist Level]**

**[IInd Level]**

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Machine Type | Available Time | Machine time | ||
---|---|---|---|---|

Product (1) | Product (2) | Product (3) | ||

Milling machine (m) | 1400 | 12 | 17 | 0 |

Lathe (l) | 1000 | 3 | 9 | 8 |

Grinder (g) | 1750 | 10 | 13 | 15 |

Jig saw (s) | 1325 | 6 | 0 | 16 |

Drill press (d) | 900 | 0 | 12 | 7 |

Band saw (b) | 1075 | 9.5 | 9.5 | 4 |

Profit (P) | 50 | 100 | 17.5 | |

Product liability (L) | 0.72 | 0.85 | 0.78 | |

Quality (Q) | 92 | 75 | 50 | |

Workers’ satisfaction (W) | 25 | 100 | 75 |

Machine Type | Available Time | Machine Time | ||
---|---|---|---|---|

Product (1) | Product (2) | Product (3) | ||

Milling machine | [1200,1400] or [1400,1600] | [10,12] or [12,14] | [15,17] or [17,19] or [19,21] | --- |

Lathe | [800,1000] or [1000,1200] or [1200,1400] | [3,5] or [5,4] | [7,9] or [9,11] or [11,13] | [6,8] or [8,10] |

Grinder | [1650,1750] or [1750,1850] or [1850,1950] or [1950,2050] | [8,10] or [10,12] or [12,14] | [13,15] or [15,17] | [15,17] or [17,19] |

Jig saw | [1225,1325] or [1325,1425] | [4,6] or [6,8] | --- | [12,14] or [14,16] or [16,18] |

Drill press | [700.900] or [900,110] or [1100,1300] | --- | [10,12] or [12,14] | [5,7] or [7,9] or [9,11] |

Band saw | [1075,1275] or [1275,1475] or [1475,1675] | [9.5,11.5] or [11.5,13.5] | [9.5,11.5] or [11.5,13.5] | [4,6] or [6,8] or [8,10] or [10,12] |

Profit | [40,50] or [50,60] or [60,70] | [90,100] or [100,110] or [110,120] | [16.5,17.5] or [17.5,18.5] | |

Product liability | [0.70,0.72] or [0.72,0.74] or [0.74,0.76] | [0.81,0.85] or [0.85,0.89] | [0.75,0.78] or [0.78,0.81] or [0.81,0.84] or [0.84,0.87] | |

Quality | [82,92] or [92,102] | [65,75] or [75,85] or [85,95] or [95,105] | [40,50] or [50,60] or [60,70] | |

Workers’ satisfaction | [15,25] or [25,35] | [90,100] or [100,110] or [110,120] or [120,130] | [65,75] or [75,85] |

Cases | FGP | Bi-Level FGP | |
---|---|---|---|

Ist Level | IInd Level | ||

I | Z_{1} = 8878.5, Z_{2} = 0.84536,Z _{3} = 10786, Z_{4} = 11185x _{1} = 28, x_{2} = 56, x_{3} = 41 | Z_{1} = 6079.50, Z_{2} = 0.77886,x _{1} = 86, x_{2} = 7, x_{3} = 47 | Z_{1} = 4929.5, Z_{2} = 0.801329, Z_{3} = 9516, Z_{4} = 8510x _{1} = 53, x_{2} = 8, x_{3} = 67 |

II | Z_{1} = 10,792, Z_{2} = 0.82057,Z _{3} = 10,634, Z_{4} = 12350x _{1} = 17, x_{2} = 75, x_{3} = 49 | Z_{1} = 5762.5, Z_{2} = 0.77532,x _{1} = 79, x_{2} = 0, x_{3} = 81 | Z_{1} = 5772.5, Z_{2} = 0.77611, Z_{3} = 11,624, Z_{4} = 8110x _{1} = 77, x_{2} = 2, x_{3} = 81 |

III | Z_{1} = 9575, Z_{2} = 0.80512,Z _{3} = 10,256, Z_{4} = 9335x _{1} = 43, x_{2} = 56, x_{3} = 28 | Z_{1} = 7437.5, Z_{2} = 0.76719,x _{1} = 110, x_{2} = 5, x_{3} = 45 | Z_{1} = 7187.5, Z_{2} = 0.76943, Z_{3} = 12,589, Z_{4} = 6885x _{1} = 102, x_{2} = 6, x_{3} = 49 |

IV | Z_{1} = 11,403, Z_{2} = 0.83419,Z _{3} = 10,929.99, Z_{4} = 10,102.83x _{1} = 48, x_{2} = 61, x_{3} = 58 | Z_{1} = 6645.50, Z_{2} = 0.79027,x _{1} = 97, x_{2} = 2, x_{3} = 83 | Z_{1} = 5697.5, Z_{2} = 0.80734, Z_{3} = 12,307, Z_{4} = 10,000x _{1} = 71, x_{2} = 3, x_{3} = 95 |

Trace Value | FGP | BL-FGP |
---|---|---|

Case I | 30,850.34 | 22,956.30 |

Case II | 33,776.82 | 25,507.28 |

Case III | 29,166.80 | 26,662.27 |

Case IV | 32,436.65 | 28,005.31 |

Case | Classical Goal Programming |
---|---|

I | Z_{1} = 7090.00, Z_{2} = 0.867600, Z_{3} = 8232.00, Z_{4} = 9325.00x _{1} = 21, x_{2} = 52, x_{3} = 48 |

II | Z_{1} = 9152.50, Z_{2} = 0.924801, Z_{3} = 9104.00, Z_{4} = 11,325.0x _{1} = 12, x_{2} = 78, x_{3} = 43 |

III | Z_{1} = 9027.50, Z_{2} = 0.904300, Z_{3} = 10,549.00, Z_{4} = 9750.00x _{1} = 47, x_{2} = 61, x_{3} = 33 |

IV | Z_{1} = 11,894.00, Z_{2} = 0.812350, Z_{3} = 10,989.00, Z_{4} = 11,000.00x _{1} = 42, x_{2} = 59, x_{3} = 54 |

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## Share and Cite

**MDPI and ACS Style**

Kamal, M.; Gupta, S.; Chatterjee, P.; Pamucar, D.; Stevic, Z.
Bi-Level Multi-Objective Production Planning Problem with Multi-Choice Parameters: A Fuzzy Goal Programming Algorithm. *Algorithms* **2019**, *12*, 143.
https://doi.org/10.3390/a12070143

**AMA Style**

Kamal M, Gupta S, Chatterjee P, Pamucar D, Stevic Z.
Bi-Level Multi-Objective Production Planning Problem with Multi-Choice Parameters: A Fuzzy Goal Programming Algorithm. *Algorithms*. 2019; 12(7):143.
https://doi.org/10.3390/a12070143

**Chicago/Turabian Style**

Kamal, Murshid, Srikant Gupta, Prasenjit Chatterjee, Dragan Pamucar, and Zeljko Stevic.
2019. "Bi-Level Multi-Objective Production Planning Problem with Multi-Choice Parameters: A Fuzzy Goal Programming Algorithm" *Algorithms* 12, no. 7: 143.
https://doi.org/10.3390/a12070143