# Using Integrated MMD-TOPSIS to Solve the Supplier Selection and Fair Order Allocation Problem: A Tunisian Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Supplier Selection Problem

#### 2.2. The Fair Order Allocation Problem

#### 2.3. Optimization Techniques in the SSFOAP

## 3. Problem Description and Mathematical Model

#### 3.1. Parameters

$i$ | Index of the supplier (I = 1,…, n) |

$d$ | Item demand |

${p}_{i}$ | Item Unit Price |

${C}_{i}$ | The capacity of supplier $i$ |

$h$ | Holding cost |

$b$ | Shortage cost |

$Q$ | Storage upper bound |

${Q}^{\prime}$ | Shortage lower bound |

$MSI$ | Meaningful suitability index |

BT | Total budget |

#### 3.2. Decision Variables

${x}_{i}$ | Number of products ordered proportionally |

${S}^{+}$ | Product inventory |

${S}^{-}$ | Product shortage |

${\delta}_{i{i}^{\prime}}^{+}$ | The positive threshold for equitable distribution of orders |

${\delta}_{i{i}^{\prime}}^{-}$ | The negative threshold for equitable distribution of orders |

#### 3.3. Mathematical Model

## 4. Integrated BWM-MMD-TOPSIS in SSFOAP

- (1)
- Phase 1: Apply the BWM-MMD-TOPSIS Method to Compute the MSI of Suppliers’ Selection for Cardinal and Ordinal Data.
- (1)
- Step 1. Establish the performance decision-making matrix.
- (2)
- Step 2. The Best–Worst Method (BWM) to evaluate criteria supplier selection.
- (3)
- Step 3. The Cardinal Data–TOPSIS step (CD-TOPSIS method).
- (4)
- Step 4. The Ordinal Data–TOPSIS step (CD-TOPSIS method).
- (5)
- Step 5. Compute the weighted Euclidean distance.
- (6)
- Step 6. Compute the relative closeness coefficients$\text{}{c}_{i}$’s.
- (7)
- Step 7. Rank the suppliers based on the decreasing values of the relative closeness coefficients c
_{i}’s. - (8)
- Step 8. Use the cutoff method to delete suppliers with MSI < ci min.

- (2)
- Phase 2: The Fair Order Allocation

#### 4.1. Problem Statement: A Real Case Study in TSE Company in Tunisia

#### 4.2. The Suggested Integrated Model

#### 4.2.1. Phase 1: Supplier Selection Problem Using BWM and MMD-TOPSIS Method

_{j}as:

- Step1: Building a Performance Decision Matrix

- Step 2: Determine the Criteria Weights by BWM

_{1}= 0.5, w

_{2}= 0.3, and w

_{3}= 0.2, and the CR = 0.1, which indicates a good degree of reliability.

- Step 3: Determine the CD-TOPSIS Step

**Step 4.2.**The meaningful normalization method for ordinal criteria

#### 4.2.2. Phase 2: Fair Order Allocation

#### Database Collect for Bi-Objective LP

#### The $\epsilon $-Constraint Method

## 5. Numerical Experiments

## 6. Discussions

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Papers | Optimization Techniques | Order Allocation | Supplier Selection |
---|---|---|---|

[13] | Multi-objective decision analysis | × | |

[53] | Fuzzy multi-objective linear programming | × | |

[54] | Fuzzy multi-objective linear programming | × | |

[20] | MMD-TOPSIS method | × | |

[24] | A mixed-integer nonlinear programming model | × | |

[25] | Genetic algorithm | × | |

[26] | Fuzzy AHP | × | |

[31] | AHP | × | |

[32] | AHP | × | |

[33] | AHP | × | |

[34] | ANP+TOPSIS + linear programming | × | × |

[35] | ANP | × | |

[36] | TOPSIS | × | |

[37] | TOPSIS | × | |

[41] | VIKOR | × | |

[40] | VIKOR | × | |

[38] | VIKOR | × | |

[39] | VIKOR | × | |

[45] | Genetic algorithm | × | |

[47] | ANP + MOLP | × | × |

[48] | Multi-objective linear programming | × | |

[55] | QFD + TOPSIS | × | |

[50] | Fuzzy TOPSIS + Goal programming | × | × |

[51] | AHP, ARAS, and MCGP | × | × |

[52] | fuzzy goal programming + IF-TOPSIS | × | × |

Ordinal Criteria | Cardinal Criteria | ||
---|---|---|---|

Suppliers/Criteria | Quality | Flexibility | Delivery |

Scap | VG | VG | 4 |

Camilec | VG | G | 5 |

Siala | VG | G | 1 |

Compto | G | G | 3 |

Dcbel | G | I | 3 |

Best Criterion | Quality | Flexibility | Delivery |
---|---|---|---|

Quality | 1 | 2 | 2 |

Worst Criterion | Quality | Flexibility | Delivery |
---|---|---|---|

Delivery | 3 | 2 | 1 |

Suppliers | S-Score | I-Score | ||
---|---|---|---|---|

Quality | Flexibility | Quality | Flexibility | |

Scap | 2 | 4 | 0 | 0 |

Camilec | 2 | 1 | 0 | 1 |

Siala | 2 | 1 | 0 | 1 |

Compto | 0 | 1 | 3 | 1 |

Dcbel | 0 | 0 | 3 | 4 |

Ordinal Data | Cardinal Data | ||
---|---|---|---|

Quality | Flexibility | Delivery | |

Scap | 1 | 1 | 0.42 |

Camilec | 1 | 0.5 | 0.28 |

Siala | 1 | 0.5 | 0.85 |

Compto | 0 | 0.5 | 0.57 |

Dcbel | 0 | 0.25 | 0.57 |

Suppliers | Separation Measures | |
---|---|---|

${\mathit{d}}^{++}$ | ${\mathit{d}}^{--}$ | |

Scap | 0.317 | 0.8688 |

Camilec | 0.4533 | 0.7573 |

Siala | 0.2382 | 0.8756 |

Compto | 0.7781 | 0.3840 |

Dcbel | 0.7985 | 0.3316 |

Suppliers | Ci | Pi |
---|---|---|

Scap | 18,000 | 2200 |

Camilec | 13,000 | 2300 |

Siala | 15,000 | 2400 |

Compto | 16,000 | 2350 |

Dcbel | 9000 | 2150 |

Suppliers | Scap | Camelic | Siala | Compto | Dcbel |
---|---|---|---|---|---|

Meaningful suitability index | 0.73 | 0.62 | 0.79 | 0.33 | 0.29 |

Optimal fair order allocation | 7961.232 | 6761.594 | 8615.580 | 3598.913 | 3162.681 |

Suppliers | Scap | Camelic | Siala |
---|---|---|---|

Meaningful suitability index | 0.73 | 0.62 | 0.79 |

Optimal fair order allocation | 10,267.76 | 8720.561 | 11,111.68 |

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**MDPI and ACS Style**

Aouadni, S.; Euchi, J.
Using Integrated MMD-TOPSIS to Solve the Supplier Selection and Fair Order Allocation Problem: A Tunisian Case Study. *Logistics* **2022**, *6*, 8.
https://doi.org/10.3390/logistics6010008

**AMA Style**

Aouadni S, Euchi J.
Using Integrated MMD-TOPSIS to Solve the Supplier Selection and Fair Order Allocation Problem: A Tunisian Case Study. *Logistics*. 2022; 6(1):8.
https://doi.org/10.3390/logistics6010008

**Chicago/Turabian Style**

Aouadni, Sourour, and Jalel Euchi.
2022. "Using Integrated MMD-TOPSIS to Solve the Supplier Selection and Fair Order Allocation Problem: A Tunisian Case Study" *Logistics* 6, no. 1: 8.
https://doi.org/10.3390/logistics6010008