# The Design of a Multi-Period and Multi-Echelon Perishable Goods Supply Network under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Review of Literature

## 3. Model Formulation

- determining the location of manufacturers, retailers, and distributors;
- establishing coordination between network levels to minimize cost and procurement time;
- creating a balance between reducing the costs and procurement time, and maximizing customer satisfaction.

#### 3.1. Mathematical Model

#### 3.1.1. Assumptions

- A time window is determined for each customer.
- A similar vehicle is not used for transportation.
- The time duration is one week.
- Manufacturers and distribution centers have limited capacity.
- Suppliers and retailers are allowed to store goods and can order more than needed.
- Various products are considered.
- Retailers and distribution centers cannot return their orders.
- Each section has a particular travel cost and distance.
- All demands of the customer must be met on the assumption that they can buy from multiple distributors.
- Fixed cost of the equipment should be considered for internal transportation costs.

#### 3.1.2. Indices

#### 3.1.3. Parameters

#### 3.1.4. Variables

#### 3.1.5. Objective Functions

#### 3.1.6. Constraints

## 4. Solution Approach

Algorithm 1 The Hybrid Algorithm |

1. Step (0): Initialize ${\hat{\mathsf{\eta}}}_{\mathrm{t}+1}=0$ 2. While Stopping index not satisfied do 3. Forward Pass 4. For $\mathrm{t}=1,\text{}2,\dots ,\text{}\mathrm{T}$ do 5. Step (1): Solve the t-stage optimization problem ${\hat{\mathrm{M}}}_{\mathrm{t}}$ 6. end for 7. Step (2): Update lower bound 8. Backward Pass 9. For $\mathrm{t}=\mathrm{T},\text{}\mathrm{T}-1,\text{}\dots ,\text{}2$ do 10. Step (3): With the stored ${\mathsf{\lambda}}_{\mathrm{j},\mathrm{t}}$ values from the forward pass, solve the t-stage Lagrangian problem ${\mathsf{\varphi}}_{\mathrm{t}}$ for optimal Lagrangian multipliers. 11. Step (4): Calculate a new Benders optimality cut for stage t using the Lagrangian multipliers and objective function value obtained from Step (3). 12. end for 13. Step (5): Solve the first-stage problem ${\hat{\mathrm{M}}}_{1}$ 14. Step (6): Calculate upper bound 15. Step (7): Increase the iteration count 16. end while 17. Step (8): Finish |

#### 4.1. Benders Sub-Problem

#### 4.2. Benders Main-Problem

## 5. Results

- The first model:

- The second model:

- The third model:

## 6. Conclusions and Discussions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The equilibrium between model stability and Z1 and Z2 functions. (

**a**) The total loss expected value; (

**b**) The expected value of cumulative maximum deficit.

Reference | Model Type | Structure | Location | Routing | Shortage | Uncertainty | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

LP | MILP | MINLP | Multi-Objective | Multi-Product | Multi-Period | Fuzzy | Stochastic | Robust | ||||

Baboli et al. (2011) [21] | ✓ | ✕ | ✕ | ✕ | ✓ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ | ✕ |

Kelle et al. (2012) [22] | ✕ | ✓ | ✕ | ✕ | ✓ | ✓ | ✕ | ✕ | ✓ | ✕ | ✓ | ✕ |

Uthayakumar et al. (2013) [23] | ✕ | ✕ | ✓ | ✕ | ✓ | ✕ | ✕ | ✕ | ✓ | ✕ | ✓ | ✕ |

Mousazadeh et al. (2015) [24] | ✕ | ✓ | ✕ | ✓ | ✓ | ✕ | ✓ | ✕ | ✕ | ✓ | ✕ | ✕ |

Zahiri et al. (2017) [25] | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✓ | ✕ | ✕ | ✓ | ✕ | ✕ |

Zahiri et al. (2018) [26] | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✓ | ✕ | ✕ | ✕ | ✕ | ✓ |

Sabouhi et al. (2018) [27] | ✕ | ✓ | ✕ | ✕ | ✓ | ✕ | ✓ | ✕ | ✕ | ✕ | ✓ | ✕ |

Singh and Goh (2019) [28] | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✕ | ✓ | ✕ | ✓ | ✕ | ✕ |

Savadkoohi et al. (2019) [29] | ✕ | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✕ | ✓ | ✓ | ✕ | ✕ |

Nasrollahi and Razmi (2019) [30] | ✕ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ | ✕ | ✓ | |

Akbarpour et al. (2020) [31] | ✕ | ✕ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ | ✓ | ✕ | ✓ | ✕ |

This Research | ✕ | ✓ | ✕ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✕ | ✓ | ✕ |

Product | Factory | Period | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | ||

1 | 1 | 19.3 | 352 | 304 | |||||||||

2 | 202 | 823 | 755 | 659 | 331 | 437 | 212 | ||||||

3 | 238 | 499 | 586 | 124 | 327 | 266 | |||||||

2 | 1 | 915 | 1132 | 650 | 991 | 228 | 194 | 53 | 92 | ||||

2 | 454 | 191 | 213 | 562 | 687 | 544 | 862 | 928 | |||||

3 | 42 | 574 | 301 | 931 | 672 | 563 | |||||||

3 | 1 | 275 | 225 | 241 | 122 | 107 | 109 | 534 | 321 | 403 | |||

2 | 200 | 970 | 768 | 830 | 160 | 90 | 300 | 620 | 593 | 550 | |||

3 | 39.6 | 102 | 780 | 250 | |||||||||

4 | 1 | 482 | 283 | 729 | 407 | 130 | 237 | 353 | |||||

2 | 767 | 516 | 655 | 626 | 930 | 280 | 427 | 387 | 833 | ||||

3 | 226 | 200 | 107 | ||||||||||

5 | 1 | 742 | 1090 | 717 | 1047 | 991 | 753 | 483 | 183 | 353 | 370 | ||

230 | 170 | 277 | |||||||||||

2 | 399 | 5.8 | 73.4 | 367 | 164 | 320 | 66.5 | ||||||

280 | 170 | ||||||||||||

3 | 70 | 170 | 150 | 333 | 598 | 277 | 20 | 349 |

Section | Product | Factory | Period (t) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

Transport | 1 | 1 | 6 | 2 | 1 | |||||||||

2 | 5 | 5 | 5 | 2 | ||||||||||

3 | 8 | 8 | 8 | 3 | ||||||||||

2 | 1 | 6 | 4 | 1 | 1 | 1 | 1 | 1 | ||||||

2 | 10 | 10 | 4 | 2 | 2 | 2 | ||||||||

3 | 22 | 17 | 10 | 10 | 10 | 10 | 10 | 7 | 7 | 5 | 4 | 4 | ||

3 | 2 | 15 | 8 | 8 | 88 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |

3 | 5 | |||||||||||||

4 | 1 | 3 | 2 | |||||||||||

5 | 1 | 6 | 7 | 9 | 9 | 6 | 4 | 4 | 4 | 3 | 3 | 3 | 3 | |

Customer | 1 | 1 | 4 | 1 | 1 | |||||||||

2 | 3 | 2 | ||||||||||||

3 | 5 | 3 | ||||||||||||

2 | 1 | 2 | 3 | 1 | ||||||||||

2 | 6 | 2 | 2 | |||||||||||

3 | 5 | 7 | 3 | 2 | 1 | |||||||||

3 | 1 | 6 | ||||||||||||

2 | 7 | |||||||||||||

3 | 15 | 5 | ||||||||||||

4 | 1 | 6 | 3 | |||||||||||

2 | 3 | 2 | ||||||||||||

5 | 1 | 3 | 2 | 1 | ||||||||||

2 | 10 | |||||||||||||

5 | 1 | 1 | 2 |

ω | Echelon Upgrade | Factory | Period t | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

500 | 2 → 5 | 2 | 4 | |||||||||||

3 → 5 | 2 | 7 | ||||||||||||

1500 | 2 → 3 | 2 | 2 | |||||||||||

2 → 3 | 3 | 11 | ||||||||||||

3 → 5 | 1 | 3 | ||||||||||||

2000 | 2 → 3 | 3 | 12 | |||||||||||

2 → 4 | 1 | 3 | ||||||||||||

3500 | 1 → 3 | 3 | 4 | |||||||||||

2 → 3 | 3 | 11 |

Iteration | Time (S) | Total Cost | Greenhouse Gases Cost | Transportation Cost | Shortage Cost | Fixed and Operational Cost |
---|---|---|---|---|---|---|

1 | 123 | 992,805 | 263,960 | 155,490 | 18,135 | 424,180 |

2 | 139 | 945,580 | 242,290 | 154,800 | 15,765 | 408,225 |

3 | 180 | 963,829.4 | 246,500 | 157,030 | 7229.4 | 423,960 |

4 | 324 | 944,579 | 253,110 | 152,060 | 2974 | 407,805 |

5 | 360 | 968,527 | 252,900 | 154,830 | 21,877 | 411,730 |

6 | 346 | 967,735 | 244,840 | 156,350 | 15,765 | 423,200 |

7 | 460 | 958,540 | 251,140 | 155,030 | 13,685 | 411,635 |

8 | 2170 | 945,948 | 242,830 | 154,180 | 4048 | 418,470 |

9 | 6410 | 934,048 | 239,610 | 150,340 | 8881 | 408,957 |

10 | 7854 | 981,056 | 251,920 | 157,850 | 23,816 | 419,070 |

Expected Value | - | 960,264.7 | 248,910 | 154,796 | 13,217.54 | 41,5723.2 |

Standard Deviation | - | 18,211.31 | 7178.2805 | 2235.85 | 7213.927 | 6760.7712 |

Iteration | OF 1 | OF 2 | OF 3 | OF 4 |
---|---|---|---|---|

1 | 1.3438 × 10^{14} | 1.1292 × 10^{11} | 9.2604 × 10^{9} | 140,628 |

2 | 3.7610 × 10^{14} | 3.4441 × 10^{11} | 5.6963 × 10^{9} | 333,747 |

3 | 3.7399 × 10^{14} | 3.4251 × 10^{11} | 5.7955 × 10^{9} | 335,087 |

4 | 3.7570 × 10^{14} | 3.4357 × 10^{11} | 5.7742 × 10^{9} | 331,030 |

5 | 3.7495 × 10^{14} | 3.4303 × 10^{11} | 5.6723 × 10^{9} | 325,840 |

6 | 3.7550 × 10^{14} | 3.4394 × 10^{11} | 5.6812 × 10^{9} | 334,992 |

7 | 3.7652 × 10^{14} | 3.4473 × 10^{11} | 5.7443 × 10^{9} | 328,672 |

8 | 3.7499 × 10^{14} | 3.4343 × 10^{11} | 5.7880 × 10^{9} | 344,037 |

9 | 3.7441 × 10^{14} | 3.4221 × 10^{11} | 5.6980 × 10^{9} | 322,564 |

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## Share and Cite

**MDPI and ACS Style**

Meidute-Kavaliauskiene, I.; Yıldırım, F.; Ghorbani, S.; Činčikaitė, R.
The Design of a Multi-Period and Multi-Echelon Perishable Goods Supply Network under Uncertainty. *Sustainability* **2022**, *14*, 2472.
https://doi.org/10.3390/su14042472

**AMA Style**

Meidute-Kavaliauskiene I, Yıldırım F, Ghorbani S, Činčikaitė R.
The Design of a Multi-Period and Multi-Echelon Perishable Goods Supply Network under Uncertainty. *Sustainability*. 2022; 14(4):2472.
https://doi.org/10.3390/su14042472

**Chicago/Turabian Style**

Meidute-Kavaliauskiene, Ieva, Figen Yıldırım, Shahryar Ghorbani, and Renata Činčikaitė.
2022. "The Design of a Multi-Period and Multi-Echelon Perishable Goods Supply Network under Uncertainty" *Sustainability* 14, no. 4: 2472.
https://doi.org/10.3390/su14042472