# A Metaheuristic Based Approach for the Customer-Centric Perishable Food Distribution Problem

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

- Most of the problems studied in the literature are bi-objective focusing on cost minimization, and freshness maximization or service level improvement.
- There are a few studies that propose multi-objective models that take into account simultaneously the economic, social, and environmental aspects.
- Those reviewed works considering the service level improvement objective, evaluate the quality of service with respect to the satisfaction of the time window constraint. However, the service level can be related to other criteria.

## 3. Model Formulation

#### 3.1. The Optimization Goal Setting

**Objective 1: Minimize the total cost**

**The fixed costs**The fixed costs of a vehicle are not related to the mileage, and represent the maintenance, and depreciation costs. Assuming that the depot has k refrigerated trucks to provide distribution services for the set of customers.F represents the fixed cost related to a vehicle. The fixed costs can be formulated as:$${C}_{1}=F\sum _{k\in K}{y}_{0}^{k}$$**The transportation costs**We denote these costs by ${C}_{2}$. They are proportional to the vehicle mileage. We consider only the fuel consumption cost and express it as:$${C}_{2}=\sum _{k\in K}\sum _{i\in V}\sum _{j\in V}{C}_{(i,j)}{x}_{(i,j)}^{k}{d}_{(i,j)}$$**The refrigeration costs**Include two types of costs: the costs incurred by the vehicle’s energy usage to maintain a specific temperature in the process of transportation, in addition to the costs of extra energy during the unloading.The refrigeration costs during transportation process denoted ${C}_{3}^{1}$ are expressed as follows:$${C}_{3}^{1}={C}_{e}\sum _{k\in K}\sum _{i\in V}\sum _{j\in V}{x}_{(i,j)}^{k}{T}_{(i,j)}$$The cost of energy supplied during the unloading ${C}_{3}^{2}$ is expressed as:$${C}_{3}^{2}={C}_{e}^{\prime}\sum _{k\in K}\sum _{i\in \mathcal{C}}{y}_{i}^{k}{U}_{i}$$$${C}_{3}={C}_{3}^{1}+{C}_{3}^{2}$$**The damage costs**The quality of perishable foods decay with the time extension, and the temperature changes during the transportation and handling process. If product quality falls to a certain level, damage costs are incurred. The quality of refrigerated goods can be expressed using the following function [28]:$${D}_{t}={D}_{0}{e}^{-\mathit{\partial}t}$$$${C}_{4}^{1}=\sum _{k\in K}\sum _{i\in \mathcal{C}}{y}_{i}^{k}P{q}_{i}\left(1-{e}^{-{\mathit{\partial}}_{1}\left({t}_{i}^{k}-{t}_{0}^{k}\right)}\right)$$The damage cost during the unloading ${C}_{4}^{2}$ is defined as:$${C}_{4}^{2}=\sum _{k\in K}\sum _{i\in \mathcal{C}}{y}_{i}^{k}P{q}_{in}\left(1-{e}^{-{\mathit{\partial}}_{2}{S}_{i}}\right)$$With ${q}_{in}$ the remaining quantity of product after servicing the customer i, the necessary time to serve is customer i is ${S}_{i}$, and ${\mathit{\partial}}_{2}$ is the spoilage rate when the vehicle is opened.The total damage cost is therefore:$${C}_{4}={C}_{4}^{1}+{C}_{4}^{2}$$Given the cost components defined above, the total cost can be formulated as:$${Z}_{1}={C}_{1}+{C}_{2}+{C}_{3}+{C}_{4}$$**Objective 2: Maximize the average freshness**

**Objective 3: Maximize the service level**

**Objective 4: Minimize the total tardiness**

#### 3.2. Mathematical Model

## 4. The Solving Strategy

## 5. General Variable Neighborhood Search Heuristic

#### 5.1. Variable Neighborhood Search (VNS)

#### 5.2. GVNS Implementation Details

Algorithm 1: GVNS |

## 6. GVNS Performance Evaluation

#### 6.1. Data Description

#### 6.2. Computational Experiments

## 7. Application Example

#### 7.1. Data and Parameter Setting

#### 7.2. Generation of Alternative Solutions

#### 7.3. Decision Maker Preferences and Ranking of Solutions

- Economic-centric (E-c): Cost ${\u2ab0}_{p}$ Average freshness ${\u2ab0}_{p}$ Service level ${\u2ab0}_{p}$ Tardiness.
- Product-centric (P-c): Average freshness ${\u2ab0}_{p}$ Cost ${\u2ab0}_{p}$ Service level ${\u2ab0}_{p}$ Tardiness.
- Customer Satisfaction-centric (C-c): Service level ${\u2ab0}_{p}$ Cost ${\u2ab0}_{p}$ Average freshness ${\u2ab0}_{p}$ Tardiness.

## 8. Conclusions and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The General Variable Neighborhood Search GVNS

#### Appendix A.1. Initial Solution Generation

#### Appendix A.2. The Neighbourhood Structures

- Relocate: In this operator the customer visit is moved from a route to another.
- GENI: Is a variant of the relocate operator which consists of placing a customer two customer nodes on the closest destination path, even though they are not consecutive.
- CROSS: The key concept behind cross exchange is to remove two edges $(k,k+1)$ and $(i-1,i)$ from the first route, and remove $(l,l+1)$ and $(j-1,j)$ from the second route. Then, by adding the edges $(i-1,j)$,$(j-1,i)$, $(k,l+1)$, and $(l,k+1)$ the segments $i-k$ and $j-l$ are swapped.
- 2-OPT: Replace two of the tour’s edges with two additional edges, and iterates until no further change is necessary.

## Appendix B. A Possibility Degree Based Approach to Rank Solutions

c_{1} | c_{2} | … | c_{n} | |

A_{1} | x_{11} | x_{12} | … | x_{1n} |

A_{2} | x_{21} | x_{22} | … | x_{2n} |

A_{m} | x_{m1} | x_{m2} | … | x_{mn} |

#### Appendix B.1. Decision Matrix Normalization

#### Appendix B.2. Intervals Calculations

#### Appendix B.3. Reference Interval and Interval’s Comparison

- 1.
- if $X\cap Y=\varnothing $$$P(X\ge Y)=\left\{\begin{array}{cc}0\hfill & \phantom{\rule{1.em}{0ex}}{x}_{r}\le {y}_{l}\hfill \\ 1\hfill & \phantom{\rule{1.em}{0ex}}{x}_{l}\ge {y}_{r}\hfill \end{array}\right.$$
- 2.
- if $X\cap Y\ne \varnothing $

#### Appendix B.4. Ranking of Alternatives

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References | Sustainability Aspect | Sustainability Measure | Objective Functions | Methods | ||
---|---|---|---|---|---|---|

Economic | Environmental | Social | ||||

[6] | * | * | Distribution cost Time window respect | Min. total cost | Heuristics | |

[7] | * | Distribution cost | Min. travel cost | Heuristics | ||

[8] | * | Distribution cost | Min. travel cost | Heuristics | ||

[9] | * | Distribution cost | Min. total distance | Heuristics+ Exact | ||

[10] | * | Distribution cost | Min. total cost | Metaheuristics | ||

[11] | * | Profitability | Max. profit | Heuristics | ||

[12] | * | Distribution cost | Min. total cost | Heuristics | ||

[13] | * | * | Distribution cost Time window respect | Min. total cost | Heuristic | |

[14] | * | * | Distribution cost Products quality | Min. total cost | Heuristics | |

[15] | * | * | Cost efficiency Time window respect | Min. total distance | Exact+ Heuristics | |

[16] | * | * | Cost efficiency Products quality Time window respect | Max. profit | Exact+ Heuristics | |

[17] | * | * | Cost efficiency Products quality | Max. profit | Exact | |

[18] | * | * | * | Cost efficiency Time window respect Carbon emission | Min. total travel time | Simulation |

[19] | * | * | Distribution cost Time window respect | Min. total cost | Exact | |

[20] | * | * | Distribution cost Time window respect | Min. total cost | Exact | |

[21] | * | * | Distribution cost Products quality | Min. total cost | Heuristic | |

[22] | * | * | Cost efficiency Products freshness Time window respect | Min. total cost Max. freshness | Mulit-objective evolutionary algorithm | |

[23] | * | * | * | Cost efficiency Products quality Time window respect Environmental impact | Min. total cost Min. carbon emissions | Metaheuristics+ Pareto optimality |

[24] | * | Distribution cost | Min. total cost | Evolutionary computation | ||

[25] | * | * | Distribution cost Time window respect | Min. total cost Min. variance of vehicles | Multi-objective Gradient Evolutionary algorithm | |

[26] | * | * | * | Cost efficiency Service level Environmental impact | Min. total cost Min. waiting time Min. carbon emissions | NSGA-II |

[27] | * | * | * | Distribution cost Products freshness Time window respect Environmental impact | Min. operational cost Min. deterioration cost Min. carbon emission Max. service level | Many-objective Gradient Evolutionary algorithm |

Sets and Indices | |

V | Set of nodes |

$\mathcal{C}$ | Set of customers |

K | Set of vehicles $k\in K$ |

i,j | Indices of nodes |

Decision Variables | |

${x}_{(i,j)}^{k}$ | A 0-1 decision variable, equal to 1 in case the truck k travels from node i to j, 0 otherwise. |

${y}_{i}^{k}$ | A 0-1 decision variable, equal to 1 in case the customer i is served by the truck k, 0 otherwise. |

${t}_{i}^{k}$ | A decision variable representing the starting service time at node i using the truck k. |

Parameters | |

${C}_{(i,j)}$ | The transportation cost from node i to j. |

F | Fixed cost associated to a vehicle. |

${T}_{(i,j)}$ | The travel time from i to j. |

${d}_{(i,j)}$ | The distance between node i and j. |

${C}_{e}$ | The cost per unit time for the refrigeration during the transportation process. |

${U}_{i}$ | The unloading time at customer i. |

${C}_{e}^{\prime}$ | The unit refrigeration cost during the unloading. |

${S}_{i}$ | The necessary time to serve customer i |

${q}_{i}$ | The demand of customer i |

P | The price per unit product |

${q}_{in}$ | The products remaining on the vehicle after serving the customer i |

Q | The loading capacity of trucks. |

K | The fleet of trucks |

${\mathit{\partial}}_{1}$ | The spoilage rate of products during the transportation process |

${\mathit{\partial}}_{2}$ | The spoilage rate of products during the unloading process |

$[{e}_{i},{l}_{i}]$ | The time window of customer i |

${T}_{g}^{i}$ | The target time of customer i |

$S{L}_{i}\left(t\right)$ | The service level for customer i if we deliver their demand at time t. |

M | Large positive constant. |

Instance | GVNS Solution | CPLEX Solution | Gap (%) | ||
---|---|---|---|---|---|

Objective | Time (s) | Objective | Time (s) | ||

R101-10 | 630.52 | 0.04 | 630.520 | 1.76 | 0 |

R101-20 | 1224.18 | 3.59 | 1224.18 | 7.15 | 0 |

R101-30 | 1665.78 | 3.64 | 1664.10 | 11.20 | 0.1 |

R101-40 | 2223.10 | 6.51 | 2223.10 | 363.00 | 0 |

R101-50 | 2600.48 | 14.36 | 2589.92 | 858.00 | 0.4 |

R101-60 | 2969.64 | 26.44 | No Sol | ||

R101-70 | 3543.04 | 35.74 | No Sol | ||

R101-80 | 3794.16 | 51.77 | No Sol | ||

R101-90 | 4159.70 | 84.99 | No Sol | ||

R101-100 | 4398.88 | 111.70 | No Sol | ||

RC107-10 | 436.32 | 0.06 | 436.32 | 1.74 | 0 |

RC107-20 | 771.70 | 1.45 | No Sol | ||

RC107-30 | 1147.52 | 4.09 | No Sol | ||

RC107-40 | 1469.18 | 8.21 | No Sol | ||

RC107-50 | 1860.02 | 20.33 | No Sol | ||

RC107-60 | 2434.92 | 45.78 | No Sol | ||

RC107-70 | 2657.12 | 180.10 | No Sol | ||

RC107-80 | 3023.60 | 249.30 | No Sol | ||

RC107-90 | 3401.92 | 358.50 | No Sol | ||

RC107-100 | 3591.20 | 589.60 | No Sol | ||

C104-10 | 1068.34 | 0.01 | 1068.34 | 0.65 | 0 |

C104-20 | 2121.10 | 5.02 | 2118.96 | 15.15 | 0.1 |

C104-30 | 3111.16 | 23.75 | No Sol | ||

C104-40 | 4279.34 | 51.13 | No Sol | ||

C104-50 | 5228.74 | 75.10 | No Sol | ||

C104-60 | 6341.10 | 69.57 | No Sol | ||

C104-70 | 7572.30 | 151.30 | No Sol | ||

C104-80 | 8591.68 | 590.80 | No Sol | ||

C104-90 | 9636.78 | 777.90 | No Sol | ||

C104-100 | 10740.60 | 943.40 | No Sol |

Parameter | Value |
---|---|

P | 20 EUR/Unit |

${\mathit{\partial}}_{1}$ | 0.002 |

${\mathit{\partial}}_{2}$ | 0.003 |

${C}_{e}$ | 0.03 EUR/unit |

${C}_{e}^{\prime}$ | 0.04 EUR/unit |

$\alpha $ | 0.8 |

Total Cost | Average Freshness | Service Level | Tardiness | |
---|---|---|---|---|

${S}_{1}$ | 18,294.37 | 62.17 | 29.02 | 1,455,295 |

${S}_{2}$ | 18,323.90 | 58.64 | 25.71 | 1,621,316 |

${S}_{3}$ | 18,256.38 | 62.02 | 30.06 | 1,498,330 |

${S}_{4}$ | 18,120.14 | 61.03 | 28.32 | 1,716,339 |

${S}_{5}$ | 18,277.78 | 62.12 | 30.71 | 1,328,918 |

${S}_{6}$ | 18,011.75 | 61.74 | 27.91 | 1,504,620 |

${S}_{7}$ | 18,179.60 | 61.08 | 27.16 | 1,467,776 |

${S}_{8}$ | 18,013.66 | 61.76 | 27.51 | 1,480,522 |

${S}_{9}$ | 17,966.30 | 61.66 | 26.87 | 1,562,182 |

Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

E-c | ${S}_{5}$ | ${S}_{3}$ | ${S}_{6}$ | ${S}_{8}$ | ${S}_{1}$ | ${S}_{4}$ | ${S}_{9}$ | ${S}_{7}$ | ${S}_{2}$ |

P-c | ${S}_{5}$ | ${S}_{1}$ | ${S}_{3}$ | ${S}_{8}$ | ${S}_{6}$ | ${S}_{7}$ | ${S}_{9}$ | ${S}_{4}$ | ${S}_{2}$ |

C-c | ${S}_{5}$ | ${S}_{3}$ | ${S}_{1}$ | ${S}_{6}$ | ${S}_{8}$ | ${S}_{7}$ | ${S}_{4}$ | ${S}_{9}$ | ${S}_{2}$ |

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## Share and Cite

**MDPI and ACS Style**

El Raoui, H.; Oudani, M.; Pelta, D.A.; El Hilali Alaoui, A.
A Metaheuristic Based Approach for the Customer-Centric Perishable Food Distribution Problem. *Electronics* **2021**, *10*, 2018.
https://doi.org/10.3390/electronics10162018

**AMA Style**

El Raoui H, Oudani M, Pelta DA, El Hilali Alaoui A.
A Metaheuristic Based Approach for the Customer-Centric Perishable Food Distribution Problem. *Electronics*. 2021; 10(16):2018.
https://doi.org/10.3390/electronics10162018

**Chicago/Turabian Style**

El Raoui, Hanane, Mustapha Oudani, David A. Pelta, and Ahmed El Hilali Alaoui.
2021. "A Metaheuristic Based Approach for the Customer-Centric Perishable Food Distribution Problem" *Electronics* 10, no. 16: 2018.
https://doi.org/10.3390/electronics10162018