# An Improved Distribution Policy with a Maintenance Aspect for an Urban Logistic Problem

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## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. Context and Motivation

_{2}emission, reducing congestion and regulating delivery traffic, which received the project partners’ approval.

#### 2.2. Parameters

- $N$: number of potential UDCs to subcontract
- $H$: number of periods
- ${C}_{t}$: cost of transportation for one product
- ${C}_{t{r}_{i}}$: cost to contract the distribution of one product out to UDC
_{i}. - ${C}_{t{p}_{i}}$: cost of pollution generated by contracting one product out to UDC
_{i}. - ${C}_{p}$: cost of one product delayed
- ${Y}_{it}$: maximal number of products to be delivered by platform i in each period t
- ${d}_{t}$: demand in each period
- ${C}_{\mathrm{max}}$: maximum capacity that can be delivered by the main platform
- ${D}_{t}$: the sum of the demand and the delayed products.
- ${L}_{p}$: the amount of decreased capacity due to a preventive maintenance action
- ${L}_{r}$: the amount of decreased capacity due to a corrective maintenance action
- ${M}_{p}$: preventive maintenance cost
- ${M}_{c}$: corrective maintenance cost
- ${\lambda}_{t}(k)$: the failure rate
- $\mathrm{\Delta}t$: the length of distribution period
- ${\lambda}_{n}(t)$: the nominal failure rate
- mu: monetary unit

- ${x}_{t}$: quantity transported in each period
- ${y}_{it}$: quantity of products subcontracted to platform i in each period t
- ${Q}_{t}$: quantity of products delayed in each period t
- $A{F}_{it}$: allocated platform i in each period t
- $P$: optimal number of preventive maintenance actions
- T: the optimal period of intervention
- ${\zeta}_{M}(x,P)$: the excepted number of failures

#### 2.3. Problem Formulation

_{i}. The cost to transport one product for the concerned UDC to the allocated urban area is ${C}_{t}$. The cost of subcontracting one product to each UDC I is ${C}_{t{r}_{i}}$; it depends on the distance between the subcontractor and the urban area.

## 3. Mathematical Model

#### 3.1. Mathematical Model of Transportation with the Subcontracting Strategy

#### 3.2. Maintenance Cost

_{p}= a × C

_{max}and L

_{r}= b × C

_{max}[18].

_{t}is given by:

## 4. Numerical Example

- Step 1.
- This consists of resolving the mathematical model of transportation using subcontracting strategy without taking into account the maintenance operations, namely the UDC delivers with the maximum capacity. To that end, we use the mixed integer solver FICO Xpress 8.0 (FICO, San Jose, CA, USA).
- Step 2.
- This consists of defining the optimal maintenance plan according to the distribution plan given by Step 1 by determining the optimal number of preventive maintenance actions and then determining the available capacities in each period. Therefore, we use the mathematical symbolic computation MATHEMATICA (11.1.1, Wolfram Research, Oxfordshire, UK).
- Step 3.
- This uses the results from Step 2 to incorporate these available capacities in our previous model in order to determine the new distribution plan. To do that, we use the same solver, FICO Xpress 8.0.

#### 4.1. Step 1: Transportation with the Subcontracting Strategy Plan

_{max}= 20. The number of periods H is equal to 10 days.

_{i}.

#### 4.2. Step 2: Maintenance Strategy

_{p}= 220 mu and M

_{c}= 1000 mu. In order to solve the maintenance cost, we use the software MATHEMATICA (11.1.1, Wolfram Research, Oxfordshire, UK).

_{p}= 1 (a = 6%) and L

_{r}= 5 (b = 35%) define capacities lost with respect to the delivery rate due respectively to the preventive maintenance actions and minimal repair actions. Table 7 shows the available delivery rate in each period taking into account the maintenance actions’ effects.

#### 4.3. Step 3: The Final Distribution Plan Using the Subcontracting Strategy and Available Capacities

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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d_{1} | d_{2} | d_{3} | d_{4} | d_{5} | d_{6} | d_{7} | d_{8} | d_{9} | d_{10} |
---|---|---|---|---|---|---|---|---|---|

25 | 17 | 35 | 35 | 22 | 25 | 16 | 20 | 17 | 25 |

Y_{1,1} | Y_{1,2} | Y_{1,3} | Y_{1,4} | Y_{1,5} | Y_{1,6} | Y_{1,7} | Y_{1,8} | Y_{1,9} | Y_{1,10} |
---|---|---|---|---|---|---|---|---|---|

4 | 5 | 1 | 7 | 2 | 5 | 2 | 3 | 0 | 1 |

Y_{2,1} | Y_{2,2} | Y_{2,3} | Y_{2,4} | Y_{2,5} | Y_{2,6} | Y_{2,7} | Y_{2,8} | Y_{2,9} | Y_{2,10} |
---|---|---|---|---|---|---|---|---|---|

0 | 5 | 3 | 8 | 4 | 6 | 9 | 1 | 2 | 0 |

Y_{3,1} | Y_{3,2} | Y_{3,3} | Y_{3,4} | Y_{3,5} | Y_{3,6} | Y_{3,7} | Y_{3,8} | Y_{3,9} | Y_{3,10} |
---|---|---|---|---|---|---|---|---|---|

7 | 0 | 8 | 0 | 2 | 5 | 3 | 1 | 2 | 2 |

Items | UDC_{1} | UDC_{2} | UDC_{3} |
---|---|---|---|

Unit service cost | 15 mu | 20 mu | 10 mu |

Unit emission cost | 2 mu | 1 mu | 4 mu |

Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Demand | 25 | 17 | 35 | 35 | 22 | 25 | 16 | 20 | 17 | 25 |

Distribution plan | 20 | 17 | 20 | 20 | 20 | 20 | 16 | 20 | 17 | 20 |

Allocation to UDC_{1} | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 |

Quantity subcontracted to UDC_{1} | 0 | 0 | 1 | 7 | 2 | 0 | 0 | 0 | 0 | 1 |

Allocation to UDC_{2} | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |

Quantity subcontracted to UDC_{2} | 0 | 0 | 3 | 8 | 1 | 0 | 0 | 0 | 0 | 0 |

Allocation to UDC_{3} | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 | 1 |

Quantity subcontracted to UDC_{3} | 5 | 0 | 8 | 0 | 2 | 5 | 0 | 0 | 0 | 2 |

Delayed quantity | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 2 |

C_{1} | C_{2} | C_{3} | C_{4} | C_{5} | C_{6} | C_{7} | C_{8} | C_{9} | C_{10} |
---|---|---|---|---|---|---|---|---|---|

19 | 18 | 16 | 19 | 18 | 16 | 19 | 18 | 16 | 19 |

Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Demand | 25 | 17 | 35 | 35 | 22 | 25 | 16 | 20 | 17 | 25 |

Distribution plan | 19 | 17 | 15 | 19 | 18 | 16 | 19 | 18 | 16 | 19 |

Allocation to UDC_{1} | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 |

Quantity subcontracted to UDC_{1} | 0 | 0 | 1 | 7 | 2 | 5 | 0 | 1 | 0 | 1 |

Allocation to UDC_{2} | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |

Quantity subcontracted to UDC_{2} | 0 | 0 | 3 | 8 | 4 | 3 | 0 | 0 | 0 | 0 |

Allocation to UDC_{3} | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |

Quantity subcontracted to UDC_{3} | 6 | 0 | 8 | 0 | 2 | 5 | 0 | 1 | 1 | 2 |

Delayed quantity | 0 | 0 | 7 | 8 | 4 | 0 | 0 | 0 | 0 | 3 |

Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Demand | 25 | 17 | 35 | 35 | 22 | 25 | 16 | 20 | 17 | 25 |

Distribution plan | 19 | 17 | 15 | 19 | 18 | 16 | 19 | 18 | 16 | 19 |

Allocation to UDC_{1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Allocation to UDC_{2} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Allocation to UDC_{3} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Delayed quantity | 6 | 5 | 24 | 40 | 44 | 53 | 50 | 52 | 53 | 59 |

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**MDPI and ACS Style**

Ndhaief, N.; Bistorin, O.; Rezg, N. An Improved Distribution Policy with a Maintenance Aspect for an Urban Logistic Problem. *Appl. Sci.* **2017**, *7*, 703.
https://doi.org/10.3390/app7070703

**AMA Style**

Ndhaief N, Bistorin O, Rezg N. An Improved Distribution Policy with a Maintenance Aspect for an Urban Logistic Problem. *Applied Sciences*. 2017; 7(7):703.
https://doi.org/10.3390/app7070703

**Chicago/Turabian Style**

Ndhaief, Nadia, Olivier Bistorin, and Nidhal Rezg. 2017. "An Improved Distribution Policy with a Maintenance Aspect for an Urban Logistic Problem" *Applied Sciences* 7, no. 7: 703.
https://doi.org/10.3390/app7070703