Journal of Combinatorial Optimization

, Volume 31, Issue 2, pp 515–532

# Multi-depot vehicle routing problem with time windows under shared depot resources

Article

## Abstract

A new variant of multi-depot vehicle routing problem with time windows is studied. In the new variant, the depot where the vehicle ends is flexible, namely, it is not entirely the same as the depot that it starts from. An integer programming model is formulated with the minimum total traveling cost under the constrains of time window, capacity and route duration of the vehicle, the fleet size and the number of parking spaces of each depot. As the problem is an NP-Hard problem, a hybrid genetic algorithm with adaptive local search is proposed to solve it. Finally, the computational results show that the proposed method is competitive in terms of solution quality. Compared with the classic MDVRPTW, allowing flexible choice of the stop depot can further reduce total traveling cost.

### Keywords

Multi-depot vehicle routing problem Time windows Hybrid genetic algorithm Adaptive local search

### Sets

$$D$$

The depot set

$$C$$

The customer set

$$V$$

The vertex set

$$A$$

The arc set

$$K$$

The vehicle set

$$K_{d }$$

The set of vehicles at depot $$d$$

### Parameters

$$q_{i}$$

The demand of customer $$i$$

$$s_{i}$$

The service time of customer $$i$$

$$e_{i}$$

The earliest start time at customer $$i$$

$$l_{i}$$

The latest start time at customer $$i$$

$$c_{ij}$$

The distance between customer $$i$$ and $$j$$

$$Q_{k}$$

The capacity of the vehicle $$k$$

$$T_{k}$$

The maximum duration of vehicle $$k$$

$$L$$

The number of vehicles

$${\vert }P_{d}{\vert }$$

The number of parking spaces at the depot $$d$$

$${\vert }K_{d}{\vert }$$

The number of vehicles in $$K_{d }$$ set

### Variables

$$x_{kij}$$

Is the 0-1 decision variable, if vehicle $$k$$ travels directly from node $$i$$ to node $$j$$, then $$x_{kij}=1$$, otherwise, $$x_{kij}=0$$

$$a_{ki}$$

The arrival time of vehicle $$k$$ at node $$i$$

$$b_{ki}$$

The start service time of vehicle $$k$$ at node $$i$$

$$\pi _k$$

The working duration of vehicle $$k$$

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