Discrete Optimization
Vehicle routing problem with time windows and a limited number of vehicles

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Abstract

This paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given (m-VRPTW). Under this scenario, a feasible solution is one that may contain either unserved customers and/or relaxed time windows.

We provide a computable upper bound to the problem. To solve the problem, we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route. We also allow time windows to be relaxed by introducing the notion of penalty for lateness. In our approach, customer jobs are inserted based on a hierarchical objective function that captures multiple objectives.

Computational results on benchmark problems show that our approach yields solutions that are competitive to best-published results on VRPTW. On m-VRPTW instances, experiments show that our approach produces solutions that are very close to computed upper bounds. Moreover, as the number of vehicles decreases, the routes become more densely packed monotically. This shows that our approach is good from both the optimality as well as stability point of view.

Introduction

Many practical transport logistics and distribution problems can be formulated as a vehicle routing problem whose objective is to obtain a minimum-cost route plan serving a set of customers with known demands. Each customer is assigned to exactly one vehicle route and the total demand of any route must not exceed the vehicle capacity.

To date, most of the proposed algorithms assume that the number of vehicles is unlimited, and the objective is to obtain a solution that either minimizes the number of vehicles and/or total travel cost. However, transport operators in the real world face resource constraints such as a fixed fleet. The question we like to ask is, if the given problem is over-constrained in the sense of insufficient vehicles, what constitutes a good solution and how may we find one?

In this paper, we provide some insights to this question. In our view point, it is desirable to have an algorithm that not only performs well given a standard VRPTW problem, but also handles over-constrained problems well in the following sense:

  • 1.

    Optimality: It returns solutions which serve (or pack) as many customers as possible as the primary objective, while optimizing standard criteria such as the number of vehicles and distance travelled.

  • 2.

    Stability: It degrades gracefully under constrainedness, i.e. when the number of vehicles is reduced, the customer packing density, defined as the average number of customers per vehicle in service, must be monotically increasing, although the total number of customers served will become less.


This paper proceeds as follows. We first introduce the problem (m-VRPTW) and a computable upper bound to the problem. We then present a tabu search approach with the following characteristics: (a) a holding list to accommodate unserved customers; (b) a mechanism that introduces new vehicles in stages so as to force denser customer packing within a route. We then extend the algorithm to a generalization of the problem with relaxed time windows.

In terms of computational results, experiments on VRPTW benchmark problems show that our approach can produce solutions that are very close to previous best-published results. What is more interesting perhaps is the performance on m-VRPTW instances. Results show that our approach produces solutions that are very close to computed upper bounds. Moreover, as the number of vehicles is reduced, the average number of customers per route is monotically increasing. This shows that our approach is good from both the optimality as well as stability point of view.

Section snippets

Literature review

The primary objective of m-VRPTW is to maximize the number of customers served, which is NP-hard, since finding it is a generalization of the multiple constrained knapsack problem. Although the classical VRPTW has been the subject of intensive research since the 80s, to our knowledge, there has been little research work on m-VRPTW.

We give some research developments in VRPTW. Solomon’s insertion heuristics [18] is the seminal work behind heuristic construction algorithms. Many efficient

Problem definition and notation

The standard VRPTW problem is defined formally as: Given an undirected graph G(V,A) where V={v0,v1,…,vn}, v0 is the depot, vi, i≠0 is a customer with demand di, time windows (ei,li) and service duration si; A={(vi,vj):ij, vi,vjV}, each arc (vi,vj) having a travel distance (time) tij; and vehicle capacity Q, find a minimum set of vertex-disjoint routes starting and ending at depot v0 such that each customer vi is served by exactly one vehicle within its time windows, ∑di for all customers vi

Upper bound for m-VRPTW

In this section, we determine an upper bound for the total number of customers that can be served by a given fixed number of vehicles. We propose an integer programming (IP) formulation. The IP formulation should be able to solve large-scale problems, yet not be overly simplified such that the gap of the bound from the optimum is too wide. We have adopted a formulation that accounts for the capacity constraints of the vehicles as well as the time constraints imposed by the latest return times

Standard two-phase method

Most of recently published VRPTW heuristics are two-phase approaches. First, a construction heuristic is used to generate a feasible and as good as possible initial solution. Then, an iterative improvement heuristic is applied to this solution. It generates successive solutions by searching the neighborhood of the current solution. In the second phase, various methods are then used to prevent the algorithms from being trapped at local optimal and to explore a larger search space.

The

Proposed algorithm

The above-mentioned two-phase method would normally have to work differently for an over-constrained problem. One way is to use the insertion heuristic to determine whether the problem instance is feasible. Following which, we have two sets of heuristics to handle separately the infeasible case and the feasible case. Another approach is to increase enough vehicles so as to serve all customers, and then, through subsequent heuristics, try to obtain a subset of the solution that maximizes the

Results and analysis

In this section, we present experimental results of applying our algorithm to solve both the standard VRPTW as well as m-VRPTW problems. In our experiments, we set the tabu length to be 100. We set the values of CountLimit and StepSize in Algorithm A to be 500 and 1 respectively. We refer to our implementation as the OV method.

Conclusion

In this paper, we considered a variant of VRPTW constrained by a limited vehicle fleet, which we believe is a more realistic problem in logistics. We presented an analytical upper bound for that formulation, and showed that our tabu search approach came fairly close to the upper bound. This algorithm is also good from the stability point of view. We also showed that the same algorithm could be used to give reasonably good results for the standard VRPTW problem. Hence, in terms of achievements,

Acknowledgements

We like to thank Prof Sun Jie for his guidance and comments on early drafts of this paper.

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