Algorithm for directing cooperative vehicles of a vehicle routing problem for improving fault-tolerance

Abstract

This paper presents a new algorithm for solving capacitated single depot vehicle routing problems without time windows in a fault-tolerant manner where the distances between points are symmetrical. The sudden breakdown of a vehicle on its pre-planned route is considered as a fault. In this case one of the other cooperating vehicles will take over the remaining tasks and perform them in the most effective way. We introduce the concept of “route change cost” as the difference between the length of the modified and the original routes of the helper vehicle, and our goal is to minimize this value. Starting from an arbitrary VRP solution, we determine the optimal direction, in terms of the route change cost, in which each route should be performed by a vehicle. A simulation with case studies was used to test the algorithm. We found that applying the proposed method, the execution of the original tasks requires routes with lengths 4.1–8.6% shorter on average than using the non-directed routes of the solution of the original vehicle routing problem in arbitrary directions.

Appendices

Appendix 1: Clarke and Wright savings method

The Clarke and Wright savings method is an efficient, popular and widespread method for creating the routes of a VRP problem. The proposed algorithm of this paper can be applied for an arbitrary VRP solution (see Sect. 3.3). We chose the Clarke and Wright savings method for initial VRP solution creation because of its simplicity and wide usage in the VRP literature. Its main steps are as follows.

Step 1::

Calculate the saving value $S(m_{i}, m_{j}) = L(d, m_{i}) + L(d, m_{j}) - L(m_{i}, m_{j})$ for every pair $(m_{i}, m_{j})$ of customers.

Step 2::

Order the saving values in descending order. This results in the “savings list”. Start the processing of the saving list with its first element.

Step 3::

For the saving value $S(m_{i}, m_{j})$ under consideration, include link $(m_{i}, m_{j})$ in a route if no route constraints will be violated, and if:

• Either, neither $m_{i}$ nor $m_{j}$ have already been assigned to a route. In this case a new route is created including both $m_{i}$ and $m_{j}$.

• Or, exactly one of the two customers ($m_{i}$ or $m_{j}$) has already been included in an existing route and that customer is not interior to that route—the concept “interior to a route” means that the customer is not adjacent to the depot—, in which case the link $(m_{i}, m_{j})$ is added to that same route.

• Or, both $m_{i}$ and $m_{j}$ have already been included in two different existing routes and neither customer is interior to its route. In this case the two routes are merged.

Step 4::

If the end of the saving list has not been reached, process its next entry returning to Step 3; otherwise, stop. If there is any customer that has not been assigned to a route, it must be assigned to its own route, which begins at the depot, visits the customer (only one) and returns to the depot.

Appendix 2: The proposed algorithm

The flowchart of our algorithm is illustrated in Fig. 8.

Fig. 8

Our algorithm /the tasks with white background belong to the input creation phase/