# An Ant Colony Algorithm for Time-Dependent Vehicle Routing Problem with Time Windows

Conference paper

## Summary

In this paper, we address the Vehicle Routing Problem with Time Windows, both time-independent and -dependent cases. In the timeindependent case, our objective is to minimize the total distance. To solve this problem, we propose an Ant Colony Optimization algorithm. Then we implement the algorithm to solve the time-dependent case where the objective is to minimize the total tour time. The time dependency is embedded in this model by using a deterministic travel speed function which is a step function of the time of the day. An experimental evaluation of the proposed approach is performed on the well-known benchmark problems Optimizing a distribution network has been and remains an important challenge both in the literature and in real-life applications and the routing of a eet of vehicles is the most widely addressed problem in a distribution network. The Vehicle Routing Problem (VRP) determines a set of vehicle routes originating and terminating at a single depot such that all customers are visited exactly once and the total demand of the customers assigned to each route does not violate the capacity of the vehicle. The objective is to minimize the total distance traveled. An implicit primary objective is to use the least number of vehicles. The Vehicle Routing Problem with Time Windows (VRPTW) is a variant of VRP in which lower and upper limits are imposed to the delivery time of each customer. The arrival at a customer outside the specified delivery time is either penalized (soft time windows) or strictly forbidden (hard time windows). The interested reader is referred to [1] for more details on VRPTW. In the Stochastic Vehicle Routing Problem, the customer demands and/or the travel times between the customers may vary. Although stochastic travel times and demand distributions have been frequently used in the literature, time-varying travel speeds and time-dependent VRPTW (TDVRPTW) have seldom been addressed. In the literature, time dependency is taken into consideration in two ways: stochastic travel times and deterministic travel times. First introduced by [2], stochastic travel times are mainly examined by [4] and [3]. [5] proposeda deterministic travel time based model in which the important nonpassing property is introduced. [6] and [7] also use deterministic travel times in a setting where the day is divided into time intervals.

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