Time dependent vehicle routing problem with a multi ant colony system

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Abstract

The Time Dependent Vehicle Routing Problem (TDVRP) consists in optimally routing a fleet of vehicles of fixed capacity when travel times are time dependent, in the sense that the time employed to traverse each given arc, depends on the time of the day the travel starts from its originating node. The optimization method consists in finding solutions that minimize two hierarchical objectives: the number of tours and the total travel time.

Optimization of total travel time is a continuous optimization problem that in our approach is solved by discretizing the time space in a suitable number of subspaces. New time dependent local search procedures are also introduced, as well as conditions that guarantee that feasible moves are sought for in constant time.

This variant of the classic Vehicle Routing Problem is motivated by the fact that in urban contexts variable traffic conditions play an essential role and can not be ignored in order to perform a realistic optimization. In this paper it is shown that when dealing with time constraints, like hard delivery time windows for customers, the known solutions for the classic case become unfeasible and the degree of unfeasibility increases with the variability of traffic conditions, while if no hard time constraints are present, the classic solutions become suboptimal.

Finally an application of the model to a real case is presented. The model is integrated with a robust shortest path algorithm to compute time dependent paths between each customer pairs of the time dependent model.

Introduction

The Vehicle Routing Problem (VRP) has been largely studied because of the importance of mobility in logistic and supply-chains management that relies on road network distribution. Many different variants of this problem have been formulated to provide a suitable application to a variety of real-world cases, with the development of advanced logistic systems and optimization tools. The features that characterize the different variants aim on one hand to take into account the constraints and details of the problem, while on the other to include different aspects of its nature, like its dynamicity, time dependency and/or stochastic aspects. The richness and difficulty of this type of problem, has made the vehicle routing an area of intense investigation.

In this paper we focus on the presence of variable traffic conditions on real road networks, like in urban environments, where these conditions can greatly affect the outcomes of the planned schedule. Accounting for variable travel times is particularly relevant when planning in presence of time constraints, such as delivery time windows. Solutions obtained without considering this variability will result in sub-optimality or unfeasibility with respect to these constraints, as it will be shown in the experimental results section.

This study is also motivated by the recent developments of real time traffic data acquisition systems. With access to these data, it is possible to include in the model dynamic and updated information, and obtain realistic and improved solutions.

The paper is organized as follow: problem formulation and review of the time dependent models; the Multi Ant Colony System is introduced for the classic VRP, and its extension to the time dependent case; the formulation of new time dependent local search procedures and related issues and discussion of issues related to the time dependency; the remainder of the paper is dedicated to computational results and its applications to a real world situation, with the use of real traffic data and integration with a Robust Shortest Path algorithm [1] to deal with realistic graphs representing the urban road network.

Section snippets

Problem description

In the classic VRP with hard time windows, VRPTW, a fleet of vehicles of uniform capacity is scheduled to visit the given set of N customers, ci, each characterized by a demand qi, a time window twi = [bi, ei], and a service time si, with routes originating and ending at a depot, whose opening and closing time [tc, tc] is specified, and a fleet of trucks of uniform capacity C is available. Each delivery can be done no later than the ending time of the customer’s time window, while if the arrival

Review of time dependent VRP models

The presence of diversified conditions of traffic at different times of the day were first taken into account by Malandraki and Daskin in [2] (for the VRP as well as for the TSP). On each arc a step-function distribution of the travel time was introduced. A mixed integer programming approach and a nearest neighbor heuristic were used in the optimization.

Another approach to the time dependent VRP is presented by Ichoua et al. in [3], where the customers are characterized by soft time windows,

Ant colony optimization

Ant Colony Optimization (ACO) was introduced by Dorigo et al. in [4], and it is based on the idea that a large number of simple artificial agents are able to build solutions via low-level based communication, inspired by the collaborative behavior of ant colonies. A variety of ACO algorithms has been proposed for discrete optimization, as discussed in [5], and have been successfully applied to the traveling salesman problem, symmetric and asymmetric [4], [6], [7], [12] the quadratic assignment

The time dependent MACS-VRPTW

It has been shown by Gambardella et al. in [15], that ACO can be used to solve the VRP with hard time windows constraints (VRPTW). This approach consists in using the algorithm called Multi Ants Colony System (MACS-VRPTW) with a hierarchy of two artificial ant colonies, each one dealing with one of the objectives of the optimization: the first colony is named ACS-VEI and deals with tour minimization while ACS-TIME minimizes distance. The two colonies co-operate by exchanging information through

Local search and other considerations

Local search procedures have been proven to be very useful in improving the quality of the solution by evaluating if small modifications can return a better solution. The two basic operations we can perform in a local search procedure applied to the VRP are: (1) insertion of a new delivery in a tour, (2) removal of a delivery from a tour. In the case of the TDVRP, since both operations generate a time shift for all the customers following an insertion or a removal, the travel times from a

Experimental results

Some experiments have been conducted to show some of the behaviors, issues and advantages of the use of this model.

Application to a real road network

In this section we present the application of the MACS-TDVRPTW to a real road network. Real data obtained from the Padua logistic district, in the Veneto region of Italy, are used in this case study.

The customers are a set of nodes that is a subset of all the nodes of the graph representing the road network of Padua. Paths connecting each pair of customers need to be calculated. Since the time dependent nature of this model, these paths are in principle also time dependent.

There are two

Conclusions

We have presented a time dependent model for the vehicle routing problem based on the MACS-VRPTW. The algorithms are supported by enhanced local search procedures, adapted to the time dependent case with a discretization model, to perform efficiently in terms of computation times and quality of the solutions found. Advantages and issues of considering a time dependent model are discussed, as well as the quality and feasibility of the solutions in various cases. In conclusion, time dependent

Acknowledgements

This work was co-funded by the European Commission IST project MOSCA: “Decision Support System For Integrated Door-To-Door Delivery: Planning and Control in Logistic Chain”, grant IST-2000-29557. The information provided is the sole responsibility of the authors and does not reflect the Community’s opinion. The Community is not responsible for any use that might be made of data appearing in this publication.

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