Journal of Advanced Transportation

Volume 2018, Article ID 9848104, 11 pages

https://doi.org/10.1155/2018/9848104

## Robust Solution Approach for the Dynamic and Stochastic Vehicle Routing Problem

Correspondence should be addressed to Marcella Bernardo; moc.liamg@gne.odranreballecram

Received 14 August 2017; Revised 8 January 2018; Accepted 30 January 2018; Published 5 March 2018

Academic Editor: Taha H. Rashidi

Copyright © 2018 Marcella Bernardo and Jürgen Pannek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The dynamic and stochastic vehicle routing problem (DSVRP) can be modelled as a stochastic program (SP). In a two-stage SP with recourse model, the first stage minimizes the a priori routing plan cost and the second stage minimizes the cost of corrective actions, performed to deal with changes in the inputs. To deal with the problem, approaches based either on stochastic modelling or on sampling can be applied. Sampling-based methods incorporate stochastic knowledge by generating scenarios set on realizations drawn from distributions. In this paper we proposed a robust solution approach for the capacitated DSVRP based on sampling strategies. We formulated the problem as a two-stage stochastic program model with recourse. In the first stage the a priori routing plan cost is minimized, whereas in the second stage the average of higher moments for the recourse cost calculated via a set of scenarios is minimized. The idea is to include higher moments in the second stage aiming to compute a robust a priori routing plan that minimizes transportation costs while permitting small changes in the demands without changing solution structure. Additionally, the approach allows managers to choose between optimality and robustness, that is, transportation costs and reconfiguration. The computational results on a generic dynamic benchmark dataset show that the robust routing plan can cover unmet demand while incurring little extra costs as compared to the preplanning. We observed that the plan of routes is more robust; that is, not only the expected real cost, but also the increment within the planned cost is lower.

#### 1. Introduction

The basic task in freight transport is to ship goods from one location to another one, which are typically represented by depots and geographically dispersed points, respectively. Hence, a combinatorial optimization problem arises, which is known as vehicle routing problem (VRP). The VRP aims to determine a set of vehicle routes to perform transportation requests with a given vehicle fleet at minimum cost, that is, to decide which vehicle handles which customer order in which sequence. In this kind of problem, one typically assumes that the values of all inputs are known with certainty and do not change. However, in today’s economy, one issue needs to be integrated: customers desire more flexibility and fast fulfillment of their orders. Besides that, the recent developments in information technology permit a growing amount of available data and both control of a vehicle fleet and management of customer orders in real-time. This context calls for real-time decision support in vehicle routing, motivating a version of the VRP, the so-called dynamic and stochastic vehicle routing problem.

The DSVRP is a generalization of the VRP, where parts or all necessary information regarding inputs is stochastic and the true values become available at runtime only. Usually, the dynamic and stochastic VRP is modelled either as a* Markov decision process* or as a* stochastic program* [1]. An MDP consists of a finite set of states, a finite set of actions, representing the nondeterministic choices, and a transition function that given a state and an action provides the probability distribution over the successor states [2]. Differently, SP determines a feasible solution for all possible outcomes [3]. All stochastic program formulations call for the determination of an a priori routing plan. Based on DSVRP models, different solution methods have been developed to address the problem. Usually, solution methods are classified into one of two families:* stochastic modelling* or* sampling*. In* stochastic modelling* approaches, the stochastic knowledge is formally included into the problem formulation, but they are highly technical in their formulation and require to efficiently compute possibly complex expected values. On the other hand,* sampling* has relative simplicity and flexibility on distributional assumptions, while its drawback is the massive generation of scenarios to accurately reflect reality [4, 5]. These approaches sample the probability distributions to generate scenarios that are used to make decisions. Different authors have proposed sampling-based approaches in the context of stochastic VRP, for instance, the Multiple Scenario Approach (MSA) proposed by Bent and Van Hentenryck [6] and Sample Average Approximation (SAA) method applied in Verweij et al. [7].

In this paper, we formulate the dynamic and stochastic capacitated vehicle routing problem, where the demands are stochastic and dynamic, as a two-stage stochastic program model. Similar to the sampling-based methods, we also make use of scenarios in the proposed robust solution approach. However, different from MSA, the scenarios are generated only once at the beginning of the planning stage and, different from SAA, we do not minimize the average of the second stage cost of a set of sample scenarios. The idea of the robust approach is to address uncertainty using higher moments calculated via scenarios, permitting the solution to be able to adapt to situations when the real demand is greater than expected. Our aim is to develop a solution approach such that the routing plan is robust against small changes in the inputs, that is, allowing to compensate for changes in the input without losing structural properties and optimality. For that, the remainder of the paper is organized as follows. Section 2 provides a brief literature review dedicated to vehicle routing problem, with an emphasis on the stochastic and dynamic vehicle routing problem. In Section 3 the problem formulation is described. This is followed in Section 4 by a description of the robust solution approach. Computational results are reported in Section 5, and, last, Section 6 concludes with a summary and an outlook for future work.

#### 2. Literature Review

The VRP is a generalization of the Traveling Salesman Problem (TSP). The TSP is a well-studied problem, in which the goal is to minimize the total distance traveled by the salesman while visiting a group of cities and returning to the first visited city. In this problem each city is visited exactly once by the salesman. The vehicle routing problem, in its turn, consists of designing a routing plan to attend to all customers with a given vehicle fleet at minimum cost. Mathematically, the VRP reads as follows.

*Definition 1 (vehicle routing problem). *Consider a set of vehicles and a fully connected graph , where is the set of vertices representing customer locations ( is the depot of vehicles) and is the set of arcs. With every arc is associated a nonnegative distance matrix . In some contexts, can be interpreted as a travel cost or as a travel time. Moreover, let be a binary variable taking the value 1 if arc is used by vehicle and the value 0 otherwise. Then, we callthe vehicle routing problem.

Within Definition 1, (2) assures that each client is visited only once. Constraint (3) guarantees that vehicles must leave the depot. The initial and termination condition are expressed by (4), insuring that the route starts and ends at the depot. Constraint (5) ensures that the same vehicle comes in and comes out for each one of its customers, and (6) guarantees that all variables are binary. Beyond the VRP classical formulation, a number of side constraints complicate the problem. These could for instance be time constraints on time windows or on capacities of the vehicles, which result in the Vehicle Routing Problem with Time Windows (VRPTW) or the most studied version of the vehicle routing problem, the capacitated vehicle routing problem (CVRP), respectively. In the CVRP a nonnegative demand () is attached to each customer and the sum of demands of any vehicle route may not exceed the vehicle capacity [8]. Thus, the constraint as follows is included in Definition 1:

In contrast to the basic definition of the VRP, most real-life applications have to be analyzed with regard to two aspects: evolution and quality of information [9]. Evolution of information refers to the fact that in some problems the available information may change during runtime. Quality of information indicates possible uncertainty on the available data [5]. Based on these aspects, there are four classes of VRP shown in Table 1.