Time and Timing in Vehicle Routing Problems

Abstract

The distribution of goods to a set of geographically dispersed customers is a common problem faced by carrier companies, well-known as the Vehicle Routing Problem (VRP). The VRP consists of finding an optimal set of routes that minimizes total travel times for a given number of vehicles with a fixed capacity. Given the demand of each customer and a depot, the optimal set of routes should adhere to the following conditions: ?? Each customer is visited exactly once by exactly one vehicle. ?? All vehicle routes start and end at the depot. ?? Every route has a total demand not exceeding the vehicle capacity. The travel times between any two potential locations are given as input to the problem. Consequently, the total travel is computed by summing up the travel time over the chosen routes. In reality, carrier companies are faced with a number of other issues not conveyed in the VRP. The research in this thesis introduces a number of realistic variants of the VRP. These variants consider the VRP as a core component and incorporate additional features. By definition the VRP is NP-hard. Throughout the years a vast amount of research was aimed at developing both exact and heuristic solution procedures. Building on this established literature, solution procedures are developed to fit the variants proposed in this thesis. The standard VRP considers that the travel time between any pair of locations is constant throughout the day. However, congestion is present in most road networks. Considering traffic congestion results in time-dependent travel times, where the travel time between two location depends not only on the distance between them but also on the time of day one chooses to traverse this distance. Time-dependent travel times are considered in Chapters 2 and 3 of this thesis. Thus, in these Chapters we incorporate the time dimension into the VRP. The standard VRP does not take into account any customer service aspect. The customers are presumed to be available to receive their goods upon arrival of the vehicles. However, a number of carrier companies quote their expected arrival time to their customers. We introduce the concept of self-imposed time windows (SITW). SITW reflect the fact that the carrier company decides on when to visit the customer and communicates this to the customer. Once a time window is quoted to a customer the carrier company strives to provide service within this time window. SITW differ from time windows in the widely studied VRP with time windows (VRPTW), as the latter are exogenous constraints. In Chapters 4 and 5 SITW are endogenous decisions in stochastic environments. Thus, in addition to the sequencings decisions required by the VRP further timing decisions are needed. This thesis extends the VRP in two major dimensions: time-dependent travel times and self-imposed time windows. In reality carrier companies are faced with various uncertainties. The presented models incorporated some of these uncertainties by addressing three stochastic aspects: (I) In Chapter 3 stochastic service times are considered. (II) In Chapter 4, stochasticity in travel time is modeled to describes variability caused by random events such as car accidents or vehicle break down. (III) Finally, in Chapter 5 the objective was to construct a long term plan for providing consistent service to reoccurring customers. Stochasticity in this thesis is treated in an a priori manner. The plan, consisting of routes and timing decisions where necessary, is determined beforehand and is not modified according to the realization of the random events. Chapter 2 addresses environmental concerns by studying CO2 emissions in a timedependent vehicle routing problem environment. In addition to the decisions required for the assignment and scheduling of customers to vehicles, the vehicle speed limit is considered. The emissions per kilometer as a function of speed, is a function with a unique minimum speed v*. However, we show that limiting vehicle speed to this v* might be sub-optimal, in terms of total emissions. We adapted a Tabu search procedure for the proposed model. Furthermore, upper and lower bounds on the total amount of emissions that may be saved are presented. Quantifying the tradeoff between minimizing travel time as opposed to CO2 emissions is an important contribution. Another important contribution lies in incorporating fuel costs in the optimization. As fuel costs are correlated with CO2 emissions, Chapter 2 shows that even in today’s cost structure limiting vehicle speeds is beneficial. Chapter 3 defines the perturbed time-dependent VRP (P-TDVRP) model which is designed to handle unexpected delays at the various customer locations. A solution method that combines disruptions in a Tabu Search procedure is proposed. In Chapter 3 we identify situations capable of absorbing delays. i.e. where inserting a delay will lead to an increase in travel time that is less than the delay length itself. Based on this, assumptions with respect to the solution structure of P-TDVRP are formulated and validated. Furthermore, most experiments showed that the additional travel time required by the P-TDVRP, when compared to the travel time required by the TDVRP, was justified. In Chapter 4 the notion of self imposed time windows is defined and embedded in the VRP-SITW model. The objective of this problem is to minimize delay costs (caused by late arrivals at customers) as well as traveling time. The problem is optimized under various disruptions in travel times. The basic mechanism of dealing with these disruptions is allocating time buffers throughout the routes. Thus, additional timing decisions are taken. The time buffers attempt to reduce potential damage of disruptions. The solution approach combines a linear programming model with a local search heuristic. In Chapter 4, two main types of experiments were conducted: one compares the VRP with VRP-SITW while the other compares VRPTW with VRPSITW. The first set of experiments assessed the increase in operational costs caused by incorporating SITW in the VRP. The second set of experiments enabled evaluating the savings in operational costs by using flexible time windows, when compared to the VRPTW. Chapter 5 extends the customer service dimension by considering the consistent vehicle routing problem. Consistency is defined by having the same driver visiting the same customers at roughly the same time. As such, two main dimensions of consistency are identified in the literature, driver- and temporal consistency. In Chapter 5, driver consistency is imposed by having the same driver visit the same customers. Furthermore, we impose temporal consistency by SITW. A stochastic programming formulation is presented for the consistent VRP with stochastic customers. An exact solution method is proposed by adapting the 0-1 integer L- shaped algorithm to the problem. The method was able to solve the majority of test instances to optimality.