Chapter 3 DYNAMIC AND STOCHASTIC VEHICLE ROUTING IN PRACTICE

4OR, 2009

Assigning and scheduling vehicle routes in a stochastic time-dependent environment is a crucial management problem. The assumption that in a real-life environment everything goes according to an a priori determined static schedule is unrealistic. Our methodology builds on earlier work in which the traffic congestion is captured in an analytical way using queueing theory. The congestion is then applied to the VRP problem. In this paper, we introduce the variability in traffic flows into the model. This allows for an evaluation of the routes based on the uncertainty involved. Different experiments show that the risk taking behavior of the planner can be taken into account during optimization. As more weight is given to the variability component, the resulting optimal route will take a slightly longer travel time, but will be more reliable. We propose a powerful objective function that is easily implemented and that captures the trade-off between the average travel time and its variance. The evaluation of the solution is done in terms of the 95th-percentile of the travel time distribution (assumed to be lognormal), which reflects well the quality of the solution in this stochastic time-dependent environment. Vehicle routing • Time-dependent travel times • Travel time reliability MSC classification (2000) 90B06 • 90B15

International Journal of Computer Mathematics - IJCM, 2004

Vehicle routing problems (VRPs) are resource management problems where the aim is to use the limited number of resources to a large number of jobs so that a maximum number of jobs can be completed with minimum cost. These problems are made complicated by the inclusion of temporal and technological constraints. These problems belong to the class of nondeterministic polynominal-time complete (NP) problems. This paper describes the application of stochastic techniques, namely simulated annealing (SA) and genetic algorithm (GA), to solve VRPs. It is found empirically that SA gives better results than GA for all randomly generated under-, critically- and over-resourced VRPs in almost all cases.

International Journal of Industrial Engineering & Production Research, 2013

This article proposes a stochastic vehicle routing problem within the frame-wok of chance constrained programming where one or more parameters are presumed to be random variables with known distribution function. The reality is that once we convert some special form of probabilistic constraint into their equivalent deterministic form then a nonlinear constraint generates. Knowing that reliable computer software for large scaled complex nonlinear programming problem with 0-1 type decision variables for stochastic vehicle routing problem is not easily available merely then the value of an approximation technique becomes imperative. In this article, theorems which build a foundation for moving toward the development of an approximate methodology for solving the stochastic vehicle routing problem are stated and proved. Using these theorems one can easily convert a nonlinear type vehicle routing problem of special type into an equivalently designed linear problem that can be solved fast ...

EURO Journal on Transportation and Logistics, 2017

Building on the work of Gendreau et al. (Eur J Oper Res 88(1):3-12; 1996), we review the past 20 years of scientific literature on stochastic vehicle routing problems. The numerous variants of the problem that have been studied in the literature are described and categorized.

European Journal of Operational Research, 1996

The purpose of this review article is to provide a summary of the scientific literature on stochastic vehicle routing problems. The main problems are described within a broad classification scheme and the most important contributions are summarized in table form.

2004

In order to solve real-world vehicle routing problems, the standard Vehicle Routing Problem (VRP) model usually needs to be extended. Here we consider the case when customers can call in orders during the daily operations, i.e. both customer locations and demand may be unknown in advance. Our heuristic approach attempts to minimize the expected number of vehicles and their travel

arXiv: Optimization and Control, 2018

This paper considers the vehicle routing problem with stochastic demands (VRPSD) under optimal restocking. We develop an exact algorithm that is effective for solving instances with many vehicles and few customers per route. In our experiments, we show that in these instances solving the stochastic problem is most relevant (i.e., the potential gains over the deterministic equivalent solution are highest). The proposed branch-price-and-cut algorithm relies on an efficient labeling procedure, exact and heuristic dominance rules, and completion bounds to price profitable columns. Instances with up to 76 nodes could be solved in less than 5 hours, and instances with up to 148 nodes could be solved in long-runs of the algorithm. The experiments also allowed new findings on the problem. Solving the stochastic problem leads to solutions up to 10% superior to the deterministic equivalent solution. When the number of routes is not fixed, the optimal solutions under detour-to-depot and optima...

XI Conference of …, 2009

Vehicle Routing Problems (VRPs) cover a wide range of well-known NP-hard problems where the aim is to serve a set of customers with a fleet of vehicles under certain constraints. Literature contains several approaches -coming from different fields like Operations Research, Artificial Intelligence and Computer Science-which try to get good (near-optimal) solutions for small-, mid-and large-size instances. The Vehicle Routing Problem with Stochastic Demands (CVRPSD) is a particular case of VRP where demands made by clients are random, which introduces uncertainty in the problem. Thus, a good aprioristic solution may become unfeasible during the delivery phase if total demand in a route exceeds total vehicle capacity. This paper presents a flexible approach for the CVRPSD, which is based on the combined use of Monte Carlo simulation and reliability indices. Our methodology provides a set of alternative solutions for a given CVRPSD instance. These solutions depend on a parameter which controls the probability of suffering route failures during the actual delivering phase. A numerical example illustrates the methodology and its potential applications.

W e consider stochastic vehicle routing problems on a network with random travel and service times. A fleet of one or more vehicles is available to be routed through the network to service each node. Two versions of the model are developed based on alternative objective functions. We provide bounds on optimal objective function values and conditions under which reductions to simpler models can be made. Our solution method embeds a branch-and-cut scheme within a Monte Carlo sampling-based procedure. Introduction We consider a stochastic vehicle routing problem (SVRP) that consists of planning optimal vehicle routes to service a number of locations in the presence of random travel and service times. The system is modeled using a network whose arcs have non-negative random travel times and whose nodes have nonnegative random service times, with distributions assumed to be known. Vehicles are uncapacitated and routes for each vehicle begin and end at a specific depot node. A route is defined as the set of arcs followed by a vehicle and the set of nodes it services. These routes are selected before knowing the random travel and service times and so that each node will be serviced. After the routes have been planned and realizations of the random travel and service times become known, the actual time to complete each route can be computed. The vehicles must follow their a pri-ori routes; no route reoptimizations are permitted. The time at which the final vehicle returns to the depot, after all nodes have been serviced, is called the completion time. We consider two models with different objectives: The first minimizes the expected completion time and the second maximizes the probability that the operation is complete on or before a prespecified target time, T. Applications of the SVRP occur in a variety of fields. For example, Lambert et al. (1993) design routes for vehicles to collect deposits from bank branches and deliver them to a central office. In this paper, we study properties of the SVRP and present solution methods. In the remainder of this section, we briefly review related work. Section 2 provides a formal problem statement. In §3, we study properties of the SVRP: relating the models under the alternative objective functions, developing bounds, and providing conditions under which reductions to simpler deterministic models can be made. Solution methods and computational results are discussed in §4, and the paper is summarized in §5. Jaillet (1985, 1988) considers a probabilistic travel-ing salesman problem (TSP) with random demand. An a priori tour is constructed that includes all potential customers, and after observing the subset that requires service, the other customers are skipped. The goal is to find a tour of minimum expected length. Bertsimas and Howell (1993) further explore this problem, and Bertsimas (1992) generalizes it to a capacitated single-vehicle routing problem with random demands. For more on a priori optimization, see Bertsimas et al. (1990). Laporte et al. (1994) formulate and solve probabilistic TSPs as stochastic integer programs with so-called simple recourse (i.e., no second-stage posterior optimization is performed). Stochastic programming formulations have been developed for SVRPs in which the demand at each node is a random variable. In these models, the assumption is that each node must be visited, and first-stage (a priori) routes for each vehicle are to be determined. The models in the literature differ primarily in how they deal with route failure (i.e., when demand on a route exceeds the vehicle's capacity). Stewart and Golden (1983), Laporte et al. (1989), and Bastian and Rinnooy Kan (1992) consider models that constrain the probability of route failure to be at most a prespecified level. Stewart and Golden (1983), Dror and Trudeau (1986), Dror et al. (1989), Laporte and Louveaux (1990), and Bastian and Rinnooy Kan (1992) formulate models in which the expected value of a recourse function is optimized. Laporte and Louveaux (1990) use recourse decisions that, after observing the demand, optimally select points in the first-stage route when the vehicle should return to the depot. The other models are stochastic programs with simple recourse. Dynamic route-reoptimization strategies are allowed in the multistage stochastic programming formulation of Dror (1993) and the Markov decision process models of Dror et al. (1989) and Dror (1993), but the authors indicated that computational solution of these models was not possible. The stochastic component of the models described previously lies in the customer demands. For incapacitated vehicles, random demands are modeled via random node service times. The arc travel times may also be random. Leipälä (1978) studies the expected length of posterior tours of TSPs with random arc lengths. Berman and Simchi-Levi (1989) consider a variant of the probabilistic TSP in which the location of the salesman's home is to be optimized on a network with a random subset of customers requiring service and random travel times. Kao (1978), Sniedovich (1981), and Carraway et al. (1989) consider a stochastic TSP in which the objective is to maximize the probability of completing the tour by a deadline when the arcs have independent and normally distributed travel times. Laporte et al. (1992) introduce the SVRP with stochastic travel and service times. In their approach, the vehicles are incapacitated, and each node must be serviced. Each vehicle has a target time by which its route should be complete: A chance-constrained formulation ensures this with prespecified probability levels, whereas a stochastic program with simple recourse penalizes the expected value by which route travel times exceed the respective targets. Laporte et al. use a branch-and-cut approach to solve instances of the SVRP on networks with 10 to 20 nodes and 2 to 5 scenarios. Lambert et al. (1993) approximately optimize collection routes through bank branches in a network with stochastic travel times by adapting the heuristic due to Clarke and Wright (1964). As in Laporte et al. (1992), the model incorporates target route completion times. They present results for networks with 28 and 44 nodes, in which random arcs access 3 or 4 of the nodes, respectively, and can take two values (all travel times are long or all are short). We use a variant of their 28-node problem in our computational work. Like most of the work described herein, our model is static in nature (i.e., we select vehicle routes before realizing the random parameters and do not subsequently reoptimize the routes). Each node in the network must be visited by a vehicle, and arc travel times and node service times are stochastic. Even though the vehicles in the models of Laporte et al. (1992) and Lambert et al. (1993) are incapacitated, they are called SVRPs instead of stochastic multiple salesmen TSPs (perhaps to avoid confusion with probabilistic TSPs). Of the models in the literature, ours is closest to those of Laporte et al. and Lambert et al.; so, we follow their naming convention. The objective functions in most of the stochastic vehicle routing problem (VRP) and TSP models depend on the total travel costs although, as noted previously, some incorporate target completion times for each vehicle. In contrast, our models' objective functions depend on the length of the longest route. Such an objective function would be most appropriate when the goal is to minimize completion time of the project.

Networks, 2022

In this paper, we propose graph-based models for several vehicle routing problems with intermediate stops: the capacitated multi-trip vehicle routing problem with time windows, the multi-depot vehicle routing problem with inter-depot routes, the arc routing problem with intermediate facilities under capacity and length restrictions and the green vehicle routing problem. In these models, the set of feasible routes is represented by a set of resource constrained paths in one or several graphs. Intermediate stops are supported by the possibility to define negative resource consumption for some arcs. The models that we propose are then solved by VRPSolver, which implements a generic branch-cut-and-price exact algorithm. Thus, a simple parameterization enables us to use several state-of-the-art algorithmic components: automatic stabilization by dual price smoothing, limited-memory rank-1 cuts, reduced cost-based arc elimination, enumeration of elementary routes, and hierarchical strong branching. For each problem, we numerically compare the proposed methodology with the best exact approach found in the literature. State-of-the-art computational results were obtained for all problems except one.