We consider a dynamic vehicle routing problem with time windows and stochastic customers (DS-VRPTW), such that customers may request for services as vehicles have already started their tours. To solve this problem, the goal is to provide a decision rule for choosing, at each time step, the next action to perform in light of known requests and probabilistic knowledge on requests likelihood. We introduce a new decision rule, called Global Stochastic Assessment (GSA) rule for the DS-VRPTW, and we compare it with existing decision rules, such as MSA. In particular, we show that GSA fully integrates nonanticipativity constraints so that it leads to better decisions in our stochastic context. We describe a new heuristic approach for efficiently approximating our GSA rule. We introduce a new waiting strategy. Experiments on dynamic and stochastic benchmarks, which include instances of different degrees of dynamism, show that not only our approach is competitive with state-of-the-art methods, but also enables to compute meaningful offline solutions to fully dynamic problems where absolutely no a priori customer request is provided.
The first D-VRP is proposed in @cite_16 , which introduces a single vehicle Dynamic Dial-a-Ride Problem (D-DARP) in which customer requests appear dynamically. Then, @cite_19 introduced the concept of immediate requests that must be serviced as soon as possible, implying a replanning of the current vehicle route. Complete reviews on D-VRP may be found in @cite_7 @cite_6 . In this section, we more particularly focus on stochastic D-VRP. @cite_6 classifies approaches for stochastic D-VRP in two categories, either based on or on . Stochastic modeling approaches formally capture the stochastic nature of the problem, so that solutions are computed in the light of an overall stochastic context. Such holistic approaches usually require strong assumptions and efficient computation of complex expected values. Sampling approaches try to capture stochastic knowledge by sampling scenarios, so that they tend to be more focused on local stochastic evidences. Their local decisions however allow sample-based methods to scale up to larger problem instances, even under challenging timing constraints. One usually needs to find a good compromise between having a high number of scenarios, providing a better representation of the real distributions, and a more restricted number of these leading to less computational effort.
@cite_19 @cite_16 @cite_6 @cite_7
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The problem of transporting patients or elderly people has been widely studied in literature and is usually modeled as a dial-a-ride problem (DARP). In this paper we analyze the corresponding problem arising in the daily operation of the Austrian Red Cross. This nongovernmental organization is the largest organization performing patient transportation in Austria. The aim is to design vehicle routes to serve partially dynamic transportation requests using a fixed vehicle fleet. Each request requires transportation from a patient's home location to a hospital (outbound request) or back home from the hospital (inbound request). Some of these requests are known in advance. Some requests are dynamic in the sense that they appear during the day without any prior information. Finally, some inbound requests are stochastic. More precisely, with a certain probability each outbound request causes a corresponding inbound request on the same day. Some stochastic information about these return transports is available from historical data. The purpose of this study is to investigate, whether using this information in designing the routes has a significant positive effect on the solution quality. The problem is modeled as a dynamic stochastic dial-a-ride problem with expected return transports. We propose four different modifications of metaheuristic solution approaches for this problem. In detail, we test dynamic versions of variable neighborhood search (VNS) and stochastic VNS (S-VNS) as well as modified versions of the multiple plan approach (MPA) and the multiple scenario approach (MSA). Tests are performed using 12 sets of test instances based on a real road network. Various demand scenarios are generated based on the available real data. Results show that using the stochastic information on return transports leads to average improvements of around 15 . Moreover, improvements of up to 41 can be achieved for some test instances. In 1991, D. J. Bertsimas and G. van Ryzin introduced and analyzed a model for stochastic and dynamic vehicle routing in which a single, uncapacitated vehicle traveling at a constant velocity in a Euclidean region must service demands whose time of arrival, location and on-site service are stochastic. The objective is to find a policy to service demands over an infinite horizon that minimizes the expected system time (wait plus service) of the demands. This paper extends our analysis in several directions. First, we analyze the problem of m identical vehicles with unlimited capacity and show that in heavy traffic the system time is reduced by a factor of 1 m2 over the single-server case. One of these policies improves by a factor of two on the best known system time for the single-server case. We then consider the case in which each vehicle can serve at most q customers before returning to a depot. We show that the stability condition in this case depends strongly on the geometry of the region. Several pol... Abstract This paper proposes a state-of-the-art branch-cut-and-price algorithm for the vehicle routing problem with stochastic demands (VRPSD). We adapt the model of Christiansen and Lysgaard [6] and formulate the VRPSD as a set partitioning model with additional constraints. Feasible routes are generated using a dynamic programming algorithm executed over a state-space graph. Our method combines 2-cycle elimination with ng -routes. In addition, our pricing problem is significantly accelerated by the introduction of a new aggregate dominance rule. To speed up the generation of negative reduced cost columns, we use a tabu search heuristic and a bidirectional labeling algorithm. We also add capacity and subset-row inequalities dynamically in order to strengthen the linear relaxation of the master problem. As extensive computational tests illustrate, our algorithm is very competitive with the one of [6] . We solve 20 additional instances from the 40-instance set considered by these authors and we considerably improve the computing times for instances already closed. We also solve 17 new instances from the literature. We consider a natural probabilistic variation of the classical vehicle routing problem (VRP), in which demands are stochastic. Given only a probabilistic description of the demand we need to design routes for the VRP. Motivated by applications in strategic planning and distribution systems, rather than resolving the problem when the demand becomes known, we propose to construct an a priori sequence among all customers of minimal expected total length. We analyze the problem using a variety of theoretical approaches. We find closed-form expressions and algorithms to compute the expected length of an a priori sequence under general probabilistic assumptions. Based on these expressions we find upper and lower bounds for the probabilistic VRP and the VRP re-optimization strategy, in which we find the optimal route at every instance. We propose heuristics and analyze their worst case performance as well as their average behavior using techniques from probabilistic analysis. Our results suggest that our approac...