Vehicle Routing Problem With Stochastic Demands Research Articles

Hardness of Pricing Routes for Two-Stage Stochastic Vehicle Routing Problems with Scenarios

On the Difficulty of Pricing Routes for Stochastic Vehicle Routing Problems Many approaches exist for dealing with the uncertainty in the vehicle routing problem with stochastic demands (VRPSD), but the most popular approach models the VRPSD as a two-stage stochastic program where a recourse policy prescribes actions that handle when the realized demands exceed the vehicle capacity. Similarly to other VRP variants, some state-of-the-art algorithms for the VRPSD use set-partitioning formulations that generate variables (routes) via a pricing problem. All of these algorithms, however, have strong assumptions on the probability distribution of customer demands, a simplification that might not be realistic in some applications. In “Hardness of Pricing Routes for Two-Stage Stochastic Vehicle Routing Problems with Scenarios,” Ota and Fukasawa examine the challenges associated with solving the pricing problem of the VRPSD when the customer demands are given by scenarios. They demonstrate that the VRPSD pricing problem is strongly NP-hard for a wide variety of recourse policies and route relaxations. This highlights the difficulty of developing efficient pricing algorithms for the VRPSD with scenario-based demand models.

An Optimized Solution to Multi-Constraint Vehicle Routing Problem

A Vehicle Routing Problem (VRP) is a Non-Polynomial Hard Category (NP-hard) problem in which the best set of routes for a convoy of vehicles is traversed to deliver goods or services to a known set of customers. In VRP, some constraints are added to improve performance. Some variations of VRP are Capacitated Vehicle Routing Problem (CVRP), Vehicle Routing Problem with Stochastic Demands (VRPSD), Vehicle Routing Problem with Time Window (VRPTW), Dynamic Vehicle Routing Problem (DVRP), and Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD) where vehicle and routes have multiple constraints. Swarm intelligence is a well-used approach to solve VRPs. Moreover, different hybrid combinations of global and local optimization techniques are also used to optimize the said problem. In this research, an attempt is made to solve CVRP with VRPSD by using two different hybridized population-based approaches, that is, the Cuckoo Search Algorithm (CSA) and Particle Swarm Optimization (PSO). The experiments showed the accuracy of the improved CVRP that is superior to one obtained by using other classical versions and better than the results achieved by comparable algorithms. Besides, this improved algorithm can also improve search efficiency.

Recent advances in vehicle routing with stochastic demands: Bayesian learning for correlated demands and elementary branch-price-and-cut

We consider the vehicle routing problem with stochastic demands (VRPSD), a stochastic variant of the well-known VRP in which demands are only revealed upon arrival of the vehicle at each customer. Motivated by the significant recent progress on VRPSD research, we begin this paper by summarizing the key new results and methods for solving the problem. In doing so, we discuss the main challenges associated with solving the VRPSD under the chance-constraint and the restocking-based perspectives. Once we cover the current state-of-the-art, we introduce two major methodological contributions. First, we present a branch-price-and-cut (BP&C) algorithm for the VRPSD under optimal restocking. The method, which is based on the pricing of elementary routes, compares favorably with previous algorithms and allows the solution of several open benchmark instances. Second, we develop a demand model for dealing with correlated customer demands. The central concept in this model is the “external factor”, which represents unknown covariates that affect all demands. We present a Bayesian-based, iterated learning procedure to refine our knowledge about the external factor as customer demands are revealed. This updated knowledge is then used to prescribe optimal replenishment decisions under demand correlation. Computational results demonstrate the efficiency of the new BP&C method and show that cost savings above 10% may be achieved when restocking decisions take account of demand correlation. Lastly, we motivate a few research perspectives that, as we believe, should shape future research on the VRPSD.

Vehicle Routing with Stochastic Demands and Partial Reoptimization

We consider the vehicle routing problem with stochastic demands (VRPSD), a problem in which customer demands are known in distribution at the route planning stage and revealed during route execution upon arrival at each customer. A long-standing open question in the VRPSD concerns the benefits of allowing, during route execution, partial reordering of the planned customer visits. Given the practical importance of this question and the growing interest on the VRPSD under optimal restocking, we study the VRPSD under a recourse policy known as the switch policy. The switch policy is a canonical reoptimization policy that permits only pairs of successive customers to be reordered. We consider this policy jointly with optimal preventive restocking and introduce a branch-cut-and-price algorithm to compute optimal a priori routing plans in this context. At its core, this algorithm features pricing routines where value functions represent the expected cost-to-go along planned routes for all possible states and reordering decisions. To ensure pricing tractability, we adopt a strategy that combines elementary pricing with completion bounds of varying complexity, and we solve the pricing problem without relying on dominance rules. Our numerical experiments demonstrate the effectiveness of the algorithm for solving instances with up to 50 customers. Notably, they also give us new insights into the value of reoptimization. The switch policy enables significant cost savings over optimal restocking when the planned routes come from an algorithm built on a deterministic approximation of the data, an important scenario given the difficulty of finding optimal VRPSD solutions. The benefits are smaller when comparing optimal a priori VRPSD solutions obtained for both recourse policies. As it appears, further cost savings may require joint reordering and reassignment of customer visits among vehicles when the context permits.

A discrete spider monkey optimization for the vehicle routing problem with stochastic demands

The Vehicle Routing Problem (VRP) is a classical NP-hard combinatorial optimization problem. In recent years, a lot of heuristic algorithms have been proposed for optimizing the problem, and many simulation and practical experiments are performed to evaluate and verify the effectiveness of different heuristic algorithms. In this paper, we focus on the Vehicle Routing Problem with stochastic demands (VRPSD) in which the customer demands follow known probability distributions. A hybrid algorithm called DSMO-GA which combines discrete spider monkey algorithm (SMO) with genetic algorithm (GA) is proposed for solving the VRPSD problem. In the proposed DSMO-GA for VRPSD, the individuals are coded as sets and sequences, and a conflict elimination method is designed to cancel the infeasible codes. Accordingly, a non adjacent 2-OPT is designed to enhance population diversity. The proposed DSMO-GA effectively integrates the advantages of DSMO and GA to further balance the global exploration and local exploitation capacity. Five groups of different experiments are conducted and the results show that DSMO-GA is valid for the VRPSD. Furthermore, the impact of various parameters on the performance of DSMO-GA are also investigated.

MIMOA: A membrane-inspired multi-objective algorithm for green vehicle routing problem with stochastic demands

Abstract Nowadays, an increasing number of vehicle routing problem with stochastic demands (VRPSD) models have been studied to meet realistic needs in the field of logistics. In this paper, a bi-objective vehicle routing problem with stochastic demands (BO-VRPSD) was investigated, which aims to minimize total cost and customer dissatisfaction. Different from traditional vehicle routing problem (VRP) models, both the uncertainty in customer demands and the nature of multiple objectives make the problem more challenging. To cope with BO-VRPSD, a membrane-inspired multi-objective algorithm (MIMOA) was proposed, which is characterized by a parallel distributed framework with two operation subsystems and one control subsystem, respectively. In particular, the operation subsystems leverage a multi-objective evolutionary algorithm with clustering strategy to reduce the chance of inferior solutions. Meanwhile, the control subsystem exploits a guiding strategy as the communication rule to adjust the searching directions of the operation subsystems. Experimental results based on the ten 120-node instances with real geographic locations in Beijing show that, MIMOA is more superior in solving BO-VRPSD to other classical multi-objective evolutionary algorithms.

THE VEHICLE ROUTING PROBLEM WITH STOCHASTIC DEMANDS IN AN URBAN AREA – A CASE STUDY

The vehicle routing problem with stochastic demands (VRPSD) is a combinatorial optimization problem. The VRPSD looks for vehicle routes to connect all customers with a depot, so that the total distance is minimized, each customer visited once by one vehicle, every route starts and ends at a depot, and the travelled distance and capacity of each vehicle are less than or equal to the given maximum value. Contrary to the classical VRP, in the VRPSD the demand in a node is known only after a vehicle arrives at the very node. This means that the vehicle routes are designed in uncertain conditions. This paper presents a heuristic and meta-heuristic approach for solving the VRPSD and discusses the real problem of municipal waste collection in the City of Niš.

Stokastik Talepli Araç Rotalama Problemi İçin Literatür Taraması

Vehicle Routing Problem (VRP) is the problem of identifying suitable routes in a way that minimizes the cost in order to serve customers in different locations from one or more depots by one or more vehicles. However, in real life problems, Stochastic Vehicle Routing Problem (SVRP) with stochastic information appears more than deterministic problems in which all parameters are known in advance. When the studies on SVRP in the literature are examined, it is found that the researchers study the most about vehicle routing problem with stochastic demand (VRPSD) in which stochastic demand takes place. In this study, a situation was examined in such a way that customer demands are not known exactly until the vehicle reaches the customer location. In VRPSD, it is accepted that the demands from customers consist of random variables with a certain probability distribution. Studies on VRPSD in the literature have been examined in detail and a classification has been made under the specified constraints. Studies on VRPSD are evaluated according to this classification, mathematical models developed for VRPSD and the proposed solution approaches for solving the problem are laid out and an effort is made to determine which problem the researchers concentrated on the most.

A Rule-Based Recourse for the Vehicle Routing Problem with Stochastic Demands

In this paper we consider the vehicle routing problem with stochastic demands (VRPSD). We consider that customer demands are only revealed when a vehicle arrives at customer locations. Failures occur whenever the residual capacity of the vehicle is insufficient to serve the observed demand of a customer. Such failures entail that recourse actions be taken to recover route feasibility. These recourse actions usually take the form of return trips to the depot, which can be either done in a reactive or proactive fashion. Over the years, there have been various policies defined to perform these recourse actions in either a static or a dynamic setting. In the present paper, we propose policies that better reflect the fixed operational rules that can be observed in practice and that also enable implementing preventive recourse actions. We define the considered operational rules and show how, for a planned route, these operational rules can be implemented using a fixed threshold-based policy to govern the recourse actions. An exact solution algorithm is developed to solve the VRPSD under the considered policies. Finally, we conduct an extensive computational study, which shows that significantly better solutions can be obtained when using the proposed policies compared with solving the problem under the classic recourse definition.

A local branching matheuristic for the multi-vehicle routing problem with stochastic demands

This paper proposes a local branching matheuristic for the vehicle routing problem with stochastic demands (VRPSD). The problem is cast in a two-stage stochastic programming model, in which routes are planned in the first stage and executed in the second stage. In this setting, a failure may occur if a vehicle does not have sufficient capacity to serve the realized demand of a customer, which is revealed only upon arrival at a customer’s location. In the event of a failure, a recourse action is performed by having the vehicle return to the depot to replenish its capacity and resume its planned route at the point of failure. Thus, the objective of the VRPSD is to minimize the sum of the planned routes cost and of the expected recourse cost. We propose a local branching matheuristic to solve the multi-VRPSD. We introduce an intensification procedure applied at each node of the local branching tree. This procedure is embedded in a multi-descent scheme for which we propose a diversification strategy. Extensive computational results demonstrate the effectiveness of our matheuristic.

An exact algorithm to solve the vehicle routing problem with stochastic demands under an optimal restocking policy

This paper examines the Vehicle Routing Problem with Stochastic Demands (VRPSD), in which the actual demand of customers can only be realized upon arriving at the customer location. Under demand uncertainty, a planned route may fail at a specific customer when the observed demand exceeds the residual capacity. There are two ways to face such failure events, a vehicle can either execute a return trip to the depot at the failure location and refill the capacity and complete the split service, or in anticipation of potential failures perform a preventive return to the depot whenever the residual capacity falls below a threshold; overall, these return trips are called recourse actions. In the context of VRPSD, a recourse policy which schedules various recourse actions based on the visits planned beforehand on the route must be designed. An optimal recourse policy prescribes the cost-effective returns based on a set of optimal customer-specific thresholds. We propose an exact solution method to solve the multi-VRPSD under an optimal restocking policy. The Integer L-shaped algorithm is adapted to solve the VRPSD in a branch-and-cut framework. To enhance the efficiency of the presented algorithm, several lower bounding schemes are developed to approximate the expected recourse cost.

A hybrid recourse policy for the vehicle routing problem with stochastic demands

In this paper, we propose a new recourse policy for the vehicle routing problem with stochastic demands (VRPSD). In this routing problem, customer demands are characterized by known probability distributions. The actual demand values are only revealed upon arriving at a customer location. The objective of the problem is to plan routes minimizing the travel cost and the expect recourse cost. The latter cost is a result of a predetermined recourse policy designed to handle route failures. Given the planned routes, such failures may occur in the event where a vehicle has insufficient capacity to serve the current customer or the next customer. In the relevant literature, there are three types of recourse policies: (i) classical, where failures at customers are handled by return trips to the depot, (ii) optimal restocking, where preventive restocking trips to the depot are performed based on optimized customer-specific thresholds, and failures are handled by return trips to the depot, and (iii) rule-based policies, where preventive restocking trips are performed based on thresholds established by preset rules, and failures are handled by performing return trips to the depot. While the first type is rather myopic, the last two types of recourse policies use simplistic comparisons of the vehicle’s residual capacity to trigger recourse actions. In this paper, we propose a more advanced rule-based recourse policy, which does not solely depend on the vehicle’s residual capacity. To do so, we first propose a taxonomy that groups rule-based policies into three classes, we then propose the first hybrid recourse policy, which simultaneously combines two of these classes, namely risk and distance. We develop an exact solution algorithm for the VRPSD with this hybrid recourse policy and conduct a broad range of computational experiments. The algorithm is able to solve instances with up to 60 customers, and for certain experimental configurations, the exact algorithm solves to optimality up to 79% of the instances. Finally, we show that when the optimal routes of the hybrid policy are operated under the optimal restocking policy they yield a marginal average cost difference.

Faster rollout search for the vehicle routing problem with stochastic demands and restocking

Rollout algorithms lead to effective heuristics for the single vehicle routing problem with stochastic demands (VRPSD), a prototypical model of logistics under uncertainty. However, they can be computationally intensive. To reduce their run time, we introduce a novel approach to approximate the expected cost of a route when executing any rollout algorithm for VRPSD with restocking. With a sufficiently large number of customers its theoretical speed-up factor is of big-o order 1/3. On a set of instances from the literature, our proposed technique applied to a known rollout algorithm and three variants thereof achieves speed-up factors that range from 0.26 to 0.34 when there are more than fifty customers, degrading only marginally the quality of the resulting routes. Our method also applies to the a priori case, in which case it is exact.

Improved Rollout Search for the Vehicle Routing Problem with Stochastic Demands

We propose two variants of a known rollout algorithm for the single vehicle routing problem with stochastic demands (VRPSD) under both the a priori (AP) and restocking (RS) strategies by expanding the search space of this algorithm. To reduce the increased computational effort, we introduce a novel approach for computing the expected cost of a route when executing a rollout algorithm, which is, however, approximate in the RS case. With a sufficiently large number of customers the theoretical speed-up factor of this approach is of big-o order 1/3. On a set of instances from the literature, our rollout algorithm variants yield better routes than the known rollout algorithm, with average expected cost improvements between 1.25% and 1.89% in the AP case and 0.64% and 0.96% in the RS case. In the AP case our second variant dominates our first variant by 0.43-0.46% on average, an observation for which we provide some theoretical support. In the RS case we do not observe any such dominance. Moreover, across all the considered rollout algorithms, our expected cost evaluation approach achieves speed-up factors that range from 0.16 to 0.38 in the AP case and 0.26 and 0.34 in the RS case, with the quality of the resulting RS routes only marginally degraded. Other VRPSD heuristics can potentially benefit from the use of this approach.

Solución del problema de ruteo de vehículos con demandas estocásticas mediante la optimización por espiral

El artículo presenta los resultados del estudio de un problema de ruteo de vehículos con demandas estocásticas (Vehicle Routing Problem with Stochastic Demands, VRPSD), en el cual la única variable estocástica es la demanda de los clientes, esta variable sigue una distribución discreta, y su valor solo es conocido cuando el vehículo llega a la ubicación del cliente. Para su solución, se implementó la metaheurística denominada Optimización por Espiral, con el enfoque a priori y la estrategia de reabastecimiento preventivo para un solo vehículo. Para mejorar el método se inicializaron las rutas mediante la heurística del vecino más cercano, y posteriormente se utilizó la mutación, un operador evolutivo, para ampliar la zona de exploración de los puntos de búsqueda. Adicionalmente, se utilizó el intercambio 2-Opt, una heurística de búsqueda local, con el fin de intensificar la búsqueda en la vecindad de soluciones óptimas encontradas. Por otra parte, se realizó un diseño de experimentos 23, con el fin de determinar la influencia de cada factor en la función objetivo. Este análisis se llevó a cabo en 8 instancias diferentes que fueron diseñadas y desarrolladas por Galván et al. [1]. Finalmente, se compararon los resultados obtenidos con los arrojados por el algoritmo híbrido EPSO, con el objetivo de probar la eficiencia y eficacia del algoritmo desarrollado. Esta comparación evidenció que el método propuesto obtiene mejores resultados en todas las instancias, con mejoras de hasta el 5,71 %.

Decomposition-based multi-objective evolutionary algorithm for vehicle routing problem with stochastic demands

Vehicle routing problem with stochastic demands (VRPSD) is a famous and challenging optimization problem which is similar to many real world problems. To resemble the real world scenario, total traveling distance, total driver remuneration, the number of vehicles used and the difference between driver remuneration are considered and formulated in the multi-objective optimization perspective. This paper aims to solve multi-objective VRPSD under the constraints of available time window and vehicle capacity using decomposition-based multi-objective evolutionary algorithm (MOEA/D) with diversity-loss-based selection method incorporates with local search and multi-mode mutation heuristics. We have also compared the optimization performance of the decomposition-based approach with the domination-based approach to study the difference between these two well-known evolutionary multi-objective algorithm frameworks. The simulation results have showed that the decomposition-based approach with diversity-loss-based selection method is able to maintain diverse output solutions.

A hybrid metaheuristic for the vehicle routing problem with stochastic demand and duration constraints

The vehicle routing problem with stochastic demands (VRPSD) consists in designing optimal routes to serve a set of customers with random demands following known probability distributions. Because of demand uncertainty, a vehicle may arrive at a customer without enough capacity to satisfy its demand and may need to apply a recourse to recover the route's feasibility. Although travel times are assumed to be deterministic, because of eventual recourses the total duration of a route is a random variable. We present two strategies to deal with route-duration constraints in the VRPSD. In the first, the duration constraints are handled as chance constraints, meaning that for each route, the probability of exceeding the maximum duration must be lower than a given threshold. In the second, violations to the duration constraint are penalized in the objective function. To solve the resulting problem, we propose a greedy randomized adaptive search procedure (GRASP) enhanced with heuristic concentration (HC). The GRASP component uses a set of randomized route-first, cluster-second heuristics to generate starting solutions and a variable-neighborhood descent procedure for the local search phase. The HC component assembles the final solution from the set of all routes found in the local optima reached by the GRASP. For each strategy, we discuss extensive computational experiments that analyze the impact of route-duration constraints on the VRPSD. In addition, we report state-of-the-art solutions for a established set of benchmarks for the classical VRPSD.

Partial-route inequalities for the multi-vehicle routing problem with stochastic demands

This paper describes an exact algorithm for a variant of the vehicle routing problem in which customer demands to be collected are stochastic. Demands are revealed upon the vehicle arrival at customer locations. As a result, a vehicle may reach a customer and does not have sufficient capacity to collect the realized demand. Such a situation is referred to as a failure. In this paper the following recourse action is then applied when failure occurs: the vehicle returns to the depot to unload and resumes its planned route at the point of failure. The capacitated vehicle routing problem with stochastic demands (VRPSD) consists of minimizing the sum of the planned routes cost and of the expected recourse cost. The VRPSD is formulated as a two-stage stochastic programming model and solved by means of an integer L-shaped algorithm. This paper introduces three lower bounding functionals based on the generation of general partial routes, as well as an exact separation procedure to identify violated cuts. Extensive computational results confirm the effectiveness of the proposed algorithm, as measured by a substantial reduction in the number of feasible solutions that have to be explicitly eliminated. This translates into a higher proportion of instances solved to optimality, reduced optimality gaps, and lower computing times.

A Hybrid Particle Swarm Optimization for Solving Vehicle Routing Problem with Stochastic Demands

As one of the most popular supply chain management problems, the Vehicle Routing Problem (VRP) has been thoroughly studied in the last decades, most of these studies focus on deterministic problem where the customer demands are known in advance. But the Vehicle Routing Problem with Stochastic Demands (VRPSD) has not received enough consideration. In the VRPSD, the vehicle does not know the customer demands until the vehicle arrive to them. This paper use a hybrid algorithm for solving VRPSD, the hybrid algorithm based on Particle Swarm Optimization (PSO) Algorithm, combines a Greedy Randomized Adaptive Search Procedure (GRASP) algorithm, and Variable Neighborhood Search (VNS) algorithm. A real number encoding method is designed to build a suitable mapping between solutions of problem and particles in PSO. A number of computational studies, along with comparisons with other existing algorithms, showed that the proposed hybrid algorithm is a feasible and effective approach for Vehicle Routing Problem with Stochastic Demands.

Paired cooperative reoptimization strategy for the vehicle routing problem with stochastic demands

In this paper, we develop a paired cooperative reoptimization (PCR) strategy to solve the vehicle routing problem with stochastic demands (VRPSD). The strategy can realize reoptimization policy under cooperation between a pair of vehicles, and it can be applied in the multivehicle situation. The PCR repeatedly triggers communication and partitioning to update the vehicle assignments given real-time customer demands. We present a bilevel Markov decision process to model the coordination of a pair of vehicles under the PCR strategy. We also propose a heuristic that dynamically alters the visiting sequence and the vehicle assignment given updated information. We compare our approach with a recent cooperation strategy in the literature. The results reveal that our PCR strategy performs better, with a cost saving of around 20–30%. Moreover, embedding communication can save an average of 1.22%, and applying our partitioning method rather than an alternative can save an average of 3.96%.