Stochastic Vehicle Routing Problem Research Articles

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Reinforcement Learning Approach to Stochastic Vehicle Routing Problem With Correlated Demands

Reinforcement Learning Approach to Stochastic Vehicle Routing Problem With Correlated Demands

Solving Vehicle Routing Problems under Uncertainty and in Dynamic Scenarios: From Simheuristics to Agile Optimization

Many real-life applications of the vehicle routing problem (VRP) occur in scenarios subject to uncertainty or dynamic conditions. Thus, for instance, traveling times or customers’ demands might be better modeled as random variables than as deterministic values. Likewise, traffic conditions could evolve over time, synchronization issues should need to be considered, or a real-time re-optimization of the routing plan can be required as new data become available in a highly dynamic environment. Clearly, different solving approaches are needed to efficiently cope with such a diversity of scenarios. After providing an overview of current trends in VRPs, this paper reviews a set of heuristic-based algorithms that have been designed and employed to solve VRPs with the aforementioned properties. These include simheuristics for stochastic VRPs, learnheuristics and discrete-event heuristics for dynamic VRPs, and agile optimization heuristics for VRPs with real-time requirements.

A Simulation-Optimization Approach for the Management of the On-Demand Parcel Delivery in Sharing Economy

This paper investigates a dynamic and stochastic vehicle routing problem with time windows that considers the use of multiple delivery options and crowd drivers, reflecting the synchromodality in the urban context. We propose a multi-stage stochastic model, and we solve the problem by using a simulation-optimization strategy. It relies on a Monte Carlo simulation and a large neighborhood search (LNS) heuristic for optimization. We conduct a case study in the medium-sized city of Turin (Italy) to measure the potential impact of integrating cargo bikes and crowd drivers in parcel delivery. Experimental results show that combining crowd drivers and green carriers with the traditional van to manage the parcel delivery is beneficial in terms of economic and environmental cost-saving, while the operational efficiency decreases. Besides, the green carriers and crowd drivers are promising delivery options to deal with online customer requests in the context of stochastic and dynamic parcel delivery. The resulting set of policies are part of the outcomes of the Logistics and Mobility Plan 2019–2021 in the Piedmont region.

Multi-objective evolutionary algorithm based on RBF network for solving the stochastic vehicle routing problem

Solving the multi-objective vehicle routing problem with stochastic demand (MO-VRPSD) is challenging due to its non-deterministic property and conflicting objectives. Most multi-objective evolutionary algorithm dealing with this problem update current population without any guidance from previous searching experience. In this paper, a multi-objective evolutionary algorithm based on artificial neural networks is proposed to tackle the MO-VRPSD. Particularly, during the evolutionary process, a radial basis function network (RBFN) is exploited to learn the potential knowledge of individuals, generate hypothesis and instantiate hypothesis. The RBFN evaluates individuals with different scores and generates new individuals with higher quality while taking into account the non-dominated relationship between individuals. Moreover, integrated with a specific non-dominated sorting strategy, i.e., ENS-SS, along with several effective heuristic operations, the proposed algorithm performs favorably for solving the MO-VRPSD. The experimental results based on the modified Solomon benchmark instances verified the effectiveness of the respective components, and the superiority to other multi-objective evolutionary algorithms.

On the stochastic vehicle routing problem with time windows, correlated travel times, and time dependency

Most state-of-the-art algorithms for the Vehicle Routing Problem, such as Branch-and-Price algorithms or meta heuristics, rely on a fast feasibility test for a given route. We devise the first approach to approximately check feasibility in the Stochastic Vehicle Routing Problem with time windows, where travel times are correlated and depend on the time of the day. Assuming jointly normally distributed travel times, we use a chance constraint approach to model feasibility, where two different application scenarios are considered, depending on whether missing a customer makes the rest of the route infeasible or not. The former case may arise, e.g., in drayage applications or in the pickup-and-delivery VRP. In addition, we present an adaptive sampling algorithm that is tailored for our setting and is much faster than standard sampling techniques. We use a case study for both scenarios, based on instances with realistic travel times, to illustrate that taking correlations and time dependencies into account significantly improves the quality of the solutions, i.e., the precision of the feasibility decision. In particular, the nonconsideration of correlations often leads to solutions containing infeasible routes.

On Dynamical Situations in Vehicle Routing Problems (DSVRP)

This paper discusses the formulation of a stochastic Vehicle Routing Problem (VRP) process, introduced to existing deterministic VRP process, directing all efforts towards the dynamical situations that occur in a typical VRP setting as it applies to everyday real-life situations. It contrasts the dichotomy between static and dynamical situations in VRP, constructs and analyzes the relationship between static and dynamical situations in Vehicle Routing Problems, formulates an expression to describe the dynamical situation in VRP and relates the problem formulation to existing VRP with a view to solving the problem as it applies to real-life situations, a major trend in the business world today. With a view to achieving these, this paper associates the merits of reactivation of route, re-optimization of operation process and projects an anticipatory demand for a dynamical situation in VRP with stochastic requests via three stages: Pre-Decision States, Decisions States and Post-Decision States. First, as a reaction to anticipatory customers’ requests, the current routing plans need to be re-optimized and current customers’ request reactivated. Second, potential future requests need to be anticipated along current decision making since life itself is dynamic. Decisions need to be made in good time. Though, the limited time frame between when a vehicle leaves and returns to the depot often prohibits extensive optimization in both dimensions rather, answer the questions that arise on how to utilize the limited time effectively and judiciously, satisfying both the current and anticipatory customers equally.

Time-dependent stochastic vehicle routing problem with random requests: Application to online police patrol management in Brussels

The Static and Stochastic Vehicle Routing Problem with Random Requests (SS-VRP-R) describes realistic operational contexts in which a fleet of vehicles has to serve customer requests appearing dynamically. Based on a probabilistic knowledge about the appearance of requests, the SS-VRP-R seeks a priori sequences of vehicle relocations, optimizing the expected responsiveness to the requests. In this paper, an existing computational framework, based on recourse strategies, is adapted to meet the objectives of the SS-VRP-R. The resulting models are applied to a real case study of the management of police units in Brussels. In this context, the expected average response time is minimized. To cope with the reality of the urban context, a time-dependent variant is also studied (TD-SS-VRP-R) in which the travel time between two locations is a function that depends on the departure time at the first location. Experiments confirm the contribution and the adaptability of the recourse strategies to a real-life, complex operational context. Provided an adequate solution method, simulation-based results show the high quality of the a priori solutions designed, even when compared to those designed by field experts. Finally, the experiments provide evidence that there is no potential gain in considering time-dependency in such an operational context.

Combining simheuristics with Petri nets for solving the stochastic vehicle routing problem with correlated demands

This paper analyzes a stochastic version of the vehicle routing problem in which customers’ demands are not only stochastic but also correlated. In order to solve this stochastic and correlated optimization problem, a simheuristic approach is combined with an adaptive demand predictor. This predictor is based on the use of machine learning methods and Petri nets. The information on real demands, provided by the vehicles as they visit the nodes of the logistic network, allows for a real-time forecast of the demand, as well as for an updated estimate of the correlation between them. A constrained prediction is provided by our hybrid algorithm, which is able to forecast an increase of 50% in the mean value of the demands of all nodes. With a very limited amount of information and reduced computational requirements, our algorithm provides a forecast with a high degree of reliability and a balanced capacity to reject false positives as well as false negatives. To illustrate its effectiveness, the methodology is applied to a wide range of benchmarks. The results show the benefits of applying this methodology in a context of correlated variation of the demands.

A vehicle routing problem with distribution uncertainty in deadlines

This article considers a stochastic vehicle routing problem with probability constraints. The probability that customers are served before their (uncertain) deadlines must be higher than a pre-specified target. It is unrealistic to expect that the perfect knowledge on the probability distributions of deadlines is always available. To this end, we propose a distributionally robust optimisation framework to study worst bounds of the problem, which exploits the moment information of the historical observations. This framework includes two steps. We first use Conditional Value-at-Risk (CVaR) as a risk approximation to the probability of missing customer deadlines. The resulting nonlinear model is then transformed into a semi-infinite mixed integer program, using the dual form of the CVaR approximation. A sample approximation approach is then used to address the computational challenges. As the standard CVaR approximation to probability constraints is rather conservative, we suggest a relaxation to the approximation and develop an iterative algorithm to find the right value of the parameter that is introduced to the relaxed CVaR constraints. The extensive numerical experiments show that the routing policies developed by the proposed solution framework are robust and able to achieve the required target, regardless of deadline distributions.

Endogenous stochastic optimisation for relief distribution assisted with unmanned aerial vehicles

Unmanned aerial vehicles (UAVs) have been increasingly viewed as useful tools to assist humanitarian response in recent years. While organisations already employ UAVs for damage assessment during relief delivery, there is a lack of research into formalising a problem that considers both aspects simultaneously. This paper presents a novel endogenous stochastic vehicle routing problem that coordinates UAV and relief vehicle deployments to minimise overall mission cost. The algorithm considers stochastic damage levels in a transport network, with UAVs surveying the network to determine the actual network damages. Ground vehicles are simultaneously routed based on the information gathered by the UAVs.A case study based on the Haiti road network is solved using a greedy solution approach and an adapted genetic algorithm. Both methods provide a significant improvement in vehicle travel time compared to a deterministic approach and a non-assisted relief delivery operation, demonstrating the benefits of UAV-assisted response.

Solving Multi-Agent Routing Problems Using Deep Attention Mechanisms

Routing delivery vehicles to serve customers in dynamic and uncertain environments like dense city centers is a challenging task that requires robustness and flexibility. Most existing approaches to routing problems produce solutions offline in the form of plans, which only apply to the situation they have been optimized for. Instead, we propose to learn a policy that provides decision rules to build the routes from online measurements of the environment state, including the customers configuration itself. Doing so, we can generalize from past experiences and quickly provide decision rules for new instances of the problem without re-optimizing any parameters of our policy. The difficulty with this approach comes from the complexity to represent this state. In this paper, we introduce a sequential multi-agent decision-making model to formalize the description and the temporal evolution of a Dynamic and Stochastic Vehicle Routing Problem. We propose a variation of Deep Neural Network using Attention Mechanisms to learn generalizable representation of the state and output online decision rules adapted to dynamic and stochastic information. Using artificially-generated data, we show promising results in these dynamic and stochastic environments, while staying competitive in deterministic ones compared to offline classical heuristics.

Two-compartment stochastic single vehicle routing problems with simultaneous pickups and deliveries from N ordered customers

ABSTRACT We consider a vehicle with two compartments that starts its route from a depot and visits N ordered customers in order to deliver new (or fresh or useful) products and to collect old (or expired or useless) products. The new products are placed in Compartment 1 and the old products are placed in Compartment 2. The quantity of new products that each customer demands and the quantity of old products that each customer returns are random variables with known joint distribution. The vehicle is allowed during its route to return to the depot in order to replenish Compartment 1 with new products and to unload the old products from Compartment 2. Under a suitable cost structure, it is possible to find the optimal restocking strategy by implementing a suitable dynamic programming algorithm. We also find the optimal restocking strategy for the corresponding infinite-time horizon problem. We further consider the same problem without the assumption that the customers are ordered. Numerical experiments indicate that the optimal restocking strategy can be computed for values of N up to 9.

The Vehicle Routing Problem with Stochastic Two-Dimensional Items

We consider a stochastic vehicle routing problem where a discrete probability distribution characterizes the two-dimensional size (height and width) as well as the weight of a subset of items to be delivered to customers. Although some item sizes and weights are not known with certainty when the routes are planned, they become known when it is time to load the vehicles, just before their departure. If it happens that not all items can be loaded in a vehicle, the items of one or more customers are put aside at a penalty or recourse cost. The objective is to minimize the sum of the routing and expected recourse costs. The problem is modeled as a two-stage stochastic program and solved with the integer L-shaped method. Some new inequalities and lower bounds are proposed. Computational results are reported on test instances specifically generated for this problem, as well as on classical instances for the deterministic case.

A Stochastic Single Vehicle Routing Problem with a Predefined Sequence of Customers and Collection of Two Similar Materials

We suppose that a vehicle visits N ordered customers in order to collect from them two similar but not identical materials. The actual quantity and the actual type of material that each customer possesses become known only when the vehicle arrives at the customer’s location. It is assumed that the vehicle has two compartments. We name these compartments, Compartment 1 and Compartment 2. It is assumed that Compartment 1 is suitable for loading Material 1 and Compartment 2 is suitable for loading Material 2. However it is permitted to load items of Material 1 into Compartment 2 and items of Material 2 into Compartment 1. These actions cause extra costs that are due to extra labor. It is permissible for the vehicle to interrupt its route and go to the depot to unload the items of both materials. The costs for travelling from each customer to the next one and the costs for travelling from each customer to the depot are known. The objective is to find the routing strategy that minimizes the total expected cost among all possible strategies for servicing all customers. A dynamic programming algorithm is designed for the determination of the routing strategy that minimizes the total expected cost among all possible strategies. The structure of optimal routing strategy is characterized by a set of critical numbers for each customer.

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.

Stochastic vehicle routing problem with heterogeneous vehicles and multiple prioritized time windows: Mathematical modeling and solution approach

This study addresses the vehicle routing problem with multiple hard prioritized time windows (VRPMPTW) with heterogeneous fleet of vehicles based on restrictions of a real-life distribution system. Three multi-objective models are proposed to address the VRPMPTW from different perspectives. The first one formulates the VRPMPTW, where travel times and transportation costs are deterministic. In the second model, a linear bi-objective model with stochastic fixed and variable transportation costs is developed to minimize total transportation costs and maximize satisfaction of customers, simultaneously. Third, a novel multi-objective stochastic VRPMPTW model, where travel times and costs are normal random variables. Due to the NP-hard essence of the problem, a binary artificial bee colony (ABC) algorithm is developed to solve the problem. The ABC algorithm is tuned via the Taguchi method. To validate the proposed algorithm, the results obtained by the ABC algorithm are compared with the optimal solutions of several small-size test problems. For the case study tackled in this paper, the ABC algorithm is utilized to solve the problem. The results of the binary ABC algorithm are approved by management experts.

Robust Solution Approach for the Dynamic and Stochastic Vehicle Routing Problem

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.

Optimization for routing vehicles of seafood product transportation

Recently, increasing usage of marine products is creating new challenges for businesses of marine products in terms of transportation that used to carry the marine products like seafood to the main warehouse. This can be a problem if the carrier fleet is limited, and there are time constraints in terms of the freshness of the marine product. There are many ways to solve this problem, including the optimization of routing vehicles. In this study, this strategy is to implement in the marine product business in Indonesia with such an expected arrangement of the company to optimize routing problem in transportation with time and capacity windows. Until now, the company has not used the scientific method to manage the routing of their vehicle from warehouse to the location of marine products source. This study will solve a stochastic Vehicle Routing Problems (VRP) with time and capacity windows by using the comparison of six methods and looking the best results for the optimization, in this situation the company could choose the best method, in accordance with the existing condition. In this research, we compared the optimization with another method such as branch and bound, dynamic programming and Ant Colony Optimization (ACO). Finally, we get the best result after running ACO algorithm with existing travel time data. With ACO algorithm was able to reduce vehicle travel time by 3189.65 minutes, which is about 23% less than existing and based on consideration of the constraints of time within 2 days (including rest time for the driver) using 28 tons capacity of truck and the companies need two units of vehicles for transportation.

Stochastic single vehicle routing problem with ordered customers and partial fulfilment of demands

ABSTRACTWe consider the problem of finding the optimal routing of a single vehicle that starts its route from a depot and delivers a product to N customers that are served according to a particular order. The vehicle during its route can return to the depot for replenishment. It is assumed a stochastic demand for each customer. The actual demand of each customer becomes known upon the vehicle's arrival at the customer's site. It is permissible to satisfy fully or to satisfy partially or not to satisfy the demand of a customer. The cost structure includes travel costs between consecutive customers, travel costs between the customers and the depot and penalty costs if a customer's demand is not satisfied or if it is satisfied partially. A dynamic programming algorithm is developed for the determination of the optimal routing policy. It is shown that the optimal routing policy has a specific threshold-type structure. Furthermore, if we consider the same problem without the assumption that the customers are ordered, numerical experiments indicate that the optimal routing strategy can be computed for N smaller or equal to nine.