Robust Vehicle Routing Research Articles

Delay-resistant robust vehicle routing with heterogeneous time windows

We consider a robust variant of the vehicle routing problem with heterogeneous time windows (RVRP-HTW) with a focus on delay-resistant solutions. Here, customers have different availability time windows for every vehicle and must be provided with a preferably tight appointment window for the planned service. Different vehicles are a possibility to model different days on which one physical vehicle can serve a customer. This is the main reason why different time windows for different vehicles are of high practical relevance. To ensure that the appointment windows are adhered to as much as possible, we introduce a new objective function that penalizes delays. Our novel approach allows us to find solutions that are robust with respect to uncertainties in travel and service times limited by a budget polytope. We present a set-partitioning model, the solution of which is based on column generation and employs a labeling algorithm that integrates robustness into the calculations and is adapted to our problem-specific constraints. In a Monte-Carlo simulation on real-life data, we evaluate this method in terms of runtime and solution quality. Our solutions show very good performance, even if the data is more uncertain than assumed for the optimization, incurring only marginal extra travel time compared to a naive deterministic planning scheme.

Robust vehicle routing with drones under uncertain demands and truck travel times in humanitarian logistics

Resource transport in the aftermath of disasters is critical, yet in the absence of sufficient historical data or accurate forecasting approaches, the development of resource transport strategies often faces the challenge of dealing with uncertainty, especially uncertainties in demand and travel time. In this paper we investigate the vehicle routing problem with drones under uncertain demands and truck travel times. Specifically, there is a set of trucks and drones (each truck is associated with a drone) collaborating to transport relief resources to the affected areas, where a drone can be launched from its associated truck at a node, independently transporting relief resources to one or more of the affected areas, and returning to the truck at another node along the truck route. For this problem, we present a tailored robust optimization model based on the well-known budgeted uncertainty set, and develop an enhanced branch-and-price-and-cut algorithm incorporating a bounded bidirectional labelling algorithm to solve the pricing problem, which can be modelled as a robust resource-constrained vehicle and drone synthetic shortest path problem. To enhance the performance of the algorithm, we employ subset-row inequalities to tighten the lower bound and incorporate some enhancement strategies to quickly solve the pricing problem. We perform extensive numerical studies to assess the performance of the developed algorithm, discuss the benefits of considering uncertainty and robustness, and analyse the impacts of key model parameters on the optimal solution. We also evaluate the benefits of the truck–drone collaborative transport mode over the truck-only transport mode through a real case study of the 2008 earthquake in Wenchuan, China.

Analysis of box and ellipsoidal robust optimization, and attention model based reinforcement learning for a robust vehicle routing problem

Analysis of box and ellipsoidal robust optimization, and attention model based reinforcement learning for a robust vehicle routing problem

Data‐Driven Robust Optimization of the Vehicle Routing Problem with Uncertain Customers

With the increasing proportion of the logistics industry in the economy, the study of the vehicle routing problem has practical significance for economic development. Based on the vehicle routing problem (VRP), the customer presence probability data are introduced as an uncertain random parameter, and the VRP model of uncertain customers is established. By optimizing the robust uncertainty model, combined with a data‐driven kernel density estimation method, the distribution feature set of historical data samples can then be fitted, and finally, a distributed robust vehicle routing model for uncertain customers is established. The Q‐learning algorithm in reinforcement learning is introduced into the high‐level selection strategy using the hyper‐heuristic algorithm, and a hyper‐heuristic algorithm based on the Q‐learning algorithm is designed to solve the problem. Compared with the certain method, the distributed robust model can effectively reduce the total cost and the robust conservatism while ensuring customer satisfaction. The improved algorithm also has good performance.

Ride-Sharing Matching Under Travel Time Uncertainty Through Data-Driven Robust Optimization

In ride-sharing services, travel time uncertainty significantly impacts the quality of matching solutions for both the drivers and the riders. This paper studies a one-to-many ride-sharing matching problem where travel time between locations is uncertain. The goal is to generate robust ride-sharing matching solutions that minimize the total driver detour cost and the number of unmatched riders. To this end, we formulate the ride-sharing matching problem as a robust vehicle routing problem with time window (RVRPTW). To effectively capture the travel time uncertainty, we propose a deep learning-based data-driven approach that can dynamically estimate the uncertainty sets of travel times. Given the NP-hard nature of the optimization problem, we design a hybrid meta-heuristic algorithm that can handle large-scale instances in a time-efficient manner. To evaluate the performance of the proposed method, we conduct a set of numeric experiments based on real traffic data. The results confirm that the proposed approach outperforms the non-data-driven one in several important performance metrics, including a proper balance between robustness and inclusiveness of the matching solution. Specifically, by applying the proposed data-driven approach, the matching solution violation rate can be reduced up to 85.8%, and the valid serving rate can be increased up to 42.3% compared to the non-data-driven benchmark.

Robust vehicle routing under uncertainty via branch-price-and-cut

This paper contemplates how branch-price-and-cut solvers can be employed along with the robust optimization paradigm to address parametric uncertainty in the context of vehicle routing problems. In this setting, given postulated uncertainty sets for customer demands and vehicle travel times, one aims to identify a set of cost-effective routes for vehicles to traverse, such that the vehicle capacities and customer time window constraints are respected under any anticipated demand and travel time realization, respectively. To tackle such problems, we propose a novel approach that combines cutting-plane techniques with an advanced branch-price-and-cut algorithm. Specifically, we use deterministic pricing procedures to generate “partially robust” vehicle routes and then utilize robust versions of rounded capacity inequalities and infeasible path elimination constraints to guarantee complete robust feasibility of routing designs against demand and travel time uncertainty. In contrast to recent approaches that modify the pricing algorithm, our approach is both modular and versatile. It permits the use of advanced branch-price-and-cut technologies without significant modification, while it can admit a variety of uncertainty sets that are commonly used in robust optimization but could not be previously employed in a branch-price-and-cut setting.

Robust counterpart open capacitated vehicle routing problem with hard time windows on waste transportation problems

Based on the problem of garbage transportation routes, this research proposes a robust counterpart open capacitated vehicle routing problems (RCOCVRP) with hard time windows. Garbage transport vehicles sometimes have delays. This model is a generalization of robust capacitated vehicle routing problems, robust open vehicle routing problems and robust vehicle routing problems with time windows. RCOCVRP with hard time windows for waste transportation is a robust optimization problem that aims to minimize the distance of vehicle routes for waste volume and time windows with uncertainty The mathematical formulation built is completed using a sequential insertion algorithm. Then, the waste transportation route obtained is completed using LINGO software. The results of this model on the problem of transporting garbage show that the total distance using the sequential insertion algorithm is more minimum than using LINGO.

The Robust Vehicle Routing Problem with Time Window Assignments

In practice, there are several applications in which logistics service providers determine the service time windows at the customers, for example, in parcel delivery, retail, and repair services. These companies face uncertain travel times and service times that have to be taken into account when determining the time windows and routes prior to departure. The objective of the proposed robust vehicle routing problem with time window assignments (RVRP-TWA) is to simultaneously determine routes and time window assignments such that the expected travel time and the risk of violating the time windows are minimized. We assume that the travel time probability distributions are not completely known but that some statistics, such as the mean, minimum, and maximum, can be estimated. We extend the robust framework based on the requirements’ violation index, which was originally developed for the case where the specific requirements (time windows) are given as inputs, to the case where they are also part of the decisions. The subproblem of finding the optimal time window assignment for the customers in a given route is shown to be convex, and the subgradients can be derived. The RVRP-TWA is solved by iteratively generating subgradient cuts from the subproblem that are added in a branch-and-cut fashion. Experiments address the performance of the proposed solution approach and examine the trade-off between expected travel time and risk of violating the time windows.

Agricultural rout planning with variable rate pesticide application in a greenhouse environment

The use of robotics in executing agricultural tasks has significantly improved productivity over the years as a result of automation in performing such activities as spray, harvesting, planting etc. In order to optimize both crop yield and quality while minimizing costs, there will be need for the application of navigation strategies. These will provide optimal as well as autonomous navigation capability which is built entirely upon field coverage plan thereby making robot navigation approach a paramount scheme. In this paper, the autonomy of an agricultural mobile robot is enhanced in a structured environment (greenhouse farm) to locate an optimum route such that the robot performs a selective and variable spray of pesticides to the plants. To realize this, a robust vehicle routing problem (VRP) scheme is designed to navigate the robot autonomously while making intelligent decisions to fulfil the pesticide demands at each node (infected plants). The improved non-dominated sorting genetic algorithm (INSGA-III) is adopted to solve this fully integer problem based on three (3) test cases carried out with 8, 32 and 56 infected plants respectively for validation. The results obtained show a trade-off solution as the Optimal INSGA-III is significantly lower than NSGA-III in terms of solution quality. On the other hand, a significant reduction in run times of between 66% and 76% and 76–93% was obtained for all test case scenarios for population sizes of 100 and 1500 respectively.

Interval travel times for robust synchronization in city logistics vehicle routing

We propose a consistent information flow to integrate Interval Travel Times (ITT) into a city logistics vehicle routing problem with satellites. The consistent information flow encompasses the derivation of ITT, the usage of ITT in robust shortest path computation and the integration of ITT into robust vehicle routing. Based on the results of the information flow, robust meeting times of first-tier and second-tier vehicles are created. We evaluate the approach with large real-world traffic data sets. We observe that the integration of ITT leads to first-tier routes that are reasonably efficient and robust.

Exact approaches to the robust vehicle routing problem with time windows and multiple deliverymen

This paper addresses the vehicle routing problem with time windows and multiple deliverymen (VRPTWMD) under uncertain demand as well as uncertain travel and service times. This variant is faced by logistics companies that deliver products to retailers located in congested urban areas, where service times are relatively long compared to travel times, and depend on the number of deliverymen assigned to each route. Differently from traditional variants, these service times show high variability, requiring an appropriate way of handling the related uncertainty. We extend two mathematical formulations to represent the VRPTWMD under uncertainty, using the robust optimization paradigm with budgeted uncertainty sets, and developed effective exact solution methods for solving each of them. The first formulation is a robust vehicle flow model solved by a tailored branch-and-cut algorithm that resorts to 1- and 2-path inequalities that we show how to effectively separate. The second formulation is a set partitioning model, for which we propose a branch-price-and-cut algorithm that relies on a robust resource-constrained elementary shortest path problem. The results of computational experiments using instances from the literature and risk analysis via a Monte Carlo simulation show the importance of incorporating uncertainties in the VRPTWMD, and indicate the sensitivity of decisions as well as cost and risk to the level of uncertainty in the input data.

Discrete scenario-based optimization for the robust vehicle routing problem: The case of time windows under delay uncertainty

In this paper we propose a robust optimization strategy for the vehicle routing problem with time windows, where the travel time is considered as uncertain. The objective is to minimize the risk of delay related to the time windows constraint. We first present a robust model in which the uncertain travel time is related to a discrete set of scenarios: each scenario can be viewed as an observation of time required to complete a current route. Such a model is based upon a mixed-integer linear programming that can handle an optimal (solution) value related to the worst observation of the total travel time over all available scenarios. Second, in order to evaluate the effects on the scenario model, a guided neighborhood search-based heuristic is adapted for solving the model and tested on a variety of instances obtained by using Solomon’s standard generator, where a total of 5040 instances are considered. Finally, we study the behavior of the adapted algorithm to solve the robust model and highlight the interaction between the number of scenarios used which are based upon uncertain events and the uncertainty of the problem.

A simulation-based solution approach for the robust capacitated vehicle routing problem with uncertain demands

ABSTRACT This article introduces a solution approach for the Stochastic Capacitated Vehicle Routing Problem (SCVRP) with uncertain demands, called Robust Simulation-Based (RoSi) approach. RoSi aims at designing route plans that can be more or less robust based on a decision-maker weight, i.e. solutions that resist demand changes with marginal additional (recourse) cost. For that, RoSi combines simulation with heuristics. It transforms a complex SCVRP into a set of deterministic ones, where well-known heuristics can be applied, computing a set of feasible solutions. These solutions are assessed by Monte Carlo simulation, and the one that deals better with demand fluctuation is selected as the final solution. The efficiency of RoSi is compared with those of three methods in the literature: Integer Linear Programming (ILP) model, Stochastic Programming with Recourse (SPR) model, and Robust Bi-Objective (RoBi) approach through numerical experiments. The results show that RoSi outperforms these methods in most scenarios.

The robust vehicle routing problem with time windows: Solution by branch and price and cut

We study the vehicle routing problem with time windows under demand uncertainty. Such a problem arises in fuel delivery to gas stations, to farms, or to production plants. We suggest a robust formulation where uncertain demand has support defined by cardinality constrained sets. The model ensures the feasibility of each route when demand values on the route change in predefined set. The range is customer specific, does not assume any probability distribution of demand, and may be estimated using historical demand data. We develop a branch-and-price-and-cut algorithm where the pricing problem is a robust resource constrained shortest path problem. We extend the rounded capacity inequalities to the robust case and suggest a separation procedure that solves a robust bin packing problem. The bound is further strengthened using 2-path and subset row inequalities and the full algorithm is tested on an extension of the benchmark Solomon instances. The robust schedules are simulated and compared to schedules obtained by solving the deterministic problem under varying levels and distributions of uncertainty. Simulation results show that the robust model provides superior protection against demand uncertainty. Over all scenarios, only 0.57% of the robust solutions turn out to be infeasible compared to 41.62% of the deterministic ones. Such protection against infeasibility comes with an 11.22% increase in routing costs on average.

The Robust Vehicle Routing Problem with Time Windows: Compact Formulation and Branch-Price-and-Cut Method

We address the robust vehicle routing problem with time windows (RVRPTW) under customer demand and travel time uncertainties. As presented thus far in the literature, robust counterparts of standard formulations have challenged general-purpose optimization solvers and specialized branch-and-cut methods. Hence, optimal solutions have been reported for small-scale instances only. Additionally, although the most successful methods for solving many variants of vehicle routing problems are based on the column generation technique, the RVRPTW has never been addressed by this type of method. In this paper, we introduce a novel robust counterpart model based on the well-known budgeted uncertainty set, which has advantageous features in comparison with other formulations and presents better overall performance when solved by commercial solvers. This model results from incorporating dynamic programming recursive equations into a standard deterministic formulation and does not require the classical dualization scheme typically used in robust optimization. In addition, we propose a branch-price-and-cut method based on a set partitioning formulation of the problem, which relies on a robust resource-constrained elementary shortest path problem to generate routes that are robust regarding both vehicle capacity and customer time windows. Computational experiments using Solomon’s instances show that the proposed approach is effective and able to obtain robust solutions within a reasonable running time. The results of an extensive Monte Carlo simulation indicate the relevance of obtaining robust routes for a more reliable decision-making process in real-life settings.

A hybrid multi-objective evolutionary optimization approach for the robust vehicle routing problem

In this paper, we propose to approximately solve the robust vehicle routing problem with a population-based method. Uncertainty can be modeled by a set of scenarios where each scenario may represent the travel costs assigned to all visited arcs of the graph associated to the problem. Unlike several existing methods that often aggregate multiple objectives into a compromise function, the goal of the proposed approach is to simultaneously optimize both the number of vehicles to use and the worst total travel cost needed. The proposed method can be viewed as a new version of an evolutionary approach which is reinforced with a “strong-diversification”. Such a strategy is based upon destroying and re-building procedures that are hybridized with a local search using a series of move operators. A number of experiments have been conducted to assess the performance of the proposed approach. Its achieved results have been tested on benchmark instances extracted from the literature and compared to those reached by the-state-of-the-art GLPK solver and one of the most recent method available in the literature. The proposed method remains competitive, where encouraging results have been obtained.

Robust vehicle routing problem with hard time windows under demand and travel time uncertainty

Due to an increase in customer-oriented service strategies designed to meet more complex and exacting customer requirements, meeting a scheduled time window has become an important part of designing vehicle routes for logistics activities. However, practically, the uncertainty in travel times and customer demand often means vehicles miss these time windows, increasing service costs and decreasing customer satisfaction. In an effort to find a solution that meets the needs of real-world logistics, we examine the vehicle routing problem with hard time windows under demand and travel time uncertainty. To address the problem, we build a robust optimization model based on novel route-dependent uncertainty sets. However, due to the complex nature of the problem, the robust model is only able to tackle small-sized instances using standard solvers. Therefore, to tackle large instances, we design a two-stage algorithm based on a modified adaptive variable neighborhood search heuristic. The first stage of the algorithm minimizes the total number of vehicle routes, while the second stage minimizes the total travel distance. Extensive computational experiments are conducted with modified versions of Solomon’s benchmark instances. The numerical results show that the proposed two-stage algorithm is able to find optimal solutions for small-sized instances and good-quality robust solutions for large-sized instances with little increase to the total travel distance and/or the number of vehicles used. A detailed analysis of the results also reveals several managerial insights for decision-makers in the logistics industry.

Heuristics for the robust vehicle routing problem with time windows

Uncertainty is frequently present in logistics and transportation, where vehicle routing problems play a crucial role. However, due to the complexity inherent in dealing with uncertainty, most research has been devoted to deterministic problems. This paper considers a robust version of the vehicle routing problem with hard time windows, in which travel times are uncertain. A budget polytope uncertainty set describes the travel times, to limit the maximum number of sailing legs that can be delayed. This makes sure that improbable scenarios are not considered, while making sure that solutions are immune to delays on a given number of sailing legs. Existing exact methods are only able to solve small instances of the problem and can be computationally demanding. With the aim of solving large instances with reduced running times, this paper proposes an efficient heuristic based on adaptive large neighborhood search. The computational study performed on instances with different uncertainty levels compares and analyzes the performance of four versions of the heuristic and shows how good quality solutions can be obtained within short computational times.

Solving the bi-objective Robust Vehicle Routing Problem with uncertain costs and demands

In this paper, a bi-objective Vehicle Routing Problem (bi-RVRP) with uncertainty in both demands and travel times is studied by means of robust optimization. Uncertain demands per customer are modeled by a discrete set of scenarios representing the deviations from an expected demand, while uncertain travel times are independent from customer demands. Then, traffic records are considered to get discrete scenarios to each arc of the transportation network. Here, the bi-RVRP aims at minimizing the worst total cost of traversed arcs and minimizing the maximum total unmet demand over all scenarios. As far as we know, this is the first study for the bi-RVRP which finds practical applications in urban transportation, e.g. , serving small retail stores. To solve the problem, different variations of solution approaches, coupled with a local search procedure are proposed: the Multiobjective Evolutionary Algorithm (MOEA) and the Non-dominated Sorting Genetic Algorithm (NSGAII). Different metrics are used to measure the algorithmic performance, the convergence, as well as the diversity of solutions for the different methods.

Models and Algorithms for Stochastic and Robust Vehicle Routing with Deadlines

We consider the vehicle routing problem with deadlines under travel time uncertainty in the contexts of stochastic and robust optimization. The problem is defined on a directed graph where a fleet of vehicles is required to visit a given set of nodes and deadlines are imposed at a subset of nodes. In the stochastic vehicle routing problem with deadlines (SVRP-D), the probability distribution of the travel times is assumed to be known and the problem is solved to minimize the sum of probability of deadline violations. In the robust vehicle routing problem with deadlines (RVRP-D), however, the exact probability distribution is unknown but it belongs to a certain family of distributions. The objective of the problem is to optimize a performance measure, called lateness index, which represents the risk of violating the deadlines. Although novel mathematical frameworks have been proposed to solve these problems, the size of the problem that those approaches can handle is relatively small. Our focus in this paper is the computational aspects of the two solution schemes. We introduce formulations that can be applied for the problems with multiple capacitated vehicles and discuss the extensions to the cases of incorporating service times and soft time windows. Furthermore, we develop an algorithm based on a branch-and-cut framework to solve the problems. The experiments show that these approaches provide substantial improvements in computational efficiency compared to the approaches in the literature. Finally, we provide a computational comparison to evaluate the solution quality of the SVRP-D and the RVRP-D. The results show that the RVRP-D produces solutions that are very competitive to those obtained by the SVRP-D with a large number of scenarios, whereas much less sensitive to the distributional uncertainty.