Resource Constrained Elementary Shortest Path Problem Research Articles

Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization

We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation-based linear-programming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resource-constrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30-minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider general-purpose algorithms for solving VRPs, including the column-generation approach for the linear programming relaxations of the integer programs of VRPs and the column-enumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a random-coloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the column-generation approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the column-enumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches.

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.

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.

Branch and Price and Cut for the Split-Delivery Vehicle Routing Problem with Time Windows and Linear Weight-Related Cost

This paper addresses a new vehicle routing problem that simultaneously involves time windows, split delivery, and linear weight-related cost. This problem is a generalization of the split delivery vehicle routing problem with time windows (SDVRPTW), which consists of determining a set of least-cost vehicle routes to serve all customers while respecting the restrictions of vehicle capacity and time windows. The travel cost per unit distance is a linear function of the vehicle weight and the customer demand can be fulfilled by multiple vehicles. To solve this problem, we propose an exact branch-and-price-and-cut algorithm, where the pricing subproblem is a variant of the resource-constrained elementary shortest path problem. We first prove that at least one optimal solution to the pricing subproblem is associated with an extreme delivery pattern, and then we design a tailored and novel label-setting algorithm to solve the pricing subproblem. Computational results show that our proposed algorithm can handle both the SDVRPTW and our new problem effectively.

The multi-vehicle traveling purchaser problem with pairwise incompatibility constraints and unitary demands: A branch-and-price approach

In this work, we study a supplier selection and routing problem where a fleet of homogeneous vehicles with a predefined capacity is available for procuring different products from different suppliers with the aim to satisfy demand at the minimum traveling and purchasing cost. Decisions are further complicated by the presence of pairwise incompatibility constraints among products, implying the impossibility of loading two incompatible products on the same vehicle. The problem is known as the Multi-Vehicle Traveling Purchaser Problem with Pairwise Incompatibility Constraints. We study the special case in which the demand for each product is unitary and propose a column generation approach based on a Dantzig–Wolfe reformulation of the problem, where each column represents a feasible vehicle route associated with a compatible purchasing plan. To solve the pricing problem we propose a hybrid strategy exploiting the advantages of two alternative exact methods, a labeling algorithm solving a Resource-Constrained Elementary Shortest Path Problem on an expanded graph, and a tailored branch-and-cut algorithm. Due to the integrality request on variables, we embed the column generation in a branch-and-bound framework and propose different branching rules. Extensive tests, carried out on a large set of instances, show that our branch-and-price method performs well, improving on average, both in quality and in computational time, solutions obtained by a state-of-art branch-and-cut approach applied to a three-index connectivity constraints based formulation.

A Multiple-Starting-Path Approach to the Resource-Constrainedkth Elementary Shortest Path Problem

The resource-constrained elementary shortest path problem (RCESPP) aims to determine the shortest elementary path from the origin to the sink that satisfies the resource constraints. The resource-constrainedkth elementary shortest path problem (RCKESPP) is a generalization of the RCESPP that aims to determine thekth shortest path when a set ofk-1shortest paths is given. To the best of our knowledge, the RCKESPP has been solved most efficiently by using Lawler’s algorithm. This paper proposes a new approach named multiple-starting-path (MSP) to the RCKESPP. The computational results indicate that the MSP approach outperforms Lawler’s algorithm.

Branch-and-price-and-cut for the multiple traveling repairman problem with distance constraints

In this paper, we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel; the resulting problem is called the multiple traveling repairmen problem with distance constraints (MTRPD). In the MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers. Each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once. The objective is to minimize the total waiting time of all customers after the vehicles leave the depot. To optimally solve the MTRPD, we propose a new exact branch-and-price-and-cut algorithm, where the column generation pricing subproblem is a resource-constrained elementary shortest-path problem with cumulative costs. An ad hoc label-setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem. Computational results show the effectiveness of the proposed method. The optimal solutions to 179 out of 180 test instances are reported in this paper. Our computational results serve as benchmarks for future researchers on the problem.

A Column Generation Algorithm for a Rich Vehicle-Routing Problem

We present an optimization algorithm developed for a provider of software-planning tools for distribution logistics companies. The algorithm computes a daily plan for a heterogeneous fleet of vehicles that depart from different depots and must visit a set of customers for delivery operations. In this rich vehicle-routing problem (VRP), we consider multiple capacities; time windows associated with depots and customers; incompatibility constraints between goods, depots, vehicles, and customers; maximum route length and duration; upper limits on the number of consecutive driving hours and compulsory drivers' rest periods; the possibility of skipping some customers and using express courier services instead of the given fleet to fulfill some orders; the option of splitting up the orders; and the possibility of open routes that do not terminate at depots. Moreover, the cost of each vehicle route is computed through a system of fees depending on the locations visited by the vehicle, the distance traveled, the vehicle load, and the number of stops along the route. We developed a column generation algorithm where the pricing problem is a particular resource-constrained elementary shortest-path problem, solved through a bounded bidirectional dynamic programming algorithm. We describe how to encode the cost function and the complicating constraints by an appropriate use of resources, and we present computational results on real instances obtained from the software company.

A method of solving a large-scale rolling batch scheduling problem in steel production using a variant of column generation

This paper presents a transparent model and a solution approach to solve a large-scale rolling batch scheduling problem. First, the problem is formulated as a multiple routes problem with multi-objective ( MRMOP). By defining a hierarchical cost structure it is natural to decompose the MRMOP into several well-studied sub-problems, i.e. the multiple routes minimum cost problem ( MRMCP), the knapsack problem ( KP) and the linear assignment problem ( LAP). Among these sub-problems the MRMCP is considered as the central one and is tackled first of all. The solution procedure for the MRMCP is based on a partial set-partitioning formulation. It makes use of a variant of column generation. Feasible column is generated as needed by solving a resource constrained elementary shortest path problem (RC-ESPP) by a mixed strategy combing an exact method and heuristics. Then a procedure called Adding-Node is introduced to implicitly solve the KP starting from the solution of the MRMCP. Finally, we solve the LAP with Hungarian algorithm to consider the total tardiness and earliness of the production. Computational results are presented compared with several promising methods on benchmark problems and production orders from Shanghai Baoshan Iron and Steel Complex. The results demonstrate the efficiency of the proposed algorithm.

New dynamic programming algorithms for the resource constrained elementary shortest path problem

AbstractThe resource constrained elementary shortest path problem (RCESPP) arises as a pricing subproblem in branch‐and‐price algorithms for vehicle‐routing problems with additional constraints. We address the optimization of the RCESPP and we present and compare three methods. The first method is a well‐known exact dynamic‐programming algorithm improved by new ideas, such as bidirectional search with resource‐based bounding. The second method consists in a branch‐and‐bound algorithm, where lower bounds are computed by dynamic‐programming with state‐space relaxation; we show how bounded bidirectional search can be adapted to state‐space relaxation and we present different branching strategies and their hybridization. The third method, called decremental state‐space relaxation, is a new one; exact dynamic‐programming and state‐space relaxation are two special cases of this new method. The experimental comparison of the three methods is definitely favorable to decrement state‐space relaxation. Computational results are given for different kinds of resources, arising from the capacitated vehicle‐routing problem, the vehicle‐routing problem with distribution and collection, and the vehicle‐routing problem with capacities and time windows. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008