Shortest Path Problem With Resource Constraints Research Articles

Partial dominance in branch-price-and-cut algorithms for vehicle routing and scheduling problems with a single-segment tradeoff

AbstractFor many variants of vehicle routing and scheduling problems solved by a branch-price-and-cut (BPC) algorithm, the pricing subproblem is an elementary shortest-path problem with resource constraints (SPPRC) typically solved by a dynamic-programming labeling algorithm. Solving the SPPRC subproblems consumes most of the total BPC computation time. Critical to the performance of the labeling algorithms and thus the BPC algorithm as a whole is the use of effective dominance rules. Classical dominance rules rely on a pairwise comparison of labels and have been used in many labeling algorithms. In contrast, partial dominance describes situations where several labels together are needed to dominate another label, which can then be safely discarded. In this work, we consider SPPRCs, where a linear tradeoff describes the relationship between two resources. We derive a unified partial dominance rule to be used in ad hoc labeling algorithms for solving such SPPRCs as well as insights into its practical implementation. We introduce partial dominance for two important variants of the vehicle routing problem, namely the electric vehicle routing problem with time windows with a partial recharge policy and the split-delivery vehicle routing problem with time windows (SDVRPTW). Computational experiments show the effectiveness of the approach, in particular for the SDVRPTW, leading to an average reduction of 20% of the total BPC computation time, with savings of 30% for the more difficult instances requiring more than 600 s of computation time.

A Dedicated Pricing Algorithm to Solve a Large Family of Nurse Scheduling Problems with Branch-and-Price

In this paper, we describe a branch-and-price algorithm for the personalized nurse scheduling problem. The variants that appear in the literature involve a large number of constraints that can be hard or soft, meaning that they can be violated at the price of a penalty. We capture the diversity of the constraints on individual schedules by seven generic constraints characterized by lower and upper bounds on a given quantity. The core of the column generation procedure is in the identification of individual schedules with minimum reduced cost. For this, we solve a shortest path problem with resource constraints (SPPRC) where several generic constraints are modeled as resource constraints. We then describe dominance rules adapted to the presence of both upper and lower bounds on the resources and leverage soft constraints to improve the dominance. We also describe several acceleration techniques for the solution of the SPPRC, and branching rules that fit the specificities of the problem. Our numerical experiments are based on the instances of three benchmarks of the literature including those of the two international nurse rostering competitions (INRC-I and INRC-II). Their objective is threefold: assess the dominance rules and the acceleration techniques, investigate the capacity of the algorithm to find provable optimal solutions of instances that are still open, and conduct a comparison with best published results. The most noticeable conclusion is that the improved solution of the SPPRC allows to solve optimally all the INRC-II instances where a four-week planning horizon is considered and 40% of the eight-week instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0019 .

A multiphase dynamic programming algorithm for the shortest path problem with resource constraints

The shortest path problem with resource constraints (SPPRC) finds a least cost path between two nodes in a network while respecting constraints on resource consumption. The problem is mainly used as a subproblem inside column generation for crew scheduling and vehicle routing problems. The standard approach for the subproblems is based on dynamic programming (DP). This class of methods is generally effective in practice when there are only a few resources, but it seems to be time-consuming for huge instances with many resources. To handle this issue, we propose a new exact primal algorithm called the multiphase dynamic programming algorithm (MPDPA) to solve the SPPRC in acyclic networks. The proposed approach splits the state-space into small disjoint subspaces. These subspaces are sequentially explored in several iterations, where each iteration builds on the previous ones, to reduce the dimension of the subspaces to explore and to quickly generate better paths. Computational experiments on vehicle and crew scheduling instances show the excellent performance of our approach compared to the standard DP method. On the one hand, MPDPA returns optimal solutions while achieving time reduction factors between 1.44 and 3.59 on average compared to DP. On the other hand, MPDPA is able to generate feasible paths with up to 90% of the optimal cost in less than 10% of the time required by standard DP. This feature is useful in column generation and may greatly reduce the computational effort, because we can stop the MPDPA solution process once columns with sufficiently negative reduced costs are obtained.

Branch-Price-and-Cut-Based Solution of Order Batching Problems

Given a set of customer orders each comprising one or more individual items to be picked, the order batching problem (OBP) in warehousing consists of designing a set of picking batches such that each customer order is assigned to exactly one batch, all batches satisfy the capacity restriction of the pickers, and the total distance traveled by the pickers is minimal. In order to collect the items of a batch, the pickers traverse the warehouse using a predefined routing strategy. We propose a branch-price-and-cut (BPC) algorithm for the exact solution of the OBP investigating the routing strategies traversal, return, midpoint, largest gap, combined, and optimal. The column-generation pricing problem is modeled as a shortest path problem with resource constraints (SPPRC) that can be adapted to handle the implications from nonrobust valid inequalities and branching decisions. The SPPRC pricing problem is solved by means of an effective labeling algorithm that relies on strong completion bounds. Capacity cuts and subset-row cuts are used to strengthen the lower bounds. Furthermore, we derive two BPC-based heuristics to identify high-quality solutions in short computation times. Extensive computational results demonstrate the effectiveness of the proposed methods. The BPC is faster by two orders of magnitude compared with the state-of-the-art exact approach and can solve to optimality hundreds of instances that were previously unsolved. The BPC-based heuristics are able to significantly improve the gaps reported for the state-of-the-art heuristic and provide hundreds of new best-known solutions. Funding: This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) [Grant 418727865]. This support is gratefully acknowledged. Supplemental Material: The e-companion is available at https://doi.org/10.1287/trsc.2023.1198 .

Approximate solution of the shortest path problem with resource constraints and applications to vehicle routing problems

<abstract><p>Vehicle routing problem (VRP) is a fundamental combinatorial optimization and integer programming problem with several important applications. The VRP is usually solved by using branch-and-bound techniques requiring solving a shortest path problem with resource constraints (SPPRC) and the determination of a lower bound, which can be computed by using column generation. The SPPRC entails finding the minimum cost elementary path in a valuated graph that is subject to constraints on resource consumption. The proposed exact solutions to this hard NP-hard problem require an excessive computation time which increases with the number of resources. In this paper, we propose a new approximate resolution of the SPPRC for acyclic and cyclic graphs. Our method is based on a Lagrangian relaxation of a subset of the constraints and using dominance only on a subset of the resources. This reduces the search space and allows users to efficiently compute solutions used to improve the column generation procedure. Extensive evaluation and comparison to the classical exact method show that the proposed algorithm achieves a good compromise between efficiency and quality of the SPPRC and the VRP solutions. Thus, our method can be used for practical large-scale VRP applications.</p></abstract>

A Primal Adjacency-Based Algorithm for the Shortest Path Problem with Resource Constraints

The shortest path problem with resource constraints (SPPRC) is often used as a subproblem within a column-generation approach for routing and scheduling problems. It aims to find a least-cost path between the source and the destination nodes in a network while satisfying the resource consumption limitations on every node. The SPPRC is usually solved using dynamic programming. Such approaches are effective in practice, but they can be inefficient when the network is large and especially when the number of resources is high. To cope with this major drawback, we propose a new exact primal algorithm to solve the SPPRC defined on acyclic networks. The proposed algorithm explores the solution space iteratively using a path adjacency–based partition. Numerical experiments for vehicle and crew scheduling problem instances demonstrate that the new approach outperforms both the standard dynamic programming and the multidirectional dynamic programming methods.

Primal column generation framework for vehicle and crew scheduling problems

AbstractThe primal adjacency‐based algorithm and the multidirectional dynamic programming algorithm are two exact methods that have recently been developed to efficiently solve the shortest path problem with resource constraints (SPPRCs). These methods are primal in the sense that they are able to produce sequences of feasible solutions using iterative exploration of the search space. Since the SPPRCs often appear as a subproblem (SP) in the solution of vehicle and crew scheduling problems (VCSP) using column generation (CG), we propose a new primal column generation framework that embeds these primal methods in a CG scheme. The primal column generation solves at each iteration a sequence of appropriate restricted SP and stops solving the SP when there is no need to continue. This approach introduces a large degree of flexibility, and allows performing good cost improvements in a very limited time. Computational experiments on VCSP instances show that the proposed approach is able to find optimal solutions while reducing the time spent solving the SP by factors of up to seven compared to the standard CG algorithm. This leads to significant improvements in the overall solution times, with an average reduction factor of 3.5.

Exact solution of the soft-clustered vehicle-routing problem

The soft-clustered vehicle-routing problem (SoftCluVRP) extends the classical capacitated vehicle-routing problem by one additional constraint: The customers are partitioned into clusters and feasible routes must respect the soft-cluster constraint, that is, all customers of the same cluster must be served by the same vehicle. In this article, we design and analyze different branch-and-price algorithms for the exact solution of the SoftCluVRP. The algorithms differ in the way the column-generation subproblem, a variant of the shortest-path problem with resource constraints (SPPRC), is solved. The standard approach for SPPRCs is based on dynamic-programming labeling algorithms. We show that even with all the recent acceleration techniques (e.g., partial pricing, bidirectional labeling, decremental state space relaxation) available for SPPRC labeling algorithms, the solution of the subproblem remains extremely difficult. The main contribution is the modeling and solution of the subproblem using a branch-and-cut algorithm. The conducted computational experiments prove that branch-and-price equipped with this integer programming-based approach outperforms sophisticated labeling-based algorithms by one order of magnitude. The largest SoftCluVRP instances solved to optimality have more than 400 customers or more than 50 clusters.

Nested branch-and-price-and-cut for vehicle routing problems with multiple resource interdependencies

This paper considers vehicle routing problems (VRPs) with multiple resource interdependencies and addresses the development and computational evaluation of an exact branch-and-price-and-cut algorithm for their solution. An interdependency between two resources means that the two resource consumptions influence one another in such a way that a tradeoff exists between them. This impacts the feasibility and/or the cost of a solution.The subproblem in branch-and-price-and-cut procedures for VRPs is very often a variant of the shortest-path problem with resource constraints (SPPRC). For the exact solution of many SPPRC variants, dynamic-programming based labeling algorithms are predominant. The tradeoffs in problems with multiple resource interdependencies, however, render the application of labeling algorithms unpromising. This is because complex data structures for managing the tradeoff curves are necessary and only weak dominance criteria are possible, so that the labeling algorithm becomes almost a pure enumeration. Therefore, we propose to solve also the SPPRC subproblem with branch-and-price-and-cut. This results in a two-level, nested branch-and-price-and-cut algorithm. We analyze different variants of the algorithm to enable the exchange of columns and also rows between the different levels.To demonstrate the computational viability of our approach, we perform computational experiments on a prototypical VRP with time windows, minimal and maximal delivery quantities for each customer, a customer-dependent profit paid for each demand unit delivered, and temporal synchronization constraints between some pairs of customers. In this problem, tradeoffs exist between cost and load and between cost and time.

Stabilized branch-and-price algorithms for vector packing problems

This paper considers packing and cutting problems in which a packing/cutting pattern is constrained independently in two or more dimensions. Examples are restrictions with respect to weight, length, and value. We present branch-and-price algorithms to solve these vector packing problems (VPPs) exactly. The underlying column-generation procedure uses an extended master program that is stabilized by (deep) dual-optimal inequalities. While some inequalities are added to the master program right from the beginning (static version), other violated dual-optimal inequalities are added dynamically. The column-generation subproblem is a multidimensional knapsack problem, either binary, bounded, or unbounded depending on the specific master problem formulation. Its fast resolution is decisive for the overall performance of the branch-and-price algorithm. In order to provide a generic but still efficient solution approach for the subproblem, we formulate it as a shortest path problem with resource constraints (SPPRC), yielding the following advantages: (i) Violated dual-optimal inequalities can be identified as a by-product of the SPPRC labeling approach and thus be added dynamically; (ii) branching decisions can be implemented into the subproblem without deteriorating its resolution process; and (iii) larger instances of higher-dimensional VPPs can be tackled with branch-and-price for the first time. Extensive computational results show that our branch-and-price algorithms are capable of solving VPP benchmark instances effectively.

A shortest-path-based approach for the stochastic knapsack problem with non-decreasing expected overfilling costs

The knapsack problem (KP) is concerned with the selection of a subset of multiple items with known positive values and weights such that the total value of selected items is maximized and their total weight does not exceed capacity. Item values, item weights, and capacity are known in the deterministic case. We consider the stochastic KP (SKP) with stochastic item weights. For this variant of the SKP we combine the chance constrained KP (CCKP) and the SKP with simple recourse (SRKP). The chance constraint allows for a violation of capacity, but the probability of a violation beyond an imposed limit is constrained. The violation of the capacity constraint is also included in the objective function in terms of a penalty function as in the SRKP. Penalty is an increasing function of the expected number of units of violation with proportionality as a special case. We formulate the SKP as a network problem and demonstrate that it can be solved by a label-setting dynamic programming approach for the shortest path problem with resource constraints (SPPRC). We develop a dominance criterion for an elimination of states in the dynamic programming approach using only the deterministic value of items along with mean and variance of the stochastic weight of items corresponding to the associated paths in the underlying network. It is shown that a lower bound for the impact of potential extensions of paths is available as an additional means to limit the number of states provided the penalty cost of expected overtime is convex. Our findings are documented in terms of a computational study.

Asymmetry matters: Dynamic half-way points in bidirectional labeling for solving shortest path problems with resource constraints faster

With their paper “Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints” [Discrete Optimization 3, 2006, pp. 255–273] Righini and Salani introduced bounded bidirectional dynamic programming (DP) as an acceleration technique for solving variants of the shortest path problem with resource constraints (SPPRC). SPPRCs must be solved iteratively when vehicle routing and scheduling problems are tackled via Lagrangian relaxation or column-generation techniques. Righini and Salani and several subsequent works have shown that bounded bidirectional DP algorithms are often superior to their monodirectional counterparts, since the former can mitigate the fact that the number of labels increases strongly with the path length. Bidirectional DP has become a quasi-standard for solving SPPRCs with general resource extension functions. In computational experiments, however, one can still observe that the number of forward and backward label extensions is very unbalanced despite a symmetric bounding of a critical resource in the middle of its feasible domain. We exploit this asymmetry in forward and backward label extensions to reduce the overall workload by introducing a so-called dynamic half-way point, which is a dynamic bounding criterion based on the current state of the simultaneously solved forward and backward DPs. Experiments with the standard and the electric vehicle routing problem with time windows as well as the vehicle routing and truck driver scheduling problem confirm that dynamic half-way points better balance forward and backward labeling and reduce the overall runtime.

A branch and price approach for deployment of multi-tier software services in clouds

This paper considers a service deployment problem that combines service placement and replication level decisions in a cloud computing context. The services are composed of multiple components that are to be placed on nodes in the private cloud of the service provider or, if the private cloud has limited capacity, partly in a public cloud. In the service delivery, the provider has to take into account the quality of service guarantees offered to his end-users. To solve the problem, we develop a branch and price algorithm, where the subproblems both are formulated as a linear mixed integer program and a shortest path problem with resource constraints (SPPRC) on a network with a special structure. The SPPRC can be solved by an exact label-setting algorithm, but to speed up the solution process, we develop a heuristic label-setting algorithm based on a reduced network and simplified dominance rule. Our results show that using the heuristic subproblem solver is efficient. Furthermore, the branch and price algorithm performs better than a previously developed pre-generation algorithm for the same problem. In addition, we analyze and discuss the differences in solutions that utilize resources in a public cloud to different degrees. By conducting this analysis we are able to identify some essential characteristics of good solutions.

Mass customization design of engineer-to-order products using Benders’ decomposition and bi-level stochastic programming

Leveraging product differentiation and mass production efficiency in mass customization basically entails a configure-to-order paradigm. In the engineer-to-order (ETO) business, however, companies build unique products in response to `foreseeable' customer specifications. The key challenge of ETO mass customization lies in the complexity of accommodating future design changes when customers are involved in customizing design specifications. This paper proposes a two-stage, bi-level stochastic programming framework to tackle ETO mass customization. At the first stage, product platform configuration is integrated with production reconfiguration, which is formulated as a shortest path problem with resource constraints (SPPRC) to optimize production delays within the capabilities of the process platform. Benders' decomposition algorithm is applied to solve this optimal configuration problem owing to its high computational efficiency. The second stage scrutinizes the optimal configuration resulting from the first stage for scaling optimization of design parameters (DPs) for each module. All DPs are differentiated by standard or customizable DPs. A bi-level stochastic program is implemented to leverage conflicting goals between the producer (leader) and consumer (follower) surpluses. As a result, ETO customization design is anchored with optimal values of standard DPs and optimal value ranges of customizable DPs. A case study of ship engine and power generator ETO design is presented, demonstrating the feasibility and potential of the ETO mass customization framework.

Constrained network-based column generation for the multi-activity shift scheduling problem

Real applications in shift scheduling often require handling rules such as multiple breaks, flexible shift lengths, overtime, multiple activities, among others. Because these rules demand a high level of flexibility, we model the problem as a Multi-Activity Shift Scheduling Problem (MASSP), where multiple activities can be scheduled in a shift. To solve the MASSP, we propose a column generation-based approach. The auxiliary problem is modeled as a Shortest Path Problem with Resource Constraints (SPPRC), where most difficult constraints are embedded in the underlying graph. To illustrate the solution approach, we present our experience solving a real-world problem from a large parking lot operator that schedules security staff and cashiers among several parking lots in Bogotá (Colombia). The results show a significant reduction on the staffing total costs and on man-hours used.

The Shortest-Path Problem with Resource Constraints and k-Cycle Elimination for k ≥ 3

The elementary shortest-path problem with resource constraints (ESPPRC) is a widely used modeling tool in formulating vehicle-routing and crew-scheduling applications. The ESPPRC often occurs as a subproblem of an enclosing problem, where it is used to generate implicitly the set of all feasible routes or schedules, as in the column-generation formulation of the vehicle-routing problem with time windows (VRPTW). As the ESPPRC problem is NP-hard in the strong sense, classical solution approaches are based on the corresponding nonelementary shortest-path problem with resource constraints (SPPRC), which can be solved using a pseudo-polynomial labeling algorithm. While solving the enclosing problem by branch and price, this subproblem relaxation leads to weak lower bounds and sometimes impractically large branch-and-bound trees. A compromise between solving ESPPRC and SPPRC is to forbid cycles of small length. In the SPPRC with k-cycle elimination (SPPRC-k-cyc), paths with cycles are allowed only if cycles have length at least k + 1. The case k = 2 forbids sequences of the form i − j − i and has been successfully used to reduce integrality gaps. We propose a new definition of the dominance rule among labels for dealing with arbitrary values of k ≥ 2. The numerical experiments on the linear relaxation of some hard VRPTW instances from Solomon’s benchmark show that k-cycle elimination with k ≥ 3 can substantially improve the lower bounds of vehicle-routing problems with side constraints. The new algorithm has proven to be a key ingredient for getting exact integer solutions for well-known hard problems from the literature.