Resource Constrained Shortest Path Problem Research Articles

First-order linear programming in a column generation-based heuristic approach to the nurse rostering problem

A heuristic method based on column generation is presented for the nurse rostering problem. The method differs significantly from an exact column generation approach or a branch and price algorithm because it performs an incomplete search which quickly produces good solutions but does not provide valid lower bounds. It is effective on large instances for which it has produced best known solutions on benchmark data instances. Several innovations were required to produce solutions for the largest instances within acceptable computation times. These include using a fast first-order linear programming solver based on the work of Chambolle and Pock to approximately solve the restricted master problem. A low-accuracy but fast, first-order linear programming method is shown to be an effective option for this master problem. The pricing problem is modelled as a resource constrained shortest path problem with a two-phase dynamic programming method. The model requires only two resources. This enables it to be solved efficiently. A commercial integer programming solver is also tested on the instances. The commercial solver was unable to produce solutions on the largest instances whereas the heuristic method was able to. It is also compared against the state-of-the-art, previously published methods on these instances. Analysis of the branching strategy developed is presented to provide further insights. All the source code for the algorithms presented has been made available on-line for reproducibility of results and to assist other researchers.

On Accuracy of Approximation for the Resource Constrained Shortest Path Problem

The paper we considers the Resource Constrained Shortest Path problem (RCSP). This problem is NP- hard extension of a well-known shortest path problem in the directed graph G = (V;E). In the RCSP problem each arc e from E has a cost w(e) and additional weight functions ri(e); i = 1; : : : ; k, which specifying its requirements from a finite set of resource. A polynomial time ϵ-approximation algorithm RevTree based on node labeling method is presented in the paper. The main advantage of the RevTree algorithm over existing ones is its ability to produce ϵ approximation of the RCSP problem in O(jV j2) time. The present paper provides a proof of complexity and aproximation of RevTree algorithm

A Branch-and-Price-and-Cut Algorithm for Resource-Constrained Pickup and Delivery Problems

We study a multicommodity, multiperiod, resource-constrained pickup-and-delivery problem inspired by the short-term planning of maintenance services at offshore wind farms. To begin a maintenance service, different types of relatively scarce servicemen need to be delivered (transported) to the service locations. We develop resource-exceeding route (RER) inequalities, which are inspired by knapsack cover inequalities, to model the scarcity of servicemen. In addition to a traditional separation approach, we present a column-dependent constraints approach so as to include the RER inequalities in the mathematical formulation. An alternative pricing strategy is developed to correctly include the column-dependent constraints. The resulting approach is broadly applicable to any routing problem that involves a set of scarce resources. We present a branch-and-price-and-cut algorithm to compare both approaches that include RER inequalities. The branch-and-price-and-cut algorithm relies on efficiently solving a new variant of the elementary resource-constrained shortest-path problem, using a tailored pulse algorithm developed specifically to solve it. Computational experiments show that the RER inequalities significantly tighten the root node relaxations. The column-dependent constraints approach then searches the branch-and-bound tree more effectively and appears to be competitive with the traditional separation procedure. Both approaches are able to solve instances of up to 92 nodes over 21 periods to optimality.

The Resource Constrained Shortest Path Problem with uncertain data: A robust formulation and optimal solution approach

The Resource Constrained Shortest Path Problem (RCSPP) models several applications in the fields of transportation and communications. The classical problem supposes that the resource consumptions and the costs are certain and looks for the cheapest feasible path. These parameters are however hardly known with precision in real applications, so that the deterministic solution is likely to be infeasible or suboptimal. We address this issue by considering a robust counterpart of the RCSPP. We focus here on resource variation and model its variability through the uncertainty set defined by Bertismas and Sim (2003, 2004), which can model the risk aversion of the decision maker through a budget of uncertainty. We solve the resulting problem to optimality through the well-known three phase approach dealing with bounds computation, network reduction and gap closing. In particular, we compute robust bounds on the resource consumption and cost by solving the robust shortest path problem and the dual robust Lagrangian relaxation, respectively. Dynamic programming is used to close the duality gap. Upper and lower bounds are used to reduce the dimension of the network and incorporated in the dynamic programming in order to fathom unpromising states. An extensive computational phase is carried out in order to assess the behavior of the defined strategy comparing its performance with the state-of-the-art. The results highlight the effectiveness of our approach in solving to optimality benchmark instances for RCSPP when Γ is not too large, tailored for the robust counterpart. For larger values of Γ, we show that the most efficient method combines deterministic preprocessing with the iterative algorithm from Bertsimas and Sim (2003). We also illustrate the failure probability of the robust solutions through Monte Carlo sampling.

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.

An exact bidirectional A⋆ approach for solving resource‐constrained shortest path problems

Bidirectional dynamic programming is an algorithm that searches for paths in a network from both the starting and the ending nodes that optimize a given objective function. In recent years, bidirectional dynamic programming has been shown to be an effective means for solving resource‐bounded shortest path problems. While many researchers have observed that bidirectional A⋆ approaches perform poor computationally, we exploit the presence of resource constraints to overcome the source of these computational challenges. Our main contribution in this paper is an exact bidirectional A⋆ algorithm for resource‐constrained shortest path problems (RCSPPs) that is capable of solving large‐sized instances that challenge the state‐of‐the‐art in the literature. We also analyze, both computationally and theoretically, the sensitivity of the algorithm's performance to its inputs.

Algorithms for non-linear and stochastic resource constrained shortest path

Resource constrained shortest path problems are usually solved thanks to a smart enumeration of all the non-dominated paths. Recent improvements of these enumeration algorithms rely on the use of bounds on path resources to discard partial solutions. The quality of the bounds determines the performance of the algorithm. The main contribution of this paper is to introduce a standard procedure to generate bounds on paths resources in a general setting which covers most resource constrained shortest path problems, among which stochastic versions. In that purpose, we introduce a generalization of the resource constrained shortest path problem where the resources are taken in a monoid. The resource of a path is the monoid sum of the resources of its arcs. The problem consists in finding a path whose resource minimizes a non-decreasing cost function of the path resource among the paths that respect a given constraint. Enumeration algorithms are generalized to this framework. We use lattice theory to provide polynomial procedures to find good quality bounds. These procedures solve a generalization of the algebraic path problem, where arc resources belong to a lattice ordered monoid. The practical efficiency of the approach is proved through an extensive numerical study on some deterministic and stochastic resource constrained shortest path problems.

Shortest Path Planning for Energy-Constrained Mobile Platforms Navigating on Uneven Terrains

Finding a shortest feasible path between two given locations is a common problem in many real-world applications. Previous studies have shown that mobile platforms would consume excessive energy when moving along shortest paths on uneven terrains, which often consist of rapid elevation changes. Mobile platforms powered by portable energy sources may fail to follow such paths due to the limited energy available. This paper proposes a new heuristic search algorithm called constraints satisfying A* (CSA*) to find solutions to such resource constrained shortest path problems. When CSA* is guided by admissible heuristics, it guarantees to find a globally optimal solution to a given constrained search problem if such a solution exists. When CSA* is guided by consistent heuristics, it is optimally efficient over a class of equally informed admissible constrained search algorithms with respect to the set of paths expanded. Test results obtained using real terrain data verify the applicability of the proposed algorithm in shortest path planning for energy-constrained mobile platforms on uneven terrains.

Multi-criteria approximation schemes for the resource constrained shortest path problem

In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist a polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problems as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.

Solving resource constrained shortest path problems with LP-based methods

In the resource constrained shortest path problem (RCSPP) there is a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resource types with finite capacities. A minimum cost source–destination directed path is sought such that the total consumption of the arcs from each resource type does not exceed the capacity of the resource. In this paper we investigate LP-based branch-and-bound methods and introduce new cutting planes, separation procedures, variable fixing, and primal heuristic methods for solving RCSPP to optimality. We provide detailed computational experiments, and a comparison to other methods in the literature.

Mathematical formulations and exact algorithm for the multitrip cumulative capacitated single-vehicle routing problem

This paper addresses the multitrip Cumulative Capacitated Single-Vehicle Routing Problem (mt-CCSVRP). In this problem inspired by disaster logistics, a single vehicle can perform successive trips to serve a set of affected sites and minimize an emergency criterion, the sum of arrival times. Two mixed integer linear programs, a flow-based model and a set partitioning model, are proposed for small instances with 20 sites. An exact algorithm for larger cases transforms the mt-CCSVRP into a resource-constrained shortest path problem where each node corresponds to one trip and the sites to visit become resources. The resulting problem can be solved via an adaptation of Bellman–Ford algorithm to a directed acyclic graph with resource constraints and a cumulative objective function. Seven dominance rules, two upper bounds and five lower bounds speed up the procedure. Computational results on instances derived from classical benchmark problems for the capacitated VRP indicate that the exact algorithm outperforms a commercial MIP solver on small instances and can solve cases with 40 sites to optimality.

An A* Label-setting Algorithm for Multimodal Resource Constrained Shortest Path Problem

Multimodal resource constrained shortest path problem consists of finding a least travel time path with multiple resources and mode chain constraints on a multimodal transportation network. Existing solution approaches based on Lagrangian relaxation technique cannot take into account the sequence constraint of used modes in a multimodal path. In this study, we propose a fast A*-based label setting algorithm to solve efficiently dynamic multimodal resource constrained shortest path problem. A multilevel graph is proposed for modeling multimodal network provided with time-dependent travel time functions and timetable information. The mode chain constraint of a multimodal path is efficiently examined in a labeled graph by applying non-deterministic finite automata method. A speed-up technique based on the A*-algorithm and Access-Node routing is proposed to effectively reduce search space in the resource constrained route searching process. The computational results show that the proposed algorithm can efficiently find constrained shortest path for realistic multimodal network.

Bounded flexibility in days-on and days-off scheduling

This article presents a flexible days-on and days-off scheduling problem and develops an exact branch and price (B&P) algorithm to find solutions. The main objective is to minimize the size of the total workforce required to cover time-varying demand over a planning horizon that may extend up to 12 weeks. A new aspect of the problem is the general restriction that the number of consecutive days on and the number of consecutive days off must each fall within a predefined range. Moreover, the total assignment of working days in the planning horizon cannot exceed some maximum value. In the B&P framework, the master problem is stated as a set covering-type problem whose columns are generated iteratively by solving one of three different subproblems. The first is an implicit model, the second is a resource constrained shortest path problem, and the third is a dynamic program. Computational experiments using both real-word and randomly generated data show that workforce reductions up to 66% are possible with highly flexible days-on and days-off patterns. When evaluating the performance of the three subproblems, it was found that each yielded equivalent solutions but the dynamic program proved to be significantly more efficient. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 678–701, 2013

A survey of resource constrained shortest path problems: Exact solution approaches

This article surveys the main contributions that have appeared in the scientific literature addressing resource constrained shortest path problems. The aim of this work is twofold: to give a structured survey of the literature on this topic; to provide a starting point for researchers who want to address the problems at hand. The study is focused on the relevant contributions dealing with exact solution approaches. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(3), 183‐200 2013

A Reference Point Approach for the Resource Constrained Shortest Path Problems

The Resource Constrained Shortest Path Problem (ℛ 𝒞 𝒮 𝒫 𝒫) is a variant of the classical shortest path problem and is of great practical importance. The aim is to find the shortest path between a given pair of nodes under additional constraints representing upper bounds on the consumption of resources along the path. In the scientific literature, different approaches have been defined to solve the ℛ 𝒞 𝒮 𝒫 𝒫. In this work we propose an innovative interactive method to address the ℛ 𝒞 𝒮 𝒫 𝒫, based on a novel search strategy of the criteria space. The performance of the proposed approach is evaluated on the basis of an extensive computational study by considering benchmark instances. A comparison with the state-of-the-art approaches developed for the ℛ 𝒞 𝒮 𝒫 𝒫 is also carried out. The computational results have shown that the developed solution strategy is competitive with the most efficient strategies known thus far.

Dynamic programming approaches to solve the shortest path problem with forbidden paths

In this paper, the shortest path problem with forbidden paths is addressed. The problem under consideration is formulated as a particular instance of the resource-constrained shortest path problem. Different versions of a dynamic programming-based solution approach are defined and implemented. The proposed algorithms can be viewed as an extension of the node selection approach proposed by Desrochers in 1988. An extensive computational test is carried out on a meaningful number of random instances with the purpose of assessing the behaviour of the developed solution approaches. A comparison with the state-of-the-art method proposed to address the problem under study is also made. The computational results are very encouraging and highlight that the proposed algorithms are very efficient.

Ant colony optimisation for a resource-constrained shortest path problem with applications in multimodal transport

Multimodal transport provides great flexibility in freight logistics which on the other hand leads to a sophisticated decision problem when choosing shipping options. Advanced analysing and planning tools are desired in supporting the decision making so as to enable the power of multimodal transport. In this paper, features and evolving patterns of a generalised multimodal network derived from industrial applications are analysed. A corresponding routing problem is defined in order to optimise the transportation cost. This problem entails selections on transport modes and paths subject to various kinds of restrictions. We show that the routing problem is a resource constrained shortest path problem and therefore is NP-hard. An ant algorithm is then developed and tested with datasets simulating real scenarios. Experimental results prove the effectiveness of our approach.

The Fixed-Charge Shortest-Path Problem

Consider a network 𝒩 =(N, A) and associate with each arc e ∈ A a fixed cost ce for using arc e, an interval [le, ue] (le, ue ∈ ℤ) specifying the range of allowable resource consumption quantities along arc e, and a per-unit cost [Formula: see text] for resource consumed along e. Furthermore, for each node n ∈ N, let Un ∈ ℤ be the maximum amount of resource consumption a path can accumulate before visiting node n. Given a source node ns ∈ N and sink node nt ∈ N, the fixed-charge shortest-path problem (FCSPP) seeks to find a minimum-cost-feasible path from ns to nt. When resource consumption is simply additive, the resource-constrained shortest-path problem (RCSPP) is a special case of FCSPP. We develop a new dynamic programming algorithm for FCSPP. The algorithm uses solutions from labeling and dominance techniques for standard RCSPPs on slightly modified problems, and it combines these solutions by exploiting the structure provided by certain classes of knapsack problems to efficiently construct an optimal solution to FCSPP. Computational experiments demonstrate that our algorithm is often several orders of magnitude faster than naive dynamic programming procedures.

Implementation of a three-stage approach for the dynamic resource-constrained shortest-path sub-problem in branch-and-price

The resource-constrained shortest-path problem (RCSP) is often used as a sub-problem in branch-and-price because it can model the complex logic by which many actual systems operate. This paper addresses two special issues that arise in such an application. First, RCSP in this context is dynamic in the sense that arc costs are updated at each column-generation iteration, but constraints are not changed. Often, only a few arc costs are updated at an iteration. Second, RCSP must be solved subject to arcs that are forbidden or prescribed as corresponding binary variables are fixed to 0 or 1 by the branching rule. A reoptimizing algorithm for dealing with a few arc-cost changes and a method for dealing with fixed arcs are proposed and incorporated into a three-stage approach, specializing it for repeatedly solving the dynamic RCSP as a sub-problem in branch-and-price. Computational tests evaluate the effectiveness of the proposed algorithms.

A three-stage approach for the resource-constrained shortest path as a sub-problem in column generation

This paper focuses on the common scenario in which the resource-constrained shortest path problem (RCSP) on an acyclic graph is a sub-problem in the context of column generation. It proposes a pseudo-polynomial time, three-stage solution approach. Stages 1 (preprocessing) and 2 (setup) are implemented one time to transform RCSP into a shortest path problem, which is solved by stage 3 (iterative solution) at each column generation iteration, each time with different arc costs. This paper analyzes certain properties related to each stage as well as algorithm complexity. Computational tests compare the performances of this method, a state-of-the-art label-setting algorithm, and CPLEX optimization software on four classes of instances, each of which involves either one or multiple resources, and show that the new method is effective. The new method outperforms the label-setting algorithm when resource limitations are tight—as can be expected in practice, and outperforms CPLEX for all tested instances. The label-setting algorithm outperforms CPLEX for all single-resource RCSP instances and almost all multiple-resource RCSP instances.