Elementary Shortest Path Problem Research Articles

A Branch-and-Price Algorithm for the Multidepot Vehicle Routing Problem with Interdepot Routes

This paper proposes a column generation algorithm for the multidepot vehicle routing problem with interdepot routes. This problem is an extension of the multidepot vehicle routing problem in which the vehicles are allowed to stop at intermediate depots along their routes to replenish. The problem can be modeled as a set covering problem in which the variables are rotations corresponding to feasible combinations of routes. We consider two pricing subproblems to generate rotations. The first one generates rotations directly by solving an elementary shortest path problem with resource constraints on a modified version of the original customer-depot network. The second one exploits the relationship between the sets of routes and rotations but results in a model with many columns. We discuss some issues related to solving this second pricing subproblem by column generation and we introduce an alternate approach to alleviate these difficulties. We show through computational experiments that the second pricing mechanism performs better than the first to compute the linear programming relaxation lower bound. We then embed it within a branch-and-bound algorithm to compute optimal integer solutions. Moreover, we assess the benefits of allowing interdepot routes in multidepot vehicle routing.

A branch-and-price approach for a multi-period vehicle routing problem

In this paper, we consider tactical planning for a class of multi-period vehicle routing problems (MPVRP). This problem involves optimizing daily product collections from several production locations over a given planning horizon. In this context, a single routing plan for the whole horizon must be prepared, and the seasonal variations in the producers’ supplies must be taken into account. Production variations over the horizon are approximated using a sequence of periods, each corresponding to a production season, while the intra-period variations are neglected. We propose a mathematical model that is based on the two-stage a priori optimization paradigm. The first stage corresponds to the design of a plan which, in the second stage, takes the different periods into account. The proposed set partitioning-based formulation is solved using a branch-and-price approach. The subproblem is a multi-period elementary shortest path problem with resource constraints (MPESPPRC), for which we propose an adaptation of the dynamic-programming-based label-correcting algorithm. Computational results show that this approach is able to solve instances with up to 60 producers and five periods.

Efficient elementary and restricted non-elementary route pricing

Column generation is involved in the current most efficient approaches to routing problems. Set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers. These formulations often produce tight lower bounds and require column generation for their pricing step. The bounds in the resulting branch-and-price are tighter when elementary routes are considered, but this approach leads to a more difficult pricing problem. Balancing the pricing with route relaxations has become crucial for the efficiency of the branch-and-price for routing problems. Recently, the ng-routes relaxation was proposed as a compromise between elementary and non-elementary routes. The ng-routes are non-elementary routes with the restriction that when following a customer, the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set. The larger the size of these sets, the closer the ng-route is to an elementary route. This work presents an efficient pricing algorithm for ng-routes and extends this algorithm for elementary routes. Therefore, we address the Shortest Path Problem with Resource Constraint (SPPRC) and the Elementary Shortest Path Problem with Resource Constraint (ESPPRC). The proposed algorithm combines the Decremental State-Space Relaxation technique (DSSR) with completion bounds. We apply this algorithm for the Generalized Vehicle Routing Problem (GVRP) and for the Capacitated Vehicle Routing Problem (CVRP), demonstrating that it is able to price elementary routes for instances up to 200 customers, a result that doubles the size of the ESPPRC instances solved to date.

A note on the NP-hardness of the separation problem on some valid inequalities for the elementary shortest path problem

In this paper, we investigate the separation problem on some valid inequalities for the s - t elementary shortest path problem in digraphs containing negative directed cycles. As we will see, these inequalities depend to a given parameter k ∈ ℕ. To show the NP-hardness of the separation problem of these valid inequalities, considering the parameter k ∈ ℕ, we establish a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s1, t1), (s2, t2) ... (sk+2, t k+2) in a digraph to the decision problem associated to the separation of these valid inequalities. Through some illustrative instances, we exhibit the evoked polynomial reduction in the cases k = 0 and k = 1.

Production scheduling for hot rolling processes with time windows in the steel industry

This paper investigates the hot rolling production scheduling problem in the steel industry. The aim of hot rolling process scheduling (HRPS) is to construct a rolling sequence which optimises a set of given criteria under constraints. This paper presents the model and solution method for hot rolling production scheduling with time windows. Three heuristic algorithms based on an elementary shortest path problem with resource constraints (ESPPRC) are developed. The methods combine exact method and heuristics to obtain near-optimal solutions for this problem. The methods are divided into two phases: in the first phase, non-dominated feasible turns are generated; in the second phase, some turns are selected and sequenced to form the scheduling. The experimental results show that the proposed mathematical model and the heuristic algorithms can be applied in practice to provide production efficiency and economic benefits for the hot rolling scheduling problem.

Exploiting Set-Based Structures to Accelerate Dynamic Programming Algorithms for the Elementary Shortest Path Problem with Resource Constraints

In this paper we consider a label-setting dynamic-programming algorithm for the Elementary Shortest Path Problem with Resource Constraints (ESPPRC). We use a pseudo resource to guarantee that labels are permanent. We observe that storing the states based on the subset of nodes visited by the associated path can improve the performance of the algorithm significantly. To this end we use a variant of a prefix tree to store the states and show by computational experiments that the performance of the dynamic programming algorithm is improved significantly when the number of undominated states is large.

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 branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.This paper proposes three mathematical models for the PDPS and a branch-and-cut-and-price algorithm to solve it. The pricing sub-problem, an Elementary Shortest Path Problem with Resource Constraints (ESPPRC), is solved with a labeling algorithm enhanced with efficient dominance rules. Three families of valid inequalities are used to strengthen the quality of linear relaxations. The method is evaluated on generated and real-world instances containing up to 193 transportation requests. Instances with up to 87 customers are solved to optimality within a computation time of one hour.

On the Generalized Elementary Shortest Path Problem: A heuristic approach

Abstract We introduce the generalized elementary shortest path problem (GESPP) where in addition to the features of the shortest path problem, nodes belong to predefined non-disjoint clusters. Each cluster is associated to a profit to the cost function, obtained if at least one element in the cluster appears in the path. Several applications can be considered as school bus routing, pricing problems, or telecommunication network design. Thus, depending on the case, clusters could be interpreted as groups of nodes with linking features as, for example, being easily reachable from each other, or some kind of coverage guarantee. We compare the GESPP to similar problems in the literature and we propose a two-phase heuristic algorithm for graphs including negative cycles. Tests on random instances with up to 100 nodes show an average gap of 0.3% to the best known solutions computed in 2.8s in average.

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported.

A note on the separation of subtour elimination constraints in elementary shortest path problems

This note proposes an alternative procedure for identifying violated subtour elimination constraints (SECs) in branch-and-cut algorithms for elementary shortest path problems. The procedure is also applicable to other routing problems, such as variants of travelling salesman or shortest Hamiltonian path problems, on directed graphs. The proposed procedure is based on computing the strong components of the support graph. The procedure possesses a better worst-case time complexity than the standard way of separating SECs, which uses maximum flow algorithms, and is easier to implement.

Column Generation for a Multitrip Vehicle Routing Problem with Time Windows, Driver Work Hours, and Heterogeneous Fleet

This study addresses a vehicle routing problem with time windows, accessibility restrictions on customers, and a fleet that is heterogeneous with regard to capacity and average speed. A vehicle can perform multiple routes per day, all starting and ending at a single depot, and it is assigned to a single driver whose total work hours are limited. A column generation algorithm is proposed. The column generation pricing subproblem requires a specific elementary shortest path problem with resource constraints algorithm to address the possibility for each vehicle performing multiple routes per day and to address the need to set the workday’s start time within the planning horizon. A constructive heuristic and a metaheuristic based on tabu search are also developed to find good solutions.

Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.

Solving elementary shortest-path problems as mixed-integer programs

Ibrahim et al. (Int Trans Oper Res 16:361---369, 2009) presented and analyzed two integer programming formulations for the elementary shortest-path problem (ESPP), which is known to be NP-hard if the underlying digraph contains negative cycles. In fact, the authors showed that a formulation based on multi-commodity flows possesses a significantly stronger LP relaxation than a formulation based on arc flow variables. Since the ESPP is essentially an integer problem, the contribution of our paper lies in extending this research by comparing the formulations with regard to the computation time and memory requirements required for their integer solution. Moreover, we assess the quality of the lower bounds provided by an integer relaxation of the multi-commodity flow formulation.

A computational study of solution approaches for the resource constrained elementary shortest path problem

We present the results of a computational investigation of solution approaches for the resource constrained elementary shortest path problem (\(\mathcal{RCESPP}\)). In particular, the best known algorithms are tested on several problem instances taken from literature. The main aims are to provide a detailed state of the art and to evaluate methods that turn out to be the most promising for solving the problem under investigation. This work represents the first attempt to computationally compare solution approaches for the \(\mathcal{RCESPP}\).

A branch-price-and-cut method for a ship routing and scheduling problem with split loads

We present a branch-price-and-cut method to solve a maritime pickup and delivery problem with time windows and split loads. The fleet of ships is heterogeneous and fixed for the planning horizon, and no common depot exists. The cargoes picked up and delivered consist of both mandatory and optional cargoes, and each cargo may be split among several ships. The objective is to design a route for each ship that will maximize the total profit from transporting all the mandatory and a subset of the optional cargoes. To solve this problem we introduce a new path-flow formulation, which we solve by branch-price-and-cut. The subproblem is a new variant of the elementary shortest path problem with resource constraints, where a multi-dimensional knapsack problem is solved to compute optimal cargo quantities. Further, we present new valid inequalities for this problem, and adaptations of existing inequalities successfully used to solve related problems in the literature. Finally, the computational results show that for certain types of instances, our solution method outperforms existing methods proposed in the literature.

시간제약이 있는 상금 획득 외판원 문제에 대한 동적 계획 접근 방법

This paper introduces one type of prize collecting travelling salesman problem with time windows (PCTSPTW), proposes a mixed integer programming model for the problem, and shows that the problem can be reduced to the elementary shortest path problem with time windows and capacity constraints (ESPPTC). Then, a new dynamic programming algorithm is proposed to solve ESPPTC quickly. Computational results show the effectiveness of the proposed algorithm.

Multi-dimensional labelling approaches to solve the linear fractional elementary shortest path problem with time windows

This paper investigates the linear fractional shortest path problem with time windows. For the specific problem, an elementary path with a minimum cost/time ratio is sought in a directed graph, where two parameters (i.e. cost and time) are associated with each arc and a time window is associated with each node. Indeed, a valid path must satisfy the time window constraints, which are assumed to be of the hard type. Multi-dimensional labelling algorithms are proposed to solve this variant of the classical shortest path problem. Extensive computational tests are carried out on a meaningful number of test problems, with the goal of assessing the behaviour of the proposed approaches. The computational study shows that the introduction of dominance rules and the adoption of a bi-directional search strategy allow the definition of solution approaches that turn out to be very effective in solving the problem under consideration.

Labelling algorithms for the elementary shortest path problem with resource constraints considering EU drivers’ rules

This paper describes how drivers’ rules according to EU social legislation can be formally modelled using the resource concept and how ‘legal’ vehicle routes and schedules can be computed by exact and heuristic labelling algorithms for solving the elementary shortest path problem with resource constraints.

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

The vehicle routing problem with multiple use of vehicles is a variant of the classical vehicle routing problem. It arises when each vehicle performs several routes during the workday due to strict time limits on route duration (e.g., when perishable goods are transported). The routes are defined over customers with a revenue, a demand and a time window. Given a fixed-size fleet of vehicles, it might not be possible to serve all customers. Thus, the customers must be chosen based on their associated revenue minus the traveling cost to reach them. We introduce a branch-and-price approach to address this problem where lower bounds are computed by solving the linear programming relaxation of a set packing formulation, using column generation. The pricing subproblems are elementary shortest path problems with resource constraints. Computational results are reported on euclidean problems derived from well-known benchmark instances for the vehicle routing problem with time windows.