Resource Constrained Shortest Path Problem Research Articles

A two-stage method for doubly resource constrained elementary shortest path problems

The resource constrained shortest path problem (RCSPP) has been extensively studied. However, its more general version, the doubly resource constrained (elementary) shortest path problem (DRCESPP), is usually not concerned and no exact solution algorithm has been found in the literature. In this work, an exact two-stage method is proposed for solving the DRCESPP. For leveraging resource lower limits, an estimation to the maximum resource consumption of all elementary source–destination paths is proposed in the first stage. This makes the network be reduced more significantly and the algorithm be terminated in the first stage. By introducing an effective search direction into the conventional Pulse algorithm, a LB-first Pulse algorithm is proposed in the second stage to find the exact solution from the reduced network output in the first stage. Extensive computational experiments on various test instances are conducted. The numerical results show that the maximum resource consumption estimation method and the improved Pulse algorithm are effective, and the proposed two-stage method outperforms the Pulse algorithm in solving complex instances.

PathWyse: a flexible, open-source library for the resource constrained shortest path problem

In this paper, we consider a fundamental and hard combinatorial problem: the Resource Constrained Shortest Path Problem (RCSPP). We describe the implementation of a flexible, open-source library for the solution of the RCSPP, called PathWyse, capable of tackling several variants of the problem. We designed PathWyse with the final user in mind, developing easy-to-use interfaces without compromising performance. We provide computational experiments on three classes of instances of the RCSPP, namely RCSPP on cyclic networks, RCSPP on large acyclic networks, and RCSPP on ad-hoc cyclic networks. We show that PathWyse is packed off-the-shelf with algorithms capable of tackling generic problems, and can exploit dedicated algorithms for specific classes. This paper represents the first step along the journey of devising and implementing a comprehensive open-source library for a large variety of RCSPPs. The current version of the library carries exact algorithms for the RCSPP but new algorithms, both heuristic and exact, will be added thanks to the flexible design. We also foresee PathWyse becoming a platform ready for data-driven and process-driven methodologies for these types of problems.

Enhanced bi-directional dynamic programming algorithm for the resource constrained shortest path problem

In this paper we propose an enhanced bi-directional dynamic programming algorithm for the Resource Constrained Shortest Path, Righini and Salani (2006). In particular we study the join procedure, one of the time consuming components of the algorithm, and propose a dominance based join that uses binary search and Pareto frontier exploration. The proposed method can be used in both node-join and arc-join procedures. The computational experiments are performed on two classes of problems both describing complete graphs with negative cost cycles. On the first set, with a single resource consumption, the proposed procedure slightly improves the overall computational time with savings up to 8.4% for the B class and just 0.4% for the most time consuming class M. On the second set, with multiple resources consumption, the proposed procedure exhibits a remarkable improvement with an average speedup of 39.2%.

ERCA*: A New Approach for the Resource Constrained Shortest Path Problem

ERCA*: A New Approach for the Resource Constrained Shortest Path Problem

Electric Vehicle Charge Scheduling with Flexible Service Operations

Operators who deploy large fleets of electric vehicles often face a challenging charge scheduling problem. Specifically, time-ineffective recharging operations limit the profitability of charging during service operations such that operators recharge vehicles off duty at a central depot. Here, high investment cost and grid capacity limit available charging infrastructure such that operators need to schedule charging operations to keep the fleet operational. In this context, flexible service operations, that is, allowing delayed or expedited vehicle departures, can potentially increase charger utilization. Beyond this, jointly scheduling charging and service operations promises operational cost savings through better utilization of time-of-use energy tariffs and carefully crafted charging schedules designed to minimize battery wear. Against this background, we study the resulting joint charging and service operations scheduling problem accounting for battery degradation, nonlinear charging, and time-of-use energy tariffs. We propose an exact branch-and-price algorithm, leveraging a custom branching rule and a primal heuristic to remain efficient during the branch-and-bound phase. Moreover, we develop an exact labeling algorithm for our pricing problem, constituting a resource-constrained shortest path problem that considers variable energy prices and nonlinear charging operations. We benchmark our algorithm in a comprehensive numerical study and show that it can solve problem instances of realistic size with computational times below one hour, thus enabling its application in practice. Additionally, we analyze the benefit of jointly scheduling charging and service operations. We find that our integrated approach lowers the amount of charging infrastructure required by up to 57% besides enabling operational cost savings of up to 5%. Funding: This work was supported by the German Federal Ministry for Economic Affairs and Energy [Grant 01MV21020B]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0272 .

Optimizing aircrew recovery considering long connections: A column generation based approach

Severe weather and air traffic flow control can often cause flight delays and disrupt the crew schedules, resulting in a too-short or a too-long connection between flights. Short connections need to be adjusted to ensure the airline’s smooth operations, while the long connection whose duration is only a bit lower than the threshold enabling the crew to leave the flying zone for a rest at airport terminals can be quite inefficient, as the crew’s ground standby is viewed as working all the time without executing any flight which adds to the shortage of crew resources. Thus, we propose a method of delaying flights to the point enabling crews to leave for a rest, which helps save the available working hours. The difficulty lies in the trade-off between the additional swap or delay cost in creating long connections and the possible global cost decrease due to savings in available working hours brought by long connections, which calls for optimization. In this paper, we introduce the crew recovery problem for one day of operation taking long connections into account. The problem is mathematically formulated as an arc-based integer programming model and a set partitioning model in order to determine the optimal flight recovery schedule and the crew reassignment. A column generation based solution approach is developed and the sub-problems consisting of the resource-constrained shortest path problem are solved by a labeling algorithm. Computational experiments based on real-world data from a major Chinese airline confirm the effectiveness and efficiency of the solution approaches. It is also demonstrated the save of available working hours resulting from adaptively creating long connections would decrease the recovery cost by about 18.87%–74.88%.

Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

A Directional Heuristics Pulse Algorithm for a Two Resources Constrained Shortest Path Problem with Reinitialization

<p style='text-indent:20px;'>In this paper, the path planning problem of UAV flight error correction is studied, and a mixed-integer linear programming (MILP) model is proposed to solve the resource constrained shortest path problem (RCSPP-R) with reinitialization, and its resource accumulation can be reinitialized to zero at the reset node. Since RCSPP-R is a NP-hard problem, Pulse algorithm is used to solve the proposed model. Aiming at the high computational cost of pulse algorithm, three pruning strategies are proposed to improve the effectiveness and efficiency of the algorithm, namely, preprocessing strategy based on direction constraint to reduce the search space, dynamic relaxation strategy to improve the quality of boundary, and heuristic search strategy to optimize the node expansion sequence. Then the time-consuming of the algorithm and the number of dominant solutions are used as evaluation indexes for comparison. Numerical results show that the proposed algorithm can deal with this problem more reliably. Specifically, compared with the Label Correction algorithm, the original Pulse algorithm and the Ant Colony algorithm, there are advantages in the number of superior solutions, and compared with the above three algorithms, the time consumption is reduced by 31.64%, 88.52% and 83.5% respectively.</p>

Development of Linear Battery Model for Path Planning with Mixed Integer Linear Programming: Simulated and Experimental Validation

Mixed Integer Linear Programs (MILPs) are often used in the path planning of both ground and aerial vehicles. Such a formulation of the path planning problem requires a linear objective function and constraints, limiting the fidelity of the tracking of vehicle states. One such state often used is the charge level of the on board battery. High-fidelity battery state estimation requires nonlinear differential equations to be solved. This state estimation is vital in path planning to ensure flyable paths, however when using a linear path planning problem cannot implement these nonlinear models. Poor accuracy in battery estimation during the path planning runs the risk of the planned path being feasible by the estimation model but in reality will deplete the battery to a critical level, resulting in a dangerous planned path. To the end of higher accuracy battery estimation within a linear framework, we test a simple linear battery model which predicts the change in state-of-charge (SOC) of a battery given a power draw, time duration, and current SOC in the context of an a-priori path planning problem. This context differentiates itself from real-time estimation. In ahead-of-time path planning, the changes to battery draw are often assumed as a series of constant power draws as opposed to rapidly changing power draw which may occur in real-time battery tracking and estimation. The linear battery model is presented and then tested against alternate models in both numerical and in experimental tests. Further, the effect of the proposed linear model on the time-to-solve a resource constrained shortest path problem is also evaluated, where two different algorithms are used to solve the path planning problem. It is seen that the linear model performs well in battery state estimation while remaining implementable in a Linear Program or MILP, with minimal effect on the time-to-solve. This provides what we consider to be a worthwhile trade-off in improved accuracy relative to increased time-to-solve.

Partial and complete replanning of an intermodal logistic system under disruptions

The exclusive and excessive use of long-distance road transportation is not suitable way to reduce the negative environmental impacts of logistics systems. Intermodal transport, combining road with other transport modes, has the potential to reduce both operating costs and carbon footprints. One of the reasons for the low share of intermodal transport is its requirement for the coordination of scheduled transport services that can result in reducing reliability in case of disruptions due to the arrival of new shipment orders, fluctuations in shipment quantities, delays, and service cancellations within the network. This calls for reliable and efficient algorithms to replan the shipments’ distribution. In this paper, the replanning problem is formulated as a path-based multi-commodity network flows. We provide two different network topologies, one of which is based on a time–space network, while the other embeds time aspect in a highly scalable alternative structure to large transportation networks. We propose a column generation method whose pricing sub-problems are presented as resource constrained shortest path problem solved via a tailored label-correcting algorithm. We look at the pros and cons of complete and partial replanning in case of disruption and provide managerial insights for intermodal networks. An extensive set of computational experiments is presented on realistic instances being generated with the consultation of our industrial partners for a logistic network including railways, waterways, and roads. The promising outcomes validate the efficiency of the proposed approach that can be easily adjusted to real-time intermodal logistic replanning.

Exact Approaches for Single Machine Total Weighted Tardiness Batch Scheduling

This paper addresses a single machine total weighted tardiness (TWT) batch-scheduling problem in which jobs have release dates, nonidentical sizes, and are compatible between each other. We propose two integer linear programming models: the first one is a time-indexed formulation (TIF), and the second is an innovative time-size-indexed formulation (TSIF). Although TIF clearly outperforms the existing formulation for the problem, TSIF is capable of producing much stronger bounds in practice. The latter also enables us to develop an efficient column-generation (CG) algorithm. The pricing subproblem corresponds to a resource-constrained shortest path problem that is solved using a bucket graph–based labeling algorithm. The solutions of such a subproblem may contain cycles (reprocessing of jobs), and thus, a memory mechanism called dynamic arc-based ng-sets is employed in the labeling with a view toward avoiding some of them. Moreover, we also implement a preprocessing scheme based on Lagrangian relaxation to perform variable fixing. Extensive computational experiments were carried out in 810 benchmark instances. The proposed CG algorithm is capable of solving instances with up to 100 jobs to optimality. In addition, we believe that this is the first exact approach for a TWT batch-scheduling variant capable of systematically solving instances with up to 50 jobs. High-quality results are also reported for three special cases of the problem—more precisely, when (i) the penalty weights are unitary, (ii) there are no release dates, and (iii) all due dates are set to zero and, hence, the objective becomes equivalent to minimizing the weighted completion time. Summary of Contribution: This paper provides the first exact algorithm for a standard variant of a batch-scheduling total weighted tardiness problem that can solve instances with up to 100 jobs to optimality, a considerable leap with respect to previous works. In particular, we propose a time-indexed formulation that has the advantage of being relatively simple to implement, and yet we show that it is not theoretically dominated by the other innovative formulation proposed in the paper referred to as the time-size-indexed formulation (TSIF). Moreover, we present a Lagrangian approach to quickly fix variables and an iterative column-generation (CG) procedure over a Dantzig–Wolfe decomposition of TSIF that combines an efficient pricing algorithm with a dynamic scheme to adjust the subproblem constraints. The proposed CG approach is capable of producing very strong bounds for the problem as well as for some of its special cases.

Location-Routing Problem of Emergency Facilities under Uncertain Demand by Branch-Price and Cut

This paper studies the location-routing problem of emergency facilities with time window under demand uncertainty. We propose a robust mathematical model in which uncertain requirements are represented by two forms: the support set defined by cardinal constraint set. When the demand value of rescue point changes in a given definition set, the model can ensure the feasibility of each line. We propose a branch and price cutting algorithm, whose pricing problem is a robust resource-constrained shortest path problem. In addition, we take the Wenchuan Earthquake as an example to verify the practicability of the method. The robust model is simulated under different uncertainty levels and distributions and compared with the scheme obtained by the deterministic problem. The results show that the robust model can run successfully and maintain its robustness, and the robust model provides better protection against demand uncertainty. In addition, we find that cost is more sensitive to uncertainty level than protection level, and our proposed model also allows controlling the robustness level of the solution by adjusting the protection level. In all experiments, the cost of robustness is that the routing cost increases by an average of 13.87%.

A column generation approach for the driver scheduling problem with staff cars

Given a set of timetabled bus trips, transport companies are faced with the challenge of finding a feasible driver schedule that covers all trips and abides by various labor union regulations. The regulations are primarily concerned with providing sufficient breaks for the drivers during the day. Practical limitations in the city network enforce drivers to travel by cars between bus stops to have breaks. Transport companies have a limited number of cars, known as staff cars, which have to be returned to their respective depots at the end of the day. The simultaneous scheduling of drivers and staff cars for the drivers is known as the driver scheduling problem with staff cars (DSPSC). It is estimated that the DSPSC accounts for 60% of a bus company’s operational expense, and this paper proposes a column generation approach that attempts to minimize operational expense. The column-generation framework iterates between a master problem, a subproblem for generating driver variables and a subproblem for generating staff car variables. The subproblem related to the drivers is formulated as a resource constrained shortest path problem, which is solved by a dynamic programming approach. Several heuristic branching strategies are explored to find integer solutions. The proposed methodology is tested on eight real-life instances from seven Northern European bus companies. A comparison with a state-of-the-art mixed integer programming (MIP) solver and an adaptive large neighborhood search (ALNS) heuristic indicate that the column generation approach provides improved solutions for six instances and the average improvement is 1.45%.

A heuristics pulse algorithm with relaxation pruning strategy for resources re-initialized UAV path planing

Path planning is the basis and prerequisite for unmanned aerial vehicle (UAV) to perform tasks, and it is important to achieve precise location in path planning. This paper focuses on solving the UAV path planning problem under the constraint of system positioning error. Some nodes can re-initiate the accumulated flight error to zero and this type of scenario can be modeled as the resource-constrained shortest path problem with re-initialization (RCSPP-R). The additional re-initiation conditions expand the set of viable paths for the original constrained shortest path problem and increasing the search cost. To solve the problem, an effective preprocessing method is proposed to reduce the network nodes. At the same time, a relaxed pruning strategy is introduced into the traditional Pulse algorithm to reduce the search space and avoid more redundant calculations on unfavorable scalable nodes by the proposed heuristic search strategy. To evaluate the accuracy and effectiveness of the proposed algorithm, some numerical experiments were carried out. The results indicate that the three strategies can reduce the search space by 99%, 97% and 80%, respectively, and in the case of a large network, the heuristic algorithm combining the three strategies can improve the efficiency by an average of 80% compared to some classical solution.

A column generation-based approach for integrated vehicle and crew scheduling on a single metro line with the fully automatic operation system by partial supervision

With the rapid development of fully automated urban metro lines, unattended train operation has become a reality, leading to significant gains in efficiency in comparison with traditional operation. To make better use of this technology and reduce operational costs, this study proposes a novel approach for collaboratively optimizing both vehicle and crew scheduling on metro lines. This approach is based on the assumption that unattended train movement without passengers is permitted, while the in-service train requires supervision by the crew. By considering the characteristics of urban metro lines, a mathematical model that considers both the meal activity and employee deadheading for the integrated vehicle and crew scheduling problem is first formulated. The model consists of a master problem for minimizing the total cost and subproblems for generating duties. The subproblem is constructed using a connection-based network and modeled as a special resource-constrained shortest path problem. To solve the model efficiently, a two-phase pricing procedure, which combines a piece generation phase and a duty generation phase, is developed to price out duties with negative reduced costs. Finally, numerical experiments based on actual conditions are conducted. The results indicate that our proposed model provides a significantly tight linear relaxation with negligible, typically zero, optimality gaps. Moreover, in comparison with the traditional integrated formulation, our proposed model and algorithm achieve improved performance in terms of solution quality and computing time.

Electric bus fleet composition and scheduling

The low energy density of batteries and the long recharging times constitute a significant barrier for electrification of public transportation (PT) systems since electric buses (EB) require too heavy and expensive batteries to achieve the operational availability of their combustion engine counterparts. New recharging technologies such as fast chargers and dynamic wireless power transfer (DWPT) emerge as promising solutions to overcome these challenges. Optimizing the bus fleet composition and the schedules is essential to take advantage of these emerging technologies and achieve electrification of PT in a cost-efficient way. To address this challenge, this paper proposes an integer (binary) programming formulation to find the optimal electric bus fleet composition and scheduling that minimizes the total procurement cost of the buses and the operating cost of the schedules. A column generation (CG) approach is devised to obtain provably high-quality solutions, for large problem instances. The success of the approach is due to a novel dynamic programming algorithm we develop to solve the generalized resource-constrained shortest path problem that needs to be solved in each CG iteration to find out new schedules to include in the model. Extensive computational studies on large real-world PT networks attest to the efficacy of the suggested methodology and reveal valuable managerial insights from a system-wide perspective.

Branch and Price for Bus Driver Scheduling with Complex Break Constraints

This paper presents a Branch and Price approach for a real-life Bus Driver Scheduling problem with a complex set of break constraints. The column generation uses a set partitioning model as master problem and a resource constrained shortest path problem as subproblem. Due to the complex constraints, the branch and price algorithm adopts several novel ideas to improve the column generation in the presence of a high-dimensional subproblem, including exponential arc throttling and a dedicated two-stage dominance algorithm. Evaluation on a publicly available set of benchmark instances shows that the approach provides the first provably optimal solutions for small instances, improving best-known solutions or proving them optimal for 48 out of 50 instances, and yielding an optimality gap of less than 1% for more than half the instances.

A Fast Exact Algorithm for the Resource Constrained Shortest Path Problem

Resource constrained path finding is a well studied topic in AI, with real-world applications in different areas such as transportation and robotics. This paper introduces several heuristics in the resource constrained path finding context that significantly improve the algorithmic performance of the initialisation phase and the core search. We implement our heuristics on top of a bidirectional A* algorithm and evaluate them on a set of large instances. The experimental results show that, for the first time in the context of constrained path finding, our fast and enhanced algorithm can solve all of the benchmark instances to optimality, and compared to the state of the art algorithms, it can improve existing runtimes by up to four orders of magnitude on large-size network graphs.

Shift Planning Under Delay Uncertainty at Air France: A Vehicle-Scheduling Problem with Outsourcing

Airlines must operate many jobs in airports, such as passenger check-in or runway tasks. In airlines’ hubs, airlines generally choose to perform these jobs with their own agents. Shift planning aims at building the sequences of jobs operated by the airline agents and has been widely studied given its impact on operating costs. The impact of delayed flights is generally not taken into account despite the propagation of flight delays along these sequences: If a flight is late, then the agents doing the corresponding jobs are delayed, and may arrive late to their next jobs and delay the corresponding flights. Since delay costs are much higher than the costs of outsourcing jobs, if the agent who is supposed to operate a job is still working elsewhere when the job begins, then airlines tend to outsource the job to their own dedicated team or to a third party. We introduce a stochastic version of the shift-planning problem that takes into account outsourcing costs due to delay. It can be seen as a natural stochastic generalization of the vehicle-scheduling problem in which delayed jobs are outsourced. We propose a column-generation approach to solve it, whose key element is the pricing subproblem algorithm, modeled as a stochastic resource-constrained shortest-path problem. Numerical results on Air France industrial instances show the benefits of using our stochastic version of the shift-planning problem and the efficiency of the solution method. Moving to the stochastic version enables Air France to reduce total operating costs by 3.5%–4.8% on instances with more than 200 jobs, and our algorithm can solve to near optimality instances with up to 400 jobs.

Joint chance constrained shortest path problem with Copula theory

In this paper, we investigate the constrained shortest path problem where the arc resources of the problem are dependent normally distributed random variables. A model is presented to maximize the probability of all constraints, while not exceeding a certain amount. We assume that the rows of the constraint matrix are dependent, so we use a marginal distribution of the Copula functions, instead of the distribution functions and the dependency is driven by an appropriate Archimedean Copula. Then, we transform the joint chance-constrained problems into deterministic problems of second-order cone programming. This is a new approach where considers the dependency between resource consumptions and connects Copulas to stochastic resource constrained shortest path problem (SRCSPP). The results indicate that the effect of marginal probability levels is considerable. Moreover, the linear relaxation of SRCSPP is generally not convex; thus we can use lower and upper bounds of the second-order cone programming approximation to solve the relaxation problem. The experimental results show that the SRCSPP with Copula theory can achieve efficient performance.