Recourse Cost Research Articles

GCFExplainer: Global Counterfactual Explainer for Graph Neural Networks

Graph neural networks (GNNs) find applications in various domains such as computational biology, natural language processing, and computer security. Owing to their popularity, there is an increasing need to explain GNN predictions since GNNs are black-box machine learning models. One way to address this issue involves using counterfactual reasoning where the objective is to alter the GNN prediction by minimal changes in the input graph. Existing methods for counterfactual explanation of GNNs are limited to instance-specific local reasoning. This approach has two major limitations of not being able to offer global recourse policies and overloading human cognitive ability with too much information. In this work, we study the global explainability of GNNs through global counterfactual reasoning. Specifically, we want to find a small set of representative counterfactual graphs that explains all input graphs. Towards this goal, we propose GCF Explainer , a novel algorithm powered by vertex-reinforced random walks on an edit map of graphs with a greedy summary . Extensive experiments on real graph datasets show that the global explanation from GCF Explainer provides important high-level insights of the model behavior and achieves a 46.9% gain in recourse coverage, a 9.5% reduction in recourse cost compared to the state-of-the-art local counterfactual explainers. We also demonstrate that GCF Explainer generates explanations that are more consistent with input dataset characteristics, and is robust under adversarial attacks. In addition, K-GCF Explainer , which incorporates a graph clustering component into GCF Explainer , is introduced as a more competitive extension for datasets with a clustering structure, leading to superior performance in three out of four datasets in the experiments and better scalability.

Fleet repositioning in the tramp ship routing and scheduling problem with bunker optimization: A matheuristic solution approach

This paper investigates an important planning problem faced by dry bulk shipping operators, referred to as the Tramp Ship Routing and Scheduling Problem with Bunker Optimization (TSRSPBO). The problem is to maximize the overall profit of a fleet of vessels by selecting cargoes and determining ship routes and schedules. We consider this problem under a set of practically relevant features such as flexibility in cargo quantities, as well as bunkering decisions on where to procure fuel and how much. As a particularly novel feature, we address the regional allocation of vessels at the end of the planning period to be well prepared for meeting (uncertain) future demand. To incorporate this, we consider the TSRSPBO as a two-stage stochastic programming problem, where cargo selection, routing, and bunkering decisions are solved in the first-stage problem, and the recourse cost of fleet repositioning is considered in the second stage. We present arc flow and path flow formulations, where the latter employs a priori generation of feasible routes as input. For solving realistically sized instances, we propose a matheuristic based on an Adaptive Large Neighborhood Search (ALNS) framework that iteratively generates columns and solves the path flow model. Computational experiments based on real data show that this matheuristic finds high-quality solutions for large test instances with 120 cargoes, 30 vessels, and ten bunker ports in less than one hour. Also, considering the TSRSPBO as a two-stage stochastic problem achieves the highest profits and is solved almost as quickly as the deterministic problem variant.

A Stochastic Drone-Scheduling Problem with Uncertain Energy Consumption

In this paper, we present a stochastic drone-scheduling problem where the energy consumption of drones between any two nodes is uncertain. Considering uncertain energy consumption as opposed to deterministic energy consumption can effectively enhance the safety of drone flights. To address this issue, we developed a two-stage stochastic programming model with recourse cost, and we employed a fixed-sample sampling strategy based on Monte Carlo simulation to characterize uncertain variables, followed by the design of an efficient variable neighborhood search algorithm to solve the model. Case study results indicate the superiority of our algorithm over genetic algorithms. Additionally, a comparison between deterministic and stochastic models suggests that considering the uncertainty in energy consumption can significantly enhance the average returns of unmanned aerial vehicle scheduling systems.

The Impact of EU Subsidies on the Competitiveness of Slovak Farms from the Perspective of Legal Form

Supporting the competitiveness of farms is one of the main priorities of the European Union’s Common Agricultural Policy. Therefore, the analysis of transfer efficiency is an important element in the policy evaluation process. The primary aim of this partial research was to evaluate the development of farms’ competitiveness in Slovakia, considering the financial support from the European Union and focusing on the legal form perspective. We assessed farm competitiveness using the Recourse Cost Ratio coefficient, which compares costs and revenues of entities in Slovakia’s agricultural sector from 2004 to 2019. The analysis revealed statistically significant differences in competitiveness among farms based on their legal forms. From these findings, it is recommended that effective farm management is crucial for enhancing competitiveness, productivity, and profitability across all farms. Despite the specifics of individual legal forms of business, it is important to clarify that it is often the skills and abilities of the management of each enterprise that contribute to better acquisition of financial resources and also strengthen the competitive position of the farm in the market.

A practical and robust approach for solving the multi‐compartment vehicle routing problem under demand uncertainty using machine learning

AbstractThe multi‐compartment vehicle routing problem is an extension of the vehicle routing problem and consists of designing a set of routes to perform the collection or delivery of different product types from customers with minimal costs. The product types are incompatible with each other and must be transported separately in multiple compartments. In practice, several uncertainties such as uncertainty in demands can arise, where the exact demand of customers is not known at the time of planning. To deal with these uncertainties, decision‐makers have to rely on robust solutions. A solution is considered robust when it can resist perturbations in every possible demand scenario as much as possible. In the day to day business of most logistic companies, historical data about each customer can be stored and used to make intelligent decisions regarding the expected demands. In this article, we propose an adaptive large neighborhood search for solving the robust multi‐compartment vehicle routing problem under demand uncertainty and present a robust solution approach for the problem in practical settings by employing machine learning. We show that by using our approach, the solutions obtained have lower recourse costs and have a lower gap between expected and actual costs, which is a favorable outcome to have in practice.

A disaggregated integer L-shaped method for stochastic vehicle routing problems with monotonic recourse

This paper proposes a new integer L-shaped method for solving two-stage stochastic integer programs whose first-stage solutions can decompose into disjoint components, each one having a monotonic recourse function. In a minimization problem, the monotonicity property stipulates that the recourse cost of a component must always be higher or equal to that of any of its subcomponents. The method exploits new types of optimality cuts and lower bounding functionals that are valid under this property. The stochastic vehicle routing problem is particularly well suited to be solved by this approach, as its solutions can be decomposed into a set of routes. We consider the variant with stochastic demands in which the recourse policy consists of performing a return trip to the depot whenever a vehicle does not have sufficient capacity to accommodate a newly realized customer demand. This work shows that this policy can lead to a non-monotonic recourse function, but that the monotonicity holds when the customer demands are modeled by several commonly used families of probability distributions. We also present new problem-specific lower bounds on the recourse that strengthen the lower bounding functionals and significantly speed up the solving process. Computational experiments on instances from the literature show that the new approach achieves state-of-the-art results.

Recourse strategy for the routing problem of mobile parcel lockers with time windows under uncertain demands

It is expected that mobile parcel lockers (MPLs) will provide more responsive and flexible service to the increasing variation of demand than fixed parcel lockers. However, the service provided by MPLs following planned routes can be degraded and even fail due to the demand uncertainty, such as demand volume and pickup time. In this study, we propose an optimization approach integrated with the recourse strategies to overcome this barrier. Three types of uncertain demands and two types of service failures are addressed in this study, with consideration of uncertain demands and time window constraint. To optimally determine the a priori routes for a set of MPLs, we propose a tailored tabu search algorithm by taking into account the recourse cost. A series of numerical experiments were conducted to examine the effectiveness of the proposed approach in enhancing route reliability with a comparison to ordinary express vehicles. The results reveal that the service reliability can be significantly improved with only a marginal increase in the operational cost by the proposed approach.

A Decision Rule Approach for Two-Stage Data-Driven Distributionally Robust Optimization Problems with Random Recourse

We study two-stage stochastic optimization problems with random recourse, where the coefficients of the adaptive decisions involve uncertain parameters. To deal with the infinite-dimensional recourse decisions, we propose a scalable approximation scheme via piecewise linear and piecewise quadratic decision rules. We develop a data-driven distributionally robust framework with two layers of robustness to address distributional uncertainty. We also establish out-of-sample performance guarantees for the proposed scheme. Applying known ideas, the resulting optimization problem can be reformulated as an exact copositive program that admits semidefinite programming approximations. We design an iterative decomposition algorithm, which converges under some regularity conditions, to reduce the runtime needed to solve this program. Through numerical examples for various known operations management applications, we demonstrate that our method produces significantly better solutions than the traditional sample-average approximation scheme especially when the data are limited. For the problem instances for which only the recourse cost coefficients are random, our method exhibits slightly inferior out-of-sample performance but shorter runtimes compared with a competing approach. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the National Science Foundation [Grants 2342505 and 2343869]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0306 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0306 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Distributionally robust facility location with uncertain facility capacity and customer demand

This study explores a capacitated facility location problem where facility capacity and customer demand are subject to uncertainties simultaneously. This problem decides the subset of facilities to open at the system design phase to serve customers during the operational phase. The objective is to minimize the total cost, including the first-stage location cost and the second-stage recourse cost, and guarantee the system’s reliability, i.e., meeting demand as much as possible when uncertainties arise. A distributionally robust optimization (DRO) framework is utilized to model the problem. A scenario-wise ambiguity set with partial distributional information of random variables is constructed, which can capture uncertainties caused by different random events or different magnitudes of the same event type and explicitly represent the correlation between facilities’ uncertain capacity and customers’ uncertain demand. We apply an adaptation policy to the DRO model and reformulate it to a mixed-integer linear programming model, which is solvable by off-the-shelf solvers. Numerical results show that the scenario-wise DRO framework can provide a better trade-off between cost and service level than the stochastic programming model and the DRO model with a marginal moment-based ambiguity set, demonstrating that the proposed scenario-wise DRO model offers a practical decision-making tool for enhancing supply chain robustness via facility location.

An Integer L-shaped algorithm for vehicle routing problem with simultaneous delivery and stochastic pickup

This paper addresses an exact algorithm for vehicle routing problem with simultaneous pickup and delivery in which customer demands to be collected are stochastic. The problem is modeled as a two-stage stochastic programming problem with recourse, in which routing decisions are made based on known delivery demand and deterministic expected pickup demand in the first stage and recourse actions are made in the second stage when stochastic pickup quantities have been revealed. However, failures happen when the load of the vehicle is insufficient to meet the observed pickup demand of a customer. Three recourse policies are proposed to help deal with the failures to proceed with the routing decisions in the first stage. The Integer L-shaped algorithmic framework is used to solve this two-stage stochastic programming problems with recourse. Furthermore, effective lower bounding of the expected recourse cost of partial routes is designed for the three recourse policies, respectively. Computational experiments on the newly generated instances compare the performance of the Integer L-shaped algorithm under the three recourse policies, and the conclusion is validated via numerous simulations. The effectiveness of the proposed lower bounding functionals is confirmed through reduced optimality gaps and lower computing times.

Generalized adaptive partition-based method for two-stage stochastic linear programs: Geometric oracle and analysis

Adaptive Partition-based Methods (APM) are numerical methods that solve, in particular, two-stage stochastic linear problems (2SLP). We say that a partition of the uncertainty space is adapted to the current first stage control xˇ if we can aggregate scenarios while conserving the true value of the expected recourse cost at xˇ. The core idea of APM is to iteratively construct an adapted partition to all past tentative first stage controls. Relying on the normal fan of the dual admissible set, we give a necessary and sufficient condition for a partition to be adapted even for non-finite distribution, and provide a geometric method to obtain an adapted partition. Further, by showing the connection between APM and the L-shaped algorithm, we prove convergence and complexity bounds of the APM methods. The paper presents the fixed recourse case and ends with elements to forgo this assumption.

A New Cooperative Recourse Strategy for Emergency Material Allocation in Uncertain Environments

Emergency material allocation is an important issue in the urgent handling of public health emergencies. This article models the relief allocation and transportation route planning as an uncertain capacitated arc routing problem, which is a classic combinatorial optimization problem that considers stochastic factors such as uncertain demand and travel time in the service. The stochastic of demand leads to the route failure that the vehicle cannot serve the tasks successfully unexpected. Most existing research uses the independent recourse strategy. That is, each vehicle takes a back-and-forth trip separately when its remaining capacity cannot meet the actual demand of the task. This leads to a considerable recourse cost. However, a few studies have considered vehicular cooperation to deal with route failure, which is beneficial for pooling the capacity of multiple vehicles. In this paper, we propose a new recourse strategy called OneFAll that lets one vehicle take charge of all the failed tasks. In this case, other vehicles can finish the service once they are full. We develop the genetic programming hyper-heuristic with the OneFAll recourse strategy for solving the uncertain capacitated arc routing problem. The experimental studies show that our proposed method outperforms the existing genetic programming hyper-heuristic with the independent recourse strategy to the uncertain capacitated arc routing problem for the ugdb and uval benchmark instances. Moreover, our strategy outperforms the recourse strategy that failed tasks are returned to the unassigned task set for any vehicle to complete. This reflects that there exists resource waste if all vehicles are involved to repair the failed routes.

Constraint generation for risk averse two-stage stochastic programs

A significant share of stochastic optimization problems in practice can be cast as two-stage stochastic programs. If uncertainty is available through a finite set of scenarios, which frequently occurs, and we are interested in accounting for risk aversion, the expectation in the recourse cost can be replaced with a worst-case function (i.e., robust optimization) or another risk-functional, such as conditional value-at-risk. In this paper we are interested in the latter situation especially when the number of scenarios is large. For computational efficiency we suggest a (clustering and) constraint generation algorithm. We establish convergence of these two algorithms and demonstrate their effectiveness through various numerical experiments.

A simulation-based solution approach for the robust capacitated vehicle routing problem with uncertain demands

ABSTRACT This article introduces a solution approach for the Stochastic Capacitated Vehicle Routing Problem (SCVRP) with uncertain demands, called Robust Simulation-Based (RoSi) approach. RoSi aims at designing route plans that can be more or less robust based on a decision-maker weight, i.e. solutions that resist demand changes with marginal additional (recourse) cost. For that, RoSi combines simulation with heuristics. It transforms a complex SCVRP into a set of deterministic ones, where well-known heuristics can be applied, computing a set of feasible solutions. These solutions are assessed by Monte Carlo simulation, and the one that deals better with demand fluctuation is selected as the final solution. The efficiency of RoSi is compared with those of three methods in the literature: Integer Linear Programming (ILP) model, Stochastic Programming with Recourse (SPR) model, and Robust Bi-Objective (RoBi) approach through numerical experiments. The results show that RoSi outperforms these methods in most scenarios.

The Vehicle Routing Problem with Stochastic Two-Dimensional Items

We consider a stochastic vehicle routing problem where a discrete probability distribution characterizes the two-dimensional size (height and width) as well as the weight of a subset of items to be delivered to customers. Although some item sizes and weights are not known with certainty when the routes are planned, they become known when it is time to load the vehicles, just before their departure. If it happens that not all items can be loaded in a vehicle, the items of one or more customers are put aside at a penalty or recourse cost. The objective is to minimize the sum of the routing and expected recourse costs. The problem is modeled as a two-stage stochastic program and solved with the integer L-shaped method. Some new inequalities and lower bounds are proposed. Computational results are reported on test instances specifically generated for this problem, as well as on classical instances for the deterministic case.

An exact method for disassembly line balancing problem with limited distributional information

As an important part in product recycling, disassembly line balancing problem (DLBP) has attracted a large amount of attention. Stochastic DLBP is now a hot and challenging research topic, due to its wide applications. This work investigates a DLBP with uncertain task times, where the distributional information is limited, i.e. only the mean values and standard deviations are given, due to the lack of data. From a two-stage perspective, a disassembly process is determined and the disassembly tasks are assigned to workstations in the first stage, and the penalty cost (i.e. the recourse cost) for exceeding the cycle time is minimised in the second stage. The objective is to minimise the expected system cost. For the problem, a two-stage distributionally robust formulation is devised, to minimise the worst-case expected system cost out of all possible probability distributions. Different from literature that focuses on approximation methods to tackle the limited distributional information, an exact method, i.e. the cutting-plane algorithm, is developed. Numerical results show that compared with the state-of-the-art solution methods, our developed cutting-plane algorithm can provide more reliable solutions in term of the robustness, especially in some extreme cases.

Risk Aversion Based Inexact Stochastic Dynamic Programming Approach for Water Resources Management Planning under Uncertainty

In this study, a dual interval robust stochastic dynamic programming (DIRSDP) method is developed for planning water resources management systems under uncertainty. As an extension of the existing interval stochastic dynamic programming (ISDP) method, DIRSDP can deal with two-stage stochastic programming (TSP)-based planning problems associated with dynamic features, input uncertainties, and multistage concerns. Compared with other optimization methods dealing with uncertainties, the developed DIRSDP method has advantages in addressing uncertainties with complex presentations and reflecting decision makers’ risk-aversion attitudes within its optimization process. Parameters in the DIRSDP model can be represented as probability distributions as well as single and/or dual intervals. Decision makers’ risk-aversion attitudes can be reflected through restricting the deviation of the recourse costs to a tolerance level. Water-allocation plans can then be developed based on the analysis of tradeoffs between the system benefit and solution robustness. The developed method is applied to a case of water resources management planning. The solutions are reasonable, indicating applicability of the developed methodology.

A local branching matheuristic for the multi-vehicle routing problem with stochastic demands

This paper proposes a local branching matheuristic for the vehicle routing problem with stochastic demands (VRPSD). The problem is cast in a two-stage stochastic programming model, in which routes are planned in the first stage and executed in the second stage. In this setting, a failure may occur if a vehicle does not have sufficient capacity to serve the realized demand of a customer, which is revealed only upon arrival at a customer’s location. In the event of a failure, a recourse action is performed by having the vehicle return to the depot to replenish its capacity and resume its planned route at the point of failure. Thus, the objective of the VRPSD is to minimize the sum of the planned routes cost and of the expected recourse cost. We propose a local branching matheuristic to solve the multi-VRPSD. We introduce an intensification procedure applied at each node of the local branching tree. This procedure is embedded in a multi-descent scheme for which we propose a diversification strategy. Extensive computational results demonstrate the effectiveness of our matheuristic.

Planning of Electric Power Systems Considering Virtual Power Plants with Dispatchable Loads Included: An Inexact Two-Stage Stochastic Linear Programming Model

In this study, an inexact two-stage stochastic linear programming (ITSLP) method is proposed for supporting sustainable management of electric power system under uncertainties. Methods of interval-parameter programming and two-stage stochastic programming were incorporated to tackle uncertainties expressed as interval values and probability distributions. The dispatchable loads are integrated into the framework of the virtual power plants, and the support vector regression technique is applied to the prediction of electricity demand. For demonstrating the effectiveness of the developed approach, ITSLP is applied to a case study of a typical planning problem of power system considering virtual power plants. The results indicate that reasonable solutions for virtual power plant management practice have been generated, which can provide strategies in mitigating pollutant emissions, reducing system costs, and improving the reliability of power supply. ITSLP is more reliable for the risk-aversive planners in handling high-variability conditions by considering peak-electricity demand and the associated recourse costs attributed to the stochastic event. The solutions will help decision makers generate alternatives in the event of the insufficient power supply and offer insight into the tradeoffs between economic and environmental objectives.

An exact algorithm to solve the vehicle routing problem with stochastic demands under an optimal restocking policy

This paper examines the Vehicle Routing Problem with Stochastic Demands (VRPSD), in which the actual demand of customers can only be realized upon arriving at the customer location. Under demand uncertainty, a planned route may fail at a specific customer when the observed demand exceeds the residual capacity. There are two ways to face such failure events, a vehicle can either execute a return trip to the depot at the failure location and refill the capacity and complete the split service, or in anticipation of potential failures perform a preventive return to the depot whenever the residual capacity falls below a threshold; overall, these return trips are called recourse actions. In the context of VRPSD, a recourse policy which schedules various recourse actions based on the visits planned beforehand on the route must be designed. An optimal recourse policy prescribes the cost-effective returns based on a set of optimal customer-specific thresholds. We propose an exact solution method to solve the multi-VRPSD under an optimal restocking policy. The Integer L-shaped algorithm is adapted to solve the VRPSD in a branch-and-cut framework. To enhance the efficiency of the presented algorithm, several lower bounding schemes are developed to approximate the expected recourse cost.