Expected Penalty Cost Research Articles

Multi-period time window assignment for attended home delivery

We study a multi-period stochastic variant of the Time Window Assignment Vehicle Routing Problem, where customers’ demands, locations, and service times are uncertain. Customers are partitioned into geographical zones, each of which has to be visited a predetermined number of times over a planning period of several days. Whenever a zone is visited, a time window is assigned. Time windows are decided before knowing customers and their demands. A fleet of homogeneous vehicles is available to serve customers each day. At a tactical level, the problem looks for a static time window assignment that minimizes the expected traveling costs plus the expected penalty costs for unserved customers. We propose a two-stage formulation and a solution approach, which relies on the Sample Average Approximation Method, while encompassing a perturbation method to assign time windows in the first stage and an Adaptive Large Neighborhood Search to optimize routes in the second stage. We experimentally evaluate three instance sets, including real ones from a Canadian company, comparing our results to lower bounds from the exact solution of a deterministic equivalent formulation over a finite number of scenarios. Our method outperforms the manual approach used by the company.

Energy scheduling for microgrids with renewable energy sources considering an adjustable convex hull based uncertainty set

In this paper, a risk-based robust energy scheduling model for microgrids is proposed to address the uncertainty of renewable energy sources (RESs). The proposed model introduces a novel adjustable convex hull based uncertainty set (ACHUS), which is a subset of the maximum uncertainty set, to quantify the uncertainty of RESs. The operational risk of an ACHUS is evaluated by a historical data-based method. This method defines the operational risk as the expected penalty cost of all historical data when the operational safety of the microgrid is guaranteed in the ACHUS. Based on the evaluation of the operational risk, a day-ahead energy scheduling model with the ACHUS for microgrids is formulated as a mixed integer nonlinear programming. By introducing auxiliary variables and constraints, the mixed integer nonlinear programming is converted into a solvable mixed integer linear programming. The effectiveness of the proposed method is validated on the IEEE 6-bus and 118-bus systems. The simulation results show that the proposed method can effectively reduce the conservativeness of uncertainty quantification and the improve the economic efficiency of the microgrid.

A Wasserstein Distance-Based Distributionally Robust Chance-Constrained Clustered Generation Expansion Planning Considering Flexible Resource Investments

This paper proposes a distributionally robust chance-constrained (DRCC) model for the clustered generation expansion planning (CGEP) of power systems. The proposed two-stage model minimizes the first-stage total cost along with the second-stage expected penalty cost with the worst-case probability distribution of renewable energy generation. A unit commitment model with flexibility constraints is embedded into the planning model, and the uncertainty is modeled via a Wasserstein distance (WD)-based ambiguity set. Demand-side resources (DSR) and concentrating solar power (CSP) plants are considered as candidates in the DRCC-CGEP model to enhance system flexibility, and the solution efficiency is improved through unit clustering. Furthermore, based on strong duality theory along with affine decision rule and conditional-value-at-risk approximation method, the proposed planning model is reformulated as a tractable mixed-integer linear programming problem. Numerical results show that the proposed WD-based DRCC-CGEP model is effective in improving the economics of the planning decisions while ensuring system reliability and maintaining computational efficiency.

Risk‐averse two‐stage stochastic programming for the inventory rebalancing of bike‐sharing systems

AbstractAs the popularity and usage of bike‐sharing systems increase, a better decision‐making model tailored for the successful operations of bike‐sharing systems is needed. This study is motivated to address operator‐based inventory rebalancing of bike‐sharing systems, and the main objective is to develop a mathematical optimization model designed to derive an optimal daily inventory rebalancing plan. Specifically, this study proposes a risk‐averse two‐stage stochastic programming to determine optimal initial inventory levels for each station to minimize operational costs for relocating bikes and expected penalty costs due to unmet requests. This study adopts the conditional value at risk to properly measure the risk associated with unmet requests to implement risk‐averse stochastic programming. Numerical experiments are conducted based on scenario data generated by empirical distributions fitted to trip data from the Houston BCycle to validate and evaluate the proposed model. The results show that the proposed model can be successfully applied to inventory rebalancing to improve the usability of bike‐sharing systems.

Multi-class pilot dispatching problem with stochastic pilotage duration

In most seaports, pilotage is compulsory for incoming and outgoing vessels to use pilotage services. Maritime pilots play a critical role in providing local knowledge and assistance to ensure the safety of vessels and seafarers. Typically, pilots are classified into multiple classes on the basis of their experiences and qualifications. Moreover, pilotage duration is usually stochastic due to various factors, including harsh weather conditions, visibility, and traffic changes. In this study, we address a multi-class pilot dispatching problem with stochastic pilotage duration. We formulate this problem as a stochastic programming model that considers the expected penalty cost for late arrival at each task's location. To solve this problem, we develop a branch-and-price algorithm that integrates column generation within the branch and bound framework. Additionally, we develop a novel labelling algorithm and several effective acceleration techniques to enhance the efficiency of the proposed algorithm. Extensive numerical experiments demonstrate the effectiveness of the proposed algorithm and show that it can handle realistic-sized instances while outperforming a metaheuristic search method.

Distributionally Robust Coordinated Expansion Planning for Generation, Transmission, and Demand Side Resources Considering the Benefits of Concentrating Solar Power Plants

Renewable energy has gradually increased its investment and penetration due to technology innovation and cost reduction, turning power system expansion planning problem (SEP) into a complex task. Under this circumstance, this paper presents a long-term distributionally robust optimization (DRO) based planning model for power systems, which minimizes the total investment and operation costs along with expected penalty cost of load shedding and wind curtailment, while considering load and wind uncertainties from the data-driven ambiguity set. Besides generating units and transmission lines, demand side resources (DSR) and concentrating solar power (CSP) plants are also considered as investment candidates to meet the load growth and effectively alleviate load and wind uncertainties. Furthermore, the principal component analysis method is used to capture the generalized moment information of historical data to reduce conservativeness of planning decisions. The proposed model is recast as a mixed integer linear programming (MILP) model via duality theory and affine decision rules. Numerical case studies illustrate the effectiveness and scalability of the proposed DRO-based coordinated expansion planning model.

A shared-mobility-based framework for evacuation planning and operations under forecast uncertainty

To meet evacuation needs from carless populations who need personalized assistance to evacuate safely, in this article we propose a ridesharing-based evacuation program that recruits volunteer drivers before a disaster strikes, and then matches volunteer drivers with evacuees once demand is realized. We optimize resource planning and evacuation operations under uncertain spatiotemporal demand, and construct a two-stage stochastic mixed-integer program to ensure high demand fulfillment rates. We consider three formulations to improve the number of evacuees served, by minimizing an expected penalty cost, imposing a probabilistic constraint, and enforcing a constraint on the conditional value at risk of the total number of unserved evacuees, respectively. We discuss the benefits and disadvantages of the different risk measures used in the three formulations, given certain carless population sizes and the variety of evacuation modes available. We also develop a heuristic approach to provide quick, dynamic and conservative solutions. We demonstrate the performance of our approaches using five different networks of varying sizes based on regions of Charleston County, South Carolina, an area that experienced a mandatory evacuation order during Hurricane Florence, and utilize real demographic data and hourly traffic count data to estimate the demand distribution.

Emergency relief network design under ambiguous demands: A distributionally robust optimization approach

This paper focuses on an emergency rescue network design problem in response to disasters under uncertainty. Considering the limited distribution information of the uncertain demands extracted from the historical data, we use the mean absolute deviation (MAD) that can derive tractable reformulations and better capture outliers and small deviations, to construct a MAD-based ambiguity set. A distributionally robust optimization model is proposed with the objective of minimizing the preparedness cost and the expected penalty cost of demand shortage under the worst-case distribution over the ambiguity set. We analyze the constructed model and provide some features such as the theoretical bounds of the objective value. For large-scale cases, we reformulate the knotty model using the linear decision rule to obtain tight and tractable problems. Computational experiments verify that the out-of-sample performance of the proposed model is better than that of the stochastic optimization model, especially for extreme cases. The MAD-based ambiguity set combined with the approximation technique can reduce the solution time and obtain high-quality solutions. Moreover, the results show that the amount of data has a significant effect on model performance. These results provide references for decision-makers in the practice of emergency response network design.

Approximation algorithm with constant ratio for stochastic prize-collecting Steiner tree problem

<p style='text-indent:20px;'>Steiner tree problem is a typical NP-hard problem, which has vast application background and has been an active research topic in recent years. Stochastic optimization problem is an important branch in the field of optimization. Compared with deterministic optimization problem, it is an optimization problem with random factors, and requires the use of tools such as probability and statistics, stochastic process and stochastic analysis. In this paper, we study a two-stage finite-scenario stochastic prize-collecting Steiner tree problem, where the goal is to minimize the sum of the first stage cost, the expected second stage cost and the expected penalty cost. Our main contribution is to present a primal-dual 3-approximation algorithm for this problem.</p>

Integrated Vehicle Routing and Service Scheduling Under Time and Cancellation Uncertainties with Application in Nonemergency Medical Transportation

In this paper, we consider an integrated vehicle routing and service scheduling problem for serving customers in distributed locations who need pick-up, drop-off, or delivery services. We take into account the random trip time, nonnegligible service time, and possible customer cancellations, under which an ill-designed schedule may lead to undesirable vehicle idleness and customer waiting. We build a stochastic mixed-integer program to minimize the operational cost plus expected penalty cost of customers’ waiting time, vehicles’ idleness, and overtime. Furthermore, to handle real-time arrived service requests, we develop K-means clustering-based algorithms to dynamically update planned routes and schedules. The algorithms assign customers to vehicles based on similarities and then plan schedules on each vehicle separately. We conduct numerical experiments based on diverse instances generated from census data and data from the Ford Motor Company’s GoRide service, to evaluate result sensitivity and to compare the in-sample and out-of-sample performance of different approaches. Managerial insights are provided using numerical results based on different parameter choices and uncertainty settings.

Parallel machine scheduling with stochastic release times and processing times

Stochastic scheduling has received much attention from both industry and academia. Existing works usually focus on random job processing times. However, the uncertainty existing in job release times may largely impact the performance as well. This work investigates a stochastic parallel machine scheduling problem, where job release times and processing times are uncertain. The problem consists of a two-stage decision-making process: (i) assigning jobs to machines on the first stage before the realisation of uncertain parameters (job release times and processing times) and (ii) scheduling jobs on the second stage given the job-to-machine assignment and the realisation of uncertain parameters. The objective is to minimise the total cost, including the setup cost on machines (induced by job-to-machine assignment) and the expected penalty cost of jobs' earliness and tardiness. A two-stage stochastic program is proposed, and the sample average approximation (SAA) method is applied. A scenario-reduction-based decomposition approach is further developed to improve the computational efficiency. Numerical results show that the scenario-reduction-based decomposition approach performs better than the SAA, in terms of solution quality and computation time.

Addressing the Bike Repositioning Problem in Bike Sharing System: A Two-Stage Stochastic Programming Model

In this paper, a bike repositioning problem with stochastic demand is studied. The problem is formulated as a two-stage stochastic programming model to optimize the routing and loading/unloading decisions of the repositioning truck at each station and depot under stochastic demands. The goal of the model is to minimize the expected total sum of the transportation costs, the expected penalty costs at all stations, and the holding cost of the depot. A simulated annealing algorithm is developed to solve the model. Numerical experiments are conducted on a set of instances from 20 to 90 stations to demonstrate the effectiveness of the solution algorithm and the accuracy of the proposed two-stage stochastic model.

An integrated car-and-ride sharing system for mobilizing heterogeneous travelers with application in underserved communities

The fast-growing carsharing and ride-hailing businesses are generating economic benefits and societal impacts in modern society, while both have limitations to satisfy diverse users, e.g., travelers in low-income, underserved communities. In this article, we consider two types of users: Type 1 drivers who rent shared cars and Type 2 passengers who need shared rides. We propose an integrated car-and-ride sharing (CRS) system to enable community-based shared transportation. To compute solutions, we propose a two-phase approach where in Phase I we determine initial car allocation and Type 1 drivers to accept; in Phase II we solve a stochastic mixed-integer program to match the accepted Type 1 drivers with Type 2 users, and optimize their pick-up routes under a random travel time. The goal is to minimize the total travel cost plus expected penalty cost of users’ waiting and system overtime. We demonstrate the performance of a CRS system in Washtenaw County, Michigan by testing instances generated based on census data and different demand patterns. We also demonstrate the computational efficacy of our decomposition algorithm benchmarked with the traditional Benders decomposition for solving the stochastic model in Phase II. Our results show high demand fulfillment rates and effective matching and scheduling with low risk of waiting and overtime.

Improving delivery performance for gamma distributed delivery time

This paper investigates strategies for improving delivery performance to the end customer in a two-stage supply chain. A cost-based analytical model is used to evaluate how the expected penalty cost resulting from early and late delivery can be reduced. The effects of the width of the delivery window and the shape and scale parameters of the gamma distributed delivery time distribution on the expected penalty cost are explored. The model can guide practitioners who are attempting to cost-justify a program to improve delivery performance. It can also be used to estimate cost reduction over several deliveries and compare cost savings with required investments into delivery performance improvement. The model overcomes the limitations of previous delivery improvement studies which did not used an optimally positioned delivery window and were limited to symmetric delivery time distributions.

Robust resource-constrained max-NPV project scheduling with stochastic activity duration

This study investigates the robust resource-constrained max-NPV project problem with stochastic activity duration. First, the project net present value (NPV) and the expected penalty cost are proposed to measure quality robustness and solution robustness from the perspective of discounted cash flows, respectively. Then, a composite robust scheduling model is proposed in the presence of activity duration variability and a two-stage algorithm that integrates simulated annealing and tabu search is developed to deal with the problem. Finally, an extensive computational experiment demonstrates the superiority of the combination between quality robustness and solution robustness as well as the effectiveness of the proposed two-stage algorithm for generating project schedules compared with three other algorithms, namely simulated annealing, tabu search, and multi-start iterative improvement method. Computational results indicate that the proactive project schedules with composite robustness not only can effectively protect the payment plan from disruptions through allocating appropriate time buffers, but also can achieve a remarkable performance with respect to the project NPV.

Solving 0–1 semidefinite programs for distributionally robust allocation of surgery blocks

We allocate surgery blocks to operating rooms (ORs) under random surgery durations. Given unknown distribution of the duration of each block, we investigate distributionally robust (DR) variants of two types of stochastic programming models using a moment-based ambiguous set. We minimize the total cost of opening ORs and allocating surgery blocks, while constraining OR overtime via chance constraints and via an expected penalty cost in the objective function, respectively in the two types of models. Following conic duality, we build equivalent 0–1 semidefinite programming (SDP) reformulations of the DR models and solve them using cutting-plane algorithms. For the DR chance-constrained model, we also derive a 0–1 second-order conic programming approximation to obtain less conservative solutions. We compare different models and solution methods by testing randomly generated instances. Our results show that the DR chance-constrained model better controls average and worst-case OR overtime, as compared to the stochastic programming and DR expected-penalty-based models. Our cutting-plane algorithms also outperform standard optimization solvers and efficiently solve 0–1 SDP formulations.

Improving delivery performance for asymmetric Laplace distributed delivery time in a two-stage supply chain

This paper investigates strategies for improving supply chain delivery timeliness when the delivery time follows an asymmetric Laplace distribution. Delivery performance is measured using a cost-based analytical model which evaluates the expected cost for early and late delivery. This paper presents a set of propositions that define the effect of changes to the parameters of the delivery time distribution on the expected penalty cost for untimely delivery when a supplier uses an optimally positioned delivery window to minimise the expected cost of untimely delivery. The scale parameter increases the expected penalty cost, skewness decreases the cost and the location parameter has no effect on the expected penalty cost. The effects are illustrated in a numerical example with real-world supplier data. The results can be used in developing strategies for improving delivery performance from a supplier’s perspective and define how the delivery time distribution parameters can be modified to decrease the expected penalty cost of untimely delivery. The paper proposes a general approach to modelling delivery performance improvement and can be applied to other delivery time distribution forms. The approach can serve as guidance for practitioners undertaking a programme to improve delivery performance.

Delivery performance improvement in two-stage supply chain

This paper investigates strategies of supply chain delivery performance improvement. The performance is measured using a cost-based analytical model which evaluates the expected penalty cost for early and late delivery. The results demonstrate strategies for improving delivery performance when a supplier uses an optimally positioned delivery window to minimize the expected penalty cost. The effect of the width of the delivery window and penalty costs for early and late deliveries on the optimal position of the delivery window and the expected penalty cost are explored in general case. In addition to that, the effect of a mean and variance parameters of a normal delivery time distribution are revised. Theoretical and managerial implications of the findings are discussed.

Screening Strategies for Patients on the Kidney Transplant Waiting List

Transplant centers screen patients on the kidney transplant waiting list to identify patients with severe conditions such as cardiovascular disease (CVD), which makes them ineligible for a transplant. We propose a model for finding screening strategies, with the objective of minimizing the sum of the expected screening cost and the expected penalty cost associated with transplanting an organ to an ineligible patient. Our results suggest that current screening guidelines, which are based only on patients’ risk for developing CVD, are significantly dominated by policies that also consider factors related to patients’ waiting time. In particular, our numerical experiments based on waiting list data in British Columbia show that compared with the current screening policy, our model-based policy results in a 35.6% reduction in the total annual cost by reducing the percentage of kidneys offered to patients with undetected CVD by 57.4% while using only 5% more screenings per year. The online appendix is available at https://doi.org/10.1287/opre.2017.1632 .

Penalties, Manipulation, and Investment Efficiency

In this study, we examine whether imposing a penalty based on an earlier positive signal and a bad realized outcome can be welfare-improving. We find that imposing a penalty helps to improve investment efficiency, but it also brings a deadweight cost of potential penalty for entrepreneurs with good projects. We show that when a good project has a much larger chance to achieve a good outcome than a bad project or when a large proportion of the penalty can be reimbursed to the investor, it is optimal to impose a penalty to deter entrepreneurs with bad projects from reporting positive signals to the greatest degree, as the benefit from reducing the investment inefficiency outweighs the expected deadweight cost. Otherwise, it is optimal not to impose any penalty. We also find that a larger penalty may induce more manipulation by a good entrepreneur. This is because, although a larger penalty directly discourages manipulation for both good and bad types by increasing the expected penalty cost, it also encourages upward manipulation by lowering the implicit financing cost upon a high signal. The encouraging effect is mostly pronounced for the good entrepreneur when the good entrepreneur has a much larger chance of success than the bad entrepreneur or when the reimbursement proportion is very high. This paper was accepted by Shiva Rajgopal, accounting.