Robust Counterpart Research Articles

Supermodularity in Two-Stage Distributionally Robust Optimization

In this paper, we solve a class of two-stage distributionally robust optimization problems that have the property of supermodularity. We exploit the explicit worst case expectation of supermodular functions and derive the worst case distribution for the robust counterpart. This enables us to develop an efficient method to obtain an exact optimal solution to these two-stage problems. Further, we provide a necessary and sufficient condition for checking whether any given two-stage optimization problem has the supermodularity property. We also investigate the optimality of the segregated affine decision rules when problems have the property of supermodularity. We apply this framework to several classic problems, including the multi-item newsvendor problem, the facility location problem, the lot-sizing problem on a network, the appointment-scheduling problem, and the assemble-to-order problem. Whereas these problems are typically computationally challenging, they can be solved efficiently under our assumptions. Finally, numerical examples are conducted to illustrate the effectiveness of our approach. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71971187], and Hong Kong Research Grants Council [Collaborative Research Fund (C6103-20GF), General Research Fund (14210821, 16204521, 16212419)]. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2023.4748 .

Robust nonparametric regression: Review and practical considerations

Nonparametric regression models offer a way to understand and quantify relationships between variables without having to identify an appropriate family of possible regression functions. Although many estimation methods for these models have been proposed in the literature, most of them can be highly sensitive to the presence of a small proportion of atypical observations in the training set. A review of outlier robust estimation methods for nonparametric regression models is provided, paying particular attention to practical considerations. Since outliers can also influence negatively the regression estimator by affecting the selection of bandwidths or smoothing parameters, a discussion of robust alternatives for this task is also included. Using many of the “classical” nonparametric regression estimators (and their robust counterparts) can be very challenging in settings with a moderate or large number of explanatory variables, so recently proposed robust nonparametric regression methods that scale well with a growing number of covariates are also discussed.

A robust optimization approach for the vehicle routing problem with cross-docking under demand uncertainty

This research addresses the Vehicle Routing Problem with Cross-docking under Demand Uncertainty (VRPCD-DU) where a set of homogeneous vehicles is used to transport orders from the suppliers to the corresponding customers via a cross-dock. VRPCD-DU considers customer demand volumes as random variables and determines a minimum cost delivery plan that is feasible for all anticipated demand realizations. A robust optimization counterpart of a deterministic VRPCD formulation is derived where support for the demand is a polyhedral set. The robust formulation can only be used to solve small VRPCD-DU instances, and an effective adaptive large neighborhood search (ALNS) algorithm is proposed for solving large instances. Extensive numerical experiments are conducted on benchmark sets for both VRPCD and VRPCD-DU. The results show that the ALNS algorithm computes new best-known solutions of the VRPCD benchmark instances. Moreover, demand uncertainty is extensively analyzed by investigating managerial insights. The price of robustness (PoR) is compared by considering various budget sets and alternative partitioning methods for the customers (i.e., “random” and “clustered”). The findings imply that a higher probability of capacity constraints’ violation is generally observed for the suppliers and that customer partitioning methods share similar PoR.

A location–transportation problem under demand uncertainty for a pharmaceutical network in Brazil

We study the integrated location–transportation problem under uncertain demand, addressing the case of a pharmaceutical logistics network in Brazil. We propose a mathematical model with multi-time scales to make decisions accordingly with their timing. The model considers practical features, such as fleet sizing, safety constraints in cargo transportation, and tax issues. We address uncertainty in decision-making by developing a robust counterpart and present solution methods based on Fix-and-Optimize heuristics. We perform computational experiments using real data from a partner company and evaluate the impact of uncertainty on the problem. The heuristics outperform the MIP model by reducing the average of logistics costs and taxes by 40%. Demand uncertainty and variability significantly influence the problem decisions. Also, the robust model reduces expected solution costs. Thus, these models and solution methods can support the decision-making process on location–transportation problems in Logistics Network Planning in contexts similar to the pharmaceutical industry in Brazil.

An optimization-based methodology equipped with lean tools to strengthen medical supply chain resilience during a pandemic: A case study from Turkey

Drawing upon economic and environmental sustainability, this study explores how developing the operational resilience of the medical supply chain (MSC) contributes to maintaining healthcare in the face of disruption risks, such as the COVID-19 pandemic. To this end, an optimization-based roadmap is proposed by employing lean tools to achieve and realize MSC resilience. A novel two-stage stochastic optimization model and robust counterpart are developed with the objective of overall cost minimization to cope with the unknowable demand uncertainty represented by scenarios. The reason behind proposing a scenario-based stochastic model is to implement preparedness strategies during the (re)design phase by making strategic and operational level decisions. That being the case, seven cases are generated based on the demand uncertainty intervals along with seven different reliability levels for sensitivity analysis. Computational experiments are conducted through a real case study to compare the centralized and decentralized distribution models in terms of efficiency and responsiveness. The results obtained by the stochastic model and robust counterpart are compared to demonstrate how strong the proposed model is. On top of that, lean tools are used to visualize and analyze the improvement opportunities to contribute to the methodology. By doing so, this paper presents novel theoretical and empirical insights regarding MSC resilience. The computational results emphasize the importance of employing a pre-disruption strategy via the proposed methodology to design a resilient MSC to be prepared for pandemic-related risk. The findings from the sensitivity analysis also verify that regardless of the disruption degree, the developed roadmap with the centralized distribution model leads to up to 40% improvements in terms of the overall cost, order lead time, emission amount, and inventory shortage metrics.

Adaptive distributionally robust hub location and routing problem with a third-party logistics strategy

In this paper, we study an essential and more realistic variant of the hub location-routing problem in which third-party logistics (3PL) are incorporated into a self-built distribution hub network. A two-stage mixed-integer programming model is developed to address the proposed hub location-routing problem with mixed logistical strategies. In the strategic stage, the hub location and allocation decisions are made before the realization of uncertain demands to minimize the total construction and transportation costs. In the operational stage, the managers make vehicle routing decisions with and without the 3PL strategy, and route planning is considered for multiple owned and 3PL vehicles on the basis of simultaneous pickup and delivery. We investigate the effects of uncertain factors on the network design of this framework, as the node demands are often unobservable or difficult to obtain accurately in practice. To address the inherent uncertainty of the problem, we propose an adaptive distributionally robust model by limiting uncertain demands to a specified ambiguity set. Moreover, we reformulate the proposed model into tractable robust counterpart forms under the ambiguity set with information about the support, mean and upper bounds on the dispersion. We design a genetic algorithm with local search to solve the reformulated model on a large scale. Finally, we conduct a case study for a leading retail enterprise in the Beijing–Tianjin–Hebei region to demonstrate the effectiveness and applicability of our model.

Optimality conditions and duality results for a robust bi-level programming problem

Robust bi-level programming problems are a newborn branch of optimization theory. In this study, we have considered a bi-level model with constraint-wise uncertainty at the upper-level, and the lower-level problem is fully convex. We use the optimal value reformulation to transform the given bi-level problem into a single-level mathematical problem and the concept of robust counterpart optimization to deal with uncertainty in the upper-level problem. Necessary optimality conditions are beneficial because any local minimum must satisfy these conditions. As a result, one can only look for local (or global) minima among points that hold the necessary optimality conditions. Here we have introduced an extended non-smooth robust constraint qualification (RCQ) and developed the KKT type necessary optimality conditions in terms of convexifactors and subdifferentials for the considered uncertain two-level problem. Further, we establish as an application the robust bi-level Mond-Weir dual (MWD) for the considered problem and produce the duality results. Moreover, an example is proposed to show the applicability of necessary optimality conditions.

A Distributionally Robust Fuzzy Optimization Method for Single-Period Inventory Management Problems

This paper investigates single-period inventory management problems with uncertain market demand, where the exact possibility distribution of demand is unavailable. In this condition, it is important to order a reliable quantity which can immunize against distribution uncertainty. To model this type of single-period inventory management problem, this paper characterizes the uncertain demand by generalized interval-valued possibility distributions. We present a novel concept about an uncertain distribution set to describe distribution perturbation characterization. First, we introduce a lambda selection of the interval-valued fuzzy variable, and the uncertain distribution set is a collection of all generalized possibility distributions of lambda selection variables. According to the uncertain distribution set, a new distributionally robust fuzzy optimization method is developed for single-period inventory management problems. Under mild assumptions, the robust counterpart of the proposed fuzzy single-period inventory management model is formulated, which is an optimization program with certain linear objectives and infinitely many integral constraints. We discuss the computational issue of integral constraints and reformulate equivalently the robust counterpart as three deterministic inventory submodels under generalized interval-valued trapezoidal possibility distributions. According to the characteristics of three submodels, a domain decomposition method is designed to find the robust optimal solution that can immunize against uncertainty in our single-period inventory management problem. Finally, some computational results demonstrate the efficiency of the proposed distributionally robust fuzzy optimization method.

Benders Decomposition Method on Adjustable Robust Counterpart Optimization Model for Internet Shopping Online Problem

In this paper, the implementation of the Benders decomposition method to solve the Adjustable Robust Counterpart for Internet Shopping Online Problem (ARC-ISOP) is discussed. Since the ARC-ISOP is a mixed-integer linear programming (MILP) model, the discussion begins by identifying the linear variables in the form of continuous variables and nonlinear variables in the form of integer variables. In terms of Benders decomposition, the ARC-ISOP model can be solved by partitioning them into smaller subproblems (the master problem and inner problem), which makes it easier for computational calculations. Pseudo-codes in Python programming language are presented in this paper. An example case is presented for the ARC-ISOP to determine the optimal total cost (including product price and shipping cost) and delivery time. Numerical simulations were carried out using Python programming language with case studies in the form of five products purchased from six shops.

A robust DEA model in the presence of uncertain integer-valued parameters

PurposeUncertainty in data, whether in real-valued or integer-valued data, may result in infeasible optimal solutions or unreliable efficiency scores and ranking of decision-making units. To handle the uncertainty in integer-valued factors in data envelopment analysis (DEA) models, this study aims to propose a robust DEA model which is applicable in the presence of such factors.Design/methodology/approachThis research focuses on the application of fuzzy interpretation of efficiency to a mixed-integer DEA (MIDEA) model. The robust optimization approach is used to address the uncertain integer-valued parameters in the proposed MIDEA model.FindingsIn this study, the authors proposed an MIDEA model without any equality constraint to avoid the arise problem by such constraints in the construction of the robust counterpart of the conventional MIDEA models. We have studied the characteristics and conditions for constructing the uncertainty set with uncertain integer-valued parameters and a robust MIDEA model is proposed under a combined box-polyhedral uncertainty set. The applicability of the developed models is shown in a case study of Malaysian public universities.Originality/valueThis study develops an MIDEA model equivalent to the conventional MIDEA model excluding any equality constraint which is crucial in robust approach to avoid restricted feasible region or infeasible solutions. This study proposes a robust DEA approach which is applicable in cases with uncertain integer-valued parameters, unlike previous studies in robust DEA field where uncertain parameters are generally assumed to be only real-valued.

Bus Bridging for Rail Disruptions: A Distributionally Robust Fuzzy Optimization Approach

Dealing with uncertain rail disruptions effectively raises a significant challenge for computational intelligence research. This article studies the bus bridging problem under demand uncertainty, where the passenger demand is represented as parametric interval-valued fuzzy variables and their associated uncertainty distribution sets. A distributionally robust fuzzy optimization model is proposed to minimize the maximum travel time and to search for the optimal scheme for vehicle allocation, route selection, and frequency determination. To solve the proposed robust model, we discuss the computational issues concerning credibilistic constraints, turning the robust counterpart model into computationally tractable equivalent formulations. The proposed approach is verified, and the resulting method is validated with a report on uncertain parameters in a real-world disrupted event of Shanghai Rail Line 1. Experimental results show that the distributionally robust fuzzy optimization approach can provide a better uncertainty-immunized solution.

Data-driven two-stage distributionally robust optimization for refinery planning under uncertainty

This work investigates the refinery planning problem under uncertainty in product prices. A novel data-driven Wasserstein distributionally robust optimization framework is proposed for handling uncertainties in the refinery-wide planning operations. A data-driven ambiguity set is constructed based on the Wasserstein metric to model the distributional uncertainty. The robust kernel density estimation technique is adopted to establish the support set to reduce the effect of the potential outliers. Based on the derived support set and ambiguity set, a data-driven two-stage distributionally robust optimization model for refinery planning is developed. Then, the robust counterpart of the proposed model is formulated to make the problem computationally tractable. Finally, a real-world case study on a petroleum refinery is presented to illustrate the effectiveness and applicability of the proposed framework.

Utility targeting in heat exchanger network with parametric uncertainties

For almost a half-decade, the methodology of Pinch Analysis (PA) and heat integration (HI) has been used to optimize the requirements of external utilities. Though much has been done, optimizations under uncertain conditions still need more work, discussion, and practical implementation. In most of the HI problems, it has been observed that variation in the minimum driving force affects the shape of the source and sink profile eventually the heat exchanger network (HEN) design. Due to changing operating conditions, human errors, and sometimes poor maintenance, it has been experienced that the process plants along with their equipment and vessels do not operate at the desired value upon which they are expected to run. The parameters with variations and uncertainties due to disturbances in the process plants need to be addressed for the desired product output. In this work, the concept of PA has been unified with robust optimization (RO) to target the utility requirements in HEN optimization where the uncertainties in the heat capacity flow rate F and supply temperatures Tin of the source streams are considered. The developed nominal forms of mathematical programming are converted into the deterministic robust counterpart equivalents to incorporate the uncertainties whose set of variations is known. RO is an optimization technique where the information of accurate probability distribution needs not be known; rather, it works in a defined set of uncertainty ensuring the feasibility of the solution. Using two examples from the literature, the developed models have been solved and results for the solutions against the range of uncertainties in parameters have been established. The produced data demonstrates, in the case of uncertain inlet temperature the rise in total utilities is 115% and 225% in examples 1 and 2 for worst case is calculated. Plots of budget parameter Γ vs the utility requirement have also been compared with results. Application of this model will further assist the plant engineers/managers in deciding the requirements of hot utilities and cold utilities under the parametric uncertainties as needed.

Robust bidding strategy of battery energy storage system (BESS) in joint active and reactive power of day-ahead and real-time markets

The most important applications of an Energy Storage System (ESS) in power systems are energy arbitrage along with procurement of Ancillary Services (ASs). In addition to economic benefits, ESS also improves network reliability and stability. In this paper, a bidding strategy model of a Battery Energy Storage System (BESS) in a Joint Active and Reactive Power Market (JARPM) in the Day-Ahead-Market (DAM) and the Real-Time-Market (RTM) using a robust framework is presented. In this study, the BESS model is considered a price-taker, with the private owner trying to maximize its profit while facing price uncertainty. In the first stage of the proposed model, JARPM is cleared in a deterministic way without the presence of BESS to determine hourly prices of active and reactive power. In the second stage, by determining the prices uncertainty interval and the robust budget for DAM and RTM, the robust bidding model aims to maximize the BESS owner's profit, using the robust counterpart formation of the objective function and dual theory. The proposed robust formulation based on a Mixed Integer Non-Linear Programming Model (MINLP) is implemented in the GAMS software environment on the IEEE-24-RTS test system. The result shows that by the participation of BEES in the market, the average price of active and reactive power is reduced by 2 % & 3 % respectively. In addition, the overall cost of DAM execution is reduced markedly. On the other hand, the proposed framework in three case studies also guarantees a suitable profit level for the private owners of BESS (30–35 $/hour).

Frailty in peripheral arterial disease

It has been consistently demonstrated that frail vascular patients have poorer outcomes compared with their robust counterparts.1 Consideration of frailty is particularly important, not only as our population continues to age but as advances in anaesthetic, surgical and endovascular techniques are enabling a broader range of interventional options for those people who may have traditionally been labelled as unsuitable for surgery.2 Frailty has implications for health and social care at the micro and macro level, and failure to consider the differing needs and natural histories of people living with frailty could result in avoidable harm and suboptimal resource allocation. National guidelines recognise the growing urgency in the need to recognise and manage frailty, yet standardised methods for this have not been identified and agreed upon.3 Despite undeniable prognostic value, the aetiology and pathophysiology of frailty in peripheral arterial disease (PAD) remains poorly understood, albeit new data are emerging. This has challenged a uniform approach to assessment and management. This editorial considers the theories of frailty and applies these to the assessment and management of patients with peripheral arterial disease.

Globalized Distributionally Robust Counterpart

Globalized Distributionally Robust Counterpart

Dosimetric impact of a robust optimization approach to mitigate effects from rotational uncertainty in prostate intensity modulated brachytherapy.

Intensity modulated brachytherapy (IMBT) is an emerging technology for cancer treatment, in which radiation sources are shielded to shape the dose distribution. The rotatable shields provide an additional degree of freedom, but also introduce an additional, directional, type of uncertainty, compared to conventional high dose-rate brachytherapy (HDR BT). We propose and evaluate a robust optimization approach to mitigate the effects of rotational uncertainty in the shields with respect to planningcriteria. A previously suggested prototype for platinum-shielded prostate 169 Yb-based dynamic IMBT is considered. We study a retrospective patient data set (anatomical contours and catheter placement) from two clinics, consisting of six patients that had previously undergone conventional 192 Ir high dose-rate brachytherapy treatment. The Monte Carlo based treatment planning software RapidBrachyMCTPS is used for dose calculations. In our computational experiments, we investigate systematic rotational shield errors of ±10 and ±20degrees, and the same systematic error is applied to all dwell positions in each scenario. This gives us three scenarios, one nominal and two with errors. The robust optimization approach finds a compromise between the average and worst-case scenariooutcomes. We compare dose plans obtained from standard models and their robust counterparts. With dwell times obtained from a linear penalty model, for 10degrees errors, the dose to urethra (D0.1cc ) and rectum (D0.1cc and D1cc ) increase with up to 5% and 7%, respectively, in the worst-case scenario, while with the robust counterpart, the corresponding increases were 3% and 3%. For all patients and all evaluated criteria, the worst-case scenario outcome with the robust approach had lower deviation compared to the standard model, without compromising target coverage. We also evaluated shield errors up to 20degrees and while the deviations increased to a large extent with the standard models, the robust models were capable of handling even such largeerrors. We conclude that robust optimization can be used to mitigate the effects from rotational uncertainty and to ensure the treatment plan quality ofIMBT. This article is protected by copyright. All rights reserved.

Robust goal programming approach for healthcare network management for perishable products under disruption

In this paper, a robust goal programming model for a healthcare network design problem under disruption is developed. The objective of the proposed multi-period, multi-echelon model is to simultaneously optimize the total deviation from the considered three goals, including minimizing network cost, maximizing network coverage, and maximizing network reliability. Distribution uncertainty of healthcare demand and facility capacity is captured. The perishability of healthcare products is regarded with limited shelf-life. Using an option contract ensures appropriate coordination with suppliers during pre and post-disaster periods. Via handling the tractable robust budget-based counterpart model, suitable strategies are proposed to present the designed network's efficiency and effectiveness. Risk mitigation strategies deal with disrupted demand and supply in the healthcare network with specific impacts on coverage, cost, and reliability of the services. The proposed cooperate coverage mechanism provides appropriate service coverage under disruption propagation. Finally, the option contract has achieved cost-efficient coordination under demand uncertainty. Furthermore, practical managerial insights are demonstrated by investigating a real-life case study and conducting extensive sensitivity analyses.

Load-flow dependent twofold technique of tolerance selection for robust estimation of power system states

Accurate estimation of the voltage magnitudes, angles, and other states of the modern power system is essential for efficient monitoring. The precision of the meter readings affects the estimating procedure. However, the majority of traditional meter readings are incorrect, and some of them, known as bad-data, are far from the true levels. The estimation performance significantly declines when such poor measurements are present, either singularly or in multiple numbers. Some estimators have post-processing capabilities but are unable to deal with flawed data directly. One of the well-known robust estimators that can weed out faulty data during the estimating process is Least Measurement Rejected (LMR). For each measurement in the LMR formulation, a tolerance value must be carefully chosen. The performance of the estimation can be greatly enhanced by the careful selection of these numbers. The use of load-flow to choose the LMR tolerance values is recommended in this paper to prepare a twofold estimation process of robust nature. Several tests and analysis were done to determine how well the suggested load flow dependent LMR (LFLMR) estimator performed. It is contrasted with well-known estimators from the literature, including Weighed Least Square (WLS), its robust counterpart (RWLS), Weighted Least Absolute Value (WLAV), Iteratively Reweighted Least Square (IRLS), and seven distinct versions of LMR. Additionally, several load profile snapshots are simulated to test the LFLMR's robustness. In order to illustrate the effectiveness of the suggested estimator and to identify and estimate the wrongly measured values of the measurement series, IEEE standard 14, 30, and 118-bus power systems with various outlier scenarios are simulated.

Robust permutation flow shop total weighted completion time problem: Solution and application to the oil and gas industry

This work proposes exact solution approaches for the m-machine robust permutation flow shop problem, where operation processing times are uncertain and vary in a given interval. Following the concept of budgeted uncertainty, the objective is to obtain a robust scheduling that minimizes the total weighted completion time of the restricted worst-case scenario, in which only a subset of operation processing times will deviate to worst-case values. We develop several robust counterpart formulations, which can be used to derive optimal solutions for medium-sized problem instances by using a Column-and-Constraint Generation algorithm. The efficacy of the solution methods is validated through experiments on three sets of randomly-generated instances. Finally, a case study for the maintenance schedule of Brazilian oil platforms is presented.