Minimax Regret Research Articles

Robust min-max (regret) optimization using ordered weighted averaging

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes the robust min–max optimization as a special case, as well as the min–max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach.

Optimal two-stage arbitraging and scheduling strategy for distributed energy resource aggregators

Given the rapid development of the distributed energy resources (DER), involving DERs into the wholesale market under the market and renewable uncertainties to achieve economic benefits is necessary but challenging. In this work, an arbitraging strategy is proposed for DER aggregators that bridge DERs with the wholesale market through energy trading. Besides, a novel self-adaptive minimax regret (MMR)-based optimal offering model is proposed for the DER aggregator to handle the uncertainties in both renewable generations and market prices. Additionally, an exact and communication-free solution methodology is proposed to resolve the formulated optimization problem with high computational efficiency. In the numerical results, the proposed methods can achieve near-to-optimal profits. Moreover, the performance of the proposed approach remains satisfactory even if the environment becomes very volatile.

Swarm optimization approaches for the minmax regret Fermat–Weber problem with budgeted-constrained uncertain point weights

Swarm optimization approaches for the minmax regret Fermat–Weber problem with budgeted-constrained uncertain point weights

Minmax regret 1-sink location problems on dynamic flow path networks with parametric weights

Minmax regret 1-sink location problems on dynamic flow path networks with parametric weights

How to Conclude a Suspended Sports League?

Problem definition: Professional sports leagues may be suspended because of various reasons, such as the recent coronavirus disease 2019 pandemic. A critical question that the league must address when reopening is how to appropriately select a subset of the remaining games to conclude the season in a shortened time frame. Despite the rich literature on scheduling an entire season starting from a blank slate, concluding an existing season is quite different. Our approach attempts to achieve team rankings similar to those that would have resulted had the season been played out in full. Methodology/results: We propose a data-driven model that exploits predictive and prescriptive analytics to produce a schedule for the remainder of the season composed of a subset of originally scheduled games. Our model introduces novel rankings-based objectives within a stochastic optimization model, whose parameters are first estimated using a predictive model. We introduce a deterministic equivalent reformulation along with a tailored Frank–Wolfe algorithm to efficiently solve our problem as well as a robust counterpart based on min-max regret. We present simulation-based numerical experiments from previous National Basketball Association seasons 2004–2019, and we show that our models are computationally efficient, outperform a greedy benchmark that approximates a nonrankings-based scheduling policy, and produce interpretable results. Managerial implications: Our data-driven decision-making framework may be used to produce a shortened season with 25%–50% fewer games while still producing an end-of-season ranking similar to that of the full season, had it been played. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0558 .

Effects of partial demand uncertainty reduction on private equity financing in small and medium-sized enterprises: A supply chain perspective.

The effect of demand uncertainty reduction (DUR) on supply chain management has received tremendous attention. From a financial perspective, studying the impact of DUR is equally significant. This study explores the relationship between DUR and private equity (PE) financing in retail enterprises within a supply chain, which comprises a dominant supplier and a subordinate retailer. This article establishes decision models for a retailer backed by PE under three market demand conditions: range, mean, and range with mean. The study further investigates the impact of partial demand uncertainty reduction (PDUR) on the retailer and PE through comparative analysis of these scenarios. To address incomplete market demand information during the decision-making process, the study employs the minimax regret criterion to construct and solve the model. An intriguing finding of this study is that contrary to intuition, PDUR not only fails to promote PE but also reduces the retailer's willingness to finance and decreases the asset size for both the retailer and PE. In addition, the better the growth potential for the retail enterprise, the more severe the negative impact brought about by PDUR. Moreover, the impact of PDUR on supplier and supply chain performance is two-fold. PDUR based on range information has a negative impact on the expected profit of the supplier and the supply chain, while PDUR based on mean information has a positive impact on their expected profit.

Coevolutionary Algorithm for Building Robust Decision Trees under Minimax Regret

In recent years, there has been growing interest in developing robust machine learning (ML) models that can withstand adversarial attacks, including one of the most widely adopted, efficient, and interpretable ML algorithms—decision trees (DTs). This paper proposes a novel coevolutionary algorithm (CoEvoRDT) designed to create robust DTs capable of handling noisy high-dimensional data in adversarial contexts. Motivated by the limitations of traditional DT algorithms, we leverage adaptive coevolution to allow DTs to evolve and learn from interactions with perturbed input data. CoEvoRDT alternately evolves competing populations of DTs and perturbed features, enabling construction of DTs with desired properties. CoEvoRDT is easily adaptable to various target metrics, allowing the use of tailored robustness criteria such as minimax regret. Furthermore, CoEvoRDT has potential to improve the results of other state-of-the-art methods by incorporating their outcomes (DTs they produce) into the initial population and optimize them in the process of coevolution. Inspired by the game theory, CoEvoRDT utilizes mixed Nash equilibrium to enhance convergence. The method is tested on 20 popular datasets and shows superior performance compared to 4 state-of-the-art algorithms. It outperformed all competing methods on 13 datasets with adversarial accuracy metrics, and on all 20 considered datasets with minimax regret. Strong experimental results and flexibility in choosing the error measure make CoEvoRDT a promising approach for constructing robust DTs in real-world applications.

The Minmax Regret Scheduling-Location Problem on Trees with Interval-Data Edge Lengths

We address in this paper a variant of the scheduling-location (ScheLoc) problem on tree networks with interval edge lengths where the total deviation of the uncertain data cannot exceed a threshold. We further use the minmax regret concept to deal with the corresponding uncertainty. In order to solve the problem, we investigate the structure of the schedule which leads to the maximum regret value at a fixed point. Then we consider the machine location belonging to a specific edge of the tree and partition the underlying edge into regions with linear maximum regret function. Finally, we develop a combinatorial algorithm that solves the minmax regret ScheLoc problem in polynomial time based on a finite dominating set approach.

Improving robustness in strategic energy planning: A novel decision support method to deal with epistemic uncertainties

This paper addresses the challenge of dealing with epistemic, i.e. non-probabilistic, uncertainties in strategic energy planning modelling. Current models have limited consideration of this type of uncertainty compared to probabilistic uncertainty, and also typically lead to overly conservative results. To address this issue, the contribution of the paper is to propose a novel decision support method which combines two decision-making methodologies into a single, internally consistent algorithm, and to show its applicability to real-size energy planning studies. Robust optimization is applied to address constraint uncertainties, while the minimax regret criterion is utilized for uncertainties in the objective function. This approach facilitates energy modelling exercises that can be more closely aligned with decision-makers' preferences for both feasibility and optimality. To demonstrate its effectiveness, the method is applied to a real-size strategic energy planning model, and the algorithm is shown to be able to provide detailed solutions in reasonable times. Ex-post evaluations confirm that this approach maintains robust optimization performance by effectively reducing the occurrence and magnitude of infeasibilities, while satisfying the minimax regret criterion across the entire range of uncertainties. Therefore, this integration preserves the distinct advantages of each methodology without any adverse effects when used together.

Robust Ordering Policies with Limited Information on Stochastic Lead Time

Lead time is a rather uncertain factor in manufacturing and inventory systems. In fact, it is always difficult to know the full information of stochastic lead time distribution. So a problem arises how a decision maker decides its ordering policy including advance ordering time, based on historically limited information. In this paper, this problem is solved under a robust-optimization framework, and the minimax regret criterion is adopted. We first derive the decision maker’s robust ordering policies only with the range or the mean of stochastic lead time distribution respectively. The corresponding maximum regret, in other words, the value of full information is then derived. It is shown that when the decision maker uses the range to make decisions, this robust ordering policy exactly minimizes the realized costs if the average of the lower and higher limits of the range equals the realized lead time. When the decision maker uses the means to make decisions, the uncertainty of lead time distribution would lead the decision maker to place the order less in advance. Besides, this uncertainty would cause significant additional costs, especially when the mean is large.

A cooperative resilience-oriented planning framework for integrated distribution energy systems and multi-carrier energy microgrids considering energy trading

Integrated distribution systems (IDSs) and multi-carrier energy microgrids (MCEMs) can play a crucial role in enhancing distribution energy systems’ overall efficiency and flexibility. By cascading energy usage and cooperating through energy trading, IDSs and MCEMs can reduce overall system costs and provide more flexibility for system operators. Adding resilience to the planning problem of IDSs can reduce planning costs in the long term, as proactive preparedness is key to coping with high-impact rare (HR) events. Adding resilience to the planning problem of IDSs can reduce the planning costs in the long term since proactive preparedness is a key necessity to cope with high-impact rare (HR) events. This paper proposes a resilience-oriented stochastic tri-level and two-stage cooperative expansion planning of IDSs and MCEMs, considering energy trading between IDSs and MCEMs. The first stage comprises two levels; the first level minimizes the investment and operation costs of IDSs and MCEMs, while the second level desires to maximize the energy exchange profit for MCEMs and thus reduce the overall costs. The second stage includes the third level problem involving two objective functions: resilience cost minimization and resilience index (RI) maximization. The multi-objective problem in the second stage is converted into a single-objective problem using the min–max regret method. The DC and AC configurations for the power distribution system (PDS) and power microgrids (PMGs) are studied to identify the optimal configuration of these networks in the expansion planning problem. A new framework is proposed based on an aggregator-agent splitting solution using the aggregator coupling coordinator unit (ACC) responsible for coordinating IDNs and MCEMs. The studied large-scale complex optimization problem is efficiently solved computationally by introducing a combined adaptive dynamic programming (ADP) and linearized alternating direction method of multipliers with parallel splitting (LADMMPSAP) algorithm. Three cases are studied to demonstrate the effectiveness of the proposed model and method. The results depict that MCEMs help reduce expansion planning costs and improve the system’s resilience. Adding resilience to the expansion planning problem enhances the resilience of the whole system and simultaneously reduces the costs by 2.7%. The expansion planning costs for the AC and DC configuration are close, and the AC is the optimal choice in all case studies. By increasing the planning horizon from 5 to 10 years, DC will be the optimal solution since network reinforcement costs and power losses are significantly lower.

Minimax regret stability in the graph model for conflict resolution

In strategic conflicts, decision-makers (DMs) do not constantly make their best decisions, which makes room for DMs to feel regretful. In this context, this work proposes the Minimax Regret Stability within the graph model for conflict resolution (GMCR) for modelling and analysing conflicts, considering that DMs would feel regret for an unsatisfactory decision. This concept does not require knowledge about the opponent’s preference in bilateral conflicts. The aim of the new solution concept is to recommend a state as stable if there exists a policy recommending to stay at that state which minimizes the maximum regret of the DM for a given conflict horizon, where a policy specifies what action a DM must choose in every conflict state. In addition, we investigate the relations among the Minimax regret stability and other classical GMCR solution concepts. Following, the influence of the horizon on the stability of states is also studied. Finally, a truckers’ strike conflict was used to illustrate how the new solution concept can provide valuable strategic insights in real-life conflict situations.

Bandwidth selection for treatment choice with binary outcomes

This study considers the treatment choice problem when the outcome variable is binary. We focus on statistical treatment rules that plug in fitted values from a nonparametric kernel regression, and show that the maximum regret can be calculated by maximizing over two parameters. Using this result, we propose a novel bandwidth selection method based on the minimax regret criterion. Finally, we perform a numerical exercise to compare the optimal bandwidth choices for binary and normally distributed outcomes.

Improved algorithms for bandit with graph feedback via regret decomposition

The problem of bandit with graph feedback generalizes both the multi-armed bandit (MAB) problem and the learning with expert advice problem by encoding in a directed graph how the loss vector can be observed in each round of the game. The mini-max regret is closely related to the structure of the feedback graph and their connection is far from being fully understood. We propose a new algorithmic framework for the problem based on a partition of the feedback graph. Our analysis reveals the interplay between various parts of the graph by decomposing the regret to the sum of the regret caused by small parts and the regret caused by their interaction. As a result, our algorithm can be viewed as an interpolation and generalization of the optimal algorithms for MAB and learning with expert advice. Our framework unifies previous algorithms for both strongly observable graphs and weakly observable graphs, resulting in improved and optimal regret bounds on a wide range of graph families including graphs of bounded degree and strongly observable graphs with a few corrupted arms.

Joint pricing and inventory management under minimax regret

We study the problem of jointly optimizing the price and order quantity for a perishable product in a single selling period, also known as the pricing newsvendor problem, under demand ambiguity. Specifically, the demand is a function of the selling price and a random factor of which the distribution is unknown. We employ the minimax regret decision criterion to minimize the worst‐case regret, where the regret is defined as the difference between the optimal profit that could be obtained with perfect/complete information and the realized profit using the decision made with ambiguous demand information. First, given the interval in which the random factor lies with high probability, we characterize the optimal pricing and ordering decisions under the minimax regret criterion and compare their properties with those in the classical models that seek to maximize the expected profit. Specifically, we explore the impact of inventory risk by comparing the optimal price and the risk‐free price and study comparative statics with respect to the degree of demand ambiguity and the unit ordering cost. We further show that the minimax regret approach avoids the high degree of conservativeness that is often incurred in the application of the commonly used max–min robust optimization approach. Second, when partial distributional information of the random factor is available, we adopt the Wasserstein distance to depict the distributional ambiguity and characterize the set of worst‐case distributions and the maximum regret given the selling price and order quantity. Third, we compare the minimax regret approaches with the traditional profit‐maximization approach in a data‐driven setting. We show via a numerical study that the minimax regret approaches outperform the traditional profit‐maximization approach, especially when the data are scarce, the demand has high volatility, and the number of exercised prices is small. Furthermore, leveraging the partial distributional information of the random factor can further improve the performance of the minimax regret approach.

Physics-informed learning under epistemic uncertainty with an application to system health modeling

This study proposes a methodology for developing deterioration models to estimate the remaining lifetime of a system using physics-informed learning. The approach consists of combining physical knowledge about the system with a set of data obtained from similar systems that have failed in the past to build a set of constraints on the evolution over time of the state of health of the system. The data consists of partial measurements of the system variables, taken from its commissioning to the present. The physical knowledge used comprises a set of differential equations that approximate the dynamics and aging of the system. In contrast to other studies, the physical model is not assumed to be accurate, but is considered to be an approximation of reality. Constraints on the temporal evolution of the state of health derived from this physical knowledge take into account its imprecision and consist of possibility distributions. In this study, the max-max, max-min and min-max regret principles are applied to extract the time evolution of the deterioration rate that best fits the constraints. The effectiveness of the proposed algorithm is evaluated by means of a comparative empirical analysis in different use cases, and the situations in which informed learning improves on purely data-driven algorithms are analyzed.

Optimal Control Model for Territory with Special (Preferential) Development Regime

The article presents the result of multicriteria optimization mathematical model to control territories with a preferential development regime (TPDR) development. TPDR are an institutional form of management of the spatial economy. TPDR should contribute to the outpacing social and economic growth of the regions. Implementing such functions, an effective methodological apparatus for decision-making support is needed. In particular, mathematical methods of optimal control can be used. The purpose of the research was to develop a complex model that allows to make an indicative predictive estimation of the main parameters of the TPDR, as well as multi-criteria optimization. The model is designed to solve the problem of choosing an acceptable control strategy from a discrete set of feasible alternatives. The research used two groups of methodological foundations: deterministic methods of predictive estimation of parameters, as well as methods of vector multicriteria optimization using a generalized criterion. The article provides an example of a numerical implementation of the TPDR control model, which was developed as a part of the study. This example illustrates the practical applicability of the model and the result of multi-objective optimization. The optimization problem was solved from the position of restrained pessimism of the decision maker based on the generalized maximin criterion, the minimax regret criterion, and the optimism-pessimism criterion. The proposed mathematical model of multicriteria optimization provides a comprehensive representation of the main economic factors of the optimal control of the TPRR and contributes the formation of a set of acceptable alternatives, as well as system analysis from the standpoint of rational management. Also, the model reveals the structural and functional content of the TPDR management mechanism and explicates the composition of significant economic parameters. This determines the importance of the model as the basis for the development of an information-analytical system for the optimal control of TPDR.

Robust identical parallel machine scheduling with two-stage time-of-use tariff and not-all-machine option

Time-of-use (TOU) tariff has been implemented in the manufacturing industry to improve energy efficiency by regulating the electricity imbalance between supply and demand. Besides, not-all-machine (NAM) option is another way of energy-saving by using only a subset of all the available machines. This study investigates a robust identical parallel machine scheduling problem with a two-stage TOU tariff and NAM option. Only interval bounds on job processing times are known. The problem is first formulated into a min–max regret model to maximise the robustness. Based on problem properties, both an iterative relaxation-based exact algorithm and a memetic differential evolution-based heuristic are developed to solve the problem. Computational experiments on 240 randomly generated instances with up to 20 jobs are conducted to evaluate the performance of the developed methods. Besides, 900 large-sized randomly generated instances with up to 150 jobs are tested for sensitivity analysis and to identify managerial insights for achieving energy-efficient schedules.

Robust Planning over Restless Groups: Engagement Interventions for a Large-Scale Maternal Telehealth Program

In 2020, maternal mortality in India was estimated to be as high as 130 deaths per 100K live births, nearly twice the UN's target. To improve health outcomes, the non-profit ARMMAN sends automated voice messages to expecting and new mothers across India. However, 38% of mothers stop listening to these calls, missing critical preventative care information. To improve engagement, ARMMAN employs health workers to intervene by making service calls, but workers can only call a fraction of the 100K enrolled mothers. Partnering with ARMMAN, we model the problem of allocating limited interventions across mothers as a restless multi-armed bandit (RMAB), where the realities of large scale and model uncertainty present key new technical challenges. We address these with GROUPS, a double oracle–based algorithm for robust planning in RMABs with scalable grouped arms. Robustness over grouped arms requires several methodological advances. First, to adversarially select stochastic group dynamics, we develop a new method to optimize Whittle indices over transition probability intervals. Second, to learn group-level RMAB policy best responses to these adversarial environments, we introduce a weighted index heuristic. Third, we prove a key theoretical result that planning over grouped arms achieves the same minimax regret--optimal strategy as planning over individual arms, under a technical condition. Finally, using real-world data from ARMMAN, we show that GROUPS produces robust policies that reduce minimax regret by up to 50%, halving the number of preventable missed voice messages to connect more mothers with life-saving maternal health information.

Minimax decision rules for planning under uncertainty: Drawbacks and remedies

It is common to use minimax rules to make planning decisions when there is great uncertainty about what may happen in the future. Using minimax rules avoids the need to determine probabilities for each future scenario, which is an attractive feature in many public sector settings. However there are potential problems in the application of a minimax approach. In this paper our aim is to give guidance for planners considering a minimax approach, including minimax regret which is one popular version of this. We give an analysis of the behaviour of minimax rules in the case with a finite set of possible future scenarios. Minimax rules will have sensitivity to the choice of a small number of scenarios. When regret-based rules are used there are also problems arising since the independence of irrelevant alternatives property fails, which can lead to opportunities to game the process. We analyse these phenomena considering cases where the decision variables are chosen from a convex set in Rn, as well as cases with a finite set of decision choices. We show that the drawbacks of minimax regret hold even when restrictions are placed on the problem setup, and we show how working with a structured set of scenarios can ameliorate the difficulty of having a final decision depend on the characteristics of just a handful of extreme scenarios.