Minimax Regret Criterion Research Articles

Using Limited Trial Evidence to Credibly Choose Treatment Dosage when Efficacy and Adverse Effects Weakly Increase with Dose.

It has become standard in medical treatment to base dosage on evidence in randomized trials. Yet it has been rare to study how outcomes vary with dosage. In trials to obtain drug approval, the norm has been to compare some dose of a new drug with an established therapy or placebo. Standard trial analysis views each trial arm as qualitatively different, but it may be credible to assume that efficacy and adverse effects weakly increase with dosage. Optimization of patient care requires joint attention to both, as well as to treatment cost. This paper develops methodology to use limited trial evidence to choose dosage when efficacy and adverse effects weakly increase with dose. I suppose that dosage is an integer t ∊ (0,1, . ,T), T being a specified maximum dose. I study dosage choice when trial evidence on outcomes is available for only K dose levels, where K < T+1. Then the population distribution of dose response is partially identified. I show that the identification region is a convex polygon. I characterize clinical and population decision making using the minimax-regret criterion. A simple analytical solution exists when T = 2. Computation is tractable when T is larger.

Robust ordering, production, and replenishment in a supply chain for innovative products: A two-echelon newsvendor problem with partial information in yield and demand

With partial yield and demand information, we investigate a two-echelon newsvendor problem in a supply chain for innovative product, in which the retailer and the supplier make sequential decisions in ordering, production, and replenishment, respectively. We adopt the minimax-regret criterion to assume the risk-averse nature and deduce the minimax-regret inventory decisions. We explore replenishment (emergency) production strategies for the supplier. Then we propose shortage penalty strategies as well as subsidized replenishment strategies for the retailer so that the supplier can maintain the required supply even under higher replenishment production costs. We further reveal the role of information uncertainty in the minimax-regret decisions and the bullwhip effect. Interestingly, the results show that the penalty strategies do not always lead to less regret for the retailer, and the information uncertainty does not necessarily lead to greater regret for the supplier. Moreover, some schemes to reduce the bullwhip effect are proposed, such as improving the accuracy of uncertain yield, increasing shortage penalty or replenishment subsidy, and decreasing replenishment cost. It is verified that the replenishment strategies can increase profits for both the retailer and the supplier under some conditions. In extensions, we reveal the supplier’s preference for emergency production and external procurement, as well as preferences for information sharing and confidentiality. Meanwhile, we find that decentralized decision-making may even outperform centralized decision-making in some situations when the minimax-regret criterion is adopted to deal with the information deficiency. Finally, the robustness of the replenishment strategies in reducing both enterprises’ regrets is checked.

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.

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.

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.

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.

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.

Remanufacturing vs. greening: Competitiveness and harmony of sustainable strategies of supply chain under uncertain yield

This paper explores the interaction between a risk-neutral supplier and a risk-averse manufacturer in a supply chain considering the remanufacturing of green defective products under scarce yield information. The minimax-regret criterion is adopted by the manufacturer in response to the risk of scarce the information. In order to overcome the conservatism of minimax-regret criterion, we improve the model by integrating ideal profit and minimax regret. Based on the Stackelberg game framework, it is revealed that there is competitiveness between the sustainable strategies. Furthermore, this paper analyzes the evolutionary equilibrium between the competitive strategies based on the evolutionary game theory. Different from the conclusions in existing research that some sustainable strategies can cause free-riding behavior, this paper finds that the supplier’s greening raw materials may lead to a negative effect on the manufacturer by increasing her minimax regret, while the manufacturer’s remanufacturing defective products may reduce the supplier’s profit. The evolutionary stable strategy of the improved model indicates that the supplier will always prefer to the greening strategy. However, it is interesting to find that higher remanufacturing production cost, lower actual yield, and production cost can motivate the manufacturer to implement the remanufacturing strategy. The numerical results illustrate the impact of some factors, such as greening difficulty, consumer preference, as well as the upper and lower bounds of yield on the evolution speed and direction. Moreover, the improved model has a higher possibility of achieving better performance. The results can promote the harmonious transformation of supply chain to sustainable development.

New decision rules under strict uncertainty and a general distance-based approach

&lt;abstract&gt;&lt;p&gt;Strict uncertainty implies a complete lack of knowledge about the probabilities of possible future states of the world. However, there is complete information about the set of alternatives under consideration, the set of future states, and the scalar evaluation of choosing every alternative if a given state occurs. The principle of insufficient reason by Laplace, the maximin rule by Wald, the Hurwicz criterion, or the minimax regret criterion by Savage are examples of decision rules under strict uncertainty. Within the context of strict uncertainty, moderate pessimism implies the existence of a decision-maker who cautiously assumes that the most favorable state will not occur when the action has been taken with no conjecture being made about the other states. The criterion of moderate pessimism proposed by Ballestero implies the use of the inverse of the range of evaluation for each state as a weight system. In this paper, we extend the notion of moderate pessimism under strict uncertainty to solve some of its limitations. First, we propose a new domination analysis that avoids removing dominated alternatives that are still relevant in the final ranking of alternatives. Second, we propose additional score functions using the inverse of the standard deviation and the mean absolute deviation instead of the range of evaluations for each future state to reduce the impact of the possible existence of outliers in the decision table. This partial result is later generalized through the concept of average deviation of a given order. Finally, we show that all the mentioned decision rules are special cases of a general ranking method based on the Minkowski distance function. We illustrate the use of distance-based decision rules through an application in the context of portfolio selection.&lt;/p&gt;&lt;/abstract&gt;

Mini-max Regret Criterion in Patient Therapeutic Healthcare Delivery

Medics are most often on the horn of a dilemma; they either make the right decision or live in regret. The evasion of the “we did our best” syndrome may be effectuated when a judicious choice is made with admissible constraints. The risk involved in decision-making is so inscrutable in situations that may lead to irreversibility as are observable in healthcare delivery. The Mini-max regret criterion depicts the benchmark required of a decision-maker (the caregiver) in cushioning the deleterious effect of taking a less optimal decision, possibly, in the presence of a better option. This work, while presenting such criterion, considered two treatment options-status quo (i.e. usual/existing) option and the inventive (innovative) option. The options were applied to effect both individualized and fractional treatments. The bound for the worst-case regret was furnished.a

Robust Optimization on Unrelated Parallel Machine Scheduling With Setup Times

The parallel machine scheduling problem has been a popular topic for many years due to its theoretical and practical importance. This paper addresses the robust makespan optimization problem on unrelated parallel machine scheduling with sequence-dependent setup times, where the processing times are uncertain, and the only knowledge is the time intervals they take values from. We propose a robust optimization model with the min-max regret criterion to formulate this problem. To solve this problem, we prove that the worst-case scenario with the maximum regret for a given solution belongs to a finite set of extreme scenarios. Based on this theoretical analysis, a procedure to obtain the maximum regret is proposed and an enhanced regret evaluation method (ERE) is designed to accelerate this process, which is of great significance to improve the efficiency of the algorithm. A multi-start decomposition-based heuristic algorithm (MDH) based on the analysis of properties is proposed to solve this problem. Computational experiments are conducted to justify the performance and robustness of these methods. Note to Practitioners—Various uncertainties may occur in the production process, which brings great challenges to production and operations management. A robust production schedule is of great significance for factories to make full use of production capacity and deal with production abnormalities. This study is motivated by an R&D and assembly task scheduling problem encountered in a high-end equipment manufacturing factory in which the processing time of each job is uncertain, and its distribution is also unknown due to limited information. In this study, with the consideration of sequence-dependent setup time and uncertain job-processing time, we view the labor groups with different skill levels as unrelated parallel machines and build a robust (min-max regret) scheduling model to formulate this problem so as to reduce the production makespan. An enhanced regret evaluation method is developed to improve the evaluation efficiency for a given solution, and a multi-start decomposition-based heuristic algorithm is proposed to solve this problem. This study can be applied in practice to release schedulers from burdensome work and provide high-quality robust schedules for this complicated production environment.

The Minmax Regret Reverse 1-Median Problem on Trees with Uncertain Vertex Weights

The classical reverse 1-median problem on trees is to adjust the edge lengths within a budget so as to reduce the 1-median function at a predetermined vertex as much as possible. This paper concerns the corresponding problem with uncertain vertex weights presented by linear functions. Moreover, we use the minmax regret criterion to measure the maximum loss of a feasible solution with respect to the worst-case scenario. The regarding problem is called the minmax regret reverse 1-median problem on trees. We first partition the set of scenarios into parts such that the optimal solution of the corresponding reverse 1-median problem does not change in each part. Then the problem can be reformulated as the minimization of a quadratic number of affine linear functions. We finally develop a greedy algorithm that solves the problem in [Formula: see text] time where n is the number of vertices in the underlying tree.

Minimax regret with imperfect ex-post knowledge of the state

We consider decision problems under complete ignorance and extend the minimax regret principle to situations where, after taking an action, the decision maker does not necessarily learn the state of the world. For example, if the decision maker only learns what the outcome is, then all she knows is that the actual state is one of the (possibly several) states that yield the observed outcome under the chosen action. We refer to this situation as imperfect ex-post information. We show that, given a choice between more information and less information, the decision maker prefers the latter. We also extend the framework to encompass the possibility of less than the extreme degree of pessimism that characterizes the minimax regret criterion.

Improved Particle Swarm Optimization Algorithms for Optimal Designs with Various Decision Criteria

Particle swarm optimization (PSO) is an attractive, easily implemented method which is successfully used across a wide range of applications. In this paper, utilizing the core ideology of genetic algorithm and dynamic parameters, an improved particle swarm optimization algorithm is proposed. Then, based on the improved algorithm, combining the PSO algorithm with decision making, nested PSO algorithms with two useful decision making criteria (optimistic coefficient criterion and minimax regret criterion) are proposed . The improved PSO algorithm is implemented on two unimodal functions and two multimodal functions, and the results are much better than that of the traditional PSO algorithm. The nested algorithms are applied on the Michaelis–Menten model and two parameter logistic regression model as examples. For the Michaelis–Menten model, the particles converge to the best solution after 50 iterations. For the two parameter logistic regression model, the optimality of algorithms are verified by the equivalence theorem. More results for other models applying our algorithms are available upon request.

Optimal Virtual Power Plant Operational Regime Under Reserve Uncertainty

Virtual power plant (VPP) has become an important resource for reserve provision owing to its fast-responding capability. In this paper, an optimal VPP operational regime considering reserve uncertainty is proposed, which includes a novel day-ahead offering strategy and a real-time dispatching model. At the day-ahead stage, the offering strategy gives the VPP’s price-dependent offers in the energy market under multiple uncertainties on market price, renewable generation, and calls of reserve deployment. A hybrid stochastic minimax regret (MMR) model is proposed to facilitate making offering decisions in the electricity market. At the real-time dispatching stage, generation scheduling can be realized based on the MMR criterion in an online fashion. To alleviate the intrinsic conservativeness of the dispatching model, a self-adaptive algorithm is also proposed to instantly modify the confidence bounds. The proposed regime is comprehensively tested through extensive case studies, which demonstrate the effectiveness of our method in obtaining operational decisions that are less conservative.

Robust Portfolio Optimization by Applying Multi-objective and Omega-conditional Value at Risk Models Based on the Mini-max Regret Criterion

Robust Portfolio Optimization by Applying Multi-objective and Omega-conditional Value at Risk Models Based on the Mini-max Regret Criterion

Code and Data Repository for An Iterated Dual Substitution Approach for Binary Integer Programming Problems under the Min–Max Regret Criterion

The software and data in this repository are a snapshot of the software and data that were used in the research reported in the paper An iterated dual substitution approach for min-max binary integer programming by W. Wu, M. Iori, S. Martello, and M. Yagiura.

An Iterated Dual Substitution Approach for Binary Integer Programming Problems Under the Min-Max Regret Criterion

We consider binary integer programming problems with the min-max regret objective function under interval objective coefficients. We propose a heuristic framework, the iterated dual substitution (iDS) algorithm, which iteratively invokes a dual substitution heuristic and excludes from the search space any solution already checked in previous iterations. In iDS, we use a best scenario–based lemma to improve performance. We apply iDS to four typical combinatorial optimization problems: the knapsack problem, the multidimensional knapsack problem, the generalized assignment problem, and the set covering problem. For the multidimensional knapsack problem, we compare the iDS approach with two algorithms widely used for problems with the min-max regret criterion: a fixed-scenario approach, and a branch-and-cut approach. The results of computational experiments on a broad set of benchmark instances show that the proposed iDS approach performs best on most tested instances. For the knapsack problem, the generalized assignment problem, and the set covering problem, we compare iDS with state-of-the-art results. The iDS algorithm successfully updates best-known records for a number of benchmark instances. Summary of Contribution: This paper proposes a heuristic framework for binary integer programming (BIP) problems with the min-max regret objective function under interval objective coefficients. We selected four representative NP-hard combinatorial optimization problems: the knapsack problem, the multidimensional knapsack problem, the set covering problem, and the generalized assignment problem. We show the effectiveness and efficiency of the approach by comparing with state-of-the-art results.

Stochastic multi-objective scheduling of a wind farm integrated with high-temperature heat and power storage in energy market

In spite of significant opportunities created by wind energy, the high uncertainty of wind farms is problematic for their operators, making their participation in the energy market quite challenging. The combination of Air-based High-Temperature Heat and Power Storage (HTHPS), which is known as a novel energy storage technology, with a wind farm as an integrated system can make new opportunities for a Generation Company (GenCo) to overcome the wind generation challenges. This paper proposes a novel Stochastic Multi-objective Optimization (SMOO) problem to determine the optimal charging and discharging scheduling and the best bidding strategy in the day-ahead energy market for an integrated energy system including a wind farm and HTHPS unit. The proposed model presents a comprehensive and coherent approach from historical data analysis as input to final decision making as output, that it can maximize Genco’s profit by avoiding incorrect commitment and energy offering. In the phase of uncertainty handling, via a scenario-based approach, the wind uncertainty is modeled by Monte-Carlo Simulation (MCS) method based on wind speed forecast errors. In the optimization phase, the Pareto optimal set is obtained by Non-dominated Sorting Genetic Algorithm-II (NSGA-II), and then, a new method is also provided to choose the best possible solution based on the entropy Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method and the minimax regret criterion. The historical data related to wind generation and energy prices provided by the Danish energy market in West Denmark is used in simulation analysis, and the results demonstrate pleasantly the effectiveness of the proposed model.