Minimax Regret Research Articles

Formulation and algorithms for the robust maximal covering location problem

Let N be the line-set and M be the column-set of a matrix {aij}, such that aij=1 if line i∈N is covered by column j∈M, or aij=0 otherwise. Besides, let bj≥0 be the benefit associated with a column j∈M. Given a constant T<|M|, the NP-Hard Maximal Covering Location Problem (MCLP) consists in finding a subset X⊆M with the maximum sum of benefits, such that |X|≤T and every line in N is covered by at least one column in X. In this study, we investigate the min-max regret Maximal Covering Location Problem, a robust counterpart of MCLP, where the benefit of each column is uncertain and modeled as an interval data. The objective is to find a robust solution that minimizes the maximal regret over all possible combinations of values for the columns benefit. This problem has applications in disaster relief. We propose a MILP formulation, an exact and 2-approximation algorithms, and test them using classical instances from the literature.

Trees on the move: using decision theory to compensate for climate change at the regional scale in forest social-ecological systems

The adaptation of social-ecological systems such as managed forests depends largely on decisions taken by forest managers who must choose among a wide range of possible futures to spread risks. We used robust decision theory to guide management decisions on the translocation of tree populations to compensate for climate change. We calibrated machine learning correlational models using tree height data collected from five common garden tests in France where Abies alba provenances from 11 European countries are planted. Resulting models were used to simulate tree height in the planting sites under current and 2050 climates (regional concentration pathway scenarios (RCPs) 2.6, 4.5, 6.0 and 8.5). Our results suggest an overall increase in tree height by 2050, but with large variation among the predictions depending on the provenance and the RCPs. We applied maximin, maximax and minimax decision rules to address outcomes under five uncertain states of the world represented by the four RCPs and the present climate (baseline). The maximin rule indicated that for 2050, the best translocation option for maximising tree height would be the use of provenances from Northwest France into all target zones. The maximax and minimax regret rules pointed out the same result for all target zones except for the ‘Les Chauvets’ trial, where the East provenance was selected. Our results show that decision theory can help management by reducing the number of options if most decision rules converge. Interestingly, the commonly suggested recommendation of using multiple provenances to mitigate long-term maladaptation risks or from ‘pre-adapted’ populations from the south was not supported by our approach.

The robust (minmax regret) assembly line worker assignment and balancing problem

Line balancing aims to assign the assembly tasks to the stations that compose the assembly line. A recent body of literature has been devoted to heterogeneity in the assembly process introduced by different workers. In such an environment, task times depend on the worker performing the operation and the problem aims at assigning tasks and workers to stations in order to maximize the throughput of the line. In this work, we consider an interval data version of the assembly line worker assignment and balancing problem (ALWABP) in which it is assumed that lower and upper bounds for the task times are known, and the objective is to find an assignment of tasks and workers to the workstations such that the absolute maximum regret among all of the possible scenarios is minimized. The relationship with other interval data minmax regret (IDMR) problems is investigated, the inapplicability of previous approximation methods is studied, regret evaluation is considered, and exact and heuristic solution methods are proposed and analyzed. The results of the proposed methods are compared in a computational experiment, showing the applicability of the method and the theoretical results to solve the problem under study. Additionally, these results are not only applicable to the problem in hand, but also to a more general class of problems.

A Five-Level MILP Model for Flexible Transmission Network Planning Under Uncertainty: A Min–Max Regret Approach

The benefits of new transmission investment significantly depend on deployment patterns of renewable electricity generation that are characterized by severe uncertainty. In this context, this paper presents a novel methodology to solve the transmission expansion planning problem under generation expansion uncertainty in a min–max regret fashion, when considering flexible network options and $n-1$ security criterion. To do so, we propose a five-level mixed integer linear programming (MILP) based model that comprises: (i) the optimal network investment plan (including phase shifters), (ii) the realization of generation expansion, (iii) the co-optimization of energy and reserves given transmission and generation expansions, (iv) the realization of system outages, and (v) the decision on optimal post-contingency corrective control. In order to solve the five-level model, we present a cutting plane algorithm that ultimately identifies the optimal min–max regret flexible transmission plan in a finite number of steps. The numerical studies carried out demonstrate: (a) the significant benefits associated with flexible network investment options to hedge transmission expansion plans against generation expansion uncertainty and system outages, (b) strategic planning-under-uncertainty uncovers the full benefit of flexible options which may remain undetected under deterministic, perfect information methods, and (c) the computational scalability of the proposed approach.

Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment

In the present paper, a robust approach is used to locate hub facilities considering network risks. An additional objective function, minimax regret, is added to the classical objective function in the hub location problem. In the proposed model, risk factors such as availability, security, delay time, environmental guidelines and regional air pollution are considered using triangular fuzzy-stochastic numbers. Then an equivalent crisp single objective model is proposed and solved by the Benders decomposition method. Finally, the results of both Benders decomposition and commercial optimization software are compared for different instances. Numerical instances were developed based on the well-known Civil Aeronautics Board (CAB) data set, considering different levels of uncertainty in parameters. The results show that the proposed model is capable of selecting nodes as sustainable hubs. Also, the results confirm that using Benders decomposition is more efficient than using classical solution methods for large-scale problems.

Optimally Protecting Elections with Uncertainty about Voter Preferences

Election control has always been an important issue of democratic institutions concerned. Considering that a voting rule is indeed susceptible to control by an external agent, it is natural to seek ways to protect elections. Much of prior work has focused on complete voter preferences and approached the problem from the perspective of the computational complexity of election control. However, it is impractical for everyone to have complete voter preferences in real-world scenarios. In addition, when given a voting rule, such as plurality which is widely used in our lives, is easy to control, how to design protection strategies to prevent the occurrence of election control is ignored. In this paper, we model the problem, where the attacker can deploy a single attack such as a denial-of-service attack to convert the voting result through deleting some voter groups, and the defender allocates the limited protection resources to prevent attacks on specific voter groups, as a Stackelberg game. Then we first use the minimax regret decision criterion for uncertainty about voter preferences in the game. We also propose heuristic algorithms to speed up computing minimax regret for the Stackelberg game. Finally, we conduct detailed experiments on both synthetic and real data, which show that our algorithms lead to much better solution quality than other algorithms in the literature.

Design of Biomass Combined Heat and Power (CHP) Systems based on Economic Risk using Minimax Regret Criterion

It is a great challenge to identify optimum technologies for CHP systems that utilise biomass and convert it into heat and power. In this respect, industry decision makers are lacking in confidence to invest in biomass CHP due to economic risk from varying energy demand. This research work presents a linear programming systematic framework to design biomass CHP system based on potential loss of profit due to varying energy demand. Minimax Regret Criterion (MRC) approach was used to assess maximum regret between selections of the given biomass CHP design based on energy demand. Based on this, the model determined an optimal biomass CHP design with minimum regret in economic opportunity. As Feed-in Tariff (FiT) rates affects the revenue of the CHP plant, sensitivity analysis was then performed on FiT rates on the selection of biomass CHP design. Besides, design analysis on the trend of the optimum design selected by model was conducted. To demonstrate the proposed framework in this research, a case study was solved using the proposed approach. The case study focused on designing a biomass CHP system for a palm oil mill (POM) due to large energy potential of oil palm biomass in Malaysia.

Shifting Interpolation Kernel Toward Orthogonal Projection

Orthogonal projection offers the optimal solution for many sampling-reconstruction problems in terms of the least square error. In the standard interpolation setting where the sampling is assumed to be ideal, however, the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited. In this paper, we propose a notion of shifting kernel toward the orthogonal projection. For a given interpolation kernel, we formulate optimization problems whose solutions lead to shifted interpolations that, while still being interpolatory, are closest to the orthogonal projection in the sense of the minimax regret. The quality of interpolation is evaluated in terms of the average approximation error over input shift. For the standard linear interpolation, we obtain several values of optimal shift, dependent on a priori information on input signals. For evaluation, we apply the new shifted linear interpolations to a Gaussian signal, an ECG signal, a speech signal, a two-dimensional signal, and three natural images. Significant improvements are observed over the standard and the 0.21-shifted linear interpolation proposed early.

A quasi-Bayesian perspective to online clustering

When faced with high frequency streams of data, clustering raises theoretical and algorithmic pitfalls. We introduce a new and adaptive online clustering algorithm relying on a quasi-Bayesian approach, with a dynamic (i.e., time-dependent) estimation of the (unknown and changing) number of clusters. We prove that our approach is supported by minimax regret bounds. We also provide an RJMCMC-flavored implementation (called PACBO, see https://cran.r-project.org/web/packages/PACBO/index.html) for which we give a convergence guarantee. Finally, numerical experiments illustrate the potential of our procedure.

A methodology for performance robustness assessment of low-energy buildings using scenario analysis

Uncertainties in building operation and external factors such as occupant behavior, climate change, policy changes etc. impact building performance, resulting in possible performance deviation during operation compared to the predicted performance in the design phase. Multiple low-energy building configurations can lead to similar optimal performance under deterministic conditions, but can have different magnitudes of performance deviation under these uncertainties. Low-energy buildings must be robust so that these uncertainties do not result in significant variations in energy use, cost and comfort. However, these uncertainties are rarely considered in the design of low-energy buildings and hence, the decision making process may result in designs that are sensitive to uncertainties and might not perform as intended. Therefore, to reduce this sensitivity, performance robustness assessment of low-energy buildings considering uncertainties should be assessed in the design phase. The probability of occurrences of these uncertainties are usually unknown and hence, scenarios are essential to assess the performance robustness of buildings. Therefore, a non-probabilistic robustness assessment methodology, based on scenario analysis, is developed to identify robust designs. Maximum performance regret calculated using the minimax regret method is used as the measure of performance robustness. In this approach, the preferred robust design is based on optimal performance and performance robustness.The proposed methodology is demonstrated using a case study with a policymaker as the decision maker. The proposed methodology can be used by designers and consultants to aid decision makers in the design phase to identify robust low-energy building designs that deliver preferred performance in the future operation.

A Semi-automatic Method to Retrieve Twitter Accounts

AbstractAlthough Twitter has become a major source of data, research still relies on heavy manual searches to find the accounts of a population of interest, such as members of Parliament, members of an association or candidates to an election. In this article, we propose a method to semi-automatically retrieve those accounts. After creating a pool of possible accounts through automatic web searches, we select a unique account for each member of a defined group. To do so, we use a simplified version of the minimax regret approach, on a score based on the account’s handle and the tweets’ content. We test this method to find the accounts of the candidates for the 2015 Spanish general elections and the members of the 57th House of Commons in the United Kingdom.

Discrete-time interval optimal control problem

ABSTRACTThis paper proposes a new approach for the discrete-time interval optimal control problem. The optimal interval solution of discrete-time interval optimal control problem is obtained using the single-level constrained interval arithmetic coupled with a dynamic programming technique. Moreover, the optimal interval solution contains the real-valued optimal solutions. To illustrate an optimal solution for the interval case, we use the minimax regret criterion in the interval inventory control problem we present.

Scheduling of recovery actions in the supply chain with resilience analysis considerations

Supply chain engineering models with resilience considerations have been mostly focused on disruption impact quantification within one analysis layer, such as supply chain design or planning. Performance impact of disruptions has been typically analysed without scheduling of recovery actions. Taking into account schedule recovery actions and their duration times, this study extends the existing literature to supply chain scheduling and resilience analysis by an explicit integration of the optimal schedule recovery policy and supply chain resilience. In particular, we compute a schedule optimal control policy and analyse the performance of this policy by varying the perturbation vector and representing the outcomes of variations in the form of an attainable set. We propose a scheduling model that considers the coordination of recovery actions in the supply chain. Further, we suggest a resilience index by using the notion of attainable sets. The attainable sets are known in control theory; their calculation is based on the schedule control model results and the minimax regret approach for continuous time parameters given by intervals. We show that the proposed indicator can be used to estimate the impact of disruption and recovery dynamics on the achievement of planned performance in the supply chain.

Finding an Optimal Decisions’ Subset by Minimaximax Regret Criterion Regarding Instability of the Decision Function

Background. A generalization of the minimax regret criterion is represented as even the best-assurance minimax regret criterion comes inconsistent under instable evaluations of decision situations. Objective. The goal is to formulate the minimaximax regret criterion. Methods. Unlike the classic one, the generalized regret criterion is minimaximax which operates over generalized regrets. These regrets are found from a generalized decision function which is defined on a Cartesian product of a decisions’ set, a set of states, and a set of metastates. Metastate describes instability of the decision function whose values change through a set of metastates. The instability destroys assurance of minimaxed regrets found classically, so regrets are found over a generalized decision function. For this, utility evaluations are subtracted from the utility maximized across a decision set, or the loss/risk minimized across a decision set is subtracted from loss/risk evaluations. Then regrets are minimized under uncertainty across two dimensions of states and metastates, that is they are minimaximaxed. Results. The suggested minimaximax regret criterion allows finding an optimal decisions’ subset with not only regarding instability of the decision function, but also with reducing the initial decisions’ set more, unlike the ultimate pes-simism criterion without regrets (minimaximax/maximinimin). This especially concerns nonnegative utility matrices with many zeros. Conclusions. A ratio of a number of optimal decisions by the without-regret maximinimin/minimaximax to a number of optimal decisions by the minimaximax regret criterion decreased by 1 can be interpreted as a gain of applying the represented minimax regret criterion generalization. This gain fundamentally depends on whether sets of decisions, states, and metastates are finite or not. If they all are finite, then the gain depends on values in a three-dimensional regret matrix and its dimensions. It is surprising but the gain may be negative, that is finding regrets may come non-effective.

Multi-objective Stackelberg game model for water supply networks against interdictions with incomplete information

Water supply networks are infrastructures pivotal to economic development and living standards, of which the increasing complexity and interdependencies have brought challenges for the protection and enhancement of water supplies. We address the problem on how to defend water supply networks with hydraulic characteristics against an interdictor by building a multi-objective Stackelberg game model with incomplete information. In this model, the defender and the interdictor, both considered as rational players, choose a subset of network components to defend or interdict based on their payoffs. The defender, with incomplete information on the interdictor's efforts, initiates to trade off the two objectives of maximizing the expected network satisfaction rate and of minimizing defense efforts, whereas the interdictor, with no information on network operational capacity, follows to trade off the objectives of minimizing the expected network efficiency and of minimizing interdiction efforts. The algorithm of determining the final optimal defense strategies is presented, which consists of three steps: (1) develop the strategy sets by the assessments of network vulnerability and resilience; (2) analyze the equilibrium through a nested heuristic genetic algorithm; and (3) determine the final optimal defense strategy based on the minimax regret approach. A case study of D-town water supply network demonstrates the practical significance of the proposed approach. Furthermore, the impacts of the incomplete information are analyzed to provide suggestions on the defense strategy making.

On scenario aggregation to approximate robust combinatorial optimization problems

As most robust combinatorial min–max and min–max regret problems with discrete uncertainty sets are NP-hard, research in approximation algorithm and approximability bounds has been a fruitful area of recent work. A simple and well-known approximation algorithm is the midpoint method, where one takes the average over all scenarios, and solves a problem of nominal type. Despite its simplicity, this method still gives the best-known bound on a wide range of problems, such as robust shortest path or robust assignment problems. In this paper, we present a simple extension of the midpoint method based on scenario aggregation, which improves the current best K-approximation result to an $$(\varepsilon K)$$ -approximation for any desired $$\varepsilon > 0$$ . Our method can be applied to min–max as well as min–max regret problems.

Adaptive robust optimization with minimax regret criterion: Multiobjective optimization framework and computational algorithm for planning and scheduling under uncertainty

Regret is defined as the deviation of objective value from the perfect information solution, and serves as an important evaluation metric for decision-making under uncertainty. This paper proposes a novel framework that effectively incorporates the minimax regret criterion into two-stage adaptive robust optimization (ARO). In addition to the conventional robustness criterion, this ARO framework also simultaneously optimizes the worst-case regret to push the performance of the resulting solution towards the utopia one under perfect information. By using a data-driven uncertainty set, we formulate a multiobjective ARO problem that generates a set of Pareto-optimal solutions to reveal the systematic trade-offs between the conventional robustness and minimax regret criteria. The resulting multi-level mixed-integer programming problem cannot be solved directly by any off-the-shelf optimization solvers, so we further propose tailored column-and-constraint generation algorithms to address the computational challenge. Two applications on process network planning and batch process scheduling are presented to demonstrate the applicability of the proposed framework and the efficiency of the proposed solution algorithms.

Climate policies under climate model uncertainty: Max-min and min-max regret

Temperature responses and optimal climate policies depend crucially on the choice of a particular climate model. To illustrate, the temperature responses to given emission reduction paths implied by the climate modules of the well-known integrated assessments models DICE, FUND and PAGE are described and compared. A dummy temperature module based on the climate denialists' view is added. Using a simple welfare-maximising growth model of the global economy, the sensitivity of the optimal carbon price, renewable energy subsidy and energy transition to each of these climate models is discussed. The paper then derives max-min, max-max and min-max regret policies to deal with this particular form of climate (model) uncertainty and with climate scepticism. The max-min or min-max regret climate policies rely on a non-sceptic view of global warming and lead to a substantial and moderate amount of caution, respectively. The max-max leads to no climate policies in line with the view of climate sceptics.

Robust aircraft sequencing and scheduling problem with arrival/departure delay using the min-max regret approach

This study considers the aircraft sequencing and scheduling problem under the uncertainty of arrival and departure delays for multiple heterogeneous mixed-mode parallel runways. To enhance runway resilience, runway operations should remain robust to mitigate the effects of delay propagation. The main objective of this research was to identify an optimal schedule by evaluating the robustness of feasible solutions under its respective worst-case scenario. A novel artificial bee colony algorithm was developed and verified by experimental results. The proposed efficient artificial bee colony algorithm can obtain close-to-optimal results with less computational effort in regard to a one-hour flight traffic planning horizon.

Heuristic algorithms for the minmax regret flow-shop problem with interval processing times

An uncertain version of the permutation flow-shop with unlimited buffers and the makespan as a criterion is considered. The investigated parametric uncertainty is represented by given interval-valued processing times. The maximum regret is used for the evaluation of uncertainty. Consequently, the minmax regret discrete optimization problem is solved. Due to its high complexity, two relaxations are applied to simplify the optimization procedure. First of all, a greedy procedure is used for calculating the criterion’s value, as such calculation is NP-hard problem itself. Moreover, the lower bound is used instead of solving the internal deterministic flow-shop. The constructive heuristic algorithm is applied for the relaxed optimization problem. The algorithm is compared with previously elaborated other heuristic algorithms basing on the evolutionary and the middle interval approaches. The conducted computational experiments showed the advantage of the constructive heuristic algorithm with regards to both the criterion and the time of computations. The Wilcoxon paired-rank statistical test confirmed this conclusion.