Max-min Operator Research Articles

A Nonlinear Programming Approach to Solving Interval-Valued Intuitionistic Hesitant Noncooperative Fuzzy Matrix Games

Initially, fuzzy sets and intuitionistic fuzzy sets were used to address real-world problems with imprecise data. Eventually, the notion of the hesitant fuzzy set was formulated to handle decision makers’ reluctance to accept asymmetric information. However, in certain scenarios, asymmetric information is gathered in terms of a possible range of acceptance and nonacceptance by players rather than specific values. Furthermore, decision makers exhibit some hesitancy regarding this information. In such a situation, all the aforementioned expansions of fuzzy sets are unable to accurately represent the scenario. The purpose of this article is to present asymmetric information situations in which the range of choices takes into account the hesitancy of players in accepting or not accepting information. To illustrate these problems, we develop matrix games that consider the payoffs of interval-valued intuitionistic hesitant fuzzy elements (IIHFEs). Dealing with these types of fuzzy programming problems requires a significant amount of effort. To solve these matrix games, we formulate two interval-valued intuitionistic hesitant fuzzy programming problems. Preserving the hesitant nature of the payoffs to determine the optimal strategies, these two problems are transformed into two nonlinear programming problems. This transformation involves using mathematical operations for IIHFEs. Here, we construct a novel aggregation operator of IIHFEs, viz., min-max operators of IIHFEs. This operator is suitable for applying the developed methodology. The cogency and applicability of the proposed methodology are verified through a numerical example based on the situation of conflict between hackers and defenders to prevent damage to cybersecurity. To validate the superiority of the proposed model along with the computed results, we provide comparisons with the existing models.

An improved IGDT approach for distributed generation hosting capacity evaluation in multi-feeders distribution system with soft open points

Unknown integrated locations caused by unpredictable consumers’ willingness and uncertainty in outputs induced by variable natural conditions are acknowledged challenges in the evaluation process of distributed generation (DG) hosting capacity (DGHC). For the former factor, this paper first proposes a novel assessment model to obtain the maximum feasible capacity no matter consumers in which locations have the possibility to stall DG. The decision variables of max operator and min operator are DG capacities and integrated locations respectively. Notably, unlike the previous works for one radial feeder, the interplay of DGHC in multi-feeders interconnected by tie lines and soft open points is also taken into consideration. After that, to combine uncertainties in integrated locations and outputs simultaneously, the above DGHC assessment model is further transformed into an improved information gap decision theory (IGDT) model with worst-case constraints for DG integrated locations. The specific solution procedure is also developed to solve this nonlinear IGDT model effectively. Case studies demonstrate that the evaluation results of DGHC based on this method always have the applicability and feasibility for any nodes in distribution system.

Some equations to identify the threshold value in the DEMATEL method

DEMATEL technique is a graphical representation method to deal with complex systems. The final analyzed cause and effect categorization would be fundamentally dependent on the threshold value setting. This research is intended to present some mathematical models for calculating the threshold value in the DEMATEL method. The min(max) operator has been intentionally used for considering three equations to identify the threshold value. Additionally, the proposed mathematical equations are gradually developed to gain more useful data to yield a threshold value as well. Particularly, the expert’s initial scoring for building the primary matrix would also be applied in one equation. Results show eliciting an expert’s opinions regarding the value of a threshold value determination leads to setting relatively high thresholds. But, there would be an equation which takes advantage of more data derived from the total influence matrix T. Moreover, a span of different threshold values is gained by making use of the Hamacher t-conorms operator which especially would cause better complexity management of the final total matrix T based on expert’s opinions. As a contribution to this research, threshold value determination is developed mathematically by making use of the direct data gained by the total matrix T. Besides combining data derived from total matrix T, the initial influence direct matrix given by experts, a simpler aggregating procedure and no need for statistical information compared to special Lenth’s method hints at this research’s novelty as well.

Multi-objectives optimization and convolution fuzzy C-means: control of diabetic population dynamic

The optimal control models proposed in the literature to control a population of diabetics are all single-objective which limits the identification of alternatives and potential opportunities for different reasons: the minimization of the total does not necessarily imply the minimization of different terms and two patients from two different compartments may not support the same intensity of exercise or the same severity of regime. In this work, we propose a multi-objectives optimal control model to control a population of diabetics taking into account the specificity of each compartment such that each objective function involves a single compartment and a single control. In addition, the Pontryagin’s maximum principle results in expansive control that devours all resources because of max-min operators and the control formula is very complex and difficult to assimilate by the diabetologists. In our case, we use a multi-objectives heuristic method, NSGA-II, to estimate the optimal control based on our model. Since the objective functions are conflicting, we obtain the Pareto optimal front formed by the non-dominated solutions and we use fuzzy C-means to determine the important main strategies based on a typical characterization. To limit human intervention, during the control period, we use the convolution operator to reduce hyper-fluctuations using kernels with different size. Several experiments were conducted and the proposed system highlights four feasible control strategies capable of mitigating socio-economic damages for a reasonable budget.

Solving Aggregate Production Planning Problems: An Extended TOPSIS Approach

Aggregate production planning (APP) was developed for solving the problem of determining production, inventory, and workforce levels to meet fluctuating demand requirements over a planning horizon. In this work, multiple objectives were considered to determine the most effective means of satisfying forecasted demand by adjusting production rates, hiring and layoffs, inventory levels, overtime work, back orders, and other controllable variables. An extended technique for order preference via the similarity ideal solution (TOPSIS) approach was developed. It was formulated to solve this complicated, multi-objective APP decision problem. Compromise (ideal solution) control minimized the measure of distance, providing which of the closest solutions has the shortest distance from a positive ideal solution (PIS) and the longest distance from a negative ideal solution (NIS). The proposed method can transform multiple objectives into two objectives. The bi-objective problem can then be solved by balancing satisfaction using a max–min operator for resolving the conflict between the new criteria based on PIS and NIS. Finally, an application example demonstrated the proposed model’s applicability to practical APP decision problems.

A NEW HYBRID ALGORITHM MPCM FOR SINGLE OBJECTIVE OPTIMIZATION PROBLEMS

One of the biggest challenges for researchers is finding optimal solutions or nearly optimal solutions for single-objective problems.
 In this article, authors have proposed new algorithm called MPCM for resolving single-objective problems. This algorithm is combined of four algorithms: Mean-Search, PSOUpdate, CRO operator and new operator call Min-Max. The authors use some parameters to balance between the local search and global search. The results demonstrate that, with the participation of Min-Max Operator, MPCM gives the good results on 23 benchmark functions. The results of MPCM will compare with three famous algorithms such as Particle Swarm Optimization (PSO), Real Code Chemical Reaction Optimization (RCCRO) and Mean PSO-CRO (MPC) for demonstration the efficiency.

Approximation by Kantorovich-type max-min operators and its applications

In this study, we construct Kantorovich variant of max-min kind operators, which are nonlinear. By using these new operators, we obtain some uniform approximation results in N-dimension (N≥1). Then, we estimate the error with the help of Hölder continuous functions and modulus of continuity. Furthermore, we give some illustrative applications to verify our theory and also investigate some shape-preserving properties of Kantorovich-type max-min Bernstein operator. Lastly, we examine the image processing implementation of our results via Kantorovich-type max-min Shepard operator.

Distance and Similarity Measures Using Max-Min Operators of Neutrosophic Hypersoft Sets With Application in Site Selection for Solid Waste Management Systems

The idea of a neutrosophic hypersoft set (NHSS) was coined by Smarandache in 2018 as a generalization of the soft set. This structure is a hybrid of a neutrosophic set with a hypersoft set. It can be a valuable structure for dealing with multi-attributes, multi-objective problems with disjoint attributive values. Similarity measures (SM) play a vital role in measuring the similarity index that how much the things are similar. Different types of similarity measures were developed in literature with different fuzzy, intuitionistic, and neutrosophic theories. It is intended to merge the neutrosophic theory with the hypersoft set theory and propose different similarity measures with the help of new proposed distances with max-min operators. Also, we proved different theorems and properties of distance and similarity measures. Then as solid waste management is a global issue, and there are some Solid Waste Management Systems (SWMS) for environment protection, so an example will be given for the site selection for SWMS to check the validity of proposed techniques. To verify the validity and superiority of the suggested work, it is contrasted to several existing methodologies, which show that decision-making issues with more bifurcation attributes provide more accurate and precise outcomes and can only be solved using this technique. In the future, the presented methodologies could be used in case studies with several qualities that are further bifurcated and multiple decision-makers. This proposed work can also be extended to many existing hypersoft set hybrids, such as Fuzzy hypersoft sets (FHSs), Intuitionistic hypersoft sets (IHSs), bipolar hypersoft sets (Bi-HSs), m-polar HSs, and Pythagorean hypersoft sets (PHSs).

New approximation properties of the Bernstein max-min operators and Bernstein max-product operators

<p style='text-indent:20px;'>In this paper we put in evidence localization results for the so-called Bernstein max-min operators and a property of translation for the Bernstein max-product operators.</p>

Graphic Representation of a Dimensional Expansion of Triangular Fuzzy Number

We calculate Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers. We prove that if the 3-dimensional result is limited to 2 dimensions, it is the same as the 2-dimensional result, which is shown as a graph. Since a 3-dimensional graph cannot be drawn, the value of the membership function is expressed with color density. We cut a 3-dimensional triangular fuzzy number by a perpendicular plane passing a vertex, and consider the cut plane as a domain. The value of the membership function for each point on the cut plane is also expressed with color density. The graph expressing the value of the membership function, defined in the plane as a 3-dimensional graph using the z -axis value instead of expressing with color density, is consistent with the results in the 2-dimensional case.

Intuitionistic fuzzy set-based time series forecasting model via delegeration of hesitancy degree to the major grade de-i-fuzzification and arithmetic rules based on centroid defuzzification

De-i-fuzzification is a process of converting the intuitionistic fuzzy set into a fuzzy set. It becomes one of the core procedures in fuzzy time series forecasting model based on the intuitionistic fuzzy set. In this paper, we propose a fuzzy time series forecasting model based on intuitionistic fuzzy set via de-i-fuzzification. The de-i-fuzzification approach used is assigning the hesitancy degree to the major grade. The data are partitioned into a few intervals using the frequency density-based method. The data in the fuzzy set form is then transformed into an intuitionistic fuzzy set using the definition of intuitionistic fuzzy set. The arithmetic rules based on centroid defuzzification is used to obtain the forecasted output. The model is implemented on the data of student enrolment at the University of Alabama. The results are then compared to forecasting method using classical fuzzy set and similar de-i-fuzzification approach using max-min operation. The proposed method outperforms the other two methods, thus supports the fact that intuitionistic fuzzy set is a generalization of a classical fuzzy set and gives better performance in forecasting.

Regular summability methods in the approximation by max-min operators

In this paper, by using nonnegative regular summability methods we improve and generalize the approximation properties of max-min operators which have been investigated systematically in our recent paper published in 2020 in this journal. We also discuss the rate of convergence in the approximation. Applications and concluding remarks at the end of the paper explain why we need such summability methods.

Fuzzy Optimization Model for Waste Load Allocation in a River with Total Maximum Daily Load (TMDL) Planning

In traditional waste load allocation (WLA) decision making, water quality-related constraints must be satisfied. Fuzzy models, however, can be useful for policy makers to make the most reasonable decisions in an ambiguous environment, considering various surrounding environments. We developed a fuzzy WLA model that optimizes the satisfaction level by using fuzzy membership functions and minimizes the water quality management cost for policy decision makers considering given environmental and socioeconomic conditions. The fuzzy optimization problem was formulated using a max–min operator. The fuzzy WLA model was applied to the Yeongsan River basin, which is located in the southwestern part of the Korean Peninsula and Korean TMDLs were applied. The results of the fuzzy model show that the pollutant load reduction should be increased in the Gwangju 1 and Gwangju 2 wastewater treatment plants (WWTPs) and in subcatchments with high pollutant load. In particular, it is necessary to perform advanced wastewater treatment to decrease the load of 932 kg ultimate biochemical oxygen demand (BODu)/day in the large-capacity Gwangju 1 WWTP and reduce the BODu emission concentration from 4.3 to 2.7 mg/L during the low-flow season. The satisfaction level of the fuzzy model is a relatively high at 0.81.

A New Complex Fuzzy Inference System With Fuzzy Knowledge Graph and Extensions in Decision Making

Context and Background:Complex fuzzy theory has a strong practical implication in many real-world applications. Complex Fuzzy Inference System (CFIS) is a powerful technique to overcome the challenges of uncertain, periodic data. However, a question is raised for CFIS: How can we deduce and predict the result in case there is little knowledge about data information and rule base? This is significance because many real applications do not have enough knowledge of rule base for inference so that the performance of systems may be low. Thus, it is necessary to have an approximate reasoning method to represent and derive final results. Motivation: Recently, the Mamdani Complex Fuzzy Inference System (M-CFIS) has been proposed with a specific inference mechanism according to the Mamdani type. A new improvement so-called the Mamdani Complex Fuzzy Inference System with Rule Reduction (M-CFIS-R) has been designed to utilize granular computing with complex similarity measures to reduce the rule base so as to gain better performance in decision-making problems. However in M-CFIS-R, testing data are checked by matching with each rule in the rule base, which leads to a high cost of computational time. Besides, if the testing data contain records that are not inferred by the rule base, the output cannot be generated. This happens in real commerce systems in which the rule base is small at the time of creation and needs to feed with new rules. Methodology: In order to handle those issues, this article first time proposes the Fuzzy Knowledge Graph to represent the rule base in terms of linguistic labels and their relationships according to the rule set. An adjacent matrix of Fuzzy Knowledge Graph is generated for inference. When a record in the Testing dataset is given, it would be fuzzified and labelled. Each component in the record is checked with the Fuzzy Knowledge Graph by the inference mechanism in approximate reasoning called Fast Inference Search Algorithm. Then, we derive the label of the new record by the Max-Min operator. Besides, we also propose four extensions of Mamdani Complex Fuzzy Inference System Rule Reduction including Sugeno Complex Fuzzy Inference Systems, Tsukamoto Complex Fuzzy Inference Systems, Complex Fuzzy Measures and Complex Fuzzy Integrals in M-CFIS-R. Results: The experiments on the UCI Machine Learning datasets show that the proposed method classifies samples as correctly as M-CFIS-R with very lower run time (6.45 times on average). The experiments are performed through many tests via 2 main scenarios. Conclusion: The proposed system has good performance in reducing the computational time of inference with acceptable accuracy. It has ability to work with systems having limited knowledge and rule base.

A Zadeh′s max-min composition operator for 3-dimensional triangular fuzzy number

We generalized triangular fuzzy numbers from ℝ to ℝ 2 . By defining parametric operations between two α-cuts, which are regions, we obtained parametric operations for two triangular fuzzy numbers defined on ℝ 2 . We also generalized triangular fuzzy numbers from ℝ 2 to ℝ 3 . By defining parametric operations between two α-cuts, which are subsets of ℝ 3 , we derived parametric operations for two triangular fuzzy numbers defined on ℝ 3 . For the calculation of Zadeh’s principle operators, the definition of parametric operations between two α-cuts, which are subsets of ℝ 3 , is critical.

Approximation by max-min operators: A general theory and its applications

In this study, we obtain a general approximation theorem for max-min operators including many significant applications. We also study the error estimation in this approximation by using Hölder continuous functions. The main motivation for this work is the paper by Bede et al. (2008) [12]. As a special case of our results, we explain how to approximate nonnegative continuous functions of one and two variables by means of the max-min Shepard operators. We also study the approximation by the max-min Bernstein operators. Furthermore, to verify the theory we display graphical illustrations.

Min-Max Predictive Control of a Five-Phase Induction Machine

In this paper, a fuzzy-logic based operator is used instead of a traditional cost function for the predictive stator current control of a five-phase induction machine (IM). The min-max operator is explored for the first time as an alternative to the traditional loss function. With this proposal, the selection of voltage vectors does not need weighting factors that are normally used within the loss function and require a cumbersome procedure to tune. In order to cope with conflicting criteria, the proposal uses a decision function that compares predicted errors in the torque producing subspace and in the x-y subspace. Simulations and experimental results are provided, showing how the proposal compares with the traditional method of fixed tuning for predictive stator current control.

Bilevel programming approaches to production planning for multiple products with short life cycles

We provide decision-making models for a manufacturer which plans to produce multiple short life cycle products with the one-shot decision theory. The obtained optimal production quantities are based on the most appropriate scenarios for the manufacturer. Since the models are the bilevel programming problems with the max–min or min–max operator in the lower levels, we propose two approaches to translate them into general single-level optimization problems such that they can be solved via the commonly used optimization methods. The effectiveness of our approaches is examined from the theoretical and computational aspects.

Supervised Learning to Aggregate Data With the Sugeno Integral

The problem of learning symmetric capacities (or fuzzy measures) from data is investigated toward applications in data analysis and prediction as well as decision making. Theoretical results regarding the solution minimizing the mean absolute error are exploited to develop an exact branch-refine-and-bound-type algorithm for fitting Sugeno integrals (weighted lattice polynomial functions, max–min operators) with respect to symmetric capacities. The proposed method turns out to be particularly suitable for acting on ordinal data. In addition to providing a model that can be used for the general data regression task, the results can be used, among others, to calibrate generalized h -indices to bibliometric data.

Takagi-Sugeno fuzzy regulator design for nonlinear and unstable systems using negative absolute eigenvalue approach

This paper introduces a Takagi-Sugeno &#x0028 T-S &#x0029 fuzzy regulator design using the negative absolute eigenvalue &#x0028 NAE &#x0029 approach for a class of nonlinear and unstable systems. The open-loop system is initially embodied by the traditional T-S fuzzy model and then, all closed-loop subsystems are combined using the proposed Max-Min operator &#x0028 discussed in Section III &#x0029 in place of traditional weighted average operator from the controller side to lessen the coupling virtually and simplify the proposed regulator design. For each virtually decoupled closedloop subsystem, the composite regulators &#x0028 i.e., primary and secondary regulators &#x0029 are designed by the NAE approach based on the enhanced eigenvalue analysis. The Lyapunov function is utilized to guarantee the asymptotic stability of the overall T-S fuzzy control system. The most popular and widely used nonlinear and unstable systems like the electromagnetic levitation system &#x0028 EMLS &#x0029 and the inverted cart pendulum &#x0028 ICP &#x0029 are simulated for the wide range of the initial conditions and the enormous variation in the disturbance. The transient and steadystate performance of the considered systems using the proposed design are analyzed in terms of the decay rate, settling time and integral errors as IAE, ISE, ITAE, and ITSE to validate the effectiveness of the proposed approach compared to the most popular and traditional parallel distributed compensation &#x0028 PDC &#x0029 approach.