Nonlinear Membership Functions Research Articles

Моделирование отклонений показателей качества объекта от нормы

The concept of a norm on the scale of an object's quality indicator in the context of determining the most preferable object or its state is explored. The notions of permissible and maximum permissible deviations from the norm are uti-lized, with maximum permissible deviations defining the boundaries of the indicator scale. The fundamental version of the norm is represented as a segment on the indicator scale that excludes these boundaries. Point and semi-interval norms (“no more”, “no less”) are considered special cases of the interval norm, where the semi-interval norm reflects the alignment of the original interval norm's boundary with the indicator scale's boundary. For certain indicators, devia-tion in one direction from the norm is not only acceptable but also beneficial. An indicator satisfying this condition is termed optimized, while an indicator with undesirable deviations from both ends of the norm is termed neutral. In the model of belonging to the norm, three classes are distinguished: “norm”, “no more” and “no less” than the norm. Piecewise linear and nonlinear membership functions for these classes are proposed. In the nonlinear model, the boundaries of the interval norm are expanded to acceptable limits, leading to the intersection of membership functions of adjacent classes. Classification into these three classes is conducted separately for neutral and optimized indicators. Interpreting deviations from the norm, both unacceptable and acceptable, necessitates introducing two fictitious clas-ses: “worse than the norm” and “better than the norm.” These classes are formed from the “not less than” and “not more than” classes. To calculate the membership function for each class across all indicators, a weighted average func-tion is employed. Aggregated indices of belonging to the classes “norm,” “better than norm,” and “worse than norm” across all indicators are referred to as indices of stability, development, and deterioration of an object, respectively. These indices are used for a comprehensive assessment of the object, termed an indicator of its condition. An example of analyzing deviations from the norm, implemented in the modified SVIR-M selection and ranking system, is provid-ed.

A framework for selecting and assessing soil quality indicators for sustainable soil management in waste dumps

The primary objective of this study was to develop soil quality indexes (SQIs) to reveal the changes in SQ during the restoration of vegetation in the reclaimed waste dumps of the Hequ open-pit coal mine. The study built an SQI evaluation model for waste dumps based on the soil management assessment framework. The total data set (TDS) consisted of nine physicochemical property indicators. The selection of the minimum data set (MDS) involved the utilization of principal component analysis (PCA) and Norm values. The SQ was comprehensively evaluated for nine indicators, taking into account the non-linear membership function and the improved Nemerow index. The findings suggested a notable disparity in the SQ between the reclaimed area and the unreclaimed area, yet the overall SQ fell short. In the TDS index system, the organic matter has the highest weight and a greater contribution to the soil quality of the waste dumps. In the MDS indicator system, the weights of organic matter and total nitrogen are both 0.5. According to Nemerow index method, the average SQIN of 5 plots is calculated to be 0.4352 ± 0.194. The average value obtained from TDS is 0.581 ± 0.236, and the average value obtained from MDS is 0.602 ± 0.351. The weighted additive method was employed to compute three SQIs, all of which yielded satisfactory outcomes. And the above evaluation methods indicate that the overall soil quality level of the waste dumps is at a moderate level. The sequence of SQ in various waste dumps was as follows: No.4lower > No.1 > No.2 > No.3 > No.4upper. Specifically, the non-linear membership function indicated that pH, available nitrogen (AN), available phosphorus (AP), surface moisture content (SMC), and bulk density (BD) were crucial in limiting SQIs in total waste dumps. The crucial limiting SQIs in unreclaimed areas were total phosphorus (TP) and total nitrogen (TN). This analysis demonstrates its efficacy in formulating strategies for the SQ evaluation and targeted soil reclamation plans of waste dumps.

A novel approach for multi-objective linear programming model under spherical fuzzy environment and its application

Every decision-making process particularly those involving real-life issues is disproportionately plagued by uncertainty. It is also unavoidable and obvious. Since its conception are several ways for representing uncertainty have been proposed by numerous academics to cope with uncertainty. Fuzzy sets and hierarchical such as picture fuzzy sets stand out among them as excellent representation techniques for modeling uncertainty. However, there are several significant drawbacks to the current uncertainty modeling techniques. Due to its vast versatility and benefits we here embrace the idea of the spherical fuzzy set, an extension of the picture fuzzy set. On the other hand amid uncertainty in real life the multi-objective plays a critical role. In this research paper determining a Multi-Objective Linear Programming Problem of Spherical fuzzy sets serves to stimulate nous. The score function corresponding to the degree positive, negative and neutral is the foundation upon which the suggested approach is developed. Additionally we apply the suggested strategy to the solution of the multi-objective linear programming problem to demonstrate its superiority through certain numerical examples. Maximization or Minimizing of the cost is the primary goal of the multi-objective linear programming problem. Using an explicitly defined score function the suggested solution transformed the Spherical Fuzzy Multi-Objective Linear Programming Problem into a Crisp Multi-Objective Linear Programming Problem (CMOLPP). We establish some theorems to show that the efficient solution of CMOLPP is likewise an efficient solution of SFMOLPP. The CMOLPP is then further simplified into a single-objective Linear Programming Problem (LPP) thus we revamp the modified Zimmermann’s approach in the environment of a nonlinear membership function with the aid of the suggested technique. It is possible to simply solve this single-objective LPP using any software or standard LPP algorithm. The suggested approach achieves the fuzzy optimum result without altering the nature of the issue. An application of the suggested approach has been used to illustrate it and its results have been distinguished from those of other preexisting methods found in the literature. To determine the importance of the suggested technique which adjudicate thorough theorem and result analysis is conducted.

Multi-objective non-linear programming problem with rough interval parameters: an application in municipal solid waste management

In dealing with the real-world optimization problems, a decision-maker has to frequently face the ambiguity and hesitancy due to various uncontrollable circumstances. Rough set theory has emerged as an indispensable tool for representing this ambiguity because of its characteristic of incorporating agreement and understanding of all the involved specialists and producing more realistic conclusions. This paper studies an application of the rough set theory for a multi-objective non-linear programming problem that originates for the management of solid wastes. Municipal solid waste management is a global problem that affects every country. Because of the poor waste management system in many nations, the bulk of municipal solid waste is disposed of in open landfills with no recovery mechanism. Hence, an effective and long term waste management strategy is the demand of the day. This research offers an incinerating, composting, recycling, and disposing system for the long-term management of the municipal solid waste. A model for the municipal solid waste management with the goal of minimizing the cost of waste transportation, cost of waste treatment and maximizing the revenue generated from various treatment facilities is developed under rough interval environment. To tackle the conflicting nature of different objectives, an approach is proposed that gives the optimistic and pessimistic views of the decision-maker for optimizing the proposed model. Also, the biasness/preference of the decision-maker for a specific objective is handled by establishing the respective non-linear membership and non-membership functions instead of the linear ones. Finally, to demonstrates the practicality of the proposed methodology, a case study is solved and the obtained Pareto-optimal solution has been compared to those obtained by the existing approaches.

Coordination of Cyclic crossover and Bat Algorithm for the Travelling Salesman Problems in Different Environments: A Simulation Approach

In this study, the features of cyclic crossover process and K-opt are incorporated in the bat algorithm (BA) to solve the Travelling Salesman Problems (TSP) in different environments. Swap operation and swap sequence are applied for the modification of the different operations of the BA to solve the TSPs. The cyclic crossover operation is applied in a regular interval of iterations on the best found solution and each solution of the final population of BA for the enhancement of the exploration as well as exploitation of the search process. K-Opt operation is applied on the population in each iteration of the BA with some probability for the exploitation. The algorithm is tested with a set of benchmark test instances of the TSPLIB. The algorithm produces exact results for a set of significantly large size problems. For the TSPs in fuzzy environment, a fuzzy simulation approach is proposed to deal with the fuzzy data having linear as well as non-linear membership functions. Also, a rough simulation process is proposed to deal with the TSPs in the rough environment where rough estimation can be done following any type of rough measure. The performance of the algorithm is compared with the state-of-the-art algorithms for the TSPs with crisp cost matrices using different statistical tools.

Multi-Objective Reliability Optimization using Fuzzy Nonlinear Programming with Interval Membership Functions and Bias Functions: A Comparison of Particle Swarm Optimization and Genetic Algorithm

Multi-objective reliability optimization is a complex problem that involves simultaneously optimizing multiple objectives while ensuring that the system meets certain reliability requirements. In this paper, we present a methodology for solving multi-objective reliability optimization problems using fuzzy nonlinear programming. The methodology involves representing the reliability of each component as a triangular interval number and each objective function as an interval membership function. Conflicts between objectives are resolved using linear and nonlinear membership functions, and exponential and quadratic membership functions are used to obtain definite biases towards the objective. The proposed methodology employs Particle Swarm Optimization (PSO) or Genetic Algorithm (GA) to solve the problem, and the approach is compared with GA for linear and nonlinear membership functions. The results indicate the effectiveness of the methodology in addressing multi-objective reliability optimization problems

Various defuzzification and ranking techniques for the heptagonal fuzzy number to prioritize the vulnerable countries of stroke disease

The idea of fuzzy numbers is determined according to their uncertainty points, which include interval values, triangular, trapezoidal, and heptagonal fuzzy numbers, etc. This article contains two types of fuzzy numbers: the heptagonal fuzzy number (HpFN) and heptagonal interval-valued fuzzy numbers (HpIVFN). For both the HpFN and HpIVFNs, the new distinct rational perspective and viewpoints are presented. First, the membership function (MF) is generalized to the linear and non-linear for observing the symmetric and asymmetric HpFN and HpIVFN. In light of this, α-cut is constructed for both linear and non-linear MF of HpFN and HpIVFN. The various types of HpIVFNs are classified along with the graphical representation. Then, new defuzzification under HpFN is determined using three methods, namely the centroid method, mean of α-cut method, and removal of area methods, and a comparison is demonstrated. A ranking method is also performed for HpFN. Using the proposed defuzzification approach in the weighted aggregates sum product assessment (WASPAS) method, an analysis is conducted to identify the countries with the highest vulnerability to stroke disease. The results indicated that world Bank upper-middle-income countries is significantly affected by stroke disease. The obtained outcome is validated through comparative and sensitivity analyses.

Ordering Fuzzy Numbers with Weighted Mean Values and Their Application to Multi-Criteria Decision-Making Problem

The most important aspect of fuzzy numbers is their ordering, which ensures a wide range of their applications in professional life and many academic applied models like linguistic decision-making and fuzzy risk analysis. Even though many researchers have presented various methods, there is still a lot of interest and scope for studies to address the weakness of methods. This paper proposes an approach for ordering generalized fuzzy numbers using weighted mean values (centroid values) of the left and the right fuzziness regions and exponential values of the altitude of the fuzzy number. The proposed method can order two or more fuzzy numbers simultaneously, irrespective of their linear or non-linear membership functions. Furthermore, the proposed method consistently orders the symmetrical fuzzy numbers, the partnered image of the fuzzy number, and the fuzzy numbers that depict the compensation of areas. The advantages of the proposed approach are demonstrated through numerical examples with various types of fuzzy numbers and comparisons with the existing techniques published in the literature. Finally, the proposed method is effectively applied to solve a linguistic multi-criteria decision-making problem related to the stock market.

Fault Diagnosis for Complex Equipment Based on Belief Rule Base with Adaptive Nonlinear Membership Function

Fault diagnosis of complex equipment has become a hot field in recent years. Due to excellent uncertainty processing capability and small sample problem modeling capability, belief rule base (BRB) has been widely used in the fault diagnosis. However, previous BRB models almost did not consider the diverse distributions of observation data which may reduce diagnostic accuracy. In this paper, a new fault diagnosis model based on BRB is proposed. Considering that the previous triangular membership function cannot address the diverse distribution of observation data, a new nonlinear membership function is proposed to transform the input information. Then, since the model parameters initially determined by experts are inaccurate, a new parameter optimization model with the parameters of the nonlinear membership function is proposed and driven by the gradient descent method to prevent the expert knowledge from being destroyed. A fault diagnosis case of laser gyro is used to verify the validity of the proposed model. In the case study, the diagnosis accuracy of the new BRB-based fault diagnosis model reached 95.56%, which shows better fault diagnosis performance than other methods.

Multi-objective optimization to the transportation problem considering non-linear fuzzy membership functions

<abstract> <p>Considering the uncertainty of transporting goods from numerous origins to diverse destinations is a critical task for the decision-maker (DM). The ultimate goal of the DM is to make the right decisions that optimize the profit or loss of the organization under the vagueness of the uncontrollable effects. In this paper, mathematical models are proposed using fuzzy non-linear membership functions for the transportation problem considering the parameters' uncertainty that can help the DM to optimize the multi-objective transportation problems (MOTP) and to achieve the desired goals by choosing a confidence level of the uncertain parameters. Based on DM's selection of the confidence level, a compromise solution of the uncertain multi-objective transportation (UMOTP) is obtained along with the satisfaction level in percent for the DM. Two non-linear fuzzy membership functions are considered: the exponential and the hyperbolic functions. Using both membership functions, the sensitivity analysis was implemented by considering different confidence levels. According to the experimental results, the hyperbolic membership function gives 100% DM's satisfaction in many instances. Moreover, it shows stability against the exponential and linear functions.</p> </abstract>

Goal programming approach for multi-objective optimization to the transportation problem in uncertain environment using fuzzy non-linear membership functions

The ultimate goal of the decision maker (DM) is to take right decisions to optimize the profit or loss of the organization when the parameters of the transportation problem are ambiguous because of some uncontrollable effects. In this paper, mathematical models are proposed using fuzzy non-linear membership functions and the inverse uncertain normal distribution has been used to eliminate the uncertainty in the parameters which will help the DM to find a compromise solution of the uncertain multi-objective transportation problem (UMOTP) and to achieve the desired goals for a chosen level of confidence for the uncertain parameters. The compromise solutions of the uncertain multi-objective transportation problem are presented to obtain the DM satisfaction if the problem becomes achievable for this preferred confidence level of the parameters. Numerical illustration is given where Linear Programming Problems (LPPs) are resolved with LINGO and the graphs are designed with the help of MATLAB 18.00. J. Bangladesh Acad. Sci. 46(1); 101-115: June 2022

Quality evaluation of the soil-root composites layer of Leymus chinensis grassland based on different degradation degrees

Soil quality index (SQI) is widely used to assess the status and use potential of soils, and it is also an important prerequisite for taking effective remedial measures for degraded land. The objectives of this study are as follows: (1) to construct a soil-root composites quality index (SRCQI) by selecting the minimum data set (MDS) and membership function that are most suitable for evaluating degraded Leymus chinensis grassland, and to analyze the relationship between the SRCQI and degraded environment; and (2) to determine the main obstacle factors of degraded Leymus chinensis grassland and propose corresponding control measures. In this study, 15 evaluation indicators representing the physical, chemical, and biological properties of the soil-root composites layer (SRCL) of Leymus chinensis grassland were measured. The MDS was determined by principal component analysis (PCA), and the accuracy of MDS was verified by Pearson correlation analysis. The suitability of three membership functions (linear membership function, nonlinear membership function, and three-effect membership function) was compared by the coefficient of variation (CV) and ANOVA. The SRCQI was constructed to evaluate the quality of Leymus chinensis grassland SRCL with different degradation degrees. An obstacle factor model was introduced to determine the dominant obstacle factors affecting degraded Leymus chinensis grassland. The results showed the following: (1) for the SRCL of Leymus chinensis grassland, the MDS could be constructed by forage biomass, bulk density, and pH; (2) based on the highest F value (102.800) and CV value (23.35%), the non-linear membership function was determined to be the optimal membership function for the quality evaluation of SRCL of Leymus chinensis grassland; (3) the average SRCQI of Leymus chinensis grassland with coverages of 70–80%, 60–70%, 50–60%, and 40–50% were 0.617, 0.532, 0.406, and 0.398, respectively, where the SRCQI gradually decreased with the aggravation of grassland degradation (coverage reduction); and (4) the main obstacle factors of degraded Leymus chinensis grassland were determined as follows: the available potassium, forage biomass, moisture content, and organic matter content were relatively low, and the obstacle degrees were 9.66%, 9.06%, 8.22%, and 8.15%, respectively. This study provides a sensitive and effective approach for the quantitative evaluation of the SRCL of Leymus chinensis grassland susceptible to compaction degradation.

Generalized techniques for solving intuitionistic fuzzy multi-objective non-linear optimization problems

This paper focuses on the methods for the efficient solution of multi-objective non-linear optimization problems with uncertain parameters represented as intuitionistic fuzzy numbers. In most of the existing techniques for such problems, generally linear membership (satisfaction) functions have been used. But every real life problem cannot be justified and modeled using the linear functions, so the efficient solution methodologies from the literature such as Zimmermann’s technique, Maximum additive operator technique, γ-operator technique have been extended in this paper by defining the non-linear membership functions in place of the linear ones. Unlike the classical versions of these techniques, the non-linear non-membership (dissatisfaction) functions have also been incorporated along with the memberships. Intuitionistic fuzzy number with non-linear grade functions has been introduced. Appropriate theorems have been proved to support the claims. Two numerical examples in the intuitionistic fuzzy environment from the field of manufacturing and transportation have been considered for the illustration of the proposed technique. The obtained results show the applicability and reliability of the suggested extensions and their comparison with the results obtained from that of the non-modified (traditional) techniques reflects its effectiveness.

A modified ACO with K-Opt for restricted covering salesman problems in different environments

In this study, the ant colony optimization (ACO) algorithm is modified with the K-Opt operation to solve the covering salesman problem (CSP) under one restriction in crisp and imprecise (fuzzy, rough) environments. A CSP involves two phases—the division of cities into groups with the selection of the visiting cities and searching of the Hamiltonian circuit through the visiting cities. But, none of the studies in the literature is made following the direct approach. Also, none of the studies in the literature gives attention to reduce the total travel distance of the unvisited cities from the visited city of a group. Moreover, there is no algorithm in the literature which provides the solution of a CSP with the specified coverage range r. Also, none has introduced any algorithm to solve CSPs in imprecise environments. Though algorithms are available to solve the Traveling Salesman Problems in the imprecise environments, the approach cannot deal with the problems involving fuzzy data with nonlinear membership functions or the problems involving rough data where the rough estimation can not be done using Lebesgue measure. The well-established algorithm for any routing problem is the ACO, but not much attention has been paid to solve the CSP using ACOs. To overcome these limitations on the studies of the ACO on the CSPs, here, an algorithm is proposed for the division of groups of the set of cities depending upon the maximum number of cities in a group and the total number of groups. Then, ACO is used to find the shortest/minimum-cost path of the problem by selecting only one visiting the city from each group without violating the restriction of the specified coverage range r of the location of the unvisited cities. K-Opt operation is applied periodically at the end of ACO operation to improve the quality of the best found solution so far by the ACO algorithm and to arrest any premature convergence. For the restricted problems, paths are searched in such a manner that the total distance/travel cost of different unvisited cities of a group from the visited city of the group should not exceed a predefined upper limit. To solve the problem in an imprecise environment, some approach is followed so that the tour is searched without transferring the imprecise optimisation problem into an equivalent crisp optimisation problem. Also, the simulation approaches in fuzzy and rough environments are proposed to deal with the CSPs with any type of estimation of the imprecise data set. Algorithm is tested with the standard benchmark crisp problems available in the literature. To test the algorithm in the imprecise environments, the imprecise instances are derived randomly from the standard crisp instances using a specified rule. Test results imply that the proposed algorithm is efficient enough in solving the CSPs in the crisp as well as in the imprecise environments.

Nonuniform piecewise membership function approximation methods based robust tracking control design of T‐S fuzzy systems

AbstractFor the stability analysis issue of a T‐S fuzzy system, improved piecewise membership functions dependent approach is proposed to reduce conservatism. Based on a least‐square method, the nonlinear membership functions are approximated by a series of decoupled polynomial functions. Then the piecewise vertices are chosen by a newly proposed derivative‐integral nonuniform division approach. For the robust tracking control task, a series of convex robust stable conditions are obtained. Finally, a mass‐spring‐damper (MSD) simulation example is adopted to illustrate the effectiveness of the designed method.

Ranking Fuzzy Numbers with Unified Integral Value and Comparative Reviews

Fuzzy numbers represent ambiguous numeric values; therefore, it is difficult to rank them according to their magnitude. In making decisions, it is important to rank the fuzzy numbers. Since fuzzy numbers in a fuzzy environment often measure alternatives, a comparison of these fuzzy numbers is, in fact, a comparison of alternatives. There are a lot of different types of methods for ranking fuzzy numbers that exist in the literature. Still, there is no single method superior to all others regarding discrimination and consistency. This paper proposes a method for ranking fuzzy numbers with a unified integral value that multiplies two discriminatory components, the mode area integral and a linear sum of the absolute values of the integrals of the left and the right limits of alpha-cut of the normalized form of a fuzzy number. The method can rank two or more fuzzy numbers simultaneously, regardless of their linear or nonlinear membership functions. Furthermore, the unified integral value consistently ranks fuzzy numbers and their images and symmetric fuzzy numbers with the same altitude. Various types of fuzzy numbers are used in examples for comparative studies and investigations.

Fuzzy optimization-based Water Distribution Network design using Self-Adaptive Cuckoo Search Algorithm

Abstract Water Distribution Network(s) (WDN) design is gaining prominence in the urban planning context. Several factors that play a significant role in design are uncertainty in data, non-linear relation of head loss & discharge, combinatorial nature of the problem, and high computational requirements. In addition, many conflicting objectives are possible and required for effective WDN design, such as cost, resilience, and leakage. Most of the research work published has used multiobjective evolutionary optimization in solving such complex WDN. However, the challenge of such population-based evolutionary approaches is that they provide multiple trade-off Pareto optimal solutions to the decision-maker who will have to choose another set of techniques to arrive at a single optimal solution. The present study employs a fuzzy optimization approach that would provide a single optimal WDN design for Hanoi and Pamapur, India. Maximization of network resilience (NR) and minimization of network cost (NC) are employed in a multiobjective context. Later, minimization of network leakages (NL) is also incorporated, leading to three objective problems. Hyperbolic membership function (HMF), exponential membership function (EMF), and non-linear membership function (NMF) are employed in Self-Adaptive Cuckoo Search Algorithm-based fuzzy optimization. HMF is found suitable to determine the best possible WDN design for chosen case studies based on the highest degree of satisfaction.

Significance of multi-objective optimization in logistics problem for multi-product supply chain network under the intuitionistic fuzzy environment

Determining the methods for fulfilling the continuously increasing customer expectations and maintaining competitiveness in the market while limiting controllable expenses is challenging. Our study thus identifies inefficiencies in the supply chain network (SCN). The initial goal is to obtain the best allocation order for products from various sources with different destinations in an optimal manner. This study considers two types of decision-makers (DMs) operating at two separate groups of SCN, that is, a bi-level decision-making process. The first-level DM moves first and determines the amounts of the quantity transported to distributors, and the second-level DM then rationally chooses their amounts. First-level decision-makers (FLDMs) aimed at minimizing the total costs of transportation, while second-level decision-makers (SLDM) attempt to simultaneously minimize the total delivery time of the SCN and balance the allocation order between various sources and destinations. This investigation implements fuzzy goal programming (FGP) to solve the multi-objective of SCN in an intuitionistic fuzzy environment. The FGP concept was used to define the fuzzy goals, build linear and nonlinear membership functions, and achieve the compromise solution. A real-life case study was used to illustrate the proposed work. The obtained result shows the optimal quantities transported from the various sources to the various destinations that could enable managers to detect the optimum quantity of the product when hierarchical decision-making involving two levels. A case study then illustrates the application of the proposed work.

Two Identification Methods for a Nonlinear Membership Function

This paper proposes two parameter identification methods for a nonlinear membership function. An equation converted method is introduced to turn the nonlinear function into a concise model. Then a stochastic gradient algorithm and a gradient-based iterative algorithm are provided to estimate the unknown parameters of the nonlinear function. The numerical example shows that the proposed algorithms are effective.

Forecasting Models of Daily Energy Generation by PV Panels Using Fuzzy Logic

This paper contains studies of daily energy production forecasting methods for photovoltaic solar panels (PV panel) by using mathematical methods and fuzzy logic models. Mathematical models are based on analytic equations that bind PV panel power with temperature and solar radiation. In models based on fuzzy logic, we use Adaptive-network-based Fuzzy Inference Systems (ANFIS) and the zero-order Takagi-Sugeno model (TS) with specially selected linear and non-linear membership functions. The use of mentioned membership functions causes that the TS system is equivalent to a polynomial and its properties can be compared to other analytical models of PV panels found in the literature. The developed models are based on data from a real system. The accuracy of developed prognostic models is compared, and a prototype software implementing the best-performing models is presented. The software is written for a generic programmable logic controller (PLC) compliant to the IEC 61131-3 standard.