Defuzzification Function Research Articles

Pythagorean Fuzzy Sets Combined with the PROMETHEE Method for the Selection of Cotton Woven Fabric

ABSTRACT Identifying and selecting the best cotton fabric from a series of available samples is a challenging multicriteria decision-making (MCDM) problem that includes fuzziness and uncertainty. Pythagorean fuzzy sets (PFSs) are widely used to manage complex and uncertain MCDM issues. The Preference Ranking Organization Method for Enrichment of Evaluation (PROMETHEE) is a widely used classical MCDM to assess and rank alternatives. In this study, we use the PROMETHEE approach to solve a real case of ranking cotton fabrics in a PFS environment, where alternatives are compared based on the PFS linguistic scales, and a score function is used as a defuzzification function. A comparative analysis is also performed with different score functions. The ranking results of the proposed PF-PROMETHEE method correlate strongly with other score functions, which indicates that the PF-PROMETHEE method is feasible and effective. The salient contributions of the PF-PROMETHEE method are as follows: (1) the method can manage uncertainty better than other methods; (2) uncertainty is evaluated by linguistic scales in the PF environment; (3) the method can effectively select cotton woven fabric; and (4) the method is applicable to a wide variety of MCDM problems in the textile industry.

Solving the Single-Valued Trapezoidal Neutrosophic Transportation Problems through the Novel Dhouib-Matrix-TP1 Heuristic

The transportation problem has been widely studied in the field of supply chain management where circulation of products with a minimal transportation cost is an important issue. This paper presents the first adaptation of the Dhouib-Matrix-TP1 heuristic to solve the transportation problem in single-valued trapezoidal neutrosophic environment. Hence, the recently developed Dhouib-Matrix-TP1 heuristic is enriched with two functions to solve the neutrosophic transportation problem. On the one hand, a defuzzification function is exploited in order to convert the single-valued trapezoidal neutrosophic numbers to crisp numbers. On the other hand, an original metric function (Average-Min) is proposed with the intention of performing the nodes selection process. With an illustration from a literature example, we show to the decision maker the multiple advantages of the novel heuristic Dhouib-Matrix-TP1 which can be easily implemented in real-life industrial transportation under neutrosophic environment.

Matrix games with dense fuzzy payoffs

A novel notion of dense fuzzy lock set was introduced as an extension of fuzzy sets. Learning experiences have a vital role in this fuzzy lock set instigation. The novel fuzzy lock set reduces the fuzziness of the situation. Occasionally, the players are bound to make little changes in their game strategies to reach their goal for some matrix game problems. It may lead to some changes in payoffs. In this scenario, a matrix game's payoffs are chosen as dense fuzzy lock sets to make the problem more realistic. This paper's prime intent is to develop a mathematical model of a matrix game that represents payoffs by triangular dense fuzzy lock sets. Initially, a new defuzzification function MagD(.) is defined to find a ranking order relation of the dense fuzzy lock sets. Then a pair of auxiliary dense fuzzy programming problems is established for two players. These two problems are transformed into two equivalent crisp linear programming problems applying the proposed defuzzification function and its linearity property. The reduced problems are solved using LINGO 17.0 software to determine each player's optimal strategies and the game values. One surprising fact of this approach is that the value of the game increases with the increment of the player's learning experience. The validity, applicability, and superiority of this proposed methodology are illustrated by considering a real-life media share problem.

Solving I-fuzzy two person zero-sum matrix games: Tanaka and Asai approach

This paper aims to design a method for solving a two person zero-sum matrix game with I-fuzzy goals and I-fuzzy pay-offs, where the symmetric triangular I-fuzzy numbers prescribe the entries of the pay-offs matrix. The most common approaches in the literature to solve matrix games with fuzzy goals and fuzzy pay-offs employ ranking function or defuzzification technique, like in Bector and Chandra (Fuzzy mathematical programming and fuzzy matrix games, vol 169. Springer, Berlin, 2005) and Vijay et al. (Fuzzy Optim Decis Making 6:299–314, 2007). Our proposed approach in this work differs from the existing approaches in the sense that it is devoid of ranking or defuzzification function. It also provides precise degrees of belief and disbelief in achieving the goals set by each player. An essential concept of ‘almost positive I-fuzzy number’ introduced by Aggarwal et al. (Notes Intuit Fuzzy Sets 23:85–101, 2017) is employed to study matrix games in I-fuzzy setting. Solving such a game is shown to be equivalent to solving a pair of crisp non-linear programming problem. In this way, our approach is unique for solving such a game. Some numerical examples are included to illustrate the proposed approach.

Exponential membership function and duality gaps for i-fuzzy linear programming problems

Fuzziness is ever presented in real life decision making problems. In this paper, we adapt the pessimistic approach tostudy a pair of linear primal-dual problem under intuitionistic fuzzy (I-fuzzy) environment and prove certain dualityresults. We generate the duality results using exponential membership and non-membership functions to represent thedecision maker’s satisfaction and dissatisfaction level. Further, two numerical examples have been given. In each ofthese illustrations, varying the values of the shape variables in the exponential membership functions, various nonlinearoptimization problems have been constructed, analyzed and solved. The duality gaps for all these optimization problemshave been computed and compared with the duality gap under the linear membership function. We found that thesegaps for the I-fuzzy linear primal-dual pair under the exponential membership functions are smaller as compared withthe linear membership functions. The main advantage of using the exponential membership function is that it has theflexibility of altering the values of the shape parameters (as per the decision-maker’s satisfaction). Finally, with thehelp of a suitable ranking defuzzification function, we have extended our approach to the I-fuzzy linear problem withfuzzy parameters and fuzzy constraints.

Fuzzy Linear Programming: Optimization of an Electric Circuit Model

A problem of a voltage division circuit is modeled in order to determine the values of the resistors, centered in away that the impedance of the resistance voltage divider is minimal. This problem is equivalent to maximizing the admittance, associated to the resistance, which is defined as the quotient of the electric current and its voltage, measured in Siemens. Three cases are analyzed for the components of the linear programming: real numbers, fuzzy numbers of type-1, and fuzzy set of type-2. The first case is considered in order to validate the other two cases. The optimal solution in the fuzzy linear programming of type-1 is obtained through a total linear order defuzzification function, defined in the trapezoidal fuzzy numbers subspace of fuzzy numbers vector space, which allows to solve the corresponding linear programming problem with real components. A discussion upon the parameter for the linear defuzzification is establish to determined the best representative of the parameters family. The α−levels representation theorem is the method to obtain the optimal solution of type-2. For each α−level is solved a fuzzy linear programming problem of type-1, using the previous methodology. Numerical simulations illustrate the results in the three cases.

Fuzzy Linear Programming: Optimization of an Electric Circuit Model

A problem of a voltage division circuit is modeled in order to determine the values of the resistors, centered in away that the impedance of the resistance voltage divider is minimal. This problem is equivalent to maximizing the admittance, associated to the resistance, which is defined as the quotient of the electric current and its voltage, measured in Siemens. Three cases are analyzed for the components of the linear programming: real numbers, fuzzy numbers of type-1, and fuzzy set of type-2. The first case is considered in order to validate the other two cases. The optimal solution in the fuzzy linear programming of type-1 is obtained through a total linear order defuzzification function, defined in the trapezoidal fuzzy numbers subspace of fuzzy numbers vector space, which allows to solve the corresponding linear programming problem with real components. A discussion upon the parameter for the linear defuzzification is establish to determined the best representative of the parameters family. The α−levels representation theorem is the method to obtain the optimal solution of type-2. For each α−level is solved a fuzzy linear programming problem of type-1, using the previous methodology. Numerical simulations illustrate the results in the three cases.

Modelling the regional vulnerability to Echinococcosis based on environmental factors using fuzzy inference system: A case study of Lorestan Province, west of Iran.

Background and aimEchinococcosis as a zoonosis disease is one of the most important parasitic helminth that is affected by many risk factors such as the environmental factors. Thus, we predicted the regional vulnerability to Echinococcosis based on environmental factors using a fuzzy inference system (FIS) in Lorestan Province.MethodsOur study was cross-sectional study on 200 patients from Lorestan Province (west of Iran) who underwent surgery for hydatidosis between October 2005 and November 2014. In order to model the vulnerability to Echinococcosis, first we determined the effective environmental variables. In the next step, the FIS was designed and implemented using MATLAB v.2012 software. Thus, definition and determination of linguistic variables, linguistic values, and their range were performed based on expert knowledge. Then, the membership functions of inputs (environmental variables) and output (vulnerability to Echinococcosis) were defined. A fuzzy rules base was formed. Also, the defuzzification of output was done using a centroid defuzzification function. To test the accuracy of the predictive model, we calculated the AUC (to this purpose, we used four different thresholds, 5%, 10%, 15%, and 20%) using IDRISI Selva v.17.0 software.ResultsBased on the results of this study, Aligoudarz and Koohdasht counties were identified as a highest and lowest risk area in Lorestan, respectively. The results showed that a predictive model was more efficient than a random model (AUC>0.5). Also, potential vulnerable areas cover 78.29% at threshold of 5%, 60.72% at threshold of 10%, 43.54% at threshold of 15%, and 39.82% at threshold of 20% of the study area.ConclusionAccording to the success of this research, we emphasized the necessity of attention to fuzzy approach to model vulnerability to hydatidosis. This approach can provide a practical economic basis for making informed preventive services decisions and the allocation of health resources.

Probabilistic Linguistic VIKOR Method to Evaluate Green Supply Chain Initiatives

To achieve the goals of sustainable economic and environmental protection, more and more firms intend to implement green supply chain (GSC) initiatives in their products. The adoption of GSC initiatives in turn influences the firms’ operations performance. Thus, the firms have to evaluate their performances carefully when implementing GSC initiatives. The performance evaluation of GSC initiatives is a laborious task, which needs to take into account many factors including the inventory level and assurance of supply purchasing-wise and the technical capability and the innovation capability manufacturing-wise, etc. This paper develops a new probabilistic linguistic VIKOR approach to support such an assessment. To do so, a new comparison method of probabilistic linguistic term sets (PLTSs) is first presented to effectively determine the probabilistic linguistic positive ideal solution and the probabilistic linguistic negative ideal solution. Next, a new defuzzification function of PLTSs is proposed to take into account the main-criteria weights and the sub-criteria weights which are represented by PLTSs. Furthermore, several probabilistic linguistic measures are introduced, such as the probabilistic linguistic group utility measure, the probabilistic linguistic individual regret measure and the probabilistic linguistic compromise measure. Finally, the compromise solution is obtained based on these three measures. The desirable advantages of the developed method are summarized as (1) it allows the evaluators to employ PLTSs to express the imprecise performances of the GSC initiatives with respect to various criteria, which greatly improves the elicitation of linguistic information; (2) it presents a probabilistic linguistic compromise solution, which is a maximum probabilistic linguistic group utility for the majority and a minimum probabilistic linguistic individual regret for the opponent. This proposed technique provides a simple and efficient decision making approach to assist the firms to make an appropriate decision in GSC management.

Short-Term Fuzzy Forecasting of Brent Oil Prices

Oil prices movements is very important macroeconomic factor for decision making. The accuracy of results fordifferent types of oil brands depends on models and algorithms. This paper evaluates the effectiveness of usingfuzzy sets to forecast daily Brent oil prices. It also contains possible modifications of the proposed method and incomparison with basic methods. The results suggest that Brent oil prices series have short memory because usinginformation about last 2-days prices shows better forecast accuracy. Forecasting based on fixed universe ofdiscourse shows better efficiency and it also proves that oil prices series has short memory. Adding theprobability of switching between linguistic terms in defuzzification function could be used to improve accuracyof predictions. Also the approach can take into consideration expert’s opinion about direction of future variation.The effective expert’s work can reduce errors of forecast from 1.5% till 0.76%. But this modification can be usedif experts correctly guess the direction of the change in trend in eight out of ten cases and more. The reasonableobtained results can be used by analysts dealing with the prediction of oil prices.

Matrix Games with Intuitionistic Fuzzy Pay-offs

In this paper two person zero sum matrix games are considered in which elements of payoff matrix are expressed with triangular intuitionistic fuzzy numbers (TIFNs). The purpose of this paper is to develop a solution methodology for solving such types of matrix games. For this, a pair of auxiliary intuitionistic fuzzy programming (IFP) models for each player are established. Then these two IFP problems are transformed into conventional crisp linear programming problems by defining a suitable defuzzification function to determine optimal strategies for each player and value of the game. The proposed method in this paper is illustrated with a numerical example to show the validity and applicability.

Particle swarm optimization-based algorithm for fuzzy parallel machine scheduling

This research extends the hybrid particle swarm optimization-based metaheuristic to solve the fuzzy parallel machine scheduling problems with bell-shaped fuzzy processing times. In this paper, we propose a discrete particle swarm optimization (DPSO) which comprises two components: a particle swarm optimization and genetic algorithm. In this paper, fuzzy arithmetic on bell-shaped fuzzy numbers is used to determine the completion time of jobs. We also use a defuzzification function to rank the fuzzy numbers. Under this ranking concept among fuzzy numbers, we plan to minimize the fuzzy makespan. An extensive numerical study on large-scale scheduling problems up to 100 jobs is conducted to assess the performance of the DPSO algorithm. The results show the proposed algorithm in comparison with lower bound to be very efficient for different structure instances.

Multi-objective fuzzy multiprocessor flowshop scheduling

This paper considers a bi-objective hybrid flowshop scheduling problems with fuzzy tasks’ operation times, due dates and sequence-dependent setup times. To solve this problem, we propose a bi-level algorithm to minimize two criteria, namely makespan, and sum of the earliness and tardiness, simultaneously. In the first level, the population will be decomposed into several sub-populations in parallel and each sub-population is designed for a scalar bi-objective. In the second level, non-dominant solutions obtained from sub-population bi-objective random key genetic algorithm (SBG) in the first level will be unified as one big population. In the second level, for improving the Pareto-front obtained by SBG, based on the search in Pareto space concept, a particle swarm optimization (PSO) is proposed. We use a defuzzification function to rank the Bell-shaped fuzzy numbers. The non-dominated sets obtained from each of levels and an algorithm presented previously in literature are compared. The computational results showed that PSO performs better than others and obtained superior results.

Defuzzification Functionals of Ordered Fuzzy Numbers

Defuzzification functionals, which play the main role when dealing with fuzzy controllers and fuzzy inference systems, for convex as well for ordered fuzzy numbers, are discussed. Three characteristic conditions for them are formulated. It is shown that most of the known defuzzification functionals satisfy them. Motivations for introducing the extended class of convex fuzzy numbers are presented, together with operations on them.

Fuzzy calculus with aplications

The aim of the artile is presenting the current state of ordered fuzzy numbers development. New model of fuzzy number was invented in 2002 to overcome drawbacks of classical (convex) fuzzy numbers. Two problems of management accounting are considered. The first relates to the management of supply and determining the optimal size of a delivery from outside, which minimalize total costs, when unit costs of delivery and storage are fuzzy. The second problem is related to determination of Internal Rate of Return (IRR) for investments in which the value of cash flow are not specified accurately. Key words and phrases: ordered fuzzy numbers, partial order relations, defuzzification functionals, management of supply, net present value (NPV), internal rate of return (IRR).

Ordered Fuzzy Numbers Approach to an Investment Project Evaluation

Abstract Aim of the paper is to propose a new tool for a decision supporting system concerning the financial project evaluation. It is based on the determination of the internal rate of return (IRR) of a investment project in which all expenditure and anticipated incomes are vague, and described by Ordered Fuzzy Numbers (OFNs). It means that the probabilistic approach is neglected in this paper and the use of the well developed arithmetics of OFNs is made to find a positive fuzzy root of a fuzzy polynomial representing the fuzzy net present value of the project. Since in the space of OFNs a partial order relation is defined together with a number of defuzzification functionals, the authors can construct a decision support system for investors helping them in acceptance procedure of most profitable investment projects.

USING ANT COLONY OPTIMIZATION TO MINIMIZE THE FUZZY MAKESPAN AND TOTAL WEIGHTED FUZZY COMPLETION TIME IN FLOW SHOP SCHEDULING PROBLEMS

The flow shop scheduling problems with fuzzy processing times are investigated in this paper. For some special kinds of fuzzy numbers, the analytic formulas of the fuzzy compltion time can be obtained. For the general bell-shaped fuzzy numbers, we present a computational procedure to obtain the approximated membership function of the fuzzy completion time. We define a defuzzification function to rank the fuzzy numbers. Under this ranking concept among fuzzy numbers, we plan to minimize the fuzzy makespan and total weighted fuzzy completion time. Because the ant colony algorithm has been successfully used to solve the scheduling problems with real-valued processing times, we shall also apply the ant colony algorithm to search for the best schedules when the processing times are assumed as fuzzy numbers. Numerical examples are also provided and solved by using the commercial software MATLAB.

Defuzzification using Steiner points

A defuzzification function assigns to each fuzzy set a crisp value in a way that this value may intuitively be understood as the “centre” of the fuzzy set. In the present paper, this vague concept is put into a mathematically rigorous form. To this end, we proceed analogously to the case of sharply bordered subsets, for which the Steiner point is frequently used. The function assigning to each convex subset its Steiner point is characterised by three properties; here, we study functions whose domains consist of fuzzy sets and which fulfil analogous properties. Although uniqueness can no longer be achieved, we give a complete characterisation of what we call Steiner points of fuzzy sets.

A new approach to universal approximation of fuzzy functions on a discrete set of points

One of the interesting, important and attractive problems in applied mathematics is approximation of functions in a given space. In this paper the problem is considered for fuzzy data and fuzzy functions using the defuzzification function of Fortemps and Roubens. Approximation of a fuzzy function on some given points ( x i , f ˜ i ) for i = 1, 2, … , m is considered by some researchers as interpolation problem. But in interpolation problem we find a polynomial of degree at most n = m − 1 where m is the number of points. But when we have lots of points ( m is very large) it is not good or even possible to find such polynomials. In this case we want to find a polynomial of arbitrary degree which is an approximation to original function. One of the works has done is regression by some researchers and we introduced a different method. In this case we have m points but we want to find a polynomial of degree at most n < m but not n = m − 1 necessarily. We introduce a fuzzy polynomial approximation as universal approximation of a fuzzy function on a discrete set of points and we present a method to compute it. We show that this approximation can be non-unique, however we choose one of them with the smallest amount of fuzziness.

Matrix games with fuzzy goals and fuzzy payoffs

A two person zero sum matrix game with fuzzy goals and fuzzy payoffs is considered and its solution is conceptualized using a suitable defuzzification function. Also, it is proved that such a game is equivalent to a primal–dual pair of certain fuzzy linear programming problems in which both goals as well as parameters are fuzzy.