Possibilistic Mean Research Articles

Practical possibilistic fuzzy pay-off method for real option valuation

This paper describes a set of proposed additions to the recently introduced possibilistic fuzzy pay-off method for real option valuation that enhances the practical usability of the method. The additions are focused on the practical usability of the method and concentrate on emphasizing the consideration of the downside-risk found in projects. Technically the additions are based on using a novel interpretation of the possibilistic mean as a proxy respectively for the weight of the downside and the upside of a possibility distribution in the context of project profitability. This interpretation of the possibilistic mean is a new theoretical contribution.The proposed new method-variant, called the “practical possibilistic fuzzy pay-off method for real option valuation”, can effectively distinguish between projects with an identical upside and non-identical downsides and allows for a more finance-theoretically comprehensive consideration of situations, where the circumstances surrounding the downside risk of a project change. Changes in the downside are reflected in the real option value. The proposed changes constitute the first variant of the possibilistic fuzzy pay-off method for real option valuation and they remarkably increase the practical usability of the method.

A novel fuzzy GRA-based possibility theory soft computing process for selection pest control methods

The study presents the effective approach that combines the ideal with non-ideal points, GRA, and possibility theory to handle uncertain information in selecting pest control methods within an MCDM problem. The proposed approach for selecting pest control methods can incorporate uncertainty information expressed by fuzzy numbers, possibilistic mean, and possibilistic variance. Decision-maker can comprehend the gaps between alternative solutions and the aspired outcomes, and it allows for the consideration of decision-making risks under a fuzzy environment. The study establishes an evaluation framework for selecting pest control methods by using the empirical study, and one obtains an appropriate pest control method for managing Tessaratoma papillosa in Taiwanese crops. Through the practical example, the proposed approach can be demonstrated the feasibility and efficiency.

The Possibilistic Mean-Variance Model with Uncertain Possibility Distributions

The possibilistic mean–variance (MV) model is the counterpart of Markowitz’s MV model in the possibility theory. This study aims to examine the possibilistic MV model when the possibility distributions of stock returns are uncertain triangular fuzzy numbers. We define an uncertainty vector and use its ellipsoidal uncertainty set in a minimax optimization problem to model this uncertainty. We also show that this minimax optimization problem reduces to a strictly convex minimization problem. Thus, unlike the possibilistic MV model, we get diversified optimal portfolios uniquely with our approach. After laying down the theoretical points of our approach, we illustrate it with a real-world example in the literature by using a software package for convex optimization. To the best of our knowledge, this is the first paper that considers uncertain possibility distributions in the possibilistic MV model.

Generalized dissemblance index as a difference of first moments of fuzzy numbers – A new perspective on the distance of fuzzy numbers

This paper investigates the formulation of the dissemblance index as a basis for the calculation of distances of fuzzy numbers and explores its potential linkages with standard and possibilistic moments of fuzzy numbers. Applying the LSC transformation introduced recently by Luukka, Stoklasa and Collan we transform the general formulation of the dissemblance index into its “probabilistic” analogy and show that the result can be interpreted as a difference of COGs of the respective fuzzy numbers (potentially with hedges applied to them). We also show that the difference of possibilistic means is a special case of the general dissemblance index, when w=1. We also propose a generalized version of the possibilistic mean of a fuzzy number and prove its properties. We discuss the implications of this relationship on the practical use of the generalized dissemblance index and investigate its performance in the task of ranking of fuzzy numbers.

Robust Versions of the Lower and Upper Possibilistic Mean - Variance Models for the One Period or Two Periods Cases

It is easy to use possibility theory in modeling incomplete information. Robust optimization is an important tool when there is parameter uncertainty. Thus, in this study, we propose robust versions of the lower and upper possibilistic mean - variance (MV) models when there are multiple possibility distribution scenarios. Here, we use entropy as a diversification constraint. In addition, we reduce these robust versions to concave maximization problems. Furthermore, we generalize them for two periods portfolio selection problem by using fuzzy addition and multiplication. On the other hand, these generalizations are not concave maximization problems. Finally, we give an illustrative example by using different solvers in Gams modeling system.

Possibilistic mean based defuzzification for fuzzy expert systems and fuzzy control—LSD for general fuzzy sets

This paper introduces a new defuzzification technique derived as a generalization of the formula for the calculation of possibilistic mean originally proposed by Carlsson and Fullér in 2001 for fuzzy numbers. Unlike the possibilistic mean, the generalized formulation allows also for the defuzzification of subnormal convex fuzzy sets and also for non-convex fuzzy sets (e.g. the outputs of Mamdani- or Larsen-type fuzzy inference). The Luukka–Stoklasa–Collan transformation introduced in 2019 is applied to generalize the possibilistic mean formula. Using this transformation an algorithm for the calculation of the possibilistic-mean-based defuzzification of a general fuzzy set with a continuous membership function on the given interval is proposed. This way the Luukka–Stoklasa Defuzzification (LSD) inspired by the possibilistic mean construction is introduced - a defuzzification that can be calculated also for fuzzy sets in general (subnormal, non-convex), not only for fuzzy numbers. As such LSD is applicable also in fuzzy expert systems and fuzzy control settings where the outputs of the inference systems can be expected to be represented by subnormal and non-convex fuzzy sets. Fast-computation formulas for LSD of piece-wise linear fuzzy sets are also provided. The applicability of LSD in the ranking of fuzzy numbers and its ability to distinguish between fuzzy numbers where other frequently used defuzzification methods do not is shown. Two more case studies are presented where LSD outperforms the chosen frequently used defuzzification methods: a fuzzy expert system for inventory control and a fuzzy cruise controller problem.

The α-weighted averaging operator

This paper introduces a new type of mean applicable in various areas of science and practice: the α-weighted averaging operator (AWA). AWA has all the properties required from a linear averaging operator and some additional ones. We discuss the applications of AWA in data aggregation in various areas including uncertainty modeling (summarization, defuzzification), multiple-criteria and multi-expert decision-making and evaluation. We prove that when applied to fuzzy numbers, the α-weighted average converges to the possibilistic mean of a fuzzy number with the increasing number of elements in its support. As such the α-weighted average is a more general aggregation operator than the original possibilistic mean. When fuzzy subsets of the real line represent the information to be aggregated, AWA provides new means for their defuzzification compatible with the possibilistic moments, but applicable to discrete and subnormal fuzzy sets. We also introduce a generalized formulation of the α-weighted averaging operator (GAWA) that can be applied in multiple-criteria and multi-expert evaluation and decision-making problems. We suggest the use of GAWA in operations research theory and applications in the context of data aggregation, multiple-criteria and group evaluation and decision-making.

Mathematical Analyses of the Upper and Lower Possibilistic Mean – Variance Models and Their Extensions to Multiple Scenarios

Possibility theory is the one of the most important and widely used uncertainty theories because it is closely related to the imprecise probability and expert knowledge. The possibilistic mean - variance (MV) model is the counterpart of the Markowitz’s MV model in the possibility theory. There are variants of the possibilistic MV model, which are called as the upper and lower possibilistic MV models. However, to the best of our knowledge, analytical solutions and exact efficient frontiers of these variants are not presented in the literature when the possibil-ity distributions are given with trapezoidal fuzzy numbers. In this study, under this assumption, we make mathemat-ical analyses of the upper and lower possibilistic MV models and derive their analytical solutions and exact efficient frontiers. Based on the max-min optimization framework, we also propose their extensions where there are multiple upper (lower) possibilistic mean scenarios. We show that the proposed extensions have the ease of use as the upper and lower possibilistic MV models. We also illustrate and compare the upper and lower possibilistic mean - variance models and their proposed extensions with an explanatory example. As we expect, we see that these extensions can be effectively used in portfolio selection by conservative investors.

Performance evaluation of possibilistic fuzzy portfolios with different investor risk attitudes based on DEA approach

By considering the stock market’s fuzzy uncertainty and investors’ psychological factors, this paper studies the portfolio performance evaluation problems with different risk attitudes (optimistic, pessimistic, and neutral) by the Data Envelopment Analysis (DEA) approach. In this work, the return rates of assets are characterized as trapezoidal fuzzy numbers, whose membership functions with risk attitude parameters are described by exponential expression. Firstly, these characteristics with risk attitude are strictly derived including the possibilistic mean, variance, semi-variance, and semi-absolute deviation based on possibility theory. Secondly, three portfolio models (mean-variance, mean-semi-variance, and mean-semi-absolute-deviation) with different risk attitudes are proposed. Thirdly, we prove the real frontiers determined by our models are concave functions through mathematical theoretical derivation. In addition, two novel indicators are defined by difference and ratio formulas to characterize the correlation between DEA efficiency and portfolio efficiency. Finally, numerical examples are given to verify the feasibility and effectiveness of our model. No matter what risk attitude an investor holds, the DEA can generate approximate real frontiers. Correlation analysis indicates that our proposed approach outperforms in evaluating portfolios with risk attitudes. At the same time, our model is an improvement of Tsaur’s work (2013) which did not study the different risk measures, and an extension of Chen et al.’s work (2018) which only considered risk-neutral attitude.

Multi-item economic production quantity model under single valued bipolar neutrosophic environment based on possibilistic theory

It is challenging for a decision-maker to decide a proper decision in severe situations of multi-aspirated real-life problems.So there is always an ambiguity in the mind of decision maker. Keeping such vagueness in mind, this paper aims to incorporate some situation parameters imprecise in nature. The imprecise parameters are taken in single-valued bipolar neutrosophic environments. Different arithmetic operations on the single-valued bipolar neutrosophic number using the (α, β) cut method are proposed in this paper. Using this we have calculated the possibility mean of single valued bipolar neutrosophic numbers. A multi-item economic production quantity model with one time only discount is considered here with some parameters in single valued bipolar neutrosophic number as a case study of our proposed work. A possibilistic mean de-fuzzification technique is used here using possibility measures. Finally, numerical illustration and sensitivity analysis is done for different variables to emphasize the excellence of our proposed work.

A hybrid decision-making method using robust programming and interval-valued fuzzy sets for sustainable-resilient supply chain network design considering circular economy and technology levels

This study introduces a new three-stage optimization approach with a circular economy perspective for sustainable-resilient supply chain network design for perishable products. The proposed model specifies the number of facilities and the number of products flowing within the supply chain in case of disruption. The three contributions of the mathematical model are considering product lifetime, monitoring financial resources, and selecting technology levels. It incorporates suppliers’ green image and circular economy rating in the supply chain network structure. Epistemic uncertainty is considered in the optimization model to deal with the unknown capacity, cost, and demand. The study includes three main stages. In the first stage, to determine suppliers' green image and circular economy rating, a new interval-valued fuzzy (IVF)-compromise decision-making method is presented based on possibilistic mean and standard deviation evaluations. In the second stage a new multi-objective mathematical model is proposed to reduce total costs, reduce environmental impacts, and increase social sustainability. To deal with uncertainties in the mathematical model, a new IVF-robust solution approach is introduced. In the third stage, the solution is incorporated with the AUGMECON2 method to produce separate Pareto-optimal solutions, offering a hybrid of trade-offs between cost, emissions, and social responsibility. A case study in the food, dairy, and drink industry is presented to show how the suggested approach might be applied. Finally, several sensitivity analyses and comprehensive comparisons of the proposed approach with the literature were performed.

A dynamic optimal solution approach for solving neutrosophic transportation problem

Neutrosophic Set (NS) allows us to handle uncertainty and indeterminacy of the data. Several researchers have investigated the Transportation Problems (TP) with various forms of input data. This paper emphasizes a dynamic optimal solution framework for TPs in a neutrosophic setting. This paper investigates a Neutrosophic Transportation Problem (NTP) in which supply, demand, and transportation cost are considered as Single-Valued Neutrosophic Trapezoidal Numbers (SVNTrNs). The weighted possibilistic mean value of their truth, indeterminacy, and facility membership function are calculated. Then, NTP is modelled as a parametric Linear Programming Problem (LPP) and solved. Further, the drawbacks of the existing approaches and advantages of the developed method are discussed. Finally, the real-time problem and numerical illustrations are presented and compared to existing solutions. This study helps the Decision-Makers (DMs) in budgeting their transportation expenses through strategic distribution.

Neutrosophic data envelopment analysis based on the possibilistic mean approach

Data envelopment analysis (DEA) is a non-parametric approach for the estimation of production frontier that is used to calculate the performance of a group of similar decision-making units (DMUs) which employ comparable inputs to produce related outputs. However, observed values might occasionally be confusing, imprecise, ambiguous, inadequate, and inconsistent in real-world applications. Thus, disregarding these factors may result in incorrect decision-making. Thus neutrosophic sets have been created as an extension of intuitionistic fuzzy sets to represent ambiguous, erroneous, missing, and inaccurate information in real-world applications. In this study, we have proposed a technique for solving the neutrosophic form of the Charnes–Cooper–Rhodes (CCR) model based on single-value trapezoidal neutrosophic numbers (SVTrNNs). The possibilistic mean for SVTrNNs is redefined and applied the Mehar approach to transforming the neutrosophic DEA (Neu-DEA) model into its corresponding crisp DEA model. As a result, the efficiency scores of the DMUs are calculated using different risk parameter values lying in [0, 1]. A numerical example is given to analyze the performance of the all India institutes of medical sciences and compared it with Abdelfattah’s ranking approach.

Mehar approach to solve neutrosophic linear programming problems using possibilistic mean

Khatter (Soft Comput 24:6847–16,867, 2020) pointed out that although several approaches are proposed in the literature to solve single-valued neutrosophic linear programming problems (SVNLPPS) (linear programming problems in which all the parameters except decision variables are either represented by single-valued triangular neutrosophic numbers (SVTNNS) or single-valued trapezoidal neutrosophic numbers (SVTrNNS)). However, all the methods for comparing single-valued neutrosophic numbers (SVNNS), used in existing approaches, are independent from the attitude of the decision-maker towards the risk. To fill this gap, Khatter (2020), firstly, proposed a method for comparing two SVNNS by considering the attitude of the decision-maker towards the risk. Then, using the proposed comparing method, Khatter (2020) proposed an approach to solve SVNLPPS. In this paper, it is pointed out that a mathematical incorrect result is considered in Khatter’s approach. Hence, it is inappropriate to use Khatter’s approach. Also, it is pointed out that some mathematical incorrect results are considered in other existing approaches for solving SVNLPPS. Hence, it is inappropriate to use other existing approaches for solving SVNLPPS. Furthermore, to resolve the inappropriateness of Khatter’s approach and other existing approaches, a new approach (named as Mehar approach) is proposed to solve SVNLPPS. Finally, correct optimal solution of some existing SVNLPPS is obtained by the proposed Mehar approach.

Multi-criteria decision making of water resource management problem (in Agriculture field, Purulia district) based on possibility measures under generalized single valued non-linear bipolar neutrosophic environment

Purulia district is a drought-prone area in West Bengal, India. Naturally, the farmers of the Purulia district could not plant the crops due to the absence of water resources. Water resource management (WRM) has the potential mechanism to address the available water resources. This paper developed a multi-criterion water resource management technique for defining the available water resources in the agriculture field. We have introduced a novel ranking method of generalized single-valued non-linear bipolar neutrosophic numbers based on possibility measures. Additionally, we have first defined the possibilistic mean of generalized single-valued non-linear bipolar neutrosophic number. Using the possibilistic mean, we have invented the positive and negative rank expositors (Sμ+,Sμ−) for authenticity, hesitate, and falsity membership function. In the present situation, water resource management in agriculture has become a big problem worldwide. Multi-criteria decision-making technique is necessary for developing this situation. Using the proposed multi-criteria decision-making method, we have solved a real water resource management problem in the Purulia district under GSVnTbN-environment. In this decision-making problem, we have considered the GSVnTbN-environment. Because during the summertime, water demand is very high, and at the end of summer, when monsoon season comes, water demand is deficient in the agriculture field of Purulia district. Sometimes the monsoon season, rainwater does not occur entirely. At that time, monsoon crop planting was tough in the agriculture field. So, this district’s water scarcity nature is a non-linear and uncertain type of nature. Therefore, we proposed an MCDM method for water resource management problems under GSVnTbN-environment.

On the relationship between possibilistic and standard moments of fuzzy numbers

In this paper we introduce a transformation of the center of gravity, variance and higher moments of fuzzy numbers into their possibilistic counterparts. We show that this transformation applied to the standard formulae for the computation of the center of gravity, variance, and higher moments of fuzzy numbers gives the same formulae for the computation of possibilistic moments of fuzzy numbers that were introduced by Carlsson and Fullér (2001) for the possibilistic mean and variance, and also the formulae for the calculation of higher possibilistic moments as presented by Saeidifar and Pasha (2009). We also present an inverse transformation to derive the formulae for standard measures of central tendency, dispersion, and higher moments of fuzzy numbers, from their possibilistic counterparts. This way a two-way transition between the standard and the possibilistic moments of fuzzy numbers is enabled. The transformation theorems are proven for a wide family of fuzzy numbers with continuous, piecewise monotonic membership functions. Fast computation formulae for the first four possibilistic moments of fuzzy numbers are also presented for linear fuzzy numbers, their concentrations and dilations.

A Hybrid Fuzzy-SCOOT Algorithm to Optimize Possibilistic Mean Semi-absolute Deviation Model for Optimal Portfolio Selection

A Hybrid Fuzzy-SCOOT Algorithm to Optimize Possibilistic Mean Semi-absolute Deviation Model for Optimal Portfolio Selection

Adjustable Security Proportions in the Fuzzy Portfolio Selection under Guaranteed Return Rates

Based on the concept of high returns as the preference to low returns, this study discusses the adjustable security proportion for excess investment and shortage investment based on the selected guaranteed return rates in a fuzzy environment, in which the return rates for selected securities are characterized by fuzzy variables. We suppose some securities are for excess investment because their return rates are higher than the guaranteed return rates, and the other securities whose return rates are lower than the guaranteed return rates are considered for shortage investment. Then, we solve the proposed expected fuzzy returns by the concept of possibility theory, where fuzzy returns are quantified by possibilistic mean and risks are measured by possibilistic variance, and then we use linear programming model to maximize the expected value of a portfolio’s return under investment risk constraints. Finally, we illustrate two numerical examples to show that the expected return rate under a lower guaranteed return rate is better than a higher guaranteed return rates in different levels of investment risks. In shortage investments, the investment proportion for the selected securities are almost zero under higher investment risks, whereas the portfolio is constructed from those securities in excess investments.

Possibilistic multiattribute decision making for water resource management problem under single‐valued bipolar neutrosophic environment

Single-valued bipolar neutrosophic (SVbN) numbers are a distinctive set of the neutrosophic set on the real line. In decision making problem, SVbN-number has a great importance when human decision is based on both positive and negative thought. All membership functions (truth, indeterminacy, and falsity) of SVbN number are contained the positive and negative parts within the range of −1 to 0 and 0 to 1. In this article, we have defined the possibilistic ranking method of single valued bipolar neutrosophic numbers. The notions of possibilistic mean and SD of SVbN number are introduced here. A new approach possibilistic multiattribute decision making is developed according to the possibilistic mean and SD. In addition, we have invented possibilistic multiattribute decision making on water resource management problem under bipolar neutrosophic environment. Finally, a numerical example is considered with single-valued trapezoidal bipolar neutrosophic informations to show the pertinence and implementation of the proposed possibilistic decision making method.

A fuzzy behavioral portfolio decision model with trapezoidal fuzzy return and aspiration

This paper deals with the stock portfolio selection problem involved with trapezoidal fuzzy number returns and multiple mental accounts. A fuzzy behavioral portfolio decision model is proposed to maximize the possibilistic mean value of portfolio return and ensure the portfolio return of each mental account exceeding the given minimum fuzzy aspiration level with a given probability. Then, some programming models are designed to solve the optimal portfolio strategy. Finally, one numerical example is given to illustrate the effectiveness of the proposed fuzzy behavioral portfolio decision approach.