The Expected Value Defuzzification Method for Pentagonal Fuzzy Number to Solve a Carbon Cost Integrated Solid Transportation Problem

In any decision-making process, imprecision is a significant issue.To deal with the ambiguous environment of collective decision-making, various tools and approaches have been created.Fuzzy set theory is one of the most recent approaches for coping with imprecision.The Fuzzy Transportation Problem (FTP) is a well-known network planned linear programming problem which exists in a variety of situations and has received a lot of attention recently.Many authors defined and solved the fuzzy transportation problem with frequently utilized fuzzy numbers such as triangular fuzzy numbers or trapezoidal fuzzy numbers.On the other hand, real-world problems usually involve more than four variables.To tackle these concerns, the pentagonal fuzzy number is applied to the problems.This article proposes an approach to solving transportation problems whose parameters are pentagonal fuzzy numbers without requiring an initial feasible solution.An algorithm based on the core and spread method and an extended MODI method is developed to determine the optimal solution to the problem.The proposed process is based on the approximation method and gives a more efficient result.An illustrated example is used to validate the model.As a result, the proposed methodology is both simpler and more computationally efficient than the existing approaches.

The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.The main objective of this study is to deal with the optimization of a fuzzy transportation problem having costs in fuzzy triangular numbers with demand and supplies as crisp numbers. It introduces a new ranking technique along with a unique approach to convert triangular fuzzy number to trapezoidal fuzzy number. Minimum demand supply algorithm is used to get optimal cost. This technique is compared with some existing methods. Two different ranking approaches based on trapezoidal fuzzy numbers are also compared. The new algorithms are found fast and better as compared to classical approaches.

The main proposal of this paper is to derive two different reduction process for a trapezoidal type-2 fuzzy number. The first reduction method is based on critical values and the second method is based on α-cut of fuzzy number. As an application a multi-objective solid transportation problem, minimizing the cost and time has been developed using trapezoidal type-2 fuzzy number as system parameters and hereby solved. Finally after solving the proposed multi-objective problem by intuitionistic fuzzy programming technique a comparison between the two proposed reduction methods are discussed briefly. The proposed models and techniques are finally illustrated by providing numerical examples at the end. Also this paper present a comparative study between the proposed method to the KM algorithm and NT method for type reduction.

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

System of simultaneous linear equations plays a vital role in mathematics, Operations Research, Statistics, Physics, Engineering and Social Sciences etc. In many applications at least some of the system's parameters and measurements are represented by fuzzy numbers rather than crisp numbers. Therefore it is imperative to develop mathematical models and numerical procedures to solve such a fuzzy linear system. The general model of a fuzzy linear system whose coefficient matrix is crisp and the right hand side column is an arbitrary fuzzy vector. In the fully fuzzy linear system all the parameters are considered to be fuzzy numbers. Since triangular fuzzy numbers is a special case of trapezoidal fuzzy numbers, hence in this paper we considered fully fuzzy linear system with trapezoidal fuzzy numbers. LU decomposition method for a crisp matrix is well known in solving linear system of equations. We discuss LU decomposition of the coefficient matrix of the fully fuzzy linear system, in which the coefficients are trapezoidal fuzzy numbers.

In conventional transportation problem (TP), supplies, demands and costs are always certain. In this paper, the author tried to categories the TP under the mixture of certain and uncertain environment and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) to solve the TP. The existing ranking procedure of Liou and Wang is used to transform the type-1 and type-3 fuzzy transportation problem (FTP) into a crisp one so that the conventional method may be applied to solve the TP. The solution procedure differs from TP to type-1 and type-3 FTP in allocation step only. Therefore, the new method called PSK method and new multiplication operation on TrFN is proposed to find the mixed optimal solution in terms of crisp numbers, TFNs and TrFNs. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. The effectiveness of the proposed method is illustrated by means of a numerical example.

In this paper, we forecasted the future of a business by using fuzzy trapezoidal numbers and the fuzzy Delphi method. This result is compared with another result obtained using the fuzzy triangular numbers and Delphi method. At last, we see that our method is more general than others. J. Bangladesh Acad. Sci. 45(2); 147-154: December 2021

Purpose – In this context, process capability indices (PCI) reveal the process zones base on specification limits (SLs). Most of the research on control charts assumed certain data. However, to measure quality characteristic, practitioners sometimes face with uncertain and linguistic variables. Fuzzy theory is one of the most applicable tools which academia has employed to deal with uncertainty. The paper aims to discuss these issues. Design/methodology/approach – In this investigation, first, fuzzy and S control chart has been developed and second, the fuzzy formulation of the PCIs such as C pm ,C pmu ,C pml , C pmk , P p , P pl , P pu , P pk are constructed when SLs and measurements are at both triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) stages. Findings – The results show that using fuzzy make more flexibility and sense on recognition of out-of-control warnings. Research limitations/implications – For further research, the PCIs for non-normal data can be conducted based on TFN and TrFN. Practical implications – The application case is related to a piston company in Konya’s industry area. Originality/value – In the previous researches, for calculating C p , C pk , C pm and C pmk indices, the base approach was calculate standard deviation for a short term variation. For calculating these indices, the variation between subgroups are being ignored. Therefore, P p and P pk indices solved this fault by mentioning long term and short term variations. Therefore these two indices calculate the actual process capability.

Ranking Fuzzy Numbers with an Area Method Using Circumcenter of Centroids

Classical Mean-Variance model pioneered by Markowitz 1952 has proposed the model to be one of the most important researches in modern finance. However the classical Markowitz’s Mean-Variance method does not match to represent the practicality of the uncertainties in stock market problem in portfolios. Seven types of extended mean-variance method of portfolio selection model based on the possibilitic return and possibilistic risk is presented in this paper. The returns of assets obey trapezoidal and pentagonal fuzzy numbers. The objective of this paper is to compare the optimal portfolio composition and performance using trapezoidal, pentagonal fuzzy numbers by considering whether the data is assumed normal, using real distribution and follow the trading day situations. A numerical example is presented to illustrate the usability of the model. The result will provide on how the investors could improve on their investment strategy.

The ranking of fuzzy numbers has an important position in fuzzy application models and systems. In this paper, a two-dimensional approach is proposed by the Monte Carlo simulation technique, and the simulation formula of the possibility degree is established. The algorithms for ranking interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers are elaborated on. Moreover, the axiomatic properties of the possibility degree formulae are investigated. It is found that the simulation formula of the possibility degree is reasonable for the satisfaction of the considered axiomatic properties. Finally, by considering a set of fuzzy numbers and the effect of a viewpoint, some comparisons are reported by carrying out numerical examples. The observations reveal that the proposed simulation method is effective to reflect the uncertainty of fuzzy numbers and coherent with the human intuition.

A fuzzy number is an important concept for handling vague, imprecise and ambiguous cases in daily life. We propose to study different types of fuzzy numbers with examples and some results on fuzzy numbers. In biological science, the growth/decay of bacteria is a very important factor in knowing the symptoms/diseases in humans and animals. The decagonal fuzzy number has proved important in studying bacteria, particularly the growth rate in bacteria, which consists of 10 points, so it is not easy to handle this situation by applying triangular or trapezoidal fuzzy numbers. Therefore, we plan to study decagonal fuzzy numbers in detail here.

A fuzzy analogue of the binomial option pricing tree model introduced by Cox et.al [1] was proposed by Yoshida [2] to study fuzzy American put option model in an uncertain environment using discrete time fuzzy stochastic process. Later, Muzzioli [3] used non overlapping triangular and trapezoidal fuzzy numbers to model the jump factors in American put option pricing model admitting impreciseness only in volatility while Xcaojian Yu [4] persumed impreciseness in both risk-free interest rate and volatility of the underlying stock involving non-overlapping trapezoidal fuzzy numbers. K. Meenakshi et al. [5] defined new fuzzy risk-neutral probability measures using general trapezoidal fuzzy numbers to study "Problem of Pricing American Fuzzy Put Option Buyer’s Model". The two up and down jump factors and the risk-free interest rate fluctuates oftenly and hence are uncertain in nature. When financial investors come across a high volatile or low volatile (up and down jump factors) fuzzy stocks, non-overlapping type of fuzzy numbers will not be sufficient to predict the underlying fuzzy stock prices as the fuzzy stock prices would go up only in the up state and would go down only in the down state of the fuzzy binomial tree. The uncertainty associated in the above stated fuzzy option pricing parameters could not be captured completely here. Such a scenario needs attention. To handle such a situation, we need to consider fuzzy numbers which are not only non - overlapping but partially or fully overlapping and/ or contained in too. Also not much consideration had been given to American fuzzy put option model involving fuzzy martingales. This paper will consider the study of the fuzzy analogue of martingale pricig theory in the context of fuzzy option pricing theory. In this study, we discuss American Fuzzy Put Option Seller’s Model (AFPOSM) based on fuzzy future contract involving general linear octagonal fuzzy numbers (GLOFN); general in the sense that they could be either overlapping or non-overlapping partially or completely and/or contained in using the fuzzy risk-neutral probability measures introduced by us. We record a computational procedure to obtain the fuzzy profit and loss (PL) values of sellers using two-period fuzzy binomial tree model wherein the fuzzy stock price and the fuzzy future price following discrete time fuzzy stochastic process. The same is performed by deploying the real stock market data obtained from the website [6] that includes American style Microsoft Corporation (MSFT) shares of the year 2018 under the bullish (the constant initial fuzzy stock price goes above the constant fuzzy strike price) scenario. Utilizing the PL values, we estimate an optimal exercise time and an optimal exercise price of sellers. Also the discounted fuzzy intrinsic values of AFPOSM on fuzzy future contract are validated. The computations carried out in AFPOSM are performed using MATLAB 2016a software.    

Ranking fuzzy numbers with integral value

Fuzzy inventory with backorder for fuzzy total demand based on interval-valued fuzzy set