Generalized ATOVIC System Based on Triangular Fuzzy Numbers for Pattern Classification

The urge to remanufacture and address environmental concerns in various industrial processes has drawn the attention of academics as well as practitioners towards Closed-loop Supply Chain Networks (CLSC). Although everchanging and complex external factors including social and economic ones, adversely impact the sustainable development of closed-loop supply chain networks. The basic aim of the research is to optimize the functioning of CLSC networks. For the above-said, two objective functions are made. The first objective is to minimize the cost of production and assembly expenses of the forward and reverse logistics. Secondly, an endeavour has been made to reduce the fixed costs associated with plants and retailers. For the sake of achieving two objective functions, two methods are employed: triangular fuzzy numbers and triangular intuitionistic fuzzy numbers. Among the two methods, triangular intuitionistic fuzzy numbers achieved the said objectives with greater optimization substantiated by statistics. This method can deal with uncertain external factors without undermining the optimization of the CLSC networks.

The inventory problem associated with trade credit is a popular topic in which interest income and interest payments are important issues. Most studies related to trade credit assume that the interest rate is both fixed and predetermined. However, in the real market, many factors such as financial policy, monetary policy and inflation, may affect the interest rate. Moreover, within the environment of merchandise storage, some distinctive factors arise which ultimately affect the quality of products such as temperature, humidity, and storage equipment. Thus, the rate of interest charges, the rate of interest earned, and the deterioration rate in a real inventory problem may be fuzzy. In this paper, we deal with these three imprecise parameters in inventory modeling by utilizing the fuzzy set theory. We develop the fuzzy inventory model based on Chang et al.'s [1] model by fuzzifying the rate of interest charges, the rate of interest earned, and the deterioration rate into the triangular fuzzy number. Subsequently, we discuss how to determine the optimal ordering policy so that the total relevant inventory cost, in the fuzzy sense, is minimal. Furthermore, we show that Chang et al.'s [1] model (the crisp model) is a special case of our model (the fuzzy model). Finally, numerical examples are provided to illustrate these results.

The fuzzy decision-making trial and evaluation laboratory (fuzzy DEMATEL) has been used to solve various multi-criteria group decision-making problems where triangular type-1 fuzzy sets are utilized in defining decision makers’ linguistic evaluation. Most of the fuzzy DEMATEL modifications are built from linguistic variables based on type-1 fuzzy sets (T1FS). Previous literature suggests that interval type-2 fuzzy sets (IT2FS) can offer an alternative that can handle vagueness and uncertainty. This paper proposes a modification fuzzy DEMATEL characterized by IT2FS for linguistic variables. Differently from the typical fuzzy DEMATEL which directly utilizes triangular type-1 fuzzy numbers, this modification introduces trapezoidal IT2 fuzzy numbers to enhance evaluation in the group decision-making environment. This new modification includes linguistic variables expressed by IT2FS and an expected value method for normalizing upper and lower memberships of IT2FS to crisp numbers. The proposed modification is applied to a case of knowledge management (KM) where eleven criteria are considered. Three experts in KM were invited to provide linguistic judgments with respect to the criteria, and the eight-step computational procedure of the proposed modification was implemented without losing the originality of the DEMATEL method. The results unveiled that ‘trust’ is the most influential criteria in KM. Therefore, trust is a phenomenon that impacts on the success of KM. Comparable results are also presented to check the feasibility of the proposed method. It is shown that the criteria weight and the causal relationship of criteria using the proposed method are consistent with the other two methods.

The fuzzy decision-making trial and evaluation laboratory (fuzzy DEMATEL) has been used to solve various multi-criteria group decision-making problems where triangular fuzzy numbers are utilized in defining decision makers’ linguistic judgements. Most of the fuzzy DEMATEL modifications are built from linguistic variables based on fuzzy sets. Recent literature suggests that Pythagorean fuzzy sets (PFS) can offer a better alternative particularly when fuzzy sets have some extent of limitations in handling vagueness and uncertainty. This paper proposes a modification fuzzy DEMATEL characterized by PFS for linguistic variables. Differently from the typical fuzzy DEMATEL which directly utilizes triangular fuzzy numbers with a single membership, this modification introduces membership and non-membership of PFS to enhance judgements in the group decision-making environment. The proposed method has a number of attractive features. It includes linguistic variables, expert’s weights, and score function, in which all of these features are expressed by PFS. The proposed modification is applied to a case of solid waste management (SWM) where ten criteria are considered in assessment. Six experts in SWM were invited to provide linguistic judgments with respect to the criteria, and the eleven-step computational procedure of the proposed method was implemented without losing the general structure of the DEMATEL method. The results unveiled that four criteria are identified as ‘cause group’ and six criteria are identified as ‘effect group’ in SWM. The grouping of criteria would help policy makers in identifying the criteria that could enhance the efficiency of SWM.

Solution of linear differential equations with fuzzy boundary values

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.

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.

Using fuzzy number for measuring quality of service in the hotel industry

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.

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.

Failure mode and effects analysis (FMEA) is one of the most popular reliability analysis tools for identifying, assessing and eliminating potential failure modes in a wide range of industries. In general, failure modes in FMEA are evaluated and ranked through the risk priority number (RPN), which is obtained by the multiplication of crisp values of the risk factors, such as the occurrence (O), severity (S), and detection (D) of each failure mode. However, the crisp RPN method has been criticized to have several deficiencies. In this paper, linguistic variables, expressed in Gaussian, trapezoidal or triangular fuzzy numbers, are used to assess the ratings and weights for the risk factors S, O and D. A new risk assessment system based on the fuzzy set theory and fuzzy rule base theory is to be applied to assess and rank risks associated to failure modes that could appear in the functioning of Turn 55 Lathe CNC. Two case studies have been shown to demonstrate the methodology thus developed. It is illustrated a parallel between the results obtained by the traditional method and fuzzy logic for determining the RPNs. The results show that the proposed approach can reduce duplicated RPN numbers and get a more accurate, reasonable risk assessment. As a result, the stability of product and process can be assured.

The nonlinear equation is a fundamentally important area of study in mathematics, and the numerical solutions of the nonlinear equations are also an important part of it. Fuzzy sets introduced by Zedeh are an extension of classical sets, which have several applications in engineering, medicine, economics, finance, artificial intelligence, decision-making, and so on. The most special types of fuzzy sets are fuzzy numbers. The important fuzzy numbers are trapezoidal fuzzy and triangular fuzzy numbers, which have several applications. In this research article, we propose an efficient numerical iterative method for estimating roots of fuzzy nonlinear equations, which are based on the special type of fuzzy number called triangular fuzzy number. Convergence analysis proves that the order of convergence of the numerical method is three. Some real-life applications are considered as numerical test problems from engineering, which contain fuzzy quantities in the parametric form. Engineering models include fractional conversion of nitrogen-hydrogen feed into ammonia and Van der Waal’s equation for calculating the volume and pressure of a gas and motion of the object under constant force of gravity. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in the literature.

Under electronic commerce, how to raise the consumers' degree of satisfaction and gain the consumers' loyalty have become the key factor relating with whether e-commerce enterprise can survive, so it's vital to evaluate status of customer satisfaction for B2C electronic-commerce enterprise. According to the investigation result by Internet, this paper brings forward the indicator system of customer satisfaction evaluation for B2C electronic-commerce enterprise on the basis of current study of home and oversea and the related reference, and establishes the performance evaluation model based on combination of linguistic variables and fuzzy triangular numbers, and also do some example research. The examples demonstrated that: evaluation model based on combination of linguistic variables and fuzzy triangular numbers can do well in evaluation.

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 this paper, we attempt to introduce an algorithm which considers not only time factor but also cost, risk, and quality criteria to determine the critical path under fuzzy environment. In this algorithm first, decision makers allocate time, cost, risk, and quality to each activity. We are lacking in data and information, so in the proposed algorithm, the ratings of each activity and the weight of each criterion are described by fuzzy numbers and linguistic variables, which can be expressed in triangular fuzzy number. Linguistic variables are applied to represent the intensity of preferences of one criterion over another. Then, we add up triangular fuzzy numbers to determine the final evaluation value of each criterion for paths. Next, we use fuzzy TOPSIS, a technique for order preferences by similarity to an ideal solution, a method proposed by the authors in another paper, to choose the best alternative. Finally, numerical example is solved to illustrate the procedure of proposed method at the end of this paper.