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In this paper we deal with fuzzy numbers that modelize uncertain quantities present in many fields of applications, such as man‐machine systems. Main attention is paid to inverse operations for fuzzy numbers which allow one to solve equations or systems of equations with fuzzy numbers. The relevance of the method proposed for the determination of parameters of fuzzy models is also stressed.

The aim of this paper is to present a fuzzy centroid‐based method to ranking customer requirements with competition consideration. The proposed method not only focuses on normal fuzzy numbers, but also considers non‐normal fuzzy numbers to capture the true customer requirements.

This paper proposes a new customer requirements ranking method using QFD that not only focuses on the voice of the customer, but also considers the competitive environment. The method uses fuzzy mathematics instead of crisp numbers; this is known as the fuzzy centroid‐based method.

A numerical example demonstrates that if the fuzzy numbers are non‐normal, previous ranking methods were shown to be incorrect and to have led to some misapplications. To avoid possible further misapplications or spread in the future, the correct centroid formula used for fuzzy numbers is derived and their simplified expressions for non‐normal fuzzy numbers are given.

Various methods have been developed to rate and rank customer needs; however, few methods consider the competitive environment. In addition, in real applications, fuzzy mathematics are usually more appropriate than crisp models. Many previous methods are misleading and have led to some misapplications if the fuzzy numbers are non‐normal. The paper contributes to theory and practice by explaining the reasons for using the fuzzy centroid‐based method.

This study aims to propose a framework for designing appropriate triangular fuzzy numbers in a fuzzy analytic hierarchy process (FAHP) for selecting a road freight transport route and investigate the appropriate fuzzy numbers determined from a reliable consensus among participating experts based on the Delphi method.

The fundamental nine-point scale was investigated and separated into nonfuzzy intervals for importance assessment. A total of 17 individual experts participated in this study, and their consensus was that appropriate fuzzy numbers could be obtained using a three-round Delphi process. Moreover, the appropriate fuzzy numbers were used instead of the primary fuzzy numbers to calculate the relative weights of the decision criteria for road freight route selection.

The results confirmed that the Delphi method can be easily and rigorously applied to define appropriate fuzzy numbers. Further, this framework can serve as a guideline for a situation wherein the input of other fuzzy multiple criteria decision-making tools must be provided.

The FAHP has been widely used to address the imprecise assessments of decision-makers. However, most existing studies on the incorporation of such techniques have not defined fuzzy numbers, which are relevant to the problem of interest. This study contributes by incorporating the Delphi method that can design an appropriate fuzzy number for road freight route selection.

As the functions of environmental regulations cannot be quantified while assessing their environmental efficiency, there has been no comprehensive evaluation of environmental efficiency. The purpose of this paper is to evaluate environmental regulations based on triangular and trapezoidal fuzzy numbers.

This paper uses L-R fuzzy numbers to transform the evaluation language into triangular fuzzy numbers, and adopts an α-level flexible slacks-based measurement model to evaluate the performance of these regulations. Trapezoidal fuzzy numbers are combined with a data envelopment analysis model, and an α-slack-based measurement (SBM) model is used to evaluate the environmental efficiency. The α-SBM model is confirmed to be stable and sustainable.

Relevant index data from 16,375 enterprises were collected to test the proposed model, and models corresponding to triangular fuzzy numbers and trapezoidal fuzzy numbers were used to evaluate their environmental efficiency. Comparative results showed that the proposed model is feasible and stable.

The main contributions of this study are twofold. First, this paper provides a valuable evaluation method for environmental regulation. Second, our research improves the practical performance of trapezoidal fuzzy data envelopment analysis and enhances its feasibility and stability.

The purpose of this paper is to present one modification of the fuzzy TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and to develop a corresponding computer program which could be used for the multicriteria decision making for problems in practice.

This method is based on the uncertainties and probabilities of input data for ratings of alternatives with respect to criteria and weights of criteria that are presented by triangular fuzzy numbers as probabilistic fuzzy values. These input data are transformed in the procedure into output data that are relevant for the ranking of alternatives and decision making.

The proposed method is based on the generalized mean and spread of fuzzy numbers that are calculated according to probability of fuzzy events due to Zadeh. Ranking of alternatives for relevant criteria performs according to relative expected closeness, coefficient of variation and relative standard deviation of distance of alternatives to the ideal solutions. The most acceptable rule is related to the minimal value of the expected relative distance to positive ideal solution, especially when the coefficient of variation of distance to this solution is small. The attached example, related to a real project, confirms these findings.

This paper proposes three novel contributions in this area. Unlike the methods proposed by other authors, the weighted fuzzy decision matrix is expressed by the matrix of generalized expected values and matrix of generalized variances. To compute elements of these two matrices, exact formulae are derived and then the modified fuzzy TOPSIS procedure is carried out.

Supplier selection is a complex multiple criteria decision (MCDM) problem which directly depends on decision makers’ choice. Some decisions are getting involved with linguistic variables and they are not mathematically operable. To solve a typical decision problem through MCDM techniques, a number or a numerical interval should be defined. The purpose of this paper is to focus on that numerical interval and in a case of supplier selection, the aim is to close the decisions to the real number that the decision maker mentions and this number is in a numerical interval.

The proposed method deals with grey relational analysis (GRA) and develops it by applying triangular fuzzy numbers. The grey numbers have two defined bounds; the proposed method defines two fuzzy bounds for each grey attribute. In the proposed method, the fuzzy membership function has been employed for each bounds of grey attribute to make them to fuzzy bounds with two undefined bounds. Also to make comparison, with employing of TOPSIS technique, both of the grey fuzzy combination decision matrix and the original grey decision matrix are obtained.

The results indicate that, except to the ideal solutions, the grey relation coefficient for each alternative is too close to each other. Indeed, they are too close to zero. Applying the proposed method in problem of supplier selection shows the difference between two selected supplier in proposed method and the original grey method.

As mentioned heretofore this paper aims to make decision makers’s decision more accurate and actually there is no other researches which used this combination method.

A new method to solve zero‐sum two‐person games with imprecise values in their matrices of pay‐offs is suggested. The natural lack of precision generated by the use of fuzzy numbers in a fuzzy game requires the use of subjective criteria by the players in the resolution model. We apply a ranking function, the Average Value, which allows the decision makers to take into account their subjectivity. The use of this function raises again the solution of the fuzzy game when two criteria, one for each player, are used.

The purpose of this paper is to investigate the performance of a turbine structure of the oil and gas Egyptian company in terms of reliability, mean time to failure (MTTF), mean time to repair (MTTR) and mean time between failures (MTBF) under fuzzy environment and working criteria. This paper examines the impact of the failure of various components on the complete turbine structure of the oil and gas system.

To overcome the problem of uncertain behavior of available data for various components, the right triangular generalized fuzzy number (RTrGFN) is proposed to be taken into the account to express the uncertainty which attains some tolerance in data. Furthermore, reliability indices are calculated with the help of the Lambda Tau method and the arithmetic operations on right generalized triangular fuzzy numbers (RTrGFN).

This paper explores the reliability of a repairable 3 out of 4 structure of turbines and along with the other parameters namely MTTF, MTTR and MTBF; under a fuzzy environment. Failure rates and repair times are expected to be exponential. The ranking of components of the structure is being found to decide the priority for maintenance.

This paper investigates the performance of the system with different spread/tolerance like 15%, 25% and 50% of crisp data. It helps to predict realistic results in the range value. To enhance the system's performance, the most important item of the system requires greater attention. For this, the authors find the sensitive part by ranking. For ranking, an extended approach has been developed to find the sensitive unit of the system by using the right triangular generalized fuzzy number. This paper explores the most and least sensitive component of the system, which helps the maintenance department to plan the maintenance action.

Improvements in quality have a very important role to play in services because such improvements add greater flexibility and give clients greater confidence in the services provided. Therefore, this paper aims to describe a framework for measuring quality in the provision of the service by using a SERVQUAL scale and fuzzy operators.

In this approach, a framework to assess quality in service provision is put forward which applies the SERVQUAL scale as an instrument of data collection. Fuzzy set theory is proposed and applied within this framework as a technique to make a quantitative assessment of the quality of the provision of services. A case study is used to illustrate how to apply this framework for this purpose. The data were processed and transformed into a fuzzy environment, and fuzzy numbers and operators were applied to the analysis of customers' assessment with regard to the dimensions of quality in the service provided.

The study demonstrated the differences between evaluations of the dimensions of quality assessed, and differences of the same customer in relation to these dimensions. The main points raised were the importance given by customers to the dimensions, and the gaps between customers' perceptions and expectations, when fuzzy numbers were used to assess levels of service quality, and to evaluating the prioritization of service quality.

This paper proposes the use of fuzzy theory within a framework by making a linguistic analysis when dealing with data collected in a SERVQUAL scale so as to assess service quality. The data were measured by examining the gap between customers' perceptions and expectations. The framework describes the phases of this assessment, and uses fuzzy operators.

Risk assessment is a very important step toward managing risks in various organizations and industries. One of the most extensively applied risk assessment techniques is failure mode and effects analysis (FMEA). In this paper, a novel fuzzy multiple-criteria decision-making (MCDM)-based FMEA model is proposed for assessing the risks of different failure modes more accurately.

In this model, the weight of each failure mode is considered instead of risk priority number (RPN). Additionally, three criteria of time, cost and profit are added to the three previous risk factors of occurrence (O), severity (S) and detection (D). Furthermore, the weights of the mentioned criteria and the priority weights of the decision-makers calculated by modified fuzzy AHP and fuzzy weighted MULTIMOORA methods, respectively, are considered in the proposed model. A new ranking method of fuzzy numbers is also utilized in both proposed fuzzy MCDM methods.

To show the capability and usefulness of the suggested fuzzy MCDM-based FMEA model, Kerman Steel Industries Factory is considered as a case study. Moreover, a sensitivity analysis is conducted for validating the achieved results. Findings indicate that the proposed model is a beneficial and applicable tool for risk assessment.

To the best of authors’ knowledge, no research has considered the weights of failure modes, the weights of risk factors and the priority weights of decision-makers simultaneously in the FMEA method.