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Fuzzy numbers are often used to represent non-probabilistic uncertainty in engineering, decision-making and control system applications. In these applications, fuzzy arithmetic operations are frequently used for solving mathematical equations that contain fuzzy numbers. There are two approaches proposed in the literature for implementing fuzzy arithmetic operations: the α-cut approach and the extension principle approach using different t-norms. Computational methods for the implementation of fuzzy arithmetic operations in different applications are also proposed in the literature; these methods are usually developed for specific types of fuzzy numbers. This chapter discusses existing methods for implementing fuzzy arithmetic on triangular fuzzy numbers using both the α-cut approach and the extension principle approach using the min and drastic product t-norms. This chapter also presents novel computational methods for the implementation of fuzzy arithmetic on triangular fuzzy numbers using algebraic product and bounded difference t-norms. The applicability of the α-cut approach is limited because it tends to overestimate uncertainty, and the extension principle approach using the drastic product t-norm produces fuzzy numbers that are highly sensitive to changes in the input fuzzy numbers. The novel computational methods proposed in this chapter for implementing fuzzy arithmetic using algebraic product and bounded difference t-norms contribute to a more effective use of fuzzy arithmetic in construction applications. This chapter also presents an example of the application of fuzzy arithmetic operations to a construction problem. In addition, it discusses the effects of using different approaches for implementing fuzzy arithmetic operations in solving practical construction problems.

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 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.

The purpose of this paper is to study a nascent theory and an emerging concept of solving a fully fuzzy linear system (FFLS) with no non negative restrictions on the triangular fuzzy numbers chosen as parameters. Two new simplified computational methods are proposed to solve a FFLS without any sign restrictions. The first method eliminates the non‐negativity constraint from the coefficient matrix while the second method eliminates the constraint of non‐negativity on the solution vector. The methods are introduced with an objective to broaden the domain of fuzzy linear systems to encompass a wide range of problems occurring in reality.

The design of numerical methods is motivated by decomposing the fuzzy based linear system into its equivalent crisp linear form which can be further solved by variety of classical methods to solve a crisp linear system. Further the paper investigates Schur complement technique to solve the crisp equivalent of the FFLS.

The results that are obtained reveal interesting properties of a FFLS. By using the proposed methods, the authors are able to check the consistency of the fuzzy linear system as well as obtain the nature of obtained solutions, i.e. trivial, unique or infinite. Further it is also seen that an n×n FFLS may yield finitely many solutions which may not be entirely feasible (strong). Also the methods successfully remove the non‐negativity restriction on the coefficient matrix and the solution vector, respectively.

Evolving methods with better computational complexity and that which remove the non‐negativity restriction jointly on all the parameters are left as an open problem.

The proposed methods are new and conceptually simple to understand and apply in several scientific areas where fuzziness persists. The methods successfully remove several constraints that have been employed exhaustively by researchers and thus eventually tend to widen the breadth of applicability and usability of fuzzy linear models in real life situations. Heretofore, the usability of FFLS is largely dormant.

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.

Supply chain performance (SCP) extent can be attributed as a function of multiple criteria/attributes. Most of the criterions/attributes being intangible in nature; SCP appraisement relies on the subjective judgment of the decision makers. Moreover, quantitative appraisement of SCP appears to be very difficult due to involvement of ill-defined (vague) performance measures as well as metrics. The purpose of this paper is to develop an efficient decision support system (DSS) to facilitate SCP appraisement, benchmarking and related decision making.

This study explores the concept of fuzzy logic in order to tackle incomplete and inconsistent subjective judgment of the decision makers’ whilst evaluating supply chain’s overall performance. Grey relational analysis has been adopted in the later stage to derive appropriate ranking of alternative companies/enterprises (in the same industry) in view of ongoing SCP extent.

In this work, a performance appraisement index system has been postulated to gather evaluation information (weights and ratings) in relation to SCP measures and metrics. Combining the concepts of fuzzy set theory, entropy, ideal and grey relation analysis, a fuzzy grey relation method for SCP benchmarking problem has been presented. First, triangular fuzzy numbers and linguistic evaluation information characterized by triangular fuzzy numbers have been used to evaluate the importance weights of all criteria and the superiority of all alternatives vs various criteria above the alternative level. Then, the concept of entropy has been utilized to solve the adjusted integration weight of all objective criteria above the alternative level. Moreover, using the concept of the grey ration grades, various alternatives have been ranked accordingly.

Finally, an empirical example of selecting most appropriate company has been used to demonstrate the ease of applicability of the aforesaid approach. The study results showed that this method appears to be an effective means for tackling multi-criteria decision-making problems in uncertain environments. Empirical data have been analysed and results obtained thereof, have been reported to exhibit application potential of the said fuzzy grey relation based DSS in appropriate situation.

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.

The aim of this study is to show in detail the theoretical and practical foundations of a new feasibility technique for cash flow forecasting (CFF) based on triangular fuzzy numbers.

One of the most complicated problems business people face is determining if they have enough cash to be able to meet all future payments of a specific period. The uncertainty of forecasting the data to solve the problem suggests that a model based on fuzzy logic tools may provide a good way to obtain new techniques to ensure the feasibility of cash flow management.

This study shows how a specific company can obtain a quantitative idea of the risk of not being able to meet payments in a specific period. This idea can be put into practice with the usual computer tools.

This work presents a technique to predict the feasibility of CFF using triangular fuzzy numbers. There are other fuzzy numbers with which we can model the study problem and that offer certain advantages over to triangular ones.

A qualitative procedure is currently used to calculate the feasibility of a CFF. This work represents a step forward since it shows how to model quantitative feasibility.

The originality and value of this contribution consists of providing a complete model for a feasibility technique of CFF, as well as several proposals to mechanize the calculations and make the results more intuitive by means of spreadsheet graphs.

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.

Multi‐sensor data fusion (MSDF) is defined as the process of integrating information from multiple sources to produce the most specific and comprehensive unified data about an entity, activity or event. Multi‐sensor object recognition is one of the important technologies of MSDF. It has been widely applied in the fields of navigation, aviation, artificial intelligence, pattern recognition, fuzzy control, robot, and so on. Hence, aimed at the type recognition problem in which the characteristic values of object types and observations of sensors are in the form of triangular fuzzy numbers, the purpose of this paper is to propose a new fusion method from the viewpoint of decision‐making theory.

This work, first divides the comprehensive transaction process of sensor signal into two phases. Then, aimed at the type recognition problem, the paper gives the definition of similarity degree between two triangular fuzzy numbers. By solving the maximization optimization model, the vector of characteristic weights is objectively derived. A new fusion method is proposed according to the overall similarity degree.

The results of the experiments show that solving the maximization optimization model improves significantly the objectivity and accuracy of object recognition.

The paper studies the type recognition problem in which the characteristic values of object types and observations of sensors are in the form of triangular fuzzy numbers. By solving the maximization optimization model, the vector of characteristic weights is derived. A new fusion method is proposed. This method improves the objectivity and accuracy of object recognition.