Interval-valued Intuitionistic Fuzzy Values Research Articles

Multiattribute decision making using probability density functions and transformed decision matrices in interval-valued intuitionistic fuzzy environments

In this paper, we propose a new method for multiattribute decision making (MADM) using probability density functions and the transformed decision matrix (TDMx) of the decision matrix (DMx) offered by the decision maker (DM) in interval-valued intuitionistic fuzzy (IVIF) environments. First, it gets the TDMx of the DMx given by the DM. Then, it computes the average value of the interval-valued intuitionistic fuzzy values (IVIFVs) appearing at each column of the TDMx. Then, it calculates the variance of each IVIFV in the TDMx. Then, it computes the standard deviation (SD) of the IVIFVs appearing at each column of the TDMx. Then, based on the obtained TDMx, the obtained average value of the IVIFVs appearing at each column of the TDMx, and the obtained SD of the IVIFVs appearing at each column of the TDMx, it gets the z-score DMx. Then, each attribute’s IVIF weight is transformed into a crisp weight between zero and one. Finally, each alternative’s weighted score is calculated using the z-score DMx and each attribute’s transformed crisp weight. The larger the weighted score of an alternative, the better the preference order (PO) of the alternative. It can overcome the shortcomings of the existing MADM methods.

Interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets and their application in decision-making

Although some statistical tools, such as mean and median, used for modelling a problem containing parameters or alternatives with multiple intuitionistic fuzzy values because these values are obtained in a specific period, decrease uncertainty, they lead to data loss. However, interval-valued intuitionistic fuzzy values can overcome such a concern. For this reason, the present study proposes the concept of interval-valued intuitionistic fuzzy parameterized interval-valued intuitionistic fuzzy soft sets (d-sets) and presents several of its basic properties. Moreover, by using d-sets, we suggest a new soft decision-making method and apply it to a problem concerning the eligibility of candidates for two vacant positions in an online job advertisement. Since it is the first method proposed in relation to this structure (d-sets), it is impossible to compare this method with another in this sense. To deal with this difficulty, we introduce four new concepts, i.e. mean reduction, mean bireduction, mean bireduction-reduction, and mean reduction-bireduction. By using these concepts, we apply four state-of-the-art soft decision-making methods to the problem. We then compare the ranking performances of the proposed method with those of the four methods. Besides, we apply five methods to a real problem concerning performance-based value assignment to some filters used in image denoising and compare the ranking performances of these methods. Finally, we discuss d-sets and the proposed method for further research.

Loan approval evaluation framework of public-private partnership project of battery storage power station under interval-valued intuitionistic fuzzy environment

The main objective of this paper is to propose a loan approval evaluation framework of public-private partnership project (PPP) of battery storage power station (BSPS) for the commercial bank to evaluate and select the optimal project to provide the loan and then promote the development of BSPS-PPP. In order to achieve this goal, this study proposed a loan approval evaluation index system which consist of 5 attributes, 14 criteria and 45 sub-criteria. The novelty of this loan evaluation index system is that it reflects the basic requirement of loan approval of the commercial bank based on the characteristics of BSPS and PPP project, the commercial banks can better prepare for possible risks of BSPS-PPP. The proposed decision model was established based on the interval-valued intuitionistic fuzzy value (IVIFV) and interval number. Its novelty is that ①The model not only satisfies the experts’ expression habit of uncertainty, but also can be handled the uncertainty in the performance evaluation value and the criteria’s weights properly; ②the proposed weight determining method based on the deviation maximization principle can not only maintain the will of decision makers as much as possible, but also decrease the uncertainty of weights. Finally, the effectiveness of loan approval evaluation framework was verified based on a case study of China. According to the application effect of the evaluation index system and the proposed model, 3 managerial implication are put forward.

Measuring the Service Quality of Urban Rail Transit Based on Interval-Valued Intuitionistic Fuzzy Model

The service quality is a key driving factor of passengers’ intention of reusing the urban rail transit. Developing and using multi-criteria evaluation models to prioritize rail transit service quality is important for promoting sustainable development of urban rail transit. This paper involves a multi-attribute group decision making model which aims to assess the service quality for the urban rail transit, the assessment is mainly build on the interval-valued intuitionistic fuzzy value. We also establish the model to identify and confirm the optimal weights of attribute with the principle of maximum entropy. Taking Tianjin urban rail transit, which contains six lines, as an example, a comprehensive analysis for the service quality of the urban rail transit lines were conducted. The research shows that the line M3 owns the highest level of service quality, while the lowest service quality level is line M9. Meanwhile, the result shows that the process of decision making is not comparatively sensitive with the weight of passengers. The results can promote the operator of urban rail transit to identify their advantages and disadvantages in specific areas of passenger service and the improvement of service quality in the future.

Measuring the Service Quality of Urban Rail Transit Based on Interval-Valued Intuitionistic Fuzzy Model

The service quality is a key driving factor of passengers' intention of reusing the urban rail transit. Developing and using multi-criteria evaluation models to prioritize rail transit service quality is important for promoting sustainable development of urban rail transit. This paper involves a multi-attribute group decision making model which aims to assess the service quality for the urban rail transit, the assessment is mainly build on the interval-valued intuitionistic fuzzy value. We also establish the model to identify and confirm the optimal weights of attribute with the principle of maximum entropy. Taking Tianjin urban rail transit, which contains six lines, as an example, a comprehensive analysis for the service quality of the urban rail transit lines were conducted. The research shows that the line M3 owns the highest level of service quality, while the lowest service quality level is line M9. Meanwhile, the result shows that the process of decision making is not comparatively sensitive with the weight of passengers. The results can promote the operator of urban rail transit to identify their advantages and disadvantages in specific areas of passenger service and the improvement of service quality in the future.

Multiattribute decision making based on the binary connection number in set pair analysis under an interval-valued intuitionistic fuzzy set environment

A new multiattribute decision-making (MADM) methodology based on set pair analysis (SPA) for an interval-valued intuitionistic fuzzy set environment is developed in this paper. The connection number, which is known as a major component of SPA, provides a quantitative analysis to integrate the certainty and uncertainty as a combined system. First, we briefly review the concepts of interval-valued intuitionistic fuzzy sets (IVIFS) and the binary connection number (BCN). Then, the transformation method of interval-valued intuitionistic fuzzy numbers into BCNs is studied. Finally, we present a new MADM method where interval-valued intuitionistic fuzzy values are used to express evaluating values of alternatives on attributes, and weights are represented with real numbers or IVIFS. Some typical examples are presented to illustrate the feasibility and validity of the proposed approach.

Possibility degree and divergence degree based method for interval-valued intuitionistic fuzzy multi-attribute group decision making

This paper investigates an order relation of interval-valued intuitionistic fuzzy values (IVIFVs) based on possibility degree and divergence degree and applies it to multi-attribute group decision making (MAGDM) problems. Firstly, based on the possibility degree of intuitionistic fuzzy values, the possibility degree and divergence degree of IVIFVs are defined to provide a new order relation of IVIFVs. Subsequently in interval-valued intuitionistic fuzzy MAGDM, an intuitionistic fuzzy linear programming model is constructed to derive decision makers' weights. Depending on the construction of membership and non-membership functions, three approaches are developed to solve the constructed intuitionistic fuzzy linear programming model, including the optimistic, pessimistic and mixed approaches. Utilizing decision makers’ weights, the collective decision matrix is obtained. Combining the possibility and divergence degrees of alternatives on attributes, the adjusted possibility distribution matrix for each attribute is constructed and proved to be an interval fuzzy preference relation. Then the attribute weights are determined by the negative distances and the positive distances of adjusted possibility distribution matrices. Using the interval-valued intuitionistic weighted arithmetic aggregation operator, the collective comprehensive values of alternatives are obtained. The ranking order of alternatives is generated according to the proposed order relation of IVIFVs. Therefore, an interval-valued intuitionistic fuzzy MAGDM method is proposed. Finally, the proposed method is implemented on some examples. Validation discussion and comparative analyses of this method is also done to show the effectiveness of the proposed method.

Interval-valued intuitionistic fuzzy aggregation methodology for decision making with a prioritization of criteria

Interval-valued intuitionistic fuzzy sets (IVIFSs), a generalization of fuzzy sets, is characterized by an interval-valued membership function, an interval-valued non-membership function.The objective of this paper is to deal with criteria aggregation problems using IVIFSs where there exists a prioritization relationship over the criteria.Based on the ${L}$ukasiewicz triangular norm, we first propose a prioritized arithmetic mean to IVIF multi-criteria decision making (MCDM) problem where there is a linear ordering among the criteria.The proposed aggregation operator overcomes the existing prioritized aggregation operator's shortcomings that it is not monotone with respect to the total order on interval-valued intuitionistic fuzzy values (IVIFVs).We also prove that it is bounded and monotone with respect to the total order on IVIFVs, and therefore is a true generalization of such operations.We finally propose an aggregation operators-based two-step procedure to IVIF MCDM in the situation that more than one criteria exist at some priority level.

Interval-valued intuitionistic fuzzy multiple attribute decision making based on nonlinear programming methodology and TOPSIS method

In this paper, we propose a new multiple attribute decision making (MADM) method based on the nonlinear programming (NLP) methodology, the TOPSIS method and interval-valued intuitionistic fuzzy values (IVIFVs). The evaluating values of the alternatives with respect to attributes and the attributes’ weights are represented by IVIFVs. The NLP methodology is applied to get the optimal attributes’ weights. The proposed MADM method can overcome the drawbacks of the existing MADM methods to deal with MADM problems using IVIFVs.

Multiattribute decision making based on probability density functions and the variances and standard deviations of largest ranges of evaluating interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiattribute decision making (MADM) method based on probability density functions (PDFs) and the variances and standard deviations of the largest ranges of evaluating interval-valued intuitionistic fuzzy values (IVIFVs). First, the proposed MADM method gets the largest range of each evaluating IVIFV in the decision matrix and calculates the average value of the largest ranges of each attribute. Then, it gets the PDF of the largest range of each evaluating IVIFV in the decision matrix, calculates the variance of each largest range and calculates the standard deviation of the largest ranges of each attribute. Then, it constructs the z-score decision matrix and gets the transformed weight of the interval-valued intuitionistic fuzzy (IVIF) weight of each attribute. Finally, it calculates the weighted score of each alternative based on the obtained z-score decision matrix and the transformed weight of the IVIF weight of each attribute. The larger the value of the weighted score, the better the preference order (PO) of the alternative. The proposed MADM method can overcome the drawbacks of the existing MADM methods.

A Consensus Reaching Model with Minimum Adjustments in Interval-Valued Intuitionistic MAGDM

This paper focuses on multiattribute group decision-making problems with interval-valued intuitionistic fuzzy values (IVIFVs) and develops a consensus reaching model with minimum adjustments to improve the consensus among decision-makers (DMs). To check the consensus, a consensus index is introduced by measuring the distance between each decision matrix and the collective one. For the group decision-making with unacceptable consensus, Consensus Rule 1 and Consensus Rule 2 are, respectively, proposed by minimizing adjustment amounts of individual decision matrices. According to these two consensus rules, two algorithms are devised to help DMs reach acceptable consensus. Moreover, the convergences of algorithms are proved. To determine weights of attributes, an interval-valued intuitionistic fuzzy program is constructed by maximizing comprehensive values of alternatives. Finally, alternatives are ranked based on their comprehensive values. Thereby, a novel method is proposed to solve MAGDM with IVIFVs. At length, a numerical example is examined to illustrate the effectiveness of the proposed method.

Multiattribute decision making based on nonlinear programming methodology, particle swarm optimization techniques and interval-valued intuitionistic fuzzy values

In this paper, we propose a new multiattribute decision making (MADM) method by applying the nonlinear programming (NLP) methodology and particle swarm optimization (PSO) techniques using interval-valued intuitionistic fuzzy values (IVIFVs) to conquer the drawbacks of Chen and Huang's MADM method (2017), which has three drawbacks, i.e., (1) multiple different preference orders (POs) of alternatives are obtained in some situations, (2) the PO of alternatives cannot be distinguished in some circumstances, and (3) the PO of alternatives cannot be obtained in some circumstances. Moreover, the proposed MADM method also can conquer the shortcomings of Chen and Chiou's MADM method (2015), Li's MADM method (2010) and Zhitao and Yingjun's method (2011).

An improved MADM method using interval-valued intuitionistic fuzzy values

In this paper, we modify Chen and Han's multiattribute decision making (MADM) method (2018) to propose a novel MADM method based on the nonlinear programming (NLP) methodology, Xu's score function (SF) (2007) and Xu's accuracy function (AF) (2007) of interval-valued intuitionistic fuzzy values (IVIFVs), the membership uncertainty (MU) index (2009) and the hesitation uncertainty (HU) index (2009) of IVIFVs, and Wang et al.’s ranking method (2009) of IVIFVs. The proposed MADM method can conquer the shortcomings of Chen and Chiou's MADM method (2015), Chen and Han's MADM method (2018), Chen and Huang's MADM method (2017), Li's MADM method (2010) and Zhitao and Yingjun's MADM method (2011).

Multi-criteria decision making method based on interval-valued intuitionistic fuzzy sets

Multi-criteria group decision making is a widely used efficient decision methodology to improve quality of the decision. In this paper, the interval-valued intuitionistic fuzzy weighted arithmetic average operator, the interval-valued intuitionistic fuzzy weighted geometric average operator, and an accuracy function of interval-valued intuitionistic fuzzy value are introduced. The proposed aggregation operators with a accuracy function is more efficient to take decision. Finally, an example is provided to illustrate the application of the developed approach. The results show that the proposed new approach is more comprehensive and flexible by comparing with the other existing aggregation operators and accuracy functions.

IFS/ER-based large-scale multiattribute group decision-making method by considering expert knowledge structure

The large-scale multiattribute group decision-making (LMGDM) consisting of at least 20 experts has been widely popular in recent years. Although lots of efforts have been spent on improving the LMGDM, the subjective factors such as experts’ domain knowledges and bounded rationalities are still not well considered. This study focuses on providing a new LMGDM method by considering expert knowledge structure. An information extraction mechanism providing three kinds of inference ways including singleton attribute inference, local integral inference and global integral inference is introduced to ensure the assessments made by each expert with interval-valued intuitionistic fuzzy values (IVIFV) to be valid. Then a transformation is introduced to derive interval-valued basic probability assignment (BPA) function from the IVIFV, based on which expert reliability and attribute weight can be both reflected by evidential reasoning (ER) discounting. A pair of nonlinear optimization models that are extended by the analytical ER rule are established to make attribute fusion for expert and group fusion for alternative. An algorithm is summarized to solve the LMGDM problems by considering expert knowledge structure. Finally, an illustrative example as well as discussions is provided to demonstrate the applicability of the proposed method and algorithm.

A Novel Method for Interval-Value Intuitionistic Fuzzy Multicriteria Decision-Making Problems with Immediate Probabilities Based on OWA Distance Operators

The goal of this work is to develop a novel decision-making method which can solve some complex decision problems that include the following three-aspect information:(1)information represented in the form of interval-valued intuitionistic fuzzy values (IVIFVs) not only intuitionistic fuzzy values (IFVs),(2)the probability information and the weighted information, and(3)the importance degree of each concept in the process of decision-making. Firstly, by integrating OWA operator, probabilistic weight (PW), and individual distance of two IVIFNs in the same formulation, we introduce two new distance operators named PIVIFOWAD operator and IPIVIFOWAD operator, respectively. Secondly, satisfaction degree of an alternative is proposed based on the positive ideal IVIFS and the negative ideal IVIFS and applied to MCDM. Finally, we use an illustrative example to show the feasibility and validity of the new method by comparing with the other existing methods.

Multiattribute decision making based on Shannon's information entropy, non-linear programming methodology, and interval-valued intuitionistic fuzzy values

In this paper, we propose a novel multiattribute decision making (MADM) methodology based on Shannon's information entropy, the non-linear programming (NLP) methodology and interval-valued intuitionistic fuzzy values (IVIFVs), where attributes’ weights and evaluating attributes’ values with respect to alternatives are expressed by IVIFVs. Several examples are used to illustrate that the proposed MADM methodology can conquer the drawbacks of Wang and Chen's MADM methodology (2018) in interval-valued intuitionistic fuzzy (IVIF) environments.

Multi-attribute group decision making based on extended TOPSIS method under interval-valued intuitionistic fuzzy environment

In this paper, we propose a novel multi-attribute group decision making (MAGDM) method under interval-valued intuitionistic fuzzy environment by integrating extended TOPSIS and linear programming methods. We assume that multiple decision makers provide the input information, namely the assessment values and attribute weights using interval-valued intuitionistic fuzzy (IVIF) values for incorporating inherent uncertainty in the MAGDM process. Furthermore, the weights of decision makers given by experts to describe their importance in group decision making are assumed as IVIF values. The advantage and disadvantage scores are employed to determine the individual measure of importance for each decision maker. Aggregation of the assessment values and attributes weights are performed using two different IVIF aggregation operators. A weighted similarity measure, based upon the optimal attributes weights obtained through linear programming method, is defined for determining relative closeness coefficients for selection of the most preferred alternative. A real-world case study is provided to illustrate the working of the proposed methodology. Moreover, a thorough comparison has been done with related existing works in order to show the advantages of the methodology.

Multiattribute decision making based on non-linear programming methodology with hyperbolic function and interval-valued intuitionistic fuzzy values

In this paper, we present a novel multiattribute decision making (MADM) method based on the non-linear programming (NLP) methodology with the hyperbolic tangent function and interval-valued intuitionistic fuzzy values (IVIFVs). Both the attributes’ weights and the decision matrix (DM) are expressed by IVIFVs. First, the proposed MADM method constructs the transformed decision matrix (TDM) of the DM. Then, it constructs a NLP model with the hyperbolic tangent function to get the optimal weights of the attributes. Then, it uses the interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator to calculate the weighted evaluating IVIFVs of the alternatives. Finally, it uses Wang et al.’s method (2009) for comparing IVIFVs to obtain the preference order (PO) of the alternatives. It can conquer the shortcomings of Chen and Huang's MADM method (2017).

A new multiple attribute decision making method based on linear programming methodology and novel score function and novel accuracy function of interval-valued intuitionistic fuzzy values

Score functions and accuracy functions of interval-valued intuitionistic fuzzy values (IVIFVs) play important roles in dealing with multiple attribute decision making (MADM) problems in interval-valued intuitionistic fuzzy (IVIF) environments. In this paper, we propose a new MADM method using the linear programming (LP) methodology and the proposed new score function and the proposed new accuracy function of IVIFVs for overcoming the drawbacks of Wang and Chen's MADM method (2017), which has the drawbacks that the preference order (PO) of alternatives cannot be distinguished in some cases and it gets an infinite number of solutions of the optimal weights of attributes when the summation values of some columns in the transformed decision matrix (TDM) are the same, such that it obtains different POs of alternatives.