The classical Analytical Hierarchy Process (AHP) has two limitations. Firstly, it disregards the aspect of uncertainty that usually embedded in the data or information expressed by human. Secondly, it ignores the aspect of interdependencies among attributes during aggregation. The application of fuzzynumbers aids in confronting the former issue whereas, the usage of Choquet Integral operator helps in dealing with the later issue. However, the application of fuzzynumbers into multi-attributedecisionmaking (MADM) demands some additional steps and inputs from decision maker(s). Similarly, identification of monotone measure weights prior to employing Choquet Integral requires huge number of computational steps and amount of inputs from decision makers, especially with the increasing number of attributes. Therefore, this research proposed a MADM procedure which able to reduce the number of computational steps and amount of information required from the decision makers when dealing with these two aspects simultaneously. To attain primary goal of this research, five phases were executed. First, the concept of fuzzy set theory and its application in AHP were investigated. Second, an analysis on the aggregation operators was conducted. Third, the investigation was narrowed on Choquet Integral and its associate monotone measure. Subsequently, the proposed procedure was developed with the convergence of five major components namely Factor Analysis, Fuzzy-Linguistic Estimator, Choquet Integral, Mikhailov‘s Fuzzy AHP, and Simple Weighted Average. Finally, the feasibility of the proposed procedure was verified by solving a real MADM problem where the image of three stores located in Sabak Bernam, Selangor, Malaysia was analysed from the homemakers‘ perspective. This research has a potential in motivating more decision makers to simultaneously include uncertainties in human‘s data and interdependencies among attributes when solving any MADM problems.
Abstract: Q-rung orthopair fuzzy set (q-ROFS) is a powerful tool to describe uncertain information in the process of subjective decision-making, but not express vast objective phenomenons that obey normal distribution. For this situation, by combining the q-ROFS with the normal fuzzy number, we proposed a new concept of q-rung orthopair normal fuzzy (q-RONF) set. Firstly, we defined the conception, the operational laws, score function, and accuracy function of q-RONF set. Secondly, we presented some new aggregation operators to aggregate the q-RONF information, including the q-RONF weighted operators, the q-RONF ordered weighted operators, the q-RONF hybrid operator, and the generalized form of these operators. Furthermore, we discussed some desirable properties of the above operators, such as monotonicity, commutativity, and idempotency. Meanwhile, we applied the proposed operators to the multi-attributedecision-making (MADM) problem and established a novel MADM method. Finally, the proposed MADM method was applied in a numerical example on enterprise partner selection, the numerical result showed the proposed method can effectively handle the objective phenomena with obeying normal distribution and complicated fuzzy information, and has high practicality. The results of comparative and sensitive analysis indicated that our proposed method based on q-RONF aggregation operators over existing methods have stronger information aggregation ability, and are more suitable and flexible for MADM problems.
The Intuitionistic fuzzy dominance matrices are mainly useful in such situations where decision makers are able to express their opinions about all the attributes in terms of fuzzy value. Insimple way,when there is no missing or unknown information,Intuitionistic fuzzy dominance matrices is proved to be more effective. This study has introduced Intuitionistic fuzzy dominance matrices for solving multiattributedecisionmaking problems in uncertain environment.We utilized one of them through a new solution procedure to solve real life decision problem which will involve more number of decision maker.
Atanassov introduce the concept of intuitionistic fuzzy set (AIFS) , which is a generalization of the concept of fuzzy set (FS) . Gau and Buehrer  introduced the concept of vague set, but Bustince and Burillo  showed that vague sets are AIFSs. Atanassov and Gargov  further generalized the AIFSs in the spirit of ordinary interval-valued fuzzy sets (IVFSs) and deﬁned the notion of an interval-valued intuitionistic fuzzy set (IVIFS), which is characterized by a membership function and a non-membership function whose values are intervals rather than exact numbers. Current research mainly focuses on basic operations and relations of IVIFSs as well as their properties . The correlation coeﬃcients of IVIFSs were systematically investigated from diﬀerent points of view [7–9]. Other aspects of IVIFSs were also investigated, such as topological properties , relationships between AIFSs, L-fuzzy sets, IVFSs and IVIFSs [11–13], and the distance of IVIFSs [14–16].
IFDM is mainly useful in such situations where decision makers are able to express their opinions about all the attributes in terms of fuzzy value. In simple way, when there is no missing or unknown information, IFDM is proved to be more effective. This study has introduced intuitionistic fuzzy dominance matrix for solving MADM problems in uncertain environment. Researchers can develop more efficient decisionmaking algorithm using IFDM. Intuitionistic rough set and vague soft set based real life decisionmaking problems which may contain more than one decision maker and to realize this procedure we also apply it to more relevant.
containing triangular fuzzynumbers, and designed the group information aggregation optimized model based on the two targets of individual evaluation value and group evaluation value, i.e., optimal distance and high similarity. Finally, he proposed an expansion VIKOR method. Huang and Luo  proposed the index weight measurement based on the triangular fuzzy number comparison possibility relation by referring to the related thoughts of maximized empowerment algorithm and the triangular fuzzy number comparison possibility theory, to solve the problem with UMADM with triangular fuzzy number as the index value. In addition, a comparison possibility degree relation method of triangular fuzzy number type UMADM is proposed to judge the advantages and disadvantages and order of the decisionmaking scheme set. By comprehensive analysis of various literatures, the researches of various scholars mainly focus on the problems with triangular fuzzy number multi-attributedecisionmaking in a single period. There have been only a few researches focusing on the problems with triangular fuzzy number multi-attributedecisionmaking of multi-period or multi-stage. In practical application, for many complicated decisionmaking problems, the original decisionmaking information of various periods should be taken into consideration. For instance, the decisionmaking information for multi-period investment decisionmaking, and validity dynamics of medical diagnosis and military system should include the time dimension. For this reason, researches of dynamic multi-attributedecisionmaking problems are of great theoretical significance and practical application value, and would hopefully become another hot research topic in the decisionmaking science field.
In 2013, Cuong and Kreinovich defined picture fuzzy set (PFS) which is a direct extension of fuzzy set (FS) and intuitionistic fuzzy set (IFS). Wang et al. (2014) proposed intuitionistic linguistic number (ILN) as a combination of IFS and linguistic approach. Motivated by PFS and linguistic approach, this paper introduces the concept of picture linguistic number (PLN), which constitutes a generalization of ILN for picture circumstances. For multi-criteria group decisionmaking (MCGDM) problems with picture linguistic information, we define a score index and two accuracy indexes of PLNs, and propose an approach to the comparison between two PLNs. Simultaneously, some operation laws for PLNs are defined and the related properties are studied. Further, some aggregation operations are developed: picture linguistic arithmetic averaging (PLAA), picture linguistic weighted arithmetic averaging (PLWAA), picture linguistic ordered weighted averaging (PLOWA) and picture linguistic hybrid averaging (PLHA) operators. Finally, based on the PLWAA and PLHA operators, we propose an approach to handle MCGDM under PLN environment.
the prospect value of each alternative. Liu et al.  proposed a multi-attributedecisionmaking method based on prospect theory, with respect to risk decisionmaking problems with interval probability in which the attribute values take the form of uncertain linguistic variables. In this method, the uncertain linguistic variables are converted to trapezoidal fuzzynumbers, and the prospect value function of the trapezoidal fuzzynumbers, based on the decision-making reference point of each attribute and the weight function of interval probability, are obtained. The prospect value of attribute for every alternative is calculated through prospect value function and the weight function; the weighted prospect value of alternative is get by using weighted average method, and all the alternatives are ranked by the expected values of the weighted prospect values. Zhang and Fan  proposed a method based on the prospect theory to solve risky multiple attributedecisionmaking problems with Decision Maker (DM's) aspirations, where attribute values and probabilities are both in the form of interval numbers. In this method, aspirations are regarded as reference points. Jiang and Cheng  proposed a decision-making analysis approach based on prospect theory to select the most desirable alternative from the available set of new product development alternatives. In this method, the prospect reference point is determined by considering the evaluation information of competing product alternatives. Hu et al.  proposed a method based on cumulative prospect theory and set pair analysis for dynamic stochastic multi-criteria decisionmaking problems in which criteria weight is unknown and criteria values are in the form of discrete random variables. Firstly, the prospect value of alternatives is calculated on all criteria at dierent periods, according to the distribution function. Then, the time series weight is derived based on the binomial distribution probability density function, and the criteria weight coecients are ascertained by the algorithm of max- imizing deviation. Finally, the concepts of identity degree, contrary degree, and set pair potential are employed and, thus, the order of alternatives can be consequently determined.
Α are called Interval Intuitionistic FuzzyNumbers (IIFNs), each of which interval of membership degree and interval of non- membership degree consist. Let the general form of IIFN shortly denoted as ( [ ] [ ] a b , , , c d ) , where
presented to rank alternatives. In this paper, we propose a method for the FMADM problem that the attribute weights are interval numbers and DM has avail preference information on alternatives is investigated. a quadratic programming model based on the minimum sum of deviation squares between the subjective and objective decision-making preference information on alternatives is firstly established to determine the attribute weights, then the overall values of alternatives are gained. Then the FPIS and FNIS of alternatives aer introduced. Based on fuzzy sets theory, the relative membership degree to which an alternative corresponds to the best alternative is obtained by means of the synthetically weighted distance between the overall values of every alternative and the ideal solution, the alternatives are ranked by using the relative membership. A practical example is lastly illustrated to show the feasibility and availability of the developed method.
Abstract. An Interval-Valued Trapezoidal Intuitionistic Fuzzy Number (IVTrIFN) is a special case of an Intuitionistic Fuzzy Set (IFS), which is dened on a real number set. From a geometric viewpoint, the expectation and expectant score of an IVTrIFN are dened using the notion of a barycenter, and a new method is developed to rank IVTrIFNs. Hereby, some generalized aggregation operators of IVTrIFNs are dened, including the generalized ordered weighted averaging operator and the generalized hybrid weighted averaging operator, which are employed to solve multi-attribute group decisionmaking problems. Using the weighted average operator of IVTrIFNs, the attribute values of alternatives are integrated into the individual comprehensive ratings, which are further aggregated into the collective one by the generalized hybrid weighted averaging operator of IVTrIFNs. The ranking orders of alternatives are then generated according to the expectation and expectant score of the collective comprehensive ratings of alternatives. A numerical example is examined to demonstrate the applicability and implementation process of the decision method proposed in this paper.
Despite a lot of methods have been developed to deal with interval and fuzzy MADM unfortunately, today research on multi-attributedecisionmaking under Z-information is scarce. The mentioned above dictated to create new approach for MADM under decision situation where decision relevant information are characterized by fuzzy uncertainty and partial reliability. For this purpose we have suggested MADM procedure based on overall criteria positive ideal and negative ideal solutions of alternatives, distance between Z-vectors and Z-information processing. Numerical example on MADM Web services selection problem demonstrates applicability and efficiency of proposed method.
A good amount of research work has been carried out in recent past for robot selection, but still it is a new concept in industry at large, so it is not unusual for an industry to be a first time robot purchaser. Knott and Getto (1982) suggested a model to evaluate different robotic systems under uncertainty, and different alternatives were evaluated by computing the total net present values of cash flows of investment, labor components, and overheads. Liang and Wang (1993) proposed a robot selection algorithm by combing the concepts of fuzzy set theory and hierarchical structure analysis. The algorithm was used to aggregate decision makers‟ fuzzy assessments about robot selection attribute .Agrawal et al. (1991) proposed a robot selection procedure to rank the alternatives in a shortlist by employing TOPSIS (technique for order preference by similarity to ideal solution) method. An expert system was also developed to assist the decision maker to establish priorities and visualize the selection process at various stages. Zhao et al. (1996) have introduced genetic algorithm (GA) for optimal Robot Selection problem for a CIM system. Khouja and Kumar (1999) used options theory and an investment evaluation procedure for selection of robots weightings, and to obtain fuzzy suitability indices. Bhangale et al. (2004) had used TOPSIS and graphical method for the selection of a robot for some pick-n-place operation.Bhattacharya et al. (2005) integrated AHP and quality function deployment (QFD) methods for solving industrial robot selection problems, while considering seven technical requirements and four alternative robots.
As the theoretical foundation of fuzzy mathematics, fuzzy measure and fuzzy integral were originally deﬁned by Choquet  in 1954. Diﬀerent from classic integral, fuzzy integral focus on non-additive cases. Based on the theory of classic mathematic, we can prove the fact that when Choquet integral with the condition of additive, Choquet integral is reduced to Lebesgue integral. Compared with sugeno integral, Choquet integral has better mathematical properties and its theoretical basis is more complete. Hence, Choquet integral is more suited to deal with the problem of fuzzy quantitative. In recent years, many scholars have done a lot of good researches in this ﬁeld and applied Choquet integral in many aspects, especially in multiple attributedecision-making problems. Xu  used Choquet integral to propose some intuitionistic fuzzy aggregation operators, extended interval value fuzzy operators under various fuzzy environment, and then applied them in intuitionistic fuzzydecisionmaking. Tan  used Choquet integral to determine attribute weight and applied it in decisionmaking problems under in- terval intuitionistic environment. Grabisch and Raufaste  investigated the statistical properties of Choquet and Sugeno integrals to deal with some sets of learning data through some procedure. Marichal  proposed an ax- iomatic approach of discrete Choquet to aggregate interacting criteria and extended the weighted arithmetic mean by considering the interaction. Huy et al. proposed a fuzzydecision support method for customer preferences analysis based on Choquet integral and applied it to plan business management and marketing strategies. Rong et al. presented a classiﬁcation method of heterogeneous fuzzy data by Choquet integral with fuzzy-valued integrand and used this method in fuzzydecision trees and fuzzy-neuro networks. On the basis of theoretical analysis, some extensions were also discussed. When using Choquet integral to solve multiple attributedecision- making problems, the most diﬃculty is how to determine fuzzy measure in a reasonable method. To date, in current literatures, fuzzy measure is given by decision makers in advance, which makes it diﬃcult to avoid the subjective preferences of decision makers. Hence, the reliability of the decision-making result is not so convinc- ing. Therefore, how to determine the fuzzy measure based on expert decision-making information has become a very important issue in theory and practice.
Multiple attributedecisionmaking (MADM) is an important part of modern decision science. It has been extensively applied to various areas, such as society, economics, management, military and engineering technology. For example, the investment decision- making, project evaluation, the economic evaluation, the personnel evaluation etc. Since the object things are fuzzy, uncertainty and human thinking is ambiguous, the majority of the multi-attributedecision-making is uncertain and ambiguous, which is called the fuzzy multiple attributedecision-making (FMADM). Since Bellmanhe and Zadeh  initially proposed the basic model of fuzzydecisionmaking based on the theory of fuzzy mathematics, FMADM has been receiving more and more attentions. Many achievements have been made on FMADM problems [2-5,7-21].
Abstract. With respect to the interval neutrosophic Multi-AttributeDecision-Making (MADM) problems, the MADM method is developed based on some interval neutrosophic aggregation operators. Firstly, the Induced Generalized Interval Neutrosophic Hybrid Arithmetic Averaging (IGINHAA) operator and the Induced Generalized Interval Neutro- sophic Hybrid Geometric Mean (IGINHGM) operator are proposed, which can weight all the input arguments and their ordered positions. Further, regarding the situation where the input elements are interdependent, the Induced Generalized Interval Neutrosophic Shapley Hybrid Arithmetic Averaging (IGINSHAA) operator and the Induced Generalized Interval Neutrosophic Shapley Hybrid Geometric Mean (IGINSHGM) operator are proposed, which are extensions of IGINHAA and IGINHGM operators, respectively, and some properties of these given operators are investigated. Furthermore, the interval neutrosophic cross entropy, which is an extension of single-valued neutrosophic cross entropy, is dened, and the models based on the interval neutrosophic cross entropy and generalized Shapley function are respectively constructed to determine the optimal fuzzy measures on the attribute and ordered sets. Finally, an approach to interval neutrosophic MADM with interactive conditions and incomplete known weight information is proposed based on these given operators, and a practical example is shown to verify the practicality and feasibility of the new approach.
The final version of SERVQUAL has five dimensions named empathy, assurance, tangibles, reliability, and responsiveness. These dimensions have some sub-criteria. The aim of this model is to measure the gap between the perception of the customers and the performance of the company. This gap shows the weaknesses and strengths of a company. Shieh, Wu, & Huang (2010) used DEMATEL method to find out the relationships among the main CSFs of service quality in Taiwan. They showed that communication and competent medical staff could increase the customer satisfaction. Büyüközkan & Çifçi (2012a) illustrated a combination model of fuzzy AHP and fuzzy TOPSIS to evaluate the service quality using strategic analysis. The contribution of their work was the employment of the internet to evaluate the service quality in Turkish hospitals. Büyüközkan, Çifçi, & Güleryüz (2011) evaluated service quality in Turkish hospitals usingfuzzy AHP. They concluded that hospitals must pay more attention to the criterion of empathy. John, Yatim, & Mani (2011) showed the way to evaluate service quality in a dental healthcare center in Malaysia. They analyzed 481 dentists using SERVQUAL model with 20 sub-criteria and the dentists provided appropriate services fulfilling most criteria. Padma, Rajendran, & Sai Lokachari (2010) evaluated servi12ce quality of Indian hospitals. They asked the attendants and patients about service quality of Indian hospitals and they informed them of the way these hospitals could improve their service quality to gain more customers’ satisfaction. Lee (2017) illustrated a HEALTHQUAL model for South Korea hospitals. In this research 389 people response to related questionnaires and the result showed that degree of improvement criterion is the highest priority among other criteria. Behdioğlu et al (2017) depicted fuzzy SERVQUAL in physiotherapy hospital. This model had 5 dimensions and 22 sub-criteria and 262 people answered to the questions. The result pointed out that there was the huge negative score in the tangibles dimension. Jabbari et al (2017) indicated healthcare services quality in Namazi hospital. They used univariate and multi-regression analysis for finding quality of services. The result showed that the level of service quality in Namazi hospital was suitable.
great interest in the IFS theory and applied it to the ﬁeld of decisionmaking. Gau and Buehrer  introduced the vague set, which is an equivalence of IFS  . Later, based on vague sets, Chen and Tan  , and Hong and Choi  utilized the minimum and maximum operations to develop some approximate technique for handling multi-attributedecisionmaking problems under fuzzy environment. Szmidt and Kacprzyk  proposed some solution concepts such as the intuitionistic fuzzy core and consensus winner in group decisionmaking with intuitionistic (individual and social) fuzzy preference relations, and proposed a method to aggregate the indi- vidual intuitionistic fuzzy preference relations into a social fuzzy preference relation on the basis of fuzzy majority equated with a fuzzy linguistic quantiﬁer. Atanassov et al.  proposed an intuitionistic fuzzy inter- pretation of multi-person multi-attributedecisionmaking, in which each decision maker is asked to evaluate at least a part of the alternatives in terms of their performance with respect to each predeﬁned attribute: the decision maker’s evaluations are expressed in a pair of numeric values, interpreted in the intuitionistic fuzzy framework: these numbers express a ‘‘positive’’ and a ‘‘negative’’ evaluation, respectively. They also proposed a method for multi-person multi-attributedecisionmaking, and presented some examples of the proposed method in the context of public relation and mass communication. Xu and Yager  developed some aggre- gation operators including the intuitionistic fuzzy weighted geometric operator, intuitionistic fuzzy ordered weighted geometric operator, and intuitionistic fuzzyhybrid geometric operator, which extend the traditional weighted geometric operator and ordered weighted geometric operator to accommodate the environment where the given arguments are IFSs. Moreover, we developed an approach, based on the intuitionistic fuzzyhybrid geometric operator, to multi-attribute
This study proposed a grey-correlation multi-decisionmaking method based on intuitionistic trapezoidal fuzzynumbers through probing into the multi-attributedecisionmaking problems with the evaluation information being the intuitionistic trapezoidal fuzzynumbers. In addition, it also described the realization steps in details and demonstrated the reasonability of this method via analysis of examples.
Where i is the supplier index, p the attribute index, Mp the preference factor or weight of p th attribute, S the number of attributes, and U i the overall supplier score. The evaluation of a preference/scaling factor is an iterative process . A similar interview processes are used to understand the weight of each attribute for the decision maker. It is based on solving simultaneous equations derived from the fact that equally preferred options must have equal overall utility values. This is accomplished by asking the decision maker to compare a consequence of one attribute at the most preferred amount and all of the other attributes at the least preferred level to a lottery yielding having probability P and probability 1- P , respectively. Probability P ∗ at which the decision maker is indifferent to this lottery is used for equating the overall utilities of two consequences to arrive at scaling factor for the attribute. This procedure also needs a number of consistency checks and iterations for refining the results which may make it time consuming and complicated. With n attributes, n_1 scaling constants need to be determined using the simultaneous equations. The last factor can be calculated from the constraint, i.e. the sum of all scaling factors equals 1. This analysis can be simplified by using software tools like Logical Decision. These kinds of decision support tools can navigate the decision maker through various steps with interactive graphical interface to arrive at final values of scaling factors .