MADM Problems with Correlation Coefficient of Trapezoidal Fuzzy Intuitionistic Fuzzy Sets

This paper discusses on the notion of trapezoidal fuzzy intuitionistic fuzzy sets (TzFIFSs) and some of the arithmetic operations of the same. Correlation coefficient of TzFIFS is proposed based on the membership, nonmembership, and hesitation degrees. The weighted averaging (WA) operator and the weighted geometric (WG) operator are proposed for TzFIFSs. Based on these operators and the correlation coefficient defined for the TzFIFS, new multiattribute decision making (MADM) models are proposed and numerical illustration is given.


Introduction
Intuitionistic fuzzy sets (IFSs) proposed by Atanassov [1][2][3] are a generalization of the concept of fuzzy sets.Atanassov and Gargov [4] expanded the IFSs, using interval value to express membership and nonmembership function of IFSs.Liu and Yuan [5] introduced the concept of fuzzy number IFSs as a further generalization of IFSs.Among the works done in IFSs, Szmidt and Kacprzyk [6][7][8] can be mentioned.To the best of our knowledge, Burillo et al. [9] proposed the definition of intuitionistic fuzzy number (IFN) and studied the perturbations of IFN and the first properties of the correlation between these numbers.Many researchers have applied the IFS theory to the field of decision making.Recently some researches [5,[10][11][12][13][14] showed great interest in the fuzzy number IFSs and applied it to the field of decision making.Based on the arithmetic aggregation operators, Xu and Yager [15], Xu and Chen [16,17], and Wang [13] developed some new geometric aggregation operators and intuitionistic fuzzy ordered weighted averaging (IFOWA) operator.Szmidt and Kacprzyk [7] proposed some solution concepts in group decision making with intuitionistic (individual and social) fuzzy preference relations.Szmidt and Kacprzyk [8] investigated the consensus-reaching process in group decision making based on individual intuitionistic fuzzy preference relations.Herrera et al. [18] developed an aggregation process for combining numerical, interval valued, and linguistic information and then proposed different extensions of this process to deal with contexts in which information such as IFSs or multigranular linguistic information can appear.Xu and Yager [15] developed some geometric aggregation operators for MADM problems.Li [19] investigated MADM problems with intuitionistic fuzzy information and constructed several linear programming models to generate optimal weights for attributes.
Multiattribute decision making (MADM) problems are of importance in most kinds of fields such as engineering, economics, and management.It is obvious that much knowledge in the real world is fuzzy rather than precise.Imprecision comes from a variety of sources such as unquantifiable information [20].In many situations decision makers have imprecise/vague information about alternatives with respect to attributes.It is well known that the conventional decision making analysis using different techniques and tools has been found to be inadequate to handle uncertainty of fuzzy data.To overcome this problem, the concept of fuzzy approach has been used in the evaluation of decision making systems.For a long period of time, efforts have been made in designing various decision making systems suitable for the arising dayto-day problems.MADM problems are widespread in reallife decision making situations and the problem is to find a 2 Advances in Decision Sciences desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative [21].In order to choose a desirable solution, the decision maker often provides his/her preference information which takes the form of numerical values, such as exact values, interval number values, and fuzzy numbers.However, under many conditions, numerical values are inadequate or insufficient to model real-life decision problems.Indeed, human judgments including preference information may be stated in intuitionistic fuzzy information, especially in trapezoidal fuzzy intuitionistic fuzzy information.Hence, MADM problems under intuitionistic fuzzy or trapezoidal fuzzy intuitionistic fuzzy environment are an interesting area of study for researchers in the recent days.
It is well known that the conventional correlation analysis using probabilities and statistics has been found to be inadequate to handle uncertainty of failure data and modeling.The method to measure the correlation between two variables involving fuzziness is a challenge to classical statistical theory.Park et al. [22] proposed the correlation coefficient of interval valued intuitionistic fuzzy sets.Robinson and Amirtharaj [23][24][25][26] proposed correlation coefficient for vague sets, interval vague sets, triangular and trapezoidal IFSs, and a revised correlation coefficient for triangular and trapezoidal IFSs using graded mean integration representation.In this paper, a novel method of correlation coefficient of trapezoidal fuzzy intuitionistic fuzzy sets (TzFIFSs) is proposed and developed by taking into account the membership, nonmembership, and the hesitation degrees of TzFIFSs.The weighted averaging (WA) and weighted geometric (WG) operators for TzFIFSs are proposed for MADM problems.Based on these operators, an approach is suggested to solve uncertain multiple attribute group decision making problems, where the attribute values are trapezoidal fuzzy intuitionistic fuzzy numbers.A new algorithm is developed to solve the MADM problems in which the correlation coefficient of TzFIFSs is used for ranking alternatives.

Arithmetic Operations for TzFIFS
Arithmetic operations of TzFIFNs are based on the arithmetic operations of TzFNs.In TzFIFN the membership and nonmembership degrees take the form of trapezoidal fuzzy number.The basic concepts related to TzFIFNs are presented in the following.
For the above operation rules, the following are true:
Then the correlation of trapezoidal fuzzy intuitionistic fuzzy numbers (TzFIFNs) is defined as follows.
For any arbitrary real number , the following relation is true: Hence, The above inequality can be written as ) .

Theorem 7. Consider K TzFIFS
Proof.Considering the inequality in the proof of Theorem 6, then the equality holds if and only if the following are satisfied for some positive real .As then it means  = 1, and therefore  = .
Proof.For all   ∈ , the following are true: If  TzFIFS (, ) = 0 for all   ∈ , then the following should be true: Conversely, when  and  are nonfuzzy sets and This property can be observed similarly for all other entries.Therefore  TrFIFS (, ) = 0.
Conversely, it can be proved by the method of contradiction.

WA and WG Operators for TzFIFNs
On the foundation of the definitions discussed by Wang [13], the weighted arithmetic operators for TzFIFNs are given as follows.
The developed model of MADM is given as follows.

Numerical Illustration
A company intends to select one person to take the position of Assistant Manager from four candidates.Five indicators (attributes) must be evaluated.They are shown as follows: (i) technical skill ( ) .

(27)
Step 1. Utilize the decision information given in the matrix R, the TzFIFWA operator, and the weighting vector of the attributes  = ( 1 ,  2 , . . .,   )  = (0.Step 3. Rank alternatives   ,  = 1, 2, . . ., , from the highest closeness value (correlation coefficient) obtained from Step 2; the result is obtained as follows: Hence the best alternative is  4 .Following the MADM model using the TzFIFWA operator, we have the following numerical results for the MADM model using TzFIFWG operator.

Conclusion
In this paper an MADM model was proposed based on the correlation coefficient of TzFIFS for ranking the alternatives together with WA and WG operators.The trapezoidal fuzzy intuitionistic fuzzy weighted averaging (TzFIFWA) operator and the trapezoidal fuzzy intuitionistic fuzzy weighted geometric (TzFIFWG) operator were used to aggregate the trapezoidal fuzzy intuitionistic fuzzy information given in a decision matrix.A numerical illustration was given to show the effectiveness of the proposed approach in using correlation coefficient of TzFIFSs because it preserved the linear relationship between the variables.