A Novel Multi-Criteria Decision Making Method based on The Ranking Values of Interval Type-2 Fuzzy Sets : An Application of a Manager Selection for a Telecommunication Company

Type-2 fuzzy sets (T2FSs), characterized by a fuzzy membership function, are much useful tool for representing the decision knowledge in the decision making process. Interval Type-2 fuzzy sets (IT2FSs) are the most commonly used T2FSs.  In this study, a method based upon ranking values of IT2FSs is used to tackle multi-criteria decision making (MCDM) problems. First, some basic concepts and arithmetic operations for IT2FSs are presented. Then, three kinds of fuzzy ranking methods, proposed by [1], based on arithmetic average (AA), geometric average (GA) and harmonic average (HA) operators to compute the ranking values of IT2FSs are applied. Finally, the outcomes of MCDM methods based on the ranking values of IT2FSs are obtained and also compared with the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method based on Type-1 fuzzy sets (T1FSs) for a numerical example.


Introduction
The object of multi-criteria decision-making (MCDM) is to select a top option from a group of alternatives by evaluating multiple criteria.So far, many approaches have been presented for MCDM problems with Type-1 fuzzy sets (T1FSs).For instance, [2] introduced a method to handle MCDM problems utilizing the similarity measure of fuzzy sets.[3] presented an extension of the TOPSIS method for group decision making under fuzzy environment.[4] proposed a method to handle MCDM problems adapted from the fuzzy preference information.[5] introduced a method for modelling the notion of risk in human decision processes.[6] proposed an application of TOPSIS with assessment aircraft training in a fuzzy setting.[7] introduced a fuzzy optimization method for MCDM.[8] proposed an approach for MCDM accompany incomplete information.[9] presented a dynamic MCDM model by using the grey number evaluations.However, fuzzy MCDM method based upon T1FSs, proposed by [10], are unsuccessful to deal with high complexity and vagueness.For that reason, Type-2 fuzzy sets (T2FSS), which can be considered as an A. Şahin, N. Yapıcı Pehlivan / A Novel Multi-Criteria Decision Making Method based on The Ranking Values of Interval Type-2 Fuzzy Sets: An Application of a Select for a Telecommunication Company 291 extension of T1FSs, were presented by [11].Recently, some methods using the T2FSs have been proposed for MCDM problem.For example, [12] proposed a new method for fuzzy multiple criteria group decision making (FMCGDM) based upon the arithmetic operations of Interval Type-2 fuzzy sets (IT2FSs).[13] proposed a method for FMCGDM by utilizing ranking values and the arithmetic operations of IT2FSs.[14] presented a FMCGDM by using the IT2 TOPSIS method.[15] presented a FMCGDM based on ranking IT2FSs.[4] presented multi-attribute group decision making models for IT2F environment.[1] introduced a new method to deal MCDM problems with merged ranking values in IT2F environment.[16] introduced a merged method of Analytical Hierarchy Process and TOPSIS method based on IT2FSs.
In this study, we use a method based on ranking values of IT2FSs to handle MCDM problems.In Section 2, we introduce some basic concepts and arithmetic operations for IT2FSs and three kinds of fuzzy ranking methods proposed by Qin and Liu (2015) for calculating the ranking values of IT2FSs with operators of arithmetic average (AA), geometric average (GA) and harmonic average (HA) and introduce MCDM method based on the ranking values of IT2FSs.In Section 3, we obtained the results of MCDM method based upon the ranking values of IT2FSs and also compared with fuzzy MCDM method rest on T1FSs for a numerical example and conlusions are given in Section 4.

Material and Method
In this section, firstly some basic definitions of T1FSs and T2FSs and arithmetic operations and ranking values for Trapezoidal IT2FSs are briefly given.After, MCDM method based on the ranking values of IT2FSs is introduced.
Definition 2.1.[10] A fuzzy set % Z in the universe of discourse X is represented by a membership function .The value of () Definition 2.2.[17].A Type-2 fuzzy set % % Z in the universe of discourse Z which can be characterized by a Type-2 membership function  and  symbolizes union over all acceptable z and u .Definition 2.3.[17].Let % % Z be a T2FS in the universe of discourse Z characterized by the Type-2 membership function .
Z is termed an Interval Type-2 fuzzy set.An IT2FS % % Z can be accepted as a specific case of T2FS and represented as follows: 1/ ( , ) Definition 2.4.[17].The upper and lower membership functions of IT2FSs are Type-1 membership functions.A trapezoidal IT2FS is illustrated as , , , , , ,

Arithmetic operations on trapezoidal interval type-2 fuzzy sets
In this section, arithmetic operations on Trapezoidal IT2FSs are given [8].

Let be trapezoidal IT2FSs
 

% L Z
Multiplication with Crisp Scalar k:

The ranking values of trapezoidal IT2FSs
Let % % Z be an IT2FSs, then three kinds of ranking of % % Z proposed by [1] are identified as follows: where; (1) ()   (( , , , ; % L jk HZ be the critera weight given by the kth decision maker, (where % % jk w is also a trapezoidal IT2 fuzzy number).We utilize assessment of the alternatives and criteria, and Eqs.( 1)-( 3) to obtain ranking values of alternatives.

Results
We use the data from [18] and the linguistic values from [15] to illustrate the MCDM method by using the ranking values of IT2FSs.Assume that a Telecommunication Company wants to select a manager from four candidates called as  The hierarchical structure of this decision problem is shown as in Fig. 1.The MCDM method based on the ranking values of IT2FSs is utilized to handle this problem and the method is given as follows.
Step 1: The decision makers evaluate the criteria by using the linguistic terms and IT2FSs given in [15] and presented in Table 1.
Step 2: The decision makers evaluate alternatives according to each criterion by using the linguistic terms and IT2FSs in [15] and given in Table 2.
Table 1.Assessments of the criteria by the decision makers Criteria Decision Makers Step 4: Construct the ranking matrix (0.9,1, = (6.30,9,9,10;1,1),(6.30,9,9,10;1,1) For example, ranking values of the Alternative 1 1 () A according to the Criterion1 Step 5: Construct the rank average agreement degree ,, AD AD AD for each criteria by different ranking values which are arithmetic average ranking value, geometric average ranking value, harmonic average ranking value.The results are shown in Table 5.
Rs % % Step 6: Compute the ranking value Results are demostrated in Table 6.

3 DM 1 (
evaluate the four relevant candidates according to five criteria that are Proficiency in identifying research areas

Table 2 .
Assessments of alternatives according to each criterion by the decision makers

Table 3 .
The weighted assessment values of the alternatives in terms of the criteria for each decision maker.