A novel approach for multiple criteria group decision making problem with unknown criteria weight information

: In this paper, an approach to multiple criteria group decision making (MCGDM) problem in which the criteria weights are unknown is proposed. The normal distribution is applied to criteria values to find the criteria weights when the decision maker weights, criteria values are taken in the form of interval valued intuitionistic fuzzy trapezoidal numbers (IVIFTNs). The effectiveness of the developed approach is verified by giving an illustrative example.


Introduction
The concept of fuzzy sets (FS) was introduced by Zadeh [12] in 1965. Later fuzzy sets were generalized by Atanassov [1], called the intuitionistic fuzzy sets (IFS). The IFS includes membership function and non-membership functions which takes the exact values [0, 1]. But it is more effective to represent membership or non-membership functions in interval form rather than exact numbers for application to real world problems. Atanassov and Gargov [2] gave the concept of interval valued intuitionistic fuzzy sets (IVIFSs) [9]. The domain in IFSs and IVIFSs is a discrete set; therefore their membership degrees and the non-membership degrees can only express fuzzy concept in terms of "excellent" or "good". To overcome this limitation, Shu et al. [5] defined intuitionistic fuzzy triangular numbers for which domain is a consecutive set. Then the extension of ITFNs that is the intuitionistic fuzzy trapezoidal numbers (IFTNs) given by Wang [8]. Later on Wan [7] introduced the concepts of interval valued intuitionistic fuzzy trapezoidal numbers (IVIFTNs), in which both membership and non-membership are in interval forms. So, it is more effective to study the MCGDM under the view of IVIFTNs.
In decision making problems, it is very important that to aggregate the overall information. The weighted averaging aggregation is a very common technique among all procedures of MCGDM. There are many aggregation operators and ordered weighted averaging operators are proposed by researchers to aggregate FNs, IFNs, IVIFNs and IVIFTNs. Guiwu et al. [3] introduced some operational laws of interval intuitionistic fuzzy trapezoidal numbers and applied interval intuitionistic trapezoidal fuzzy ordered weighted geometric (IITFOWG) and interval valued intuitionistic trapezoidal fuzzy hybrid geometric (IITFHG) operators for decision making. Later on, interval valued intuitionistic trapezoidal fuzzy weighted geometric operator (IVITFWG), interval valued intuitionistic trapezoidal fuzzy ordered weighted operator (IVITFOWG), and interval valued intuitionistic trapezoidal fuzzy weighted hybrid operators (IVITFWH) and their properties introduced by Wu and Liu [9]. If weights of the criteria are completely unknown, there are criteria-independent and criteria-dependent approaches to find the weights. Xu [10] proposed normal distribution based method which includes the number of criteria but independent on the values of criteria. Hence, in this paper we proposed a criteriadependent approach to find the criteria weights using normal distribution. Subsequently, the interval-valued intuitionistic fuzzy trapezoidal weighted averaging operator (IVIFTWA) is used to find the best alternative among the given.

Preliminaries
In this section, we briefly introduce some basic concepts related to interval valued intuitionistic trapezoidal fuzzy numbers and their arithmetic operations.

Definition 1 [1]: Intuitionistic Fuzzy
Set. An intuitionistic fuzzy set over universe of discourse X is of the form: A= {〈 , ( ), ( ) where A µ denotes membership function, and A υ denotes non-membership function, with the condition 0 ( ) ( ) 1, for all . X x ∈ Definition 2 [2]: Interval valued intuitionistic fuzzy set. An interval-valued intuitionistic fuzzy set in A over X is an object having the form: is called an interval-valued intuitionistic fuzzy trapezoidal set (IVIFTS). [ , , , , iii)

Proposed method to find the best alternative when the criteria weights are unknown in advance
In multicriteria decision making problems, different weights will be assigned to criteria depending on the problem. Sometimes, the criteria weight is completely unknown in advance.
There are criteria-independent and criteria-dependent approaches to find the weights criteria [4]. In this section, the method to find the weight of the criteria and best alternative is given in step wise. In the present approach, the given criteria values are used together with normal distribution to obtain the criteria weights, given in Definition 10. As the approach is criteria dependent, it can relieve the influence of unfair arguments on the decision by assigning low weights to those false and biased.
is respectively the value index and ambiguity index of the criteria i .
Step 3: Obtain the aggregated values for each criteria from decision matrix by Step 5: Apply the IVITFWA operator with the criteria weights obtained in Step 4 and the decision matrix obtained in Step 2 . Then we get the collective overall preference values j t of each alternative m j A j ..., , 3 , 2 , 1 , = .
Step 6: Calculate the corresponding value index for j t that is Step 5 and rank the best alternative using the proposed ranking method.

Numerical example
In this section, the proposed method is applied to find the best green supplier for a food company presented by Wu and Liu [4]. After pre-evaluation, three suppliers (alternatives) are selected for further evaluation.
The assessments of three suppliers by three decision makers based on each criterion are given respectively in Table 1, Table 2 and Table 3.   Step . Hence 2 A is the best alternative.
The result obtained by proposed method agrees with the Wu and Liu [9]. Moreover, the present approach orders the alternatives strictly when compared to [9].