Multi-attribute decision making using grey relational projection method based on interval type-2 trapezoidal fuzzy numbers

Abstract

Here we present a method to deal with multi-attribute decision making(MADM)problems when the attribute values are modeled in the form of interval type-2 trapezoidal fuzzy numbers (IT2FNs), and the attribute weights are completely unknown. Grey Relation Projection Method (GRPM), which is a combination of “grey relational analysis method" and the “projection method" is employed for ranking the alternatives. The linguistic information is modeled in the form of “interval type-2 trapezoidal fuzzy numbers" (IT2TFN) which are able to capture both the intra personal and inter personal uncertainties associated with a linguistic term. Information Entropy Method (IEM) is used for calculating unknown attribute weights. Lastly, an illustrative example is provided as a verification of the developed approach.

Keywords

Multi-attribute decision making (MADM)interval type-2 trapezoidal fuzzy number (IT2TFN)Information Entropy method (IEM)grey relational projection method (GRPM)

1 Introduction

There are many uncertain problems in the world. Fuzzy set’s idea is given by Zadeh [5] to deal with uncertainties. Fuzzy set theory gives a new concept for decision making in many fields. In many complicated problems in the fields of engineering, economics, social and biological sciences, we come across the situations where we have to choose one object from different choices. Neither any analytical (or numerical) nor any statistical approach is often helpful in these situations due to the reason that every person has his/her own choice. Weaver [49] refers to them as the problems of organized complexity. The concept of fuzzy sets (FSs) by Zadeh [5] in 1965 is considered as an evolution for dealing such type of problems, as fuzzy sets do not have precise boundaries like the typical crisp sets.The main purpose of MADM is to select most suitable candidate from alternatives according to information about attribute weights and values, which is provided by decision makers [4, 56], but information given in form of linguistic terms about attribute value is fuzzy or uncertain. To deal such uncertain situation many method have been develop to dealing with MADM problems on T1FS [56], we can cover more uncertainty and provide more flexibility we apply IT2FS instead of T1FS [8 , 56].

Atanassov and Gargov [2, 3] extended this idea of FSs by introducing the concept of intuitionistic fuzzy set (IFS) and interval valued intutionistic fuzzy set (IVIFS) Firstly, T2FS introduced by Zadeh [5] which is extension of T1FS. In T1FS membership degree [25] is crisp [0,1] and in T2FS membership degree of is T1FS in [0,1]. So T2FS provides more degree of freedom to deal vagueness and uncertainty of real and practical world problems as compared to T1FS. Dinga and Wanga [20] define a IVTIFS. Joshi and Kumar [21] give the idea of Intuitionistic fuzzy parameterized fuzzy soft set theory(IFPFSST). Sajjad et al. [1] define a new extension of TOPSIS method based on Pythagorean hesitant fuzzy sets(PHFS) with incomplete weight information by using maximum deviation method. IT2FS are more capable then any other fuzzy sets of dealing with imperfect and uncertain information in real world application [10, 25]. IT2FS is applied in many practical fields such as decision making and ranking with trapezoidal IT2FS.

Zang at al [55] deal with intuitionists Type 1 trapezoidal fuzzy number(IT1TFN) with completely unknown information about attribute weights and using GRPM for ranking the alternatives. Wu and Mendel [11, 12] using IT2 and linguistic weighted averaging deal with the multiple criteria hierarchical group decision making (MCHGDM) problem. Chen and Lee [8, 23] deal for fuzzy MAGDM based on arithmetic operation and ranking value of IT2FS. Wang et al., [4, 45] deal with MADM problem where attribute value are given in the form of IT2TFN and partially information about attribute weight. Uncertain practical problem of MADM problems are evaluate in term of linguistic variables because decision maker are not always confident about their decision and performance about theirs judgment. In practical decision making problems or in mathematical modeling decision maker express their judgment evaluate in term of linguistic variables because they are not certain of theirs decision or performance.They often use degree of uncertain in term of linguistics term. There are certain linguistic scale are used 3,4 and 5 term scale. In this paper use 10 term scale [0-10].

The main fauces of this paper is to apply GRPM for ranking the alternatives by using IT2TFN along by using IEM for unknown attribute weight. For this purpose firstly introduce some operational laws about IT2TFN, new expected values is define and also define Hamming distance between two IT2TFN. Then IEM is used to determine unknown weights. GRPM is used to rank the alternatives. At the and verify the whole development approach with example.

2 Preliminaries

Definition 2.1. A T1FS $\tilde{F}$ [35] in the universe of discourse X can be characterized by its membership function μ_F (x) and represented as follows: $F = {(x, μ_{F}^{-} (x)) | \forall x \in X, \forall x \in X, μ_{F}^{-} (x) \in [0, 1]}$ (1)

Definition 2.2. A T2FS $\tilde{F}$ [17] in the universe of discourse U can be represented by a type-2 membership function $μ_{F}^{-}$ shows as follows: $\begin{matrix} \tilde{F} = {((x, u), μ_{F}^{-} (x, u)) | \forall x \in U, \forall u \in J_{x} \subseteq \\ [0, 1], 0 \leq μ_{F}^{-} (x, u) \leq 1} \end{matrix}$ (2) whereJ_x denotes an interval in [0, 1] The T2FS $\tilde{A}$ also can represent as follows; $\tilde{F} = \int_{x = U} \int_{U = j_{x}} \frac{μ_{F}^{-} (x, u)}{(x, u)}$ (3)

Definition 2.3. [17] Let F be a T2FS in the universe of discourse X denoted by the type-2 membership function μ_F . If all the secondary grades $μ_{F}^{-} (x, u)$ of F are equal to 1, then F is called an interval type-2 fuzzy set. It symbolically shown as: $\tilde{F} = \int_{x = U} \int_{U = j_{x}} \frac{1}{(x, u)}$ (4)

Definition 2.4. Expected value J is defined [48] as for $\begin{matrix} \tilde{F} = (\tilde{F^{u}}, \tilde{F^{u}}) = (a_{1}^{u}, a_{2}^{u}, a_{3}^{u}, a_{4}^{u} : \\ h (\tilde{F})^{u} a_{1}^{l}, a_{2}^{l}, a_{3}^{l}, a_{4}^{l} : h (\tilde{F})^{l}) \end{matrix}$ (5) $J_{λ} (F) = \frac{K_{1} + K_{2}}{4}$ (6) where $k_{1} = h (\tilde{F})^{u} [(1 - λ) (a_{1}^{u} + a_{2}^{u}) + λ (a_{3}^{u} + a_{4}^{u})]$ and $k_{2} = h (\tilde{F})^{l} [(1 - λ) (a_{1}^{l} + a_{2}^{l}) + λ (a_{3}^{l} + a_{4}^{l})]$

Definition 2.5. Hamming distance between two IT2TFN is define as: $\begin{matrix} \tilde{F} = (\tilde{F^{u}}, \tilde{F^{u}}) = (a_{1}^{u}, a_{2}^{u}, a_{3}^{u}, a_{4}^{u} : h (\tilde{F})^{u} \\ a_{1}^{l}, a_{2}^{l}, a_{3}^{l}, a_{4}^{l} : h (\tilde{F})^{l}) \end{matrix}$ (7) $\begin{matrix} \tilde{G} = (\tilde{G^{u}}, \tilde{G^{u}}) = (b_{1}^{u}, b_{2}^{u}, b_{3}^{u}, b_{4}^{u} : h (\tilde{G})^{u} \\ b_{1}^{l}, b_{2}^{l}, b_{3}^{l}, b_{4}^{l} : h (\tilde{G})^{l}) \end{matrix}$ (8) $\begin{matrix} d (\tilde{F}, \tilde{G}) = (\sum_{p}^{4} (a_{p}^{u} - b_{p}^{u})^{2} + \sum_{p}^{4} (a_{p}^{l} - b_{p}^{l})^{2} + \\ | (h (\tilde{F})^{u}) - h (\tilde{G})^{u} | + | (h (\tilde{F})^{l}) - (h (\tilde{G})^{l}) |)^{\frac{1}{2}} \end{matrix}$ (9)

3 MADM mathod based on the IT2FN

To make process uniform and remove the different physical aspect on decision solution so first standardize the decision matrix.

where R is standardized decision matrix. $\begin{matrix} R = [r_{pq}]_{m \times n} = \tilde{F} = (a_{pq}^{u}, a_{pq}^{u}, a_{pq}^{u}, a_{pq}^{u} : h (\tilde{F})^{u} \\ a_{pq}^{l}, a_{pq}^{l}, a_{pq}^{l}, a_{pq}^{l} : h (\tilde{F})^{l}) \end{matrix}$ (10) where p represent row and q represent column. we can evaluate into two type criteria [52] such as cost type and benefit type criteria such as for Cost type criteria;- $r_{pq}^{\overset{´}{v}} = \frac{{max}_{q} (a_{pq}^{4}) - a_{pq}^{\overset{´}{v}}}{{max}_{q} (a_{pq}^{4}) - {min}_{q} (a_{pq}^{4})} \overset{´}{v} = 1, 2, 3, 4$ (11) Benefit type criteria;- $r_{pq}^{\overset{´}{v}} = \frac{a_{pq}^{\overset{´}{v}} - {max}_{q} (a_{pq}^{4})}{{max}_{q} (a_{pq}^{4}) - {min}_{q} (a_{pq}^{4})} \overset{´}{v} = 1, 2, 3, 4,$ (12)

4 Information Entropy Method (IEM)

Many optimizing techniques which deal with the situation where we have either partial information or complete information about attribute weights are given to determine the attribute weights.There are few methods such as Information Entropy Method IEM [18 , 51], Maximizing Deviation Method(MDM) [43 , 47], which deal with completely unknown information. Therefore, we will use IEM to deal with completely unknown weights. These methods have very complex structure and can’t deal properly with uncertainty. To avoid such objective factor we use IEM for determining unknown attribute weight. Entropy is basic idea of thermodynamics which mean measure of disorder and randomness. Shannon [31, 32] introduce the idea of information entropy and connect with communication theory which is known is Shannon Entropy Weighting Technique (S.E.W.T). The S.E.W.T is one of the best methods for weighting attributes. The idea of entropy is mostly used in many other fields such as engineering ad economy.Entropy should be zero for incomplete information, but if some information is given then entropy should be greater than zero In the S.E.W.T [31, 32], attribute weights would be less, if value of entropy corresponding to attribute is large, and attribute weights would be greater, if value of entropy corresponding to attribute is small, and weight should be higher in a case where entropy value is small or vise versa.The following are the characteristics of Information Entropy Method:

the given method is easy and simple to use

the person making the decision and the circle of model should become nearer for insufficient information

the value of all criteria produced by the entropy method should be more divergent.

Entropy method only applicable for crisp number and its fail to deal with IT2FS or any other fuzzy number so firstly we convert IT2FN into crisp number we use espected value. The formula of espected value J is given below

J_{λ} (F) = \frac{K_{1} + K_{2}}{4}

(13) where k₁=

h (\tilde{F})^{u} [(1 - λ) (a_{1}^{u} + a_{2}^{u}) + λ (a_{3}^{u} + a_{4}^{u})]

and

k_{2} = h (\tilde{F})^{l} [(1 - λ) (a_{1}^{l} + a_{2}^{l}) + λ (a_{3}^{l} + a_{4}^{l})]

where

f_{pq} = \frac{J ({\tilde{r}}_{pq})}{\sum_{p = 1}^{m} J ({\tilde{r}}_{pq})}

(14) where

v = \frac{1}{lnm}

and suppose if f_pq = 0 then 0ln0 = 0 Calculate the entropy value of the attribute C_i

E_{q} = - v \sum_{p = 1}^{m} f_{pq} \ln f_{pq} (1 \leq p \leq m) (1 \leq q \leq n)

(15) calculate the entropy weight w_q

w_{q} = \frac{1 - E_{q}}{\sum_{q = 1}^{m} (1 - E_{q})} (1 \leq q \leq n)

(16) Zhou et al. [53] improved his formula because for slightly difference between attribute may lead multiple change in entropy weight so this formula is unreasonable.so new formula to determine attribute weight is define as below

w_{q} = \frac{\sum_{q = 1}^{m} E_{q} + 1 - 2 \times E_{q}}{\sum_{q = 1}^{m} (\sum_{q = 1}^{m} E_{q} + 1 - 2 \times E_{q})} (1 \leq q \leq n)

(17)

5 Grey relational Projection Method (GRPM)

GRPM is translating the alternative performance into an ideal target sequence called comparable sequences, which is also called Grey Relational Grading. So firstly, we have to define a reference sequence, then calculate grey relation coefficient between reference sequence and ideal target sequence and, at last, if the ideal target sequence has highest grey relational grade, it will be the best choice among others.

Grey relational generating

The purpose of ideal target sequence or comparable sequence is to make the process uniform, because some attribute value is very small, which has very little influence on performance which is negligent. And some attributes have very large influence on performance, which therefore causes an incorrect result. So the first step of GRA is to make sequence into Positive ideal target sequence PITS and Negative NITS of standardized matrix. Grey System Theory(GST) has been presented by Deng [13]. It is a quantitative evaluation technique used to analyze alternatives, and applied in different field like data-processing, decision-making, modeling, prediction, and system analysis [13 , 37]. Basically, it is a very helpful tool deals with systems-analysis that is categorized by incomplete information, thereby assigning to the complex relations between elements, things, and behaviors of systems. GRA, which is usually adopted to solve the complicated correlation problems between different variables and multiple factors. Also, some of the MADM problems like hiring decision in particular field [27] etc, can be solved by using the GRA. In this process of solving the MADM problems the total range of attribute values that is considered responsible for every alternative is combined into one single value. The original problem is reduced to a single attribute decision making problem. We can also compare the alternative with multiple attributes once we are done with the GRA process. The SAW and TOPSIS methods are also concerned with the process of combining multi-attribute values into a single value.

From normalized matrix PITS of IT2TFN is define as $\begin{matrix} r_{q}^{+} = ([r_{q}^{1} +, r_{q}^{2} +, r_{q}^{3} +, r_{q}^{4} +]); h (A_{p}) \\ = ([\underset{︸}{max_{p}} (r_{pq}^{1}), \underset{︸}{max_{p}} (r_{pq}^{2}), \underset{︸}{max_{p}} (r_{pq}^{3}), \underset{︸}{max_{p}} (r_{pq}^{4})]; \underset{︸}{max_{p}} h (A_{p})) \end{matrix}$ (18) and NITS of IT2TFN are defined as $\begin{matrix} r_{q}^{-} = ([r_{q}^{1} -, r_{q}^{2} -, r_{q}^{3} -, r_{q}^{4} -]); h (A_{p}) \\ = & ([\underset{︸}{max_{p}} (r_{pq}^{1}), \underset{︸}{max_{p}} (r_{pq}^{2}), \underset{︸}{max_{p}} (r_{pq}^{3}), \underset{︸}{max_{p}} (r_{pq}^{4})]; \underset{︸}{max_{p}} h (A_{p})) \end{matrix}$ (19)Grey Relational Coefficient(GRC) Calculation

GRC calculation is used for determining how close sequence is to PITS. The larger the GRC is,the closer to PITS. The GRC calculated by formula $ζ_{pq}^{+} = \frac{S^{+} + λ T^{+}}{d_{pq}^{+} + λ S^{+}}$ (20) where $\begin{matrix} d (\tilde{F}, \tilde{G}) = (\sum_{p}^{4} (a_{p}^{u} - b_{p}^{u})^{2} + \sum_{p}^{4} (a_{p}^{l} - b_{p}^{l})^{2} + \\ | (h (\tilde{F})^{u}) - h (\tilde{G})^{u} | + | (h (\tilde{F})^{l}) - (h (\tilde{G})^{l}) |)^{\frac{1}{2}} \end{matrix}$ (21) $T^{+} = \underset{p}{\min_{︸}} \underset{q}{\min_{︸}} d_{pq}^{+} S^{+} = \underset{p}{\max_{︸}} \underset{q}{\max_{︸}} d_{pq}^{+}$ (22)λɛ [01] represents resolution coefficient. Generally its value is 0.5 if the value λ > 0.5 the decision maker is optimistic attitude towards problem. But if λ < 0.5 the decision maker has pessimistic attitude towards problem. And if the λ = 0.5 decision maker has moderate attitude towards problem. Therefore we use λ = 0.5 to deal with the situation weather it is neither optimistic nor pessimistic. Similarly, the GRC calculation of each alternative from NITS is given as: $ζ_{pq}^{-} = \frac{T^{-} + λ S^{-}}{d_{pq}^{-} + λ S^{-}}$ (23) also $\begin{matrix} d (\tilde{F}, \tilde{G}) = (\sum_{p}^{4} (a_{p}^{u} - b_{p}^{u})^{2} + \sum_{p}^{4} (a_{p}^{l} - b_{p}^{l})^{2} + \\ | (h (\tilde{F})^{u}) - h (\tilde{G})^{u} | + | (h (\tilde{F})^{l}) - (h (\tilde{G})^{l}) |)^{\frac{1}{2}} \end{matrix}$ (24) $T^{-} = \underset{p}{\min_{︸}} \underset{q}{\min_{︸}} d_{pq}^{-} S^{-} = \underset{p}{\max_{︸}} \underset{q}{\max_{︸}} d_{pq}^{-}$ (25) Calculating the grey relational grade of each alternative from PITS and NITS using the following equation, respectively: $ζ_{p}^{+} = \sum_{q = 1}^{n} w_{q} ζ_{pq}^{+}$ (26) $ζ_{p}^{-} = \sum_{q = 1}^{n} w_{q} ζ_{pq}^{-}$ (27)

Rank the alternative

The basic rule of the GRM is that, from the PITS we chose "greatest degree of grey relation" and NITS we should chose the “smallest degree of grey relation”.

6 Projection method

Definition 6.1.γ = (γ₁, γ₂, γ₃ . . . . . . γ_n) and δ = (δ₁, δ₂, δ₃ . . . . . . δ_n) be two vectors, Then cosine of angles between γ and δ is define as $\cos (γ, δ) = \frac{\sum_{q = 1}^{n} γ_{q} δ_{q}}{\sqrt{\sum_{q = 1}^{n} γ_{q}^{2}} \sqrt{\sum_{q = 1}^{n} δ_{q}^{2}}}$ (28)

Definition 6.2. let γ = (γ₁, γ₂, γ₃ . . . . . . γ_n) $∥ γ ∥ = \sqrt{\sum_{q = 1}^{n} γ_{q}^{2}}$ (29) A vector consist of two parts, magnitude and direction and cos γ, δ measure the direction between two vectors. The closeness level of two vectors can be estimated by the projection method. Projection is define as

Definition 6.3. suppose γ = (γ₁, γ₂, γ₃ . . . . . . γ_n) and δ = (δ₁, δ₂, δ₃ . . . . . . δ_n) be two vectors, then projection of γ onto vector δ can be define as $\begin{matrix} P (γ) = ∥ γ ∥ \cos (γ, δ) = \\ \sqrt{\sum_{q = 1}^{n} γ_{q}^{2}} \frac{\sum_{q = 1}^{n} γ_{q} δ_{q}}{\sqrt{\sum_{q = 1}^{n} γ_{q}^{2}} \sqrt{\sum_{q = 1}^{n} δ_{q}^{2}}} \\ = \frac{\sum_{q = 1}^{n} γ_{q} δ_{q}}{\sqrt{\sum_{q = 1}^{n} δ_{q}^{2}}} \end{matrix}$ (30) The larger the value of P (γ) is more close to vector δ the vector γ is

For a MADM problem, if the vector δ gets the PITS, and the vector a takes separately for each alternative, we can get each alternatives projection onto the PITS. The larger the projection value is,the better the alternative is.

7 Grey Relation Projection Method (GRPM)

GRPM is combination of GRM and Projection Method(PM) in which GRM analyze the data change in the trend curve, it is the a measure of curvature matching, and the PM can express the position relationship in the data curve between each alternative and to PITS and NITS. Therefore, both methods have their own importance.

Calculated the WGCP method of alternative Al onto the PITS $\begin{matrix} P (γ) = ∥ γ ∥ \cos (γ, δ) = \\ \sqrt{\sum_{q = 1}^{n} w_{q}^{2} ζ_{pq}} \frac{\sum_{q = 1}^{n} w_{q} ζ_{pq}^{+} \times w_{q}}{\sqrt{\sum_{q = 1}^{n} w_{q}^{2} ζ_{pq}} \sqrt{\sum_{q = 1}^{n} w_{q}^{2}}} \\ = \frac{\sum_{q = 1}^{n} w_{q}^{2} ζ_{pq}^{+}}{\sqrt{\sum_{q = 1}^{n} w_{q}^{2}}} = \sum_{q = 1}^{n} (\frac{w_{q}^{2}}{\sum_{q = 1}^{n} w_{q} ζ_{pq}^{+}} \times ζ_{pq}^{+}) \end{matrix}$ (31) where $\bar{w}$ = $\frac{w_{q}^{2}}{\sqrt{\sum_{q = 1}^{n} w_{j}^{2}}}$ $s_{p}^{+} = \sum_{q = 1}^{n} (\bar{w_{q}} \times ζ_{pq}^{+})$ (32) $s_{p}^{-} = \sum_{q = 1}^{n} (\bar{w_{q}} \times ζ_{pq}^{-})$ (33) $\begin{matrix} {cc}_{p} & = \frac{s_{p}^{+}}{s_{p}^{+} + s_{p}^{-}} \\ = \frac{\sum_{q = 1}^{n} (\bar{w_{q}} \times ζ_{pq}^{+})}{\sum_{q = 1}^{n} (\bar{w_{q}} \times ζ_{pq}^{+}) + \sum_{q = 1}^{n} (\bar{w_{q}} \times ζ_{pq}^{-})} \end{matrix}$ (34)

8 Example

We consider a decision making situation where a university wants to hire a professor. The criterion under which an applicant is to be evaluated includes C₁: Academics, C₂: Communication skills, C₃: Experience, C₄: Administrative abilities. Let us consider the situation that the weights for these attribute are completely unknown.

First of all define a linguistic terms in term of IT2TFN in Table 1.

Table 1
LINGUISTIC TERMS OF IT2TFS

Linguistic term IT2FS

AbsolutelyLow (AL) $\begin{matrix} (0 & 0 & 0 & 2 & 1) \\ (0 & 0 & 0 & 1 & 0.1) \end{matrix}$

Low (L) $\begin{matrix} (0 & 1 & 1 & 3 & 1) \\ (0 & 1 & 1 & 2 & 0.2) \end{matrix}$

SlightyLow (SL) $\begin{matrix} (1 & 3 & 3 & 5 & 1) \\ (2 & 3 & 3 & 4 & 0.3) \end{matrix}$

Equelly (E) $\begin{matrix} (2 & 3 & 4 & 5 & 1) \\ (2 & 3 & 5 & 5 & 0.5) \end{matrix}$

SlightyHigh (SH) $\begin{matrix} (5 & 7 & 7 & 9 & 1) \\ (6 & 7 & 7 & 8 & 0.9) \end{matrix}$

High (H) $\begin{matrix} (7 & 9 & 9 & 10 & 1) \\ (8 & 9 & 9 & 10 & 0.8) \end{matrix}$

AbsolutelyHigh (AH) $\begin{matrix} (9 & 10 & 10 & 10 & 1) \\ (8 & 9 & 10 & 10 & 0.9) \end{matrix}$

Linguistic term	IT2FS
AbsolutelyLow (AL)	$\begin{matrix} (0 & 0 & 0 & 2 & 1) \\ (0 & 0 & 0 & 1 & 0.1) \end{matrix}$
Low (L)	$\begin{matrix} (0 & 1 & 1 & 3 & 1) \\ (0 & 1 & 1 & 2 & 0.2) \end{matrix}$
SlightyLow (SL)	$\begin{matrix} (1 & 3 & 3 & 5 & 1) \\ (2 & 3 & 3 & 4 & 0.3) \end{matrix}$
Equelly (E)	$\begin{matrix} (2 & 3 & 4 & 5 & 1) \\ (2 & 3 & 5 & 5 & 0.5) \end{matrix}$
SlightyHigh (SH)	$\begin{matrix} (5 & 7 & 7 & 9 & 1) \\ (6 & 7 & 7 & 8 & 0.9) \end{matrix}$
High (H)	$\begin{matrix} (7 & 9 & 9 & 10 & 1) \\ (8 & 9 & 9 & 10 & 0.8) \end{matrix}$
AbsolutelyHigh (AH)	$\begin{matrix} (9 & 10 & 10 & 10 & 1) \\ (8 & 9 & 10 & 10 & 0.9) \end{matrix}$

Decision matric is given as Table 2 by decision maker in term of linguistic terms decision information is given in term of IT2TFN where q represent alternative and p represent attribute Standardize decision matric R produce by formula (12) benefit type criteria in Table 4; Calculate the entropy weights w from decision matrix R first calculate expected value J for using Equation (13) $J = (\begin{matrix} 0.0638 & 0.5521 & 0.8922 & 0.1950 \\ 0.0638 & 0.1936 & 0.5313 & 0.9037 \\ 0.2687 & 0.8939 & 0.7719 & 0.6725 \\ 0.7975 & 0.1299 & 0.5313 & 0.9037 \\ 0.0263 & 0.6146 & 0.1484 & 0.1950 \end{matrix})$ the attribute values of weight becomes by using Equation (17) $w = (\begin{matrix} 0.2868 & 0.2344 & 0.2250 & 0.2537 \end{matrix})$

Table 2

Decision matric

U	m ₁	m ₂	m ₃	m ₄
Al ₁	L	SH	AH	SL
Al ₂	L	E	SH	AH
Al ₃	E	AH	H	L
Al ₄	H	SL	SH	AH
Al ₅	AL	SH	M	SL

Table 3

Attribute value of alternatives

U	m ₁	m ₂	m ₃	m ₄
Al ₁	(0 1 1 3: 1)(0 0 0 1: 0.1)	(5 7 7 9: 1)(6 7 7 8: 0.7)	(9 10 10 10: 1)(8 9 10 10: 0.9)	(1 3 3 5: 1)(2 3 3 4: 0.3)
Al ₂	(0 1 1 3: 1)(0 0 0 1: 0.1)	(2 4 4 5: 1)(2 3 5 5: 0.5)	(5 7 7 9: 1)(6 7 7 8: 0.7)	(9 10 10 10: 1)(8 9 10 10: 0.9)
Al ₃	(2 3 4 5: 1)(2 3 5 5: 0.5)	(9 10 10 10: 1)(8 9 10 10: 0.9)	(7 9 9 10: 1)(8 9 9 10: 0.8)	(0 1 1 3: 1)(0 1 1 2: 0.2)
Al ₄	(7 9 9 10: 1)(8 9 9 10: 0.8)	(1 3 3 5: 1)(2 3 3 4: 0.3)	(5 7 7 9: 1)(6 7 7 8: 0.7)	(9 10 10 10: 1)(8 9 10 10: 0.9)
Al ₅	(0 0 0 1: 1)(0 0 0 1: 0.1)	(5 7 7 9: 1)(6 7 7 8: 0.9)	(2 3 4 5: 1)(2 3 5 5: 0.5)	(1 3 3 5: 1)(2 3 3 4: 0.3)

Table 4

Standardize decision matric

U	m ₁	m ₂	m ₃	m ₄
Al ₁	(0 0.1 0.1 0.3:1)(0 0 0 0.1:0.1)	(0.4444 0.6667 0.6667 0.8889: 1)(0.5 0.6250 0.6250 0.75: 0.7)	(0.8750 1 1 1: 1)(0.75 0.8750 1 1: 0.9)	(0.1 0.3 0.3 0.5 1)(0.2 0.3 0.3 0.4:0.3)
Al ₂	(0 0.1 0.1 0.3: 1)(0 0 0 0.1: 0.1)	(0.1 1 1 1 0.2222 0.3333 0.4444: 1)(0 0.1250 0.3750 0.3750: 0.5)	(0.3750 0.6250 0.6250 0.8750: 1)(0.5 0.6250 0.6250 0.75: 0.7)	(0.9 1 1 1: 1)(0.8 0.9 1 1:0.9)
Al ₃	(0.2 0.3 0.4 0.5: 1)(0.2 0.3 0.5 0.5: 0.5)	(0.8889 1 1 1: 1)(0.75 0.8750 1 1: 0.9)	(0.6250 0.8750 0.8750 1: 1)(0.75 0.8750 0.8750 1: 0.8)	(0 0.1 0.1 0.3: 1)(0 0.1 0.10 2: 0.2)
Al ₄	(0.7 0.9 0.9 1: 1)(0.8 0.9 0.9 1: 0.8)	(0 0.2222 0.2222 0.4444: 1)(0 0.1250 0.1250 0.250: 0.3)	(0.3750 0.6250 0.6250 0.8750: 1)(0.5 0.6250 0.6250 0.75: 0.7)	(0.9 1 1 1: 1)(0.8 0.9 1 1: 0.9)
Al ₅	(0 0 0 0.2: 1)(0 0 0 0.1: 0.1)	(0.4444 0.6667 0.6667 0.8889: 1)(0.5 0.6250 0.6250 0.75: 0.9)	(0 0.1250 0.25 0.3750: 1)(0 0.1250 0.3750 0.3750: 0.5)	(0.1 0.3 0.3 0.5: 1)(0.2 0.3 0.3 0.4: 0.3)

Table 5

Positive ideal target solution and negative ideal target solution

U	m ₁	m ₂	m ₃	m ₄
$\tilde{r^{+}}$	$\begin{matrix} (0.7 0.9 0.9 1 : 1) \\ (0.8 0.9 0.9 1 : 1) \end{matrix}$	$\begin{matrix} (0.8889 1 1 1 : 1) \\ (0.75 0.8750 1 1 : 0.9) \end{matrix}$	$\begin{matrix} (0.8750 1 1 1 : 1) \\ (0.75 0.8750 1 1 : 0.9) \end{matrix}$	$\begin{matrix} (0.9 1 1 1 : 1) \\ (0.8 0.9 1 1 : 0.9) \end{matrix}$
$\tilde{r^{-}}$	$\begin{matrix} (0 0 0 0.2 : 1) \\ (0 0 0 0.1 : 0.8) \end{matrix}$	$\begin{matrix} (0 0.2222 0.2222 0.4444 : 1) \\ (0 0.1250 0.1250 0.2500 : 0.9) \end{matrix}$	$\begin{matrix} (0 0.1250 0.2500 0.3750 : 1) \\ (0 0.1250 0.3750 : 0.9) \end{matrix}$	$\begin{matrix} (0 0.1 0.1 0.3 : 1) \\ (0 0.1 0.1 0.2 : 0.2) \end{matrix}$

Calculate (PITS) and (NITS) of the IT2TFN, it appear as follows by Equations (18,19). To find the GRC with the help of PITS and NITS of each alternative by using the following equation respectively. It is shown as follows by Equations (26, 27) $[ζ_{pq}^{+}]_{4 \times 5} = (\begin{matrix} 0.4250 & 0.6933 & 1 & 0.4841 \\ 0.4252 & 0.4761 & 0.6774 & 1 \\ 0.5463 & 1 & 0.8704 & 0.4203 \\ 0.9249 & 0.4438 & 0.6774 & 1 \\ 0.4147 & 0.7251 & 0.4597 & 0.4841 \end{matrix})$ $[ζ_{pq}^{-}]_{4 \times 5} = (\begin{matrix} 0.8826 & 0.7236 & 0.5780 & 0.8704 \\ 0.8826 & 0.9654 & 0.7547 & 0.5335 \\ 0.7997 & 0.5635 & 0.6139 & 0.9117 \\ 0.5255 & 0.9372 & 0.7547 & 0.5335 \\ 0.8890 & 0.7433 & 1 & 0.8704 \end{matrix})$ Determined the Weighted Grey relation Projections (WGRP) of alternative s_p onto the PITS and NITS, as follows $s_{1}^{+} = 0.31534 s_{2}^{+} = 0.34237 s_{3}^{+} = 0.31497$ $s_{4}^{+} = 0.42939 s_{5}^{+} = 0.32138$ $s_{1}^{-} = 0.34794 s_{2}^{-} = 0.33104 s_{3}^{-} = 0.29367,$ $s_{4}^{-} = 0.22961 s_{5}^{-} = 0.35375$ The value of relative closeness is ${cc}_{i} = \begin{matrix} (0.47542 & 0.50841 & 0.51750 & 0.65158 & 0.47603) \end{matrix}$ Ranking according to relative closeness is cc₄ > cc₃ > cc₂ > cc₅ > cc₁

9 Conclusion

The presented method can be utilized to deal with multi-attribute decision making (MADM) problems when the attribute values are modeled in the form of interval type-2 trapezoidal fuzzy numbers (IT2FNs), and the attribute weights are completely unknown. IT2TFN used for modeling linguistic information have the ability to capture both the intra personal and inter personal uncertainties associated with a linguistic term. The IEM has been employed to evaluate the unknown attribute weights. Lastly, the selection process of a candidate in university has been illustrated to justify the developed approach. The proposed technique though computational more demanding has the ability to resolve decision making problems than can be solved using techniques involving ordinary fuzzy sets.

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