A grey relational projection method for multi attribute decision making based on three trapezoidal fuzzy numbers

Abstract

Multi-criteria decision making (MCDM) problems have been solved involving various types of fuzzy sets. We know that interval type-2 fuzzy sets (IT2FSs) are the most representative known fuzzy sets since they have the ability to capture both type of linguistic uncertainties associated with a word namely, the intra-personal and inter-personal uncertainties respectively. Here for MCDM problems, we will use the three trapezoidal fuzzy numbers (TT2FNs) which are more effective in capturing the uncertainty than IT2FSs, just like triangular fuzzy numbers has a better representational power than simple interval numbers. Moreover, Entropy method is employed for evaluating the values of unknown attribute weights. The ranking method employed here is the grey correlation projection method (GRPM), obtained by joining grey relational method (GRM) and projection method (PM) respectively. Lastly an example will be given to check the productivity of the suggested method.

Keywords

Multi-criteria decision making (MCDM)three trapezoidal fuzzy number (TT2FN)Entropy method (EM)grey correlation projection method (GRPM)

1 Introduction

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 [45] refers to them as the problems of organized complexity. The concept of fuzzy sets (FSs) by Zadeh [54] 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. 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). For solving complex decision making problem, a multiattribute decision making model is presented [5]. Also practical MCDM approach combining hesitant and interval type-2 sets is adopted and put forwarded for permeating the service quality of airline companies [6]. Since then different aggregation operators have been suggested in different applications such as [8 , 57], involving group decision making based on IFSs. Many MCDM models involving IFSs are based on optimization theory, where the optimization techniques are used to find the attribute weights in case of partial information or no information on weights. Decision making under such environments have been implemented in many practical applications like [17 , 57] involving intuitionistic trapezoidal fuzzy numbers (ITFNS). The problems of organized complexity have been very effectively dealt by modeling attribute values in the form of "type-1 trapezoidal fuzzy numbers (T1TFNs)" and ITFNs. But according to Mendel [24, 25], these types of fuzzy numbers lack the ability of effective modeling of linguistic uncertainties. He proposed that since "words mean different things to different people" therefore only IT2FSs are suitable for modeling linguistic uncertainties. IT2FSs are computationally simple and therefore have widely used by many researchers to cope with uncertain situations. [1 , 39] are some examples where different authors have applied IT2FSs for their desired optimal results. Real life decision making problems which cannot be tackled using typical analytical techniques can be handled using different types of fuzzy sets depending upon the situation. Examples are [4 , 44]. Here we will make use of three trapezoidal fuzzy numbers (TT2FNs) in a MCDM situation, which are more effective in capturing the uncertainty than IT2FSs, just like triangular fuzzy numbers has a better representational power than simple interval numbers. Moreover, Entropy method is employed for evaluating the values of unknown attribute weights. The ranking method employed here is the grey correlation projection method (GRPM), obtained by joining grey relational method (GRM) and projection method (PM) respectively. Lastly an illustrative example will be provided to check the credibility of the suggested method.

2 Preliminaries

In the tailing para we will acquaint some fundamental ideas analogous to TT2FNs.

Definition 2.1. [37] A T2Fs ${\bar{A}}^{★}$ is said to be TT2FNs if:

The "Footprint of uncertainty" (FOU) of ${\bar{A}}^{★}$ is a IT2TFNs, which is represented as 〈A^U, A^L〉 where, $\begin{matrix} 〈 A^{U}, A^{L} 〉 = 〈 [a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}, h (A^{U})], \\ [a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}, h (A^{L})] 〉 \end{matrix}$ (1)

The principal membership function of ${\bar{A}}^{★}$ should be a TITFN, represented as A^M, where $A^{M} = 〈 a_{1}^{M}, a_{2}^{M}, a_{3}^{M}, a_{4}^{M}, h (A^{M}) 〉;$

For any 0 ≤ α ≤ 1 its α plane is given as: ${\bar{A}}_{α}^{★} = 〈 a_{1}^{U} + α (a_{1}^{M} - a_{1}^{U}), a_{2}^{U} + α (a_{2}^{M} - a_{2}^{U}), a_{3}^{U} + α (a_{3}^{M} - a_{3}^{U}), a_{4}^{U} + α (a_{4}^{M} - a_{4}^{U}), h (A^{U}) + α [h (A^{M}) - h (A^{U})]; a_{1}^{L} + α (a_{1}^{M} - a_{1}^{L}), a_{2}^{L} + α (a_{2}^{M} - a_{2}^{L}), a_{3}^{L} + α (a_{3}^{M} - a_{3}^{L}), a_{4}^{L} + α (a_{4}^{M} - a_{4}^{L}), h (A^{L}) + α [h (A^{M}) - h (A^{L})] 〉$ In the definition discussed above TT2FNs is analyzed by three TITFNs i.e UMF, the principle membership function, LMF and is represented as: $\begin{matrix} 〈 A^{U}, A^{L} 〉 = (〈 [a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}, h (A^{U})], \\ [a_{1}^{M}, a_{2}^{M}, a_{3}^{M}, a_{4}^{M}, h (A^{M})], \\ [a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{4}^{L}, h (A^{L})]); 〉 \end{matrix}$ (2)

Definition 2.2 [37] Let us consider that ${\bar{A}}_{1}^{★}$ and ${\bar{A}}_{2}^{★}$ be the two positive TT2FNs, and ζ > 0, then the operations on arithmetics are given as:

ADDITION ${\bar{A}}_{1}^{★} + {\bar{A}}_{2}^{★} = 〈 A_{1}^{U} + A_{2}^{U}; A_{1}^{M} + A_{2}^{M}; A_{1}^{L} + A_{2}^{L} 〉$ $= 〈 a_{11}^{U} + a_{21}^{U}, a_{12}^{U} + a_{22}^{U}, a_{13}^{U} + a_{23}^{U}, a_{14}^{U} + a_{24}^{U} \frac{h (A^{U}) . | | A_{1}^{U} | | + h (A_{2}^{U} . | | A_{2}^{U} | |)}{| | A_{1}^{U} | | + | | A_{2}^{U} | |}; a_{11}^{M} + a_{21}^{M}, a_{12}^{M} + a_{22}^{M}, a_{13}^{M} + a_{23}^{M}, a_{14}^{M} + a_{24}^{M}, \frac{h (A^{M}) . | | A_{1}^{M} | | + h (A_{2}^{M} | | A_{2}^{M} | |)}{| | A_{1}^{M} | | + | | A_{2}^{M} | |}; a_{11}^{L} + a_{21}^{L}, a_{12}^{L} + a_{22}^{L}, a_{13}^{L} + a_{23}^{L}, a_{14}^{L} + a_{24}^{L}, \frac{h (A^{L}) . | | A_{1}^{L} | | + h (A_{2}^{L} | | A_{2}^{L} | |)}{| | A_{1}^{L} | | + | | A_{2}^{L} | |} 〉$ where, $| | A_{j}^{U} | | = \frac{a_{11}^{U} + a_{12}^{U} + a_{13}^{U} + a_{14}^{U}}{4}$ , $| | A_{j}^{M} | | = \frac{a_{11}^{M} + a_{12}^{M} + a_{13}^{M} + a_{14}^{M}}{4}$ , $| | A_{j}^{L} | | = \frac{a_{11}^{L} + a_{12}^{L} + a_{13}^{L} + a_{14}^{L}}{4}$ , where (j = 1, 2)

Scalar multiplication $ζ {\bar{A}}^{★}_{1} = 〈 ζ A_{1}^{U}; ζ A_{2}^{M}; ζ A_{3}^{L} 〉$ $= 〈 ζ a_{11}^{U}, ζ a_{12}^{U}, ζ a_{13}^{U}, ζ a_{14}^{U}, ζ h (A_{1}^{U}), ζ a_{11}^{M}, ζ a_{12}^{M}, ζ a_{13}^{M}, ζ a_{14}^{M}, ζ h (A_{1}^{M}), ζ a_{11}^{L}, ζ a_{12}^{L}, ζ a_{13}^{L}, ζ a_{14}^{L}, ζ h (A_{1}^{L}) 〉$

Multiplication ${\bar{A}}_{1}^{★} \times {\bar{A}}_{2}^{★} = 〈 A_{1}^{U} \times A_{2}^{U}, A_{1}^{M} \times A_{2}^{M}, A_{1}^{L} \times A_{2}^{L} 〉$ $= 〈 a_{11}^{U} . a_{21}^{U}, a_{12}^{U} . a_{22}^{U}, a_{13}^{U} . a_{23}^{U}, a_{14}^{U} . a_{24}^{U}, h (A_{1}^{U}) . h (A_{2}^{U}); a_{11}^{M} . a_{21}^{M}, a_{12}^{M} . a_{22}^{M}, a_{13}^{M} . a_{23}^{M}, a_{14}^{M} . a_{24}^{M}, h (A_{1}^{M}) . h (A_{2}^{M}); a_{11}^{L} . a_{21}^{L}, a_{12}^{L} . a_{22}^{L}, a_{13}^{L} . a_{23}^{L}, a_{14}^{L} . a_{24}^{L}, h (A_{1}^{L}) . h (A_{2}^{L}) 〉$

Exponentiation $(A_{1}^{- ★})^{ζ} = 〈 (A_{1}^{U})^{ζ}; (A_{1}^{M})^{ζ}, (A_{1}^{L})^{ζ} 〉$ $= 〈 (a_{11}^{U})^{ζ}, (a_{12}^{U})^{ζ}, (a_{13}^{U})^{ζ}, (a_{14}^{U})^{ζ}, (h (A_{1}^{U}))^{ζ}; (a_{11}^{M})^{ζ}, (a_{12}^{M})^{ζ}, (a_{13}^{M})^{ζ}, (a_{14}^{M})^{ζ}, (h (A_{1}^{M}))^{ζ}; (a_{11}^{L})^{ζ}, (a_{12}^{L})^{ζ}, (a_{13}^{L})^{ζ}, (a_{14}^{L})^{ζ}, (h (A_{1}^{L}))^{ζ} 〉$ Let $\begin{matrix} \bar{A} = (A^{U}, A^{M}, A^{L}) = ([a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}, h (A_{i}^{U})], \\ [a_{1}^{M}, a_{2}^{M}, a_{3}^{M}, a_{4}^{M}, h (A_{i}^{M})], [a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{3}^{L}, h (A_{i}^{L})]) \end{matrix}$ (3) then expected value is defined as: $E_{ζ} = h (A) \times [\frac{(1 - ζ) (a_{1} + a_{2}) + ζ (a_{3} + a_{4})}{2}]$ (4) Here ζ ∈ [0, 1] it shows decision makers optimistic attitude, and is said to be index of optimism. ζ > 0.5 shows decision makers optimistic attitude, ζ < 0.5 shows decision makers pessimistic attitude and ζ = 0.5 if decision maker is moderate. $\begin{matrix} \bar{A} = (A^{U}, A^{M}, A^{L}) = ([a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}, h (A_{i}^{U})], \\ [a_{1}^{M}, a_{2}^{M}, a_{3}^{M}, a_{4}^{M}, h (A_{i}^{M})], [a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{3}^{L}, h (A_{i}^{L})]) \end{matrix}$ (5) then expected value is defined as: $E_{ζ} ({\bar{A}}^{*}) = \frac{E_{ζ} (A^{U}) + E_{ζ} (A^{M}) + E_{ζ} (A^{L})}{3}$ (6)

Here ζ ∈ [0, 1] it shows decision makers optimistic attitude, and is said to be index of optimism. ζ > 0.5 shows decision makers optimistic attitude, ζ < 0.5 shows decision makers pessimistic attitude and ζ = 0.5 if decision maker is moderate.

3 Weight determination by using Entropy method

Uncertainty of a system is measured more effectively using Entropy. As soon as irregularity increases, the value of entropy also rises. Hence there is a direct relationship between Entropy and irregularity. Previously we have discussed in detail the main definitions of FS, FN, T1FNs, T2FNs, IT2FNs and a new type of T2FNs known as TT2FNs. As in this paper the situation is to deal with the case where attribute weights are completely unknown. Various techniques are used to calculate unknown attribute weights. These techniques are Optimization technique, Maximizing deviation method (MDM), Entropy method (EM) and many others. Within that paper EM is utilized to deduce the attribute weights that are not known using Three Trapezoidal Fuzzy numbers.

3.1 MCDM on basis of three trapezoidal fuzzy numbers

In a few MADM issues, consider there are p alternatives A = {L₁, L₂, L₃, . . . . , L_p}, n decision criteria C = {C₁, C₂, C₃, . . . . , C_q}, and W = {w₁, w₂, w₃, . . . . , w_q}, is weight vector of attributes, wherew_y ∈ [0, 1], $\sum_{x = 1}^{n} w_{y} = 1$ , but appropriate value of w_y is not known. For the criteriac_y the attribute value of the alternativeA_x is a three trapezoidal fuzzy number $\begin{matrix} a_{xy} = ([a_{xy}^{U}, a_{xy}^{U}, a_{xy}^{U}, a_{xy}^{U}, h (A_{x}^{U})], \\ [a_{xy}^{M}, a_{xy}^{M}, a_{xy}^{M}, a_{xy}^{M}, h (A_{x}^{M})], \\ [a_{xy}^{L}, a_{xy}^{L}, a_{xy}^{L}, a_{xy}^{L}, h (A_{x}^{L})]) \end{matrix}$ (7) where $h (A_{x}^{U})$ denote the height of upper trapezoidal fuzzy number, $h (A_{x}^{M})$ denotes the height of principal trapezoidal fuzzy number and $h (A_{x}^{L})$ denotes the height of lower trapezoidal fuzzy number on criteria c_j, where 0 ≤ h (A_x) ≤1. Following are the certain steps taken regarding Decision Matrix (DM) on the basis of three trapezoidal fuzzy numbers. First of all take a DM of any suitable order then follow the steps given below.

3.2 Standardization of the DM

To avoid the consequence from various physical measurements to decision outcomes, at first the normalization of matrix for decision making is done. Let us consider that R = (s_xy) _p×q denotes the decision matrix where $s_{xy}^{z} = ([s_{xy}^{1}, s_{xy}^{2}, s_{xy}^{3}, s_{xy}^{4}, h A_{x}])$ , then the methods to normalize the decision matrix in case when we come across two distinct kinds of criteria namely cost type and benefit type are given as under:

3.3 For cost type of criteria

$s_{xy}^{p} = \frac{\max (a_{xy}^{1}) - a_{xy}^{k}}{\max (a_{xy}^{4}) - \min (a_{xy}^{1})} z = 1, 2, 3, 4$ (8)

3.4 For benefit type of criteria

$s_{xy}^{p} = \frac{a_{xy}^{k} - \min (a_{xy}^{1})}{\max (a_{xy}^{4}) - \min (a_{xy}^{1})} z = 1, 2, 3, 4$ (9)

3.5 Conversion of three trapezoidal fuzzy numbers to crisp numbers

As in this paper we are dealing with TT2FNs to determine the unknown attribute weights. So, these three trapezoidal fuzzy numbers are translated into crisp form through use of the formula of expected value. As three trapezoidal fuzzy numbers contains upper membership values, principal membership values and lower membership values respectively. The formula of expected value to be used is shown as under: $\begin{matrix} Let \bar{A} = (A^{U}, A^{M}, A^{L}) = ([a_{1}^{U}, a_{2}^{U}, a_{3}^{U}, a_{4}^{U}, h (A_{x}^{U})], \\ [a_{1}^{M}, a_{2}^{M}, a_{3}^{M}, a_{4}^{M}, h (A_{x}^{M})], [a_{1}^{L}, a_{2}^{L}, a_{3}^{L}, a_{3}^{L}, h (A_{x}^{L})]) \end{matrix}$ (10) then expected value is defined as: $E_{ζ} = h (A) \times [\frac{(1 - ζ) (a_{1} + a_{2}) + ζ (a_{3} + a_{4})}{2}]$ (11) Using this formula we find the expected value of upper membership values, principal membership values and lower membership values in form of crisp number. After that we find average of these three numbers to have a single crisp value. In this way three trapezoidal number is converted into a crisp number. Formula to find average is then given as: $E_{ζ} (A) = \frac{E_{ζ} (A^{u}) + E_{ζ} (A^{M}) + E_{ζ} (A^{l})}{3}$ (12) where ζ ∈ [0, 1] is the indicant of optimism, and it represent DM is optimistic if ζ > 0.5, pessimistic if ζ < 0.5 and ζ = 0.5 for moderate DM. Thus matrix of I_xy is obtained in the form of crisp entries, and measure of insanity of the system.

3.6 Determining attribute weights

Attribute weights are calculated by various ways; like maximizing deviation method (MDM) [36], information entropy method (IEM) [26, 27] and a multitude of optimizing methods here we employ “Information Entropy method" (IEM) to findout the weights of attribute. Shannon [26, 27] for the very first time gave concept of entropy which is one of the notion of thermodynamics. It is also related to communication theory. Entropy is assumed to be comparable to uncertainty. Many researches has been made in other branches using the concept of entropy, like administrative sciences [15, 52] and engineering automation [29, 31]. In these areas of development entropy is used to quantify the disorderly and uneven distribution and measure of chaos within the system. The quality of adequate information is measured by entropy which is perfect, solution of uncertainty from the concept of entropy the entropy would be greater if the degree system chaos is higher which is understandable by the concept of entropy.

Uncertainty of a system is measure more effectively using Entropy. As soon as irregularity increases, the value of entropy also rises. Hence there is a direct relationship between Entropy and irregularity. We will explain idea of entropy with the help of an example. It is observed that entropy of earth is increasing day by day, as a result earth is loosing its order tremendously from very first day of its formation. Entropy is usually related with information. Hence as irregularity increases probability decreases as a result the information will decrease.

Entropy value should be larger if for all alternatives terms of attribute should have less difference under one attribute, and hence least information is provided for ranking alternatives and it plays only a minor part in preference system. If difference is large for all alternatives in between their attribute values, then the entropy value of such attribute is less, and hence plays vital role in finding the best alternative. Thus small weight would be given if entropy value of an attribute is large and large weight would be assigned if entropy value is small. While working on three trapezoidal fuzzy number classical entropy method fails but is usually appropriate for attribute values that takes shape of crisp entries.

3.7 Calculation of entropy value of attribute

The entropy value of an attribute is calculate using: $H_{y} = - k \times \sum_{x = 1}^{n} f_{xy} \ln f_{xy} (1 \leq x \leq p, 1 \leq y \leq q)$ (13) where $k = \frac{1}{lpq}$ and if f_xy = 0,then 0 × ln0 = 0 and $f_{xy} = \frac{I (r_{xy})}{\sum_{x = 1}^{q} I_{(} r_{xy})} (1 \leq x \leq p), (1 \leq y \leq q)$ (14) The entropy weight is calculated as: $w_{y} = \frac{(1 - H_{y})}{\sum_{x = 1}^{q} (1 - H_{y})}$ (15) Above formula is used to determine weight vector Zhou et al. [58] has supposed the actuality that while H_y → 1 (y = 1, 2, 3, . . . . . . . . n), there arises a collective diversity for entropy weights, because of slight difference between assorted attributes. This formula was then found unreasonable.zhou et al. then recommended a refined formula for measuring weight, which is defined as: $w_{y} = \frac{\sum_{y = 1}^{n} H_{y} + 1 - 2 \times H_{y}}{\sum_{y = 1}^{q} (\sum_{y = 1}^{q} H_{y} + 1 - 2 \times H_{y})} (1 \leq y \leq q)$ (16)

4 Ranking of alternatives on basis of GCPM

Deng [10] put forth the grey theory, its adequate mathematical way to pledge the system scrutiny portrayed by inadequate information. In grey system the relational analysis is a form of perceptible scrutiny for interpreting alternatives. Precarious relations like relation among things among elements and and behaviors etc are referred by the grey relation. Grey theory is broadly connected in fields for example frameworks examination, information handling, demonstrating and expectation, as well as control and basic leadership. GRA is a significant piece of grey relational analysis, that is reasonable for taking care of issues with arduous relationships between various components and factors. GRA has been effectively connection clarifying the variety of MADM issues, example hiring election, employing choice the reclamations making arrangements for power appropriation organizations, recognition of silicon disk cutting imperfections and so on.

MADM problems can be solved by GRA by joining the complete territory of attribute terms actually supposed for every alternative into individual term. Thus initial complication will be reduced into single attribute decision making problem, so after GRA process alternatives with various attributes can be correlated easily. Thus approach enroute in SAW and TOPSIS is somewhat analogous to the procedure of joining attribute terms into an individual term. The attribute values of all alternatives are translated into comparability sequence by discarding the outcome from different physical dimensions. The grey relational coefficient is calculated between all the comparability sequence and reference sequence. Lastly based on grey relational coefficient(GRC), GRG is calculated between reference sequence and every comparability sequence.

4.1 The fuzzy positive ideal solution (FPIS) and negative ideal solution (FNIS)

Trapezoidal FPIS for a standardized trapezoidal fuzzy decision matrix is as follows: $s^{+} = (s_{1}^{+}, s_{2}^{+}, . . . . . . . . . ., s_{q}^{+})$ (17) where $s_{y}^{+}$ is evaluated as: $\begin{matrix} s_{y}^{+} = ([s_{y}^{1} +, s_{y}^{2} +, s_{y}^{3} +, s_{y}^{4} +]); h (A_{x}) \\ = ([\underset{︸}{max_{x}} (s_{xy}^{1}), \underset{︸}{max_{x}} (s_{xy}^{2}), \underset{︸}{\max_{x}} (s_{xy}^{3}), \\ \underset{︸}{max_{x}} (s_{xy}^{4})]; \underset{︸}{max_{x}} h (A_{x})) \end{matrix}$ (18)

Also the trapezoidal FNIS can be evaluated using: $s^{-} = (s_{1}^{-}, s_{2}^{-}, . . . . . . . . . ., s_{q}^{-})$ (19) where each of $s_{y}^{-}$ can be calculated using the following relation, Fuzzy positive ideal solution can also be find by another method directly as: $s_{y}^{+} = ([1, 1, 1, 1]; 1)$ and $s_{y}^{-} = ([0, 0, 0, 0]; 0)$

4.2 Calculation of Grey Relational Coefficient (GRC)

Using the FPIS, GRC of each alternative is calculated. The from FPIS the GRC of each alternative is defined as: $χ_{xy}^{+} = \frac{F^{+} + ρ E^{+}}{d_{xy}^{+} + ρ E^{+}}$ (20) where hamming distance is defined as $\begin{matrix} d (\bar{A}, \bar{B}) = (\sum_{i = 1}^{m} (a_{i}^{U} - b_{i}^{U})^{2} + \\ (a_{i}^{M} - b_{i}^{M})^{2} + (a_{i}^{L} - b_{i}^{L})^{2}) (+ h ((A_{x})^{U} - h (B_{x})^{U}) + \\ h ((A_{x})^{M} - h (B_{x}^{M}) + h ((A_{x})^{L} - h (B_{x}^{L}))^{\frac{1}{2}} \end{matrix}$ (21) $F^{+} = \underset{︸}{min_{x}} \underset{︸}{min_{y}} d_{xy}^{+}$ $E^{+} = \underset{︸}{max_{x}} \underset{︸}{max_{y}} d_{xy}^{+}$ and σ denotes the resolution coefficient, and σ ∈ (0, 1). Generally its value is 0.5. Likewise, from FNIS the GRC of each alternative is given as: $χ_{xy}^{-} = \frac{F^{-} + ρ E^{-}}{d_{xy}^{-} + ρ E^{-}}$ (22) $F^{-} = \underset{︸}{min_{x}} \underset{︸}{min_{y}} d_{xy}^{-}$ $E^{+} = \underset{︸}{max_{x}} \underset{︸}{max_{y}} d_{xy}^{-}$ and σ represents the resolution coefficient, and σ ∈ (0, 1). Generally its value is 0.5.

4.3 Calculation of Grey Relational Grade (GRG)

From FPIS the GRG is evaluated by the formula given below: $χ_{x}^{+} = \sum_{y = 1}^{q} w_{y} χ_{xy}^{+}$ (23) From FNIS the GRG is evaluated by the formula given below: $χ_{x}^{-} = \sum_{y = 1}^{q} w_{y} χ_{xy}^{-}$ (24)

4.4 Ranking of alternatives

The idea of GRA method is that the optimal alternatives ought to have the "grey relation’s highest degree from the FPIS and "gray relation’s least degree from the FNIS.

5 Projection Method

Definition 5.1. let β = (β₁, β₂, . . . , β_q) andγ = (γ₁, γ₂, . . . , γ_q) are two vectors, then between vectors β and γ the cosine of angle is as under: $cos (β, γ) = \frac{\sum_{y = 1}^{q} (β_{y} γ_{y})}{\sqrt{\sum_{y = 1}^{n} β_{y}^{2}} \times \sqrt{\sum_{y = 1}^{q} γ_{y^{2}}}}$ (25) obviously, the range of cosine of angle included is: 0 < cos(β, γ) ≤1, and the direction of angles β and γ is more concordant if value of cos(β, γ) is higher.

Definition 5.2. Let us suppose that β = (β₁, β₂, . . . , β_q), then $| | β | | = \sqrt{\sum_{y = 1}^{q} β_{y}^{2}}$ represents the norm of the vector β.

Direction and norm are two major parts by which a vector is composed. The cosine of angle that is cos(β, γ) determines whether their directions are accordant, but cannot ponder upon magnitude of norm. The similarity between two vectors can only be quantified through the projection method if we want to consider the cosine of angle and norm magnitude together. Projection method can be defined as under:

Definition 5.3. [41, 42] Let us consider that β = (β₁, β₂, . . . , β_q) and γ = (γ₁, γ₂, . . . , γ_q) are the two vectors, then the projection of vector β over γ will defined as: $\begin{matrix} p (β) = | | β | | cos (β, γ) = \sqrt{\sum_{y = 1}^{q} β_{y}^{2}} \times \frac{\sum_{y = 1}^{q} (β_{y} γ_{y})}{\sqrt{\sum_{y = 1}^{q} β_{y}^{2}}} \times \\ \sqrt{\sum_{y = 1}^{q} γ_{y^{2}}} = \frac{\sum_{y = 1}^{q} (β_{y} γ_{y})}{\sqrt{\sum_{y = 1}^{n} β_{y}^{2}}} \end{matrix}$ (26) The vector γ is more closer to vector β if value of p (β) should be higher. Each alternatives projection onto FPIS is obtained for a MCDM problem. Vector γ gets FPIS and vector β takes individually for every alternative.

5.1 Grey correlation projection method

In trend curve the change in the data is evaluated by GRA method, and actually it is the degree of closeness to shape of curve. The role courting in statistics curve between every alternative and ideal solution is asserted by projection method. Grey correlation and projection method collectively form GCP method. Both methods are characterized by their own. For each alternative and positive ideal solution Grey correlation coefficient matrix (GCCM) is given as: $χ^{+} = (\begin{matrix} χ_{11}^{+} & χ_{12}^{+} & . . . & χ_{1 q}^{+} \\ χ_{21}^{+} & χ_{22}^{+} & . . . & χ_{2 q}^{+} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ χ_{p 1}^{+} & χ_{p 2}^{+} & . . . & χ_{pq}^{+} \end{matrix})$ and between PIS and PIS, WCCM is: $χ_{\circ}^{+} = (χ_{\circ 1}^{+}, χ_{\circ 2}^{+}, . . . . . . . ., χ_{\circ q}^{+}) = (\underset{︸}{1, 1, 1, . . . . ., 1})$

5.2 Weighed grey correlation matrix (WGCM)

Suppose that their is a (WGCM) Y which can be obtained by GCCM (matrix) and weight vector denoted by w, then: ${\tilde{χ}}^{+} = (\begin{matrix} w_{1} χ_{11}^{+} & w_{2} χ_{12}^{+} & . . . & w_{q} χ_{1 q}^{+} \\ w_{1} χ_{21}^{+} & w_{2} χ_{22}^{+} & . . . & w_{q} χ_{2 q}^{+} \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ w_{1} χ_{p 1}^{+} & w_{2} χ_{p 2}^{+} & . . . & w_{q} χ_{pq}^{+} \end{matrix})$ and the WGCM between PIS with PIS: $\begin{matrix} Y_{\circ}^{+} = (w_{1} χ_{\circ 1}^{+}, w_{2} χ_{\circ 2}^{+}, . . . . . . . ., w_{3} χ_{\circ q}^{+}) \\ = (\underset{︸}{w_{1}, w_{2}, w_{3}, . . . . ., w_{q}}) \end{matrix}$ Therefore for an alternative $A_{x}^{+}$ onto the FPIS s⁺ the weighed GCP is obtained. For xth row Y_x in WGCD matrix Y the weighed grey correlation projection (WGCP) of alternative $A_{x}^{+}$ , onto the FPIS is obtained by using 26 as under: $\begin{matrix} P_{x}^{+} = | | Y_{x} | | cos (Y_{x}, Y_{\circ}) = \sqrt{\sum_{y = 1}^{q} (w_{y} χ_{xy}^{+})^{2}} \times \\ \frac{\sum_{y = 1}^{n} ((w_{y} χ_{xy}^{+}) \times w_{y})}{\sqrt{\sum_{y = 1}^{q} (w_{y} χ_{xy}^{+})^{2}} \times \sqrt{\sum_{y = 1}^{q} w_{y}^{2}}} \\ = \frac{\sum_{y = 1}^{q} (w_{y}^{2} χ_{xy}^{+})}{\sqrt{\sum_{y = 1}^{q} w_{y}^{2}}} = \sum_{y = 1}^{q} (\frac{w_{y}^{2}}{\sqrt{\sum_{y = 1}^{q} w_{y}^{2}}} \times χ_{xy}^{+}) \end{matrix}$ (27) For grey correlation projection, let its weight be ${\bar{w}}_{y} = \frac{w_{y}^{2}}{\sqrt{\sum_{y = 1}^{q} w_{y}^{2}}}$ , then $P_{x}^{+} = \sum_{y = 1}^{q} ({\bar{w}}_{y} \times χ_{xy}^{+})$ (28) Thus for an alternative A_x onto the FNIS s^- the WGCP is obtained. For xth row Y_x in WGCD matrix Y the WGCP of alternative A_x, onto the FNIS is obtained by following the same procedure and using the formula discussed earlier for $P_{x}^{+}$ : $P_{x}^{-} = \sum_{y = 1}^{q} ({\bar{w}}_{y} \times χ_{xy}^{-})$ (29) where ${\bar{w}}_{y} = \frac{w_{y}^{2}}{\sqrt{\sum_{y = 1}^{q} w_{y}^{2}}}$

Table 1

Defining the linguistics terms and assigning TT2FNs to each of them

Linguistic term	TT2TFNS
HIS	(0, 0, 0.12, 1.95, 1; 0, 0, 0.08, 0.98, 1; 0, 0, 0.03, 0.64, 1)
IS	(0, 0, 0.62, 2.61, 1; 0, 0, 0.48, 1.48, 1; 0, 0, 0.07, 0.97, 1)
FPIS	(0.57, 1.98, 3.23, 4.39, 1; 0.98, 2.48, 2.98, 3.98, 1; 2.27, 2.68, 2.68, 3.19; 0.42)
O	(3.57, 4.73, 5.48, 6.89, 1;3.98, 4.98, 4.98, 5.98, 1; 4.84, 5.01, 5.01, 5.12, 0.27)
PS	(5.36, 7.48, 8.73, 9.79, 1; 6.48, 7.98, 8.48, 9.48, 1; 7.77, 8.20, 8.20, 8.79, 0.45)
S	(7.35, 9.39, 10, 10, 1; 8.48, 9.48, 10, 10, 1; 8.70, 9.89, 10, 10, 1)
HS	(8.66, 9.89, 10, 10, 1; 8.98, 9.93, 10, 10, 1; 9.59, 9.59, 10, 10, 1)

5.3 Calculation of relative closeness to FPIS and ranking of alternatives

Estimation of GCP of all the alternatives onto FPIS or alternatives are ranked by projection to FNIS. If the value of similarity measure of an alternative onto FPIS is higher then such alternative with higher value will be closer to FPIS, hence is said to be the best alternative. If the value of projection of an alternative onto FNIS is smaller it would be more far to FNIS and hence is said to be the best alternative. So to consider predominantly both projections on FPIS and FNIS, we consider the relative closeness of each alternative to FPIS, which is given as: $\begin{matrix} {CC}_{x} = \frac{p_{x}^{+}}{p_{x}^{+} + p_{x}^{-}} = \\ \frac{\sum_{y = 1}^{q} ({\bar{w}}_{y} \times χ {xy}^{+})}{\sum_{y = 1}^{n} ({\bar{w}}_{y} \times ξ {xy}^{+}) + \sum_{y = 1}^{q} ({\bar{w}}_{y} \times χ {xy}^{-})} \end{matrix}$ (30)

6 Example

Assume that there are five different locations for constructing a bridge for defence purpose, set of alternatives is denoted as A = {L₁, L₂, L₃, L₄, L₅}. These locations and bridge conditions are to be evaluated under four criteria that are bearing capacity, length of place, wall piers, and topography and set of these four attributes is denoted by C = {C₁, C₂, C₃, C₄}. Since weight vector in this case is completely unknown, (which is given by decision maker). For conveying the estimation information in linguistics form, a DM makes use of linguistic term set.

Table 2
Evaluation values given by decision maker

Alternatives length seasonless bearing capacity topography

L ₁ PS S HS HS

L ₂ S PS HS S

L ₃ HS S O HS

L ₄ HS S S PS

L ₅ HS HS S O

Alternatives	length	seasonless	bearing capacity	topography
L ₁	PS	S	HS	HS
L ₂	S	PS	HS	S
L ₃	HS	S	O	HS
L ₄	HS	S	S	PS
L ₅	HS	HS	S	O

Table 3

Evaluation values given by decision maker

Alternative⇒	C ₁	C ₂	C ₃	C ₄
L ₁	$\begin{matrix} (5.36, 7.48, 8.73, 9.79, 1) \\ (6.48, 7.98, 8.48, 9.48, 1) \\ (7.77, 8.20, 8.20, 8.79, 0.45) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (3.57, 4.73, 5.48, 6.89, 1) \\ (3.98, 4.98, 4.98, 5.98, 1) \\ (4.84, 5.01, 5.01, 5.12, 0.27) \end{matrix}$
L ₂	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (5.36, 7.48, 8.73, 9.79, 1) \\ (6.48, 7.98, 8.48, 9.48, 1) \\ (7.77, 8.20, 8.20, 8.79, 0.45) \end{matrix}$	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$
L ₃	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (3.57, 4.73, 5.48, 6.89, 1) \\ (3.98, 4.98, 4.98, 5.98, 1) \\ (4.84, 5.01, 5.01, 5.12, 0.27) \end{matrix}$	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$
L ₄	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (5.36, 7.48, 8.73, 9.79, 1) \\ (6.48, 7.98, 8.48, 9.48, 1) \\ (7.77, 8.20, 8.20, 8.79, 0.45) \end{matrix}$
L ₅	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (8.66, 9.89, 10, 10, 1) \\ (8.98, 9.93, 10, 10, 1) \\ (9.59, 9.59, 10, 10, 1) \end{matrix}$	$\begin{matrix} (7.35, 9.39, 10, 10, 1) \\ (8.48, 9.48, 10, 10, 1) \\ (8.70, 9.89, 10, 10, 1) \end{matrix}$	$\begin{matrix} (3.57, 4.73, 5.48, 6.89, 1) \\ (3.98, 4.98, 4.98, 5.98, 1) \\ (4.84, 5.01, 5.01, 5.12, 0.27) \end{matrix}$

Eq. 12 is employed to calculate the expected values of each entry of the above matrix as under: I_xy= $[\begin{matrix} 6.5510 & 9.4408 & 9.7500 & 2.1096 \\ 9.4408 & 6.5510 & 9.7500 & 6.3792 \\ 9.7500 & 9.4408 & 3.8329 & 6.5375 \\ 9.7500 & 9.4408 & 9.4408 & 3.9377 \\ 9.7500 & 9.7500 & 9.4408 & 2.1096 \end{matrix}]$ Further eqs. 13, 14 and 16 respectively, are employed for evaluation of the attribute weights as following: f_xy= $[\begin{matrix} 0.144800 & 0.211566 & 0.230963 & 0.100105 \\ 0.208675 & 0.146806 & 0.230963 & 0.302711 \\ 0.215509 & 0.211566 & 0.090795 & 0.310225 \\ 0.215509 & 0.211566 & 0.223639 & 0.186855 \\ 0.215509 & 0.218495 & 0.223639 & 0.100105 \end{matrix}]$

H_y = $[\begin{matrix} 1.1535 & 1.1540 & 1.1287 & 1.0813 \end{matrix}]$

W_y= $[\begin{matrix} 0.24630 & 0.24622 & 0.25011 & 0.25737 \end{matrix}]$ For TT2FNs we will evaluate FPIS relative to each of the attributes by making use of the eqs. 17, 18 as:

and FNIS relative to each attribute by using 19, as:

By using eqs. 20, we will evaluate the values of GRC of each alternative from FPIS as: $χ_{xy}^{+}$ = $[\begin{matrix} 0.55714 & 0.82082 & 1.00000 & 0.37472 \\ 0.82082 & 0.55714 & 1.00000 & 0.82005 \\ 1.00000 & 0.82082 & 0.33333 & 0.85310 \\ 1.00000 & 1.00000 & 0.82082 & 0.62082 \\ 1.00000 & 1.00000 & 0.82082 & 0.37472 \end{matrix}]$ Similarly, by using equations 22, we will evaluate the values of GRC of each alternative from FNIS as: $χ_{xy}^{-}$ = $[\begin{matrix} 1.00000 & 0.66984 & 0.36347 & 0.95870 \\ 0.66984 & 1.00000 & 0.36347 & 0.42008 \\ 0.60890 & 0.66984 & 0.98755 & 0.40776 \\ 0.60890 & 0.66984 & 0.37990 & 0.51561 \\ 0.60890 & 0.60890 & 0.37990 & 0.95870 \end{matrix}]$ After calculating the grey relational coefficient we will obtain values for $P_{i}^{+}$ and $P_{i}^{-}$ by employing equations 27 and 29 as: $P_{1}^{+} = 0.33763$ , $P_{2}^{+} = 0.40082$ , $P_{3}^{+} = 0.39414$ , $P_{4}^{+} = 0.39819$ , $P_{5}^{+} = 0.39107$ $P_{1}^{-} = 0.37788$ , $P_{2}^{-} = 0.29911$ , $P_{3}^{-} = 0.32784$ , $P_{4}^{-} = 0.26580$ , $P_{5}^{-} = 0.32470$

Table 4

Evaluation values given by decision maker

{\tilde{s_{1}}}^{+}

\begin{matrix} [(8.66000, 9.89000, 10.00000, 10.00000, 1.00000) \\ (8.98000, 9.93000, 10.00000, 10.00000, 1.00000) \\ (9.59000, 9.95000, 10.00000, 10.00000, 1.00000)] \end{matrix}

Table 5

Evaluation values given by decision maker

{\tilde{s_{2}}}^{+}

\begin{matrix} [(8.66000, 9.89000, 10.00000, 10.00000, 1.00000) \\ (8.98000, 9.93000, 10.00000, 10.00000, 1.00000) \\ (9.59000, 9.95000, 10.00000, 10.00000, 1.00000)] \end{matrix}

Table 6

Evaluation values given by decision maker

{\tilde{s_{3}}}^{+}

\begin{matrix} [(8.66000, 9.89000, 10.00000, 10.00000, 1.00000) \\ (8.98000, 9.93000, 10.00000, 10.00000, 1.00000) \\ (9.59000, 9.95000, 10.00000, 10.00000, 1.00000)] \end{matrix}

Table 7

Evaluation values given by decision maker

{\tilde{s_{4}}}^{+}

\begin{matrix} [(0.00000, 0.00000, 0.00000, 0.00000, 0.00000) \\ (8.98000, 9.93000, 10.00000, 10.00000, 1.00000) \\ (9.59000, 9.95000, 10.00000, 10.00000, 1.00000)] \end{matrix}

Table 8

Evaluation values given by decision maker

{\tilde{s_{1}}}^{-} =

\begin{matrix} [(5.36000, 7.48000, 8.73000, 9.79000, 1.00000) \\ (6.48000, 7.98000, 8.48000, 9.48000, 1.00000) \\ (7.77000, 8.20000, 8.20000, 8.79000, 1.00000)] \end{matrix}

Table 9

Evaluation values given by decision maker

{\tilde{s_{2}}}^{-} =

\begin{matrix} [(5.36000, 7.48000, 8.73000, 9.79000, 1.00000) \\ (6.48000, 7.98000, 8.48000, 9.48000, 1.00000) \\ (7.77000, 8.20000, 8.20000, 8.79000, 1.00000)] \end{matrix}

Table 10

Evaluation values given by decision maker

{\tilde{s_{3}}}^{-} =

\begin{matrix} [(3.57000, 4.74000, 5.48000, 6.89000, 1.00000) \\ (3.98000, 4.98000, 4.98000, 5.98000, 1.00000) \\ (4.84000, 5.01000, 5.01000, 5.12000, 1.00000)] \end{matrix}

Table 11

Evaluation values given by decision maker

{\tilde{s_{4}}}^{-} =

\begin{matrix} (0.00000, 0.00000, 0.00000, 0.00000, 0.00000) \\ (3.98000, 4.98000, 4.98000, 5.98000, 1.00000) \\ (4.84000, 5.01000, 5.01000, 5.12000, 1.00000) \end{matrix}

Lastly, 30 is employed to calculate the relative closeness of each alternative to FPIS as: CC₁ = 0.47187, CC₂ = 0.57266, CC₃ = 0.54591, CC₄ = 0.59969, CC₅ = 0.54637 The ranking order of alternatives is obtained in accordance to the relative closeness to FPIS and ranking order thus obtained is: $L_{4} > L_{2} > L_{5} > L_{3} > L 1$ Five different locations for constructing a bridge for defence purpose and various bridge conditions were considered in this example. These locations and bridge conditions were evaluated under four criteria that were bearing capacity, length of place, wall piers and topography. Thus out of all these locations, location four is found to be the best in accordance to above mentioned conditions for the construction of a bridge.

7 Conclusion

For MCDM problems, though computationally more demanding but a more representative TT2FN has been used involving three TITFNs. It is therefore more effective in capturing the uncertainty than IT2FSs, just like triangular fuzzy numbers has a better representational power than simple interval numbers. We have considered the environment where there is no information on attribute weights. Therefore, Entropy method has been employed for evaluating the values of attribute weights for which the attribute values were changed into crisp numbers by using the expected value of a TT2FN. Grey correlation projection method (GRPM) has been employed to rank the alternatives. An illustrative example elaborates the practicality of the method by considering various locations for a bridge to be constructed for security purposes. Work is in progress to extend the proposed technique for other decision making problems. In future we expect that TT2FN can be employed for solving models where we want to capture the inter-personal uncertainties involved in them.

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A grey relational projection method for multi attribute decision making based on three trapezoidal fuzzy numbers

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 MCDM on basis of three trapezoidal fuzzy numbers

3.3 For cost type of criteria

3.7 Calculation of entropy value of attribute

4.1 The fuzzy positive ideal solution (FPIS) and negative ideal solution (FNIS)

5 Projection Method

5.2 Weighed grey correlation matrix (WGCM)

Table 2 Evaluation values given by decision maker Alternatives length seasonless bearing capacity topography L 1 PS S HS HS L 2 S PS HS S L 3 HS S O HS L 4 HS S S PS L 5 HS HS S O

References

Table 2
Evaluation values given by decision maker

Alternatives length seasonless bearing capacity topography

L ₁ PS S HS HS

L ₂ S PS HS S

L ₃ HS S O HS

L ₄ HS S S PS

L ₅ HS HS S O