Sciresol Sciresol https://indjst.org/author-guidelines Indian Journal of Science and Technology 0974-5645 10.17485/IJST/v13i31.115 An extension of grey relational analysis for interval-valued fuzzy soft sets Touqeer Muhammad touqeer.fareed@uettaxila.edu.pk 1 Saeed M Nauman 1 Salamat Nadeem 2 Mustahsan Muhammad 3 Basic Sciences Department, University of Engineering and Technology Taxila Pakistan Khwaja Fareed University of Engineering and Information Technology RYK Pakistan Department of Mathematics, The Islamia University of Bahawalpur Bahawalpur Pakistan 13 31 2020 Abstract

Background: Neither any analytical (nor numerical) nor any statistical approach is often helpful in a situation where every person has his/her own choice. To cope with such situations usually we have to use fuzzy sets in combination with soft sets, which consist of predicates and approximate value sets as their images. Material: Choice values and comparison table techniques are two common decision-making techniques, which often don’t result in same preference order or optimal choice. To overcome this kind of situation in decision-making problems, grey relational analysis method is used to get on a final decision. Method: Here we have used grey relational analysis method involving “interval-valued fuzzy soft sets” and “AND operation” to deal with such kind of problems. Our approach is also a correction to the method used by Kong et al. 1, where the Uni-Int technique is incorrectly used in case of soft sets instead of fuzzy soft sets. Result: The proposed method is effective in seeking an optimal choice in the case when common decision-making techniques fail to get on a final decision. Conclusion: By using grey relational analysis, a suitable method to choose one object from different choices has been proposed. It overcomes the grayness in decision-making problems for getting on a final decision when one gets too many options and finds it difficult to choose an optimal choice.

Keywords Fuzzy soft set Grey relational analysis Un-Int interval-valued fuzzy soft set None
Introduction

Many problems of real life cannot be solved used typical Mathematical rules involving methods based on precise reasoning. Weaver 2 has categorized the different nature of problems of real life as problems of “organized simplicity” and “disorganized complexity”. The first type of the two involves analytical problems which can be solved used calculus while the second type of problems refer to the statistical approaches for dealing with physical problems at molecular level that involve numerous variables and﻿ randomness of high degree. These problems are highly complementary to one another, under certain situations if one works then other fails. Majority of real life problems lie between these two which Weaver has named as the problems of organized complexity. The introduction of computer technology during world war II helped to deal many problems resembling the problems of organized complexity but it was realized that there are certain limits in dealing with complexity, which are unable to be overcome by either any computer technology or human abilities. Generally to deal with any type of problem, we have to construct a model based on reality aspects or some artificial objects. In construction of any model the factors affecting its usefulness are the credibility of the model, its complexity and uncertainty involved in it. Allowing more uncertainty will help in overcoming the complexity of the model and increasing its credibility. Therefore, the challenge was to develop techniques which can be used to estimate allowable uncertainty for such type of resulting models. The concept of fuzzy sets by Zadeh 3 in 1965 is considered as an evolution for dealing with uncertainty as his concept of fuzzy sets are the sets which do not have price boundaries like the typical sets have.

Though the theory of fuzzy sets has served as the best tool for dealing with uncertainties but scarcity of criterion for modeling different linguistic uncertainties limits its use as is pointed out by Molodtsov 4. To provide a rich platform for parameterizations by overcoming the deficiencies in the fuzzy set theory, Molodtsov 5 introduced the idea of soft sets as a generalization of fuzzy sets. Fuzzy set theory in connection with soft sets have proved to be one of the most effective tool for dealing with uncertain situations some of which are discussed here. Maji et al. 6 propounded the perception of fuzzy soft sets and application of soft sets in a decision-making problems. Theoretic approach regarding “fuzzy soft set” offered by Roy et al. 7. Maji et al. 8 deliberate reduct soft set’s notion and discussed soft set’s postulate. The abstraction of decision-making through comparison table technique discussed by Roy et al. He made decision-making very useful by constructing comparison table technique. Kong et al. 1 analyzed choice values and score values as evaluation bases to make a decision by discussing a counter example. After that Cagman et al. 9 introduced postulate of soft matrix and Uni-Int technique to make a decision for a problem. Un-Int technique facilitate decision maker to works on small number of attributes instead of larger number of attributes for soft set. He constructed Un-int technique for “AND”, “OR”, (AND), (OR) products. He also considered an example of 48 candidate and analyzed it with the help of Un-int technique for “AND” product. To make a decision with uncertain problems Jiang et al. 10, 11 presented “semantic” methodology by using “ontology” and properties of “intuitionistic fuzzy soft set”. Feng et al. 12, 13 enhanced idea by presenting an adjustable approach for “level soft set” and “interval-valued soft set”. Yang et al. 14 has given the conception of “interval-valued fuzzy soft set” “AND operation” and different applications. Majumdar et al. 15 gave the conception of generalized fuzzy soft sets. The soft set’s algebra was presented by Zhan et al. 16. Xu et al. 17 introduced the conception of vague soft sets and its properties. By relating different parameters, Ali et al. 18 introduced some advanced operations of soft set’s concept. Kong et al. 19 gave an algorithm to overcome the problems of adding parameters and suboptimal choices.

Mostly decision-making techniques involve “choice values” technique or “score values” technique for ranking of alternatives which often don’t result in same preference order. To overcome this kind of situation, grey relational analysis method is used to get on a final decision. Here we have used grey relational analysis method involving “interval-valued fuzzy soft sets” and “AND operation” to deal with such kind of problems. Our approach is also a correction to the method used by Kong et al. 1, where the Uni-Int technique is incorrectly used in case of soft sets instead of fuzzy soft sets. Moreover, the technique has been extended to “intuitionistic fuzzy soft sets” and “interval-valued intuitionistic fuzzy soft sets” by imposing different thresholds on different criterion using level soft sets.

Preliminaries

Definition 2.1. 7 For a universal set U and a parameters set E. Let (L, E) be a set of all fuzzy in U then a combination (L, E) is known as fuzzy soft set over , where L is a mapping as described below.

L:EL(U)

Definition 2.2. 7 If we have two fuzzy soft sets (L,T1) and (M,T2) over a universe set U. Then we define (L,T1) AND (M,T2) is a fuzzy soft set denoted by (L,T1)  (M,T2) defined as (L,T1)  (M,T2) = H, T1 X T2 where H(α,β)=L(α) ~ M(β),αT1 and βT2 , ~ is a operation “fuzzy intersection” of two fuzzy soft sets.

Choice value technique

The choice value 6 of a participant/alternative ptiPt is cvi, given by cvi=Σk pikwhere pik are the entries in the given table. We illustrate the idea by discussing an example.

Suppose we have three participants, and we want to select a participant by using choice values technique. Here pt3 is the best choice.

 Pt x1 x3 x5 Choice values pt1 0.1 0.2 0.3 cv1 = 0.6 pt2 0.2 0.3 0.4 cv2 = 0.9 pt3 0.8 0.7 0.5 cv3 = 2.0
Comparison table technique (or Score value technique)

A square table 7 where both the rows and columns involve alternatives/objects is called comparison table. Here each alternative/object is compared with every other alternative/object in the universal set U. For a comparison table involving n object pt1 , pt2 ,...ptn , let cik = the count of attributes such that degree of membership grade of pti ≥ that of ptk .

It can be observed that cik ∈ {0, 1, 2, ...n} and cik = n if i=k. Thus cik indicates an integral number for which pti dominates ptk for all ptk ∈ U. In comparison table technique we use score of an alternative for their ranking process for which we need to calculate the row sum (ri) and column sum (tk) of each alternative computed as ri = k=1ncik and tk = k=1ncik respectively. Here ri is the count of total attributes of U and ti is the count of total attributes for which ptk is dominated by all the members of U. Then the score ji of an alternative/object pti is calculated as

ji=ri-ti
Grey Relational Analysis algorithm

Step 1.

In first step we input the choice value sequence {cv1 , cv2 ,..., cvn } and score value sequence {j1 , j2 ,..., jn }.

Step 2.

“Grey relational generating”

cvi'=cvi-mincvimaxcvj-mincvi,,  j 'vi=jvi-minjvimaxjvi-minjviwhere 1=1,2,3,4,,n

Step 3.

In this step we reorder the sequence as

cv1', j1',cv2', j2',,cm', jn'

Step 4.

“Difference information”

cvmax=Maxcvi',jmax=MaxΔji',Δcvi'=cvmax-cvi',Δji'=jmax-ji' Δmax=MaxΔcv',Δji',Δmin=MinΔcv',Δji', where i=1,2,,n

Step 5.

“Grey relational coefficient”

γcv,cvi=Δmin+χ*ΔmaxΔcij'+χΔ*max and γj,ji=Δmin+χ*ΔmaxΔji'+χ*Δmax

Where χ is called “distinguishing coefficient” and χ0,1. Its principle is to amplify or shorten the amplitude of “grey relative coefficient”.

Step 6.

γpti=w1*γcv,cvj+w2*γj,ji

Where w1 and w2 are weights of evaluation factor and w1+w2=1

Step 7.

“Decision making”.

ptk is the optimal choice, where ptk=max γptr. If decision makers wish to select more than one participants then they will select the participants according to the maximum number of grey relational grade.

Grey relational analysis for interval-valued fuzzy soft sets (IVFSS)

Definition 6.1. 20 For a universal set and parameter set E,T1E, a combination (L, T1) is known as IVFSS over , where F is a mapping such that

L:T1P(U)

An IVFSS is a parameterized family of interval-valued fuzzy subsets of U,sT1,L(s) is referred as the interval fuzzy value set of parameter s. Clearly L(s) is written as L(s)=x,Ls-,Ls+:XU where Ls+ and Ls- be the upper and lower membership’s degrees of x to L(s)respectively.

Example 6.2. Let’s imagine a business organization needs to fill a vacant position. There are 10 participants who applied legally for vacant position. The organization have chosen two decision makers, one is from the panel of directors and second one is from the office of human development. They wish to select a participant to fill a vacant position. They separately judge the desired qualities that are required to fill a vacant position by using grey algorithm based on “AND” operation.

Consider the set of participants Pt=pt1,pt2,pt10 which may be characterized by the set of parameters E=x1,x2,x3,x4,x5,x6. The parameters xiwhere i = 1,2, ... 6, signifies “experience”, “computer knowledge”,“training”,“young age”,“higher education” “good health” respectively.

Step 1.

First decision maker considered the set of parameters, T1=x1,x3,x4 and second decision considered the set of parameters, T2=x2,x3,x4,x6, where T1,T2E .

Step 2.

In this step decision makers assign membership grades to their desired parameters as in the table given below.

<inline-formula id="if-c91abdcd9896"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced><mml:mrow><mml:mi mathvariant="normal">L</mml:mi><mml:mo>,</mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>
 Pt x1 x3 x4 pt1 [0.3,0.5] [0.48,0.5] [0.3,0.8] pt2 [0.4,0.6] [0.54,0.58] [0.5,0.54] pt3 [0.64,0.68] [0.72,0.76] [0.43,0.47] pt4 [0.62,0.68] [0.51,0.55] [0.78,0.82] pt5 [0.88,0.92] [0.71,0.77] [0.39,0.41] pt6 [0.78,0.82] [0.84,0.88] [0.63,0.67] pt7 [0.69,0.72] [0.71,0.78] [0.58,0.62] pt8 [0.52,0.58] [0.61,0.69] [0.78,0.82] pt9 [0.49,0.55] [0.68,0.7] [0.48,0.52] pt10 [0.78,0.82] [0.41,0.47] [0.32,0.38]

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 Pt x2 x3 x4 x6 pt1 [0.1,0.3] [0.48,0.5] [0.31,0.81] [0.58,0.62] pt2 [0.45,0.55] [0.54,0.58] [0.52,0.56] [0.32,0.34] pt3 [0.5,0.7] [0.72,0.76] [0.45,0.48] [0.69,0.71] pt4 [0.68,0.72] [0.51,0.55] [0.79,0.83] [0.87,0.93] pt5 [0.31,0.35] [0.71,0.77] [0.39,0.41] [0.63,0.68] pt6 [0.45,0.55] [0.84,0.88] [0.64,0.68] [0.93,0.98] pt7 [0.88,0.92] [0.71,0.78] [0.58,0.63] [0.78,0.82] pt8 [0.58,0.62] [0.61,0.69] [0.78,0.82] [0.87,0.93] pt9 [0.43,0.45] [0.68,0.7] [0.48,0.52] [0.52,0.58] pt10 [0.47,0.51] [0.41,0.47] [0.33,0.39] [0.3,0.36]

Step 3.

Now we will find AND product LT1 MT2 of the fuzzy soft sets L, T1and M, T2. Here we observe that if we perform the AND product of the above fuzzy soft sets then we will get 3×4=12 parameters of the form eik , where eik=aibki=1,2,3 and k=1,2,3,4 , but here we need fuzzy soft set for the parameters R=x12,x32,x33,x44,x46 . So we will get the resultant of L, T1and M, T2say K, R after performing the AND operation as follows.

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 Pt x12 x32 x33 x44 x46 pt1 [0.1,0.3] [0.1,0.3] [0.48,0.5] [0.3,0.8] [0.3,0.62] pt2 [0.4,0.55] [.45,0.55] [0.54,0.58] [0.5,0.54] [0.32,0.34] pt3 [0.5,0.68] [0.5,0.7] [0.72,0.76] [0.43,0.47] [0.43,0.47] pt4 [0.62,0.68] [0.51,0.55] [0.51,0.55] [0.78,0.82] [0.78,0.82] pt5 [0.31,0.35] [0.31,0.35] [0.71,0.77] [0.39,0.41] [0.39,0.41] pt6 [0.45,0.55] [0.45,0.55] [0.84,0.88] [0.63,0.67] [0.63,0.67] pt7 [0.69,0.72] [0.71,0.78] [0.71,0.78] [0.58,0.62] [0.58,0.62] pt8 [0.52,0.58] [0.58,0.62] [0.61,0.69] [0.78,0.82] [0.78,0.82] pt9 [0.43,0.45] [0.43,0.45] [0.68,0.7] [0.48,0.52] [0.48,0.52] pt10 [0.47,0.51] [0.41,0.47] [0.41,0.47] [0.32,0.38] [0.3,0.36]

Step 4.

We find the optimistic reduct fuzzy soft set for the table.

Optimistic reduct FSS of the table 9
 Pt x12 x32 x33 x44 x46 Choice value pt1 0.3 0.3 0.5 0.8 0.62 cv1 =2.52 pt2 0.5 0.55 0.58 0.54 0.34 cv2 =2.56 pt3 0.68 0.7 0.76 0.47 0.47 cv3 =3.08 pt4 0.68 0.55 0.55 0.82 0.82 cv4 =3.42 pt5 0.35 0.35 0.77 0.41 0.41 cv5 =2.29 pt6 0.55 0.55 0.88 0.67 0.67 cv6 =3.32 pt7 0.72 0.78 0.78 0.62 0.62 cv7 =3.52 pt8 0.58 0.62 0.69 0.82 0.82 cv8 =3.53 pt9 0.45 0.45 0.7 0.52 0.52 cv9 =2.64 0.51 0.47 0.47 0.38 0.36 cv10 =2.19

Step 5.

After computing “AND” product we will compose the comparison table of optimistic reduct FSS.

Comparison table of optimistic reduct FSS
 Pt pt1 pt2 pt3 pt4 pt5 pt6 pt7 pt8 pt9 pt10 pt1 5 2 2 0 2 1 2 0 2 3 pt2 3 5 1 2 3 2 0 0 3 4 pt3 3 4 5 3 4 2 0 3 3 5 pt4 5 4 3 5 4 4 2 3 4 5 pt5 3 2 1 1 5 0 0 1 1 3 pt6 4 5 3 2 5 5 3 1 5 5 pt7 4 5 5 3 5 2 5 3 5 5 pt8 5 5 2 4 4 4 2 5 4 5 pt9 3 2 2 1 4 0 0 1 5 3 pt10 2 1 0 0 2 0 0 0 2 5

Step 6.

Now we will calculate the score values ji=ri-ti where ri denote the column sum and ti denote the row sum as calculated in table 3.6 as follows.

Score values table
 Pt Row sum Column sum Score value pt1 19 37 j1=-18 pt2 23 35 j2=-12 pt3 32 24 j3= 8 pt4 39 21 j4= 18 pt5 17 38 j5=-21 pt6 38 20 j6= 18 pt7 42 14 j7= 28 pt8 40 17 j8= 23 pt9 21 34 j9=-13 pt10 12 43 j10=-31

According to choice values pt8 is the optimal choice but score values shows that pt7 is the best choice. To choose which answer is the best one we use Grey algorithm which have following steps.

Step 1.

From the tables we write the choice value sequence cvi={2.52, 2.56, 3.08, 3.42, 2.29, 3.32, 3.52, 3.53, 2.64, 2.19}

and sore value sequence ji={-18, -12, 8, 18, -21, 18, 28, 23, -13, -31}.

Step 2.

“Grey relational generating”,

We find the values through grey relational generating

c'vi=cvi-mincvimaxcvi-mincvi,  jvi'=jvj-minjvjmaxjvj-minjvj, where i=1,2,3,,10 are

cvi'={0.23,0.28,0.66,0.92,0.07,0.84,0.99,1,0.34,0} and j'i={0.22,0.32,0.66,0.83,0.17,0.83,1,0.92,0.30,0}

Step 3.

In this step we reorder the sequence as cv1', j1', cv1', j1',,cvn', jn' and we get cv1', j1'={0.23,0.22}, cv2', j2'={0.28,0.32}, cv3', j3'={0.66,0.66}, cv4', j4'={0.92,0.83}, cv5', j5'={0.07,0.17}, cv6', j6'={0.84,0.83}, cv7', j7'={0.99,1}, cv8', j8'={1,0.92}, cv9', j9'={0.34,0.30}, cv10', j10'={0,0}.

Step 4.

“Difference information”

To find Δmax=MaxΔcv',Δji' and Δmin=MinΔcv',Δji' we have cvmax=Maxcvi'=1 and jmax=MaxΔji'=1. Calculated values are Δcvi '{0.77,0.72,0.34,0.08,0.93,0.16,0.01,0,0.66,1} and Δji'={0.78,0.68,0.34,0.17,0.83,0.17,0,0.08,0.7,1} So Δmax=1 and Δmin=0

Step 5.

In this step we will find “grey relative coefficient” through

γcv,cvi=Δmin+χ*ΔmaxΔCvi'+χ*Δmax and γj,ji=Δmin+χ*ΔmaxΔji'+χ*Δmax

Where χ is called “distinguishing coefficient” and χ0,1. Its principle is to amplify or shorten the amplitude of “grey relative coefficient”. Here we took χ=0.52. Calculated values are γcv, cvi={0.40, 0.42, 0.60, 0.87, 0.36, 0.76, 0.98, 1, 0.44, 0.84} γj, ji={0.40, 0.43, 0.60, 0.75, 0.38, 0.75, 1, 0.87, 0.43, 0.34}

Step 6.

In this step we find the “grey relational grade” through

γpti=w1* γcv, cvi+w2* γj, ji, where w1 and w2 are weights of evaluation factor and w1 + w2 = 1 but in this thesis w1=w2=0.5. Calculated values are γpt1=0.4, γpt2=0.42, γpt3=0.6, γpt4=0.81, γpts=0.37, γpt6=0.76, γpt7=0.99, γpt8=0.94, γpt9=0.44, γpt10=0.59.

Step 7.

“Decision making”

After analysis, we observe that pt7 is optimal choice. If we select γui0.5 then selected participants according to the maximum are pt7=0.99, pts=0.94, pt4=0.81, pt6=0.76, pt3=0.61, pt10=0.59.

Neutral reduct fuzzy soft set based decision-making

Step 1.

“Neutral redut” fuzzy soft set with choice values.

Neutral reduct fuzzy soft set
 Pt x12 x32 x33 x44 x46 Choice value pt1 0.2 0.2 0.49 0.55 0.47 cv1 =1.91 pt2 0.48 0.5 0.56 0.52 0.33 cv2 =2.39 pt3 0.59 0.6 0.74 0.45 0.45 cv3 =2.83 pt4 0.65 0.53 0.53 0.8 0.8 cv4 =3.31 pt5 0.33 0.33 0.74 0.4 0.4 cv5 =2.2 pt6 0.5 0.48 0.86 0.65 0.65 cv6 =3.14 pt7 0.7 0.74 0.74 0.6 0.6 cv7 =3.38 pt8 0.55 0.6 0.65 0.8 0.8 cv8 =3.4 pt9 0.44 0.44 0.69 0.5 0.5 cv9 =2.57 pt10 0.49 0.44 0.44 0.35 0.33 cv10 =2.05

Step 2.

Neutral reduct fuzzy soft set’s comparison table.

Comparison table of neutral reduct fuzzy soft set
 Pt pt1 pt2 pt3 pt4 pt5 pt6 pt7 pt8 pt9 pt10 pt1 5 2 2 0 2 0 0 0 1 3 pt2 3 5 1 1 3 1 0 0 3 4 pt3 3 4 5 2 5 2 1 3 3 5 pt4 5 4 3 5 4 4 2 3 4 5 pt5 3 2 1 1 5 0 1 1 1 3 pt6 5 4 3 1 5 5 3 1 5 5 pt7 5 5 5 3 5 2 5 3 5 5 pt8 5 5 3 4 4 4 2 5 4 5 pt9 4 2 2 1 4 0 0 1 5 4 pt10 2 2 0 2 0 0 0 0 2 5

Step3.

Now we will calculate the score values ji=ri-ti where ri denote the column sum and ti denote the row sum as calculated in the table as follows.

Score values table
 Pt Row sum Column sum Score value pt1 15 40 j1= -25 pt2 21 35 j2= -14 pt3 33 25 j3= 8 pt4 40 20 j4= 20 pt5 18 37 j5= -19 pt6 37 18 j6= 19 pt7 43 14 j7= 29 pt8 41 17 j8= 24 pt9 23 33 j9= -10 pt10 13 44 j10= -31

According to choice values pt8 is the optimal choice but score values shows that pt7 is the best choice. To choose which answer is the best one we use grey algorithm which have following steps.

Step 1.

From the tables we write the choice value sequence cvi {1.91, 2.39, 2.83, 3.31, 2.2, 3.14, 3.38, 3.4, 2.57, 2.05}

and score value sequence ji={-25, -14, 8, 20, -19, 19, 29, 24, -10, -31}.

Step 2.

Values generated through “grey relational generating” are cvi'={0, 0.32, 0.62, 0.94, 0.19, 0.83, 0.99, 1, 0.44, 0.09} and j'i={0.1, 0.28, 0.65, 0.83, 0.2, 0.83, 1, 0.92 , 0.35, 0}.

Step 3.

In this step we reorder the sequence cv1', j1'={0.0,0.1}, cv2', j2'={0.32,0.28}, cv3', j3'={0.62, 0.65},  cv4', j4'={0.94,0.85}, cv5', j5'={0.19,0.2}, cv6', j6'={0.83, 0.83} cv7', j7'={0.99,1}, cv8', j8'={1,0.92}, cˇ9', j9'={0.44, 0.35}, cv10', j10'={0.09, 0}.

Step 4 .

Δmax=MaxΔcvi',Δji'=1 and Δmin=MinΔcvi',Δji'=0

Step 5.

For χ=0.52“grey relative coefficient” γcv, cvi={0.34, 0.43, 0.58, 0.9, 0.39, 0.75, 0.98, 1, 0.48, 0.36} γj, ji={0.37, 0.42, 0.6, 0.78, 0.39, 0.75, 1, 0.87, 0.44, 0.34}

Step 6.

γpt1=0.36, γpt2=0.42, γpt3=0.59, γpt4=0.84, γpts=0.39, γpts=0.39, γpt6=0.75, γpt7=0.99, γpt8=0.94, γpt9=0.46, γpt10=0.35.

Step 7.

According to grey relational analysis pt7 is the optimal choice. If there are more than one seats then we select the participants according to maximum numbers of grades for example if we take γ(pti ) ≥ 0.5 then selected participants are pt7, pt6, pt4, pt8, pt3.

Mid-level soft set based decision-making

Mid-level soft set of neutral reduct fuzzy soft set. Thresholds for parameters {x12, x32, x33, x44, x46} are {0.49, 0.49, 0.64, 0.56, 0.53} respectively. Mid-level soft set’s tabular representation with choice values.

Mid level soft set
 Pt x12 x32 x33 x44 x46 Choice value pt1 0 0 0 0 0 cv1 =0 pt2 0 1 0 0 0 cv2 =1 pt3 1 1 1 0 0 cv3 =3 pt4 1 1 0 1 1 cv4 =4 pt5 0 0 1 0 0 cv5 =1 pt6 1 0 1 1 1 cv6 =4 pt7 1 1 1 1 1 cv7 =5 pt8 1 1 1 1 1 cv8 =5 pt9 0 0 1 0 0 cv9 =1 pt10 1 0 0 0 0 cv10 =1

Here choice values shows that pt7 and pt8 are the best choices but score value from table 3.9 of neutral reduct fuzzy soft set shows that pt7 is the optimal decision. To overcome this confusion we use grey algorithm.

Step 1.

From the tables we write the choice value sequence cvi = {0, 1, 3, 4, 1, 4, 5, 5, 1, 1} and score value sequence ji={-25, -14, 8, 20, -19, 19, 29, 24, -10, -31}.

Step 2.

We compute “grey relational generating” cvi'={0, 0.2, 0.6, 0.8, 0.2, 0.8, 1, 1, 0.2, 0.2} and ji'={0.1, 0.28, 0.65, 0.83, 0.2, 0.83, 1, 0.92, 0.35, 0}.

Step 3.

In this step we reorder the sequence cv1', j1'={0.0,0.1}, cv2', j2'={0.2,0.28}, cv3', j3'={0.6,0.65},  cv4', j4'={0.8,0.85}, cv5', j5'={0.2,0.2}, cv6', j6'={0.8,0.83},  cv7', j7'={1,1}, cv8', j8'={1,0.92}, cˇ9', j9'={0.2,0.35}, cv10', j10'={0.2,0}.

Step 4.

Δmax=MaxΔcv',Δji'=1 and Δmin=MinΔcvi',Δji'=0

Step 5.

For χ=0.52 generated values of “grey relative coefficient” are γcv, cvi={0.34, 0.39, 0.56, 0.72, 0.39, 0.72, 1, 1, 0.39, 0.39} γj, ji={0.37, 0.42, 0.6, 0.78, 0.39, 0.75, 1, 0.87, 0.44, 0.34}

Step 6.

γpt1=0.36, γpt2=0.40, γpt3=0.58, γpt4=0.75, γpt5=0.39,  γpt6=0.74, γpt7=1, γpt8=0.94, γpt9=0.42, γpt10=0.36.

Step 7.

According to grey relational analysis pt7 is the optimal choice. If there are more than one seats then we select the candidate according to maximum numbers of grades for example if we take γpti0.54 then selected participants are pt7, pt8, pt4, pt6, pt3. ﻿

Pessimistic reduct fuzzy soft set based decision-making

Step 1.

Pessimistic reduct FSS of LT1MT2 with choice values.

Pessimistic reduct fuzzy soft set of <inline-formula id="if-f4ff12b55673"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mfenced><mml:mrow><mml:msub><mml:mi mathvariant="normal">L</mml:mi><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:msub><mml:mo>∧</mml:mo><mml:msub><mml:mi mathvariant="normal">M</mml:mi><mml:msub><mml:mi mathvariant="normal">T</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:msub></mml:mrow></mml:mfenced></mml:math></inline-formula>
 Pt x12 x32 x33 x44 x46 choice value pt1 0.1 0.1 0.48 0.3 0.3 cv1 =1.28 pt2 0.4 0.45 0.54 0.5 0.32 cv2 =2.21 pt3 0.5 0.5 0.72 0.43 0.43 cv3 =2.58 pt4 0.62 0.51 0.51 0.78 0.78 cv4 =3.20 pt5 0.31 0.31 0.71 0.39 0.39 cv5 =2.11 pt6 0.45 0.45 0.84 0.63 0.63 cv6 =3.00 pt7 0.69 0.71 0.71 0.58 0.58 cv7 =3.27 pt8 0.52 0.58 0.61 0.78 0.78 cv8 =3.27 pt9 0.43 0.43 0.68 0.48 0.48 cv9 =2.50 pt10 0.47 0.41 0.41 0.32 0.3 cv10 =1.91

Step 2.

Table of comparison for pessimistic reduct fuzzy soft set presented as follows.

Comparison table of pessimistic reduct fuzzy soft set
 pt pt1 pt2 pt3 pt4 pt5 pt6 pt7 pt8 pt9 pt10 pt1 5 0 0 0 0 0 0 0 0 2 pt2 5 5 1 1 3 1 0 0 2 4 pt3 5 4 5 1 5 2 1 1 3 5 pt4 5 4 4 5 4 4 2 3 4 5 pt5 5 2 0 1 5 0 1 1 1 3 pt6 5 5 3 1 5 5 3 1 5 4 pt7 5 5 5 5 5 2 5 3 5 5 pt8 5 5 4 4 4 4 2 5 4 5 pt9 5 3 2 1 4 0 0 1 5 4 pt10 4 1 0 0 2 1 0 0 1 5

Step 3.

Tabular representation of Score values presented in table 3.13.

Score values table
 Pt Row sum Column sum Score value pt1 7 49 j1=-42 pt2 22 34 j2=-12 pt3 32 24 j3= 8 pt4 40 19 j4= 21 pt5 19 37 j5=-18 pt6 37 19 j6= 18 pt7 45 14 j7= 31 pt8 42 15 j8= 27 pt9 25 30 j9=-5 pt10 14 42 j10=-28

Here choice values shows that pt7 and pt8 are the best choices but score value table of neutral reduct fuzzy soft set shows that pt7 is the optimal decision. To overcome this confusion we use grey algorithm.

Step 1.

From the tables we write the choice value sequence cvi={1.28, 2.21, 2.58, 3.2, 2.11, 3, 3.27, 3.27, 2.5, 1.91}and sore value sequence ji={-42, -12, 8, 21, -18, 18, 31, 27, -5, -28}.

Step 2.

we compute “grey relational generating” cvi'={0, 0.47, 0.65, 0.96, 0.42, 0.86, 1, 1, 0.61, 0.32} and ji'={0, 0.41, 0.68, 0.86, 0.33, 0.82, 1, 0.94, 0.51, 0.19}

Step 3.

In this step we reorder the sequence.

cv1', j1'={0,0}, cv2', j2'={0.47,0.41}, cv3', j3'={0.65,0.68},  cv4', j4'={0.96,0.86}, cv5', j5'={0.42,0.33}, cv6', j6'={0.86,0.82},  cv7', j7'={1,1}, cv8, j8'={1,0.94}, cv9', j9'={0.61,0.51}, cv10',j10'={0.32,0.19}.

Step 4.

Δmax=MaxΔcvi',Δji'=1 and Δmin=MinΔcvi',Δji'=0

Step 5.

For χ=0.52 values of “grey relative coefficient” are

γcv,  cvi={0.34, 0.50, 0.60, 0.93, 0.47, 0.79, 1, 1, 0.57, 0.43} γj, ji={0.34, 0.47, 0.62, 0.79, 0.44, 0.74, 1, 0.90, 0.51, 0.39}

Step 6.