Many problems of real life are not certain which cannot be solved used typical Mathematical rules involving methods based on precise reasoning. In

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

In

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 by

In decision making problems, we often come across situations where we get different optimal choices while using different techniques, like choice value technique and score value technique. To overcome this greyness we use grey relational analysis to get on an optimal solution. Similar situation has been tackled in this article where we resolve the ambiguous situation of different choices based on choice value technique and score value technique. In our work we extend the idea of grey relational analysis by constructing an algorithms that is composed of grey relational analysis and AND operation for intuitionistic fuzzy soft set and interval-valued intuitionistic fuzzy soft set.

Definition 2.1.

Definition 2.2.

The choice value_{ik} 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 (Supplementary table 1).

A square table

It can be observed that _{ti }dominates p_{tk} for all p_{tk} ∈ U. In comparison table technique we use score of an alternative for their ranking process for which we need to calculate the row sum (r_{i}) and column sum (t_{k}) of each alternative computed as _{i }is the count of total attributes of U and t_{i} is the count of total attributes for which p_{tk} is dominated by all the members of U. Then the score j_{i} of an alternative/object p_{ti }is calculated as

Step I.

In first step we input the choice value sequence {c_{v1} , c_{v2} ,..., c_{vn} } and score value sequence {j_{1} , j_{2} ,..., j_{n} }.

Step II.

“Grey relational generating”

Step III.

In this step we reorder the sequence as

Step IV.

“Difference information”

Step V.

“Grey relational coefficient”

where χ is called “distinguishing coefficient” and

shorten the amplitude of “grey relative coefficient”.

Step VI.

“Grey relational grade”

_{1} + w_{2} = 1.

Step VII.

“Decision making”.

p_{tk} is the optimal choice, where

Definition 6.1.

Generally for

Consider the set of participants

Step I.

First decision maker considered the set of parameters,

Step II.

In this step decision makers assign membership grades and non-membership grades to their desired parameters as in table given below. (Supplementary Table 2 and Table 3)

Step III.

Now we will find AND product

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

Step IV.

We find Top-bottom level soft set with choice values, with corresponding parameters values are

Step V.

Top-bottom level soft set’s comparison table.(Supplementary Table 6)

Step VI.

Now we will calculate the score values _{i} denote the row sum of the above comparison table.(Supplementary Table 7)

According to choice values

Step I.

From the tables we write the choice value sequence

Step II.

“Grey relational generating”,

We find the values through grey relational generating

Step III.

In this step we reorder the sequence as

Step IV.

“Difference information”

To find

Step V.

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

Where

Step VI.

In this step we find the grey relational grade through _{1} and w_{2 }are weights of evaluation factor and w_{1} + w_{2} = 1 but in this the- sis w_{1} =w_{2}=0.5. Calculated values are

Step VII.

“Decision making”

After analysis, we observe that pt9 is optimal choice. If we select

Bottom-bottom level soft set of

Here is the case where all the choice values are zero but score values of

Step I.

From the tables we write the choice value sequence

Step II.

We compute “grey relational generating’’

Step III.

In this step we order the sequence as

Step IV.

Step V.

For

Step VI.

Calculated values of “grey relational grade”, for

Step VII.

According to grey relational analysis pt9 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

Definition 8.1.

For any

Consider the set of participants is

Step I.

First decision maker considered the set of parameters,

Step II.

In this step decision makers assign membership grades to their desired parameters as follows. (Supplementary Table 9 and Table 10)

Step III.

In this step we will find AND product

Step IV.

Optimistic-optimistic reduct intuitionistic fuzzy soft set of

Step V.

Mid-level soft set of optimistic-optimistic reduct intuitionistic fuzzy soft set with choice values. For

Step VI.

Here we compute the comparison table of optimistic-optimistic reduct interval valued intuitionistic fuzzy soft set. (Supplementary Table 14)

Step VII.

Now we will calculate the score values

According to choice values there are four participants

Step I.

From the tables we write the choice value sequence

Step II.

“Grey relational generating”

Step III.

In this step we reorder the sequence as

Step IV.

“Difference information”

To find

Step V.

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

Here

Step VI.

In this step we find the grey relational grade through _{1} and w_{2} are weights of evaluation factor and w_{1} + w_{2} = 1 but in this thesis w_{1}=w_{2 }=0.5. Calculated values are

Step VII.

“Decision making”

After analysis, we observe that p_{t7} is optimal choice. If we select

Step I.

Tabular representation of

Step II.

Mid level soft for the above pessimistic-pessimistic reduct interval-valued intuitionistic fuzzy soft set with choice values. For

Step III.

Comparison table of pessimistic-pessimistic reduct interval valued intuitionistic fuzzy soft set. (Supplementary Table 18)

Step IV.

Now we will calculate the score values _{i} denote the row sum as calculated in the table as follows. (Supplementary Table 19)

According to the choice values p_{t6} but score values shows that p_{t7 }is the best choice. To overcome this confusion of optimal decision in between choice values and score values we apply grey algorithm having following steps.

Step I.

From the tables we write the choice value sequence

Step II.

Values generated from “grey relational generating” are

Step III.

In this step we reorder the sequence.

Step IV.

Step V.

For

Step VI.

“Grey relational grade”, for

Step VII.

According to grey relational analysis p_{t6} 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

Here we have dealt with one of the ambiguous situations arising in solving a problem from the class of organized complexity by making use of grey relational analysis technique. By using IFSS and IVIFSS and level soft sets for imposing desired thresholds on different criterion, we arrived at different optimal choices by using Choice value technique and comparison table technique. To resolve the problem of preference order, grey relational analysis method with AND operation was used to get on a suitable selection.