^{1}, where the Uni-Int technique is incorrectly used in case of soft sets instead of fuzzy soft sets.

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.

^{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.

^{7 }If we have two fuzzy soft sets (L,T_{1}) and (M,T_{2}) over a universe set U. Then we define (L,T_{1}) AND (M,T_{2}) is a fuzzy soft set denoted by

The choice value ^{6} of a participant/alternative

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

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 |

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 _{t1} , p_{t2} ,...p_{tn}_{ik} = the count of attributes such that degree of membership grade of p_{ti} ≥ that of p_{tk} .

It can be observed that c_{ik} ∈ {0, 1, 2, ...n} and c_{ik} = n if i=k. Thus c_{ik} indicates an integral number for which p_{t}i dominates p_{t}k 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

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} }.

“Grey relational generating”

In this step we reorder the sequence as

“Difference information”

“Grey relational coefficient”

Where

“Grey relational grade”

Where w1 and w2 are weights of evaluation factor and

“Decision making”.

p_{tk} is the optimal choice, where

^{20} For a universal set _{1}) is known as IVFSS over

An IVFSS is a parameterized family of interval-valued fuzzy subsets of

Consider the set of participants

First decision maker considered the set of parameters,

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

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] |

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]

Now we will find AND product _{ik ,} where

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] |

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

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 |

After computing “AND” product we will compose the 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 |

Now we will calculate the score values

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 p_{t8} is the optimal choice but score values shows that p_{t7} is the best choice. To choose which answer is the best one we use Grey algorithm which have following steps.

From the tables we write the choice value sequence

and sore value sequence

“Grey relational generating”,

We find the values through grey relational generating

In this step we reorder the sequence as

“Difference information”

To find

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

Where

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

“Decision making”

After analysis, we observe that

“Neutral redut” fuzzy soft set with choice values.

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 |

Neutral reduct fuzzy soft set’s comparison table.

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 |

Now we will calculate the score values

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 p_{t8} is the optimal choice but score values shows that p_{t7} is the best choice. To choose which answer is the best one we use grey algorithm which have following steps.

From the tables we write the choice value sequence

and score value sequence

Values generated through “grey relational generating” are

In this step we reorder the sequence

For

“grey relational grade” for w_{1}=w_{2}=0.5

According to grey relational analysis p_{t7} 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(p_{t}i ) ≥ 0.5 then selected participants are p_{t7,} p_{t6}, p_{t4,} p_{t8,} p_{t3}.

Mid-level soft set of neutral reduct fuzzy soft set. Thresholds for parameters {x_{12}, x_{32}, x_{33}, x_{44}, x_{46}} are {0.49, 0.49, 0.64, 0.56, 0.53} respectively. Mid-level soft set’s tabular representation with choice values.

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 p_{t7} and p_{t8} are the best choices but score value from table 3.9 of neutral reduct fuzzy soft set shows that p_{t7} is the optimal decision. To overcome this confusion we use grey algorithm.

From the tables we write the choice value sequence

We compute “grey relational generating”

In this step we reorder the sequence

For

“grey relational grade”, for w_{1}=w_{2}=0.5

According to grey relational analysis p_{t7} 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 _{t7,} p_{t8,} p_{t4,} p_{t6,} p_{t3}.

Pessimistic reduct FSS of

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 |

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

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 |

Tabular representation of Score values presented in table 3.13.

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 p_{t7} and p_{t8} are the best choices but score value table of neutral reduct fuzzy soft set shows that p_{t7} is the optimal decision. To overcome this confusion we use grey algorithm.

From the tables we write the choice value sequence

we compute “grey relational generating”

In this step we reorder the sequence.

For

“Grey relational grade”, for w_{1}=w_{2}=0.5

According to grey relational analysis p_{t7} 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 _{t7,} p_{t8,} p_{t4,} p_{t6,}p_{t3.}

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 IVFSS 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 was used to get on a suitable selection. The approach used here is a correction to the method used by Kong et al. ^{19}, where the Uni-Int technique was incorrectly used in case of soft sets instead of fuzzy soft sets.