Solution to a Soft Fuzzy Group Decision-Making Problem Involving a Soft Fuzzy Number Valued Information System

In this paper, we introduce an operation union on the collection of soft fuzzy numbers [1] related to multi-parameter set and elucidated with a hypothetical example. For a given soft fuzzy number valued information system ( ), we define a strict partial ordering and a fuzzy number valued utility function on the initial universal set relating to each attribute, which in turn yields a utility soft information corresponding to each entity. We also define a finite collection of soft fuzzy number valued information systems, soft fuzzy number valued hierarchical information systems and their corresponding soft unions. A group decision-making problem with individual attribute set for decision makers wherein the perceptions are expressed using soft fuzzy numbers is modelled involving the collection of soft fuzzy number valued information systems. Such a problem is called soft fuzzy group decision-making problem. A new procedure to solve the problem of finding importance (weights) of the decision makers in such a situation is also proposed in which utility soft information plays a major role. An algorithm is developed to solve the same. Validation of the methodology is shown with an illustration of real life situation.


Introduction
In real life situations, we come across problems that comprises of exact, imprecise or uncertain, simple or complex information that needs to be analysed for various requirements. We begin with the collection of facts that are available to us and end at a stage where we are equipped with models and methodologies applicable to handle the existing scenario. The search for new models to interpret the knowledge acquired and methods to handle any situations in various fields of subjects is an ongoing research process. To process and analyse any aspect of an entity or a collection of entities, there is a need to find an appropriate model which will enable us to capture all aspect of its nature without loss of information. In such situations, the information expressed in terms of multi-parameter sets is of great importance. A collection of soft fuzzy number valued information systems is one such model introduced in this paper which is used as tools to model several characteristics, CONTACT Arul Roselet Meryline S. roselet.meryline2020@gmail.com uncertainty, impreciseness, etc. in complex situations leading to the development of the soft fuzzy group decision-making problem (SFGDMP).

Literature Survey
In many real life situations, more than one individual or decision makers or experts were involved in a decision-making process. To handle various kinds of problems that arose in group decision-making (GDM), different approaches were developed and studied by several researchers. In most of the GDM problems agreed attribute set were considered by the decision makers. Hwang and Lin [2] in their book (Part III) had presented an overall view of methods and techniques in group participation analysis till 1987 wherein some sections had been dedicated to GDM problems in classical (non-fuzzy) setup that involved individual attribute set corresponding to decision makers. In the literature of GDM, problems involving individual attribute set were further studied by few researchers (to cite [3][4][5]).
In 1965, Zadeh [6] formulated fuzzy sets, which in an imprecise environment captured the inexactness present in a system. Decision-making in fuzzy environment was initiated by Bellman and Zadeh [7], which paved the way to the development of several methods to solve multi-attribute decision-making problems. Zadeh [8] elucidated the concept of linguistic variables to handle situations that involved less preciseness in humanistic systems which was further studied by several researchers using appropriate kind of fuzzy numbers. Jean and Andrew [9] in 1973 applied social preferences as fuzzy binary relation in GDM problems and by the end of the decade researchers [10][11][12] dealt with multipleaspect decision-making in the presence of uncertainty wherein weights and ratings were represented as fuzzy variables. GDM in the fuzzy environment was further studied in several directions and it was recorded in the collection of papers [13] edited by Kacprzyk and Fedrizzi, published in 1990. Evaluation or selection of alternatives under multiple attribute set is one type of problem in GDM. Over the years, many researchers developed various methods in solving GDM problems (to cite a few [14][15][16]).
A detailed literature survey on multiple attribute GDM problems in both classical and fuzzy environment due to Kabak and Ervural [17] was recorded in the year 2017.
On the other hand, the concept of a soft set as a mathematical tool for dealing uncertainty was introduced by Molodtsov [18] in 1999 to be a parameterised family of subsets of some universal set U. Combination of soft sets with fuzzy sets was studied to capture the nature of entities in the problem in hand. In 2001, Maji and Roy [19] defined a fuzzy soft set to be a soft set in which the set of all subsets of U were replaced by the collection of fuzzy sets on U. With the on set of the new millennium, Biswas et al. [20] had applied soft sets in decision-making problems. GDM involving fuzzy soft set theory was studied by few researchers (to cite a few [21,22]).
In 2012 and 2013, Samantha and Das [23] defined the soft real set in which the initial universal set was considered to be R, the set of real numbers, where they studied the properties in-depth and applied it to decision-making problems. Beaula and Raja introduced fuzzy soft numbers [24] as a fuzzy set over soft real number [25][26][27][28][29]. In 2018, the authors [30] had defined a real measure on soft real set for comparison purposes and applied in a multi-attribute decision-making problems. In 2019, the concept of soft fuzzy numbers (combining fuzzy numbers and soft set), fuzzy number valued measure on soft fuzzy numbers and soft fuzzy number valued information systems ( IS) was introduced and studied by the authors [1] wherein a decision-making problem to handle in-depth information was considered.

Motivation
Down the century, many researchers had considered the importance of decision makers in GDM problems and the problem of determining the same led to new research avenues. Determination of the objective weights of decision makers was considered as one such avenue and in 2019, an overview of various methods was reviewed by Kabak and Koksalmis [31].
The objective methods so far developed were applicable only for certain types of problems in which the decision matrix of each decision maker was considered with agreed attribute set. A procedure for determining the importance of decision makers for each alternative in a GDM problem with individual attribute set had not been considered yet.
In this paper, for the first time, we record a new formulation involving collection of soft fuzzy number valued information systems and a new methodology to solve the problem of determining the importance of decision makers in such a situation. Using these SFGDMP involving collection of soft fuzzy number valued information systems is discussed.

Outline of the Paper
The paper is systematised as follows: in Section 2, we provide needed prerequisites and some results needed for further study. A hypothetical example is introduced to study the different situations involved in the paper. In Section 3, we discuss the union of soft fuzzy numbers related to multi-parameter sets and property of the fuzzy number valued measure on the same with respect to the weights of the parameters are studied. In Section 4, we define a finite collection of soft fuzzy number valued information systems and is discussed in detail. Soft union on the collection of soft fuzzy number valued information systems and soft fuzzy number valued hierarchical information systems are defined. In Section 5, we deal with the mathematical formulation for a SFGDMP problem and an algorithm to solve the same. In Section 6, the methodology proposed is applied and discussed as a case study based on a real life situation, using a secondary data collected from websites. Also conclusion of the paper are recorded.
(3) Upper semi continuous, i.e. for all t ∈ R and c > 0 The collection of such fuzzy numbers is denoted as F(R).
, a partial ordering on F(R) was defined bỹ Definition 2.3: [32] A fuzzy numberÃ is said to be non-negative ifÃ(t) = 0 for t < 0 . The collection of all non-negative fuzzy numbers is denoted as F * (R). For rest of the paper, we consider only F * (R).
Definition 2.4: [33] For any two fuzzy numbersÃ 1 , the arithmetic operations ⊕ on collection of fuzzy numbers F * (R) were expressed using resolution identity due to Ref. [8] as follows: Definition 2.5: [34] The scalar multiplication of anyÃ ∈ F * (R) by a non-negative real number λ was defined as λ(Ã) = ∪ α∈ [0,1] Definition 2.6: [35] The distance d between any two fuzzy numbersÃ 1 ,Ã 2 ∈ F * (R) was defined by d(Ã 1 ,Ã 2 ) = sup Fuzzy numbers that were very often used in various real life applications were triangular and trapezoid fuzzy numbers. Recently, in 2013, the concept of linear octagonal fuzzy number was introduced by Malini and Kennedy [36], which was found to be more useful for solving real life problems. Definition 2.7: [36] A fuzzy numberÃ was said to be a linear octagonal fuzzy number denoted by (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ; k) where a 1 ≤ a 2 ≤ a 3 ≤ a 4 ≤ a 5 ≤ a 6 ≤ a 7 ≤ a 8 ∈ R with membership functionÃ(x) given bỹ where 0 ≤ k ≤ 1.

Remark 2.1:
A linear octagonal fuzzy number would look like Figure 1.
The α -cut of a linear octagonal fuzzy number was computed as follows: Definition 2.8: [36] LetÃ be an octagonal fuzzy number. The measure onÃ was defined by Remark 2.2: [36] Any two linear octagonal fuzzy numbersÃ andB could be compared using the following:

Remark 2.3:
Linear octagonal fuzzy numbers yield better results for the choices of k < 0.5 [15,37]  A soft real number denoted (F, E) was defined as a particular soft real set which is a singleton soft real set that has been identified with the corresponding soft element. For comparing any two soft real numbers, we define the following: where w j are the weights assigned to the parameters e j such that l j=1 w j = 1.

Remark 2.5:
The measure defined in Definition 2.12 is applied in the example cited in the Remark 4.1.

Definition 2.13: [1]
A soft fuzzy number was defined as a mapping f : where E is the parameter set. The collection of soft fuzzy numbers was denoted asF * (R)(E).

Remark 2.6:
The soft fuzzy number considered in Definition 2.13 is used in Example 2.1.

Remark 2.7:
If the fuzzy number associated with the parameters is linear octagonal fuzzy numbers then the corresponding soft fuzzy numbers are called soft linear octagonal fuzzy number.
To understand the concept of soft fuzzy numbers and various concepts introduced in Sections 3, 4 and 5, we consider the following hypothetical example.   Let the verbal assessment of the three judges be recorded using the linguistic terms 'Excellent (EX)', 'Very good (VG)' and 'Good (G)'.
We consider a situation that records the result regarding elegance of one of the contestants given by the three judges based on various related 'phrases' as listed in Table 1.
We express the assessment of the three judges as soft fuzzy numbers ( f , E), ( g, E) and ( h , E) with the 'phrases' as parameter set E = {e 1 , e 2 , e 3 }, where e 1 = phrase 1, e 2 = phrase 2 and e 3 = phrase 3.
The various linguistic terms considered are represented by linear octagonal fuzzy numbers (see Definition 2.7) given in Table 2.
The soft linear octagonal fuzzy numbers are The scalar multiplication was defined by λ( f , E) = {λ f (e)e ∈ E} any non-negative real number λ.
where w j ≥ 0 are weights of the parameters in E with l j=1 w j = 1. Here note that ⊕ represents sum of fuzzy numbers. Definition 2.16: [1] Let M : F(R) → R, M(Ã) denote the defuzzified value of a fuzzy num-berÃ ∈ F(R) based on any suitable defuzzification method under consideration. Then any two soft fuzzy numbers ( f , E), ( g, E) ∈F * (R(E)) are related by the relation ' < ' given by .
is a partially ordered set.
Note that though we have considered element ( f , E), we do not distinguish between two elements ( f 1 , E), ( f 2 , E) which have the same measureM, rather we consider instead of individual element's ( f , E) equivalence classes of elements of same measureM without explicit mention.

Definition 2.17: [1] A soft fuzzy number valued information system is a quadruple
is the parameter set associated with attribute a j , l j representing the number of parameters in E j and if ˜I : We shall define order on collection of alternatives based on soft information as follows: Remark 2.8: From Definition 2.17, we obtain soft fuzzy number valued function on U for each a ∈ A and denote collection of such function by Ũ (F * (R)(E)). IS can be viewed as mapping F : A → Ũ (F * (R)(E)). Such a mapping defines a soft fuzzy number soft set denoted ( F , A).

Proposition 2.2:Ũ a is an order preserving function.
Proof: To proveŨ a is order preserving, we need to prove the conditioñ Choose an arbitrary u ≺ a v and fix it. For all s a u and t a v, we havẽ Hence condition (2.1) holds Definition 2.20: For a soft fuzzy number valued information system Then ( f u , A) will be a soft fuzzy number called soft utility information of u in IS.

Definition 2.21: [1] A soft fuzzy number valued hierarchical information system is a quintu-
where H a j denote the concept hierarchy tree of attribute a j for j = 1, 2, . . . , n.
the collection of parameter sets associated with l j leaf nodes of the concept hierarchy tree and ˜I a j : is a function such that ˜I a j (u i , a j ) consists of corresponding collection of soft fuzzy numbers in all levels of concept hierarchy tree.

Properties of Soft Fuzzy Numbers Related to Multi-parameter Sets
In this section, we define union on collection of soft fuzzy numbers related to multiparameter sets and discuss it with an example. Also study the behaviour of fuzzy number valued measure over the operation introduced.
. . , l n denote the corresponding number of parameters in E 1 , . . . , E n andF * (R)(E) be the collection of soft fuzzy numbers related to multiparameter set E.

Definition 3.1: The soft union denoted˜ of any finite collection of soft fuzzy numbers
and ( f˜ , H) is a soft fuzzy number related to parameter set H.  Table 3, then the evaluation of the judges I, II, and III are represented as soft linear octagonal fuzzy numbers ( Using the operation defined in Definition 3.1, the combined evaluation is obtained as a soft linear octagonal fuzzy number  Properties ofM related to multi-parameters sets involving soft union are studied in the following proposition: the associated weights of the parameters, we havẽ Proof: Considering Definition 3.1 for j = 1, 2, the soft union of ( f 1 , ,e ∈ E 1 ∩ E 2 . . Therefore, which is a α -cut of the fuzzy number which yieldsM .
Without loss of generality, we suppose that E 1 ∩ E 2 consist of one parameter (say), e = e 1,l 1 = e 2,l 2 , then the weights associated with the parameters in H˜ are {( Then using operations on fuzzy numbers, for each e, we have and 2) and using Definition 2.2, we obtain and hence, which in turn yieldM

Remark 3.2:
Using induction principle, we can extend the above proposition for any finite collection of soft fuzzy numbers related to multi-parameter sets.

Soft Union on Collection of Soft Fuzzy Number Valued Information Systems
Collection of soft fuzzy number valued information systems are introduced and soft union on the collection are dealt in this section.
A finite collection of soft fuzzy number valued information system { IS p } q p=1 , for some q ∈ Z + , can be obtained from Definition 2.17 and is given by Definition 4.1.

Definition 4.1:
A collection of soft fuzzy number valued information system { IS q p } p=1 , for some q ∈ Z + , are defined as the quadruple      Here, the attribute set is common for all the judges, i.e. A 1 = A 2 = A 3 = A. A = {a 1 , a 2 }, where a 1 = elegance and a 2 = intelligence . The parameter set associated with a 1 is same for all the judges (from Example 2.1) while a 2 is evaluated through personalised questions and is different for each judge (from Example 3.2). Therefore, we The evaluations of the three judges corresponding to the attributes are given by ˜I p and are represented as corresponding soft linear octagonal fuzzy number valued information systems given in Table 7.
The assessment of Judge I:

Remark 4.1:
Need for a soft fuzzy scenario in an information system. Information systems are best suited to model complex situations involving attributes. In the fuzzy information system, the objects related to qualitative attributes are imprecise in nature and quantification of these using linguistic variables is evaluated with linguistic values (fuzzy numbers) describing fuzziness in such a system.
In Example 2.1, suppose the information considered is based on only fuzziness (not including the parameters), such a system gives only peripheral information and does not yield a foolproof model. I.e. suppose evaluation by the three judges (p = 1, 2, 3) corresponding to the attributes A = {a 1 , a 2 }, where a 1 = Elegance and a 2 = Intelligence , are recorded in Table 8 as information systems S p = (U, A, V, ρ p ), where ρ p : U × A → V, V the set of linear octagonal fuzzy numbers that are used to describe the linguistic variables 'EX', 'VG' and 'G' as in Table 2.
The problem of ranking the contestants by the individual judges for this fuzzy information (fuzzy decision matrix) is solved using a fuzzy simple additive weighting method   and the ranking order (Definition 2.8) using the defuzzification measure of linear octagonal fuzzy numbers is shown in Table 9. Again in Example 2.1, suppose we consider the evaluation by the three judges on a fivepoint scale (crisp), incorporating parameters as recorded in Table 10.
The evaluation of Judge I (say) for the five contestants corresponding to the attribute a 1 are Similarly, the evaluations for judge I corresponding to attribute a 2 can be evaluated. Along lines, the evaluation of other judges for the attributes a 1 , a 2 can be evaluated which are soft real numbers. On computing the real measure (see Definition 2.12) on soft real numbers in the soft real number valued information system, we have crisp decision matrices corresponding to the three judges as in Table 12.
The ranking order of the contestants corresponding to individual judge based on the crisp simple additive weighting method is shown in Table 13.
In this case, repetition of the ranks has occurred due to the fact that the intricacies in expressing the qualitative information are not captured appropriately.
Note that the contestants are not ranked in the same way and uniquely in Tables 9 and  13. Hence, there is a need for a new model incorporating intricate points such as attributes, collection of sub-attributes, parameters, collection of sub-parameters and impreciseness which occur in the natural scenario. The answer to this is exhibited in this paper having the soft fuzzy number valued information system to model such a scenario (see Table 18) wherein we infer from columns 2, 4 and 6 that the contestants are ranked uniquely.

Remark 4.1:
Consideration of only fuzzy information, the attribute set is an agreed set. The problem of determining the importance of the judges and then their combined evaluation to select the best contestant in such a situation could be done by any available methods (see Section 1.1). But these methods cannot be applied to the situation wherein parameters are considered to capture the in-depth information. Also through Remark 4.1, we have insisted the need of soft fuzzy information systems in choosing the beat contestant. Hence, new methodology is needed to handle a SFGDMP.

Remark 4.2:
Using this information in the process of selecting the best pageant by combined evaluation by all the three judges is cited in Section 5. We define a mapping ˜I˜ : a l ), a 1 ∈ A p , &a l / ∈ A q , p = q, for any q, , a l ∈ ∩ q 1 p=1 A p , for some q 1 ≤ q and the quadruple (U, A, F * (R)(E), ˜I˜ ) or in short IS˜ is the soft union of soft fuzzy number valued information systems. Note that IS˜ is a soft fuzzy number valued information system. H 2 a j and , i.e. common only at nodes. a j ), i.e. common at both nodes and leaf values.  a j ) consists of corresponding collection of soft fuzzy numbers in all levels of concept hierarchy tree.

Definition 4.4: SI H
Call A = ∪ 2 p=1 A p = {a l } s l=1 for s ≤ n 1 + n 2 + · · · + n p , H A l = {H a l /a l ∈ A} with H a l = H p a l , a l ∈ A p and E A = {E a l }, where , a 1 ∈ A 1 &a l / ∈ A 2 , E l ls parameter sets associated with leaf nodes in E 1 a l , , a 1 ∈ A 2 &a l / ∈ A 1 , E 2 ls parameter sets associated with leaf nodes in E 2 a l , , a l ∈ Att A 1 ∩A 2 , E p ls parameter sets associated with leaf nodes in {E p a l } 2 p=1 , is the collection of soft fuzzy numbers associated with E A .
We define a mapping ˜I˜ H A : U × A → F * (R)(E A ) given bỹ , a l ∈ Att A 1 ∩ A 2 and the quintuple (U, A, F * (R)(E A ), ˜I˜ H A ) or in short IS˜ H A is the soft union of soft fuzzy number valued hierarchical information systems. Note that IS˜ H A is a soft fuzzy number valued hierarchical information system.

SFGDMP
In multi-attribute GDM problems, importance (weights) to decision makers play a major role, wherein combined evaluation of decision makers are involved for selecting the optimal entity (alternative). Assigning or determining importance of decision makers is problem context. Subjective importance to the decision makers are assigned based on the expertise level (knowledge, experience, etc.). Difficulty of considering subjective importance may arise in many situations (expertise level not known, to avoid partiality in assigning weights, etc.). On the other hand, deriving objective importance of decision makers using the data provided (in the form assessments, evaluations, perception, etc.) is modelled using appropriate mathematical models.
In this section, we introduce SFGDMP as a situation in decision-making wherein the perception of each decision maker describing some aspect Q of collection the entities based on corresponding individual attribute sets associated with underlying parameters. In such a situation, to handle the problem of determining the importance to decision makers, we have developed a new mathematical formulation which in turn is used to rank the entities. An algorithm is proposed to solve the same and validated with a real life situation.
The problem in hand is to (1) frame a suitable mathematical model for SFGDMP, (2) find a methodology to compute appropriate importance to decsion makers for each, (3) alternative as the attributes are different for different decision makers, (4) measure the aspect by combining the evaluations of the decision makers and (5) choose the best entity.

Problem Description
Consider the SFGDMP involving a finite number of decision makers evaluating a finite collection of entities based on different characteristics features with associated parameter sets to choose the best entity. The problem is mathematically formulated as the collection of { IS p } , the collection of soft fuzzy numbers related to multi-parameter set, for i = 1, 2, . . . , m, j = 1, 2, . . . , n p and p = 1, . . . , q, represents the perception of q decision makers about the entities in relation to different characteristics (individual attribute set A p ) features describing the aspect Q with associated parameter sets {E p }.
Let The problem is (1) to determine λ i p such that q p=1 λ i p = 1, (2) to compute the aspect value of the ith entity Q p i and combined aspect value Q C i incorporating λ i p for p = 1, . . . , q and (3) to choose the optimum entity based on Q C i .

Methodology
As the attribute set is different for each of the decision maker, a new procedure is adapted wherein objective importance to the decision makers are determined as real numbers for each entity. Here, fuzzy number valued measure on utility soft information is obtained as fuzzy numbers corresponding to the various entities in individual and combined decision makers information system. We consider the closeness of each decision maker's decision to that of the combined decision in a fuzzy setup to derive the weights. Applying this methodology, we propose an algorithm to solve the problem.

Procedure
Step 1: For each p, compute the corresponding fuzzy number valued measureŨ p (u i ) of utility soft information of each entity u i by performing the following: Step 1 i: If the information consists of IS p , then consider the following steps or if the information consists of IS H p , go to Step 1 v.
Step Step 1 iv.: Go to Step 1 x.
Step 1 v: For IS H p , perform the following: Step 1 vi.: Input the soft fuzzy number for each i = 1, . . . , m corresponding to each sub-characteristic feature at the leaf nodes.
Step 1 vii.: Obtain the fuzzy number valued measure for the inputs.
Step 1 viii.: Compute Step I.ii.ii recursively back tracking till we reach Step 1 ix.: Go to Step 1 x.
Step 1 x: For each p, compute the following to obtain fuzzy numbersŨ p (u 1 ), . . . , U p (u m ).
Step 1 xi.: For each u i , construct utility soft information ( f Step 1 xii.: Step 1 xiii.: Go to Step 2. Step 2: To construct utility soft information in equally combined IS˜ or IS H˜ and compute fuzzy number valued measureŨ(u i ) on it, perform the following: Step 2 i.: If the information is recorded as IS p , do the following steps or if IS H p go to Step 2 ii.
Step 2 iii.: Evaluate soft union of soft fuzzy number valued information systems obtained in previous step SI˜ = (U, A, F * (R)(E), ˜I˜ ).
Step 2 v.: Go to Step 2 x.
Step 2 vi.: If the information is recorded as IS H p , do the following steps: Step 2 vii.: Obtain for each p, (1/q)( ˜I H Ap ).
Step 2 viii.: Compute the soft union IS H˜ = (U, A, F * (R)(E A ), ˜I H A˜ ).
Step 2 ix.: Perform Step 1 vi. to Step 1 ix. for the obtained IS H˜ in the pervious step.
Step 2 x.: Go to Step 2 xi.
and t i are the assigned weights based on the measure M(U p (u i )) and M(U(u i )) such that Step 1: For p = 1, 2, 3, fuzzy number valued measure of utility soft information is computed as follows: Step 1 i.: Since the information for p = 1 is in the form IS 1 , perform the following steps: Step 1 ii.: Input the soft fuzzy numbers corresponding to a 1 and a 2 from Table 7.
Step 1 iii.:M(( f Table 15.M( f Step 1 vii.: The fuzzy number valued measure on ( f , A 1 ) for each i is computed and is shown in Table 15.
Since for p = 2, 3, the information are recorded as IS 2 and IS 3 as given in Table 7; by performing Step 1 i. to Step 1 xii.,M( f , A) are computed and tabulated in columns 3 and 4 of Table 15. Go to Step 2.
Step 2: The utility soft information for the equally combined soft fuzzy number valued information system is obtained by performing the following steps: Step 2 i.: For p = 1, 2, 3, (1/3) IS p are obtained.
Step 2 ii.: Using Definition 4.3, the soft union of {(1/3) IS p } 3 p=1 are computed and given by SI˜ 3 are the associated parameter sets in E 1 , E 2 and E 3 , and the mapping ˜I˜ : U × A → F * (R)(E) is given by  Table 16.
Step 2 xi. Fuzzy numbersŨ(PT 1 ), . . . ,Ũ(PT 5 ) are computed by performing Step 2 xii., Step 2 xiii. and Step 2 xiv. wherein the utility soft information ( f , A)] ∈ F * (R) and the equally combined value of the aspect for each contestant Q c e i are computed and given in Table 8. We go to the following step:   Table 18. M(Ũ p (PT i )) and assigined weights. Step 3: Using Definition 2.6, the closeness of each judge's decision to that combined is calculated, wherein the weights t i and t p i are assigned based on the measure given in Tables 17 and 18  Step 4: The normalised importance are calculated and given by Step 5: Using these λ i 1 , λ i 2 , λ i 3 values performing Step 2 to Step 2 iv., we obtain [λ p ( ˜I (PT i , a l ))] for p = 1, 2, 3, l = 1, 2 and SI˜ = (U, A, F * (R)(E), ˜I˜ ), where a l )], a l ∈ A, l = 1, 2. The utility soft information ( g (i) , A) is constructed for each PT i in the soft fuzzy number valued information SI˜ obtained in the previous step and the fuzzy number valued measure is computed by performing Step 6 i. and Step 6 ii., respectively, and given in Table 19.  Table 19.
Step 7: The combined evaluations of the judges yield the contestant PT 5 to be ranked first.

Choosing the Best Car Brand Based on Safety Aspect: A SFGDMP Model
In this section, we consider the problem of choosing a car based on the aspect safety by combining the evaluations provided from websites www.nhtsa.gov and www.iihs.org for the list of cars brands {C 1 , C 2 , C 3 , C 4 , C 5 } (see Ref. [1]).
We quantify the star ratings, where an individual assessment based on the stars, i.e. highest number of stars accounting to better safety, is analysed as linguistic terms good, acceptable, marginal, poor with suitable comparative quantification. The linear octagonal fuzzy numbers representing the linguistic terms and the quantified ratings of the car brands are considered as given in Table 20.
The websites are considered to be the two experts and we have modelled their required information as collection of hierarchical soft fuzzy number valued information systems { IS H p } 2 p=1 wherein the safety features are considered as attribute sets and the corresponding tests conducted as parameter sets. The problem of determining the importance of experts and incorporating the same for combining their evaluations to choose the car brand with optimum level of safety are modelled as SFGDMP and solved by performing the Algorithm proposed in Section 5.

Problem Description and Solution
We consider a situation where two people in a family (say) P 1 and P 2 want to buy a car for a common usage. They are concerned about the safety measures, so P 1 and P 2 gather information from websites www.iihs.org and www.nhtsa.gov, respectively, to make a combined decision in the choice of a car.  Suppose the person P 1 gathered information based on the problem (Case 2) worked in the earlier paper [1], then the required information is formulated as hierarchical soft fuzzy number valued information system }} where e 2 311 is rollover resistance level test. The various tests are considered as the set of parameters and the ratings of the car brands are recorded as soft octagonal fuzzy numbers for i = 1, 2, . . . , 5 as follows:  Table 21.  Using the values from Table 22, the closeness of the decision of each expert to that of the combined was computed and the importance of the experts are obtained: Incorporating the obtained weights, the combined evaluations of the experts, the corresponding fuzzy number valued measure on utility informationŨ diff (C i ) and combined value of safety Q c i are computed and tabulated in the last two columns of Table 22. Based on the procedure, it was found that C 5 is the best brand based on the safety measure among the five car brands of choice. Remark 6.1: In this real life situation, equal weights are assigned to the parameters involved. Weights to the various safety features are assigned according to the needs of persons P 1 , P 2 and their individual perspectives. From columns 4 and 6 of Table 22, we note that the change in ranking of car brands is due to implementation of derived objective importance to the experts for each car brands, the weights assigned for attributes and parameters.

Conclusion and Future Study
In this paper, theoretical development of soft fuzzy numbers and soft fuzzy number valued information systems are studied in detail. The knowledge pertaining to collection of entities involved, based on E are mathematically modeled as collection of IS or IS H and is used as a tool to solve SFGDMP using an algorithm developed. It is established that this type of problems cannot be effectively solved, but for the usage of soft fuzzy number valued information systems. In Section 6, we have limited our work for 5 midsize car brands based on information from two websites only wherein the qualitative information is quantified using the above developed model.
In the problems for each of the aspects there will be different attributes and parameters (defining some or all of the attributes). To cite a few: It is possible to consider similar problems using the model developed and we are continuing to work on some of them. It is also possible to develop models involving fuzzy analogue of soft fuzzy number valued information systems in different scenario. fuzzy logic and math education. She has published 26 research papers in international journals with Google scholar 592 citations. To her credit, one of the research paper contributing to fundamental work in the area of fuzzy normed linear space was published in the journal of Fuzzy Sets and Systems that has 460 citations and there are works based on fuzzy normed linear space of the Felbin's type. Her current research interests are fuzzy analysis, fuzzy optimisation and soft fuzzy sets.