Decision making algorithmic techniques based on aggregation operations and similarity measures of possibility intuitionistic fuzzy hypersoft sets

: Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while deﬁning fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, ﬁrstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND / OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.


Introduction
Fuzzy sets theory (FST) [1] and intuitionistic fuzzy set theory (IFST) [2] are considered apt mathematical modes to tackle many intricate problems involving various uncertainties, in different mathematical disciplines. The former one emphasizes on the degree of true belongingness of a certain object from the initial sample space whereas the later one accentuates on degree of true membership and degree of non-membership with condition of their dependency on each other. FST and IFST have some sort of complexities which bound them to tackle issue including uncertainty efficiently. The main cause of these hurdles is potentially, the deficiency of the parametrization tool. It requests a mathematical tool free from all such obstacles to handle such issues. This inadequacy is settled down with the introduction of soft set theory (SS-Theory) [3] which is another defined class of subsets of the universe of discourse. The researchers [4][5][6][7][8][9][10][11][12][13] examined and explored some rudimentary properties, operations, laws of SS-Theory with applications in decision making.
The existing studies based on fuzzy soft sets (FS-sets), intuitionistic fuzzy soft sets (IFS-sets), etc., are widely used in different environments to solve decision-making problems. However, under certain cases, these existing approaches fail to classify the objects based on their possibility degrees. In other words, the existing studies have treated the possibility degree of each element as one. However, in many practical applications, different persons may have their possibility degrees which differ from one related to each object. To address this issue in decision-making process, Alkhazaleh et al. [14] introduced the concept of possibility of FS-sets by assigning a possibility degree to each number of FS-set. However, in this set, there is a complete lack of degree of non-membership during the analysis. To tackle it and address more appropriately, a concept of possibility intuitionistic fuzzy soft set (PIFSset) was introduced by Bashir et al. [15]. PIFS-set is more generalized than the existing FS-sets, IFS-sets, and other sets. In PIFS-set, a degree of possibility of each component is assigned to degrees of intuitionistic fuzzy soft numbers (IFS-Ns) during evaluating the object. In numerous daily-life problems, parametric values are additionally subdivided into disjoint attribute-values sets. The existing SS-Theory is deficient for managing such sort of attributive-valued sets. Hypersoft set theory (HS-Theory) [16] is developed to make SS-Theory in accordance with attributive-valued sets to handle real life situations. Certain rudimentary properties, operations, laws, relations and points of HS-Theory are explored in [17][18][19] for appropriate indulgent and further use in various fields. The utilizations of HS-Theory in decision making are premeditated in [20][21][22][23][24] and the gluing investigations of HS-Theory with concave, convex, complex sets and parameterization are discussed in [25][26][27]. Kamacı [28] made very valuable research on hybrid structures of hypersoft sets and rough sets. Recently Kamacı et al. [29] investigated some new hybridized structures i.e., n-ary fuzzy hypersoft expert sets which are the extensions of existing structures (i.e., n-ary fuzzy soft expert sets).

Research gap and motivation
The following points may lead to motivation of proposed study.
1) The possibility fuzzy soft set (PFS-set), introduced by Alkhazaleh et al., is a gluing concept of FS-Theory and SS-theory with attachment of a possibility degree to each approximate element of fuzzy soft set. They also calculated similarity measures between two PFS-sets and applied PFSset to decision-making problem for clinical diagnosis. In 2012, Bashir et al. [30] extended the concept of PFS-set with introduction of possibility fuzzy soft expert set (PFSE-set) to adequate it with the consideration of multi-expert opinions. In 2014, Zhang et al. [31] characterized possibility multi-fuzzy soft set (PMFS-set) by assigning a possibility grade to each approximate member of multi-fuzzy soft set. In 2017, Kalaiselvi et al. [32] presented decision-making approach in sports via PFS-set. In 2018, Ponnalagu et al. [33] investigated more on PFSE-set with applications. In 2019, Garg et al. [34] and Khalil et al. [35] discussed decision-making algorithms based on PIFS-set and possibility m-polar fuzzy soft sets (PMPFS-set) respectively. 2) In certain real world scenarios, distinct parameters are further classified into disjoint sets having sub-parametric values ( Figure 1 depicts the vivid comparison of soft set model and hypersoft set model. It presents the optimal selection of a mobile with the help of suitable parameters in case of soft set and suitable sub-parametric values in case of hypersoft set). In decisionmaking, the jury may endure some sort of tendency and proclivity while paying no attention to such parametric categorization during the decision. Soft set theory and its hybridized models have constraints regarding the consideration of such parametric categorization. Along these lines another construction requests its place in writing for tending to such obstacle, so hypersoft set is conceptualized to handle such situations. It has made the whole decision-making process more flexible and reliable. Also, it not only fulfills the requirements of existing soft-like literature for multi-attributive approximate functions but also supports the decision makers to make decisions with deep inspection.  fuzzy set with such situation Atanassov [2] introduced intuitionistic fuzzy set as a generalization of fuzzy set. It provides due status to both membership and non-membership degrees of an alternative.
Definition 2.4. (properties of intuitionistic fuzzy set) [2] If Y 1 and Y 2 are two intuitionistic fuzzy sets then for allb ∈ Z, we have Fuzzy set and intuitionistic fuzzy set depict some kind of insufficiency regarding the consideration of parameterization tool. In order to manage this limitation, Molodtsov [3] developed soft set as a new mathematical tool to tackle uncertainties and vagueness in the data.
and W is a subset of set of attributes X.
Definition 2.8. (intuitionistic fuzzy soft set) [13] A pair (Ψ IF , V) is said to be fuzzy soft set over Z, where Ψ IF : V → P(IF ) and P(IF ) is a collection of all intuitionistic fuzzy subsets over Z, V ⊆ X.
In many real-world scenarios the classification of attributes into sub-attributive values in the form of sets is necessary. The existing concept of soft set is not sufficient and incompatible with such scenarios so Smarandache [16] introduced hypersoft sets to address the insufficiency of soft set and to handle the situations with multi-argument approximate function (MAAF). Definition 2.9. (hypersoft set) [16] Let Z = {ẑ 1 ,ẑ 2 , ...,ẑ n } be an initial universe and X = {p 1 ,p 2 , ...,p n } be a set of parameters. The respective attribute-valued non-overlapping sets of each element of X are Q 1 = {q 11 ,q 12 , ...,q 1n } Q 2 = {q 21 r) is a n-tuple element of Q and r = n i=1 |Q i |, | • | denotes set cardinality then a MAAF is a mapping and defined as where P(Z) denotes the power set of Z, V ⊆ Q with k ≤ r. The pair (ξ H , V) is known as hypersoft set.
Definition 2.10. (fuzzy and intuitionistic fuzzy hypersoft set) [16] Fuzzy hypersoft set and intuitionistic fuzzy hypersoft set are hypersoft sets defined over fuzzy universe and intuitionistic fuzzy universes respectively i.e., in Eq (2.1) P(Z) is replaced with P(F ) and P(IF ).

Possibility intuitionistic fuzzy hypersoft set (PIFHS-set)
Consider the daily life scenario of clinical study to diagnose heart diseases in patients, doctors (decision-makers) usually prefer chest pain type, resting blood pressure, serum cholesterol etc., as diagnostic parameters. After keen analysis, it is vivid that these parameters are required to be further partitioning into their sub-parametric values i.e., chest pain type (typical angina, atypical angina, etc.), resting blood pressure (110 mmHg, 150 mmHg, 180 mmHg, etc.) and serum cholesterol (210 mg/dl, 320 mg/dl, 430 mg/dl, etc.). Patients are advised to visit medical laboratories for test reports regarding indicated parameters. As the efficiency of medical instruments in laboratories varies which leads different observations (data) for each patient. It is quite ambiguous for any decision-maker (i.e., medical practitioner) to assess the exact role of each parameter in diagnosis. Again this assessment is judged by relevant medical field specialist on the basis of standard values of each sub-attribute in accordance with WHO or some local medical body by considering a possibility degree within [0,1]. The literature has no suitable model to deal (i) sub-attribute values in the form of disjoint sets, and (ii) possibilitic data collectively. In order to meet the demand of literature, the novel model PIFHS-set is being characterized. The case (i) is addressed by considering multi-argument approximate function which considers the cartesian product of attribute-valued disjoint sets as its domain and then maps it to power set of initial universe (collection of intuitionistic fuzzy sets) and the case (ii) is tackled by assigning a possibility degree with each intuitionistic fuzzy number attached with each sub-parameter.
... × A n , A i are disjoint attribute-valued sets corresponding to distinct attributes a i , i = 1, 2, ..., n respectively, (ii) F : A → (I×I) Z and µ : A → I Z , I Z and (I×I) Z are the collections of all fuzzy and intuitionistic fuzzy subsets of Z respectively, (iii) (F(α)(u) is the degree of membership of u ∈ Z in F(α), (iv) µ(α)(u) is the degree of possibility of membership of u ∈ Z in F(α).
so F µ (α i ) can be written as: For convenience, PIFHS-set is denoted by F µ and collection of all PIFHS-sets is denoted by Ω pi f hss . The pictorial representations of existing structure PIFS-set (possibility intuitionistic fuzzy soft set) and proposed structure PIFHS-set are presented in Figure 2.  The best selection may be evaluated by observing the attributes i.e., a 1 = Camera Resolution (Mega pixels), a 2 = storage (Giga Bytes), and a 3 = Battery power (mAh). The attribute-valued sets corresponding to these attributes are: <0.7,0.2> , 0.5 Then F µ is a PIFHS-set over (Z, A). Its matrix representation is Example 3.6. Considering data in example 3.2, we have Definition 3.7. (possibility null IFH-set) A PIFHS-set F µ is said to be a possibility null intuitionistic fuzzy hypersoft set, denoted by Example 3.8. Considering matrix notation of F µ given in example 3.2, we have Example 3.10. Considering matrix notation of F µ given in example 3.2, we have where c is a intuitionistic fuzzy complement.
Example 3.12. Considering matrix notation of F µ given in example 3.2, we have

Set theoretic operations of PIFHS-sets
In this section, definitions and properties of set theoretic operations like union, intersection, ANDoperation and OR-operation of PFS-sets are developed and are explained with the help of suitable examples.

Definition 4.1. (union and intersection of PIFHS-sets) Let
∈ Ω pi f hss . Then the following properties hold: ∈ Ω pi f hss . Then the following properties hold: ∈ Ω pi f hss . Then the following properties hold: can be proved in a similarly way as in (i).
∈ Ω pi f hss with matrix notations as

Application of AND-operation and OR-operation of PIFHS-sets
In this section, two algorithms are proposed for decision making to have right decision regarding best selection of certain material/product using AND-operation and OR-operation of PIFHS-sets. Moreover, proposed algorithms are explained with illustrated examples.
Example 5.1. Suppose John wants to buy a washing machine from the market. There are five kinds of washing machines w 1 , w 2 , w 3 that form the universe of discourse Z = {w 1 , w 2 , w 3 , w 4 , w 5 }. Such selection is made on the basis of three parameters y 1 = power (watts), y 2 = voltage, y 3 = capacity in kg, which depict their performances. The attribute-valued sets corresponding to these parameters arë Step 1 Construct PIFHS-sets W ζ , L ξ according to Experts.
Step 2 Calculate AND-operation V δ of PIFHS-sets constructed in step 1.
Step 3 Present V δ in matrix notation with reduced fuzzy numerical grades Step 4 Mark the highest numerical grade ρ r f in each row of matrix.
Step 5 Calculate score S = Sum of the products of ρ r f i with the corresponding possibility σ i .
Step 2 Calculate OR-operation X of PIFHS-sets constructed in step 1.
Step 3 Present X in matrix notation with reduced fuzzy numerical grades Step 4 Mark the highest numerical grade ρ r f in each row of matrix.
Step 5 Calculate score S = Sum of the products of ρ r f i with the corresponding possibility σ i .
The flow chart of Algorithm 2 is presented in Figure 6. Step 1 This step is same as in step 1 of algorithm 1.
Step 2 Now W ζ ∨ L ξ = X where X (p 1 , p 1 ) = <0.58,0.14> , 0.8 , Step 3 Matrix notation of X is given as Step 4 Reduced fuzzy numerical grades ρ r f (see Figure 7) and possibilities σ are given in Table 2. The value of ρ and σ against the pairs (p i , p i ), i = 1, 2, 3 are considered as 0 as both the parameters are the same. Step 5 and Step 6 As score of d 5 is maximum (as depicted in Figure 8), therefore it is selected.   The comparison of ranking of Dryers by both methods has been presented in Table 3 and Figure  9. It can be observed that mean score of AND-operation is 0.1892 and of OR-operation is 0.9964. Since 0.1892 & 0.9964 ∈ [0, 1] but the value 0.1892 is more reliable and precise as compared to 0.9964 therefore it can be concluded that OR-operation is not consistent and requires some weight to be applied.  Figure 9. Comparison analysis of score values for AND-operation and OR-operation.

Similarity measures between PIFHS-sets
Similarity measures have broad applications in many fields like pattern recognition, image processing, region extraction, coding theory, etc. We are frequently intrigued to know whether two examples or pictures are alike or nearly alike or possibly how much they are alike. Many researchers have discussed the similarity measures between fuzzy sets, fuzzy numbers, and vague sets. Majumdar and Samanta [38][39][40] have employed the concept of similarity measures to soft sets, fuzzy soft sets, and generalised fuzzy soft sets. In this section, we introduce a measure of similarity between two PFSSs with partial modification in similarity methods adopted by Majumdar and Samanta [38][39][40]. The set theoretic approach has been employed in this regard because it is very popular and easier for computations. Definition 6.1. (similarity measures b/w PIFHS-sets) Similarity measure between two PIFHS-sets W ζ and L ξ , denoted by S(W ζ , L ξ ), is defined as follows: and here W i j (ε) and L i j (ε) are reduced fuzzy numerical grades with Definition 6.2. (Radical similarity b/w PIFHS-sets) Let W ζ , L ξ ∈ Ω pi f hss then W ζ and L ξ are said to be radically similar if S(W ζ , L ξ ) ≥ 1 2 . Proposition 6.3. Let W ζ , L ξ , H λ ∈ Ω pi f hss then the following holds: Proof. The proofs of (i)-(v) are straightforward and follow from Definition 6.1. Hence, the similarity between the two PIFHS-sets W ζ and L ξ is given by S(W ζ , L ξ ) = 0.90×0.9231 0.831 which means W ζ and L ξ are radically similar.

Application of similarity between PIFHS-sets in recruitment pattern recognition
In this subsection, we will try to estimate the possibility that a candidate having some qualification and experience, is suitable for a job in a company or not. For this we first propose an algorithm based on the concept of similarity measures b/w PIFHS-sets. The methodology is that a model PIFHS-set for Standard recruitment is constructed and ordinary PIFHS-sets are designed for the candidate. Then we calculate the similarity measure of this set with model PIFHS-set to observe pattern recognition. If PIFHs-set of a candidate is significantly similar to model PIFHS-set then that candidate is recommended for selection. The input variables are sub-parameters with PIFHS-numbers as approximate elements with hypothetical observation. Step 1 Construct PIFHS-set W ζ according to Company Experts Team.
Step 2 Construct PIFHS-set L ξ according to External Experts.
Step 3 Represent PIFHS-sets W ζ & L ξ in matrix notations having reduced fuzzy numerical grades as entries.
Step 7 Check the nature of similarity in accordance with Definition 6.2.
<0,1> , 1 , and its matrix representation is given as Now we present matrix representation of W ζ with reduced fuzzy numerical grades as <0.5,0.2> , 0.9 , and its matrix representation is given as Now we present matrix representation of L ξ with reduced fuzzy numerical grades as Step 4 and Step 5 Now we calculate similarity between W ζ and L ξ according to Definition 6.1 Step 6 and Step 7 Hence, the similarity between the two PIFHS-sets W ζ and L ξ is given by S(W ζ , L ξ ) = 0.9189 × 0.6207 0.5704 > 1 2 which means W ζ and L ξ are radically similar. Therefore candidate is recommended for the job in the company.

Comparison analysis
In literature, certain decision-making algorithmic approaches have already been discussed by [14,15,[30][31][32][33][34] that based on fuzzy-like and intuitionistic fuzzy-like soft set theories. Decision making is badly affected due to omission of some features. For example, in recruitment process, it is insufficient to consider qualification, experience, age as only attributes because candidates may have different qualifications, experiences and ages so it is much appropriate to further classify these attributes into their disjoint attributive sets as we have done in Example 6.5. All existing decision making models have used single set of attributes but in proposed model, distinct attributes are further partitioned into disjoint attributive sets. The consideration of such sets will make the decision making process more reliable and trust-worthy. We present a comparison analysis of our proposed structure with the relevant existing structures [14,15] in Table 4. The Figure 11 depicts the generalization of proposed structure. Figure 11. Generalization of proposed structure i.e., PIFHS-set.

Merits of proposed structure
In this subsection, some merits of the proposed structure (i.e., PIFHS-set) are highlighted, which are given below: (i) The introduced approach took the significance of the idea of possibility alongside the IFHSset to deal with current decision-making issues. The considered possibility degree mirrors the possibility of the existence of the level of acknowledgment and excusal; along these lines, this association has tremendous potential in the genuine depiction inside the space of computational incursions. (ii) As the proposed structure emphasizes on in-depth study of attributes (i.e., further partitioning of attributes) rather than focussing on attributes merely therefore it makes the decision-making process better, flexible and more reliable. (iii) It contains all the characteristics and properties of the existing structures i.e., PFHS-set, PHS-set, FHS-set, HS-set, PIFS-set, PFS-set, PS-set, FS-set and S-set so it is not unreasonable to call it the generalized form of all these structures.

Conclusions
The key features of this work can be summarized as follows: with the help of daily life problems. (iv) Similarity between PIFHS-sets is formulated and successfully applied to real world decision making problem. A comparison of similarity for proposed model is made with some existing models. (v) Authors have carve out a conceptual framework for a generalized model i.e., PIFHS-set to deal decision-making real life problems by considering hypothetical data. The authors are devoted to discuss some case studies based on PIFHS-set by using real data. Moreover, other types of similarity measures, entropies and aggregation operators will be investigated for studying multicriteria decision-making problems. (vi) The proposed study may also be extended for developing the following hybridized models: -Possibility Interval-valued Intuitionistic fuzzy hypersoft set -Possibility Neutrosophic hypersoft set -Possibility Interval-valued Neutrosophic hypersoft set -Possibility Interval-valued fuzzy hypersoft set -Possibility fuzzy hypersoft expert set etc.
with applications in decision-making.