Open Journal of Discrete Applied Mathematics

Pythagorean fuzzy multiset and its application to course placements

Paul Augustine Ejegwa
Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria.; ejegwa.augustine@uam.edu.ng; Tel.: +2347062583323

Abstract

The concept of fuzzy set theory is of paramount relevance to tackling the issues of uncertainties in real-life problems. In a quest to having a reasonable means of curbing imprecision, the idea of fuzzy sets had been generalized to intuitionistic fuzzy sets, fuzzy multisets, Pythagorean fuzzy sets among others. The notion of intuitionistic fuzzy multisets (IFMS) came into the limelight naturally because there are instances when repetitions of both membership and non-membership degrees cannot be ignored like in the treatment of patients, where each consultations are key in diagnosis and therapy. In IFMS theory, the sum of the degrees of membership and non-membership is less than or equals one at each levels. Supposing the sum of the degrees of membership and non-membership is greater than or equal to one at any level, then the concept of Pythagorean fuzzy multisets (PFMS) is appropriate to handling such scenario. In this paper, the idea of PFMS is proposed as an extensional Pythagorean fuzzy sets proposed by R. R. Yager. In fact, PFMS is a Pythagorean fuzzy set in the framework of multiset. The main objectives of this paper are to expatiate the operations under PFMSs and discuss some of their algebraic properties with some related results. The concepts of level sets, cuts, accuracy and score functions, and modal operators are established in the setting of PFMSs with a number of results. Finally, to demonstrate the applicability of the proposed soft computing technique, a course placements scenario is discussed via PFMS framework using composite relation defined on PFMSs. This soft computing technique could find expression in other multi-criteria decision-making (MCDM) problems.

Keywords:

Course placement, fuzzy set, fuzzy multiset, intuitionistic fuzzy set, pythagorean fuzzy set, pythagorean fuzzy multiset.

1. Introduction

Fuzzy set theory proposed by Zadeh [1] has achieved a huge impact in many fields to handle uncertainty/vagueness. Due to the vast majority of imprecise and vague information in real-life problems, different extensions of fuzzy set have been developed by some researchers. Yager [2] applied the idea of multiset [3], which is an extension of set with repeated elements in a collection to propose fuzzy multiset. Consequently, a fuzzy multiset allows repetition of membership degrees of elements in multiset framework. In fact, fuzzy multiset generalizes fuzzy set [4].

The concept of intuitionistic fuzzy sets (IFS) was proposed and studied in [5, 6, 7] as a generalization of fuzzy sets. The main advantage of the IFS is its ability to cope with the hesitancy that may exist due to information impression. This is achieved by incorporating a second function, along with the membership function, μ of the conventional fuzzy set, called non-membership function, ν. The idea of IFS has found expression in many cases like medical diagnosis, career placements, pattern recognition and other MCDM problems [8, 9, 10, 11, 12, 13].

However robust the notion of IFS is, there are circumstances where μ+ν1 unlike the situation captured in IFS (where, μ+ν1). The shortcoming in IFS naturally led to the introduction of a concept, called Pythagorean fuzzy sets (PFSs) by Yager [14]. PFS is a tool to deal with vagueness considering the membership grade, μ and non-membership grade, ν satisfying the condition μ+ν1. As a generalized set, PFS has close relationship with IFS. This idea can be used to characterize the uncertain information more sufficiently and accurately than IFS. PFSs have been applied in many areas, like the one discussed in [15].

In [16], the concepts of IFS and fuzzy multiset were combined to proposed intuitionistic fuzzy multisets (IFMS) as the generalization of IFS in multiset framework or the extension of fuzzy multisets by incorporating count non-membership functions, CN={ν1,...,νn} in addition to the count membership functions, CM={μ1,...,μn} captured in fuzzy multisets. Some operations and modal operators on IFMS have been studied in [17, 18]. Due to the resourcefulness of IFMS, it has been applied in many real-life problems as seen in [19, 20, 21, 22, 23, 24, 25, 26, 27].

The motivation of this paper follows from the ideas of PFSs [14] and IFMS [16]. The paper proposes Pythagorean fuzzy multisets (PFMSs), studies its properties and also, its application to course placements. PFMS is either the incorporation of IFMS in PFS setting or PFS in multiset framework.

The paper is organized by presenting some mathematical preliminaries such as fuzzy sets, fuzzy multisets, IFSs, IFMSs and PFSs in Section 2. Moreover, Section 3 covers the concept of PFMS and explicates the ideas of level sets, accuracy and score functions in the setting of PFMS. Also, the idea of cuts in PFMSs context is discussed with some results, and some modal operators on PFMSs are proposed with some deduced theorems. In Section 4, the application of PFMSs in course placements are discussed through composite relation defined on PFMSs. Finally, Section 5 summarises the paper and gives some useful conclusions.

2. Preliminaries

2.1. Fuzzy sets

Definition [1] Let X be a nonempty set. A fuzzy set A of X is characterized by a membership function μA:X[0,1]. That is, μA(x)={1,ifxX0,ifxX(0,1)ifxis partly inX Alternatively, a fuzzy set A of X is an object having the form A={x,μA(x)xX}orA={μA(x)xxX}, where the function μA(x):X[0,1] defines the degree of membership of the element, xX.

2.2. Fuzzy multisets

Definition 2. [2] Assume X is a set of elements. Then, a fuzzy bag/multiset A drawn from X can be characterized by a count membership function CMA such that CMA:XQ, where Q is the set of all crisp bags or multisets from the unit interval I=[0,1].
A fuzzy multiset can also be characterized by a high-order function. In particular, a fuzzy multiset A can be characterized by a function CMA:XNIorCMA:X[0,1]N, where I=[0,1] and N=N{0}.
It follows that CMA(x) for xX is given as CMA(x)={μ1A(x),μ2A(x),...,μnA(x),...}, where μ1A(x),μ2A(x),...,μnA(x),...[0,1] such that μ1A(x)μ2A(x)...μnA(x)..., whereas in a finite case, we write CMA(x)={μ1A(x),μ2A(x),...,μnA(x)}, for μ1A(x)μ2A(x)...μnA(x).
A fuzzy multiset A can be represented in the form A={CMA(x)xxX}orA={x,CMA(x)xX}.

2.3. Intuitionistic fuzzy sets

Definition 3. [5] Let a nonempty set X be fixed. An IFS A of X is an object having the form A={x,μA(x),νA(x)xX} or A={μA(x),νA(x)xxX}, where the functions μA(x):X[0,1]andνA(x):X[0,1] define the degree of membership and the degree of non-membership, respectively of the element xX to A, which is a subset of X, and for every xX, 0μA(x)+νA(x)1. For each A in X, πA(x)=1μA(x)νA(x) is the intuitionistic fuzzy set index or hesitation margin of x in X. The hesitation margin πA(x) is the degree of non-determinacy of xX, to A and πA(x)[0,1]. The hesitation margin is the function that expresses lack of knowledge of whether xX or xX. Thus, μA(x)+νA(x)+πA(x)=1.

2.4. Intuitionistic fuzzy multisets

Definition 4. [16] Let X be a nonempty set. An IFMS A drawn from X is of the form A={CMA(x)x,CNA(x)x|xX} where CMA(x)=μ1A(x),...,μnA(x),... and CNA(x)=ν1A(x),...,νnA(x),... are the count membership and count non-membership degrees defined by the functions CMA:XN[0,1]andCNA:XN[0,1] such that 0CMA(x)+CNA(x)1, where N=N{0}. }

If the count membership functions and count non-membership functions have only nterms (i.e. finite), then n is called the dimension of A. Consequently A={μ1A(x),...,μnA(x)x,ν1A(x),...,νnA(x)xxX} for i=1,...,n. For each IFMS A of X, CHA(x)=1CMA(x)CNA(x) is the intuitionistic fuzzy multisets index or count hesitation margin of x in A, where CHA(x)=π1A(x),...,πnA. The hesitation margin πiA(x) for each i=1,...,n is the degree of non-determinacy of xX to A and πiA(x)[0,1]. The count hesitation margin is the function that expresses lack of knowledge of whether xA or xA. Thus, μiA(x)+νiA(x)+πiA(x)=1 for each i=1,...,n.

2.5. Pythagorean fuzzy sets

Definition 5. [14] Let X be a universal set. Then, an PFS A of X is a set of ordered pairs defined by A={x,μA(x),νA(x)xX} or A={μA(x),νA(x)xxX}, where the functions μA(x):X[0,1]andνA(x):X[0,1] define the degree of membership and the degree of non-membership, respectively of the element xX to A, which is a subset of X, and for every xX, 0(μA(x))2+(νA(x))21. Supposing (μA(x))2+(νA(x))21, then there is a degree of indeterminacy of xX to A defined by πA(x)=1[(μA(x))2+(νA(x))2] and πA(x)[0,1]. In what follows, (μA(x))2+(νA(x))2+(πA(x))2=1. Otherwise, πA(x)=0 whenever (μA(x))2+(νA(x))2=1. }

3. Pythagorean fuzzy multisets

Definition 6. Let X be a nonempty set. Then, an PFMS A drawn from X is of the form A={CMA(x)x,CNA(x)x|xX} or A={x,CMA(x),CNA(x)|xX} where CMA(x)=μ1A(x),...,μnA(x) and CNA(x)=ν1A(x),...,νnA(x) are the count membership and count non-membership degrees defined by the functions CMA:XN[0,1]andCNA:XN[0,1] such that 0[CMA(x)]2+[CNA(x)]21, where N=N{0}.

For each PFMS A of X, CHA(x)=1[CMA(x)]2[CNA(x)]2 is the count hesitation margin of x in A, where CHA(x)=π1A(x),...,πnA. The count hesitation margin CHA(x) is the degree of non-determinacy of xX to A and CHA(x)[0,1]. The count hesitation margin is the function that expresses lack of knowledge of whether xA or xA. Thus, [CMA(x)]2+[CNA(x)]2+[CHA(x)]2=1. We denote the set of all PFMS over X by PFMS(X). Table 1 explains the difference between IFMS and PFMS.
Table 1. IFMS and PFMS.
IFMS PFMS
CM+CN1 CM+CN1 or CM+CN1
0CM+CN1 0CM2+CN21
CH=1(CM+CN) CH=1[CM2+CN2]
CM+CN+CH=1 CM2+CN2+CH2=1

Example Let A be an PFMS of X={x,y} such that CMA(x)=0.7,0.5,0.4 CNA(x)=0.3,0.5,0.6 CMA(y)=0.8,0.6,0.4 CNA(y)=0.4,0.5,0.5. That is A={0.7,0.5,0.4,0.3,0.5,0.6x,0.8,0.6,0.4,0.4,0.5,0.5y}. Then CHA(x)=0.6481,0.7071,0.6928 CHA(y)=0.4472,0.6245,0.7681.

For easy computational purpose, an PFMS can be converted to PFS by taking the mean values of the count membership degrees, count non-membership degrees and count hesitation margin, respectively. That is, an PFMS A in Example 1 becomes an PFS A={0.5333,0.4667x,0.6,0.4667y}.

Definition 7. Two PFMSs A and B are said to be equal or comparable if CMA(x)=CMB(x),CNA(x)=CNB(x) xX.

Definition 8. Let A,BPFMS(X), then A is contained in B denoted by AB if CMA(x)CMB(x)andCNA(x)CNB(x)xX. We say A is properly contained in B, that is, AB if AB and AB. It means CMA(x)CMB(x) and CNA(x)CNB(x) but CMA(x)CMB(x) and CNA(x)CNB(x)xX.

Definition 9. Let X and Y be nonempty sets and let f:XY be a mapping. Suppose APFMS(X) and BPFMS(Y), respectively. Then

  • (i) the inverse image of B under f, denoted by f1(B), is an PFMS of X defined by f1(B)={CMf1(B)(x)x,CNf1(B)(x)xxX}, where CMf1(B)(x)=CMB(f(x)) and CNf1(B)(x)=CNB(f(x))xX.
  • (ii) the image of A under f, denoted by f(A), is an PFMS of Y defined by f(A)={CMf(A)(y)y,CNf(A)(y)yyY}, where CMf(A)(y)={xf1(y)CMA(x),f1(y)0,otherwise, and CNf(A)(y)={xf1(y)CNA(x),f1(y)0,otherwise, for each yY. This is called the extension principle of PFMS.

Theorem 10. Let APFMS(X). Suppose that CHA(x)=0, then the following hold:

  • (i) |CMA(x)|=|(CNA(x)+1)(CNA(x)1)|.
  • (ii) |CNA(x)|=|(CMA(x)+1)(CMA(x)1)|.

Proof. Suppose xX and APFMS(X). Then we prove (i) and (ii). Since CHA(x)=0 for every xX, we have
(CMA(x))2+(CNA(x))2=1 \\ (CMA(x))2=(CNA(x))21\\ (CMA(x))2=(CNA(x)+1)(CNA(x)1)\\ |(CMA(x))2|=|(CNA(x)+1)(CNA(x)1)|\\ |CMA(x)|2=|(CNA(x)+1)(CNA(x)1)|\\ |CMA(x)|=|(CNA(x)+1)(CNA(x)1)|,\\ which proves (i). The proof of (ii) is similar to that of (i).

3.1. Some operations under PFMSs

Definition 11. For any two PFMSs A and B drawn from X, the following operations hold.

  • (i) ComplementAc={CNA(x)x,CMA(x)xxX}
  • (ii) Union AB={max(CMA(x),CMB(x))x,min(CNA(x),CNB(x))xxX}
  • (iii) Intersection AB={min(CMA(x),CMB(x))x,max(CNA(x),CNB(x))xxX}.

Definition 12. Let A,BPFMS(X). Then, the addition of A and B is defined as AB={(CMA(x))2+(CMB(x))2(CMA(x))2(CMB(x))2x,CNA(x)CNB(x)x|xX}, and the multiplication of A and B is defined as AB={CMA(x)CMB(x)x,(CNA(x))2+(CNB(x))2(CNA(x))2(CNB(x))2x|xX}.

Proposition 1. Let A,B,CPFMS(X), then the following properties follow.

  • (i)Complementary law(Ac)c=A
  • (ii) Idempotent laws AA=AAA=A
  • (iii)Commutative laws AB=BAAB=BAAB=BAAB=BA
  • (iv) Associative laws (AB)C=A(BC)(AB)C=A(BC)(AB)C=A(BC)(AB)C=A(BC)
  • (v)Distributive lawsA(BC)=(AB)(AC)A(BC)=(AB)(AC)A(BC)=(AB)(AC)A(BC)=(AB)(AC)A(BC)=(AB)(AC)A(BC)=(AB)(AC) Distributive laws hold for both sides (right and left).
  • (vi)DeMorgan laws (AB)c=AcBc(AB)c=AcBc(AB)c=AcBc(AB)c=AcBc
  • (vii) Absorption lawsA(AB)=AA(AB)=A

Proof. Straightforward, so we omit.

Theorem 13. Let A,BPFMS(X) such that A=Bc and B=Ac, then

  • (i) (AcB)(ABc)=(AcBc)(AB),
  • (ii) (AcB)(ABc)=(AcBc)(AB).

Proof. Since A=Bc and B=Ac, we show that the left hand side (LHS) is equal to the right hand side (RHS). Now, (AcB)(ABc)=(BB)(AA)=AB. Similarly, (AcBc)(AB)=(BA)(AB)=AB. Thus, LHS=RHS, and hence (i) is proved. The proof of (ii) is similar to (i), so we omit.

Definition 14. Let APFMS(X). Then, the level/ground set of A is defined by A={xX|CMA(x)>0,CNA(x)<1}. Certainly, A is a subset of X. }

Proposition 2. Suppose A and B are PFMSs of a non-empty set X, then

  • (i) (AB)=AB,
  • (ii) (AB)=AB.

Proof. Straightforward, so we omit.

3.2. Accuracy and score functions of PFMS

Definition 15. Let APFMS(X). Then the score function, s of A is defined by s(A)=Σni=1[(CMA(xi))2(CNA(xi))2], where s(A)[1,1].

Definition 16. Let APFMS(X). Then the accuracy function, a of A is defined by a(A)=Σni=1[(CMA(xi))2+(CNA(xi))2] for a(A)[0,1].

Theorem 17. Let APFMS(X). Then the following hold xX:

  • (i) s(A)=0CMA(xi)=CNA(xi).
  • (ii) s(A)=1|CNA(xi)|=|(1+CMA(xi))(1CMA(xi))|.
  • (iii s(A)=1CMA(xi)=(CNA(xi)+1)(CNA(xi)1).

Proof.

  • (i) Suppose s(A)=0. Then (CMA(xi))2=(CNA(xi))2 implies CMA(xi)=CNA(xi) xiX.
    Conversely, assume CMA(xi)=CNA(xi). Then (CMA(xi))2=(CNA(xi))2 xiX. Thus (CMA(xi))2(CNA(xi))2=0. Hence s(A)=0.
  • (ii) Suppose s(A)=1. Then
    1(CMA(xi))2=(CNA(xi))2
    (1+CMA(xi))(1CMA(xi))=(CNA(xi))2
    |(1+CMA(xi))(1CMA(xi))|=|CNA(xi)|2
    |CNA(xi)|=|(1+CMA(xi))(1CMA(xi))| xiX.
    Conversely, assume |CNA(xi)|=|(1+CMA(xi))(1CMA(xi))|. So we get
    |CNA(xi)|2=|1(CMA(xi))2|
    (CNA(xi))2=1(CMA(xi))2 or |CNA(xi)|2=|1(CMA(xi))2|
    (CNA(xi))2=1(CMA(xi))2. Take (CNA(xi))2=1(CMA(xi))2 (CMA(xi))2(CNA(xi))2=1 s(A)=1.
  • (iii) Suppose s(A)=1. Then
    (CNA(xi))21=(CMA(xi))2
    (CNA(xi)1)(CNA(xi)+1)=(CMA(xi))2
    CMA(xi)=(CNA(xi)1)(CNA(xi)+1).
    Conversely, suppose CMA(xi)=(CNA(xi)1)(CNA(xi)+1). Then
    (CMA(xi))2=(CNA(xi))21 (CMA(xi))2(CNA(xi))2=1 s(A)=1.

Theorem 18. Let APFMS(X). Then the following statements hold xX:

  • i) a(A)=1CHA(xi)=0.
  • (ii) a(A)=0|CMA(xi)|=|CNA(xi)|.

Proof.

  • (i) Suppose a(A)=1. So we have (CMA(xi))2+(CNA(xi))2=1, that is, CHA(xi)=0 since CHA(xi)=1[(CMA(xi))2+(CNA(xi))2].
    Conversely, assume that CHA(xi)=0. Then it follows that (CMA(xi))2+(CNA(xi))2=1a(A)=1.
  • (ii)] Suppose a(A)=0. Then (CMA(xi))2=(CNA(xi))2 or (CNA(xi))2=(CMA(xi))2 |CMA(xi)|2=|CNA(xi)|2 |CMA(xi)|=|CNA(xi)|.

3.3. Some properties of PFMSs

3.3.1. (α,β)cuts of PFMS

Definition 19. Let APFMS(X). Then for α,β[0,1], the sets A[α,β] and A(α,β) defined by A[α,β]={xXCMA(x)α,CNA(x)β} and A(α,β)={xXCMA(x)>α,CNA(x)<β} are called strong and weak upper α,βcuts of A.
Similarly, the sets A[α,β] and A(α,β) defined by A[α,β]={xXCMA(x)α,CNA(x)β} and A(α,β)={xXCMA(x)<α,CNA(x)>β} are called strong and weak lower α,βcuts of A.

Remark 1. Let APFMS(X) and take any α,β[0,1] such that A[α,β] and A[α,β] exist. Then, it follows that

  • (i) A(α,β)A[α,β] and A(α,β)A[α,β].
  • (ii) A[α,β]=B[α,β],A(α,β)=B(α,β),A[α,β]=B[α,β] and A(α,β)=B(α,β) iff A=B.

For the purpose of this work, we shall be restricted to strong cuts of PFMS since A(α,β)A[α,β] and A(α,β)A[α,β].

Proposition 3. Let A,BPFMS(X) and α,β,α1,α2,β1,β2[0,1]. Then we have

  • (i) A[α1,β1]A[α2,β2] iff α1α2 and β1β2,
  • (ii)AB iff A[α,β]B[α,β].

Proof.

  • (i) Let xA[α1,β1]CMA(x)α1 and CNA(x)β1. Since α1α2CMA(x)α1α2. Also, β1β2CNA(x)β1β2. Hence, A[α1,β1]A[α2,β2].
    Conversely, for A[α1,β1]A[α2,β2], it is clear that α1α2 and β1β2.
  • (ii) We know that ABCMA(x)CMB(x) and CNA(x)CNB(x) xX. For xA[α,β] and xB[α,β] CMB(x)CMA(x)α and CNB(x)CNA(x)β. So, A[α,β]B[α,β].
    The converse is straightforward.

Corollary 1. Let A,BPFMS(X) and α,β,α1,α2,β1,β2[0,1]. Then the following hold.

  • (i) A[α1,β1]A[α2,β2] iff α1α2 and β1β2.
  • (ii) AB iff A[α,β]B[α,β].

Proof. It follows from Proposition 3.

Proposition 4. Let APFMS(X). For any \alpha_1,\beta_1\alpha_2,\beta_2\in [0,1] such that \alpha_1\leq \alpha_2 and \beta_1\geq \beta_2, we have

  • (i) A_{(\alpha_2,\beta_2)}\subseteq A_{[\alpha_2,\beta_2]}\subseteq A_{(\alpha_1,\beta_1)} and
  • (ii) A^{(\alpha_1,\beta_1)}\subseteq A^{(\alpha_2,\beta_2)}\subseteq A^{[\alpha_2,\beta_2]}.

Proof. Combining Definition 19 and Remark 1, the proof follows.

Proposition 5. Let A,B\in PFMS(X) and \alpha,\beta\in [0,1]. Then

  • (i)(A\cap B)_{[\alpha,\beta]} =A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]},
  • (ii)(A\cup B)_{[\alpha,\beta]} =A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}.

Proof.

  • (i) If A,B\in PFMS(X)\Rightarrow A\cap B\subseteq A and A\cap B\subseteq B. By Proposition 3, (A\cap B)_{[\alpha,\beta]}\subseteq A_{[\alpha,\beta]} and (A\cap B)_{[\alpha,\beta]}\subseteq B_{[\alpha,\beta]} \Rightarrow (A\cap B)_{[\alpha,\beta]}\subseteq A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}.
    Again, suppose x\in A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\Rightarrow x\in A_{[\alpha,\beta]} and x\in B_{[\alpha,\beta]}, then \begin{eqnarray*}A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}&=& \left\lbrace x\in X\mid CM_A (x)\geq \alpha,\: CN_A (x)\leq \beta \right\rbrace \cap \left\lbrace x\in X\mid CM_B (x)\geq \alpha, \: CN_B (x)\leq \beta\right\rbrace \\ &=& \left\lbrace x\in X\mid min[CM_A (x)\geq \alpha,CM_B (x)\geq \alpha],\: max[CN_A (x)\leq \beta,CN_B (x)\leq \beta]\right\rbrace\\ & =& \left\lbrace x\in X\mid min[CM_A (x),CM_B (x)]\geq \alpha, \: max[CN_A (x),CN_B (x)]\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid CM_{A\cap B} (x)\geq \alpha,\: CN_{A\cap B} (x)\leq \beta\right\rbrace\\ &\subseteq& (A\cap B)_{[\alpha,\beta]}.\end{eqnarray*}
    Consequently, x\in A_{[\alpha,\beta]}\cap B_{[\alpha,\beta]}\Rightarrow x\in (A\cap B)_{[\alpha,\beta]}. Hence (A\cap B)_{[\alpha,\beta]} =A_{[\alpha,\beta]} \cap B_{[\alpha,\beta]}.
  • (ii) For A,B\in PFMS(X), it is clear that A\subseteq A\cup B and B\subseteq A\cup B. By Proposition 3, implies A_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]} and B_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]}, that is, A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}\subseteq (A\cup B)_{[\alpha,\beta]}.
    Also, x\in (A\cup B)_{[\alpha,\beta]}\Rightarrow CM_{(A\cup B)}(x)\geq \alpha and CN_{(A\cup B)}(x)\leq \beta, that is,
    \begin{eqnarray*}(A\cup B)_{[\alpha,\beta]} &=& \left\lbrace x\in X\mid CM_{(A\cup B)}(x)\geq \alpha, CN_{(A\cup B)}(x)\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid max[CM_A(x),CM_B(x)]\geq \alpha, min[CN_A(x),CN_B(x)]\leq \beta\right\rbrace \\ &=& \left\lbrace x\in X\mid max[CM_A(x), CM_B(x)]\geq \alpha, min[CN_A(x), CN_B(x)]\leq \beta\right\rbrace\\ &=& \left\lbrace x\in X\mid CM_A(x)\geq \alpha, CN_A(x)\leq \beta\right\rbrace \cup \left\lbrace x\in X\mid CM_B(x)\geq \alpha, CN_B(x)\leq \beta\right\rbrace\\ &\subseteq& A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}.\end{eqnarray*}
    Thus x\in A_{[\alpha,\beta]} and x\in B_{[\alpha,\beta]}. Hence (A\cup B)_{[\alpha,\beta]}=A_{[\alpha,\beta]}\cup B_{[\alpha,\beta]}.

Corollary 2. Let A,B\in PFMS(X) and \alpha,\beta\in [0,1]. Then

  • (i) (A\cap B)^{[\alpha,\beta]} =A^{[\alpha,\beta]}\cap B^{[\alpha,\beta]},
  • (ii) (A\cup B)^{[\alpha,\beta]} =A^{[\alpha,\beta]}\cup B^{[\alpha,\beta]}.

Proof. Straightforward from Proposition 5.

Proposition 6. Suppose \left\lbrace A_i\right\rbrace_{i\in I}\in PFMS(X) and \alpha,\beta\in [0,1], then

  • (i)](\bigcap_{i\in I}A_i)_{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)_{[\alpha,\beta]},
  • (ii)](\bigcup_{i\in I}A_i)_{[\alpha,\beta]}=\bigcup_{i\in I}(A_i)_{[\alpha,\beta]},
  • (iii)](\bigcap_{i\in I}A_i)^{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)^{[\alpha,\beta]},
  • (iv)](\bigcup_{i\in I}A_i)^{[\alpha,\beta]}=\bigcup_{i\in I}(A_i)^{[\alpha,\beta]}.

Proof. (i) Let C=\bigcap_{i\in I}A_i, then CM_C(x)=\bigwedge_{i\in I}CM_{A_{i}}(x) and CN_C(x)=\bigvee_{i\in I}CN_{A_{i}}(x) \forall x\in X. Thus \begin{eqnarray*} C_{[\alpha,\beta]} & = & \left\lbrace x\in X\mid CM_{C}(x)\geq \alpha, CN_{C}(x)\leq \beta \right\rbrace\\ & = & \left\lbrace x\in X\mid (\bigwedge_{i\in I}CM_{A_{i}}(x))\geq \alpha, (\bigvee_{i\in I}CN_{A_{i}}(x))\leq \beta\right\rbrace\\ & = & \left\lbrace x\in X\mid \bigwedge_{i\in I}CM_{A_{i}}(x)\geq \alpha, \bigvee_{i\in I}CN_{A_{i}}(x)\leq \beta \right\rbrace\\ & = & \bigcap_{i\in I}(A_i)_{[\alpha,\beta]}. \end{eqnarray*}Hence (\bigcap_{i\in I}A_i)_{[\alpha,\beta]}=\bigcap_{i\in I}(A_i)_{[\alpha,\beta]}.
(ii)-(iv) follow similarly.

Remark 2. Suppose A,B,C\in PFMS(X) such that B\subseteq C. Then for \alpha,\beta\in [0,1], we have

  • (i) (A\cap B)_{[\alpha,\beta]}\subseteq (A\cap C)_{[\alpha,\beta]},
  • (ii) (A\cup B)_{[\alpha,\beta]}\subseteq (A\cup C)_{[\alpha,\beta]},
  • (iii) (A\cap B)^{[\alpha,\beta]}\subseteq (A\cap C)^{[\alpha,\beta]},
  • (iv)(A\cup B)^{[\alpha,\beta]}\subseteq (A\cup C)^{[\alpha,\beta]}.

Proposition 7. Let f be a function from X to Y, A\in PFMS(X) and B\in PFMS(Y), respectively. Then, for any \alpha,\beta\in [0,1], we have

  • (i)f(A_{[\alpha,\beta]})\subseteq (f(A))_{[\alpha,\beta]},
  • (ii)f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]},
  • (iii) f(A_{(\alpha,\beta)})\subseteq f(A_{[\alpha,\beta]})\subseteq (f(A))_{[\alpha,\beta]},
  • (iv)f^{-1}(B_{(\alpha,\beta)})\subseteq f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]}.

Proof.

  • (i) Let y\in f(A_{[\alpha,\beta]}), then \exists \: x\in A_{[\alpha,\beta]} such that f(x)=y and CM_A(x)\geq \alpha, CN_A(x)\leq \beta. Consequently, we get CM_A(f^{-1}(y))\geq \alpha, \alpha\in [0,1] \: \textrm{implies}\: CM_{f(A)}(y)\geq \alpha, \alpha\in [0,1],
    Similarly, CN_A(f^{-1}(y))\leq \beta, \beta\in [0,1] \: \textrm{implies}\: CN_{f(A)}(y)\leq \beta, \beta\in [0,1], and so, y\in (f(A))_{[\alpha,\beta]}. Hence, f(A_{[\alpha]})\subseteq (f(A))_{[\alpha]}.
  • (ii) For every x, x\in f^{-1}(B_{[\alpha,\beta]})\Leftrightarrow f(x)\in B_{[\alpha,\beta]} \Leftrightarrow CM_B(f(x))\geq \alpha and CN_B(f(x))\leq \beta. Thus CM_{f^{-1}(B)}(x)=CM_B(f(x))\geq \alpha and CN_{f^{-1}(B)}(x)=CN_B(f(x))\leq \beta that is, x\in (f^{-1}(B))_{[\alpha,\beta]}. Hence, f^{-1}(B_{[\alpha,\beta]})=(f^{-1}(B))_{[\alpha,\beta]}.
  • (iii) Since A_{(\alpha,\beta)}\subseteq A_{[\alpha,\beta]}, then f(A_{(\alpha,\beta)})\subseteq f(A_{[\alpha,\beta]}). Hence, the result follows from (i).
  • (iv) Also, B_{(\alpha,\beta)}\subseteq B_{[\alpha,\beta]} and so, f^{-1}(A_{(\alpha,\beta)})\subseteq f^{-1}(A_{[\alpha,\beta]}) by the same reasons as in (iii). The proof is completed by (ii).

Corollary Suppose f is a function from X to Y. If A\in PFMS(X) and B\in PFMS(Y), respectively, then for at least one \alpha, \beta\in [0,1],

  • (i)f(A^{[\alpha,\beta]})\subseteq (f(A))^{[\alpha,\beta]},
  • (ii)f^{-1}(B^{[\alpha,\beta]})=(f^{-1}(B))^{[\alpha,\beta]},
  • (iii) f(A^{(\alpha,\beta)})\subseteq f(A^{[\alpha,\beta]})\subseteq (f(A))^{[\alpha,\beta]},
  • (iv)f^{-1}(B^{(\alpha,\beta)})\subseteq f^{-1}(B^{[\alpha,\beta]})=(f^{-1}(B))^{[\alpha,\beta]}.

Proof. Similar to Proposition 7.

3.3.2. Some modal operators on PFMS
Now, we propose and explicate some modal operators on PFMS, which transform every PFMS to fuzzy multiset. These modal operators are similar to the operators necessity and possibility defined in some modal logics.

Definition 20. Let A\in PFMS(X). Then we define the following operators:

  • (i) the necessity operator \square A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace
  • (ii) the possibility operator \lozenge A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace.

Remark 3. If A is an ordinary fuzzy multiset, then \square A=A=\lozenge A. An ordinary fuzzy multiset A can also be written in PFMS setting as A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace or A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace.

Theorem 21. Let A\in PFMS(X). Then the following properties hold:

  • (i) \overline{\square \bar{A}}=\lozenge A
  • (ii) \overline{\lozenge \bar{A}}=\square A
  • (iii) \square \square A=\square A
  • (iv) \lozenge \lozenge A=\lozenge A
  • (v)\square \lozenge A=\lozenge A
  • (vi) \lozenge \square A=\square A.

Proof. Let x\in X. Using Definition 20, we have

  • (i) \begin{eqnarray*} \overline{\square \bar{A}} & = & \overline{\square \overline{\left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace}}\\ & = & \overline{\square \left\lbrace \left\langle x, CN_A(x),CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \overline{ \left\lbrace \left\langle x, CN_A(x),\sqrt{1-(CN_A(x))^2} \right\rangle |x\in X\right\rbrace}\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A. \end{eqnarray*}
  • (ii) \begin{eqnarray*} \overline{\lozenge \bar{A}} & = & \overline{\lozenge \overline{\left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace}}\\ & = & \overline{\lozenge \left\lbrace \left\langle x, CN_A(x),CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \overline{ \left\lbrace \left\langle x, \sqrt{1-(CM_A(x))^2}, CM_A(x) \right\rangle |x\in X\right\rbrace}\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}
  • (iii) \begin{eqnarray*} \square \square A & = & \square \square \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \square \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}
  • (iv) \begin{eqnarray*} \lozenge \lozenge A & = & \lozenge \lozenge \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A. \end{eqnarray*}
  • (v) \begin{eqnarray*} \square \lozenge A & = & \square \lozenge \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \square \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{1-(\sqrt{1-(CN_A(x))^2)})^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{1-(1-(CN_A(x))^2)} \right\rangle |x\in X\right\rbrace \end{eqnarray*} \begin{eqnarray*} & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, \sqrt{(CN_A(x))^2)} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge A.\end{eqnarray*}
  • (vi) \begin{eqnarray*} \lozenge \square A & = & \lozenge \square \left\lbrace \left\langle x, CM_A(x),CN_A(x) \right\rangle |x\in X\right\rbrace\\ & = & \lozenge \left\lbrace \left\langle x, CM_A(x)), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(\sqrt{1-(CM_A(x))^2)})^2}, \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{1-(1-(CM_A(x))^2)}, \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, \sqrt{(CM_A(x))^2}, \sqrt{1-(CM_A(x))^2)} \right\rangle |x\in X\right\rbrace\\ & = & \left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2} \right\rangle |x\in X\right\rbrace\\ & = & \square A. \end{eqnarray*}

Corollary 4. Let A\in PFMS(X). Then the following properties hold:

  • (i)\overline{\square \bar{A}}=\square \lozenge A= \lozenge \lozenge A
  • (ii) \overline{\lozenge \bar{A}}=\lozenge \square A= \square \square A.

Proof. Straightforward from Theorem 21.

Theorem 22. Let A,B\in PFMS(X). Then the following properties hold:

  • (i) \square(A\cap B)=\square A\cap \square B
  • (ii) \lozenge(A\cap B)=\lozenge A\cap \lozenge B
  • (iii) \square(A\cup B)=\square A\cup \square B
  • (iv) \lozenge(A\cup B)=\lozenge A\cup \lozenge B.

Proof. The results are straightforward from Definitions 11 and 20, so we omit the proofs.

Theorem 23. Let A,B\in PFMS(X). Then the following properties hold:

  • (i) \overline{\square \overline{(A\cap B)}}=\lozenge (A\cap B)
  • (ii) \overline{\lozenge \overline{(A\cap B)}}=\square (A\cap B)
  • (iii) \overline{\square \overline{(A\cup B)}}=\lozenge (A\cup B)
  • (iv) \overline{\lozenge \overline{(A\cup B)}}=\square (A\cup B)
  • (v) \square \square (A\cap B)=\square (A\cap B)
  • (vi) \lozenge \lozenge (A\cap B)=\lozenge (A\cap B)
  • (vii) \square \square (A\cup B)=\square (A\cup B)
  • (viii) \lozenge \lozenge (A\cup B)=\lozenge (A\cup B)
  • (ix) \square \lozenge (A\cap B)=\lozenge (A\cap B)
  • (x) \lozenge \square (A\cap B)=\square (A\cap B)
  • (xi) \square \lozenge (A\cup B)=\lozenge (A\cup B)
  • (xii) \lozenge \square (A\cup B)=\square (A\cup B).

Proof. Synthesizing Theorems 21 and 22, the proofs follow.

Corollary 5. Let A,B\in PFMS(X). Then the following properties hold:

  • (i) \overline{\square \overline{(A\cap B)}}=\lozenge \lozenge (A\cap B)=\square \lozenge (A\cap B)
  • (ii) \overline{\lozenge \overline{(A\cap B)}}=\square \square (A\cap B)=\lozenge \square (A\cap B)
  • (iii) \overline{\square \overline{(A\cup B)}}=\lozenge \lozenge (A\cup B)=\square \lozenge(A\cup B)
  • (iv) \overline{\lozenge \overline{(A\cup B)}}=\square \square(A\cup B)=\lozenge \square(A\cup B).

Proof. Straightforward from Theorem 23.

Theorem 24. Let A\in PFMS(X). Then \square A\subset A\subset \lozenge A.

Proof. Recall that for A=\left\lbrace \left\langle x, CM_A(x), CN_A(x)\right\rangle |x\in X\right\rbrace, we have \square A=\left\lbrace \left\langle x, CM_A(x), \sqrt{1-(CM_A(x))^2}\right\rangle |x\in X\right\rbrace and \lozenge A=\left\lbrace \left\langle x, \sqrt{1-(CN_A(x))^2}, CN_A(x)\right\rangle |x\in X\right\rbrace. Also, A\subset B\Leftrightarrow A\subseteq B and A\neq B, and A\subseteq B\Leftrightarrow CM_A(x)\leq CM_B(x) and either CN_A(x)\geq CN_B(x) (or CN_A(x)\leq CN_B(x)) \forall x\in X. To prove that \square A\subset A, it is sufficient to show that \sqrt{1-(CM_A(x))^2}\geq CN_A(x). From Definition 6, we have \begin{eqnarray*} (CM_A(x))^2+(CN_A(x))^2\leq 1 & \Rightarrow & (CN_A(x))^2\leq 1-(CM_A(x))^2\\ & \Rightarrow & CN_A(x)\leq \sqrt{1-(CM_A(x))^2}, \end{eqnarray*} that is \sqrt{1-(CM_A(x))^2}\geq CN_A(x) \forall x\in X. Thus \square A\subset A. Again, we show that A\subset \lozenge A. To see this, it is enough to prove that CM_A(x)\leq \sqrt{1-(CN_A(x))^2}. By Definition 6, We get \begin{eqnarray*} (CM_A(x))^2+(CN_A(x))^2\leq 1 & \Rightarrow & (CM_A(x))^2\leq 1-(CN_A(x))^2\\ & \Rightarrow & CM_A(x)\leq \sqrt{1-(CN_A(x))^2}\; \forall x\in X. \end{eqnarray*} Hence, A\subset \lozenge A, and the proof is complete.

4. Composite relation on PFMS and its application in course placements

4.1. Composite relation defined on PFMS

In what follows, we define the composite relation on PFMS.

Definition 25. Let X and Y be two non-empty sets. A Pythagorean fuzzy multi-relation (PFMR), R from X to Y is a PFMS of X\times Y characterised by the count membership function, CM_R and count non-membership function, CN_R. A PF multi-relation or PFMR from X to Y is denoted by R(X\to Y).

Definition 26. Let A\in PFMS(X). Then the max-min-max composition of R(X\to Y) with A is a PFMS B of Y denoted by B=R\circ A, such that its count membership and count non-membership functions are defined by CM_B(y)=\bigvee_{x}(min[CM_A(x), CM_R(x,y)]) and CN_B(y)=\bigwedge_{x}(max[CN_A(x), CN_R(x,y)]) \forall x\in X and y\in Y, where \bigvee=maximum, \bigwedge=minimum.

Definition 27. Let Q(X\to Y) and R(Y\to Z) be two PFMRs. Then the max-min-max composition R\circ Q is a PFMR from X to Z such that its count membership and count non-membership functions are defined by CM_{R\circ Q}(x,z)=\bigvee_{y}(min[CM_Q(x,y), CM_R(y,z)]) and CN_{R\circ Q}(x,z)=\bigwedge_{y}(max[CN_Q(x,y), CN_R(y,z)]) \forall (x,z)\in X\times Z and \forall y\in Y.

Remark 4. From Definitions 26 and 27, the max-min-max composition B or R\circ Q is calculated by B=CM_B(y)-CN_B(y)CH_B(y) \forall y\in Y or R\circ Q=CM_{R\circ Q}(x,z)-CN_{R\circ Q}(x,z)CH_{R\circ Q}(x,z) \forall (x,z)\in X\times Z.

Proposition 8. If R and S are two PFMRs on X\times Y and Y\times Z, respectively. Then

  • (i) (R^{-1})^{-1}=R,
  • (ii) (S\circ R)^{-1}=R^{-1}\circ S^{-1}.

4.2. Composite relation on PFMS in course placement

We apply the notion of PFMR as follows. Let S=\left\lbrace s_1,...,s_l\right\rbrace, C=\left\lbrace c_1,...,c_m\right\rbrace \: \textrm{and}\: A=\left\lbrace a_1,...,a_n\right\rbrace be finite set of subjects related to the courses, finite set of courses and finite set of applicants, respectively. Assume there are two PFMRs, R(A\to S) and U(S\to C) such that R=\left\lbrace \left\langle (a,s), CM_R(a,s), CN_R(a,s)\right\rangle| (a,s)\in A\times S \right\rbrace and U=\left\lbrace \left\langle (s,c), CM_U(s,c), CN_U(s,c)\right\rangle| (s,c)\in S\times C \right\rbrace, where CM_R(a,s) signifies the grade to which the applicant, a passes the related subject requirement, s, and CN_R(a,s) signifies the grade to which the applicant, a does not pass the related subject requirement, s.
Similarly, CM_U(s,c) signifies the grade to which the related subject requirement, s determines the course, c, and CN_U(s,c) signifies the grade to which the related subject requirement, s does not determine the course, c.
The composition, T of R and U is given as T=R\circ U. This describes the state in which the applicants, a_i with respect to the related subjects requirement, s_j fit the courses, c_k. Thus: CM_T(a_i,c_k)=\bigvee_{s_j\in S}\left\lbrace min[CM_R(a_i,s_j), CM_U(s_j,c_k)]\right\rbrace and CN_T(a_i,c_k)=\bigwedge_{s_j\in S}\left\lbrace max[CN_R(a_i,s_j), CN_U(s_j,c_k)]\right\rbrace \forall a_i\in A and c_k\in C, where i,j and k take values from 1,...,n. The career placement can be attained if the value of T given by T=CM_T(a_i,c_k)-CN_T(a_i,c_k) CH_T(a_i,c_k), as calculated from R and U for the placements of a_i into any c_k with respect to s_j is the greatest, greater than 0.5.
4.2.1. Case study
Let A=\left\lbrace \textrm{Eli, Ella, Avi, Joe, Jones}\right\rbrace be the set of applicants for the course placements, C=\left\lbrace \textrm{medicine, pharmacy, surgery, anatomy, physiology}\right\rbrace be the set of courses the applicants are competing for, and S=\left\lbrace \textrm{English Lang., Maths, Biology, Physics, Chemistry, Health Sci.}\right\rbrace be the set of subjects' requirement to the set of courses.
Suppose the PFMR, R(A\to S) is given in Table 3. This data in Pythagorean fuzzy multi-values are supposedly drawn after the applicants sat for a multiple choice qualification assessments on the listed subjects within a specified time, for two different times where the first and second assessments are closely related to checkmate the effect of test contingencies.
The first entries are the membership values signifying the Pythagorean fuzzy multi-values of the marks allocated to the questions the applicants answered, and the second entries are the non-membership values signifying the Pythagorean fuzzy multi-values of the marks allocated to the questions failed. Converting the PFMSs in Table 2 to PFSs for easy computation, the results in Table 3 are obtained.
Table 2. R(A\to S).
R English Maths Biology Physics Chemistry Health
Eli \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.3\right\rangle \left\langle 0.5,0.5\right\rangle \left\langle 0.6,0.2\right\rangle
\left\langle 0.5,0.3\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.5,0.3\right\rangle \left\langle 0.5,0.1\right\rangle
Ella \left\langle 0.5,0.3\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.3\right\rangle \left\langle 0.4,0.5\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.7,0.1\right\rangle
\left\langle 0.4,0.3\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.7,0.3\right\rangle \left\langle 0.6,0.2\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.7,0.3\right\rangle
Avi \left\langle 0.7,0.3\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.7,0.3\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.4,0.5\right\rangle \left\langle 0.6,0.3\right\rangle
\left\langle 0.6,0.2\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.5,0.2\right\rangle \left\langle 0.4,0.5\right\rangle \left\langle 0.5,0.5\right\rangle \left\langle 0.7,0.3\right\rangle
Joe \left\langle 0.6,0.4\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.7,0.2\right\rangle
\left\langle 0.6,0.3\right\rangle \left\langle 0.6,0.1\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.2\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.7,0.2\right\rangle
Jones \left\langle 0.8,0.1\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.7,0.1\right\rangle \left\langle 0.6,0.1\right\rangle \left\langle 0.8,0.1\right\rangle
\left\langle 0.6,0.2\right\rangle \left\langle 0.5,0.1\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.7,0.1\right\rangle \left\langle 0.4,0.2\right\rangle \left\langle 0.8,0.1\right\rangle
Table 3. R(A\to S).
R English Maths Biology Physics Chemistry Health
Eli \left\langle 0.55,0.30\right\rangle \left\langle 0.60,0.30\right\rangle \left\langle 0.60,0.30\right\rangle \left\langle 0.50,0.35\right\rangle \left\langle 0.50,0.40\right\rangle \left\langle 0.55,0.15\right\rangle
Ella \left\langle 0.45,0.30\right\rangle \left\langle 0.70,0.25\right\rangle \left\langle 0.60,0.30\right\rangle \left\langle 0.50,0.35\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.20\right\rangle
Avi \left\langle 0.65,0.25\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.60,0.25\right\rangle \left\langle 0.45,0.45\right\rangle \left\langle 0.45,0.50\right\rangle \left\langle 0.65,0.30\right\rangle
Joe \left\langle 0.60,0.35\right\rangle \left\langle 0.70,0.15\right\rangle \left\langle 0.60,0.30\right\rangle \left\langle 0.55,0.25\right\rangle \left\langle 0.65,0.25\right\rangle \left\langle 0.70,0.20\right\rangle
Jones \left\langle 0.70,0.15\right\rangle \left\langle 0.60,0.15\right\rangle \left\langle 0.65,0.30\right\rangle \left\langle 0.70,0.10\right\rangle \left\langle 0.50,0.15\right\rangle \left\langle 0.80,0.10\right\rangle
PFMR, U(S\to C) is the School bench-mark for admission into the mentioned courses in Pythagorean fuzzy values. The data is in Table 4.
Table 4. U(S\to C).
U medicine pharmacy surgery anatomy physiology
English \left\langle 0.8,0.1\right\rangle \left\langle 0.9,0.1\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.7,0.3\right\rangle \left\langle 0.8,0.2\right\rangle
Maths \left\langle 0.7,0.2\right\rangle \left\langle 0.8,0.1\right\rangle \left\langle 0.5,0.3\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.5,0.3\right\rangle
Biology \left\langle 0.9,0.1\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.9,0.1\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.9,0.1\right\rangle
Physics \left\langle 0.6,0.3\right\rangle \left\langle 0.5,0.2\right\rangle \left\langle 0.5,0.4\right\rangle \left\langle 0.6,0.3\right\rangle \left\langle 0.6,0.2\right\rangle
Chemistry \left\langle 0.8,0.2\right\rangle \left\langle 0.7,0.2\right\rangle \left\langle 0.7,0.3\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.7,0.2\right\rangle
Health \left\langle 0.8,0.1\right\rangle \left\langle 0.8,0.2\right\rangle \left\langle 0.7,0.3\right\rangle \left\langle 0.9,0.1\right\rangle \left\langle 0.8,0.2\right\rangle
The values of CM_{R\circ U}(a_i,c_k) and CN_{R\circ U}(a_i,c_k) of the composition T=R\circ U follow:
Table 5. CM_{R\circ U}(a_i,c_k) and CN_{R\circ U}(a_i,c_k).
CM, CN medicine pharmacy surgery anatomy physiology
Eli \left\langle 0.60,0.15\right\rangle \left\langle 0.60,0.20\right\rangle \left\langle 0.60,0.30\right\rangle \left\langle 0.60,0.15\right\rangle \left\langle 0.60,0.20\right\rangle
Ella \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.30\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.20\right\rangle
Avi \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.65,0.25\right\rangle \left\langle 0.65,0.25\right\rangle \left\langle 0.65,0.25\right\rangle
Joe \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.15\right\rangle \left\langle 0.70,0.30\right\rangle \left\langle 0.70,0.20\right\rangle \left\langle 0.70,0.20\right\rangle
Jones \left\langle 0.80,0.10\right\rangle \left\langle 0.80,0.15\right\rangle \left\langle 0.70,0.30\right\rangle \left\langle 0.80,0.10\right\rangle \left\langle 0.80,0.20\right\rangle
After computing the values of the count of hesitation margins which signify the marks loss due to the hesitation in answering questions within the specified time, T is calculated as show in Table 6.
Table 6. T=CM_T-CN_T CH_T.
T medicine pharmacy surgery anatomy physiology
Eli 0.4821 0.4451 0.3775 0.4821 0.4451
Ella 0.5629 0.5629 0.5056 0.5629 0.5629
Avi 0.5629 0.5629 0.4706 0.4706 0.4706
Joe 0.5629 0.5953 0.5056 0.5629 0.5629
Jones 0.7408 0.7129 0.5056 0.7408 0.6869

The course placements are carried out on the basis of which of the applicant has the greatest T such that T >0.5. However, if an applicant is suitable to study more than one courses based on the value of T, then the applicant would be allowed to make a personal choice within the range of the courses he/she has the greatest T such that T >0.5.

From Table 6, the following placements are made: Eli is not suitable to read any of the courses; Ella is suitable to read any of medicine, pharmacy, anatomy and physiology; Avi is suitable to read either medicine or pharmacy; Joe is suitable to read pharmacy and Jones is suitable to read either medicine or anatomy. Suppose there is only one slot for medicine, it would be given to Jones. Also, if one applicant is to ready pharmacy, it would be Joe. Notwithstanding, Jones is very suitable to read any of the courses ahead of all the applicants.

5. Conclusion

We have proposed the idea of PFMSs as the generalization of PFSs such that each of membership degree, non-membership degree and hesitation margin of PFS, are allowed to repeat as count membership degrees, count non-membership degrees and count hesitation degrees. It was shown that PFMS is either the incorporation of IFMS into PFS setting or PFS in multiset framework. We have discussed some algebraic properties of PFMS and proposed the analogs of the modal logic operators "necessity" and "possibility" with some results. Also the ideas of level sets, cuts, accuracy and score functions were established in the setting of PFMS with a number of results. The notion of composite relation is proposed in PFMSs and applied to practical decision-making problem of course placements in higher institution. PFMS as a soft computing technique can find expression in image enhancement techniques for better image quality, other multi-criteria decision-making (MCDM) problems or multi-attribute decision-making (MADM) problems among other potential applications.

Acknowledgments

The author would like to thank the Editor -in-chief for his technical comments and the reviewers for their insightful contributions.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

The authors declare no conflict of interest.

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