# Fuzzy Soft Multiset Theory

Authors: Shawkat Alkhazaleh; Abdul Razak Salleh
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.


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## Body

## 1. Introduction

Most of the problems in engineering, medical science, economics, environments, and so forth have various uncertainties. Molodtsov [1] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. After Molodtsov’s work, some different operations and application of soft sets were studied by Chen et al. [2] and Maji et al. [3, 4]. Furthermore, Maji et al. [5] presented the definition of fuzzy soft set and Roy and Maji [6] presented the applications of this notion to decision-making problems. Alkhazaleh et al. [7] generalized the concept of fuzzy soft set to possibility fuzzy soft set and they gave some applications of this concept in decision making and medical diagnosis. They also introduced the concept of fuzzy parameterized interval-valued fuzzy soft set [8], where the mapping is defined from the fuzzy set parameters to the interval-valued fuzzy subsets of the universal set and gave an application of this concept in decision making. In 2012 Alkhazaleh and Salleh [9] introduced the concept of generalised interval-valued fuzzy soft set and studied its properties and application. Alkhazaleh and Salleh [10] introduced the concept of soft expert sets where the user can know the opinion of all experts in one model and gave an application of this concept in decision-making problem. Salleh et al. in 2012 [11] introduced and studied the concept of multiparameterized soft set as a generalization of Molodtsov’s soft set. Alkhazaleh et al. [12] as a generalization of Molodtsov’s soft set, presented the definition of a soft multiset and its basic operations such as complement, union, and intersection. Salleh and Alkhazaleh [13] studied the application of soft multiset in decision making problem. In 2011 Salleh gave a brief survey from soft sets to intuitionistic fuzzy soft sets [14]. In this paper we give the definition of a fuzzy soft multiset, a more general concept, which is a combination of fuzzy set and soft multiset and studied its properties. We also introduce its basic operations, namely, complement, union, and intersection, and their properties. An application of this theory in a decision-making problem is given.


## 2. Preliminaries

In this section, we recall some basic notions in soft set theory, soft multiset theory, and fuzzy soft set. Molodtsov defined soft set in the following way. LetU be a universe and E a set of parameters. Let P(U) denote the power set of U and A⊆E.Definition 2.1 (see [1]).
A pair(F,A) is called a soft set over U, where F is a mapping
(2.1)F:A⟶P(U).
In other words, a soft set over U is a parameterized family of subsets of the universe U. For ε∈A,F(ε) may be considered as the set of ε-approximate elements of the soft set (F,A).Definition 2.2 (see [5]).
LetU be an initial universal set and let E be a set of parameters. Let IU denote the power set of all fuzzy subsets of U. Let A⊆E. A pair (F,E) is called a fuzzy soft set over U, where F is a mapping given by
(2.2)F:A→IU.All the following definitions are due to Alkhazaleh and Salleh [12].Definition 2.3.
Let{Ui:i∈I} be a collection of universes such that ⋂i∈IUi=∅ and let {EUi:i∈I} be a collection of sets of parameters. Let U=∏i∈IP(Ui) where P(Ui) denotes the power set of Ui, E=∏i∈IEUi and A⊆E. A pair (F,A) is called a soft multiset over U, where F is a mapping given by F:A→U.In other words, a soft multiset overU is a parameterized family of subsets of U. For ε∈A,F(ε) may be considered as the set of ε-approximate elements of the soft multiset (F,A). Based on the above definition, any change in the order of universes will produce a different soft multiset.Definition 2.4.
For any soft multiset(F,A), a pair (eUi,j,FeUi,j) is called a Ui-soft multiset part ∀eUi,j∈ak and FeUi,j⊆F(A) is an approximate value set, where ak∈A,k={1,2,…,n},i∈{1,2,…,m},andj∈{1,2,…,r}.Definition 2.5.
For two soft multisets(F,A) and (G,B) over U, (F,A) is called a soft multisubset of (G,B) if(1)
A⊆Band(2)
∀eUi,j∈ak,(eUi,j,FeUi,j)⊆(eUi,j,GeUi,j),where ak∈A,k={1,2,…,n},i∈{1,2,…,m}, and j∈{1,2,…,r}.This relationship is denoted by(F,A)⊆̃(G,B). In this case (G,B) is called a soft multisuperset of (F,A).Definition 2.6.
Two soft multisets(F,A) and (G,B) over U are said to be equal if (F,A) is a soft multisubset of (G,B) and (G,B) is a soft multisubset of (F,A).Definition 2.7.
LetE=∏i=1mEUi, where EUi is a set of parameters. The NOT set of E denoted by ⌉E is defined by
(2.3)⌉E=∏i=1m⌉EUi,
where ⌉EUi={⌉eUi,j=noteUi,j,∀i,j}.Definition 2.8.
The complement of a soft multiset(F,A) is denoted by (F,A)c and is defined by (F,A)c = (Fc,⌉A), where Fc:⌉A→U is a mapping given by Fc(α)=U-F(⌉α), ∀α∈⌉A.Definition 2.9.
A soft multiset(F,A) over U is called a seminull soft multiset, denoted by (F,A)≈Φi, if at least one of the soft multiset parts of (F,A) equals ∅.Definition 2.10.
A soft multiset(F,A) over U is called a null soft multiset, denoted by (F,A)Φ, if all the soft multiset parts of (F,A) equal ∅.Definition 2.11.
A soft multiset(F,A) over U is called a semiabsolute soft multiset, denoted by (F,A)≈Ai if (eUi,j,FeUi,j)=Ui for at least one i, ak∈A,ak∈A,k={1,2,…,n}, i∈{1,2,…,m},andj∈{1,2,…,r}.Definition 2.12.
A soft multiset(F,A) over U is called an absolute soft multiset, denoted by (F,A)A, if (eUi,j,FeUi,j)=Ui, ∀i.Definition 2.13.
The union of two soft multisets(F,A) and (G,B) over U, denoted by (F,A)∪̃(G,B), is the soft multiset (H,C) where C=A∪B, and ∀ε∈C,
(2.4)H(ε)={F(ε),ifε∈A-B,G(ε),ifε∈B-A,F(ε)∪G(ε),ifε∈A∩B.Definition 2.14.
The intersection of two soft multisets(F,A) and (G,B) over U, denoted by (F,A)∩̃(G,B), is the soft multiset (H,C) where C=A∪B, and ∀ε∈C,
(2.5)H(ε)={F(ε),ifε∈A-B,G(ε),ifε∈B-A,F(ε)∩G(ε),ifε∈A∩B.


## 3. Fuzzy Soft Multiset

In this section, we introduce the definition of a fuzzy soft multiset, and its basic operations such as complement, union, and intersection. We give examples for these concepts. Basic properties of the operations are also given.Definition 3.1.
Let{Ui:i∈I} be a collection of universes such that ⋂i∈IUi=∅ and let {EUi:i∈I} be a collection of sets of parameters. Let U=∏i∈IFS(Ui) where FS(Ui) denotes the set of all fuzzy subsets of Ui, E=∏i∈IEUi and A⊆E. A pair (F,A) is called a fuzzy soft multiset over U, where F is a mapping given by F:A→U.In other words, a fuzzy soft multiset overU is a parameterized family of fuzzy subsets of U. For ε∈A,F(ε) may be considered as the set of ε-approximate elements of the fuzzy soft multiset (F,A). Based on the above definition, any change in the order of universes will produce a different fuzzy soft multiset.Example 3.2.
Suppose that there are three universesU1, U2, and U3. Suppose that Mr. X has a budget to buy a house, a car and rent a venue to hold a wedding celebration. Let us consider a fuzzy soft multiset (F,A) which describes “houses,” “cars,” and “hotels” that Mr. X is considering for accommodation purchase, transportation purchase, and a venue to hold a wedding celebration, respectively. Let U1 = {h1,h2,h3,h4,h5}, U2 = {c1,c2,c3,c4} and U3 = {v1,v2,v3}.
Let{EU1,EU2,EU3} be a collection of sets of decision parameters related to the above universes, where
(3.1)EU1={eU1,1=expensive,eU1,2=cheap,eU1,3=wooden,eU1,4=ingreensurroundings},EU2={eU2,1=expensive,eU2,2=cheap,eU2,3=sporty},EU3={eU3,1=expensive,eU3,2=cheap,eU3,3=inKualaLumpur,eU3,4=majestic}.
Let U=∏i=13FS(Ui),E=∏i=13EUi and A⊆E, such that
(3.2)A={a1=(eU1,1,eU2,1,eU3,1),a2=(eU1,1,eU2,2,eU3,1),a3=(eU1,2,eU2,2,eU3,1),a4=(eU1,4,eU2,3,eU3,2),a5=(eU1,4,eU2,2,eU3,2),a6=(eU1,2,eU2,2,eU3,2)}.
Suppose that
(3.3)F(a1)=({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7}),F(a2)=({h10.2,h20.4,h30.8,h40.5,h50},{c10.4,c20.5,c30.8,c40.5},{v10.4,v20.4,v30.3}),F(a3)=({h10.7,h20.7,h30.1,h40.8,h50.7},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2}),F(a4)=({h10.9,h20.5,h30.5,h40.2,h50.7},{c10,c20.2,c30.7,c40.6},{v10.8,v20.7,v30.9}),F(a5)=({h10.9,h20.5,h30.5,h40.2,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7}),F(a6)=({h10.7,h20.7,h30.1,h40.8,h50.7},{c10.8,c20.6,c30.3,c40.5},{v10.8,v20.7,v30.6}).
Then we can view the fuzzy soft multiset (F,A) as consisting of the following collection of approximations:
(3.4)(F,A)={(a1,({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(a2,({h10.2,h20.4,h30.8,h40.5,h50},{c10.4,c20.5,c30.8,c40.5},{v10.4,v20.4,v30.3})),(a3,({h10.7,h20.7,h30.1,h40.8,h50.7},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2})),(a4,({h10.9,h20.5,h30.5,h40.2,h50.7},{c10,c20.2,c30.7,c40.6},{v10.8,v20.7,v30.9})),(a5,({h10.9,h20.5,h30.5,h40.2,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7})),(a6,({h10.7,h20.7,h30.1,h40.8,h50.7},{c10.8,c20.6,c30.3,c40.5,},{v10.8,v20.7,v30.6}))}.
Each approximation has two parts: a predicate and an approximate value set.We can logically explain the above example as follows: we know thata1=(eU1,1,eU2,1,eU3,1) where eU1,1= expensive house, eU2,1= expensive car, and eU3,1= expensive venue. Then,(3.5)F(a1)=({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7}).
We can see that the membership value for house h1 is 0.2, so this house is not expensive for Mr. X; also we can see that the membership value for house h3 is 0.8, this means that the house h3 is expensive, and since the membership value for house h5 is 0, then this house is absolutely not expensive. Now, since the first set is concerning expensive houses, then we can explain the second set as follows: the membership value for car c1 is 0.8, so this car is expensive (this car maybe not expensive if the first set is concerning cheap houses), also we can see that the membership value for car c3 is 0.4, this means that this car is not so expensive for him, and since the membership value for car c4 is 0.6, then this car is quite expensive. Now, since the first set is concerning expensive houses and the second set is concerning expensive cars, then we can also explain the third set as follows: since the membership value for venue v1 is 0.8, so this venue is expensive (this venue maybe not expensive if the first set is concerning cheap houses or/and the second set is concerning cheap cars), also we can see that the membership value for venue v2 and v3 is 0.7, this means that this venue is almost expensive. So depending on the previous explanation we can say the following.If{h1/0.2,h2/0.4,h3/0.8,h4/0.5,h5/0} is the fuzzy set of expensive houses, then the fuzzy set of relatively expensive cars is {c1/0.8,c2/0.5,c3/0.4,c4/0.6}, and if {h1/0.2,h2/0.4,h3/0.8,h4/0.5,h5/0} is the fuzzy set of expensive houses and {c1/0.8,c2/0.5,c3/0.4,c4/0.6} is the fuzzy set of relatively expensive cars, then the fuzzy set of relatively expensive hotels is {v1/0.8,v2/0.7,v3/0.7}. It is clear that the relation in fuzzy soft multiset is a conditional relation.Definition 3.3.
For any fuzzy soft multiset(F,A), a pair (eUi,j,FeUi,j) is called a Ui-fuzzy soft multiset part ∀eUi,j∈ak and FeUi,j⊆F(A) is a fuzzy approximate value set, where ak∈A, k = {1,2,…,n}, i∈{1,2,…,m}, andj∈{1,2,…,r}.Example 3.4.
Consider Example3.2. Then
(3.6)(eU1,j,FeU1,j)={(eU1,1,{h10.2,h20.4,h30.8,h40.5,h50}),(eU1,1,{h10.2,h20.4,h30.8,h40.5,h50}),(eU1,2,{h10.7,h20.7,h30.1,h40.8,h50.7}),(eU1,4,{h10.9,h20.5,h30.5,h40.2,h50.7}),(eU1,4,{h10.9,h20.5,h30.5,h40.2,h50.7}),(eU1,2,{h10.7,h20.7,h30.1,h40.8,h50.7})}
is a U1-fuzzy soft multiset part of (F,A).Definition 3.5.
For two fuzzy soft multisets(F,A) and (G,B) over U, (F,A) is called a fuzzy soft multisubset of (G,B) if (a)
A⊆B and(b)
∀eUi,j∈ak,(eUi,j,FeUi,j) is a fuzzy subset of (eUi,j,GeUi,j),where ak∈A,k={1,2,…,n},i∈{1,2,…,m}andj∈{1,2,…,r}.This relationship is denoted by(F,A)⊆̃(G,B). In this case (G,B) is called a fuzzy soft multisuperset of (F,A).Definition 3.6.
Two fuzzy soft multisets(F,A) and (G,B) over U are said to be equal if (F,A) is a fuzzy soft multisubset of (G,B) and (G,B) is a fuzzy soft multisubset of (F,A).Example 3.7.
Consider Example3.2. Let
(3.7)A={a1=(eU1,1,eU2,1,eU3,1),a2=(eU1,2,eU2,3,eU3,1),a3=(eU1,4,eU2,3,eU3,3),a4=(eU1,3,eU2,1,eU3,1)},B={b1=(eU1,1,eU2,1,eU3,1),b2=(eU1,1,eU2,2,eU3,1),b3=(eU1,2,eU2,3,eU3,1),b4=(eU1,5,eU2,4,eU3,2),b5=(eU1,4,eU2,3,eU3,3),b6=(eU1,3,eU2,1,eU3,1)}.
Clearly A⊆B. Let (F,A) and (G,B) be two fuzzy soft multisets over the same U such that
(3.8)(F,A)={(a1,({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(a2,({h10.7,h20.7,h31,h40.8,h50.3},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2})),(a3,({h11,h20.8,h30.7,h40,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7})),(a4,({h10.8,h20.6,h30.1,h40.5,h51},{c10.5,c20.3,c30.1,c40.2},{v10.5,v20.3,v30.4}))},(G,B)={(b1,({h10.3,h20.4,h30.9,h40.7,h50.2},{c10.8,c20.6,c30.6,c40.7},{v10.9,v20.7,v30.9})),(b2,({h10.4,h20.6,h30.8,h40.7,h50.3},{c11,c20.9,c30.9,c40.8},{v10.8,v20.5,v30.4})),(b3,({h10.8,h20.9,h31,h40.8,h50.6},{c10.8,c20.8,c30.5,c40.5},{v10.7,v20.6,v30.5})),(b4,({h10.9,h20.6,h30.7,h40.5,h50.8},{c10.3,c20.5,c30.7,c40.7},{v10.8,v20.8,v31})),(b5,({h11,h20.9,h30.8,h40.4,h50.7},{c10.8,c20.9,c30.5,c40.5},{v10.5,v20.6,v30.8})),(b6,({h10.8,h20.7,h31,h40.9,h50.6},{c10.8,c20.7,c30.6,c40.5},{v10.8,v20.9,v30.9}))}.
Therefore, (F,A)⊆̃(G,B).Definition 3.8.
The complement of a fuzzy soft multiset(F,A) is denoted by (F,A)c and is defined by (F,A)c = (Fc,A), where Fc:A→U is a mapping given by Fc(α)=c(F(α)),∀α∈A where c is any fuzzy complement.Example 3.9.
Consider Example3.2. By using the basic fuzzy complement which is c(x)=1-x, we have
(3.9)(F,A)c={(a1,(F(a1))),(a2,(F(a2))),(a3,(F(a3))),(a4,(F(a4)))(a5,(F(a5))),(a6,(F(a6)))}={(a1,({h10.8,h20.6,h30.2,h40.5,h51},{c10.2,c20.5,c30.6,c40.4},{v10.2,v20.3,v30.3})),(a2,({h10.8,h20.6,h30.2,h40.5,h51},{c10.6,c20.5,c30.2,c40.5},{v10.6,v20.6,v30.7})),(a3,({h10.3,h20.3,h30.9,h40.2,h50.3},{c10.2,c20.4,c30.7,c40.5},{v10.5,v20.6,v30.8})),(a4,({h10.1,h20.5,h30.5,h40.8,h50.3},{c11,c20.8,c30.3,c40.4},{v10.2,v20.3,v30.1})),(a5,({h10,h20.2,h30.3,h41,h50.3},{c10.3,c20.2,c30.5,c40.6},{v10.5,v20.5,v30.3})),(a6,({h10.3,h20.3,h30.9,h40.2,h50.3},{c10.2,c20.4,c30.7,c40.5},{v10.2,v20.3,v30.4}))}.Definition 3.10.
A fuzzy soft multiset(F,A) over U is called a seminull fuzzy soft multiset, denoted by (F,A)≈Φi, if at least one of a fuzzy soft multiset parts of (F,A) equals ∅.Example 3.11.
Consider Example3.2. Let us consider a fuzzy soft multiset (F,A) which describes “stone houses,” “cars,” and “hotels” with
(3.10)A={a1=(eU1,3,eU2,1,eU3,1),a2=(eU1,3,eU2,3,eU3,1),a3=(eU1,4,eU2,3,eU3,3)}.
Then a seminull fuzzy soft multiset (F,A)≈Φ1 is given as
(3.11)(F,A)≈Φ1={(a1,({h10,h20,h30,h40,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(a2,({h10,h20,h30,h40,h50},{c11,c20.8,c30.6,c40.8},{v10.6,v20.5,v30.4})),(a3,({h10,h20,h30,h40,h50},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2}))}.Definition 3.12.
A fuzzy soft multiset(F,A) over U is called a null fuzzy soft multiset, denoted by (F,A)Φ, if all the fuzzy soft multiset parts of (F,A) equal ∅.Example 3.13.
Consider Example3.2. Let us consider a fuzzy soft multiset (F,A) which describes “stone houses,” “very cheap classic cars,” and “hotels in Kajang” with
(3.12)A={a1=(eU1,3,eU2,2,eU3,3),a2=(eU1,3,eU2,1,eU3,3)}.
Then a null fuzzy soft multiset (F,A)Φ is given as
(3.13)(F,A)Φ={(a1,({h10,h20,h30,h40,h50},{c10,c20,c30,c40},{v10,v20,v30})),(a2,({h10,h20,h30,h40,h50},{c10,c20,c30,c40},{v10,v20,v30}))}.Definition 3.14.
A fuzzy soft multiset(F,A) over U is called a semi-absolute fuzzy soft multiset, denoted by (F,A)≈Ui, if (eUi,j,FeUi,j)=Ui for at least one i, ak∈A,k={1,2,…,n},i∈{1,2,…,m},andj∈{1,2,…,r}.Example 3.15.
Consider Example3.2. Let us consider a fuzzy soft multiset (F,A) which describes “wooden houses,” “cars,” and “hotels” with
(3.14)A={a1=(eU1,3,eU2,1,eU3,1),a2=(eU1,3,eU2,3,eU3,1),a3=(eU1,3,eU2,3,eU3,3)}.
Then a semi-absolute fuzzy soft multiset (F,A)≈U1 is given as
(3.15)(F,A)≈U1={(a1,({h11,h21,h31,h41,h51},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(a2,({h11,h21,h31,h41,h51},{c11,c20.8,c30.6,c40.8},{v10.6,v20.5,v30.4})),(a3,({h11,h21,h31,h41,h51},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2}))}.Definition 3.16.
A fuzzy soft multiset(F,A) over U is called an absolute fuzzy soft multiset, denoted by (F,A)U, if (eUi,j,FeUi,j)=Ui, ∀i.Example 3.17.
Consider Example3.2. Let us consider a fuzzy soft multiset (F,A) which describes “wooden houses,” “expensive classic cars,” and “hotels in Kuala Lumpur” with
(3.16)A={a1={eU1,3,eU2,1,eU3,3},a2={eU1,3,eU2,3,eU3,3}}.
Then an absolute fuzzy soft multiset (F,A)U is given as
(3.17)(F,A)U={(a1,({h11,h21,h31,h41,h51},{c11,c21,c31,c41},{v11,v21,v31})),(a2,({h11,h21,h31,h41,h51},{c11,c21,c31,c41},{v11,v21,v31}))}.Proposition 3.18.
If(F,A)is a fuzzy soft multiset over U, then (a)
((F,A)c)c=(F,A),(b)
(F,A)≈Φic=(F,A)≈Ui,(c)
(F,A)Φc=(F,A)U,(d)
(F,A)≈Uic=(F,A)≈Φi,(e)
(F,A)Uc=(F,A)Φ.Proof.
The proof is straightforward.


## 4. Union and Intersection

In this section we define the operation of union and intersection and give some examples by using the basic fuzzy union and intersection.Definition 4.1.
The union of two fuzzy soft multisets(F,A) and (G,B) over U, denoted by (F,A)∪̃(G,B), is the fuzzy soft multiset (H,C), where C=A∪B, and ∀ε∈C,
(4.1)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,⋃(F(ε),G(ε))ifε∈A∩B,
where ⋃(F(ε),G(ε))=s(FεUi,j,GεUi,j),∀i∈{1,2,…,m} with s as an s-norm.Example 4.2.
Consider Example3.2. Let
(4.2)A={a1=(eU1,1,eU2,1,eU3,1),a2=(eU1,2,eU2,3,eU3,1),a3=(eU1,4,eU2,3,eU3,3),a4=(eU1,3,eU2,1,eU3,1)},B={b1=(eU1,1,eU2,1,eU3,1),b2=(eU1,1,eU2,2,eU3,1),b3=(eU1,2,eU2,3,eU3,1),b4=(eU1,5,eU2,4,eU3,2),b5=(eU1,2,eU2,3,eU3,2),b6=(eU1,1,eU2,3,eU3,2)}.
Suppose (F,A) and (G,B) are two fuzzy soft multisets over the same U such that
(4.3)(F,A)={(a1,({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(a2,({h10.7,h20.7,h31,h40.8,h50.3},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2})),(a3,({h11,h20.8,h30.7,h40,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7})),(a4,({h10.8,h20.6,h30.1,h40.5,h51},{c10.5,c20.3,c30.1,c40.2},{v10.5,v20.3,v30.4}))},(G,B)={(b1,({h10.3,h20.4,h30.9,h40.7,h50.2},{c10.8,c20.6,c30.6,c40.7},{v10.9,v20.7,v30.9})),(b2,({h10.4,h20.6,h30.8,h40.7,h50.3},{c11,c20.9,c30.9,c40.8},{v10.8,v20.5,v30.4})),(b3,({h10.8,h20.9,h31,h40.8,h50.6},{c10.8,c20.8,c30.5,c40.5},{v10.7,v20.6,v30.5})),(b4,({h10.9,h20.6,h30.7,h40.5,h50.8},{c10.3,c20.5,c30.7,c40.7},{v10.8,v20.8,v31})),(b5,({h11,h20.9,h30.8,h40.4,h50.7},{c10.8,c20.9,c30.5,c40.5},{v10.5,v20.6,v30.8})),(b6,({h10.8,h20.7,h31,h40.9,h50.6},{c10.8,c20.7,c30.6,c40.5},{v10.8,v20.9,v30.9}))}.
By using the basic fuzzy union (maximum) we have
(4.4)(F,A)∪̃(G,B)=(H,D)={(d1,({h10.3,h20.4,h30.9,h40.7,h50.2},{c10.8,c20.6,c30.6,c40.7},{v10.9,v20.7,v30.9})),(d2,({h10.8,h20.9,h31,h40.8,h50.6},{c10.8,c20.8,c30.5,c40.5},{v10.7,v20.6,v30.5})),(d3,({h11,h20.8,h30.7,h40,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5,h51},{c10.5,c20.3,c30.1,c40.2},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7,h50.3},{c11,c20.9,c30.9,c40.8},{v10.8,v20.5,v30.4})),(d6,({h10.9,h20.6,h30.7,h40.5,h50.8},{c10.3,c20.5,c30.7,c40.7},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4,h50.7},{c10.8,c20.9,c30.5,c40.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9,h50.6},{c10.8,c20.7,c30.6,c40.5},{v10.8,v20.9,v30.9}))},
where
(4.5)D={d1=a1=b1,d2=a2=b3,d3=a3,d4=a4,d5=b2,d6=b4,d7=b5,d8=b6}.Proposition 4.3.
If(F,A),  (G,B), and  (H,C) are three fuzzy soft multisets over  U, then  (a)
(F,A)∪̃((G,B)∪̃(H,C))=((F,A)∪̃(G,B))∪̃(H,C),(b)
(F,A)∪̃(F,A)=(F,A),(c)
(F,A)∪̃(G,A)≈Φi=(R,A), where R is defined by (4.1)(d)
(F,A)∪̃(G,A)Φ=(F,A),(e)
(F,A)∪̃(G,B)≈Φi=(R,D), where D=A∪B and R is defined by (4.1)(f)
(F,A)∪̃(G,B)Φ={(F,A)ifA=B,(R,D)otherwise, where D=A∪B,(g)
(F,A)∪̃(G,A)≈Ui=(R,A)≈Ui,(h)
(F,A)∪̃(G,A)U=(G,A)U,(i)
(F,A)∪̃(G,B)≈Ui={(R,D)≈UiifA=B,(R,D)otherwise, where D=A∪B,(j)
(F,A)∪̃(G,B)U={(G,B)UifA=B,(R,D)otherwise, where D=A∪B.Proof.
The proof is straightforward.Definition 4.4.
The intersection of two fuzzy soft multisets(F,A) and (G,B) over U, denoted by (F,A)∩̃(G,B), is the fuzzy soft multiset (H,C), where C=A∪B, and ∀ε∈C,
(4.6)H(ε)={F(ε)ifε∈A-B,G(ε)ifε∈B-A,⋂(F(ε),G(ε))ifε∈A∩B,
where ⋂(F(ε),G(ε))=t(FεUi,j,GεUi,j), ∀i∈{1,2,3,…,m} with t as a t-norm.Example 4.5.
Consider Example4.2. By using the basic fuzzy intersection (minimum) we have
(4.7)(F,A)∩̃(G,B)=(H,D)={(d1,({h10.2,h20.4,h30.8,h40.5,h50},{c10.8,c20.5,c30.4,c40.6},{v10.8,v20.7,v30.7})),(d2,({h10.7,h20.7,h31,h40.8,h50.3},{c10.8,c20.6,c30.3,c40.5},{v10.5,v20.4,v30.2})),(d3,({h11,h20.8,h30.7,h40,h50.7},{c10.7,c20.8,c30.5,c40.4},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5,h51},{c10.5,c20.3,c30.1,c40.2},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7,h50.3},{c11,c20.9,c30.9,c40.8},{v10.8,v20.5,v30.4})),(d6,({h10.9,h20.6,h30.7,h40.5,h50.8},{c10.3,c20.5,c30.7,c40.7},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4,h50.7},{c10.8,c20.9,c30.5,c40.5},{v10.5,v20.6,v30.8})),(c8,({h10.8,h20.7,h31,h40.9,h50.6},{c10.8,c20.7,c30.6,c40.5},{v10.8,v20.9,v30.9}))},
where
(4.8)D={d1=a1=b1,d2=a2=b3,d3=a3,d4=a4,d5=b2,d6=b4,d7=b5,d8=b6}.Proposition 4.6.
If(F,A),  (G,B), and  (H,C) are three fuzzy soft multisets over  U, then  (a)
(F,A)∩̃((G,B)∩̃(H,C))=((F,A)∩̃(G,B))∩̃(H,C),(b)
(F,A)∩̃(F,A)=(F,A),(c)
(F,A)∩̃(G,A)≈Φi=(R,A)≈Φi, where R is defined by (4.6)(d)
(F,A)∩̃(G,A)Φi=(R,A)Φi, where R is defined by (4.6)(e)
(F,A)∩̃(G,B)≈Φi={(R,D)≈ΦiifA⊆B,(R,D)otherwise, where D=A∪B,(f)
(F,A)∩̃(G,B)Φ={(R,D)ΦifA⊆B,(R,D)otherwise, where D=A∪B,(g)
(F,A)∩̃(G,A)≈Ui=(R,D), where D=A∪B and R is defined by (4.6)(h)
(F,A)∩̃(G,A)U=(F,A),(i)
(F,A)∩̃(G,B)≈Ui=(R,D), where D=A∪B and R is defined by (4.6)(j)
(F,A)∩̃(G,B)U={(F,A)ifA⊇B,(R,D)otherwise, where D=A∪B and R is defined by 4.6.Proof.
The proof is straightforward.


## 5. Fuzzy Soft Set-Based Decision Making

We begin this section with a novel algorithm designed for solving fuzzy soft set-based decision-making problems, which was presented in [6].


### 5.1. Roy and Maji’s Original Algorithm Using Scores

Roy and Maji  [6] used the following algorithm to solve a decision-making problem. (a)
Input the fuzzy soft sets(F,A),(G,B), and (H,C).(b)
Input the parameter setP as observed by the observer.(c)
Compute the corresponding resultant fuzzy soft set(S,P) from the fuzzy soft sets (F,A),(G,B),and(H,C) and place it in tabular form.(d)
Construct the comparison table of the fuzzy soft set(S,P) and compute ri and ti for oi,∀i.(e)
Compute the score ofoi,∀i.(f)
The decision isSk if Sk=maxiSi.(g)
Ifk has more than one value, then any one of ok may be chosen.


### 5.2. A Fuzzy Soft Multiset Theoretic Approach to Decision-Making Problem

In this section we suggest the following algorithm to solve fuzzy soft multisets-based decision-making problem, which is a generalization of the algorithm given by Salleh and Alkhazaleh in [13]. We note here that we will use the abbreviation RMA for Roy and Maji’s a Algorithm.(a)
Input the fuzzy soft multiset(H,C) which is introduced by making any operations between (F,A)and(G,B).(b)
Apply RMA to the first fuzzy soft multiset part in(H,C) to get the decision Sk1.(c)
Redefine the fuzzy soft multiset(H,C) by keeping all values in each row where Sk1 is maximum and replacing the values in the other rows by zero, to get (H,C)1.(d)
Apply RMA to the second fuzzy soft multiset part in(H,C)1 to get the decision Sk2.(e)
Redefine the fuzzy soft multiset(H,C)1 by keeping the first and second parts and apply the method in step (c) to the third part.(f)
Apply RMA to the third fuzzy soft multiset part in(H,C)2 to get the decision Sk3.(g)
The decision is(Sk1,Sk2,Sk3).


### 5.3. Application in a Decision-Making Problem

LetU1={h1,h2,h3,h4}, U2={c1,c2,c3}, and U3={v1,v2,v3} be the sets of “houses,” “cars,” and “hotels”, respectively. Let {EU1,EU2,EU3} be a collection of sets of decision parameters related to the above universes, where
(5.1)EU1={eU1,1=expensive,eU1,2=cheap,eU1,3=wooden,eU1,4=ingreensurroundings},EU2={eU2,1=expensive,eU2,2=cheap,eU2,3=sporty},EU3={eU3,1=expensive,eU3,2=cheap,eU3,3=inKualaLumpur,eU3,4=majestic}.Let(5.2)A={a1=(eU1,1,eU2,1,eU3,1),a2=(eU1,2,eU2,3,eU3,1),a3=(eU1,4,eU2,3,eU3,3),a4=(eU1,3,eU2,1,eU3,1)},B={b1=(eU1,1,eU2,1,eU3,1),b2=(eU1,1,eU2,2,eU3,1),b3=(eU1,2,eU2,3,eU3,1),b4=(eU1,4,eU2,3,eU3,2),b5=(eU1,2,eU2,3,eU3,2),b6=(eU1,1,eU2,3,eU3,2)}.
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let there be two observations (F,A) and (G,B) by two experts Y1 and Y2, respectively. Let(5.3)(F,A)={(a1,({h10.2,h20.4,h30.8,h40.5},{c10.8,c20.5,c30.4},{v10.8,v20.7,v30.7})),(a2,({h10.7,h20.7,h30.1,h40.5},{c10.8,c20.6,c30.3},{v10.5,v20.4,v30.2})),(a3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(a4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4}))},(G,B)={(b1,({h10.3,h20.4,h30.9,h40.7},{c10.8,c20.6,c30.6},{v10.9,v20.7,v30.9})),(b2,({h10.4,h20.6,h30.8,h40.7},{c11,c20.9,c30.9},{v10.8,v20.5,v30.4})),(b3,({h10.8,h20.9,h30.3,h40.8},{c10.8,c20.8,c30.5},{v10.7,v20.6,v30.5})),(b4,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(b5,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(b6,({h10.8,h20.7,h31,h40.9},{c10.8,c20.8,c30.5},{v10.8,v20.9,v30.9}))}.By using the basic fuzzy union we have(5.4)(F,A)∪̃(G,B)=(H,D)={(d1,({h10.3,h20.4,h30.9,h40.7},{c10.8,c20.6,c30.6},{v10.9,v20.7,v30.9})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10.8,c20.8,c30.5},{v10.7,v20.6,v30.5})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c11,c20.9,c30.9},{v10.8,v20.5,v30.4})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10.8,c20.8,c30.5},{v10.8,v20.9,v30.9}))}.
Now we apply RMA to the first fuzzy soft multiset part in (H,D) to take the decision from the availability set U1. The tabular representation of the first resultant fuzzy soft multiset part will be as in Table 1.Table 1
Tabular representation:U1-fuzzy soft multiset part of (H,D).
U1d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8h10.30.810.80.40.910.8h20.40.90.80.60.60.60.90.7h30.90.30.70.10.70.70.81h40.70.800.50.70.50.40.9The comparison table for the first resultant fuzzy soft multiset part will be as in Table2.Table 2
Comparison table:U1-fuzzy soft multiset part of (H,D).
U1h1h2h3h4h18555h23845h33486h43338Next we compute the row-sum, column-sum, and the score for eachhi as shown in Table 3.Table 3
Score table:U1-fuzzy soft multiset part of (H,D).
Row-sum(ri)Column-sum(ti)Score(Si)h123176h220200h321201h41724−7From Table3, it is clear that the maximum score is 6, scored by h1.Now we redefine the fuzzy soft multiset(H,D) by keeping all values in each row where h1 is maximum and replacing the values in the other rows by zero:(5.5)(H,D)1={(d1,({h10.3,h20.4,h30.9,h40.7},{c10,c20,c30},{v10,v20,v30})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10,c20,c30},{v10,v20,v30})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c10,c20,c30},{v10,v20,v30})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10,c20,c30},{v10,v20,v30}))}.Now we apply RMA to the second fuzzy soft multiset part in(H,D)1 to take the decision from the availability set U2. The tabular representation of the second resultant fuzzy soft multiset part of (H,D)1 will be as in Table 4.Table 4
Tabular representation:U2-fuzzy soft multiset part of (H,D)1.
U2d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8c1000.80.500.80.80c2000.60.300.80.80c3000.30.100.50.50The comparison table for the second resultant fuzzy soft multiset part of(H,D)1 is as in Table 5.Table 5
Comparison table:U2-fuzzy soft multiset part of of (H,D)1.
U2c1c2c3c1888c2688c3448Next we compute the row-sum, column-sum, and the score for eachci is shown in Table 6.Table 6
Score table:U2-fuzzy soft multiset part of (H,D)1.
Row-sum(ri)Column-sum(ti)Score(Si)c124186c222202c31624−8From Table6, it is clear that the maximum score is 6, scored by c1.Now we redefine the fuzzy soft multiset(H,D)1 by keeping all values in each row where c1 is maximum and replacing the values in the other rows by zero: (5.6)(H,D)2={(d1,({h10.3,h20.4,h30.9,h40.7},{c10,c20,c30},{v10,v20,v30})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10,c20,c30},{v10,v20,v30})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c10,c20,c30},{v10,v20,v30})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10,c20,c30},{v10,v20,v30}))}.Now we apply RMA to the third fuzzy soft multiset part in(H,D)2 to take the decision from the availability set U3. The tabular representation of the third resultant fuzzy soft multiset part of (H,D)2 is as in Table 7.Table 7
Tabular representation:U3-fuzzy soft multiset part of (H,D)2.
U3d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8v1000.50.500.80.50v2000.50.300.80.60v3000.70.4010.80The comparison table for the second resultant fuzzy soft multiset part of(H,D)2 is as in Table 8.Table 8
Comparison table:U3-fuzzy soft multiset part of of (H,D)2.
U3v1v2v3v1875v2784v3788Next we compute the row-sum, column-sum, and the score for eachvi as shown in Table 9.Table 9
Score table:U3-fuzzy soft multiset part of (H,D)2.
Row-sum(ri)Column-sum(ti)Score(Si)v12022−2v21923−2v323176From Table9, it is clear that the maximum score is 6, scored by v3.Then from the above results the decision for Mr. X is(h1,c1,v3).


## 5.1. Roy and Maji’s Original Algorithm Using Scores

Roy and Maji  [6] used the following algorithm to solve a decision-making problem. (a)
Input the fuzzy soft sets(F,A),(G,B), and (H,C).(b)
Input the parameter setP as observed by the observer.(c)
Compute the corresponding resultant fuzzy soft set(S,P) from the fuzzy soft sets (F,A),(G,B),and(H,C) and place it in tabular form.(d)
Construct the comparison table of the fuzzy soft set(S,P) and compute ri and ti for oi,∀i.(e)
Compute the score ofoi,∀i.(f)
The decision isSk if Sk=maxiSi.(g)
Ifk has more than one value, then any one of ok may be chosen.


## 5.2. A Fuzzy Soft Multiset Theoretic Approach to Decision-Making Problem

In this section we suggest the following algorithm to solve fuzzy soft multisets-based decision-making problem, which is a generalization of the algorithm given by Salleh and Alkhazaleh in [13]. We note here that we will use the abbreviation RMA for Roy and Maji’s a Algorithm.(a)
Input the fuzzy soft multiset(H,C) which is introduced by making any operations between (F,A)and(G,B).(b)
Apply RMA to the first fuzzy soft multiset part in(H,C) to get the decision Sk1.(c)
Redefine the fuzzy soft multiset(H,C) by keeping all values in each row where Sk1 is maximum and replacing the values in the other rows by zero, to get (H,C)1.(d)
Apply RMA to the second fuzzy soft multiset part in(H,C)1 to get the decision Sk2.(e)
Redefine the fuzzy soft multiset(H,C)1 by keeping the first and second parts and apply the method in step (c) to the third part.(f)
Apply RMA to the third fuzzy soft multiset part in(H,C)2 to get the decision Sk3.(g)
The decision is(Sk1,Sk2,Sk3).


## 5.3. Application in a Decision-Making Problem

LetU1={h1,h2,h3,h4}, U2={c1,c2,c3}, and U3={v1,v2,v3} be the sets of “houses,” “cars,” and “hotels”, respectively. Let {EU1,EU2,EU3} be a collection of sets of decision parameters related to the above universes, where
(5.1)EU1={eU1,1=expensive,eU1,2=cheap,eU1,3=wooden,eU1,4=ingreensurroundings},EU2={eU2,1=expensive,eU2,2=cheap,eU2,3=sporty},EU3={eU3,1=expensive,eU3,2=cheap,eU3,3=inKualaLumpur,eU3,4=majestic}.Let(5.2)A={a1=(eU1,1,eU2,1,eU3,1),a2=(eU1,2,eU2,3,eU3,1),a3=(eU1,4,eU2,3,eU3,3),a4=(eU1,3,eU2,1,eU3,1)},B={b1=(eU1,1,eU2,1,eU3,1),b2=(eU1,1,eU2,2,eU3,1),b3=(eU1,2,eU2,3,eU3,1),b4=(eU1,4,eU2,3,eU3,2),b5=(eU1,2,eU2,3,eU3,2),b6=(eU1,1,eU2,3,eU3,2)}.
Suppose Mr. X wants to choose objects from the sets of given objects with respect to the sets of choice parameters. Let there be two observations (F,A) and (G,B) by two experts Y1 and Y2, respectively. Let(5.3)(F,A)={(a1,({h10.2,h20.4,h30.8,h40.5},{c10.8,c20.5,c30.4},{v10.8,v20.7,v30.7})),(a2,({h10.7,h20.7,h30.1,h40.5},{c10.8,c20.6,c30.3},{v10.5,v20.4,v30.2})),(a3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(a4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4}))},(G,B)={(b1,({h10.3,h20.4,h30.9,h40.7},{c10.8,c20.6,c30.6},{v10.9,v20.7,v30.9})),(b2,({h10.4,h20.6,h30.8,h40.7},{c11,c20.9,c30.9},{v10.8,v20.5,v30.4})),(b3,({h10.8,h20.9,h30.3,h40.8},{c10.8,c20.8,c30.5},{v10.7,v20.6,v30.5})),(b4,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(b5,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(b6,({h10.8,h20.7,h31,h40.9},{c10.8,c20.8,c30.5},{v10.8,v20.9,v30.9}))}.By using the basic fuzzy union we have(5.4)(F,A)∪̃(G,B)=(H,D)={(d1,({h10.3,h20.4,h30.9,h40.7},{c10.8,c20.6,c30.6},{v10.9,v20.7,v30.9})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10.8,c20.8,c30.5},{v10.7,v20.6,v30.5})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c11,c20.9,c30.9},{v10.8,v20.5,v30.4})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10.8,c20.8,c30.5},{v10.8,v20.9,v30.9}))}.
Now we apply RMA to the first fuzzy soft multiset part in (H,D) to take the decision from the availability set U1. The tabular representation of the first resultant fuzzy soft multiset part will be as in Table 1.Table 1
Tabular representation:U1-fuzzy soft multiset part of (H,D).
U1d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8h10.30.810.80.40.910.8h20.40.90.80.60.60.60.90.7h30.90.30.70.10.70.70.81h40.70.800.50.70.50.40.9The comparison table for the first resultant fuzzy soft multiset part will be as in Table2.Table 2
Comparison table:U1-fuzzy soft multiset part of (H,D).
U1h1h2h3h4h18555h23845h33486h43338Next we compute the row-sum, column-sum, and the score for eachhi as shown in Table 3.Table 3
Score table:U1-fuzzy soft multiset part of (H,D).
Row-sum(ri)Column-sum(ti)Score(Si)h123176h220200h321201h41724−7From Table3, it is clear that the maximum score is 6, scored by h1.Now we redefine the fuzzy soft multiset(H,D) by keeping all values in each row where h1 is maximum and replacing the values in the other rows by zero:(5.5)(H,D)1={(d1,({h10.3,h20.4,h30.9,h40.7},{c10,c20,c30},{v10,v20,v30})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10,c20,c30},{v10,v20,v30})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c10,c20,c30},{v10,v20,v30})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10,c20,c30},{v10,v20,v30}))}.Now we apply RMA to the second fuzzy soft multiset part in(H,D)1 to take the decision from the availability set U2. The tabular representation of the second resultant fuzzy soft multiset part of (H,D)1 will be as in Table 4.Table 4
Tabular representation:U2-fuzzy soft multiset part of (H,D)1.
U2d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8c1000.80.500.80.80c2000.60.300.80.80c3000.30.100.50.50The comparison table for the second resultant fuzzy soft multiset part of(H,D)1 is as in Table 5.Table 5
Comparison table:U2-fuzzy soft multiset part of of (H,D)1.
U2c1c2c3c1888c2688c3448Next we compute the row-sum, column-sum, and the score for eachci is shown in Table 6.Table 6
Score table:U2-fuzzy soft multiset part of (H,D)1.
Row-sum(ri)Column-sum(ti)Score(Si)c124186c222202c31624−8From Table6, it is clear that the maximum score is 6, scored by c1.Now we redefine the fuzzy soft multiset(H,D)1 by keeping all values in each row where c1 is maximum and replacing the values in the other rows by zero: (5.6)(H,D)2={(d1,({h10.3,h20.4,h30.9,h40.7},{c10,c20,c30},{v10,v20,v30})),(d2,({h10.8,h20.9,h30.3,h40.8},{c10,c20,c30},{v10,v20,v30})),(d3,({h11,h20.8,h30.7,h40},{c10.8,c20.6,c30.3},{v10.5,v20.5,v30.7})),(d4,({h10.8,h20.6,h30.1,h40.5},{c10.5,c20.3,c30.1},{v10.5,v20.3,v30.4})),(d5,({h10.4,h20.6,h30.8,h40.7},{c10,c20,c30},{v10,v20,v30})),(d6,({h10.9,h20.6,h30.7,h40.5},{c10.8,c20.8,c30.5},{v10.8,v20.8,v31})),(d7,({h11,h20.9,h30.8,h40.4},{c10.8,c20.8,c30.5},{v10.5,v20.6,v30.8})),(d8,({h10.8,h20.7,h31,h40.9},{c10,c20,c30},{v10,v20,v30}))}.Now we apply RMA to the third fuzzy soft multiset part in(H,D)2 to take the decision from the availability set U3. The tabular representation of the third resultant fuzzy soft multiset part of (H,D)2 is as in Table 7.Table 7
Tabular representation:U3-fuzzy soft multiset part of (H,D)2.
U3d1,1d1,2d1,3d1,4d1,5d1,6d1,7d1,8v1000.50.500.80.50v2000.50.300.80.60v3000.70.4010.80The comparison table for the second resultant fuzzy soft multiset part of(H,D)2 is as in Table 8.Table 8
Comparison table:U3-fuzzy soft multiset part of of (H,D)2.
U3v1v2v3v1875v2784v3788Next we compute the row-sum, column-sum, and the score for eachvi as shown in Table 9.Table 9
Score table:U3-fuzzy soft multiset part of (H,D)2.
Row-sum(ri)Column-sum(ti)Score(Si)v12022−2v21923−2v323176From Table9, it is clear that the maximum score is 6, scored by v3.Then from the above results the decision for Mr. X is(h1,c1,v3).


## 6. Conclusion

In this paper we have introduced the concept of fuzzy soft multiset and studied some of its properties. The operations complement, union, and intersection have been defined on the fuzzy soft multisets. An application of this theory is given in solving a decision-making problem.



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*Source: 350603-2012-06-11.xml*