Research Article Fuzzy Soft Multiset Theory

Abstract and Applied Analysis Volume 22 Article ID 6 2 pages doi:/22/6 Research Article Fuzzy Soft Multiset Theory Shawkat Alkhazaleh and Abdul Razak Salleh School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Selangor DE 6 UKM Bangi Malaysia Correspondence should be addressed to Shawkat Alkhazaleh shmk79@gmailcom Received 2 January 22; Revised 22 March 22; Accepted 28 March 22 Academic Editor: P J Y Wong Copyright q 22 S Alkhazaleh and A R Salleh This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited In 999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty Alkhazaleh et al in 2 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 Introduction Most of the problems in engineering medical science economics environments and so forth have various uncertainties Molodtsov 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 Furthermore Maji et al 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 22 Alkhazaleh and Salleh 9 introduced the concept of generalised interval-valued fuzzy soft set and studied its properties and application Alkhazaleh and Salleh 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 22 introduced and studied the

2 Abstract and Applied Analysis concept of multiparameterized soft set as a generalization of Molodtsov s soft set Alkhazaleh et al 2 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 studied the application of soft multiset in decision making problem In 2 Salleh gave a brief survey from soft sets to intuitionistic fuzzy soft sets 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 Let U be a universe and E asetof parameters Let P U denote the power set of U and A E Definition 2 see Apair F A is called a soft set over U where F is a mapping F : A P U 2 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 22 see LetU be an initial universal set and let E be a set of parameters Let I U denote the power set of all fuzzy subsets of U LetA E Apair F E is called a fuzzy soft set over U where F is a mapping given by F : A I U 22 All the following definitions are due to Alkhazaleh and Salleh 2 Definition 2 Let U i : i I be a collection of universes such that i I U i and let E Ui : i I be a collection of sets of parameters Let U i I P U i where P U i denotes the power set of U i E i I E U i and A E Apair 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 over U 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 For any soft multiset F A apair e Ui jf is called a U eui j i-soft multiset part e Ui j a k and F eui F A is an approximate value set where a j k A k 2n i 2m and j 2r

Abstract and Applied Analysis Definition 2 For two soft multisets F A and G B over U F A is called a soft multisubset of G B if A B and 2 e Ui j a k e Ui jf eui j e U i jg eui j where a k A k 2ni 2m andj 2r This relationship is denoted by F A G B In this case G B is called a soft multisuperset of F A Definition 26 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 27 Let E m i E U i where E Ui is a set of parameters The NOT set of E denoted by E is defined by E m E Ui i 2 where E Ui e Ui j not e Ui j i j Definition 28 The complement of a soft multiset F A is denoted by F A c and is defined by F A c F c A where F c : A U is a mapping given by F c α U F α α A Definition 29 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 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 A soft multiset F A over U is called a semiabsolute soft multiset denoted by F A Ai if e Ui jf U eui j i for at least one i a k A a k A k 2n i 2m and j 2r Definition 22 A soft multiset F A over U is called an absolute soft multiset denoted by F A A if e Ui jf eui j U i i Definition 2 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 F ε if ε A B H ε G ε if ε B A F ε G ε if ε A B 2

Abstract and Applied Analysis Definition 2 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 F ε if ε A B H ε G ε if ε B A 2 F ε G ε if ε A B 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 Let U i : i I be a collection of universes such that i I U i and let E Ui : i I be a collection of sets of parameters Let U i I FS U i where FS U i denotes the set of all fuzzy subsets of U i E i I E U i and A E Apair 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 over U 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 2 Suppose that there are three universes U U 2 andu 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 U h h 2 h h h U 2 c c 2 c c and U v v 2 v Let E U E U2 E U be a collection of sets of decision parameters related to the above universes where E U e U expensive e U 2 cheap e U wooden e U in green surroundings E U2 e U2 expensive e U2 2 cheap e U2 sporty E U e U expensive e U 2 cheap e U in Kuala Lumpur e U majestic Let U i FS U i E i E U i and A E such that A a e U e U2 e U a 2 e U e U2 2e U a e U 2e U2 2e U a e U e U2 e U 2 a e U e U2 2e U 2 a 6 e U 2e U2 2e U 2 2

Abstract and Applied Analysis Suppose that F a F a 2 F a F a F a F a 6 2 h 2 h 8 h h 2 h 2 h 8 h h h 2 h h 8 h 9 h 2 h h 2 h 9 h 2 h h 2 h h 2 h h 8 h 8 c 2 c c 6 c 2 c 8 c 8 c 2 6 c c c 2 2 c c 6 c 2 8 c c 8 c 2 6 c c 8 v 2 v v 2 v v 2 v 2 8 v 2 v 9 v 2 v 8 v 2 v 6 ) ) ) ) ) ) Then we can view the fuzzy soft multiset F A as consisting of the following collection of approximations: F A a a 2 a a a a 6 2 h 2 h 8 h h 2 h 2 h 8 h h h 2 h h 8 h 9 h 2 h h 2 h 9 h 2 h h 2 h h 2 h h 8 h 8 c 2 c c 6 c 2 c 8 c 8 c 2 6 c c c 2 2 c c 6 c 2 8 c c 8 c 2 6 c c 8 v 2 v v 2 v v 2 v 2 8 v 2 v 9 v 2 v 8 v 2 v 6 )) )) )) )) )) )) Each approximation has two parts: a predicate and an approximate value set We can logically explain the above example as follows: we know that a e U e U2 e U where e U expensive house e U2 expensive car and e U expensive venue Then F a 2 h 2 h 8 h h 8 c 2 c c 6 8 v 2 v )

6 Abstract and Applied Analysis We can see that the membership value for house h is 2 so this house is not expensive for Mr X; also we can see that the membership value for house h is 8 this means that the house h is expensive and since the membership value for house h is 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 c is 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 c is this means that this car is not so expensive for him and since the membership value for car c is 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 v is 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 v 2 and v is this means that this venue is almost expensive So depending on the previous explanation we can say the following If h /2h 2 /h /8h /h / is the fuzzy set of expensive houses then the fuzzy set of relatively expensive cars is c /8c 2 /c /c /6 andifh /2 h 2 /h /8h /h / is the fuzzy set of expensive houses and c /8c 2 /c / c /6 is the fuzzy set of relatively expensive cars then the fuzzy set of relatively expensive hotels is v /8v 2 /v / It is clear that the relation in fuzzy soft multiset is a conditional relation Definition For any fuzzy soft multiset F A apair e Ui jf is called a U eui j i-fuzzy soft multiset part e Ui j a k and F eui F A is a fuzzy approximate value set where a j k A k 2n i 2m andj 2r Example Consider Example 2 Then ) e U jf eu e j U 2 h 2 h 8 h h ) e U 2 h 2 h h 8 h ) e U 9 h 2 h h 2 h ) e U 2 h 2 h 8 h h ) e U 9 h 2 h h 2 h ) e U 2 h 2 h h 8 h ) 6 is a U -fuzzy soft multiset part of F A Definition 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 e Ui j a k e Ui jf is a fuzzy subset of e eui j U i jg eui j where a k A k 2ni 2m and j 2r

Abstract and Applied Analysis 7 This relationship is denoted by F A G B In this case G B is called a fuzzy soft multisuperset of F A Definition 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 7 Consider Example 2 Let A a e U e U2 e U a 2 e U 2e U2 e U a e U e U2 e U a e U e U2 e U B b e U e U2 e U b 2 e U e U2 2e U b e U 2e U2 e U 7 b e U e U2 e U 2 b e U e U2 e U b 6 e U e U2 e U Clearly A BLet F A and G B be two fuzzy soft multisets over the same U such that F A G B a a 2 a a b b 2 b b b b 6 2 h 2 h 8 h h h 2 h h 8 h h 2 8 h h h 8 h 2 6 h h h h 2 h 9 h h 2 h 2 6 h 8 h h 8 h 2 9 h h 8 h 6 9 h 2 6 h h h 8 8 c 2 c c 6 8 c 2 6 c c h 2 9 h 8 h h 8 h 2 h h 9 h 6 c 2 8 c c 8 v 2 v v 2 v 2 c 2 c c 2 8 c 2 6 c 6 c c 2 9 c 9 c 8 8 c 2 8 c c c 2 c c 8 c 2 9 c c 8 c 2 c 6 c )) )) v 2 v )) v 2 v )) 9 v 2 v )) 9 8 v 2 v v 2 6 v 8 v 2 8 v )) )) )) v 2 6 v )) 8 8 v 2 9 v )) 9 8 Therefore F A G B

8 Abstract and Applied Analysis Definition 8 The complement of a fuzzy soft multiset F A is denoted by F A c and is defined by F A c F c A where F c : A U is a mapping given by F c α c F α α A where c is any fuzzy complement Example 9 Consider Example 2 By using the basic fuzzy complement which is c x x we have F A c a F a a 2 F a 2 a F a a F a a F a a 6 F a 6 a 8 h 2 6 h 2 h h a 2 a a a a 6 8 h 2 6 h 2 h h h 2 h 9 h 2 h h 2 h h 8 h h 2 2 h h h h 2 h 9 h 2 h 2 c 2 c 6 c 6 c 2 c 2 c 2 c 2 c c c 2 8 c c c 2 2 c c 6 2 c 2 c c 2 v 2 v 6 v 2 6 v v 2 6 v 8 2 v 2 v v 2 v 2 v 2 v )) )) )) )) )) )) 9 Definition 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 Consider Example 2 Let us consider a fuzzy soft multiset F A which describes stone houses cars and hotels with A a e U e U2 e U a 2 e U e U2 e U a e U e U2 e U Then a seminull fuzzy soft multiset F A Φ is given as F A Φ a a 2 a h 2 h h h h 2 h h h h 2 h h h 8 c 2 c c 6 c 2 8 c 6 c 8 8 c 2 6 c c 8 v 2 v 6 v 2 v v 2 v 2 )) )) ))

Abstract and Applied Analysis 9 Definition 2 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 Consider Example 2 Let us consider a fuzzy soft multiset F A which describes stone houses very cheap classic cars and hotels in Kajang with A a e U e U2 2e U a 2 e U e U2 e U 2 Then a null fuzzy soft multiset F A Φ is given as F A Φ a a 2 h 2 h h h h 2 h h h c 2 c c c 2 c c v 2 v v 2 v )) )) Definition A fuzzy soft multiset F A over U is called a semi-absolute fuzzy soft multiset denoted by F A Ui if e Ui jf eui j U i for at least one i a k A k 2n i 2m and j 2r Example Consider Example 2 Let us consider a fuzzy soft multiset F A which describes wooden houses cars and hotels with A a e U e U2 e U a 2 e U e U2 e U a e U e U2 e U Then a semi-absolute fuzzy soft multiset F A U is given as F A U a a 2 a h 2 h h h h 2 h h h h 2 h h h 8 c 2 c c 6 c 2 8 c 6 c 8 8 c 2 6 c c 8 v 2 v 6 v 2 v v 2 v 2 )) )) )) Definition 6 A fuzzy soft multiset F A over U is called an absolute fuzzy soft multiset denoted by F A U if e Ui jf eui j U i i Example 7 Consider Example 2 Let us consider a fuzzy soft multiset F A which describes wooden houses expensive classic cars and hotels in Kuala Lumpur with A a e U e U2 e U a 2 e U e U2 e U 6

Abstract and Applied Analysis Then an absolute fuzzy soft multiset F A U is given as F A U a a 2 h 2 h h h h 2 h h h c 2 c c v 2 v )) c 2 c c v 2 v )) 7 Proposition 8 If F A is a fuzzy soft multiset over U then a F A c c F A b F A c Φ i F A Ui c F A c Φ F A U d F A c U i F A Φi e F A c U F A Φ Proof The proof is straightforward 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 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 F ε if ε A B H ε G ε if ε B A F ε G ε if ε A B where F ε G ε s F G εui j ε Ui i 2m with s as an s-norm j Example 2 Consider Example 2 Let A a e U e U2 e U a 2 e U 2e U2 e U a e U e U2 e U a e U e U2 e U B b e U e U2 e U b 2 e U e U2 2e U b e U 2e U2 e U 2 b e U e U2 e U 2 b e U 2e U2 e U 2 b 6 e U e U2 e U 2

Abstract and Applied Analysis Suppose F A and G B are two fuzzy soft multisets over the same U such that F A G B a a 2 a a b 2 h 2 h 8 h h h 2 h h 8 h h 2 8 h h h 8 h 2 6 h h h h 2 h 9 h h 2 8 c 2 c c 6 8 c 2 6 c c b 2 h 2 6 h 8 h h b b 8 h 2 9 h h 8 h 6 9 h 2 6 h h h 8 b h 2 9 h 8 h h b 6 8 h 2 h h 9 h 6 c 2 8 c c 8 v 2 v v 2 v 2 c 2 c c 2 8 c 2 6 c 6 c c 2 9 c 9 c 8 8 c 2 8 c c c 2 c c 8 c 2 9 c c 8 c 2 c 6 c )) )) v 2 v )) v 2 v )) 9 v 2 v )) 9 8 v 2 v )) v 2 6 v )) 8 v 2 8 v )) v 2 6 v )) 8 8 v 2 9 v 9 )) By using the basic fuzzy union maximum we have F A G B H D d d 2 d d d h 2 h 9 h h 2 8 h 2 9 h h 8 h 6 h 2 8 h h h 8 h 2 6 h h h h 2 6 h 8 h h 8 c 2 6 c 6 c 8 c 2 8 c c c 2 c 2 8 c c c c 2 c 2 9 c 9 c 8 9 v 2 v 9 v 2 6 v v 2 v )) )) )) v 2 v )) 8 v 2 v ))

2 Abstract and Applied Analysis d 6 d 7 d 8 9 h 2 6 h h h 8 h 2 9 h 8 h h 8 h 2 h h 9 h 6 c 2 c c 8 c 2 9 c c 8 c 2 c 6 c 8 v 2 8 v )) v 2 6 v )) 8 8 v 2 9 v )) 9 where D d a b d 2 a 2 b d a d a d b 2 d 6 b d 7 b d 8 b 6 Proposition If F A G B and H C are three fuzzy soft multisets over Uthen 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 wherer is defined by d F A G A Φ F A e F A G B Φi R D whered A B and R is defined by f F A G B Φ FA if A B RD 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 RD Ui if A B RD otherwise where D A B j F A G B U Proof The proof is straightforward GB U if A B RD otherwise where D A B Definition 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 Band ε C F ε if ε A B H ε G ε if ε B A 6 F ε G ε if ε A B where F ε G ε t F G εui j ε Ui i 2 m with t as a t-norm j

Abstract and Applied Analysis Example Consider Example 2 By using the basic fuzzy intersection minimum we have F A G B H D d d 2 d d d d 6 d 7 c 8 2 h 2 h 8 h h h 2 h h 8 h h 2 8 h h h 8 h 2 6 h h h h 2 6 h 8 h h 9 h 2 6 h h h 8 h 2 9 h 8 h h 8 h 2 h h 9 h 6 8 c 2 c c 6 8 c 2 6 c c c 2 8 c c c 2 c c 2 c 2 9 c 9 c 8 c 2 c c 8 c 2 9 c c 8 c 2 c 6 c 8 v 2 v v 2 v 2 v 2 v v 2 v 8 v 2 v 8 v 2 8 v v 2 6 v 8 8 v 2 9 v 9 )) )) )) )) )) )) )) )) 7 where D d a b d 2 a 2 b d a d a d b 2 d 6 b d 7 b d 8 b 6 8 Proposition 6 If F A G B and H C are three fuzzy soft multisets over Uthen 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 wherer is defined by 6 d F A G A Φi R A Φi wherer is defined by 6 RD Φi if A B e F A G B Φi where D A B RD otherwise f F A G B Φ RD Φ if A B where D A B RD otherwise g F A G A Ui R D whered A B and R is defined by 6 h F A G A U F A i F A G B Ui R D whered A B and R is defined by 6 j F A G B U FA if A B RD otherwise where D A B and R is defined by 6

Abstract and Applied Analysis Proof The proof is straightforward 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 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 set P 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 r i and t i for o i i e Compute the score of o i i f The decision is S k if S k max i S i g If k has more than one value then any one of o k may be chosen 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 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 S k c Redefine the fuzzy soft multiset H C by keeping all values in each row where S k is maximum and replacing the values in the other rows by zero to get H C d Apply RMA to the second fuzzy soft multiset part in H C to get the decision S k2 e Redefine the fuzzy soft multiset H C 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 S k g The decision is S k S k2 S k

Abstract and Applied Analysis Application in a Decision-Making Problem Let U h h 2 h h U 2 c c 2 c andu v v 2 v be the sets of houses cars and hotels respectively Let E U E U2 E U be a collection of sets of decision parameters related to the above universes where E U e U expensive e U 2 cheap e U woodene U in green surroundings E U2 e U2 expensive e U2 2 cheap e U2 sporty E U e U expensive e U 2 cheap e U in KualaLumpure U majestic Let A a e U e U2 e U a 2 e U 2e U2 e U a e U e U2 e U a e U e U2 e U B b e U e U2 e U b 2 e U e U2 2e U b e U 2e U2 e U 2 b e U e U2 e U 2 b e U 2e U2 e U 2 b 6 e U e U2 e U 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 Y and Y 2 respectively Let F A G B a 2 h 2 h 8 h 8 c 2 c 8 v 2 v )) a 2 h 2 h h 8 c 2 6 c v 2 v )) 2 a h 2 8 h h 8 c 2 6 c v 2 v )) a 8 h 2 6 h h c 2 c v 2 v )) b h 2 h 9 h 8 c 2 6 c 6 9 v 2 v )) 9 b 2 h 2 6 h 8 h c 2 9 c 9 8 v 2 v )) b 8 h 2 9 h h 8 8 c 2 8 c v 2 6 v )) b 9 h 2 6 h h 8 c 2 8 c 8 v 2 8 v ))

6 Abstract and Applied Analysis b h 2 9 h 8 h 8 c 2 8 c v 2 6 v )) 8 b 6 8 h 2 h h 9 8 c 2 8 c 8 v 2 9 v )) 9 By using the basic fuzzy union we have F A G B H D d h 2 h 9 h 8 c 2 6 c 6 9 v 2 v )) 9 d 2 8 h 2 9 h h 8 8 c 2 8 c v 2 6 v )) d h 2 8 h h 8 c 2 6 c v 2 v )) d 8 h 2 6 h h c 2 c v 2 v )) d h 2 6 h 8 h c 2 9 c 9 8 v 2 v )) d 6 9 h 2 6 h h 8 c 2 8 c 8 v 2 8 v )) d 7 h 2 9 h 8 h 8 c 2 8 c v 2 6 v )) 8 d 8 8 h 2 h h 9 8 c 2 8 c 8 v 2 9 v )) 9 Now we apply RMA to the first fuzzy soft multiset part in H D to take the decision from the availability set U The tabular representation of the first resultant fuzzy soft multiset part will be as in Table The comparison table for the first resultant fuzzy soft multiset part will be as in Table 2 Next we compute the row-sum column-sum and the score for each h i as shown in Table From Table it is clear that the maximum score is 6 scored by h

Abstract and Applied Analysis 7 Table : Tabular representation: U -fuzzy soft multiset part of H D U d d 2 d d d d 6 d 7 d 8 h 8 8 9 8 h 2 9 8 6 6 6 9 h 9 8 h 8 9 Table 2: Comparison table: U -fuzzy soft multiset part of H D U h h 2 h h h 8 h 2 8 h 8 6 h 8 Table : Score table: U fuzzy soft multiset part of H D Row-sum r i Column-sum t i Score S i h 2 7 6 h 2 2 2 h 2 2 h 7 2 7 Now we redefine the fuzzy soft multiset H D by keeping all values in each row where h is maximum and replacing the values in the other rows by zero: H D d d 2 d d d d 6 d 7 d 8 h 2 h 9 h 8 h 2 9 h h 8 h 2 8 h h 8 h 2 6 h h h 2 6 h 8 h 9 h 2 6 h h h 2 9 h 8 h 8 h 2 h h 9 c 2 c c 2 c 8 c 2 6 c c 2 c c 2 c 8 c 2 8 c 8 c 2 8 c c 2 c v 2 v v 2 v )) )) v 2 v v 2 v v 2 v )) 8 v 2 8 v v 2 6 v 8 v 2 v )) )) )) )) ))

8 Abstract and Applied Analysis Table : Tabular representation: U 2 -fuzzy soft multiset part of H D U 2 d d 2 d d d d 6 d 7 d 8 c 8 8 8 c 2 6 8 8 c Table : Comparison table: U 2 -fuzzy soft multiset part of of H D U 2 c c 2 c c 8 8 8 c 2 6 8 8 c 8 Now we apply RMA to the second fuzzy soft multiset part in H D to take the decision from the availability set U 2 The tabular representation of the second resultant fuzzy soft multiset part of H D will be as in Table The comparison table for the second resultant fuzzy soft multiset part of H D is as in Table Next we compute the row-sum column-sum and the score for each c i is shown in Table 6 From Table 6 it is clear that the maximum score is 6 scored by c Now we redefine the fuzzy soft multiset H D by keeping all values in each row where c is maximum and replacing the values in the other rows by zero: H D 2 d d 2 d d d d 6 d 7 d 8 h 2 h 9 h 8 h 2 9 h h 8 h 2 8 h h 8 h 2 6 h h h 2 6 h 8 h 9 h 2 6 h h h 2 9 h 8 h 8 h 2 h h 9 c 2 c c 2 c 8 c 2 6 c c 2 c c 2 c 8 c 2 8 c 8 c 2 8 c c 2 c v 2 v v 2 v )) )) v 2 v v 2 v v 2 v )) 8 v 2 8 v v 2 6 v 8 v 2 v )) )) )) )) )) 6

Abstract and Applied Analysis 9 Table 6: Score table: U 2 -fuzzy soft multiset part of H D Row-sum r i Column-sum t i Score S i c 2 8 6 c 2 22 2 2 c 6 2 8 Table 7: Tabular representation: U -fuzzy soft multiset part of H D 2 U d d 2 d d d d 6 d 7 d 8 v 8 v 2 8 6 v 8 Table 8: Comparison table: U -fuzzy soft multiset part of of H D 2 U v v 2 v v 8 7 v 2 7 8 v 7 8 8 Table 9: Score table: U -fuzzy soft multiset part of H D 2 Row-sum r i Column-sum t i Score S i v 2 22 2 v 2 9 2 2 v 2 7 6 Now we apply RMA to the third fuzzy soft multiset part in H D 2 to take the decision from the availability set U The tabular representation of the third resultant fuzzy soft multiset part of H D 2 is as in Table 7 The comparison table for the second resultant fuzzy soft multiset part of H D 2 is as in Table 8 Next we compute the row-sum column-sum and the score for each v i as shown in Table 9 From Table 9 it is clear that the maximum score is 6 scored by v Then from the above results the decision for Mr X is h c v 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

2 Abstract and Applied Analysis Acknowledgments The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the research grants UKM-ST-6-FRGS-29 and UKM-DLP- 2-8 References D Molodtsov Soft set theory first results Computers & Mathematics with Applications vol 7 no - pp 9 999 2 D Chen E C C Tsang D S Yeung and X Wang The parameterization reduction of soft sets and its applications Computers & Mathematics with Applications vol 9 no -6 pp 77 76 2 S Alkhazaleh and A R Salleh Generalised interval-valued fuzzy soft set Journal of Applied Mathematics vol 22 Article ID 87 8 pages 22 P K Maji A R Roy and R Biswas An application of soft sets in a decision making problem Computers & Mathematics with Applications vol no 8-9 pp 77 8 22 P K Maji R Biswas and A R Roy Fuzzy soft sets Journal of Fuzzy Mathematics vol 9 no pp 89 62 2 6 R Roy and P K Maji A fuzzy soft set theoretic approach to decision making problems Journal of Computational and Applied Mathematics vol 2 no 2 pp 2 8 27 7 S Alkhazaleh A R Salleh and N Hassan Possibility fuzzy soft set Advances in Decision Sciences vol 2 Article ID 7976 8 pages 2 8 S Alkhazaleh A R Salleh and N Hassan Fuzzy parameterized interval-valued fuzzy soft set Applied Mathematical Sciences vol no 67 pp 6 2 9 S Alkhazaleh and A R Salleh Generalised interval-valued fuzzy soft set Journal of Applied Mathematics vol 2 Article ID 87 8 pages 22 S Alkhazaleh and A R Salleh Soft expert sets Advances in Decision Sciences vol 2 Article ID 77868 2 pages 2 A R Salleh S Alkhazaleh N Hassan and A G Ahmed Multiparameterized soft set Journal of Mathematics and Statistics vol 8 no pp 92 97 22 2 S Alkhazaleh A R Salleh and N Hassan Soft multisets theory Applied Mathematical Sciences vol no 72 pp 6 7 2 A R Salleh and S Alkhazaleh An application of soft multiset theory in decision making in Mathematics Extended Abstract th Saudi Science Conference pp 6 8 April 22 A R Salleh From soft sets to intuitionistic fuzzy soft sets: a brief survey in Proceedings of the International Seminar on the Current Research Progress in Sciences and Technology ISSTech )Universiti Kebangsaan Malaysia-Universitas Indonesia Bandung Indonesia October 2

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