






































2 Soft Multiset Topology.dvi


Abstract

The aim of this paper is to introduce the notion of soft multi-set topology (SMS-topology) defined on

a soft multi-set (SMS). Soft multi-set and soft multi-set topology are fundamental tools in computa-

tional intelligence, which have a large number of applications in soft computing, fuzzy modeling and

decision-making under uncertainty. The idea of power whole multi-subsets of a SMS is defined to explore

various rudimentary properties of SMS-topology. Certain properties of SMS-topology like SMS-basis,

SMS-subspace, SMS-interior, SMS-closure and boundary of SMS are explored. Furthermore, the multi-

criteria decision-making (MCDM) algorithms with aggregation operators based on SMS-topology are

established. Algorithm i (i = 1, 2, 3) are developed for the selection of best alternative for biopesticides,

for the selection of best textile company, for the award of performance, respectively. Some real life ap-

plications of the proposed algorithms in MCDM problems are illustrated by numerical examples. The

the reliability and feasibility of proposed MCDM techniques is shown by comparison analysis with some

existing techniques.

Keywords: Soft multi-sets; soft multi-set topology; aggregation operators, algorithms; MCDM.

1 Introduction
Modeling and handling uncertainties has become an issue of great importance in the solution of sophisticated

problems originating in a vast range of various fields such as computational intelligence, artificial intelligence,

data analysis, information fusion, image processing, signal processing, environmental sciences and medical

sciences. Mathematical models like multi-sets (Blizard, 1989), fuzzy sets (Zadeh, 1965), soft sets (Molodtsov,

1999) and rough sets (Pawlak, 1982) are fundamental tools for uncertainty, hesitancy and vagueness in the

real life circumstances. The researchers have been developed some extension of fuzzy sets like intuitionistic

fuzzy sets (Atanassov, 1986), bipolar fuzzy sets (Zhang, 1994), Pythagorean fuzzy sets (Yager, 2013; Yager

and Abbasov, 2013) and q-rung orthopair fuzzy sets (Yager, 2017) which have a large number of applications

70

Decision Making: Applications in Management and Engineering  
Vol. 3, Issue 2, 2020, pp. 70-96. 
ISSN: 2560-6018 
eISSN: 2620-0104  

 DOI: https://doi.org/10.31181/dmame2003070r 

 

CERTAIN PROPERTIES OF SOFT MULTI-SET TOPOLOGY 
WITH APPLICATIONS IN MULTI-CRITERIA DECISION 

MAKING 

Muhammad Riaz 1*, Naim Çagman 2, Nabeela Wali 3 and Amna Mushtaq 4 
 

1 Department of Mathematics, University of the Punjab, Lahore, Pakistan.  
2 Department of Mathematics, Tokat Gaziosmanpasa University, Tokat, Turkey.  

3 Department of Mathematics, University of the Punjab, Lahore, Pakistan.  
4 Department of Mathematics, University of the Punjab, Lahore, Pakistan.  

 
* Corresponding author. 
 E-mail addresses: mriaz.math@pu.edu.pk (M. Riaz), naim.cagman@gop.edu.tr (N. 
Çagman), nabeelawali.math@gmail.com (N. Wali), amna44mushtaq@gmail.com (A. Mushtaq) 
 
 

 

Original scientific paper 
 
 

 
 
 
Received: 5 June 2020;  
Accepted: 19 August 2020;  
Available online: 12 September 2020. 



in computational intelligence, decision making under uncertainty and many other fields of science and engi-

neering. Indeed, the real power of these sets are in their ability to handle and manipulate verbally-stated

information into mathematical modeling and seeking feasible solutions to complicated real life problems.

Additionally, fuzzy sets and its extensions are strong mathematical models to solve real world problems

which can not be solved by classical mathematical techniques.

Fuzzy sets, extensions of fuzzy sets, rough sets, soft sets and hybrid structures of these sets have been studied

by many researchers like Ali, (2009,2011); Cagman et al., (2011); Chen (2005); Feng et al., (2010,2011,2018);

Garg and Rani, (2019); Hashmi et al., (2019); Karaaslan and Hunu, (2020); Kumar and Garg, (2018), Maji

et al., (2002,2003); Naeem et al., (2019), Peng and Yang (2015), Peng et al., (2017), Pie and Miao (2005),

Roy and Maji (2007); Riaz et al., (2019);, Riaz and Hashmi (2019); Riaz and Tehrim, (2019); Shabir and

Naz (2011); Zhang and Xu (2014); Zhan et al., (2015,2019); and Zhang (1994).

Multi-set theory and soft multi-set theory have been studied by many researchers including Alkhazaleh

et al. (2011); Babitha and John (2013); Balami and Ibrahim (2013); Girish and John (2009,2019); Kumar

and Naisal (2016); Mukherjee et al. (2014); Syropoulos (2001) and Tokat and Osmanoglu (2011,2013).

A large number of MCDM methods have been developed by the researchers under rough sets, fuzzy sets and

soft sets. But these methods do not deal with real life situations under the universe of soft multi-sets. Due to

repetition of objects or objects have multiplicity more than one and variety of attributes under consideration

in the universe of soft multi-sets it is necessary to develop novel MCDM approaches. The goal of this

article is deal with these challenges and to extend the notion of soft multi-sets and soft multi-set topology

towards MCDM problems. The topological and algebraic structures of soft multi-sets have large number

of applications in soft computing, decision-making, data analysis, data mining, expert systems, information

aggregation and information measures.

The remaining article is arranged as follows: In section 2, we use power whole multi-subsets of a SMS to

introduce some basic concepts of SMS-theory. In section 3, we present some new results of SMS-topology

and certain properties including basis, subspace, interior, closure and boundary of soft multi-sets (SMSs).

In Section 4, we present Algorithm 1, Algorithm 2 and Algorithm 3 for the selection of best alternative for

biopesticides, for the selection of best textile company, for the award of performance, respectively. We also

present applications of SMS-topology for MCDM by using proposed algorithms. At the end, the sum up of

this research studies is given in the in Section 5.

2 Preliminaries

In this section, we study few primary rudiments of multi-sets (MSs) and soft multi-sets (SMSs).

Definition 2.1. ”A multi-set (MS) over Z is just a pair < Z, f >, where f : Z → W is a function, Z is a

crisp set and W is a set of whole numbers. Also in order to avoid any confusion we will use square brackets

for multi-sets and braces for sets. Multiset A is given by A =< Z, f >= [k1
z1
, k2
z2
, ..., kn

zn
], where z1 occuring

k1 times, z2 occuring k2 times and so on (Syropoulos, 2001).

Definition 2.2. Let A =< Z, f > and B =< Z, g > be two multi-sets. Multiset A is a submulti-set of B,

71

Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



denoted by A ⊆ B if for all z ∈ A, f(z) ≤ g(z) (Syropoulos, 2001).

Definition 2.3. A submulti-set A =< Z, f > of B =< Z, g > is a whole submulti-set of B with each

element in A having full multiplicity as in B. i.e. f(z) = g(z), for every z in A (Babitha and John (2013))

Definition 2.4. Let [Z]n denotes the set of all MSs whose elements are in Z such that no element in a

multi-set appears more than n times. Let A ∈ [Z]n be a multi-set. The power whole multi-set of A denoted

by PW(A) is defined as the set of all whole sub MSs of A. The cardinality of PW(A) is 2m, where m is the

cardinality of the support set (root set) of A” (Babitha and John (2013)).

In the sequel, H indicates to universal multi-set, E is a set of attributes or parameters , PW(H) is a power

whole multi-set of H and A ⊆ E.

Example 2.5. Let
...
M = [2/r, 1/y, 1/k] be a multi-set. Then the set of all sub MSs of M is

PW(A) =

{
...
M1 = [0/r, 0/y, 0/k],

...
M2 = [0/r, 0/y, 1/k],

...
M3 = [0/r, 1/y, 0/k],

...
M4 = [0/r, 1/y, 1/k],

...
M5 = [1/r, 0/y, 0/k],

...
M6 = [1/r, 0/y, 1/k],

...
M7 = [1/r, 1/y, 0/k],

...
M8 = [1/r, 1/y, 1/k],

...
M9 = [2/r, 0/y, 0/k],

...
M10 = [2/r, 0/y, 1/k],

...
M11 = [2/r, 1/y, 0/k],

...
M12 = [2/r, 1/y, 1/k]

}

and card(P(M)) = (2 + 1)(1 + 1)(1 + 1) = 12.

Furthermore, the power whole multi-set is given by

PW(M) = {
...
M1,

...
M2,

...
M3,

...
M4,

...
M9,

...
M10,

...
M11,

...
M12}

and its cardinality is given by card(PW(M)) = 23 = 8.

Definition 2.6. ”A soft multi-set (SMS) ΩA on the universal multi-set H is defined by the set of all ordered

pairs ΩA = {(ν, ΩA(ν)) : ν ∈ E, ΩA(ν) ∈ PW(H)}, where ΩA : E → PW(H) such that ΩA(ν) = ∅ if ν /∈ A.

Throughout this paper, SM(H) denotes the family of all SMSs over H with attributes from E. Now, we

elaborate the definition of soft multi-set by the succeeding example” (Babitha and John (2013)).

Example 2.7. Let H = [ 2
r1
, 4
r2
, 3
r3
, 5
r4
, 7
r5
, 6
r6
, 9
r7
] be the universal multi-set of classrooms,

E = {comfortable, air conditioned, well decorated, flipped classroom}

and A = E. Then the SMS ΩA is given by

ΩA = {(comfortable, [
2
r1
, 5
r4
]), (air conditioned, [ 6

r6
, 9
r7
]),

(well decorated, [ 2
r1
, 4
r2
]), (flipped classroom, [ 3

r3
, 7
r5
, 9
r7
])}.

Definition 2.8. ”Let ΩA ∈ SM(H). If ΩA(ν) = ∅, ∀ ν ∈ E, then ΩA is called an empty or null SMS,

denoted by Ωφ (See Babitha and John (2013)).

72

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Definition 2.9. Let ΩA ∈ SM(H). Then ΩA is said to be A-universal SMS, denoted by ΩÂ, if ΩA(ν) = H,

∀ ν ∈ A. If A = E, then A-universal soft multi-set is said to be an universal or absolute SMS, denoted by

Ω
Ê
” (Babitha and John (2013)).

Definition 2.10. Let ΩA, ΩB ∈ SM(H). Then, ΩA is a soft multi subset of ΩB, denoted by ΩA⊆̂ΩB, if

ΩA(ν) ⊆ ΩB(ν) for all ν ∈ E” (Babitha and John (2013)).

Definition 2.11. Let ΩA, ΩB ∈ EM(H). Then, the union ΩA∪̂ΩB, the intersection ΩA∩̂ΩB, the difference

ΩA\̂ΩB of ΩA and ΩB are defined by the approximate functions ΩA∪̂B(ν) = ΩA(ν) ∪ ΩB(ν), ΩA∩̂B(ν) =

ΩA(ν) ∩ ΩB(ν), ΩA\̂B(ν) = ΩA(ν) ⊖ ΩB(ν), respectively, and the complement Ω
c
A of ΩA is defined Ω

c
A(ν) =

H ⊖ ΩA(ν), for all ν ∈ E. Note that (Ω
c
A)

c = ΩA and Ω
c
φ = ΩÊ.

Definition 2.12. A soft multi-set ΩA over H is called soft multi-set point (SMS-point), if there is exactly

one ν ∈ A, such that ΩA(ν) 6= ∅ and ΩA(µ) = ∅, ∀µ ∈ A \ {ν}. The SMS-point ΩA is in the SMS δA, if for

the element ν ∈ A, ΩA(ν) ⊆ δA(ν).

Example 2.13. Let H = [ 2
a
, 3
b
, 4
c
], A = {ν, µ} = E. Let ΩA = {(ν, [

2
a
])} and δA = {(ν, [

2
a
, 3
b
]), (µ, [3

b
, 4
c
])}.

Since ΩA(ν) = [
2
a
] ⊆ [ 2

a
, 3
b
] = δA(ν) and ΩA(µ) = ∅ ∀µ ∈ A \ {ν}.

Therefore, ΩA is a SMS-point of SMS δA. , where

Proposition 2.14. Let ΩA, ΩB ∈ SM(H). Then

(i) (ΩA∪̂ΩB)
c = σcA∩̂σ

c
B,

(ii) (ΩA∩̂ΩB)
c = σcA∪̂σ

c
B.

3 Soft Multi-Set Topology

Different approaches have bee studied by the researchers to define soft multi-set topology (SMS-topology)

(Mukherjee et al. (2014), and Tokat and Osmanoglu (2011,2013)). In this section, we introduce the notion

of SMS-topology on a soft multi-set and its analogous properties by using the concept of power whole sub

multi-sets to use the full multiplicity or zero multiplicity of each objects.

Definition 3.1. Let ΩA be a SMS over H. The soft power whole multi-set of the SMS ΩA is denoted by

P̃W(ΩA) and is defined as

P̃W(ΩA) = {ΩAi : ΩAi⊆̃ΩA, i ∈ I}

and its cardinality is given by

|P̃W(ΩA)| = 2
∑

i∈N
|Xi|,

where |Xi| is the cardinality of the support set Xi of approximation image multi-set
...
Mi with respect to

parameter ëi, where i ∈ N.

Example 3.2. Let H = [ 5
a
, 4
b
, 3
c
], E = {ë1, ë2, ë3}, A = {ë1, ë2} ⊆ E and a soft multi-set over H is

ΩA = {(ë1, [
5

a
,
4

b
]), (ë2, [

4

b
,
3

c
])}.

Then |P̃W(ΩA)| = 2
|X1|+|X2| = 22+2 = 24 = 16, where |X1| = 2, since X1 = {a, b} and |X2| = 2, since

X2 = {b, c}.

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Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



The soft power whole multi-set of the soft multi-set ΩA is given by P̃W(ΩA) = {ΩA1, ΩA2, · · ·, ΩA16}, where

ΩA1 = Ω∅,

ΩA2 = {(ë1, [
5
a
])},

ΩA3 = {(ë1, [
4
b
])},

ΩA4 = {(ë1, [
5
a
, 4
b
])},

ΩA5 = {(ë2, [
4
b
])},

ΩA6 = {(ë2, [
3
c
])},

ΩA7 = {(ë2, [
4
b
, 3
c
])},

ΩA8 = {(ë1, [
5
a
]), (ë2, [

4
b
])},

ΩA9 = {(ë1, [
5
a
]), (ë2, [

3
c
])},

ΩA10 = {(ë1, [
5
a
]), (ë2, [

4
b
, 3
c
])},

ΩA11 = {(ë1, [
4
b
]), (ë2, [

4
b
])},

ΩA12 = {(ë1, [
4
b
]), (ë2, [

3
c
])},

ΩA13 = {(ë1, [
4
b
]), (ë2, [

4
b
, 3
c
])},

ΩA14 = {(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
])},

ΩA15 = {(ë1, [
5
a
, 4
b
]), (ë2, [

3
c
])},

ΩA16 = ΩA.

Example 3.3. Let H = [ 1
2
, 1
3
, 2
4
, 3
5
, 2
6
, 5
7
, 1
8
, 5
9
, 4
10
] and E = {ë1, ë2, ë3, ë4, ë5, ë6} where

ë1 denotes divisibility by 2,

ë2 denotes divisibility by 3,

ë3 denotes divisibility by 4,

ë4 denotes divisibility by 5,

ë5 denotes divisibility by 6,

ë6 denotes divisibility by prime numbers.

Let A = {ë3, ë4, ë5} ⊆ E and a soft multi-set over H is

ΩA = {(ë3, [
2

4
,
1

8
]), (ë4, [

3

5
,
4

10
]), (ë5, [

2

6
])}.

Then |P̃W(ΩA)| = 2
|X1|+|X2|+|X3| = 22+2+1 = 25 = 32,

where |X1| = 2, since X1 = {4, 8}, |X2| = 2, since X2 = {5, 10} and |X3| = 1, since X3 = {6}.

The soft power whole multi-set of the SMS ΩA is given by P̃W(ΩA) = {ΩA1, ΩA2, · · ·, ΩA32}, where

ΩA1 = Ω∅,

ΩA2 = {(ë3, [
2
4
])},

ΩA3 = {(ë3, [
1
8
])},

ΩA4 = {(ë3, [
2
4
, 1
8
])},

ΩA5 = {(ë4, [
3
5
])},

ΩA6 = {(ë4, [
4
10
])},

ΩA7 = {(ë4, [
3
5
, 4
10
])},

ΩA8 = {(ë5, [
2
6
])},

ΩA9 = {(ë3, [
2
4
]), (ë4, [

3
5
])},

ΩA10 = {(ë3, [
2
4
]), (ë4, [

4
10
])},

ΩA11 = {(ë3, [
2
4
]), (ë4, [

3
5
, 4
10
])},

ΩA12 = {(ë3, [
1
8
]), (ë4, [

3
5
])},

74

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



ΩA13 = {(ë3, [
1
8
]), (ë4, [

4
10
])},

ΩA14 = {(ë3, [
1
8
]), (ë4, [

3
5
, 4
10
])},

ΩA15 = {(ë3, [
2
4
]), (ë5, [

2
6
])},

ΩA16 = {(ë3, [
1
8
]), (ë5, [

2
6
])},

ΩA17 = {(ë3, [
2
4
, 1
8
]), (ë5, [

2
6
])},

ΩA18 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
])},

ΩA19 = {(ë3, [
2
4
, 1
8
]), (ë4, [

4
10
])},

ΩA20 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
, 4
10
])},

ΩA21 = {(ë4, [
3
5
]), (ë5, [

2
6
])},

ΩA22 = {(ë4, [
4
10
]), (ë5, [

2
6
])},

ΩA23 = {(ë4, [
3
5
, 4
10
]), (ë5, [

2
6
])},

ΩA24 = {(ë3, [
2
4
]), (ë4, [

3
5
]), (ë5, [

2
6
])},

ΩA25 = {(ë3, [
2
4
]), (ë4, [

4
10
]), (ë5, [

2
6
])},

ΩA26 = {(ë3, [
2
4
]), (ë4, [

3
5
, 4
10
]), (ë5, [

2
6
])},

ΩA27 = {(ë3, [
1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])},

ΩA28 = {(ë3, [
1
8
]), (ë4, [

4
10
]), (ë5, [

2
6
])},

ΩA29 = {(ë3, [
1
8
]), (ë4, [

3
5
, 4
10
]), (ë5, [

2
6
])},

ΩA30 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])},

ΩA31 = {(ë3, [
2
4
, 1
8
]), (ë4, [

4
10
]), (ë5, [

2
6
])},

ΩA32 = ΩA.

Definition 3.4. ”Let ΩA be a soft multi-set over universal multi-set H. A SMS-topology on a soft multi-set

ΩA, denoted by τ̃, is a collection of soft multi subsets of ΩA having the following properties:

(i) Ω∅, ΩA ∈ τ̃.

(ii) Union of any number of members of τ̃ belongs to τ̃

i.e. {ΩAi⊆̃ΩA : i ∈ I ⊆ N}⊆̃τ̃ ⇒
⋃̃

i∈I
ΩAi ∈ τ̃.

(iii) Intersection of finite number of members of τ̃ belongs to τ̃

i.e. {ΩAi⊆̃ΩA : 1 ≤ i ≤ n, n ∈ N}⊆̃τ̃ ⇒
⋂̃

1≤i≤nΩAi ∈ τ̃.

Then a SMS topological space is denoted by (ΩA, τ̃)” (Mukherjee et al. (2014), and Tokat and Osmanoglu

(2011,2013)).

Example 3.5. Let H = [ 5
a
, 4
b
, 3
c
], E = {ë1, ë2, ë3}, A = {ë1, ë2} ⊆ E and a soft multi-set over H is

ΩA = {(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
, 3
c
])} as given in Example 3.2. Then

τ̃1 = {Ω∅, ΩA}, τ̃2 = P̃W(ΩA),

and τ̃3 = {Ω∅, {(ë1, [
4
b
])}, {(ë1, [

4
b
]), (ë2, [

3
c
])}, {(ë1, [

5
a
, 4
b
]), (ë2, [

4
b
])}, ΩA}

are three SMS topologies on the soft multi-set ΩA.

Likewise τ̃4 = {Ω∅, {(ë1, [
5
a
])}, {(ë1, [

4
b
])}, ΩA} is not a SMS-topology on ΩA.

Example 3.6. Take soft multi-set (SMS)

ΩA = {(ë3, [
2

4
,
1

8
]), (ë4, [

3

5
,
4

10
]), (ë5, [

2

6
])}.

which is same as given in Example 3.3. So that

τ̃1 = {Ω∅, ΩA},

τ̃2 = {Ω∅, ΩA24, ΩA26, ΩA30, ΩA} or

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Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



τ̃2 = {Ω∅, {(ë3, [
2
4
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, {(ë3, [

2
4
]), (ë4, [

3
5
, 4
10
]), (ë5, [

2
6
])},

{(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, ΩA} and

τ̃3 = P̃W(ΩA)

are SMS topologies on the SMS ΩA.

Throughout this work, we use the following definition of complement in a SMS topological space.

Definition 3.7. The soft multi complement Ωc̃B of a soft multi subset ΩB in a SMS topological space (ΩA, τ̃)

is defined as Ωc̃B = ΩA\̃ΩB.

Definition 3.8. Let τ̃ be a SMS-topology then each of its element is called soft open multi-set (SOMS) and

the complement of each soft open multi-set is called called a soft closed multi-set.

Example 3.9. Let τ̃2 be the SMS-topology which considered in Example 3.6.

Since ΩA24 = {(ë3, [
2
4
]), (ë4, [

3
5
]), (ë5, [

2
6
])} is a soft open multi-set. Then Ωc̃A24 = {(ë3, [

1
8
]), (ë4, [

4
10
])} is a soft

closed multi-set.

Remark. The union of two SMS-topologies on a SMS ΩE may not be a SMS-topology on ΩE.

Example 3.10. Let H = [ 2
g
, 4
h
, 6
i
], E = {ë1, ë2}, and τ̃1 = {Ω∅, ΩẼ, Ω1E , Ω2E , Ω3E , Ω4E },

τ̃2 = {Ω∅, ΩẼ, Ω5E , Ω6E , Ω7E , Ω8E } be two SMS topologies on ΩẼ where Ω1E , Ω2E , Ω3E , Ω4E , Ω5E , Ω6E , Ω7E
and Ω8E are SMSs over H defined as follows:

Ω1E = {(ë1, [
4
h
]), (ë2, [

2
g
])},

Ω2E = {(ë1, [
4
h
, 6
i
], (ë2, [

2
g
, 4
h
])},

Ω3E = {(ë1, [
2
g
, 4
h
]), (ë2, X)},

Ω4E = {(ë1, [
2
g
, 4
h
]), (ë2, [

2
g
, 6
i
])},

Ω5E = {(ë1, [
4
h
]), (ë2, [

2
g
])},

Ω6E = {(ë1, [
4
h
, 6
i
], (ë2, [

2
g
, 4
h
])},

Ω7E = {(ë1, [
2
g
, 4
h
]), (ë2, [

2
g
, 4
h
])},

Ω8E = {(ë1, [
4
h
]), (ë2, [

2
g
, 6
i
])}.

Now, we define

τ̃ = τ̃1∩̃τ̃2

= {Ω1E , Ω2E , Ω3E , Ω4E , Ω5E , Ω6E , Ω7E , Ω8E }.

If we take Ω2E ∪̃Ω7E = HE. Then

hE(ë1) = f2E (ë1) ∪ f7E (ë1) = [
4
h
, 6
i
] ∪ [ 2

g
, 4
h
] = H

hE(ë2) = f2E (ë2) ∪ f7E (ë2) = [
2
g
, 4
h
] ∪ [ 2

g
, 4
h
] = [ 2

g
, 4
h
]

but HE /∈ τ̃. Thus τ̃ is not a SMS-topology on ΩẼ.

Definition 3.11. Let (ΩA, τ̃) be a SMS topological space and B̃⊆̃τ̃. If every element of τ̃ can be written as

a union of members of B̃, then B̃ is called a soft multi basis for the SMS-topology τ̃.

Example 3.12. Let H = [ 5
a
, 4
b
, 3
c
], E = {ë1, ë2, ë3}, A = {ë1, ë2} ⊆ E and a SMS over H is ΩA =

{(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
, 3
c
])}. Let τ̃2 = P̃W(ΩA). Then B̃ = {Ω∅, ΩA2, ΩA3, ΩA5, ΩA6} or

B̃ = {Ω∅, {(ë1, [
5

a
])}, {(ë1, [

4

b
])}, {(ë2, [

4

b
])}, {(ë2, [

3

c
])}}

is a soft multi basis for the SMS-topology τ̃2.

76

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Example 3.13. Consider the SMS-topology τ̃3 that is given in Example 3.6. Since τ̃3 = P̃W(ΩA). Then

B̃ = {Ω∅, ΩA2, ΩA3, ΩA5, ΩA6, ΩA8} is a soft multi basis for the SMS-topology τ̃3.

Definition 3.14. Let (ΩA, τ̃ΩA) be a SMS topological space and ΩB is contained in ΩA. Let τ̃ΩB be the

collection of ΩBi such that ΩBi = ΩAi∩̃ΩB where each ΩAi are contained in τ̃ΩA.

Then τ̃ΩB is called a soft multi subspace topology or soft multi relative topology on ΩB. Hence (ΩB, τ̃ΩB ) is

soft multi subspace of (ΩA, τ̃ΩA).

Theorem 3.15. Let (ΩA, τ̃ΩA) be a SMS topological space and ΩB⊆̃ΩA. Then a soft multi subspace topology

τ̃ΩB on ΩB is a SMS-topology.

Proof. (i) Since ΩB⊆̃ΩA and Ωφ⊆̃ΩA. Then clearly Ωφ and ΩB are contained in τ̃ΩB this is so because

Ωφ∩̃ΩB = Ωφ and ΩA∩̃ΩB = ΩB, where Ωφ, ΩA are in τ̃ΩA.

(ii)-(iii) Since τ̃ΩA SMS topology, then by the given relations

⋂̃n
i=1

(ΩAi∩̃ΩB) = (
⋂̃n

i=1
ΩAi)∩̃ΩB,

⋃̃
i∈I

(ΩAi∪̃ΩB) = (
⋂̃

i∈I
ΩAi)∩̃ΩB

τ̃ΩB is the SMS topology on ΩB.

Example 3.16. Let us consider SMS-topology τ̃3 on ΩA as given in Example 3.5. Let ΩB = ΩA10 =

{(ë1, [
5
a
]), (ë2, [

4
b
, 3
c
])}, and τ̃3 = {Ω∅, {(ë1, [

4
b
])}, {(ë1, [

4
b
]), (ë2, [

3
c
])}, {(ë1, [

5
a
, 4
b
]), (ë2, [

4
b
])}, ΩA}

then τ̃ΩB = {Ω∅, ΩA6, ΩA9, ΩA10}. So (ΩB, τ̃ΩB ) is soft multi subspace of (ΩA, τ̃3).

Example 3.17. Let us consider the SMS-topology τ̃2 that is given in Example 3.6. Let ΩB = ΩA11 =

{(ë3, [
2
4
]), (ë4, [

3
5
, 4
10
])} then τ̃ΩB = {Ω∅, ΩA9, ΩA11}. So (ΩB, τ̃ΩB ) is soft multi subspace of (ΩA, τ̃2).

Definition 3.18. ”Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then the soft multi interior of

ΩB, denoted by Int(ΩB) or Ω
◦
B, is the soft multi union of all soft open multi subsets of ΩB”.

Example 3.19. Let us consider the SMS-topology τ̃3 given in Example 3.5. If ΩB = ΩA13 = {(ë1, [
4
b
]), (ë2, [

4
b
, 3
c
])},

then Ω◦B = Ω∅∪̃ΩA3∪̃ΩA12 = ΩA12.

Example 3.20. Let us consider the SMS-topology τ̃2 given in Example 3.6. If ΩB = ΩA17 = {(ë3, [
2
4
, 1
8
]), (ë5, [

2
6
])},

then Ω◦B = Ω∅.

Definition 3.21. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then the soft multi closure of ΩB,

denoted by Cl(ΩB) or ΩB, is the soft multi intersection of all soft closed super multi-sets of ΩB.

Example 3.22. Let us consider the SMS-topology τ̃3 given in Example 3.5. If ΩB = ΩA10 = {(ë1, [
5
a
]), (ë2, [

4
b
, 3
c
])},

then Ωc̃A3 = {(ë1, [
5
a
]), (ë2, [

4
b
, 3
c
])} = ΩB and Ω

c̃
φ = ΩA are soft closed super multi-sets of ΩB. Hence

ΩB = ΩA∩̃ΩB = ΩB.

77

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Example 3.23. Let us consider the SMS-topology τ̃2 given in Example 3.6. If ΩB = ΩA3 = {(ë3, [
1
8
])}, then

Ωc̃A24 = ΩA13, Ω
c̃
A26

= ΩA3 and Ω
c̃
φ = ΩA are soft closed super multi-sets of ΩB. Hence ΩB = ΩA3∩̃ΩA13∩̃ΩA =

ΩA3.

Theorem 3.24. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) (Ω◦B)
◦ = Ω◦B

(ii) ΩB⊆̃ΩC ⇒ Ω
◦
B⊆̃Ω

◦
C

(iii) Ω◦B∩̃Ω
◦
C = (ΩB∩̃ΩC)

◦

(iv) Ω◦B∪̃Ω
◦
C⊆̃(ΩB∪̃ΩC)

◦.

(v) (ΩB) = ΩB

(vi) ΩC⊆̃ΩB ⇒ ΩC⊆̃ΩB

(vii) (ΩB∩̃ΩC)⊆̃ΩB∩̃ΩC

(viii) (ΩB∪̃ΩC) = ΩB∪̃ΩC.

(ix) Ω◦B⊆̃ΩB⊆̃ΩB

Proof. The proof follows by Definition 3.18 and Definition 3.21.

Example 3.25. Let U = [ 2
g
, 4
h
, 6
i
], E = {ë1, ë2} and

τ̃ = {Ω∅, ΩẼ, Ω1E , Ω2E , Ω3E , ..., Ω7E }, where

Ω1E = {(ë1, [
2
g
, 4
h
]), (ë2, [

2
g
, 4
h
])},

Ω2E = {(ë1, [
4
h
]), (ë2, [

2
g
, 6
i
])},

Ω3E = {(ë1, [
4
h
, 6
i
]), (ë2, [

2
g
])},

Ω4E = {(ë1, [
4
h
]), (ë2, [

2
g
])},

Ω5E = {(ë1, [
2
g
, 4
h
]), (ë2, U)},

Ω6E = {(ë1, U), (ë2, [
2
g
, 4
h
])},

Ω7E = {(ë1, [
4
h
, 6
i
]), (ë2, [

2
g
, 4
h
])}.

Then (Ω
Ẽ
, τ̃) is a soft multi-set topological space.

Let ΩE and Ω̈E are defined as follows:

ΩE = {(ë1, [
2
g
, 6
i
]), (ë2, ∅)},

Ω̈E = {(ë1, [
4
h
, 6
i
]), (ë2, [

2
g
, 4
h
])}.

Then ΩE∩̃Ω̈E = {(ë1, [
6
i
]), (ë2, ∅)}.

Now, ΩE = ΩẼ∩̃Ω
c̃
2E

∩̃Ωc̃4E = Ω
c̃
2E

and Ω̈E = ΩẼ.

Therefore ΩE∩̃Ω̈E = ΩE.

Also ΩE∩̃Ω̈E = ∩̃{ΩẼ, Ω
c̃
1E

, Ωc̃2E , Ω
c̃
4E

, Ωc̃5E } = Ω
c̃
5E

.

So ΩE∩̃Ω̈E⊆̃ΩE∩̃Ω̈E but ΩE∩̃Ω̈E*̃ΩE∩̃Ω̈E.

Hence, ΩE∩̃Ω̈E 6= ΩE∩̃Ω̈E.

Definition 3.26. Let (ΩA, τ̃) be a soft multi-set topological space and ΩB⊆̃ΩA. The soft multi interior of

soft multi complement of ΩB is called the soft multi exterior of ΩB and is denoted by Ext(ΩB) or Ω
ẽ
B. In

other words, ΩẽB = (Ω
c̃
B)

◦.

Example 3.27. From Example 3.5, we take SMS-topology τ̃3. Then for

78

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



ΩB = ΩA14 = {(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
])}, then Ωc̃A14 = {(ë2, [

3
c
])} = ΩA6. Hence Ω

ẽ
B = (Ω

c̃
B)

◦ = Ωφ, (because null

soft multi-set is the only soft open multi subset contained in Ωc̃B).

Example 3.28. From Example 3.6, we take SMS-topology τ̃2. Then for

ΩB = ΩA30 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, then Ωc̃B = {(ë4, [

4
10
])} = ΩA6. Hence Ω

ẽ
B = (Ω

c̃
B)

◦ = Ωφ,

(because null soft multi-set is the only soft open multi subset contained in Ωc̃B).

Theorem 3.29. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) (ΩB∪̃ΩC)
ẽ = (ΩB)

ẽ∩̃(ΩC)
ẽ,

(ii) (ΩB)
ẽ∪̃(ΩC)

ẽ⊆̃(ΩB∩̃ΩC)
ẽ.

Proof. (i) (ΩB∪̃ΩC)
ẽ = ((ΩB∪̃ΩC)

c̃)◦ = (Ωc̃B∩̃Ω
c̃
C)

◦ = (Ωc̃B)
◦∩̃(Ωc̃C)

◦ = (ΩB)
ẽ∩̃(ΩC)

ẽ

(ii) (ΩB)
ẽ∪̃(ΩC)

ẽ = (Ωc̃B)
◦∪̃(Ωc̃C)

◦ ⊆̃(Ωc̃B∪̃Ω
c̃
C)

◦ = (Ω◦B∪̃(Ω
c̃
C)

◦ = (ΩB∩̃ΩC)
ẽ.

Definition 3.30. ”Let (ΩA, τ̃) be a SMS topological space. A soft multi point α ∈ ΩA is said to be a soft

multi interior point of the soft multi-set ΩA if there is a soft open multi-set ΩB such that α ∈ ΩB⊆̃ΩA.

Moreover, If α is soft multi interior point of the soft multi-set ΩA then ΩA is called soft multi neighborhood

(or soft multi open neighborhood) of α. Thus ν̃(α) = {ΩB : ΩB ∈ τ̃} is the family of soft multi neighborhoods

of α” (Mukherjee et al. (2014), and Tokat and Osmanoglu (2011,2013)).

Example 3.31. Let ΩA = {(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
, 3
c
])} be the soft multi-set as given in Example 3.5 and

τ̃3 = {Ω∅, {(ë1, [
4
b
])}, {(ë1, [

4
b
]), (ë2, [

3
c
])}, {(ë1, [

5
a
, 4
b
]), (ë2, [

4
b
])}, ΩA} be a SMS-topology on the soft multi-set

ΩA.

Let α = (ë1, [
5
a
, 4
b
]) ∈ ΩA then α ∈ ΩA14⊆̃ΩA, where ΩA14 = {(ë1, [

5
a
, 4
b
]), (ë2, [

4
b
])} is the soft multi open

neighborhood of α. Similarly α ∈ ΩA⊆̃ΩA this shows that ΩA is multi soft neighborhood of α. Thus

ν̃(α) = {ΩA14, ΩA} is the family of soft multi neighborhoods of α.

Example 3.32. Let ΩA = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
, 4
10
]), (ë5, [

2
6
])} be the soft multi-set as given in Exam-

ple 3.6 and τ̃2 = {Ω∅, {(ë3, [
2
4
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, {(ë3, [

2
4
]), (ë4, [

3
5
, 4
10
]), (ë5, [

2
6
])}, {(ë3, [

2
4
, 1
8
]), (ë4, [

3
5
]),

(ë5, [
2
6
])}, ΩA} be a SMS-topology on the SMS ΩA. Let α = (ë3, [

2
4
, 1
8
]) ∈ ΩA then α ∈ ΩA30⊆̃ΩA, where

ΩA30 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])} is the soft multi open neighborhood of α. Similarly α ∈ ΩA⊆̃ΩA

this shows that ΩA is soft multi neighborhood of α. Thus ν̃(α) = {ΩA30, ΩA} is the family of soft multi

neighborhoods of α.

Theorem 3.33. Let τ̃ be a SMS topology on SMS ΩA. Then a subset ΩB of ΩA is said to be open if and

only if it is neighborhood of each of its own soft multi point.

Proof. Let ΩB be soft multi open subset of ΩA. Then for each soft multi point λ in ΩB, we have λ∈̃ΩB⊆̃ΩB.

This shows that ΩB is a neighborhood of each of its own soft multi point.

Conversely, suppose that ΩB is a neighborhood of each of its own soft multi point. Then for each soft multi

point λ∈̃ΩB there exists soft multi open set ΩUλ such that λ∈̃ΩUλ⊆̃ΩB.

This shows that ΩB = ∪̃{λ}⊆̃∪̃ΩUλ⊆̃ΩB.

Thus we get ΩB = ⊆̃∪̃ΩUλ. This proves that ΩB is soft multi open set.

79

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Definition 3.34. ”Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA and α ∈ ΩA. If every multi soft

neighborhood of α soft multi intersects ΩB in some soft multi points other than α itself, then α is called a

soft multi limit point of ΩB. The collection of all soft multi limit points of ΩB is denoted by Ω
′
B.

In other words, if (ΩA, τ̃) is a SMS topological space and ΩB⊆̃ΩA and α ∈ ΩA, then α ∈ Ω
′
B ⇔ ΩC∩̃(ΩB\̃{α}) 6=

Ωφ for all ΩC ∈ ν̃(α)”.

Example 3.35. Consider example 3.31. If ΩB = ΩA14 and α = (x1, [
5
a
, 4
b
]) ∈ ΩA, then α ∈ Ω

′
B, since

ΩA∩̃(ΩB\̃{α}) 6= Ωφ and ΩA14∩̃(ΩB\̃{α}) 6= Ωφ.

Example 3.36. Consider Example 3.32. If ΩB = ΩA30 and α = (ë3, [
2
4
, 1
8
]) ∈ ΩA, then α ∈ Ω

′
B, since

ΩA∩̃(ΩB\̃{α}) 6= Ωφ and ΩA30∩̃(ΩB\̃{α}) 6= Ωφ.

Theorem 3.37. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then, ΩB∪̃Ω
′
B = ΩB.

Proof. If α ∈ ΩB∪̃Ω
′
B, then α ∈ ΩB or α ∈ Ω

′
B. In this case, if α ∈ ΩB, then α ∈ ΩB. If α ∈ Ω

′
B, then

ΩC∩̃(ΩB\̃{α}) 6= Ωφ for all ΩC ∈ ν̃(α), and so ΩC∩̃ΩB 6= Ωφ for all ΩC ∈ ν̃(α); hence, α ∈ ΩB. Conversely,

if α ∈ ΩB, then α ∈ ΩB or α ∈ Ω
′
B. In this case, if α ∈ ΩB, it is trivial that α ∈ ΩB∪̃Ω

′
B. If α /∈ ΩB, then

ΩC∩̃(ΩB\̃{α}) 6= Ωφ for all ΩC ∈ ν̃(α). Therefore, α ∈ Ω
′
B, so α ∈ ΩB∪̃Ω

′
B. Hence ΩB∪̃Ω

′
B = ΩB.

Theorem 3.38. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then, ΩB is soft closed multi-set if

and only if Ω′B⊆̃ΩB.

Proof. ΩB = ΩB ⇔ ΩB∪̃Ω
′
B = ΩB ⇔ Ω

′
B⊆̃ΩB.

Theorem 3.39. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) Ω′B⊆̃ΩB

(ii) ΩB⊆̃ΩC ⇒ Ω
′
B⊆̃Ω

′
C

(iii) (ΩB∩̃ΩC)
′⊆̃Ω′B∩̃Ω

′
C

(iv) (ΩB∪̃ΩC)
′ = Ω′B∪̃Ω

′
C

(v) ΩB is a soft closed multi-set ⇔ Ω
′
B⊆̃ΩB.

Proof. The proof is straightforward.

Theorem 3.40. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) (Ωc̃B) = (Ω
◦
B)

c̃

(ii) (ΩB)
c̃ = (Ωc̃B)

◦

(iii) Ω◦B = ((Ω
c̃
B))

c̃

(iv) ΩB = ((Ω
c̃
B)

◦)c̃

(v) (ΩB\̃ΩC)
◦⊆̃ΩB

◦\̃ΩC
◦
.

Proof. (i) Let α ∈ ΩB such that α /∈ Ω
◦
B. Then, for each soft multi open neighborhood of ΩC of α, ΩC soft

multi intersects Ωc̃B. Otherwise, for some soft multi open neighborhood ΩC of α, ΩC∩̃Ω
c̃
B = Ωφ or ΩC⊆̃ΩB.

Since Ω◦B is the largest soft open multi-set in ΩB, therefore α ∈ ΩC⊆̃Ω
◦
B, which is a contradiction. Therefore,

α ∈ (Ωc̃B). Hence, (Ω
◦
B)

c̃⊆̃(Ωc̃B).

80

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Conversely, suppose α ∈ Ωc̃B, then by Definition 3.34, α ∈ Ω
c̃
B or α is a soft multi limit point of Ω

c̃
B. If

α ∈ Ωc̃B, then α ∈ (Ω
◦
B)

c̃. In the second case, α /∈ Ω◦B. Otherwise, by the definition of soft multi limit point,

Ω◦B∩̃Ω
c̃
B 6= Ωφ, which is false. Therefore, (Ω

c̃
B)⊆̃(Ω

◦
B)

c̃.

Combining, we get (i).

(ii) Clearly

(ΩB)
c̃ = (

⋂̃
ΩAi ⊇̃ΩB,Ω

c̃
Ai

∈τ̃
ΩAi)

c̃ =
⋃̃
Ωc̃Ai = (Ω

c̃
B)

◦.

(iii) and (iv) are directly obtained by taking the complements of (i) and (ii), respectively.

(v) (ΩB\̃ΩC)
◦ = (ΩB∩̃Ω

c̃
C)

◦ = Ω◦B∩̃(Ω
c̃
C)

◦ = Ω◦B∩̃(ΩC)
c̃ ⊆̃Ω◦B∩̃(Ω

◦
C)

c̃ = Ω◦B\̃Ω
◦
C.

Definition 3.41. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. The soft multi frontier or boundary

of ΩB is denoted by Ωr(ΩB) or Ω
b̃
B and is defined as Ω

b̃
B = ΩB∩̃Ω

c̃
B. Stated differently, the soft multi points

that do not belong to soft multi interior and exterior of ΩB are in Ω
b̃
B.

Example 3.42. From Example 3.5, we take SMS-topology τ̃3, then for ΩB = ΩA14 = {(ë1, [
5
a
, 4
b
]), (ë2, [

4
b
])},

then Ωc̃A14 = {(ë2, [
3
c
])} = ΩA6. Hence Ω

b̃
B = ΩB∩̃Ω

c̃
B = ΩA∩̃ΩA6 = ΩA6.

Example 3.43. Let us consider the SMS-topology τ̃2 given in Example 3.6.

If ΩB = ΩA30 = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, then Ωc̃A30 = {(ë4, [

4
10
])} = ΩA6. Hence Ω

b̃
B = ΩB∩̃Ω

c̃
B =

ΩA∩̃ΩA6 = ΩA6.

Theorem 3.44. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) Ωb̃B⊆̃ΩB

(ii) Ωb̃B = (Ω
c̃
B)

b̃

(iii) Ωb̃B = ΩB\̃Ω
◦
B.

Proof. (i) The proof is clear by definition of a soft multi boundary.

(ii) Take as given α ∈ Ωb̃B ⇔ ΩC∩̃ΩB 6= Ωφ and ΩC∩̃ΩB
c̃ 6= Ωφ for all ΩC ∈ ν̃(α) ⇔ ΩC∩̃ΩB

c̃ 6= Ωφ and

ΩC∩̃(ΩB
c̃)c̃ 6= Ωφ for all ΩC ∈ ν̃(α). Hence Ω

b̃
B = (Ω

c̃
B)

b̃.

(iii) By using the definitions of a soft multi closure and a multi soft interior, we have

ΩB\̃Ω
◦
B = ΩB∩̃(Ω

◦
B)

c̃ = ΩB∩̃(
⋃̃

ΩBi ⊆̃ΩB,ΩBi ∈τ̃
ΩBi)

c̃ = ΩB∩̃(
⋂̃
ΩBi

c̃) = ΩB∩̃(ΩBi
c̃) = Ωb̃B.

Theorem 3.45. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then,

(i) (Ωb̃B)
c̃ = Ω◦B∪̃(Ω

c̃
B)

◦ = Ω◦B∪̃Ω
ẽ
B

(ii) ΩB = ΩB∪̃Ω
b̃
B

(iii) Ω◦B = ΩB\̃Ω
b̃
B.

Proof. (i) Ω◦B∪̃(Ω
c̃
B)

◦ = ((Ω◦B)
c̃)c̃∪̃(((Ωc̃B)

◦)c̃)c̃ = [(Ω◦B)
c̃∩̃((Ωc̃B)

◦)c̃ ]c̃ = [Ωc̃B∩̃ΩB ]
c̃ = (Ωb̃B)

c̃.

(ii) ΩB∪̃Ω
b̃
B = ΩB∪̃(ΩB∩̃Ω

c̃
B) = [ΩB∪̃ΩB ]∩̃[ΩB∪̃Ω

c̃
B ] = ΩB∩̃[ΩB∪̃Ω

c̃
B ] = ΩB∩̃ΩA = ΩB.

(iii) ΩB\̃Ω
b̃
B = ΩB∩̃(Ω

b̃
B)

c̃ = ΩB∩̃(Ω
◦
B∪̃(Ω

c̃
B)

◦) (by (i)) = [ΩB∩̃Ω
◦
B]∪̃[ΩB∩̃(Ω

c̃
B)

◦] = Ω◦B∪̃Ωφ = Ω
◦
B.

Remark. From Theorem 3.45, it follows that ΩA = Ω
◦
B∪̃Ω

ẽ
B∪̃Ω

b̃
B.

Theorem 3.46. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then,

(i) Ωb̃B∩̃Ω
◦
B = Ωφ

(ii) Ωb̃B∩̃Ω
ẽ
B = Ωφ.

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Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



Proof. (i) Ωb̃B∩̃Ω
◦
B = (ΩB∩̃Ω

c̃
B)∩̃Ω

◦
B = ΩB∩̃(Ω

◦
B)

c̃∩̃Ω◦B = Ωφ.

(ii) Ωb̃B∩̃Ω
ẽ
B = (Ω

c̃
B)

◦∩̃(ΩB∩̃Ω
c̃
B) = (Ω

c̃
B)

◦∩̃ΩB∩̃Ω
c̃
B = (ΩB)

c̃∩̃ΩB∩̃Ω
c̃
B = Ωφ.

Theorem 3.47. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then,

(i) ΩB is soft open multi-set ⇔ ΩB∩̃Ω
b̃
B = Ωφ

(ii) ΩB is soft closed multi-set ⇔ Ω
b̃
B⊆̃ΩB.

(iii) ΩB is both soft open multi-set and soft closed multi-set ⇔ Ω
b̃
B = ∅.

Proof. (i) Let ΩB is soft open multi-set. Then Ω
◦
B = ΩB. Thus ΩB∩̃Ω

b̃
B = Ω

◦
B∩̃Ω

b̃
B = Ωφ (by Theorem

3.46(i)).

Conversely, let ΩB∩̃ΩB = Ωφ. Then, ΩB∩̃[ΩB∩̃Ω
c̃
B] = Ωφ, ΩB∩̃Ω

c̃
B = Ωφ, or Ω

c̃
B⊆̃Ω

c̃
B, which implies that

Ωc̃B is soft closed multi-set and hence, ΩB is soft open multi-set.

(ii) Let ΩB is soft closed multi-set. Then ΩB = ΩB. Now, Ω
b̃
B = ΩB∩̃Ω

c̃
B⊆̃ΩB = ΩB, or Ω

b̃
B⊆̃ΩB and

conversely.

(iii) We know that ΩB is open ⇔ (ΩB)
◦ = ΩB and ΩB is closed ⇔ ΩB = ΩB. Also by Theorem 3.45, we

obtain ΩB = ΩB∪̃Ω
b̃
B and Ω

◦
B = ΩB\̃Ω

b̃
B. This completes the proof.

Theorem 3.48. Let (ΩA, τ̃) be a SMS topological space and ΩB, ΩC⊆̃ΩA. Then,

(i) [ΩB∪̃ΩC]
b̃⊆̃[Ωb̃B∩̃Ω

c̃
C]∪̃[Ω

b̃
C∩̃Ω

c̃
B]

(ii) [ΩB∩̃ΩC]
b̃⊆̃[Ωb̃B∩̃ΩC ]∪̃[Ω

b̃
C∩̃ΩB ].

Proof. Proof is obvious.

Theorem 3.49. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then,

((Ωb̃B)
b̃)b̃ = (Ωb̃B)

b̃.

Proof. (i) ((Ωb̃B)
b̃)b̃ = (Ωb̃B)

b̃∩̃((Ωb̃B)
b̃)c̃ = (Ωb̃B)

b̃∩̃((Ωb̃B)
b̃)c̃ (1)

Now, consider ((Ωb̃B)
b̃)c̃ = [(Ωb̃B)∩̃(Ω

b̃
B)

c̃ ]c̃ = (Ωb̃B∩̃(Ω
b̃
B)

c̃)c̃ = (Ωb̃B)
c̃∪̃((Ωb̃B)

c̃)c̃.

Therefore,

(((Ωb̃B)
b̃)c̃) = [(Ωb̃B)

c̃∪̃((Ωb̃B)
c̃ )c̃ ] = ((Ωb̃B)

c̃ )∪̃(((Ωb̃B)
c̃)c̃) = ΩC∪̃((ΩC)c̃) = ΩA (2)

where ΩC = ((Ω
b̃
B)

c̃ ).

From (1) and (2), we have ((Ωb̃B)
b̃)b̃ = (Ωb̃B)

b̃∩̃ΩA = (Ω
b̃
B)

b̃.

Definition 3.50. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Then ΩB is said to be a soft clopen

multi-set if ΩB is both soft open and soft closed multi-set.

Example 3.51. Since Ωφ and ΩA are always present in τ̃, so Ωφ and ΩA are soft open multi-sets. Moreover,

Ωφ and ΩA are also soft closed multi-sets since Ω
c̃
φ = ΩA and Ω

c̃
A = Ωφ. In fact, these two soft multi-sets are

soft open and soft closed multi-sets simultaneously. Hence, Ωφ and ΩA are soft clopen multi-sets.

Example 3.52. Let us consider the SMS-topology τ̃3 given in Example 3.6. Let ΩB, ΩC∈̃τ̃3, where ΩB =

{(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])}, and ΩC = {(ë4, [

4
10
])}.

Then Ωc̃C = {(ë3, [
2
4
, 1
8
]), (ë4, [

3
5
]), (ë5, [

2
6
])} = ΩB. Hence ΩB is a soft clopen multi-set.

82

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Theorem 3.53. Let (ΩA, τ̃) be a SMS topological space and ΩB⊆̃ΩA. Ω
b̃
B = Ωφ if and only if ΩB is soft

clopen multi-set.

Proof. Suppose that Ωb̃B = Ωφ. First we prove that ΩB is a soft closed multi-set. Consider

Ωb̃B = Ωφ ⇒ ΩB∩̃(Ω
c̃
B) = Ωφ ⇒ ΩB⊆̃((Ω

c̃
B)

c̃) = Ω◦B⊆̃ΩB ⇒ ΩB⊆̃ΩB ⇒ ΩB = ΩB.

This implies that ΩB is a soft closed multi-set. Now we now prove that ΩB is a soft open multi-set. Consider

Ωb̃B = Ωφ ⇒ ΩB∩̃(Ω
c̃
B) = Ωφ or ΩB∩̃(Ω

◦
B)

c̃ = Ωφ ⇒ ΩB⊆̃Ω
◦
B

Ω◦B = ΩB. This implies that ΩB is a soft open multi-set. Conversely, suppose that ΩB is a soft clopen

multi-set. Then,

Ωb̃B = ΩB∩̃(Ω
c̃
B) = ΩB∩̃(Ω

◦
B)

c̃ = ΩB∩̃Ω
c̃
B = Ωφ.

4 MCDM based on SMS-topology

There are different kinds of decision-making methods for selection of a best alternative. Sometimes it is quite

difficult to select an appropriate decision-making method with similar situation in our real life problems.

However, MCDM method based on SMS-topology plays a enthusiastic role in our daily life and this is very

helpful in selection of a best alternative. MCDM is the thought process of selecting a logical choice from

the available options. The concept of aggregation operators in the framework of soft sets and fuzzy soft sets

have been introduced by Çağman et al. (2011). We used the notion of aggregation operators to compute

aggregate fuzzy soft sets and aggregate multi-sets.

4.1 MCDM for selection of best alternative of biopesticides

A big challenge to the agricultural department is to enlarge the production and to meet the demands of

the increasing world population without destroying the environment. In modern agricultural exercises, the

check of pests is generally completed by means of the extreme usage of agrochemicals, which is source of

ambient pollution and the improvement of repellent pests. But biopesticides can proffer a best substitute to

synthetic pesticides empowering safer check of pest communities. It is always a challenging task for a farmer

to choose a best agrochemicals for biopesticides. Every farmer has to face many difficulties to save his fields

from pests. For these challenging tasks various components are take into examination by the farmer either

searching for agrochemicals in order to provide safety from pests attack, improve the soil quality, increase

the quantity of crops, enhance the quality of crops. Major components of biopesticides include microbial

pesticides, biochemical pesticides and biological control agent. The examples of biopesticides include insects,

virus, bacteria, fungi, protozoan, and nematodes. Table 1 gives the comparison of merits and demerits of

biopesticides and chemicals.

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Biopesticides Chemicals pesticides

Environmentally intelligent farming Conflicting to intelligent farming

Cheaper, affordable Costly, expansive

Warmly to non-target genus Dangerous to non-target genus

Do not cause pollution Serious pollution to the environment

Pests never develop resistance Pests eventually become resistance

Expanding market inclination Reduce market inclination

Fight their intended pests End up affecting non target species

Derived from living organisms Contain non-living organism

Table 1: Comparison analysis of biopesticides and chemicals

Algorithm 1 The selection of best alternative for biopesticides

Step 1: Input a suitable parameter set S and universal multi-set H.

Step 2: Input SMSs ΩA and ΩB over H.

Step 3: Construct SMS-topology τ̂ containing ΩA and ΩB as soft open MSs in τ̂.

Step 4: Compute the aggregate fuzzy soft sets by using the formula,

ΓA = {(µi, ΓA(µi)) : µi ∈ S}, where ΓA(µi) = {
ki/|ΩA(µi)|

ωi
: ki
ωi

∈ ΩA(µi)}.

Step 5: Find resultant fuzzy soft set ΓA ∨ ΓB = ΓA×B by applying ’OR’ operation on ΓA and ΓB.

Step 6: Use comparison table of ΓA ∨ ΓB to calculate row-sum (ri) and column-sum (ti) for ωi, ∀ i.

Step 7: Calculate the resulting score Ri of ωi, ∀ i.

Step 8: Optimal choice is ωj that has max{Ri}.

Step 9: Compute the SMS boundary of soft open multi-sets.

Step 10: Here non-null SMS boundary of SMS that contains
kj
ωj

is a decision set.

Figure 1 shows a brief flow-chart of Algorithm 1.

Assume that a farmer wants to safe his fields from pests by using leading alternative of biopesticides

without damaging the sustainability of environment.

Let H = [ 30
ω1

, 25
ω2

, 28
ω3

, 30
ω4

] be the universe of some plants, where

ω1 = Sheesham (Dalbergia sissoo),

ω2 = Safeda (Eucalyptus),

ω3 = Sukh Chain (Pongamia pinnata),

ω4 = Neem (Azadirachta indica)

and the multiplicity of ωi (i = 1, 2, 3, 4) denotes the number of plants corresponding to ωi. Consider the set

of attributes S = {µ1, µ2, µ3, µ4}, where

µ1 = provide safety from pests attack,

µ2 = improve the soil quality,

µ3 = increase the quantity of crops,

84

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Start

input a multi-set
H

input a parameter set
S

input SMSs
ΩA, ΩB

Constuct SMS-topology τ̂
s.t ΩA, ΩB ∈ τ̂

Compute ΓA = {(µi, ΓA(µi)) : µi ∈ S},

where ΓA(µi) = {
ki/|ΩA(µi)|

ωi
: ki
ωi

∈ ΩA(µi)},

Find ΓA ∨ ΓBConstruct the comparison table of ΓA ∨ ΓB

Calculate score Ri = ri − ti Choose ωj that has max{Ri}

If ΩA(µ) = ∅,
∀ µ ∈ S, ΩA ∈ τ̂

Yes No

Ωb̂A = Ωφ Ω
b̂
A 6= Ωφ

Select that Ωb̂A that contains
kj
ωj

Stop

∀ ΩA ∈ τ̂

Figure 1: Graphical representation of Algorithm 1

µ4 = enhance the quality of crops.

We here use the following algorithm to choose the best alternative of agrochemicals for biopesticides without

damaging the environment to safe the fields from pests.

Two decision makers (DMs) Ω1 and Ω2 presented the report to farmer on plant production by using tra-

ditional farming system. Let the DMs Ω1 and Ω2 select two sets of attribute A = {µ1, µ2, µ3, µ4} and

B = {µ1, µ2, , µ3}, respectively. Then DMs construct two SMSs named as ΩA and ΩB over H given by

ΩA = {(µ1, [
30
ω1

, 25
ω2

, 30
ω4

]), (µ2, [
25
ω2

, 28
ω3

, 30
ω4

]), (µ3, [
30
ω4

]), (µ4, H)} and

ΩB = {(µ1, [
30
ω1

, 25
ω2

]), (µ2, [
25
ω2

, 28
ω3

]), (µ3, [
30
ω4

])}.

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The first SMS ΩA can be written as:

ΩA µ1 µ2 µ3 µ4

ω1 30 0 0 30

ω2 25 25 0 25

ω3 0 28 0 28

ω4 30 30 30 30

The second SMS ΩB can be written as:

ΩB µ1 µ2 µ3

ω1 30 0 0

ω2 25 25 0

ω3 0 28 0

ω4 0 0 30

Here we make a SMS-topology on ΩA as τ̂ = {Ωφ, ΩA, ΩB}, where Ωφ is an empty SMS. Now we find the

aggregate fuzzy soft sets ΓA and ΓB given by

ΓA = {(µ1, {
0.35
ω1

, 0.29
ω2

, 0.35
ω4

}), (µ2, {
0.30
ω2

, 0.33
ω3

, 0.36
ω4

}), (µ3, {
1
ω4

}), (µ4, {
0.26
ω1

, 0.22
ω2

, 0.24
ω3

, 0.26
ω4

})} and

ΓB = {(µ1, {
0.54
ω1

, 0.45
ω2

}), (µ2, {
0.47
ω2

, 0.52
ω3

}), (µ3, {
1
ω4

})}.

The fuzzy soft set ΓA can be written as:

ΓA µ1 µ2 µ3 µ4

ω1 0.35 0 0 0.26

ω2 0.29 0.30 0 0.22

ω3 0 0.33 0 0.24

ω4 0.35 0.36 1 0.26

The fuzzy soft set ΓB can be written as:

ΓB µ1 µ2 µ3

ω1 0.54 0 0

ω2 0.45 0.47 0

ω3 0 0.52 0

ω4 0 0 1

We apply here ’OR’ operation on ΓA and ΓB, then we get 4 ∗ 3 = 12 attributes of the form µij =

(µi, µj), ∀ i = 1, 2, 3, 4 and j = 1, 2, 3. We find the fuzzy soft set for the set of attributes A × B =

{µ11, µ12, µ13, µ21, µ22, µ23, µ31, µ32, µ33, µ41, µ42, µ43}. After applying ’OR’ operation we get fuzzy soft set

ΓA ∨ ΓB given as:

86

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



ΓA ∨ ΓB = {(µ11, {
0.54
ω1

, 0.45
ω2

, 0
ω3

, 0.35
ω4

}), (µ12, {
0.35
ω1

, 0.47
ω2

, 0.52
ω3

, 0.35
ω4

}), (µ13, {
0.35
ω1

, 0.29
ω2

, 0
ω3

, 1
ω4

}),

(µ21, {
0.54
ω1

, 0.45
ω2

, 0.33
ω3

, 0.36
ω4

}), (µ22, {
0
ω1

, 0.47
ω2

, 0.52
ω3

, 0.36
ω4

}), (µ23, {
0
ω1

, 0.30
ω2

, 0.33
ω3

, 1
ω4

}),

(µ31, {
0.54
ω1

, 0.45
ω2

, 0
ω3

, 1
ω4

}), (µ32, {
0
ω1

, 0.47
ω2

, 0.52
ω3

, 1
ω4

}), (µ33, {
0
ω1

, 0
ω2

, 0
ω3

, 1
ω4

}),

(µ41, {
0.54
ω1

, 0.45
ω2

, 0.24
ω3

, 0.26
ω4

}), (µ42, {
0.26
ω1

, 0.47
ω2

, 0.52
ω3

, 0.26
ω4

}), (µ43, {
0.26
ω1

, 0.22
ω2

, 0.24
ω3

, 1
ω4

})}.

Now the tabular form of ΓA ∨ ΓB is written as:

ΓA ∨ ΓB µ11 µ12 µ13 µ21 µ22 µ23 µ31 µ32 µ33 µ41 µ42 µ43

ω1 0.54 0.35 0.35 0.54 0 0 0.54 0 0 0.54 0.26 0.26

ω2 0.45 0.47 0.29 0.45 0.47 0.30 0.45 0.47 0 0.45 0.47 0.22

ω3 0 0.52 0 0.33 0.52 0.33 0 0.52 0 0.24 0.52 0.24

ω4 0.35 0.35 1 0.36 0.36 1 1 1 1 0.26 0.26 1

Now we find the comparison-table of fuzzy soft set ΓA ∨ ΓB by using the algorithm which is given by Roy

and Maji in (2007). The comparison-table is given below.

ω1 ω2 ω3 ω4

ω1 12 6 6 5

ω2 6 12 6 6

ω3 6 7 12 3

ω4 9 6 9 12

Here we calculate the column-sum (ti) and row-sum (ri) after that we calculate the score (Ri) for each

ωi, i = 1, 2, 3, 4.

row-sum (ri) column-sum (ti) score (Ri = ri − ti)

ω1 29 33 -4

ω2 30 31 -1

ω3 28 33 -5

ω4 36 26 10

Table 2: Tabular form of score score (Ri = ri − ti)

From Table 2, we see that the topmost score is 10 which is gained by ω4. Which shows that neem plant is

selected to safe the fields from pests. Now problem is that where to grow the neem plants to protect the

field from pests. To solve this problem, we find the SMS boundary of soft open multi-sets.

If the SMS boundary of at least one soft open multi-sets is not a null SMSs and contains 30
ω4

in non-null

µ-approximate elements, ∀ µ ∈ S, then neem plants can grow on the corners of the field. If the SMS boundary

of all soft open multi-sets are null SMSs, then neem plants cannot grow on the corners of the field.

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Now compute the SMS boundary of Ωφ, ΩA and ΩB given as:

Ωb̂φ = Ωφ, Ω
b̂
A = Ωφ and Ω

b̂
B = ΩB∩̂Ω

c
B = ΩA∩̂Ω

c
B = Ω

c
B = {(µ1, [

30
ω4

]), (µ2, [
30
ω4

]), (µ4, H)}.

Which shows that Ωb̂B contains
30
ω4

in non-null µ-approximate elements ∀ µ ∈ S. So farmer decides to

grow neem plants on the corners of field.

The attention should be given to grow neem plants as a reassuring choice to exchange agrochemicals in

agriculture pest control. Neem can conduce to acceptable development and the determination of pest control

problems in agriculture which can be best alternative to plant fertilizer.

The proposed Algorithm 1 is used in the environment of SMSs information for the selection of best alternative

of biopesticides and the results are compared as indicated in the Table 3.

Method Ranking of alternatives The optimal alternative

Algorithm 1 (Proposed) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4

Algorithm (Çağman et al., 2011) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4

Algorithm (Riaz et al., 2019) ω4 ≻ ω2 ≻ ω1 ≻ ω3 ω4

Table 3: Comparison of final ranking with existing methods using Algorithm 1.

4.2 MCDM by using SMS-topology for the selection of best textile company

We present two modified algorithms based on SMS-topology for a decision-making problem. At the end,

we show the comparison of ranking of objects obtained by Algorithm 2 and Algorithm 3. Furthermore we

present another interesting application in agriculture for decision-making to find the optimal choice by using

SMS-topology and boundaries of soft open multi-set.

Algorithm 2 The selection of best textile company

Input:

Step 1: Consider a universe of multi-set (MS) U.

Step 2: A set E of attributes.

Step 3: Construct SMS FA and FB.

Output:

Step 4: Write SMS-topology τ̃ in which FA and FB are open SMSs in τ̃.

Step 5: Write the aggregate multi-sets of all open SMSs by using the formula, F ∗A = [
F ∗A(Ωi)

Ωi
: Ωi ∈ X],

where F ∗A(Ωi) = ΣjΩij.

Step 6: Add F ∗A and F
∗
B to find decision MS.

Step 7: Select the object with greatest multiplicity determined by max F ∗A⊕B(σ).

88

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



Start Stop

input a Multi-Set
U

input a Parameter Set
E

input Soft Msets

Choose that σ that has max F ∗A⊕B(σ)

Construct SMS

Compute F ∗A = [
F ∗A(Ωi)

Ωi
: Ωi ∈ X]

where F ∗A(Ωi) = ΣjΩij , ∀ FA ∈ τ̃

Add F ∗A and F
∗
B that is F

∗
A ⊕ F

∗
B

FA and FB

topology τ̃ s.t FA, FB ∈ τ̃

Figure 2: Graphical representation of Algorithm 2

Graphical representation of Algorithm 2 is shown in the Figure 2. Here we introduce another algorithm for

SMS-topology in decision-making.

Now we give Algorithm 3 and compare the optimal decision obtained by Algorithm 2.

Algorithm 3 The award of performance

Input:

Step 1: Consider a universe of multi-set U.

Step 2: A set E of attributes.

Step 3: Construct SMSs FA and FB.

Output:

Step 4: Write SMS-topology τ̃ containing FA and FB as open SMSs in τ̃.

Step 5: Find the cardinal MSs of all open SMSs by using the formula,

cFA = [
cFA(λi)

λi
: λi ∈ E], where cFA(λi) = ΣiΩij.

Step 6: Find the aggregate multi-sets by using the formula,
...
MF ∗

A
=

...
MFA ∗ M

t
cFA

, → (1)

where
...
MFA,

...
McFA and

...
MF ∗

A
are representation matrices of FA, cFA and F

∗
A, respectively.

Step 7: Adding F ∗A and F
∗
B to find decision mset.

Step 8: Select the object that has greatest multiplicity i.e. max F ∗A⊕B(σ).

A brief sketch of Algorithm 3 is given in the Figure 3.

Assume that government of a country is interested to give the ”award of performance” to best textile

company of country to appreciate the contribution of the company. Let U = [ 2
Ω1

, 2
Ω2

, 1
Ω3

, 1
Ω4

, 1
Ω5

, 1
Ω6

, 1
Ω7

]

be the multi-set of big textile companies of the state, and the multiplicity of Ωi, i = 1, 2, ..., 7 denotes the

89

Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



Start Stop

input a Multi-Set
U

input a Parameter Set
E

input Soft Msets

Choose that σ that has max F ∗A⊕B(σ)

Construct SMS-topology τ̃

Compute cFA = [
cFA(λi)

λi
: λi ∈ E],

where cFA(λi) = ΣiΩij., ∀ FA ∈ τ̃

Add F ∗A and F
∗
B that is F

∗
A ⊕ F

∗
B

FA and FB

Find F ∗A and F
∗
B

s.t FA, FB ∈ τ̃

Figure 3: Graphical representation of Algorithm 3

number of branches of company Ωi that are selected for the award. Let X = {Ω1, Ω2, Ω3, Ω4, Ω5, Ω6, Ω7} be

the support set of U.

The set of parameters is given as E = {λ1, λ2, λ3, λ4, λ5} where

λ1 = best hosiery,

λ2 = best export,

λ3 = healthy working environment,

λ4 = use of modern technology,

λ5 = expert workers.

We here use the following Algorithm 2 to select the best company of the state for the ”award of performance.

The DMs Ω1 and Ω2 construct two squads named as squad-Ω1 and squad-Ω2, respectively. Then they choose

two sets of attributes A = {λ1, λ2, λ3} and B = {λ1, λ2} and use them to construct soft multi-sets (SMSs)

FA and FB over U given by

FA = {(λ1, [
2
Ω1

, 2
Ω2

, 1
Ω3

, 1
Ω4

]), (λ2, [
2
Ω1

, 2
Ω2

, 1
Ω6

, 1
Ω7

]), (λ3, [
2
Ω1

, 2
Ω2

, 1
Ω5

, 1
Ω6

, 1
Ω7

])} and

FB = {(λ1, [
2
Ω1

, 2
Ω2

, 1
Ω4

]), (λ2, [
2
Ω2

, 1
Ω6

, 1
Ω7

])}.

The 1st SMS FA can be written as

90

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



FA λ1 λ2 λ3

Ω1 2 2 2

Ω2 2 2 2

Ω3 1 0 0

Ω4 1 0 0

Ω5 0 0 1

Ω6 0 1 1

Ω7 0 1 1

The 2nd SMS FB can be written as

FB λ1 λ2

Ω1 2 0

Ω2 2 2

Ω3 0 0

Ω4 1 0

Ω5 0 0

Ω6 0 1

Ω7 0 1

Now we construct a SMS-topology as

τ̃ = {Fφ, FA, FB, FẼ},

where Fφ and FẼ are empty soft and absolute soft msets, respectively.

Write aggregate multi-sets of all open SMSs given by

F ∗A = [
6
Ω1

, 6
Ω2

, 1
Ω3

, 1
Ω4

, 1
Ω5

, 2
Ω6

, 2
Ω7

],

F ∗B = [
2
Ω1

, 4
Ω2

, 1
Ω4

, 1
Ω6

, 1
Ω7

],

F ∗φ = [
0
Ω1

, 0
Ω2

, 0
Ω3

, 0
Ω4

, 0
Ω5

, 0
Ω6

, 0
Ω7

]

and F ∗
Ẽ
= [ 10

Ω1
, 10
Ω2

, 5
Ω3

, 5
Ω4

, 5
Ω5

, 5
Ω6

, 5
Ω7

].

In order to evaluate decision multi-set, The DMs added the sets F ∗A and F
∗
B.

Thus F ∗A⊕B(σ) = F
∗
A(σ) + F

∗
B(σ), ∀ σ ∈ X.

Thus F ∗A ⊕ F
∗
B = [

8
Ω1

, 10
Ω2

, 1
Ω3

, 2
Ω4

, 1
Ω5

, 3
Ω6

, 3
Ω7

].

Since max F ∗A⊕B(σ) = 10 which shows that Ω2 has the highest multiplicity, so Ω2 is chosen for the ”award

of performance”.

Next we use Algorithm 3 on the same data as above and then compare the optimal results.

The DMs Ω1 and Ω2 consider SMSs (data same as above) FA and FB over U given by

FA = {(λ1, [
2
Ω1

, 2
Ω2

, 1
Ω3

, 1
Ω4

]), (λ2, [
2
Ω1

, 2
Ω2

, 1
Ω6

, 1
Ω7

]), (λ3, [
2
Ω1

, 2
Ω2

, 1
Ω5

, 1
Ω6

, 1
Ω7

])}

and FB = {(λ1, [
2
Ω1

, 2
Ω2

, 1
Ω4

]), (λ2, [
2
Ω2

, 1
Ω6

, 1
Ω7

])}.

Again consider first SMS FA given as

91

Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



FA λ1 λ2 λ3

Ω1 2 2 2

Ω2 2 2 2

Ω3 1 0 0

Ω4 1 0 0

Ω5 0 0 1

Ω6 0 1 1

Ω7 0 1 1

Now consider second SMS FB given as

FB λ1 λ2

Ω1 2 0

Ω2 2 2

Ω3 0 0

Ω4 1 0

Ω5 0 0

Ω6 0 1

Ω7 0 1

Now we make a SMS-topology as

τ̃ = {Fφ, FA, FB, FẼ},

where Fφ and FẼ are empty soft and absolute soft msets, respectively.

Here we find the cardinal msets of all soft open msets given by

cFA = [
6
λ1
, 6
λ2
, 7
λ3
],

cFB = [
5
λ1
, 4
λ2
],

cFφ = [
0
λ1
, 0
λ2
, 0
λ3
, 0
λ4
, 0
λ5
]

and cF
Ẽ
= [ 9

λ1
, 9
λ2
, 9
λ3
, 9
λ4
, 9
λ5
].

The aggregated multi-set F ∗A is calculated by first decision maker by using (1),

...
MF ∗

A
=




2 2 2

2 2 2

1 0 0

1 0 0

0 0 1

0 1 1

0 1 1






6

6

7


 =




38

38

6

6

7

13

13




that means, F ∗A = [
38
Ω1

, 38
Ω2

, 6
Ω3

, 6
Ω4

, 7
Ω5

, 13
Ω6

, 13
Ω7

].

Furthermore, the aggregate multi-set for FB is calculated by second decision maker,

92

 
Certain properties of soft multi-set topology with applications in multi-criteria decision making 



...
MF ∗

B
=




2 0

2 2

0 0

1 0

0 0

0 1

0 1




[
5

4

]
=




10

18

0

5

0

4

4




which is, F ∗B = [
10
Ω1

, 18
Ω2

, 0
Ω3

, 5
Ω4

, 0
Ω5

, 4
Ω6

, 4
Ω7

].

Now we find the final decision multi-set by adding F ∗A and F
∗
B only.

Thus F ∗A⊕B(σ) = F
∗
A(σ) + F

∗
B(σ), ∀ σ ∈ X.

Thus F ∗A ⊕ F
∗
B = [

48
Ω1

, 56
Ω2

, 6
Ω3

, 11
Ω4

, 7
Ω5

, 17
Ω6

, 17
Ω7

].

Since max F ∗A⊕B(σ) = 56 which shows that Ω2 has the greatest multiplicity, so Ω2 is chosen for the ”award of

performance”. It is interesting to note that Algorithm 2 and Algorithm 3 provides the same optimal decision.

The proposed Algorithm 2 and Algorithm 3 are used in the environment of soft multi-sets information

systems for the award of performance and the results are compared with existing methods as indicated in

the Table 4.

Method Ranking of alternatives The optimal alternative

Algorithm 2 (Proposed) Ω2 ≻ Ω1 ≻ Ω6 = Ω7 ≻ Ω4 ≻ Ω5 ≻ Ω3 Ω2

Algorithm 3 (Proposed) Ω2 ≻ Ω1 ≻ Ω6 = Ω7 ≻ Ω4 ≻ Ω5 ≻ Ω3 Ω2

Algorithm (Çağman et al., 2011) Ω2 ≻ Ω1 ≻ Ω6 ≻ Ω7 ≻ Ω4 ≻ Ω5 ≻ Ω3 Ω2

Algorithm (Riaz et al., 2011) Ω2 ≻ Ω1 ≻ Ω6 ≻ Ω7 ≻ Ω4 ≻ Ω5 ≻ Ω3 Ω2

Table 4: Comparison of final ranking by using Algorithm 2 and Algorithm 3

The comparison analysis of final ranking determined by Algorithm 2, Algorithm 3, Çağman et al. (2011)

and Riaz et al. (2011) is also shown by multiple bar chart in the Figure 4.

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Riaz et al./Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 70-96 

 

 



Figure 4: Multiple bar chart view of final ranking

5 Conclusion

The algebraic and topological structures of soft multi-sets (SMSs) are quite different from traditional crisp

sets. Moreover the MCDM methods developed under rough sets, fuzzy sets and soft sets do not deal with

real life situations under the universe of soft multi-sets. Due to the repetition of objects in the universe of

soft multi-sets there is a need to develop novel MCDM methods. The goal of this article is deal with these

challenges and to extend the notion of SMS-topology towards MCDM problems. We initiated the idea of

SMS-topology which is defined on soft multi-sets for a fixed set of attributes. We used the idea of power

whole multi-subsets of a soft multi-set in the construction of SMS-topology. The notions of SMS-basis, SMS-

subspace, SMS-interior, soft multi-set closure and boundary of soft multi-set are introduced. Additionally,

the concept of SMS-topology is extended to develop novel multi-criteria decision-making (MCDM) methods.

To meet these objectives, Algorithm 1, Algorithm 2 and Algorithm 3 are presented for the selection of best

alternative for biopesticides, for the selection of best textile company and for the award of performance,

respectively. The aggregation operators are used to compute aggregate fuzzy soft sets and aggregate multi-

sets. Based on proposed MCDM methods some real life applications are justified by illustrative examples.

Soft multi-sets and SMS-topology have large number of applications in soft computing, decision-making,

data analysis, data mining, expert systems, information aggregation and information measures.

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