2460-0245Journal of the Indonesian Mathematical SocietyJIMS2460-02452086-8952IndoMS10.22342/jims.v32i1.1798Extended Fuzzy Binary Soft Sets and Their Applications in Multi Parameter Decision Making ProblemsPatilP. G.Indiapgpatil@kud.ac.inAngdiBhadramma M.Indiageetagma@gmail.comElluruVyshakhaIndiavyshakh2720@gmail.comRizalJosejrizal04@unib.ac.idDepartment of MathematicsKarnatak University DharwadDepartment of MathematicsJain College of Engineering and TechnologyCorresponding author: Vyshakha Elluru. Email: vyshakh2720@gmail.com0302202605012026321MARCH1163108202429112025Copyright (c) 2026 Journal of the Indonesian Mathematical Society2026Journal of the Indonesian Mathematical Societyhttps://creativecommons.org/licenses/by-nc-nd/4.0/This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Multi-criteria decision-making (MCDM) problems involve evaluating and selecting alternatives based on multiple criteria. This article aims to solve MCDM problems by extending the definition of fuzzy binary soft sets to two parameter sets, which are called extended fuzzy binary soft sets. Operations such as "AND" and "MaxMin" are defined and illustrated with examples. Additionally, an algorithm is presented to solve MCDM problems using extended fuzzy binary soft sets. Finally, an application of the proposed algorithm for decision-making is discussed.
fuzzy setsfuzzy soft setsextended fuzzy soft setsfuzzy binary soft setsextended fuzzy binary soft setsFile created by JATS Editorhttps://jatseditor.comissue-created-year20261. INTRODUCTION
Multi-criteria decision-making (MCDM) problems arise in a wide range of applications, including business, engineering, public policy and social sciences, where decision-makers need to consider various factors to arrive at an optimal decision. To handle such situations, Zadeh [1] introduced the concept of fuzzy set theory. Atanassov [2] generalized the fuzzy set theory to intuitionistic fuzzy set theory to handle more uncertainty precisely. Under these environments, researchers discussed several types of approaches to solving decision-making problems [3], [4], [5], [6], [7], [8], [9].
Since there is an absence of parametrization in these tools, Maldtsov [10] introduced the concept of soft theory. Soft sets emerged as a powerful tool for modelling uncertainty in a better way. Maji et al. [11] discussed MCDM problems using soft sets. Fuzzy sets were combined with soft sets to deal with more uncertainty and called fuzzy soft sets by Roy and Maji [12]. Many interesting applications of fuzzy soft set theory were discussed by many researchers [13], [14], [15], [16], [17], [18], [19].
Soft sets can deal with only one universal set. But as data becomes vaguer, it requires more powerful tool to deal with it. As a result, Acikgoz and Tas [20] defined binary soft sets and studied their properties. Fuzzy sets were combined with binary soft sets called the fuzzy binary soft sets by Metlida and Subhashini [21] and studied their properties. The application of fuzzy binary soft sets in MCDM problems was discussed by Patil et al. [22].
Some situations require more parameters to decide on uncertain things. To deal with such situations Anil and Patil [13] defined extended fuzzy soft sets and discussed their application in MCDM problems which deal with two parameter sets but only one universal set. Some decision-making problems involve two independent sets. The opinion of the decision makers is very crucial in making decision and there can be error while combining all the decision maker’s opinion. Hence, it is very necessary to account the set of decision makers and there is a need to relate both sets and rank them in a pair. Also, in some cases, instead of decision makers there will be another parameter set which associates with the data. To address these kind of situations, this article defines with fuzzy binary soft sets with two parameter sets and is called extended fuzzy binary soft sets to solve MCDM problems. "AND" and “MaxMin” operations on extended fuzzy binary soft sets are defined and illustrated with some examples. An algorithm to solve MCDM problems is presented and illustrated with application in deciding the college-course combination.
This article is arranged into 6 sections. In section 2, basic concepts are discussed. Section 3 deals with extended fuzzy binary soft sets and their operations with examples. Section 4 gives an application of extended fuzzy binary soft sets in MCDM problems. The result and discussion are presented in section 5. The conclusion is given in section 6.
2. Preliminaries
Definition 2.1.[1]Let X be a Universal set and A be a function defined by
Then, the set is called a fuzzy subset of X.
Definition 2.2.[10]A pair is called a soft set over a universal set U if F is a mapping of E, a set of parameters into the set of all subsets of set
Definition 2.3.[12]Let be the set of fuzzy subsets of U , a pair is called a fuzzy soft set over U , where is a mapping given by,
A fuzzy soft set (FSS) is a mapping from parameters to
Definition 2.4.[16]Resultant matrix is a square matrix in which object names universal set label rows and columns, and the entries are where is the membership value of the object and parameter.
Definition 2.5.[13]Suppose X is an initial universe and E and K are primary and secondary set of parameters. Let denote family of all fuzzy sets over X and denote family of all fuzzy soft sets over X with respect to the parameter set . For any , a pair denoted by is called extended fuzzy soft set over , where is a mapping given by
defined by for any
Definition 2.6.[13]The cartesian "AND" product of two extended fuzzy soft sets and over a common universe denoted by , is defined as and , where
Definition 2.7.[20]Let and be two universal sets. E be a set of parameters, . Let F be a function defined by
Then, the set is called Binary Soft Set over and .
Definition 2.8.[21]Let and be two universal sets, E be the set of parameters, and . Let F be a function defined by
where and are a set of all fuzzy sets of and , respectively. Then is called fuzzy binary soft set over and .
Definition 2.9.[22]Let be fuzzy binary soft set over and . Let and be expanded matrices of then the operator of M (F, A) and with respect to the parameter e is denoted by and defined by
Definition 2.10.[22]Let and be expanded matrices of fuzzy binary soft set over and . An extended resultant matrix is a resultant matrix in which rows and columns are labeled with order pair elements of and
3. Extended Fuzzy Binary Soft Sets
In this section, the definition of fuzzy binary soft sets is extended to two parameter sets and is called extended fuzzy binary soft sets.
Definition 3.1.Let and be two initial universal sets, E and P be two parameter sets. Let be set of all fuzzy binary soft sets over and with respect to the parameter set E. For any is denoted by and is called extended fuzzy binary soft set over , where is a mapping given by and defined by
Example 3.2.Let and be two initial universal sets. and be two parameter sets. Let and and are extended fuzzy binary soft sets given by ,
The tabular representation of the example 3.2 is given in Table 1.
Definition 3.3.Let and be two initial universal sets. Let E and P be two parameters sets and . Let and be two extended fuzzy binary soft sets (EFBSSs) over common universe and . Then, is said to be extended fuzzy binary soft subset if
Example 3.4.Let be two initial universal sets. Let and be two parameter sets. Let and . Let and be two EFBSSs over defined
Tabular representation of extended fuzzy binary soft set
0.2
0.3
0.6
0.4
0.6
0.4
0.7
0.8
0.5
0.4
0.8
0.7
0.7
0.4
0.8
0.3
0.4
0.5
0.6
0.5
0.7
0.6
0.6
0.4
0.2
0.4
0.3
0.7
0.4
0.2
0.6
0.7
0.2
0.4
0.4
0.5
0.4
0.5
0.6
0.5
0.4
0.8
0.7
0.8
0.6
0.4
0.8
0.7
where
Proposition 3.5.Let and be two initial universal sets. Let E and be two parameter sets and Let be FBSS and be EFBSS over common universe . Then if and only for some .
Theorem 3.6.Let be two universal sets, E and P be two parameter sets and . Let be FBSS and be EFBSS over common universe Then if and only if and .
Proof. Suppose . This implies and A contain only one element precisely p.
Since and A contain only one element,
Hence,
Conversely, If then A contains only one element, precisely p. Since (F, E) ∈ (F ∗, A) and A contain only one element
Hence
Definition 3.7.Union of two and over common universe is EFBSS and
Example 3.8.Consider the example 3.4. Then , where
Definition 3.9. Intersection of two EFBSSs and over common universe is EFBSS and
Example 3.10.Consider the example 3.4. Then , where ,
Proposition 3.11.Let and be three EFBSSs over common universe . Then
if and only if (G∗, B) ⊆ (F ∗, A)
if and only if
Definition 3.12.The cartesian product of two extended fuzzy binary soft sets and over common universal sets denoted by defined as and , where
Example 3.13.Consider the example 3.2. The AND product on and is given by,
The extended fuzzy binary soft set is given in Table 2.
Extended fuzzy binary soft set H _ { C } ^ { * }
0.2
0.3
0.3
0.4
0.2
0.3
0.3
0.4
0.2
0.2
0.6
0.4
0.2
0.2
0.6
0.4
0.2
0.3
0.4
0.4
0.2
0.3
0.6
0.4
0.2
0.4
0.3
0.7
0.4
0.4
0.6
0.5
0.4
0.2
0.6
0.7
0.4
0.4
0.7
0.8
0.2
0.4
0.4
0.5
0.6
0.4
0.7
0.7
0.2
0.4
0.3
0.7
0.4
0.4
0.6
0.5
0.4
0.2
0.6
0.7
0.4
0.4
0.7
0.7
0.2
0.4
0.4
0.5
0.5
0.4
0.8
0.7
0.2
0.4
0.3
0.3
0.4
0.4
0.6
0.3
0.4
0.2
0.6
0.3
0.4
0.4
0.7
0.3
0.2
0.4
0.4
0.3
0.6
0.4
0.8
0.3
0.2
0.4
0.3
0.5
0.4
0.5
0.6
0.5
0.4
0.2
0.6
0.5
0.4
0.5
0.6
0.5
0.2
0.4
0.4
0.5
0.4
0.4
0.6
0.5
0.2
0.4
0.3
0.4
0.4
0.5
0.5
0.4
0.4
0.2
0.5
0.4
0.4
0.6
0.5
0.4
0.2
0.4
0.4
0.4
0.6
0.4
0.5
0.4
Definition 3.14.The cartesian "OR product of two extended fuzzy binary soft sets and over common universal sets denoted by defined as and , where
Example 3.15.Consider the example 3.2. The AND product on and is given by,
The extended fuzzy binary soft set is given in Table 3.
Extended fuzzy binary soft set K _ { D } ^ { * }
0.2
0.4
0.6
0.7
0.4
0.5
0.6
0.5
0.4
0.3
0.6
0.7
0.4
0.8
0.7
0.8
0.2
0.4
0.6
0.5
0.6
0.4
0.8
0.7
0.6
0.4
0.7
0.8
0.6
0.5
0.7
0.8
0.6
0.4
0.7
0.8
0.6
0.8
0.7
0.8
0.6
0.4
0.7
0.8
0.6
0.4
0.7
0.8
0.5
0.4
0.8
0.7
0.5
0.5
0.8
0.7
0.5
0.4
0.8
0.7
0.5
0.8
0.8
0.8
0.5
0.4
0.8
0.7
0.6
0.4
0.8
0.7
0.7
0.4
0.8
0.7
0.7
0.5
0.8
0.5
0.7
0.4
0.8
0.7
0.7
0.8
0.8
0.8
0.7
0.4
0.8
0.5
0.7
0.4
0.8
0.7
0.4
0.5
0.6
0.7
0.4
0.5
0.6
0.5
0.4
0.5
0.6
0.7
0.4
0.8
0.7
0.8
0.4
0.5
0.6
0.5
0.6
0.5
0.8
0.7
0.7
0.6
0.5
0.7
0.7
0.6
0.6
0.5
0.7
0.6
0.6
0.7
0.7
0.8
0.7
0.8
0.7
0.6
0.5
0.5
0.7
0.4
0.8
0.7
Definition 3.16. The MaxMin operator on products of two extended fuzzy binary soft sets and is given by
Example 3.17.Consider the example 3.13. The MaxMin operator is given
The extended fuzzy binary soft set is given in Table 4.
The extended fuzzy binary soft set is given in Table 5.
Extended fuzzy binary soft set F _ { C }
0.2
0.4
0.3
0.4
0.4
0.2
0.6
0.4
0.2
0.2
0.4
0.4
0.2
0.4
0.3
0.5
0.4
0.2
0.6
0.7
0.2
0.4
0.4
0.5
0.2
0.4
0.3
0.5
0.4
0.2
0.6
0.7
0.2
0.4
0.4
0.5
Extended fuzzy binary soft set F _ { D }
0.2
0.3
0.3
0.3
0.2
0.2
0.6
0.3
0.2
0.3
0.6
0.3
0.4
0.4
0.6
0.5
0.4
0.4
0.6
0.5
0.4
0.4
0.6
0.5
0.4
0.4
0.5
0.4
0.4
0.4
0.5
0.4
0.5
0.4
0.5
0.4
4. Application in Decision Making Problems
Suppose Mr. V wants to choose the best course in the best college. For this, he has a choice of four colleges and four courses. Let be the set of colleges and be the set of courses to be selected with respect to some parameters . For this, he has hired two pairs of experts say and . Consider the parameter set as experts, . The experts assigned the scores between 0 to 100 given Table 6. The data is converted on the scale of 0 to 1. This section provides an algorithm to solve MCDM problem and illustrated with example.
The scores assigned by experts
94
94
87
86
66
74
58
72
37
50
31
38
40
100
96
96
73
89
57
56
49
95
54
63
96
78
73
41
36
85
56
43
95
57
32
73
78
43
91
93
70
60
91
74
90
48
38
63
69
76
34
93
60
70
72
96
66
100
86
56
56
67
51
94
77
80
37
44
88
61
72
60
65
71
85
59
74
53
42
99
91
48
63
92
47
83
87
52
39
56
76
68
95
66
81
90
38
67
57
82
45
93
58
73
64
86
41
96
46
89
62
50
84
75
55
43
94
40
69
98
51
79
97
78
84
69
49
70
4.1. Algorithm to solve MCDM problems:
Since, the problems involves two universal sets and more than one parameter set, there is a need of systematic method to combine all the alternative regarding to each parameter to make a decision. In this subsection, an algorithm to solve the MCDM problem is defined and its implication is discussed.
Step 1: Input extended fuzzy binary soft sets and
Step 2: Apply “AND” operator between and to obtain .
Step 3: Apply “AND” operator on expanded matrices of extended fuzzy binary soft set to obtain
Step 4: Apply MaxMin operator on to get two extended fuzzy binary soft sets and
Step 5: Apply MaxMin operator between and to get four fuzzy sets.
Step 6: Find the extended resultant matrix.
Step 7: Find row sum of all the rows.
Step 8: Highest row sum is given rank 1.
Implementation of algorithm
Step 1: Based on the scores of the experts (Table 6), let and be extended fuzzy binary soft sets given by
which are shown Table 7.
Applying step 2 to step 6, the extended resultant matrix is obtained which is given in the Tabel 8.
Step 7: Calculate the row sum (Table 9.
Step 8: rank them accordingly.
Extended fuzzy binary soft sets F _ { A } ^ { * } and F _ { B } ^ { * }
0.37
0.50
0.31
0.38
0.40
1
0.96
0.96
0.73
0.89
0.57
0.56
0.49
0.95
0.54
0.63
0.96
0.78
0.73
0.41
0.36
0.85
0.56
0.43
0.77
0.80
0.37
0.44
0.88
0.61
0.72
0.60
0.65
0.71
0.85
0.59
0.74
0.53
0.42
0.99
0.91
0.48
0.63
0.92
0.47
0.83
0.87
0.52
0.39
0.56
0.76
0.68
0.95
0.66
0.81
0.90
0.95
0.57
0.32
0.73
0.78
0.43
0.91
0.93
0.70
0.60
0.91
0.74
0.90
0.48
0.38
0.63
0.69
0.76
0.34
0.93
0.60
0.70
0.72
0.96
0.66
1
0.86
0.56
0.56
0.67
0.51
0.94
0.38
0.67
0.57
0.82
0.45
0.93
0.58
0.73
0.64
0.86
0.41
0.96
0.46
0.89
0.62
0.50
0.84
0.75
0.55
0.43
0.94
0.40
0.69
0.98
0.51
0.79
0.97
0.78
0.84
0.69
0.49
0.70
Extended resultant matrix:
(u₁,v₁)
(u₁,v₂)
(u₁,v₃)
(u₁,v₄)
(u₂,v₁)
(u₂,v₂)
(u₂,v₃)
(u₂,v₄)
(u₃,v₁)
(u₃,v₂)
(u₃,v₃)
(u₃,v₄)
(u₄,v₁)
(u₄,v₂)
(u₄,v₃)
(u₄,v₄)
(u₁,v₁)
0
-0.02
0.10
-0.09
-0.22
-0.09
-0.14
-0.44
0.19
0.15
0.25
0.06
-0.08
-0.10
-0.04
-0.17
(u₁,v₂)
0.02
0
0.12
-0.07
-0.20
-0.07
-0.12
-0.42
0.21
0.17
0.27
0.08
-0.06
-0.08
-0.02
-0.15
(u₁,v₃)
-0.10
-0.12
0
-0.19
-0.32
-0.19
-0.24
-0.54
0.09
0.05
0.15
-0.04
-0.18
-0.20
-0.14
-0.27
(u₁,v₄)
0.09
0.07
0.19
0
-0.13
0
-0.05
-0.35
0.28
0.24
0.34
0.15
0.01
-0.01
0.05
-0.08
(u₂,v₁)
0.22
0.20
0.32
0.13
0
0.13
0.08
-0.22
0.41
0.37
0.47
0.28
0.14
0.12
0.18
0.05
(u₂,v₂)
0.09
0.07
0.19
0
-0.13
0
-0.05
-0.35
0.28
0.24
0.34
0.15
0.01
-0.01
0.05
-0.08
(u₂,v₃)
0.14
0.12
0.24
0.05
-0.08
0.05
0
-0.30
0.33
0.29
0.39
0.20
0.06
0.04
0.10
-0.03
(u₂,v₄)
0.44
0.42
0.54
0.35
0.22
0.35
0.30
0
0.63
0.59
0.69
0.50
0.36
0.34
0.40
0.27
(u₃,v₁)
-0.19
-0.21
-0.09
-0.28
-0.41
-0.28
-0.33
-0.63
0
-0.04
0.06
-0.13
-0.27
-0.29
-0.23
-0.36
(u₃,v₂)
-0.15
-0.17
-0.05
-0.24
-0.37
-0.24
-0.29
-0.59
0.04
0
0.10
-0.09
-0.23
-0.25
-0.19
-0.32
(u₃,v₃)
-0.25
-0.27
-0.15
-0.34
-0.47
-0.34
-0.39
-0.69
-0.06
-0.10
0
-0.19
-0.33
-0.35
-0.29
-0.42
(u₃,v₄)
-0.06
-0.08
0.04
-0.15
-0.28
-0.15
-0.20
-0.50
0.13
0.09
0.19
0
-0.14
-0.16
-0.10
-0.23
(u₄,v₁)
0.08
0.06
0.18
-0.01
-0.14
-0.01
-0.06
-0.36
0.27
0.23
0.33
0.14
0
-0.02
0.04
-0.09
(u₄,v₂)
0.10
0.08
0.20
0.01
-0.12
0.01
-0.04
-0.34
0.29
0.25
0.35
0.16
0.02
0
0.06
-0.07
(u₄,v₃)
0.04
0.02
0.14
-0.05
-0.18
-0.05
-0.10
-0.40
0.23
0.19
0.29
0.10
-0.04
-0.06
0
-0.13
(u₄,v₄)
0.17
0.15
0.27
0.08
-0.05
0.08
0.03
-0.27
0.36
0.32
0.42
0.23
0.09
0.07
0.13
0
Row sum of extended resultant matrix
College-Course Pair
Row sum
College-Course Pair
Row sum
-0.64
-3.68
-0.32
-3.04
-2.24
-4.64
0.8
-1.6
2.88
0.64
0.8
0.96
1.6
0
6.4
2.08
5. Result and Discussion
The performance of the algorithm is illustrated with an example of ranking college-course combination based on the reports by experts (considered as set of second parameters) considering four parameters. The ranking strategy is based on the value of row sum as given in Table 9 The pair with highest row sum is ranked 1. For the example under discussion, the pair got highest score with value 6.4, indicating that college has the best performance in course . Also, has highest score across multiple courses and . This shows, for any course, the college is the best choice. Further, is the best course in all the colleges. Hence, college is the most preferred college in overall with strong performance across multiple courses, where as appears to be least preferred. Similarly, is the most preferred course and is the least preferred course. The diferences in the scores between the college-course combination suggests that students may have diferent preferences and priorities when choosing a college and course. Some may prioritize specific course, while others may focus on the overall performance of the college. This article provide the solution for all the situations.
6. Conclusion
This article aims to solve MCDM problem involving multiparameter sets by extending the definition of fuzzy binary soft sets for two parameter sets. operator and MaxMin operators are defined on extended fuzzy binary soft sets. An algorithm to solve decision making problems was presented. The algorithm is illustrated with an example, providing the solution to choosing the college-course combination. This MCDM approach provides a quantitative way to evaluate and compare the performance of college-course combinations. This approach can be extended for more than two parameter sets and can be improved by considering additional factors, such as cost, availability and some personal preference while making final decision.
Acknowledgement.
The authors thank the anonymous reviewers for their valuable suggestions that helped improve the article.
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