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Abstract
This paper is concerned with the reduction of soft sets and fuzzy soft sets. Firstly, the problems of suboptimal choice and added parameter set of soft sets are analyzed. Then, we introduce the definition of normal parameter reduction in soft sets to overcome these problems. In addition, a heuristic algorithm of normal parameter reduction is presented. Two new definitions, parameter important degree and decision partition, are proposed for analyzing the algorithm of normal parameter reduction. Furthermore, the normal parameter reduction is also investigated in fuzzy soft sets.
... Selanjutnya diperkenalkan konsep normal parameter reduction (NPR) oleh Kong dkk. [3] dan suatu algoritma NPR juga disajikan di dalamnya. Namun algoritma ini sulit dipahami dan melibatkan banyak perhitungan, karena menggunakan perhitungan terhadap nilai derajat kepentingan parameternya. ...
... Oleh karena itu, pada penelitian ini akan dikaji kembali tentang reduksi parameter pada soft set yang terdapat pada [3] dan [5]. Kajian ini cukup menarik karena terdapat beberapa perbedaan antara [3] dan [5]. ...
... Oleh karena itu, pada penelitian ini akan dikaji kembali tentang reduksi parameter pada soft set yang terdapat pada [3] dan [5]. Kajian ini cukup menarik karena terdapat beberapa perbedaan antara [3] dan [5]. ...
Soft set merupakan salah satu teori matematika yang mengkaji tentang ketidakpastian dalam suatu pengambilan keputusan dalam kehidupan sehari-hari.Di sisi lain, sering kali terjadi pengambilan keputusan dengan melibatkan data yangbanyak. Terdapat kemungkinan beberapa data tersebut ada yang dapat diabaikan tanpa mengubah keputusan awalnya. Usaha untuk mengabaikan beberapa informasi yang berlebihan tanpa mengubah hasil keputusan awalnya disebut dengan reduksi parameter. Pada penelitian ini, dikaji kembali konsep reduksi parameter oleh Kong, dkk tahun 2008 yang menggunakan suatu metode dimana derajat kepentingan digunakan sebagai acuan untuk mereduksi parameter dalam pengambilan keputusan. Selanjutnya metode ini akan dibandingkan dengan algoritma alternatif yang mana proses reduksinya cukup dengan menghitung kelipatan dari banyaknya objek dalam pengambilan keputusan.
... In this section, we analyse the method of normal parameter reductions and their algorithms proposed by Kong et al., [33] and Ma et al., [34]. ...
... (See [33]) Given a S-Set (F, E), with E = {e 1 , e 2 , ..., e m }, and U ={h 1 , h 2 , ..., h n }, then the decision partition as given in equation 3.1 and the decision partition deleted e i are respectively given as ...
... Using the parameter importance degree the authors in [33] presented the algorithm for parameter reduction as in Fig 1. (1) Input the S-Set (F, E) and its parameter set E ; ...
Soft set has been introduced to deal with uncertainty involved in many real life problems. However , most of the time, these decision-making problems involve less important and redundant parameters, which make the decision making process more complex and challenging.Therefore, in this study the concept of reduct of a soft set is discussed and a new algorithm is developed for normal parameter reduction (NPR) base on the unit similarity matrix. Finally, the propose algorithm is compared with previous parameter reduction algorithms in terms of computational complexity.
... However, in this case, we may also need a new reduction, as the new parameters may change the decision order of decision-making problems. To overcome these two drawbacks, Kong et al. [25] introduced the concept of normal parameter reduction (NPR) of (fuzzy) soft sets and presented its heuristic algorithm. NPR can reduce the number of parameters without changing the entire ranking (or decision) order of decision alternatives. ...
... NPR can reduce the number of parameters without changing the entire ranking (or decision) order of decision alternatives. However, the algorithm of NPR, as proposed in [25], was based on the parameter importance degree, which was hard to compute and involved a great amount of computation. Therefore, Ma et al. [26] proposed the new efficient normal parameter reduction algorithm (NENPR) for soft sets to reduce the computational complexity of the NPR process. ...
... • We propose a new algorithm for NPR using σ-algebraic soft sets that not only overcomes the existing problems of IRSSs method, but also makes the reduction process more simple and convenient. • We provide a comparative study to show that the proposed algorithm has less computational complexity and workload as compared to the previous algorithm of Kong et al. [25]. • We present an application of the proposed algorithm in a real-life decision-making problem. ...
The soft set is one of the key mathematical tools for uncertainty description and has many applications in real-world decision-making problems. However, most of the time, these decision-making problems involve less important and redundant parameters, which make the decision making process more complex and challenging. Parameter reduction is a useful approach to eliminate such irrelevant and redundant parameters during soft set-based decision-making problems without changing their decision abilities. Among the various reduction methods of soft sets, normal parameter reduction (NPR) can reduce decision-making problems without changing the decision order of alternatives. This paper mainly develops a new algorithm for NPR using the concept of σ-algebraic soft sets. Before this, the same concept was used to introduce the idea of intersectional reduced soft sets (IRSSs). However, this study clarifies that the method of IRSSs does not maintain the decision order of alternatives. Thus, we need to develop a new approach that not only keeps the decision order invariant but also makes the reduction process more simple and convenient. For this reason, we propose a new algorithm for NPR using σ-algebraic soft sets that not only overcome the existing problems of IRSSs method but also reduce the computational complexity of the NPR process. We also compare our proposed algorithm with one of the existing algorithms of the NPR in terms of computational complexity. It is evident from the experimental results that the proposed algorithm has greatly reduced the computational complexity and workload in comparison with the existing algorithm. At the end of the paper, an application of the proposed algorithm is explored by a real-world decision-making problem.
... To solve the reduction issue, Kong et al. [20] defined and developed the heuristic technique for normal parameter reductions (NPRs) in FSS. The NPR soft set algorithm, as proposed in [20], was complex to grasp, required numerous computations, and depended on the dispensability. ...
... To solve the reduction issue, Kong et al. [20] defined and developed the heuristic technique for normal parameter reductions (NPRs) in FSS. The NPR soft set algorithm, as proposed in [20], was complex to grasp, required numerous computations, and depended on the dispensability. To lessen its computational complexity, this approach was further investigated by several authors; see [21][22][23][24]. ...
Issues in daily life, where making the best decisions is crucial, are frequently encountered. But, in the majority of these situations, the best course of action is uncertain. We must take into account a number of parameters in order to find the best possible solution to these difficulties. The best mathematical instrument for this is fuzzy soft set FSS theory in decision making. Nutrition is the process of supplying cells and organisms with the nutrients they need to grow and thrive and to sustain life. A healthy diet has the potential to prevent or mitigate numerous prevalent health issues. The purpose of this paper is to select a burning problem for the nutrition of students and successfully apply the FSS theory in decision making. We aim to prove that the approach to decision-making problems with imprecise data via FSSs is more accurate than other types of approaches, and we present a new approach to the FSS model and its applications in decision-making problems.
... Ali [1] studied another point of view on parameter reduction in soft sets. Kong et al. [13] first proposed the idea of normal parameter reduction of soft sets in [10], which was intended to address the problem of suboptimal selection. However, the idea is too abstract and the procedure is difficult to understand and takes a long time. ...
... However, the idea is too abstract and the procedure is difficult to understand and takes a long time. An improved approach is provided in [13], while Ma et al. [14] studied the normal parameter reduction. Xie [42] investigated parameter reduction by attribute reduction in information systems. ...
For a mathematical model to describe vague (uncertain) problems effectively, it must have the ability to explain the links between the objects and parameters in the problem in the most precise way. There is no suitable model that can handle such scenarios in the literature. This deficiency serves as motivation for this study. In this article, the bipolar hypersoft set (abbreviated, BHSS) is considered since the parameters and their opposite play a symmetrical role. We present a novel theoretical technique for solving decision-making problems using BHSS and investigate parameter reductions for these sets. Algorithms for parameter reduction are provided and explained with examples. The findings demonstrate that our suggested parameter reduction strategies remove unnecessary parameters and still retain the same decision-making options.
... The combination of an interval-valued fuzzy set and a soft set was introduced by Yang et al. [29]. In fuzzy soft-sets, Kong et al. [30] developed the typical parameter reduction and demonstrated that Roy and Maji's [22] technique is not practical in most situations. Çaǧman and Kartaş [31] introduced the concept of intuitionistic fuzzy soft set and successfully applied it in decision-making problems. ...
... if q IFSEs ( 1 ) ⊂ q IFSEs ( 2 ), then (30) contradicts that (q IFSEs , S) is a strictly convex IFSEs. Now if we take q IFSEs ( 1 ) = q IFSEs ( 2 ) and δ ∈ [0, α], ...
A new area of research called intuitionistic fuzzy soft expert sets is expected to overcome the drawbacks of intuitionistic fuzzy soft sets in terms of eligibility for soft expert-argument approximate functions. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, (, v)-convexity (concave) sets, strongly (, v)-convexity (concave) sets, strictly (, v)-convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and result on the basis of these specified notions. Additionally, by using a theoretic-based analytical approach, various elements of computational geometry, such as the convex hull and convex cone, are theorized, and some pertinent results are generalized.
... The characterization devaluation of soft sets analyzed by Chen at al. [10]. The main objective of Kong at al. [16,17,18] is to show a devaluation of soft sets and fuzzy soft sets. They also explained algorithm of normal parameter reduction. ...
... et al.[18] said that Roy's innovation has error, Kong gave correct innovation, in it they compare choice values of different objects.Rule 4.1:Take the set of interval-valued fuzzy soft sets. (Ĩv f 1 , Ω 1 ) and (Ĩv f 2 , Ω 2 ) demonstrate in Tables 3.1 and 3.2 are under deliberation.Here, we obtain the value of sets (Ĩv f 1 , Ω 1 ) and (Ĩv f 2 , Ω 2 ) from Tables 3.1 and 3.2. ...
Molodtsov was a father of soft set approach. We can’t easily settle the membership degree in some practical application. So it must be much better to describe interval-valued data instead of explaining membership degree. In this paper, we introduce the latest approach of the interval-valued fuzzy soft set by combining the interval-valued fuzzy set and soft set models. This approach successfully follows distributive, associative and DeMorgan’s laws as well. In the end, a decision problem is solved by this approach.
... [36] presented another view of parameterization reduction using soft sets and its applications to improve the reduction model proposed by [35]. Subsequently, [37] presented the idea of the normal parameter reduction of soft sets and its algorithm to improve the work by [36] in maintaining the consistency of all decisions across all levels. Then, [38] further improved the work by [37] by proposing a new efficient normal parameter reduction algorithm of soft sets and reported a better computational time performance than the previous works. ...
... Subsequently, [37] presented the idea of the normal parameter reduction of soft sets and its algorithm to improve the work by [36] in maintaining the consistency of all decisions across all levels. Then, [38] further improved the work by [37] by proposing a new efficient normal parameter reduction algorithm of soft sets and reported a better computational time performance than the previous works. Finally, [39] redefined soft set operations and constructed a UniInt decision-making model. ...
Conflict situations in football have become a significant issue because they affect the players, supporters, referees, management team, the governing body of football, and the government. As time passes, the discovery of conflicts within the football industry has also become increasingly diverse; both affected in-game or out of the game. In 2015, Indonesia had no activity in football for almost a year when the International Federation of Association Football (FIFA) banned the Football Association of Indonesia (PSSI) from competing in international competitions until the conflict among their internal agents was resolved. The agents involved in this highly controversial ban include the Commission 10 of the Parliament of Indonesia, the National Sports Committee of Indonesia (KONI), the Indonesian President, and the Ministry of Youth and Sport of Indonesia. Conflict resolution strategies outside the football games are delicate and more challenging to overcome due to the involvement of the government and various governing bodies. This opens to higher unpredictability in modeling the conflict situations, hence a lower possibility of a successful conflict resolution model strategy. In addressing this gap, this paper proposes a new Computational Intelligence approach based on the Soft Set Theory, where an alternative algorithm is derived from modeling the conflict situations. We then delineated the proposed algorithm for an instructional example of the Indonesian football conflict situation in 2015 concerning the Indonesia Football Super League. The results showed that the proposed algorithm successfully handled conflict and recommended the Indonesian football agents involved, including PSSI and FIFA.
... Another important direction for soft set study is the parameter reduction problem of soft sets. Based on different conditions, several kinds of parameter reduction problems of soft sets or fuzzy soft sets [31,32,[34][35][36][37][38][39][40][41][42][43][44] have been defined and discussed. The parameter reduction problems of soft sets actually deal with a kind of important structure for the parameter domain. ...
... According to [38], we have the following concept about normal parameter reduction of soft sets. Note that we don't require the minimality condition for normal parameter reductions as defined in [32]. ...
This paper aims to propose a novel algorithm for computing all normal parameter reductions of the soft set (# NPRS for short). Firstly, a weight vector is assigned to objects of the soft set domain. Then, a necessary condition for a normal parameter direction can be derived. A parameter subset is a solution only if the total value of the weighted sum of corresponding parameter approximations is a multiple of a constant number, which is equal to the sum of weights. Based on this necessary condition, we can figure out all potential solutions by using integer partition technique. It needs only to screen out the right ones at last. Experimental results are listed and compared when weight vectors are UNA, BIN, TER, OCT, DEC, DUO and HEX. Comparison results show that our method has a better performance for solving # NPRS.
... Maji et al. [29] proposed a reduction of SS using the rough set approach. Kong et al. [30] incorporated new parameters to SS theory to account for suboptimal decision making. Ma and Qin [31] described parameter value reduction to maintain full decision-making flexibility while simultaneously reducing computational effort and increasing reduction success rates. ...
... Parameter reduction can remove unnecessarily existing parameters in the parameter set so that the remaining indispensable parameters in the parameter set maintain the same descriptive or decision-making ability as the original complete parameter set, e.g., normal parameter reduction [38], which is an adjustable parameter reduction approach for fuzzy soft sets based on the three-way decision [39]. Han et al. [40] transformed the soft-set parametric reduction problem into a 0-1 linear programming problem and developed a model for the pseudo-parametric reduction problem of soft sets, which was more efficient with a larger number of parameters. ...
Bio-inspired thin-wall structures with excellent mechanical properties, high-energy absorption capabilities, and a desirable lightweight level have been extensively applied to the passive safety protection of transportation and aerospace. Collaboration matching and the selection of optional structures with different bionic principles considering the multiple attribute evaluation index and engineering preference information have become an urgent problem. This paper proposes a parameter reduction-based indifference threshold-based attribute ratio analysis method under an interval-valued neutrosophic soft set (IVNS-SOFT) to obtain the weight vector of an evaluation indicator system for the selection of bionic thin-wall structures, which can avoid the problem of an inadequate subjective evaluation and reduce redundant parameters. An IVNS-SOFT-based multi-attributive border approximation area comparison (MABAC) method is proposed to obtain an optimal alternative, which can quantify uncertainty explicitly and handle the uncertain and inconsistent information prevalent in the expert system. Subsequently, an application of five bio-inspired thin-wall structures is applied to demonstrate that this proposed method is valid and practical. Comparative analysis, sensitivity analysis, and discussion are conducted in this research. The results show that this study provides an effective tool for the selection of bionic thin-wall structures.
... In particular, Maji et al. [23,24] studied several operations on soft sets and applied their findings to decision-making problems in the literature. Several writers, including Chen [6], Pei and Miao [32], Zou and Xiao [47], and Kong et al. [22], have discovered significant characteristics of soft sets. Soft semirings, soft ideals, and idealistic soft semirings were all investigated by Feng et al. [17]. ...
... Research on including all above mentioned hybrid notions has been active recently, and significant advancements have been made including the use of fundamental theory [11], theory in abstract algebra [12], and for data analysis [13] and especially in decision-making [14]. Aktas and Cagman [15] started the use of in algebra. ...
The release of harmful materials into the environment is called pollution and the harmful materials are called pollutants. There are four basic categories of pollution: land, water, noise, and air pollution. All forms of pollution often have severe consequences on human health as well as the environment and wildlife. There are certain decision-making scenarios like the phenomenon of voting where we have to utilize the third grade called abstinence grade along with membership grade and non-membership grade. Many remarkable fuzzy structures like the intuitionistic fuzzy set, Pythagorean fuzzy set and q-rung orthopair fuzzy set can never discuss abstinence grades that show their flaws. Moreover, we can observe that the parametrization tool is a remarkable instrument used in soft set theory and all above-mentioned structures fail to cover the parametrization as well. Moreover, Einstein operations comprise Einstein product and Einstein sum, which serve as excellent substitutes for algebraic product and algebraic sum. So keeping in view the characteristics of the parametrization tool, the more advanced structure of the picture fuzzy soft set and Einstein operational rules, in this article, we have established Einstein operational laws for picture fuzzy soft numbers. Moreover, we have elaborated the basic notion of Einstein-weighted average operators and Einstein-weighted geometric aggregation operators. Furthermore, we have discussed the basic properties of these introduced notions. Moreover, we have discussed the algorithm for the application of these aggregation operators in the identification of types of pollution that mostly affect the environment. We have provided a comparison of these introduced works for the superiority of these introduced conceptions.
... Kovkov et al. (2007) used optimization problems to reflect the idea of soft sets and also concentrated on validating the idea of the estimated description of objects. The clue for a standard decrease of parameters in soft sets is taken into account to find solutions to the problems of suboptimal choice and supplemented parameter sets of soft sets (Kong et al., 2008). ...
Parameter reduction without performance degradation is a promising task in decision-making problems. For instance, a great challenge exists in constructing cost functions in gaming theory. Nevertheless, soft set theory handles all its drawbacks conveniently through a new tool for the choice function mathematically. In this paper, we propose an algorithm (SSPRDM) for parameter reduction of soft sets through discernibility matrices, and it is implemented in detecting the risk factor of heart disease problems by using six types of machine learning techniques. The parameters are extracted from the heart disease patient data by the SSPRDM algorithm, and then six machine learning techniques (LDA, KNN, SVM, CART, NB, RF) are performed in the prediction of risk factors for heart disease. The experimental results showed that the present hybrid approach provides an accuracy of 88.46% in the Random Forest technique, whereas the same Random Forest classifier provides an accuracy of 69.23% in the prediction of risk factors of cardiovascular disease (CVD) diagnosis in the earlier work which is a drastic improvement. Moreover, out of 18 parameter reductions, the core component is identified as Total Cholesterol, which is to be considered in all types of CVD diagnosis, whereas Sugar-Fasting (C), Total-Cholesterol (G), and HDL-Cholesterol (I) are the core components identified in three parameter reductions ABCEGHI, ACFGIJ, and BCFGIJK.
... Later, Maji et al, (2001) created the fuzzy structure of the soft set by merging the fuzzy set and the soft set. In order to solve decision-making problems, Kong et al, (2008) Kong et al, (2009) used the soft set theoretic technique. Majumdar and Samanta, (2011), have investigated the problem of soft and fuzzy soft sets in terms of similarity measure. ...
Green Supply Chain Management (GSCM) is essential to ensure environmental compliance and commercial growth in the current climate. Businesses constantly look for fresh concepts and techniques for ensuring environmental sustainability. To keep up with the new trends in environmental concerns related to company management and procedures, Green Supplier Selection (GSS) criteria are added to the traditional supplier selection processes. This study aims to identify general and environmental supplier selection criteria to provide a framework that can assist decision-makers in choosing and prioritizing appropriate green supplier selection. The development and implementation of decision support systems aimed to solve these difficulties at a rapid rate. In order to manage inaccurate data and simulate decision-making problems. Fuzzy sets introduced by Zadeh, are a useful technique to handle the imperfectness and uncertainty in different problems. Although fuzzy sets can handle incomplete information in different real worlds problems, but its cannot handle all type of uncertainty such as incomplete and indeterminate data. Therefore different extensions of fuzzy sets such as intuitionistic fuzzy, pythagorean fuzzy and q-rung orthopair fuzzy sets introduced to address the problems of uncertainty by considering the membership and non-membership grade. However, these concepts have some shortcomings in the handling uncertainty with sub-attributes. To overcome this difficulties Khan et al. developed the structure of q-rung orthopair fuzzy hypersoft sets by combining q-rung orthopair fuzzy sets with hypersoft sets. A remarkable and beneficial research work is done in the field of q-rung orthopair fuzzy hypersoft sets, and then we think about the application. In this paper, we use the structure of q-rung orthopair fuzzy hypersoft in multi-criteria supplier selection problems. For this, we present aggregation operator to solve multi-criteria decision-making (MCDM) problems with q-rung orthopair fuzzy hypersoft (q-ROFH) information, known as ordered weighted geometric aggregation operator. Since the uncertainty and vagueness is an unavoidable feature of multi-criteria decision-making problems, the proposed structure can be a useful tool for decision making in an uncertain environment. Further, the expert opinions were investigated using the multi-criteria decision-making (MCDM) technique, which helped identify interrelationship and causal preference of green supplier evaluation aspects that used aggregation operators. Finally, a numerical example of the proposed method for the task of Green Supplier Selection is presented.
... (Maji, P. K. et al., 2001 and have done a theoretical study on the soft set theory in more detail and contributed towards the fuzzification of the notion of it and described the application of soft set theory to a decision making problem using rough sets. Recently (Kong, Z. et al., 2008 and) applied the soft set theoretic approach in decision making problems. Soft fuzzy set was studied by (Yao et al., 2008) followed by intuitionistic fuzzy soft set defined by (Xu Yong et al., 2010). ...
The soft set theory plays a key role for dealing with uncertainty, fuzziness and vagueness. The concept of fuzzy soft set which can be seen as a new mathematical approach to vagueness is used in many applications including reliability evaluation, multi criteria decision making and medical diagnosis problems. Later, it is generalized to an interval-valued intuitionistic fuzzy soft set. In this attempt we present the definition and operations of an interval-valued intuitionistic fuzzy soft set. Furthermore, based on the analysis of several operations on interval-valued intuitionistic fuzzy soft set in the study, we provide some notions such as the restricted intersection and restricted union of two interval-valued intuitionistic fuzzy soft sets for selection of appropriate hospital for patient affected by specific dieses.
... Parameter reduction is one of important research issues involving applications of these tools that deal with uncertainty [37,38]. Kong et al. [39] first proposed the normal parameter reduction of fuzzy soft set theory. Ma et al. [40] proposed an efficient distance-based parameter-reduction algorithm for this model. ...
A fuzzy soft set is a mathematical tool used to deal with vagueness and uncertainty. Parameter reduction is an important issue when applying a fuzzy soft set to handle decision making. However, existing methods neglect newly added parameters and have higher computational complexities. In this paper, we propose a new S-Score table-based parameter-reduction approach for fuzzy soft sets. Compared with two existing methods of parameter reduction for a fuzzy soft set, our method takes newly added parameters into account, which brings about greater flexibility and is beneficial to the extension of fuzzy soft sets and a combination of multiple fuzzy soft sets. Additionally, our method accesses fewer elements from the dataset, which results in lower computation compared with the two existing approaches. The experimental results from two applications show the availability and feasibility of our approach.
... Multiple assessment bases were supported by the findings, which are more easily accepted and reasonable in one's mind. Kong et al. [8] looked into the issue of suboptimal choice and added a few soft settings to the parameter set. Akram et al. [9] investigated a variety of situations, ranging from rough to soft. ...
A novel approach of perceiving from imperfect multi-observer data is provided in this study. During a parametric sense, the method includes constructing a comparison table for higher cognitive processes from an FSS. The notion of an FSS with Grey relational analysis is backed by a novel method. The new algorithm's evaluation grounds are diverse. The findings demonstrate that the proposed method is effective in addressing choice issues, particularly FSS decision problems.KeywordsObject identificationComparison tableDecision-making problemGrey theoryChoice-value
... In particular, Maji et al. [20,21] studied several operations on soft sets and applied their findings to decision-making problems in the literature. Several writers, including Chen [22], Pei and Miao [23], Zou and Xiao [24], and Kong et al. [25], have discovered significant characteristics of soft sets. Soft semirings, soft ideals, and idealistic soft semirings were all investigated by Feng et al. [26]. ...
We developed the operators ideal in this article by extending s-soft reals and a particular space of sequences with soft real numbers. The criteria necessary for the Nakano sequence space of soft real numbers given with the definite function to be prequasi Banach and closed are investigated. This space's (R) and normal structural features are illustrated. Fixed points have been introduced for Kannan contraction and nonexpansive mapping. Finally, we investigate whether the Kannan contraction mapping has a fixed point in the prequasi operator ideal with which it is linked. By examining some real-world instances and their applications, it is demonstrated that there exist solutions to nonlinear difference equations.
... Maji [27] explored some fundamental properties of S f Ss and also defined its algebraic operations inclusive of restricted and extended union, intersection, and difference. Chan [28], Zou and Xiao [29], Ali [30], Kong [31], Cagman and Enginoglu [32,33], and many other pedagogues contributed to the development and improvement of the S f S theory. Moreover, Maji [34] combined the proficient theories of both S f Ss and fuzzy sets and developed an innovative concept of fuzzy soft sets (FS f Ss). ...
The main objective of this research study is to amplify the schematic representation of human reasoning by launching the most generalized fantastic theories of bipolar type-2 fuzzy set (BT2FS) and bipolar type-2 fuzzy soft set (BT2FS f S). These incredible models are exclusively developed for the simultaneous capturing of both polarity and abstruseness inherent in equivocal interpretations. The proposed BT2FS f S theory renders an outstanding parameterized framework that skilfully wipes out the high-order uncertainty of imprecise knowledge-based systems. First, we keenly provide the formal structure of both proposed models along with the deep exploration of their elementary properties. Moreover, we explore rudimentary set-theoretical operations of developed frameworks inclusive of equality, subset, complement, union, and intersection with their noticeable results. We brilliantly formulate a highly proficient algorithm using proposed theory to disentangle the real-world multiattribute decision-making conundrums with two-sided ambiguous information. Finally, we holistically scrutinize an empirical analysis for the selection best way to cultivate the dessert city to demonstrate the remarkable accountability of the proposed methodology.
... Their approach is for finding optimal decisions on a general Boolean dataset and the difference among all objects according to the parameter in the parameter set does not influence the final decision. Kong et al. (2008) proposed the concept of Normal Parameter Reduction (NPR) to find optimal decisions as well as suboptimal decision. proposed algorithms for parameter reduction based on invariability of the optimal choice object, to efficiently decrease the number of parameters used for evaluation and cut down the work load. ...
Soft set theory is a mathematical approach to vagueness introduced by Molodtsov. This is a parameterized family of subsets defined over a universal set using a set of parameters. So far, more than one attempt has been made to define this concept because of inefficiency of previous definitions. Out of these the one put forth by Tripathy et al is the most appropriate one and so we use it in this thesis. The characteristic function approach due to Tripathy et al. is used to define all the notions in this thesis. In this thesis, many hybrid models obtained as combination of soft set with other uncertainty models like interval valued fuzzy soft sets, interval valued intuitionistic fuzzy soft set, interval valued hesitant fuzzy soft set and interval valued intuitionistic hesitant fuzzy soft set are defined using membership function approach. Decision making in real life contexts selected as the application domain for the study. In fact, six algorithms are proposed in this thesis to deal with the decision making problem. These algorithms can be categorized into two broad categories; individual decision making and group decision making. Also, we have shown through different application areas the process of decision making for illustration as well as the applicability of our algorithms in various directions. After defining the new models we have also studied some of their properties so that further research can be done on these models, both from the algebraic front and application front. During the development of the algorithms, we have introduced a new concept by categorizing the parameters into positive and negative classes. This notion makes the algorithms more realistic. Also, our score function is found to be appropriate as it does not have the drawbacks in most of the earlier algorithms developed for decision making using soft sets or their hybrid models. The models developed in our work can be applied to some other application areas like game theory, medical diagnosis and data clustering.
... In addition, they proposed an application of SSs to solve the uncertainty problems encountered in a near ideal way. Chen et al. [16] and Kong et al. [34] worked on parameter reduction and normal parameter reduction methods for SSs, respectively. Then, Ali et al. [7] defined some new operations between SSs and Qin and Hong [43] defined concept of soft equality. ...
In this paper, the extension of N-soft sets, which is a very important mathematical model in non-binary evaluations to overcome uncertainty, under neutrosophic logic are studied and neutrosophic N-soft sets are introduced and are motivated. This new mathematical model, which deals with neutrosophic logic and N-soft set, which have been studied extensively in recent years to overcome uncertainty, aims to express the uncertainty situations encountered in the best way and thus approach the ideal in decision making. Moreover, some fundamental properties, products and useful operations are given for this new mathematical model. Then, we defined distance measures between two neutrosophic N-soft sets and expressed similarity measures based on decision making problem. Finally, an application is given that illustrates how uncertainty situations can be expressed in a decision-making problem by using the suggested similarity measures.
... There were some deficiencies in the decision-making methodology of the soft set model presented by Maji et al. [35]. Chen et al. [11] and Kong et al. [27] launched parameterization and normal attribute reductions for the soft set model to remove these flaws. Later, Ma et al. [31] presented another novel normal attribute reduction approach for soft sets to enhance [11,35]. ...
The interval-valued q-rung orthopair fuzzy sets and soft sets are two different uncertainty theories to cope with incomplete and uncertain information in several real-world multi-attribute decision-making (MADM) situations. This study develops a novel hybrid model called interval-valued q-rung orthopair fuzzy soft sets (IVqROFSSs, for brevity) to generalize the interval-valued q-rung orthopair fuzzy set model and to address the decision-makers preference information more effectively in complicated MADM processes. Afterward, some basic useful properties of the proposed model are explored, including subset relation, complement, union, intersection, the ‘AND’ operation, and the ‘OR’ operation. Further, four kinds of attribute reduction techniques for IVqROFSSs are presented. An algorithm for each reduction approach is developed and explained through an illustrative numerical example which verifies that developed reduction methods remove the redundant attributes by preserving the ranking order of decision objects unchanged. Later on, an application is proposed, that is, site selection for a wind power plant, to explain the developed model’s reliability and its reduction approaches. Finally, the proposed hybrid model and its attribute reduction methods are compared with some existing models, and their attribute reduction approaches, respectively.
... For redundant information in practical MCDM problems, some effective parameter reduction methods of soft sets have been proposed to solve complex MCDM problems with complete information (Maji and Roy, 2002;Chen et al., 2005;Kong et al., 2008;Khan and Zhu, 2020), which not only simplifies MCDM problems, but also improves the decision-making efficiency. However, these methods cannot be applied to MCDM problems with incomplete information. ...
Multiple criteria decision making (MCDM) problems in practice may simultaneously contain both redundant and incomplete information and are difficult to solve. This paper proposes a new decision-making approach based on soft set theory to solve MCDM problems with redundant and incomplete information. Firstly, we give an incomplete soft set a precise definition. After that, the binary relationships of objects in an incomplete soft set are analyzed and some operations on it are provided. Next, some definitions regarding the incomplete soft decision system are also given. Based on that, an algorithm to solve MCDM problems with redundant and incomplete information based on an incomplete soft set is presented and illustrated with a numerical example. The results show that our newly developed method can be directly used on the original redundant and incomplete data set. There is no need to transform an incomplete information system into a complete one, which may lead to bad decision-making due to information loss or some unreliable assumptions about the data generating mechanism. To demonstrate its practical applications, the proposed method is applied to a problem of regional food safety evaluation in Chongqing, China.
... Pei and Miao [19] have shown that soft sets are a special class of information systems. Kong et al. [9] introduced the reduction and algorithm of soft sets to normal parameters. Zou and Xiao [28] discussed the soft data analysis approach. ...
In this paper, a new environment namely, intuitionistic fuzzy hy-persoft set (IFHSS) is defined. We introduce some fundamental operators of intuitionistic fuzzy hypersoft sets such as subset, null set, absolute set, complement , union, intersection, equal set etc. Validity and application are presented with appropriate examples. For greater precision and accuracy, in the future, proposed operations in decision making processes such as personal selection, management issues and others will play a vital role.
... Several additional investigations on the reduction of parameters in soft sets have been reported. For instance, Chen et al. (2005) and Kong et al. (2008) launched parameterization and normal parameter reductions for soft sets, respectively, to overcome the deficiencies of decision-making applications in (Maji and Roy 2002). Ma et al. (2011) developed a novel efficient normal parameter reduction method in order to further improve (Chen et al. 2005;Maji and Roy 2002;Pawlak and Skowron 2007). ...
This paper formalizes a novel model that is able to use both interval representations, parameterizations, partial memberships and multi-polarity. These are differing modalities of uncertain knowledge that are supported by many models in the literature. The new structure that embraces all these features simultaneously is called interval-valued multipolar fuzzy soft set (IVmFSS, for short). An enhanced combination of interval-valued mpolar fuzzy (IVmF) sets and soft sets produces this model. As such, the theory of IVmFSSs constitutes both an interval-valued multipolar-fuzzy generalization of soft set theory; a
multipolar generalization of interval-valued fuzzy soft set theory; and an interval-valued
generalization of multi-polar fuzzy set theory. Some fundamental operations for IVmFSSs, including intersection, union, complement, ‘‘OR’’, ‘‘AND’’, are explored and investigated through examples. An algorithm is developed to solve decision-making problems having
data in interval-valued m-polar fuzzy soft form. It is applied to two numerical examples. In addition, three parameter reduction approaches and their algorithmic formulation are proposed for IVmFSSs. They are respectively called parameter reduction based on optimal choice, rank based parameter reduction, and normal parameter reduction. Moreover, these outcomes are compared with existing interval-valued fuzzy methods; relatedly, a comparative analysis among reduction approaches is investigated. Two real case studies for the selection of best site for an airport construction and best rotavator are studied.
... Base on that, several related operations have been proposed, a new theory system has been constructed, and a new decision-making method has been presented [37]. At present, soft set theory is widely used in parameter reduction and decision making [38], and a large number of methods for parameter reduction [39][40][41][42][43] and decision making [44][45][46] have been developed. ...
To process a huge amount of data, computing resources need to be organized in clusters that can be scaled out easily. Still, traditional SQL databases built on the relational data model are difficult to be put to use in such clusters, which has motivated the movement named NoSQL. However, NoSQL databases have their limits by using their own data models. In this paper, the original soft set theory is extended, and a new theory system called n-tier soft set is brought up. We systematically constructed its concepts, definitions, and operations, establishing it as a novel soft set algebra. And some features of this algebra display its natural advantages as a data model which could combine the logicality of the SQL model (also known as the relational model) and the flexibility of NoSQL models. This data model provides a unified and normative perspective logic for organizing and manipulating data, combines metadata (semantic) and data to form a self-described structure, and combines index and data to realize fast locating and correlating.
... Maji et al. [17,18] worked on this theory and gave the first practical application in decision-making problems. Chen et al. [3], Kong et al. [14] and then Ma et al. [16] gave their approaches for reduction of problems in soft sets. Pei and Miao [23] also contributed in the development of soft set theory. ...
The purpose of this paper is to introduce the concept of soft proximity bases and subbases. We determine the relation between proximity bases (subbases) and soft proximity bases (subbases). Further, we have demonstrated that the set of all soft proximities forms a complete lattice. Also, we substantiate a few results analogous to the ones that hold for soft proximity spaces.
This multifaceted thesis contributes significantly to mathematical literature, providing
innovative solutions to various problems across different mathematical domains. The findings
presented herein open avenues for further exploration and application in diverse scientific
contexts.
This research addresses common solutions under uncertainty, applying the fractional
Adams- Bashforth method to both fractional differential equations and integral-type equations in
the context of topological space like complex-valued controlled metrics. The perspective on
Rough Metric and Variational Inequalities contributes novel fixed-point results for Compact
Rough topological space. Emphasizing the significance of alternative models in addressing
complex phenomena, the study provides implicit methodologies for iterative strategies. The
introduction of soft compatible maps explores their applications in soft S-metric spaces. This
research study establishes a common fixed-point theorem for soft self-maps and utilizes fixed
point principles to demonstrate existence and uniqueness solutions. Exploration of generalized
results with partially ordered topological metric spaces incorporates various dominated distance
functions and mappings inspired by Caristi and Ciric.
Introduction of new theorems in topological space like soft metric spaces generalizes
soft Ciric and Caristi fixed-point theorems. The study examines soft open and soft closed sets
within soft topological structures. Investigation of complex-valued double controlled metric
spaces expands and generalizes findings within the framework of topological metric space. The
study establishes complex-valued fixed-point theorems and applies them to a Fredholm type
integral equation. Introduction of symmetrical, sequentially dense soft sets and exploration of
isometry in the context of partially ordered soft topological spaces. The study provides insights
into the convergence of soft topological spaces with a contraction condition for soft fixed point
theory. Our thesis fall under the following different areas:
Our Thesis
• Our study Focuses on the application of Complex Valued topological Metric Spaces
(CVMS), demonstrating common fixed results and addressing second-order nonlinear boundary
value problem using greens function,
This study provides solutions without assuming the continuity of multivalued mappings.
• Explores generalized results with partially ordered topological space, incorporating
various dominated distance functions and mappings inspired by Caristi and Ciric.
• Introduces new theorems in soft topological space, generalizing soft Ciric and Caristi
fixed-point theorems. The study examines soft open and soft closed sets within soft topological
structures.
• Investigates complex-valued double controlled metric spaces, expanding and
generalizing findings within this framework. The study establishes complex-valued fixed-point
theorems and applies them to a Fredholm type integral equation.
• Introduces and soft compatible maps and explores their applications in soft S-
topological space. The research study establishes a common fixed-point theorem for soft self-
maps and utilizes fixed point principles to demonstrate existence and uniqueness solutions, Our
study gives extended version of Ciric type contraction theorem for this we taken vector valued
metric space. This generalization gives us extension of Perov’s contraction theorem.
• Introduces symmetrical, sequentially dense soft sets and explores isometry in the
context of partially ordered soft topological spaces. The study provides insights into the
convergence of soft topological spaces with a contraction condition for soft fixed point theory.
• Extends the notion of Banach’s contraction principle and -contraction mapping to soft
fuzzy metric spaces, ensuring the existence and uniqueness of soft fixed points. Throughout this
study we taken under consideration absolute soft set, soft point as a restriction of our results and
we successfully applied Continuity of soft-t-norm under soft topological space.
• Work presents a common fixed-point solution for Urysohn integral equations
utilizing weakly compatible mappings in CVMS and establishing common solutions to integral
equations.
• Presents a unique perspective on Rough Metric and Variational Inequalities under
topological space, contributing novel fixed-point results for Compact Rough Metric Spaces. The
study emphasizes significance of alternative models in addressing complex phenomena,
providing implicit methodologies for iterative strategies. Furthermore we given fixed point
results for general variational inequalities which help to find uniqueness for solution, our
research provides a numerical example for practical illustration.
• Extend complex-valued integro-differential and integral equations
within the framework of controlled metric spaces. A novel extension of Fisher and Banach
contraction theorems is introduced, addressing common solutions under uncertainty. The
fractional Adams-Bashforth method is applied to both fractional differential equations and
integral-type equations in the context of complex-valued controlled topological metric space
Our thesis contributes significantly to mathematical literature Specially in the domain of
Soft, Rough, Complex valued, Ciric caristi all types of topological space, providing existence
solutions to various problems across different mathematical domains using fixed point. The
findings presented herein open avenues for further exploration and application in diverse
scientific contexts.
Keywords
Complex-valued fixed point theorems, Complex valued metric space (CVMS), Boundary
value Problem, Topological Multivalued Mapping, g.l.b. property, Contractive condition and
completeness, class function, Urysohn integral equation, Topological complete CVMS, Atangana
Baleanu Fractional integral operator, Fredholm type integral equations, Fuzzy Volterra integro-
differential equation(FVIdE), Cauchy sequence, Contractive condition and completeness,
Complex valued double controlled metric like spaces, Topological space.
Rough Sets, Rough Metric Spaces, General variational inequalities and convergence,
iterative methods, convergence criteria, projection iterative process, split variational, equilibrium
Conditions, split equilibrium problem, variational inequality.
Soft open and closed sets, soft continuous Topological function, soft topology, Soft
metric space, Soft fixed point, Soft Ciric and caristi mapping, Soft-t-norm, Soft fuzzy metric,
altering distance, Banach’s contraction principle.
Partially ordered metric space, Dominated distance function, Lower Semicontinuous
map in topological space, Totally ordered maximal subset, Invariant point and complete metric
space, Ciric and Caristi Common fixed point theorem under topological space, -Ciric
contraction, S-complete vector-valued topological space.
2020 Mathematics Subject Classification
45J05, 47H10, 54H25, 46S40, 26D10, 26A33, 26A51, 54D10, 54D15, 54D99, 47H09,
24H25, 34A08, 46N99, 30D35, 45J05, 47H10, 54D15, 54A05.
Selection of a most suitable cipher system among several such systems is an important problem of the customers which help to acquire a right cipher system for meeting the requirements of information security to safeguard the vital data. Any mistake in the selection of a right cipher system is very harmful as it may lead to compromise the security of vital information. In the paper, we propose a multi-criterion fuzzy soft decision making methodology to select one of the most suitable cipher system. Proposed methodology uses fuzzy soft set, fuzzy soft matrix, dominancy matrix and S-fuzzy set to compute score values and take a right decision. It exhibits out performance over existing methods as shown by explaining some examples. We apply proposed methodology in selecting a most suitable * cipher system from a set of systems with respect to the choice of desired security attributes required by the customers. Subject Classification: 62H86.
Soft set theory, initially introduced through the seminal article “Soft set theory—First results” in 1999, has gained considerable attention in the field of mathematical modeling and decision-making. Despite its growing prominence, a comprehensive survey of soft set theory, encompassing its foundational concepts, developments, and applications, is notably absent in the existing literature. We aim to bridge this gap. This survey delves into the basic elements of the theory, including the notion of a soft set, the operations on soft sets, and their semantic interpretations. It describes various generalizations and modifications of soft set theory, such as N-soft sets, fuzzy soft sets, and bipolar soft sets, highlighting their specific characteristics. Furthermore, this work outlines the fundamentals of various extensions of mathematical structures from the perspective of soft set theory. Particularly, we present basic results of soft topology and other algebraic structures such as soft algebras and σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}-algebras. This article examines a selection of notable applications of soft set theory in different fields, including medicine and economics, underscoring its versatile nature. The survey concludes with a discussion on the challenges and future directions in soft set theory, emphasizing the need for further research to enhance its theoretical foundations and broaden its practical applications. Overall, this survey of soft set theory serves as a valuable resource for practitioners, researchers, and students interested in understanding and utilizing this flexible mathematical framework for tackling uncertainty in decision-making processes.
Purpose of the study: This research paper proposes the use of soft mapping techniques to model the relationship between crop treatment and crop yield, with the goal of analyzing and recommending the best treatment options for crops. Soft mapping combines fuzzy logic and neural networks to create a more accurate and robust model that considers uncertain or ambiguous inputs. Methodology: The model can be trained using data on past crop yields, treatment options, and other relevant factors such as climate and soil quality. By taking into account the inherent uncertainty and ambiguity in the input data, the soft mapping model can provide more accurate predictions and recommendations for the best treatment options for a given crop and environmental conditions. Main Findings: The findings of this research could have important implications for the agricultural industry, particularly in the context of sustainable agriculture and food security. Applications of this study: The proposed approach has the potential to significantly improve the analysis and decision-making processes in agriculture, helping farmers to make more informed decisions about crop treatments and ultimately increasing crop yields.
Fuzzy soft set as a tool to deal with uncertainty can effectively handle decision making problems. However, there are many redundant parameters in the decision making process. In order to remove redundant parameters to improve the efficiency of decision making, different parameter reduction algorithms for fuzzy soft sets based on different decision criteria have been proposed. This paper focuses on the problem of parameter reduction of fuzzy soft sets based on choice value criteria. The restrictions of the strict conditions about parameter reduction lead to a very low applicability of some previous algorithms based on choice value criteria. To address this limitation, we introduce a flexible definition of parameter reduction for fuzzy soft sets. Further a difference-based parameter reduction algorithm for fuzzy soft sets is proposed. Compared with some previous algorithms based on choice value criteria, the proposed algorithm not only has wider applicability, but also can reduce more redundant parameters making the found parameter reduction with a lower cardinality, and it is easier to find the parameter reduction of fuzzy soft sets.
The present investigation is concerned with the estimation of the upper bound to the H4 (p) Hankel determinant for a subclass of p-valent functions in the open unit disc E={z:|z|<1}. This work will motivate the researchers to work in the direction of investigation of fourth Hankel determinant for several other subclasses of univalent and multivalent functions.
In early December 2019, a new virus named “2019 novel coronavirus (2019-nCoV)” appeared in Wuhan, China. The disease quickly spread worldwide, resulting in the COVID-19 pandemic. In the current work, we will propose a novel fuzzy soft modal (i.e., fuzzy-soft expert system) for early detection of COVID-19. The main construction of the fuzzy-soft expert system consists of five portions. The exploratory study includes sixty patients (i.e., forty males and twenty females) with symptoms similar to COVID-19 in (Nanjing Chest Hospital, Department of Respiratory, China). The proposed fuzzy-soft expert system depended on five symptoms of COVID-19 (i.e., shortness of breath, sore throat, cough, fever, and age). We will use the algorithm proposed by Kong et al. to detect these patients who may suffer from COVID-19. In this way, the present system is beneficial to help the physician decide if there is any patient who has COVID-19 or not. Finally, we present the comparison between the present system and the fuzzy expert system.
Data analysis is widely used in various fields, and data is often uncertain, which increases the difficulty of solving problems. Soft set method is a good mathematical tool to deal with uncertain information, but it can’t handle unknown data well. The existing approach for data filling in incomplete soft sets can fill data with high accuracy, however it involves a great amount of computation. In this paper a novel approach called simplified approach for data filling in incomplete soft sets (SDFIS) is proposed based on total values of association degrees, which is simpler and easier to understand. The comparison is carried out from several different aspects, and the comparison results show that the proposed approach has low complexity and good scalability. The experiments based on UCI data are performed, and the experiment results show that the proposed approach has almost the same accuracy as the existing approach, but it consumes less time.
Association rule (AR) mining is a common method to find the associations between objects in a transaction or in a data set, which is simple logical rules, determine the critical relationship among the objects. The rule generation is difficult if it satisfies a predefined threshold value. To trace the relationship is very important for domains to reveal crucial and hidden information from the data set. Some critical relationship of the objects may appear rarely and to trace them using classical methods are difficult one. The apriori algorithm (AA) with hashing technique along with the PDI algorithm is used to find the appropriate ARs from the students’ dataset. Since soft set is a tool for handling imprecise parameterized data, the soft set concept is used in optimization process to get optimized ARs. In this paper, the ARs are optimized using soft set for further analysis.
In this study, we first define expansion and reduction of the soft sets that are based on the linguistic modifiers. By using these new notions, we then construct a decision-making method called soft reduction method, which selects a set of optimum alternatives. We finally present an example which shows that the methods can be successfully applied to many problems containing uncertainties.
According to the 2017 World Health Organization (WHO) report, a large part of Egypt suffers from kidney disease. In recent years, kidney disease has become a leading cause of death worldwide, according to the report provided by the National Kidney Research Foundation. In the present paper, we will propose a novel method to predict kidney disease depend on the seven symptoms (i.e., Nephron Functionality, Blood Sugar, Systolic and Diastolic Blood Pressure, Alcohol Intake, Weight, and Age). Further, we design the new expert system (i.e., a soft fuzzy expert system or a fuzzy soft expert system) based on five basic steps to help the researcher and specialist doctor to predict kidney disease. After an exploratory study, which took it from the 60 patients (i.e., thirty men and thirty females) showed symptoms similar to kidney disease they have been conducting this study through (Nephrology Department, Damanhour Teaching Hospital, Beheira Governorate, Egypt). Accordingly, the user system provides helpful and reliable diagnostic results to predict kidney disease or kidney failure. Lastly, we present the comparison between the soft fuzzy expert system and Shakil et al.’s fuzzy expert system.
Bipolar soft set is an extended model of soft set. However, as in soft set theory, this theory can also leave mismatched objects in the universe set for a parameter set. In this paper, our aim is not to leave unmatched objects in the universe set for each parameter set. Thus, we have information about every object in the universe set and it becomes easier to get better approaches to uncertainty problems. For this, we have defined N-polar soft sets and some properties and several theoretical operations of N-polar soft sets are given. In addition to these, an algorithm is given on how to determine the value N for any uncertainty problem. Next, two decision-making algorithms based on N-polar soft sets are proposed. Finally, a detailed analysis was made by comparing some of the important algorithms previously studied with the algorithms we suggested in our paper.
A new model named multi-granularity belief interval-valued soft set is introduced in this paper. Some basic properties about it are presented and illustrated. The improved concepts of the soft belief value and soft belief degree are proposed, which provided an easier and better compared horizontally and vertically method among the different objects and different parameters. An algorithm for decision-making problems on multi-granularity belief interval-valued soft set is put forward and its validity is proved by the application of an example. Moreover, the newly proposed algorithm is compared with existing method to indicate its extensive application.
Due to the digitization of information, organizations have abundant data in databases. Large-scale data are equally
important and complex hence gathering, storing, understanding, and analyzing data is a problem for organizations. To extract
information from this superfluous data, the need for dimensionality reduction increases. Soft set theory has been efficaciously
applied and solved problems of dimensionality, which saves the cost of computation, reduces noise, and redundancy in data.
Different methods and measures are developed by researchers for the reduction of dimensions, in which some are probabilistic,
and some are non-probabilistic. In this paper, a non-probabilistic approach is developed by using soft set theory for dimensionality
reduction. Further, an algorithm of dimensionality reduction through bipartite graphs is also described. Lastly, the proposed
algorithm is applied to a case study, and a comparison of results indicates that the proposed algorithm is better than the existing
algorithms.
To contribute to soft topology, we originate the notion of soft bioperatorsγ andγ. Then, we apply them to analyze soft (γ,γ)-open sets and study main properties. We also prove that every soft (γ,γ)-open set is soft open; however, the converse is true only when the soft topological space is soft (γ,γ)-regular. After that, we define and study two classes of soft closures namely Cl (γ,γ) and τ (γ,γ)-Cl operators, and two classes of soft interior namely Int (γ,γ) andτ (γ,γ)-Int operators. Moreover, we introduce the notions of soft (γ,γ)-g.closed sets and soft (γ,γ)-T 1 2 spaces, and explore their fundamental properties. In general, we explain the relationships between these notions, and give some counterexamples.
In this paper the concept of soft α ( γ,β ) -continuous functions in soft topological spaces has been introduced and discussed. Further, the soft α ( γ,β ) -open functions, soft α ( γ,β ) -closed functions, and soft α ( γ,β ) -homeomorphisms have been introduced and analyzed. Also, the idea of soft α ( γ,β ) -contra continuous functions, soft ( α γ , β s )-contra continuous functions are introduced and analyzed. Further, the relationship and among soft α ( γ,β ) -contra continuous functions with other forms of continuous functions, and soft α ( γ,β ) -open functions, soft α ( γ,β ) -closed functions with another forms of, open, closed functions, soft α ( γ,β ) -contra continuous functions with other forms of contra continuous functions has been detailed.
Soft set theory is a mathematical approach to vagueness introduced by Molodtsov. This is a parameterized family of subsets defined over a universal set using a set of parameters. We introduce the notion of characteristic function of a soft set, which helps us in defining the basic operations on soft sets concisely. We rectified the definition of complement of a soft set and the earlier definition of complement is now called as the negation of a mul tiset. Soft multiset is a notion which allows multiple occurrences of elements in a model. So far, more than one attempt has been made to define this concept. Out of these the one put forth by Majumdar is the most appropriate one and so we use it in this paper. We defined the concepts of complement of a soft multiset, null
soft multiset and absolute soft multiset and introduced many operations on soft multisets like the union and intersection of soft multisets and cardinality of soft multisets. Also, we defined the concepts of addition and deletion of elements from a soft multiset. Two new operations, called the addition and difference of two soft multisets are introduced. We establish several properties of these operations on soft multisets. The Cartesian product of soft multiset is been discussed which leads to soft multiset relations. Some of the properties of soft multiset relation and equivalence soft multi set relation were also introduced with their respective examples. Building an engineering application is the typical task in all fields. Following these hybrid concepts many research put forth their work to develop an application. One of such applications is decision making which was initiated by Maji et al. We present a new approach to decision making based on soft multiset concept which is applicable in normal real life scenarios.
The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future.
Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper, we consider some category-theoretic properties of the category of all soft sets, with morphisms between them. We also characterize several kinds of epimorphisms and monomorphisms.
We investigate in this paper approximate operations on sets, approximate equality of sets, and approximate inclusion of sets. The presented approach may be considered as an alternative to fuzzy sets theory and tolerance theory. Some applications are outlined.
Rough sets – theoretical aspects of resaoning about data by PawlakZdzislaw, Kluwer Academic, The Netherlands, 1991, pp 229, £56.00, ISBN 0-792-31472-7. - Volume 9 Issue 2 - Simon Parsons
In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.
In this paper, we focus our discussion on the parameterization reduction of soft sets and its applications. First we point out that the results of soft set reductions offered in [1] are incorrect. We also observe that the algorithms used to first compute the reduct-soft-set and then to compute the choice value to select the optimal objects for the decision problems in [1] are not reasonable and we illustrate this with an example. Finally, we propose a reasonable definition of parameterization reduction of soft sets and compare it with the concept of attributes reduction in rough sets theory. By using this new definition of parameterization reduction, we improve the application of a soft set in a decision making problem found in [1].
A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.
The problem of decision making in an imprecise environment has found paramount importance in recent years. A novel method of object recognition from an imprecise multiobserver data has been presented here. The method involves construction of a Comparison Table from a fuzzy soft set in a parametric sense for decision making.
In this paper, we apply the theory of soft sets to solve a decision making problem using rough mathematics.
Molodtsov introduced the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper we introduce the basic properties of soft sets, and compare soft sets to the related concepts of fuzzy sets and rough sets. We then give a definition of soft groups, and derive their basic properties using Molodtsov’s definition of the soft sets.
pdf contains a preliminary version of the book