Abstract
In this paper, we study some basic concepts of fuzzy sets, soft sets, fuzzy soft sets, and interval-valued fuzzy soft sets. Secondly, we study the interval-valued fuzzy soft max-min decision-making function and developed the graphical model for the Interval-valued fuzzy soft max-min decision making (IVFSMmDM) method by using Interval-valued fuzzy soft max-min decision-making function. Finally, we used IVFSMmDM for faculty selection in the education department and observed that is the best teacher for teaching by using hypothetical data.
Keywords: Fuzzy set, Soft set,Interval-valued fuzzy soft set, IvfsMmDM function.
Received: 10 January 2020 / Revised: 13 February 2020 / Accepted: 17 March 2020/ Published: 8 April 2020
Contribution/ Originality
This paper contributes a new graphical model for IVFSMmDM by using interval-valued fuzzy soft max-min decision-making function and used for faculty selection in the education department
1. INTRODUCTION
L. A. Zadeh revolutionized the Mathematical approach to answer the problems of uncertainties with the Concept of the fuzzy set (FS) in 1965 [1]. Later this concept leads to many further concepts such as Hesitant FSs, Intuitionistic FSs, Neutrosophic Sets, Linguistic Term Sets, and Hesitant Fuzzy Linguistic term sets, etc. This stream of knowledge flourished when Molodstov formulated the concept of soft set (SS) in 1999 [2]. This idea of SS was channelized by many other Mathematicians who took further actions to establish the foundations. In [3] Maji et al in 2003 defined the ideas such as subset, superset, equality of SSs, union, intersection AND and OR operations of SSs. Pei and Miao also contributed to redefining these ideas in Daowu and Miao [4]. In Ali, et al. [5] some more operations were also defined by Ali, et al. [5]. Cağman and Enginoğlu [6] developed the concept of the soft matrix with different properties and operations. In these days, mathematician plays a vital role in the SS fuzzification. A new perception of the fuzzy soft set (FSS) was introduced in Maji, et al. [7] after the fuzzification of SS with different types and properties. They also reviewed the SS theory which was given by Molodtsov and used this theory for decision making in Roy and Maji [8]. The work on FSS theory was extended and introduced a new concept from FSS which is known as a fuzzy soft aggregation operator and its cardinal set [9]. They introduced the cardinal matrix of the cardinal set also construct a decision algorithm and used this method in decision making successfully. Some limitations were faced in the work of Maji, et al. [7]. which were attensonoid by Chen, et al. [10] by constructing a new definition of parameterization reduction and applied this definition in decision making. Ahmad and Kharal [11] extended the work of Maji, et al. [7]; Roy and Maji [12] with illustrations and counter illustrations and improved their work, also they extended the concept of FSS by defining some new definition such as the union of arbitrary FSS, the intersection of arbitrary FSS and proved De-Morgan inclusion on FSS.
In Majumdar and Samanta [13] the author’s generalized the concept of FSS and proposed a new theory of generalized FSS with some properties and used generalized FSS for decision-making problem, also use this theory for the diagnosis of Pneumonia. Dayan, et al. [14]. also used this theory for decision making in Dayan, et al. [14]. In Celik and Yamak [15] fuzzy soft set theory used for medical diagnosis through Sanchez’s approach by using fuzzy arithmetic operations. Many researchers extend the concept of studied FS and SS and developed TOPSIS models for fuzzy sets and soft sets and used for decision making and medical diagnoses [16-20]. The work on FSM was extended by Borah, et al. [21] they forwarded the FSM theory with some definitions and properties; they proposed a special type of product of FSMs which is known as T-product of FSMs and used this product in decision-making problem after construct algorithm.
The theory of IVFS was forwarded by Yushi in 2010 and discussed the properties of IVFS [22]. He proposed a new concept on IVFS which is known as a cut set of IVFS with some properties and introduced three decomposition theorems of IVFS. A most important topic of fuzzy algebra is the fuzzy matrix (FM) in this matrix elements are belonging to the unit interval [0, 1]. FM becomes the interval-valued fuzzy matrix (IVFM) when we generalized the FM elements in subintervals of [0, 1]. The membership value of rows and columns is crisp in IVFM, but there are many challenges are faced in real-life problems. To solve those challenges [23] presented the idea of IVFM with the help of interval-valued fuzzy rows and columns and defined some basic definitions with some binary operators. He also defined complement and density of interval-valued fuzzy rows and columns with verification of few important properties. Yang, et al. [24] worked on IVFS and SS and proposed a new theory which is known as interval-valued fuzzy soft set (IVFSS) with different types and properties. They define different operations on IVFSS such as complement, AND-operation and OR-operation on IVFSS with examples also prove the De-Morgan laws, associative laws and distributive laws on IVFSS and successfully used this theory in decision making to show the validity of IVFSS. In Chetia and Das [25] an algorithm developed on IVFSS and used for medical diagnoses by extending Sanchez’s approach. The author’s studied the fuzzy set theory and used the trapezoidal fuzzy numbers for disease identification by using Sanchez’s approach [26]. The work on IVFSS was extended by Shawkat and Abdul [27] they introduced generalized IVFSS with basic operations such as union, intersection, and complement also proved some properties related to these operations. They also proposed AND-operation, OR-operation similarity measure on generalized IVFSS and use this similarity measure in medical diagnoses for decision making. Dayan and Zulqarnain studied the IVFSM and developed some new operations and properties on IVFSM in Dayan and Zulqarnain [28].
Rajarajeswari and Dhanalakshmi worked on IVFSM and proposed new definitions on IVFSM with examples and properties in Rajarajeswari [29]. A decision-making method on IVFSM developed in Zulqarnain and Saeed [30] which is known as the “Interval Valued fuzzy soft max-min decision-making method” and used the developed method for decision making. They introduced some new operations on IVFSM such as arithmetic mean, weighted arithmetic mean, geometric mean, weighted geometric mean, harmonic mean and weighted harmonic mean with some properties of IVFS-matrices in decision making. In Prabhavath and Sarala [31] Sarala and Prabharathi worked on IVFSS and extended Sanchez`s approach for medical diagnoses by using IVFSM with illustrations. They proposed some new definitions on IVFSM such as union and intersection of IVFSM with their examples. They also proved the commutative laws, associative laws and construct an algorithm for medical diagnoses. The author’s used IVFSM for decision making in Zulqarnain and Saeed [32]; Sarala and Prabhavathi [33]. The author’s defined some new operations on IVFSM and used for decision making [34].
In this research, we study fuzzy sets, soft sets, fuzzy soft sets, and interval-valued fuzzy soft matrices. We constructed a graphical model for the IVFSMmDM method by using the IVFSMmDM function. Finally, we use this for decision making for the selection of faculty in the educational department.
2. PRELIMINARIES
Definition 1.1[1]
Definition 1.2 [2]
A pair (F, A) is called an SS over M if A is any subset of E, and there exists a mapping from A to P (M) is F, P (M) is the parameterized family of subsets of the M but not a set.
Definition 1.3 [6]
In other words, an SS is uniquely represented by its corresponding SM. So, we can define a function which maps SS to SM i.e.
Definition 1.4 [7]
A pair (F, A) is called FSS over M, and there exists a mapping from A to P (M) is F, P (M) is the collection of fuzzy subsets of M.
Definition 1.5 [35]
A pair (F, A) is called FSS in the fuzzy soft class (M, E). Then (F, A) is represented in a matrix form such as
Definition 1.8 [24]
A pair (F, A) is called IVFSS over M where F is a mapping such that
Definition 1.10 [34]
Definition 1.12 [34]
3. INTERVAL VALUED FUZZY SOFT MAX-MIN DECISION MAKING METHOD
In this section, we present the “Interval Valued Fuzzy Soft Max-min Decision Making (IVFSMmDM)” function and construct a graphical model for decision making by using the IVFSMmDM function in the following Figure 1.
Definition 2.1 [30]
Definition 2.2 [30]
Figure-1 . Graphical Model for IVFSMmDM.
Source: Zulqarnain and Saeed [30].
4. APPLICATION IVFSMMDM METHOD FOR FACULTY SELECTION
We construct the IVFSMs of both committee members according to their selected parameters.
Step 3
Take And-product of both developed IVFSMs
Step 4
We get IVFSM by using IVFSMmDM as follows
5. CONCLUSION
We studied some basic concepts of fuzzy sets, soft sets, fuzzy soft sets, and interval-valued fuzzy soft sets in this work with some properties and operations. A graphical model also constructed for IVFSMmDM by using the IVFSMmDM function. Finally, we use the proposed model in decision making for the selection of teachers.
Funding: This study received no specific financial support. |
Competing Interests: The authors declare that they have no competing interests. |
Acknowledgement: All authors contributed equally to the conception and design of the study. |
REFERENCES
[1] L. A. Zadeh, "Fuzzy sets," Infortion and Control, vol. 8, pp. 338–353, 1965.
[2] D. A. Molodtsov, "Soft set theory — first results soft set," Computers & Mathematics with Applications, vol. 37, pp. 19–31, 1999. Available at: https://doi.org/10.1016/s0898-1221(99)00056-5.
[3] P. K. Maji, R. Biswas, and A. R. Roy, "Soft set theory," Computers & Mathematics with Applications, vol. 45, pp. 555–562, 2003.
[4] D. Daowu and P. Miao, "From soft sets to information systems," Proceedings of the IEEE International Conference on Granular Computing, vol. 2, pp. 617–621, 2005.
[5] M. I. Ali, F. Feng, X. Liu, W. Keun, and M. Shabir, "On some new operations in soft set theory," Computers & Mathematics with Applications, vol. 57, pp. 1547–1553, 2009. Available at: https://doi.org/10.5120/9538-3975.
[6] N. Cağman and S. Enginoğlu, "Soft matrix theory and its decision making," Computers & Mathematics with Applications, vol. 59, pp. 3308–3314, 2010. Available at: https://doi.org/10.1016/j.camwa.2010.03.015.
[7] P. K. Maji, A. R. Roy, and R. Biswas, "Fuzzy soft sets," The Journal of Fuzzy Mathematics, vol. 9, pp. 589–602, 2001.
[8] A. R. Roy and P. K. Maji, "An application of soft sets in a decision making problem," Computers & Mathematics with Applications, vol. 44, pp. 1017–1083, 2002.
[9] N. Cagman, S. Enginoglu, and F. Citak, "Fuzzy soft set theory and its applications,” Iran," Journal Fuzzy Systems, vol. 8, pp. 137–147, 2011.
[10] D. Chen, E. C. C. Tsang, D. S. Yeung, and X. Z. Wang, "The parameterization reduction of Soft sets and its applications," Computers & Mathematics with Applications, vol. 49, pp. 757–763, 2005. Available at: https://doi.org/10.1016/j.camwa.2004.10.036.
[11] B. Ahmad and A. Kharal, "On fuzzy soft sets," Advances in Fuzzy Systems, vol. 2009, pp. 1–6, 2009.
[12] A. R. Roy and P. K. Maji, "A fuzzy soft set theoretic approach to decision making problems," Computers & Mathematics with Applications, vol. 203, pp. 412–418, 2007.
[13] P. Majumdar and S. K. Samanta, "Generalised fuzzy soft sets," Computers & Mathematics with Applications, vol. 59, pp. 1425–1432, 2010. Available at: https://doi.org/10.1016/j.camwa.2009.12.006.
[14] F. Dayan, M. Zulqarnain, and H. Naseer, "A ranking method for students of different socio economic backgrounds based on generalized fuzzy soft sets," International Journal of Science and Research, vol. 6, pp. 691–694, 2017.
[15] Y. Celik and S. Yamak, "Fuzzy soft set theory applied to medical diagnosis using fuzzy arithmetic operations," Journal of Inequalities and Applications, vol. 2013, pp. 1–9, 2013.
[16] M. Zulqarnain, F. Dayan, and M. Saeed, "Topsis analysis for the prediction of diabetes based on general characteristics of humans," International Journal of Pharmaceutical Sciences and Research, vol. 9, pp. 2932-2939, 2018.
[17] M. Zulqarnain, "Application of TOPSIS method in decision making via soft set," Biomedical Journal of Scientific & Technical Research, vol. 24, pp. 18208–18215, 2020.
[18] M. Zulqarnain and F. Dayan, "Selection of best alternative for an automotive company by intuitionistic fuzzy TOPSIS method," International Journal of Science and Technology, vol. 6, pp. 126–132, 2017.
[19] R. M. Zulqarnain, S. Abdal, B. Ali, L. Ali, F. Dayan, M. I. Ahamad, and Z. Zafar, "Selection of medical clinic for disease diagnosis by using TOPSIS method," International Journal of Pharmaceutical Sciences Review and Research, vol. 61, pp. 22-27, 2020.
[20] F. Dayan and M. Zulqarnain, "Choose best criteria for decision making via fuzzy topsis method," Mathematics in Computer Science, vol. 2, pp. 113-119, 2017. Available at: https://doi.org/10.11648/j.mcs.20170206.14.
[21] M. J. Borah, T. J. Neog, and D. K. Sut, "Fuzzy soft matrix theory and its decision making," International Journal of Modern Engineering and Research, vol. 2, pp. 121–127, 2012.
[22] W. Zeng and Y. Shi, "Note on interval-valued fuzzy set," Lecture Notes in Computer Science, vol. 3613, pp. 20–25, 2005.
[23] M. Pal, "Interval-valued fuzzy matrices with interval-valued fuzzy rows and columns," Fuzzy Information and Engineering, vol. 7, pp. 335–368, 2015. Available at: https://doi.org/10.1016/j.fiae.2015.09.006.
[24] X. Yang, T. Young, J. Yang, Y. Li, and D. Yu, "Combination of interval-valued fuzzy set and soft set," Computers & Mathematics with Applications, vol. 58, pp. 521–527, 2009.
[25] B. Chetia and P. K. Das, "An application of interval-valued fuzzy soft sets in medical diagnosis," International Journal of Contemporary Mathematical Sciences, vol. 5, pp. 1887–1894, 2010.
[26] R. M. Zulqarnain, X. L. Xin, B. Ali, S. Abdal, A. Maalik, L. Ali, M. I. Ahamad, and Z. Zafar, "Disease identification using trapezoidal fuzzy numbers by Sanchez’s approach," International Journal of Pharmaceutical Sciences Review and Reaserch, vol. 61, pp. 13-18, 2020.
[27] A. Shawkat and R. S. Abdul, "Generalised interval-valued fuzzy soft set," Journal of Applied Mathematics, pp. 1–18, 2012.
[28] F. Dayan and M. Zulqarnain, "On generalized interval valued fuzzy soft matrices," Mathematical and Computer Modelling, vol. 3, pp. 1–9, 2018.
[29] P. D. P. Rajarajeswari, "Interval-valued fuzzy soft matrix theory," Annals of Pure and Applied Mathematics, vol. 7, pp. 61–72, 2014.
[30] M. Zulqarnain and M. Saeed, "A new decision making method on interval valued fuzzy soft matrix (IVFSM)," British Journal of Mathematics & Computer Science, vol. 20, pp. 1–17, 2017.
[31] M. Prabhavath and N. Sarala, "An application of interval valued fuzzy soft matrix in medical diagnosis," IOSR Journal of Mathematics, vol. 11, pp. 1–6, 2015.
[32] M. Zulqarnain and M. Saeed, "An application of interval valued fuzzy soft matrix in decision making problem," Science International, vol. 28, pp. 2261–2264, 2016.
[33] M. Sarala and N. Prabhavathi, "An application of interval valued fuzzy soft matrix in decision making problem," International Journal of Mathematics Trends and Technology, vol. 21, pp. 21–30, 2015.
[34] M. Zulqarnain, M. Saeed, and M. F. Tbassum, "Comparison between fuzzy soft matrix (FSM) and interval valued fuzzy soft matrix (IVFSM) in decision making," Science International, vol. 28, pp. 4277–4283, 2016.
[35] J. Yang and C. Ji, "Fuzzy soft matrices and their applications," Part 1 Lecture Notes in Computer Science, vol. 7002, pp. 618–627, 2011.
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