Samsiah Abdul Razak\[1\]\*, Ini Imaina Abdullah2, Nurazila Yahaya3 123 College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Cawangan Perak, Kampus Tapah, 35400 Tapah Road, Perak, Malaysia
ARTICLE INFO ABSTRACT

Article history:

Received XX Month 2024

Revised XX Month 2024

Accepted XX Month 2024

Online first

Published 1 September 2024

Maji et al. expanded on soft set theory by introducing fuzzy soft set theory, offering a versatile approach for tackling problems marked by uncertainty and fuzziness, while effectively modelling and representing data. The authors developed a matrix representation within this fuzzy soft set framework and explored various properties of these matrices. Despite this, existing applications of interval-valued fuzzy soft matrices in group decision-making often assume equal importance for all criteria, which fails to capture the true preferences of decision-makers.

This study proposes a novel approach to group decision-making through the Interval Valued Fuzzy Soft Max-min Decision-Making Method (IVFSMmDM), which considers the varying importance of each criterion, followed by using the Fuzzy Soft Max-min decision-making technique to prioritize decisions. The integration of these methods provides a more accurate and practical decision-making framework.

The effectiveness of IVFSMmDM is illustrated through a detailed numerical example in the context of manpower recruitment, involving the selection of 7 programmers for a software development organization’s team. The results indicate that Programmer 5 was chosen, achieving the highest-ranking value of (0.019, 0.021). This highlights the practical utility and effectiveness of the Interval Valued Fuzzy Soft Max-min Decision-Making Method in real-world decision-making scenarios.

Keywords:

Fuzzy Soft set

Fuzzy Soft Matrix

Interval Valued Fuzzy Soft Max- min Decision-making (IVFSMmDM)

Fuzzy Analytic Hierarchy Process (FAHP)

Lambda-max method

Manpower recruitment

DOI:

10.24191/jcrinn.v9i2
# introduction Numerous challenges encountered in fields such as engineering, management, social sciences, and medicine often involve data characterized by inherent uncertainties, rather than being precise or deterministic. To navigate these complexities, traditional concepts such as probability, fuzzy sets, intuitionistic fuzzy sets, interval mathematics, and rough sets have been employed. Nevertheless, these approaches present certain limitations. In response to these shortcomings, Molodtsov introduced soft set theory in 1999, which provides a framework that remains unaffected by the parameterization challenges faced by other methodologies. Since its introduction, soft set theory has gained traction and found extensive applications in diverse domains, including decision–making and data mining. Significant contributions have been made by researchers such as Maji et al. (2002), Kong et al. (2021), Qin et al. (2021), and Feng et al. (2016, 2020), who have applied the TOPSIS method for decision-making within the context of soft sets. Furthermore, studies by Zulqarnain (2020), Maharana (2021), and Tripathy et al. (2019) have concentrated on parameter reduction and the advancement of new mathematical tools pertinent to soft set theory. Additionally, Sai (2020) has offered a thorough overview of its applications in decision-making, highlighting the considerable importance of soft set theory in this area. Regenerate response fuzzy soft sets extend soft sets by incorporating fuzzy numbers, a concept developed by Zadeh in 1965 to handle uncertainties in real-life situations. These sets are characterized by a membership function that assigns a grade of membership to each object. Liu and Kwon (2007) further expanded the concept by considering parameters as fuzzy hedges or fuzzy parameters and defining operations on fuzzy soft groups. Shagari and Azam (2020) proposed a novel algorithm for decision-making in a fuzzy soft set environment, enhancing object discrimination and inference. Qin et al. (2021) presented a new approach to decision-making using interval-valued fuzzy soft sets, incorporating a contrast table to address extreme values and outliers. In 2019 Khalil and Hassan, introduced the idea of inverse fuzzy soft sets and their application in decision-making, providing more mathematical insight for decision-makers. Das et al. (2022) defined various practical operations on fuzzy soft sets, including the algebraic sum, bounded sum, and Einstein product, and investigated the basic properties of these new operations. These studies collectively contribute to the development of decision-making techniques based on soft set theory. Cagman and Enginoglu (2013) advanced the practical use of fuzzy soft sets by defining fuzzy soft matrices, particularly for decision-making under uncertainty. They redefined the four products in soft matrices as fuzzy soft matrix products and investigated their properties. Using the fuzzy soft max-min decision function and the "And" product of fuzzy soft matrices, they developed the Fuzzy Soft Max-min Decision-Making (FSMmDM) method tailored for scenarios with two decision makers. Subsequent studies by Khalil and Hassan (2019) and Razak et al. (2013) further explored the use of FSMmDM in different contexts. Razak et al. (2017) introduced a hierarchical approach to fuzzy soft matrices and group decision-making, while Khalil developed inverse fuzzy soft sets for decision-making applications. Enginoğlu and Memis (2020) refined the criteria weighted FSMmDM approach with two new algorithms that enhanced its complexity and efficiency. These contributions highlight the flexibility and effectiveness of FSMmDM in managing uncertainty and vagueness in decision-making. The concept of Interval-Valued Fuzzy Soft Sets (IVFSS) has been explored for various applications. Shanthi and Gaynthri (2020) introduced the normalized Euclidean distance between IVFSS establishing it as a metric, while Lambodharan (2019) discussed IVFSS operations and their properties, including principal disjunctive and conjunctive normal forms. Zulqarnain (2017,2020) applied IVFSS in a medical context for patient identification. In 2021, Silambarasan introduced Hamacher operations, scalar multiplication, and exponentiation for interval valued fuzzy matrices, enhancing their algebraic properties. Interval-valued fuzzy soft sets (IVFSS) have become an effective tool for managing uncertainty in decision-making processes. This approach integrates interval-valued fuzzy sets with soft set theory, providing a more adaptable way to represent membership degrees (Abdul & Musheer, 2020). Interval-valued fuzzy soft sets (IVFSS) have been applied to various real-world evaluation systems, such as those for apartments, universities, and stock markets (Qin & Ma, 2018; Sooraj & Tripathy, 2017). Researchers have developed comprehensive models for these evaluation systems using IVFSS, incorporating steps like data collection, decision-making, parameter reduction, and data set integration (Qin & Ma, 2018). Further expansions, like interval-valued hesitant fuzzy soft sets, have been proposed to enhance the model’s effectiveness (Sooraj & Tripathy, 2017). Recent progress includes the formulation of interval-valued fuzzy soft preorderings and equivalence relations, as well as the creation of scoring functions for multi-group decision-making challenges (Ali & Kılıçman, 2021). The previous research based on IVFSS already applied to decision-making problems, however, the combination of determination criteria weight by using proper the method with IVFSS remains relatively unexplored. To address this gap, this study proposes a novel method, the IVFSMmDM, which integrates the fuzzy AHP Lambda-max method for more objective criteria weighting and efficient resolution of decision-making problems. The fuzzy AHP method is straightforward to compute and provides a definite value directly from experts, but it doesn't fully capture the human thinking style. The findings of this study will significantly contribute to addressing group decision-making problems. We developed a graphical model for the IVFSMmDM method using the IVFSMmDM function incorporating with Fuzzy AHP Lambda-Max method. ## Preliminaries ### 1.1.1. Fuzzy Soft Matrix A fuzzy set represents a group of elements characterized by a membership grade ranging continuously from 0 to 1, inclusive. A triangular fuzzy number, represented by the 3–tuple (l, m, u), is a convex and normal fuzzy set with the highest membership grade of 1. It is characterized by the membership function defined as: ![](66cd6a53ad816_media/media/image1.wmf) Maji et al. (2022) defined fuzzy soft set theory as a generalization of standard soft sets in the following manner: **Definition 1:** Let be an initial universe set andbe a set of all parameters. Let *F* (*U*) denote the set of all fuzzy sets in Then is called a fuzzy soft set over where and is a mapping given by *F*. In general, for every,is a fuzzy set inand it is called a fuzzy value set of parameter *x*. If everyis a crisp subset of, then is degenerated to be the standard soft set. Cagman and Enginoglu (2012) developed a fuzzy soft decision-making method by the following definition. **Definition 2:** Let be a fuzzy soft set over , where be an initial universe set, be a set of parameters, and. For and, there exists a membership degree , then all the membership degrees will be presented as in Table 1: **Table 1. Evaluation of membership degrees fuzzy soft matrices** | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | The matrix is called interval fuzzy soft matrix of over (Basu et al. 2014). **Definition 3:** Let. The *And*-product betweenandis defined by,, wheresuch that ### 1.1.2. **Interval Valued Fuzzy Soft Matrix** **Definition 4:** Let be an interval valued fuzzy soft set over . Then a subset of is uniquely defined by which is called a relation form of . Now the relationis characterized by the membership function such that where Int (\[0,1\]) stands for the set of all closed sub-intervals of \[0,1\] and denotes the interval–valued fuzzy membership degree of the object associated with the parameter . Now if the set of universes be an initial universe set, be a set of parameters then can be presented by a table in the following form **Table 2. Evaluation of membership degrees interval value fuzzy soft matrices** | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | where . If , then from Table 2 we can define a matrix which is called an interval fuzzy soft matrix or simply IVFS-matrix of order corresponding to the interval-valued fuzzy soft set over . An interval-valued fuzzy soft set is uniquely characterized by the matrix ## *1.1.2. Fuzzy Soft Max–min Decision Making Method* ## Cagman and Enginoglu (2012) introduced a Fuzzy Soft Max–min Decision Making Method by using And–product and defined as follows: **Definition 5:** *Let* for all *. Then Max-min decision function, denoted Mm, is defined as follows:* , where The one column soft matrix is called max-min decision fuzzy soft matrix. **Proposition 1:** The operators for and are defined as follows, for**,:** **Definition 6:** Let. The *And*-product between and is defined by **,,** where such that # METHODOLOGY The Analytic Hierarchy Process (AHP), introduced by Saaty in 1980, is one of the most popular and widely used techniques for determining criteria weight. It is highly flexible and can accommodate various types of **multi-criteria** decision**-**making (MCDM) methods, making it useful as an input for ranking alternatives (Liu, Kwon & Kang, 2014). AHP combine the evaluation results and expert opinions with a sophisticated decision**-**making process to create a straightforward, elementary hierarchy. Moreover, AHP can effectively manage both qualitative and quantitative data and is easy to compute. It also provides a reliable method for checking the consistency of the evaluation criteria and the alternatives selected by the decision maker. Thus, it can reduce bias in decision making (Lixiong, Liang & Minzhong, 2010). All the evaluation methods among the criteria will be analysed through pairwise comparisons using the AHP. However, AHP’s inability to adequately handle the evaluations uncertainty and imprecision in which the human judgement is represented in terms of fuzzy numbers (Cheng et al. 2009). Fuzzy sets can be aligned through pairwise comparisons as an extension of AHP to overcome this limitation. Han et al. (2020) applied AHP for road selection and obtained a result that preserved the original road network structure. Li and Mo (2020) emphasized the value of problem investigation creative direction and communication presentation by implementing AHP in the teaching assessment system. To create a priority list of counties based on hazard and exposure vulnerability, Guo (2020) utilized AHP in earthquake risk assessment. França (2020) employed AHP to map environmental fragility mapping and provide a hierarchy of critical environment criteria. The concept of Analytic Hierarchy Process (AHP) was initially extended to fuzzy AHP in 1983 by Van Laarhoven and Pedrycz. This approach effectively addresses uncertainty and vagueness inherent in subjective performance and decision makers' experiences in solving hierarchical problems. The Lambda-max method, a key component of fuzzy hierarchical analysis, was introduced by Csutora & Buckley (2001) as a technique for determining fuzzy weights. Razak et al. (2012,2013,2017) have utilized fuzzy AHP (Lambda–max Method) in determining the criteria weight for the main and sub criteria in solving the group decision making problem. The proposed method involves two procedures. The first procedure involves determining the criteria weight using Lambda-max method obtained from Razak et al. (2017). The second procedure addresses solving group decision making problems. This paper employs IVFSMmDM incorporating together with criteria weight. The details of both procedures are given below. ## **Criteria Weight Determination** The Lambda-max method (Csutora and Buckley, 2012) is used in determining criteria weight. The procedure of the Lambda-max method involves 4 steps as follows: **Step 1:** Apply. To obtain the positive matrix of decision maker, let , , and let to obtain the lower bound and upper bound positive matrices of decision maker *s*, and .Calculate the weight vector based on the weight calculation procedure in AHP, , **Step 2:** To minimize the fuzziness of the weight, choose two constants, and, as follows: **(1)** and the upper bound and lower bound of the weight is defined as: , , **(2)** so the lower bound and upper bound weight vectors are , **Step 3:** By combining the upper bound, the middle bound and lower bound weight vectors, the fuzzy weight matrix for decision maker *s* can be obtained and is defined as **Step 4**: Calculate local fuzzy weights and global fuzzy weight with repetition from step 1 until step 3. ## **Interval Valued Fuzzy Soft Max-min Decision Making (IVFSMmDM) Method** Zulqarnain et al. (2020) introduced an IVFSMmDM by using *And*–product. They then defined max-min decision function as follows: **Definition 7:** Letfor all and IVFS max**-**min decision function defined as follows where , such that is known as IVFSMmDM function. **Definition 8:** Let be an initial universe and. Then a subset of can be obtained by usingas in the following expression which is called an optimum set of . *Now using definition 8 and 9, the IVFSMmDM method can be developed by the following algorithm:* **Step 1**: Choose the feasible subsets of the set of parameters. **Step 2:** Use the matrix form to construct the *ivfs*–matrix for each set of parameters. **Step 3:** Find the *And*–product for the *ivfs*–matrices. **Step 4**: Find a max-min decision *ivfs*–matrix. **Step 5:** Find an optimum set of*.* > . **2.3 Development of IVFSMmDM Method with Criteria Weight** Our proposed decision–making procedure, IVFSMmDM, is outlined as follows: **Step 1:** Assess the membership value of each alternative concerning each criterion in the decision–making problem. **Step 2:** Use the matrix form to create the interval value fuzzy soft matrices for each set of criteria. **(3)** *whereis an interval fuzzy soft matrix of decision maker k, m represents the number of alternatives involved and n refers to the parameters/criteria.* **Step 3:** Multiply the matrix from step 2 by the criteria weight *wa* and compute the values for each alternative and then construct the resulting matrix. **(4)** **Step 4:** Determine the *And-*product of interval fuzzy soft matrices **(5)** *The resulting* fuzzy soft *matrix will have a size*, *where there are n blocks of elements in the matrix.* **Step 5:** Calculate , where First, we find , to find we need to find for every , If and ,is then, to find for every , If and ,is then, to find for every , If and ,is Find the minimum of And – product between andfor each n block on interval value of elements above. , where the elements of is in term of Interval numbers **Step 6:** Find the max – min decision fuzzy soft matrix, **(6)** ***Step 7:** Find an optimum set of* **(7)** **Step 8:** Find an optimum fuzzy set according to # NUMERICAL EXAMPLE: Manpower Recruitment problem As an illustration, we revisit numerical illustration of manpower recruitment by Chaudhuri et al (2013) as an example for this paper. In this research we use an interval – valued fuzzy numbers to describe the membership degree. Two staffs’ members from the Human Resources Department, referred to as *A* and *B*, are involved as decision-makers. There are eight criteria considered as a parameter and seven programmers to be recruited by a Software Development Organization. The recruitment process for a Software Development Organization considers eight criteria as parameters and evaluates seven programmers. The criteria weights are calculated using the Lambda – max method, and the IVFSMmDM method is applied to prioritize the seven programmers in this decision – making problem. Let be a set of seven programmers to be recruited by a Software Development Organization by the Human Resources Manager as a possible alternative. The set of parameters , where e1, e2, e3, e4, e5, e6, e7, and e8, represent the parameters “hardworking”, “disciplined”, “honest”, “obedient”, “intelligence”, “innovative”, “entrepreneurial attitude”, and “aspirant” respectively. Intelligence and innovation reflect the programmer's creative mindset, while hard work and discipline signify their punctuality. Honesty and obedience indicate the integrity in the programmer's behaviour, and an entrepreneurial attitude along with being aspirant highlight their exploratory nature. **A.** **Criteria weight for each decision maker** Table 3 showed the criteria weight by two decision makers calculate by using Lambda-max method from each decision makers. The criteria weight obtains from (Razak et al. 2017). | **Table 3:** Criteria weight by every decision maker | | | | ---------------------------------------------------- | ------------------------------------- | ------------------------------------- | | **CRITERIA** | **DM 1 (*WA*)** | **DM 2 (*WB*)** | | **C1** | 0.024 | 0.025 | | **C2** | 0.031 | 0.208 | | **C3** | 0.220 | 0.175 | | **C4** | 0.057 | 0.036 | | **C5** | 0.061 | 0.122 | | **C6** | 0.109 | 0.082 | | **C7** | 0.177 | 0.053 | | **C8** | 0.322 | 0.299 | ## **B. IVFSMmDM Calculation** **Step 1:** Assessment of membership degrees by each decision maker **Step 2:** Construct interval valued fuzzy soft evaluation in Step 1 into matrix form, where andrefer to decision maker 1 and decision maker 2 respectively. **Step 3:** Integrate the criteria weights for each decision maker into the fuzzy soft matrix. These yields: The calculation for this step is: The calculation for this step is: **Step 4:** By applying the *And-*product, the product of interval valued fuzzy soft matrices between and is obtained as follows: *In this step we obtained* fuzzy soft *matrix of size*, *consisting of n blocks of elements each. The matrix size , transform into a matrix size .* **Step 5**: Calculate , where First, we find , to find we need to find for every , If and ,is If and ,is **Step 6:** Find the Max – min decision interval valued fuzzy soft matrix. The calculation for this matrix come from step 5 as follows: ***Step 7:** Find the maximum set interval valued fuzzy soft matrix* *The calculation for this step is come from matrix in Step 6 as follows:* **Step 8:** Finally, we find an optimum fuzzy set according to as: It is clear that represent the best choice of programmer within the universal set. Consequently, the human resources department will choose programmer 5 to join the Software Development Organization’s team. # conclusioN # This research introduced a novel decision–making approach called the IVFSMmDM. The method was designed to address the limitations of existing group decision–making techniques, particularly in contexts characterized by uncertainty and imprecise data. By integrating the Lambda–max method, IVFSMmDM enables more objective and precise determination of criteria weights, which are crucial for evaluating alternatives in decision-making processes. A key contribution of this research is the application of IVFSMmDM in a real-world scenario manpower recruitment for a software development organization. The method's efficacy was demonstrated through a detailed numerical example involving the selection of programmers. The results identified Programmer 5 as the optimal choice, with a maximum interval value of the overall priority vector calculated as (0.019, 0.021). This result highlights the method's robustness and practical utility in managing complex decision-making scenarios with multiple criteria and decision-makers. This outcome not only underscores the robustness of the IVFSMmDM but also supports previous studies, such as those by Maji et al. (2002) and Kong et al. (2021), which have shown the effectiveness of advanced decision-making frameworks within soft set theory contexts. The findings reinforce the method's practical utility for addressing complex decision–making challenges involving multiple criteria and decision makers, aligning with the conclusions of prior research that emphasized the necessity of innovative approaches in the field. # The study also indicates that IVFSMmDM has potential applications across a range of decision-making # contexts where data is uncertain or vague. Its adaptability and flexibility suggest opportunities for further # development and refinement to extend its use. Future research could explore additional enhancements to # the IVFSMmDM method, potentially by incorporating new factors or integrating it with other decision # making techniques to increase its precision and applicability. Advancing these research avenues could help # IVFSMmDM evolve, offering even greater accuracy and versatility in decision-making processes # characterized by uncertainty. # # Acknowledgements/Funding # Conflict of interest statement # References Ali, M., & Kılıçman, A. (2021). On interval-valued fuzzy soft preordered sets and associated applications in decision- making. Mathematics, 9(23), 3142. Abdul, R., & Musher, A. (2020). Application Of Interval Valued Fuzzy Set and Soft Set. Jnanabha 50(2), 114-121. https://doi.org/10.58250/jnanabha.2020.50 Cagman, N., & Enginoglu, S. (2012). Fuzzy soft matrix theory and its application in decision making. Iranian Journal of fuzzy systems, 9(1), 109-119. https://ijfs.usb.ac.ir/article\_229\_b6d292816ccde2e91d91920714cb6245.pdf Chaudhuri, A., De, K., & Chatterjee, D. (2013). Solution of the decision-making problems using fuzzy soft relations. arXiv preprint arXiv:1304.7238. https://doi.org/10.48550/arXiv.1304.7238 Csutora, R., & Buckley, J. J. (2001). Fuzzy hierarchical analysis: the Lambda – Max method. Fuzzy sets and Systems, 120(2), 181-195. https://doi.org/10.1016/S0165-0114(99)00155-4. Csutora, R., & Buckley, J. J. (2001). Fuzzy hierarchical analysis: the Lambda – Max method. Fuzzy sets and Systems, 120(2), 181-195. https://doi.org/10.1016/S0165-0114(99)00155-4 Das, A. K., Granados, C., & Bhattacharya, J. (2022). Some new operations on fuzzy soft sets and their applications in decision-making. Songklanakarin Journal of Science and Technology, 44(2), 440-449. https://sjst.psu.ac.th/journal/44-2/21.pdf Enginoğlu, S., & Memiş, S. (2020). A new approach to the criteria-weighted fuzzy soft max-min decision-making method and its application to a performance-based value assignment problem. Journal of New Results in Science, 9(1), 19-36. https://dergipark.org.tr/en/pub/jnrs/issue/53974/709375 Feng, F., Cho, J., Pedrycz, W., Fujita, H., & Herawan, T. (2016). Soft set-based association rule mining. Knowledge-Based Systems, 111, 268-282. https://doi.org/10.1016/j.knosys.2016.08.020 Feng, F., Wang, Q., Yager, R. R., Alcantud, J. C. R., & Zhang, L. (2020). Maximal association analysis using logical formulas over soft sets. Expert Systems with Applications, 159, 113557. https://doi.org/10.1016/j.eswa.2020.113557 França, L. C. D. J., Mucida, D. P., Santana, R. C., de Morais, M. S., Gomide, L. R., & Bateira, C. V. D. M. (2020). AHP approach applied to multi-criteria decisions in environmental fragility mapping. De Jesus França, L. C., Piuzana Mucida, D., Campos Santana, R., Santos de Morais, M., Rezende Gomide, L., & de Meneses Bateira, C. V. (2020). AHP approach applied to multi-criteria decisions in environmental fragility mapping. Floresta, 50(3). Guo, X., & Kapucu, N. (2020). Assessing social vulnerability to earthquake disaster using rough analytic hierarchy process method: A case study of Hanzhong City, China. Safety science, 125, 104625. https://doi.org/10.1016/j.ssci.2020.104625 Han, Y., Wang, Z., Lu, X., & Hu, B. (2020). Application of AHP to road selection. ISPRS International Journal of Geo-Information, 9(2), 86. https://doi.org/10.3390/ijgi9020086 Khalil, A. M., & Hassan, N. (2019). Inverse fuzzy soft set and its application in decision making. International Journal of Information and Decision Sciences, 11(1), 73-92. https://doi.org/10.1504/IJIDS.2019.096630 Kong, Z., Zhao, J., Wang, L., & Zhang, J. (2021). A new data filling approach based on probability analysis in incomplete soft sets. Expert Systems with Applications, 184, 115358 https://doi.org/10.1016/j.eswa.2021.115358 Lambodharan,V., Anitha. N. (2019). Interval Valued Fuzzy Soft Sets and Fuzzy Connectives. International journal of recent technology and engineering, 8(4), 2277-3878. https://citeseerx.ist.psu.edu/document?repid=rep1\&type=pdf\&doi=22332842c0cd25b2b32d420b038852f170f70983 Li, X., & Mo, X. (2020, October). Application of AHP based on mathematical operational research in teaching evaluation system. In Journal of Physics: Conference Series (Vol. 1650, No. 3, p. 032015). IOP Publishing. https://doi.org/10.1088/1742-6596/1650/3/032015 Liu, Y. W., & Kwon, Y. J. (2007, August). A Fuzzy AHP approach to evaluating e-commerce websites. In 5th Acis International Conference on Software Engineering Research, Management & Applications (Sera 2007) (pp. 114-124). IEEE. https://doi.org/10.1109/SERA.2007.16 Maharana, M., & Mohanty, D. (2021). An application of soft set theory in decision making problem by parameterization reduction. Soft Computing, 25(5), 3989-3992. https://doi.org/10.1007/s00500-020-05420-7 Maji, P. K., Roy, A. R., & Biswas, R. (2002). An application of soft sets in a decision-making problem. Computers & Mathematics with Applications, 44(8-9), 1077-1083. https://core.ac.uk/download/pdf/82179651.pdf Qin, H., Fei, Q., Ma, X., & Chen, W. (2021). A new parameter reduction algorithm for soft sets based on chi-square test. Applied Intelligence, 51, 7960-7972. https://doi.org/10.1007/s10489-021-02265-x Qin, H., Wang, Y., Ma, X., & Wang, J. (2021). A Novel Approach to Decision Making Based on Interval-Valued Fuzzy Soft Set. Symmetry, 13(12), 2274. Qin, H., & Ma, X. (2018). A complete model for evaluation system based on interval-valued fuzzy soft set. IEEE Access, 6, 35012-35028. Razak, S. A., & Mohamad, D. (2012). An Application of Soft Matrices in Group Decision Making Problems (Aplikasi Matrik Lembut dalam Masalah Pembuatan Keputusan secara Berkumpulan). Menemui Matematik (Discovering Mathematics), 34(1), 33-40. https://myjms.mohe.gov.my/index.php/dismath/article/view/13672 Razak, S. A., & Mohamad, D. (2013). A decision-making method using fuzzy soft sets. Malaysian Journal of Fundamental and Applied Sciences, 9(2). https://doi.org/10.11113/mjfas.v9n2.91 Razak, S. A., Mohamad, D., & Abdullah, I. I. (2017). A group decision making problem using hierarchical based fuzzy soft matrix approach. International Journal of Advanced and Applied Sciences, 4(10), 26-32. Saaty, T. L. (1980). The analytic hierarchy process (AHP). The Journal of the Operational Research Society, 41(11), 1073-1076. Sai, B. V. S. T. (2020). Applications Of Soft Set in Decision Making Problems. Global Journal of Pure and Applied Mathematics, 16(2), 305-324. http://dx.doi.org/10.37622/GJPAM/16.2.2020.305-324 Shagari,S. M., Azam.A, (2020). An algorithm for fuzzy soft set-based decision-making approach. Yugoslav journal of operations research 30(00):26-26. https://doi.org/10.2298/YJOR190715026M Shanthi, S. A., & Gayathri, M. (2020). Interval valued picture fuzzy soft set-in pattern recognition. Adv. Mathematics: Scientific J., 9(4), 2019-2029. https://doi.org/10.37418/amsj.9.4 Silambarasan, I. (2021). Hamacher operations of Fermatean fuzzy matrices. Applications and Applied Mathematics: An International Journal (AAM), 16(1), 15. Sooraj, T. R., & Tripathy, B. K. (2017). Interval valued hesitant fuzzy soft sets and its application in stock market analysis. In Artificial Intelligence and Evolutionary Computations in Engineering Systems: Proceedings of ICAIECES 2016 (pp. 755-764). Springer Singapore. Tripathy, B. K., Sooraj, T. R., & Mohanty, R. N. (2019). Application of soft set-in game theory. In Advanced Methodologies and Technologies in Artificial Intelligence, Computer Simulation, and Human-Computer Interaction (pp. 421-435). IGI global. Van Laarhoven, P. J., & Pedrycz, W. (1983). A fuzzy extension of Saaty's priority theory. Fuzzy sets and Systems, 11(1-3), 229-241. https://doi.org/10.1016/S0165-0114(83)80082-7 Cheng, C. H., Wang, J. W., & Huang, K. C. (2009). Fuzzy hierarchical TOPSIS for supplier selection. Applied soft computing, 9(1), 377-386. https://doi.org/10.1016/j.asoc.2008.04.014 Zulqarnain, R. M., Abdal, S., Maalik, A., Ali, B., Zafar, Z., Ahamad, M. I., & Dayan, F. (2020). Application of TOPSIS method in decision making via soft set. Biomed J Sci Tech Res, 24(3). http://dx.doi.org/10.26717/BJSTR.2020.24.004045 Zulqarnain, R. M., Xin, X., Saeed, M., Ahmad, N., Dayan, F. Bilal Ahmad. (2020). Application of Interval Valued Fuzzy Soft. International Journal of Mathematical Research, 9(1),11-19. http://dx.doi.org/10.18488/journal.24.2020.91.11.19 Zulqarnain, R. M., Xin, X. L., Saeed, M., & Ahmed, N. (2020). Application of interval valued fuzzy soft max-min decision making method in medical diagnosis. International Journal of Pharmaceutical Sciences Review and Research, 62(1), 56-60. Min\_Decision\_Making\_Method\_in\_Medical\_Diagnosis/links/5eef7bcf92851ce9e7f7a51e/Application-of-Interval-Valued-Fuzzy-Soft-Max-Min-Decision-Making-Method-in-Medical-Diagnosis.pdf Zulqarnain, M., & Saeed, M. (2017). A new decision making method on interval valued fuzzy soft matrix (IVFSM). British Journal of Mathematics & Computer Science, 20(5), 1-17. 1. \* Corresponding author. *E-mail address*: (Add the e-mail in the final camera-ready submission)