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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. introduced fuzzy soft set theory as a generalization of soft set theory, presenting a well-known and practical method for addressing problems characterized by uncertainty and fuzziness, and effectively representing data. In the fuzzy soft set framework, a matrix form has been established and some of its properties have been examined. However, current applications of interval-valued fuzzy soft matrices in group decision-making assume equal importance weights for criteria, which does not accurately reflect the true opinions of decision-makers on each criterion. This study aims to propose a method for group decision-making using an interval-valued fuzzy soft max-min decision-making method that incorporates the weight of each criterion. The Lambda-Max method is employed to calculate the weight for each criterion, and the Fuzzy Soft Max-Min decision-making method is used to prioritize the group decision-making problem. By integrating these methods, the proposed approach, termed the Interval-Valued Fuzzy Soft Max-Min Decision-Making Method (IVFSMmDM), provides a robust framework for addressing complex decision scenarios. The efficacy of this method is demonstrated through a numerical example related to manpower recruitment, showcasing its practical relevance and effectiveness in real-world applications.

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 Most of our real-world problems in engineering, management, social sciences, and medicine often involve data that are not necessarily crisp or deterministic due to various uncertainties. To address these ambiguities, concepts such as probability, fuzzy sets, intuitionistic fuzzy sets, interval mathematics, and rough sets have been traditionally employed. However, these methods have limitations. Recognizing this, Molodtsov introduced Soft Set Theory in 1999, offering a framework unaffected by the parameterization inadequacies of other theories. Soft Set Theory has since been widely applied in decision-making, data mining, and other fields (Maji et al (2002), Kong et a. (2021) Qin et al. (2021), Feng et al. (2016,2020), Zulqarnain, 2020) applied the TOPSIS method in decision-making using soft sets, while Maharana (2021) and Tripathy et al. (2019) explored reducing parameters and developing new mathematical tools within soft set theory. Sai (2020) provided a comprehensive overview of its applications in decision-making of how soft set theory is used in decision-making, further emphasizing its significance in this field. 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, introduced the idea of inverse fuzzy soft sets and their application in decision-making, providing more mathematical insight for decision-makers. Das et al.in 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) defined fuzzy soft matrices to enhance the practical application of fuzzy soft sets, particularly in decision-making under uncertainty. They redefined the four products in soft matrices as fuzzy soft matrix products and explored their properties. By employing the fuzzy soft max-min decision function and the "And" product of fuzzy soft matrices, they formulated the fuzzy soft max-min decision-making (FSMmDM) method for scenarios involving two decision-makers. Khalil (2019) and Razak et al. (2013) further examined the application of fuzzy soft max-min decision-making in various contexts. Razak et al. (2017) proposed a hierarchical fuzzy soft matrix and fuzzy soft set group decision-making process, while Khalil introduced inverse fuzzy soft sets and applied them to decision-making. Enginoğlu and Memis (2020) enhanced the criteria-weighted FSMmDM approach by proposing two new algorithms that improved complexity and running time. These studies underscore the versatility and effectiveness of the fuzzy soft max-min decision-making method in addressing uncertainty and vagueness in decision-making processes. 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 IVPFSS 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. Ali (2021) extended this work by establishing interval-valued fuzzy soft preordering and proposing crisp preordering sets for multi-group decision-making.In 2021, Silambarasan introduced Hamacher operations, scalar multiplication, and exponentiation for interval-valued fuzzy matrices, enhancing their algebraic properties. The previous research based on IVFSS already applied in decision making problem, however, the combination of determination criteria weight by using proper method with IVFSS remain relatively unexplored. To address this gap, this study proposes a novel method, the IVFSMmDM, which integrates 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. ## 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 - 3-tuple (l, m, u), is a convex and normal fuzzy set (highest membership grade of 1). It is characterized by the membership function defined as: ![](667e1dc6408fb_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,\[x\] 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: **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 between and is defined by,, where such that ### 1.1.2. **Interval Valued Fuzzy Soft Matrix** **Definition 4:** Let be an initial universe set and be a set of all parameters. Let denotes the set of all fuzzy sets in Then is called a fuzzy soft set over where and is a mapping given by . In general, for every, is a fuzzy set inand it is called fuzzy value set of parameter *x*. If for every is a crisp subset of, then is degenerated to be the standard soft set. **Definition 5:** The concept of IVFSM 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 (\[0,1\]) stand 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 fuzzy 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* **Definition 6 \[25\]:** Let. The *And*-product between and is defined by **,** where such that **Proposition 1:** The operators for and are defined as follows, for**,:** **Definition 7:** Let. The *And*-product between and is defined by **,,** where such that # METHODOLOGY The Analytical 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). AHPs combine the evaluation results and expert opinions with a sophisticated decision-making process to create a straightforward, elementary hierarchy. Furthermore, AHP can handle both qualitative and quantitative data with equal effectiveness and is simple to compute Additionally, it offers a precise mechanism for verifying the consistency of the evaluation measures and the alternatives chosen by the decision maker. Thus, it can reduce bias in decision making (Lixiong, Liang & Minzhong, 2010). all the evaluation method among the criteria will analysed in pair-wise comparison in AHP. However, AHP’s inability to adequately handle the evaluations uncertainty and imprecision in which the human judgment is represented in fuzzy numbers Cheng et al. (2009). Fuzzy sets can be synced using pair-wise comparison as an extension of AHP to get over this drawback. Han et all. (2020) used AHP for road selection, achieved a result that maintained the original road network structure. Li and Mo (2020) emphasized the value of problem investigation creative direction and communication presentation by implementing AHP in teaching assessment system. To create a priority list of counties based on hazard, exposure ad 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 (2021) as a technique for determining fuzzy weights. 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) has been utilized fuzzy AHP (Lambda-Max Method) in determining the criteria weight for the main and sub-criteria in solving the group decision making problem. There are two procedures involved in this paper. The criteria weight in this paper is a secondary data (2021) calculate by using Lambda-Max method. Second is decision making method by using IVFSMmDM incorporating together with criteria weight. ## **Criteria Weight Determination** 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:** In order 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 \[35\] introducing an Interval Valued fuzzy soft (*fs*) max – min decision making method by using *And* – product. They then defined max – min decision function as follows: **Definition 8:** Letfor all and IVFS max-min decision function defined as follows where where such that is known as IVFSMmDM function. **Definition 9:** 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 **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 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. Let be a set of seven programmers to be 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 behavior, and an entrepreneurial attitude along with being aspirant highlight their exploratory nature.**Criteria Weight for Each Decision Maker** Table 1 showed the criteria weight from each decision makers are secondary data calculate by using Lambda-max method \[19\]. | **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 | ## ## **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. This 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 # In this paper, we have introduced the interval-valued fuzzy soft max-min decision-making method, which effectively incorporates determination criteria weight to address group decision-making challenges. Utilizing the lambda max method introduced by csutora and buckley (2001) we demonstrated its application in determining the weight for each criterion and subsequently solving group decision-making problems through the interval-valued fuzzy soft max-min approach. A numerical example focusing on a manpower recruitment problem illustrated the practicality and efficacy of this method. The results highlighted the maximum interval value of the overall priority vector as (0.019, 0.021), indicating the method’s robustness. Moreover, we suggest that this method can be applied to various decision-making scenarios characterized by uncertainty or vague data. 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