Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 212 https://internationalpubls.com The Development of IV-VIKOR Method Under Hesitant Fuzzy Sets in MCDM Rafi Raza1, Ahmad Termimi Ab Ghani2*, Lazim Abdullah3 1,2,3Faculty of Computer Science and Mathematics, University Malaysia Terengganu, 210300 Kuala Nerus, Kuala Terengganu, Malaysia. *Corresponding author: termimi@umt.edu.my Article History: Received: 13-08-2024 Revised: 27-09-2024 Accepted: 14-10-2024 Abstract: Due to its solution correctness and ease of evaluation, the VIKOR approach has already established itself as a popular multi-criterion decision-making (MCDM) method. This strategy focuses on picking and ranking from a list of viable options in order to assist the decision-maker in determining a final course of action. Additionally, it looks for a middle ground to resolve a problem with competing demands. It produces a ranking list of compromises based on a precise measurement of distance from the ideal response. The hesitant fuzzy set (HFS) has furthermore come under increased study because of its potency and ability in capturing ambiguity and uncertainty. This paper introduces the inversion of VIKOR method (IV-VIKOR) to take uncertain fuzzy conditions into account. These novel concepts serve as the foundation for an IV- VIKOR method that we suggest, and a real-world application is given to demonstrate how well our method works when dealing with MCDM problem with hesitant fuzzy preference data. Keywords: MCDM, F-VIKOR method, Decision Making, Hesitant fuzzy sets. 1 Introduction Finding the best option from a list of possibilities while taking into account the decision maker’s expectations and various considerations is the act of decision making. Every choice is made inside an environment, which is the set of facts, options, standards, and preferences that are at hand at the time, the choice needs to be made. The diversity of criteria established for evaluating the options is the decision-making process’s most challenging component. Therefore, MCDM is the process of choosing amongst several, frequently at odds criteria. In daily life, MCDM issues occur frequently Taherdoost and Madanchian [1]. As more decision-makers from various groups participate in the process, the problem of decision-making becomes increasingly complicated. Despite the constant prevalence of MCDM issues, the field of MCDM as a whole is rather young, having only been for around 30 years. The rapid development of computer technology, which permits the methodical analysis of difficult MCDM problems, is closely related to the growth of the MCDM discipline [2]. But MCDM is becoming more and more significant and helpful in assisting business decision-making as more and more individuals use information technology and produce massive volumes of data. The advantage of MCDM is that it strives to remove emotion from the decision-making process while providing a balanced assessment of how appropriate each alternative. It also prevents one component from dominating the others. The majority of MCDM issues fall into one of two categories; that is a finite an Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 213 https://internationalpubls.com infinite number of solutions. The number of potential solutions is typically constrained in tasks involving selection and evaluation. The typical way to represent an MCDM problem is a decision matrix as can be written as given below: [ π‘Ž11 π‘Ž12 … π‘Ž1π‘š π‘Ž21 π‘Ž22 . . . π‘Ž2π‘š π‘Ž31 π‘Ž32 . . . π‘Ž3π‘š ∢ ∢ ∢ ∢ π‘Žπ‘›1 π‘Žπ‘›2 ... π‘Žπ‘›π‘š ] where each entry π‘Žπ‘–π‘— denotes how effectively the π‘–π‘‘β„Ž alternative meets the π‘—π‘‘β„Ž criterion. Therefore, a family of tools known as MCDM techniques has developed in response to the requirement for a formalised strategy to aid decision making in scenarios containing many variables [3]. This facilitates in determining the optimal course of action. How attribute data should be evaluated in order to make the right choice is determined by an MCDM technique. To far, a number of MCDM approaches have been put out and used with success to address complex decision-making issues that arise in various management and engineering fields. The VIKOR method has become a very well-liked among those involved in decision-making because of its straightforward and understandable computational steps [4]. First establishing the β€œpositive ideal solutions (PIS)” and β€œnegative ideal solutions (NIS)” forms the foundation of VIKOR technique. When there are conflicting and incomparable criteria at play, it concentrates on assessing and choosing among a small number of plausible options. Depending on how closely the response adheres to the ideal response, a multi-criteria ranking index is evaluated. The ranking of the compromise options can be determined by comparing the relative closeness measure to the ideal alternative after each compromise option has been assessed in light of each criterion. The compromise option thus represents a viable alternative that is the most remote from the unfavorable ideal solution but the least nearer the favorable ideal solution [5, 6]. Establishing the VIKOR technique in the presence of HFSs and an inversion of the VIKOR method is the primary goal of this article. An MCDM method for complex structures called VIKOR was first introduced by Opricovic [7] with the intention of dealing with crisp information. Subsequently, it was extended to rationalize a wider range of informational settings. Finding a compromise solution to the MCDM problem that fulfils the criteria of β€œmaximum group utility” and β€œminimal individual regret” among opponents is the primary goal of this method. Opricovic used the following two defining essential elements of a feasible compromise solution. 𝐿𝑝 measure and an aggregation function. 𝐿𝑝 metric 𝐿𝑖 𝑝 = {βˆ‘[ 𝐹𝑖𝑗 βˆ’ 𝐹𝑗 βˆ— 𝐹𝑗 βˆ— βˆ’ 𝐹𝑗 βˆ’] 𝑝𝑛 𝑗=1 } 1 𝑝 Where 𝐹𝑗 βˆ— = max 𝑖 (𝐹𝑖𝑗) and 𝐹𝑗 βˆ’ = min 𝑖 (𝐹𝑖𝑗). But in this chapter, the IV-VIKOR method based on 𝐿𝑝 metric can be written as 𝐿𝑝 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 214 https://internationalpubls.com 𝐿𝑖 𝑝 = {βˆ‘[(𝐹𝑖𝑗 βˆ’ 𝐹𝑗 βˆ—) Γ— (𝐹𝑗 βˆ— βˆ’ 𝐹𝑗 βˆ’)] 𝑝 𝑛 𝑗=1 } 1 𝑝 Where 𝐹𝑗 βˆ— = max 𝑖 (𝐹𝑖𝑗) and 𝐹𝑗 βˆ’ = min 𝑖 (𝐹𝑖𝑗). Despite the false premise that a decision-maker shows exact instances of the determining preferences circumstances, traditional MCDM methods use crisp information to evaluate the alternatives and qualities in typical unexpected and uncertain situations. Because priorities are a very vague and ambiguous aspect of human character [8], as a result, a decision-maker should not use precise information to assess the tendency of his issue. Bellmen and Zadeh [9] were the first to use fuzzy numbers (FNs) for the MCDM approaches. Fuzzy membership values are chosen from the range [0, 1] while making decisions in the FS domain. One of the best tools for making decisions is the fuzzy set theory, which Zadeh [10] first developed. Many writers have since used the FS theory to illustrate real-world situations. The fuzzy VIKOR was introduced by Wang and Chang [11] to capture MCDM. Chang [12] classified Taiwanese hospitals based on their amenities and services using the F-VIKOR method. Other F-VIKOR method applications to address MAGDM issues with supplier selection were put forth [13-15]. Opricovic [16] modified the conventional F-VIKOR method of managing the Mlava River’s water resources. An expansion of the VIKOR method for trapezoidal FNs was introduced by Ju and Wang [17]. Wang, Liang and Ho [18] created an F-VIKOR method using triangular FNs as linguistic variables to determine the best software company. To evaluate the Saudi Arabian power networks for investment, Taylan, Alamoudi [19] used various decision-making methods, including F- VIKOR, fuzzy AHP, as well as fuzzy TOPSIS. All of these VIKOR method variations could only handle the data in a way that benefited the object in question in accordance with FS’s capabilities. As a result, these techniques are ineffective when there is a high level of discontent. Zadeh [20] introduced the fuzzy sets theory. It can deal with ambiguous and imprecise information. Whenever working with insufficient and fuzzy data when several or so more sourcesof uncertainty are present simultaneously, the classic fuzzy set has a number of drawbacks Rodriguez, Martinez and Herrera [21]. Torra and Narukawa were firstly develop the HFS Torra and Narukawa [22], Torra [23] providing up new directions for future studies on decisionmaking under unclear circumstances. This generalised type of FS founded on aforementioned extensional forms of FSs. The HFS illustrates fuzziness by presenting all possible values when calculating an element’s membership degree in a certain set, despite the fact that it does not offer an accurate membership function. The article’s remaining sections are organized as follows. The FS, HFSs, distance between hesitant fuzzy element inverse of any element in Section 2 along with a quick overview of some fundamental ideas that are used throughout the article. We present the VIKOR approach of HFSs for MCDM issues in Section 3. In Section 4, we go through how the IV-VIKOR method was developed. An illustration of the suggested technique is provided in Section 5 for clarity. The main findings of the paper are then briefly explored. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 215 https://internationalpubls.com 2 Preliminaries Definition 2.1 [24] Fuzzy set Suppose 𝑋 is the universal set. A fuzzy set 𝑋 is defined by πœ‡π΅(οΏ½ΜƒοΏ½): 𝑋 β†’ [0,1] is the membership function of 𝐡, and it can be written as given below. 𝐡 = {(οΏ½ΜƒοΏ½, πœ‡π΅(οΏ½ΜƒοΏ½)), οΏ½ΜƒοΏ½πœ–π‘‹} Where πœ‡π΅(π‘₯ Μƒ) is the degree of membership of π‘₯ Μƒ. In 𝐡 each pair (οΏ½ΜƒοΏ½, πœ‡π΅(οΏ½ΜƒοΏ½)) is a singleton set. Definition 2.2 [22] Hesitant Fuzzy set Suppose 𝑋 is the universal set; then an HFS as 𝐡 on 𝑋 is defined by a function 𝜌𝐡(οΏ½ΜƒοΏ½) that 𝑋 returns to a subset of [0,1]. Xia and Xu [25] indicated the HFS mathematically as given below. 𝐡 = {βŒ©οΏ½ΜƒοΏ½, 𝜌𝐡(οΏ½ΜƒοΏ½)βŒͺ|οΏ½ΜƒοΏ½πœ–π‘‹} Where 𝜌𝐡(οΏ½ΜƒοΏ½) is defined as a set of membership degrees for an element under a subset of[0,1], indicating the membership degree of an element οΏ½ΜƒοΏ½πœ–π‘‹. Xia and Xu [25] called 𝜌 = 𝜌𝐡(οΏ½ΜƒοΏ½) hesitant fuzzy element (HFE). A hesitant fuzzy element (HFE) is a finite and non-empty subset of [0, 1]. Example 2.1 Let 𝑋 = {οΏ½ΜƒοΏ½, οΏ½ΜƒοΏ½, οΏ½ΜƒοΏ½} be a universal set, 𝜌𝐡(οΏ½ΜƒοΏ½) = {0.1,0.9,0.6}, 𝜌𝐡(οΏ½ΜƒοΏ½) = {0.6,0.7}, 𝜌𝐡(οΏ½ΜƒοΏ½) = {0.7,0.3}, and then HFSs can be written as. 𝐡 = {< π‘Ž,Μƒ {0.1,0.9,0.6} >, < 𝑏,Μƒ {0.6,0.7} >,< 𝑐,Μƒ {0.7,0.3} >} Definition 2.3 [26] Hesitant Normalized Hamming Distance Suppose β„Ž1and β„Ž2 be two HFSs on 𝑋 = {π‘₯1, π‘₯2, π‘₯3, … , π‘₯𝑛} then the hesitant normalized Hamming distance measure between β„Ž1 and β„Ž2 is defined as. 𝑑(β„Ž1, β„Ž2) = 1 𝑙 βˆ‘|β„Ž1𝜎(𝑗) βˆ’ β„Ž2𝜎(𝑗)| 𝑙 𝑗=1 Where 𝑙 is the number of the elements in the β„Ž, in most cases, 𝑙1 β‰  𝑙2 and for convenience, let 𝑙 = max (𝑙1, 𝑙2). 3 Extended VIKOR Method Under HFSs Opricovic [16] adopted the F-VIKOR method to solve problems in an uncertain way where the criteria and weight characterize fuzzy sets [27]. Step 1. Determine PIS and NIS as below π΄βˆ— = {β„Ž1 βˆ— , β„Ž2 βˆ— , β„Ž3 βˆ— , … , β„Žπ‘› βˆ— }, Where β„Žπ‘– βˆ— = ⋃ β„Žπ‘–π‘— π‘š 𝑖 π΄βˆ’ = {β„Ž1 βˆ’, β„Ž2 βˆ’, β„Ž3 βˆ’, … , β„Žπ‘› βˆ’}, Where β„Žπ‘– βˆ’ = β‹‚ β„Žπ‘–π‘— , 𝑗 = 1,2,3, … ,π‘š.π‘š 𝑖 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 216 https://internationalpubls.com Step 2. Compute 𝑆𝑖 and 𝑅𝑖 as below 𝑆𝑖 = βˆ‘π‘€π‘— ||β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—|| ||β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’|| 𝑛 𝑗=1 Calculate regret measure 𝑅𝑖 = max 𝑗 (𝑆𝑖) = 𝑀𝑗 ||β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—|| ||β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’|| , where 𝑖 = 1,2,3, … ,π‘š. Step 3. Evaluate 𝑄𝑖 as given below 𝑄𝑖 = 𝑣 𝑆𝑖 βˆ’ π‘†βˆ’ π‘†βˆ— βˆ’ π‘†βˆ’ + (1 βˆ’ 𝑣) 𝑅𝑖 βˆ’ π‘…βˆ’ π‘…βˆ— βˆ’ π‘…βˆ’ And 𝑣 is introduced weight, where π‘†βˆ— = min 𝑖 (𝑆𝑖), π‘†βˆ’ = max 𝑖 (𝑆𝑖), π‘…βˆ— = min 𝑖 (𝑅𝑖) and π‘…βˆ’ = max 𝑖 (𝑅𝑖) 𝑖 = 1,2,3, … ,π‘š. Step 4. Classify the alternatives, categorizing through 𝑆, 𝑅, and 𝑄 values from largest to smallest. The outcome is in three grades. Step 5. Provide the alternative (𝐴1), which is graded top by the smallest value 𝑄, as a compromise solution if the two factors persist, as given below. C1. Acceptable advantage. 𝑄(𝐴2) βˆ’ 𝑄(𝐴1) β‰₯ 𝐷𝑄 Where (𝐴2) is the alternative with 2𝑛𝑑 position in the grading list 𝑄, 𝐷𝑄 = 1 π‘šβˆ’1 where π‘š denotes the possible alternatives. C2.''Acceptable stability in decision making". An alternative (𝐴1) should also possess the highest position 𝑆 or 𝑅. In a decision-making procedure, such as voting by majority rule 𝑣 > 0.5 or by consensus 𝑣 β‰ˆ 0.5 or with veto 𝑣 < 0.5, this compromise solution is stable. As suggested, the following compromise solution is if one of the requirements is not fulfilled. β€’ Alternative (𝐴1) and (𝐴2) if only C2 is not satisfied, or β€’ Alternatives (𝐴𝑖) where 𝑖 = 1,2,3, … ,π‘š. if C1 is not satisfied; (π΄π‘š) It is determined by the relation 𝑄(π΄π‘š) βˆ’ 𝑄(𝐴1) ≀ 𝐷𝑄 by maximum π‘š. 4 An IV-VIKOR method from the extended VIKOR method under HFSs IV-VIKOR method means an inversion of the VIKOR method in the HFS information. The values of the difference of PIS and NIS should more appropriately be considered as hesitant fuzzy elements in this method because, in the hesitant fuzzy VIKOR method, the difference of the values of PIS and NIS is in division form, but in this chapter, as in multiplication. Because of this, in the current research, we extend the IV-VIKOR under the HFSs approach to solve the MCDM problem. The following structure is used for hesitant fuzzy elements. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 217 https://internationalpubls.com Step 1. Determine PIS and NIS. π΄βˆ— = {β„Ž1 βˆ— , β„Ž2 βˆ— , β„Ž3 βˆ— , … , β„Žπ‘› βˆ— }, Where β„Žπ‘– βˆ— = ⋃ β„Žπ‘–π‘— π‘š 𝑖 π΄βˆ’ = {β„Ž1 βˆ’, β„Ž2 βˆ’, β„Ž3 βˆ’, … , β„Žπ‘› βˆ’}, Where β„Žπ‘– βˆ’ = β‹‚ β„Žπ‘–π‘— , 𝑗 = 1,2,3, … ,π‘š.π‘š 𝑖 Step 2. Compute 𝑆𝑖 and 𝑅𝑖 as below 𝑆𝑖 = βˆ‘π‘€π‘—β€–(β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—). (β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’)β€– 𝑛 𝑗=1 Calculate regret measure 𝑅𝑖 = max 𝑗 (𝑆𝑖) = 𝑀𝑗‖(β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—). (β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’)β€– where 𝑖 = 1,2,3, … ,π‘š. Step 3. Evaluate 𝑄𝑖 as given below 𝑄𝑖 = 𝑣(𝑆𝑖 βˆ’ π‘†βˆ’)(π‘†βˆ— βˆ’ π‘†βˆ’) + (1 βˆ’ 𝑣)(𝑅𝑖 βˆ’ π‘…βˆ’). (π‘…βˆ— βˆ’ π‘…βˆ’) And 𝑣 is introduced weight, where π‘†βˆ— = min 𝑖 (𝑆𝑖), π‘†βˆ’ = max 𝑖 (𝑆𝑖), π‘…βˆ— = min 𝑖 (𝑅𝑖) and π‘…βˆ’ = max 𝑖 (𝑅𝑖) 𝑖 = 1,2,3, … ,π‘š. Step 4. Classify the alternatives, categorizing through 𝑆, 𝑅, and 𝑄 values from largest to smallest. The outcome is in three grades. Step 5. Provide the alternative (𝐴1), which is graded top by the smallest value 𝑄, as a compromise solution if the two factors persist, as given below. C1. Acceptable advantage. 𝑄(𝐴2) βˆ’ 𝑄(𝐴1) ≀ 𝐷𝑄 Where (𝐴2) is the alternative with 2𝑛𝑑 position in the grading list by 𝑄,𝐷𝑄 = 1 π‘šβˆ’1 where π‘š denotes the possible alternatives. C2.''Acceptable stability in decision making". An alternative should also possess the highest position from 𝑆 or𝑅. In a decision-making procedure, such as voting by majority rule 𝑣 > 0.5 or by consensus 𝑣 β‰ˆ 0.5 or with veto 𝑣 < 0.5, this compromise solution is stable. As suggested, the following compromise solution is if one of the requirements is not fulfilled. β€’ Alternative (𝐴1) and (𝐴2) if only C2 is not satisfied, or Alternatives 𝐴𝑖 where 𝑖 = 1,2,3, … ,π‘š. if C1 is not satisfied; (π΄π‘š) it is determined by the relation 𝑄(π΄π‘š) βˆ’ 𝑄(𝐴1) β‰₯ 𝐷𝑄 by maximum π‘š. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 218 https://internationalpubls.com 5 Numerical Example Suppose we have to find out the best airlines as 𝐴, 𝐡, 𝐢, and 𝐷 with criteria weight 𝑀 = (0.21,0.32,0.41,0.06) on the basis of multiple criteria as given, πœ…1: Customer service, πœ…2: Comfort and Space, πœ…3: Great price, great deals, πœ…4: Inflight meals. Table 1. Hesitant fuzzy decision matrix (vi) Alternatives 𝐢1 𝐢2 𝐢3 𝐢4 𝐴 (0.3, 0.4) (0.1, 0.2, 0.4) (0.1, 0.2, 0.3,0.5) (0.4, 0.5, 0.6) 𝐡 (0.1, 0.2, 0.3) (0.1, 0.2, 0.3) (0.2, 0.3) (0.2, 0.3) 𝐢 (0.1, 0.2, 0.3) (0.2, 0.4, 0.7) (0.3, 0.4, 0.5) (0.1, 0.8) 𝐷 (0.3, 0.4) (0.4, 0.5, 0.6, 0.7) (0.4, 0.5, 0.6) (0.5, 0.6, 0.7, 0.8) IV-VIKOR method from extended VIKOR method under HFS. Step 1. Determined PIS and NIS as below. π΄βˆ— = [0.4, 0.7, 0.6, 0.8] π΄βˆ’ = [0.1, 0.1, 0.1, 0.1] Step 2. Computed 𝑆𝑖 and 𝑅𝑖 as below 𝑆𝑖 = βˆ‘π‘€π‘—β€–(β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—). (β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’)β€– 𝑛 𝑗=1 𝑆1 = 0.17035, 𝑆2 = 0.20345, 𝑆3 = 0.1195, 𝑆4 = 0.08605. Calculated regret measure 𝑅𝑖 = max 𝑗 (𝑆𝑖) = 𝑀𝑗‖(β„Žπ‘— βˆ— βˆ’ β„Žπ‘–π‘—). (β„Žπ‘— βˆ— βˆ’ β„Žπ‘— βˆ’)β€– 𝑅1 = 0.0896, 𝑅2 = 0.096, 𝑅3 = 0.0512, 𝑅4 = 0.0478. Step 3. Evaluated 𝑄𝑖 as given below 𝑄𝑖 = 𝑣(𝑆𝑖 βˆ’ π‘†βˆ’)(π‘†βˆ— βˆ’ π‘†βˆ’) + (1 βˆ’ 𝑣)(𝑅𝑖 βˆ’ π‘…βˆ’). (π‘…βˆ— βˆ’ π‘…βˆ’) 𝑄1 = 0.0059, 𝑄2 = 0.0080, 𝑄3 = 0.00204, 𝑄4 = 0.000. Step 4. Classified the alternatives, categorizing through 𝑆, 𝑅, and 𝑄 values, from largest to smallest. The outcome is in three grades, as listed in Table 2. Step 5. Provide the alternative (𝐴), which is graded top by smallest values by 𝑄, as a compromise solution if the factor persists, as given below. C1. Acceptable advantage. C1. 𝑄(𝐡) βˆ’ 𝑄(𝐴) ≀ 1 3 satisfied. It means that the alternative 𝐴 is a compromise solution. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 219 https://internationalpubls.com Table 2. Ranking the alternatives through the IV-VIKOR method under hesitant fuzzy sets Alternatives 𝑆 𝑅 𝑄 Ranking 𝐴 0.17035 0.0896 0.00594 1 𝐡 0.20345 0.096 0.0080 2 𝐢 0.1195 0.0512 0.00204 4 𝐷 0.08605 0.0478 0.000 3 Conclusion. In this study, a novel concept for IV-VIKOR technique under hesitant fuzzy information was devised, and multi-criteria issues with various and incommensurable criteria were resolved, notably accounting for the complicated subjective nature of the decision maker. The difference between the PIS and NIS in this IV-VIKOR method under HFSs is in multiplication form, however in the VIKOR method under HFSs, is it in division form. This approach demonstrates the VIKOR method’s inversion because we multiply their difference in this way rather than dividing it. Complex MCDM problems can be solved with this strategy extremely effectively. References. [1] Taherdoost, H. and M. Madanchian, Multi-criteria decision making (MCDM) methods and concepts. Encyclopedia, 2023. 3(1): p. 77-87. [2] Yin, S., et al., Practical teaching method innovation decisions related to labor and reading at agricultural colleges based on entropy-fuzzy AHP combination weights. AIMS Mathematics, 2023. 8(4): p. 7641-7661. 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