An application of interval valued neutrosophic soft multisets in MCDM

Abstract

Keywords: Accuracy Function; Interval Valued Neutrosophic Sets; Interval Valued Neutrosophic Soft Sets; Interval Valued Neutrosophic Soft Multisets; WSM; WPM; TOPSIS

Introduction

Lotfi A. Zadeh [1] introduced Fuzzy sets and Fuzzy logic to study the nature of an object that cannot be described clearly. It assigns membership values ranging between 0 to 1 for those elements to characterize their nature. Thereafter, it can be extended to Interval valued fuzzy sets [2] which allocate intervals as membership values. In real world, there are innumerable situations based on both truth membership and false membership for proper description of an object in uncertain situations. To handle those situations Intuitionistic fuzzy sets [3] were developed. Later, it was extended to Interval valued intuitionistic fuzzy sets [4] by K. Atanassov. F. Smarandache [5] extended fuzzy sets, classical set theory and intuitionistic set theory in the name of Neutrosophic sets to talk about indeterminacy nature of an object through an indeterminacy membership function.

Later, Wang et al. [6] extended neutrosophic sets to Interval valued neutrosophic sets which is more flexible to deal in case of real-life problems. Molodtsov [7] initiated a new mathematical tool called Soft sets for handling uncertain information. It can easily handle objects with uncertain nature by the help of parameter sets which is one of the major benefits while using soft sets. Several researchers have studied soft sets. P.K. Maji gave an application of soft sets in a decision making problem with the aid of rough set theory [8]. Chen et al. [9] initiated the new concept called parameterization reduction of soft sets which reduced the attributes to enhance the application of soft sets. Çağman-Enginoğlu defined soft matrices to solve problems without the help of fuzzy soft sets or rough sets and they defined some operations on soft matrices [10]. Also, they presented a soft max-min decision making algorithm to select an optimal decision from the

*Corresponding author. *E-mail address: kowsalyachinnaraj176@gmail.com This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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set of alternatives. Subsequently they defined the product of soft sets and presented a uni-int decision making method in [11]. Feng et al. [12] devised new decision making methods namely uni-𝑖𝑛𝑡𝑘, 𝑢𝑛𝑖 − 𝑖𝑛𝑡𝑡, and 𝑖𝑛𝑡𝑚 − 𝑖𝑛𝑡𝑛. Later, Fuzzy soft sets [13] and intuitionistic fuzzy soft sets [14] were introduced as an extension of fuzzy sets and intuitionistic fuzzy sets. Later, Neutrosophic soft sets were introduced as an extension of neutrosophic sets by P.K. Maji [15]. It was a new mathematical approach for describing indeterminate and incomplete information through parameter sets and membership functions. I. Deli et al. [16] introduced Neutrosophic soft matrix and some operations on them and further solved a decision making problem using Neutrosophic soft matrices. I. Deli [17] proposed Interval valued neutrosophic soft sets as a generalization of soft sets, fuzzy soft sets, intuitionistic fuzzy soft sets, interval valued intuitionistic fuzzy soft sets and neutrosophic soft sets.

underdeveloped areas. In medical field, Z. Yong [31] used WSM method to analyze Breast cancer. C.A. Alban-Perez et al. [32] applied WSM technique in decision making to improve the structure of a complete street in Colombia. Weighted Product Method (WPM) is a simplified model of WSM which is used in multi-dimensional decision making problems. Particularly, Sathiyaraj et al. [33] used WPM technique to evaluate drinking water quality in Salem district. Many industrial decision making problems were dealt by WSM and WPM. Hwang and Yoon [34] defined one of the primary methods, called TOPSIS (Technique for Order Performance by Similarity to Ideal Solution). TOPSIS technique is to provide an optimal solution by measuring distance from PIS (Positive Ideal Solution) and NIS (Negative Ideal Solution) for each alternative. PIS is the most preferred solution and NIS is the least preferred solution as decided by decision makers. Triantaphyllou and Lin [35] used fuzzy arithmetic operator to define the fuzzy version of TOPSIS method, which shows fuzzy relative closeness. Chen [36] applied TOPSIS method to fuzzy group decision making by defining the crisp Euclidean distance between fuzzy numbers. Jahanshahloo et al. [37] extended the TOPSIS approach for decision making problems involving interval data. Chen and Tsao [38] expanded the TOPSIS method to solve MADM problems involving interval valued fuzzy data and compared the result by Hamming distance and Euclidean distance. M. Imtiaz et al. [39] used the accuracy function in the TOPSIS method to solve the MCDM problem in Octagonal intuitionistic fuzzy environment. Additionally, in the medical field TOPSIS, WSM and WPM techniques were used for selecting LASER as an efficient surgical instrument [40].

Research Gap and Motivation Alkhazaleh et al. [18] introduced soft multisets to expand soft sets in multiple dimensions. Soft multisets are used to deal with multiple universes at a time. Many researchers have dealt with soft multisets ( [19], [20]). Further, Alkhazaleh and Salleh [21] defined fuzzy soft multisets and solved a decision making problem for fuzzy soft multisets. A.E. Coskun [22] employs soft matrices on soft multisets to explore the use of soft matrices in decision making. I. Deli et al. [23] initiated Neutrosophic soft multisets that handle uncertain and indeterminate information in multiple universes. Neutrosophic vague soft multisets and Weighted neutrosophic soft multisets were introduced by A. Al-Quran & N. Hassan [24] and C. Granados et. Al [25] respectively as well as some real-life applications were discussed. Multi-criteria decision making (MCDM) provides a method for decision making in a practical and familiar situation in which multiple criteria are taken into consideration. The aim of this technique is to help decision makers where a large number of alternatives exist. MCDM attempts to choose a best alternative among the set of given alternatives. There are lot of methods used to solve MCDM problems to name a few, AHP, TOPSIS, ELECTRE and PROMETHEE, but they are all unique in their own way. Many researchers used hybrid structures of fuzzy set theory, soft set theory, rough set theory and multiset theory to solve MCDM problems in uncertain environment ( [26], [27], [28], [29]). Weighted Sum Method (WSM) and Weighted Product Method (WPM) are more frequently used in MCDM. In both the methods alternatives are being compared with one another according to weights and criteria. Weighted Sum Method (WSM) allocates weights to each criteria according to their importance then calculate weighted sum for each alternative and finally choose the best one. D. Handoko et al. [30] applied WSM method to determine the allocation of special funds to primary and secondary schools located in

Contribution The first section begins with a brief Introduction followed by the study framework of Interval valued neutrosophic soft multisets. In section 2, some fundamental definitions and concepts are discussed. In section 3, we propose new algorithms using accuracy function which employs WSM, WPM and TOPSIS techniques to solve problems using Interval valued neutrosophic soft multisets. In section 4, we make a comparison of some fundamental sets with IVNSMS and additionally the results of all three methods are compared and analyzed to find the best alternative in all given universes. Finally in section 5, we conclude our research work.

Preliminaries

Definition 1 [5] Let 𝑈 be a space of points (objects), with a generic element in 𝑈 denoted by u. A neutrosophic set 𝐴 in 𝑈 is characterized by a truth-membership function 𝑇𝐴, an indeterminacy-membership function 𝐼𝐴 and a falsity- membership function 𝐹𝐴. 𝑇𝐴(u); 𝐼𝐴(u) and 𝐹𝐴(u) are real standard or nonstandard subsets of [0, 1].

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There is no restriction on the sum of 𝑇𝐴(u); 𝐼𝐴(u) and 𝐹𝐴(u), so 0 ≤ 𝑠𝑢𝑝𝑇𝐴(𝑢) + 𝑠𝑢𝑝𝐼𝐴(𝑢) + 𝑠𝑢𝑝𝐹𝐴(𝑢) ≤ 3. Definition 2 [6] Let 𝑈 be a space of points (objects), with a generic element in 𝑈 denoted by u. An interval valued neutrosophic set (IVN-set) 𝐴 in 𝑈 is characterized by truth-membership function 𝑇𝐴, an indeterminacy-membership function 𝐼𝐴 and a falsity-membership function 𝐹𝐴. For each point 𝑢 ∈ 𝑈; 𝑇𝐴, 𝐼𝐴 and 𝐹𝐴 ⊆ [0,1]. Thus, an IVN-set over U can be represented by the set

Neutrosophic SOFT Multisets In Decision Making

Algorithm 1: Weighted Sum Method WSM technique is a frequently used technique in MCDM, which is strong in single dimension problems. Now, we propose a WSM algorithm to solve problem in multiple universes.

2. To use the accuracy function

Here, (𝑇𝐴(𝑢), 𝐼𝐴(𝑢), 𝐹𝐴(𝑢)) is called interval valued neutrosophic number for all 𝑢 ∈ 𝑈 and all interval valued neutrosophic numbers over U will be denoted by 𝐼𝑉𝑁(𝑈). Definition 3 [17] Let 𝑈 be an initial universe set, 𝐼𝑉𝑁(𝑈) denotes the set of all interval valued neutrosophic sets of 𝑈 and 𝐸 be a set of parameters that describe the elements of 𝑈. An interval valued neutrosophic soft sets over 𝑈 is a set defined by a set valued function 𝛶𝐾 representing a mapping 𝑣𝐾: 𝐸 → 𝐼𝑉𝑁(𝑈).

It can be written a set of ordered pairs 𝛶𝐾 = { (𝑥, 𝑣𝐾(𝑥)) ∶ 𝑥 ∈ 𝐸}.

Here, an interval valued neutrosophic set 𝜈𝐾 is called approximate function of the interval valued neutrosophic (ivn)- soft sets 𝛶𝐾. And 𝑣𝐾(𝑥) is called x-approximate value of 𝑥 ∈ 𝐸. Definition 4 [41] Let 𝑥 = ([𝑇𝐿, 𝑇𝑈], [𝐼𝐿, 𝐼𝑈], [𝐹𝐿, 𝐹𝑈]) be an Interval neutrosophic number and the accuracy function 𝑎(𝑥) of an Interval neutrosophic number can be defined as follows:

In this section we try to solve a decision making problem using Interval valued neutrosophic soft multisets. If a person wants to purchase a personal computer, he has lot of choices for the same. By using the following algorithm and by setting required number of parameters, we shall arrive at a wise conclusion.

4. Construct the normalized decision matrix 𝑅𝑖.

For beneficial attributes (criteria of benefits): (2) For non-beneficial attributes (criteria of cost): (3)

5. Construct the weighted normalized decision matrix

Definition 5 [18] Let {𝑈𝑖: 𝑖 ∈ 𝑈} be a collection of universes such that ⋂𝑖∈𝐼 𝑈𝑖 = ∅ and let {𝐸𝑈𝑖: 𝑖 ∈ 𝐼} be a collection of sets of parameters, 𝑈 = ∏𝑖∈𝐼 𝑃(𝑈𝑖) where 𝑃(𝑈𝑖) denotes the powerset of 𝑈𝑖, 𝐸 = ∏𝑖∈𝐼 𝐸𝑈𝑖 and 𝐴 ⊆ 𝐸. A pair (I, A) is called a soft multiset over U given by the mapping 𝐼: 𝐴 → 𝑈. Definition 6 [42] Let {𝑈𝑖: 𝑖 ∈ 𝑈} be a collection of universes such that ⋂𝑖∈𝐼 𝑈𝑖 = ∅ and let {𝐸𝑈𝑖: 𝑖 ∈ 𝐼} be a collection of sets of parameters, 𝑈 = ∏𝑖∈𝐼 𝐼𝑉𝑁(𝑈𝑖) where 𝐼𝑉𝑁(𝑈𝑖) denotes the set of all Interval valued neutrosophic sets of 𝑈𝑖, 𝐸 = ∏𝑖∈𝐼 𝐸𝑈𝑖 and 𝐴 ⊆ 𝐸. An Interval valued neutrosophic soft multiset (IVNSMS) over U is the pair (I, A) given by the mapping 𝐼: 𝐴 → 𝑈. It can be represented by, (I, A) = {(ak, 〈[inf TI(𝑢), supTI(u)], [inf II(u), supII(u)], [inf FI(u), supFI(u)]〉) ∶ ak ∈ A ⊆ E, u ∈ U}.

8. Continuing this procedure to all 𝐼𝑉𝑁𝑆𝑀-set parts

(all universes), finally we obtain the optimal decision (𝐴1, 𝐴2, … . , 𝐴𝑛).

Algorithm 2: Weighted Product Method WPM technique is a more efficient technique in MCDM. In this technique, alternatives are evaluated by

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multiplying the normalized data to the power of the corresponding weight criteria.

5. Construct the weighted normalized decision matrix

on every element of (𝐼, 𝐴) to get the matrix , 𝑘 = 1,2, … 𝑛 and (𝑖, 𝑗) = |𝑈1| + |𝑈2|+. . . +|𝑈𝑖|.

6. Determine the Positive Ideal Solution (PIS) and

where 𝐾+ is the parameter set of benefit type and 𝐾− is the parameter set of cost type.

8. Continuing this procedure to all 𝐼𝑉𝑁𝑆𝑀-set parts

(remaining universes), finally we obtain the optimal decision (𝐴1, 𝐴2, … . , 𝐴𝑛)

Algorithm 3: TOPSIS Method The TOPSIS method is a straight-forward method in MCDM. The main idea of the TOPSIS method is to rank the optimal solution that is closer to PIS and far from NIS.

2. To use the accuracy function

on every element of (𝐼, 𝐴) to get the matrix , 𝑘 = 1,2, … 𝑛 and (𝑖, 𝑗) = |𝑈1| + |𝑈2|+. . . +|𝑈𝑖|.

9. Continuing the above steps 3 to 8 for all 𝐼𝑉𝑁𝑆𝑀-set

parts (all universes), finally we obtain the optimal decision (𝐴1, 𝐴2, … . , 𝐴𝑛).

Example 1. Suppose that a person Mr. X wants to buy a computer, a printer and an UPS for his personal work within his budget. Let (𝐼, 𝐴) be an 𝐼𝑉𝑁𝑆𝑀𝑆(𝑈) which describes “computers,” “printers” and “UPS for PC” respectively that Mr. X is considering a good branded computer, printer for document work and an UPS for battery backup. Let 𝑈1 = {𝑐1, 𝑐2, 𝑐3, 𝑐4} be the universe for branded computers, 𝑈2 = {𝑝1, 𝑝2, 𝑝3} be the universe for printers and 𝑈3 = {𝑢1, 𝑢2, 𝑢3} be the universe for UPS. Let {𝐸𝑈1, 𝐸𝑈2, 𝐸𝑈3 } be a collection of parameters which describes above universes, where 𝐸𝑈1 = {𝑒𝑈1,1 = Intel core i3 processor; 𝑒𝑈1,2 = 16GB RAM; 𝑒𝑈1,3 = AMD processor; 𝑒𝑈1,4 = SSD Storage; 𝑒𝑈1,5 = OS window 11}

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𝐸𝑈2 = {𝑒𝑈2,1 = Inkjet; 𝑒𝑈2,2 = multi function; 𝑒𝑈2,3 = single function; 𝑒𝑈2,4 = built in wifi; 𝑒𝑈2,5 = good printing speed} 𝐸𝑈3 = {𝑒𝑈3,1 = Output 600VA; 𝑒𝑈3,2 = Backup time upto 40mins; 𝑒𝑈3,3 = Output socket 3 nos}.

Let and 𝐴 ⊆ 𝐸 such that 𝐴 = {𝑎1 = (𝑒𝑈1,1, 𝑒𝑈2,1, 𝑒𝑈3,1), 𝑎2 = (𝑒𝑈1,3, 𝑒𝑈2,2, 𝑒𝑈3,2), 𝑎3 = (𝑒𝑈1,2, 𝑒𝑈2,1, 𝑒𝑈3,1), 𝑎4 = (𝑒𝑈1,4, 𝑒𝑈2,3, 𝑒𝑈3,3), 𝑎5 = (𝑒𝑈1,5, 𝑒𝑈2,5, 𝑒𝑈3,2)}. Suppose that Mr. X wants to choose a combination of computer, printer and UPS for PC from the set of given objects with respect to the set of choice parameters. WSM Algorithm: Let Table 1. represents the Interval valued neutrosophic soft multiset (𝐼, 𝐴). Conversation of 𝐼𝑉𝑁𝑆 to 𝐼𝑉𝑁𝑁 by using accuracy func-

Table 3 represents the weight criteria given by expert. Table 3. The weighted criteria a1

Construct the normalized decision matrix for 𝑈1 by using Equation (2).

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Now, let us construct the weighted normalized decision matrix for 𝑈1 and calculate the score of alternatives using Equations (4) and (5).

For the 𝑈2-part, the weighted normalized decision matrix is constructed and the score of each alternative is calculated as given below. Hence, 𝑝1 > 𝑝2 > 𝑝3 is the ranking order of 𝑈2 alternatives and 𝑝1 is the best alternative in the universe of printers. Next, let us consider the 𝑈3 part in 𝑀. The normalized decision matrix for 𝑈3 is computed as follows. Repeating the procedure as above for 𝑈3-part, we get the score of alternatives as shown below. Here, the ranking order of 𝑈3 alternatives is 𝑢1 > 𝑢3 >

Hence, the ranking order of 𝑈1 alternatives is 𝑐2 > 𝑐1 > 𝑐4 > 𝑐3 and 𝑐2 is the best alternative in the universe of computers. Now, consider the 𝑈2 part in 𝑀.

The normalized decision matrix for 𝑈2 is constructed in the following table. Table 8. 𝑈2 normalized decision matrix U2 p1 p2 p3

𝑢2, where 𝑢1 is the best alternative in the universe of UPS. Thus, from the above rankings Mr. X can choose the combination (𝑐2, 𝑝1, 𝑢1). (i.e) Mr. X chooses computer 𝑐2, printer 𝑝1 and UPS 𝑢1 for his personal work. Next, we use WPM algorithm to solve the above Example 1.

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Table 13. Tabular representation of Normalized decision matrix Ui U1

WPM Algorithm Conversation of 𝐼𝑉𝑁𝑆 to 𝐼𝑉𝑁𝑁 by using accuracy function is identical to WSM method. Next, we use Equation (7) and construct the normalized decision matrix and display it in Table 13. Now, we use Equation (9) to construct the weighted normalized decision matrix for 𝑈1 and calculate the score of alternatives. The resultant matrix is given below. Hence, the ranking order of 𝑈1 alternatives is 𝑐2 > 𝑐1 > 𝑐4 > 𝑐3, where 𝑐2 is the best alternative in the universe of computers.

Now, construct the weighted normalized decision matrix for 𝑈2 and calculate the score of alternatives as executed in 𝑈1-part. The ranking order of 𝑈2 alternatives is 𝑝1 > 𝑝2 > 𝑝3, where 𝑝1 is the best alternative in the universe of printers. We shall follow the same steps for 𝑈3 and calculate the score of alternatives and determine the ranking order. Hence, 𝑢1 > 𝑢3 > 𝑢2 is the ranking order of 𝑈3 alternatives and 𝑢1 being the best alternative in the universe of UPS.

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Thus, from the above rankings Mr. X can choose the combination (𝑐2, 𝑝1, 𝑢1). Now, we shall apply TOPSIS method and solve Example 1.

TOPSIS Method Conversation of 𝐼𝑉𝑁𝑆 to 𝐼𝑉𝑁𝑁 by using accuracy function is identical to WSM method and the same was presented in Table 2. Let us consider the 𝑈1-part of 𝑀 and construct the normalized decision matrix using Equation (12). The resulting matrix is given in Table 17.

Constructing the weighted normalized decision matrix for 𝑈1 by using weight vector for each criteria, we get the following matrix. Table 18. Weighted normalized decision matrix for 𝑈1 U1

Hence, the ranking order of 𝑈1 alternatives based on relative closeness coefficient is 𝑐2 > 𝑐1 > 𝑐4 > 𝑐3 and 𝑐2 is the best alternative in the universe of computers. Now, consider the 𝑈2-part of 𝑀 and construct the normalized decision matrix as done in 𝑈1-part.

Now, let us determine PIS and NIS as follows: 𝑋+ = {0.18303, 0.17253, 0.05181, 0.05372, 0.11138} and 𝑋− = {0.11161, 0.12686, 0.04737, 0.04700, 0.09325} Using the PIS and NIS, let us calculate the separation measures and closeness coefficient and represent it in Table 19. Table 19. Table of relative closeness coefficient

The weighted normalized decision matrix for 𝑈2 is constructed by using weight vector for each criteria. Hence, we get the following matrix. Table 21. Weighted normalized decision matrix for 𝑈2

The PIS and NIS are determined as follows: 𝑋+ = {0.20056, 0.17943, 0.06237, 0.07008, 0.12013} and 𝑋− = {0.13697, 0.16003, 0.05346, 0.04672, 0.11307} The separation measures and closeness coefficient are calculated using PIS and NIS. Table 22. Table of relative closeness coefficient U2 p1 p2 p3

Here, the ranking order of 𝑈2 alternatives based on relative closeness coefficient is 𝑝1 > 𝑝2 > 𝑝3, where 𝑝1 is the best alternative in the universe of printers. We shall repeat the same procedure for 𝑈3-part of 𝑀 and obtain the following matrices.

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Comparison

The PIS and NIS are as follows: 𝑋+ = {0.21764, 0.18890, 0.06162, 0.06146, 0.12425} and 𝑋− = {0.14332, 0.14482, 0.05306, 0.05463, 0.11082} Now, we have the following table using PIS and NIS. Table 25. Table of relative closeness coefficient U3 u1

The ranking order of 𝑈3 alternatives based on relative closeness coefficient is 𝑢1 > 𝑢3 > 𝑢2, and 𝑢1 being the best alternative in the universe of UPS. Thus, from the above rankings Mr. X can choose the combination (𝑐2, 𝑝1, 𝑢1).

The decision making problem in Example 1 is converted to 𝐼𝑉𝑁𝑆𝑀𝑆 and solved by proposed WSM, WPM and TOPSIS methods. By using WSM, WPM and TOPSIS techniques we calculate the rank of all alternatives in all universes. Now, let us compare the results obtained by WSM, WPM and TOPSIS. In this comparison, rank of all alternatives in each universe are presented in the following table. From the comparative study, each alternative in the available universe achieves the same rank in all the methods. According to the results, the combination (𝑐2, 𝑝1, 𝑢1) is the first choice for Mr. X. Otherwise he goes with the second choice (𝑐1, 𝑝2, 𝑢3) and the worst choice is either (𝑐3, 𝑝3, 𝑢2) or (𝑐4, 𝑝3, 𝑢2). In this section, we compare the theory of IVNSMS with other existing theories like Soft sets (SS), Neutrosophic soft sets (NSS), Interval valued neutrosophic soft sets (IVNSS), Soft multisets (SMS) and Neutrosophic soft multisets (NSMS) to show that IVNSMS are more adaptable and generalized than other existing sets. The comparison analysis is developed by their characters like domain, co-domain, universe, and membership functions. In Table 27, a comparison analysis of the IVNSMS with other above-mentioned sets has been executed. Now, we observe that the set IVNSMS is a generalized form of all above mentioned sets. The main advantage of IVNSMS is that it describes real world problems involving multiple universes, one at a time. Another merit of the set is that it can address information ranges from minimum to maximum with the help of interval membership functions. The application of IVNSMS can assist people pursue a right choice out of capable choices in uncertain and incomplete data conditions.

Table 26. Alternative rank comparison using WSM, WPM and TOPSIS Universes U1

Topsis

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Ethics

In this paper, we have proposed WSM, WPM and TOPSIS methods to solve MCDM problems for Interval valued neutrosophic soft multisets and a decision making problem is illustrated. We have demonstrated the significance of IVNSMS by comparing it with other relevant models. In addition, we have compared the results and concluded that three methods prefer the same combination to Mr. X. These techniques were more adaptable and effective in solving decision making problems and were very useful to rank the alternatives in multiple universes at the same time. These algorithms can be used in many practical problems like decision making and personal selection problems. The major advantage of IVNSMS is that it can be used to maximize the benefits and minimize the cost by considering only choice parameters. In future work, we will provide many applications of these methods and propose a distance based TOPSIS technique to handle MCDM problems for Interval valued neutrosophic soft multisets.

There are no ethical issues with the publication of this manuscript.

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