Measure of Distance and Similarity for Single Valued Neutrosophic Sets with Application in Multi-attribute

Transcription

1 Measure of Distance and imilarity for ingle Valued Neutrosophic ets ith pplication in Multi-attribute Decision Making *Dr. Pratiksha Tiari * ssistant Professor, Delhi Institute of dvanced tudies, Delhi, India

2 BTRCT single valued neutrosophic sets (vns) is a particular case of neutrosophic set, hich can handle scientific and engineering applications in real orld. Distance, similarity, entropy and cross entropy measures of vns are major tool is to solve various real orld problems In the paper, an average distance measure based on Hamming and Hausdorff distance is defined for vns. imilarity measure is proposed on the basis of defined averaged distance using the relationship beteen distance and similarity measures. Further, one more similarity measure is defined and its validity is proved. Then a multi-attribute decision making is also established under single valued neutrosophic environment, in hich attribute values are assigned to each course of action by the decision makers/experts and attribute eights are knon, hereas the optimal point is defined using the decision matrix using the proposed method. lso, an algorithm is presented to determine the ranking of available course of actions on the basis of values of measure of similarity amongst optimal point and the available course of actions. Lastly, an example is used to demonstrate the application of proposed similarity measures in multi-attribute decision making. 09

3 INTRODUCTION Neutrosophic set developed by marand ache (1999) is a ne evolving instrument for uncertain data processing. It has the potential of being a frameork for analysis of uncertain data sets hich include big data sets. Neutrosophic sets are generalization of interval valued intuitionistic fuzzy sets. Because of the fuzziness and uncertainty of many practical problems in the real orld, it is applicable to a ide range of practical problems. Zadeh (1965) developed fuzzy set theory hich is a huge success in various areas involving uncertainty. In fuzzy set theory, non-membership value of an element is defined normally as complement of its membership value from one, but practically it is not so. This situation is dealt by higher order fuzzy sets proposed by tanassov (1986) and is termed as intuitionistic fuzzy sets (IFs). It is found to be highly useful in dealing vagueness and hesitancy originated from inadequate information. It characterizes three characteristic functions for membership and nonmembership and hesitancy respectively for an element of the universe of discourse here sum of all the three functions is 1. Therefore, IFs are much more flexible and practical than fuzzy sets in dealing ith vagueness and uncertainty problems. Hoever, fuzzy sets, IFs, cannot handle indeterminate information and inconsistent information that exist in the real orld and e need further structures such as neutrosophic sets hich is a poerful generalization of classic sets, fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval valued intuitionistic fuzzy sets, paraconsistent sets, dialetheist set, paradoxist sets, and tautological sets. Like intuitionistic fuzzy sets,neutrosophic sets are also characterized by three functions: truth-membership, indeterminacy-membership, and falsity-membership, but they are represented independently. The neutrosophic sets generalize the above-mentioned sets from a philosophical point of vie and its functions are real standard or nonstandard subsets of ] 0, 1+[, and there is no restriction on the sum of three. Thus, it ill be difficult to apply these in real scientific and engineering areas. Thus, Wang et al. (2010) derived single-valued neutrosophic set (vn), hich is a particular case of neutrosophic set. It can describe and handle indeterminate information and inconsistent information. For example, hen e ask a customer for his/ her opinion about a statement, he/she may say that the possibility of agreeing to a statement is 0.6 and disagreeing is 0.4 and not sure is 0.2. For a vn notation it can be represented as (0.6, 0.4, 0.2) hereas it cannot be dealt by intuitionistic fuzzy sets as sum of membership, non-membership and hesitation value is not 1. Therefore, the notion of a neutrosophic set is more general and overcomes the aforementioned issues. Various researcher studied different measures such as entropy, cross entropy, distance and similarity for fuzzy and intuitionistic fuzzy sets that ill help in decision making. The concept of similarity is primarily significant in practically every field. Many methods have been proposed for measuring the degree of similarity beteen fuzzy and intuitionistic fuzzy sets. But these methods are not suitable for dealing ith the similarity measures of neutrosophic set. Fe researchers have dealt ith the similarity measures for neutrosophic set. This paper deals ith the similarity measures for single valued neutrosophic sets and demonstrates its use in multi-attribute decision making, using an illustrative example. IC CONCEPT ND RELTED WORK Neutrosophic sets emerged as a tool to deal uncertain data. It has the potential to become a general frameork for uncertainty analysis in data. The neutrosophic set is a part of neutrosophy and generalizes F, IvF, IF, and IvIF from a philosophical point of vie. marandache (1999) derived neutrosophic set hich is defined as follos: Definition 1: Let X be a universe of discourse, a neutrosophic set in X is characterized by a quadruple x, T (x) i.e. (x, truth-membership function, indeterminacymembership function, falsity-membership function), here x X. The functions T (x): X ] 0, 1 + [, I (x),: X ] 0, 1 +[, and F (x): X ] 0, 1 +[. Thus there is no restriction on the sum so 0 sup T (x) + sup I (x) + sup F (x) 3+. Wang et al. (2011) proposed a subclass of neutrosophic sets termed as single valued neutrosophic sets hich are easier to apply to real scientific and engineering problems and is defined as Definition 1.1: Let X universe of discourse, a single valued neutrosophic set (vn) in X is characterized by quadruple x, T (x) i.e. (x, truth-membership function, indeterminacy-membership function, falsity-membership function), here x X, T (x) [0, 1] and 0 T (x) + I (x) + F (x) 3. Definition 2 (et operations on single valued neutrosophic sets): Let and B be to vns defined by quadruple x, T (x) and x, TB(x), IB B(x) respectively, here x X then set operations are defined as follos: 1) B = { x, T (x) T (x), I (x) I (x) F (x) /x X} B B B 2) B = {+x, T (x) T (x), I (x) I (x) F (x) /x X} B B B 3) 4) 5) B Definition 3 (Distance beteen to single valued neutrosophic sets): For any to vns and B, a real valued function D: vns (X) vns () [0, 1] is termed as a distance measure of vns on X, if it satisfies the belo mentioned axioms: 1) Distance beteen any to vns and B is zero if = B. 10

4 2) Distance measure is symmetrical.r.t to any to vns and B. Ye also presented one more similarity measure as 3) For any three vns, B and C such that B C, e have D(, C) D(, B) and D(, C) D(B, C). Distance beteen Fs as presented by (Kacprzyk, 1997). Its extension as proposed by tanassov in 1999 as to dimensional distances hereas third parameter hesitancy degree in distance as introduced by zmidt and Kacprzyk (2000) for intuitionistic fuzzy sets. Yang & Chiclana (2012) proved three dimensional distance consistency over to dimensional distances. Grzegorzeski (2004) and Park et al. (2007) gave distance measure for IvFs and IvIFs respectively. Broumi and marandache (2013) presented eighted Hausdorff distance measure based on Hausdorff distance. Ye (2014) presented eighted distance measures for vns based on Hamming and Euclidean Distances given belo: Weighted Hamming Distance here i is eight corresponding to x i such that eighted Euclidean Distance D 2 (, B) nd Ye (2015) proposed improved measures of cosine similarity measures for simplified neutrosophic sets including single valued cosine similarity and interval valued neutrosophic cosine similarity. He also introduced corresponding eighted similarity measures and applied it to medical diagnoses. ydoğdu (2015) introduced to measures of similarity for vns and developed single entropy measure for the same and applied it to neutrosophic multi-criteria decision making. Ye &marandache (2016) introduced refined single-valued neutrosophic sets, developed similarity measure of the same and applied it to multi criteria decision making. zmidt and Kacprzyk (2009) constructed Hausdorff distance beteen Intuitionistic Fuzzy ets based on the Hamming metric and particularly give importance to the consistency of the metric used and the essence of the Hausdorff distances. Next section deals ith ne eighted distance and similarity measures based on averaged distance measure based on Hamming and Hausdorff distance defined as here i is eight corresponding to x i such that Definition 4 (imilarity beteen vns) : Let and B be any to vns, a real valued function : IvIs () IvIFs () [0, 1] is defined as a measure of similarity for IvIFs on X, if it satisfies folloing axioms: 1) Measure of similarity beteen any to IvIFs is 1 iff = B. 2) Measure of similarity is symmetric.r.t. any to IvIFs. 3) For any three IvIFs, B and C such that B C. We have (, C) (, B) and (, C) (B, C). From axiomatic definition of distance and similarity measures it is clear that (, B) = 1 D(, B) here and B are vns, D and are distance and similarity measure for vns respectively. Broumi and marandache (2013) presented several similarity measures for vns. Ye (2014) presented similarity measures based on equation (2) and (3) as Distance, similarity measures and optimal point is proposed in the folloing sections under single valued neutrosophic environment. Distance and imilarity Measures for vns: This section defines eighted distance and similarity measures based on equation (4) as follos: Definition 5 Distance Measure: Consider to vns and B, represented by quadruple x, T (x) and x, T B(x), I (x) respectively in a universe of discourse X = {x, x,, B B 1 2 x n}. Weighted distance measure D W(, B) based on averaged distance measure is defined as follos: here i is eight corresponding to xi such that Theorem 1: D(, B) is a valid measure of distance beteen to vns and B. Proof: Consider to vns and B represented by quadruple x i, T (x i), I (x i), F (x i) and x i, T B(x i), I B(x i), F B(x i) respectively in a universe of discourse = {x, x,, x }. It is easy to see that 1 2 n D (,B) [0, 1] satisfies axioms (1) and (2) defined in Definition W 3. In order to prove axiom (3), consider three vns, B and C such that B C. 11

5 Then T (x ) + I (x ) + F (x ) T (x ) + I (x ) + F (x )...(11) B i B i B i C i C i C i lso, T (x )=T (x ) T (x )=T (x ) T (x ) = T (x ) T (x ) = T (x )...(12) B i i B i B i C i i C i C i I (x ) = I (x ) I (x ) = I (x ) = I (x ) I (x ) = I (x )...(13) B i i B i i i C i C i nd F (x ) = F (x ) F (x ) = F (x ) F (x ) = F (x ) F (x ) = F (x )...(14) B i i B i B i C i i C i C i dding equations (12), (13) and (14) T (x ) + I (x ) + F (x ) T (x ) + I (x ) + F (x )...(15) B i B i B i C i C i C i Thus D (, B) and D (B, C) D (, C). Hence D (, B) is a valid measure of distance....(16) ccording to the relationship beteen the distance and similarity measures, similarity measure corresponding to D (, B) is derived as follos imilarly, (, C) (B, C). Thus (, B) is a valid measure of similarity beteen to vns. here i is eight corresponding to x i such that Obviously, e can easily prove that (, B) satisfies the axioms mentioned in Definition 4. Furthermore, e can also propose another measure of similarity for vns. Theorem 2: (, B) is a valid measure of similarity beteen to vns and B. Proof: Consider to vns and B represented by quadruple x i, T (x i), I (x i), F (x i) and x i, T B(x i), I B(x i), F B(x i) respectively in a universe of discourse = {x 1, x 2,, x n}. It is easy to see that (, B) [0, 1] satisfies axioms (1) and (2) defined in Definition 4. In order to prove axiom (3), assume that, B and C are three vns such that B C. Then T (x i) T B(x i) T C(x i), I (x i) > I B(x i) I C(x i) and F (x i) F B(x i)) F C(x i). Thus T (x ) = T (x ) T (x ) = T (x ) = T (x ) T (x ) = T (x )...(8) B i i B i i i C i C i I (x )=T (x ) I (x ) = I (x ) I (x ) = I (x ) I (x ) = I (x )... (9) B i i B i B i C i i C i C i nd F (x ) = F (x ) F (x ) = F (x ) = F (x ) F (x ) = F (x )...(10) B i i B i i i C i C i dding equations (8), (9) and (10) Corollary: Weighted similarity measure corresponding to equation (7) can be defined as...(17) Next, section defines an optimal point and presents an algorithm that helps to solve multi-attribute decision making problems. pplication of VNs imilarity Measures to Multi-attribute Decision Making This section proposes definition of optimal point and uses proposed similarity measures to dra inferences in multi attribute decision making under single valued neutrosophic environments. Multi-attribute decision making in an organization involves various attributes along ith various decision takers. Recently, it has become more complex. To take any decision, a manager needs to have relevant information and decent analytical skills. In some practical situations the course of action involves incomplete and indeterminate information, hich is stated in terms of vns. imilarity measures can be used as a tool to identify best course of action by determining the similarity beteen each course of action and ideal decision criteria/ideal point. Ideal point does not exist in reality. In order to identify 12

6 the substitute of ideal point e have defined optimal point as follos: = { x, max[t(x )], max[i(x )], min[f(x )], I}...(18 * i i i i The best course of action can be identified on the basis of similarity values beteen optimal point and available course of actions. Larger the value of similarity more closer it is to the optimal point. Let us consider a multi-attribute decision making problem involving a set of options P = {P 1, P 2, , P m} to be considered on the basis of attributes C = {C, C, , C }. 1 2 n ssume that the eight of an attribute C (j = 1, 2,..., n), j entered by the decision-maker, is [0, 1], j = 1,..., n and j Corresponding to each option P, I = 1,..., m and i attribute C, j = 1,2,..., n, the values of three function T(xi), j I (x ), and F (x ) denoted by single valued neutrosophic value i i d ij(x ij), hich is derived from evaluation of each course of action based on each attribute. To identify best course of action, similarity beteen each course of action and identified optimal course of action is calculated. Higher the value of similarity measure closer it is ith the optimal value. Course of action ith highest value is identified as best course of action. Multi-attribute decision process can be summarized as follos: tep 1: Identify eight corresponding to each attribute. tep 2: Formulate decision matrix corresponding to each attribute provided by the decision maker. tep 3: Identify optimal point using equation (18) using decision matrix obtained in step 2. tep 4: Calculate the similarity value (P, ) or (P, ) or j * j * (P, ) by using equation (6) or (7) or (17). j * tep 5: Rank and identify the best alternatives on the basis eighted similarity measure value. Next subsection explains the procedure of application of similarity measure in multi-attribute decision making using a numerical example. I LLUTRTIVE EXMPLE Here multi-attribute decision making problem is adopted from Ye (2014) to demonstrate the procedure of multi-attribute decision making. Consider four suppliers P = {P 1, P 2, P 3, P 4} hich are concerned ith a manufacturing company, that ants to select the best global supplier according to the core competencies of suppliers. The core competencies of suppliers are evaluated on the basis of four attributes: (i) C is the level of technology innovation; (ii) C is 1 2 the control ability of the flo; (iii) C is the ability of 3 management; (iv) C 4 is the level of service. Then, the eight T vector for the four criteria is = (0.3, 0.25, 0.25, 0.2). The decision matrix of the suppliers is made according to the four evaluating criteria. Therefore, the single valued neutrosophic decision matrix of the suppliers is as follos: To identify the most desirable supplier, e calculate the similarity of each supplier ith optimal identified values. Optimal solution is identified using equation (18) as follos = { 0.6, 0.3, 0.1, 0.5, 0.2, 0.3, 0.9, 0.3, 0.1, 0.7, 0.3, 0.1 } * (*, 1 ) = , (*, 2 ) = , (*, 3) = , (*, 4) = , optimal solution is identified as the alternative ith maximum value of similarity measure. o, the manufacturing company should order from supplier 2 ith preference order,,. ccording to distance similarity measure, (, ) = , (, ) = , (, ) = * 1 * 2 * , (, ) = , optimal solution is identified as * 4 the alternative ith maximum value of similarity measure. o, the manufacturing company should order from supplier 3 ith preference order,, It can be easily observed that the cross entropy measures proposed by Ye (2014) and Ye (2016) provide ranking 1, 3, 2, 4 and 3, 1, 2, 4 respectively to the suppliers. But Ye (2014) and Ye (2016) had taken optimal solution as 1,0,0 hich is quite unrealistic. o, the ranking provided by the proposed similarity measure is more reasonable as the optimal solution considered in this paper is more realistic. Comparison ith some Existing Measures of imilarity In this section, e evaluate the performance of suggested similarity measure ith fe existing measures of similarity mentioned belo, here and B are vns: To revie the performance of similarity measures let us consider an example. Consider the folloing four vns = { x, 0.1,0.2,0.2 }, = { x,0.2,0.4,0.4 }, = { x,0.1,0.2,0.3 } and = { x,0.0,0.0,0.0 }. 4 Table 1 shos the comparison of proposed measures ith some existing measures of similarity beteen to vns. 13

7 TBLE 1 Comparison beteen similarity measures B B1 J D C Cot Cot Not Defined It is clear from Table1 that and are unresonable in C determining similarity beteen,, and,. Where as, B1 and are zero for. Thus,, and are able to J D 3 4 B cot cot1 determine similarity in a better ay. C ONCLUION Distance and similarity measure are significant research area in neutrosophic information theory as they are efficient tools to deal ith uncertain and insufficient information. In this paper e have derived averaged distance measure and derived similarity measure using the proposed distance measure and the relation beteen distance and similarity measure under single valued neutrosophic environment. Further, another similarity measure is proposed for vns. Next, a method is derived to define ideal /optimal point using the existing information as previous literature used 1,0,0 as optimal point hich does not exists in reality. Further, proposed similarity measures along ith defined optimal point are used to solve multiattribute decision making problem, hich helps in providing rank to all the available alternatives and helps in identifying the best one. n illustrative example is used to demonstrate its application. Finally the proposed similarity measures are compared ith some existing similarity measures. Proposed similarity measure is proved to be better than some existing measures. BIBLIOGRPHY I. tanassov, K. (1999). Intuitionistic Fuzzy ets. Heidelberg, Ne York: Physica-Verlag. ii. ydoğdu,. (2015). On imilarity and Entropy of ingle Valued Neutrosophic ets. Gen. Math. Notes, 29(1), iii. Broumi,., & marandache, F. (2013). everal imilarity Measures of Neutrosophic ets. Neutrosophic ets and ystems,, I, iv. Grzegorzeski, P. (2004). Distance Beteen Intuitionistic Fuzzy ets and/or Interval-valued Fuzzy ets Based in the Hausdorff Metric. Fuzzy ets and ystems, 48, v. Kacprzyk. (1997). Multistage Fuzzy Control. Chichester: Wiley. vi. Park, J. H., Lin, K. M., Park, J.., & Kun, Y. C. (2007). Distance beteen Interval-valued Intuitionistic Fuzzy ets. Journal of Physics: Conference eries, 96. vii. marandache, F. (1998). Unifying Field in Logics. Neutrosophy: NeutrosophicProbability, et and Logic. Rehoboth: merican Research Press. viii. marandache, F. (2005). Neutrosophic et, a Generalisation of the Intuitionistic Fuzzy ets. International Journal of Pure pplied Mathematics, 24, ix. zmidt, E., & Kacprzyk, J. (2000). Distance beteen Intuitionistic Fuzzy ets. Fuzzy ets and ystems, 114(3), x. Wang, H., Zhang, Y., underraman, R., & marandache, F. (2011). ingle Valued Neutrosophic ets. Fuzzy ets, Rough ets and Multivalued Operations and pplications,, 3(1), xi. Yang, Y., & Chiclana, F. (2012). Consistency of 2D and 3D Distance of Intuitionistic Fuzzy ets. Expert ystems ith pplications, 39(10), xii. Ye, J. (2014). Clustering Methods Using Distance-Based imilarity Measures of ingle-valued Neutrosophic ets. Journal of Intelligent ystems, xiii. Ye, J. (2014). imilarity Measures beteen Interval Neutrosophic ets and their Multicriteria Decision-Making Method. Journal of Intelligent and Fuzzy ystems, 26, xiv. Ye, J. (2015). Improved Cosine imilarity Measures of implified Neutrosophic ets for Medical Diagnoses. rtificial Intelligence in Medicine, 63, xv. Ye, J. (2015). Improved Cross Entropy Measures of ingle Valued Neutrosophic ets and Interval Neutrosophic ets and Their Multicriteria Decision Making Methods. Cybernetics and Information Technologies, 15(4), xvi. Ye, J., & marandache, F. (2016). imilarity Measure of Refined ingle-valued Neutrosophic ets and Its Multicriteria Decision Making Method. Neutrosophic ets and ystems, 12,