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Multi-attribute Cognitive Decision Making via Convex Combination of Weighted Vector Similarity Measures for Single-Valued Neutrosophic Sets

Multi-attribute Cognitive Decision Making via Convex Combination of Weighted Vector Similarity... Similarity measure (SM) proves to be a necessary tool in cognitive decision making processes. A single-valued neutrosophic set (SVNS) is just a particular instance of neutrosophic sets (NSs), which is capable of handling uncertainty and imprecise- ness/vagueness with a better degree of accuracy. The present article proposes two new weighted vector SMs for SVNSs, by taking the convex combination of vector SMs of Jaccard and Dice and Jaccard and cosine vector SMs. The applications of the proposed measures are validated by solving few multi-attribute decision-making (MADM) problems under neutro- sophic environment. Moreover, to prevent the spread of COVID-19 outbreak, we also demonstrate the problem of selecting proper antivirus face mask with the help of our newly constructed measures. The best deserving alternative is calculated based on the highest SM values between the set of alternatives with an ideal alternative. Meticulous comparative analysis is presented to show the effectiveness of the proposed measures with the already established ones in the literature. Finally, illustrative examples are demonstrated to show the reliability, feasibility, and applicability of the proposed decision-making method. The comparison of the results manifests a fair agreement of the outcomes for the best alternative, proving that our proposed measures are effective. Moreover, the presented SMs are assured to have multifarious applications in the field of pattern recognition, image clustering, medical diagnosis, complex decision-making problems, etc. In addition, the newly constructed measures have the potential of being applied to problems of group decision making where the human cognition- based thought processes play a major role. Keywords Similarity measure · Neutrosophic set · Single-valued neutrosophic set · Convex vector similarity measure · Multi-attribute decision making Introduction to psychologists, the decision making process can be understood by considering both individual judgments and Human beings remain constantly in a state of making deci- taking into account both rational and irrational aspects of sions due to the intrinsic nature of their mind. They are natu- behavior. Thus, for proper representation of decision maker ral decision makers since every action ultimately results in interests, we are bound to consider those cognitive aspects. a decision, no matter how significant it might be. Several This implies that decisions considering cognitive aspects cognitive factors like people’s level of expertise, behavioral are comparatively better and they closely depict the deci- style, and decision maker’s credibility have a huge psycho- sion maker’s preferences. Human cognition-based methods logical impact on the decision making process. According or techniques not only help the decision makers in express- ing their preferences regarding a certain scenario but also helps visualize people’s intentions or underlying thought * Palash Dutta processes. palashdutta@dibru.ac.in It is worth mentioning that various disciplines in the field Gourangajit Borah of operations research, economics, management science, gjit1993@gmail.com etc., flourished with convincing outcomes when the notion 1 of MADM was introduced to their researchers in respective Department of Mathematics, Dibrugarh University, domains. However, it is noticed that the decision makers Dibrugarh 786004, Assam, India Vol.:(0123456789) 1 3 1020 Cognitive Computation (2021) 13:1019–1033 involved in MADM problems are unable to come up with the In the recent 5 years, SVNSs have seen widespread proper justification of the involved decision parameters, due applications in the realm of cognitive decision-making. For to reasons like lack of information about the public domain, instance, Chai et al. [35] enriched the literature of SVNSs by poor information processing capabilities, complexity of the proposing certain novel similarity measures. Saqlain et al. scenario, shortage of time, etc. This results in incorrect pref- [36] proposed the concept of tangent similarity measure for erence ordering of alternatives. We encounter a wide litera- single and multi-valued hypersoft sets under a neutrosophic ture on MADM problems, where the attribute values take the setting. Likewise, Qin and Wang [37] proposed certain form of crisp numbers [3], fuzzy numbers [4], interval-valued entropy measures for SVNSs with applications to MADM fuzzy numbers [5], interval-valued intuitionistic fuzzy num- problems. Basset et al. [38] gave the form for cosine similar- bers [6], and so on. ity measure for treatment of bipolar disorder diseases with For the first time, neutrosophic sets (NSs) were developed the help of bipolar neutrosophic sets. Tan and Zhang [39] by Smarandache [1, 2], which are capable of dealing with illustrated the decision-making procedure which is helpful in imprecise or unclear information. These sets are character- outrage/havoc assessment of typhoon disaster havoc by appli- ized by three independent functions namely, truth, indeter- cation of Refined SVNSs. Ye [40] carried out fault analysis minacy, and falsity membership functions. Noteworthy that, of a steam turbine with the help of cotangent function-based fuzzy sets and intuitionistic fuzzy sets can only deal with SMs for SVNSs to maximize its efficiency. Moreover, Ye [41] partial or incomplete information, but on the other hand, discussed certain bidirectional projection measures of SVNSs NSs tackle inconsistent information to a pretty decent extent. for their applications in mechanical design. Mondal et al. [42] The widespread application of NSs to MADM problems, demonstrated a MADM strategy based on hyperbolic sine where the decision makers express the ratings of alterna- similarity measures for SVNSs. Thereafter, certain entropy tives with the help of NSs is gaining huge attention among and cross-entropy measures for SVNSs were proposed by Wu researchers recently [7–9]. Single-valued neutrosophic set et al. [43]. Ye and Fu [44] in their paper, showed the useful- (SVNSs) were first proposed by Wang et al. [10], where ness of tangent function-based SM for the treatment of the he discussed some of their preliminary ideas and the arith- multi-period medical condition. In the same year, Ye [45] metic operations between SVNSs. Furthermore, Wang real- demonstrated how a dimension root SM can be applied for ized that interval numbers could better represent the truth, the diagnosis of faults in hydraulic turbines. Garg and Nancy the indeterminacy, and the falsity degree of a particular [46] proposed certain novel ideas of biparametric distance statement, over the classical non-fuzzy or crisp numbers. measures for SVNSs. The concept of SVNSs when applied Hence, Wang et al. [11], proposed interval neutrosophic to graphs, or precisely, “single-valued neutrosophic graphs”, sets (INSs). Thereafter, various methods were developed was a significant and enticing theory which was proposed by for MADM problems involving SVNSs and INSs, such as Broumi et el. [47]. Unlike simple graphs, a stronger version TOPSIS method [12, 13], weighted aggregation operators of it known as hypergraphs had many remarkable applica- [14–18], subsethood measure [19], inclusion measures [20], tions in the literature. In simple graphs where a single edge and outranking method [21, 22]. As a result, MADM prob- can connect exactly two vertices only, but a hyperedge in a lems are tackled with an efficient and significant tool known hypergraph can connect a set of vertices. In this context, Yu as similarity measure (SM) [23–27]. The highest weighted et al. [48] applied the learning method involving hypergraphs SM value between the set of alternatives and the positive to model the pair-wise coherency between images. Another ideal alternative corresponds to the deserving or best alter- term for it is ‘transductive image classification’. Moreover, native. Broumi and Smarandache [28], considered NSs and Yu et al. [49] applied the complex concept of multimodal hence defined the Hausdorff distance measure between two hypergraphs to propose a sparse coding technique for the such sets. Later on, it was Majumdar and Samanta [29] who prediction of click data and image re-ranking, and as a con- utilized the concepts of membership degrees, matching func- sequence of which we get minimum margin for an era in tion, and distance measure, for defining certain SMs between a text-based image search. The application of hypergraphs two SVNSs. The correlation coec ffi ient for SVNSs was then tends to improve the visual efficacy which was established improved by Ye [30]. Moreover, Ye [31] also introduced by Yu et al. [50] in their journal article. Moreover, in their vector SMs for SVNSs and INSs, where SVNS acts as a article, they also proposed a novel ranking model which takes vector in three dimensions. Also, Ye [32] utilized the vector into account the click features and visual features. Later, a concept in improving the cosine SM, so that it can be applied significantly effective and novel framework by the name of to problems of medical decision-making. It was Broumi and Muli-task Manifold Deep Learning (M2DL) for estimating Smarandache [33] who tackled pattern recognition problems face poses via multi-modal information was introduced by by extension of the cosine SM for SVNSs onto INSs. Con- Hong et al. [51]. sequently, Pramanik et al. [34] proposed hybrid vector SM Therefore, it is clear that in the context of decision mak- for SVNSs and INSs. ing, the psychological aspects play a significant role. The 1 3 Cognitive Computation (2021) 13:1019–1033 1021 process through which we “perceive”, “interpret”, and “gen- of truth, indeterminacy, and falsity. The cognitive decision- erate” our responses towards the thought process of people making problems are innate to possession of factors like undergoing social interactions is termed as Social Cognition. vagueness, error, contradiction, and redundancy data. In this Social functioning outputs have a strong linkage to social context, NSs which are capable of handling such imprecise- cognition, such that when such aspects are not considered ness or vagueness shall prove to be a good fit while solving the quality of decisions is degraded, while people achieve multi-attribute decision-making and multi-attribute group an enhanced degree of satisfaction with decisions when such decision-making problems. SVNSs are a particular sub- factors are considered. A person needs to first understand class of NSs, which we will be using to tackle a MADM the process of social interactions to handle social cognition method. INSs are also a similar subclass of NSs, which are tasks. For example, suppose a family decides to celebrate exempted in this study. There is a common practice of apply- the birthday of one of its members and it is trying to figure ing the vector SMs in decision-making problems [57–59]. out the best restaurant in the city for the celebration. Here, However, motivated by the works of Ye [56] and Pramanik the intention of the family members comes into play rather et al. [34], we propose two convex vector SM by taking the than only the preferences towards the alternatives and their convex combination of Jaccard and Dice SMs and Jaccard respective criteria. Now, the person whose birthday needs and cosine SMs, respectively. In this article, our proposed to be celebrated is the crucial one who needs to be content measures for SVNSs are developed extending the concept with the decision. Ironically, if some other members opt for of variation coefficient similarity method [56], under the a restaurant according to their liking which eventually makes neutrosophic setting. We discuss some basic properties of the person (whose birthday is to be celebrated) unhappy, the newly constructed measures and thereby demonstrating then the decision will be considered to be a bad/inappropri- their valid structural formulation. The application of our ate one. proposed measures is shown while tackling few practical However, there exist certain cognitive limitations in peo- MADM scenarios. The final yielded outcomes, affirms the ple’s minds like initial impression and emotional satisfaction accuracy and good fit of our proposed measures. at the time of making decisions, which restrict them from making rational decisions in MCDM scenarios. For instance, suppose there is a real estate dealer who is interested in buy- Structure of the Paper ing a house from four options that are presented to him/her by the estate agency. Now, if the dealer enters one of these Hence, the rest of the paper is set out as follows: in “Pre- houses and suddenly feels uncomfortable, then no matter liminaries and Existing Methods”, we give a brief overview how good the deal of the house is at that price, the dealer of the concepts related to neutrosophic sets, SVNSs, and will restrict himself from buying it since his feeling does not also, we review certain existing weighted vector SMs for allow him to do so. Consider another scenario, where a per- SVNSs. In “Proposed Method”, our proposed definitions of son wants to buy a second-hand truck. Even though the truck weighted convex vector similarity measures are discussed, might be in excellent working condition and the car dealer while the subsequent “MADM based on Proposed Similar- offers him a good deal, but if the person who is interested in ity Measure for SVNSs” firstly elaborates the procedure to buying somehow feels that the dealer is tricking him, then solve any MADM problem based on the proposed method. he would not buy it. Here, the first impression of the person Then, to validate the applicability of our proposed meas- restricts him from making a final decision. Similarly, there ures, we illustrate few practical MADM scenarios which for are many such examples. From the literature, it is evident instance are optimum profit for an investment company and that MCDM techniques including cognitive aspects are close the selection of proper face masks, to prevent the spread of to inexistent. However, few researchers have tried to estab- the COVID-19 pandemic. In addition, they are supported lish the linkage of social cognition into decision making by meticulous comparative analysis showing the feasibility problems (for details please refer to Bisdorff [52], Carneiro of our proposed measures with the existing methods in the et al. [53], Homenda et al. [54], and Ma et al. [55]). literature. Finally, “Conclusion” provides the conclusions for the article. Motivation of Our Work Uncertain or imprecise information is an indispensable fac- Preliminaries and Existing Methods tor in most real-world application problems. In this regard, NSs are well equipped with handling inconsistent informa- Here, in this section, we briefly review some of the concepts tion and indeterminate decision data with a better perspec- of NSs, SVNSs, and their basic arithmetic operations, the tive over fuzzy sets and intuitionistic fuzzy sets. It is due ideas of which will be necessitated in the subsequent sec- to their characteristic independent membership functions tions of our study. 1 3 1022 Cognitive Computation (2021) 13:1019–1033 Definition 1 (Neutrosophic Sets) [1, 2] 2. Inclusion: A SVNS P is said to be a subset of another SVNS Q , that is P ⊆ Q if and only if T (x) ≤ T (x), P Q Suppose we consider U to be a space of objects (points) I (x) ≥ I (x), F (x) ≥ F (x); ∀x ∈ U. P Q P Q and let us denote the generic element in U by x . Then a 3. Equality: For equality between two SVNSs P and Q neutrosophic set P in U is characterized by three inde- to hold, we must have that both sets must be a subset pendent functions, a truth membership function T (x) , a n of each other, that is, P ⊆ Q and P ⊇ Q . Or in other indeterminacy membership function I (x) , and a falsity words, we can also say that, P = Q if and only if membership function F (x) . The three functions are stand- T (x) = T (x), I (x) = I (x), F (x) = F (x); ∀x ∈ U. P P Q P Q P Q − + ard and non-standard subsets of the interval 0, 1 , and the 4. Addition: The addition operation between two P ⊕ Q = x, T following condition is satisfied, SVNSs P and Q is defined by ⟨ (x) +T (x) − T (x)T (x), I (x)I (x), F (x)F (x) x ∈ U , Q P Q P Q P Q − + 0 ≤ sup T (x) + sup I (x) + sup F (x) ≤ 3 (1) P P P which is not the traditional additive rule. Here, the first component is the algebraic sum of truth degrees minus Noteworthy that, neutrosophic sets introduced by their product, the second component is the product of Smarandache [1] were more from a philosophical point the indeterminacy degree for the two sets, and the third of view and they had very scarce applications in the field component is the product of their falsity degrees. of science and engineering. As a better successor to it, 5. Multiplication: The multiplication operation Wang et al. [10] introduced a subclass of the neutrosophic between two SVNSs P and Q is defined by P ⊗ Q sets, which are called single-valued neutrosophic sets = x, T (x)T (x), I (x) + I (x) − I (x)I (x), F (x)+ P Q P Q P Q P (SVNSs). F (x) − F (x)F (x) x ∈ U , where the components Q P Q swap their arithmetic as in the addition case. That is, Definition 2 (Single-Valued Neutrosophic Sets) [10] the first component is the product of their truth degrees, the second component being the algebraic sum minus Suppose we consider U be a space of objects (points) the product of their indeterminacy degrees, and it is the and let us denote the generic element in U by x . Then a algebraic sum minus their product of falsity degrees in single-valued neutrosophic set P in X is characterized by the third component. three independent functions, a truth membership function 6. Union: The union of two SVNSs P and Q is defined by T (x) , an indeterminacy membership function I (x) , and a P P P ∪ Q = x, T (x) ∨ T (x), I (x) ∧ I (x), F (x) ∧ F P Q P Q P Q falsity membership function F (x) . We denote the SVNS (x)⟩�∀x ∈ U} . The resulting SVNS has the first compo- P as � � nent as the maximum of their truth degrees, the sec- P = ⟨x, T (x), I (x), F (x)⟩�x ∈ X , w h e re T (x) , I (x) , P P P P P ond component as the minimum of their indeterminacy F (x)∈[0, 1]; x ∈ U. degrees, and the third component is the minimum of Also, the following inequality is satisfied by the sum of their respective falsity degrees as well. T (x) , I (x) , and F (x), P P P 7. Intersection: The intersection of two SVNSs P and Q is 0 ≤ T (x) + I (x) + F (x) ≤ 3 (2) defined by  P ∩ Q = x, T (x) ∧ T (x), I (x) ∨ I (x), P P P P Q P Q F (x) ∨ F (x) ∀x ∈ U . Broadly speaking, the inter- P Q For the sake of simplicity, let us consider, P = section is the reverse case for union operation since ⟨T (x), I (x), F (x)⟩ as the SVNS in U. P P P the maximum function in the case of union operation becomes the minimum function here, and vice-versa. Definition 3 (Arithmetic operations between SVNSs) [10, 16] Existing Non‑weighted Vector Similarity Measures For any two SVNSs P = ⟨T (x), I (x), F (x)⟩ and Q = P P P SMs greatly enhance the valuable output efficiency in decision- T (x), I (x), F (x) considered in a finite universe U , t he Q Q Q making processes. Many experts from time to time have formu- arithmetic operations between them were proposed in previ- lated several fruitful definitions of SMs based on distances and ous studies [10, 16] as follows: vectors. Hence, in the following sequel, we recall the definitions of Jaccard [57], Dice [58], and cosine [59] similarity measures. 1. Complement: The complement of SVNS P is denoted by These SMs are structurally simple, easy to compute, and mod- C C P and defined as  P = ⟨F (x),1 − I (x), T (x)⟩ , where P P P est, which enables the decision-makers to determine the differ- the first component is the falsity membership degree for ent similarity value options at ease. P , the second component is 1 minus the indeterminacy Let M = m , m , ..., m and N = n , n , ..., n be two s 1 2 s 1 2 s degree for P , and the third is the truth membership grade -dimensional vectors having non-negative co-ordinates. Then, for P. 1 3 Cognitive Computation (2021) 13:1019–1033 1023 C3. C (M, N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) Definition 4 Between any two vectors M = m , m , ..., m SM i i 1 2 s for every m ∈ M and n ∈ N and N = n , n , ..., n , the Jaccard similarity measure [57] i i 1 2 s is defined as Remark 1 The common property that each of these simi- m n i i M.N i=1 larity measures hold is that they assume values within the J (M, N) = = SM s s s 2 2 ∑ ∑ ∑ ‖M‖ + ‖N‖ − M.N unit interval [0, 1] . Jaccard and Dice SMs are undefined for 2 2 m + n − m n i i i i both m = 0 and n = 0 , whereas cosine similarity measure is i=1 i=1 i=1 i i (3) undefined for either m = 0 or n = 0 , for i = 1, 2, ..., s. i i � � s s ∑ ∑ 2 2 where ‖M‖ = m and ‖N‖ = n are called the Definition 7 Between two vectors M = m , m , ..., m and i i 1 2 s i=1 i=1 N = n , n , ..., n , the variation coefficient similarity meas- 1 2 s Euclidean norms of M and N , and the inner product of vec- ure [56] is defined as tors M and N is given by M.N = m n . i i i=1 2M.N M.N V (M,N) =  + (1 − ) CF 2 2 ‖M‖‖N‖ The above-mentioned similarity measure satisfies the fol- ‖M‖ + ‖N‖ lowing properties: s s ∑ ∑ 2 m n m n i i i i J1. 0 ≤ J (M, N) ≤ 1 SM i=1 i=1 ⇒ V (M, N) =  + (1 − ) � � CF s s s J2. J (M, N) = J (N, M) ∑ ∑ ∑ s s SM SM 2 2 ∑ ∑ m + n − m n 2 2 i i i i m n J3. J (M, N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) SM i i i=1 i=1 i=1 i i i=1 i=1 for every m ∈ M and n ∈ N i i (6) It satisfies the following properties: Definition 5 Between two vectors M = m , m , ..., m 1 2 s and N = n , n , ..., n , the Dice similarity measure [58] is 1 2 s V1. 0 ≤ V (M,N) ≤ 1 CF defined as V2. V (M,N) = V (N,M) CF CF V3. V (M,N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) CF i i 2 m n i i for every m ∈ M and n ∈ N 2M.N i=1 i i D (M, N) = = (4) SM s s 2 2 ∑ ∑ ‖M‖ + ‖N‖ 2 2 m + n i i Some Weighted Vector Similarity Measures of SVNSs i=1 i=1 It satisfies the following properties: In multiple-criteria decision making methods, criteria weights have a much larger influence on the outcomes D1. 0 ≤ D (M,N) ≤ 1 yielded by a decision process and also on the ranking of SM D2. D (M,N) = D (N,M) alternatives. The reason being that it takes into account the SM SM D3. D (M,N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) relative importance of each criterion concerning the set of SM i i for every m ∈ M and n ∈ N alternatives chosen. Consequently, any such process which i i does not consider the respective criteria weightage and sets identical weights of importance for them thus loses its logi- Definition 6 Between two vectors M = m , m , ..., m cal importance. Thus, decision-makers from their best of 1 2 s and N = n , n , ..., n , the cosine similarity measure [59] knowledge try to allocate weights to each criterion involved 1 2 s is defined as in a decision process. Hereby, we list down three existing definitions of weighted vector SMs. Suppose we consider two SVNSs P and Q in a three- m n i i i=1 M.N dimensional space defined by C (M, N) = = � � SM (5) ‖M‖.‖N‖ s s P = T x , I x , F x x ∈ U and Q = T ∑ ∑ P i P i P i i Q 2 2 m . n i i x , I x , F x x ∈ U . i Q i Q i i i=1 i=1 Then, we can dene fi weighted vector similarity as follows: It satisfies the following properties: Definition 8 Let U be a universe of discourse defined by U = x , x , ..., x , wher e P = T x , I x , F x C1. 0 ≤ C (M, N) ≤ 1 1 2 r P i P i P i SM C2. C (M, N) = C (N, M) SM SM 1 3 1024 Cognitive Computation (2021) 13:1019–1033 P2. J (P, Q) = J (Q, P); D (P, Q) = D (Q,P); x ∈ U and Q = T x , I x , F x x ∈ U be WSM WSM WSM WSM i Q i Q i Q i i C (P,Q) = C (Q,P) two SVNSs. WSM WSM P3. J (P, Q) = 1; D (P, Q) = 1; C (P, Q) = 1 if WSM WSM WSM and only if P = Q , which implies Let w ∈[0, 1] be the weight of every element x (i = 1, i i 2, ..., r) , so that w = 1. T x = T x , I x = I x , F x = F x , for i=1 P i Q i P i Q i P i Q i Then, the weighted Jaccard similarity measure [30] every x ∈ X (i = 1, 2, ..., r). between P and Q is defined as �� �� ���� �� �� T x T x + I x I x + F x F x P i Q i P i Q i P i Q i J (P, Q) = w � � WSM i � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ i=1 (7) T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ Definition 9 Let U be a universe of discourse defined by Remark 3 Now, J (P, Q), D (P, Q) for SVNSs WSM WSM U = x , x , ..., x , wher e P = T x , I x , F x P = T x , I x , F x x ∈ U and Q = T x , 1 2 r P i P i P i P i P i P i i Q i x ∈ U and Q = T x , I x , F x x ∈ U be I x , F x x ∈ U , are undefined when P = ⟨0, 0, 0⟩ i Q i Q i Q i i Q i Q i i two SVNSs. and Q = ⟨0, 0, 0⟩ , i.e., for T = I = F = 0 and P P P T = I = F = 0 given i = 1, 2, ..., r . On the other hand, Q Q Q Let w ∈[0, 1] be the weight of every element x (i = 1, C (P,Q) is undefined for P = ⟨0, 0, 0⟩ or Q = ⟨0, 0, 0⟩ , i i WSM i.e., when T = I = F = 0 or T = I = F = 0 for P P P Q Q Q 2, ..., r) , so that w = 1. i=1 i = 1, 2, ..., r. Then, the weighted Dice similarity measure [30] between P and Q is defined as 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i D (P, Q) = w (8) WSM i 2 2 2 2 2 2 i=1 T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q Definition 10 Let U be a universe of discourse defined by Proposed Method U = x , x , ..., x , wher e P = T x , I x , F x 1 2 r P i P i P i x ∈ U and Q = T x , I x , F x x ∈ U be Based on the enormous potential of SVNSs which help raise i Q i Q i Q i i two SVNSs. the utility of cognitive decision-making process, we pro- pose two weighted convex vector SMs (WCVSMs) which Let w ∈[0, 1] be the weight of every element x (i = 1, are dependent on the coefficient parameter. The SMs have i i arranged in such a way that their structure represents a con- 2, ..., r) , so that w = 1. i=1 vex combination. The idea for such formulation came up as Then, the weighted cosine similarity measure [30] the vector SMs are empirically established to produce feasi- between P and Q is defined as ble and rational outcomes on their own, so there is nothing T x T x + I x I x + F x F x P i Q i P i Q i P i Q i C (P, Q) = w WSM i (9) i=1 2 2 2 2 2 2 T x + I x + F x . T x + I x + F x i i i i i i P P P Q Q Q Remark 2 It is noteworthy that Eqs. (7), (8), and (9) satisfy wrong with constructing a function out of those SMs. The sole intention is to serve the decision-making domain with P1. 0 ≤ J (P, Q) ≤ 1; 0 ≤ D (P, Q) ≤ 1; 0 ≤ C efficient similarity measures which are capable of produc- WSM WSM WSM (P, Q) ≤ 1 ing convincing outcomes by providing a global evaluation 1 3 Cognitive Computation (2021) 13:1019–1033 1025 JD JD JC JC framework for each alternative with respect to each criterion. P2. S (P, Q) = S (Q, P); S (P, Q) = S (Q, P) W W W W JD JC Hence, the two proposed measures are listed below. P3. S (P, Q) = 1; S (P, Q) = 1 when P = Q , i.e., W W Definition 11 Let us consider U be a universe of T x = T x , I x = I x , F x = F x , for P i Q i P i Q i P i Q i discourse defined by U = x , x , ..., x , wher e P = every x ∈ X (i = 1, 2, ..., r) 1 2 r T x , I x , F x x ∈ U and Q = T x , I P i P i P i i Q i Q x , F x x ∈ U are two SVNSs. Proof: i Q i i Also, let w ∈[0, 1] be the weight of every element (P1). From eqns. (7) and (8), we find that for Jaccard and Dice similarity measures of SVNSs, 0 ≤ J (P, Q) ≤ 1 and x (i = 1, 2, ..., r) , such that w = 1. WSM i i i=1 0 ≤ D (P,Q) ≤ 1 for all i = 1, 2, ..., r . Now, Eq. (10) can WSM Then the two weighted convex vector similarity measures be written as follows, between SVNSs are proposed as follows: �� �� ���� �� �� ⎡ T x T x + I x I x + F x F x ⎤ P i Q i P i Q i P i Q i ⎢ � � ⎥ � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ ⎢ i=1 ⎥ T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ JD ⎢ ⎥ (10) S (P, Q) = ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ � 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i + (1 − ) w � � �� ⎥ � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i ⎣ P P P Q Q Q ⎦ and �� �� ���� �� �� � T x T x + I x I x + F x F x ⎡ ⎤ P i Q i P i Q i P i Q i � � ⎢ ⎥ � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ i=1 ⎢ T x + I x + F x + T x + I x + F x ⎥ i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ ⎢ ⎥ JC (11) S (P, Q) = ⎢ ⎥ ⎢ ⎥ �� �� ���� �� �� ⎢ T x T x + I x I x + F x F x ⎥ P i Q i P i Q i P i Q i + (1 − ) w ⎢ � �⎥ i � � � � �� �� ��� �� �� �� ⎢ i=1 ⎥ 2 2 2 2 2 2 T x + I x + F x . T x + I x + F x ⎢ ⎥ i i i i i i P P P Q Q Q ⎣ ⎦ Our proposed measure(s) satisfy the following JD S (P, Q) =J (P, Q) + (1 − )D (P, Q) W W (12) proposition, ≤  + (1 − ) = 1 Proposition 1 The two proposed weighted convex vector Since, J P, Q ≥ 0 and D P,Q ≥ 0 , so does the ( ) ( ) WSM WSM JD similarity measure (WCVSM) of SVNSs between P and Q WCVSM, S (P, Q) ≥ 0 for all  ∈[0, 1]. satisfy the properties given below: Thus, the first property is satisfied. Similarly, we can JC prove 0 ≤ S (P, Q) ≤ 1. (P2). From Eq. (10), JD JC P1: 0 ≤ S (P, Q) ≤ 1; 0 ≤ S (P, Q) ≤ 1 W W 1 3 1026 Cognitive Computation (2021) 13:1019–1033 �� �� ���� �� �� � T x T x + I x I x + F x F x ⎡ ⎤ P i Q i P i Q i P i Q i ⎢ � � ⎥ � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎡ ⎤ ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ JD ⎢ ⎥ S (P, Q) = ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ � ⎥ 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i ⎢ + 1 −  w ⎥ ( ) � � �� � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i ⎣ P P P Q Q Q ⎦ �� �� ���� �� �� ⎡ T x T x + I x I x + F x F x ⎤ Q i P i Q i P i Q i P i ⎢ � � ⎥ �� �� �� � �� �� ��� ⎡ 2 2 2 2 2 2 ⎤ ⎢ i=1 ⎥ T x + I x + F x + T x + I x + F x i i i i i i Q Q Q P P P ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ Q i P i Q i P i Q i P i ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ � ⎥ 2 T x T x + I x I x + F x F x Q i P i Q i P i Q i P i ⎢ + (1 − ) w ⎥ �� � � �� �� �� � �� �� ��� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i Q Q Q P P P ⎣ ⎦ JD = S (Q, P) set C (j = 1, 2, ..., y) are depicted by single valued neutro- Similar results are also obtained from Eq.  (11 ), JC JC sophic element of the form s = T , I , F . Here, T indi- ij ij ij ij ij S (P, Q) = S (Q, P) , which proves the second property. W W cates the membership degree, I denotes the indeterminacy ij (P3). If T x = T x , I x = I x and F x P i Q i P i Q i P i degree and F indicates the non-membership degree for the ij = F x , for i = 1, 2, ..., r , then the value of J (P, Q) = Q i WSM alternative A with respect to the attribute C . Thus, for i j 1, D (P,Q) = 1 and C (P,Q) = 1 . Therefore, from WSM WSM JD JC i = 1, 2, ..., x; j = 1, 2, ..., y we have, 0 ≤ T + I + F ≤ 3 ij ij ij Eq. (10), the value of S (P, Q) = 1 and S (P, Q) = 1 . This W W and T ∈[0, 1], I ∈[0, 1], F ∈[0, 1]. ij ij ij concludes the proof. Let us consider that the alternative A (i = 1, 2, ..., x) takes single-valued neutrosophic values and has the following rep- Remark 4 Now for P = T x , I x , F x x ∈ U P i P i P i i resentation, A = s , s , ..., s , for i = 1, 2, ..., x; i i1 i2 ix and Q = T x , I x , F x x ∈ U , the convex sim- Q i Q i Q i i � � �� ilarity measure value is assumed to be zero for P = ⟨0, 0, 0⟩ = ⟨T , I , F ⟩, ⟨T , I , F ⟩, ..., T , I , F (13) i1 i1 i1 i2 i2 i2 iy iy iy and Q = ⟨0, 0, 0⟩. There are certain steps to follow while selecting the best alternative amongst a set of alternatives which are as follows, MADM Based on Proposed Similarity Measure for SVNSs Step 1 Determination of the ideal solution We consider a multi-attribute decision-making problem with It is a very common procedure in MADM to utilize the x set of alternatives and y set of attributes, where the values concept of an ideal alternative/solution. And, realizing a of the attributes are represented by SVNSs. Let perfectly ideal solution in the real world is an abstract A = A , A , ..., A be a finite collection of alternatives and 1 2 x idea as there does not exist any such. However, in order to C = C , C , ..., C be the finite collection of attributes. 1 2 y construct a useful theoretical framework and to carry out Also, let the weight vector be denoted by the mathematical calculations, we incorporate the concept w = w , w , ..., w corresponding to the set of attributes 1 2 y of an ideal solution. It facilitates the set of alternatives C (j = 1, 2, ..., y) such that w = 1 and w ≥ 0 . We denote under study to be ranked based on the degree of similarity j j j j=1 (closeness) or non-similarity (farness) from the ideal solu- the decision matrix by D = s , where the preference ij x×y tion. Thus, we need to determine the SVNS-based ideal values of the alternatives A (i = 1, 2, ..., x) over the attribute solution. 1 3 Cognitive Computation (2021) 13:1019–1033 1027 Let Y denote the entire collection of two types of attrib- From Eqs.  (10) and (11), the WCVSMs between ute, which are namely profit/benefit type attribute ( B ) and the ideal alternative A and the alternative A for the cost-type attribute ( L ). Then the ideal solution (IS), i = 1, 2, ..., x;  ∈[0, 1] are given by ∗ ∗ ∗ T T + I I + F F ij ij ij ⎡ j j j ⎤ � � ⎢ � � � � � � � � ⎥ 2 2 2 � � � � � � 2 2 2 ⎡ ⎤ i=1 ∗ ∗ ∗ ⎢ ⎥ T + I + F + T + I + F ij ij ij j j j ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � � ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ ⎥ − T T + I I + F F ij ij ij � � ⎣ j j j ⎦ JD ∗ ⎢ ⎥ (16) S A , A = ⎢ ⎥ ⎢ � � ⎥ ∗ ∗ ∗ ⎢ ⎥ 2 T T + I I + F F � ij ij ij j j j ⎢ ⎥ + (1 − ) w �� � � ⎢ � � � � � � � � ⎥ 2 2 2 � � � � � � 2 2 2 i=1 ∗ ∗ ∗ ⎢ ⎥ T + I + F + T + I + F ij ij ij j j j ⎣ ⎦ ∗ ∗ ∗ T T + I I + F F ⎡ ⎤ ij ij ij j j j ⎢ � � ⎥ � � � � � � � � 2 2 2 � � � � � � 2 2 2 ⎢ ⎥ i=1 ⎡ ∗ ∗ ∗ ⎤ T + I + F + T + I + F ij ij ij j j j ⎢ ⎢ ⎥ ⎥ ⎢ � � ⎥ ⎢ ⎥ ⎣ ⎦ − T T + I I + F F ⎢ ⎥ P ij P ij P ij � � JC ∗ ⎢ ⎥ (17) S A , A = ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ T T + I I + F F ij ij ij j j j ⎢ ⎥ + (1 − ) w � � � ⎢ � � � ⎥ � � � � � � � � 2 2 2 i=1 � � � � � � 2 2 2 ⎢ ⎥ ∗ ∗ ∗ T + I + F . T + I + F ij ij ij ⎢ j j j ⎥ ⎣ ⎦ where the ideal solution A takes respective forms accord- ∗ ∗ ∗ ∗ A = s , s , ..., s , is that solution of the decision matrix 1 2 x ing to the nature of the attribute as depicted in Eqs. (14) D = s which is defined as ij x×y and (15). ∗ ∗ ∗ ∗ (a) s = T , I , F = max T , min I , min F j ij ij ij j j j i i i Step 3 We rank the alternatives. for benefit − type attribute (B) (14) The ranking of the alternatives could be easily deter- and mined according to the values obtained from Eqs.  (14) and (15). The decreasing order of the WCVSMs gives the ∗ ∗ ∗ ∗ (b) s = T , I , F = max T , min I , min F required ranking or preference ordering of the alternatives. ij ij ij j j j j i i i for cost − type attribute (L) Optimum Profit Determination by an Investment (15) Company Step 2 We evaluate the WCVSM between the ideal alterna- We consider a multi-attribute decision making problem (adapted tive and each alternative. from Ye [7]), in which an investment company is interested in finding out the best suitable alternative amongst a set of four Table 1 Decision matrix with single-valued neutrosophic entries alternatives: (1) A is a computer company, (2) A is an arms 1 2 Criteria →/ C C C 1 2 3 company, (3)A is a car company, and (4) A is a food company. 3 4 Alternatives ↓ Three criteria(s) are taken into consideration by the investment A ⟨0.5, 0.3, 0.2⟩ ⟨0.5, 0.2, 0.3⟩ ⟨0.3, 0.2, 0.3⟩ company based on which alternatives are evaluated, which are A ⟨0.4, 0.3, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ ⟨0.7, 0.0, 0.1⟩ (1) C is the environmental impact analysis, (2) C is the growth 1 2 A ⟨0.2, 0.2, 0.5⟩ ⟨0.4, 0.2, 0.3⟩ ⟨0.4, 0.2, 0.3⟩ analysis, and (3) C is the risk factor analysis. The decision A ⟨0.5, 0.2, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ maker assesses the four possible alternatives with respect to the 1 3 1028 Cognitive Computation (2021) 13:1019–1033 Table 2 Differ ent WCVSM Proposed measures Pairs of alternatives Best alter- Ranking order values for different values of native JD ∗ ∗ ∗ ∗ for our first defined measure (A , A ) (A , A ) (A , A ) (A , A ) WCVSM value ( S ) 1 2 3 4 = 0.15 0.7666 0.9172 0.8905 0.8875 A A > A > A > A 2 2 3 4 1 = 0.30 0.7478 0.9093 0.8796 0.8771 A = 0.55 0.7163 0.8962 0.8615 0.8598 A = 0.80 0.6849 0.8831 0.8433 0.8425 A = 0.95 0.6661 0.8752 0.8324 0.8322 A attributes based on the SVNS values provided. The SVNS-based Based on the outcomes obtained for different values of decision matrix D = d are presented in Table 1. as shown in Tables 2 and 3, the alternative A turns out to ij 2 4×3 The weight vector is given by W = w ,w ,w = be the best suitable alternative. 1 2 3 {0.40, 0.25, 0.35} such that Comparative Analysis w = 1 (18) Here, we provide a comparison of the outputs obtained j=1 via our proposed convex vector similarity measures with some of the existing similarity measures on the illus- Step 1 Identification of the attribute-type. trated MADM scenario. The comparison results along with their evaluated similarity values are depicted in Here, the attributes C and C are benefit-type attribute, Table 4. It is very obvious from Table 4 that our results 2 3 while C is identified as cost-type attribute. for evaluation of the best alternative are in agreement with Ye’s vector similarity measure method [31], Ye’s Step 2 Determination of the ideal solution (IS). improved cosine similarity measure [32], and hybrid vector similarity measure by Pramanik et al. [34], for With the help of Eqs. (14) and (15), the ideal solution for SVNSs. Even the ranking order of the alternatives the given decision matrix D = d can be determined as obtained with our proposed measures coincides exactly ij 4×3 � � with that of Ye’s SMs ([31, 32]), whereas the alterna- A = ⟨0.2, 0.3, 0.5⟩, ⟨0.6, 0.1, 0.2⟩, ⟨0.7, 0.0, 0.1⟩ (19) tives A and A , interchange their places under Pramanik 3 4 et al.’s method. Furthermore, the ranking order evaluated by subset-hood Step 3 Evaluation of the weighted convex vector similarity measure method [19], improved correlation coefficient [ 30] measures. is demonstrated in Table  5. According to [19] and [30], the alternative A is the second-best choice amongst the We evaluate the weighted convex vector similarity measures set of alternatives, whereas it is the best choice according with the help of the Eqs. (12–14), and the outcomes obtained to previous studies [31, 32, 34], and our presented meas- for various values of  are presented in Tables 2 and 3. ures. Hence, the validity and feasibility of our measures is established. Step 4 Ranking of the alternatives. Table 3 Differ ent WCVSMs Proposed measures Pairs of alternatives Best alter- Ranking order values for different values of native ∗ ∗ ∗ ∗ JC for our second defined measure (A , A ) (A , A ) (A , A ) (A , A ) WCVSM value ( S ) 1 2 3 4 = 0.15 0.7846 0.9198 0.9036 0.8780 A A > A > A > A 2 2 3 4 1 = 0.30 0.7628 0.9115 0.8905 0.8692 A = 0.55 0.7263 0.8976 0.8687 0.8545 A = 0.80 0.6899 0.8838 0.8469 0.8399 A = 0.95 0.6681 0.8755 0.8339 0.8311 A 1 3 Cognitive Computation (2021) 13:1019–1033 1029 Table 4 Comparison of Similarity measure method Measure value Ranking order WCVSM for SVNSs with different SMs ∗ ∗ A > A > A > A C A , A  [31] C A ,A = 0.7689 2 4 3 1 W i W 1 C A ,A = 0.9281 W 2 C A ,A = 0.8975 W 3 C A ,A = 0.8979 W 4 ∗ ∗ A > A > A > A Hyb A , A  [34] Hyb A , A = 0.7912 2 3 4 1 w i w 1 ( = 0.1) Hyb A , A = 0.9433 w 2 Hyb A , A = 0.9036 w 3 Hyb A , A = 0.9019 w 4 ∗ ∗ A > A > A > A WSC A , A [32] WSC A , A = 0.9401 2 4 3 1 2 i 2 1 WSC A , A = 0.9804 2 2 WSC A , A = 0.9691 2 3 WSC A , A = 0.9761 2 4 JD ∗ JD ∗ A > A > A > A S A , A  (Proposed) ( = 0.1 5) S A ,A = 0.7666 2 3 4 1 W i W 1 JD ∗ S A ,A = 0.9172 W 2 JD ∗ S A ,A = 0.8905 W 3 JD ∗ S A ,A = 0.8875 W 4 JC ∗ JC ∗ A > A > A > A S A , A  (Proposed) ( = 0.1 5) S A , A = 0.7846 2 3 4 1 W i W 1 JC ∗ S A , A = 0.9198 W 2 JC ∗ S A , A = 0.9036 W 3 JC ∗ S A , A = 0.8780 W 4 Further, it is necessary to assign attribute weights to each Appropriate Mask Selection to Prevent COVID‑19 attribute since different people have different respiratory Outbreak conditions. For instance, a person having high respiratory complications will for obvious reasons put more weightage The demand for face masks has seen an unprecedented on the filtration capability attribute of the mask, to minimize spike as a result of the havoc and outrage caused by the the chances for transmission of COVID-19 disease. COVID-19 pandemic. Several types of masks which are Thus, instead of considering equal weights for the normally available in the market are, namely, disposable attributes A , A , A and A , we consider the weigh vector medical masks ( M ), normal non-medical masks ( M ), sur- 1 2 3 4 1 2 to be W = {0.6, 0.1, 0.1, 0.2} . We proceed in a step-wise gical masks ( M ), gas masks ( M ), thick-layered medical 3 4 manner which is illustrated below. protective masks ( M ), and N95 masks or particulate res- pirators ( M ). People interested in buying an appropriate Step 1 Identification of the attribute-type. mask keep the following four attributes in mind, namely, high filtration capability ( A ), ability to re-utilize or re-use Here, attributes A , A , and A are of benefit-type, while ( A ), material texture or quality ( A ), and rate of leak- 1 2 3 2 3 A is a cost-type attribute. age ( A ). The attribute values are determined based on the evaluation index provided by people for each type of mask Step 2 Determination of the ideal solution (IS). and are presented via SVNSs as shown in Table 6. ∗ ∗ ∗ ∗ ∗ The ideal solution M = M ,M ,M ,M is constructed 1 2 3 4 Table 5 Comparison of the ranking order by proposed method with using the formulae given below, other existing methods ∗ ∗ ∗ ∗ M = T , I , F = max T , min I , min F ij ij ij j j j j i  i i Ranking order ∗ ∗ ∗ ∗ Existing methods for MADM with SVNS for benefit-type attribute, and   M = T , I , F = min j j j j Subset-hood measure method [19] A > A > A > A 4 2 1 3 T , max I , max F for cost-type attribute, and where ij ij ij i i Improved correlation coefficient [30] A > A > A > A 4 2 1 3 i = 1, 2, ..., 6 ; j = 1, 2, 3, 4. Therefore, with the help of above Pramanik et al. hybrid dice similarity measure A > A > A > A 2 4 1 3 two equations, the ideal solution for the given decision matrix [34] R = r is evaluated as, ij Proposed method A > A > A > A 6×4 2 3 4 1 1 3 1030 Cognitive Computation (2021) 13:1019–1033 Table 6 Decision matrix R = r , for different mask types and their attribute values in terms of SVNSs ij 6×4 Attributes → mask A A A A 1 2 3 4 types ↓ M ⟨0.0698, 0.5731, 0.4246⟩ ⟨0.5320, 0.0234, 0.0493⟩ ⟨0.0813, 0.2139, 0.3334⟩ ⟨0.6213, 0.0910, 0.0740⟩ M ⟨0.0634, 0.4217, 0.4429⟩ ⟨0.1246, 0.1930, 0.2222⟩ ⟨0.2216, 0.0816, 0.0727⟩ ⟨0.1891, 0.5163, 0.6491⟩ M ⟨0.0810, 0.4070, 0.3996⟩ ⟨0.3116, 0.4218, 0.4119⟩ ⟨0.0836, 0.4890, 0.4514⟩ ⟨0.0912, 0.3914, 0.3823⟩ M ⟨0.3716, 0.3716, 0.3017⟩ ⟨0.1136, 0.0886, 0.0914⟩ ⟨0.1969, 0.1471, 0.1524⟩ ⟨0.0202, 0.0742, 0.0781⟩ M ⟨0.3821, 0.4061, 0.4063⟩ ⟨0.3052, 0.5353, 0.5249⟩ ⟨0.5893, 0.2041, 0.1981⟩ ⟨0.3013, 0.0926, 0.0717⟩ M ⟨0.5542, 0.1823, 0.1800⟩ ⟨0.1919, 0.3228, 0.3617⟩ ⟨0.3816, 0.3014, 0.2961⟩ ⟨0.2918, 0.3814, 0.2223⟩ Therefore, with the help of above two equations, the ideal solu- For  = 0.8, tion for the given decision matrix R = r is evaluated as, ij 6×4 JD ∗ JD ∗ S M , M = 0.3664, S M , M = 0.5032, 1 2 W W � � JD ∗ JD ∗ ⟨0.5542, 0.1823, 0.1800⟩, ⟨0.5320, 0.0234, 0.0493⟩, S M , M = 0.4731, S M , M = 0.6015, ∗ 3 4 W W M = JD ∗ JD ∗ ⟨0.5893, 0.0816, 0.0727⟩, ⟨0.0202, 0.5163, 0.6491⟩ S M , M = 0.6100, S M , M = 0.8193 5 6 W W (20) JC ∗ By our first proposed measure S M , M , Step 3 Determining the weighted convex vector similarity For  = 0.1, measures. JC ∗ JC ∗ S M , M = 0.4652, S M , M = 0.6295, 1 2 W W By multiplication of the respective weight to each attribute, we JC ∗ JC ∗ S M , M = 0.5912, S M , M = 0.8520, 3 4 W W obtain the weighted vector similarity measure values as, JC ∗ JC ∗ S M , M = 0.7149, S M , M = 0.8756 JD ∗ 5 6 W W By our first proposed measure S M , M , For  = 0.1, For  = 0.4, JD ∗ JD ∗ S M , M = 0.4580, S M , M = 0.5987, 1 2 W W JC ∗ JC ∗ S M , M = 0.4235, S M , M = 0.5783, 1 2 W W JD ∗ JD ∗ S M , M = 0.5759, S M , M = 0.6775, 3 4 JC ∗ JC ∗ W W S M , M = 0.5421, S M , M = 0.7613, 3 4 W W JD ∗ JD ∗ S M , M = 0.6860, S M , M = 0.8623 5 6 JC ∗ JC ∗ W W S M , M = 0.6727, S M , M = 0.8528 5 6 W W For  = 0.4, For  = 0.8, JD ∗ JD ∗ S M , M = 0.4188, S M , M = 0.5578, 1 2 JD ∗ JD ∗ W W S M , M = 0.3680, S M , M = 0.5100, 1 2 W W JD ∗ JD ∗ S M , M = 0.5319, S M , M = 0.6449, 3 4 JD ∗ JD ∗ W W S M , M = 0.4765, S M , M = 0.6403, 3 4 W W JD ∗ JD ∗ S M , M = 0.6534, S M , M = 0.8439 5 6 JD ∗ JD ∗ W W S M , M = 0.6164, S M , M = 0.8223 5 6 W W Table 7 Similarity values SM methods Similarity values between pairs of masks Best mask Worst mask obtained for different mask ∗ ∗ ∗ ∗ ∗ ∗ types under different methods M ,M M ,M M ,M M ,M M ,M M ,M 1 2 3 4 5 6 J [30] 0.3402 0.4759 0.4438 0.5798 0.5883 0.8071 M M VSM 6 1 D [30] 0.4711 0.6124 0.5906 0.6883 0.6969 0.8684 M M VSM 6 1 C [30] 0.4791 0.6466 0.6076 0.8823 0.7290 0.8833 M M VSM 6 1 Hyb [34] 0.4771 0.6380 0.6034 0.8338 0.7210 0.8795 M M W 6 1 ( = 0.25) JD S ( = 0.1) 0.4580 0.5987 0.5759 0.6775 0.6860 0.8623 M M 6 1 JD S ( = 0.4) 0.4188 0.5578 0.5319 0.6449 0.6534 0.8439 M M 6 1 JD 0.3664 0.5032 0.4731 0.6015 0.6100 0.8193 M M S ( = 0.8) 6 1 JD 0.4652 0.6295 0.5912 0.8520 0.7149 0.8756 M M S ( = 0.1) 6 1 JD 0.4235 0.5783 0.5421 0.7613 0.6727 0.8528 M M S ( = 0.4) 6 1 JD 0.3680 0.5100 0.4765 0.6403 0.6164 0.8223 M M S ( = 0.8) 6 1 1 3 Cognitive Computation (2021) 13:1019–1033 1031 Moreover, the similarity measure results obtained under have recently become one of the research hotspots for research- various existing measures are also presented in Table 7. ers from all over the globe. In our study, we investigate and Step 4 Ranking of the masks. propose certain similarity measures for SVNSs since the concept Based on the highest similarity measure value obtained of similarity has a big influence on MADM problems. It is to between the set of masks and the ideal solution (mask), be noted that elements that are regarded as similar are viewed we find that M (N95-mask) is the appropriate mask or the from different perspectives of parameters like closeness, prox- best buying option to help minimize the transmission rate imity, resemblances, distances, and dissimilarities. Moreover, in of the COVID-19 pandemic. decision-making problems, human beings as decision-makers It is evident from Table  7 that our evaluation for the scrutinize several criteria before making a final decision. So, best suitable mask coincides with the outcomes obtained considering the relative importance of weights becomes a neces- with other similarity measures as well. This demonstrates sity. Most often, the weights are considered in such a way that the credibility and validity of our newly proposed similar- their sum is equal to one. While comparing two objects, we nor- ity measures. mally are interested in knowing whether the objects are identical or partially (approximately) identical or at least identical to what Computational‑time Analysis degree. This instinct compels us to investigate and address some desirable properties about the form of similarity measures for Based on conclusive evidence obtained from experimental SVNSs under consideration. data and owing to their “simple” and “easy-to compute” structure, the vector SMs are found to be very much effective. This implies that the calculations involved consume a sub- Conclusion stantially less amount of time which provided the decision- makers with the surplus advantage of time. Decision making in humans can be described as a cognitive pro- It is noteworthy that since our newly constructed measures cess that mainly focuses on the data which is given as input and are devised with the help of the Jaccard, Dice, and cosine on the cognitive capabilities of people. By cognitive capabili- vector SMs, so the computation time is much smaller in our ties, we mean the ability by which we know how the available case too. However, the only difference is that the time taken information is further processed. Unlike machines which pro- for our calculations is twice that when vector SMs are con- cess the given information in a binary form, whereas humans do sidered alone. But even then, a very miniature amount of time not think similarly. Rather people’s opinions can be expressed/ is spent, and adopting powerful software like MATLAB for measured on a specific evaluation scale. Mostly the fluctuations calculation purposes, just eases our load and provides instant or deviations observed in decision making problems are due results in the blink of an eye. For obvious reasons, that addi- to behavioral biases among people, which are deviations from tional amount of time taken via our constructed measures is rational standards while processing arguments. Further, few compensated by the accurate and efficient results evaluated. other factors like time factor, overestimation of negative com- The main advantage of our proposed measures over the ments, perception differences among people regarding positive, existing methods in the literature is the fact that they can and negative information also contribute towards irrational deci- not only accommodate the SVN environment, but they can sions being made. Often people make such non-rational deci- also capture the indeterminate information supplied by the sions in the process of trying to avoid losses. Therefore, proper decision-makers, automatically. knowledge of psychology is required to understand how people choose between different courses of action. Hence, we can say Importance of the Study that a significant target of cognitive psychology is to elaborate the mental state-of-art processes that define human behavior. NSs are developed from a philosophical point of view and as a In the same vein, the similarity measure concept is one of generalization to many sets like classical set, fuzzy set, intuition- the prime concepts in human cognition. The role of similar- istic fuzzy set, interval-valued fuzzy set, interval-valued intui- ity measures is so crucial in decision-making domain that tionistic fuzzy set, tautological set, paradoxist set, dialetheist set, it has diverse applications in the field of machine learn- and paraconsistent set. But, due to their scarce real-scientific ing, taxonomy, case-based reasoning, recognition, ecology, and engineering applications, a particular subclass of NSs was physical anthropology, automatic classification, psychology, developed known as SVNSs. SVNSs have the unique ability citation analysis, information retrieval, and many more. In to imitate the ambiguous nature of subjective judgments pro- this regard, one of the efficient and significant tools for the vided by the decision-makers and are suitable for dealing with measurement of similarity between two objects is the vec- uncertain, imprecise, and indeterminate information which are tor SMs. Jaccard, Dice, and cosine SMs are the ones that prevalent in multiple-criteria decision analysis. Thus, SVNSs are mostly sought for. But each element of the universe has provide a significant and powerful mathematical framework and some inherent weight associated with them, so to provide 1 3 1032 Cognitive Computation (2021) 13:1019–1033 an order of importance among the elements, we opt for Declarations weighted vector SMs over the non-weighted vector SMs. Ethical Approval This article does not contain any studies with animals Hence, in this article, we have proposed two weighted con- performed by any of the authors. vex vector SMs for SVNSs. SVNSs being a subclass of NSs are considered here, due to their efficiency in tackling Conflict of Interest The authors declare no competing interests. imprecise, incomplete, and inconsistent information. They provide the decision-makers with an additional probability to capture the indeterminate information which normally References exists in almost all real-world phenomena. Thereafter, a MADM method is discussed in detail using the proposed 1. Smarandache F. 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Introduction to modern information set for diagnosis of bipolar disorder diseases. Artif Intell Med. retrieval. McGraw-Hill, Auckland. 1983. 2019. https:// doi. org/ 10. 1016/j. artmed. 2019. 101735 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Cognitive Computation Pubmed Central

Multi-attribute Cognitive Decision Making via Convex Combination of Weighted Vector Similarity Measures for Single-Valued Neutrosophic Sets

Cognitive Computation , Volume 13 (4) – May 21, 2021

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Pubmed Central
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© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
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1866-9956
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10.1007/s12559-021-09883-0
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Abstract

Similarity measure (SM) proves to be a necessary tool in cognitive decision making processes. A single-valued neutrosophic set (SVNS) is just a particular instance of neutrosophic sets (NSs), which is capable of handling uncertainty and imprecise- ness/vagueness with a better degree of accuracy. The present article proposes two new weighted vector SMs for SVNSs, by taking the convex combination of vector SMs of Jaccard and Dice and Jaccard and cosine vector SMs. The applications of the proposed measures are validated by solving few multi-attribute decision-making (MADM) problems under neutro- sophic environment. Moreover, to prevent the spread of COVID-19 outbreak, we also demonstrate the problem of selecting proper antivirus face mask with the help of our newly constructed measures. The best deserving alternative is calculated based on the highest SM values between the set of alternatives with an ideal alternative. Meticulous comparative analysis is presented to show the effectiveness of the proposed measures with the already established ones in the literature. Finally, illustrative examples are demonstrated to show the reliability, feasibility, and applicability of the proposed decision-making method. The comparison of the results manifests a fair agreement of the outcomes for the best alternative, proving that our proposed measures are effective. Moreover, the presented SMs are assured to have multifarious applications in the field of pattern recognition, image clustering, medical diagnosis, complex decision-making problems, etc. In addition, the newly constructed measures have the potential of being applied to problems of group decision making where the human cognition- based thought processes play a major role. Keywords Similarity measure · Neutrosophic set · Single-valued neutrosophic set · Convex vector similarity measure · Multi-attribute decision making Introduction to psychologists, the decision making process can be understood by considering both individual judgments and Human beings remain constantly in a state of making deci- taking into account both rational and irrational aspects of sions due to the intrinsic nature of their mind. They are natu- behavior. Thus, for proper representation of decision maker ral decision makers since every action ultimately results in interests, we are bound to consider those cognitive aspects. a decision, no matter how significant it might be. Several This implies that decisions considering cognitive aspects cognitive factors like people’s level of expertise, behavioral are comparatively better and they closely depict the deci- style, and decision maker’s credibility have a huge psycho- sion maker’s preferences. Human cognition-based methods logical impact on the decision making process. According or techniques not only help the decision makers in express- ing their preferences regarding a certain scenario but also helps visualize people’s intentions or underlying thought * Palash Dutta processes. palashdutta@dibru.ac.in It is worth mentioning that various disciplines in the field Gourangajit Borah of operations research, economics, management science, gjit1993@gmail.com etc., flourished with convincing outcomes when the notion 1 of MADM was introduced to their researchers in respective Department of Mathematics, Dibrugarh University, domains. However, it is noticed that the decision makers Dibrugarh 786004, Assam, India Vol.:(0123456789) 1 3 1020 Cognitive Computation (2021) 13:1019–1033 involved in MADM problems are unable to come up with the In the recent 5 years, SVNSs have seen widespread proper justification of the involved decision parameters, due applications in the realm of cognitive decision-making. For to reasons like lack of information about the public domain, instance, Chai et al. [35] enriched the literature of SVNSs by poor information processing capabilities, complexity of the proposing certain novel similarity measures. Saqlain et al. scenario, shortage of time, etc. This results in incorrect pref- [36] proposed the concept of tangent similarity measure for erence ordering of alternatives. We encounter a wide litera- single and multi-valued hypersoft sets under a neutrosophic ture on MADM problems, where the attribute values take the setting. Likewise, Qin and Wang [37] proposed certain form of crisp numbers [3], fuzzy numbers [4], interval-valued entropy measures for SVNSs with applications to MADM fuzzy numbers [5], interval-valued intuitionistic fuzzy num- problems. Basset et al. [38] gave the form for cosine similar- bers [6], and so on. ity measure for treatment of bipolar disorder diseases with For the first time, neutrosophic sets (NSs) were developed the help of bipolar neutrosophic sets. Tan and Zhang [39] by Smarandache [1, 2], which are capable of dealing with illustrated the decision-making procedure which is helpful in imprecise or unclear information. These sets are character- outrage/havoc assessment of typhoon disaster havoc by appli- ized by three independent functions namely, truth, indeter- cation of Refined SVNSs. Ye [40] carried out fault analysis minacy, and falsity membership functions. Noteworthy that, of a steam turbine with the help of cotangent function-based fuzzy sets and intuitionistic fuzzy sets can only deal with SMs for SVNSs to maximize its efficiency. Moreover, Ye [41] partial or incomplete information, but on the other hand, discussed certain bidirectional projection measures of SVNSs NSs tackle inconsistent information to a pretty decent extent. for their applications in mechanical design. Mondal et al. [42] The widespread application of NSs to MADM problems, demonstrated a MADM strategy based on hyperbolic sine where the decision makers express the ratings of alterna- similarity measures for SVNSs. Thereafter, certain entropy tives with the help of NSs is gaining huge attention among and cross-entropy measures for SVNSs were proposed by Wu researchers recently [7–9]. Single-valued neutrosophic set et al. [43]. Ye and Fu [44] in their paper, showed the useful- (SVNSs) were first proposed by Wang et al. [10], where ness of tangent function-based SM for the treatment of the he discussed some of their preliminary ideas and the arith- multi-period medical condition. In the same year, Ye [45] metic operations between SVNSs. Furthermore, Wang real- demonstrated how a dimension root SM can be applied for ized that interval numbers could better represent the truth, the diagnosis of faults in hydraulic turbines. Garg and Nancy the indeterminacy, and the falsity degree of a particular [46] proposed certain novel ideas of biparametric distance statement, over the classical non-fuzzy or crisp numbers. measures for SVNSs. The concept of SVNSs when applied Hence, Wang et al. [11], proposed interval neutrosophic to graphs, or precisely, “single-valued neutrosophic graphs”, sets (INSs). Thereafter, various methods were developed was a significant and enticing theory which was proposed by for MADM problems involving SVNSs and INSs, such as Broumi et el. [47]. Unlike simple graphs, a stronger version TOPSIS method [12, 13], weighted aggregation operators of it known as hypergraphs had many remarkable applica- [14–18], subsethood measure [19], inclusion measures [20], tions in the literature. In simple graphs where a single edge and outranking method [21, 22]. As a result, MADM prob- can connect exactly two vertices only, but a hyperedge in a lems are tackled with an efficient and significant tool known hypergraph can connect a set of vertices. In this context, Yu as similarity measure (SM) [23–27]. The highest weighted et al. [48] applied the learning method involving hypergraphs SM value between the set of alternatives and the positive to model the pair-wise coherency between images. Another ideal alternative corresponds to the deserving or best alter- term for it is ‘transductive image classification’. Moreover, native. Broumi and Smarandache [28], considered NSs and Yu et al. [49] applied the complex concept of multimodal hence defined the Hausdorff distance measure between two hypergraphs to propose a sparse coding technique for the such sets. Later on, it was Majumdar and Samanta [29] who prediction of click data and image re-ranking, and as a con- utilized the concepts of membership degrees, matching func- sequence of which we get minimum margin for an era in tion, and distance measure, for defining certain SMs between a text-based image search. The application of hypergraphs two SVNSs. The correlation coec ffi ient for SVNSs was then tends to improve the visual efficacy which was established improved by Ye [30]. Moreover, Ye [31] also introduced by Yu et al. [50] in their journal article. Moreover, in their vector SMs for SVNSs and INSs, where SVNS acts as a article, they also proposed a novel ranking model which takes vector in three dimensions. Also, Ye [32] utilized the vector into account the click features and visual features. Later, a concept in improving the cosine SM, so that it can be applied significantly effective and novel framework by the name of to problems of medical decision-making. It was Broumi and Muli-task Manifold Deep Learning (M2DL) for estimating Smarandache [33] who tackled pattern recognition problems face poses via multi-modal information was introduced by by extension of the cosine SM for SVNSs onto INSs. Con- Hong et al. [51]. sequently, Pramanik et al. [34] proposed hybrid vector SM Therefore, it is clear that in the context of decision mak- for SVNSs and INSs. ing, the psychological aspects play a significant role. The 1 3 Cognitive Computation (2021) 13:1019–1033 1021 process through which we “perceive”, “interpret”, and “gen- of truth, indeterminacy, and falsity. The cognitive decision- erate” our responses towards the thought process of people making problems are innate to possession of factors like undergoing social interactions is termed as Social Cognition. vagueness, error, contradiction, and redundancy data. In this Social functioning outputs have a strong linkage to social context, NSs which are capable of handling such imprecise- cognition, such that when such aspects are not considered ness or vagueness shall prove to be a good fit while solving the quality of decisions is degraded, while people achieve multi-attribute decision-making and multi-attribute group an enhanced degree of satisfaction with decisions when such decision-making problems. SVNSs are a particular sub- factors are considered. A person needs to first understand class of NSs, which we will be using to tackle a MADM the process of social interactions to handle social cognition method. INSs are also a similar subclass of NSs, which are tasks. For example, suppose a family decides to celebrate exempted in this study. There is a common practice of apply- the birthday of one of its members and it is trying to figure ing the vector SMs in decision-making problems [57–59]. out the best restaurant in the city for the celebration. Here, However, motivated by the works of Ye [56] and Pramanik the intention of the family members comes into play rather et al. [34], we propose two convex vector SM by taking the than only the preferences towards the alternatives and their convex combination of Jaccard and Dice SMs and Jaccard respective criteria. Now, the person whose birthday needs and cosine SMs, respectively. In this article, our proposed to be celebrated is the crucial one who needs to be content measures for SVNSs are developed extending the concept with the decision. Ironically, if some other members opt for of variation coefficient similarity method [56], under the a restaurant according to their liking which eventually makes neutrosophic setting. We discuss some basic properties of the person (whose birthday is to be celebrated) unhappy, the newly constructed measures and thereby demonstrating then the decision will be considered to be a bad/inappropri- their valid structural formulation. The application of our ate one. proposed measures is shown while tackling few practical However, there exist certain cognitive limitations in peo- MADM scenarios. The final yielded outcomes, affirms the ple’s minds like initial impression and emotional satisfaction accuracy and good fit of our proposed measures. at the time of making decisions, which restrict them from making rational decisions in MCDM scenarios. For instance, suppose there is a real estate dealer who is interested in buy- Structure of the Paper ing a house from four options that are presented to him/her by the estate agency. Now, if the dealer enters one of these Hence, the rest of the paper is set out as follows: in “Pre- houses and suddenly feels uncomfortable, then no matter liminaries and Existing Methods”, we give a brief overview how good the deal of the house is at that price, the dealer of the concepts related to neutrosophic sets, SVNSs, and will restrict himself from buying it since his feeling does not also, we review certain existing weighted vector SMs for allow him to do so. Consider another scenario, where a per- SVNSs. In “Proposed Method”, our proposed definitions of son wants to buy a second-hand truck. Even though the truck weighted convex vector similarity measures are discussed, might be in excellent working condition and the car dealer while the subsequent “MADM based on Proposed Similar- offers him a good deal, but if the person who is interested in ity Measure for SVNSs” firstly elaborates the procedure to buying somehow feels that the dealer is tricking him, then solve any MADM problem based on the proposed method. he would not buy it. Here, the first impression of the person Then, to validate the applicability of our proposed meas- restricts him from making a final decision. Similarly, there ures, we illustrate few practical MADM scenarios which for are many such examples. From the literature, it is evident instance are optimum profit for an investment company and that MCDM techniques including cognitive aspects are close the selection of proper face masks, to prevent the spread of to inexistent. However, few researchers have tried to estab- the COVID-19 pandemic. In addition, they are supported lish the linkage of social cognition into decision making by meticulous comparative analysis showing the feasibility problems (for details please refer to Bisdorff [52], Carneiro of our proposed measures with the existing methods in the et al. [53], Homenda et al. [54], and Ma et al. [55]). literature. Finally, “Conclusion” provides the conclusions for the article. Motivation of Our Work Uncertain or imprecise information is an indispensable fac- Preliminaries and Existing Methods tor in most real-world application problems. In this regard, NSs are well equipped with handling inconsistent informa- Here, in this section, we briefly review some of the concepts tion and indeterminate decision data with a better perspec- of NSs, SVNSs, and their basic arithmetic operations, the tive over fuzzy sets and intuitionistic fuzzy sets. It is due ideas of which will be necessitated in the subsequent sec- to their characteristic independent membership functions tions of our study. 1 3 1022 Cognitive Computation (2021) 13:1019–1033 Definition 1 (Neutrosophic Sets) [1, 2] 2. Inclusion: A SVNS P is said to be a subset of another SVNS Q , that is P ⊆ Q if and only if T (x) ≤ T (x), P Q Suppose we consider U to be a space of objects (points) I (x) ≥ I (x), F (x) ≥ F (x); ∀x ∈ U. P Q P Q and let us denote the generic element in U by x . Then a 3. Equality: For equality between two SVNSs P and Q neutrosophic set P in U is characterized by three inde- to hold, we must have that both sets must be a subset pendent functions, a truth membership function T (x) , a n of each other, that is, P ⊆ Q and P ⊇ Q . Or in other indeterminacy membership function I (x) , and a falsity words, we can also say that, P = Q if and only if membership function F (x) . The three functions are stand- T (x) = T (x), I (x) = I (x), F (x) = F (x); ∀x ∈ U. P P Q P Q P Q − + ard and non-standard subsets of the interval 0, 1 , and the 4. Addition: The addition operation between two P ⊕ Q = x, T following condition is satisfied, SVNSs P and Q is defined by ⟨ (x) +T (x) − T (x)T (x), I (x)I (x), F (x)F (x) x ∈ U , Q P Q P Q P Q − + 0 ≤ sup T (x) + sup I (x) + sup F (x) ≤ 3 (1) P P P which is not the traditional additive rule. Here, the first component is the algebraic sum of truth degrees minus Noteworthy that, neutrosophic sets introduced by their product, the second component is the product of Smarandache [1] were more from a philosophical point the indeterminacy degree for the two sets, and the third of view and they had very scarce applications in the field component is the product of their falsity degrees. of science and engineering. As a better successor to it, 5. Multiplication: The multiplication operation Wang et al. [10] introduced a subclass of the neutrosophic between two SVNSs P and Q is defined by P ⊗ Q sets, which are called single-valued neutrosophic sets = x, T (x)T (x), I (x) + I (x) − I (x)I (x), F (x)+ P Q P Q P Q P (SVNSs). F (x) − F (x)F (x) x ∈ U , where the components Q P Q swap their arithmetic as in the addition case. That is, Definition 2 (Single-Valued Neutrosophic Sets) [10] the first component is the product of their truth degrees, the second component being the algebraic sum minus Suppose we consider U be a space of objects (points) the product of their indeterminacy degrees, and it is the and let us denote the generic element in U by x . Then a algebraic sum minus their product of falsity degrees in single-valued neutrosophic set P in X is characterized by the third component. three independent functions, a truth membership function 6. Union: The union of two SVNSs P and Q is defined by T (x) , an indeterminacy membership function I (x) , and a P P P ∪ Q = x, T (x) ∨ T (x), I (x) ∧ I (x), F (x) ∧ F P Q P Q P Q falsity membership function F (x) . We denote the SVNS (x)⟩�∀x ∈ U} . The resulting SVNS has the first compo- P as � � nent as the maximum of their truth degrees, the sec- P = ⟨x, T (x), I (x), F (x)⟩�x ∈ X , w h e re T (x) , I (x) , P P P P P ond component as the minimum of their indeterminacy F (x)∈[0, 1]; x ∈ U. degrees, and the third component is the minimum of Also, the following inequality is satisfied by the sum of their respective falsity degrees as well. T (x) , I (x) , and F (x), P P P 7. Intersection: The intersection of two SVNSs P and Q is 0 ≤ T (x) + I (x) + F (x) ≤ 3 (2) defined by  P ∩ Q = x, T (x) ∧ T (x), I (x) ∨ I (x), P P P P Q P Q F (x) ∨ F (x) ∀x ∈ U . Broadly speaking, the inter- P Q For the sake of simplicity, let us consider, P = section is the reverse case for union operation since ⟨T (x), I (x), F (x)⟩ as the SVNS in U. P P P the maximum function in the case of union operation becomes the minimum function here, and vice-versa. Definition 3 (Arithmetic operations between SVNSs) [10, 16] Existing Non‑weighted Vector Similarity Measures For any two SVNSs P = ⟨T (x), I (x), F (x)⟩ and Q = P P P SMs greatly enhance the valuable output efficiency in decision- T (x), I (x), F (x) considered in a finite universe U , t he Q Q Q making processes. Many experts from time to time have formu- arithmetic operations between them were proposed in previ- lated several fruitful definitions of SMs based on distances and ous studies [10, 16] as follows: vectors. Hence, in the following sequel, we recall the definitions of Jaccard [57], Dice [58], and cosine [59] similarity measures. 1. Complement: The complement of SVNS P is denoted by These SMs are structurally simple, easy to compute, and mod- C C P and defined as  P = ⟨F (x),1 − I (x), T (x)⟩ , where P P P est, which enables the decision-makers to determine the differ- the first component is the falsity membership degree for ent similarity value options at ease. P , the second component is 1 minus the indeterminacy Let M = m , m , ..., m and N = n , n , ..., n be two s 1 2 s 1 2 s degree for P , and the third is the truth membership grade -dimensional vectors having non-negative co-ordinates. Then, for P. 1 3 Cognitive Computation (2021) 13:1019–1033 1023 C3. C (M, N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) Definition 4 Between any two vectors M = m , m , ..., m SM i i 1 2 s for every m ∈ M and n ∈ N and N = n , n , ..., n , the Jaccard similarity measure [57] i i 1 2 s is defined as Remark 1 The common property that each of these simi- m n i i M.N i=1 larity measures hold is that they assume values within the J (M, N) = = SM s s s 2 2 ∑ ∑ ∑ ‖M‖ + ‖N‖ − M.N unit interval [0, 1] . Jaccard and Dice SMs are undefined for 2 2 m + n − m n i i i i both m = 0 and n = 0 , whereas cosine similarity measure is i=1 i=1 i=1 i i (3) undefined for either m = 0 or n = 0 , for i = 1, 2, ..., s. i i � � s s ∑ ∑ 2 2 where ‖M‖ = m and ‖N‖ = n are called the Definition 7 Between two vectors M = m , m , ..., m and i i 1 2 s i=1 i=1 N = n , n , ..., n , the variation coefficient similarity meas- 1 2 s Euclidean norms of M and N , and the inner product of vec- ure [56] is defined as tors M and N is given by M.N = m n . i i i=1 2M.N M.N V (M,N) =  + (1 − ) CF 2 2 ‖M‖‖N‖ The above-mentioned similarity measure satisfies the fol- ‖M‖ + ‖N‖ lowing properties: s s ∑ ∑ 2 m n m n i i i i J1. 0 ≤ J (M, N) ≤ 1 SM i=1 i=1 ⇒ V (M, N) =  + (1 − ) � � CF s s s J2. J (M, N) = J (N, M) ∑ ∑ ∑ s s SM SM 2 2 ∑ ∑ m + n − m n 2 2 i i i i m n J3. J (M, N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) SM i i i=1 i=1 i=1 i i i=1 i=1 for every m ∈ M and n ∈ N i i (6) It satisfies the following properties: Definition 5 Between two vectors M = m , m , ..., m 1 2 s and N = n , n , ..., n , the Dice similarity measure [58] is 1 2 s V1. 0 ≤ V (M,N) ≤ 1 CF defined as V2. V (M,N) = V (N,M) CF CF V3. V (M,N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) CF i i 2 m n i i for every m ∈ M and n ∈ N 2M.N i=1 i i D (M, N) = = (4) SM s s 2 2 ∑ ∑ ‖M‖ + ‖N‖ 2 2 m + n i i Some Weighted Vector Similarity Measures of SVNSs i=1 i=1 It satisfies the following properties: In multiple-criteria decision making methods, criteria weights have a much larger influence on the outcomes D1. 0 ≤ D (M,N) ≤ 1 yielded by a decision process and also on the ranking of SM D2. D (M,N) = D (N,M) alternatives. The reason being that it takes into account the SM SM D3. D (M,N) = 1 for M = N , i.e., m = n (i = 1, 2, ..., s) relative importance of each criterion concerning the set of SM i i for every m ∈ M and n ∈ N alternatives chosen. Consequently, any such process which i i does not consider the respective criteria weightage and sets identical weights of importance for them thus loses its logi- Definition 6 Between two vectors M = m , m , ..., m cal importance. Thus, decision-makers from their best of 1 2 s and N = n , n , ..., n , the cosine similarity measure [59] knowledge try to allocate weights to each criterion involved 1 2 s is defined as in a decision process. Hereby, we list down three existing definitions of weighted vector SMs. Suppose we consider two SVNSs P and Q in a three- m n i i i=1 M.N dimensional space defined by C (M, N) = = � � SM (5) ‖M‖.‖N‖ s s P = T x , I x , F x x ∈ U and Q = T ∑ ∑ P i P i P i i Q 2 2 m . n i i x , I x , F x x ∈ U . i Q i Q i i i=1 i=1 Then, we can dene fi weighted vector similarity as follows: It satisfies the following properties: Definition 8 Let U be a universe of discourse defined by U = x , x , ..., x , wher e P = T x , I x , F x C1. 0 ≤ C (M, N) ≤ 1 1 2 r P i P i P i SM C2. C (M, N) = C (N, M) SM SM 1 3 1024 Cognitive Computation (2021) 13:1019–1033 P2. J (P, Q) = J (Q, P); D (P, Q) = D (Q,P); x ∈ U and Q = T x , I x , F x x ∈ U be WSM WSM WSM WSM i Q i Q i Q i i C (P,Q) = C (Q,P) two SVNSs. WSM WSM P3. J (P, Q) = 1; D (P, Q) = 1; C (P, Q) = 1 if WSM WSM WSM and only if P = Q , which implies Let w ∈[0, 1] be the weight of every element x (i = 1, i i 2, ..., r) , so that w = 1. T x = T x , I x = I x , F x = F x , for i=1 P i Q i P i Q i P i Q i Then, the weighted Jaccard similarity measure [30] every x ∈ X (i = 1, 2, ..., r). between P and Q is defined as �� �� ���� �� �� T x T x + I x I x + F x F x P i Q i P i Q i P i Q i J (P, Q) = w � � WSM i � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ i=1 (7) T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ Definition 9 Let U be a universe of discourse defined by Remark 3 Now, J (P, Q), D (P, Q) for SVNSs WSM WSM U = x , x , ..., x , wher e P = T x , I x , F x P = T x , I x , F x x ∈ U and Q = T x , 1 2 r P i P i P i P i P i P i i Q i x ∈ U and Q = T x , I x , F x x ∈ U be I x , F x x ∈ U , are undefined when P = ⟨0, 0, 0⟩ i Q i Q i Q i i Q i Q i i two SVNSs. and Q = ⟨0, 0, 0⟩ , i.e., for T = I = F = 0 and P P P T = I = F = 0 given i = 1, 2, ..., r . On the other hand, Q Q Q Let w ∈[0, 1] be the weight of every element x (i = 1, C (P,Q) is undefined for P = ⟨0, 0, 0⟩ or Q = ⟨0, 0, 0⟩ , i i WSM i.e., when T = I = F = 0 or T = I = F = 0 for P P P Q Q Q 2, ..., r) , so that w = 1. i=1 i = 1, 2, ..., r. Then, the weighted Dice similarity measure [30] between P and Q is defined as 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i D (P, Q) = w (8) WSM i 2 2 2 2 2 2 i=1 T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q Definition 10 Let U be a universe of discourse defined by Proposed Method U = x , x , ..., x , wher e P = T x , I x , F x 1 2 r P i P i P i x ∈ U and Q = T x , I x , F x x ∈ U be Based on the enormous potential of SVNSs which help raise i Q i Q i Q i i two SVNSs. the utility of cognitive decision-making process, we pro- pose two weighted convex vector SMs (WCVSMs) which Let w ∈[0, 1] be the weight of every element x (i = 1, are dependent on the coefficient parameter. The SMs have i i arranged in such a way that their structure represents a con- 2, ..., r) , so that w = 1. i=1 vex combination. The idea for such formulation came up as Then, the weighted cosine similarity measure [30] the vector SMs are empirically established to produce feasi- between P and Q is defined as ble and rational outcomes on their own, so there is nothing T x T x + I x I x + F x F x P i Q i P i Q i P i Q i C (P, Q) = w WSM i (9) i=1 2 2 2 2 2 2 T x + I x + F x . T x + I x + F x i i i i i i P P P Q Q Q Remark 2 It is noteworthy that Eqs. (7), (8), and (9) satisfy wrong with constructing a function out of those SMs. The sole intention is to serve the decision-making domain with P1. 0 ≤ J (P, Q) ≤ 1; 0 ≤ D (P, Q) ≤ 1; 0 ≤ C efficient similarity measures which are capable of produc- WSM WSM WSM (P, Q) ≤ 1 ing convincing outcomes by providing a global evaluation 1 3 Cognitive Computation (2021) 13:1019–1033 1025 JD JD JC JC framework for each alternative with respect to each criterion. P2. S (P, Q) = S (Q, P); S (P, Q) = S (Q, P) W W W W JD JC Hence, the two proposed measures are listed below. P3. S (P, Q) = 1; S (P, Q) = 1 when P = Q , i.e., W W Definition 11 Let us consider U be a universe of T x = T x , I x = I x , F x = F x , for P i Q i P i Q i P i Q i discourse defined by U = x , x , ..., x , wher e P = every x ∈ X (i = 1, 2, ..., r) 1 2 r T x , I x , F x x ∈ U and Q = T x , I P i P i P i i Q i Q x , F x x ∈ U are two SVNSs. Proof: i Q i i Also, let w ∈[0, 1] be the weight of every element (P1). From eqns. (7) and (8), we find that for Jaccard and Dice similarity measures of SVNSs, 0 ≤ J (P, Q) ≤ 1 and x (i = 1, 2, ..., r) , such that w = 1. WSM i i i=1 0 ≤ D (P,Q) ≤ 1 for all i = 1, 2, ..., r . Now, Eq. (10) can WSM Then the two weighted convex vector similarity measures be written as follows, between SVNSs are proposed as follows: �� �� ���� �� �� ⎡ T x T x + I x I x + F x F x ⎤ P i Q i P i Q i P i Q i ⎢ � � ⎥ � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ ⎢ i=1 ⎥ T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ JD ⎢ ⎥ (10) S (P, Q) = ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ � 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i + (1 − ) w � � �� ⎥ � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i ⎣ P P P Q Q Q ⎦ and �� �� ���� �� �� � T x T x + I x I x + F x F x ⎡ ⎤ P i Q i P i Q i P i Q i � � ⎢ ⎥ � �� �� ��� �� �� �� ⎡ 2 2 2 2 2 2 ⎤ i=1 ⎢ T x + I x + F x + T x + I x + F x ⎥ i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ ⎢ ⎥ JC (11) S (P, Q) = ⎢ ⎥ ⎢ ⎥ �� �� ���� �� �� ⎢ T x T x + I x I x + F x F x ⎥ P i Q i P i Q i P i Q i + (1 − ) w ⎢ � �⎥ i � � � � �� �� ��� �� �� �� ⎢ i=1 ⎥ 2 2 2 2 2 2 T x + I x + F x . T x + I x + F x ⎢ ⎥ i i i i i i P P P Q Q Q ⎣ ⎦ Our proposed measure(s) satisfy the following JD S (P, Q) =J (P, Q) + (1 − )D (P, Q) W W (12) proposition, ≤  + (1 − ) = 1 Proposition 1 The two proposed weighted convex vector Since, J P, Q ≥ 0 and D P,Q ≥ 0 , so does the ( ) ( ) WSM WSM JD similarity measure (WCVSM) of SVNSs between P and Q WCVSM, S (P, Q) ≥ 0 for all  ∈[0, 1]. satisfy the properties given below: Thus, the first property is satisfied. Similarly, we can JC prove 0 ≤ S (P, Q) ≤ 1. (P2). From Eq. (10), JD JC P1: 0 ≤ S (P, Q) ≤ 1; 0 ≤ S (P, Q) ≤ 1 W W 1 3 1026 Cognitive Computation (2021) 13:1019–1033 �� �� ���� �� �� � T x T x + I x I x + F x F x ⎡ ⎤ P i Q i P i Q i P i Q i ⎢ � � ⎥ � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎡ ⎤ ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i P P P Q Q Q ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ P i Q i P i Q i P i Q i ⎦ JD ⎢ ⎥ S (P, Q) = ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ � ⎥ 2 T x T x + I x I x + F x F x P i Q i P i Q i P i Q i ⎢ + 1 −  w ⎥ ( ) � � �� � �� �� ��� �� �� �� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i ⎣ P P P Q Q Q ⎦ �� �� ���� �� �� ⎡ T x T x + I x I x + F x F x ⎤ Q i P i Q i P i Q i P i ⎢ � � ⎥ �� �� �� � �� �� ��� ⎡ 2 2 2 2 2 2 ⎤ ⎢ i=1 ⎥ T x + I x + F x + T x + I x + F x i i i i i i Q Q Q P P P ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ ⎥ ⎢ ⎥ − T x T x + I x I x + F x F x ⎣ Q i P i Q i P i Q i P i ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � �� �� ���� �� ��� ⎢ � ⎥ 2 T x T x + I x I x + F x F x Q i P i Q i P i Q i P i ⎢ + (1 − ) w ⎥ �� � � �� �� �� � �� �� ��� 2 2 2 2 2 2 ⎢ ⎥ i=1 T x + I x + F x + T x + I x + F x i i i i i i Q Q Q P P P ⎣ ⎦ JD = S (Q, P) set C (j = 1, 2, ..., y) are depicted by single valued neutro- Similar results are also obtained from Eq.  (11 ), JC JC sophic element of the form s = T , I , F . Here, T indi- ij ij ij ij ij S (P, Q) = S (Q, P) , which proves the second property. W W cates the membership degree, I denotes the indeterminacy ij (P3). If T x = T x , I x = I x and F x P i Q i P i Q i P i degree and F indicates the non-membership degree for the ij = F x , for i = 1, 2, ..., r , then the value of J (P, Q) = Q i WSM alternative A with respect to the attribute C . Thus, for i j 1, D (P,Q) = 1 and C (P,Q) = 1 . Therefore, from WSM WSM JD JC i = 1, 2, ..., x; j = 1, 2, ..., y we have, 0 ≤ T + I + F ≤ 3 ij ij ij Eq. (10), the value of S (P, Q) = 1 and S (P, Q) = 1 . This W W and T ∈[0, 1], I ∈[0, 1], F ∈[0, 1]. ij ij ij concludes the proof. Let us consider that the alternative A (i = 1, 2, ..., x) takes single-valued neutrosophic values and has the following rep- Remark 4 Now for P = T x , I x , F x x ∈ U P i P i P i i resentation, A = s , s , ..., s , for i = 1, 2, ..., x; i i1 i2 ix and Q = T x , I x , F x x ∈ U , the convex sim- Q i Q i Q i i � � �� ilarity measure value is assumed to be zero for P = ⟨0, 0, 0⟩ = ⟨T , I , F ⟩, ⟨T , I , F ⟩, ..., T , I , F (13) i1 i1 i1 i2 i2 i2 iy iy iy and Q = ⟨0, 0, 0⟩. There are certain steps to follow while selecting the best alternative amongst a set of alternatives which are as follows, MADM Based on Proposed Similarity Measure for SVNSs Step 1 Determination of the ideal solution We consider a multi-attribute decision-making problem with It is a very common procedure in MADM to utilize the x set of alternatives and y set of attributes, where the values concept of an ideal alternative/solution. And, realizing a of the attributes are represented by SVNSs. Let perfectly ideal solution in the real world is an abstract A = A , A , ..., A be a finite collection of alternatives and 1 2 x idea as there does not exist any such. However, in order to C = C , C , ..., C be the finite collection of attributes. 1 2 y construct a useful theoretical framework and to carry out Also, let the weight vector be denoted by the mathematical calculations, we incorporate the concept w = w , w , ..., w corresponding to the set of attributes 1 2 y of an ideal solution. It facilitates the set of alternatives C (j = 1, 2, ..., y) such that w = 1 and w ≥ 0 . We denote under study to be ranked based on the degree of similarity j j j j=1 (closeness) or non-similarity (farness) from the ideal solu- the decision matrix by D = s , where the preference ij x×y tion. Thus, we need to determine the SVNS-based ideal values of the alternatives A (i = 1, 2, ..., x) over the attribute solution. 1 3 Cognitive Computation (2021) 13:1019–1033 1027 Let Y denote the entire collection of two types of attrib- From Eqs.  (10) and (11), the WCVSMs between ute, which are namely profit/benefit type attribute ( B ) and the ideal alternative A and the alternative A for the cost-type attribute ( L ). Then the ideal solution (IS), i = 1, 2, ..., x;  ∈[0, 1] are given by ∗ ∗ ∗ T T + I I + F F ij ij ij ⎡ j j j ⎤ � � ⎢ � � � � � � � � ⎥ 2 2 2 � � � � � � 2 2 2 ⎡ ⎤ i=1 ∗ ∗ ∗ ⎢ ⎥ T + I + F + T + I + F ij ij ij j j j ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � � ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ ⎥ − T T + I I + F F ij ij ij � � ⎣ j j j ⎦ JD ∗ ⎢ ⎥ (16) S A , A = ⎢ ⎥ ⎢ � � ⎥ ∗ ∗ ∗ ⎢ ⎥ 2 T T + I I + F F � ij ij ij j j j ⎢ ⎥ + (1 − ) w �� � � ⎢ � � � � � � � � ⎥ 2 2 2 � � � � � � 2 2 2 i=1 ∗ ∗ ∗ ⎢ ⎥ T + I + F + T + I + F ij ij ij j j j ⎣ ⎦ ∗ ∗ ∗ T T + I I + F F ⎡ ⎤ ij ij ij j j j ⎢ � � ⎥ � � � � � � � � 2 2 2 � � � � � � 2 2 2 ⎢ ⎥ i=1 ⎡ ∗ ∗ ∗ ⎤ T + I + F + T + I + F ij ij ij j j j ⎢ ⎢ ⎥ ⎥ ⎢ � � ⎥ ⎢ ⎥ ⎣ ⎦ − T T + I I + F F ⎢ ⎥ P ij P ij P ij � � JC ∗ ⎢ ⎥ (17) S A , A = ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ T T + I I + F F ij ij ij j j j ⎢ ⎥ + (1 − ) w � � � ⎢ � � � ⎥ � � � � � � � � 2 2 2 i=1 � � � � � � 2 2 2 ⎢ ⎥ ∗ ∗ ∗ T + I + F . T + I + F ij ij ij ⎢ j j j ⎥ ⎣ ⎦ where the ideal solution A takes respective forms accord- ∗ ∗ ∗ ∗ A = s , s , ..., s , is that solution of the decision matrix 1 2 x ing to the nature of the attribute as depicted in Eqs. (14) D = s which is defined as ij x×y and (15). ∗ ∗ ∗ ∗ (a) s = T , I , F = max T , min I , min F j ij ij ij j j j i i i Step 3 We rank the alternatives. for benefit − type attribute (B) (14) The ranking of the alternatives could be easily deter- and mined according to the values obtained from Eqs.  (14) and (15). The decreasing order of the WCVSMs gives the ∗ ∗ ∗ ∗ (b) s = T , I , F = max T , min I , min F required ranking or preference ordering of the alternatives. ij ij ij j j j j i i i for cost − type attribute (L) Optimum Profit Determination by an Investment (15) Company Step 2 We evaluate the WCVSM between the ideal alterna- We consider a multi-attribute decision making problem (adapted tive and each alternative. from Ye [7]), in which an investment company is interested in finding out the best suitable alternative amongst a set of four Table 1 Decision matrix with single-valued neutrosophic entries alternatives: (1) A is a computer company, (2) A is an arms 1 2 Criteria →/ C C C 1 2 3 company, (3)A is a car company, and (4) A is a food company. 3 4 Alternatives ↓ Three criteria(s) are taken into consideration by the investment A ⟨0.5, 0.3, 0.2⟩ ⟨0.5, 0.2, 0.3⟩ ⟨0.3, 0.2, 0.3⟩ company based on which alternatives are evaluated, which are A ⟨0.4, 0.3, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ ⟨0.7, 0.0, 0.1⟩ (1) C is the environmental impact analysis, (2) C is the growth 1 2 A ⟨0.2, 0.2, 0.5⟩ ⟨0.4, 0.2, 0.3⟩ ⟨0.4, 0.2, 0.3⟩ analysis, and (3) C is the risk factor analysis. The decision A ⟨0.5, 0.2, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ ⟨0.6, 0.1, 0.2⟩ maker assesses the four possible alternatives with respect to the 1 3 1028 Cognitive Computation (2021) 13:1019–1033 Table 2 Differ ent WCVSM Proposed measures Pairs of alternatives Best alter- Ranking order values for different values of native JD ∗ ∗ ∗ ∗ for our first defined measure (A , A ) (A , A ) (A , A ) (A , A ) WCVSM value ( S ) 1 2 3 4 = 0.15 0.7666 0.9172 0.8905 0.8875 A A > A > A > A 2 2 3 4 1 = 0.30 0.7478 0.9093 0.8796 0.8771 A = 0.55 0.7163 0.8962 0.8615 0.8598 A = 0.80 0.6849 0.8831 0.8433 0.8425 A = 0.95 0.6661 0.8752 0.8324 0.8322 A attributes based on the SVNS values provided. The SVNS-based Based on the outcomes obtained for different values of decision matrix D = d are presented in Table 1. as shown in Tables 2 and 3, the alternative A turns out to ij 2 4×3 The weight vector is given by W = w ,w ,w = be the best suitable alternative. 1 2 3 {0.40, 0.25, 0.35} such that Comparative Analysis w = 1 (18) Here, we provide a comparison of the outputs obtained j=1 via our proposed convex vector similarity measures with some of the existing similarity measures on the illus- Step 1 Identification of the attribute-type. trated MADM scenario. The comparison results along with their evaluated similarity values are depicted in Here, the attributes C and C are benefit-type attribute, Table 4. It is very obvious from Table 4 that our results 2 3 while C is identified as cost-type attribute. for evaluation of the best alternative are in agreement with Ye’s vector similarity measure method [31], Ye’s Step 2 Determination of the ideal solution (IS). improved cosine similarity measure [32], and hybrid vector similarity measure by Pramanik et al. [34], for With the help of Eqs. (14) and (15), the ideal solution for SVNSs. Even the ranking order of the alternatives the given decision matrix D = d can be determined as obtained with our proposed measures coincides exactly ij 4×3 � � with that of Ye’s SMs ([31, 32]), whereas the alterna- A = ⟨0.2, 0.3, 0.5⟩, ⟨0.6, 0.1, 0.2⟩, ⟨0.7, 0.0, 0.1⟩ (19) tives A and A , interchange their places under Pramanik 3 4 et al.’s method. Furthermore, the ranking order evaluated by subset-hood Step 3 Evaluation of the weighted convex vector similarity measure method [19], improved correlation coefficient [ 30] measures. is demonstrated in Table  5. According to [19] and [30], the alternative A is the second-best choice amongst the We evaluate the weighted convex vector similarity measures set of alternatives, whereas it is the best choice according with the help of the Eqs. (12–14), and the outcomes obtained to previous studies [31, 32, 34], and our presented meas- for various values of  are presented in Tables 2 and 3. ures. Hence, the validity and feasibility of our measures is established. Step 4 Ranking of the alternatives. Table 3 Differ ent WCVSMs Proposed measures Pairs of alternatives Best alter- Ranking order values for different values of native ∗ ∗ ∗ ∗ JC for our second defined measure (A , A ) (A , A ) (A , A ) (A , A ) WCVSM value ( S ) 1 2 3 4 = 0.15 0.7846 0.9198 0.9036 0.8780 A A > A > A > A 2 2 3 4 1 = 0.30 0.7628 0.9115 0.8905 0.8692 A = 0.55 0.7263 0.8976 0.8687 0.8545 A = 0.80 0.6899 0.8838 0.8469 0.8399 A = 0.95 0.6681 0.8755 0.8339 0.8311 A 1 3 Cognitive Computation (2021) 13:1019–1033 1029 Table 4 Comparison of Similarity measure method Measure value Ranking order WCVSM for SVNSs with different SMs ∗ ∗ A > A > A > A C A , A  [31] C A ,A = 0.7689 2 4 3 1 W i W 1 C A ,A = 0.9281 W 2 C A ,A = 0.8975 W 3 C A ,A = 0.8979 W 4 ∗ ∗ A > A > A > A Hyb A , A  [34] Hyb A , A = 0.7912 2 3 4 1 w i w 1 ( = 0.1) Hyb A , A = 0.9433 w 2 Hyb A , A = 0.9036 w 3 Hyb A , A = 0.9019 w 4 ∗ ∗ A > A > A > A WSC A , A [32] WSC A , A = 0.9401 2 4 3 1 2 i 2 1 WSC A , A = 0.9804 2 2 WSC A , A = 0.9691 2 3 WSC A , A = 0.9761 2 4 JD ∗ JD ∗ A > A > A > A S A , A  (Proposed) ( = 0.1 5) S A ,A = 0.7666 2 3 4 1 W i W 1 JD ∗ S A ,A = 0.9172 W 2 JD ∗ S A ,A = 0.8905 W 3 JD ∗ S A ,A = 0.8875 W 4 JC ∗ JC ∗ A > A > A > A S A , A  (Proposed) ( = 0.1 5) S A , A = 0.7846 2 3 4 1 W i W 1 JC ∗ S A , A = 0.9198 W 2 JC ∗ S A , A = 0.9036 W 3 JC ∗ S A , A = 0.8780 W 4 Further, it is necessary to assign attribute weights to each Appropriate Mask Selection to Prevent COVID‑19 attribute since different people have different respiratory Outbreak conditions. For instance, a person having high respiratory complications will for obvious reasons put more weightage The demand for face masks has seen an unprecedented on the filtration capability attribute of the mask, to minimize spike as a result of the havoc and outrage caused by the the chances for transmission of COVID-19 disease. COVID-19 pandemic. Several types of masks which are Thus, instead of considering equal weights for the normally available in the market are, namely, disposable attributes A , A , A and A , we consider the weigh vector medical masks ( M ), normal non-medical masks ( M ), sur- 1 2 3 4 1 2 to be W = {0.6, 0.1, 0.1, 0.2} . We proceed in a step-wise gical masks ( M ), gas masks ( M ), thick-layered medical 3 4 manner which is illustrated below. protective masks ( M ), and N95 masks or particulate res- pirators ( M ). People interested in buying an appropriate Step 1 Identification of the attribute-type. mask keep the following four attributes in mind, namely, high filtration capability ( A ), ability to re-utilize or re-use Here, attributes A , A , and A are of benefit-type, while ( A ), material texture or quality ( A ), and rate of leak- 1 2 3 2 3 A is a cost-type attribute. age ( A ). The attribute values are determined based on the evaluation index provided by people for each type of mask Step 2 Determination of the ideal solution (IS). and are presented via SVNSs as shown in Table 6. ∗ ∗ ∗ ∗ ∗ The ideal solution M = M ,M ,M ,M is constructed 1 2 3 4 Table 5 Comparison of the ranking order by proposed method with using the formulae given below, other existing methods ∗ ∗ ∗ ∗ M = T , I , F = max T , min I , min F ij ij ij j j j j i  i i Ranking order ∗ ∗ ∗ ∗ Existing methods for MADM with SVNS for benefit-type attribute, and   M = T , I , F = min j j j j Subset-hood measure method [19] A > A > A > A 4 2 1 3 T , max I , max F for cost-type attribute, and where ij ij ij i i Improved correlation coefficient [30] A > A > A > A 4 2 1 3 i = 1, 2, ..., 6 ; j = 1, 2, 3, 4. Therefore, with the help of above Pramanik et al. hybrid dice similarity measure A > A > A > A 2 4 1 3 two equations, the ideal solution for the given decision matrix [34] R = r is evaluated as, ij Proposed method A > A > A > A 6×4 2 3 4 1 1 3 1030 Cognitive Computation (2021) 13:1019–1033 Table 6 Decision matrix R = r , for different mask types and their attribute values in terms of SVNSs ij 6×4 Attributes → mask A A A A 1 2 3 4 types ↓ M ⟨0.0698, 0.5731, 0.4246⟩ ⟨0.5320, 0.0234, 0.0493⟩ ⟨0.0813, 0.2139, 0.3334⟩ ⟨0.6213, 0.0910, 0.0740⟩ M ⟨0.0634, 0.4217, 0.4429⟩ ⟨0.1246, 0.1930, 0.2222⟩ ⟨0.2216, 0.0816, 0.0727⟩ ⟨0.1891, 0.5163, 0.6491⟩ M ⟨0.0810, 0.4070, 0.3996⟩ ⟨0.3116, 0.4218, 0.4119⟩ ⟨0.0836, 0.4890, 0.4514⟩ ⟨0.0912, 0.3914, 0.3823⟩ M ⟨0.3716, 0.3716, 0.3017⟩ ⟨0.1136, 0.0886, 0.0914⟩ ⟨0.1969, 0.1471, 0.1524⟩ ⟨0.0202, 0.0742, 0.0781⟩ M ⟨0.3821, 0.4061, 0.4063⟩ ⟨0.3052, 0.5353, 0.5249⟩ ⟨0.5893, 0.2041, 0.1981⟩ ⟨0.3013, 0.0926, 0.0717⟩ M ⟨0.5542, 0.1823, 0.1800⟩ ⟨0.1919, 0.3228, 0.3617⟩ ⟨0.3816, 0.3014, 0.2961⟩ ⟨0.2918, 0.3814, 0.2223⟩ Therefore, with the help of above two equations, the ideal solu- For  = 0.8, tion for the given decision matrix R = r is evaluated as, ij 6×4 JD ∗ JD ∗ S M , M = 0.3664, S M , M = 0.5032, 1 2 W W � � JD ∗ JD ∗ ⟨0.5542, 0.1823, 0.1800⟩, ⟨0.5320, 0.0234, 0.0493⟩, S M , M = 0.4731, S M , M = 0.6015, ∗ 3 4 W W M = JD ∗ JD ∗ ⟨0.5893, 0.0816, 0.0727⟩, ⟨0.0202, 0.5163, 0.6491⟩ S M , M = 0.6100, S M , M = 0.8193 5 6 W W (20) JC ∗ By our first proposed measure S M , M , Step 3 Determining the weighted convex vector similarity For  = 0.1, measures. JC ∗ JC ∗ S M , M = 0.4652, S M , M = 0.6295, 1 2 W W By multiplication of the respective weight to each attribute, we JC ∗ JC ∗ S M , M = 0.5912, S M , M = 0.8520, 3 4 W W obtain the weighted vector similarity measure values as, JC ∗ JC ∗ S M , M = 0.7149, S M , M = 0.8756 JD ∗ 5 6 W W By our first proposed measure S M , M , For  = 0.1, For  = 0.4, JD ∗ JD ∗ S M , M = 0.4580, S M , M = 0.5987, 1 2 W W JC ∗ JC ∗ S M , M = 0.4235, S M , M = 0.5783, 1 2 W W JD ∗ JD ∗ S M , M = 0.5759, S M , M = 0.6775, 3 4 JC ∗ JC ∗ W W S M , M = 0.5421, S M , M = 0.7613, 3 4 W W JD ∗ JD ∗ S M , M = 0.6860, S M , M = 0.8623 5 6 JC ∗ JC ∗ W W S M , M = 0.6727, S M , M = 0.8528 5 6 W W For  = 0.4, For  = 0.8, JD ∗ JD ∗ S M , M = 0.4188, S M , M = 0.5578, 1 2 JD ∗ JD ∗ W W S M , M = 0.3680, S M , M = 0.5100, 1 2 W W JD ∗ JD ∗ S M , M = 0.5319, S M , M = 0.6449, 3 4 JD ∗ JD ∗ W W S M , M = 0.4765, S M , M = 0.6403, 3 4 W W JD ∗ JD ∗ S M , M = 0.6534, S M , M = 0.8439 5 6 JD ∗ JD ∗ W W S M , M = 0.6164, S M , M = 0.8223 5 6 W W Table 7 Similarity values SM methods Similarity values between pairs of masks Best mask Worst mask obtained for different mask ∗ ∗ ∗ ∗ ∗ ∗ types under different methods M ,M M ,M M ,M M ,M M ,M M ,M 1 2 3 4 5 6 J [30] 0.3402 0.4759 0.4438 0.5798 0.5883 0.8071 M M VSM 6 1 D [30] 0.4711 0.6124 0.5906 0.6883 0.6969 0.8684 M M VSM 6 1 C [30] 0.4791 0.6466 0.6076 0.8823 0.7290 0.8833 M M VSM 6 1 Hyb [34] 0.4771 0.6380 0.6034 0.8338 0.7210 0.8795 M M W 6 1 ( = 0.25) JD S ( = 0.1) 0.4580 0.5987 0.5759 0.6775 0.6860 0.8623 M M 6 1 JD S ( = 0.4) 0.4188 0.5578 0.5319 0.6449 0.6534 0.8439 M M 6 1 JD 0.3664 0.5032 0.4731 0.6015 0.6100 0.8193 M M S ( = 0.8) 6 1 JD 0.4652 0.6295 0.5912 0.8520 0.7149 0.8756 M M S ( = 0.1) 6 1 JD 0.4235 0.5783 0.5421 0.7613 0.6727 0.8528 M M S ( = 0.4) 6 1 JD 0.3680 0.5100 0.4765 0.6403 0.6164 0.8223 M M S ( = 0.8) 6 1 1 3 Cognitive Computation (2021) 13:1019–1033 1031 Moreover, the similarity measure results obtained under have recently become one of the research hotspots for research- various existing measures are also presented in Table 7. ers from all over the globe. In our study, we investigate and Step 4 Ranking of the masks. propose certain similarity measures for SVNSs since the concept Based on the highest similarity measure value obtained of similarity has a big influence on MADM problems. It is to between the set of masks and the ideal solution (mask), be noted that elements that are regarded as similar are viewed we find that M (N95-mask) is the appropriate mask or the from different perspectives of parameters like closeness, prox- best buying option to help minimize the transmission rate imity, resemblances, distances, and dissimilarities. Moreover, in of the COVID-19 pandemic. decision-making problems, human beings as decision-makers It is evident from Table  7 that our evaluation for the scrutinize several criteria before making a final decision. So, best suitable mask coincides with the outcomes obtained considering the relative importance of weights becomes a neces- with other similarity measures as well. This demonstrates sity. Most often, the weights are considered in such a way that the credibility and validity of our newly proposed similar- their sum is equal to one. While comparing two objects, we nor- ity measures. mally are interested in knowing whether the objects are identical or partially (approximately) identical or at least identical to what Computational‑time Analysis degree. This instinct compels us to investigate and address some desirable properties about the form of similarity measures for Based on conclusive evidence obtained from experimental SVNSs under consideration. data and owing to their “simple” and “easy-to compute” structure, the vector SMs are found to be very much effective. This implies that the calculations involved consume a sub- Conclusion stantially less amount of time which provided the decision- makers with the surplus advantage of time. Decision making in humans can be described as a cognitive pro- It is noteworthy that since our newly constructed measures cess that mainly focuses on the data which is given as input and are devised with the help of the Jaccard, Dice, and cosine on the cognitive capabilities of people. By cognitive capabili- vector SMs, so the computation time is much smaller in our ties, we mean the ability by which we know how the available case too. However, the only difference is that the time taken information is further processed. Unlike machines which pro- for our calculations is twice that when vector SMs are con- cess the given information in a binary form, whereas humans do sidered alone. But even then, a very miniature amount of time not think similarly. Rather people’s opinions can be expressed/ is spent, and adopting powerful software like MATLAB for measured on a specific evaluation scale. Mostly the fluctuations calculation purposes, just eases our load and provides instant or deviations observed in decision making problems are due results in the blink of an eye. For obvious reasons, that addi- to behavioral biases among people, which are deviations from tional amount of time taken via our constructed measures is rational standards while processing arguments. Further, few compensated by the accurate and efficient results evaluated. other factors like time factor, overestimation of negative com- The main advantage of our proposed measures over the ments, perception differences among people regarding positive, existing methods in the literature is the fact that they can and negative information also contribute towards irrational deci- not only accommodate the SVN environment, but they can sions being made. Often people make such non-rational deci- also capture the indeterminate information supplied by the sions in the process of trying to avoid losses. Therefore, proper decision-makers, automatically. knowledge of psychology is required to understand how people choose between different courses of action. Hence, we can say Importance of the Study that a significant target of cognitive psychology is to elaborate the mental state-of-art processes that define human behavior. NSs are developed from a philosophical point of view and as a In the same vein, the similarity measure concept is one of generalization to many sets like classical set, fuzzy set, intuition- the prime concepts in human cognition. The role of similar- istic fuzzy set, interval-valued fuzzy set, interval-valued intui- ity measures is so crucial in decision-making domain that tionistic fuzzy set, tautological set, paradoxist set, dialetheist set, it has diverse applications in the field of machine learn- and paraconsistent set. But, due to their scarce real-scientific ing, taxonomy, case-based reasoning, recognition, ecology, and engineering applications, a particular subclass of NSs was physical anthropology, automatic classification, psychology, developed known as SVNSs. SVNSs have the unique ability citation analysis, information retrieval, and many more. In to imitate the ambiguous nature of subjective judgments pro- this regard, one of the efficient and significant tools for the vided by the decision-makers and are suitable for dealing with measurement of similarity between two objects is the vec- uncertain, imprecise, and indeterminate information which are tor SMs. Jaccard, Dice, and cosine SMs are the ones that prevalent in multiple-criteria decision analysis. Thus, SVNSs are mostly sought for. But each element of the universe has provide a significant and powerful mathematical framework and some inherent weight associated with them, so to provide 1 3 1032 Cognitive Computation (2021) 13:1019–1033 an order of importance among the elements, we opt for Declarations weighted vector SMs over the non-weighted vector SMs. Ethical Approval This article does not contain any studies with animals Hence, in this article, we have proposed two weighted con- performed by any of the authors. vex vector SMs for SVNSs. SVNSs being a subclass of NSs are considered here, due to their efficiency in tackling Conflict of Interest The authors declare no competing interests. imprecise, incomplete, and inconsistent information. They provide the decision-makers with an additional probability to capture the indeterminate information which normally References exists in almost all real-world phenomena. Thereafter, a MADM method is discussed in detail using the proposed 1. Smarandache F. 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