Medical treatment options selection using extended TODIM method. with single valued trapezoidal neutrosophic numbers

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1 Medical treatment otions selection using extended TODIM method with single valued traezoidal neutrosohic numbers Hong-yu Zhang, Pu Ji, Jian-qiang Wang School of Business, Central South University, Changsha 008, China Abstract: The selection rocess of medical treatment otions is a multi-criteria decision-making (MCDM) one, and single valued traezoidal neutrosohic numbers (SVTNNs) are useful in deicting information and fuzziness in selection rocesses. Some comarison methods for SVTNNs have been develoed and alied in MCDM roblems. However, some defects exist in these comarison methods. Furthermore, few studies have been focused on the distance measurements for SVTNNs. Moreover, the revious MCDM methods under SVTNN environments assume that decision makers are erfectly rational. Nevertheless, in ractical roblems like the selection of medical treatment otions, decision makers bounded rationality should be considered due to the comlexity of human cognition. To address the above deficiencies, in this aer, an imroved comarison method and several distance measurements for SVTNNs are defined. Furthermore, a novel MCDM method for medical treatment otions selection is established based on an acronym in Portuguese of interactive and MCDM method (TODIM method) with SVTNNs to consider the risk reference of hysicians. In addition, a numerical examle of the selection of medical treatment otions is rovided in order to verify the roosed method and the influence of the arameter. Finally, a comarative analysis is conducted to demonstrate the feasibility of the roosed method. Keywords: single valued traezoidal neutrosohic number; comarison method; distance measurement; TODIM; selection of medical treatment otions Introduction As a common but significant rocess in clinical medicine, the selection of medical treatment otions is, Corresonding author. Tel.: ; fax: address: jqwang@csu.edu.cn.

2 indeed, a multi-criteria decision-making (MCDM) roblem, and it can be summarized as a hysician makes the decision which medical treatment otion is most robable for a articular atient considering several factors like the survival rate and the robability of a recurrence. Due to the comlexity of human cognition, a lot of imrecise, uncertain, incomlete information which cannot be deicted by cris values may exist in the selection roblems of medical treatment otions []. For instance, a hysician may not sure how much the survival rate of a treatment otion is. To handle fuzziness and uncertainty, fuzzy logic and fuzzy sets (FSs) are introduced. FSs were roosed by Zadeh [2] in 965. After that, many extensions of FSs have been develoed. For examle, Turksen [] extended FSs by utilizing an interval value to deict the degree of membershi, and defined the interval valued fuzzy sets (IVFSs). On the basis of FSs and IVFSs, Atanassov and Gargov [, 5] introduced the notion of non-membershi, and resented the intuitionistic fuzzy sets (IFSs) and the interval valued intuitionistic fuzzy sets (IVIFSs). In some situations, eole may be hesitant in selecting single values to exress their references regarding an individual object. To deal with these situations, Torra [6] utilized several single values to reflect the degree of membershi, and develoed the hesitant fuzzy sets (HFSs). Moreover, all these extensions of FSs have been alied in decision-making [7-9] with further extensions still being develoed [0, ] and alied [2-6]. Neutrosohic sets (NSs), which were defined by Smarandache [7, 8], are imortant extensions of FSs. Unlike FSs and IFSs, NSs make use of the degrees of truth, indeterminacy, and falsity to describe fuzzy information in decision-making rocesses. Moreover, the values of these three degrees lie in ] 0, [. It is obvious that NSs are difficult to be alied in actual decision-making roblems. Therefore, Wang and Smarandache [9] defined the single valued neutrosohic sets (SVNSs) whose degrees of truth, indeterminacy, and falsity are between 0 and. In addition, many other extensions of NSs have been roosed [20-22]. For instance, the simlified neutrosohic sets (SNSs) were develoed by Ye [2], and the interval valued 2

3 neutrosohic sets (IVNSs) were defined by Wang and Smarandache [2]. Furthermore, NSs have been effectively alied in many fields, such as decision-making [25-0] and image segmentation [, 2]. Recently, the above extensions of NS have been alied in medical decision-making roblems to deict fuzzy information. Such as, SNSs are utilized by Ye [] and SVNSs are used by Ye and Fu [] to deal with medical decision-making roblems. In some ractical medical decision-making roblems like the selection of medical treatment otions, truth, indeterminacy, and falsity may exist simultaneously in evaluation information rovided by hysicians. These three factors can be accurately and fully deicted by the degrees of truth, indeterminacy, and falsity in NSs. However, SNSs and SVNSs are cris sets. That is to say, they utilize cris values to denote the degrees of truth, indeterminacy, and falsity, which may result in a loss of information. To overcome the above shortcoming, some researchers studied the extensions of NSs in the field of the continuous sets [5]. In articular, Deli and Subas [6] extended NSs into the domain of continuous sets, and defined the single valued traezoidal neutrosohic numbers (SVTNNs). In SVTNNs, the degrees of truth, indeterminacy and falsity are traezoidal fuzzy numbers rather than single values. Comaring to discrete sets, continuous sets like SVTNNs can deict more information and the fuzziness in decision-making rocesses. SVTNNs are more suitable to be utilized to reresent the fuzzy information in medical decision-making roblems like the selection of medical treatment otions than the extensions of NSs in the domain of cris sets like SNSs and SVNSs. Furthermore, the comarison method for SVTNNs has been defined in some studies. For instance, Ye [7] gave the definition of the score function, and resented a comarison method for SVTNNs. Other than the comarison method roosed by Ye [7], the comarison method for SVTNNs defined by Deli and Subas [6] utilized a different score function and introduced the notion of accuracy function.

4 Many kinds of continuous sets including SVTNNs have been widely used in solving decision-making method [8-]. For examle, Zhang and Jin [] roosed a grey relational rojection method for decision-making roblems based on intuitionistic traezoidal fuzzy number. Dong and Wan [5] develoed a new method for decision-making roblems in which the criteria values are triangular intuitionistic fuzzy numbers. SVTNNs are also an imortant kind of tools in constructing decision-making methods. As for decision-making methods with SVTNNs, Deli and Subas [6] resented a decision-making method based on the roosed aggregation oerator with the arameter of SVTNNs. Similarly, Ye [7] also constructed decision-making methods utilizing aggregation oerators of SVTNNs, but the aggregation oerators in Ref. [7] are the weighted arithmetic averaging oerator and the weighted geometric averaging oerator for SVTNNs. Distance measurements, which are imortant tools in decision-making, have been studied under various fuzzy environments. For instance, researchers have studied distance measurements of various FSs in the domain of cris sets: distance measurements of FSs [6-9], distance measurements of IFSs [50-5], distance measurements of HFSs [5-57], distance measurements of NSs [58-60]. As for distance measurements of FSs in the domain of continuous sets, Önüt and Soner [6] and Chen [62] defined the Euclidean distance between two triangular fuzzy numbers. Fu [6] extended the distance defined by Önüt and Soner [6] and Chen [62] by introducing a arameter, and develoed a generalized distance measurement of triangular fuzzy numbers. Distance measurements of traezoidal intuitionistic fuzzy numbers have also been defined. Wan [6] gave the definition of Hamming distance and Euclidean distance between two traezoidal intuitionistic fuzzy numbers based on Hausdorff metric. Furthermore, distance measurements have been alied in lenty of extant fuzzy decision-making methods [57, 58, 60, 65, 66]. An acronym in Portuguese of interactive and decision-making method named Tomada de decisao interativa

5 e multicritévio (TODIM method) has also been used in decision-making roblems to consider the risk reference of decision makers. TODIM method was roosed by Gomes and Lima [67, 68] on the basis of rosect theory [69]. However, the TODIM method roosed by Gomes and Lima [67, 68] can only deal with cris values. To overcome this deficiency, Krohling and Souza [70] originally extended TODIM method into fuzzy environments. On the basis of the fuzzy TODIM method develoed by Krohling and Souza [70], many researchers have studied TODIM methods for decision-making roblems under various fuzzy environments. For examle, Lourenzutti and Krohling [7] established a decision-making method based on the roosed intuitionistic fuzzy TODIM method. Moreover, Li et al. [72] further extended TODIM method into interval valued intuitionistic fuzzy environments, and constructed an extended TODIM method for decision-making roblems. TODIM methods for decision-making roblems have also been roosed under various other discrete fuzzy set environments, like hesitant fuzzy environments [7], and multi-valued neutrosohic environments [7]. TODIM methods under fuzzy continuous set environments have been studied and alied in decision-making. For instance, Tseng et al. [75] roosed a triangular fuzzy number TODIM method for decision-making roblems and alied it to evaluate green suly chain ractices. Similarly, Tosun and Akyüz [76] utilized the triangular fuzzy number TODIM method to solve decision-making roblems in the selection of sulier. Unlike the TODIM method roosed by Tseng et al. [75], Gomes et al. [77] combined Choquet integral with TODIM methods under triangular fuzzy number environments, and resented a decision-making method for decision-making roblems where criteria are interdeendent and the decision makers are bounded rational. Furthermore, a TODIM method under traezoidal intuitionistic fuzzy number environments has been develoed by Krohling et al. [78] to tackle decision-making roblems. SVTNNs can comrehensively denote fuzzy, uncertain, and imrecise information in medical 5

6 decision-making roblems like the selection of medical treatment otions. Fuzzy TODIM method takes into account the risk reference of decision makers, and it is a significant vehicle in decision-making roblems. However, few studies have been focused on the distance measurement of SVTNNs or the fuzzy TODIM method under SVTNN environments. In addition, the aforementioned comarison methods for SVTNNs have some drawbacks, i.e. some results obtained by these comarison methods may be not consistent with reality in some cases (the details will be discussed in Section 2). To overcome these deficiencies, this aer roosed an imroved comarison method and several distance measurements for SVTNNs. Furthermore, TODIM method was extended to SVTNN environments, and a decision-making method for medical treatment otions selection was constructed based on the TODIM method to take into consideration the risk reference of hysicians. The structure of this aer is organized as follows. In Section 2, we introduce several relevant concets of SVTNNs. In Section, an imroved comarison method for SVTNNs is develoed to overcome the deficiencies of the extant comarison method. Moreover, some distance measurements are resented in Section. In Section, a novel decision-making method for medical diagnosis roblems with SVTNNs is constructed based on TODIM method. Furthermore, a numerical examle for medical treatment otions selection is given and the influence of the arameter is discussed in Section 5. Furthermore, a comarative analysis is rovided in Section 5 to verify the feasibility of the roosed method. Finally, Section 6 concludes the aer. 2 Preliminaries This section reviews some basic concets of SVTNNs. And these concets will be used in the reminder of this aer. Definition [9]. Let X be a sace of oints (objects), with a generic element in X denoted by x. An 6

7 SVNS A in X is characterized by a truth-membershi function T ( x ), an indeterminacy-membershi function IA( x ) and a falsity-membershi function FA ( x ). The functions T ( x ), IA( x ) and FA ( x ) are singleton subintervals/subsets in the real standard [0,]. That is TA ( x): X [0,], I A( x): X [0,] and FA ( x): X [0,]. Then, a simlification of A is denoted by A { x, T ( x), I ( x), F ( x) x X}. A A A It is a subclass of NSs. And the sum of TA ( x ), IA( x ) and FA ( x ) satisfies that 0 su T ( x) su I ( x) su F ( x) A A A Definition 2 [6, 7]. An SVTNN a a, a, a, a, T a,, 2 A A I a F a is a secial NS on the real number set R, whose truth-membershi function Ta x, indeterminacy-membershi function I x and a falsity-membershi Fa x are given as follows: When 0 I F a a T a x x x x a T a a2 a a x a2 T a a2 x a a xt a a a a x a,,, 0 otherwise. a2 x I a x a a2 a a x a2 I a a2 x a x a I aa x a a a x a,,, otherwise. a2 x F ax a a2 a a x a2 F a a2 x a x a F aa x a a a x a,,, otherwise. is called a ositive SVTNN, denoted by a 0. a, a a, a, a, a, T a, I a, F a Similarly, when 0 2 a, a a, a, a, a, T a, I a, F a is called a negative SVTNN, denoted by 2 a 0. When 0 a a2 a a and T a, I a, Fa 0,, a a a a a T a I a and,,,,,, 2 Fa is called a normalized SVTNN, which is used for this aer. It should be noted that a, a 2, a and a are not zero simultaneously. 7

8 Definition [6, 7]. Let a a, a, a, a, T a, I a, F a and b b, b, b, b, T b, I b, F b be two SVTNNs and 0, then 2 2 () a b a b, a b, a b, a b, T a T b T at b, I ai b,faf b ; 2 2 (2) a b a b, a b, a b, a b, T at b, I a I b I ai b,fa F b F a F b ; 2 2 () a a, a2, a, a, T a, I a, F a ; and () a a, a2, a, a, T a, I a, F a. Furthermore, some comarison methods for SVTNNs are defined in order to comare two SVTNNs. Definition [7]. Let a a, a, a, a, T a, I a, F a defined as: be an SVTNN. The score function of a is 2 sa a a2 a a 2 T a I a F a. 2 Definition 5 [7]. Let a a, a, a, a, T a, I a, F a and b b, b, b, b, T b,, 2 two SVTNNs. When sa sb, a b; when sa sb, a b. 2 I b F b be However, a shortcoming exists in Definition 5, and it is illustrated in the following examle. Examle. Let a 0.,0.,0.6,0.7,0.2,0.,0.6 and b 0.6,0.7,0.8,0.9,0.,0.8,0.5 be two SVTNNs, it is clear that a b. However, according to Definitions and 5, sa sb 0.2, and a b, which does not conform to our intuition. Moreover, Deli and Subas [6] defined a novel score function and the accuracy function for SVTNNs, and they also resented a new comarison method for SVTNNs. Definition 6 [6]. Let a a, a, a, a, T a, I a, F a be an SVTNN. The score function of a is 2 defined as: sa a a2 a a 2 T a I a F a, 6 8

9 and the accuracy function of a is defined as: ha a a2 a a 2 T a I a F a. 6 Definition 7 [6]. Let a a, a, a, a, T a, I a, F a and b b, b, b, b, T b,, 2 two SVTNNs. The comarison method for a and b can be defined as:. When sa sb, a b; 2. When sa sb and ha hb. When sa sb and ha hb, a b; and, a b. 2 I b F b be Examle 2. Utilizing Definition 7 to comare a and b in Examle, we can obtain that sa sb 0.5 ha 0., hb Therefore, it is true that a b., This comarison method solves the roblem in Examle. Nevertheless, another drawback exists in this comarison method and the following examle resents the drawback. Examle. Let a 0.,0.,0.6,0.7,0.,0.9,0 and b 0.,0.,0.5,0.6,0.,0.6,0 be two SVTNNs, it is clear that a b. However, according to Definitions 6 and 7, sa sb 0.5 a b, which is contradictory to our intuition. Lemma (Minkowski s inequality [56]) Let numbers, and. Then x x x and,,, n 2, ha hb 0.5 and y, y2,, y n be two sequences of real n n n xi yi xi yi i i i. Comarison method and distance measurements for SVTNNs In this section, we roose the novel score and accuracy functions for SVTNNs. Subsequently, an imroved comarison method for SVTNNs is defined on the basis of the score and accuracy functions. Moreover, several distance measurements for SVTNNs are defined based on the rior distance measurements under 9

10 various fuzzy environments. Definition 8. Let a a, a, a, a, T a, I a, F a be an SVTNN. A score function SVTNN a can be defined as: 2 s a s a of the a a 2 2 a T a I a F a a. () 2 The bigger the value of sa is, the better the SVTNN a will be. Definition 9. Let a a, a, a, a, T a, I a, F a be an SVTNN. An accuracy function SVTNN a can be defined as: 2 ha of the The bigger the value of a a2 a a hn T a I a T a F a I a. (2) sa is, the better the SVTNN a will be. According to the score and accuracy functions for SVTNNs, we can obtain the following comarison method. Definition 0. Let a a, a, a, a, T a, I a, F a and b b b b b T b I b 2 SVTNNs. The comarison method for a and b can be defined as: () a b if and only if a b, a2 b2, a b, a b 0,,,,,, 2 Fb be two, T a T b, I a I b, and F a F b (2) a b if and only if a b and b a (i.e. a b, a2 b2, a b, a b I b, and F a F b I a ); () When a b and a b, a b; () When neither a b nor b a (5) When neither a b nor b a (6) When a b exists and sa sb, sa sb and ha hb It is obvious that a b when a b., a b; exists, sa sb and ha hb, a b., a b; and Examle. Let a 0.,0.,0.6,0.7,0.,0.9,0 and ;, T a T b, b 0.,0.,0.5,0.6,0.,0.6,0 be two SVTNNs.

11 By alying the comarison method in Definition 0, since 0. 0., 0. 0., , , 0. 0., and 0 0, neither b a nor b a exists. Furthermore, sa 0.75 and sb Here sa sb, thus a b. As can be seen from Examle, the imroved comarison method overcomes the defect of the comarison method in Definition 7. In addition, we define several distance measurements for SVTNNs as follows. Definition. Let a a, a, a, a, T a, I a, F a and b b b b b T b I b 2,,,,,, 2 SVTNNs. The distances between a and b can be defined by the following equations: Fb be two () The Hamming distance: d a b a b a2 b2 a b a b at a bt b 2 2 a T a b T b 2 a T a b T b a T a b T b, a I a b I b a I a b I b a I a b I b 2 2 a I a b I b a F a b F b a F a b F b 2 a F a b F b a F a b F b () (2) The Euclidean distance: d2 a b a b a2 b2 a b a b at a bt b , a T a b T b 2 a T a b T b a T a b T b a I a b I b a I a b I b a I a b I b a I a b I b a F a b F b a F a b F b 2 a F a b F b a F a b F b () () The Chebyshev distance:

12 d a, b max a b,2 a2 b2,2 a b, a b, at a bt b, 2 2 a T a b T b,2 a T a b T b, a T a b T b,,2,2,,,2 2 2, 2 2 a I a b I b a I a b I b a I a b I b 2 2 a I a b I b a F a b F b a F a b F b 2 a F a b F b, a F a b F b (5) () The generalized distance: d a b a b a2 b2 a b a b at a bt b 2, a T a b T b 2 a T a b T b a T a b T b a F a b F b a F a b F b a I a b I b a I a b I b a I a b I b a I a b I b a F a b F b a F a b F b (6) where. Esecially, when, the generalized distance in Eq. (6) reduces to the Hamming distance in Eq. (); when 2, the generalized distance in Eq. (6) reduces to the Euclidean distance in Eq. (); when, the generalized distance in Eq. (6) reduces to the Chebyshev distance in Eq. (5). Theorem. Let that a, b and c be three SVTNNs, then above distance measurements d a, b k,2,, satisfy the following roerties: () d a b 0, ; k d a, b 0 if and only if a b; (2) k () d a, b d b, a k k ; and () d a, c d a, b d b, c. k k k Proof. Let a a, a, a, a, T a, I a, F a, b b, b, b, b, T b, I b, F b 2 2 and c c c 2 c c T c I c F c. The roof of the generalized distance measurements d,,, 2, k,,, a b is shown as follows. () According to Definition 2, 0 a a2 a a, 0 b b2 b b,, I a, F a 0,, T a and T b, I b, F b 0,. Since, it holds that a b a b a T a bt b,,,

13 ,,, b F b, af a b F b 0, a T a b T b a I a b I b a I a b I b a F a, and a b,2 a b,2 a T a b T b,2 a T a b T b,2 a I a b I b,2 a I a b I b, 2 a2f a b2 F b,2 af a bf b 0,2. Therefore, a b 2a2 b2 2a b a b 2 2 a T a bt b a T a b T b a T a b T b a T a b T b a I a b I b a I a b I b 2 a I a b I b a I a b I b a F a b F b 2 a F a b F b Thus, according to Eq. (6), 2 a F a b F b a F a b F b 0,2 roerty () holds. (2) According to Definition 0, if a b, we have that a b, a2 b2, a b, a b I b, and F a F b. Thus, I a d a, b 0. d a, b 0,, i.e. the, T a If d a, b 0, we can obtain that a b 2a b 2a b a b a T a bt b 2 2 Tb, 2 a T a b T b 2 a T a b T b a T a b T b a I a b I b 2 a I a b I b a I a b I b a I a b I b a F a b F b 2 a F a b F b 2 a F a b F b 2 2 a F a bf b 0. Hence, it is true that a b, a2 b2, a b, a b. Moreover, according to Definition 2, a, a 2, a and a are not zero simultaneously, we have that Ta Tb, I a I b, and Fa F b. Therefore, according to Definition 0, a b holds. () It is clear that a b b a, 2 a2 b2 2 b2 a2, 2 a b 2 b a, a b b a,, 2a T a b T b 2b T b a T a, 2 a T a b T b a T a bt b bt b a T a , a T a b T b b T b a T a, a I a b I b b I b a I a 2 b T b a T a,, 2a I a b I b 2b I b a I a, a I a b I b 2a I a b I b 2b I b a I a b I b a I a, a F a b F b b F b a F a, 2a F a b F b 2b F b a F a, and a F a b F b b F b a F a 2a F a b F b 2b F b a F a to Eq. (6), d a, b d b, a.. Therefore, according

14 () According to Eq. (6), we have that d a, c a c 2 a b 2 a b a c a T a c T c a T a b T b a T a b T b a T a c T c a I a c I c a I a b I b 2 a I a b I b a I a c I c a F a c F c 2 a F a b F b a F a b F b a F a c I c. According to the Minkowski s inequality in Lemma, d a, c a b b c 2 a2 b2 b2 c2 2 a b b c a b b c at a bt b bt b c T c a T a b T b b T b c T c a T a b T b b T b c T c a T a b T b b T b c T c a I a b I b b I b c I c a I a b I b b I b c I c a I a b I b b I b c I c a I a b I b b I b c I c a F a b F b b F b c F c a F a b F b b F b c F c a F a b F b b F b c F c af a b F bb F b cf c a b 2a2 b2 2a b a b at a bt b a T a b T b a T a b T b a T a b T b a I a b I b a I a b I b 2 a I a b I b a I a b I b a F a b F b 2 a F a b F b af a b F b af a bf b c b 2 c2 b2 2 c b c c c T c bt b c T c b T b c T c b T b c T c b T b c I c b I b c I c b I b 2 c I c b I b c I c b I b c F c b F b 2 c F c b F b c F c b F b c F c b F b d a, c d c, b. Thus, the generalized distance measurement satisfies the roerty that d a, c d a, b d b, c. Furthermore, as mentioned in Definition, when, the generalized distance in Eq. (6) reduces to the Hamming distance in Eq. (); when 2, the generalized distance in Eq. (6) reduces to the Euclidean distance in Eq. (); when, the generalized distance in Eq. (6) reduces to the Chebyshev distance in Eq.

15 (5). Since the generalized distance satisfies the roerties ()-(), the Hamming distance d Euclidean distance d2 a, b, and the Chebyshev distance, Therefore, Theorem holds. d a b satisfy the roerties ()-()., a b, the The TODIM method with SVTNNs for medical treatment otions selection Here we resent a TODIM method for medical treatment otions selection with SVTNNs. Assume there are m medical treatment otions A{ A, A,, } 2 A m which are evaluated by the hysicians concerning n criteria C{ C, C,, } 2 C n. The evaluation values rovided by the hysicians are transformed into SVTNNs, and U reresents the evaluation value for the treatment otions A i i,2,, m under the ij criterion C j j,2,, n. The decision matrix, which is transformed from the evaluation values rovided by the hysicians, can be denoted as U : U U U U U U U U U U 2 n n m m2 mn Since each criterion has distinct weight, the weight vector of criteria is w ( w, w2,, w ) T n, where wj 0. j,2,, n and n wj. j In the following art, the TODIM method to rank the medical treatment otions and select the best one is roosed based uon the roosed comarison method and distance measurements. The rocedure of the method is as follows: Ste : Normalize the decision matrix. Benefit criterion and cost criterion may exist in an individual MCDM roblem simultaneously. To unify all criteria, it is necessary to normalize the evaluation value under the cost criterion. It should be noted that 5

16 normalization is needless if all criteria are benefit ones. If the criterion C j is a cost one, the evaluation value ij ij ij ij ij ij ij ij, 2,,,,, U u u u u T u normalized utilizing the following equation: I u F u of the treatment otion A i under the criterion C j should be ij ij ij ij ij ij ij Nij U, U,U2, U, T u, I u, F u. (7) It is obvious that the normalized values N ij are also SVTNNs. Ste 2: Obtain score values. Utilizing the score function in Eq. (), we can obtain the score value ij of the treatment otion A i concerning the criterion C j. s N i,2,, m; j,2,, n Ste : Obtain accuracy values. Utilizing the accuracy function in Eq. (2), we can obtain the accuracy value hn ij i,2,, m; j,2,, n of the treatment otion A i concerning the criterion C j. Ste : Obtain distance matrices. The generalized distance measurement is utilized here seeing that the Hamming, Euclidean, and Chebyshev distance measurements are secial cases of the generalized distance measurement. Based on the generalized distance in Definition, we can obtain the distance d i,2,, m; r,2,, m; j,2,, n between j ir two treatment otions A i and A r concerning the criterion C j. Ste 5: Obtain artial dominance matrices. The distance matrix j under the criterion C j is comosed of artial dominance degrees j ir i,2,, m; r,2,, m; j,2,, n of the treatment otion A i over the treatment otion A r concerning the criterion C j and j ir can be obtained by the following formula: 6

17 j judir, N N orn N n ju j j 0, N N orn N n j judir j, N N orn N t ju ij rj ij rj ir ij rj ij rj ij rj ij rj, (8) where j ju and u max j u j,2,, n. The comarison relation between N ij and N rj can be obtained according to the comarison method in Definition 0 utilizing the score values and accuracy values. If Nij N rj or Nij Nrj, it reresents a gain; if Nij Nrj, it is breakeven; if Nij N rj or N ij Nrj, it reresents a loss. The arameter t in the situation where ij rj N N or Nij Nrj is the decay factor of the loss and t 0. Ste 6: Obtain the final dominance matrix. The final dominance matrix is comosed of dominance degrees. The dominance degree ir i,2,, m; r,2,, m denotes the degree that the treatment otion A i is better than the treatment otion A r and can be obtained by: ir n j ir. (9) j Ste 7: Calculate the global value of each treatment otion. The global value i,2,, m of the treatment otion A i can be obtained by: i m m I mini. (0) max Iir min Iir i m i m r r ir ir i m r r i m m Ste 8: Rank the treatment otions. The treatment otions can be ranked based on the global value of each treatment otion. The bigger the 7

18 global value of an individual treatment otion is, the better the otion will be. 5 A numerical examle for medical treatment otions selection In this section, a numerical examle for medical treatment otions selection with SVTNNs is used to demonstrate the alicability of the roosed method. We utilize the roosed method to tackle a selection roblem of medical treatment otions. Assume that a hysician wants to find a medical treatment otion for a articular atient with verruca lantaris. There are five medical treatment otions: () A is carbon dioxide laser; (2) A 2 is high frequency theraeutic instrument; () A is microwave theraeutic instrument; () A is cryotheray; and (5) A 5 is aoxesis. They will be evaluated by the hysician from the three asects: () C is the robability of a cure; (2) C 2 is severity of the side effects; and () C is cost. The weight vector of the three criteria is suosed as 0.,0.,0. T. Furthermore, Table shows the decision matrix U which is transformed from the evaluation values of the hysician. Table. The decision matrix transformed from the evaluation values of the exerts A A 2 A A C C 2 C 0.,0.,0.5,0.7,0.5,0.,0. 0.,0.5,0.7,0.8,0.5,0., ,0.,0.8,0.9,0.9,0., ,0.5,0.6,0.9,0.8,0.2,0. 0.2,0.,0.6,0.8,0.,0.2,0. 0.,0.,0.7,0.8,0.5,0.,0.8 0.,0.5,0.8,0.8,0.7,0.2,0.5 0.,0.2,0.5,0.7,0.2,0.5,0.8 0.,0.5,0.6,0.7,0.8,0.2,0.6 0.,0.5,0.8,0.8,0.7,0.2, ,0.,0.7,0.8,0.9,0.8, ,0.,0.,0.6,0.5,0.,0.2 A 5 0.,0.6,0.7,0.8,0.,0.5,0.6 0.,0.5,0.7,0.9,0.9,0.7,0.5 0.,0.,0.5,0.6,0.9,0., The stes of the roosed method Ste : Normalize the decision matrix. Since the criterion C is a benefit one while the criteria C 2 and C are cost ones, the decision matrix needs to 8

19 be normalized utilizing Eq. (7). Table 2 lists the normalized decision matrix. Table 2. The normalized decision matrix C C 2 C A 0.,0.,0.5,0.7,0.5,0.,0. 0.2,0.,0.5,0.6,0.5,0.,0.7 0.,0.2,0.7,0.8,0.9,0.,0.5 A 2 0.2,0.5,0.6,0.9,0.8,0.2,0. 0.2,0.,0.6,0.8,0.,0.2,0. 0.2,0.,0.6,0.7,0.5,0.,0.8 A 0.,0.5,0.8,0.8,0.7,0.2,0.5 0.,0.5,0.8,0.9,0.2,0.5,0.8 0.,0.,0.5,0.6,0.8,0.2,0.6 A 0.,0.5,0.8,0.8,0.7,0.2, ,0.,0.7,0.8,0.9,0.8,0.7 0.,0.6,0.7,0.8,0.5,0.,0.2 A 5 0.,0.6,0.7,0.8,0.,0.5,0.6 0.,0.,0.5,0.7,0.9,0.7,0.5 0.,0.5,0.6,0.7,0.9,0.,0.6 Ste 2: Obtain score values. Utilizing Eq. (), we can obtain the score value of an individual treatment otion concerning each criterion, and these score values are shown in Table. Table. The score value of each treatment otion concerning each criterion C C 2 C A A A A A Ste : Obtain accuracy values. Utilizing Eq. (2), we can obtain the accuracy value of each treatment otion concerning each criterion. 9

20 Table. The accuracy value of each treatment otion concerning each criterion C C 2 C A A A A A Ste : Obtain distance matrices. Here for ease of comutation, we assume that. Utilizing Eq. (6), the distance matrix D j j,2, concerning the criterion C j can be obtained as: D , D , and D Ste 5: Obtain artial dominance matrices. Utilizing Eq. (8) (Suose t [79]), the artial dominance matrix j j,2, concerning the criterion C j can be obtained as: ,

21 , and Ste 6: Obtain the final dominance matrix. Utilizing Eq. (9), the final dominance matrix can be obtained as: Ste 7: Calculate the global value of each treatment otion. The global value i of the treatment otion A i can be obtained as:, , 0, , and Ste 8: Rank the treatment otions. Since 2 5, the ranking order of the five treatment otions is A A2 A5 A A. Thus, the best treatment otion is A. 5.2 The influences of the arameter In this subsection, we investigate and discuss the influences of the arameter t in Eq. (8) and in the generalized distance measurement in Eq. (6) in detail. Firstly, the influence of t on the shae of the rosect value function is verified and discussed. Fig. deicts the rosect value functions using different values of t, i.e. t [79] and t 2.5 [70]. It can be seen from 2

22 artial dominance degree Section 5. that the score values of two SVTNNs in the decision matrix are same if and only if these two SVTNNs are same. That is to say, the disarity between any two SVTNNs in this numerical examle can be reflected by the difference between their score values. Thus, in Fig., the horizontal axis is the difference between any two score values concerning the same criterion and the vertical axis is the corresonding artial dominance degrees difference between score values t= t=2.5 Fig.. Prosect value functions using t and t 2.5 As resented in Fig., the shae of the rosect value function in the third quadrant is influenced by the value of the arameter t while that in the first quadrant is immune to the value of t. Moreover, the artial dominance obtained by t 2.5 is bigger than the corresonding artial dominance obtained by t when the difference of score values is negative. The reason for this henomenon is given as follows. The value of t denotes the degree that the losses are attenuated when t. The greater the value of t is, the bigger the degree that the losses are attenuated will be. Thus, it is reasonable that the shae of the rosect value function when t is deeer than that when t 2.5. Secondly, we investigate the influence of the arameter t on the ranking order of the treatment otions by obtaining the ranking orders with different values of t. As the value of t changes from 0.00 to 50, the corresonding ranking order of the treatment otions can be obtained. Table 5 lists the value of t, the 22

23 corresonding global values, and the ranking order of the treatment otions. Table 5. Ranking orders of the treatment otions with different values of t t global value i Ranking order t 0.00 t 0.0 t 0. t 0.5 t t 20 t 2 t 25 t 50, , 0, 5 0., , 0, 5 0.9, , , 5 0.5, , 0.029, , , , , , , , , 0.080, , 2 0, , , 2 0, 0.085, A A2 A5 A A A A2 A5 A A A A2 A5 A A A A2 A5 A A A A2 A5 A A A A2 A5 A A A A2 A5 A A A2 A A5 A A A2 A A5 A A As shown in Table 5, the ranking order of the treatment otions may be different when the value of t 2

24 changes. When t 0.0 the ranking orders are same and the best treatment otion is A and the worst one is A. The ranking orders are same when t changes from 0. to 2. The best treatment otion when 0.t 2 is same to that when t 0.0 while the worst one when 0.t 2 is different from that when t 0.0. And when 0.t 2, A is the worst treatment otion. Moreover, the best treatment otion becomes A 2 and the worst one remains A when t 25. The reason for these differences is rovided as follows. From Eq. (8), we can see that the losses are amlified when t and they are attenuated when t. When t 0.0, the losses are amlified, and eventually the global value of A becomes smaller than that of A. When 0.t, the losses are amlified but the degree of amlification is smaller than the degree when t 0.0, which makes the global value of A become bigger than that of A. When t 2, the losses are attenuated, and the global value of A remains bigger than that of A. When t 25, the losses are attenuated and the degree of attenuation is bigger than the degree when t 2, which makes the global value of A 2 becomes bigger than that of A. Thus, it is reasonable that the value of t influences the ranking order of the treatment otions. Thirdly, we exlore the influence of the arameter on the final ranking order of the treatment otions with different values of. Table 6 shows the value of, the corresonding global values (suose t ). Table 6. Global values of treatment otions with different values of global value i Ranking order 2, , , 5 0.5, , 0.000, 5 0. A A2 A5 A A A A2 A5 A A 2

25 6, , 0.008, 5 0.0, , , , , 0.028, A A2 A5 A A A A2 A5 A A A A2 A5 A A As resented in Table 6, the ranking order of the treatment otions remains same when the value of changes. In other words, the ranking order of the treatment otions is unacted on the value of the arameter. The best treatment otion is A and the worst one is A. In generally, the arameter t in Eq. (8) influences the shae of the rosect function in the third quadrant. Furthermore, the arameter t may influence the ranking order obtained by the roosed method while the arameter in the generalized distance measurement in Eq. (6) cannot. The roosed method can be deemed as a flexible one considering the influence of t. 5. Comarative analysis We conduct a comarative analysis to verify and discuss the feasibility of the roosed method in this subsection. Two cases are included in this comarative analysis. In the first case, the roosed method is comared to the method that was outlined in Ref. [78] using traezoidal intuitionistic fuzzy information. In the second case, the roosed method is comared to the methods that were roosed by Deli and Subas [6] under SVTNN environments. Case : comarative analysis under traezoidal intuitionistic fuzzy environments. Krohling et al. [78] take into account the risk reference of decision makers, and develoed a MCDM method for roblems with traezoidal intuitionistic fuzzy information based on fuzzy TODIM method. The 25

26 roosed method and the method roosed by Krohling et al. [78] are utilized to solve MCDM roblems in Ref. [78]. Table 7 lists the ranking orders of the roosed TODIM method and the method roosed by Krohling et al. [78] when t and t 2.5. Table 7. The ranking orders of different methods Method Ranking order The best treatment otion(s) The worst treatment otion(s) Method t [78] A2 A5 A A A 2 A A Method 2 t 2.5 [78] The roosed method t The roosed method t 2.5 A2 A5 A A A A 2 A A2 A A A A 5 A 2 A 5 A2 A A A A 5 A 2 A 5 From Table 7, the best treatment otion of these four methods is A 2 while the worst ones are different. A is the worst treatment otion of the method roosed by Krohling et al. [78] when t and t 2.5 while A 5 is the worst one of the roosed method when t and t 2.5. Furthermore, the ranking orders of the method roosed by Krohling et al. [78] when t and t 2.5 are same while the roosed method obtains two different ranking orders when t and t 2.5. The reasons for these differences are illustrated as follows. the comarison methods and distance measurements utilized in the MCDM method roosed by Krohling et al. [78] and the roosed MCDM method are greatly distinct even though both of these two MCDM methods consider the risk reference of decision makers. Furthermore, the roosed method makes use of the intuitionistic fuzzy index besides the degrees of membershi and non-membershi while the method roosed by Krohling et al. [78] only utilizes and. It is reasonable that these two MCDM methods may obtain different ranking order when the value of t is same. 26

27 The ranking order of the roosed method meets decision makers actual references better comared to that of the method roosed by Krohling et al. [78]. Moreover, as resented in Section 5.2, the value of t may influence the ranking order of the roosed method. When t, the losses are attenuated. However, the degrees of attenuation are different when t and t 2.5. Therefore, the ranking orders of the roosed method are different when t and t 2.5. The ranking orders of the method roosed by Krohling et al. [78] are same when t and t 2.5. It is because that the difference between the degrees of attenuation when t and t 2.5 is not big enough to make the ranking orders become distinct. Overall, the roosed method can be effectively used to solve MCDM roblems under traezoidal intuitionistic fuzzy environments, and the ranking orders obtained by the roosed method are closer to decision makers actual references than those obtained by the extant traezoidal intuitionistic fuzzy methods like the method roosed by Krohling et al. [78]. Case 2: comarative analysis under SVTNN environments. Deli and Subas [6] constructed a MCDM method by utilized a single valued traezoidal neutrosohic weighted aggregation oerator with the arameter and the comarison method in Definition 7. The roosed method and the method roosed by Deli and Subas [6] are utilized to solve the selection roblem of medical treatment otions (a MCDM roblem) in Section 5.. Table 8 lists the ranking orders of the roosed TODIM method t and the method roosed by Deli and Subas [6]. Table 8. The ranking orders of different methods under SVTNN environments Method Ranking order The best treatment otion(s) Method [6] A A A5 A2 A The worst treatment otion(s) A A Method 2 [6] A A A5 A2 A A A 27

28 The roosed method A A2 A5 A A A A From Table 8, the best treatment otion of the method roosed by Deli and Subas [6] when 2 is A while the worst one is A. A is the best treatment otion of the method roosed by Deli and Subas [6] when and the roosed TODIM method while A is the worst treatment otion of these two methods. Furthermore, different ranking orders of the other three treatment otions (i.e. A2, A, A 5 ) are obtained by these two methods. The reasons for the differences are that the aggregation oerator calculates the weighted arithmetic average value of elements when while the aggregation oerator obtains the weighted arithmetic average value of the square values of elements when 2. It is clear that the aggregation oerator with 2 enlarges the differences between treatment otions while the aggregation oerator with does not. Thus, different ranking orders are obtained by these two methods. Moreover, in ractice, decision makers are bounded rational, it is indisensable to consider their risk reference in decision-making rocess. The roosed TODIM method takes into account the decision makers bounded rationality while the methods roosed by Deli and Subas [6] assume that the decision makers are erfectly rational. Therefore, the ranking orders of the three methods are different. In addition, it is true that the roosed method is effective in tackling the selection roblems of medical treatment otions with SVTNNs. Generally seaking, the roosed method is alicable and feasible to tackle MCDM roblems like the selection roblems of medical treatment otions under traezoidal intuitionistic fuzzy environments and SVTNN environments. However, the extant traezoidal intuitionistic fuzzy methods cannot solve MCDM roblems under SVTNN environments. From this ersective, the roosed method is a flexible one. Furthermore, the roosed method overcomes the shortcomings exist in the rior MCDM methods with SVTNNs, and its ranking order better corresonds with decision makers real references than those obtained by the revious MCDM methods. 28

29 6 Conclusion SVTNNs can be used to reflect fuzzy information in the selection of medical treatment otions. Moreover, TODIM methods under fuzzy environments can be utilized to construct MCDM methods to consider the risk reference of decision makers. In this aer, an imroved comarison method for SVTNNs was roosed to overcome the deficiencies of the rior comarison methods. Furthermore, we develoed several distance measurements for SVTNNs. Subsequently, a TODIM method for the selection roblems of medical treatment otions with SVTNNs was constructed on the basis of the roosed comarison method and distance measurements. In addition, a numerical examle for medical treatment otions selection was rovided to illustrate the rocess of the roosed method and the influence of the arameter was discussed. Finally, a comarative analysis was conducted to verify and discuss the feasibility of the roosed method. The rominent characteristics of this aer are that the roosed method for medical treatment otions selection utilizes SVTNNs, which can deict much information and fuzziness in selection rocesses. Moreover, the comarison method utilized in the roosed method cover the defects of the extant comarison methods. Furthermore, the roosed method for medical treatment otions selection extends TODIM method to SVTNN environments, and considers the risk attitude of hysicians which makes the decision-making results closer to hysicians actual references than the revious methods. There are two directions for future research: first, the roosed MCDM method can be alied to solve ractical issues in other fields, such as urchasing decision-making, and tourism destination selection. Second, the roosed MCDM method does not take into consideration the interrelationshis among criteria. It would be interesting to imrove the roosed method by introducing indeendent aggregation oerators. Acknowledgement 29

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