An application of neutrosophic sets in medical diagnosis by Ryan Elias

An application of neutrosophic sets in medical diagnosis Athar Kharalyz National University of Sciences and Technology (NUST), H-12, Islamabad, PAKISTAN atharkharal@gmail.com Abstract. In this paper we study the Sanchez’s approach for medical diagnosis and extend this concept with the notion of neutrosophic set theory (which is a generalization of both Fuzzy Set Theory and Intuitionistic Fuzzy Set Theory).

Introduction

Neutrosophic logic, introduced by Smarandache [18] is a generalization of fuzzy logic and several related systems. Neutrosophic sets and logic are the foundation for many theories which are more general than their classical counterparts in fuzzy, intuitionistic fuzzy and interval valued frameworks. This framework has found practical applications in a variety of di¤erent …elds, such as relational database systems, semantic web services [19], …nancial data set detection [9], new economies growth and decline analysis [10], image segmentation [4, 6], image denoising [5] and threshholding [2], binary classi…cation through neural networks [11], multiresolution wavelet based image segmentation [15] and multi criteria decision making [8]. Such multiplicity of applications also prompt towards further exploration of neutrosophic sets from a practical and theoretical point of view. Thus U. Rivieccio has documented di¤erent aspects of the potentials of neutrosophic sets and logic in his seminal work [12]. Automated medical diagnosis is of considerable interest and practical application in circumstances where access of human means of diagnosis are either denied or are very di¢ cult e.g. in telemedicine, space medicine and rescue services. Consequently, starting from the early time of Arti…cial Intelligence (days of ELISA) medical diagnosis has received ample attention from both computer science and computer applicable mathematics research community. After the inception of fuzzy sets in 1965, attempts were renewed to apply the then new framework of fuzzy logic to the problem of diagnosis. Sanchez [14] was the …rst one who applied his method of resolution of composition for fuzzy relational equations [13] to the problem of medical diagnosis. This method was further extended using intuitionistic fuzzy sets by De, Biswas and Roy [3]. Kharal [7] used another intuitionistic fuzzy variant of Sanchez’s method to …nd best suited drug for patients in an automated medical diagnosis environment. In this paper we extend the work of Sanchez in the framework of netrosophic sets. The method presented is more ‡exible and better equipped to handle the uncertainty, typically found in medical diagnostic reasoning. The method of neutrosophic medical diagnosis involves neutrosophic relations. Organization of the work is as follows: Section Paper writing started on 7 October 2011 author. Ph: 0092 333 6261309 z A version of this manuscript has also been submitted to arXiv.org on ??. y Corresponding

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20112

2 presents the necessary prelimnaries of neutrosophic sets and relations. Section 3 presents the methodology of Sanchez in terms of neutrosophic sets. Main method, its generalization to n number of patients and proposal for accumulation of neutrosophic knowledge-base for diagnosis have been given in this section. At the end of this section, the algorithm and a ‡ow chart have been given for computer implementation of the method. Lastly Section 5 presents an illustrative example for rendering the computational details of the present method more clear. The paper concludes with a Conclusion section. 2.

Neutrosophic Sets and Relations

Neutrosophic set permits one to incorporate indeterminacy, hesitation and/or uncertainty independent of the membership and non-membership information. Thus the notion of neutrosophic set is a generalization of fuzzy, intuitionistic fuzzy and interval-valued sets. De…nition 1. Let U be a nonempty …xed set. A neutrosophic set A upon U is an object given as x TA (x) ; IA (x) ; FA (x)

x : x2U TA (x) ; IA (x) ; FA (x)

;

where TA (x) ; IA (x) ; FA (x) ; called neutrosophic components, are Dsubintervals or union E of subintervals of unit interval [0; 1]. Thus a neutrosophic set A = TA (x);IAx(x);FA (x) in

U can be identi…ed to ordered triplets < TA ; IA ; FA > in I X I X I X or to elements in (I I I)X . Equivalently a neutrosophic set A is a function given as 3

A : U ! [0; 1] : x belongs to A in the following way: it is t% true, i% indeterminate, The element t;i;f and f % false,where t varies in TA (x), i varies in IA (x), and f varies in FA (x). Statically, TA ; IA and FA are membership sets, but dynamically TA ; IA and FA are functions depending on known and/or unknown parameters. Neutrosophic components may overlap, as well. In this paper we shall use neutrosophic sets whose TA ; IA and FA values are single points in [0; 1] instead of subintervals/subsets in [0; 1] : Consequently, in the sequel, we shall write an element in neutrosophic set A simply as TA ;IxA ;FA instead of writing it as

x j t 2 TA (x) ; i 2 IA (x) ; f 2 FA (x) : t; i; f Now we give the basic de…nitions of operations on neutrosophic sets [17]. We will concentrate on the case when the neutrosophic components are real values within unit interval instead of subintervals or subsets of the unit interval. De…nition 2. Let U be a universe and A and B are neutrosophic sets upon U: For x x TA ;IA ;FA 2 A and TB ;IB ;FB 2 B we de…ne 1. Complement of A as Ac =

x jT =1 T; I; F

TA ; I = 1

IA ; F = 1

FA ;

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20113

2. Intersection of A and B as A\B =

x j T = TA TB ; I = IA IB ; F = FA FB T; I; F

;

3. Union of A and B as A[B =

x j T = TA + TB T; I; F

TA TB ; I = IA + IB

IA IB ; F = FA + FB

FA FB

4. Di¤erence of A and B as AnB =

x j T = TA T; I; F

TA TB ; I = IA

IA IB ; F = FA

FA FB

;

5. A is subset of B as A

B () 8

y x 2 A; 2 B; TA TA ; IA ; FA TB ; IB ; FB

TB and FA

FB :

The set of operators given above are not unique for de…ning respective operations between neutrosophic sets. Some other schemes of operators may also be found in literature. However, in this paper we shall limit ourselves to only the above-de…ned operations. De…nition 3. A neutrosophic relation R from a set U to another set V is a neutrosophic set in U V denoted and de…ned as RU V =

(x; y) : x 2 U; y 2 V TR (x; y) ; IR (x; y) ; FR (x; y)

where TR : U have

V ! [0; 1] ; IR : U

(x; y) TR (x; y) ; IR (x; y) ; FR (x; y)

V ! [0; 1] and FR : U

V ! [0; 1] ; equivalently we

RU V : U

V ! [0; 1] :

A neutrosophic relation R=

(xi ; yj ) : xi 2 U; yj 2 V TR (xi ; yj ) ; IR (xi ; yj ) ; FR (xi ; yj )

may also be denoted using matrix notation as: y1 x1 R=

x2 .. . xm

where

[T11 ; I11 ; F11 ]

6 6 6 [T21 ; I21 ; F21 ] 6 6 6 6 .. 6 . 6 4 [Tm1 ; Im1 ; Fm1 ]

[T12 ; I12 ; F12 ]

[T1n ; I1n ; F1n ]

[T22 ; I22 ; F22 ] .. . [Tm2 ; Im2 ; Fm2 ]

7 7 [T2n ; I2n ; F2n ] 7 7 7 7 7 .. 7 . 7 5 [Tmn ; Imn ; Fmn ]

Tij = TR (xi ; yj ) ; Iij = IR (xi ; yj ) ; Fij = FR (xi ; yj ) : Use of square brackets, as elements of above matrix, denotes the vector nature of matrix entries i.e. [TR (xi ; yj ) ; IR (xi ; yj ) ; FR (xi ; yj )] :

;

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20114

De…nition 4. Let QU V = n (yj ;zk ) TR (yj ;zk );IR (yj ;zk );FR (yj ;zk )

(xi ;yj ) TQ (xi ;yj );IQ (xi ;yj );FQ (xi ;yj )

: xi 2 U; yj 2 V

and RV W =

: yj 2 V; zk 2 W be two neutrosophic relations. The maxE D ;zk ) is the neutrosophic reav-min composition R Q = TR Q (xi ;zk );IR (xQi(x i ;zk );FR Q (xi ;zk ) lation in U W , de…ned by the membership, indeterminacy and the non-membership functions, respectively, as: i _h ^ TRQ (xi ; zk ) = TQ (xi ; yj ) TR (yj ; zk ) ; yj

_ 1 IRQ (xi ; zk ) = (IQ (xi ; yj ) + IR (yj ; zk )) ; 2 yj i ^h _ FRQ (xi ; zk ) = FQ (xi ; yj ) FR (yj ; zk ) : yj

8 (xi ; zk ) 2 U

W and 8yj 2 V: o n (x ;y ) De…nition 5. Let QU V = TQ (xi ;yj );IQ (xi i ;yj j );FQ (xi ;yj ) : xi 2 U; yj 2 V and RV W = n o (yj ;zk ) : y 2 V; z 2 W be two neutrosophic relations. The maxj k TR (yj ;zk );IR (yj ;zk );FR (yj ;zk ) D E ;zk ) is the neutrosophic reav-min composition R Q = TR Q (xi ;zk );IR (xQi(x i ;zk );FR Q (xi ;zk ) lation in U W , de…ned by the membership, indeterminacy and the non-membership functions, respectively, as: _ 1 (TQ (xi ; yj ) + TR (yj ; zk )) ; 2 y

TRQ (xi ; zk )

IRQ (xi ; zk )

_ 1 = (IQ (xi ; yj ) + IR (yj ; zk )) ; 2 y

FRQ (xi ; zk )

^ 1 (FQ (xi ; yj ) + FR (yj ; zk )) : = 2 y j

8 (xi ; zk ) 2 U

W and 8yj 2 V: 3.

Sanchez’s Scheme in Terms of Neutrosophic Sets

By using neutrosophic sets we …rst generalize and adapt Sanchez’s scheme for medical diagnosis ([13], [14]) to the problem of medical diagnosis. Suppose there are l patients with m symptoms and n possible diseases. Explicitely we have S P D

= fsj ; j = 1; 2; : : : ; mg ; sj = pi = : sj 2 S ; i = 1; 2; : : : ; l ; T; I; F = fdk ; k = 1; 2; : : : ; ng :

Being part of a generalized method association, indeterminacy and non-association degrees may be interpreted variously. For example …rst number of a triplet is the degree to which a patient thinks the symptom to be important, second number indicates di¤erence

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20115

of opinion between patient-doctor and/or between two di¤erent doctors, and the third is the degree of importance attached to it by the doctor. Patient may attach a great value to his aching abdomen but for doctor it is just a byresult of a larger problem. Determination of symptoms of the patient and their associations, non-association and indetermination to her by routine case-taking. Thus mathematically, a patient is a neutrosophic set, say pi ; on the set of symptoms S: These neutrosophic sets would help to construct a matrix representing the neutrosophic relations CP S ; relating patient to symptoms, i.e. 2

s2 (T12 ; I12 ; F12 )

6 6 6 (T21 ; I21 ; F21 ) 6 6 6 6 .. 6 . 6 4 (Tl1 ; Il1 ; Fl1 )

CP S =

s1 (T11 ; I11 ; F11 )

.. . pl

sm 3 (T1m ; I1m ; F1m ) 7 7 (T2m ; I2m ; F2m ) 7 7 7 7 7 .. 7 . 7 5 (Tlm ; Ilm ; Flm )

(T22 ; I22 ; F22 ) .. .

(Tl2 ; Il2 ; Fl2 )

Analogous to Sanchez’s notion of "Medical Knowledge" we de…ne Neutrosophic Knowledge (NSK for short) as a neutrosophic relation K from the set of symptoms S to the set of diseases D (i.e., on S D) which reveals the degree of association, indetermination and the degree of non-association between symptoms and diseases. s1 KSD =

s2 .. . sm

d1 (T11 ; I11 ; F11 )

d2 (T12 ; I12 ; F12 )

6 6 6 (T21 ; I21 ; F21 ) 6 6 6 6 .. 6 . 6 4 (Tm1 ; Im1 ; Fm1 )

dn (T1n ; I1n ; F1n )

(T22 ; I22 ; F22 ) .. .

(Tm2 ; Im2 ; Fm2 )

7 7 (T2n ; I2n ; F2n ) 7 7 7 7 7 .. 7 . 7 5 (Tmn ; Imn ; Fmn )

Determination of patient-disease relational strength through composition of neutrosophic relations. The max-min-max or max-average composition RP D of CP S with the neutrosophic relation KSD denoted by R = KoC (may interestingly be read as, Result = Knowledge applied to Case) signi…es the patient-disease relation as a neutrosophic set on D with the membership, indeterminacy and non-membership functions, respectively, given by i _h ^ TA (s) TR (s; d) ; R (d) = s2S

IR (d) R (d)

_ 1 (IC (s) + IK (s; d)) ; 2 s2S i ^h _ = FA (s) FR (s; d)

s2S

8d 2 D:

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20116

Computation of selection index SK for …nal decision. Any one of the following selection indices may be used: 1 SK 2 SK 3 SK

= TK

FK :IK FK + IK = TK 2 T K + FK = IK 2

The algorithm describing neut-MeDis is given as follows: Name Input

: :

neut-MeDis Netrosophic set of alternatives, Neutrosophic set of criteria

Output

Ordered list of alternatives, most preferred as the …rst element

A brief ‡ow-chart for this module is given below:

4. Discussion on the Method 4.1. Generality of the method. The generality of method includes the fuzzy and intuitionistic fuzzy cases as a special ones. In fuzzy sets we have (x) = 1 (x) and in

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20117

the case of intuitionistic fuzzy sets we have allows us to select non-association (x)

(x) as

(x) :

This implies that if only the traditional analysis is to be adhered to, one should use the mechanical formula for the non-association i.e. (x) = 1

(x) :

(1)

Clearly, in this case one deals with a subclass of IFS’s i.e. fuzzy sets and, in fact, would be under-utilizing this method. While using (1), this should also be kept in mind that it is bit debatable if an association of (say) 33% mechanically implies a two-third non-association. 4.2. Collection of Nesutrosophic Knowledge. Codi…cation of neutrosophic knowledge as an neutrosophic relation. It would be achieved by the help of a medical expert or some other dependable source of medical knowledge.This neutrosophic relation from S to D of medical knowledge is denoted as KSD : For a given K and C, the relation R = KoC can be computed. Conversely, from the knowledge of C and K, one may also compute a re…ned version of the neutrosophic relation K (knowledge) such that the following hold true: 1. selection index is greatest, and 2. the equality R = KoC is retained. This re…ned version of K will be a more signi…cant neutrosophic relation attributing higher degrees of association and lower degrees of non-association of symptoms as well as lower degrees of indeterminations/hesitations to the diseases. Thus this is one of the many possible approaches towards Accumulated Neutrosophic Medical Knowledge Base (ANMKB). From a re…ned version of K, one may infer diseases from symptoms in the sense of a triplet of values, one being the degree of association, one for indetermination and third one the degree of non-association. In case, the medical expert is not satis…ed with the results, K is modi…ed. This improvement of K over a period of time would establish a symptom in therapeutic index which would be the compilation of all such K 0 s: A computer-based knowledge acquisition module can be used for this purpose. This module would use the method presented in this work as a means to accumulate medical knowledge, more precisely, the symptom-disease associations, indetermination and nonassociations, from di¤erent experts. It is important to note that this ‡ow chart and specially the loop based upon expert satisfaction is not part of this neutrosophic set method presented in this paper, instead it is the mechanism to elicit ANMKB which if gathered and compiled would yield a new therapeutic index in which symptoms, for example, may look as follows: Pain in abdomen :

Diahorrhea(:6;:2;:5) ; Umblical Hernia(:8;:5;:2) ;

4.3. Choice of composition method. The choice of composition method lies with the user. If user’s knowledge of medical case is to be considered, then De…nition ?? should be employed. If one decides to depend upon the knowledge of expert solely, then choice of De…nition 4 seems more correct.

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20118

2 1 4.4. Choice of Selection Index. SK is more discriminatory as compared to SK : 3 Whereas SK is …rmer on giving judgement and seldom comes up with a tie decision. In case of a tie decision in step 4, hesitations are called in to get …nal judgement and the drug with least hesitation is selected.

Illustrative Example

To see an application of the method, in the following we give an example (patient’s personal details have been omitted due to medical privacy laws) : Example 6. Give an example focusing upon neutrosophic relations and their compositions only. QXY x1 x2 x3 x4 RY Z y1 y2 y3

y1 [0:7; 0:4; 0:1]

6 6 6 [0:6; 0:5; 0:3] 6 6 6 6 [0:8; 0:4; 0:2] 6 4 [0:4; 0:6; 0:3]

z1 [0:9; 0:6; 0:7]

6 6 6 [0:5; 0:3; 0:3] 6 4 [0:8; 0:8; 0:9]

y2 [0:8; 0:6; 0:7] [0:6; 0:5; 0:2] [0:5; 0:1; 0:5] [0:5; 0:4; 0:8]

y3 3 [0:4; 0:8; 0:5] 7 7 [0:7; 0:9; 0:0] 7 7 ; 7 7 [1:0; 0:5; 1:0] 7 7 5 [0:5; 0:6; 0:9]

z2 [0:9; 1:0; 0:5]

z3 [0:9; 0:2; 0:8]

[0:4; 0:6; 0:6]

[0:4; 0:5; 0:3]

[0:7; 0:8; 0:3]

[0:8; 0:1; 0:8]

z4 3 [0:6; 0:2; 0:3] 7 7 [0:9; 0:5; 0:8] 7 7 5 [0:3; 0:4; 0:5]

TXZ is the maximum of the minimum of outer product of TXY and TY Z ; it is calculated as TXZ (x; z) y1 y2 y3 max

min! 2 (x1 ; z1 ) 6 0:7 6 6 0:5 6 4 0:4 0:7

(x1 ; z2 ) 0:7 0:4 0:4 0:7

avg ! 2 (x1 ; z1 ) 6 0:5 6 6 0:45 6 4 0:8 0:8

(x1 ; z2 ) 0:7 0:6 0:8 0:8

(x1 ; z3 ) 0:7 0:4 0:4 0:7

(x1 ; z4 ) 0:6 0:8 0:3 0:8

(x2 ; z1 ) 0:6 0:5 0:7 0:7

(x2 ; z2 ) 0:6 0:4 0:7 0:7

(x2 ; z3 ) 0:6 0:4 0:7 0:7

(x2 ; z4 ) 0:6 0:6 0:3 0:6

(x3 ; z1 ) 0:8 0:5 0:8 0:8

(x3 ; z2 ) 0:8 0:4 0:7 0:8

(x3 0 0 0 0

(x3 ; z1 ) 0:5 0:2 0:65 0:65

(x3 ; z2 ) 0:7 0:35 0:65 0:7

(x3 0 0 0 0

IXZ is the maximum of the average of outer product of TXY and TY Z ; it is calculated as IXZ (x; z) y1 y2 y3 max

(x1 ; z3 ) 0:3 0:55 0:45 0:55

(x1 ; z4 ) 0:3 0:55 0:6 0:6

(x2 ; z1 ) 0:55 0:4 0:85 0:85

(x2 ; z2 ) 0:75 0:55 0:85 0:85

(x2 ; z3 ) 0:35 0:5 0:5 0:5

(x2 ; z4 ) 0:35 0:5 0:65 0:65

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 20119

FXZ is the minimum of the maximum of outer product of TXY and TY Z ; it is calculated as FXZ (x; z)

max ! 2 (x1 ; z1 ) 6 0:7 6 6 0:7 6 4 0:9 0:7

y1 y2 y3 min

(x1 ; z2 ) 0:5 0:7 0:5 0:5

(x1 ; z3 ) 0:8 0:7 0:8 0:7

(x1 ; z4 ) 0:3 0:8 0:5 0:3

(x2 ; z1 ) 0:7 0:3 0:9 0:3

(x2 ; z2 ) 0:5 0:6 0:3 0:3

(x2 ; z3 ) 0:8 0:3 0:8 0:3

(x2 ; z4 ) 0:3 0:8 0:5 0:3

Hence the matrix of composition of relations QXY and RT Z is given as R Q x1 x2 x3 x4

z1 [0:7; 0:8; 0:7]

6 6 6 [0:7; 0:85; 0:3] 6 6 6 6 [0:8; 0:65; 0:5] 6 4 [0:5; 0:7; 0:7]

z2 [0:7; 0:8; 0:5] [0:7; 0:85; 0:3] [0:8; 0:7; 0:5] [0:5; 0:8; 0:5]

Calculation of all three selection indices yield S1 x1 x2 x3 x4

2z1 z2 0:14 6 0:45 6 4 0:48 0:01

z2 0:05 6 0:13 6 4 0:23 0:20

z3 z4 3 [0:7; 0:55; 0:7] [0:8; 0:6; 0:3] 7 7 [0:7; 0:5; 0:3] [0:6; 0:65; 0:3] 7 7 7 7 [0:8; 0:3; 0:5] [0:6; 0:45; 0:3] 7 7 5 [0:5; 0:45; 0:8] [0:5; 0:5; 0:3]

z3 z4 0:30 0:32 0:62 0:45 0:55 0:41 0:45 0:65 0:47 0:10 0:14 0:35

S2 x1 x2 x3 x4

2z1

z3 z4 0:05 0:08 0:13 0:30 0:20 0:40 0:15 0:13

S3 x1 x2 x3 x4

2z1

z3 z 4 0:20 0:35 0:05 0:30

z2 0:10 6 0:35 6 4 0:00 0:10

0:15 0:00 0:35 0:20

7 ; 7 5

3 0:35 0:13 7 7; 0:23 5 0:10

3 0:05 0:20 7 7 0:00 5 0:10

So one easily decides that patient xi has disease zj under selection index Sk from following table S1 S2 S3 MajorityVote x1 z4 z4 z3 z4 x2 z3 z3 z3 z3 x3 z3 z3 z3 z3 z4 x4 z4 z4 z3 Conclusion 7. In this paper we have presented a new method for handling medical diagnosis problems, where the characteristics of the patients and symptoms are represented by

(x3 ; z1 ) 0:7 0:5 1 0:5

(x3 ; z2 ) 0:5 0:6 1 0:5

(x3 0 0

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 201110

neutrosophic sets. The proposed method allows the degree of satis…ability, non satis…ability and indeterminacy of each alternative with respect to a set of criteria to be represented by neutrosophic sets, respectively. Furthermore the proposed method allows the decision maker to assign the degree of satis…ability, non satis…ability and indeterminacy of the symptoms to a vague concept “importance”. An example is presented to illustrate the neutrosophic medical diagnosis process. From these we can see that the proposed method di¤ers from previous approaches for medical diagnosis not only due to the fact that the proposed method uses neutrosophic set theory, but also due to the degree of importance of the symptoms are not constant and the calculation is simpler. A two-progend approach has been presented in this paper. On one hand a scheme to develop a new therapeutic index, establishing the associations, hesitations/indeterminacies and non-associations of a disease with a given symptom have been presented. On the other hand a method to fully exploit the bene…ts of such a new index has been presented. For this purpose NS theory has been used to adapt Sanchez’s approach for medical diagnosis. Three di¤erent selection indices and two types of NSR compositions have been discussed in the sequel. The method is then described in detail and applied to two example cases. Besides providing a methodology, we also propose an automated mechanism to accumulate medical knowledge in a systematic manner: As the …eld of neutrosophic sets is a new one, attempt has been made to present a short but comprehensive …rst introduction as part of preliminaries of neutrosophic sets and relations. References [1] K.T. Atanassov, Intuitionistic fuzzy sets, in: VII ITKR’s Session, So…a, June 1983 (Deposed in Central Sci. Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian). [2] H.D. Cheng, Y. Guo, A new neutrosophic approach to image thresholding, New Mathematics and Natural Computation 4 (3) (2008) 291–308. [3] S.K. De, R. Biswas, A.R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117 (2001) 209-213. [4] Y.Guo, H.D.Cheng, New neutrosophic approach to image segmentation, Pattern Recognition 42 (2009) 587–595. [5] Y. Guo, H.D. Cheng, A new neutrosophic approach to image denoising, New Mathematics and Natural Computation 5 (3) (2009) 653–662. [6] Y. Guo, H.D. Cheng, A new neutrosophic approach to image segmentation, Pattern Recognition 42 (2009) 587–595. [7] Athar Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy (2009) 98, 35–39. [8] Athar Kharal, A neutrosophic multicriteria decision making method, (submitted). [9] M. Khoshnevisan, S. Bhattacharya, A short note on …nancial data set detection using neutrosophic probability, in: F. Smarandache (Ed.), Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, University of New Mexico, 2002, pp. 75–80.

An application of neutrosophic sets in medical diagnosis Paper writing started on 7 October 201111

[10] M. Khoshnevisan, S. Singh, Neurofuzzy and neutrosophic approach to compute the rate of change in new economies, in: F. Smarandache (Ed.), Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Neutrosophic Set, Neutrosophic Probability and Statistics, University of New Mexico, 2002, pp. 56–62. [11] P.Kraipeerapun, C.C.Fung, Binary classi…cation using ensemble neural networks and interval neutrosophic sets, Neurocomputing 72 (2009) 2845–2856. [12] U.Rivieccio, Neutrosophic logics: Prospects and problems, Fuzzy Sets and Systems 159 (2008) 1860–1868. [13] E. Sanchez, Resolution of composition of fuzzy relation equations, Information and Control 30 (1976) 38-48. [14] E. Sanchez, Solutions in composite fuzzy relation equation. Application to Medical diagnosis in Brouwerian Logic, in: M.M. Gupta, G.N. Saridis, B.R. Gaines (Eds.), Fuzzy Automata and Decision Process, Elsevier, North-Holland, 1977. [15] A. Sengur, Y. Guo, Color texture image segmentation based on neutrosophic set and wavelet transformation, Computer Vision and Image Understanding 115 (2011) 1134– 1144. [16] Florentin Smarandache, "A Unifying …eld in logics. Neutrosophy: neutrosophic probability, set, and logic", Multiple-Valued Logic : An International Journal, 8(3) (2002) 385-438. [17] Florentin Smarandache, Neutrosophic Set – A Generalization of the Intuitionistic Fuzzy Set [18] F. Smarandache, AUnifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press, Rehoboth, NM, 1999. [19] F. Smarandache, R. Sunderraman, H. Wang, Y. Zhang, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, HEXIS Neutrosophic Book Series, No. 5, Books on Demand, Ann Arbor, MI, 2005.