Operations on type2 fuzzy sets


 Juniper Dixon
 2 years ago
 Views:
Transcription
1 Fuzzy Sets and Systems (00) Operations on type fuzzy sets Nilesh N. Karnik, Jerry M. Mendel Signal and Image Processing Institute, Department of Electrical EngineeringSystems, 3740 McClintock Ave., EEB400, University of Southern California, Los Angeles, CA , USA Received 4 August 999; received in revised form 0 April 000; accepted May 000 Abstract In this paper, we discuss set operations on type fuzzy sets (including join and meet under minimum=product tnorm), algebraic operations, properties ofmembership grades oftype sets, and type relations and their compositions. All this is needed to implement a type fuzzy logic system (FLS). c 00 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy relations; Fuzzy numbers; Type fuzzy sets. Introduction The concept ofa type fuzzy set was introduced by Zadeh [30] as an extension ofthe concept ofan ordinary fuzzy set (henceforth called a type fuzzy set). Such sets are fuzzy sets whose membership grades themselves are type fuzzy sets; they are very useful in circumstances where it is dicult to determine an exact membership function for a fuzzy set; hence, they are useful for incorporating linguistic uncertainties, e.g., the words that are used in linguistic knowledge can mean dierent things to dierent people []. A fuzzy relation of higher type (e.g., type) has been regarded as one way to increase the fuzziness of a relation, and, according to Hisdal, increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner [5]. According to John, Type fuzzy sets allow for linguistic grades ofmembership, thus assisting in knowledge representation, and they also oer improvement on inferencing with type sets [6]. Type sets can be used to convey the uncertainties in membership functions of type sets, due to the dependence of the membership functions on available linguistic and numerical information. Linguistic information (e.g., rules from experts), in general, does not give any information about the shapes of the membership functions. When membership functions are determined or tuned based on numerical data, the uncertainty in the numerical data, e.g., noise, translates into uncertainty in the membership functions. In all such cases, information about the linguistic=numerical uncertainty can be incorporated in the type framework. In [8], Liang and Mendel demonstrated (using real data) that a type fuzzy set, a Gaussian with xed mean and Corresponding author. Tel.: + (3) ; fax: + (3) address: (J.M. Mendel) /0/$  see front matter c 00 Elsevier Science B.V. All rights reserved. PII: S (00)
2 38 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) uncertain standard deviation (std), is more appropriate to model the frame sizes of I=P=B frames in MPEG VBR video trac than is a type Gaussian membership function. In this paper, A is a type fuzzy set, and the membership grade (a synonym for the degree of membership) of x X in A is A (x), which is a crisp number in [0; ]. A type fuzzy set in X is A, and the membership grade of x X in A is A (x), which is a type fuzzy set in [0; ]. The elements ofthe domain of A (x) are called primary memberships of x in A and the memberships ofthe primary memberships in A (x) are called secondary memberships of x in A. The latter dene the possibilities for the primary membership, e.g., A (x) can be represented, for each x X as A (x) =f x(u )=u + f x (u )=u + + f x (u m )=u m = i f x (u i )=u i ; u i J x [0; ]: When the secondary MFs are interval sets, we call them interval type fuzzy sets. The operations of interval type fuzzy sets are studied in [9,6]. Example. Fig. (a) shows a type set in which the primary membership grade for each point in the set is a Gaussian type set. The means ofall these Gaussian type sets, follow a Gaussian and the standard deviations ofthese Gaussians decrease as the mean decreases. Uncertainty in the primary membership grades ofthe type MF consists ofa shaded region (the union ofall primary membership grades), that we call the footprint of uncertainty ofthe type MF (e.g., see Fig. (a)). The principal membership function, i.e., the set ofprimary memberships having secondary membership equal to, is indicated with a thick line. This principal membership function is a Gaussian because ofthe way the set is constructed. Intensity ofthe shading is approximately proportional to secondary membership grades. Darker areas indicate higher secondary memberships. The at portion from about.5 to 3.5 appears because primary memberships cannot be greater than (since primary memberships, themselves, are possible membership values, they have to be in [0; ]) and so the Gaussians have to be clipped. The domain ofthe membership grade corresponding to x =4 is also shown. Fig. (b) shows the secondary membership function for x = 4. Many other examples oftype sets can be found in []. The concept of a principal membership function also illustrates the fact that a type fuzzy set can be thought ofas a special case ofa type fuzzy set. We can think ofa type fuzzy set as a type fuzzy set whose membership grades are type fuzzy singletons, having secondary membership equal to unity for only one primary membership and zero for all others, i.e., we can think of the principal membership function of a type set as an embedded type set. A type fuzzy set can also be thought of as a fuzzy valued function, which assigns to every x X a type fuzzy membership grade. In this sense, we will call X the domain of the type fuzzy set. Mizumoto and Tanaka [3] have studied the settheoretic operations oftype sets, properties ofmembership grades ofsuch sets, and, have examined the operations ofalgebraic product and algebraic sum for them [4]. Nieminen [5] has provided more details about algebraic structure oftype sets. Dubois and Prade [4] and Kaufmann and Gupta [4] have discussed the join and meet operations between fuzzy numbers under minimum tnorm. Dubois and Prade [ 4] have also discussed fuzzy valued logic and have given a formula for the composition oftype relations as an extension ofthe type supstar composition, but this formula is only for minimum tnorm. Karnik and Mendel [,3] have provided a general formula for the extended supstar composition oftype relations. Type fuzzy sets have been used in decision making [,9], solving fuzzy relation equations [7], survey processing [], preprocessing ofdata [8], and modeling uncertain channel states [,3]. In engineering applications of fuzzy logic, minimum and product tnorm are usually used, so we focus on product and minimum tnorm in this paper, and leave the extensions ofour results to other tnorm to others. Recently, a type FLS [,3] has been developed which uses either tnorm; however, in order to implement such a
3 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig.. (a) Pictorial representation ofa Gaussian type set. The standard deviations ofthe secondary Gaussians decrease by design, as x moves away from 3. The secondary memberships in this type fuzzy set are shown in (b), and are also Gaussian. type FLS, one needs algorithms for settheoretical operations on type sets (which are given in Section ), algebraic operations for type sets (which are given in Section 3), an understanding of set properties of membership grades (which is provided in Section 4) and algorithms for type fuzzy relations and their compositions (which are given in Section 5). So, the main purpose ofthis paper is to provide the foundation for type FLSs.. Settheoretic operations on type sets Consider two fuzzy sets of type, A and B, in a universe X. Let A (x) and B (x) be the membership grades (fuzzy sets in J x [0; ]) ofthese two sets, represented, for each x, as A (x)= u f x(u)=u and B (x)= w g x(w)=w, respectively, where u; w J x indicate the primary memberships of x and f x (u); g x (w) [0; ] indicate the secondary memberships (grades) of x. Using Zadeh s Extension Principle [30,4,], the membership grades for union, intersection and complement of type fuzzy sets A and B have been dened as follows [3]: Union: A B A B (x) = A (x) B (x) = (f x (u)?g x (w))=(u w); () u w Intersection: A B A B (x) = A (x) B (x) = (f x (u)?g x (w))=(u?w); () u w Complement: A A (x) A (x) = f x (u)=( u); (3) u
4 330 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) where represents the max tconorm and? represents a tnorm. The integrals indicate logical union. In the sequel, we adhere to these denitions, and, as in [3], we refer to the operations, as join, meet and negation, respectively. If A (x) and B (x) have discrete domains, the integrals in () (3) are replaced by summations. In (), ifmore than one calculation ofu and w gives the same point u w, then in the union we keep the one with the largest membership grade. We do the same in (). Suppose, for example, u w = and u w =. Then within the computation of() we would have f x (u )?g x (w )= + f x (u )?g x (w )= (4) where + denotes union. Combining these two terms for the common is a type computation in which we can use a tconorm. We choose to use the maximum as suggested by Zadeh [30] and Mizumoto and Tanaka [3]. Using the denitions in () (3), we examine the operations of join, meet and negation in more detail, both under minimum ( ) and product tnorms (which are the most widely used tnorms in engineering applications offuzzy sets and logic). We consider only convex, normal membership grades. Our goal is to obtain algorithms that let us compute the join, meet and negation. These algorithms will be needed later to compute type relations. Since these operations are performed on membership grades of type sets, which themselves are type sets, the results discussed next are in terms oftype sets. The type sets, F i, in Sections..4, can be thought ofas membership grades ofsome type sets, A i, for some arbitrary input x 0. For example, (4) in Fig. (b) is a type set associated with x =4... Join and meet under minimum tnorm Our major result for join and meet under minimum tnorm is given in: Theorem. Suppose that we have n convex; normal; type real fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively. Let v ;v ;:::;v n be real numbers such that v 6v 6 6v n and f (v )=f (v )= =f n (v n )=. Then; using max tconorm and min tnorm; n i= f i(); v ; n Fi() = n i= i=k+ f i(); v k 6 v k+ ; 6k6n ; (5) n i= f i(); v n ; and n i= f i(); v ; n Fi() = k i= i= f i(); v k 6 v k+ ; 6k6n ; n i= f i(); v n : (6) See Appendix A for the proofoftheorem. Fig. shows an example ofan application oftheorem for the case n = 4. Observe that Eqs. (5) and (6) are very easy to program. Dubois and Prade present the result in Theorem just for the case n =, in the context offuzzication ofmax and min operations. Though their method ofproofis very similar to ours, they prove the result for a special case, where f and f have at most three points ofintersection and one needs to keep track of the points ofintersection off and f to use their theorem. We reprove this theorem in a general setting in Appendix A. We believe that our statements of F F () and F F () are more amenable to computer
5 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig.. An illustration oftheorems and : Join and meet operations between Gaussians: (a) Participating Gaussians; (b) join under minimum tnorm (solid line) and product tnorm (thick dashed line); and (c) meet under minimum tnorm. implementations than those ofdubois and Prade. Generalization to the case n is also dicult using Dubois and Prade s result, but is provided by us in Theorem. As a consequence oftheorem, we have the following interesting result: Corollary. If we have n convex; normal; type fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively; such that f i ()=f ( k i ); and 0=k 6k 6 6k n ; then; using max tconorm and min tnorm; n i= F i = F n and l n i= F i = F. The prooffollows by a direct application oftheorem to the n type sets described in the statement of the corollary. Fig. 3 shows an example ofunion and intersection oftype sets, using the results in this section... Join under product tnorm For type fuzzy sets satisfying the requirements of Theorem, join operation under product tnorm gives exactly the same result as (5) with product replacing the minimum. Theorem. Suppose that we have n convex; normal; type real fuzzy sets F ;:::;F n characterized by membership functions f ;:::;f n ; respectively. Let v ;v ;:::;v n be real numbers such that v 6v 6 6v n and f (v )=f (v )= = f n (v n )=. Then; the membership function of n i= F i using max tconorm and
6 33 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 3. Union and Intersection of two type fuzzy sets using the pictorial representation introduced in Fig., and, for maximum tconorm and minimum tnorm: (a) Participating sets; (b) union; and (c) intersection. product tnorm; can be expressed as n i= f i(); v ; n i= Fi () = n i=k+ f i(); v k 6 v k+ ; 6k6n ; n i= f i(); v n : (7) The proofproceeds in the same way as the proofofpart (I) in Theorem (Appendix A), with the minimum operation replaced by product. See [] for details. Fig. shows an example ofan application oftheorem for the case n =4..3. Meet under product tnorm The meet operation under the product tnorm between two convex normal type sets F and F can be represented as F F = [f (v)f (w)]=(vw): (8) v w Ref. [4] gives a similar result while discussing multiplication of fuzzy numbers (see Section 3 for algebraic operations on fuzzy sets). If is an element of F F, then the membership grade of can be found by nding all the pairs {v; w} such that v F ; w F and vw = ; multiplying the membership grades of v and w in each pair; and then nding the maximum ofthese products ofmembership grades. The possible admissible {v; w} pairs whose
7 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) product is are {v; =v} (v R; v 0) for 0 and {v; 0} or {0;w}, where v; w R for = 0, i.e., ( F F () = sup f (v)f v R;v 0 v [ ] F F (0) = sup f (v)f (0) v R = [ f (0) sup f (v) v R = f (0) f (0): ] ) ; R; 0; [ ] sup f (0)f (w) w R ] [ f (0) sup f (w) w R (9) Since the meet operation under product tnorm is commutative (see Section 4), we get the same result whether we substitute =w = v or =v = w. As is apparent from (9), the result is very much dependent on functions f and f and does not easily generalize like the join and meet operations considered earlier, and generally, it is very dicult to obtain a closedform expression for the result of the meet operation. Even ifboth the fuzzy sets involved have Gaussian membership functions, it is dicult to obtain a nice closedform expression for the result of the meet operation. To determine the membership ofa particular point in F G, we nd all the pairs {v; w} such that v R; w R and vw = ; and multiply the memberships ofeach pair. The membership grade of is given by the supremum ofthe set ofall these products. For example, if = 0, all the pairs {v; w} that give 0 as their product are v and 0=v (v R;v 0). So, the membership grade of0 is given by the supremum of the set ofall the products f(v)g(0=v) (v R;v 0). Fig. 4(c) shows how f(v)g(0=v) looks for F and G in Fig. 4(a). Clearly, it is no easy matter to represent (9) visually. Because ofthe complexity of(9) for general membership functions (MFs), we concentrate on Gaussian MFs. For a derivation of F F when both F and F are Gaussian type sets, see []. It is very complicated, does not lead to a closedform expression, and does not generalize to more than two MFs. Since the meet operation under product tnorm is heavily used in the development ofa type FLS (because it is widely used in type FLSs), here we develop a Gaussian approximation to this operation that will make it practical. Ifthere are n Gaussian fuzzy sets F ;F ;:::;F n with means m ;m ;:::;m n and standard deviations ; ;:::; n, respectively, then, under the product tnorm, F F F n () e (=)(( mm mn)= ) : (0) where (i =;:::;n) = m i + + j m i + + n m i : () i;i i;i j i; i n For the derivation ofthis approximation, see Appendix B. One important feature ofthis approximation is that it is Gaussian; hence, it is reproducible, i.e., Gaussians remain Gaussians. Fig. 5 shows some examples ofthe meet operation under product tnorm, and the approximation in (0). See [9] for a triangular approximation for triangular type sets, i.e., type sets in which all the secondary membership functions are triangular. A popular type fuzzy set is the interval type set, i.e., the type set all of whose secondary MFs are interval sets. Note that an interval [l; r] can also be represented in terms ofits mean and spread as [m s; m+s],
8 334 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 4. An example showing how f(v)g(0=v) looks for the membership functions f and g oftype sets F and G, respectively, in (a); (b) f(v), which is the same as that in (a) and g(0=v); and, (c) the product f(v)g(0=v). where m =(l + r)= and s =(r l)=. The meet ofinterval sets under product=minimum tnorm has been given by Schwartz [6] and Kaufmann and Gupta [4], and is discussed in detail in Liang and Mendel [7]..4. Negation From the denition ofthe negation operation, it follows that Theorem 3. If a type real fuzzy set F has a membership function has a membership function f( ). The prooffollows by substituting = u in (3). 3. Algebraic operations Just as the tconorm and tnorm operations have been extended to the membership grades oftype sets using the Extension Principle, algebraic operations between type sets have been dened using the Extension Principle [4,4,]. A binary operation dened for crisp numbers, can be extended to two type sets, F = v f (v)=v and F = w f (w)=w, as follows: F F = v [f (v)?f (w)]=(v w) () w
9 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 5. Some examples ofthe meet between Gaussians under product tnorm (solid line) and the approximation in (0) (thick dashed line). Means and standard deviations ofparticipating Gaussians in each case are also shown on the gure. In each case, the solid curve was obtained numerically, using (9). where? indicates the tnorm used. We will mainly be interested in multiplication and addition oftype sets. Observe, from (8) and (), that multiplication of two type sets under product tnorm is the same as the meet ofthe two type sets under product tnorm; therefore, all our earlier discussion about the meet under product tnorm applies to the multiplication oftype sets under product tnorm. Theorem 4. Given n interval type numbers F ;:::;F n ; with means m ;m ;:::;m n and spreads s ;s ;:::;s n ; their ane combination n i= if i + ; where i (i =;:::;n) and are crisp constants; is also an interval type number with mean n i= im i + ; and spread n i= i s i. See Appendix C for the proofoftheorem 4. Theorem 5. Given n type Gaussian fuzzy numbers F ;:::;F n ; with means m ;m ;:::;m n and standard deviations ; ;:::; n ; their ane combination n i= if i + ; where i (i =;:::;n) and are crisp constants; is also a Gaussian fuzzy number with mean n i= im i + ; and standard deviation ; where n i= = i i if product tnorm is used; n i= i i if minimum tnorm is used: See Appendix D for the proofoftheorem 5. (3)
10 336 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Table Properties ofmembership grades. In the type case, we assume convex and normal membership grades; and for type laws,, and replace,? and 6, respectively. For product tnorm, the laws that are satised in the type case, but not in the type case, are highlighted. Results for the minimum tnorm are taken from [3] Settheoretic laws Minimum tnorm Product tnorm Type Type Type Type Reexive A 6 A Yes Yes Yes Yes Antisymmetric A 6 B ; B 6 A Yes Yes Yes Yes A = B Transitive A 6 B ; B 6 C Yes Yes Yes Yes A 6 C Idempotent A A = A Yes Yes Yes No A? A = A Yes Yes No No Commutative A B = B A Yes Yes Yes Yes A? B = B? A Yes Yes Yes Yes Associative ( A B ) C Yes Yes Yes Yes = A ( B C ) ( A? B )? C Yes Yes Yes Yes = A? ( B? C ) Absorption A? ( A B )= A Yes Yes No No A ( A? B )= A Yes Yes Yes No Distributive A? ( B C ) Yes Yes Yes No =( A? B ) ( A? C ) A ( B? C ) Yes Yes No No =( A B )? ( A C ) Involution A = A Yes Yes Yes Yes De Morgan s Laws A B = A? B Yes Yes No No A? B = A B Yes Yes No No Identity A 0= A Yes Yes Yes Yes A? = A Yes Yes Yes Yes A = Yes Yes Yes Yes A? 0 = 0 Yes Yes Yes Yes Complement A A Yes Yes Yes Yes (Failure) A?A 0 Yes Yes Yes Yes 4. Properties of membership grades Mizumoto and Tanaka [3] discuss properties ofmembership grades oftype sets in detail, under the minimum tnorm and maximum tconorm. We examined these properties under product tnorm and maximum tconorm for convex, normal membership grades. Our ndings are summarized in Table. See Chapter 3 of [] for details, which we have omitted here because they are rather straightforward. In proving the identity law, we use the concepts of0 and membership grades for type sets [3]; they are represented as =0 and =, respectively, which is in accordance with our earlier discussion about type sets being a special case oftype sets. An element is said to have a zero membership in a type set ifit has a secondary membership equal to corresponding to the primary membership of0, and ifit has all other secondary memberships equal to 0. Similarly, an element is said to have a membership grade equal to in a type set, ifit has a secondary membership equal to corresponding to the primary membership of and ifall other secondary memberships are zero. All our operations on type sets collapse to their type counterparts, i.e., ifall the type sets are replaced by their principal membership functions (assuming principal membership function can be dened for all of
11 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) them), all our results remain valid. We, therefore, conclude that if there are any settheoretic laws that are not satised by type fuzzy sets, we can safely say that type sets will not satisfy those laws either; however, the converse ofthis statement may not be true. If any condition is satised by type sets, it may or may not be satised by type sets. Observe, from Table, that the {max, product} tconorm=tnorm pair, for type sets, does not satisfy idempotent, absorption, distributive, De Morgan s and complement laws; hence, ifthe design ofa type fuzzy logic system (controller) involves the use of these laws, it will be in error. Observe also that some laws that are satised in the type case under {max, product} tconorm=tnorm pair are not satised in the type case. 5. Type fuzzy relations, their compositions, and cartesian product 5.. Composition of type fuzzy relations Let X ;X ;:::;X n be n universes. A crisp relation in X X X n is a crisp subset ofthe product space. Similarly, a type fuzzy relation in X X X n is a type fuzzy subset of the product space and a type fuzzy relation in X X X n is a type fuzzy subset of the product space. We concentrate on binary type fuzzy relations. Unions and intersections oftype relations on the same product space can be obtained exactly in the same manner as unions and intersections ofordinary type sets are obtained. If R(U; V ) and S(U; V ) are two type relations on the same product space U V, their union and intersection are determined as R S (u; v)= R (u; v) S (u; v) and R S (u; v)= R (u; v) S (u; v). Consider two dierent product spaces, U V and V W, that share a common set and let R(U; V ) and S(V; W ) betwocrisp relations on these spaces. The composition ofthese relations is dened [5] as a subset T (U; W )ofu W such that (u; w) T ifand only if(u; v) R and (v; w) S. This can be expressed as a max min, maxproduct or, in general, as the supstar composition R S (u; w) = sup v V [ R (u; v)? S (v; w)] where? indicates any suitable tnorm operation. The validity ofthe supstar composition for crisp sets is shown in [8]. If R and S are two crisp relations on U V and V W, respectively, then the membership for any pair (u; w); u U and w W, is ifand only ifthere exists at least one v V such that R (u; v)= and S (v; w) =. In [8], it is shown that this condition is equivalent to having the supstar composition equal to. When we enter the fuzzy domain, set memberships (which are crisp numbers for type sets, and type fuzzy sets for type sets) belong to the interval [0; ] rather than just being 0 or. So, now we can think ofan element as belonging to a set ifit has a nonzero membership in that set. In this respect, the aforementioned condition on the composition of relations can be rephrased as follows (recall the concept of 0 and membership grades in the case oftype sets discussed in Section 4): IfR ( R)andS( S) are two type (type) fuzzy relations on U V and V W, respectively, then the membership for any pair (u; w), u U and w W, is nonzero ifand only ifthere exists at least one v V such that R (u; v) 0 (R (u; v) 0) and S(v; w) =0 ( S (v; w) =0). It can be easily shown that in the type case, this condition is again equivalent to the supstar composition, R S (u; w) = sup [ R (u; v)? S (v; w)] (4) v V and, in the type case, it is equivalent to the following extended version of the supstar composition. R S (u; w) = v V [ R (u; v) S (v; w)]: (5)
12 338 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Proofs of(4) and (5) are very similar. We just give the proofof(5) next. In the proof, we use the following method. Let A be the statement R S (u; w) =0, and B be the statement there exists at least one v V such that R (u; v) =0 and S (v; w) =0. We prove that A i B by rst proving that B A (which is equivalent to proving A B, i.e., necessity of B) and then proving that A B (which is equivalent to proving B A, i.e., suciency of B). Proof. (Necessity) Ifthere exists no v V such that R (u; v) =0 and S (v; w) =0, then this means that either R (u; v)==0 or S (v; w)==0 (or both are zero) for every v V. So, assuming that R (u; v) and S (v; w) are normal, then from the identity law ( A =0==0), R (u; v) =0==0 or=0 S (v; w)==0, so that we have that R (u; v) S (v; w)==0 for every v V, and therefore, in (5), v V =0==0. (Suciency) Ifthe extended supstar composition is zero, then, from (5), v V [ R (u; v) S (v; w)]==0: For each value of v, the term in the bracket is a type fuzzy set. Suppose, for example, that there are two such terms, say P and Q, so that P Q ==0. Observe, from (), that for two normal membership grades, P = u f(u)=u and Q = v g(v)=v, P Q ==0 u (6) [f(u)?g(v)]=(u v) ==0: (7) v For u v =0; u= v = 0, in which case, P = Q ==0. This means that each ofthe two terms in the bracket of(5) equals /0. The extension ofthis analysis to more than two terms is easy. We conclude, therefore, that R (u; v) S (v; w)==0 for every v V, which means that for every v V, either R (u; v) or S (v; w) (or both) is /0. Consequently, there is no v V such that R (u; v) =0 and S (v; w) =0. Dubois and Prade [3,4] give a formula for the composition of type relations under minimum tnorm as an extension ofthe type supmin composition. Their formula is the same as (5); however, we have demonstrated the validity of(5) for product as well as minimum tnorms. Example. Consider the type relation u is close to v onu V, where U = {u ;u } and V = {v ;v ;v 3 } are given as: U = {; }, V = {; 7; 3} and close (u; v) = u u ( v v v 3 0:9 0:4 0: 0: 0:4 0:9 ) : (8) Now consider another type fuzzy relation v is bigger than w onv W where W = {w ;w } = {4; 8}, with the following membership function: bigger (v; w) = w w v 0 0 v 0:6 0 : v 3 0:7 (9)
13 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) As is well known, the statement u is close to v and v is bigger than w indicates the composition ofthese two type relations, which can be found by using (4) and the min tnorm, as follows: close bigger (u i ;w j )=[ close (u i ;v ) bigger (v ;w j )] [ close (u i ;v ) bigger (v ;w j )] [ close (u i ;v 3 ) bigger (v 3 ;w j )] (0) where i =; and j =; ; 3. Using (0), we get close bigger (u; w) = u u ( w w 0:4 0: 0:9 0:7 ) : () Now, let us consider the type relations u is close to v and v is bigger than w which are obtained by adding some uncertainty to the type relations in (8) and (9). Let the membership grades be as follows: close ] (u; v) = u u v 0:3=0:8+=0:9 +0:7= 0:5=0+=0: v 0:7=0:3+=0:4 +0:=0:5 0:7=0:3+=0:4 +0:=0:5 v 3 0:5=0+=0: 0:3=0:8+=0:9 +0:7= () and bigger ] (v; w) = v v v 3 w =0+0:6=0: 0:4=0:5+=0:6 +0:9=0:7 0:7=0:9+= w =0+0:=0: =0+0:8=0: +0:=0: : 0:5=0:6+=0:7 +0:7=0:8 (3) The composition ofthe type relations u is close to v and v is bigger than w can be found by using (5), as follows: close ] bigger ] (u i;w j )=[ close ] (u i;v ) bigger ] (v ;w j )] [ close ] (u i;v ) bigger ] (v ;w j )] [ close ] (u i;v 3 ) bigger ] (v 3;w j )] (4) where i =; and j =;. Using (4), (), (3), () and (), we have (using min tnorm and max tconorm) u close ] bigger ] (u; w) = u w w 0:7=0:3+=0:4 0:3=0:8+=0:9 +0:7= 0:5=0+=0: +0:=0: 0:5=0:6+=0:7 +0:7=0:8 : (5) Comparing (5) and (), we observe that the results in (5) are indeed quite similar to the results of the type supstar composition in (), i.e., in the type results, primary memberships corresponding to the memberships in the type results, have unity secondary memberships. Eq. (5) provides additional uncertainty information to account for the secondary MFs.
14 340 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Next, consider the case where one ofthe relations involved in the composition is just a fuzzy set. The composition ofthe type set R U and type fuzzy relation S(U; V ) is given as [6] R S (v) = sup [ R (u)? S (u; v)]; (6) u U which is a type fuzzy set on V. Similarly, in the type case, the composition ofa type fuzzy set in R U and a type relation S(U; V ) is given by R S (v) = u U [ R (u) S (u; v)]: Eq. (7) plays an important role as the inference mechanism of a rule where antecedents or consequents are type fuzzy sets, and is the fundamental inference mechanism for rules in type FLSs (e.g., [0,,7]). 5.. Cartesian product If U and V are domains oftype fuzzy sets F and G, respectively, characterized by certain membership functions, then the Cartesian product of these membership functions will be supported over U V, and each point {(u; v); u U; v V } in this plane will have a membership grade (a crisp number in [0; ]) which is the result ofa tnorm operation between the membership grade ofu in F and the membership grade of v in G. Now consider the case oftype sets F and G. The membership grades ofevery element in F and G are now type fuzzy sets. So, now the tnorm operation in the type case has to be replaced by the meet operation; therefore, when we nd the Cartesian product, it is still supported over U V as in the type case; but now each point (u; v) U V will have a membership grade which is again a type fuzzy set (result ofa meet operation between membership grades of u and v). (7) 6. Conclusions We have discussed settheoretic operations for type sets, properties ofmembership grades oftype sets, and type relations and their compositions, and cartesian product under minimum and product tnorms. We have also developed easily implementable algorithms to perform settheoretic and algebraic operations oftype sets. When actual results are dicult to generalize or implement, we have developed practical approximations. All our results reduce to the correct type results when all the type uncertainties collapse to type uncertainties. All ofthe results in this paper are needed to implement type FLSs [,7], and should be ofuse to other researchers who explore applications of type fuzzy sets. Software to implement some of the algorithms developed herein are online at the URL: mendel=software. Appendix A. Proof of Theorem In the proofoftheorem, given next, we represent fuzzy sets F i as F i = v f i(v)=v (i =;:::;n), where v, in general, varies over the real line. Ifa real number w 0 is not in F i, f i (w 0 ) is zero. Proof. First we prove (5), and then we prove (6). (I) We rst prove (5) for the case n =. The join operation between F and F can be expressed, as F F = [f (v) f (w)]=(v w): (A.) v w
15 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Let us see what operations are involved here. For every pair ofpoints {v; w}, such that v F and w F, we nd the maximum of v and w and the minimum oftheir memberships, so that v w is an element of F F and f (v) f (w) is the corresponding membership grade. Ifmore than one {v; w} pair gives the same maximum (i.e., the same element in F F ), we use the maximum ofall the corresponding membership grades as the membership ofthis element. So, every element ofthe resulting set is obtained as a result of the max operation on one or more {v; w} pairs, and its membership is the maximum ofall the results ofthe min operation on memberships of v and w. If F F, the possible {v; w} pairs that can give as the result ofthe maximum operation are {v; } where v ( ;] and {; w} where w ( ;]. The process ofnding the membership of in F F can be broken into three steps: () nd the minima between the memberships ofall the pairs {v; } such that v ( ;] and then nd their supremum; () do the same with all the pairs {; w} such that w ( ;]; and, (3) nd the maximum ofthe two suprema, i.e., (F F )() = () () (A.) where and () = sup {f (v) f ()} = f () sup f (v) v ( ;] v ( ;] (A.3) () = sup {f () f (w)} = f () sup f (w): (A.4) w ( ;] w ( ;] We break into the following three ranges: v, v 6 v and v (see Fig. 6). Recall that f (v )= and f (v ) =. Also, observe that convexity of F i (i =; ) is equivalent to the condition that f i is monotonic nondecreasing in ( ;v i ] and monotonic nonincreasing in [v i ; ). = v : See Fig. 6(a). Since f and f both are monotonic nondecreasing in ( ;v ], sup v ( ;] f (v)=f () and sup w ( ;] f (w)=f (); therefore, from (A.) (A.4), we have (F F )() =f () f (); v : (A.5) v 6 = v : See Fig. 6(a). Recall that f (v )= and that f is monotonic nondecreasing in ( ;v ]. Therefore, sup v ( ;] f (v) = and sup w ( ;] f (w)=f (). Using these facts in (A.) (A.4), we have (F F )() =f () [f () f ()] = f (); v 6 v (A.6) where we have made use ofthe fact that a [a b]=a, for real a and b. = 3 v : For in this range [see Fig. 6(a)], both f and f have already attained their maximum values; therefore, sup v ( ;] f (v) = sup w ( ;] f (w) =. Consequently, from (A.) (A.4), we have (F F )() =f () f (); v : (A.7) From (A.5) (A.7), we get f () f (); v ; (F F )() = f (); v 6 v ; f () f (); v : (A.8) Observe that, since F and F both are convex and normal with f (v )=f (v )=: () f f is monotonic nondecreasing in ( ;v ]; () f (v ) f (v )=f (v ); (3) f is monotonic nondecreasing in [v ;v ]; and, (4) f f is monotonic nonincreasing in [v ; ). From these three observations, we see that (F F ) is
16 34 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Fig. 6. An example oftwo general membership functions, f and f, that satisfy the requirements of Theorem. Observe that for set F, any ofthe points at which f attains its maximum value ofunity may be chosen as v. We arbitrarily chose v =:8: (a) The three possibilities: v, v 6 v, 3 v. (b) Result ofthe join operation. (c) Result ofthe meet operation. The tnorm used is min. monotonic nondecreasing in ( ;v ] and monotonic nonincreasing in [v ; ), which implies that F F is convex [see Fig. 6(b)]. Also, F F is normal with (F F )(v )=. Next, we prove (5) for n. Using the associative law [3], we have F F F 3 =(F F ) F 3. Since, as explained in the previous paragraph, F F is also convex and normal, from (A.8) we have [recall that f 3 (v 3 ) = and that v 3 v ] (F F )() f 3 (); v ; (F F F 3)() = f 3 (); v 6 v 3 ; (A.9) (F F )() f 3 (); v 3 : Since v 6v 3, (A.8) and (A.9) can be rewritten as f () f (); v ; f (F F )() = (); v 6 v ; f () f (); v 6 v 3 ; f () f (); v 3 ; (F F )() f 3 (); v ; (F F )() f 3 (); v 6 v ; (F F F 3)() = f 3 (); v 6 v 3 ; (F F )() f 3 (); v 3 : (A.0) (A.)
17 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Substituting for (F F )() from (A.0) into (A.), we have f () f () f 3 (); v ; f (F F F 3)() = () f 3 (); v 6 v ; f 3 (); v 6 v 3 ; f () f () f 3 (); v 3 ; (A.) 3 i= f i(); v ; 3 = i=k+ f i(); v k 6 v k+ ; 6k6; 3 f i (); v 3 : i= (A.3) It is straightforward to show that F F F 3 is also a convex and normal set. Therefore, (A.8) can be applied again to obtain (F F F 3 F 4). Continuing in this fashion, we get (5). (II) For the case, n =, the meet operation between F and F can be represented as F F = v [f (v) f (w)]=(v w): (A.4) w The only dierence between this operation and the join operation in (A.) is that every F F is the result ofthe minimum operation on some pair {v; w}, such that v F and w F ; the possible such pairs being {v; } where v [; ) and {; w} where w [; ). The process ofnding the membership of in F F can be broken into three steps: () nd the minima between the memberships ofall the pairs {v; } such that v [; ) and then nd their supremum; () do the same with all the pairs {; w} such that w [; ); and, (3) nd the maximum ofthe two suprema. By proceeding in a manner very similar to that in part (I) ofthe proof, we get the result in (6). For details, see []. Appendix B. Solving for the Gaussian meet approximation Consider two Gaussian type sets F and F characterized by membership functions f and f, having means m and m and standard deviations and. Since we will perform the join and meet operations on the membership grades oftype sets when computing unions and intersections oftype sets, the type sets involved in the operations will have [0; ] as their domains. The membership functions of these type sets may, therefore, be clipped (i.e., any portion of the membership function lying outside [0; ] is cut o); and, this clipping may aect the result ofthe meet operation. This constraint needs to be considered in the derivation ofthe actual result ofthe meet between two Gaussian type sets under the product tnorm; however, we will ignore it for simplicity, while deriving the Gaussian approximation. For a detailed discussion about the eects ofthis constraint on the actual result, as well as on the Gaussian approximation and its upper and lower bounds, see []. Applying (9) for Gaussian membership functions, we have F F () = sup v R ;v 0 { [ (v exp ) ( ) ]} m =v m + ; R ; 0; (B.)
18 344 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) { F F (0) = exp ( m ) } { exp ( ) } m : (B.) Finding F F () for 0, requires minimization of ( ) ( ) ( ) ( ) v m =v m v m m v J (v) = + = + : (B.3) v Observe that J is nonconvex and dicult to minimize. Therefore, we simplify the problem by replacing the v in the denominator ofthe second term ofj (v) by a constant k. This gives us the following objective function to minimize: ( ) ( ) v m m v H(v) = + : (B.4) k Observe that H is convex (H == +m = 0). Therefore, equating H to zero and assuming that the inmum is obtained at v = v,weget ( v )( m ) + ( m v )( ) m k Substituting (B.5) into (B.4), we get inf v H(v) = ( k =0 v = m + m k m + k : (B.5) m m m + k ) : (B.6) Recall that the type sets involved in the meet operation have [0; ] as their domains. Since the constant k in (B.4) replaces v that varies between 0 and, it seems apparent that some value of k between 0 and will give a good approximation to the actual result ofthe meet operation. Observe, from (B.5), that replacing k by m, makes the Gaussian approximation commutative [i.e., ifwe interchange {m ; } and {m ; },we still get the same result] just as the actual meet operation is (see Section 4). Therefore, by replacing k with m, we obtain the Gaussian approximation. ( F F () exp m m m + m ) : (B.7) Generalization ofthis result to the meet of n Gaussian type sets is straightforward and gives the result in (0). For a detailed derivation ofbounds on the approximation error between (0) and the actual value of F F F n (), see []. Appendix C. Proof of Theorem 4 Consider F i =[m i s i ;m i + s i ]. Multiplying F i by a crisp constant i (== i ) yields (see ()) i F i = =( i v); v [m i s i ;m i + s i ]: (C.) v
19 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) Adding a crisp constant (==) to i F i yields i F i + = =( i v + ); v [m i s i ;m i + s i ]: (C.) v Substituting w = i v +, (C.) gives us i F i + = =w; w [ i m i + i s i ; i m i + + i s i ]: (C.3) w Since F i can be represented as [l i ;r i ], where l i = m i s i and r i = m i + s i, then [ n n ] n n n F i = m i s i ; m i + s i : (C.4) i= i= i= i= i= Using (C.3) and (C.4), we get the result in Theorem 4. Appendix D. Proof of Theorem 5 We follow a method similar to the one used in the proofoftheorem 4 in Appendix C. We prove the theorem in two parts: (a) we prove that i F i + is a Gaussian fuzzy number with mean i m i + and standard deviation i i ; and (b) we prove that n i= F i is a Gaussian fuzzy number with mean n i= m i and standard deviation, where n i= = i ifproduct tnorm is used; (D.) n i= i ifminimum tnorm is used: (a) Consider { F i = exp ( ) }/ v mi v: (D.) v i Multiplying F i by a constant i (== i ) and adding a constant (==) to i F i, yields [see ()] [ { i F i + = exp ( ) } ]/ v mi? ( i v + ) v i { = exp ( ) }/ v mi ( i v + ): (D.3) v i Let i v + = v ; this gives v =(v )= i, which when substituted into (D.3), leads to { i F i + = exp [ (v ] }/ )= i m i v v i { = exp [ v ] }/ ( i m i + ) v i i v (D.4)
20 346 N.N. Karnik, J.M. Mendel / Fuzzy Sets and Systems (00) which shows that i F i + is a Gaussian fuzzy number with mean i m i + and standard deviation i i. Note that this result does not depend on the kind oftnorm used, since i and are crisp numbers. (b) Consider F and F, with means m and m and standard deviations and, respectively. The sum ofthese two fuzzy numbers can be expressed as [see ()] { F + F = ( ) } { v m? exp ( ) }/ w m [v + w] (D.5) v F w F exp where? indicates the chosen tnorm. (i) Product tnorm: In this case, (D.5) reduces to { F + F = ( ) } { v m exp ( ) }/ w m [v + w]: (D.6) v F w F exp If is an element of F + F, the membership grade of in F + F can be obtained by considering all the {v; w} pairs such that v F and w F and v + w =, multiplying the memberships of v and w in every pair, and, choosing the maximum ofall these membership products. In other words, { [ (v (F+F )() = sup exp ) ( ) ]} m ( v) m + : (D.7) v Let v maximize the expression on the RHS of(d.7). v is obtained by minimizing ( ) [ ] v m ( v) m J (v) = + : (D.8) Since J is convex (J == += 0), equating the rst derivative of J to zero, we get ( v )( ) [ m ( v ) ) m + ]( =0; ( v + ) = m + m ; v = m +( m ) + : Substituting (D.9) into (D.7), we get [ ] (F+F )() = exp (m + m ) + : (D.9) (D.0) This result generalizes easily to the sum of n Gaussian type sets, F ;:::;F n ; hence, n i= F i, under product tnorm, is a Gaussian fuzzy number with mean n i= m n i and standard deviation i= i. (ii) Minimum tnorm: In this case, (D.5) reduces to { F + F = ( ) } { v m exp ( ) }/ w m [v + w]: (D.) v F w F exp
The Algebra of Truth Values of Type2 Fuzzy Sets: A Survey
Noname manuscript No. (will be inserted by the editor) The Algebra of Truth Values of Type2 Fuzzy Sets: A Survey Carol L. Walker, Elbert A. Walker New Mexico State University hardy@nmsu.edu, elbert@nmsu.edu
More informationOn closedform solutions of a resource allocation problem in parallel funding of R&D projects
Operations Research Letters 27 (2000) 229 234 www.elsevier.com/locate/dsw On closedform solutions of a resource allocation problem in parallel funding of R&D proects Ulku Gurler, Mustafa. C. Pnar, Mohamed
More informationA New Operation on Triangular Fuzzy Number for. Solving Fuzzy Linear Programming Problem
Applied Mathematical Sciences, Vol. 6, 2012, no. 11, 525 532 A New Operation on Triangular Fuzzy Number for Solving Fuzzy Linear Programming Problem A. Nagoor Gani PG and Research Department of Mathematics
More informationFuzzy regression model with fuzzy input and output data for manpower forecasting
Fuzzy Sets and Systems 9 (200) 205 23 www.elsevier.com/locate/fss Fuzzy regression model with fuzzy input and output data for manpower forecasting Hong Tau Lee, Sheu Hua Chen Department of Industrial Engineering
More informationOptimization of Fuzzy Inventory Models under Fuzzy Demand and Fuzzy Lead Time
Tamsui Oxford Journal of Management Sciences, Vol. 0, No. (6) Optimization of Fuzzy Inventory Models under Fuzzy Demand and Fuzzy Lead Time ChihHsun Hsieh (Received September 9, 00; Revised October,
More informationProject Management Efficiency A Fuzzy Logic Approach
Project Management Efficiency A Fuzzy Logic Approach Vinay Kumar Nassa, Sri Krishan Yadav Abstract Fuzzy logic is a relatively new technique for solving engineering control problems. This technique can
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationPortfolio selection based on upper and lower exponential possibility distributions
European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department
More informationSubspaces of R n LECTURE 7. 1. Subspaces
LECTURE 7 Subspaces of R n Subspaces Definition 7 A subset W of R n is said to be closed under vector addition if for all u, v W, u + v is also in W If rv is in W for all vectors v W and all scalars r
More informationThe Characteristic Polynomial
Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem
More informationA FUZZY LOGIC APPROACH FOR SALES FORECASTING
A FUZZY LOGIC APPROACH FOR SALES FORECASTING ABSTRACT Sales forecasting proved to be very important in marketing where managers need to learn from historical data. Many methods have become available for
More informationApplications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The settheoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
More informationOptimization under fuzzy ifthen rules
Optimization under fuzzy ifthen rules Christer Carlsson christer.carlsson@abo.fi Robert Fullér rfuller@abo.fi Abstract The aim of this paper is to introduce a novel statement of fuzzy mathematical programming
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationNotes from February 11
Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationIncorporating Evidence in Bayesian networks with the Select Operator
Incorporating Evidence in Bayesian networks with the Select Operator C.J. Butz and F. Fang Department of Computer Science, University of Regina Regina, Saskatchewan, Canada SAS 0A2 {butz, fang11fa}@cs.uregina.ca
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationA set is a Many that allows itself to be thought of as a One. (Georg Cantor)
Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains
More informationWhy Product of Probabilities (Masses) for Independent Events? A Remark
Why Product of Probabilities (Masses) for Independent Events? A Remark Vladik Kreinovich 1 and Scott Ferson 2 1 Department of Computer Science University of Texas at El Paso El Paso, TX 79968, USA, vladik@cs.utep.edu
More informationFUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM
International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 3448 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT
More informationOPRE 6201 : 2. Simplex Method
OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationSummary. Operations on Fuzzy Sets. Zadeh s Definitions. Zadeh s Operations TNorms SNorms. Properties of Fuzzy Sets Fuzzy Measures
Summary Operations on Fuzzy Sets Zadeh s Operations TNorms SNorms Adriano Cruz 00 NCE e IM/UFRJ adriano@nce.ufrj.br Properties of Fuzzy Sets Fuzzy Measures One should not increase, beyond what is necessary,
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
More informationMaximum likelihood estimation of mean reverting processes
Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For
More informationTypes of Degrees in Bipolar Fuzzy Graphs
pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 48574866 HIKRI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department
More informationOn Development of Fuzzy Relational Database Applications
On Development of Fuzzy Relational Database Applications Srdjan Skrbic Faculty of Science Trg Dositeja Obradovica 3 21000 Novi Sad Serbia shkrba@uns.ns.ac.yu Aleksandar Takači Faculty of Technology Bulevar
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationInduction Problems. Tom Davis November 7, 2005
Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationFuzzy Implication Rules. Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey
Fuzzy Implication Rules Adnan Yazıcı Dept. of Computer Engineering, Middle East Technical University Ankara/Turkey Fuzzy IfThen Rules Remember: The difference between the semantics of fuzzy mapping rules
More informationFuzzy Decision Making in Business Intelligence
Master Thesis Mathematical Modeling and Simulation Thesis no:20096 October 2009 Fuzzy Decision Making in Business Intelligence Application of fuzzy models in retrieval of optimal decision Asif Ali, Muhammad
More informationFuzzy Numbers in the Credit Rating of Enterprise Financial Condition
C Review of Quantitative Finance and Accounting, 17: 351 360, 2001 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. Fuzzy Numbers in the Credit Rating of Enterprise Financial Condition
More informationA Direct Numerical Method for Observability Analysis
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method
More informationFuzzy sets I. Prof. Dr. Jaroslav Ramík
Fuzzy sets I Prof. Dr. Jaroslav Ramík Fuzzy sets I Content Basic definitions Examples Operations with fuzzy sets (FS) tnorms and tconorms Aggregation operators Extended operations with FS Fuzzy numbers:
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationLinguistic Preference Modeling: Foundation Models and New Trends. Extended Abstract
Linguistic Preference Modeling: Foundation Models and New Trends F. Herrera, E. HerreraViedma Dept. of Computer Science and Artificial Intelligence University of Granada, 18071  Granada, Spain email:
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationInfinitely Repeated Games with Discounting Ù
Infinitely Repeated Games with Discounting Page 1 Infinitely Repeated Games with Discounting Ù Introduction 1 Discounting the future 2 Interpreting the discount factor 3 The average discounted payoff 4
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationThis article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.
IEEE/ACM TRANSACTIONS ON NETWORKING 1 A Greedy Link Scheduler for Wireless Networks With Gaussian MultipleAccess and Broadcast Channels Arun Sridharan, Student Member, IEEE, C Emre Koksal, Member, IEEE,
More informationDomain representations of topological spaces
Theoretical Computer Science 247 (2000) 229 255 www.elsevier.com/locate/tcs Domain representations of topological spaces Jens Blanck University of Gavle, SE801 76 Gavle, Sweden Received October 1997;
More informationThe Relation between Two Present Value Formulae
James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 6174. The Relation between Two Present Value Formulae James E. Ciecka,
More informationRegression Using Support Vector Machines: Basic Foundations
Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationUsing Generalized Forecasts for Online Currency Conversion
Using Generalized Forecasts for Online Currency Conversion Kazuo Iwama and Kouki Yonezawa School of Informatics Kyoto University Kyoto 6068501, Japan {iwama,yonezawa}@kuis.kyotou.ac.jp Abstract. ElYaniv
More informationExtracting Fuzzy Rules from Data for Function Approximation and Pattern Classification
Extracting Fuzzy Rules from Data for Function Approximation and Pattern Classification Chapter 9 in Fuzzy Information Engineering: A Guided Tour of Applications, ed. D. Dubois, H. Prade, and R. Yager,
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationOn the Algebraic Structures of Soft Sets in Logic
Applied Mathematical Sciences, Vol. 8, 2014, no. 38, 18731881 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/ams.2014.43127 On the Algebraic Structures of Soft Sets in Logic Burak Kurt Department
More informationconstraint. Let us penalize ourselves for making the constraint too big. We end up with a
Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the
More informationForecasting of Economic Quantities using Fuzzy Autoregressive Model and Fuzzy Neural Network
Forecasting of Economic Quantities using Fuzzy Autoregressive Model and Fuzzy Neural Network Dušan Marček 1 Abstract Most models for the time series of stock prices have centered on autoregressive (AR)
More informationOn the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 10361040, 2012 ISSN 15493636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
More informationA DECISION SUPPORT SYSTEM FOR MATERIAL AND MANUFACTURING PROCESS
A DECISION SUPPORT SYSTEM FOR MATERIAL AND MANUFACTURING PROCESS SELECTION 1 Ronald E. Giachetti Manufacturing Systems Integration Division National Institute of Standards and Technology Bldg. 304 Rm.
More informationTopologybased network security
Topologybased network security Tiit Pikma Supervised by Vitaly Skachek Research Seminar in Cryptography University of Tartu, Spring 2013 1 Introduction In both wired and wireless networks, there is the
More informationBy choosing to view this document, you agree to all provisions of the copyright laws protecting it.
This material is posted here with permission of the IEEE Such permission of the IEEE does not in any way imply IEEE endorsement of any of Helsinki University of Technology's products or services Internal
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationThe Conference Call Search Problem in Wireless Networks
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationDegrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationFuzzy Continuous Resource Allocation Mechanisms in Workflow Management Systems
Fuzzy Continuous Resource Allocation Mechanisms in Workflow Management Systems Joslaine Cristina Jeske, Stéphane Julia, Robert Valette Faculdade de Computação, Universidade Federal de Uberlândia, Campus
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationFuzzy Logic based Preprocessing for Fuzzy Association Rule Mining
Fuzzy Logic based Preprocessing for Fuzzy Association Rule Mining by Ashish Mangalampalli, Vikram Pudi Report No: IIIT/TR/2008/127 Centre for Data Engineering International Institute of Information Technology
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationOn nding the generalized inverse matrix for the product of matrices
PU.M.A. Vol. 6 2005, No. 3, pp. 997 On nding the generalized inverse matrix for the product of matrices X. M. Ren Department of Mathematics, Xi'an University of Architecture Technology, Xi'an, 70055, P.R.China.
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationAN APPLICATION OF INTERVALVALUED INTUITIONISTIC FUZZY SETS FOR MEDICAL DIAGNOSIS OF HEADACHE. Received January 2010; revised May 2010
International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN 13494198 Volume 7, Number 5(B), May 2011 pp. 2755 2762 AN APPLICATION OF INTERVALVALUED INTUITIONISTIC
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationThe Collapsing Method of Defuzzification for Discretised Interval Type2 Fuzzy Sets
The Collapsing Method of Defuzzification for Discretised Interval Tpe Fuzz Sets Sarah Greenfield, Francisco Chiclana, Robert John, Simon Coupland Centre for Computational Intelligence De Montfort niversit
More informationSEARCHING AND KNOWLEDGE REPRESENTATION. Angel Garrido
Acta Universitatis Apulensis ISSN: 15825329 No. 30/2012 pp. 147152 SEARCHING AND KNOWLEDGE REPRESENTATION Angel Garrido ABSTRACT. The procedures of searching of solutions of problems, in Artificial Intelligence
More informationFactoring Patterns in the Gaussian Plane
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
More informationAlternative proof for claim in [1]
Alternative proof for claim in [1] Ritesh Kolte and Ayfer Özgür Aydin The problem addressed in [1] is described in Section 1 and the solution is given in Section. In the proof in [1], it seems that obtaining
More informationProblems often have a certain amount of uncertainty, possibly due to: Incompleteness of information about the environment,
Uncertainty Problems often have a certain amount of uncertainty, possibly due to: Incompleteness of information about the environment, E.g., loss of sensory information such as vision Incorrectness in
More informationOn productsum of triangular fuzzy numbers
On productsum of triangular fuzzy numbers Robert Fullér rfuller@abo.fi Abstract We study the problem: if ã i, i N are fuzzy numbers of triangular form, then what is the membership function of the infinite
More informationCardinality of Fuzzy Sets: An Overview
Cardinality of Fuzzy Sets: An Overview Mamoni Dhar Asst. Professor, Department Of Mathematics Science College, Kokrajhar783370, Assam, India mamonidhar@rediffmail.com, mamonidhar@gmail.com Abstract In
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationInterval Neutrosophic Sets and Logic: Theory and Applications in Computing
Haibin Wang Florentin Smarandache YanQing Zhang Rajshekhar Sunderraman Interval Neutrosophic Sets and Logic: Theory and Applications in Computing Neutrosophication truthmembership function Input indeterminacy
More informationTwostep competition process leads to quasi powerlaw income distributions Application to scientic publication and citation distributions
Physica A 298 (21) 53 536 www.elsevier.com/locate/physa Twostep competition process leads to quasi powerlaw income distributions Application to scientic publication and citation distributions Anthony
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationA Novel Binary Particle Swarm Optimization
Proceedings of the 5th Mediterranean Conference on T33 A Novel Binary Particle Swarm Optimization Motaba Ahmadieh Khanesar, Member, IEEE, Mohammad Teshnehlab and Mahdi Aliyari Shoorehdeli K. N. Toosi
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationLoad balancing of temporary tasks in the l p norm
Load balancing of temporary tasks in the l p norm Yossi Azar a,1, Amir Epstein a,2, Leah Epstein b,3 a School of Computer Science, Tel Aviv University, Tel Aviv, Israel. b School of Computer Science, The
More informationDuality in General Programs. Ryan Tibshirani Convex Optimization 10725/36725
Duality in General Programs Ryan Tibshirani Convex Optimization 10725/36725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T
More informationAPPLICATION SCHEMES OF POSSIBILITY THEORYBASED DEFAULTS
APPLICATION SCHEMES OF POSSIBILITY THEORYBASED DEFAULTS ChurnJung Liau Institute of Information Science, Academia Sinica, Taipei 115, Taiwan email : liaucj@iis.sinica.edu.tw Abstract In this paper,
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationOpen and Closed Sets
Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationPossibilistic programming in production planning of assembletoorder environments
Fuzzy Sets and Systems 119 (2001) 59 70 www.elsevier.com/locate/fss Possibilistic programming in production planning of assembletoorder environments HsiMei Hsu, WenPai Wang Department of Industrial
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More information