Fuzzy number division and the multi-granularity

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 65, No. 4, 2017 DOI: 10.1515/bpasts-2017-0055

Fuzzy number division and the multi-granularity phenomenon A. PIEGAT* and M. PLUCIŃSKI Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, 49 Żołnierska St., 71-210 Szczecin, Poland

Abstract. The paper presents difficulties connected with fuzzy and interval division. If operations such as fuzzy addition, subtraction and multiplication provide as a result one compact, multidimensional granule, then a result of the fuzzy division can consists of few separated granules. Such results are more difficult to use in next calculations. The paper shows that the number of solution granules can be higher than 2 and that in certain problems division does not occur explicitly. In certain problems, separation of particular solution granules can be considerable. The paper also shows how to realize the fuzzy division when its denominator contains zero. Most types of fuzzy arithmetics forbid such operation. However, the paper shows that it is possible. Multidimensional fuzzy RDM arithmetic and horizontal membership functions which facilitate detecting of solution granules are also described. The considered problems are visualized by examples. Key words: granular computing, fuzzy arithmetic, fuzzy calculations, RDM fuzzy arithmetic, horizontal membership function.

1. Introduction Fuzzy arithmetic (FA) [1–7] is an extension of the interval arithmetic (IA) [4, 8]. It extends a calculation domain from standard intervals to fuzzy intervals or fuzzy numbers. Both kinds of arithmetic are very important for the uncertainty theory [9], granular computing [4], grey systems [10] and computing with words [11, 12]. IA and FA are necessary for solving linear and nonlinear systems of equations with uncertain coefficients [4]. Such systems describe real economic, engineering, medical, environmental protection (and many other) problems [13–16]. FA is a basis for intuitionistic fuzzy arithmetic [17, 18]. It is also applied in rough-set problems to increase the solution quality [19]. FA meets with considerable interest of scientists and has been developed for many years. Numerous FA methods have been elaborated, e.g. L-R fuzzy arithmetic [20, 21], FA based on discretized fuzzy numbers and on the extension principle of Zadeh [22], FA using decomposed fuzzy membership functions (standard IA with α-cuts) [22], advanced FA based on transformation method [23], constrained FA [22] etc. An overview of FA methods can be found in [3]. Some of FA methods allow to find a solution of simpler problems analytically, but more complicated ones must be solved numerically. All the time, we observe the emergence of new FA methods, so the question arises: what is the reason of such situation? Are existing methods not sufficiently effective? The motivation of this paper is to present a phenomenon which can occur during the division of fuzzy numbers, intervals and other granular models of uncertain data. This phenomenon consists in a multi-granular division result and it does not occur in a division of conventional crisp numbers. The multi-granu*e-mail: [email protected] Manuscript submitted 2016-01-26, revised 2016-07-08, 2016-10-26 and 2016-12-30, initially accepted for publication 2017-01-20, published in August 2017.

larity phenomenon occurs always when the uncertain divisor contains zero. If the divisor does not contain zero, then the division result always consists of a single granule. Examination of the problem has shown that if the multi-granularity occurs then the result usually consists of 2 granules, which is intuitively understandable. However, the division result can sometimes consist of more granules, e.g. of 4. The next surprising observation is that the distance between component granules, which intuitively should be infinitely small, can sometimes be quite considerable. The multi-granularity of division result has been observed thanks to the application of a new type of fuzzy arithmetic that was called multidimensional fuzzy RDM arithmetic (MD-F-RDM arithmetic). In papers on other existing types of fuzzy arithmetic, the multi-granularity phenomenon has not been investigated because of a simple reason: the division by the fuzzy divisor containing zero is not allowed. Moreover, mathematical properties of these arithmetic types do not allow for such analysis. MD-F-RDM, which is shortly presented in Section 2, allows for this kind of division. The paper is organized as follows: Section 1 describes new, horizontal membership functions (MFs) that are the basis of MD-F-RDM arithmetic. Other fuzzy arithmetic types use conventional, vertical MFs. Horizontal MFs considerably facilitate fuzzy calculations. A characteristic feature of MD-F-RDM arithmetic is that its calculation result is not a typical fuzzy set, but a multidimensional fuzzy solution set (MDFS-set). Such a set has properties of a complete, algebraic solution set and gives MD-F-RDM arithmetic properties which other fuzzy arithmetic types do not have. These properties are mentioned in Section 3. An additional property that has not been described in this section is the restoration property, owing to which MD-F-RDM arithmetic is able to satisfy the forward-backward calculation test. In Section 4, various examples of multi-granularity have been described. Among them,

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Bull. Pol. Ac.: Tech. 65(4) 2017 Unauthenticated Download Date | 8/25/17 12:47 PM

A. Piegat and M. Pluciński

examples of multi-granularity greater than 2. Observation of the multi-granularity was possible thanks to MD-F-RDM arithmetic, which uses in calculations not only borders of intervals but also their insides. In Section 4, a real problem in which the multi-granularity can occur is also presented. It is the A. known Piegat, M. Pluci´nski A. Piegat, M. Pluci´nski system-balance problem of Leontief in which an economic country-system is described by a set of linear equation. Since fuzzy solution set can be multi-granular. Piegat, M. Pluci´nski at solution least a part system parameters is uncertain, its A. determifuzzy setofcan be multi-granular. Further on, a short description of the best known and frenant is on, also auncertain and can contain theand fuzzy Further short description of thezero. bestHence, known frePiegat, M. 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Their purpose isapproximate to approximate and [bways bway. then: onthe the set of real numbers R. Their purpose isto approximate Fuzzy arithmetic deals with fuzzy numbers (FN) real, and fuzzy theset set offuzzy . Their purpose is common defining the extended operations are based on osed interval denoted by [a,Ab] is setUncertain of real numbers given 1 ,given 2 ]of Given a[a, set defined on and ato real µof∈ the set of real numbers R. Their purpose isa to closed interval denoted by b]number isbe the set real values can modelled in numbers aonvarious AFig. very b] = ∈ Rthe :∈ ≤ xR ≤ (1) 1. Trapezoidal function b]b][a, =set {x R :{x areal ≤ xnumbers ≤approximate b}b} (1) [0, the crisp A∈µ{x =values R :variables. A(x) ≥ µ }[a,is called µ -cut by1], (1). ations closed intervals [8]. If two intervals [aintervals , 2a]2,] ations on closed we have [a1 , a(4) 1b atwo ⊕intervals [bwe bhave ][aIf =membership ,two athe [a intervals (FI) [24]. 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Their purpose is toµ Fuzzy arithmetic deals with fuzzy numbers (FN) and fuzzy Given a fuzzy set A defined on and number μ 2 [0, 1], principle set theory [20]. Given fuzzy set A defined on aa real real number 1 ,approximate Given aof fuzzy set A defined on R and a≤ real number ∈real Given aa(FI) fuzzy set Areal defined on Rand and ∈the Fuzzy arithmetic deals fuzzy numbers (FN) and [a, b] =set {xwith ∈ R := a{x xR ≤ closed interval by [a, b]µa=denoted {x ∈ number Rfuzzy : (1) a[a,≤2b]xµis≤ b} set of (1) A. The crisp set S(A) ∈ R :b} A(x) > 0} is called the Fig. Trapezoidal membership , aFig. ]1.⊗on [bclosed [ min(a ⊗function b1we , afunction btwo ⊗ bopera[areal 1ations 2numbers 1 , bgiven 2] = 1is 1 ⊗arithmetic 2 , a2intervals 1 , a2 ⊗ b2 ), intervals [8]. ] 1. Trapezoidal membership support of A. When max A(x) = 1, A is called a normal intervals (FI) [24]. FN and FI are special fuzzy sets defined on the set of real numbers R. 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If we have two intervals [a , a ] and [b , b ] then: 2oper1 intervals 2 Fig. 1.are on the set of real Their purpose is to Fuzzy arithmetic deals with fuzzy numbers (FN)on and fuzzy crisp set S(A) =  x 2  :  A(x) > 0 is on called the of A. µtions fuzzy defined1 ⊗ in membership terms real, not precisely values of Given a∈S(A) fuzzy set A defined R and asupport real number b1, a1of ⊗arithmetic b2 , afunction max(a ∈ of A. The crisp set = {x R :variables. > 0} is the Fuzzy arithmetic deals fuzzy numbers (FN) and fuzzy {with }called fuzzy set. The crisp set S(A) =numbers {xknown, R : R. A(x) >∈ 0} is:A(x) the 21⊗ b 1 , a2 ⊗ b2 )] , Trapezoidal of A. The crisp set S(A) = {x∈purpose R A(x) >approximate 0}b]called is= [a, {x R : and a ]≤ x[2min(a ≤ b} (1) ⊗ nA. the set ofdeals real numbers R. Their is tofuzzy approximate Fig. 1.,baTrapezoidal function ,⊗ atwo ,membership a1b2,2a⊗ b1, a , a2]function ⊗ b2 )] , max(a intervals (FI) [24]. 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FN and FI are fuzzy sets µspecial x∈R x 2  1a,2a 2⊗]b2(4) x∈R A fuzzy interval A is a fuzzy set on whose a ] ⊗ [b , b ] = [ min(a ⊗ b , a ⊗ b , a ⊗ b , ), (5) , a ] ⊕ [b , b ] = [a ⊕ b , a ⊕ b ] , [a 1 2 1 2 1 1 1 2 2 1 1 2 1 2 1 1 2 2 ations on closed intervals [8]. If we have two intervals [a support of A. When max A(x) = 1, A is called a normal A. Piegat, M. Pluci´ n ski 1 2 Fig. 1. Trapezoidal membership function al, not precisely known, values of variables. x∈R on the set of real numbers R. Their purpose is to approximate and [b , b ] then: A fuzzy interval A is a normal fuzzy set on R whose µ (5) 1tomax(a 2approximate ⊗ b , a ⊗ b , a ⊗ b , b )] , zythe set. ⊗ b , a ⊗ , a ⊗ b , a ⊗ b )] , max(a on the set of real numbers R. Their purpose is Fuzzy arithmetic deals with fuzzy numbers (FN) and fuzzy A fuzzy interval A is a normal fuzzy set on whose μ-cuts and [b , b ] then: Given a fuzzy set A defined on R and a real number µ ∈ [0, 1], the crisp set A = {x ∈ R : A(x) ≥ µ } is called -cut fuzzy set. of A. The crisp set S(A) = {x ∈ R : A(x) > 0} is called the 1 1 1 2 2 1 2 2 [24]. FN and FI are special fuzzy sets defined 1 1 1 2 1 2 2 µ 1 2 nFI) set of real numbers R. Their purpose is to approximate , a2 ]closed ⊕ [b1, bintervals = [a1 ⊕[8]. b⊗ aIf b2⊗ ]have , bTrapezoidal (4), membership [a1on cuts for all µ ∈ (0, 1] are closed intervals of real numbers and 2 ] max(a 1, b 2 ,⊕ Fig. 1. fun and [b , b ] then: a , a ⊗ b a ⊗ b )] fuzzy set. 1 2 ations we two intervals [a , a ] ⊗ [b , b ] = [ min(a ⊗ b , a ⊗ b , a ⊗ , a ⊗ b ), [a Given a fuzzy set A defined on R and a real number µ ∈ 1 1 1 2 2 1 2 2 ations on closed intervals [8]. 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Their purpose is to approximate ]intervals ⊕ {+, [bdefined =⊗ [a1If ⊕{×, b1 , have a÷} [a1 , a⊕2sets al, not precisely known, of variables. solution set can be whose support is A(x) 1 for one 1 , b−}, 2 ] [8]. 2 ⊕ and 2 ] , 0intervals , b ] where: ∈ ∈ ∈ / [b , a2 ]intervals ⊗For [b ,fuzzy b12,approximate ]b[8]. min(a boperations ,ba12, a⊗1 b⊗1,b[a a2defined A fuzzy interval Aintervals a{x fuzzy set on whose µµclosed -[a and b1and ]atwo then: then: 1real 1[b 1⊗ 2, a⊗ 2]), s, 1], for all µ ∈ (0,support 1] are closed real numbers and the set = {x ∈is R :normal A(x) ≥ µ }of called -cut 1 ,b 2, [a 2 ] [⊕ (4) (5) b21,by a2 ⊗ b2and )] , (5) max(a on of real numbers R. Their purpose isthen: to support isAfuzzy bounded. A(x) = 1 for exactly one x 2  then Given aknown fuzzy set A defined on R, ∈ and a∈ number support of A. When max A(x) =on 1,set isisreal called normal fuzzy set. of A.crisp The crisp set S(A) = ∈of R : the A(x) > 0} called the µ avalues set Athe defined RA and ais real number cuts for all µ ∈ (0, 1] are closed intervals whose isdefined bounded. When = 1numbers foraRµations exactly one on If intervals ,=2a1b⊗ 1⊗ x∈R 1⊗ ,µ−}, a= , b11[b,⊗ = b2[b ]1,,(4) (4) [a 2÷. and [b ,⊕ band ]{×, 1÷} 2∈] intervals 2have 1B 1 , a, 2b ⊕ ecisely of variables. b]A if ⊗ = ÷. ⊕ {+, −}, ∈ and /1we [b ther on, aGiven short description ofWhen and fre2For Given fuzzy set A(0, onbest Rinterval and aA(x) real number 2,] a µ and ∈number. x ∈aknown, R, then this special fuzzy is called a where: fuzzy ],intervals ifb22⊗ where: ∈ {+, ⊗0[b ∈∈ {×, ÷} and 0b⊕ ∈ /operations fuzzy intervals A and B defined by (4) and (5) 1 2 , a ] ⊗ [b , b ] = [ min(a ⊗ b ⊗ , a ⊗ b a ⊗ b ), [a ations on closed [8]. If we have two ose support is bounded. When A(x) = 1 for exactly one 1 2 1 2 1 1 1 2 2 1 2 cuts for all µ ∈ 1] are closed intervals of real numbers and ⊗ b , a ⊗ b , a ⊗ b , a ⊗ b )] , max(a this special fuzzy interval is called a fuzzy number. A convereal, not precisely known, values of variables. A. The crisp set S(A) = {x ∈ R : A(x) > 0} is called the support of A. When max [0, 1], the crisp set A A(x) = 1, A is called a normal = {x ∈ R : A(x) ≥ µ } is called µ -cut fuzzy set. whose support is bounded. When A(x) = 1 for exactly one A fuzzy interval A is a normal fuzzy set on R whose µ (5) are realised for each µ -cut, µ ∈ [0, 1], which is the interval and 1 1 1 2 2 1 2 2 µ [0, 1], the crisp set A x∈R = {x ∈ R : A(x) ≥ µ } is called µ -cut x ∈ R, then this special fuzzy interval is called a fuzzy number. and [b , b ] then: ,] adefined ](5) ⊕ 1[b⊕ ,∈ ], = [a ⊕2if]b,1⊗, a= b2] , (4) For fuzzy A and intervals B⊕ operations by{×, and 1is intervals 2 Forwhere: 1by 12 b 2 ⊕÷. ly used standard fuzzy arithmetic [24] based onµnumber. standard a, 2bB]⊗µ ⊕ [b ,,[a baµ⊕ =2[0, [a bb/21(4) a1,2and ⊕ [amin(a fuzzy set A defined on R∈µand a A(x) real number µcalled ∈interval , b ] ∈ {+, −}, ∈ and 0 [b fuzzy A and operations (5) 1 ,defined 1(4) 21÷} ,R, 1], the crisp set A = {x R : ≥ } is µ -cut A convenient way of expressing any fuzzy A the are realised for each -cut, ∈ 1], which is the interval and µ then this special fuzzy interval is called a fuzzy , a ] ⊗ [b , b ] = [ ⊗ b ⊗ b , a ⊗ b a ⊗ b ), [a and [b , b ] then: , a ] ⊕ [b ] = [a b , a ⊕ b ] , (4) [a 11 1R 2µcan 1 denoted 2[0, 1 bµ ⊗is , ainterval a2and b12,µa2), )] ,(5) 2 max(a nient way of expressing any fuzzy interval A is the canonical Given aset fuzzy set Anumber. defined on and real µ11∈ fuzzy set. of A. crisp S(A) = ∈realised Rcalled : -isA(x) >and 0} isa called the whose support is bounded. When =on for exactly one AA. fuzzy interval Aof is a(0, normal R whose µ R, then this special fuzzy interval is called a1fuzzy cuts for all µS(A) ∈expressing 1]The are closed intervals of real numbers 1 number 21], 21bµ 1µb 12 2⊗2[b( 2⊗ pport A. When max A(x) 1, Aany is:set called a {x normal 1), 11⊗ 2 ,and 22 be as [a( )] are for each -cut, µ ∈ which Axof∈convenient way fuzzy interval A the The crisp = ∈fuzzy RA(x) A(x) > 0} is the al [8] will be presented. are realised for each µ= -cut, µ,⊕ ∈2µthe 1], which the b12,)]. ]aand if ⊗(4) ÷. where: ⊕ ∈ {+, −}, ⊗ ∈a( {×, ÷} and 0 µ∈ /interval [bb( crisp set Acrisp = {x ∈S(A) R :Aset A(x) ≥R µ= }[0, is called µreal -cut For fuzzy intervals and operations defined by and (5) 1 ,(4) µ convenient way ofset expressing any fuzzy interval Aof ison the aA ][0, ⊗ [b ,⊕ b)] ]and = [is,min(a baµ ⊗ b,= b), , a2 A.arithmetic The =x∈R {x ∈ :{x A(x) > 0} isR called the canonical form: be denoted as ), a( µ [b( ), b( )]. 1[a( 1B 2b 1b⊗ 1b 2⊗ a≥), ]SFA ⊕ [b , b ] [a b , a ] , [a form: 1 ,can 2[a 1 2 1 1 2 2 support of A. When max 1], the crisp set A A(x) = 1, A is called a normal = {x ∈ R : A(x) µ } is called µ -cut A convenient way of expressing any fuzzy interval A is the whose support is bounded. When A(x) = 1 for exactly one A fuzzy interval is a normal fuzzy set whose µ cuts for all µ ∈ (0, 1] are closed intervals numbers and , a ] ⊗ [b , b ] = [ min(a ⊗ b a ⊗ , ⊗ a2 ,21,a⊕ ⊗ (5) µ x ∈ R, then this special fuzzy interval is called a fuzzy number. x∈R ⊗ b , a ⊗ b , a ⊗ b , a ⊗ b )] max(a can be denoted as [a( µ a( µ )] and [b( µ ), b( µ )]. 1 2 1 2 1 1 1 2 2 12), zzy set. , a ] ⊕ [b , b ] = [a bb12and , 1a2 ⊕ [a does not posses the inverse element of addition 1 1 1 2 2 1 2 support of A. When max A(x) = 1, A is called a normal canonical form: 1 2 1 2 certain values can be modelled in a various way. A very , a ] ⊗ [b , b ] = [ min(a ⊗ b , a ⊗ b , a ⊗ b , a ⊗ b [a x∈R , b ] if ⊗ = ÷. where: ⊕ ∈ {+, −}, ⊗ ∈ {×, ÷} and 0 ∈ / [b can be denoted as [a( For fuzzy intervals A B operations defined by (4) and (5) µ ), a( µ )] and [b( µ ), b( µ )].  onical form: 1 2 1 2 1 1 1 2 2 1 2 2 1 2 crisp set S(A) = {x ∈ R : A(x) > 0} is called the are realised for each µ -cut, µ ∈ [0, 1], which is the interval and pport of A. When max = 1, A is called a normal does not posses the inverse element addition a1 ⊗ b2 , a2 ⊗ and b1 , a max(a x∈R fuzzy set. offuzzy A.is The crisp set = {x R : posses A(x)SFA > 0} is called the whose support is[8]. When A(x) =on 1[a, for exactly canonical xform: ∈∈R, then this special isb) called aand fuzzy number. cuts all µ (0, 1]bounded. are closed intervals of real numbers 1 ⊗ b1 ,of SFA does not the inverse element of addition and A convenient way expressing any fuzzy interval Aone is ∈the  A fuzzy interval A is aof normal set R whose µ fA for ∈ , S(A)  ar isfor anmax interval Its definition asinterval  multiplication with properties (6). ⊗ b1b,11[b( ,of a2if ⊗ b= ,and a)]. ⊗ b1b)] ,1(5) a,,2a1⊗ max(a (5) [bSFA ]does [intervals min(a b{×, , aB÷} ⊗ b[a ⊗ [a fuzzy set. L (x) are realised for not the inverse µ), -cut, µ ∈ [0, 1], which is the interval and ,b ]ba ÷. where: ⊕ ∈=be {+, −},posses ⊗ ∈ and ∈ [b For A and operations defined (5) 1]⊗ 1by 2),µ 2 and 1 ,-a2 ] ⊗ 1 , bcan 2fuzzy 1⊗ 1each 1µ 21,µ 2/ 1b[b 2,a  ,0,aaelement ⊗ ]addition =b(4) [2min(a ⊗bb22)], a,2 A.form When A(x) = 1, isfuzzy called axxfollows: normal  denoted as [a( a( )] and ), b( 21 2µ 1⊗ ⊗ b a ⊗ , ⊗ b , a ⊗ b max(a x∈R zzy set.  support of A. When max A(x) = 1, A is called a normal f (x) for ∈ [a, b) , A convenient way of expressing any fuzzy interval A is the whose support is bounded. When A(x) = 1 for exactly one A fuzzy interval A is a normal fuzzy set on R whose µ x ∈ R, then this special fuzzy interval is called a fuzzy number. f (x) for x ∈ [a, b) , 1 1 2 2 1 2 2 multiplication with properties (6).  multiplication with properties (6). L x∈R L    canonical form: d interval by1] [a,are b] isclosed the of real given uts for alldenoted ∈ (0, intervals numbers fset fornumbers xof ∈fuzzy [a,∈b) , c] , on Rand are for each µand -cut, ∈(6). the interval and  multiplication with properties can be denoted as [a( Forrealised fuzzy intervals Anot Bµaoperations defined byµ⊗ (4) (5) µ[0, ),1], a(aµwhich )] andis,[b( ), b(and µ )].  Lis Aµ fuzzy interval Acuts a fuzzy normal  (5) 1(x) for xreal   max(a b+ ,of  SFA does posses and ⊗ b(6) fuzzy set. A R, convenient way expressing fuzzy interval A is the 1⊗ 1 ,(−X 1⊗ 2, ⊗ canonical form: x∈ then this special fuzzy called fuzzy number. for all,any µis ∈ (0, 1]a[b, are closed intervals ofµ -real ⊕ numbers and  b2b]2max(a if)]⊗ ÷.b(5) where: ∈ 0{+, −}, ∈ {×, ÷} 0 b(6) ∈ /1 X [baelement X )bthe =2 ,and 0inverse ·12,(1/X) = 1=1addition A fuzzy interval A is = aof normal set on Rset whose µwhose - are 1 , a1 ⊗ 2, a 1A(x) for x1∈interval [b, c]   (2) for x ∈ [b, c] , ).hose X + (−X ) = , X · (1/X ) = 1 realised for each SFA does not posses the inverse element of addition and µ -cut, µ ∈ [0, 1], which is the interval and , b ] if ⊗ where: ⊕ ∈ {+, −}, ⊗ ∈ {×, ÷} and 0 ∈ / [b can be denoted as [a( µ ), a( µ )] and [b( µ ), b( µ )]. support is bounded. When A(x) = 1 for exactly one 1 2 1 for x ∈ [b, c] , X + (−X ) = 0 , X · (1/X) = 1 (6) A(x) = (2) cuts for all µ ∈ (0, 1] are closed intervals of real numbers and  whose fxLfx∈ (x) for xxreal ∈bounded. [a, b) µ (2) A(x) XR + (−X )= 0properties X {×, · (1/X) =and 1 0∈ (6) convenient way of expressing any fuzzy interval AWhen isand the support is A(x) (2) =For 1 for exactly one A a (2) normal fuzzy set on whose µ -, B⊗ A is aform: normal fuzzy set on R whose µd] -, ,[a,  for ∈for (c, multiplication with (6). (5) fuzzy intervals A and operations defined by (4) and (5) R utsinterval forAcanonical all ∈ (0, 1] are closed intervals of numbers   A(x) == f for (c, d] , , b ] if ⊗ = ÷. where: ⊕ ∈ {+, −}, ∈ ÷} / [b  R [a, b] = {x ∈ R : a ≤ ≤ b} (1)  1 2 f (x) x ∈ b) ,  SFA does not posses the inverse element of addition and  multiplication with properties (6). can be denoted as [a( For fuzzy intervals A and B operations defined by µ ), a( µ )] and [b( µ ), b( µ )]. Lfor   x(c,∈ (c, d]1,1] ∈µR,∈canonical then special fuzzy interval is aµd] fuzzy number. The sub-distributivity law and cancellation law for addition(4 , b22] ]ifif ⊗ = ÷. where: ⊕law {+, ⊗ ∈ {×, ÷} 00 2 ∈ // [b [b11, b    whose is bounded. When A(x) for exactly onerealised  The sub-distributivity andnumbers cancellation for addition  fR∈fRR, forcalled x=∈special ,= where: © 2  +, ¡µ ,-cut,  2  £, ¥ and  = ¥. {−}, }law {µlaw } and form: xWhen this fuzzy interval is called a fuzzy number. cuts for all ∈and (0, areone closed intervals of∈For real and    (0, 1]this aresupport closed intervals real numbers are for each ∈ [0, 1], which is the interval and   hose support is bounded. A(x) 1 for exactly 1L0(x)ofthen for x ∈ [b, c] , The sub-distributivity law and cancellation law for addition otherwise,  fuzzy intervals A and B operations defined by (4) 0 otherwise, f for x ∈ [a, b) , The sub-distributivity and cancellation law for addition are realised for each SFA does not posses the inverse element of addition and µ -cut, µ ∈ [0, 1], which isand int where: ⊕ ∈ {+, −}, ⊗ ∈ {×, ÷} and 0the ∈ / (5) [b  multiplication with properties (6).  (−X = 0operations ·⊗ =by 1(FN) xis∈bounded. [b, c] ,the if = ÷. where: ⊕(2)∈For {+, −}, ⊗ ∈exactly {×, ÷} and ∈ /B  multiplication does hold in X SFA. Therefore, convenient of expressing any fuzzy interval A is and does not hold in SFA. Therefore, applicafuzzy intervals A+ and B0)operations defined by (4) andand (5) A(x) = ∈ this R, way then this special interval isafor called aand fuzzy zzy deals with fuzzy numbers and fuzzy For fuzzy intervals A and defined X + (−X) =[b 0,1applica,, b2µ]),X Xb( ·(1/X) (1/X) = 11 (4) (6)(6) 1 A convenient way of expressing anynumber. fuzzy interval A is the whose support When A(x) =not 1multiplication for one 0fuzzy otherwise,  0(x) otherwise, A(x) = (2) isx bounded. When A(x) = 1 for exactly one  can be denoted as [a( µ ), a( µ )] and [b( µ )]. ∈port R,arithmetic then special fuzzy interval is called fuzzy number. and multiplication does not hold in SFA. Therefore, applicaf and multiplication does not hold in SFA. Therefore, applicafor x ∈ [a, b) , Fig. 1. Trapezoidal membership function are realised for each  µ -cut, µ ∈ [0, 1], which is the interval and multiplication with properties (6). can be denoted as [a( For fuzzy intervals A and B operations defi µ ), a( µ )] and [b( µ ), b( µ )].  L1f R that for ∈defined (c, , interval for xany ∈x [b, c]and , d] tion possibilities of SFA   fuzzy intervals and B defined by (4) (5) ffunction for xspecial ∈is (c, d]fuzzy ,For ere x ∈where R, fconvenient increasing nonical form: tion possibilities are limited. FI are special fuzzy (5) areare realized for each μ-cut, is interval the interval are realised for each µoperations -cut, µ law ∈ [0, which the Xfuzzy + (−X) =of0 ,SFA Xμ 2 [0, 1], ·1], (1/X) =which 1 is and (6)forand RR,sets L isx aFN canonical form: xais ∈ then this is called aAlimited. number.   A[24]. way ofis expressing fuzzy Ainterval isand the ∈real-valued R,and fLexpressing is= afunction real-valued that increasing  A(x) (2)   The sub-distributivity and cancellation law addition nals this(FI) special fuzzy interval called fuzzy number.  –of tion possibilities ofposses SFAperfect are limited. The sub-distributivity law and cancellation law addition SFA does not the inverse element addition and convenient way of any fuzzy interval A is the are realised for [b( each SFA does not posses the inverse element of add µ -cut, µfor ∈ [0, 1], which  tion possibilities of SFA are limited. where ∈ R, f is a real-valued function that is increasing and – can be denoted as [a( µ ), a( µ )] and µ ), b( µ(6) )]. 1 for x ∈ [b, c] , Currently known methods are not and can be imwhere x ∈ R, f is a real-valued function that is increasing and L    ht-continuous, f is a real-valued function that is decreasing L f for x ∈ (c, d] , are realised for each µ -cut, µ ∈ [0, 1], which is the interval and R x 2  e set ofright-continuous, realwhere numbers R. Their purpose is to approximate , f is a real-valued function that is increasing and and can be denoted as [a (μ), a (μ)] and [b (μ), b (μ)]. 0 otherwise, X + (−X) = 0 , X · (1/X) = 1 R A convenient way of expressing any fuzzy interval A is the  Currently known methods are not perfect and can be im0 otherwise, can be denoted as [a( µ ), a( µ )] and [b( µ ), b( µ )]. L canonical form: f is a real-valued function that is decreasing – – A(x) = (2)  R and multiplication does not hold in SFA. Therefore, applicaf Currently known methods are not perfect and can be imand multiplication does not hold in SFA. Therefore, applica(x) for x ∈ [a, b) , The sub-distributivity law and cancellation law for addition ent way of expressing any fuzzy interval A is the  multiplication with properties (6). can be denoted as [a( µ ), a( µ )] and [b( ),and b(µ  f (x) for x ∈ [a, b) , L decreasing fRfR multiplication with properties right-continuous, abe function that is proved. multi-dimensional, relative distance measure nonical form: Currently known methods are and can beµ imLRis SFA does not the inverse element of addition ations onThe closed intervals [8]. If we haveposses two intervals [anot a2 ]perfect left-continuous. fL and left right borright-continuous, fcan a called real-valued that iscan decreasing for xand ∈ function (c, d] , that  1 ,relative right-continuous, fis is0real-valued a real-valued function is decreasing SFA does not posses the inverse element of addition addition and not precisely known, values of variables. R canonical form:   be denoted as [a( µ ), a( µ )] and [b( µ(6). ), b( µlimited. )]. otherwise, Rff proved. The multi-dimensional, distance measure  SFA does not posses the inverse element of and   and left-continuous. and f can be called left and right bor L R and multiplication does not hold in SFA. Therefore, applicaproved. The multi-dimensional, relative distance measure tion possibilities of SFA are The sub-distributivity law and cancellation law for addition  SFA does not posses the inverse elemen  tion possibilities of SFA are limited. fuzzy arithmetic (MD RDM-FA)The is anmulti-dimensional, alternative method(6). of relative distance measure orm: and left-continuous. fLfR where x 1∈ R, ffLsuch and can be≤number called and borisfffor function that isfor increasing and [b, c] , proved. fL is aon real-valued function is increasing and of A. a, b,and c,xset d∈ are real numbers that a∈ bcalled ≤called ≤µleft d.1 and [b ,xb2∈ ] then:  for x= ∈left [a, b) ,right multiplication with [b, c] ,c,that 1border LaR(x) and left-continuous. be left and right borleft-continuous. and fareal-valued can and right multiplication with properties (6). ven awhere fuzzy AR, defined and real ∈ (−X) = 01alternative , and X · (1/X ) = 1 of 0L otherwise, Land RR  SFA does posses the inverse element of addition Xwith + (−X )are =hold 0properties ,methods XXare ·+ (1/X) = (6) A(x) (2) fadLare (x) for xxcan ∈ be [a, b) fuzzy arithmetic (MD RDM-FA) iswith an method multiplication properties (6).  der of A. a, b,f c, are real numbers such that aFA ≤that bffor ≤ cxdecreasing ≤ d.d] is known are not perfect and bebeimand multiplication does not in Therefore, applica(x) for xnot ∈arithmetic [a, b)Currently ,by tion possibilities of SFA limited. fuzzy (MD RDM-FA) isSFA. an alternative method A(x) = (2) proposed and developed Polish scientists from Szczecin  multiplication properties (6). L  where x ∈ R, is real-valued function that is increasing and der of A. a, b, c, d real numbers such that a ≤ b ≤ c ≤ d. right-continuous, f is a real-valued function is Currently known methods not perfect andofcan can im-of en b = c, A a fuzzy number. When f and f are lin L R  L R  right-continuous, f is a real-valued function that is decreasing f ∈ (c, ,  fuzzy arithmetic (MD RDM-FA) is an alternative method R :are b,{x d are real numbers such that a ∙ b ∙ c ∙ d. When the crisp Aµfa,La, = ∈ R A(x) ≥ µ } is called µ -cut R   der set ofA. A. b,c, c, d real numbers such that a ≤ b ≤ c ≤ d.  1 for x ∈ [b, c] , (x) for x ∈ [a, b) , f for x ∈ (c, d] , ,proposed a25]. ⊕ Its [b ,known b2of ]= [a ⊕ bmulti-dimensional, ,by asub-distributivity ⊕ b)not ], 0 (4) [a18,  multiplication with properties Rais 1bor2∈]FA 1basic 1 (6). 1limited. 2are 2= proposed and developed by Polish scientists from Szczecin  Currently methods perfect and can be imFA and developed Polish scientists from Szczecin proved. The relative distance measure tion possibilities SFA are The law and cancellation law for [5, 6,1fand 7,lin11,right 13, concepts and the idea of, horX + (−X X · (1/X ) = 1 (6)  When b = c, A is fuzzy number. When f and are lin 1 for x ∈ [b, c] , functions we obtain a special type of fuzzy intervals called When = c, A is a fuzzy number. When f and left-continuous. f where x ∈ R, f and f are and f can be called left and is a real-valued function that is increasing right-continuous, f a real-valued function that is decreasing for x [b, c] ,  L R L R L R L R proved. The multi-dimensional, relative distance measure A(x) = (2)  The sub-distributivity law and cancellation law for addition left-continuous. fLfuzzy fnumber. can called left and right borX(MF) + (−X) = 0, Xby · (1/X =scientists 1(6) b = c, A is a fuzzy number. and fRA(x) are linear functions FA proposed The and crisp set S(A) = {xis∈aR :and A(x) > 0} be isfL called the X) + (−X) = 0from , (6) X · (1/X ) and developed Polish Szczecin 0fL= otherwise, RWhen  When b = c, A When and f are lin(2) A(x) = (2) R  izontal membership functions were proposed by Piegat Currently known methods are not perfect and can be imand multiplication does not hold in SFA. Therefore proved. The multi-dimensional, relative distance measure [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of horfuzzy arithmetic (MD RDM-FA) is an alternative method of f for x ∈ (c, d] , [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of hor0 otherwise, 1max for x= ∈ c] pezoidal fuzzy intervals which dominant in applications. and left-continuous. and can betype called left and right borear functions we obtain atype special of intervals of a,faare c, dffor are real numbers such that ≤ bcalled d.fuzzy right-continuous, real-valued function iscalled decreasing RRof  ear functions we a1,[b, special ofnormal fuzzy ,≤ [b b2multiplication =+(c, [ min(a ⊗ b01does a1 ⊗ bX2 ,·ahold b1in, = aSFA. ⊗ bis2),Therefore, [a Lb, RA(x)  1fa 1c, ≤ 2⊗ 21  for x]X ∈ d]arithmetic ,sub-distributivity (MD RDM-FA) an alternative method of (−X )111, = ,,be (1/X) (6) law  fisRare x,type ∈ (c, d] ,that we obtain a special intervals trapezoidal rt A(x) ofder A. When Anumbers isfuzzy called afuzzy and not applicaRa2c] ⊗  of A. a,der b, c,fA. dobtain real such that aintervals ≤ bcalled ≤≤ d.  x∈R [5, 6, 7, 13, 18, 25]. Its basic concepts the of horThe law and cancellation for addition = (2)  ear functions we obtain afis special type ofathat fuzzy intervals called [25]. Further on, these concepts will shortly presented.  FA proposed and developed by Polish scientists from Szczecin proved. The multi-dimensional, relative distance measure tion possibilities of SFA are limited. The sub-distributivity law and cancellation fuzzy arithmetic (MD RDM-FA) is an alternative method of idea izontal membership functions (MF) were proposed byand Piegat y trapezoidal fuzzy interval A is fully characterized byreal-valued the  The sub-distributivity law and cancellation law for addition izontal membership functions (MF) were proposed by Piegat When b = c, A a fuzzy number. When f and left-continuous. f where x ∈ R, f and f are linand can be called left and right boris function that is increasing and der of A. a, b, c, d are real numbers such a ≤ b ≤ c ≤ d. trapezoidal fuzzy intervals which are dominant in applications.  L R L R L trapezoidal fuzzy intervals which are dominant in applications.  0 otherwise, fRintervals, x ∈are (c, d] , that b(fuzzy , a1 developed ⊗are b2does ,limited. aand bcancellation ahold ⊗be b2in )]were , SFA. max(a fuzzy dominant inisapplications. Any trapThe sub-distributivity law law proposed for addition set. When FA proposed and by scientists from Szczecin 0and otherwise, 1⊗ 1SFA 2 ⊗ not 1 , can 2Polish tion possibilities of bfbd) c, A isfuzzy awhich fuzzy number. When f≤Lof and fby are linizontal membership functions (MF) by Piegat and multiplication Therefore, applicahere xtrapezoidal ∈of R, 0for otherwise, afuzzy real-valued function Rintervals Fig. 1function shows athat trapezoidal MF interval). It  L= intervals which are dominant in applications. druple (a, b, c, of real numbers occurring in the Currently known methods are not perfect and ca and does not hold inhorSFA. Th FA proposed and developed by25]. Polish scientists from Szczecin [25]. Further on, concepts will be shortly presented. [5,Further 6, 7,these 11, 13, 18, basic concepts and the idea of fuzzy arithmetic (MD RDM-FA) isIts anmultiplication alternative method of sub-distributivity law and cancellation law for addition When =is c, Anormal a fuzzy number. When fincreasing fthe are linAny trapezoidal interval A on is fully characterized the ear functions we obtain aAcharacterized special called der A. a, b, c, dis are real numbers such that bfuzzy ≤ ≤ d. right-continuous, fspecial isLacharacterized aand real-valued is decreasing  and multiplication does not hold in SFA. Therefore, applica[25]. on, these concepts will be shortly presented. R cThe Rtype Any trapezoidal fuzzy interval is fully by the ezoidal fuzzy interval A is fully by quadruple and multiplication do not hold in SFA. Therefore, application fuzzy interval A is a fuzzy set R whose µ (5) [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea ofdistance horCurrently known methods are not perfect and can be im0 otherwise, noted that triangular or rectangular MFs are a special case of ear functions we obtain a special type of fuzzy intervals called [25]. Further on, these concepts will be shortly presented. tion possibilities of SFA are limited. ght-continuous, f is a real-valued function that is decreasing onical form (3). See also Fig. 1. FA proposed and developed by Polish scientists from Szczecin proved. The multi-dimensional, relative tion possibilities of SFA are limited. [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of horFig. 1 shows a trapezoidal MF (fuzzy interval). It can be izontal membership functions (MF) were proposed by Piegat Any trapezoidal fuzzy interval A is fully characterized by the R where x ∈ R, f is a real-valued function that is increasing and When b b, = is c, Aclosed isof areal number. When fR, left-continuous. fthe where x ∈fuzzy and faRthe linand freal-valued can called and right is function that is increasing and ear functions we areal special type of called (a, b, c,Lobtain d)fuzzy of numbers occurring in special trapezoidal fuzzy intervals which dominant inbe applications. multiplication does not hold in SFA. Therefore, applicaLare Lfand Rare Lintervals tionleft possibilities ofSFA SFA limited. Fig. 1borshows aare trapezoidal MF (fuzzy interval). byIt Piegat can be (a, c, numbers occurring special canonical possibilities of are limited. or alltrapezoidal µquadruple (0, intervals ofnumbers real numbers here xquadruple ∈∈R, ad) real-valued function that isin increasing and (a, b, c, d) ofand real occurring inand the special L are this function. membership were proposed f1] proved. The multi-dimensional, distance measure fuzzy intervals which are dominant in applications. known methods notcan perfec noted that triangular orknown rectangular are a(fuzzy special case ofanarealternative [25]. Further on, these be shortly presented. 6,function 7,aizontal 11, 13, 18, 25]. Its concepts and the idea of horfuzzy (MD RDM-FA) is m izontal membership functions (MF) were proposed by Piegat 1 shows a trapezoidal MF interval). It be Currently methods are not perfect and can be imnd left-continuous. f and f can be called left and right bor,concepts bCurrently ]MFs ifrelative ⊗(MF) =will ÷. where: ⊕ ∈[5, {+, −}, ⊗Fig. ∈ {×, ÷} and 0basic ∈ /functions [barithmetic Any trapezoidal fuzzy interval A is fully characterized by the ear functions we obtain a special type of fuzzy intervals called der of A. a, b, c, d are real numbers such that ≤ b ≤ c ≤ d. right-continuous, f is a real-valued that is decreasing trapezoidal fuzzy intervals which are dominant in applications. canonical form (3). See also Fig. 1. L R 1 2 quadruple (a, b, c, d) of real numbers occurring in the special right-continuous, f is a real-valued function that is decreasing R tion possibilities of SFA are limited. R (x(3) − a)/(b − a) for ∈ b)exactly , that is Currently known methods are not and beimim-case noted that triangular or rectangular MFs arecan apresented. special of form Fig. 1). Currently known methods are notperfect perfect and be fL canonical is ais real-valued function that increasing and  eR,support bounded. When A(x) =xFig. 1is[a, for one ght-continuous, fR(see is(3). afuzzy real-valued function decreasing form See also 1. Vertical models of MFs are used in classic fuzzy systems.  [25]. Further on, these concepts will be shortly fuzzy arithmetic (MD RDM-FA) is an alternative method of FA proposed and developed by Polish scientists from proved. The multi-dimensional, relative [25]. Further on, these concepts will be shortly presented. this function. Fig. 1 shows a trapezoidal MF (fuzzy interval). It can be izontal membership functions (MF) were proposed by Piegat  Any trapezoidal interval A is fully characterized by the For fuzzy intervals A and B operations defined by (4) and (5) noted that triangular rectangular MFs are special case of proved. Theare multi-dimensional, relative When = c, Athat isnumber. aafuzzy number. fLknown left-continuous. fcLand and fRleft are linand fWhen can be called and right borAny fuzzy interval A is fully characterized byCurrently the quadruple b, c, d)bFig. of real numbers occurring in special fuzzy are dominant in er of trapezoidal A. b,ac, dfuzzy areintervals real(a, numbers such ≤applications. bleft ≤ ≤ d.  Rthe canonical form (3). See also 1. and left-continuous. and f∈and can be called right bormethods notorperfect anddistance can bemeasure im-adistance Lwhich R proved. The multi-dimensional, relative fuzzy measure ,nuous, then this interval is called a[b, fuzzy proved. The multi-dimensional, relative distance measure this function. 1real-valued for x that c]for ,decreasing fa, istrapezoidal function is They express a vertical dependence µtriangular =[5, f trapezoidal (x):  Rspecial nd left-continuous. ffuzzy and fRfear can be called left and right bor(xform − a)/(b − a) x1. ∈b, [a, b) ,are L Fig. 1 shows a MF (fuzzy interval). be noted that or rectangular MFs are a concepts special of [25]. Further on, these concepts will be shortly presented. 6, 11, 13, 18, 25]. Its basic and the ide fuzzy arithmetic (MD RDM-FA) iscan an alter Fig. 1fuzzy shows a∈bdeveloped trapezoidal MF (fuzzy interval). Itsystems. can be Itcase Vertical models of MFs are7, used ininterval classic fuzzy are realised for each µ -cut, µ [0, 1], which is the and FA proposed and by Polish scientists from Szczecin  (3) A(x) = Any trapezoidal interval A is fully characterized by the functions we obtain a special type of fuzzy intervals called der of A. a, c, d real numbers such that a ≤ ≤ c ≤ d. quadruple (a, b, d) of real numbers occurring in the special canonical (3). See also Fig. quadruple (a, b, c, d) of real numbers occurring in the special this function. arithmetic (MD RDM-FA) is an alternative method of When b = c, A is a fuzzy number. When f and f are lin  L R der of A. a,fx)/(d b, c, dbe are realxleft numbers such that ≤ ≤The d.arithmetic (x − a)/(b − a) for xthe ∈ [a,acb) , bd.≤ c fuzzy (MD RDM-FA) is an alternative method of FA pro-of nvenient way of expressing any fuzzy interval A is≤  proved. multi-dimensional, relative distance measure  arithmetic (MD RDM-FA) is an alternative method Vertical models of MFs are used in classic fuzzy systems. ntinuous. f and can called and right bor(d − − c) for ∈ (c, d] ,   L R er of canonical A. a, b, c, d are real numbers such that a b ≤ ≤  1 for x ∈ [b, c] , FA proposed and developed by Polish scienti noted that triangular or rectangular MFs are a special case of They express a vertical dependence this function. Fig. 1 shows a trapezoidal MF (fuzzy interval). It can be izontal membership functions (MF) were proposed µ = f (x): can be denoted as [a( µ ), a( µ )] and [b( µ ), b( µ )].  noted that triangular or rectangular MFs are a special case of  (x − a)/(b − a) for x ∈ [a, b) ,  When b = c, A is a fuzzy number. When f and f are linquadruple (a, b, c, d) of real numbers occurring in the special trapezoidal fuzzy intervals which are dominant in applications. canonical form See also Fig. 1. [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of horL R form (3). See also Fig. 1. Vertical models of MFs are used in classic fuzzy systems. FA proposed and developed by Polish scientists from Szczecin   r functions we obtain a special type of fuzzy intervals called (xThey −(MD a)/(b − a)developed for xis∈Polish [a,dependence b) ,scientists posed andexpress developed by from Szczecin [5–7,  bA(x) =anumbers c, is1anumber. fuzzy number. When fL[b, fR(3) are lin= A fuzzy arithmetic RDM-FA) an alternative method of   (x for xand ∈ b) , noted xinterval ∈ c] FA and by Polish scientists from Szczecin arectangular vertical µ = f case (x): a,ical b, form: c, d are such that aa)/(b ≤ for b− ≤ c∈≤ d.  0form otherwise.  When b When = c, Areal fuzzy When ffuzzy and fobtain linthat triangular MFs a13, special ofbasic [25]. Further on, these concepts will be shortly prese [5, 6,addition 7,are 11, 25]. Its concepts an this function. Vertical models of MFs are used in18, classic fuzzy systems. See does not posses the inverse element of and    Lfor R[b, Any trapezoidal ASFA is fully characterized by theorIts ear functions we a,,[a, special type of fuzzy intervals called canonical (3). Fig. 1. (d1which −also x)/(d −− c) xa)in (c, d] ,are proposed this function.  isintervals  izontal membership functions (MF) were proposed by Piegat   A(x) fuzzy = (3) (3) for x ∈ c] They express a vertical dependence µ = f (x): [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of hor11, 13, 18, 25]. basic concepts and the idea of horizontal apezoidal are dominant applications. 1 for x ∈ [b, c] ,   ear functions we obtain a special type of fuzzy intervals called FA proposed and developed by Polish scientists from Szczecin (x − a)/(b − a) for x ∈ [a, b) ,   (x − a)/(b − a) for x ∈ [a, b) , [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of horc, A is a fuzzy number. When f and f are lin1 for x ∈ [b, c] , f (x) for x ∈ [a, b) ,  They express a vertical dependence this function. Fig. 1 shows a trapezoidal MF (fuzzy interval). izontal membership functions (MF) were pr Vertical models of MFs are used in classic fuzzy systems. µ = f (x):  multiplication with properties (6).  (3) L R A(x) = −= a)/(b − a) for xofx∈real b)  L rAfunctions we aA(x) special type intervals called (7) µ (x) = (dshorthand − x)/(d −of c)fuzzy for ∈[a, (c,numbers d], , which quadruple (a, b, c, d)trapeoccurring inVertical the special trapezoidal fuzzy intervals are dominant in applications.    models of∈ MFs are used in classic fuzzy systems.  = (a,trapezoidal b, c, d) beobtain used as aintervals notation of 0(x otherwise.  [25]. Further on, these concepts will be shortly presented. membership functions (MF) were proposed bywere Piegat [25]. Fur  (3)  izontal membership functions (MF) proposed bybe Piegat  ny trapezoidal interval Afuzzy iswhich fully characterized by the  (d − x)/(d − c) for x (c, d] ,   fuzzy are dominant in applications. (x − a)/(b − a) for x ∈ [a, b) , (d − x)/(d − c) for x ∈ (c, d] , fuzzy     [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of hor (x − a)/(b − a) for x ∈ [a, b) , 1 for x ∈ [b, c] , noted that triangular or rectangular MFs are a specia [25]. Further on, these concepts will short They express a vertical dependence  Vertical models of MFs are used in classic fuzzy systems. µ = f (x):  izontal membership functions (MF) were proposed by Piegat ns we obtain a special type of intervals called  b1=  1 aX for xpresented. ∈µ[b, c] f, (x): Any trapezoidal fuzzy characterized by the (3). See 1.(3) (dAc] − x)/(d − c) for (c, d] , A isFig. 1When x ∈ [b, apezoidal fuzzy intervals which are dominant in will  dal fuzzy intervals. ccanonical in (3), is,form usually called   for xinapplications. ∈also [b,xFig. c]∈,interval They vertical =be ther these be 0for otherwise. Xfully + (−X) = 0 , aconcepts · (1/X) = 1shortly (6)  A(x) = 1[25]. shows trapezoidal MF (fuzzy interval).    (x −dependence a)/(b − a)will forPiegat x ∈It[a, b) , be(fuzzy in (7) µon, (x)express Further on, these concepts shortly presented. uadruple (a, b, c,which d) numbers occurring the A(x) = (2)  0Further otherwise. Any trapezoidal fuzzy interval Aapplications. is characterized by the  (x − a)/(b − a) for b)can ,MF  Let A number. = (a, b, c,of d) be used as afully shorthand notation of trape1real for xotherwise. ∈ [b, c]c, ,d), special  They express a= vertical dependence this function. shows afor µFig. =for f1be (x): izontal membership functions (MF) were proposed by (3) A(x) = fully  0 quadruple (a, b, of real numbers occurring in the special [25]. on, these concepts will shortly presented. (d − x)/(d − c) for x ∈ (c, d] lny fuzzy intervals are dominant in     ngular fuzzy  1 c] , (d − x)/(d − c) x ∈ (c, d]trapezoidal , x x∈∈[a,[b, trapezoidal fuzzy interval A is characterized by the  0 otherwise. f for x ∈ (c, d] ,   R   (3) A(x) =   noted that triangular or rectangular MFs are a special case ofItMFs   (dd)− x)/(d − c) for x− (c, d] ,called (x a)/(b − a) for 1.x ∈on, [a, b) ,Fig.  1a(x) shows aVertical MF (fuzzy can be   nonical form (3). See also Fig. 1. (x − a)/(b −noted a) for xtriangular [a, b) , xor (7)  µ = 1trapezoidal forare x∈interval). ∈ [b, , be zoidal fuzzy intervals. When in (3), A is∈ usually  The sub-distributivity law and cancellation law addition that rectangular are  quadruple (a, b, c,is of numbers in the special models of∈interval). MFs used in,c]classic fuzzy    [25]. concepts will shortly presented. 1be for [b, c]can canonical (3). also Fig. (d x)/(d −bc) for x form ∈occurring (c, d] , See  The mostly fuzzy arithmetic isreal called fuzzy   Formula (7) expresses a unique dependence of thefor ‘vertiLet Aused = (a, b, c, be used as= acstandard shorthand notation ofFurther trapeFig. 1these shows trapezoidal MF (fuzzy It(c, zoidal fuzzy interval Ad) fully characterized by the   0− otherwise. uadruple (a, b, c, d) of real numbers occurring in the special   0 otherwise.  (7)of µ (x) = 0 otherwise,   (x − a)/(b − a) for x ∈ [a, b) , (d − x)/(d − c) for x ∈ d] ,  (7) µ (x) =  this function.    Let A = (a, b, c, d) be used as a shorthand notation of trape(x − a)/(b − a) for x ∈ [a, b) , noted that triangular or rectangular MFs are a special case  Let A = (a, b, c, d) be used as a shorthand notation of trape1 for x ∈ [b, c] , and multiplication does not hold in SFA. Therefore, applicatriangular fuzzy number.  They express a vertical dependence this function. µ = f (x): 0 arithmetic otherwise.  form See alsooperations Fig. 1. 1 variable for xc) ∈are [b, ,Tech. hmetic (SFA). Inreal SFA, basic on real variable µshows fromthat the ‘horizontal’ x.−− However,    Fig.called 1 noted aµ trapezoidal MF interval). Itfor can (d(d x)/(d −(7) c) for ∈(c, (c, d] x)/(d x x∈be d] , , of intervals. When bfor =inotherwise. cthe in (3), A, is(xcal’ usually 0(3).  triangular rectangular MFs ac] special case (a, b,zoidal c,canonical d) offuzzy numbers occurring special  498 Bull. Pol. Ac.: 65(4) (fuzzy b)(3)  A(x) = nonical form (3). See also Fig. (x −d) a)/(b − a)1. ∈=(3), [a, b) (7) =    − a)/(b − a)MF forSFA x(x) ∈ [a, , ofor Vertical models MFs are used inxspecial classic fuzzy systems. for x MFs ∈‘verti[b, c]2017 ,used in class Let A a= (a, b, c, be used as a bisshorthand notation of trapezoidal fuzzy intervals. When bcalled cTwo in (3), A is usually called tion possibilities of are limited. 0unique otherwise.  Vertical models of are The mostly used fuzzy arithmetic standard fuzzy  Formula (7) expresses a1 dependence of the zoidal fuzzy intervals. When = cisx in A is isusually called  this function.  mbers are extended to operations on fuzzy intervals. 1 for ∈ [b, c] , a model of borders only. Function (7) does not model eorm x ∈ R, f is real-valued function that increasing and  (d − x)/(d − c) for x ∈ (c, d] ,   L  (d − x)/(d − c) for x ∈ (c, d] ,   noted that triangular or rectangular MFs are a case of  triangular fuzzy number. this function. (3). See also Fig. 1.be 0 otherwise. (7) , µfrom (x) = triangular    0 otherwise. Unauthenticated (7) µ (x) =   Let A = (a, b, c, d) used as a shorthand notation of trape(x − a)/(b − a) for x ∈ [a, b) , (x − a)/(b − a) for xsystems ∈ Let A = (a, b, c, d) be used as a shorthand notation of trape1 for x ∈ [b, c] , zoidal fuzzy intervals. When b = c in (3), A is usually called Currently known methods are not perfect and can be imfuzzy number.   They express a vertical dependence arithmetic (SFA). In SFA, basic arithmetic operations on real 1 for x ∈ [b, c] ,  µ [a, = fb)(x) They express a vertical dependence cal’ variable µ the ‘horizontal’ variable x. However, (7) µ = f (x): mmon ways of defining the extended operations are based on an this interior of MF. If suchVertical function used in calculatriangular number. continuous, fR isafuzzy real-valued function that isxdecreasing models of MFs are used in classic fuzzy    (d− −c)fuzzy x)/(d − for x PM ∈ (c, d] , the  (d0(7) −isx)/(d for x| c) ∈ (c,dependence d] , fuzzy (x − a)/(b − a) for ∈ [a, b) , 0called otherwise.  function. Download Date 8/25/17 12:47   otherwise. (3) A(x) =   The mostly used fuzzy arithmetic is standard fuzzy Vertical models of MFs are used in classic systems. (3) A(x) = Formula expresses a unique of ‘verti     zoidal intervals. = c[a, inand (3), A(c, isborusually called triangular proved. The relative distance measure  fuzzy intervals. = cthe (3), A usually called numbers are extended toWhen operations on fuzzy intervals. The used fuzzy arithmetic is,is∈called standard  Formula expresses unique ofof the ‘vertiamulti-dimensional, model MF only.are Function (7) does model  µ -cutzoidal representation of fuzzy and on tions, only of MF (without its(7) applied. 1It dependence forthe x ∈‘verti[b, c] , ft-continuous. f and fnumber. canintervals be called right (d − −borders c)isfuzzy for xFormula ∈of (c, d]  1When for x [b, c]x)/(d ,Two Lfuzzy Rmostly The mostly used fuzzy is called standard fuzzy They express ainterior) vertical dependence (7) expresses aaotherwise. unique dependence µotherwise. =not f (x):  (d − x)/(d − c)bxarithmetic for xin ∈ d]  (xfuzzy − a)/(b − a) for ∈bleft b) ,extension 0,borders   Vertical models of MFs are used in classic fuzzy systems.  0function  µ (x) =also 1 for x ∈ [b, c] , arithmetic (SFA). In SFA, basic arithmetic operations on real They express a vertical dependence µ = f (x): cal’ variable µ from the ‘horizontal’ variable x. However,   (3) A(x) = (x − a)/(b − a) for(7) nciple of fuzzy set theory [20].  Let A = (a, b, c, d) be used as a shorthand notation trapedecreases accuracy of achieved results. Vertical MFs intriangular fuzzy number.  fuzzy arithmetic (MD RDM-FA) is an alternative method of arithmetic (SFA). In SFA, basic arithmetic operations on real The mostly used fuzzy arithmetic is called standard fuzzy common ways of defining the extended operations are based on Formula (7) expresses a unique dependence of the ‘vertical’ variable an interior of MF. If such is used in fuzzy calculaµ from the ‘horizontal’ variable x. However, (7)  triangular number. A. a, b, c, d=arefuzzy real numbers such that a ≤ b ≤ c ≤ d. (x − a)/(b − a) for x ∈ [a, b) ,     (d − x)/(d − c) for x ∈ (c, d]x,∈ (3) A(x) arithmetic (SFA). In SFA, basic arithmetic operations on real 0 otherwise. cal’ variable µ from the ‘horizontal’ variable x. However, (7)    

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functions fuzzy arithmetic (MD RDM-FA) is an alternative method of it FA proposed and developed by Polish scientists from SzczecinHowever However the the question question arises, arises, whether whether it is is possible possible to to create create −1 (µ ). At first glance, it an inverse (‘horizontal’) model x = f However the question arises, whether it is possible to create −1 [5, 6, 7, 11, 13, 18, 25]. Its basic concepts and the idea of horan inverse (‘horizontal’) model x = f −1 it (µ ). At first glance, seems impossible because the dependence would not be unaman inverse (‘horizontal’) model x = f ( µ ). At first glance, division the multi-granularity phenomenon izontal membership functions (MF) Fuzzy were number proposed by and Piegat seems impossible because the dependence would not be unam- it it not aa function then. However, it will seems and impossible dependence not be [25]. Further on, these concepts will be shortly presented. biguous biguous and it would wouldbecause not be be the function then.would However, it unamwill be shown that such inverse model can be created. biguous and it would not be a function then. However, it will Fig. 1 shows a trapezoidal MF (fuzzy interval). It canbe beshown that such inverse model can be created. Figure 1 shows a trapezoidal MF (fuzzy interval). It can be be biguous and it would not bemodel a function then. However,µit. will Let us consider aa horizontal cut MF on the shown that such inverse can be created. noted that triangular or rectangular MFs are a special case of Letbe usshown consider horizontal cut of ofcan MF on the level level µ . FurFurnoted that triangular or rectangular MFs are a special case of that such inverse model be created. ther on, this cut will be called µ -cut (not α -cut as usually), Let us consider a horizontal cut of MF on the level µ . Furthis this function. ther on, this cut will be called µ -cut (not α -cut as usually), function. Let us consider a horizontal cut of MF on the level μ. FurFig. 2a. Variable ,, α ∈ [0, 1], will be called ther on, this cutα be called µ -cut (not αRDM-variable -cut as usually), Vertical models of MFs areareused fuzzysystems. systems. Fig. 2a. αxxwill αxxbe ∈ called [0, 1], μ-cut will be Vertical models of MFs usedinin classic classic fuzzy ther Variable on, this cut will (notcalled α-cut RDM-variable as usually), as (RDM --Fig. 2a. relative distance determines aaRDM-variable relative Fig. Variable αxα, α, measure). [0, 1], It will becalled calledRDM-variable x ∈ TheyThey express a vertical dependence = f (x): express a vertical dependenceµμ = f(x): in 2a. Variable α  2 [0, 1], will be (RDM relative distance measure). It determines relative disdisx x ∗ tance of point x ∈ [x ( µ ), x ( µ )] from the origin of the local (RDM relative distance measure). It determines a relative dis∗ L R – relative distance measure). It determines a relative distance(RDM of point x ∈∗ [x ( µ ), x ( µ )] from the origin of the local  L R ¤ coordinate system, Fig. 2. tance of point x ∈ [x ( µ ), x ( µ )] from the origin of the local tance ofsystem, point xFig.  2 [xL2.L(μ), xRR(μ)] from the origin of the local (x − a)/(b − a) for x ∈ [a, b) ,  coordinate    1 The RDM-variable α the coordinate system, 2. coordinate system (Fig. 2). for x ∈ [b, c] , The RDM-variableFig. αxx introduces introduces the local local Cartesian Cartesian coorcoor µ (x) = (7) The RDM-variable αx x introduces introduces the local Cartesian coor(7) The dinate system into the interior of an interval. The left boundRDM-variable α the local Cartesian coordinate system into the interior of an interval. The left bound (d − x)/(d − c) for x ∈ (c, d] ,  dinate system into the interior of an interval. The left boundary  ary x ( µ ) of MF and the right boundary x ( µ ) are expressed dinate system into the interior of an interval. The left boundR (µ ) are expressed  ary xxLL ((μ) µ )ofofMF MFand and the right boundary R expressed by (8). the right boundary xR(μ) xare 0 otherwise. Lx ( µ by (8). ary ) of MF and the right boundary xR (µ ) are expressed L by (8). by (8). Formula (7) expresses a unique ‘verti- xxL = a + (b − a)µ , xxR = d − (d − c)µ (8) (8) Formula (7) expresses a uniquedependence dependence of of the the ‘ver(8) L = a + (b − a) µ , R = d − (d − c)µ cal’ tical’ variable µ from the ‘horizontal’ variable x. However, variable μ from the ‘horizontal’ variable x. However, (7)(7) xL = a + (b − a)µ , xR = d − (d − c)µ (8) α the ((µ is a is model of of MF Function(7)(7) does model a model MFborders borders only. only. Function does not not model an The The RDM-variable αx transforms leftboundary boundary xxLL(μ) αxx transforms transforms thetheleft left boundary µ )) The RDM-variable RDM-variable into the boundary xxαRxx((Rµ ). Contour line x( ,, α of transforms the left boundary xL ( µ ) RDM-variable interior of MF. is used in fuzzy calculations, into right the right boundary (μ). x(μ, α an interior MF.IfIfsuch suchfunction function is used in fuzzy calculax) into The the right boundary ). Contour Contourline line x(µ µ αofxx ))constant of conconR µ stant α values is determined by (9). into the right boundary x ( µ ). Contour line x( µ , α ) of cononly borders of MF its interior) are applied. It reduces determined by (9). R by (9). x tions, only borders of (without MF (without its interior) are applied. It ααx xxvalues stant valuesis is determined the accuracy of achieved results. Vertical MFs also stant αx values is determined by (9). decreases accuracy of achieved results. Vertical MFsincrease also in(9) x( computational effort in fuzzy calculations. Hence, an idea of µ ,, α αxx )) = = xxLL + + (x (xRR − − xxLL))α αxx ,, α αxx ∈ ∈ [0, [0, 1] 1](9) (9) x(µ crease computational effort in fuzzy calculations. Hence, an µ , α ) = x + (x − x ) α , α ∈ [0, 1] (9) x( x L R L x x horizontal MFs was conceived. µ ,, α set of points lying at equal relaThe contour line x( x )) is idea of horizontal MFs was conceived. µ α is set of points lying at equal relaThe contour line x( x However the question arises whether it is possible to create contour linex( x(μ, α is situated at).an equal tive distance α the of xxL ((µ A µ ,left αxx))boundary is set setofofpoints points lying at relaTheThe contour line αxx from from of MF MF ). equal A more more LxLµ an inverse (‘horizontal’) model x = f –1(μ). At first glance, tive it distance relative distance αx the fromleft theboundary left boundary of MF (μ). A more precise form (10) of (9) can be called a horizontal MF. tive distance α from the left boundary of MF x ( µ ). A more x of (9) can be called a horizontal LMF. Bull. Pol. would Ac.: Tech. 2016 precise formform (10)(10) seems impossible because the dependence not XX(Y) be unamprecise of (9) can be called a horizontal MF. precise form (10) of (9) can be called a horizontal MF. xx = [a + (b − a) µ = [a + (b − a)µ ]] (10) (10) x [(d = [a−+a)(b−−(da)−µc] + b − a)µ ]αx , αx ∈ [0, 1] + + [(d − a) − (d − c + b − a)µ ]αx , αx ∈ [0, 1] (10)(10) + [(d − a) − (d − c + b − a)µ ]αx , αx ∈ [0, 1] The µ ,, α αxx )) is is aa function function of of two two varivariThe horizontal horizontal MF MF xx = = ff ((µ ables and exists in 3D-space, is αx It ) is aunique. function of two varihorizontal MF x = Fig. fFig. (µ , 3. ablesThe and exists in 3D-space, 3. It is unique. The horizontal MF x = f(μ, αx) is a function of two variables ables and exists in 3D-space, Fig. 3. It is unique. and exists in 3D-space (Fig. 3). It is unique.

Bull. Pol. Ac.: Tech. XX(Y) 2016 Bull. Pol. Ac.: Tech. XX(Y) 2016 Bull. Pol. Ac.: Tech. XX(Y) 2016

Fig. 3. Horizontal MF x = (1 + 2μ) + (4 ¡ 3μ)αx , αx 2 [0, 1], corresponding to vertical function shown in Fig. 2

Fig. 2. Visualization of the horizontal approach to fuzzy membership functions

The horizontal MF x = f(μ, αx) defines not a single value of the variable x, but a set of possible values of x corresponding to a given μ-cut level. It defines an information granule and hence it will be denoted as x gr. Formula (10) describes the trapezoidal MF. However, it can be adapted to triangular MF by setting b = c and to rectangular MF by setting a = b and c = d.

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Bull. Pol. Ac.: Tech. 65(4) 2017 Unauthenticated Download Date | 8/25/17 12:47 PM

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sponding to vertical function MF shown Fig. 3. Horizontal x =in(1Fig. + 22µ ) + (4 − 3µgr )αgr 1], correx , )α= x ∈[2[0, s(z , 92span − µaddition ](17) ,gr µ ∈ [0, ., 9addition (17) )∈ =is [2 + 31] µthe − 2 µThe ] result. , result. µ ∈re[0, 1] .isaddition (17) the span (17) The not the readdition The (17) is The span isnot result. not The addition addition result. The addition re- o sponding to function shown in Fig. 2 s(z )span = [2 + 3+ µ ,3The 9µnot − µthe ]2,s(z µ(17) [0, 1] .the (17) unction shown invertical Fig. 2 to information about the maximal uncertainty The span (17) isis not the addition The addition reThe span (17) is not the addition result. Theresult. addition The horizontal MF x = f ( µ , α ) defines not a single value of The span (17) not addition result. The addition responding vertical function shown in Fig. 2 x The span (17) is The not span the addition (17) is not result. the addition The addition re-(16 Th MF x = (1 + 2 µ ) + (4 − 3 µ ) α , α ∈ [0, 1], corresult has the form of 4D-function sult (16). has The the form span of is only 4D-function a 2DTheThe horizontal MF xThe (µf, (αµx ,)αdefines not a single value of x MF x=xf= sult has the form of sult 4D-function has the form (16). of 4D-function The span is only (16). a The 2Dspan is only a 2Dhorizontal horizontal ) MF defines x = not f ( µ a , α single ) defines value not of a single value of xvalues of x corresponding x sult has the form of 4D-function (16). The span is only aa2Dsult has the form ofaddition 4D-function (16). The span isreonly a2 the variable x, 2but ax set of possible sult has the form of 4D-function (16). The span is only 2DThe span (17) is not the addition result. The reThe horizontal MF = f ( µ , α ) defines not a single value of The span (17) is not the addition result. The addition sult has the form sult of 4D-function has the form (16). of 4D-function The span is (16). only The a 2Dspan function shown in Fig. x Fsp the the variable x, but athe seta variable of possible values of x(17) corresponding information about the maximal information uncertainty of about the the result. maximal unce The span isvalue notabout the addition result. The addition reFig. 4.the MFresult. of the information information the maximal about uncertainty the maximal of the uncertainty result. of the result. variable x, but set of possible x, but a values set of of possible x corresponding values of x corresponding Fig information about the maximal uncertainty of the result. information about the maximal uncertainty of The horizontal MF x = f ( µ , α ) defines not a single of MF x = f (variable µa ,given αx ) defines single value of Fig x -cut level. ItMF defines an information granule and information about the maximal uncertainty of the result. sult has form of 4D-function (16). The span is only a 2Dtheto x,µ but anot seta of possible values of x the corresponding sult has the form of 4D-function (16). The span is only a 2Dinformation about information the maximal about uncertainty the maximal of the uncertainty result. of th The horizontal x = f ( µ , α ) defines not a single value of (2 (22) x to a given µ -cut level. It defines an information granule and sult has the form of 4D-function (16). The span is only a 2DA. Piegat M. Plucińskigranule andSubtraction of two independent fuzzy inter (22 to aitgiven to a given µ -cut level. an of information defines granule an and information grµ. -cut the variable x, but adenoted of possible values xItcorresponding (22 a setto ofhence possible values ofset xlevel. corresponding will be asIt xdefines Formula (10) describes theand uncertainty Fig. 4. MF of the span repr information about the maximal of the result. gr Fig a given µ -cut level. It defines an information granule and information about the maximal uncertainty of the result. the variable x, but a set of possible values of x corresponding gr gr Fig. 4. MF of the span represe belevel. denoted as xwill .an Formula (10) describes the the information about maximal uncertainty the result. MF =hence f (given µ , αitx )an defines nothence aItsingle value of hence it-cut will be denoted as xand be as(10) x .the . denoted Formula describes Formula (10) describesofthe to adefines µwill defines information granule and vel.xhence It information granule trapezoidal However, itxitgr can adapted triangular MF by (22) X − Y Subtraction = Zfuzzy :fuzzy xgr(22) (intervals µintervals , αofx )two − ygrindependent (µ , αy ) = zgr(22 (µ it will be denoted . Itbe Formula describes the Subtraction ofoftwo independent fu to a MF. given µ -cut as level. defines anto information granule and Subtraction of two Subtraction independent two fuzzy independent intervals trapezoidal MF. However, it can be adapted(10) to triangular MF by

a sethence of possible values ofHowever, x describes corresponding Subtraction of fuzzy intervals gr Subtraction two independent fuzzy intervals MF. trapezoidal can MF. be However, adapted can triangular be adapted MF to bytriangular MFindependent byofindependent Subtraction oftwo two independent fuzzy intervals it .will denoted ascan xgrit . are Formula (10) describes enoted asBoundaries xtrapezoidal the setting b Formula = cbeand to(10) rectangular MF by setting aitto = bformulas and cthe = d.Subtraction Subtraction Subtraction of two fuzzy fuzzy interva trapezoidal MF. However, it be adapted to triangular MFdescribes by of these functions linear. To derive of of twotwo independent fuzzy hence it will be denoted as xgr .by Formula (10) the µintervals , independent αxgr, intervals α setting b = c and to rectangular MF by setting a = b and c = d. y ∈gr[0, 1] gr gr gr evel. It defines an information granule and gr gr gr gr gr gr setting b = c and setting to rectangular b = c and MF to rectangular setting a = MF b and by setting c = d. a = b and c = d. trapezoidal MF. However, it can be adapted to triangular MF by X − Y = Z : x X − Y = Z : x ( µ , α ) − y ( µ , α ) = z ( µ , α ( , µ α , α ) ,α gr gr gr Subtraction of two independent fuzzy intervals wever, it can be adapted to triangular MF by gr gr gr Boundaries of these functions are linear. To derive formulas X − Y = Z : x X − Y = Z : x ( ( µ , α ) − y ( µ , α ) µ = , α z ) − ( µ y , α ( µ , α , ) = z ( µ , α , α ,−, yα gr)(µ Subtraction of two independent fuzzy intervals x y x y gr gr(, µ x= x y(µ xα y z(µgr,(α x )(,µyx,,))α forbgr nonlinear boundaries, e.g.MF ofare Gauss type, for thed. XXMF setting = cFormula and to rectangular setting aformulas =derive and cindependent = −−YYderive ZZX: :− xformulas ((µ= ,y,α ))x−− ,gr )gr == µµygr ,),αα= ,gr,α Subtraction ofb two fuzzy intervals trapezoidal MF. However, itby can befunctions adapted toare triangular by Y Z : α ) − y z gr x y x y Boundaries of these functions linear. To formulas x x y ,(µ , α = x µ α y ( µ , α ) z ( α ) , (18) denoted as x . (10) describes the x y x y hip (18) (18) Boundaries of these Boundaries functions of are these linear. To derive linear. formulas To X X − Y = Z : x − Y = Z : x ( ( µ , α ) − y ( µ , α µ ) , α = ) z − ( y µ , α ( µ , α ) , = z x y x x y y setting b = c and to rectangular MF by setting a = b and c = d. (18) rectangular MF by setting a = b and c = d. ( for nonlinear boundaries, e.g. of Gauss type, formulas for the leftmembership and right should be linear. determined and usedformulas in (8).b µ hip to ∈ [0, 1]1] , αytrapezoidal ∈(18) [0, (18) 1] M µ ,µαgr , αare example, if gr X andµ Y Boundaries of these are To derive gr ,)α − ,y= αgryd. ∈ [0, 1] α,(yxα ,∈ ∈ [0, xα yα x(18) setting bboundary =boundaries, cfunctions and toe.g. rectangular MF by setting aGauss = and oach bership fuzzy gr gr xc− yFor for nonlinear boundaries, of Gauss type, formulas for the ,,the [0, 1] µ ,= α it can be adapted to triangular MF by µ , α , α ∈ [0, 1] X − Y = Z : x ( µ , α ( µ , α ) z µ , α , α ) , gr gr gr x for nonlinear for nonlinear e.g. boundaries, of Gauss type, e.g. formulas of for type, the formulas for X Y = Z : x ( µ , α ) − y ( µ , α ) = z ( µ , α , α ) , x y x y x y , ∈ [0, 1] µ , α α x y x y x y powever, Boundaries of these functions are linear. To derive formulas e functions are linear. Toofderive formulas X type, − =formulas Zand : x used ∈, [0,result 1] µ ,is αxgiven , αy ∈by: [0, 1] (µfor , in α (8). ) −formulas y (µ , αy ) = z (µµ,,ααxthen αyy )the and right boundary should determined x,,α forleft nonlinear boundaries, e.g. of be Gauss (18) Boundaries functions areY linear. To derive (18) left andMF right boundary should be determined andand used inxthe (8). po rectangular by setting athese = b should and c= (18) left and right boundary left and right boundary bed.determined should be used determined inyif∈ (8). and used in (8). µ , α , α [0, 1] for nonlinear boundaries, e.g. of Gauss type, formulas for the For example, if X and Y are trapezoidal For example, MF if(14) X and and Y(15) are trap , ∈ [0, 1] µ , α α daries, e.g. of Gauss type, formulas for the membership x For example, X For and example, Y are trapezoidal if X and Y MF are (14) trapezoidal and (15) MF (14) and (15) x y µ , α , α ∈ [0, 1] left and right boundary should be determined and used in (8). for nonlinear boundaries, e.g. of Gauss type, formulas for the For example, if X and Y are trapezoidal MF (14) and (15) x y For example, if X and Y are trapezoidal MF (14) and ( ForFor example, if ifXifXand YYexample, are trapezoidal MF (14) and (15) gr se functions are linear. To derive formulas For example, X For and Y are trapezoidal if X and Y MF are (14) trapezoidal and (15) MF ate example, and are trapezoidal MF (14) and (15) 2. Operations of MD RDM fuzzy arithmetic left and right boundary should be determined and used in (8). z = µ + (4 − 3 µ ) α − (3 − 2 µ ) α , µ , α , then the result is given by: then the result is given by: dary should be determined and used in (8). then the result is given then the by: result is given by: x y x 2. Operations of MD RDM Fuzzy Arithmetic ate left and right boundary should beFor determined and used in (8). then the result is by: then the result isthe given by:is(15) rcreate itit is possible toGauss create then the result isgiven given by: ndaries, e.g. of type, formulas for the 2. Operations of MD RDM Fuzzy Arithmetic example, if X and Y are trapezoidal MF (14) and then the result is given by: For example, if X and Y are trapezoidal MF (14) and (15) then the result is then given by: result given by: e, 2. Operations 2. of Operations MD RDM of Fuzzy MD Arithmetic RDM Fuzzy Arithmetic e1 it 2. Operations For example, if X and are trapezoidal MF (14) and (15) A. Piegat, M. Y Pluci´ ski e, gr grn gr MD RDM Fuzzy Arithmetic gr a fuzzy gr ance, itshould At first glance, LetLet xgr = = f f ((μ, α µ ,itαof be a used horizontal MF representing a(4 fuzzy be a horizontal MF representing inndary be determined and in (8). et (µ ).2. grx then the result is+ given by: z)µ =µ (3 2yare ,1] . (3 − 2 µ )α µresult + (4 3))µα α µ∈ α ,µ(19) α (4 α − 3∈ µ )span α[0, grz xx)) then the is2− given gr amwe interested the representa z z = µ − 3 µ ) α − µ(3 + − (4 µµ − 3α)µ ,)− α− µ− ,If3− α ,)− α − 2α [0, α− α,in , yαµ ∈ 1] .(19) (19) xby: xα x1] gr gr gr x= y(4 x(3 x(3 y[0, Let x = f ( µ , α ) be a horizontal MF representing a fuzzy then the result is given by: Operations of MD RDM Fuzzy Arithmetic A. Piegat, M. Pluci´ n ski z = + (4 − 3 µ − 2 µ ) ,)y= µyµ,µ2,µ.,α 1] .− x e to create z = − (3 ,)[0, ∈ [0, 1],.α ( + µ α µ+ )x,,− α ,[0, α[0, α of MD RDM Fuzzy Arithmetic gr gr gr x yµ yx,y∈ Let x = f ( µ , α Let ) be x a = horizontal f ( µ , α ) MF be a representing horizontal MF a fuzzy representing a fuzzy gr amx yα x1] z = − (3 − 2 ,,α ∈ .yy(19) (19) µ + (4 − 3 µ ) α µ ) α αxxµ x x x y,3 gr nce unamwould not be unam2. Operations of MD RDM Fuzzy Arithmetic terval X (11) and y  = f(μ, α ) be a horizontal MF representing z z = = − (3 − 2 , (3 , − 2 ∈ 1] , . (19) µ + (4 − 3 µ ) α µ + (4 − ) α µ ) α α µ α µ , α interval X (11) and y = f ( µ , α ) be a horizontal MF repregr x y x y x y y y t gr Let x = f ( µ , α ) be a horizontal MF representing a fuzzy will 4D-subtraction result, then it can be determin (19) x of span s(z ) is determined. It can be found with known methgr gr interval X (11) and y = f ( µ , α ) be a horizontal MF repregr gr glance, it gr(3 − 2 µ )αMF gr gr interval Xgr (11) and interval y (12). (11) y=a = =Xa ffuzzy (µy ,MF αand be horizontal f ((4 µ ,− αare MF be areprehorizontal reprewill gr µ are y )zrepresenting y3)fuzzy Let x = f ( µ , α ) be a horizontal a a fuzzy interval Y (12). µ + µ ) α − , µ , α , α ∈ [0, 1] . (19) gr gr be a horizontal MF representing it then. will However, it will z = µ + (4 − 3 µ ) α − (3 − 2 µ ) α , , α , α ∈ [0, 1] . (19) x x y x y senting a fuzzy interval Y If we are interested in the span If representation we interested s(z in the span r ) of the gr x y x y gr If we interested If in we the are span interested representation in the span s(z representation s(z ) of the ) of the gr interval and yf (gr µbe ,found αay )horizontal bezwith a= horizontal MF µ +MF (4 meth− 3 µ )repreαx − (3IfIf 2 µare )are αy,interested µwe , αxare , αyinterested [0, 1] .grin (19) of MD RDM Fuzzy Arithmetic x fuzzy =interval µ= ,can αYxf )(be representing a−we fuzzy of function examination: in∈ span representation s(z ))of s(z ) isXLet determined. It known If the representation spangr representation s(z ) of senting a(11) fuzzy (12). l-span gr the grgrs(z we interested inthe the span representation s(z of the tods be grunamsenting a interval senting Y a (12). fuzzy interval Y (12). If we are interested If we in are the interested span in the span representatio ) of the interval X (11) and y = f ( µ , α ) be a horizontal MF repres(z ) = [ min z , max z ] = [−3 + 3 µ , 4 dbe y = f ( µ , α ) be a horizontal MF reprecreated. y 4D-subtraction result, then it can 4D-subtraction be determined result, from: then it can be y 4D-subtraction result, 4D-subtraction then it can be result, determined then itgrthen can from: be determined from: l of senting aXexamination: fuzzy interval Yand (12). urinterval X[arepresenting ygr =fuzzy (µ , αIfy )we be a horizontal MF repre4D-subtraction result, then it can be determined ds function αxcan ,αy be determined αxfrom: ,s(z αy gr 4D-subtraction result, itbe from: : xgr = + (b ] fgr grof the grMF x(11) x−a x )aµ 4D-subtraction result, then it can be determined from: ver, it will ) be a horizontal ) If we are interested in the span representation are interested in the span representation s(z ) of the urgr gr gr gr gr If we are interested in the span representation s(z ) of the 4D-subtraction result, 4D-subtraction then it can result, determined then it can from: be determined senting a fuzzy interval Y (12). X : x = [a + (b − a ) µ ] (12). MF onY the level µerval . Furµ . :Furxmax If are interested representation s(z )grof the gr (xµ[a , αx x+ ,xα(b zµ ][a (µxwe ,+αx(b ,α )]a.x )µgr] in the span )= [Xmin xza fuzzy X xax )(12). = = y ), x :− x y− gr gr gr gr ly), gr senting interval grs(z gr gr -d ygr =s(z 4D-subtraction result, then itgr,zgr4 can be determined ,α+ )αx= be a[(d αrepres(z )be = []result, min ,+ max ]= + 3),µ3= ,µ− 4[,.from: min − 2]µ2z,gr ]µ,,] max gr grz :, α xusually), +x(b a−x )MF µα]xx,Y− y[a y cx + −,xaα− (d bx,− a)]x ).µ ]s(z αresult, 4D-subtraction result, then it ,can determined from: gr gr gr∈ ygr µ [0, s(z ) =then [(11) min zcan max z)[gr= = [ min [−3 z , 3 max µ z − ] 2 = µ[−3 ]determined [−3 ,+ + 42+ − ,−,α2 µz ] ,] = [− ly), grf (Xµas gr gr 4D-subtraction then it can be from: xhorizontal x )), x ,,(11) gr gr (not ually), α -cut s(z ) = min z , max z ] = [−3 3gr µ 41] µ s(z ) = [ min z , max z ] = [−3 3 µ , 4 4D-subtraction it be determined from: gr gr gr gr gr s(z z ( µ , α max z ( µ , α α ) = [ min α α α α , , α α α , − a ) − (d − c + b − a ) µ ] α , + [(d (11) -+ s(z ) = [ min z , max z ] = [−3 + 3 µ , 4 − 2 µ ] , x y x y x y x y x y x y α , α α , α α , α α , α x x x x x x x ble X : x = [a + (b − a ) µ ] x y x y x y x y − a ) − (d − c + − b a ) − − a (d ) µ − ] α c , + b − a ) µ ] α , + [(d + [(d (11) (11) (b − a ) µ ] gr ,y,αmax) z=αx][,αmin = [−3 z + ,(20) max 3 µ ,z4 −] = 2 µ[−3 ], + 3µ , 4 − x xx xα ,xα,αy[yminαzαxα x x(b xx] x x xx x x s(zx )ααx= , µ .x Y ,αs(z x (20) terval (12). el Furαx ,α+ ,αy x+ (20) y X ble + x(d a= ) [a −1] as(z )xxµgr])α= [(d (11) (20) αx gr ,αy x ,xαyαyx ,gr αyx ,αy αy αgr µ :, α ( gr grgr e called RDM-variable x ,αy xx− x[0, x −xc− x axb)xµ− x , [ min x∈ (20)(20) ),eariable dis, max z ] = [−3 + 3 µ , 4 − 2 µ ] , grx )(16) gr z(11) gr s(z In the case of discussed example, extrema of lie not µ , α ∈ [0, 1] ) = [ min z , max z ] = [−3 + 3 µ , 4 − 2 µ ] , µ ∈ [0, 1] . µ ∈ [0, 1] . (19) corresponds to α The span (20) of z x µ ∈ [0, 1] . µ ∈ [0, 1] . − a ) − (d − c + b − a µ ] α , + [(d sdisusually), µ , α ∈ [0, 1] µ , α ∈ [0, 1] − a ) − (d − c + b − a ) µ ] α , (11) s(z = [ min z , max z ] = [−3 + 3 , 4 − 2 µ ] , x x x x x x x Fig. 5. Span r x x x x x x x µ ∈ [0, 1] . α , α α , α µ ∈ [0, 1] . ,αy[0, 1]α.x ,αy gr etermines − aextrema − (dx − − +the [(dxresult µαx∈ +dis(bx but − aaxon )relative µ ] discussed x )domain. x + bminimum xlie gr ex-ve µdis,= αx[a∈ of [0, 1] αx ,a αxyx)µy]ααxx ,,αyx y (11) gr . µ ∈ [0, 1] .z and µα(20) ∈ [0,1,1]α. y = 0 for(20) cal (20) inside In the case ofboundaries example, ofcThe (16) not min = max z + (b − a ) µ ] Y : y M-variable (20) gr x y y y cal ,y= αgr = ∈y + [0, 1]y (b [0,origin 1] − (dof gr.gr(19) corresponds gr (19) )yµagr1] ]= Y−the : cYy µ m ecorresponds the local gr span y[0, µ= [0, . ofThe -l∈ [a µ ][ay The +to(bαyminimum aµαy )∈ ]∈1.1] :local Y∈,ya:− µ(20) ∈ [0, 1] to α 0,α0, α = correspo (20) of (20) of zSpan α bαxxy[a − )y(b µ,+ ]result αx−xthe delative Fig. 5. represen grz y )(11) (19) corresponds (19) αzxgrcorresponds =(19) 0,The αcorresponds = 1αto for = =1for 1for for= 1 The (20) zThe span z grto xα= yαSpan =+ =a(b 0the and maximum = grx + side onY boundaries of domain. x − abut x )disxto xµ y span x0, y= x− y span [0, . 1] gr (19) corresponds toto = = 11representati The span (20) of zof Fig. 5. to α 0,for α The span (20) of = [a + − a ) µ ] : yα x y gr , − a ) − (d − c + b − a ) µ ] α [(d (12) yfor (19) corresponds α = 0, α The span (20) of z y y y y y y y y y y x y x= gr gr gr gr gr gr (19) corresponds (19) to α corresponds = 0, α = 1 to αDiv = The span (20) of The z span (20) of z gr + [(dy − ay) − (dy − cy + by − amin Division X /Y of t x y xD lSpan , (12) , ) µ ] α (12) min z min z and α = 1, α = 0 for max . and α = 1, α = 0 for max gr gr ory− z)= min zµ=gr 1, for =(12) 1, zygr . y= =for 0αfor max z gr . zxgr . grmax xα yα y of granule (16) is−]given by: rresponds 0gr[(d and the αayy(d + (b aayy)maximum :]αyx ==αy[a= ,x z= − )µ (d cµto −αbax yy= )− µyy and − ]1. αcy y,α+xgrmin b= azαgr ]min α0and + [(d x1, (b[0, − )toµ4D-result (12) gr y+ yy− Diz ∈ 1]athe yα ya− yy]+ y− yy)and ylocal yY z α α = 0 max z . the and α 1, = 0 for max xf+ gr gr gr orx y min and = 1, α = 0 for max z . = [a + (b − ) Y : y x (20) yz x(19) (19) corresponds to α = 0, α = 1 for The span (20) of z y(dy −ycy +yby − The span of (19) corresponds to α  = 0, α  = 1 for , − a ) − a ) µ ] α + [(d gr (12) npan e coorlocal Cartesian coorcorresponds to α = 0, α = 1 for The span (20) of z µ , α ∈ [0, 1] min z min z and α = 1, α = and 0 for α max = 1, z α . = 0 for max z . y y y y y x y x y y x y x y gr (19) corresponds to αx =Multiplication 0, αy = 1 for The span (20), of z (12) of two independent fuzzy in ndthe(d4D-result (16) given 1] µy − ,α -− aof gr y ,y∈ X/Y Z : ind xDi gr gr = 1, aµy[0, )]∈ (d +by: bαmin +bgranule [(d Division of= two 1] ∈cza1] [0, 1]b]α− ,∈ gr gr ,isµ − a− )α α−− yαx = 1, αy min yµ y − c(12) yµ yy− y )µ ndy) − y[0, minz and . . Division 0zfor max . 1, ααyy = 0 gr grα.zxαx= boundThe left and = 0 for formaxz max zgr X/YX /Y of two indep a] ,y) − − a1, ]α=y ,0=for )+µ =, [2 + 3[0, µ+ ,1] 9[(d µ(d . and (17) (12) ayy)s(z µ ]cgrybound-- + (b y2− yz[0, y+ yx = y )µ yterval. y− α ∈ min and α α max z sed y y sian gr grexpressed sed , 1] 2 µ ] , fuzzy µ[2two ,α ∈ [0, Xintervals ·Y Multiplication = Z : fuzzy xgr (µ ,intervals αintervals ·/Y ytwo (µZ,independent α zgr (µ pressed ry xRacoor(1]µ ) are Multiplication oftwo independent s(z ) c=of +yy− 3∈ µa[0, ,y9µ − µ (12) ∈ [0, 1] . Multiplication (17) x )Xof yx)gr=(µ of Multiplication two independent oftwo two fuzzy independent fuzzy Addition independent intervals -y − ,)/y αµ = : gr ∈ [0, 1] , α x )g ) − (d − + b ) µ ] α , ddeft Multiplication of independent fuzzy intervals y y y y y Multiplication of two independent fuzzy intervals ( µ , α X /Y = Z : x Multiplication ofoftwo independent fuzzy intervals boundAddition ofis independent fuzzy intervals Multiplication two independent fuzzy intervals Addition oftwo two independent fuzzy intervals xinter Multiplication of Multiplication two independent of two fuzzy independent intervals fuzzy Addition of two Addition independent of two fuzzy independent intervals fuzzy intervals The span (17) not the addition result. The addition red ∈ [0, µ , αgrxgr , α ∈gr[0, 1] gr gr gr gr gr gr gr For example, if∈ Addition µygr , αα ααy[0 gr X ·Y = gr X(17) +form Yofis =two Z : independent xthe 1]the (µaddition , αx ) +(16). yfuzzy , αintervals )span = zaddition , αXxgr·Y ,areα2D) ,two expressed gr (µgr gr :µgrof xgr X ·Y x,,xα ,µ αindependent )gr µ(xα,µ)α),·yα )fuzzy zgr= (Z,µ(αy:,µα ,gr ,)yα,)µ (xµy,,∈ (y,µ)(intervals (xαµ ysult ygr yof Multiplication independent fuzzy gr gr = Zindependent : xMultiplication Xx= ·Y =Z:X(xZ :,(intervals xµtwo (gr µ·Y ,α ) ·Zyfuzzy α = , xzα )((ygrµ ·µ yµ ,,(αα )z(gr = zα ,(yα ,),·α intervals (8) xα yF of The is(()zµ only Thehas span not result. The x ,gr x·gr yy y,= gr, α X+ Yof =Y Z=4D-function : independent xZgr: (xµgr +xY)yfuzzy (yµ )α =µy ),zα , ) , , (αµXx ,)+ ,Multiplication αxintervals µ α Addition two X α ) · y ( , = ( µ α ) , of two ·Y = Z : x µ , ) = z µ , , α ) , dependent fuzzy intervals y x y gr gr gr gr gr x y x y (13) x y x y (8) µ , α ) · y ( µ , α ) = z ( µ , α , α ) , X ·Y = Z : x ( X = Z : + α + ( µ , ( = + ( y µ , ( α µ , α ) , = z ( µ , α , α ) , th (21) then the division x y x y x intervals x = y Z : x (µ X gr of two independent gr Addition , α·Y y Z(:µofx, αthe = , α(21) zxmultiplication ) ·(yµ , α (µx , α(21) = z(21) (µthe ,the α(r X ·Y d has −(8) (dthe −+ x ) ·= y()µ4D yy) ,(21) X Yµabout = Z :4D-function x(8) (gr,µα, xα,xα)y+∈ y[0, (1] µThe , αy )span =offuzzy zgr (µ , αx ,aα2D,xgr y (13) x4.,grαMF c) µmaximal Fig. of the span representation result information the uncertainty the result. (21) lt form of (16). is only (13) (13) y )(13) µ , α , α ∈ [0, 1] µ , α , α ∈ [0, 1] For example, if X and For example, if X and Y are trapezoidal ) gr µ , α ∈ [0, 1] µ , α , α ∈ [0, 1] x y x y gr gr gr gr gr gr (21) y x y gr ,µα αα)xyxgr example, if X and YM (µ ,α µα ,y[0, α∈1] ,)[0, ∈yµ 1]z, α(For ·Y =[0, µ ,)α·(22) , α= = α , ,yα gr (13) Y(µfuzzy = x µ + (,1] µ[0, , α1]y )µgr = (µ = Zy ):zµ xgr (xµµrepresentation ,µα αyx,yx,x∈ )∈ µα )[0, = µ∈ , α[0, ) ,grresult xy(1] xy)gr· X x·[0, µ, ,(αzα ,,∈ α(αyµy[0, ,the ,,Xresult. α 1] ,))αx ) +X y+about , αZX = z(= (,µ µxα,,,:xα gr x:,Z y:)(x,µ ,(α x, α x,(,,α y ),∈α y ):+ y ),∈ g )µ X(αzµx·Y = Z= xα (yMF µ·Y ,(αµof µ ndependent intervals α ,α 1] , α4D 1]ydivision (formation µ x,Fig. y )the yy) ,the 4. span of the x multiplication ythen Y Z x ) + y z ( ) , α α µ , α α the maximal uncertainty of the result zis then multiplication result z is expressed x y x y he x left ( µ ) boundary x ( µ ) (21) grzgrthe µ , α , α ∈ [0, 1] (13) (21) (8) L L x y (13) For example, if X is trapezoidal MF (1,3,4,5): µ , α , α ∈ [0,(13) then MF the division result on= x z (21) ) 1] For example, if X is trapezoidal MF (1,3,4,5): µ , α , α ∈ [0, 1] For example, if X and Y are trapezoidal For example, if (14) X and and Y (15) are trap µ , α , α ∈ [0, 1] x y on(22) αof , conα [0, ifFor X1]For andexample, YFor areifexample, trapezoidal Xyand and Yare MF are (14) trapezoidal and (15) MF (14) and (15) y ∈)x( For example, ifµ trapezoidal example, if1]MF X(1,is3,(1,3,4,5): trapezoidal MF αx , α [0, ur line ,grαFor example, Xifxifand YYifYgr are trapezoidal MF (14) and (15) y ∈(1,3,4,5): X and are MF (14) and ( example, Xytrapezoidal trapezoidal MF 4, For 5):µ ,example, For example, trapezoidal MF (14) and (15) µFor [0, ,α +µy1] (xµ) ,of α )= zxgr ,Xis α,is ,α ) y, ∈MF )-(xµ x ,yα yconxα For example, are trapezoidal MF (14) and (15) gr Ytrapezoidal grX trapezoidal grif grare gr if X is(= (1,3,4,5): For example, ifXand X For and example, Y if and Yby: MF are (14) trapezoidal and grifisXexpressed gr ary xxL (For µ ) example, gr gr = x · y = [(1 + 2 µ ) + (4 − 3 )/y αMF ] e(= z Subtraction of two independent fuzzy intervals (1 + 2 µ ) + (4 − 3 µ ) α , (14) x then the multiplication result z then the multiplication result z(15) is expressed is gr xgr gr (13) then the multiplication then the result multiplication z result z by: is expressed by: gr x gr = x z gr by:gr µ gr gr then the multiplication result z is expressed by: gr gr3 µ )αx , then the multiplication result z is expressed by: For example, if X is trapezoidal MF (1,3,4,5): then the multiplication result z is expressed = (1 + 2 µ ) + (4 − (14) x - xis, αtrapezoidal Xα MF (1,3,4,5): gr gr µ the multiplication result z and is expressed by:(14) = x(15) /y = z and X−and Yα trapezoidal MF (15)expressed (1is+trapezoidal 2intervals µ ) +For = − MF 3(1example, µ+ )(1,3,4,5): α2xµ , )if+Xif(4and 3then )are ,example, (14) x = x(4For ∈ [0,of 1] For For if X and Y(14) are trapezoidal MF then the multiplication then the result multiplication result xare example, ) ofyconubtraction fuzzy x(9) Yµ(14) (15) −by: 2z µ )isαyexpressed ] , µ , αx ,by α ·z[(1is + µ) + Y istwo = (1 +if(1,2,3,4): 2X µ ) + (4 −gr3 µ )example, αx ,,(14) xgr gr independent gr grtrapezoidal gr gr gr grMF (14) and gr gr (3gr gr x gr gr(14) grresult gr and trapezoidal MF gr gr (9) X − Y = Z : x ( µ , α ) − y ( µ , α ) = z ( µ , α , α ) , = · y = [(1 + 2 = x ] · y = [(1 + 2 µ ) + (4 − 3 µ ) α µ ) +repres (4 − z z then the multiplication z is expressed by: gr gr gr gr gr gr y (4 x, y z = x (14) the multiplication is−(4 expressed ·result y then =z z[(1 + = 2grµ·x)zy gr + ·= y(4gr −= 3·+ [(1 µyresult )2grα + µz gr )(4+ − 3αµ(4 )xα−xby: ]3 µ )α ]The span z is (9) ∈and [0, (1 +MF 2 µis)(1,2,3,4): + − 3 µthe )α xµ gr gr x)]2+ Y1] trapezoidal MF (1,2,3,4): xtrapezoidal =α (1 + 2 µand ) +isgr −trapezoidal 3(9) )xα=x ,and (14) xmultiplication then expressed by: = x [(1 µ 3 µ ) ] x = [(1 + 2 µ ) + µ , α , α X is MF (1,3,4,5): gr gr Y(4 is Ygr trapezoidal MF (1,2,3,4): x x x = x · y = [(1 + 2 ] µ ) + (4 − 3 µ ) α z (18) = (1 + 2 µ ) + (4 − 3 µ ) α , (14) x gr gr (22) ) =2xµ ) ·+y (4= + ] ∈(T −[(1 3 µx(22) )α2x]µ ) + (4(22) − 3µ µ, α)(22) α,xα z = x · y = [(1 z + gr elaX −and Y =Yand Zis: xYtrapezoidal y[0, ((1,2,3,4): µ ,(1, αyµ ) )= µ ,2αµx)gr ,α αyy,) , gr xgr · [(1 + MF (22) µ(,µ α,xα, xα)y− 1] )µ+ (3 µ2yα)µ(3 α]),yα ] (22) ,y] 1] ,µα,,·xα , xα,∈ µ∈ [0, )by +1] (3 .− 2 µx(22) )is α. yysho ] ,bo ·2[(1 + [(1 +∈ = (1 + (3(− (15) y∈ Formula the full 4D-result )ints is trapezoidal MF 2, 3,+z4): µ )2+ − )3gr αµ+ ]α ,+ )(3µ + ,(3 α− ,)2α)+ ∈ [0, ,µµ,µ.)αdescribes α [0, 1] .(22) ·µ+ [(1 gr yα ela(26) and gr)(3 gr y)µ x− y[0, gr gr (18) [(1 − 2 µ 1] . · al relalying at equal relaand Y is trapezoidal MF (1,2,3,4): µ − 2 α ] , µ , α , α ∈ [0, 1] · [(1 + by = x · y = [(1 + µ + (4 − ] z gr gr gr µ ) + (3 − 2 µ ) α , (15) = (1 + y lore MF (1,2,3,4): (9) y x y = x · y = [(1 + 2 µ ) + (4 − 3 µ ) α ] z y x y x µ ) + (3 − 2 µ ) α ] , , , α ∈ [0, 1] . · [(1 + y by x = (1 + 2 µ )µ+ (4 − 3 µ ) α , (14) y x y = (1 + µ ) + (3 − 2 µ (1 ) α + µ , ) + (3 − 2 µ ) (15) α , (15) y y The span representatio x µ ) + (3 − 2 µ ) α ] µ , ) + µ (3 , α − , 2 α µ ) ∈ α [0, ] , 1] . µ , α , α = x · y = [(1 + 2 µ ) + (4 − 3 µ ) α ] · [(1 + · [(1 + z y y and Y is trapezoidal MF (1,2,3,4): gr -ore y interested x span y the y 2D simplified x , αyy∈ [0, 1] + µ ) + (3 − 2 µ )α , αxmore tion. If we(22) are in (22) The representation s (22)x gry ∈ = (1 (15) Ar-of more MFthen xL (µzgr ). ,isA given grby:Y are trapezoidal MFy(14) ·and gr s(z )4F= µ ) + (3 − 2 µ ) α ] , µ , α , α ∈ [0, 1] . [(1 + For example, if X and (15) gr by (26) and is shown in Formula (22) describes the full Formula 4D-result (22) of the describes multiplicathe full µ ) + (3 − 2 µ ) α ] , µ , α , α ∈ [0, 1] . · [(1 + y x y by gr gr Formula (22) describes Formula the (22) full 4D-result describes the of the full multiplica4D-result of the multiplicaFormula (22) describes the full 4D-result of the multiplica , (15) y = (1 + µ ) + (3 − 2 µ ) α , (15) y x y eal=MF is given by: then z (1rela+ µ ) + (3 − 2 µ ) α , (15) gr y equal µ ) + (3 − 2 µ ) α ] , µ , α , α ∈ [0, 1] . · [(1 + this result in the form of a span s(z ) then for y by (26) and isof shown in Fig. (1,2,3,4): Formula (22) describes 4D-result of multiplicay xdescribes y (22) the by: yz =is(1 given then z is given then Formula describes full 4D-result the multipl + (3 − 2 µ )αy , (15) + µ )by: Formula (22) thefull full 4D-result ofthe the multiplicahorizontal grMF. (22) describes Formula the (22) full describes the of full the 4D-result multiplicaofst ethen grresult is if given by: z= the is given by: For example, X and Y(4are trapezoidal MF (14) and (15) tion. Ifwe we interested 2D tion. simplified If4D-result we are interested in of the ofof 2D tion.Formula IfIf we are interested inin the 2D simplified representation of tion. If we are interested tion. in weare the are 2D interested simplified inthe representation the 2D simplified ofrepresentation representation gr (2 + 3 µ ) + − 3 µ ) α + (3 − 2 µ ) α , µ , α , α ∈ [0, 1] . z ). Athen more gr be used. gr x y x y tion. If are interested in the 2D simplified representation tion. If we are interested in the 2D simplified representation gr s(z ) = [ min gr gr gr gr tion. If we are interested in the 2D simplified representation of is given by: then z gr gr4D-result = (2 + 3 µ ) + (4 − 3 µ ) α + (3 − 2 µ ) α , µ , α , α ∈ [0, 1] . z :10) Formula (22) describes the full 4D-result of the multiplicagr gr gr x y x y Formula (22) describes the full of the multiplicagr gr tion. If we are interested tion. If we in the are 2D interested simplified in the representation 2D simplified of = (1 + µ ) + (3 − 2 µ ) α , (15) s(z )should [ min (2 3by: µ )given + − 3(2µ+ )α3xµ+)(3 +Formula − (42−µ3)α µ(22) ,describes α(3x ,in − αy2the ∈ µ(16) )[0, αthis ,1]full .µof , α4D-result ,inα ∈ [0, 1] .of z is= then zthen is + by: result inin the form a span s(z )(23) then formula (23) should y z(4 = y),αresult xµ+ ythis xspan ythe en gr the given αxz,rep α this result form ofaof amultiplicaspan s(z this result informula form of a=should span s(z ) then (23) is by: zgiven the of the grresult gr y this form result a the s(z form a span s(z ) then formula ) should then formula (23) grthe MF. gr α , α this result in the form of span s(z ) then formula (23) should = (2 + 3 µ ) + (4 − 3 µ ) α + (3 − 2 µ ) α , µ , α , α ∈ [0, 1] . z this result in the form of a span s(z ) then formula (23) sho gr s(z ) then grformulagr(23)grshouldx y = 10) gr x y x y this result in the form of a span (16) z(10) = ) α − (3 − 2 µ ) α , µ , α , α ∈ [0, 1] . (19) tion. If we are interested in the 2D simplified representation of gr µ + (4 − 3 µ(10) tion. If we are interested in the 2D simplified representation of s(z ) = [ min z , max z ] be used. x y x y this result in the form this result of a span in the s(z form of a span s(z ) then formula (23) ) then should formu (16) (16) be used. be used.  (2(3 +4D-solution 3 µ2)µ+ 3,µα)xα −1] 2 µ. )αspace µbe ,are αused. ,interested αy ∈ [0,(16) 1]be tion. Ify,we in.bethe 2D simplified representation of be[0, used. 4)y: − 3zµx )∈ α=x[0, + − )α(4 ,(4 αx y+ ∈(3 [0, gr The exists which cannot y, − 1+ αx ,αy used. be grαx ,αy + 3µ µ(16) )2+ 3µ µin )α αthe (3 −[0, 2 µ1] )xin , µ ,α 1]gr . ingrused. z4D-solution be used. xin y(19) x, α ybe (16) in space which cannot this result the form of a∈ span s(z ) then formula (23) z),gr =αµ ,exists ,+ ∈(16) .αwhich +The (4 1] − 3 µ= )α(2 µ(16) )exists α− ,(16) α this result the form ofused. a(23) span s(zshould ) then formula (23) should be used. be x − (3 − y4D-solution xthe ythe =1 + 2 µ The 4D-solution The space exists in the cannot space be which cannot be (16) this result in the form of a span s(z ) then formula should (16) gr gr gr gr gr gr seen. Therefore we can be interested in its low dimensional ari-If we gr = The gr gr gr gr The 4D-solution (16) exists in [0, the1]space which cannot bebezgr − (16) interested the span representation s(zlow )interested of s(z s(z )2= )= []min , max zgrgr ] ] = [(1 [ min zµ ),, (5 max zµ )(4 ] 4−−4µµgr grz s(z s(z µ )(1 + − solution )T ) the = [ min ,grmax z)grmin = [ gr min , max seen. Therefore can interested its dimensional aribe used. gr gr,zmax (4 − 3 µ(10) )αare + (3 −Therefore 2variµ )αinywe ,(16) µwe ,α ,be α ∈ .wein used. seen. Therefore be can in be its low dimensional its low oa] (16) function variof two xseen. xcan ythe s(z )= zy,zgr s(z [gr zy])z] z,= max zgrz]+ gr[dimensional gr gr gr αxα ,[αxzmin αxz,gr αy] so αitx ,α αmin ,αmax The 4D-solution exists in interested the space which cannot be be used. [ min z max y,) xs(z y αxform ,αin α= exists in the space which cannot be α α so representations. Frequently, 2D-representation in the y s(zαs(z y ,αz gr x ,)αy= x cannot be pr ) = , ] [ min , max α , α seen. Therefore we can be interested in its low dimensional -4D-subtraction , α α , α The 4D-solution (16) exists in the space which cannot be x x ,α yy xα y,α (23) result, then it can be determined from: If we are interested in the span representation s(z x y ) of the x y so α α , α representations. Frequently, the 2D-representation in the form x y soo (16) αx gr ,form αy αx ,αy αx gr ,αy αx ,αy The solution granule gr the form representations. representations. Frequently, the 2D-representation Frequently, the 2D-representation in in the is unique. gr seen. Therefore we be interested in itsgrlow dimensional e-two can be interested in itsitcan low dimensional sh s(z )[= min ,be max zµgrPluci´ ] )+= The solution granule of gr gr= inµ1] a .simpli [= min z+ ,2max z+ ])] [(1 + 2µ = 2µshown )(1 + ), (5 − µ∈), )(4 − µ.[(1 )] ,+ µµµ∈ )(1 + 1] .), (5 − µt representations. Frequently, the 2D-representation the form The 4D-solution (16) inbethe space cannot [(1 2y ]s(z [(1 )(1 µ ), (5 − )(4 µ,)(1 − µ µ ,), (5 µ − µ[0, )(4 1] − µ )] ,cannot ∈[0, [0, seen. Therefore we can interested in[in dimensional variD-subtraction result, then can beexists determined from: n= ski sho s(z ) =which min z,A. ,zPiegat, max z,αgrM. αits αlow = [(1 + 2 µ )(1 + µ ), (5 − µ )(4 − µ )] , µ ∈ [0, 1] . αx+ sh n (16) exists in the space which cannot be = [(1 + 2 µ )(1 + µ (5 − µ )(4 − )] , µ ∈ [0, 1] gr gr gr α , α x y α α so it be presented x y x y = [(1 + 2 µ )(1 + µ ), (5 − µ )(4 − µ )] , µ ∈ [0, 1] . so representations. Frequently, the 2D-representation in the form α , α α , α A. Piegat, M. Pluci´ n ski s(z seen. ) = 2D-representation [representations. min z , max z ] = [−3 + 3 µ , 4 − 2 µ ] , equently, the in the form = [(1 + 2 = [(1 + 2 µ )(1 + µ ), (5 − µ µ )(4 )(1 − + µ µ )] ), , (5 − µ ∈ µ )(4 [0, 1] − . µ )] , x y x y µ so it cannot be presented in µ -coordinate. Fig Therefore we,αcan be interested in its2D-representation low dimensional rep-in the 3 form (23) Fig. 4 shows the MF of the representa Frequently, the (23) (23) µ αx ,αgr y y µ-c(we s(z cangr be interested inαxits 3 µ ), (23) shown in∈ aspan simplified wa sh (23) 3Figure 4 = [(1 + 2methµ )(1 + (5= −)(4 µ )(4 µ)(1 )]3 ,µ µ), ∈(51] [0, 1] . representation (20) gr )z is,determined. [(1 +µ2−)] µ + µ − µ )(4 − µ )] , µ [0, 1] . )= [s(z min max zgrlow ] = dimensional [−3 + 3be µ ,found 4 − 2 µwith ] , [(1 µi resentations. Frequently, the 2D-representation inknown the form of shows the MF of the span of the shown in a simplified way, (23) µ = 1 values. of span It can = + 2 µ )(1 + µ ), (5 − µ − , ∈ [0, . plication result. 3 µ α , α α , α gr y∈ µ= xdetermined. requently, thexs(z 2D-representation incan form gr [0, µ -coordinate. Fig. 6 pre µs(z .y µ of span )isis1] Itthe can be found with known meth(20) Fig. 5. Span representation of the 4D division result (25) span ) determined. It be found with known methods multiplication result. (23) (23) 3 Fig. 4 shows the MF of the span Fig. representation 4 shows the of MF the of multithe span rA µ -coordinate. Fig. 6 presen 3 Fig. 4 shows the MF Fig. of 4 the shows span the representation MF of the span of the representation multiof the multiAs Fig. 6 show ods of function examination: (23) Fig. 4 shows the MF of the span representation of the multiFig. 4 shows the MF of the span representation of the mu 3 Fig. 4 shows the MF of the span representation of the multiµ = 1 values. µfa odsofoffunction µ function ∈ [0, 1] . examination: examination: 4result. shows the Fig. MF of 4ofshows the span representation MF the span the multiFig.Fig. 5. Span representation the 4Dthe division (25)ofrepresentatio plication result. plication re µresult. =of1result values. plication result.plication plication results from theresu gr result. plication result. 4of the = 0,,4ααshows = )] 1 .the for MFplication The spans(z (20) gr of z (19) gr corresponds to 3αxgr result. res y, α Division X/Y of two independent fuzzy intervals, 0 2 / Y Fig. of the span representation of the multiAs Fig. 6 shows, the s Fig. 4 shows the MF span representation of the multiplication result. plication result. z ( µ , α , α ), max z ( µ ) = [ min x y x the y MF ofDivision Fig. 4grshows the span X representation of the multi-fuzzy intervals, As Fig. 6 0shows, gr and αs(z /Y of two independent ∈ / Y the solu αy 0corresponds αxplication zgr (max µ , αzxgr , α.to ),xα,max αx1, αfor min y = z0, ( min αα(19) for αµy ,result. = Thezspan (20)x = ofgr1, z)gr= yα y )].. yx[,= results from the fact that t plication result. res αx ,αy αx ,αy plication result. results from 4fuzzy intervals, 4 Division of two 0∈ /the Y fact that the gr independent gr gr 444 4 X/YX /Y in zgr and αx = 1, αy = 0 for max zgr . 4 = Z : x (µ4, αx )/y (µ , αy ) = z (µ , αx , αy ) , In theIncase of discussed example, extrema (16) liebut not 4 case extrema of (16) do notoflie (24)(24) gr In thethe case ofdiscussed, discussed example, extrema ofinside, (16) lie not (µ , α ) = zgr (µ , αx , αy ) , X /Y = Z : xgrµ(,µα,xα, xα)/y 4 insideonbut on boundaries of the result domain. The minimum4 y ∈ [0, 1] y 4The boundaries ofindependent the result domain. minimum corresponds Multiplication of two fuzzy intervals (24) inside but to onαboundaries the domain. minimum µ , αx , αy ∈ [0, 1] corresponds = maximum 0 of and theresult maximum to αThe αy =of1. For example, to αx = α andαythe to αx = αy = 1. The x= x =span y = 0 ultiplication of two independent fuzzy intervals if X and Y are trapezoidal MF (14) and (15) corresponds to, granule α and XSpan ·Y =ofZ4D-result : xgr αxx )=· granule yαgry(16) (= µ ,0α ) =the zgr (maximum µ , αxby: , αy )to , αx = αy = 1. For example, if X and (µ the is(16) by: Y are trapezoidal MF (14) and (15) ygiven the 4D-result is given is given by: MF (14) and (15) then the division For example, if Xresult and zYzgrgrare trapezoidal (21) gr granule (16) gr is given by: 4D-result then the division result is given by: X ·Y Span = Z :ofxgrthe ( µ , α ) · y ( µ , α ) = z ( µ , α , α ) , x y x y µ , αgrx , αy ∈ [0, 1] gr then the division result z is given (21) (17) = [0, [2 + 3µ , 9 − 2µ ] , µ ∈ [0, 1] ..(17) (1 + 2by: µ ) + (4 − 3 µ )αx µ , αs(z , αy)gr∈ 1]+ x s(z ) = [2 3 µ , 9 − 2 µ ] , µ ∈ [0, 1] . (17) , zgr = xgr /ygr = For example, if X and Y are trapezoidal MF (14) and (15) (1 + )+ )α)α (12+µµ ) +(4(3−−3µ 2µ xy gr gr gr (25) , z = x /y = The span (17) is not theis addition result. The addition result span (17) isresult not the addition result. The addition regr then theThe multiplication z expressed by: For example, if X and Y are trapezoidal MF (14) and (15) (1 + µ ) + (3 − 2 µ ) α The span (17)4D-function is not the (16). addition result. Thea 2D-inforaddition re(25) y (25) µ , αx , αy ∈ [0, 1] . has the the form The The span is only sult has formof of 4D-function (16). span is only a 2D- gr is expressed en the result z by: sultmation has gr the form of 4D-function (16). The span is only a 2Dgr multiplication gr [0, 1] x, α y ∈representation Fig.span 4. MF µ of, α the span the 4D multiplication about themaximal maximal ofresult. the result. = x · y about = [(1the + 2µ ) + (4uncertainty −uncertainty 3 µ )αx] of the zinformation gr.) of the of The result expressedresult Fig. 4.representation MF of the spans(z representation of the(25) 4D is multiplication resul about the maximal uncertainty of the result. (22) gr information gr gr (22) y +=µ[(1 µ )2+ z = x · ·[(1 gr5. x]x , αy ∈ [0, 1] . ) ++(32− µ )(4 αy− ] , 3 µµ)α, α by (26) and is shown in Fig. The span representation s(z ) of the result (25) is expressed (22) (22) · [(1 + µ ) + (3 − 2 µ )αy] , µ , αx , αy ∈ [0, 1] . by (26) and is shown in Fig. 5. 500 gr Pol. Ac.: Tech. 65(4) 2017 Subtraction of two independent fuzzy of intervals s(zgr ) = [ min zgr , max zBull. ] Formula (22) describes the full 4D-result the multiplicaαx ,αgr αx ,αgr Subtraction of two independent fuzzy intervals y y Unauthenticated gr s(z ) = [ min z , max z ]  Download tion. If we(22) are describes interestedthe in the simplified representation Formula full2D 4D-result of the multiplica-of Date  | 8/25/17 12:47 PM (26) αx ,α1y + α , α x y X − Y = Z : xgr (grµ , αx ) − ygrgr(grµ , αy ) = zgr (grµ , αx , αy ) , 2 µ 5 − µ  thisIfresult the form on. we are simplified Xin−interested Y= Z : xofina(the µspan , α2D − y ) (then µ , αformula ) = z ((23) µ , αxshould , αofy ) , (18) = , µ ∈ [0, 1] . , x )s(z yrepresentation (26) µµ 1 +42−µµ 5 − (18) , αy ∈ 1+ gr 1] beresult used.in the form µ is of, α a xspan s(z[0, ) then formula (23) should

t m .

Fuzzy number division and the multi-granularity phenomenon

)

-

Fig. 4. MF of the span representation of the 4D multiplication result (22)

)

)

Fig. 4. MF of the span representation of the 4D multiplication result (22)

)

e

)

r

)

Fig. 5. Span representation of the 4D division result (25)

Division X/Y of two independent fuzzy intervals, 0 ∈ /Y X /Y = Z : xgr (µ , αx )/ygr (µ , αy ) = zgr (µ , αx , αy ) , µ , αx , αy ∈ [0, 1]

f d

)

-

(24) 3. Mathematical properties of MD RDM

For example, if X and Y are trapezoidal MF (14) and (15) then the division result zgr is given by: Fig. 5. Span representation result (1 + 2ofµthe ) +4D (4division − 3 µ )α x (25)

zgr = xgr /ygr =

)

)

Fig. 6. Simplified view of the 4D-solution granule (25) z gr(x, y, z) in 3D-space X£Y£Z, without μ-coordinate

(1 + µ ) + (3 − 2 µ )αy

,

µ , αx , αy ∈ [0, 1] .

The span representation s(z gr) of the result (25) is expressed



gr

gr

s(z ) = [ min z , max z ] αx ,αy αx ,αy   1 + 2µ 5 − µ = , , 4−µ 1+µ

µ ∈ [0, 1] .

Commutativity For any fuzzy intervals X and Y, equations (27) and (28) are true.

(25)

gr ) of the result (25) is expressed The byspan (26) representation and is shown in s(z Fig. 5. by (26) and is shown in Fig. 5. gr

fuzzy arithmetic

(26) (26)

The solution granule of the division (25) is 4-dimensional, The solution granuleinofitsthefull division is 4-dimensional, so it cannot be presented space.(25) However, it can be so it cannot be presented in its full space. However, it can be shown in a simplified way, in the X ×Y × Z 3D-space, without shown in a simplified way, in the X£Y£Z 3D-space, without µ -coordinate. Fig. 6 presents surfaces forforconstant = 0and and μ-coordinate. Figure 6 presents surfaces constant µμ = 0 µ = 1μ = 1 values. values. As Fig. shows, the the solution granule This As6Fig. 6 shows, solution granule(25) (25)isisuniform. uniform. This results fromfrom the fact thatthat thethe divisor does Asitit results the fact divisor doesnot notcontain contain zero. zero. As



X + Y 

= 

Y + X (27)

XY 

= 

YX (28)

Associativity For any fuzzy intervals X, Y and Z, equations (29) and (30) are true.

X + (Y + Z) 

= 

(X + Y ) + Z (29)



X(YZ) 

= 

(XY)Z (30)

Neutral element of addition and multiplication In MD RDM FA, there exist additive and multiplicative neutral elements such as the degenerate interval 0 and 1 for any interval X.

will be shown further on, division results can be discontinuous and multi-granular in more complicated cases. Bull. Pol. Ac.: Tech. XX(Y) 2016

X + 0 

= 

0 + X = X (31)

X ¢1 

= 

1¢ X = X (32) 501

Bull. Pol. Ac.: Tech. 65(4) 2017 Unauthenticated Download Date | 8/25/17 12:47 PM

tral elements such as the degenerate interval 0 and 1 for any interval X .

tion cost c1 per 1 kg of the product is uncertain because it depends on current prices of components changing from month X +0 = 0+X = X (31) to month and which are results of negotiations with many supA. Piegat and M. Pluciński pliers. Price p1 of 1 kg of the product is also uncertain because X ·1 = 1·X = X (32) it is also a result of negotiations with customers (supermarkets, shops, dealers). It also depends on volume of purchased Inverse elements Inverse elements Example 1. A company produces a food product. The producproduct. Profit Pr of the company is equal to the difference In MD RDM FA, In MD RDM FA,fuzzy fuzzyinterval: interval: tion cost c1 per 1 kg of the product is uncertain because it between incomes and Pr =changing In − Cs. from The month global income depends on current prices of costs: components −X : −xgr = − [a + (b − a)µ ] − [(d − a) can be calculated as: In = n · p , where n is the number 1 to month and on negotiations with many suppliers. Price p1 of of kilothe product. Generalbecause costs of − µphenomenon (d − a + b − c)]αx , αx ∈ [0, 1] , 1 kggrams of the of product is also uncertain it isproduction also a resultare equal the on the and multi-granularity multi-granularity the multi-granularity phenomenon phenomenon dnd the multi-granularity phenomenon

of negotiations with customers (supermarkets, shops, dealers). It also depends on volume of purchased product. Profit Pr of 5 the company is equal to the difference between incomes and gr =[a [a = + (b [a−a) + − a) a) µ(b µ ][(d + µ[(d −]−+ a) − [(d a) − a) XX:X:xgr :xgrx= Xgr=:[a x+ +(b (b− a) µ] + ]−+ [(d a) costs: Pr = In ¡ Cs. The global income can be calculated as: A. P In = n ¢ p , where n is the number of kilograms of the product. 1 (d µa+(d ab+ bac)] − +α c)] bα − , α c)] , α α ∈ , [0, ∈ α 1] [0, . ∈ 1] [0, . 1] . −−µ−µ(dµ − (d− −a− + b−− c)] , α ∈ [0, 1] . x x x x xx x x A. Piegat, M. Pluci´nski General costs of = production to Cs = 100 000 + n ¢ c to Cs 100 000 are + nequal · c1, where the number 100 000 desc Piegat, 1, A. IfIfparameters Ifparameters If parameters ofofof two two of fuzzy two fuzzy intervals fuzzy intervals intervals XXXand Xand and YXYare are Yand are equal: Yequal: equal: are equal: fuzzy intervals and are Ifparameters parameters oftwo two fuzzy intervals equal: where the number 100 000 describes fixed costs of the company. fixed costs of the company. to Cs 100 + cHence, the number 100 000 describesby equation: 1, where A. Piegat, M. Plucin  = a the interval ¡Y isis000 the the company is profit expressed = ,, x= bba= by= ,yb,byb,cxyy,xc,=cx= cx= bcy= ,yc,ycyc,dxyxd,=x= d = d ,ydy,dy,d,then ,=then dthe then the interval interval the=interval −Y −Y is−Y thenis· the axax= aa= = interval −Y isthe the x xa= ya,a xyyxb x = b x = c xcd y= xthen ythen y ,the to Cs = 100 000the + profit ncompany · c1, Pr where the Pr number 100 000bydescribes ya,b Hence, is expressed equation fixed costs of the company. A. Piegat, M. Pluci´nski additive inverse interval when inner RDM-variables additive additive additive inverse inverse interval inverse interval interval ofofof X ofX,X,X when ,ofwhen Xalso ,also when also inner inner also RDM-variables inner RDM-variables RDM-variables additive inverse interval , when also inner RDM-variables fixed costs of the company. Hence, the company profit by Prnis= p1 − (100 000 + n ·100 c1) = n ·describes (p − c1) − 100 00 to Cs = 100 000Pr + ·expressed cn1,· where theequation: number 000 are equal: α  = α . It means full coupling (correlation) of both x y are are equal: equal: areαequal: α = α = α . α It = . means It α means . It full means full coupling coupling full coupling (correlation) (correlation) (correlation) of both of both of both are equal: α = α . It means full coupling (correlation) of both x x x y y xy y Hence, the company profit Pr is expressed by1 equation: costs company. 100 000 + nof where describes uncertain values xxand and modelled intervals. 1·, cthe uncertain uncertain uncertain values values xvalues and y ymodelled xyymodelled and modelled y modelled byby intervals. intervals. by intervals. uncertain values xand byby intervals. Pr to = Cs n · p=1 fixed − (100 000 +· cnthe n the · year, (p1number −the c1)company −100 100000 000managers . In next plan to achieve ge 1) = Pr = n · p − (100 000 + n · c ) = n · (p Assuming that 0 2 / X, a multiplicative inverse element of the Hence, the company profit Pr is expressed equation: fixed costs of the company. 1 1 1 − c1 ) − 100 000 . Assuming Assuming Assuming that that 0 0∈ /0∈ X∈ /X, , aX0amultiplicative ,∈ /amultiplicative multiplicative X , a multiplicative inverse inverse element inverse element element ofofthe ofthe theof the profit ‘about 5 500 000’ definedby Assuming that /that inverse element by the triangular fuzzy nu fuzzy interval isisequal inMD MD FA: the next year, thePrcompany plan to achieve general the next year,the theIncompany company managers plan tomanagers achieve general Hence, profit is expressed by equation: fuzzy fuzzy interval fuzzy interval interval XXisX Xisequal equal X isinin equal inMD MD RDM inRDM RDM MDFA: RDM FA: InFA: fuzzy interval equal RDM FA: next company to000 achieve general (5 300year, 000,the 5·managers 700 000). Company eval Prprofit = nIn·defined pthe (100 000 +5defined n500 · c1000, )= nthe (p c1) plan − 100 .experts 1− 1− ‘about 5 500 000’ by triangular fuzzy number profit ‘about 5 500 000’ by the triangular fuzzy number 1 11 1 1 11 1 11 1 profit ‘about 5 500 000’ defined by the triangular fuzzy number the cost as ‘about 1’ defined by the triangular fuzzy nu Pr = n · p − (100 000 + n · c ) = n · (p − c ) − 100 000 . 1 1(5 300 000, 1 5 700 000). 1 Company 1evaluated 5 500 000, (5 300 000, 5, ,500 000, Company experts : : : gr= , 5 700 =: = gr = In, the next year,000). the company managers plan toexperts achieveevaluated general (5 300 000, 5 500 000, 5 700 000). Company experts evaluated x+[a [a + (b a) XX Xxgr µa) ]+ ]+ − µ− ]a) + − [(d a) − (d(d µa) − (d − a− µ+ a(d b+ bc)] a− + αc)] b α − c)] α x xXgr [a[a +(b+ (b−(b −a)− a) µ ]µ− +[(d [(d[(d a)− −µ− µ − a+ b−− − c)] α (0.9, 1, 1.1) and the price p as ‘about 1.2’ expressed b xx x xthe cost c1 as ‘about 1’ defined by the triangular fuzzy number number 1 the cost ‘about 1’the defined the triangular number profit ‘about 5 500by 000’ defined by fuzzy the fuzzy In cthe next year, company managers plan totriangular achieve general 1 as the cost c as ‘about 1’ defined by the triangular fuzzy number Fig. 1.2, How many kilograms n of by the(1,product shoul (0.9, 1, 1.1) and the price p1 as ‘about 1.2’ expressed 1.2, 11.4). αα α∈ ∈ α [0, ∈ [0,1] 1] . .[0, 1] . x x∈ x[0, x. 1] (0.9, 1, 1.1)‘about and the price p1defined as000, ‘about 1.2’ expressed byexperts (1, (5 300 000, 5 500 5by 700 000). Company evaluated profit 51.4). 500 000’ the triangular fuzzy number How 1, many ofin thethe should1.2’ the company (0.9, 1.1)kilograms and the nprice pproduct as ‘about expressed by (1, company produce year? 1 next 1.2, 1.4). How n1’ ofyear? the product should thefuzzy number themany cost asin‘about defined by theexperts triangular 300 000, 500ckilograms 5the 700 000). Company evaluated 1000, Ifparameters parameters oftwo fuzzy intervals are equal: produce next IfIfparameters Ifparameters If parameters ofofof two two of fuzzy two fuzzy intervals fuzzy intervals intervals XXXand Xand and YXY(5 are Yand are equal: Yequal: equal: are 5equal: 1.2, 1.4). How many kilograms n ofcalculation the product should the fuzzy intervals and are To get the answer, following should be real The results are giv company produce the next year? (0.9, 1.1) anddefined the price p1the asfollowing ‘about fuzzy 1.2’ expressed by (1, the cost c1 asin1, ‘about 1’ by the triangular number  = a , b  = b , c  = c , d  = d , then the interval 1/Y is To obtain the answer, calculation should be y x y x y x y axax= aax=xa= a , a b = , = b a b = , b , b c = , = c b c = , , c c d , = = d c d = , d , d then = , then d the , then the interval interval the interval 1/Y 1/Y is 1/Y is is company produce in the next year? a , b = b , c = c , d = d , then the interval 1/Y is y yx yx x x y y yx yx x x y y y xy x x xy y yx y y ˜ ˜ ˜10 6 6 6 ˜ ·the 8 ·−105.5) ·+8.5 To get the answer, following calculation be realised: −‘about 100should 000 (5.3 ·product 10 , 5.5 ·D10=,65.7 1.2, 1.4). How many kilograms n of the should (0.9, 1, 1.1) and the price p Pr as 1.2’ expressed by (1, the multiplicative inverse interval ofX only when also inner the the multiplicative multiplicative the multiplicative inverse inverse inverse interval interval interval ofof ofXXonly X ofonly X when only when also when also inner also innerrealized: inner To getnthe be realised: the multiplicative inverse interval only when also inner =1 answer, following = calculation should ˜ ˜ 5.5 500.1.0, 1.1) The r company produce the·pnext 1.2, 1.4). How many product the RDM-variables are equal: αequal:  = α .α It means fullItcoupling (cor6of 6) + 10 cyear? (1,5should 1.2, 1.4) − ·(0.9, xα y= 1n− 1the Pr − 100 000 (5.3 ·kilograms 106in, 5.5 10 , 5.7 · 10 RDM-variables RDM-variables RDM-variables are are equal: equal: are α =x=α = . Itmeans α Itmeans full means full coupling coupling full coupling RDM-variables are equal: α full coupling x xα y .yα x.yIt y .means 6, 5.5 · 106, 5.7Determinant 6) + 105 D = (5.3 · 10 · 10 Pr − 100 000 D = a6˜ n by =company . =the The results relation) of both uncertain values x and y modelled intervals. produce To get answer, calculation should be realised: in the next following year? (correlation) (correlation) (correlation) ofofboth ofboth both of uncertain both uncertain uncertain values values xvalues and xand and y yxmodelled ymodelled and modelled y modelled by by ininby in- n = = 1.0, 1.1) (correlation) uncertain values x by inp − c (1, 1.2, 1.4) − (0.9, 1 1 ˜. 5. ˜ ˜ (8.2, 8.5, conta Such full or partial correlation of uncertain variables occurs · ˜5 =8.8) 85.5 − Denominator of the formula (36) contains 0D1.1) in a6 ·hidden To getinthe answer, following calculation should be realised: p − c (1, 1.2, 1.4) − (0.9, 1.0, 6 6 6 5 1 1 (36) tervals. tervals. tervals. Such Such full Such full or partial or full partial or correlation partial correlation correlation of uncertain of uncertain of uncertain variables variables variables Pr − 100 000 (5.3 · 10 , 5.5 · 10 , 5.7 · 10 ) + 10 tervals. Such full or partial correlation of uncertain variables ˜ of(36) many real problems. the solution the Dei 5.5 · 500. Example division n= .˜ described = 1 corresponds to Case 3 of (36)

isadditive an additive inverse element offuzzy fuzzy interval: isis an isan an isadditive an additive inverse inverse element inverse element ofof offuzzy fuzzy ofinterval: fuzzy interval: interval: additive inverse element interval: Bull. Pol. Ac.: Tech. XX(Y) 2016 element

occurs occurs in occurs inmany many inreal many real problems. problems. real problems. occurs inmany real problems.

6)way. DenominatorPr of−the formula in6,a1.4) hidden 100 000 (5.3 ·contains 106, 5.5 10 5.7 ·− 10(0.9, + 1.0, 105 1.1) p1 − c(36) (1,·01.2, 1

on in the pa Denominator of theon. formula (36) contains. 0further in a hidden way. (8.2, paper further Determina z) (x, y, z) in )rin inin Examplen1=corresponds to = Case 3(1, of1.2, division described in the (36) Sub-distributivity Sub-distributivity Sub-distributivity law law Sub-distributivity law p1 − c1Example Sub-distributivity lawlaw 1.4) − (0.9, 1.0, 1.1) 1 corresponds to Case 3 of division described in theso the (8.2, 8.5, 8.8) paper further on. of the formula (36) contains 0 in a hidden (36) way. The sub-distributivity law holds in MD RDM FA: The The sub-distributivity sub-distributivity The sub-distributivity law law holds holds lawinin holds MD inMD MD RDM inRDM RDM MDFA: RDM FA: FA:Denominator The sub-distributivity law holds FA: paperExample further on.2 An oil rafinery produces Example 3 The feo regular super furthe theand solution Denominator of the formula 0 described in a hidden Example 1 corresponds tocontains Case(36) 3 of division the Denominator of the formula (36) 0contains in a hidden way. inway. bank of the gasolin river a line using two production lines. Line 1 produces further on in X(YXX + Z)  =  + Example 1produces corresponds to division Case division described in the 2 An oil rafinery regular and3 of super gaso(YX + (Y+Z) + XZ) (Y Z) ==+ XY =XY Z) XY ++ = X+ ZXZ. X .Example X Z . Example (33) (33) on. (Y XXY Z. Z .+(33) (33) paper further us ontinuous uous 1(33) corresponds to 2Case 3oilof described in the and ous Example An rafinery produces regular super gasoon the northe cording to an older process and line quay 2 according to an Exam paper further line using production lines.on.Line 1 produces gasoline acpapertwo further on.line using two production lines. Line 1 produces gasoline actance l = 1 km. process. The older process produces about 6.0 units of re bank Example 3 T The The consequence The consequence consequence of of this this of law law this is a is law possibility a is possibility a possibility of of formulas formulas of formulas B1B2 The consequence of this law is a possibility of formulas The consequence of this law is the possibility of formula cording to anExample older process line 2 according a newer 2 An and oil rafinery produces to regular and super gasocording to an older process and line 2 according to a newer not suitable for mo and about 5.5 units of super gasoline in one run. The quay bank of the rn transformations. transformations. transformations. They They dodo They do not not change do change not the change the calculation calculation the calculation result. result. result. transformations. They not change the calculation result. transformations. They do not change the calculation result. Example 2.production An oil rafinery produces regular and super gasoline process. The older process produces about 6.0 unitsand regular line using lines. Line 1ofproduces gasoline acExample 2 An oiltwo rafinery produces regular super gasoM uzzy yzyFuzzy process. The older process produces about 6.0 units ofabout regular depends on the wa produces about 8.5 units of regular and 8n using two process production lines. Line 1 produces gasoline according tance quay on the Cancellation Cancellation Cancellation law law for for law addition addition for addition and and multiplication multiplication and multiplication Cancellation law for addition and multiplication and about 5.5cording units gasoline in one newer tosuper an older process and line The 2 according to a newer line using two of production lines. Line 1 run. produces gasoline acabout 5.5and units super gasoline in one Thelferry newer Cancellation law for addition and multiplication to anand older process lineof 2 according to a newer process. The km/h. The h of super gasoline. Production results arerun. uncertain beca not tance = B1B2 su process produces about 8.5process unitsprocess ofand regular about 8 6.0 units Cancellation Cancellation Cancellation laws laws (34) laws (34) andand (34) and (35) (35) and hold (35) hold ininMD hold inMD MD RDM inRDM RDM MD FA: RDM FA:process. Cancellation laws (34) (35) FA: The older produces about regular cording toFA: an older line 2and according to aunits newer Cancellation laws (34)and (35)hold hold in MD RDM FA: older process produces about 6.0 units of regular andTof about 5.5suitable process produces about 8.5 units and 8 The units = 1about hour. df types of crude oils used in of theregular production (supplied by depen not of super gasoline. Production results are uncertain because 3 The and 5.5 units of super gasoline in oneofrun. newer process. Theabout older process produces about 6.0 units regular units of super gasoline in one run. The newer process produces of super gasoline. Production results are uncertain because 3 specify an angle β a companies) are of different quality. XX+X+Z+ +=ZYY+Y+ =Z+ ZY X+X= ZX=⇒ Y=Y, Y X, ,= Yof, crude (34) (34) (34) Z ZX== Z⇒ ⇒⇒ (34) dependskm/h on th oils used in the production (supplied byThe few process produces about 8.5 units of regular and aboutgasoline. 8 units and about 5.5about units ofunits super gasoline in one run. newer X + Z  =  Y + Z ) X = Y,types (34) re dare ) (28) are are 8.5 of regular and about 8 units of super types ofUncertain crude oilsresults used incan thebeproduction (supplied byThe few ferry shoul describedthat bythe following T =trife 1 km/h. XXZX X=ZYYZY ⇒ Z⇒⇒ XYX= ZX=⇒ Y=Y. Y X. .companies) = Y . processare (35) (35) (35) Z Z== Z= (35) of different quality. of super gasoline. resultsand areabout uncertain because produces about 8.5Production units of regular 8 units Production results areofuncertain oils3 companies) are differentbecause quality.3 types ofBcrude B=2 .1 hour. fuzzy numbers: speci T 1 and Uncertain be production described bythe following XZ  =  YZ ) X = Y. (35) used inProduction the (supplied by fewtriangle companies) are of few diftypes ofcan crude oils used in (supplied of superresults gasoline. results areproduction uncertain because 3 by Uncertain results can be described by following triangle A vertical compo thatang t specify an 7) (27) (27) 27) ˜ by quality. fuzzy numbers: about(supplied 5.5 = 5.5 = (5.2, are of in different quality. types ofcompanies) crude oils used the production few 5.5, 5.8) , 4.4.4. Multi-granularity Multi-granularity 4. Multi-granularity phenomenon phenomenon phenomenon in division division in division of of Multi-granularity phenomenon inin division ofofferent fuzzy numbers: the condition: B an that the ferry 1 Uncertain can bedescribed described by triangle 8) (28) (28)fuzzy results can by following companies)Uncertain are of different quality. 28) ˜following ˜results about =the 6.0 = (5.7,triangle 6.0,B6.3) fuzzy intervals fuzzy intervals intervals fuzzy intervals about 5.5 = 5.5 = (5.2,be5.5, 5.8) , 6.0 and, B2 .A v ˜ fuzzy numbers: 4. Multi-granularity phenomenon in division about 5.5 = 5.5 = (5.2, 5.5, 5.8) fuzzyresults numbers: Uncertain can be described by following triangle , 1

˜ = (7.8, 8.0, 8.2) , the coc ˜ = (5.7, 6.0, about = 8.0 about 6.0 = 6.0 6.3) , 8.0 As ititwas itwas As was presented it presented was intervals presented ininsection insection section in 2, section 2,result result 2,inin result the inthe the form inform the of only ofonly only of one only one one AsAs presented 2,result form ofform one of fuzzy ˜ = (5.7, 6.0, 6.3) , A vertical fuzzy numbers: about 6.0 = 6.0 ˜ about 5.5 = = (5.2, 5.5, 5.5 ferry ˜5.8) condition granule granule granule isisachieved isachieved achieved is achieved during during dividing during dividing dividing fuzzy fuzzy intervals fuzzy intervals intervals XXand Xand and YXY, about Y , Y8.0 , = 8.0 ˜ = (7.8, 8.0, about granule during dividing fuzzy intervals ,and = 8.5 = ,(8.2,As 8.5,the 8.8) . (apa 0) (30) and (30) 8.2) , 8.5 30) ˜ ˜ about 8.0 = 8.0 = (7.8, 8.0, 8.2) , As it was presented in Section 2, result in the form of only one about 5.5 = 5.5 = (5.2, 5.5, 5.8) , with a speed V1 , a ˜ 0 0∈ /0∈ /Y. Y 0.However, ∈ /However, Y . However, ititwill itwill will be itbebe shown will shown be further shown further on, further on, that that on, the the that solution solution the solution /Y∈ .However, shown further on, that the solution 6.0 = 6.0 = (5.7, 6.0, 6.3) , ˜about about 8.5 = 8.5 = 8.8) .the ˜company predict In the(8.2, next8.5, month to getsatisfy contrac granule is achieved during dividing fuzzy intervals X and Y, V should th ˜ As about = 8.58.0, . isismulti-granular ismulti-granular multi-granular is multi-granular ififaifadenominator adenominator denominator if a denominator set contains contains set contains 0.0.0. 0. setset contains about 6.0 = 6.0 (5.7, 6.0, 6.3) ,= (8.2, ˜ 8.5 about 8.0= 8.0 =500 (7.8, 8.2) ,8.5, 28.8) production of=about units of regular and about 350 un 0 2 / Y. However, it will be shown further on that In thethe solution is 9) (29) (29) 29) As thewith ferry next month the company predict to˜ get contracts for Three Three examples Three examples examples below below describe below describe describe problems problems problems ininwhich inwhich which indivision which division division Three examples below describe problems division ˜month about 8.0 =gasoline, 8.0 8.2)8.5, , predict In the next the8.0, get described contracts by for( multi-granular if a denominator set contains 0. super where uncertain values =(7.8, 8.5 =company (8.2, 8.8) . to are about 8.5=and V with a speed production of about 500 units of regular about 350 units of 0) (30) (30) 30) 2 sh bybyby fuzzy fuzzy by denominator fuzzy denominator denominator containing containing containing 0 occurs 0 occurs 0 and occurs and in which in and which in soluwhich solusolufuzzy denominator containing 0 occurs and in which solu˜about Three examples below describe problems in which division production 500 8.5, units8.8) of regular and about 350 units of numbers: aboutgle 8.5fuzzy =of8.5 =are (8.2, Thus, the totalsati sp V2for should super gasoline, values described by. to trianIn theInuncertain next month the the company predict contracts for tions tions tions will bebedenominator multi-granular. be will multi-granular. be multi-granular. tions will multi-granular. bywill fuzzy containing 0 occurs and in which solu- where the next month company predicts toget get contracts super gasoline, where uncertain values are described by trian˜ gle fuzzyInnumbers: about =contracts 500 = (480, 500, 520) ofthe about 500500 units of of regular about 350 of , theproduction nextproduction month predict to500 getand for units tions will be multi-granular. of company about units regular and about 350 units of gle fuzzy numbers: uative neu- neueusuper gasoline, where uncertain values are described by trianproduction of about 500 units of regular and about 350 units of ˜ 2 .+ VTh 2 Example Example Example 1 1A1Acompany Acompany company 1 A company produces produces produces a afood afood food product. aproduct. food product. product. The The producproducTheabout producExample produces The produc˜ = (480, 500, about V2x 500 = 500 520)350 , ˜ = 350 = (330,V350, 2 = 370) 2y ny 1any for tion any any about 500 = 500 = (480, 500, 520) , Thus, the t gle fuzzy numbers: super gasoline, where uncertain values are described by triantion cost cost tion c1c1per ccost c per 1 kg 1 per kg of 1 the of kg the product of product the product is uncertain is uncertain is uncertain because because because it deit deit detion cost per 1 kg of the product is uncertain because it de1 1 ˜Let=us(330, about 350 = 350 350, . the denote by370) x1 –Bull. number of production runs of 502 Pol. Ac.: Tech. 65(4) 2017 ˜ gle fuzzy numbers: about 350 = 350 = (330, 350, 370) . pends pends on current current prices prices ofprices ofcomponents components changing changing from from month month pends on current of components changing from month pends onon current prices ofcomponents changing from month ˜ the course V aboutprocess) 500 = 500 500, 520) , and angl (older and=by(480, x2 –Unauthenticated the number of runs of line 22(n= Let us denote by x – the number of production runs of line 1 2 to month to month and and which which are are results results of negotiations of negotiations with with many many supsupto month and which are results of negotiations with many supto month and which are results of negotiations with many sup1 ˜ 1) (31) (31) = V 31) about = 500 = 500, 520) , of1370) Let500 us denote by x350 number production of line 1 2x process). runs of line and.PM x2 runsruns of2 line 2Vam about 350After =(480, = (330, 350, Download Date | 8/25/17 12:47 1˜x–1the (older process) and by x – the number of runs of line 2 (newer pliers. pliers. Price Price p1pPrice p pliers. p of 1 of kg 1 kg of the 1 of kg the product of product the product is also is also uncertain is uncertain also uncertain because because because pliers. Price of 1 kg of the product is also uncertain because 2 β tg 1 11 and th ˜ ˜ ˜ (older350 by x2350, –will the370) number runs of line 2 (newer 2) (32) (32) ofprocess) regular gasoline be 6x 8.5x of super gas 32) about =and 350xand = (330, . 1 +of 2 and process). After xLet of line 1 runs of line 2 amount ititisitisalso isalso italso aisaresult also aresult result aofresult negotiations negotiations of negotiations with with customers with customers customers (supermar(supermar(supermar– the number of production runs of line 1 us denote by x ofof negotiations with customers (supermar1 runs 2 1 and the cours ˜ After ˜ x runs of line 1 and x runs of line 2 amount process).

process. The older process produces 6.0of6.0, units of ,regular process produces about 8.5 regular and about 8specify units an specify an angle angle ββ and and aa speed speedVVT2T in relation relation to to water, water, assuming assuming 1 2 in 1in ˜about ˜production cess produces about 6.0 units regular about 6.0 = 6.0 =units (5.7, 6.3) of super gasoline. Production are uncertain because 3few that the ferry should reach the point Btolocated between types types of of crude crude oils oils used used in inofresults the the production (supplied (supplied by few about 6.0 = 6.0 = (5.7, 6.0, 6.3) , by depends on level and varies thebank range ∈points [2, 4] We m Tthe = water 1βinhour. The distance the isVassuming d1 = 1 km. companies) aresuper of different quality. depends on the water level and varies the range V ∈ [2, 4] specify an angle and a speed V in relation to water, and about 5.5 units of super gasoline in one run. The newer of gasoline. Production results are uncertain because 3 that that the the ferry ferry should should reach reach the the point point B B located located between between points points 1 2 As the ferry (apart from its own move) is floated by the water As the ferry (apart from its own move) is floated by the super in of one run. The newer ypes ofgasoline crude are oils used in the production (supplied by few ˜ B and B . companies) companies) are of different different quality. quality. km/h. The ferry has to reach thea opposite bank within timewater anreach angle βwithin and speed V2 in relation topoints water, assum ˜ regular 1 2 specify about 8.0 =of 8.0 =by (7.8, 8.0,about 8.2) ,8ferry about 8.0 = 8.0 = (7.8, 8.0, 8.2) Uncertain results can beoils described following km/h. The,triangle reach the opposite bank time that the ferry should the point B located between process produces about 8.5 units and unitshas to types of crude used in the production (supplied by few B B and and B B . . 1 1 2 2 with a speed V , a horizontal component of the relative speed t 8.5 units of regular and about 8 units with a speed V , a horizontal component of the relative speed 1 companies) are ofresults different quality. Uncertain Uncertain results can can be be described described by by following following triangle triangle should satisfy A vertical component of the relative speed V 1 T = 1 hour. The distance to the bank is d = 1 km. We must that the ferry should reach the point B located between poi 2 ˜ ˜ = about 8.5 =8.5 8.5 =(8.2, (8.2, 8.5, 8.8) . The 3distance uzzy numbers: T 8.5, = 1 8.8) hour. to the is d =the 1 km. We must speed B1 and B of super gasoline. results are uncertain because companies) arebeof8.5 different about = .triangle should satisfy satisfy A A vertical component component of of the thecondition: relative relative speed VV22 should 2 . bank V2vertical should satisfy next condition: duction are Production uncertain because 3quality. V should satisfy the next 2 Uncertain results can described by Fuzzy following fuzzy fuzzyresults numbers: numbers: the condition: specify an angle β and a speed V in relation to water, assuming B and B . number division and the multi-granularity phenomenon 2 1relation 2to water, specify an β and a the speed V2 in component A vertical of the relative speed V2 should satisfy ypes of oils used in= production (supplied by few results can be 5.5, described byangle following triangle the condition: condition: 1 of d assuming ˜company Incrude theUncertain next month thethe predict to get contracts for ed in numbers: the production (supplied by few about 5.5 5.5 = (5.2, 5.8)to , get uzzy In the next month the company predict contracts for that the ferry should reach the point Bllthe located between A vertical component V(38) 2 should = 1 .··relative = V = 11= d d ˜ ˜ + V )T = cos α . speedpoints (39)sati (V 2y(Vbetween 2x+ V11)T that the ferry should reach the point B located points about about 5.5 5.5 = = 5.5 5.5 = = (5.2, (5.2, 5.5, 5.5, 5.8) 5.8) , , the condition: companies) are of different quality. fuzzy numbers: 2x AB =11 = =AB 11..cos α . (38) (38)(39) VV2y =Td = productionof ofabout about 500 units ofregular regular and,about about350 350units unitsof of rent production quality. 2y = ˜˜ =of 500 units and B and B . the condition: about 6.0 = 6.0 (5.7, 6.0, 6.3) 1 2 TT= 11= d1 . about 5.5be = 5.5 = 5.5, 5.8) , B,,2 . triangle B and results can described by following ˜˜ (5.2, 1are about 6.0 6.0 = = 6.0 6.0 = =values (5.7, (5.7, 6.0, 6.3) 6.3) (38) Vfrom = 1is2be ˜ 6.0, ofmove) theferry ferry can be calculated as: Thus, the total speed V22relative super gasoline, where uncertain values described by trian- As 2y an Uncertain besuper described byabout following triangle super gasoline, where uncertain values are described byby triangle As thecomponent ferry (apart from its move) floated by theas: of the can calculated the total speed Vown about 5.5 = 5.5 = (5.2, 5.5, 5.8) , triangasoline, where uncertain are described theThus, ferry (apart itsthe is=floated by water should satisfy A vertical ofshould V ˜ 1own T = 1by . the (3 V2y = speed about 8.0 = 8.0 = (7.8, 8.0, 8.2) , ˜ satisfy A vertical component of the relative speed V uzzygle numbers: 2 As As the the ferry ferry (apart (apart from from its its own own move) move) is is floated floated by the the water water about 6.0 = 6.0 = (5.7, 6.0, 6.3) ,  ˜˜ =  fuzzynumbers: numbers: water withV1a speed V1, a horizontal component of therelative fuzzy numbers: 1relative T of the about about 8.0 8.0about = = 8.0 8.0 = (7.8, (7.8, 8.0, 8.0, 8.2) 8.2),,6.0, 6.3) , gle fuzzy with a speed , a horizontal component speed ˜ the condition:     6.0 = 6.0 = (5.7, 2 2  2 condition: ˜˜ = (8.2, 8.5,the Asspeed the ferry (apart from the itsd own move) is of floated by the with with aa speed speed VV112,, aasatisfy horizontal horizontal component of the the2relative relative speed speed about 8.5 = 8.8) ., cos lAB V2  next 1condition: ··cos ddwater lcomponent AB 8.0 = 8.5 8.0 = 8.0, 8.2) 5.58.5 5.5 5.5, 5.8) ˜˜500 satisfy next condition: As ferry (apart from its1ααown move) is floated the wa 2the 22 = 1should dV ˜(5.2, 2 should ˜ = (5.2,about about about 8.5 = == 8.5 8.5 = == (8.2, (8.2, 8.5, 8.8) 8.8) ..8.0,,8.2) − V = V + (40) + V = V ˜(7.8, ˜ 8.5, = . (38) = about = (480, 500, 520) , , VV 5.5 = 5.5 5.5, 5.8) ,500 − V = V + ,,by(40) + V 2 1 2y about 500 (480, 520) 500 about 8.0 = 8.0 =500, (7.8, with a speed V , a horizontal component of the relative speed 2 1 2x 2y V22 should should satisfy satisfy the2ynext next condition: condition: 1 2x1 .the = (38) = with = 2y V T T 1 T ˜ T T a speed V , a horizontal component of the relative spe In the˜next month company predict to get contracts for aboutthe 8.5 = 8.5 = (8.2, 8.5, 8.8) . 6.0 ,350 6.0= 350 6.0, 6.3) , 370) . 1 )T = l · cos α ..(39) 1 should satisfy the ˜˜(5.7, ˜ = + Vcondition: (39) (V2xnext T about (330, 350, 6.0 =In (5.7,month 6.0, 6.3) 1 AB In6.0 the the=next next month the the350 company company predict predict to to get get370) contracts contracts for for V2 As = 350 = (330, 350, . about about 8.5 = 8.5 (8.2, 8.5, 8.8) . the ferry (apart from its own move) is by the water +V )T = = lwater lAB ·is ·is cos cos ααfloated .. as: (39) (39) (V Vmove) should satisfy the next production of about 5008.0 units of about 350 units of from its 2x 2x +V 1)T ABcondition: 2course ˜ regular and the course angle of1by the ferry defined as: and the angle of the ferry defined = 8.0 = predict (7.8,and 8.0, 8.2) ,contracts As the ferry (apart own is(V floated the Inproduction the month the company to get forof ˜ next production of ofabout about about 500 500 units of of regular regular and and about about 350 350 units units of with 8.0 8.0 = (7.8, 8.0, 8.2) ,units + V )T = l · cos α . (39) (V Let usLet denote by x – the number of production runs of line 1 of the ferry can be calculated as: Thus, the total speed V a speed V , a horizontal component of the relative speed uper=gasoline, where uncertain values are described by trian2x 1 AB – the number of production runs of line 1 Let us denote by x 1 In the next month the company predict to get contracts for 2 1 1by x1of Thus, total Vof ofthe the ferry can calculated as: us denote number of production runs 1 Thus, Fuzzy number division and the multi-g with a. 350 speed V1trian,of a line horizontal component ofspeed theVVrelative speed 2 (V ˜– the production of about 5008.5 units regular and about units of + Vl1lAB )T lbe · cos α . and (3 of ferry ferry can can be be calculated as: as: the Thus, the thethe total total speed speed super gasoline, where where uncertain uncertain values are are described described by by sin about = = values (8.2, 8.5, 8.8) 2xFuzzy AB 22VV 2ythe ··= sin ααcalculated AB ˜ gasoline, 2y  number division multi-g (older process) and by x28.5 the number of runs of line 2trian(newer Vunits satisfy thetg next condition: 8.5super = 8.5 = (8.2, 8.5, 8.8) .xby gle fuzzy numbers: (older process) and by the number of runs of line 2 (newer 2–x–500 production of about units of regular and about 350 of 2 should . (41) β = = tg (older process) and – the number of runs of line 2 (newer Fuzzy number division and     . (41) β = = 2 V should satisfy the next condition:   2 2 of the ferry can be calculated as: Thus,the total speed V2 2 uper gasoline, where uncertain values are described by triangle gleprocess). fuzzy fuzzy numbers: numbers: lAB ·cos cos α 22 calculated as:  −  VV2x ·of α 2x 1TT · coslα lAB dcan After x11 runs of line x2values runs of line amount 2ferry 2−VV1   process). After xruns line 1and andxto x22runs runs ofline line amount After xthe of line 11for and of 222for amount the Thus, the total speed VAB In process). the next month company predict get contracts super gasoline, uncertain described byVtrian2 + 2 = 1 runs ˜ of  Vthe V2x V2y , (40) ··cos cos αα2·−cos llAB ddbe he predict to get contracts about 500 =where 500 = (480, 500, 520) ,are glecompany fuzzy numbers: +V1division )T lABFuzzy . +  division (39) (V 2= 1α AB= 2x Fuzzy number and multi-granularity phenomen 2 2 2 2  number and the mult ˜ ˜ ˜ ˜ ˜ ˜ ˜ + V )T = l · cos α . (39) (V  equals 2value of regular regular gasoline willwill be 6x1= 8.5x of gasoline  ,,,(40) ofgle gasoline be 6= x+  + 8 and ofsuper gasoline −V −V = VV +V +Vof = + + (40) (40) V1V22 = Tdthe about about 500 500 = = 500 500 (480, 500, 520) 520) ,super , unitsgasoline α·T always equals and the value of lAB sin TheAB value of l= of gasoline will be 6x 8.5x and of 1+ 2about production ofregular about 500 units of regular and.5x 350 of 2x fuzzy numbers: 11 2 11 1(480, 2and α always and of l The value l AB··sin AB·· 2500, 2x 2x 2y 2y AB   cos α l 0 units of regular and about 350 units of AB ˜ 2 2 T T T T  2 2 about 350 = 350 = (330, 350, 370) . ˜ ˜ ˜ ˜ ˜ ˜ 115+ .5x  + 8 xHow How runs ofline line1520) and oftrianline are ˜18x −lAB V1· cos = (40)d into + Vbe = +8x many runs x11x(330, line 11and and line222 are areVThus, 5.5x α3],Fig. =many 500 500, , xxx2..2by 2. 500 1of 2 of cos αVferry must be included the interval [2,+calculated Fig.7.7.,Taking Taking into the ferry can be as: the total speed V2 ofin many x(480, of line 2.. How uper5.5x gasoline, where uncertain values are ˜= ˜ triancos α must included the interval [2, 2x 2y 2about 2calculated 2in about about 350 350 = =runs 350 350 = = (330, 350, 350, 370) 370) ˜ described of the can be the V22the uncertain values are described by course of the isas: defined as:3], − T= TV1 + = V + Vferry , (4 V2angle about 500 = 500 =Thus, (480, 500,total 520)speed , and 2x 2y necessary to fulfil the contract commitments? necessary to realise the contract commitments? ˜ andaccount the the course course angle of of the the ferry ferry is is get defined defined as: account allangle known values, we get new new interval dependencies necessary to the contract commitments? gleLet fuzzy numbers: T as: T all known values, we interval dependencies us denote byrealise x1 350 – the production runs about =number 350 = of (330, 350, 370) . of line 1 and    2 2  ˜ of  and theequations course angle ofand the ferry isdefined as: d The results and xhave have not to beintegers integers because Let Let us us results denote denote by by the number of of production production runs runs of of line line. 1it 1itit is about = 350 = (330, 350, 370) V2y · defined sin α as: 1 the 2 350 11x–– 2(40) 2lis ·(41): cos lAB  The results and not to be integers because is lthe and the ferry ABα from The xx11xxxand xx22 ˜number have not to be is from equations (40) and 2α+angle older process) and by number of runs of line 2because (newer ·course cos d (41): 2 –=the VV2y β2 course tg llAB ··sin sin αα is+.defined as: AB − V = V V = , (41) (40) V 2y= AB about 500 500 = (480, 500, 520) , and the angle of the ferry 2 2 possible to organize fractional runs that use raw materials in 2 1 Let us denote by x – the number of production runs of line 1 ˜ 2x 2y (older (older process) process) and and by by x x – – the the number number of of runs runs of of line line 2 2 (newer (newer 1 = V +2x = (40) = V − V 22 , 1 and 00 =possible 500After = (480, 500, 520) possible toxorganise organise fractional runs thatuse use raw materials inan an (41) (41) =V = tg tg1ββ = lAB ·l,cos α −αV1T .. T 2 line  T to fractional runs that raw materials in 2x + V2y process). runs of line x runs of 2 amount Let us denote by x – the number of production runs of line 1    V · sin 1 2 1     T T 2y  an appropriate proportion. To determine results x212 and x2, fol2 11 VV2x ··cos cos αα2−V −V T T..(41) ˜number 2x= 22llAB ABAB older process) and by x – the of runs of line 2 (newer process). process). After After x x runs runs of of line line 1 1 and and x x runs runs of of line line amount amount about 350 = 350 = (330, 350, 370) . 2 β = (41) tg 1 1 2 2 V [2, 3] 1 l · sin α appropriate proportion. To determine results x and x , follow˜ ˜ ˜ [2, 3] 1 2y proportion. determine results xof x2of 2, follow50regular =appropriate 350lowing = (330, 350, .To 11and of gasoline will370) beand 6x 8.5x and of super gasoline (older process) x2to–x˜˜2be the number line 2and (newer always 1V and value of The of angle lAB · sin 1˜by 2·2+ solved: AB Vα α=− TABthe V2value = − [2, 4] +·equals = ([2, 3]− −[2, [2, 4]) .l4]) βlAB= tg the of the ferry iscos defined as: ˜1+has 2x 1and V = − [2, 4] + = ([2, 3] 2course process). After xequation runswill ofsystem line and ofof line 2runs amount of ofing regular regular gasoline gasoline will be beto 6x 6x + 8.5x 8.5x and and of super super gasoline gasoline 1system 2 runs α α always always equals equals 1 1 and the the value value of of l l ··+11,, (4 · · sin sin The The value value of of l l 1of 1+ 22 and and the course angle of the ferry is defined as: ing equation system has to be solved: AB AB AB AB ˜ Let ˜ 1 1 V l · cos α − V T equation has be solved: 1 1 2x AB 1 + 8x . How many runs x line 1 x of line 2 are 5.5x by x – the number production runs of line 1 After x runs of line 1 and x runs of line 2 amount cos α must be included in the interval [2, 3], Fig. 7. Taking into 1˜˜ us denote 2 process). 1 2 1 1 2 ˜ ˜ ˜8x ˜ of number of 1line ofthe regular will beruns 6x  +gasoline 8x How How many many runs runs xx18.5x of of 2line and and xx22of ofgasoline line line 22 are are cos 5.5x 5.5x always 1α3], and the 7. value of lAB · (42) Theααvalue of 1+ the ˜and11of cos must mustvalue be belincluded included in in the thealways interval interval [2, [2, 3],1Fig. Fig. 7. Taking Taking into into 11 + 22..production 1line lequals ·equals sin AB · sinVα (42)of l ABnew ˜ ˜super necessary toofrealise commitments? The of lAB  2y ¢ sinα and the value ofvalue older process) and by6˜gasoline x˜contract –8.5 the of of 2 (newer V ·The sin αvalue regular will be 6x11runs + 8.5x of2 super gasoline account alllAB known values, we get interval dependencies ˜ xnumber ˜ line α always equals 1 and the · sin of l 2of 2y 2ofand . (41) β = = tg ˜ ˜2 –necessary ˜ AB A 6 8.5 500 x x the number of runs line 2 (newer 500 x + 8x . How many runs of line and x line are 5.5x 1 necessary to to realise realise the the contract contract commitments? commitments? cos α must be included in the interval [2, 3], Fig. 7. Taking into 1 1 Pluci´ 2nski 1 2 1 . (41) β = = tg account account all all known known values, values, we we get get new new interval interval dependencies dependencies 1 V l · cos α − V T at, M. ˜ x1x+ ˜ 2x.2of =1 ofofline .and (37) lare   ¢ cosα must beVand included in the in interval [2, Fig. 7. Taking After 2x(41): AB 1 3], The results and have not toruns be integers because it isof(37) = process). runs line 1amount and x2 runs line 2..(37) amount AB 8x How many x 1 x line 2 5.5x from equations (40) must be included the interval [2, 3], Fig. 7. Taking i . (43) tg β = 1 2 V l · cos α − T . (43) tg β = 2x AB 1 ˜ ˜ ˜ of line 1 and x runs of line 2 necessary realise commitments? ˜ contract ˜ because 5.5 350 The The to results results and and xx22 8˜have not not to be be integers integers because itit is is from 2 xx11the all known values, we get dependencies from equations equations (40) (40) and and (41): (41): 5.5 [2,new 3] −1interval [2, 4]the into account all values, we obtain new interval depen˜8have ˜xx22to [2, 3] − [2, 4] value possible to gasoline organise fractional runs that materials in an account of regular will be 6x + 8.5x andraw of350 super gasoline necessary to realise the contract commitments? ·known sin α always equals and value of lABdependenc · The of lAB account all known values, we get new interval 1not 2use ˜ results ˜ organise     l be 6x + 8.5x and of super gasoline The x and x have to be integers because it is    · sin α always equals 1 and the value of l · The value of l possible possible to to organise fractional fractional runs runs that that use use raw raw materials materials in in an an 2 2 1 2 from equations (40) and (41): 1 2 AB AB es dencies from equations (40) and (41): ˜ ˜         [2, 3] 1 appropriate proportion. Toruns determine results x1toxand xintegers , follow+ 8x . How many x of line 1 and of 2 are 5.5x   The results x and x have not be because it is 2line 2 2 2 2 cos α must be included in the interval [2, 3], Fig. 7. Taking into from equations (40) and (41): 1 2 1 2 1 2 y runs line 1 fractional and are x2 To of determine lineby2that possible organise runs use raw in an Vin − [2, + = Bull. ([2,Pol. 3] − [2,Tech. 4])22XX(Y) + 1 , 2016 [2, [2,3] 3] 11 into cosand α included theinterval [2, 3],4] Fig. Taking appropriate appropriate proportion. proportion. To determine results results xxmaterials and and xxbe follow1 of The results given xare x2 = D /D where: 2= 11must 22,,2followBull. Pol. Ac.: Tech. 1 = D 1/D 66 xto 2 7. ng equation system has to be solved:  necessary topossible realise the contract commitments? to organise fractional runs that use raw materials an  VVin − − [2, [2,4] 4] = + + 11get2new = = ([2, ([2, 3] 3]Ac.: − −[2, [2, 4]) 4])2XX(Y) + +1, 1, 2016 1known account all values, we interval dependencies 22 = ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ contract commitments?   [2, 3] appropriate proportion. To determine results x and x , followall we get new interval dependencies ing ing equation equation system system has has to toDbe solved: solved: D = 6  ¢ 8  ¡ 5.5 ¢ 8 .5,  = 50 0 ¢ 8 ¡ 35account 0 ¢ 8  = 6 ¢ 35values, 0 ¡  1 .5 and 2 D2known 2 2 1 1 1 1 1be     2 Vfollow− [2,and 4] + = ([2, 1, [2, 3] 1 3] − [2, 4]) + The results and x˜2proportion. have not to integers because is x2 , from (42) appropriate Tobe determine results x1 itand 2 = equations (40) (41): 1  ˜ from  solved: ˜ xbe not integers because ¡ 5to .5 ¢ 50 ng equation system has to be equations (40) and (41): 2 have 12 = 14]1 + 6˜˜ 0. 8.5 500 x1 itis  V − [2, = ([2, 3] − [2, 4])2 + (42) (42) ˜ ˜ ˜ ˜ ˜ ˜  possible to organise fractional runs that use raw materials in an ing equation system has to be solved: 6 6 8.5 8.5 500 500 x x = . (37) 1 1       1 1 Determinant D = (5.7, 6, 6.3) ¢ (7.8, 8, 8.2) ¡ (5.2, 5.5, 5.8) ¢ (42) ctional runs that an (43) β=  112 . (42) ˜use raw ˜˜   = = (37) 2   [2,  23]  tgtg 5.5 8˜˜ To xx2 in  materials  350  (37) (43) (43) tgββ2 = =[2, 3] [2, 4] .. 1− appropriate proportion. determine results x˜˜1 and..values. x2[2, , follow6˜ 5.5 500 (4 ˜˜ 8.5 ˜ ˜ 10follow¢ (8.2, 8.5, 8.8) contains as one of possible Therefore 2 1 5.5 8 8 x x 350 350 3] 1 To determine results x and x , 2 2 ˜ ˜ ˜ 1 2 8.5= [2, [2,4]) 3] 3]2− − [2, [2, 4] 4]. ([2,13] − [2, 4]) + V2 =+ − [2, 4]β3]=− +[2, = 1, al 6 solved: x˜1V2 =. 500 (37) (43) tg − [2, 4] = ([2, + 1 , ng equation system has to be ˜ ˜ 1 1subsets the solution corresponds 5 presented . 5.5 of8 the problem x2 350 =to Case Fig.(37) 8. Two granules representing of possible solutions; the s to be solved: [2, 3] . (4 tg−β[2, =4]of 1 granules ˜1  on in the paper   be 5.5 ˜ and  8˜it will x2multi-granular. 350 Fig. 8.granule TwoFig. representing subsets possible solutions; the 6er   further Bull. Pol.angle Ac.: Tech. XX(Y) 2016 [2, 3] − [2, 4] (42)  β < upper corresponds to the acute course of the ferry ( 8. Two granules representing subsets of possible solutions; t ˜ ˜ ˜ 6 6 Bull. Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 (42) d 6 500 500 x1 < ferry (β granule corresponds to thetoacute course anglecourse of the90angle ferry ˜.5 ˜ 8.5 ◦ ) (β the x1 theupper corresponds the1obtuse angle β > 90◦◦ ) and corresponds to thePol. acute = . (37) upper 1lowergranule 6er Bull. Ac.:((β Tech. XX(Y) 2016 ◦ ) 9. of . (43) tg β = . (43) Fig. Fuzzy numbe = . (37) ˜ ˜ ˜ ) and the lower corresponds to the obtuse angle > 90 90 Fig. 8. Two granules representing subsets of possible solutions; the Fig. 8. Two granules representing subsets of possible solutions; the ◦ ◦) 8 7.ferry x2departs from 350 .the lower [2, (43) tgβ = Example 3.˜Fig. The point A on 6 5.5350 Bull. Pol. Ac.:(βTech. XX(Y) 2 3] − [2, 4] ˜ ) and corresponds to the obtuse angle > 90 90 An illustration to Example 3 the southern 8 x 2 3] − [2, 4] angle of 1, β < angle of the ferry (β < upperofgranule corresponds to [2, thegranule acute course corresponds to the the ferry acute(course bank of the river and has to get to any place the concrete quay upper ◦ and the lower corresponds ◦ ) and ◦ )2 and Formulas and (43)(43) the relative V to the obtuse angle (βdetermine >determine 90 ) obtuse the(42) lower corresponds to◦the angle (βspeed >XX(Y) 90V Formulas (42) and the relative speed on the northern bank between points 90 B1 )and B2, the distance 90 6e Bull. Pol. Ac.: Tech. 2016 Formulas (42) and (43) determine the relative speed V22 and and Bull. Pol. Ac.: Tech. XX(Y) 2016 Formulas (42) and (43) determine the relative speed V2 a the course angle β of the ferry and it can be seen that The lresults are given by x = D /D and x = D /D where: northern bank2 of the2river is not the course angle β of the ferry and it can be seen that theythey are 1 1 B1B2 = 1 km. The remaining the course angle β of the ferry and it can be seen that they ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ the course angle β of the ferry and it can be seen D = suitable 6 · 8 − 5.5 8.5, D1 =Fig. 7. 500 ·An 8 −average 350 · 8.5 and of D2the = 6 · 350 − for·mooring, speed river decoupled. The RDM arithmetic, in in contrast to the standard in- that th are coupled. Thethe RDM arithmetic, contrast to the standard d: Formulas (42) and (43)coupled. determine relative speed V2in and Formulas (42)RDM and (43) determine the relative speed V2 and are The arithmetic, contrast to the standard ˜ ˜ range V1 2 [2, 4] km/h. terval arithmetic, allows for taking into account such couplings. are coupled. The arithmetic, in contrast to the standa interval arithmetic, allows forRDM the taking into account of such 5.5 · pends 500. on the water level and varies in the the course angle β of theThe ferry and it 3], can seen that they the course angle β describing ofbethe and it into can be that they interval arithmetic, allows forferry the account of such The ferry hasDto=reach bank within time T = 1 hour. interval [2, the taking concrete quay onseen theinto northern interval arithmetic, allows for the taking account couplings. The interval [2, 3], describing concrete quay on of su Determinant (5.7,the 6,opposite 6.3) · (7.8, 8, 8.2) − (5.2, 5.5, 5.8) · are coupled. The RDM arithmetic, in contrast to the standard are coupled. The RDM arithmetic, in contrast to the standard couplings. The interval [2, 3], describing the concrete quay on to the bank d = 1 We must specifyTherefore an angle bank between points B1 interval and B2, B is[2, defined couplings. The 3], describing the as: concrete quay the northern bank between points and B2as: , is defined (8.2,The 8.5,distance 8.8) contains 0 asisone ofkm. possible values. 1 such interval arithmetic, allows for the taking into account of arithmetic, allows for the taking into accountas: of such Fig. 1 6) theinterval northern bank between points B and B , is defined β and a speed V in relation to water, assuming that the ferry 1 2 2 the northern bank between points B1 and B2 , is definedFig. as: the solution of the problem corresponds to Case The 5 presented couplings. interval [2, 3], describing the concrete quay on couplings. The interval [2, 3], describing the concrete quay y. (44)on XB− 1a lAB · cos α = 2 + αB1B2, αB1B2 ∈ [0, 1],,(44) should reach the point B located between points B1 and B2. further on in the paper and it will be multi-granular. cos αB= αB1B2 αB1B2 ∈, [0, , ∈ [0,as: (44) lABB·bank the northern bank between points ,2is·+cos defined α =, 2as: + α1]B1B2 1] , XB−(4a l2AB the northern points B defined e 1 and between B1B2B 1αand 2 , is and the interval describing values of the river speed is: Fig. 10. Decomposition ofFig th and the interval describing values theriver riverspeed speedis: is: describing values of,ofthe cos α and = 2 the + αinterval ,lABthe α 1] (44) lAB ·southern and interval values is: XTh · cos α∈=[0,2describing + ,αB1B2 αB1B2Xof ∈B−the [0, 1]river , B+ speed (44) and X Example 3 The ferry departs from point A on the B1B2 B1B2 B− Th (45) V1 = 2 + 2αV 1 , αV 1 ∈ [0, 1] . bank of the river and has to get to any place of the concrete =river 2 +speed 2α ,is: αV2of [0, 1]α..(45) 1 describing 1α∈ and the interval describing of V the o= 2+ ∈ [0, is: 1] . (45) (4 VV 1values and values the interval the 1 , river V 1speed quay on the northern bank between points B1 and B2 , the dis- Finally, possible values of1V2 and tgβV are described by equaThe quotient X /X can T cB A and tg β Vare described equaFinally, possible values described by equa2is+ 2αVFinally, αFinally, 1] (45) tance lB1B2 = 1 km. The remaining northern bank of theVriver = 2.+of2V αof2Vvalues αof [0, 1]tg. β areby (45) by equ Vvalues and described possible 1 = tions: 1, Vpossible 1∈ 1[0, 1V,2 and Vtgβ 1∈ 2are Fr er tions:tions: Fr  tions: XA /XBX= (X not suitable for mooring, Fig. 7. An average speed of the river B− , V2 and ar 2 tg β are described by equaFinally, possible values of and tg β are described by equaFinally, possible values of V 2  X , − V = (2 + α ) + (2 + 2 α ) + 1, α , α ∈ [0, 1] , B depends on the water level and varies in the range V1 ∈ [2, 4]2 B1B2 V1 2 B1B2 V 1 2 From er V2tions: = (2 + α= ++(2α+ 2α)V1 )(2++1,2α α)B1B2 ,1, αV 1 α∈ [0, 1] , (10) tions: B1B2)(2 equation and F X V + + , α ∈ [0, 1 (46) A B1B2 V1 B1B2 V 1 km/h. The ferry has to reach the opposite bank  within time  2 (46) XA ts X be found: − , XB+ can(46) X 1 1 B B− T = 1 hour. The distance to the bank isV d== 1 (2 km. We must Vtg+β X(4 (47) αV1(2)2++α1, αV21α∈ 1] 1, , ) 1+ (2,+ )2= + α1B1B2 , αV.1 ∈1[0, 1] , 2 B1B2 3 Fig. 7. An illustration to Example 3 + αB1B2) + (2 2 =2= B1B2α V1 XBB− 1[0, ) − (2 + 2 α ) α − 2 α = . (47) V1 B1B2 V 1 specify an angle β and a speed V2 in relation to water, assuming tgβ = (2 +tgαβB1B2 XA : 1 xA = (1 .+ µX)(4+ = = ) − (2 + 2αV1 ) (46) + (46) (2 + αB1B2 w + αB1B2) − (2 +α2B1B2 αV1− ) 12ααVB1B2 − 2αV 1 XBB+ that the ferry should reach the point B located between points1Formulas (46) and (2 1 the (47) also show that the speed V2 and xB−the = −3 + (0 XB− :V X tgβ = . (47) ..(47) tgβ = (46)=and (47) also show = the speed (47) Formulas that 2 and B1 and B 2. and t Formulas (46) and show that the speed V A vertical component of the relative speed V2 should (2 + αsatisfy ) − (2β+are 2αV1 ) + ααB1B2 2(2α coupled. The be α calculated according (2 ) −speed + 2(47) αV1 )also − 2 α 2 + B1B2angle B1B2 V 1should e B1B2X + : V 1 x + = 0 + (1 angle β areangle coupled. speed should be calculated B A the vertical component of the relative speed V2 should satisfy condition: TaX β areThe coupled. The speed shouldB be according calculated accordi to the chosen course angle. and the Formulas (46) andto(47) also show that the speed V2 show and the, αXBTa (46) and also thatthe thespeed speedV Vand theFormulas chosen course angle. 2µ ,the αisXA −, the condition: Formulas (46) and(47) (47) also show that 2 topoint the chosen course angle. From theshould of calculated view of this paper, the most interesting angle β are coupled. The speed be according d 1 X angle β are coupled. The speed should be calculated according A From the point of view of this paper, the most interesting is angle(47). β are coupled. The speed should begranules calculated according From point of of this paper, the most interesting V2y = = = to 1..(38) (38) formula Fig. 8the presents twoview separate of itsinto possiXAT Taking account (49 the chosen course angle. to the chosen course angle. T 1 to the chosen angle. formula (47). Fig.course 8(47). presents two separate granules ofgranules its possi-of its pos formula Fig. 8 presents two separate ble solutions. A positive value of tgβ means the movement of From the point of view of this paper, the most interesting is From the point of view of this paper, the most interesting is ble solutions. A positive value of tg β means the movement of As the ferry (apart from its own move) is floated by the water /XaBnegative : xthe ble solutions. A positive value of tgwhile βXAmeans movement B = xA /xB A /x the ferry in the direction of the river current, formula (47). Fig. 8 presents two separate granules of its possiformula (47). Fig. 8 presents two separate granules of its possithe ferry in the direction of the river current, while a negative with a speed V1 , a horizontal component of the relative speed themovement ferry in the direction of the river current, while a negati value means in the opposite direction. (1 Bull. Pol. Ac.: Tech. 65(4) 2017 of movement tgβ Ameans thethe movement blevalue solutions. positive value of tgβof means the movement value means in opposite =503 of − V2 should satisfy the next condition: ble solutions. A positive value means movement in thedirection. opposite direction. −3 + (0 + Unauthenticated the ferry in the direction the river while a negative theofferry in thecurrent, direction of the river current, while a negative or 4.1. Case 1 Let us assume that two triangular fuzzy Download Date | 8/25/17 12:47 PMnumbers: α . (39) (V2x + V1)T = lAB · cos value means movement inCase themeans opposite (1 + value movement the two opposite direction. 4.1. 1 Let usdirection. assumeinthat triangular fuzzy numbers: of Case us0,assume that triangular fuzzy XA = (1, 2,4.1. 3) and XB1=Let (−3, 1) have to two be divided, Fig. 9.∪ numbe + + (1 and(1,X2, be have divided, 9. 0 Fig. B = Thus, the total speed V2 of the ferry can be calculated as: XA = (1, 2,X3) = n3) (−3, and X0, 1) = have (−3, to 0, 1) to beFig. divided,

Fig. 9. Fuzzy numbers XA (abo Fig. 8. Two granules representing subsets of possible solutions; the upper granule corresponds the acute angle of the ferry (β < A. to Piegat and M.course Pluciński Fig. 9. Fuzzy numbers X (about 2) and X (about Fig. Fig. Fig. Fig. Fig. 9. 9. 9. 9. 9. Fig. 9. Fig. Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy Fuzzy 9. 9.Fuzzy numbers Fuzzy numbers numbers numbers numbers numbers numbers numbers XXXXX (about (about (about (about (about (about (about 2) 2) 2) 2) 2) 2)and and and and and and 2) 2) XXXXX and and (about (about (about (about (about (about (about 0) 0) 0) 0) 0) 0) 0) 0) 0) BXX AXX BX B BB BB BB(about ◦possible AX AAAAA AA (about Fig. 8. Two granules representing subsets of possible solutions; g. .8. .8. 8. 8. 8. Fig. 8. Fig. Two Two Two Two Two Two 8.8. granules granules granules granules Two granules granules Two granules granules representing representing representing representing representing representing representing representing subsets subsets subsets subsets subsets subsets subsets subsets of of of of of of possible possible possible possible of ofthe possible possible solutions; solutions; solutions; solutions; solutions; solutions; solutions; solutions; the the the the the thethethe theto the Fig. )possible and lower corresponds obtuse angle (numbers β9.>Fuzzy 90X◦ )X(about 90 Fig. Fig. 9. 9. Fuzzy Fuzzy Fig. numbers numbers (about 2)Fig. 2) X and and (about 9.XBFuzzy X(about (about 2) numbers and 0) 0) XB (about XA (about 0) 2) and X B A A A 8. Two Two Fig. granules granules 8. Two representing representing granules Fig. representing subsets subsets 8. Two of of possible granules subsets possible solutions; of representing solutions; possible the solutions; the subsets of the possible solutions; the β((β < upper granule corresponds to the acute course angle of the ferry (< β(β(ferry β(βββ< < < < < β β90 > ) )) 90 )))and )and and and and and the the )the the )the the and and lower lower lower lower lower lower the the lower corresponds corresponds lower corresponds corresponds corresponds corresponds corresponds corresponds to to to to to tothe the the the the theobtuse to obtuse obtuse to obtuse obtuse obtuse the the obtuse obtuse angle angle angle angle angle angle (angle ((β angle (β(β(βββ> > > > > 90 (90 90 (90 β 90 β)> )90 )> )))90 90 90 90 ◦ ) and ◦obtuse ◦(42) ◦obtuse and d thethe lower lower corresponds corresponds the lower to corresponds to the ) and obtuse the to angle the angle lower obtuse (β (corresponds > β> 90angle 90 ) ◦ ) (β to > the 90 ) determine angle (the β >relative 90◦ ) 90 90the Formulas and (43) speed V2 and

the course angle β of the ferry and it can be seen that they Formulas (42) and (43) determine relative speed V and Formulas Formulas Formulas ormulas Formulas ormulas Formulas Formulas (42) (42) (42) (42) (42) (42) and and and (42) and and (42) and (43) (43) (43) (43) (43) and (43) and determine determine determine (43) determine determine (43) determine determine determine the the the the the thethe relative relative relative relative relative relative the the relative relative speed speed speed speed speed speed speed speed VVVVV V and and and and and and and 2VV 2RDM 22222and 22arithmetic, are coupled. The in contrast the standard mulas (42) Formulas (42) and and (43) (42) determine determine and (43) Formulas the determine the relative relative (42) the and speed speed relative (43) V Vdetermine and and speed V2the andrelative speed Vto 2that 2that 2 and he course angle β(43) of the ferry and it can be seen they eormulas course course course course course the course the course course angle angle angle angle angle angle angle angle ββββββ of of of of of ofβ the the β the the the the of of ferry ferry ferry ferry ferry ferry the the ferry and and ferry and and and and itititit and it it and can can can can can can it it be be be be can be be can seen seen seen seen seen seen be be that seen that that seen that that they they they that they they that they they they interval arithmetic, allows for the taking into account course urse angle the angle course angle the course angle β The βRDM of of the the ferry βferry of and the and itin ferry itcan can be and be seen it βto seen can of that the that be they ferry seen they that and it they can be seen that they of such coupled. RDM arithmetic, in contrast to the standard eare coupled. coupled. coupled. coupled. coupled. coupled. are are coupled. coupled. The The The The The The RDM RDM RDM The RDM RDM The RDM RDM arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, in in in in in contrast contrast contrast contrast contrast contrast in in contrast contrast to to to to to the the the the the the to to standard standard standard standard standard standard the the standard standard couplings. The interval [2, 3],contrast describing thestandard concrete quay on coupled. upled. are The coupled. The RDM RDM arithmetic, The arithmetic, RDM are coupled. in arithmetic, in contrast contrast The to inaccount to RDM the contrast the standard standard arithmetic, tosuch the standard in to the nterval arithmetic, allows for the taking into account of such erval rval erval rval erval rval interval interval arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, arithmetic, allows allows allows allows allows allows allows allows for for for for for for the the the the the the for for taking taking taking taking taking taking the the taking taking into into into into into into account account into account account into account account account of of of of of of such such such such such of of such such the northern bank between points B and B , is defined 1 2 al val arithmetic, arithmetic, interval allows arithmetic, allows for for the allows interval the taking taking for arithmetic, into the into taking account account allows into ofquay of such account for such the taking ofon suchinto account of such as: couplings. The interval [2, 3], describing the concrete quay uplings. plings. uplings. plings. uplings. plings. couplings. couplings. The The The The The The interval interval interval interval The interval interval The interval interval [2, [2, [2, [2, [2, [2, 3], 3], 3], 3], 3], 3], [2, describing [2, describing describing describing describing describing 3], 3], describing describing the the the the the the concrete concrete concrete concrete concrete concrete the the concrete concrete quay quay quay quay quay on quay on on quay on on onon on Fig. 10. Decomposition of the fuzzy nu plings. ngs. The couplings. The interval interval The [2, [2, 3], interval 3], describing couplings. describing [2, 3], the describing The the concrete concrete interval the quay [2, quay concrete 3], on on describing quay on the concrete quay on · cos α = 2 + α , α ∈ [0, 1] , (44) l he northern bank between points B and B , is defined as: XB− and enorthern northern northern northern northern northern the thenorthern northern bank bank bank bank bank bankbetween bank between between bank between between between between between points points points points points points points points B B B B B B B1B and and and and and B B B and B ,is ,is is is BB defined defined defined defined defined ,isisdefined defined as: as: as: as: as: as: as: as: B1B2 Fig. B1B2 1B 2is 10. Decomposition of the fuzzy number XBXinto B+ two components 111111and 1B 22and 22,2,2,,is 2defined 2,AB Fig. 10. Decomposition of the fuzzy number X into two components northern rthernthe bank bank northern between between bank points points between theBnorthern B, 1is between points and B2B,bank defined and defined B2 as: , isas: definedB1as: and B ,Decomposition is defined as:of Fig. Fig. Fig. Fig. Fig. Fig. 10. 10. 10. 10. 10. Fig. 10. Fig. Decomposition Decomposition Decomposition Decomposition 10. 10. Decomposition Decomposition of of of of ofthe the the the the the of fuzzy fuzzy of fuzzy fuzzy fuzzy fuzzy the the fuzzy number number fuzzy number number number number X X X X X X into into into into into into X X two two two two two into two into components components components components components components two two components components B 1Band 1 points 2is 2Decomposition XBnumber and Xnumber B B B B B B B B B Fig. Fig. 10. 10. Decomposition Decomposition Fig. 10. of of thethe fuzzy fuzzy Fig. number ofnumber 10. the fuzzy Decomposition XBXinto into two two components XBcomponents ofinto the two fuzzy components number XB in and interval describing values the river speed is: ·α cos α 2= + α ,α αB1B2 ∈ [0, ,1] (44) Bnumber + − X X ·l··AB ·cos ·cos ·cos cos cos ααα ·α α ·cos = cos = = = = =222α 2= 2α 2+ + + + + = + ααααα α 2B1B2 2B1B2 + +B1B2 α ,,α ,,,B1B2 ,B1B2 α ααα α ,B1B2 ,B1B2 αthe α ∈ ∈ ∈ ∈ ∈ ∈[0, [0, [0, [0, [0, [0, 1] 1] ∈ 1] 1] ∈ 1] 1] ,,[0, ,,[0, ,1] , 1] ,, (44) (44) (44) (44) (44) (44) (44) (44) lllAB lAB lAB lAB lcos lAB + + + + + +BX − − − − − − − − X XXXX X XBof and and and and and and X XX XX and and XBB++Decomposition B AB AB AB B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 BX BBB B B Band BX B BB B B · cos · cos αα =lAB = 2+ 2· cos + αB1B2 ααB1B2 = , 2, α +B1B2 α αB1B2 ∈ ,∈ [0, · cos [0, 1] αB1B2 1] α , ,= ∈2 + [0, (44) α1] (44) , ,XBX−αB(44) [0, 1] , (44) lABlAB lAB − and + + − + − + X X and X∈ X and X and X B1B2 B1B2 B1B2 The quotient X /X can be prese B BB B B B B A (45) Vis: the interval describing values of the river speed 1= dand ddthe the the the the and the and interval interval interval interval interval interval the the interval interval describing describing describing describing describing describing describing describing values values values values values values values values of of of of of ofthe the the the the the of of river river river river river river the the speed speed river speed speed river speed speed speed speed is: is: is: is: is: is: is: is:2 + 2αV 1 , αV 1 ∈ [0, 1] . The quotient X /Xbe can be presented in the form: ethe interval interval and the describing describing intervalvalues describing values andofthe of thevalues the interval river river of speed describing speed the is: river is: speed values is:of theThe river speed is: A B The quotient X /X be presented in the form: XA /XB = (XA /XB− ) ∪ The The The The The quotient quotient quotient quotient The quotient quotient The quotient quotient XXXXX X /X /X /X X X can can /X can can /X be be be be can be can presented presented presented presented presented presented be be presented presented in in in in in inthe the the the the the in form: in form: form: form: form: form: the the form: form: B can AB BB BBB BB A AAA/X A/X A/X Acan Acan tgβThe are described equaFinally, values ofThe VThe 2= ,α ∈ [0, .possible (45) V= α α α ,,αα α α ∈ ∈ α ∈ ∈ α [0, [0, [0, 1] ∈ 1] 1] ∈ 1] 1] ..[0, ..[0, .1] . 1] 1].. (45) (45) (45) (45) (45) (45) (45) (45) = = = 22V 2= 212+ + + + = + 2222α 2+ 2α 2α VVVVV V 2 and 12 V 1Vα V∈ 1[0, quotient quotient XAX/X quotient be XAby be /X presented presented can quotient beinpresented in thethe form: Xform: canform: be presented in th 1V 1111= 1= 1+ V V+ V2 V+ V 1V 11α 1,1,2 1,,2 Vα 1α V 1V,VV,V 1V 11α 111∈ V[0, V[0, 111] B can B can B The B the A /X A /Xin ∈1V∈ [0, ,[0, 1]α 1] . VV.11 ∈=[0, 1]2. (45) αV(45) αV 1 (45) ∈ [0, 1] . (45) 2+ 2+ 2α2VVα 2+ V1V= 1= 11V,= 1 ,2α+ Vα12Vα 1tions: 1, X /X = /X ) ∪ (X /X ) . (48) . (48) − + From equation (10) and Fig. 9, XXXXX X X X /X /X /X = = = = /X = = /X (X (X (X (X = /X = /X /X /X /X (X (X ) ) /X ) ) /X ) ∪ ) ∪ ∪ ∪ ∪ ∪ (X (X (X (X (X (X ) ) ∪ /X /X ∪ /X /X /X /X (X (X ) ) /X ) ) /X . ) . ) . . . . ) ) . . (48) (48) (48) (48) (48) (48) (48) (48) − − − − − − − − + + + + + + + + B(X AB A A B B BB BBB B(X BAAA A AAA/X A/X A/X A A A(X A A/X A A A A A A A A A A BBBBBB BB BBBBBB BB and tg β are described by equaFinally, possible values of V and and and and and tg tg tg and tg βand βββββ are are are are are tg are tgβdescribed described βdescribed described described described are are described described by by by by by byequaequaequaequaequaequaby by equaequaFinally, Finally, Finally, inally, Finally, inally, Finally, Finally, possible possible possible possible possible possible possible possible values values values values values values values values of of of of of ofVVVV V of  2VV XAX/X (XX (X /X /X )(X (X ∪A(X /X /X ∪ ./X =Bbe (X )(48) . /X (48) (48) −∪ −+)B)X +. ) + − ) ∪ (X 2V 2222and 2of 2tg 2tg B= B= B/X A /X AA/X A A/X AB A(X Acan A A /XB+ BB−B)= B B X , X + found: − B B and tgβ tgβare and described described tgβ values arebydescribed by equabytgequaβFrom areFrom by equaally, nally, possible possible Finally, values values possible ofof V2V values Finally, of V2are possible ofequaV2 and 2described 2 and ions: equation (10) and Fig. 9, horizontal MFs of sets X , ns: ns: s: ns: s: ns: tions: tions: From From From From From From equation From equation equation equation equation equation equation equation (10) (10) (10) (10) (10) (10) and and (10) and and (10) and and Fig. Fig. Fig. Fig. and Fig. Fig. and 9, 9, 9, Fig. 9, 9, Fig. 9, horizontal horizontal horizontal horizontal horizontal horizontal 9, 9, horizontal horizontal MFs MFs MFs MFs MFs MFs of MFs of of MFs of of of sets sets sets sets sets sets of of X X X X sets X X sets X X , , , , , , , , V2 = (2 + αB1B2) + (2 + 2αV1 ) + 1, ∈ [0, 1] Fig. 9, , Fromαequation and horizontal MFs of setsAAAAAX A AA, AA B1B2 , αV 1(10)  s:  tions:     tions:  From From equation equation From (10) (10) equation and and Fig. Fig. (10) 9,(46) From 9, and horizontal horizontal Fig. equation 9, MFs horizontal MFs (10) ofxof sets and sets MFs X Fig. horizonta XA− , Aof ,9,)sets , µ AX X , X + can be found: − ¡ + X : = (1 + µ + 2(1 X , X can be found: X X X X X X , , , , X , X , X X X X + + + + + + , can can , can can X can can X + be + be be be be can be can found: found: found: found: found: found: be be found: found: − − − − − − − − A A B B B B B B B B B B B B B B B B B B 2  = (2 + α 22222 2) + 22 1, α V ) + (2 + 2 α , α ∈ [0, 1] , X X X X , X , X + can + can be , be X found: found: + can be found: , X + can be found: − − − − = = = = = (2 = (2 (2 = (2 (2+ + + + + +Fig. ααα (2 αα (2 α + + α ) ) α ) ) ) + ) + + + + + (2 (2 (2 (2 (2 (2 ) + ) + + + + + 2 2 2 (2 2 α 2 (2 α 2 α α α α + + ) ) 2 ) ) 2 ) ) α + α + + + + + 1, 1, 1, ) 1, 1, ) 1, + + α α α 1, α α 1, α α , , α , α , α , α , α α α ∈ , ∈ ∈ , ∈ α ∈ ∈ α [0, [0, [0, [0, [0, [0, 1] 1] ∈ 1] 1] ∈ 1] 1] , , [0, , , [0, , , 1] 1] , , 2VV B1B2 V1 B1B2 V 1 2= 22(2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 V1 V1 V1 V1 V1 V1 V1 V1 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 B1B2 V V V V V 1 V 1 1 1 1 1 V V 1 1 1 2 B BtheB B B B 1 B B 8. Two granules 2representing subsets of possible solutions; xB− = −3 + (0− + 3)µ XB−,: VV1 = (2(2 +V+ α = α )(2 +)granule + (2(2 α+B1B2 + 2corresponds α2V1 )α+ )2(2 )= +2 + + 1,21, (2 αV1 α+B1B2 α )tg α2acute +, = α 1,,V)course α1+ α ∈ [0, +[0, 1]2,(46) α 1] ,(46) ,1(46) )∈ + [0, 1,1]XXXX,X α::X::B1B2 ∈(1 [0, 1] β .+)2(1 (47) 2 B1B2 B1B2 B1B2 B1B2 V∈ 1(2 B1B2 V V1 1= :AA,:xxα x= = (1 µ+ + 2(1 µ− )XA α (49) X :):AXX= :xxAxAxAVAAA= = = = x(1 (1 x(1 (1 + = + + = + + µµ(1 µµ+ (1 µµ ),))))+ )+ + + µ 2(1 2(1 µ 2(1 2(1 2(1 ))+ − + − − − − − 2(1 µ 2(1 µµµ− µµ )))α )α )− α )α α α µXA µ,,),XA ,),α ,αXA (49) (49) (49) (49) (49) (49) (49) (49) A upper to the of the ferry (46) (46) (46) (46) (46) (46) A AAAAA A(1 A+ XA XA XA XA XA,, (2 + αangle ) − (2 + 2 α α − 2 α + − µ )α(49) B1B2 X(46) : xAxX= = (1 : (1 ++ µx(46) )µA+V)=1+ 2(1 (1 2(1 − +−− µX )µ +):XA α2(1 , −−− ,X − x−BAµ ):−α (1 ,xµB+) + (49) 2(1 (49) (46)(β > 90°) AX:A V1 AAB1B2 Aα XA XA+ −= (β  0, variables αbeXAnoticed ,beαXB ,α ,inthe and the denominator It should noticed in the considered domain It should that thenominator considered domain of RDM RDM XCthat C1 : xC = (1 − µ )(−1 + 4αXC), αXC ∈ [0, ) , of−2 formula (61) do not change their sign, i.e.: 1 + 2 α > 0, XA + 3 α < 0, −1 + 4 α < 0. So, all possible division re4 XCnominator α, XA , αdo ,the the nominator andi.e.: the denominator of variables formula not change their sign, 1 + 2αXA > 0, be si XB XC αXB αXB ,+ α, α and the denominator variables XA(61) XC −2sults +of3formula α −1 4granule α, XC < 0.Z1their So, all possible division XB < 0, 1 contained in the are positive. Analysis ofrefor(61) do not change sign, i.e.: 1 + 2α  > 0, −2 + 3 α < 0, −1 + 4 α < 0. So, all possible reof formula (61)indothe notgranule change αXA > 0, XB XC their sign, i.e.: 1 + 2division XA C2 : xC = (1 − µ )(−1 + 4αXC), αXC ∈ ( , 1] . 56) sults contained Z are positive. Analysis of for1 ¡2 + 3α   1+ 2α2XA 1+ αXA 3 gr gr 0, B2C1Z4Z:4 z:gr , , A : xA = (1 + µ ) + 3(1 − µ )αXA, µ , αXA ∈ [0, 1] , e-re1 + 2αXA 4 z4= = [−2 ++ 3α3XB ][−1 ++ 4α4XC ] ,] [−2 αµ ][−1 αXC Z : z = XB e4 4 (67) (1 + µ ) + 2(1 − ) α r(67) XA number division and (64) the multi-granularity phenomenon for1 [−2 +Fuzzy 3αXB + 4α XC ]1 zgr = 2 ][−1 (67) XB1 : xB1 = (2 + µ ) + 3(1 − µ )αXB, αXB ∈ [0, ) , (69) 2 1 , (67) of (69) [(1 −αµXA )(−2 + 3 α )][(1 − µ )(−1 + 4 α )] er-of 3 , 1], , 1] . ∈ [0, 1], α ∈ ( α ∈ ( αXA ∈ [0, 1],XBXB αXB ∈2( , 1], XC α ∈XC 1( , 1] . 3 3 ,31], αXCXC 4 ,41] . of αXA ∈ [0, 1], α ∈ ( ∈ ( XB 2 1 1 Fig. 18. Decomposition of membership function of fuzzy 4 XB2 : xB2 = (2 + µ ) + 3(1 − µ )αXB, αXB ∈ ( , 1]value XB ∈ [0, 1], αXB ∈ ( , 1], αXC 3∈ [0, ) . e-de- αXAThis component granule is shown in Fig. 16 and it can be = � 1/3 into two parts, x = � 3, α This component 3granule is shown 4 in Fig. 16 and it can be 3 B XB e- seen This component granule is shown 16 and it can be that it it differs from other component granules. other This component granule is component shownininFig. Fig. 16 and For it For can be seen that differs from other granules. other that that itµ differs from other component granules. For other other 10 seen Bull.formula Pol. Ac.:(68) Tech.we XX(Y) seen differs from other component granules. For have:2016 fractional -levels, distances between component granules areare After inserting horizontal MFs into fractional µit-levels, distances between component granules After inserting horizontal MFs into formula (68) we have: fractional µ -levels, distances between component granules are fractional μ-levels, distances between component are different from granule distances forfor thethe level 0. 0.For thethe different from granule distances levelµ µ=granules = For different from granule distances for the level µ = 0. For the different from granule distances for the level μ = 0. For the (1 + µ ) + 3(1 − µ )αXA level µ = 1, the division result is undetermined. Because in in level µ = 1, the division result is undetermined. Because Z1 : zgr = , level µ = 1, the division result is undetermined. Because in 1 level μ = 1, the division result is undetermined. Because in real system only one value ofof each variable αXA , α,XB , α,XC D 2) real system only one value each variable αXA αXB αXC 62) thethe the therefore real system only one value ofresult each αXA , αXB real system only one value of each variable ,α 2) the XB,, α XC XA XC 1 is is true, also one division zvariable = xAxα /(x true, therefore also one division result z= /(x )α exists. B xB Cx)Cexists. A/(x µ , αXA ∈ [0, 1], αXB ∈ [0, ) , is true, therefore also one division resultz = z = x x ) exists. A B C is true, therefore also one division result x /(x x ) exists. B A C This result is a single point that can lie only in one of the four 3 (70) This result isisa a single single point that can canlie lieonly onlyininone one the four This result point that of of the four (70) This result is a single point that can lie only in one of the four component granules Z1ZZ–1  ¡ Z Z–4Z. 4 .. (1 + µ ) + 3(1 − µ )αXA component granules gr component granules 1 4 Z : z = , component granules Z1 – Z4 . 2 2 D en in 1 n 4.4. 4.4. Case 4.Now, Now, let us examine the division case with a deCase 4 4Now, letlet usus examine the division case with a deis 4.4. Case examine the division case with a dethis µ , αXA ∈ [0, 1], αXB ∈ ( , 1] , Case 4in in Now, let us examine thepolynomial division case with (68). a(68). denominator in which the second-order polynomial occurs, is 4.4. 3 nominator which thethe second-order occurs, el nominator which second-order polynomial occurs, (68). evel Value of the variable x is uncertain and is expressed by the in which the second-order polynomial occurs, (68). B el nominator Value ofof thethe variable xBxBis isuncertain and is isexpressed byby the where: Value variable uncertain and expressed the where: fuzzy value XB given byistriple (2, 3, 5). Fuzzy values A and X B Value of the variable x uncertain and is expressed by the Fig. Z1 and Z2 of the division result B fuzzy value X given by triple (2,3,5). Fuzzy values A and X fuzzy valueBX given by triple (2,3,5). Fuzzy values A andBXB 19. Two component granules 2 BFig. 17. are shown in µ )+3(1− µ ) α ] −10[(2+ )+3(1− µ )αXB ]+21 . D = [(2+ = � 3, αX B �= µ 1/3, z∈ / [−0.25, 0.2] determined by formula (71), x XB fuzzy valueinXin given by triple (2,3,5). Fuzzy values A and XB B 2 B Fig. areare shown Fig. 17.17. shown D = [(2 + μ) + 3(1 ¡ μ)α ]  ¡ 10[(2 + μ) + 3(1 ¡ μ)α ] + 21. xA xB are shown in Fig. 17. 3) 63) (1,(1, AA 2, 2, 4)4) Bull. Ac.: Tech. XX(Y) 2016 3) AsPol. can formula (70), (70), the division resultresult con, , XBX= (2,(2, 3, 3, 5)(68) Z= Asbe canseen be from seen from formula the division 5) (68) (68) Z= A = = 2 2(1, 2, 4) B= 10X + 21 = XB2X− , X = (2, 3, 5) (68) Z= BB − 10X + 21 B sists of two granules Z and Z which exist in 4D-space B B consists of two granules 1 Z1 and2 Z 2 which exist in 4D-space B B XB − 10XB + 21 , α), ), hence they cannot be precisely visualz = fz = f (µ , α(μ, α αXB they cannot be precisely visualized. XA , XA XBhence = 3 and and The denominator in in formula (68) has two roots: xxBx = 3 at 3However, and ized. However, The denominator formula (68) has two roots: that16. Simplified B= they can be visualized in a simplified way in The denominator in formula (68) has two roots: Fig. visualization of 5D result granules Z – Z of the they can be visualized in a simplified way in 2DB 1 4 =The 3diviand denominator in formula (68) has two roots: at xBxThe = 7. = 7. The fuzzy value XBX the root xxB=x = 3. = 3.xB3. The 7. The fuzzy value contains the root = diviB= B The fuzzy value Xcontains contains the root The divi2D-space for particularly chosen membership levels μ. For For the x= µ 0 division Z in a 3D subspace the cut level BA/(BC) B for BB space for particularly chosen membership levels µ . the xBsion =result 7.result Theisfuzzy value XB for contains the root xMF =of3.of The diviBMF indeterminate this value, sososo should for this value, asion resultis isindeterminate indeterminate for this value, MF of X XBBXshould formula (70) takes form: B should ma- sion levellevel µ =μ = 0 0 formula (70) takes thetheform: sion result is indeterminate for this value, so MF of X should adecomposed into two parts asas shown ininin Fig. 18.18. B be decomposed into two parts shown Fig. 18. decomposed into two parts as shown Fig. lt bebe sult decomposed into two horizontal parts as shown in Fig. 18. 1 + 3αXA lt be Formula gr (69) presents MFs ofof fuzzy values A,A, XB1 Formula (69) presents horizontal MFs fuzzy values XB1 , Z1 : z1 = 2 Formula (69) presents horizontal MFs of fuzzy values A, X 9αXB − 18αXB + 5 B1 and XB2 . . and XB2 and XB2 . 1 µ , αXA ∈ [0, 1], αXB ∈ [0, ) , AA : x:Ax= (1(1 ++ µ )µ+) + 3(1 −− µ )µα)XA , ,µ ,µα,XA ∈∈ [0,[0, 1] 1] , , 3(1 αXA αXA A= 3 (71) A : xA = (1 + µ ) + 3(1 − µ )αXA, µ , αXA ∈ [0, 1] , (71) 1 1 (1 + µ ) + 3(1 − µ )αXA 4) gr 64) ) , ) , (69) : x:B1xB1 == (2(2 ++ µ )µ+) + 3(1 −− µ )µα)XB , ,αXB ∈∈ [0,[0, XB1 3(1 α α X 1 XB XB B1 Z , : z = 2 (69) 4) 2 2 − 18α XB1 : xB1 = (2 + µ ) + 3(1 − µ )αXB, αXB ∈ [0, 3 )3, 9αXB XB + 5 (69) 1 31 1 XB2 : x:B2xB2 =functions (2(2 ++ µ )µof +) + 3(1 −values µ )µα)XB ,and ∈occurring ( 1(, 1], 1]in Fig. 17. Membership fuzzy XBXB XB2 = 3(1 − αAXB ,αXB α ∈ µ , αXA ∈ [0, 1], αXB ∈ ( , 1] . XB2 : xB2 = (2 + µ ) + 3(1 − µ )αXB, αXB ∈ ( 3 ,31] he division (68) 3 3 Component granules Z1 Zandand Z2Zofofthe are Bull. Pol. Ac.: Tech. XX(Y) 2016 Bull. Pol. XX(Y) 2016 Component granules thedivision division result result are Fig. 17. Membership functions of fuzzy values A Ac.: and XTech. 1 2 B occurring in Bull. Pol. Ac.: Tech. XX(Y) 2016 shown in Fig. 19. shown in Fig. 19. the division (68) The true, crisp result of the division (70) lies in one of two granules: Z1 or Z2 . In the real problem we have sometimes 508 Ac.: Tech. 65(4) 2017 some additional knowledge, e.g. thatBull. the Pol. division result is negUnauthenticated ative. Then the true result can be located only in Z2 , which Download Date | 8/25/17 12:47 PM means decreasing the uncertainty.

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Fuzzy number division and the multi-granularity phenomenon

Fig. 19. Two component granules Z1 and Z2 of the division result / [−0.25, 0.2] determined by formula (71), xB �= 3, αX B �= 1/3, z ∈

As can be seen from formula (70), the division result consists of two granules Z1 and Z2 which exist in 4D-space z = f (µ , αXA , αXB ), hence they cannot be precisely visualized. However, they can be visualized in a simplified way in 2DA. Piegat, M. Pluci´nski Fig. 2 space for particularly chosen membership levels µ . For the tion o Fig. 20. Membership functions of uncertain coefficients A , B 1 1, C level µ = 0 formula (70) takes the form: Fig. 20. Membership functions of uncertain coefficients A , B , C ,

1 21. 1 Visualisatio 1 Fig. A2 , B2 , C2 occurring in the equation system (72) A2 , B2 , C2 occurring in the equation system (72) tion of the denomina Z1 : Fig. 20. Membership functions Fig. 20. of Membership uncertain coefficients functionsAof B1 , C1 , coefficients A1 , B1 , C 1 , uncertain Fig. A2 , B2 , C2 occurring in theAequation B2 , C2 system occurring (72)in equationcoefficients system (72) MFs of the uncertain can21.beVisualisatio determine 2 ,Horizontal Horizontal MFs of uncertain coefficients can be determined 1 tion of the denomina as: , 20. Membership µ , αXA ∈ [0, 1], αXB ∈ [0, )Fig. functions of Membership uncertain coefficients B1 , C1 , coefficients A1 , B1 , C Fig. 20. functionsAof 1 , uncertain as: 3 (71) A , B , C occurring in the equation system (72) A , B , C occurring in the equation system MFs of uncertain : xA1ofcan =uncertain (1be+determined µ ) +coefficients 3αA1(1(72) − µcan ) , be determine 2 2 2 Horizontal 2 coefficients 2 A1MFs (1 + µ ) + 3(1 − µ )α 2 Horizontal

gr z1

1 + 3αXA , = 2 9αXB − 18αXB + 5

XA A : x = (1 + µ ) + 3αA1(1 − µ ) , , as: as: 1 A1 B : x = (3 + 2 − 18α 2 µ ) + 3α (1 − µ ) , 1 9αXB + 5 XB + B1 2 µ ) + 3αB1(1 − µ ) ,B1 B1 : xB1 = (3 Horizontal MFs of uncertain coefficients can be determined Horizontal MFs of uncertain A A : x = (1 + µ ) + 3 α (1 : x − µ = ) , (1 + ) )+coefficients −−µcan 1 A1 A1 = (15 +µµ A1(1(1 +32αα µ) ,) ,be determine CA1 1 11 : xC1 n + µ ) + 2αC1(1 − µ ) C1 , C1 : xC1 = (15 . µ , αXA ∈ [0, 1], αXB ∈ ( , 1]as: as: xxA2 −= µ )(3 , + B1 : xB1 = (3 + 2 µ ) + 3A Bα21B1:: (1 B1 = (2 + 2µµ) )++33αα (1(1−−µµ) ,) , (73 A2B1 Fig. 19. Two component granules Z1 and Z23of the division result : x = (2 + µ ) + 3 α (1 − µ ) , A (73) 2 A2 A2 A : x = (1 + µ ) + 3 α (1 − µ ) , A : x = (1 + µ ) + 3 α (1 − µ ) , determined by formula (71), x  6 =  3, α  6 =  1/3, z 2 / [¡0.25, 0.2] 1 A1 A1 1 A1 A1 xxC1 −= µ )(15 , + µ ) + 2α (1(1−−µµ) ,) , Component granules Z1 andB Z2 ofXBthe division result Care 1 : xC1 = (15 + µ ) + 2C Bα12C1:: (1 B2 = (7 + µ ) + 2αB2C1 (7:+ µ−)of= + 2,α+ )α , (1 B+2 2: µx)B2+= A. Piegat, M.Fig. Pluci´ nMembership ski B2(1 − 20. functions uncertain A− Bµ1µ ,)C : x = (3 3 α (1 µ ) B shown in Fig. 19. x (3 )+)+µ3+(73) 3α B 1,− 1 B1 B1 1 B1 B1 : xx− =) ,(21 (2 ++2µ3µ)coefficients (1 ,) 1,, A2, A2 : xA2 = (2 + µ ) + 3αC AA2 (73 2 (1 A2µ= A2 : µ 4 α 2 C2 C2(1 − µ ) , B , C occurring in the equation system 2 xC2 2 = (21 + 3 µ ) + 4αC2(1 − µ )(72) : , C 2 The true, crisp result of the division (70) lies in one of C two : x = (15 + µ ) + 2 α (1 − µ ) , : x = (15 + µ ) + 2 α (1 − µ ) , C 12,C1 C1 : A1 x− ) ,(7 + µ ) + 2α ,C1 (1 −∈µ[0, ) , 1] . B12 : xC1 B B2 = (7 + µ ) + 2α B2 µB2 α(1 , αµ= α TheZtrue, crisp result of the division (70) lies in one of two B1 , αC1 , αA2 , αB2B2 granules: µ+, α , α3B1 , α(1 , αµA2 ,,(2 αB2+, α ∈3(73) [0, 1](1C2 . − µ), 1 or Z2 . In the real problem we have sometimes A1 C1x− C2 : x = (2 µ ) + α ) A : = µ ) + α A (73 A2 = (21 + 3 µ ) +C4 A2 A2− Z1 or Z2. In a reale.g. proble,m wedivision sometimes haveissome =µ(21 ) , + 3 µ ) + 4A2αC2(1 − µ ) , C22 : xC2 22α:C2x(1 C2 somegranules: additional knowledge, that the result negThe determinant D of the equation system (72) has the form additional knowledge, e.g. that the division result is negative. The D ofx− the equation (72) has the ofB2 the system (72) has form x , αdeterminant = (7 +,µα)A2+,D 2αα (1 ) ,,(7 B + )A2 +, gr 2ααB2 (1C2 −∈µthe )gr, form B 2, α B2 µ2, α: The αdeterminant µ ,equation ∈ αµ= [0, α 1]C1 .grgr ,µαsystem ,α 1] . ative. Then the true result can be located only in Z2 , which A1B2 B1 ,D C1 B2 A1 C2x = ¡ x xA1 xB2 −and x:A2 and solutions xB2gr and x[0, gr B1B1 1can 2 can be formu Then the true result can be located only in Z2, which means D = x x x solutions z and z can be formulated A1 B2 A2 B1 D = x x − x x and solutions x and x be formu1 2 A1 B2 A2 B1 1 2 = (21 + as: 3 µ ) +C42α:C2x(1 ) , + 3 µ ) + 4αC2(1 − µ ) , C2 : xC2 means decreasing the uncertainty. =µ(21 C2− decreasing the uncertainty. The determinant D oflated the The equation determinant system D (72) of the has equation the form system (72) has the form latedas: as: gr ,− gr gr ∈ [0, gr gr gr gr 1] . , α , α , α , α ∈ [0, 1] . µ , α , α , α , α α , α x x x xC2 A1 B1 C1 A2 B2 C2 A1 B1 C1 A2 B2 C2 21. Visualisation of 2-granularity of x = x ∪ funcB2 C1 B1 gr D = xA1 xB2 µ−, α D = x xFig. x and solutions x − x x and x x and can solutions be formux and be formu A2 B1 A1 B2 1 A2 B1 1 1,1 1 x 1,2, asx2thecan 2− x x x x x = B2 C1 B1 C2 gr 4.5. 4.5. Case 5 Solving linear equation systems is a frequent task 1 numerator ,N of the solution (75) Case 5. Solving linear equation systems is a frequent task tion of the denominator D and the x = xhas x the−form xA2system xB1 (72) has theFig. lated as: as: B A1equation 1 system D oflated the equation The determinant of form 2 xA1 xD − the xA2 xB1B2 Fig. in 20.practical Membership functions uncertain coefficientsoccurring A1The , B1determinant , Cin (74 problems [26,of 27, 28].Coefficients Frequently, coefficients ocB2(72) 1 , the in practical problems [26–28]. grx gr x grx gr (74) x − x (74) A1 C2 A2 C1 gr x x x − x x − x x B2 C1 B1 C2 B2 C1 B1 C2 nomin gr gr D = x x − x x and solutions x and x can be formuD = x x − x x and solutions x and x can be formu x x − x x A2 , Bcurring , C occurring in the equation system (72) A1 B2 A2 B1 A1 B2 A2 B1 . 2 A1 xC2 = A2 C1 2 2equations often uncertain. A typical Typical exampleexample is an economin theare equations are uncertain. is an 1 x1 = xgr =1 x ,x21 2 = xxA1 xxB2.− xxA2 xB1 ,Fig. 22. Visualisatio 2 xA2 xA1as: xB2 − as:balance lated A2 xB1 ical plan plan modelmodel for the for nextthe period as e.g. balance model of LexB1A1 xB2 − xA2A1xB1B2 −(74) economical next period aslated e.g. (74 gr gr nominator D and the xA1 xxB2 xC2 − xA2 xC2 gr − xxB1 ontief for state economy Solving [29]. a linearSolving equationasystem A1 x C2 A2 x C1 gr xC1 xC2 the f x really have charac model of MFs Leontief for state[29]. economy linear B2 xcheck C1 − xwhether B1 xC1 gr grx1− and gr = xTo grsolutions Horizontal of uncertain coefficients can be determined 2 x x . = . gr really have characwhether x21To=check ,x21x1z=grand , character 22 really requires division by the systemby determinant which also isD To check − solutions xA2solutions xxB1 xxxzB2 − equa Fig. 22.them Visualisatio B2 A1 B1 terxxA1 ofxxwhether multi-granules it 1isxxand sufficient toxxhave examine for th system requires division the systemDdeterminant − xxA2 A1 B2 − xitA2 B1 A1 B2 A2 B1 as: equation ter of multi-granules is sufficient to examine them for theof multi(74) (74 uncertain and can contain negative numbers, zero and positive of multi-granules it is sufficient to examine them for the memthe form nominator D and the gr grµ gr gr gr membership level = 0. For this level solutions (74) take th x x − x x which also is uncertain and can contain negative numbers, zero x x − x x A1 C2 A2 C1 A1 C2 A2 C1 gr really havesolutions and take x(74) really have charac To checkLet whether To check xµ1 =and whether solutions xcharacD membership level 0. xFor this take the numbers. solutions. bership μ = 0. For this level solutions the form: x2solutions = level 2 .x 1 (74) 2 .equation =level A1 : xSuch (1 + µ ) results + 3αA1in(1multi-granular − µ), A1 = situation system (7 2 form: and positive numbers. Such situation results inter multi-granular xB2for − xthe B2 − xA2 A1 A2 of multi-granules isxA1 sufficient ofxmulti-granules toxB1 examine it isxthem sufficient toxB1 examine them for αth us consider equation (72)(1 with coefficients form: itter : xB1 =consider (3 + 2system µequation ) + 3α − µuncertain ) (72) , B1 Let B1 system solutions. us with uncertain multiDform =(74) (14of + 3α) A1th) gr gr gr (7 + 2 α )(15 + 2 α ) − (3 + 3 αthe )(21 + αtake membership level membership level µ = 0. For this level solutions µ = 0. (74) For this take level the solutions determined by triangle MFs. B1 C2 and really To check whether x1)(15 and x+gr really charac To whether xcharac0) coefficients (7 + αcheck +x22B2 αC1 )solutions − (3have +C1 3α αC2) have , x2gr = B2 1B1)(21 2 4equation gr solutions Be MFs. system xC1 = (15 +by µ )triangle + 2αC1(1 − µ ) , form: C1 : determined 1 x1 = itform: , ,them (1 + 3examine αA1)(7 + 2sufficient αB2)for − (2 + 3examine αA2)(3 αB1 )A2 ,(7 αA1+ α3B1 , αfor α ter of multi-granules is sufficient to them the ter of multi-granules it is to th (1 + 3αA1)(7 + 2αB2) − (2 + 3αA2)(3 + 3αB1) RDM 3 α (1 − µ ) , A2 : xA2 = (2 +A1µx)1+ (73) D =(74) (12 3α) A1 ) + B1A2 x2 = C1 (1 +level 32αsolutions 42α4C1 ) −)level (2 3αA2 )(15 + αthe (7 +level 2α µ)(15 (7level + For 2αC1this )− (3 + 3B2 αµ)(21 )(15 )(21 + αthis (3 )(21 4+ membership = 0. (74) take the+ solutions membership 0.+ For take th× C1 B1= C2 B1Because C2 αB1,de (1 +x3gr αA1)(21 + 4A1 αC2 ) − (2 +C2 3α (72) (72) B2 xgr , gr+ 2αgr A2)(15 C1 ) gr 21 = 1 = gr 2 α (1 − µ ) , B2 : xB2 = (7 +A2µx)1+ form: x , = form: (2 3αRDM-variables, )(3 αfunc(1 + 3α2A1 )(721. + 2αB2) −(1 (2++33αα )(3++23ααB2 + B2B2 x2 = C2 αA1 α3B1 , α)value A1 B1 A2)(7 B1 A2 ,itα Fig. 2-granularity of))x−1 )(3 =+ x+ x1,2 as+,the 1,13∪ (1 Visualisation + 3αA1)(7 +of2α αA2 B2) − (2 + 3αA2 B1 ) tion of the denominator D and the numerator N of the solution (75) α × α × α .f : x = (21 + 3 µ ) + 4 α (1 − µ ) , C (7 + 32αA1 )(15 + 42αα (3 )(21 (1 (1 )− + )(21 24C1 α∈ ) (3 (21]+.(75) )(15 2αthe (7 23α,3B2 )(15 +,+ 24ααC2 )− 3αB1 )(21 +A24 )(21 ,α ,+ αC1 ααA2 αB2 [0, α)B2 2 particular C2 C2 be expressed gr B2 C1 B1,)(15 C2 C2 A1 A2 C2 C1 A2B1 C1 C2 gr A1 B1 (2 B2de Because coefficients can inA= the Fig.MFs 20. of Membership functions of uncertain coefficients , B1form , C1 , αA1 , αB1 , αxC1 , α , α , α ∈ [0, 1] . , = , x 1 A2 B2 C2 1 2 2 1 value for all comb MFs of, particular coefficients be[0, expressed the form (12 + (2++33ααA1A2)(7 )(3++23ααB2 (75 α α(1, αC1 , α(3, αcan . 16,in17), it A2)(3 + 3αB1 )amin B1))− (2 + 3αRDM-variables, of2 ,triples: AA1 2, 4), BA2 5, 6),∈C A =3αA1)(7 + 2αB2) −(1 B1 ,in B2 ,system C2 1= 1= 1 =1](15, A B2 , Cµ2,occurring the, α equation (72) (75) of triples: A1 = (1, 2, 4), B1 = (3, 5, 6), C1 = (15, 16, 17), α for values 0 a B2 α × α × α 21 . (2, 3, 5), B2 = (7, 8, 9), C2 = (21, 24, 25), Fig. 20. αgrA1 , αB1 (1,+ 3 α )(21 + 4 α ) − (2 + 3 α )(15 + 2 α ) (1 + 3 α )(21 + 4 α ) − (2 + 3 α )(15 + 2 α ) B1 A2 B2 gr α , α , α , α , ∈ α [0, , α 1] . , α , α , α ∈ [0, 1] . C2 B1 C1 A1A2 A2 B2 C2 C2C1 A2 C1 .m gr C1 A1A2 B2 Let A1 C2 The determinant D of the equation (72) hasFig. 20. the A2 = (2, 3, 5), B2 = (7, 8, 9), C2system  = (21, 24, 25), us notice, that the nominator and denominator of x gr x2 form , = x , = amination are give 1 i for comb that the and denominator of)(3 x1+ in gr (1 + 3Let αA1us )(7notice, + 22αare )not −(1 (2 )(3++They )−(75) ++nominator 33ααA1A2)(7 23ααB2B1 )are (2 + 3αvalue 3αall B2 A2 B1 ) (75 MFs of uncertain coefficients can be determined D = xA1 xB2 Horizontal − xA2xMFs solutions xgr and x can be formum (75) independent. coupled by RDM-variable Horizontal of uncertain coefficients determined B1 and 1 2 for D values a (75) independent. They are coupled by 1] RDM-variables = D(0α grαB2min A1 αA1 , αB1 , α11 , αare ,not αα , ∈ [0, 1] . α , α , α , α , α , α ∈ [0, . gr gr latedas: as: as: and α . Similar coupling occurs in x by RDM-variable C1 A2 B2 C2 A1 B1 C1 A2 B2 C2 B2 and that 2amination are Let us notice, nominator us notice, denominator the nominator xby and ofgr give x1 i in denominator αB1that and the αB2B1 .Let Similar coupling occurs in xofgr RDM-variables 1 gr 2 (75) αA D =x D(hav αA1 and αA2 . To check whether solutions xmax and gr gr −) x+B13xαC2 (1 − µ ) ,(75) are not independent. C1µ 1RDM-variable 2 (75 gr They are not coupled independent. by RDM-variables They are A1 :xgrxA1==xB2 (1x+ It αA1 andLet α(75) . notice Toare check whether solutions xcoupled and xby have A1, A2 us that the nominator and denominator of z in 1 2 gr gr gr 1 min D = D( α 1 gr of the deFig. 22.coupling Visualisation ofSimilar x2 xgr = by xcoupling ∪ x2,2 as the gr function grA xA1 xB2 − xA2 xB1 α and α . Similar α and occurs α . in RDM-variables occurs in x by RDM-variable 2,1 B1 B2 B1 B2 (75)the are nominator not arenominator coupled RDM-variables 2They 2 notice, that and that denominator of x1byand in Letindependent. us notice, the denominator x1 i B1 : xB1 = (3 + 2 µ ) + 3αB1(1 − µ ) , Let us (74) gr the gr D =of gr Dwhether and theαnumerator Nxof solution (75) gr have αth D(hav 12 xA1 xC2 − xA2 xC1 A Itxmax can be xseen α and α .12nominator ToαB1 check α and solutions .coupling To check and whether xzgr solutions and gr and α . Similar occurs in by RDM-variables A1 A2 A1 A2 B2 1 2 1 2 (75) are not independent. They are coupled by RDM-variables (75) are not independent. They are coupled by RDM-variable 2 . (1 − µ ) , C1 :x2xC1== (15 + µ ) + 2αC1 gr gr gr gr xA1 xB2 − xA2 xB1 αA1 coupling and αA2and . To solutions z1 and the αB1 and αB2 . Similar occurs inwhether x2 by RDM-variables αB1 αcheck coupling occurs in zx22 have by RDM-variable B2 . Similar A2 : xA2 = (2 + µ (73) (73)theform gr denominator gr gr gr form of multi-granules or not, the denominator D of the of multi-granules or not, the D of the equagr) + 3αA2 gr (1 − µ ) ,12 12 be xseen th αA1 characand αA2 . To check solutions x1 and x2 have αA1whether and αA2 . To check whether solutions Itx1can and To check whether solutions x1 and x2 really have 2 hav tion system (75) for the level μ = 0 should be examined. equation system (75) for the level µ = 0 should be examined. : x = (7 + µ ) + 2 α (1 − µ ) , B 2 B2 B2 ter of multi-granules it is sufficient to examine them for the D = (1 xC20.=For (21this + 3level µ ) + solutions 4αC2(1 −(74) µ12 ) , take the C2 µ: = 12+ 3αA1)(7 + 2αB2) − (2 + 3αA2)(3 + 3αB1) ,(76) membership level (76) form: µ , αA1 , αB1 , αC1 , αA2 , αB2 , αC2 ∈ [0, 1] . αA1 , αB1 , αA2 , αB2 , ∈ [0, 1]

Z2 : zgr 2 =

(7determinant + 2αB2)(15 D + of 2αthe − (3 + 3αsystem + 4αhas C1 ) equation B1)(21(72) C2 )the form Because the denominator D is a monotonic function of The xgr , 1 = gr gr (1 + 3 α )(7 + 2 α ) − (2 + 3 α )(3 + 3 α )be formu-RDM-variables, its extremes lie in corners of its domain αA1 × A1 B2 A2 B1 D = Bull. xA1 xPol. x x and solutions x and x can B2 − A2 B1 Ac.: Tech. 65(4) 2017 509 1 2 αB1 × αA2 × αB2 . To detect these extremes, the denominator lated (1 as:+ 3αA1)(21 + 4αC2) − (2 + 3αA2)(15 + 2αC1) gr Unauthenticated x2 = , value for all combinations of RDM αA1 , αB1 , αA2 , Download Date | variables 8/25/17 12:47 PM (1 + 3αA1)(7 + )− (2−+x3B1αxA2 xB2 xC1 C2)(3 + 3αB1 ) gr2αB2 x1 = , gr gr gr α for values 0 and 1 must be calculated. Results of this exB2 Fig. 22. Visualisation of x = x ∪ x −.xA2 xB1 αA1 , αB1 , αC1 , αA2 , αB2 , αC2xA1 ∈ x[0, B21] 2 2,1 2,2 as the function of the de(74)amination are given in (77).

A. Piegat and M. Pluciński

Fuzzy number division and the multi-granularity phenomenon couplings exists between the numerator and the denominator of system solutions. Consciousness of the solutions’ multigranularity is very important for correct and precise solving uncertain problems. Fig. 21. Visualization of 2-granularity of x1gr = x1,g1r [ x1,gr2 as the function of the denominator D and the numerator N of the solution (75)

Because the denominator D is a monotonic function of RDM-variables, its extremes lie in corners of its domain αA1£αB1£αA2£αB2. To detect these extremes, the denominator value for all combinations of RDM variables αA1, αB1, αA2, αB2 for values 0 and 1 must be calculated. Results of this examination are given in (77).

minD = D(αA1 = 0, αB1 = 1αA2 = 1, αB2 = 0) = ¡23

(77) Fig. 23. Visualisation of the set of possible solutions (75) for maxD = D(α  = 1, α B1 = 0αA2 = 0, αB2 = 1) = 30 αA1 , αB1 , αC1 , αA2 , αA1 B2 , αC2 ∈ [0, 1] in the space X1 × X2 ; no separation is visible It can be seen that D 2 [¡23, 30]. Occurrence of zero in the interval of the denominator D means that the solution interval of equation the denominator means that xthe gr gof r the gr solution of the system is Dtwo-granular: and gr gr 1  = x gr1,1 [ x 1, 2 gr gr gris two-granular: x gr equation system = x ∪ x and x2 = 1 1,1 1,2 x2  = x2,1 [ x2,2. gr xgr 2,1 ∪ x2,2 .

xgr 1,1 = f 1,1 (αA1 , αB1 , αC1 , αA2 , αB2 , αC2 ) ,

xgr 1,2

for D(αA1 , αB1 , αA2 , αB2 ) ∈ [−23, 0) = f1,2 (αA1 , αB1 , αC1 , αA2 , αB2 , αC2 ) ,

(78) (78)

for D(αA1 , αB1 , αA2 , αB2 ) ∈ (0, 30] The 3D-projection from the 7D-space of the two-granular 3D-projection from the 7D-space of the two-granulargr solutionThe xgr 1 isgrshown in Fig. 21. Projection of the solution xgr 2 z1 ischaracter, shown in Fig. Fig. 21. is of solution the similar 22. Projection of the solution x2 of similar character InisFigs. 21 and 22 one(Fig. 22). can distinctly see the separation of In Figs. 21 and 22 one can distinctly see the separation of component-solution granules. One can also see that compocomponent-solution granules. One can also see that component gr nent granules granulesx gxrgr 1,1 andgrx1,2 are various functions. 1,1 and x1,2 are different functions. WhenWhen the solution of the equation the solution of the equationsystem system(72) (72) is is presented presented in a too low-dimensional or inappropriate space (as inFig. 23), Fig. 23) in a too low-dimensional or inappropriate space (as in then the theseparation separation of solution granules is not visible. of solution granules is not visible. However,Howsuch presentation can be found many in papers ever,inappropriate such inappropriate presentation can beinfound manyconpauncertain equation systems, e.g in [26]. pers cerning concerning uncertain equation systems, e.g in [26].

5. Conclusions 510 Solving problems described by uncertain mathematical models which explicitly or sometimes implicitly (as fuzzy equation systems) contain the division operation can provide multi-

R EFERENCES Fig. 22. Visualization of x2gr = x2,g1r  [ x2,gr2 as the function of the denominator D and the numerator N of the solution (75)

[1] D. Dubois and H. Prade, “Operations on fuzzy numbers”, International Journal of Systems Science, 9(6), 613–626 (1978). [2] P. Dutta, H. Boruah and T. Ali, “Fuzzy Arithmetic with and without using α -cut method: A Comparative Study”, International Journal of Latest Trends in Computing, 2(1), 99–107 (2011). [3] J. Fodor and B. Bede, “Arithmetics with fuzzy numbers: a comparative overview”, Proceeding of 4th Slovakian-Hungarian Joint Symposium on Applied Machine Intelligence, 54–68, Herlany, Slovakia (2006). [4] W. Pedrycz, A. Skowron and V. Kreinovich, “Handbook of granular computing”, John Wiley & Sons, Chichester (2008). [5] A. Piegat and M. Landowski, “Two interpretations of multidimensional RDM interval arithmetic: multiplication and division”, International Journal of Fuzzy Systems, 15(4), 486–496 (2013). [6] A. Piegat and M. Pluci´nski, “Fuzzy number addition with the application of horizontal membership functions”, The Scientific World Journal, 1–16, Article ID: 367214, Hindawi Publishing Corporation (2015). [7] K. Tomaszewska, “The application of horizontal membership Fig. 23.function Visualization of the set of possible solutions (75) for αA1, to fuzzy arithmetic operations”, Journal of Theoretical αB1, αC1and , αA2Applied , αB2, αC2 2 [0, 1] in Science, the space 8(2), X1£ X3–10 separation is 2; no (2014). Computer visible [8] R.E. Moore, R.B. Kearfott and M.J. Cloud, “Introduction to interval analysis”, Society for Industrial and Applied Mathematics, Philadelphia (2009). 5. [9] Conclusions B. Liu, “Uncertainty theory”, Springer Verlag, Berlin, Heidelberg (2010). Solving problems by uncertain mathematical [10] Q.X. Li anddescribed S.F. Liu, “The foundation of the grey models matrix and the which explicitly or sometimes implicitly (like fuzzy equation grey input-output analysis”, Applied Mathematical Modelling, systems) contain the division 32(3), 267–291 (2008).operation can provide multi-granular, solutions consisting of two or more com-with the [11]not A.compact Piegat and M. Pluci´ nski, “Computing with Words ponent Use granules which are distinctly separated. Sometimes, of Inverse RDM Models of Membership Functions”, Inthe granule separation is considerable, as has been shown in Sciternational Journal of Applied Mathematics and Computer examples. The multi-granularity hinders problem solving, esence, 25(3), 675–688 (2015). pecially when the multi-granular of onewith sub-model [12] L.A. Zadeh, “Fuzzy logic solution = computing words”, IEEE has to be introduced in the next submodel in which division Transactions on Fuzzy Systems, 4(2), 103–111 (1996). [13] K. Tomaszewska and A. Piegat, “Application of the horizontal membership function to the uncertain displacement calculation Pol. aAc.: Tech.load”, 65(4) 2017 of a composite massless rodBull. under tensile Soft ComUnauthenticated puting in Computer and Information Science, 63–72, Eds.: A. Download Date | 8/25/17 12:47 PM Wili´nski, I. El Fray and J. Peja´s, Springer (2015). [14] J. Smoczek, “Interval arithmetic-based fuzzy discrete-time

Fuzzy number division and the multi-granularity phenomenon

also occurs. Fuzzy RDM arithmetic allows for detecting the multi-granularity and allows for taking into account possible couplings (correlations) existing between particular problem variables, as in the case of fuzzy system equations where such couplings exists between the numerator and the denominator of system solutions. Being aware of the multi-granularity of the solutions is very important for correct and precise solving of uncertain problems.

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