 American Journal of Operations Research, 2013, 3, 299-306 http://dx.doi.org/10.4236/ajor.2013.32027 Published Online March 2013 (http://www.scirp.org/journal/ajor) Trapezoidal Approximation of a Fuzzy Number Preserving the Expected Interval and Including the Core B. Asady Department of Mathematics, Islamic Azad University, Arak Branch, Arak, Iran Email: babakmz2002@yahoo.com, b-asadi@iau-arak.ac.ir Received October 11, 2012; revised November 15, 2012; accepted November 28, 2012 ABSTRACT In this paper, we introduce a method to obtain the nearest trapezoidal approximation of fuzzy numbers so that preserv-ing condition s expect interval and include the core of a fuzzy n umber. Keywords: Trapezoidal Approximation; Fuzzy Number; Fuzzy Partition; Fuzzy Approximation; Trapezoidal Fuzzy Numbers 1. Introduction :0,1AR I which satisfies: Trapezoidal fuzzy intervals are often used in practice. An interesting problem is to approximate general fuzzy in-tervals by means of trapezoidal ones, so as to simplify calculations. The investigations in this area were started by Ma et al.  that proposed the symmetric triangular approximation. Actually, symmetric triangular approxi-mation is a particular case of the trapezoidal approxima-tion that was discussed by many authors including Abbasbandy, Amirfakhrian , Abbasbandy and Asady , Coroianu , Ban [5-8], Grzegorzewski [9-11], Grzegorzewski and Mrowka [12,13], Grzegorzewski, K. Pasternak-Winiarska [14,15] and Yeh [16-21] and other methods same as [22-27]. Although, there are many scholars who have investigated interval, triangular and trapezoidal approximation of fuzzy numbers, but the re-sult of approximation is not always a fuzzy number, sometimes it is not a fuzzy set. For example Grzegor-zewski and Mrowka proposed in  a method to find the nearest (with respect to a well-known metric between fuzzy numbers) trapezoidal approximation operator that preserving the expected interval. Unfortunately, there was a gap in the suggested solution which was later im-proved by Grzegorzewski and Mrowka  and finally solved by Ban  and Yeh . In this paper, we com-bine the ideas proposed in papers [3,12], so that preserv-ing conditions expect interval and include the core of a fuzzy number. In Section 2, we recall some fundamental results on fuzzy nu mbers. Trap ezo idal app rox imation and examples are in Section 3. 2. Preliminaries Definition 1. (cf. ) A fuzzy number is a fuzzy set like 1) A is the strictly quasi-convex, 2) 0Ax outside some interval ,cdcabd, 3) There are real numbers a, b such that  and a) x is monotonic increasing on A,ca , b) x is monotonic decreasing on A,bd, 1,Axaxb. c) The set of all these fuzzy numbers is denoted by FR. The -cut, 20,1, of a fuzzy number A is a crisp set defined as  :AxRAx .  Every -cut of a fuzzy number A is a closed interval ,AALAU, where  inf :,sup :.ALxRA xAUxRA x   The set of all elements that have a nonzero degree of membersh ip in A is called the support of A , i.e. supp: 0AxRAx . Definition 2. (complement) The complement of a fuzzy number A is defined as 1.AxAx A,The pair of functions LAU,,,aaaa R gives a parametric representation of fuzzy number A (see ). Another important type of fuzzy numbers was introduced in  as follows. Let 1234 such that A fuzzy number defined as 1234.aaaaCopyright © 2013 SciRes. AJOR B. ASADY 300 11223344fiffiffxaaxaaxaaxaxa,0,lr,,,1214430i1i0ilrxaaaAxaxaa where is denoted by 1234Aaaaa1234,,, lr and if by lrAaaaa r. Its parametric form is 3,,0,1LUlraaaaaa   1lr,,, .111214 4,AAA A popular fuzzy number is the trapezoidal fuzzy num-ber, completely characterized by Equation (1) when , denoted by 1234Aaaaa We denoted by FR the set of all fuzzy numbers and FTR ,LAU the set of all trapezoidal fuzzy numbers. Definition 3. For arbitrary fuzzy numbers A, AA ,LBUBB and B,  the quantity  1010,DAB A22ddLLUUBAB (2) is the distance between A and B, [3,24,25]. The function ,DAB is a metric in FR and ,FRD is a complete metric space. The expected interval EIA of a fuzzy number ,AL AUAA1100dLuAA introduced independently by Dubois and Prade  and Heilpern . It is defined by  d,,AEI AEAE (3) Grzegorzewski  shows that the interval EIA is the nearest interval to the fuzzy number A. Hence in  the expected value of a fuzzy number A defined follow-ing as  EA AE12EV A (4) and B. Asady and M. Zendehnam in  show that EV A: 1RAx is the best approximation of a fuzzy number A. Finally, let us recall that . coreAx3. Trapezoidal Approximation of Fuzzy Numbers Suppose we want to approximate a fuzzy number by a trapezoidal fuzzy number. Thus, we use an operator :TFRFTR which transforms fuzzy numbers into family of trapezoidal fuzzy number. Abbasbandy and Asady  considered a trapezoidal approximation that includes the -cut superset, i.e., 1core coreTA TA (5) Grzegorzewski and Mrówka  said that an operator :TFRFTRTA fulfills the criterion if for any fuzzy number A its output valu e preserves the expected interval, i.e., 2TEIAEITA (6) In this part, we combine ideas proposed in the papers [3,12] to obtain the nearest trapezoidal fuzzy number respect to the original fuzzy number so that it preserves T1 and T2 conditions. Since, any x belongs to the core A of a fuzzy number A if and only if it does not belong to the complement of A corexAxA Therefore, for preserving of these points we consider core coreATA. Also, we are going to preserve the expected interv al of the fuzzy number that th is additional requirement by the significant role of the expected inter-val is in many situations an d applications (see, e.g., [5-11, 33]. Additionally, the propose approach can provide de-cision makers with a new alternative to trapezoidal ap-proximation of fuzzy nu mbers. Now, given a fuzzy number B with -cut set ,BL BU, the problem is to find a trapezoidal fuzzy number ,,,TB tttt 1234 which is the nearest to B with respect to metric D and preserves the condi-tions T1 and T2 i.e. we have  12011220min ,ddLLUUDBTBB TBBTB (7) Subject to 23,1,1,LUttB B (8)  11341200,d,d,22 LUtttt BB  (9) As, in order to minimize ,DBTB it suffices to minimize function Copyright © 2013 SciRes. AJOR B. ASADY 301  21211201440,,LUFhhD BTBBttBt2123dd,ttt2211LtB h2321UtB h122d,Lt432d,Ut (10)    11211100112003140,0,4d6d,2d6 d,1,12 d,LLLLUUUhhhtB BtB BtBtB Bwith respect to h1 and h2. Subject to (11) and (12) and 10tB (13) and 10tB (14) Clearly, with note to Equations (8), (9), we can say the conditions (11)-(14) are suitable for finding the nearest trapezoidal fuzzy number TBcoreB to a fuzzy number B with the conditions as the expected interval and are preserved. Theorem 1: Let B with TBcore-cut set ,BUBL  1d321d,LLB, be a fuzzy number and 11010LdBB  1d321d,UUB (15) 12010UdBB200dd (16) Case 1: If 1 and  then, optimal solution of problem (10) is 120hh  1010d,dLLUUBB and consequently, we have 1234121,1,12LUtBtBtBtB20  0d (17) Case 2: If and d1 then   10d (18)  and d then Case 3: If 20  122110211300114000, ,12 d,1,2d6d,4d6 d,LLLUUUUhhdtB BtBtB BtB B 10d20 (19) Case 4: If and d then    112 211100112001130011400,4d6d,2d6d,2d6 d,4d6 d,LLLLUUUUhdhdtB BtB BtB BtB B   (20) F12,Proof: In order to minimize hh it suffices to minimize function  12 122, ,Fhh fhh with the par-tial derivatives  12111211 0012211222 0041d321d3,41d3,21d3LL LUU Ufhhhhh BBBfhhhhh BBB   (21) Copyright © 2013 SciRes. AJOR B. ASADY Copyright © 2013 SciRes. AJOR 302 212212122221212,,,fhhhfhhhfhhhh2112224434430,hdhd0hh (22) system (10) is 12 and the point 0,0 mini-mizes the function F in the Equation (10). Because 12,Fhh0d is lower bounded function and has only a critical point. Also, with note to Equations (11)-(14) and optimal solution formula (17) is correct. 122) If 1 and 2d are satisfied then the solu-tions of the system (10) are 0hh01020hCase 1: , , hCase 2: 11hd20h , Moreover, since 221222212121220,00,0,0,0fhfhfdhfdh120,0,80,340,3dd therefore case 2 such that 11, 2 minimizes the function F. Because the Hessian matrix for it is posi-tive definite and for case 1 is not positive definite. Also, with note to Equations (11)-(14) and hd 0h11hd20h,  Formulae (18) are satisfied. 3) Proof is similar with 2) 4) If 1 and 2 are satisfied then the solu-tions of the system (10) are given by 0d0d1020h10hCase 1: , ; hCase 2: , 22hd; Case 3: 11hd20h, ; Case 4: 11hd, 22Because the Hessian matrix for of case 4 is positive definite and for others cases are not positive definite, then the solution of case 4 is minimizer and formulae (20) is correct. 2 Now we compare our method with the other works [3,7,20] in following exa mples. hd. Example 1.  Let us consider the fuzzy number 1,2, 3,302A, in the parametric forms, 1,3027,0,1 .LUAA   By substitute abov e equations on the (2) and (3) equa-tions, we obtain the expect interval and core of the fuzzy number A same as follows 5,12, core2,33EI AA Also, trapezoidal approximation of it by the other methods and proposed method is shown in Table 1. Clearly, in Table 1, by Ban and Yeh methods, we have EIAEI TAcore core. and TAA and, by Abbasbandy and Asady method, get EIAEI TAcore core. and ATA Consequently, with note above results proposed method preserved T1 and T2 conditions (see Figures 1-4). Example 2. Consider the fuzzy number 41 310,10,10,14BcoreTA that the parametric form of it is as follows Table 1. Comparative results of example 1. EI TA The trapezoidal approximation of fuzzy numbers A Authors 53 5,123 53,53,53,673 Ban 71 78,35 35 5,123 4735,7135,7835,72635 Yeh 2,3 33 240,20 20 13 10,2,3,219 10 Abbasbandy Asady 2,3 5,123 3 4,2,3,21 Proposed method B. ASADY 303 Figure 1. The trapezoidal approximation of A by Ban method. Figure 2. The trapezoidal approximation of A by Yeh method. Figure 3. The trapezoidal approximation of A by Abbas-bandy and Asady method. Figure 4. The trapezoidal approximation of A by prposed method. 14 31020, 144BL BU.   Clearly, by Equations (2) and (3) the expect interval and core of it are same as follows 6,13and core56510, .EI BB Also, trapezoidal approximation of the fuzzy number B by the other methods and proposed method is shown in Table 2. Clearly, in Table 2 by Ban and Yeh methods, core coreBTB and even core coreBTB (see Figures 5 and 6). But, for Abbasbandy and Asady method, we get EIBEI TB. Finally, the results of proposed method are  EIAEI TAcore core.Aand TAzR Therefore, we can say that proposed method preserves T1 and T2 conditions in ex-ample 2. (please see Figures 7 and 8). Theorem 3. Trapezoidal approximation satisfies the following properties. A1) Translation invariance: , FR : then we have TA zTAzzR. , A2) Scale invariance: FR : then we have .TzA zTA Proof. Proof is similar to proof of Theorem 12 in . Theorem 4. Trapezoidal approximation satisfies the identity property. 1234,,,Bbbbb be a trapezoidal fuzzy Proof. Let number. By Equations (2) and (3), we get 2112ttd, 4322ttd. Therefore, d1 and d2 are poitive and conditions in the s Copyright © 2013 SciRes. AJOR B. ASADY 304 Table 2. Comparative results of Example 2. coreTB EIproximation of fuzzy numbers B TB The trapezoidal apAuthors 16915 6,13  15 11 15,16915,16915,221TB B Ban 410 54,39 5 6,13  5 90 39,410 39,54 5,78YTBYeh 17 ,1335610, 5  3,10,565,745AAbbasbandy, Asady 4TB 5610, 5 6,13 2,10,565,745P Proposed method TB of B by BaFigure 5. The trapezoidal approximation n Figure 7. The trapezoidal approximation of B by Abbas-bandy and Asady method. method. of B by Yemethod. Figure 6. The trapezoidal approximation h Figure 8. The trapezoidal approximation of B by proposed method. Copyright © 2013 SciRes. AJOR B. 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