Joural of Scetfc esearch & eports 3(5): 0-7, 04; rtcle o. JS.04.5.00 SCIENCEDOMIN teratoal www.scecedoma.org New Method for Decso Mag Based o Soft Matrx Theory Zhmg Zhag * College of Mathematcs ad Computer Scece, Hebe Uversty, Baodg 0700, Hebe Provce, P.. Cha. uthor s cotrbuto Ths whole wor was carred out by the author ZZ. Short esearch rtcle eceved 30 th March 04 ccepted 8 th May 04 Publshed 3 rd July 04 BSTCT ms: The am of ths paper s to provde a ote o Soft matrx theory ad ts decso mag. Study Desg: I a recet paper [N. Cagma, S. Egoglu, Soft matrx theory ad ts decso mag, Computers ad Mathematcs wth pplcatos 59 (0) (00) 3308-334], Çaðma ad Egoðlostructed a soft max-m decso mag method whch selected optmum alteratves from the set of the alteratves. Place ad Durato of Study: I ths paper, we show by a example that Çagma ad Egoglu s method s very lely to get a empty optmum set. Methodology: Furthermore, we preset a ew approach to soft set based decso mag. esults: We gve a llustratve example to show the advatage of the developed method. Cocluso: The developed method ths paper ca effectvely mprove the method proposed Cagma ad Egoglu paper. Keywords: Soft sets; soft matrx; products of soft matrces; soft max-m decso mag. *Correspodg author: E-mal: zhmgzhag@ymal.com;
Zhag; JS, rtcle o. JS.04.5.00. INTODUCTION The soft set theory, orgally proposed by Molodtsov [], s a geeral mathematcal tool for dealg wth ucertaty. Sce ts appearace, soft set theory has a wde applcato may practcal problems, especally the use of soft sets decso mag. Maj ad oy [] frst troduced the soft set to the decso mag problems wth the help of rough sets [3]. By usg a ew defto of soft set parameterzato reducto, Che et al. [4] mproved the soft sets based decso mag []. Çaðma ad Egoðlu [5] defed soft matrces ad costructed a soft max-m decso mag method whch selected optmum alteratves from the set of the alteratves. It should be oted that the Çaðma ad Egoðlu s method has ts heret lmtato. There exst some soft set based decso problems whch Çaðma ad Egoðlu s method s very lely to get a empty optmum set. The purpose of ths paper s to pot out the lmtato of Çaðma ad Egoðlu s method by usg a example. Moreover, to overcome ths lmtato, we preset a ew approach to soft set based decso mag problems ad gve a llustratve example.. PELIMINIES I the curret secto, we wll brefly recall the otos of soft sets [] ad soft matrces [5]. Throughout ths paper, let U be a tal uverse of objects ad E the set of parameters relato to objects U. Parameters are ofte attrbutes, characterstcs, or propertes of P U deote the power set of U ad E. objects. Let ( ) Defto. [5]. soft set ( f ) where f : E P( U ) such that ( ) Defto. [5]. Let ( ) defed by, E o the uverse U s defed by the set of ordered pars { } ( f, E) ( e, f ( e) ) : e E, f ( e) P( U ), =, f e = f e. f E be a soft set over U. The a subset of U E s uquely {(, ) :, ( )} = u e e u f e whch s called a relato form of ( f, E ). The characterstc fucto of s wrtte by χ : { 0,}, χ ( u, e) U E If U = { u u L u }, E { e e e } ( u e) ( u e),,, = 0,,.,,, m =,, L, ad E, the the ca be preseted by a Table as the followg form
Zhag; JS, rtcle o. JS.04.5.00 e e L e u χ ( u, e ) χ ( ) u, e L χ (, u e ) u χ (, ) χ (, ) L χ ( u e ) u e u e M M M O M u, e u, e χ u m ( ) m If aj χ ( u, e j ) χ χ ( ) L ( u, e ) =, we ca defe a matrx m, m a a L a a a a a L j = M M O M m am L am whch s called a soft matrx of the soft set ( f E ) over U. ccordg to ths defto, a soft set ( ), It meas that a soft set ( ),, f E s uquely characterzed by the matrx j. f E s formally equal to ts soft matrx j. The set of all soft matrces over U wll be deoted by m. Defto.3 [5]. Let,[ ] where cp m { aj, b } a b j m. The d-product of j, [ ] : = such that ( ) Defto.4 [5]. Let,[ ] where cp max { aj, b } a b p = j +. ad [ ] j b = c p j m. The Or-product of j, [ ] : = such that ( ) Defto.5 [5]. Let,[ ] by a b p = j +. ad [ ] j b = c p j m. The d-not-product of j, [ ] : = such that ( ) where cp m { aj, b } p = j +. j b = c p b s defed by b s defed by ad [ ] b s defed
Zhag; JS, rtcle o. JS.04.5.00 Defto.6 [5]. Let,[ ] by where cp max { aj, b } a b j m. The Or-Not-product of j, [ ] : = such that ( ) p = j +. j b = c p ad [ ] 3. ÇÐMN ND ENGINOÐLU S METHOD ND ITS LIMITTION b s defed I [5], Çaðma ad Egoðlostructed a soft max-m decso mag (mdm) method by usg soft max-m decso fucto. The method selected optmum alteratves from the set of the alteratves. I the curret secto, we troduce the Çaðma ad Egoðlu s method ad show ts lmtato by a example. Defto 3. [5]. Let c p m {,,, } { } I = p c < p for all, :, p 0, ( ) I = L. The soft max-m decso fucto, deoted Mm, s defed as follows where Mm : m m = I, Mm c max { } p t t { cp} m, f I, p I = 0, f I =. The oe colum soft matrx Mmc p s called max-m decso soft matrx. Defto 3. [5]. Let U = { u, u, L, um} be a tal uverse ad Mm c p = [ d] subset of U ca be obtaed by usg [ d ] as the followg way whch s called a optmum set of U. opt :, [ ] ( U ) = d { u u U d = }. The a By usg above deftos, Çaðma ad Egoðlostructed a mdm method by the followg algorthm. lgorthm 3. [5]. Step : Choose feasble subsets of the set of parameters, Step : costruct the soft matrx for each set of parameters, Step 3: fd a coveet product of the soft matrces, Step 4: fd a max-m decso soft matrx, Step 5: fd a optmum set of U. 3
Zhag; JS, rtcle o. JS.04.5.00 It s worth otg that Çaðma ad Egoðlu s method has ts heret lmtato. There exst some soft set based decso problems whch lgorthm 3. s very lely to get a empty optmum set. To llustrate ths lmtato, we cosder the followg example. Example 3.. Suppose that a marred couple, Mr. X ad Mrs. X, come to the real estate U = u, u, u, u, u s a set of fve houses uder the aget to buy a house. ssume that { } 3 4 5 cosderato of Mr. X ad Mrs. X to purchase, whch may be characterzed by a set of parameters E = { e, e, e3, e4}. For =,,3, 4, the parameters e stad for expesve, beautful, located the gree surroudgs ad coveet traffc, respectvely. Mr. X e, e, e B = e, e, e, respectvely, to ad Mrs. X cosder set of parameters, = { } ad { } 4 3 4 evaluate the caddates. fter a careful evaluato, Mr. X ad Mrs. X costruct the followg two soft matrces over U accordg to ther ow parameters, respectvely, 0 0 0 0 j = 0 0 0 0 0 0 0 0 [ ] b 0 0 0 0 0 = 0 0 0 0 0 0 Followg we shall select a house by usg the mdm method. Here, we use d-product sce both Mr. X ad Mrs. X's choces have to be cosdered. We ca obta a product of the soft matrces a b by usg d-product as follows j ad [ ] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 j [ b ] = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 We ca fd a max-m decso soft matrx as 0 0 Mm( j [ b ]) =0 0 0 ( j ) Fally, we ca fd a empty optmum set of U accordg to Mm a [ b ] opt Mm a j b ( [ ]) ( U ) =. Followg let us aalyze the algorthm 3.. Suppose that U { u, u,, um} objects ad E = { e e L e } s a set of parameters. Let, [ ] c,,, = L s a set of m a b j m. p s a product 4
Zhag; JS, rtcle o. JS.04.5.00 of a j ad [ b ] Thus, we ca obta that. It s clear that opt ( ) ( U ) cp Mm opt {,, L, },,ad,. ( ) ( U ) = { u } U I c p I c p = p Mm s a oempty set f ad oly f there exst a object u U ad {,, L, } such that I ad c p = for all p I. It s easy to see that the codto, uder whch opt ( ) ( U ) s a oempty set, s so restrctve that t may lmt the Mm cp applcato of algorthm 3. some practcal problems. I other words, Çagma ad Egoglu s method s very lely to get a empty optmum set some decso mag problems. 4. NEW PPOCH TO SOFT SET BSED DECISION MKING To overcome the lmtato of the algorthm 3., the curret secto we shall preset a ew approach to soft set based decso mag problems. Ths approach s based o the followg cocept called the uo of soft matrces. Defto 4. [5]. Let j, b j m. The the soft matrx c j s called the uo of j ad b lgorthm 4.. b j, deoted j j Step : Iput the (resultat) soft matrces Step : Compute the uo %U, f c max {, } j aj bj c j of Step 3: Compute the choce value = for all ad j. Step 4: The optmal decso s to select u f a j ad b b j. c = %U b. a j j j j j = cj, =,, L, m. j= c c = max c. m Step 5: If has more tha oe value the ay oe of u may be chose. To llustrate ths dea, let us recosder the example 3.. Example 4.. Let U { u, u, u3, u4, u5} parameter sets E = { e, e, e3, e4}, = { e, e, e4} ad B { e, e3, e4} ad [ b ] are show as the example 3.. The uo of j ad [ ] = be a set of fve houses uder the cosderato. The 0 j %U [ b ] = 0. 0 0 0 Followg we ca compute the choce value c ( 5 ) as follows: =. Two soft matrces j b are gve as follows. 5
Zhag; JS, rtcle o. JS.04.5.00 Table. Choce values U Choce values = 4 = 3 3 = 3 3 4 = 4 5 = 3 5 From the above Table, t s clear that the maxmum choce value s { c } c max = = 4. Therefore, accordg to the algorthm 4., u could be selected as the optmal house. Maj ad oy [5] preseted a approach to soft set based decso mag. It s oted that Maj ad oy s method dffers from our method. Maj ad oy s method wors oly for a soft set, cotrast, our method ca wor for several soft sets. Our method frst performs a uo operato o these soft sets ad the compute the choce value of each object from the assocated soft set. Therefore, our paper has a broader rage of applcatos tha Maj ad oy s method. 5. CONCLUSION I ths paper, we show by a example that Çagma ad Egoglu s method [5] s very lely to get a empty optmum set. Furthermore, we preset a ew approach to soft set based decso mag ad gve a llustratve example. CKNOWLEDGEMENTS The authors tha the aoymous referees for ther valuable suggestos mprovg ths paper. Ths wor s supported by the Natoal Natural Scece Foudato of Cha (Grat Nos. 6073 ad 6375075) ad the Natural Scece Foudato of Hebe Provce of Cha (Grat Nos. F0000 ad 00033). COMPETING INTEESTS uthor has declared that o competg terests exst. EFEENCES. Molodtsov D. Soft set theory-frst results. Computers & Mathematcs wth pplcatos. 999;37:9-3.. Maj PK, oy. applcato of soft sets a decso mag problem. Computers & Mathematcs wth pplcatos. 00;44:077-083. 3. Pawla Z. ough sets. Iteratoal Joural of Computer & Iformato Sceces. 98;:34-356. 5 6
Zhag; JS, rtcle o. JS.04.5.00 4. Che D, Tsag ECC, Yeug DS, Wag X. The parameterzato reducto of soft sets ad ts applcatos. Computers & Mathematcs wth pplcatos. 005;49:757-763. 5. Çaðma N, Egoðlu S. Soft matrx theory ad ts decso mag. Computers & Mathematcs wth pplcatos. 00;59:3308-334. 04 Zhag; Ths s a Ope ccess artcle dstrbuted uder the terms of the Creatve Commos ttrbuto Lcese (http://creatvecommos.org/lceses/by/3.0), whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal wor s properly cted. Peer-revew hstory: The peer revew hstory for ths paper ca be accessed here: http://www.scecedoma.org/revew-hstory.php?d=578&d=&ad=574 7