A New Method for Decision Making Based on Soft Matrix Theory

Transcription

1 Joural of Scetfc esearch & eports 3(5): 0-7, 04; rtcle o. JS SCIENCEDOMIN teratoal 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) ], 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;

2 Zhag; JS, rtcle o. JS 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

3 Zhag; JS, rtcle o. JS 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

4 Zhag; JS, rtcle o. JS 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

5 Zhag; JS, rtcle o. JS 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 { } 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 { } 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, j = [ ] b = 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 [ ] j [ b ] = 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

6 Zhag; JS, rtcle o. JS 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 ] = Followg we ca compute the choce value c ( 5 ) as follows: =. Two soft matrces j b are gve as follows. 5

7 Zhag; JS, rtcle o. JS Table. Choce values U Choce values = 4 = 3 3 = = 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 ad ) 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: Pawla Z. ough sets. Iteratoal Joural of Computer & Iformato Sceces. 98;:

8 Zhag; JS, rtcle o. JS Che D, Tsag ECC, Yeug DS, Wag X. The parameterzato reducto of soft sets ad ts applcatos. Computers & Mathematcs wth pplcatos. 005;49: Çaðma N, Egoðlu S. Soft matrx theory ad ts decso mag. Computers & Mathematcs wth pplcatos. 00;59: Zhag; Ths s a Ope ccess artcle dstrbuted uder the terms of the Creatve Commos ttrbuto Lcese ( 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: 7