A Satisfaction Degree of Optimal Value for. Grey Linear Programming*

Transcription

1 A atsacton Degree o Optmal Value or Grey Lnear Programmng* Yunchol Jong a a Center o Natural cence Unversty o cences Pyongyang DPR Korea Abstract. Ths paper consders the grey lnear programmng and ntroduces a new satsacton degree o optmal value or the postoned lnear programmng o the grey problem. The -satsacton degree seems to relect the real meanng o the postoned optmal values. By selectng accordng to the atttude o decson maker towards the satsacton degree an approprate optmal soluton can be obtaned or the grey lnear programmng problem. An eample s gven to show the meanng o the new satsacton degree. Keywords: Grey lnear programmng satsacton degree. Grey lnear programmng The problem o lnear programmng wth grey parameters (LPGP) ] s dened by ma = A s. t. where C X X b X C = ( c L c ( )) A( ) = b( ) = ( b ) T c n ( ) a m n ( ) c c ] a ( ) a a ] b( ) b b ] b m L = L m = L n. We assume that c b a = L m = L n. Denton.. uppose that α β γ ] = L m = L n and let the whte values o grey parameters be respectvely as ollows c b = α c + ( α ) c = β b + ( β ) b = L n = L m * Ths paper was supported n part by Nanng Unversty o Aeronautcs and Astronautcs. E-mal: yuncholong@yahoo.com

2 Then a γ ) = L m = L n. ( ) = γ a + ( a ma = A s. t. X C X X b s called a postoned programmng o LPGP where C = ( c L c ( )) A( ) = b = b n ( ) a m n ( ) T b m L. Ths problem and ts optmal value s denoted by LP( α β γ ) = L m = L n) ( α β γ ) = L m = L n) respectvely. Theorem. ]. For a postoned programmng o a LPGP whenα α = L n we and have ( α β γ ) = L m = L n) ( β γ ) = L m = L n) α. Theorem. ]. For a postoned programmng o a LPGP whenβ β = L m we have ( α β γ ) = L m = L n) ( β γ ) = L m = L n) α. Theorem.3 ]. For a postoned programmng o a LPGP whenγ γ = L m = L n we have ( α β γ ) = L m = L n) ( β γ ) = L m = L n) α. Denton.. ] Assume that or every = L m = L n α α β = β γ = γ. = Then LP( β = LP( α β γ ) = L m = L n) α s called a ( α β -postoned lnear programmng o LPGP and ( α β = ( α β γ ) = L m = L n) called a ( α β -postoned optmal value o LP( α β. s

3 Denton.3. ] When α θ β = θ γ = θ or every = L m = L n LP( θ ) = = LP( α β θ -postoned optmal value o LP( θ ). s called a θ -postoned programmng and ( θ ) = ( α β s called a. atsacton degree o optmal value Accordng to Theorem..3 the optmal value o a postoned programmng s an ncreasng uncton wth the postoned coecentsα β = L n = L m and an decreasng uncton wth the postoned coecents ( α β γ = L m = L n. Thereore s ncreasng wthα β and decreasng wthγ. Thus we have the ollowng theorem. Theorem.. For( α β we have ( β ) ( = ( ) α γ ) =. Denton.. ]. The optmal value = ( ) and ( ) = s called a crtcal optmal value and an deal optmal value o LPGP respectvely. Based on the act o Theorem.4 a pleased degree o optmal value s dened as ollows. α β γ α β γ o LP( α β Denton.. ] For ( )-postoned optmal value ( ) µ = + ( α β ( α β s called a pleased degree o ( α β. ( α β Denton.3.] Gven a grey target = µ ] optmal soluton s called a pleased soluton o LPGP. D µ ( α β D Remark. From Theorem. and Denton. we can see that ( ) µ ( ) = + = + = > ( ) the correspondng and ( ) µ ( ) = + = + = < ( )

4 µ ( α β whch means that the pleased degree o the crtcal value s greater than and the pleased degree o the deal value s less than. Thus t s desrable to ntroduce a new satsacton degree µ ( α β such that µ ( ) = µ ( ) = and µ ( α β γ ) ]. Denton.4. For ( α β -postoned optmal value ( α β o LP( α β ( α β ( α β ( ) ( α β µ ( α β = + ( ) + s called a -satsacton degree o optmal value ( α β. Then µ ( α β and µ ( α β γ ) and an optmstc satsacton degree respectvely. Remark. The optmstc satsacton degree µ ( α β ( ) o LP( α β where s called a pessmstc satsacton degree s ust the same as ( α β ( α β + whch eplans more ntutvely the meanng o the optmstc satsacton degree. Denton.5. Gven a grey target D = µ ] µ ( α β D the correspondng optmal soluton s called a ( µ ) -satsactory soluton o LPGP. Proposton.. For the -satsacton degree µ ( α β µ ; () ( α β () µ ( ) =; () µ ( ) =. we have (Proo) By Theorem. we have ( α β and ( α β have ( α β µ and () and (). Eample.(Eample.5. o ]) ma = c + c where s. t. 3 ( ) ( ) b b b 3 68] c 95] c 35] ] ( ) 5 35] ( ) Thus we 7] 35].53.5] 8] 3 b b 836] b 733]. 3 3

5 The deal value crtcal value and some postoned optmal values or ths problem are gven n Table. The correspondng optmal solutons are pleased solutons or µ =.5. Table. Postoned optmal values and ts pleased degree ( α β () () (.6.6.6) (.7.9.5) (.5.9.4) (.7.5.3) ( α β µ ( α β The correspondng -satsacton degrees are shown n Table. Table. -satsacton degrees µ (.6) µ (.7.9.5) µ (.5.9.4) µ (.7.5.3) As shown n the table the optmal soluton correspondng to (.6) s not( µ ) satsactory soluton or every ] are less than µ =.5. I satsactory soluton or every - because all o the -satsacton degrees µ (.6) (.6 s ( µ ) µ =.4 the optmal soluton correspondng to ) -.8 ]. From Table t can be seen that the satsacton degree was more senstve to the change oγ or the eample problem and that γ should be selected n (.5) to obtan a satsactory soluton orµ = Concluson In ths paper we consdered the grey lnear programmng and ntroduced a new satsacton degree o optmal value or the postoned lnear programmng o the grey problem. The µ α β γ conorms wth the concept o the deal optmal value and -satsacton degree ( )

6 crtcal optmal value because µ ( ) = or crtcal value ( ) µ = or deal value µ α β γ. Thereore the -satsacton degree seems to relect the real and ( ) ] meanng o the postoned optmal values. By selectng accordng to the atttude o decson maker towards the satsacton degree an approprate optmal soluton can be obtaned or the grey lnear programmng problem. Reerences ] eng Lu Y Ln (6). Grey Inormaton. Theory and Practcal Applcatons London prnger-verlag ] eng Lu Yaoguo Dang Jerey Forrest (9). On postoned soluton o lnear programmng wth grey parameters Proceedngs o the 9 IEEE Internatonal Conerence on ystems Man and Cybernetcs an Antono TX UA - October 9