Journal of American Science, 2011;7(1)

Transcription

1 Jornal o Aercan Scence, 0;7() Interactve Coprose Stablty o Mlt-obectve Nonlnear Prograng probles Kasse, M. ()*, El-Benna, A. (), and El-Badry, N. () () Matheatcs departent, Faclty o Scence, Tanta nversty () Matheatcs departent, Faclty o Scence, Daetta Branch, Mansora nversty Abstract: Ths paper presents a solton ethod or lt-obectve nonlnear prograng (MONP) probles and stablty o ths solton. The ethod, oers a practcal solton to MONP probles by dervng the coprose weghts and cobnng dgent wth an atoatc optzaton technqe n zzy decson akng. Ths s acheved by sng the ethod and algorth o coprose prograng and the ethod o coprose weghts and we obtan the stablty or the solton n each step o the algorth. A nercal exaple llstrates varos aspects o the reslts developed n ths paper. A aple procedre or ths algorth s ntrodced. [Kasse, M., El-Benna, A., and El-Badry, N., Interactve Coprose Stablty o Mlt-obectve Nonlnear Prograng probles. Jornal o Aercan Scence 0;7():-9]. (ISSN: ). Keywords: MONP; Stablty; Interactve decson akng; Coprose weghts; Mebershp nctons.. Introdcton Most decson probles have ltple obectves conlctng aong theselves. The solton or sch probles can only be obtaned by tryng to get coproses based on noraton provded by the decson aker (DM). Several ethods have been developed to solve ltobectve decson akng (MODM) probles, see [0]. In [5,8] soe o these ethods are based on pror noraton reqred ro the DM. Ths noraton ay be n the ro the desred acheveent levels o the obectve nctons and the rankng o the levels ndcatng ther portance, sch as n goal prograng. It ay also be n the or o weghts showng the portance o the obectves. The dsadvantages wth these ethod s that the DM cannot easly provded ths pror noraton snce he has no dea abot the solton process o the proble. Other ethods, called nteractve ethods, have been developed n order to overcoe ths dsadvantage. There are two categores o nteractve ethods. Interactve ethods o the rst type reqre the DM to provde soe trade-os aong the attaned vales o the obectve nctons n order to deterne the new solton [4]. The nteractve ethods o the second type reqre the DM to provde soe preerence noraton by coparng the varos ecent soltons n the space o the obectve nctons or the decson varables. The qantty and coplexty o the noraton reqred ro the DM n sch ethods are portant actors aectng the chances o reachng the best coprose solton. In [, 7] an nteractve lnear ltple obectve ethod, called nteractve coprose prograng (ICP) were ntrodced. The notons o the solvablty set, stablty set o the rst knd and stablty set o the second knd, and analysed these concepts or paraetrc convex nonlnear prograng probles were ntrodced n [6, 9]. Ths paper presents an nteractve stablty coprose prograng ethod or solvng MONP probles by sng the coprose weghts ro the pay-o table and zzy ebershp ncton or each obectve ncton. An llstratve exaple s gven to clary the obtaned reslts.. Proble Forlaton et s consder the MONP proble: ( MONP ) : ax ( ( x ), ( x ),..., ( x )) sbect to n x X = { x R g ( x ) 0, =,,..., k} where ( x ), =,...,, and g ( x ), =,..., k, are convex real valed nctons whch belong to class C. The correspondng scalarzaton proble s ( MONP ) λ ax λ ( x ) = sbect to x X, λ = λ,..., λ 0, λ 0, =,,...,, and where = λ =. et ( x ) be the th obectve ncton and x be the ax possble vales o x are the n possble vales ( x) and o ( x ) ond nder the constrants, respectvely. To obtan the coprose solton o the MONP proble, nd the solton whch has a n x. dstance wth respect to the deal solton Ths reqres noralzaton o the obectve nctons and approprate choce or the dstance easre. The solton ond n ths way s a redced set o all ecent solton. The set o coprose solton ay be large, and also the choce o weghts by the DM ay be dclt. edtor@aercanscence.org

2 Jornal o Aercan Scence, 0;7() these dcltes cold be redced by cobnng the basc deas or the ethods o coprose prograng and coprose weghts.. Coprose Weghts Here, we ntrodce a ethod based on the ollowng two an deas: Frst, the DM cold state hs preerence aong soe alternatve soltons ore easly the vales o obectve nctons were easred on the soe scale varyng between zero and one. Ths cold be done by eployng the ebershp ncton or the obectve nctons concept n the coprose prograng. In order to elct a ebershp μ x ro the DM or each o the ncton obectve nctons ( x ) n MONP probles, we rst calclate the ndvdal n and ax o each obectve ncton x nder the gven constrants. By takng accont o the calclated ndvdal n and ax o each obectve ncton together wth the rate o ncrease o ebershp o satsacton, DM st deterne hs sbectve ebershp μ x whch s a strctly onotone ncton ncreasng ncton wth respect to ( x ). Here, t s assed that μ ( x ) = 0 or 0 μ x = or ( x ) x a where represents the vale o the vale o ebershp ncton ( x ) and, x sch that μ s a. In ths ethod, the ollowng denton o the ebershp nctons s sed or scalng: ( x ) μ ( x ) =, () where ( x ) are the obectve nctons, are the ax possble vales o ( x ), =,,..., and are the n possble vales o ( x ) satsyng the constrants x X.The μ ( x ) are dened as the ebershp nctons o ( x ) to the possble vale ( x ). The correspondng scalarzaton proble s: ax μ + ( x ) λ μ ( x ) = = () sbect to x X. The second an dea, one o the an drawbacks o the nteractve ethods s the dclty o gettng the weghts o the obectve nctons ro the DM even vales o obectve nctons are presented to h on the sae scale. In ths ethod, the coprose weghts o obectve nctons can be obtaned by constrctng the pay-o table dsplayng vales o obectve nctons at x,..., x, where x solves ax ( x ), =,...,, sbect to x X. A payo table s x where = ( x ) and = or each =,...,, =,..., k and the coprose weghts λ, =,,..., can be obtaned ro the pay-o atrx by the orla, αa e λ =, =,,...,, αa e = a ˆ * α = ln, a =, =,,...,. a a = a () where ˆ = ax ( x ) s the ax entry n row. 4. Stablty set o the rst knd [7] Denton. The solvablty set o proble MONP s dened by λ B = λ R+ ax λ ( x ) exsts, x X = where R + s the nonnegatve orthant o the vector paraeter λ. Denton. Sppose that B wth a correspondng optal pont x, then the stablty set o the rst knd o proble (MONP) λ correspondng to x s dened by S ( x ) = λ B λ ( x ) = ax λ ( x ). x X = = It s clear that the stablty set o the rst knd s the set o all paraeters correspondng to an optal solton o the scalarzng proble. k et λ S ( x ) then there exst R sch that ( x, ) solves the ollowng Khn-Tcker proble: ( x ) g ( x ) λ + = 0, α =,,..., n, x x = α J α edtor@aercanscence.org

3 Jornal o Aercan Scence, 0;7() g ( x ) 0, g ( x ) = 0, =,,..., k, = 0, J {,,..., k}, 0, {,,..., k} J, that eans, we order the ncton g ( x ), =,,,k, n sch a way that,,..., s g x = { } 0, { +,..., } g ( x ) < 0. s k Consder the syste o eqatons s ( x ) g ( x ) λ + = 0, = xα = x (I) α α =,,..., n. It represent n lnear hoogenos eqatons n +s nknowns λ, =,,...,, and, =,,..., s, whch can be solved explctly. Sppose that λ 0, =,,...,, and 0, =,,,s, solve the above syste o eqatons, then t s clear that ( x, ) solves the Khn-Tcker proble, where =, =,,..., s, = 0, =s+,,k, and hence S ( x ) λ. et s dene the set s P λ, = λ, R + λ, solves the syste (I), { + } s where R + and R + are the nonnegatve orthants o s the R vector λ space, and R vector space, respectvely. Then { + } = λ ( λ ) ( λ ) S x R, P,. (II) g x < = k then t s easy to see that I 0,,,...,, S ( x ) can be wrtten n the ollowng or: ( x ) = λ + λ = α = S x R 0,,,..., n. = x α 5. Interactve coprose algorth In ths ethod, the solton process by solvng sple nonlnear prograng probles to nd the ax and n possble vales o obectve nder the gven constrants. The coprose weghts o the obectve nctons are deterne ro the Eq.() and eployed n the proble () we have ax μ + ( x ) λ μ ( x ) = = sbect to x X μ + x s the coposte ncton o where μ ( x ) and t deternes the ( ) + th solton. The steps o the algorth can be sarzed as ollows: Step. Deterne, or all =,,, as ollows: () ax ( x ) sbect to x X, The soltons o ths proble are x and whch are known as the "deal solton". () n ( x ) sbect to x X, The solton are x and whch are known as the "ant -deal solton". Step. Deterne the ebershp nctons correspondng the solton x, =,,..., as n the relaton (). Constrct the pay-o table where x solves n ( x ),,,..., = x sbect to x X, = x or each =,,, =,,,k, =, and constrct zzy atrx. μ x μ μ x x μ μ μ μ μ μ μ Step : The coprose weghts λ, =,,..., can be ond ro αa e λ =, =,,...,, αa e = a α = = ˆ = a a a * ln, a,,,...,. = ˆ = ax x s the ax entry n row. Step 4: By sng ths weghts, we establsh the new coprose solton x +, ro the proble (). Step 5. Deterne the stablty set o the rst knd correspondng to ths solton as n relatons (I) and (II). Step 6: Deterne the ebershp obectve 4 edtor@aercanscence.org

4 Jornal o Aercan Scence, 0;7() nctons o the new solton o the proble n step 4, μ +.Add ths coln to table o zzy n step. Step 7: Ask the DM whether he preers one solton strctly over all the other -soltons he does go to step 8, otherwse ask h hs least preerred solton aong all the others. Then replace ths preerred solton by the new ond n step 6 and go to step. Step 8: Stop. 6. Nercal exaple et s consder the ollowng proble n x = x + x, = + n x x 5 x, sbect to x + x 5, x 0, x The solton o ths exaple wll be obtaned sng a Maple progra: Step. (I) ax ( x ) = x + x, sbect to x + x 5, x 0, x x = 0.5, 4.97, = 5.5. solton (II) ax x = x 5 + x, sbect to x + x 5, x 0, x x = 0,5, = 0. solton (III) n x = x+ x sbect to x + x 5, x 0, x x = 0,0, = 0. solton (IV) n x = x 5 + x, sbect to x + x 5, x 0, x x = 5,0, = 0. solton Step.the correspondng pay- o table s = + = 5, where ( x ) x x ( 5,0) ( x ) ( x ) x = 5 + = 5. ( 0,0) the correspondng zzy atrx s μ x x Step. Sbsttte o pay- o table n relaton () to obtan the correspondng coprose weghts λ = and λ = Step 4. The new coposte ebershp ncton s n μ ( x ) = n x x + ( x 5) + x Sbect to x + x 5, x 0, x The solton s x = ,0, x = , ( x ) = Step 5. The set o all paraeters whch corresponds to ths solton s dened by the stablty set o the rst knd n the ollowng or: S ( x ) = { λ λ = 0, λ = 0, > 0 Step 6. ( ) } I x μ = = = , I I ( x ) μ = = = I 0 0 Thereore, the new zzy atrx s μ x x x Step 7. Present the three solton to DM he s certan that one o the s the best solton o the proble (not only preerred regardng the other two), stop. Else, ask DM whether he preers one solton over the two soltons. Sppose that he wold not, and hs least preerred solton wold be solton. Ths solton s then replaced by solton retrn to Step. The new pay-o table s By sng relaton () we obtan the coprose weghts λ = and λ = We note that these weghts are ot o the range o paraeters whch were dened n the above set S ( x ) so we st have the next solton. The new coposte ebershp ncton s n μ ( x ) = n 5.5 x + x ( x 5) + x 0 sbect to x + x 5, 5 edtor@aercanscence.org

5 Jornal o Aercan Scence, 0;7() x 0, x whch solton s x = , 0, x = 4.809, x = , the correspondng stablty set o the rst knd s S ( x ) = { λ λ = 0, λ = 0, > 0} We have the correspondng ebershp ncton n the or μ = , μ =. Thereore the zzy atrx s μ x x x Sppose the DM wold preer the new solton over these soltons. Go to Step 8. Step 8. Stop. The best coprose solton o ths proble wold be x = ( , 0 ), = ( , ), μ = , Conclson An nteractve stablty coprose prograng ethod, sng a zzy approach and a pay-o table. In ths ethod, no pror noraton s reqred ro the DM and the coprose weghts o the obectve nctons are deterned ro the pay-o table and zzy atrx. The ethod does not reqre sgncantly ore data than pre nonlnear prograng and the scale o lt-obectve proble by sng sbstttng the obectve nctons by the ebershp ncton and to obtan coprose weghts by the grades o ebershp o the crrent vectors n each teraton n the "close deal" zzy set. The proposed algorth prograed by sng Maple progra. 8. Reerence [] Chankong, V. and Haes, Y. Y., "Mltobectve Decson Makng Theory and Methodology". Elsever Scence, New York, 98. [] El-Sayed, H. M. "A ned Interactve Approach For Solvng Mltple-Obectve Nonlnear Prograng and Copter Code ", Proceedngs o the rst nternatonal conerence on operatons research and ts applcatons, Caro 994. [] Evren, R. "Interactve coprose prograng", Jornal o the Operatonal Research Socety 8 () (987) 6-7. [4] Georon, A., Dyer, J. and Fnbred, A. "An nteractve approach or lt-crtera optzaton wth an applcaton to the operaton o an acadec departent", Manageent Scence 9 (97) [5] Ignzeo, J. "Goal Prograng and Extensons", Heath, exngton, MA, 976. [6] Kasse, M. "Interactve Stablty o Mltobectve Nonlnear Prograng Probles wth Fzzy Paraeters n the Constrants", Fzzy Sets and Systes 7 (995) 5-4. [7] Kasse, M. "Interactve Stablty o Vector Optzaton Probles", Eropean Jornal o Operatonal Research 4 (00) [8] ee, S. "Goal Prograng or decson Analyss", Aerbach, Phladelpha, PA, 97. [9] Osan, M. and El-Benna, A. "stablty o ltobectve nonlnear prograng probles wth zzy paraeters", Matheatcs and Copter Slaton 5 (99) -6. [0] Zeleny, M. "Mltple Crtera Decson Makng", McGraw-Hll, New York, Appendx A aple progra or solvng lt-obectve nonlnear prograng (MONP) probles and stablty o ths solton. > restart: wth(optzaton): wth(maplets[eleents]): wth(groebner): > aplo:=maplet(["enter The Type o The Proble", [Btton("Mnze",Shtdown("Mnze")), Btton("Maxze",Shtdown("Maxze")) ]]): d:=maplets[dsplay](aplo):d:=parse(d); a := Maplet([["Enter No o Vector Spaces", TextFeld['TF']()], [Btton("ok",Shtdown(['TF']))]]): n:=maplets[dsplay](a):n:=parse(n[]); q:="a"::=0: whle(q="a") do :=+: aplet := Maplet([["Enter an Obectve ncton ", TextFeld[TF]()], [Btton['b']("ok",Shtdown( [TF]))]]): t[]:=maplets[dsplay](aplet); aplet := Maplet([[abel("Enter Another obectve ncton?:")], [Btton['B']("Ok",Shtdown("a"))], [Btton['B']("No",Shtdown())]]): q:= Maplets[Dsplay](aplet): > :=; > or ro to do 6 edtor@aercanscence.org

6 Jornal o Aercan Scence, 0;7() []:=parse(t[][]); > q:="a"::=0: whle(q="a")do :=+: apl := Maplet([["Enter Yor Constrants", TextFeld[TF]()], [Btton['b']("ok",Shtdown( [TF]))]]): t[]:=maplets[dsplay](apl); apl := Maplet([ [abel("enter Another Constrant?: ")], [Btton['B']("Ok", Shtdown("a"))],[Btton['B']("No", Shtdown())]]): q:= Maplets[Dsplay](apl): > k:=; > or ro to k do g[]:=parse(t[][]); > or ro to do or ro to k do > Q[]:=Maxze([], {g[]}, asse=nonnegatve); P[]:=Mnze([], {g[]}, asse=nonnegatve); > Q[];P[]; > or ro to do or ro to + do > R[,]:=rhs(Q[][][ ]); > # Mnzaton. > S[,]:=rhs(P[][][ ]); > nnn:=proc(r,s,q,p,,g,n,,k,m) local R,,,K,M,A,alpha,FN,MF,x,,,,4,5,6, aplet: global S,M,ada,Z,eq,ss,eq,rr,s,s,la, alph,kk,,: or ro to do > # Maxzaton. R[,]:=Q[][]; > # Mnzaton. > S[,]:=P[][]; or ro to + do R[,]:=R[,]; S[,]:=S[,]; > Z:=Matrx(..,..+): > or ro to do or ro to do 5[]:=[]:4[]:=[]: = then kk:=: whle(kk<=) do []:=sbs(x[kk]=s[,kk+],5[]): 5[]:=[]:kk:=kk+: Z[,]:=5[]: else kk:=: whle(kk<=) do []:=sbs(x[kk]=s[,kk+],4[]): 4[]:=[]:kk:=kk+: Z[,]:=4[]: end : Z; or ro to do > M[]:=([] S[,])/(R[,] S[,]): > M:=Matrx(..,..+): > or ro to do > or ro to do > M[,]:=(Z[,] S[,])/(R[,] S[,]); > M[,+]:=R[,]; > M; > x:=array(..): or ro to do x[]:=z[,]; or ro to do (x[]<z[,]) then x[]:=z[,]; end ; x; ada:=array(..): A:=Array(..):or ro to do > A[]:=x[] Z[,]; > > alpha:=ln(abs(add((a[]/a[]),=..)))/ (A[] A[ ]); > or ro to do > ada[]:=exp(alpha*a[])/ add(exp(alpha*a[k]),k=..); > > add(ada[],=..); > FN:=add(ada[]*M[],=..); > or ro to k do MF:=d(FN, {g[k]}, asse=nonnegatve); > or ro to n+ do S[+,]:= rhs(mf[][ ]); la:=vector(,sybol=la): 7 edtor@aercanscence.org

7 Jornal o Aercan Scence, 0;7() :=Vector(k,sybol=): > or alph ro to n do s:=add(d([],x[alph])*la[],=..); s:=add(d(lhs(g[]),x[alph])*[],=..k); > eq[alph]:=s+s; > or ro to do > :=: 6[]:=[]: whle(<=) do []:=sbs(x[]=s[+,+],6[]); 6[]:=[]::=+: Z[,+]:=6[]; > Z; > or ro to do > M[,+]:=M[,+]; M[,+]:=(Z[,+] S[,])/(R[,] S[,]); > M:=prnt("M=",M); end proc: > nnn(r,s,q,p,,g,n,,k,m); RS:=Array(..n):RS:=Array(..n): or alph ro to n do rr:=eq[alph]:rr:=eq[alph]: :=: whle(<=) do rr:=sbs([la[]=ada[],x[]=s[+,+]],rr): r:=sbs([x[]=s[+,+]],rr): rr:=rr: rr:=r: :=+: RS[alph]:=rr; RS[alph]:=rr; RS;#RS; syst:= [seq(rs[alph],alph=..n)]; > var:=[seq([],=..k)]; bs:=0; prntlevel :=4: IsProper(syst)=tre then B:=solve(syst,var); or ro to k do rhs(b[][])<0 then bs:=bs+ end : bs>= then "not stable" else "stable" end ; else "Syste s not stable" ; end ; > aplet := Maplet([["agree these vales?"], [Btton['q']("&OK", Shtdown("yes")), Btton['q']("&No",Shtdown("No"))]]): ss:=maplets[dsplay](aplet): or ro to n+ do prnt("x",+,"=",s[+,]);prnt("pay Table=",Z);prnt("ada=", ada); > whle ss="no" do apl:=maplet([["enter the no o x yo want to replace wth",textfeld['tf']()],[btton("ok",shtdown (['TF']))]]): d:=maplets[dsplay](apl): d:=parse(d[]): nassgn('ss');nassgn('m'); nassgn('ada'); nassgn('z');nassgn('eq');nassgn('m'); > or ro to do > R[d,+]:=S[+,+]; > S[d,+]:=S[+,+]; or ro to d do S[,+]:=S[,+]; or ro d+ to do S[,+]:=S[,+]; > nnn(r,s,q,p,,g,n,,k,m); > aplet := Maplet([["agree these vales?"], [Btton['q']("&OK", Shtdown("yes")), Btton['q']("&No",Shtdown("No"))]]): ss:=maplets[dsplay](aplet): > RS:=Array(..n):RS:=Array(..n): or alph ro to n do rr:=eq[alph]:rr:=eq[alph]: :=: whle(<=) do rr:=sbs([la[]=ada[],x[]=s[+,+]],rr): r:=sbs([x[]=s[+,+]],rr): rr:=rr: rr:=r: :=+: RS[alph]:=rr; RS[alph]:=rr; RS;#RS; syst:= [seq(rs[],=..n)]: > var:=[seq([],=..k)]: bs:=0; > IsProper(syst)=tre then B:=solve(syst,var); or ro to k do rhs(b[][])<0 then bs:=bs+ end : bs>= then "not stable" else "stable" end ; else "Syste s not stable" ; end ; > ss="yes" then apl4:=maplet([["whch one yo preere, x(",textfeld['tf'](),")"], 8 edtor@aercanscence.org

8 Jornal o Aercan Scence, 0;7() [Btton("ok",Shtdown(['TF']))]]): dd:=maplets[dsplay](apl4): d:=parse(dd[]): > end : > or ro to do > zz[]:=sbs({x[]=s[d,],x[]=s[d,]},[]); > l:=d; > prnt("m=",m); > or ro to do > prnt("x=",s[l,+]); or ro to do prnt("n=",zz[]); > or ro to do > prnt("m=",m[,l]); > prnt("pay Table=",Z);prnt("ada=",ada); // edtor@aercanscence.org