Laplace Distribution And Probabilistic (b i ) In Linear Programming Model

Transcription

1 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model Al Khaleel T. AL-Zad Mstry Of Hgher Edcato Ad Scetfc Research Spervso & Scetfc Evalato Astract The theory of proalstc programmg may e coceved several dfferet ways. As a method of programmg t aalyses the mplcatos of proalstc varatos the parameter space of lear or olear programmg model. The geeratg mechasm of sch proalstc varatos the ecoomc models may e de to complete formato aot chages demad, prodcto ad techology, specfcato errors aot the ecoometrc relatos presmed for dfferet ecoomc agets, certaty of varos sorts ad the coseqeces of mperfect aggregato or dsaggregatg of ecoomc varales. I ths Research we dscss the proalstc programmg prolem whe the coeffcet s radom varale wth gve Laplace dstrto. 4 -: Lear Programmg Lear Programmg ofte represets allocato prolem whch lmted resorces are allocated to a mer of ecoomc actvtes. To provde ths terpretato we wrte the LP model as follows: Max( or Mm) sectto.... a m Z = c χ + c χ c χ χ + amχ a χ, χ χ... χ 0, a χ + a χ a χ a χ + a χ a χ m χ ( or ) ( or ) ( or ) m

2 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model From the ecoomc stadpot,lear programmg seeks the est allocato of lmted resorces to specfc ecoomc actvtes. I the geeral LP model, there are actvtes whose levels are χ, χ, χ represeted y There are also m resorces whose maxmm or mmm avalaltes are gve y,,,, m each t of actvty cosmes a amot a of resorce.ths meas that the qatty a χ represets the total sage y all actvtes of resorce ad = hece caot exceed. The oectve fcto c = χ represets a measre of the cotrto of the dfferet actvtes. I the maxmzato case, c represets the proft per t of actvty, whereas the case of mmzato, c represets the cost per t. We ote that the worth of a actvty caot e dged terms of the oectve coeffcet c oly, the actvty's cosmpto of the lmted resorces s also a mportat factor, ecase all the actvtes of the model are competg for lmted resorces, the relatve cotrto of a actvty depeds o the oth ts oectve coeffcet c ad ts cosmpto of the resorces a. Ths a actvty wth very hgh t proft may rema at the zero level ecase of ts excessve se of lmted resorces. -: Proalstc Lear Program A stochastc or proalstc program s a programmg prolem whch some or all of the prolem data s radom. For sch a program, we mst therefore defe the cocepts of feasle ad optmal soltos that wll properly accot for the radom atre of the prolem. A stochastc lear program ca e wrtte as: M S. t Z = = = a χ χ 0 c χ ( =,..., m) ( =,..., )

3 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model where the coeffcets c, a ad are radom varales wth gve proalty dstrtos. There are two types of decso rles for determg the optmal vales of the decso varales χ. The type of decso rles that determe the optmal vales of χ efore the actal" vales of the radom elemets ecome kow are called zero order rles. The other type of decso rles are kow as ozero order rles. I these rles, we wat for the vales of the radom elemets to ecome kow efore determg χ, t decde advace how the kowledge of the sample vales of the radom elemets s gog to e sed. The kow vales of the decso varales may e assmed determstc. If ths s the case, a decso rle s called a oradomzed decso rle. Sce the radom varatos the parameters of a prolem dce radom varatos the optmal vales of varales χ, we ca have a chace mechasm to determe the optmal vales of χ. The rles goverg sch a mechasm are called radomzed decso rles. I ths case, χ are treated as radom varales, ad coseqetly we may fd ther proalty dstrtos. ow, we tr or atteto to a mportat class of stochastc programmg prolems, called the chace-costraed prolems. These prolems were tally stded y A. Chares ad W. W. Cooper. I a stochastc programmg prolem, some costrats may e determstc ad the remag may volve radom elemets. O the other had, a chace-costraed programmg prolem, the latter set of costrats s ot reqred to always hold, t these mst hold smltaeosly or dvdally wth gve proaltes. I other words, we are gve asset of proalty measres dcatg the extet of volato of the radom costrats, The geeral chace-costraed lear program s of the form: M S. t Z = = a P ( r = c χ a χ ) χ ( =,..., m ) ( = m...() +,..., m)...() χ 0 ( =,..., ) where, 0 < <. for all = m +, m, ad Pr( ) meas the proalty of the evet paretheses. Costrats () are determstc the sese that the coeffcets volved are determstc. Costrats () are Stochastc ca e volated wth a proalty less tha -

4 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model -:Determstc Eqvalets of Proalstc Costrats I program for aove, costrats () are determstc, ad ths eed ot e altered. Frther, the chace costrats gve y (),.e., Pr ( aχ ) ( = m +,..., m)...() are ot determstc. We shall therefore fd ther determstc eqvalets: Oly. are radom varales. Let the dstrto fcto of e: The: F ( z) = F ( z) = P ( z) a χ F ( ) = r...() Eqato () s eqvalet to the ostochastc lear costrat. -4: Laplace Dstrto, Ths s a cotos proalty dstrto. It s amed after a Frech mathematca. Wkpeda pots ot that t s also kow as a dole expoetal dstrto, ecase t remds oe of a expoetal dstrto "splced together ack-to-ack." Ths dstrto s characterzed y locato parameter ϕ (ay real mer) ad scale parameter λ (has to e greater tha zero) parameters. The proalty desty fcto of Laplace(ϕ,λ) s: f ( χ ϕ, λ ) = exp( λ χ ϕ ) λ The cmlatve desty fcto looks eve more mpressve, yet rather easy to tegrate ecase of the asolte vale the formla: χ ϕ F ( χ ϕ, λ ) = exp( ), whe ( χ ϕ ) λ ad F ( χ ϕ, λ ) = ϕ χ exp( ), λ whe ( χ f ϕ )

5 4 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model The expoetal dstrto's proalty desty fcto s defed for χ > 0: χ Expoeta l( ) : f ( χ λ) = exp( ), χ f 0 λ λ λ Ulke the expoetal, the Laplace s defed - <χ <. If ϕ = 0, the the proalty desty fcto for Laplace o χ > 0 s eqal to / of the proalty of the expoetal. I Fgre (,), we llstrate ths fact y plottg the proalty desty of the Laplace o ( 5,5) sde-y-sde wth a expoetal dstrto, ad oe ca oserve that f the expoetal s dvded y half, the t s eqal to the Laplace: Fgre : Laplace p.d.f where ϕ =0, λ= f(x) x

6 5 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model Fgre :Expoetal p.d.f where λ= f(x) x The expected vale of a Laplace dstrto s: E(x) = ϕ As the case of other symmetrcal dstrtos, sch as the ormal ad the logstc dstrtos. Laplace's locato s the same as ts mea, meda, ad mode. The varace s: Var ( χ ) = λ From the Fgre, we see that the scale parameter determes the wdth of the dstrto. From Fgre 4, t s apparet that chagg the locato smply shfts the proalty desty crve to the rght or to the left.

7 6 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model Fgre: Laplace Desty Fcto where ϕ =4 f(x) 0.0 λ= λ=4 λ= x Fgre 4: Laplace Desty Fcto where λ = f(x) ϕ =- ϕ = ϕ =5 ϕ = x

8 7 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model -5: Illstratory Example Max Z = 5χ + 4χ S. T P (χ + 5χ ) 0.60 r P (0.χ + 0.χ ) 0.40 r P ( χ + χ ) 0.0 r χ χ, χ 0, Lap(,) Lap(,4) Exp() *The Proalstc Costrat χ + 5χ ) s eqvalet: P r ( χ + 5 χ Where : χ ϕ exp( ) dχ = 0.60 λ λ χ exp( ) dχ = χ [ exp( ) ] = 0.60 [ exp( ) 0] = 0.60 L + = L0.60 = ( L0.60 L0.5) + = 4.45 The the determstc Costrat s: χ + 5χ 4. 45

9 8 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model *The Proalstc Costrat 0.χ + 0.χ ) s eqvalet : P r ( 0.χ + 0.χ F ( ) = Where : χ ϕ exp( ) dχ = 0.40 λ λ χ exp( ) dχ = χ [ exp( ) ] = [ 0 exp( ) ] = L = L = 4( L(/ ) L0.40) + =.65 The the determstc Costrat s:.χ + 0.χ *The Proalstc Costrat ( χ + χ ) 0. 0 s eqvalet : P r + χ χ Where : χ exp( ) dχ = 0.0 λ λ χ exp( ) dχ = 0.0 χ [ exp( ) ] = 0.0 [ exp( ) 0] = 0.0 = L0.0 = ( L0.0) = Exp() = Lap(0,)

10 9 Laplace Dstrto Ad Proalstc ( ) I Lear Programmg Model The the determstc Costrat s: χ + χ χ + 5χ χ + 0.χ.65 χ + χ χ χ, χ 0 The determstc Programmg Prolem s: Max Z = 5χ + 4χ S. T, Coclsos. Lear determstc Programmg ofte represets allocato prolem whch lmted resorces are allocated to a mer of ecoomc actvtes.. Proalstc program s a programmg prolem whch some or all of the prolem data s radom, may e de to complete formato aot chages demad, prodcto ad techology.. Proalstc programmg s the most realstc of determstc programmg dealg wth prolems of lfe. 4. solvg Proalstc programmg prolem eed to kow mathematcal dstrto to radom parameter. 5. Laplace dstrto s kow as a dole expoetal dstrto, ecase t remds oe of a expoetal dstrto "splced together ack-toack". Refereces. Kamo,.s Mathematcal Programmg, Afflated East-West Press Prvate Lmted.. adaraah, R " Relalty for Laplace Dstrto", math. Pro. Eg.0,pp Stockte, ramta & Johso, pal Laplace Dstrto ( 4. Taha, Hamady A Operato Research, Macmlla Plshg Compay (ew york).