A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming

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1 Aerca Joural of Operatos Research, 4, 4, Publshed Ole Noveber 4 ScRes A Pealty Fucto Algorth wth Obectve Paraeters ad Costrat Pealty Paraeter for ult-obectve Prograg Zhqg eg, Ru She, Jag College of Busess ad Adstrato, Zheag Uversty of Techology, Hagzhou, Cha Eal: egzhqg@zuteduc, sheru@zuteduc, ag9@6co, Receved 8 August 4; revsed Septeber 4; accepted 6 October 4 Copyrght 4 by authors ad Scetfc Research Publshg Ic Ths wor s lcesed uder the Creatve Coos Attrbuto Iteratoal Lcese (CC BY Abstract I ths paper, we preset a algorth to solve the equalty costraed ult-obectve prograg (P by usg a pealty fucto wth obectve paraeters ad costrat pealty paraeter Frst, the pealty fucto wth obectve paraeters ad costrat pealty paraeter for P ad the correspodg ucostrat pealty optzato proble (UPOP s defed Uder soe codtos, a Pareto effcet soluto (or a wealy-effcet soluto to UPOP s proved to be a Pareto effcet soluto (or a wealy-effcet soluto to P The pealty fucto s proved to be eact uder a stable codto The, we desg a algorth to solve P ad prove ts covergece Fally, uercal eaples show that the algorth ay help decso aers to fd a satsfactory soluto to P Keywords ult-obectve Prograg, Pealty Fucto, Obectve Paraeters, Costrat Pealty Paraeter, Pareto Wealy-Effcet Soluto Itroducto ult-obectve prograg s a portat odel solvg vector optzato probles ay ethods had bee gve to fd solutos to ultobectve prograg [] It s well-ow that the pealty fucto s oe of effcet ethods studyg ultobectve prograg For eaple, 984, Whte [] preseted a eact pealty fucto for ultobectve prograg Suaga, azeed ad Kodo [3] appled pealty fucto How to cte ths paper: eg, ZQ, She, R ad Jag, (4 A Pealty Fucto Algorth wth Obectve Paraeters ad Costrat Pealty Paraeter for ult-obectve Prograg Aerca Joural of Operatos Research, 4,

2 Z Q eg et al forulato to teractve ultobectve prograg probles Rua ad Huag [4] studed wea caless ad wea stablty of eact pealty fuctos for ultobectve prograg By pealty fucto, Lu [5] derved ecessary ad suffcet codtos wthout a costrat qualfcato for e-pareto optalty of ultobectve prograg, ad the geeralzed e-saddle pot for Pareto optalty of the vector Lagraga Huag ad Yag [6] gave olear Lagraga for ultobectve optzato to dualty ad eact pealzato Chag ad L [7] solved terval goal prograg by usg S-shaped pealty fucto Atcza [8] studed the vector eact l pealty ethod for odfferetable cove ultobectve prograg probles Huag, Teo ad Yag [9] dscussed caless of eact pealzato vector optzato wth coe costrats Huag [] proved caless of eact pealzato costraed scalar set-valued optzato eg, She ad Jag [] defed a obectve pealty fucto based o obectve weght for ultobectve optzato proble ad preseted a teractve algorth Ths paper defes a pealty fucto wth obectve paraeters ad costrat pealty paraeter whch dffers fro a obectve pealty fucto [] Because t s alost ot possble for decso aers (Ds to obta all effcet solutos to P, t s sgfcat to preset a effcet algorth of P so that Ds fds a easy ad satsfactory soluto to the P Luque, Ruz ad Steuer poted out that a effcet algorths ot oly help decso aers lear ore about effcet solutos, but also avgate to a fal soluto as qucly as possble [] Ths paper presets a algorth by odfyg every obectve paraeter of pealty fucto so that a fal soluto s easly ad qucly obtaed I Secto, we troduce a pealty fucto wth the obectve paraeters ad costrat pealty paraeter, ad ts algorth I Secto 3, we gve uercal results to show that the proposed algorth s effcet Pealty Fucto wth Obectve Paraeters ad Costrat Pealty Paraeter I ths paper we cosder the followg equalty costraed ult-obectve prograg: P f = f, f,, f q st g (, =,,,, where f : R R {, g : R R {, for J = {,,, q, I = {,,, We deote the feasble set of P ( by X = { R g (, I As usual, X wealy-effcet soluto f there s o X such that f ( < f ( for all J, e f ( f ( X s called a Pareto effcet soluto f there s o X such that f ( f ( for all J f ( < f ( for at least oe J, e f ( f ( Let fuctos Q: R R { ad P: R R { satsfy where Q( t Let l = ad t where (,,, q Q t = f ad oly f t Q t > f ad oly f t > Q t > Q t f ad oly f t > t > P t = f ad oly f t, P t > f ad oly f t > P t > P t f ad oly f t > t > ( ρ ( ρ ( F,, = Q f P g, =,,, q, I ( s called a Pareto < ad = s a obectve paraeter ad ρ > s the costrat pealty paraeter Let (,,, q = ad the pealty fucto of ( be defed as: 33

3 ( ρ = ( F( ρ F( ρ Fq( q ρ F,,,,,,,,,,, Cosder the followg ucostrat pealty optzato proble: For R, let de set ( F( P, ρ,, ρ, st R ( ρ = { ( ρ = ( ρ = { ( ρ > J J F J,,,,, for, J,, J F,,, for J Z Q eg et al We have J = J (,, ρ J (,, ρ Theore Suppose that for gve (, ρ, s a Pareto wealy-effcet soluto to P (, ρ The the followg three assertos hold: If ( J,, ρ, the s a feasble soluto to (P, f ( for all J (,, ρ ad f ( > for all J (,, ρ If J (,, ρ = ( e (,, F ρ >, the there s o X such that f ( < f ( 3 If (,, F ρ > ad s a feasble soluto to (P, the soluto to (P Proof The cocluso s obvous fro the deftos of P ad Q Suppose that there be a X f f have Whe such that < Whe ( s a Pareto wealy-effcet f for soe J, we ( ( ( ρ ( ( Q f = Q f < Q f P g = f > for soe J, we have ( ( ρ ( Q f < Q f Q f P g Hece, (,, ρ < (,, ρ = F F, the 3 Accordg to, the cocluso holds Theore Suppose that for a gve (, ρ, followg three assertos hold: s ot a Pareto wealy-effcet soluto to P (, ρ s a Pareto effcet soluto to P (, ρ The the If ( J,, ρ, the s a feasble soluto to (P, f ( for all J (,, ρ ad f ( > for all J (,, ρ If J (,, ρ ( e (,, F ρ >, the there s o X such that f ( f ( 3 If (,, F ρ > ad s a feasble soluto to (P, the (P Proof The cocluso s obvous fro the deftos of P ad Q Suppose that there be a X f f have Whe such that Whe ( s a Pareto effcet soluto to f for soe J, we ( ρ ( Q f = Q f < Q f P g = f > for soe J, we have 333

4 Z Q eg et al ( ( ( ( ρ ( ( Q f Q f Q f P g Hece, F(,, ρ F(,, ρ =, the s ot a Pareto effcet soluto to P (, ρ 3 Accordg to, the cocluso holds Based o Theore, we develop a algorth to copute a effcet soluto to (P The algorth solves the proble P (, ρ sequetally, ad s called ultobectve Pealty Fucto Algorth (PFA for short PFA Algorth: Step : Choose X, ρ >, < f for each J Let =, ad f = J,, ρ N > ad X Step : Solve F(, where = (,,, q soluto R Step 3: If ( J,, ρ Step Otherwse, ( ρ, for each J, let F,, >, go to Step 4 Let be a Pareto wealy-effcet =, ρ = Nρ, : = ad go to Step 4: If s ot feasble to (P, for each J, let =, ρ = Nρ, : = ad go to Step Otherwse, stop ad s a Pareto wealy-effcet soluto to (P I the PFA algorth, t s assued that for each J < f ( ca always be obtaed The covergece of the PFA algorth s proved the followg theore For soe J, let ( whch s called a Q-level set (,, S( L, f s bouded X { S L, f = L Q f, =,,, S L f s bouded f, for ay gve L > ad a coverget sequece Theore 3 Suppose that Q, f ( J ad g ( I S( L, f s bouded for all J If { (,,, Let { are cotuous o R, ad the Q-level set be the sequece geerated by the PFA algorth = s a fte sequece (e, the PFA algorth stops at the -th terato, the s a Pareto wealy-effcet soluto to (P If { s a fte sequece, the { effcet soluto to (P Proof For all J, t s clear that the sequece { Therefore, { coverges to s bouded ad ay lt pot of t s a Pareto wealy- decreases wth =, =,, ( for all J If the PFA algorth terates at the th terato ad the secod stuato of Step 4 occurs, by Theore, s a Pareto wealy-effcet soluto to (P < f for We frst show that the sequece { X Sce { coverges to all X ad all > If s bouded Fro the PFA algorth, we have for all J, there s a such that < f ( for all X for each >, we have ( Q f > for all J Hece, we 334

5 F,, ρ > for all > By Theore, there s a J such that have ( So, Therefore, there s a L > such that f f, =,, ( Q f Q f, =,, Q( f ( Q( f ( L Sce S( L, f s bouded, the sequece { Sce s a Pareto wealy-effcet soluto to P (, ρ > such that We have Whe <, =,, ( ρ Z Q eg et al s bouded Wthout loss of geeralty, we assue ( Q f P g Q f = = ( ρ, we have P g ( = ( P g Q f Q f ρ = Hece, soluto to (P, there s a X such that f ( f ( Fro, there s soe such that So, we have f ( f ( δ, for soe, there are fte X If s ot a Pareto wealy-effcet { f f q δ = =,,, < Let f f < f f, =,,, q <, whch by Theore s a cotradcto Hece, s a Pareto wealy-effcet soluto to (P Theore 3 eas that the PFA algorth s coverget theory Now, we dscuss the eactess of the q pealty fucto for (P If there are a R ad ρ such that a Pareto wealy-effcet soluto to P,ρ for < ad ρ > ρ, the (P s also a Pareto wealy-effcet soluto to ( F(,, ρ s called a eact pealty fucto Let (P(s be a perturbed proble of (P gve by where s ( s s s ( P( s f ( = f(, f(,, fq ( st g ( s, =,,,, =,,, Slar to that for a costraed pealty fucto [], we defe stablty Defto Let be ay feasble soluto to (P ad s ay feasble soluto to (P(s for each s R If there s a such that for J where s P( s P = ( Q f Q f s ρ =, the (P s stable We have a eact result of the pealty fucto Theore 4 Let fucto s, < ad ρ > ρ be a optal soluto to (P If (P s stable, (,, ρ P (3 F s a eact pealty 335

6 Z Q eg et al F s ot a eact pealty fucto Let s a Pareto wealy-effcet soluto to (P(s Accordg to the defto of stablty, we obta that there s a satsfyg Proof Suppose that (,, ρ ( Q f Q f s Ths ples that there s soe < soe < such that that Thus, ρ such that s, < ad ρ > ρ (4 P f > for J The, there always ests s ot a Pareto wealy-effcet soluto to (P(, e there s soe such ( ρ ( ρ F,, < F,, = Q f, J ( ( ρ ( ( Q f P g < Q f, J I Suppose that s a feasble soluto to (P If f ( < for J, we have f ( ( f Otherwse f f ( for J, fro Q( f ( < Q f (, f ( f ( shows that < < <, whch s ot a Pareto wealy-effcet soluto to (P A cotradcto occurs Hece, s ot a feasble soluto to (P ad P( f ( > I Τ Let s = ( s s s wth s g (,,, =, =,,,, ad (P(s' The, there s soe such that f ( f ( ad ( Therefore, whch shows that where s P( s P I s ( ( s ( ( Q f Q f s be a Pareto wealy-effcet soluto to f f Thus, s ( ( s ρ ( ( ( ρ ( I I = F ( ρ < Q( f ( Q f P s Q f P s (,,, Q f Q f > ρ s s P = Ths equalty cotradcts to (4 Hece, (P s stable whch yelds a cotradcto wth the assupto ad proves that (,, ρ 3 Nuercal Eaples F s a eact pealty fucto I the PFA algorth, t s ot easy to solve ultobectve proble ( ρ, F,, Let R ( ρ = F( ρ F( ρ Fq( q ρ F,,,,,,,, It s easly ow that a optal soluto to the proble (,, ρ R solutos to the proble F(,, ρ Hece, we replace the proble (,, ρ R R of the PFA algorth wth the proble F(,, ρ Let Q ( t > for we have R F s a Pareto wealy-effcet F the Step t > Whe f ( <, 336

7 Hece, whe (,, ( each J( F (,, ρ F (,, ρ ( = = Q f < decreases, the -th obectve (,, ρ, F ρ wll decrease too For fed (,, ρ F l = F (,, ρ Z Q eg et al So, we ay obta dfferet Pareto wealy-effcet solutos at gve dfferet (,,, q, we ca cotrol the -th obectve value F (,, ρ cotrollg We have appled the PFA algorth to several eaples prograed by atlab 65 The a of uercal eaples s to chec the covergece of the algorth ad to cotrol chages obectves Eaple Cosder the followg proble: Let pealty fucto F ( P f ( { , =, 4 st 3 6,, (,, ρ = a{, a{ 4, Let the startg pot (, (, Clearly, f { { { ρa 3 6, ρa, ρa, =, ρ =, N = ad costrat error = { { { e a 3 6, a, a, e =, s a feasble soluto We tae dfferet paraeters (, By the PFA algorth, the results are show Table I Table, whe or decreases, the frst obectve value f(, or f(, decrease too Obectve paraeter ca cotrol chage of each obectve fucto It helps decso aers lear about the chage of each obectve fucto ad choose a satsfactory soluto as qucly as possble Eaple Cosder the proble: ( P f (, = {,, st We wat to fd a soluto that three obectves are as sall as possble wth the frst ad secod obectve value less tha ad the thrd obectve value less tha 5 Let pealty fucto F (,, ρ = a{, a{, a {, Let the startg pot (, (, { ρ { ρa 8 8, a 3, { { { ρa 4, ρa, ρa, =, ρ =, N = ad costrat error 337

8 Z Q eg et al Table Nuercal results wth dfferet obectve paraeters (, e( (, ( f (,, (, f 5 (4, 4 (955488, 75 (55974, (4, 4 (4439, 476 (, (4, 4 (54867, 9 (86, 43 Table Nuercal results wth dfferet obectve paraeters (,, 3 e( (, ( f (,, (,, 3(, f f 5 (,, (3958, (47439, 48558, (,, (5347, 396 (54448, 3984, (,, (48979, 339 (388, 66854, (,, (44495, (755, 3366, = { { { { { We tae dfferet paraeters (,, 3 e a 8 8, a 3, a 4, a, a, the PFA algorth ad get the results show Table I Table, we fd a satsfactory soluto (, ( 44495, (,, 3 4 Cocluso = whe tag dfferet I ths paper, we defe a pealty fucto wth obectve paraeters ad costrat pealty paraeter for P ad the correspodg ucostrat pealty optzato proble Uder soe codtos, we prove that a Pareto effcet soluto (or a wealy-effcet soluto to UPOP s a Pareto effcet soluto (or a wealyeffcet soluto to P, ad the pealty fucto s eact uder a stable codto We preset the PFA algorth to solve the ult-obectve prograg wth equalty costrats by usg the olear pealty fucto wth obectve paraeters Wth ths algorth, we ay fd a satsfactory soluto Acowledgets We tha the Edtor ad the referee for ther coets The research s supported by the Natoal Natural Scece Foudato of Cha uder grut 739 ad 9793 Refereces [] Sawarag, Y, Naayaa, H ad Tao, T (985 Theory of ultobectve Optzato Acadec Press, Lodo [] Whte DJ (984 ultobectve Prograg ad Pealty Fuctos Joural of Optzato Theory ad Applcatos, 43, [3] Suaga, T, azeed, A ad Kodo, E (988 A Pealty Fucto Forulato for Iteractve ultobectve Prograg Probles Lecture Notes Cotrol ad Iforato Sceces, 3, -3 [4] Rua, GZ ad Huag, XX (99 Wea Caless ad Wea Stablty of ultobectve Prograg ad Eact Pealty Fuctos Joural of atheatcs ad Syste Scece,, [5] Lu JC (996 ε -Pareto Optalty for Nodfferetable ultobectve Prograg va Pealty Fucto Joural of atheatcal Aalyss ad Applcatos, 98, [6] Huag, XX ad Yag, XQ ( Nolear Lagraga for ultobectve Optzato to Dualty ad Eact Pealzato SIA Joural o Optzato, 3, [7] Chag, C-T ad L, T-C (9 Iterval Goal Prograg for S-Shaped Pealty Fucto Europea Joural of 338

9 Z Q eg et al Operatoal Research, 99, 9- [8] Atcza, T ( The Vector Eact l Pealty ethod for Nodfferetable Cove ultobectve Prograg Probles Appled atheatcs ad Coputato, 8, [9] Huag, XX, Teo, KL ad Yag, XQ (6 Caless ad Eact Pealzato Vector Optzato wth Coe Costrats Coputatoal Optzato ad Applcatos, 35, [] Huag, XX ( Caless ad Eact Pealzato Costraed Scalar Set-Valued Optzato Joural of Optzato Theory ad Applcatos, 54, [] eg, ZQ, She, R ad Jag, ( A Obectve Pealty Fuctos Algorth for ultobectve Optzato Proble Aerca Joural of Operatos Research,, [] Luque,, Ruz, F ad Steuer, RE ( od-fed Iteractve Chebyshev Algorth (ICA for Cove ultobectve Prograg Europea Joural of Op-Eratoal Research, 4,

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