Eplotg the Margal Profts of Costrats wth Evolutoary Mult-objectve Optmzato Techques Ya Zheyu Zh We Kag Lsha Laboratory of Software Egeerg Departmet of Computer Scece Laboratory of Software Egeerg Wuha Uversty Rutgers Uversty Wuha Uversty Wuha 43007, P.R. Cha Pscataway, NJ 08854, USA Wuha 43007, P.R. Cha Abstract May real-world search ad optmzato problems aturally volve costrat hadlg. Recetly, qute a few heurstc methods were proposed to solve the olear costraed optmzato problems. However, the costrat-hadlg approaches these methods have some drawbacks. I ths paper, we gave a Multobjectve optmzato problem based (MOP-based) formula for costraed sgle-objectve optmzato problems. We proposed a way to solve them by usg mult-objectve evolutoary algorthms (MOEAs). By smulato epermets, we fd ths approach for costrat hadlg ot oly ca fd the costraed optmalty, but also ca provde the decso maker (DM) wth a group of trade-off solutos wth slghtly costrat volato ad meawhle wth substatal ga the objectve fucto. Ths ca eable the DM to have more freedom to choose hs preferred soluto ad therefore eplot more profts the marg of costrat volatos, where the costrat volatos are small or acceptable. Keywords: costrat hadlg, costraed optmzato problems, evolutoary mult-objectve optmzato, decso makg.. Itroducto May real-world search ad optmzato problems aturally volve costrat hadlg. By judgg whether the objectve fuctos ad costrats are lear or olear, we ca classfy the costraed optmzato problems to two categores: lear costrat programmg ad olear costrat programmg. Smple algorthm [] ca hadle the former effcetly, whle t faces dffcultes whe dealg wth the latter, sce the olear objectve fuctos ad olear costrats make the problem harder. Therefore, may heurstc methods were proposed to solve the olear oes. I these heurstc methods, the classcal ways to hadle costrats s to covert the objectve fucto ad costrats to a weghted sum of objectves (pealtyfucto approach) [] ad the try to fd the feasble ad optmal solutos by optmzg the weghted sum fucto. However, ths costrat-hadlg method has some drawbacks. It s dffcult to f a weght vector for successful workg ad mproper weght vector may lead the search process to local optmalty rather tha the global oes. Furthermore, may real-world problems, some costrat ca be soft [3], that s, a soluto wth a permssble costrat volato ca stll be cosdered f there s a substatal ga the objectve fucto, whch are ot take to accout by the pealty-fucto approach. Thus, we eed more fleble methods to provde the decs o makers (DM) wth more caddate solutos, cludg the solutos wth slghtly costrat volato but meawhle wth substatal ga the objectve fucto. I addto, some dffcult real world problems, the olear costrats make most of, eve all of the search space feasble. We call ths pheomeo overcostraed. To solve the over-costraed problems, there should be some compromses oe or several costrats based o the DM s eperece ad other codtos. To solve ths kd of problems, we must release or compromse some costrats. It s the DM kowg the backgroud ad the real mea of the problem who should determe whch costrats compromse ad how much they eed compromse. Therefore we should provde the DM a group of trade-off solutos rather tha just oe to make effcet decso. We fd that the objectve hadlg for mult-objectve optmzato problems (MOPs) s somewhat smlar to the costrat hadlg for costraed sgle-objectve optmzato problems (SOPs). The tradtoal ways of objectve hadlg MOP geerally covert a MOP to a SOP by sum approach ad solve t wth classcal optmzato techques. Ths sum approach solvg MOPs faces the smlar dffcultes as that faced wth the pealty-fucto approach we descrbed above. Wth the developmet of MOP optmzato techques, especally evolutoary mult-objectve optmzato (EMO), we may cosder: ca we covert the costrat SOPs to MOPs ad solve them wth the state-of-the-art EMO techques.
I ths paper, we gave a MOP-based formula of costraed sgle-objectve optmzato problems. The we proposed the way to solve them by usg oe of the curret mult-objectve evolutoary algorthms. By smulato epermets, we fd ths approach for costrat hadlg ot oly ca fd the costraed optmalty, but also ca preset the decso maker a group of trade-off solutos wth slghtly costrat volato ad meawhle wth substatal ga the objectve fucto. Ths ca eable the DM to have more freedom to choose hs preferred soluto ad therefore eplot more profts the marg of costrat volatos, where the costrat volatos are small or acceptable.. Costraed sgle-objectve optmzato I a Costraed SOP, there est a sgle objectve fucto ad a umber of costrats []: Mmze Subject to: f ( ) g ( ) 0, h ( ) = 0, k j ( L) j =,,..., J k = J +, J +,..., K ( U ) =,,..., () Wthout loss of geeralty, we assume that the objectve fucto f () s mmzed. For a mamzato problem, the dualty prcple ca be used to covert the problem to a mmzato problem. I most dffcult ad real-world problems, the costrats g j () ad h k () are olear ad make most of the search space feasble. Ths causes dffculty eve fdg a sgle feasble soluto. Therefore, may tradtoal methods [4], cludg cojugate gradet method, Newto terato method, ad moder heurstc methods, cludg smulated aealg [5], Tabu search, geetc algorthms [6] etc., were proposed to solve the olear costraed optmzato problems. I these methods, the classcal ways to hadle costrats s to covert the objectve fucto ad costrats to a weghted sum of objectves ad the try to fd the feasble ad optmal solutos by optmzg t. Ths method s largely kow as the pealty-fucto approach, where the orgal objectve fucto f () ad all costrats are added together wth a weght vector cosstg of pealty parameters, as follows [4]: Mmze X = (,,..., R, r P(X,R,r) = f() + + are pealty k = j+ R r h (X) < g (X) > ), D, =,,..., j = parameters () Where g (X) g (X) > 0 < g(x) >= 0 g(x) 0 h (X) h (X) > 0 h(x) = h (X) h(x) 0 However, ths costrat hadlg method has some drawbacks cludg, but ot lmted to, the followg: (a) The qualty of the solutos s heavly depedet o a weght vector (also called pealty parameters). The compoets of the weght vector determe a fed path from aywhere the search space towards the costraed mmum. Sometmes, stead of covergg to the true costraed mmum, the path termates to a local mmum. Thus, for suffcetly olear problems, ot all weght vectors wll allow a smooth covergece towards the true costraed mmum. Ofte, the user has to epermet wth varous weght vectors to solve the costraed optmzato problem. (b) I may real-world problems, some costrats ca be soft, that s, a soluto wth a permssble costrat volato ca stll be cosdered f there s a substatal ga the objectve fucto, whch s ot take to accout by the sum approach. I some dffcult real world problems, the olear costrats make all of the search space feasble. To solve the over-costraed problems, the DM should compromse oe or several costrats based o hs eperece ad other codtos. Therefore we should provde the decso makers a group of trade-off solutos rather tha just oe to make effcet decso. However, the pealty-fucto costrat hadlg method caot tackle ths kd of problems. 3. Solvg costraed SOPs wth EMO 3.. The MOP-based descrpto of costraed SOPs A costraed SOP defed () ca be posed as a MOP of mmzg the objectve fucto ad mmzg all costrat volatos. Mmze f(x) Mmze < g (X) > =,,...,j Mmze h(x) = j +,j +,...,k X = (,,..., ), D, =,,..., (3)
We ca defe: f(x) f(x) = < g () > h () The (3) s equvalet to (4): Mmze = 0 =,,...,j = j +,j +,...,k F(X) (f (X),f (X),...,f (X)) X = (,,..., ), D, =,,..., 0 Usg ths way, problem () s coverted to problem (4), whch s a typcal MOP wthout costrats. 3.. Evolutoary mult-objectve optmzato (EMO) for Solvg costraed SOPs Mult-objectve optmzato (MO) methods, as the ame suggests, deal wth fdg optmal solutos to multple objectve optmzato problems (MOPs). I a MOP, the preseces of coflctg objectves gve rse to a set of optmal solutos (called Pareto optmal Solutos [3]), stead of a sgle optmal soluto. Thus, t becomes essetal that a Mult-objectve optmzato algorthm fd a wde varety of Pareto optmal solutos, stead of just oe of them. Evolutoary algorthms (EAs) are a atural choce for solvg MOPs because of ther populatos approach. A umber of Pareto-optmal solutos ca be captured a EA populato, thereby allowg the DM to fd dverse multple Pareto-optmal solutos oe smulato ru. I addto, the good search abltes of EAs ca guaratee ther good performace whe dealg wth problems havg ocove search spaces ad other dffcult search spaces [7]. A lot of successful Mult-objectve evolutoary algorthms (MOEAs), such as VEGA [8], NPGA [9], MOGA [0], NSGA [], SPEA [] have bee proposed for EMO. To solve the costraed optmzato problem defed by (), we frst eed to covert t to (4) by the way we have descrbed above. We the try to fd the good uform solutos to appromate the Pareto frot [7] (Pareto frot s the mage of the set of optmal solutos) of ths MOP by usg MOEAs. After we preset these solutos, cludg the optmal soluto wthout costrat volato ad the solutos that volate oe or some costrats margally or by a large etet, to the DM, he wll choose the BEST oe from these caddate solutos based o hs eperece, hs preferece or other subjectve or objectve factors. It s easy to prove that the Pareto frot of (4) deftely cludes the optmal soluto of (). Besdes ths soluto, t also comprses the solutos wth costrat volatos ad meawhle wth some gas the objectve fucto. k (4) 3.3. Advatages of usg EMO solvg costraed optmzato problems Costrat hadlg wth EMO has some advatages comparg wth other costrat hadlg methods: (a) The Pareto frot of (4) cludes the optmal soluto of (), f () has a feasble optmal soluto. Therefore, we ca coclude that the optmal soluto set we get by pealty-fucto approach s a subset of the soluto set we get by EMO approach. (b) By EMO, the costrat-hadlg problem ca be solved a atural way. There s o eed of ay pealty parameters ad pealzed objectve fucto. (c) Besdes the optmal soluto of (), Ths method ca provde the DM wth the trade-off solutos that volate oe or some soft costrats margally or by a large etet but have substatal ga the objectve fucto. I fgure, f the costrat s hard, that s, o volato s permtted; the the oly soluto s o pot A, where the costrat volato s zero. If the costrat s f() Costrat volato Fgure. A eample for the Pareto frot (trade-off) of objectve fucto ad costrat volato soft, the the DM ca choose hs preferable soluto, whch ca make the most profts for hm, amog the whole Pareto frot. The DM wll carry out ths posteror decso makg, based o hs eperece, hs preferece or other subjectve or objectve factors. Wth the vsualzato of the caddate solutos as fgure, the trade-off of the objectve fucto ad the soft costrats ca by easly carred out by the DM. If we use pealty-fucto approach for costrat hadlg, the the decso process s pror. For the pror decso makg process, we must decde the weght parameters before optmzato, whch s very dffcult for both decso makers ad optmzators. (d) The good search abltes of EAs ca guaratee ther good performace whe tacklg the problems wth
ocove search spaces ad other dffcult search spaces. (e) It ca hadle over-costraed problems. I fgure, f the mmum of the costrat volato s above zero, the the whole search space s feasble (ths more lkely happes, f the problem has more costrats ad there are o pots where every costrats equals zero). To solve ths kd of problems, we must release or compromse some costrats. Ths compromse decso process s problem-depedet. Wth the group of odomated solutos preseted by ths method, the DM, who has more kowledge of the real mea of the problem tha the computer scetsts ca determe whch costrats compromse ad how much they eed to compromse ad therefore choose the best soluto. Whe practcal cosderg, we are usually terested the solutos, whch are based towards the rego where all costrat volatos are small. I fgure, although the DM ca choose hs favorte soluto from the whole Pareto frot, he may be especally terested the solutos wth the rego AB, where the costrat volato are small. Thus, we ca use a algorthm, whch s bas fdg Pareto optmal solutos, amely, whch ca fd dese solutos towards the rego wth small costrat volatos ad lak solutos towards the rego wth large costrat volatos. 4. Epermets ad aalyses To help ga further sght o the effectveess of ths EMO costrat hadlg approach, a seres of epermetal smulatos were ru. 4.. Test fuctos Test fucto (TF) s a et remely hard ad famous costraed optmzato problem, kow as Bump problem []. Mamze f ( ) = s. t. where = 0 < 4 cos ( ) = = 0.75 < 0,, = = 7.5 cos ( ) (5) Test fucto (TF) s a costraed optmzato problem wth two soft costrats from [5,6]. Mmze s. t. where f ( X ) = ( h(x) = ) g( X ) = 0.5 + ( + = 0 [.8,0.8], 4.. Implemetato detals ) + 0 [ 0.4,0.9] (6) Here, we adopted SEEA [3] as the MOEA for EMO. I TF, we otce that although t has two costrats, the secod costrat s effectve, that s, t has o effect o the feasble search space. We coverted the frst costrat to a objectve; meawhle, we coverted the problem to a mmzato problem by the dualty prcple. Therefore, TF was coverted to (7) by the meas we descrbed secto 3.: M f M f where ( ) = 0.75 ( ) = 0 0 < < 0, 4 cos ( ) = = = f otherwse.. cos ( ) (0.75 = = ) > 0 As we dscussed secto 3.3, here we tegrated the bas techques to SEEA. Sce t s easy for ths tegrato, we wll ot epla t detal. Please refer to [3] for the detals. The other parameters are set as followg: (problem s dmeso) =, Ma geerato=0000; Populato=50; Number of parets for mult-paret crossover = 6; σ =0.04. share TF has two soft costrats, both of whch are effectve. Therefore we coverted TF to a three objectve problem: Mmze f( X ) = ( ) + ( ) Mmze f ( X ) = h( X ) 0 g(x) 0 Mmze f 3(X) = g(x) otherwse where [.8,0.8], [ 0.4,0.9] For TF, we used SEEA drectly wthout bas techques. The other parameters are set as followg: (8) (7)
Ma geerato=0000; Populato=80; Number of parets for mult-paret crossover = 6; σ =0.038. share -0.35-0.4 4.3. Epermetal results ad dscussos Fgure s the Pareto frot of (7) we got wth bas SEEA a smulato ru. I Fgure, f () s TF s objectve fucto ad f () s the frst costrat volato of TF. If the DM s especally terested the solutos, whose costrat volato s less tha ε = 0.03, there wll be fve caddate solutos, whose objectve fucto values ad costrat volatos are as followg, for DM to choose. Soluto : f= -0.364999780, f= 0.00000000000000 Soluto : f= -0.3656564706854, f= 0.0035793360 Soluto 3: f= -0.37878744599, f= 0.05778703894 Soluto 4: f= -0.37933409043999, f= 0.070380563788 Soluto 5: f= -0.37987044678696, f= 0.078656488056 f -0.45-0.5-0.55-0.6-0.65-0.7 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 f Fgure. Pareto frot of (7) A I curret lteratures [,4], whe =, the best soluto of Bump problem (5) foud wth pealty-fucto method s 0.36497974587066. Although the DM ca use ths ocostrat-volated soluto, he wll ot kow how much he ca ga f slght costrat volatos to a certa eted are permtted. Whle our costrat hadlg method ca fd a group of caddate solutos. We ca fd that although soluto -5 have costrat volatos (sce f>0), they meawhle have substatal gas the objectve fuctos. Basg o the real mea of ths problem, the DM ca make trade-off betwee costrat volatos ad objectve fucto gas ad therefore choose hs preferable solutos freely to ga more profts. The ma dfferece betwee TF ad TF s that TF has two soft costrats. Therefore we coverted TF to (8), a three objectve problem. Cosequetly, The Pareto frot of (8) we got wth SEEA s of three dmesos, so we plot t wth three two-dmeso graphs. Our costrat hadlg method ca gve the vvd relatoshp betwee the objectve fucto ad the costrats. Fgure 3 s the relatoshp betwee the objectve fucto ad the frst costrat of TF, fgure 4 s the relatoshp betwee the objectve fucto ad the secod costrat ad fgure 5 s the relatoshp betwee the frst ad the secod costrat. From fgure 3-5, we otce vsually that at pot A, although the frst costrat(sce f >0) s volated, there s a substatal Fgure 3. Trade-off frot betwee f ad f A Fgure 4. Trade-off frot betwee f ad f 3 A Fgure 5. Trade-off frot betwee f ad f 3
ga the objectve fucto, at the same tme the secod costrat (f 3 =0) s ot volated. Therefore, we cosder pot A may be very allurg to the DM. Besdes pot A, the other solutos wth dfferet etet costrat volatos ad dfferet etet objectve gas also ca be preseted to the DM. All these caot be realzed wth pealty-fucto costrat hadlg method. 5. Cocluso ad future work I ths paper, we frst aalyzed the drawbacks of pealtyfucto costrat hadlg method for olear costrat programmg. The we preseted a ew costrat hadlg method based o EMO. Epermetal tests demostrated that ths costrat hadlg method ot oly ca fd the costraed optmalty, but also ca provde the decso maker wth a group of trade-off solutos wth slghtly costrat volato ad meawhle wth substatal ga the objectve fucto, Ths eables the DM to have more freedom to choose hs preferred soluto ad therefore eplot more profts the marg of costrat volatos, where the costrat volatos are small or acceptable. I addto, the good search abltes of EAs ca guaratee ther good performace whe tacklg the problems wth ocove search spaces ad other dffcult search spaces. I the ear future, we ted to tegrate our EMObased costrat hadlg to a decso support system, whch ca offer DM vsualzato decso support for lear ad olear costrat programmg. We also ted to do some epermets of tacklg overcostraed problems wth ths method ad therefore eted our system order to maage over-costraed problems. Refereces [] LU Kacheg. Sgle-objectve, mult-objectve ad teger programmg. Tsghua Uversty Press,999(ch). [] Mchalewz Z. Geetc Algorthms +Data Structures = Evoluto Programs. Sprger-Verlag, Berl. 99 [3] Deb,K. Mult-Objectve Optmzato usg Evolutoary Algorthms. Joh Wley&Sos,Ltd Baffs Lae,Chchester,West Susse,PO9,IUD,Eglad. 00 [4] YUAN Yaag, Su Weyu. Theory ad method of optmzato. Scece Press,999(ch). [5] WU Zhyua, SHAO Huhe,WU Xyu. Aealg Accuracy Pealty-fucto Based Nolear Costraed Optmzato Method wth Geetc Algorthms. Cotrol ad Decso, 998,Vol.3 No.,pp36-40(ch). [6] Homafar A, Charlee X Q, La SH. Costraed optmzato va geetc algorthms. Smulato. 994,6 (4) :4-54. [7] Davd A.VaVeldhuze, Gary B. Lamot. Multobjectve Evolutoary Algorthms: Aalyzg the Stateof-the-Art. Evolutoary Computato, 000, 8():5-47. [8] Schaffer,J. Multple objectve optmzato wth vector evaluated geetc algorthms. Proceedg of the Frst Iteratoal Coferece Geetc Algorthms, 985, page 93-000, Lawrece Erlbaum, Hllsdale, New Jersey. [9] Hor,j, Nafplots,N., Goldberg,D.E. A Nche Pareto Geetc Algorthm for Mult-objectve Optmzato. Proceedgs of the frst IEEE Coferece Evolutoary Computato, 994, pages 8-87, IEEE Press, Pscataway, New Jersey. [0] Carlos Foseca, Peter J. Flemg Mult-objectve Optmzato ad Multple Costrat Hadlg wth Evolutoary Algorthms -Part :A Ufed Formulato. IEEE Trasactos o Systems, Ma ad Cyberetcs- Part A: Systems ad Humas, 998, 8(): 6-37. []N.Srvas, Kalyamoy Deb. Mult-objectve Optmzato Usg Nodomated Sortg I Geetc Algorthms, Evolutoary Computato, 995, (3) -48. [] Ztzler,E., Thele, L. Mult-objectve Evolutoary Algorthms: A Comparatve Case Study ad the Stregth Pareto Approach. IEEE Trasactos o Evolutoary Computato, 999, 3(4), 57-7. [3] Ya Zheyu, Kag Lsha, Bob (R I) Mckay. SEEA For Mult-Objectve Optmzato: Reforcg Eltst MOEA Through Mult-Paret Crossover, Steady Elmato ad Swarm Hll Clmbg, Proceedgs of the 4th Asa-Pacfc Coferece o Smulated Evoluto Ad Learg, 00, Vol. I, pages -6, Sgapore. [4] GUO Tao,KANG L-sha,LI Ya. A New Algorthm for Solvg Fucto Optmzato Problems wth Iequalty Costrats. J Wuha Uv(Nat Sc ED),999,45(5B):77-775(Ch).