PARETO BAYESIAN OPTIMIZATION ALGORITHM FOR THE MULTIOBJECTIVE 0/1 KNAPSACK PROBLEM

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1 PARETO BAYESIAN OPTIMIZATION ALGORITHM FOR THE MULTIOBJECTIVE 0/ KNAPSACK PROBLEM Josef Schwarz Jří Očenáše Brno Unversty of Technology Faculty of Engneerng and Computer Scence Department of Computer Scence and Engneerng CZ Brno, Bozetechova 2 Tel.: , fax: 44270, e-mal: schwarz@dcse.fee.vutbr.cz, Tel.: , fax: 44270, e-mal: ocenase@dcse.fee.vutbr.cz Abstract: Ths paper deals wth the utlzng of the Bayesan optmzaton algorthm (BOA) for the Pareto b-crtera optmzaton of the 0/ napsac problem. The man attenton s focused on the ncorporaton of the Pareto optmalty concept nto classcal structure of the BOA algorthm. We have modfed the standard algorthm BOA for one crteron optmzaton utlzng the nown nchng technques to fnd the Pareto optmal set. The experments are focused manly on the b-crtera optmzaton because of the vsualzaton smplcty but t can be extended to multobectve optmzaton, too. Key Words: Knapsac problem, multobectve optmzaton, Pareto set, evolutonary algorthms, Bayesan optmzaton algorthm, nchng technques. Introducton Practcal problems are often characterzed by several non-commensurable and often competng obectves. Whle n the case of sngle-obectve optmzaton the optmal soluton s smply dstngushable, ths s not true for multobectve optmzaton. The standard approach to solve ths dffculty les n fndng all possble trade-offs among the multple, competng obectves. These solutons are optmal, nondomnated, n that there are no other solutons superor n all obectves. These so called Pareto optmal solutons le on the Pareto optmal front. There are many papers that present varous approaches to fnd Pareto optmal front almost based on the classcal evolutonary algorthms. The Nched Pareto Genetc Algorthm (NPGA) combnes tournament selecton and the concept of Pareto domnance []. A wde revew of basc approaches and the specfcaton of orgnal Pareto evolutonary algorthms nclude the dssertatons [2], [3] where the last one descrbes the orgnal Strength Pareto Evolutonary Algorthm (SPEA). An nterestng approach usng nondomnated sortng n genetc algorthm (NSGA) s publshed n [4]. The classcal genetc algorthms (GA) have the common dsadvantage - the necessty of the settng the parameters le crossover, mutaton and selecton rate and the choce of sutable type of genetc operators. That s why we have analyzed and used one of the Estmaton of Dstrbuton Algorthms (EDAs) called probablstc model-buldng genetc algorthms, too. The crossover and mutaton operators used n standard GA are replaced by probablty estmaton and samplng technques. We have focused on the Bayesan optmzaton algorthm publshed n the basc ssue n [5], [6]. Recently we have publshed our experence wth ths algorthm n [7], [8] where sngle crteron and b-crtera optmzaton of hypergraph bsectonng was descrbed. In ths paper we have focused on the b-obectve optmzaton of napsac problem whch belongs to the well nown test benchmars. 2 Multobectve optmzaton A general multobectve optmzaton/maxmzaton problem (MOP) can be descrbed as a vector functon f that maps a tuple of n parameters to a tuple of m obectves [3]: max y = f(x)=( (x), (x),, f m (x)) () subect to g(x) = (g (x), g 2 (x),..., g (x)) <=0 subect to x = (x, x 2,,..,x n ) X y = (y, y 2,,.,y m ) Y, where x s called decson vector, X s the parameter space, y s the obectve vector, Y s the obectve space and the constrant vector g(x) <=0 determnes the set of feasble solutons/set X f. The set of solutons of MOP ncludes all decson vectors for whch the correspondng obectve vectors can not be mproved n any dmenson wthout degradaton n another - these vectors are called Pareto optmal set. The dea of Pareto optmalty s based on the Pareto domnance.

2 For any two decson vectors a, b t holds a f b (a domnates b) ff f(a)>f(b), a f = b (a wealy domnates b) ff f(a)>=f(b), a ~ b (a s ndfferent to b) ff a, b are not comparable A decson vector a domnates decson vector b ( a f b) ff f (a ) f (b) for =,2,.., m wth f (a )> f (b) for at least one. The vector a s called Pareto optmal f there s no vector b whch domnates vector a n parameter space X. In obectve space the set of nondomnated solutons les on a surface nown as Pareto optmal front. The goal of the optmzaton s to fnd a representatve samplng of solutons along the Pareto optmal front. The way how to do t les n eepng the dversty usng some of the nchng technques. Standard BOA s able to fnd mostly one optmal soluton at the end of the optmzaton process, when the whole populaton s saturated by phenotype-dentcal ndvduals. 3 Problem specfcaton Generally, the 0/ napsac problem conssts of set of tems, weght and proft assocated wth each tem, and an upper bound of the capacty of the napsac. The tas s to fnd a subset of tems whch maxmzes the sum of the profts n the subset, yet all selected tems ft nto the napsac so as the total weght does not exceed the gven capacty. Ths sngle obectve problem can be extended to multobectve problem by allowng more than one napsac. Formally, the multobectve 0/ napsac problem s defned n the followng way: Gven a set of n tems and a set of m napsacs, wth p, proft of tem accordng to napsac w, weght of tem accordng to napsac c capacty of napsac fnd a vector x = (x, x 2,..., x n ) {0,} n, such that x = ff tem s selected and f(x)=( (x), (x),, f m (x)) s maxmum, where (2) f (x) = n p, * x (3) = and for whch the constrant s fulflled n {,2 K, m}: w * x c (4) =, The complexty of the problem solved depends on the values of napsac capacty. Accordng to [3] we used the napsac capactes stated by the equaton: n c = 0.5 w, (5) = The encodng of soluton nto chromosome s realzed by bnary strng of the length n. To satsfy the constrants (4) t s necessary to use repar mechansm on the generated offsprng to be feasble one. The repar algorthm removes tems from the soluton step by step untl the capacty constrants are fulflled. The order n whch the tems are deleted s determned by the maxmum proft/weght rato per tem; the maxmum proft/weght rato q s gven by the equaton q m p = max w =,, The tems are consdered n ncreasng order of the q,.e., tem wth the lowest proft per weght unt s removed frst. Ths mechansm respects the capacty constrants whle decreasng the overall proft as lttle as possble. 4 Pareto optmal BOA In our Pareto BOA algorthm we replaced the ftness assgnment and replacement step of standard BOA by the dversty-preservng nchng method based on the promsng Pareto technque utlzng a new strength crteron for the evaluaton process [3]. The followng specfcaton descrbes the whole reproducton process of our algorthm. Let us note that although we solved b-obectve optmzaton, our algorthm s able to solve m-obectve optmzaton problems. Our Pareto BOA algorthm can be descrbed by the followng steps: Step : Intalzaton: Generate an ntal populaton P 0 of sze N randomly. Step 2: Ftness assgnment: Evaluate the ntal populaton. (6)

3 Step 3: Selecton: Select the parent populaton as the best part of current populaton by 50% truncaton selecton. Step 4: Model constructon: Estmate the dstrbuton of the selected parents usng Bayesan networ constructon. Step 5: Offsprng generaton: Generate new offsprng (accordng to the dstrbuton assocated to the Bayesan networ). Step 6: Nondomnated set detecton and ftness assgnment: Current populaton and offsprng are oned, nondomnated solutons are found, evaluated and stored at the top of the new populaton. Then domnated offsprng and parents are evaluated separately. Step 7: Replacement: The new populaton s completed by offsprng and the best part of current populaton, so the worst ndvduals from current populaton are canceled to eep the sze of the populaton constant. Step 8: Termnaton: If predefned number of generatons N g s reached or stoppng crteron s satsfed then the last Pareto front s saved, else go to Step 3. The most mportant part of our Pareto algorthm descrbed above s the procedure (step 6) for detecton of nondomnated (current Pareto front) and domnated solutons and sophstcated ftness calculaton. The procedure for current nondomnated and domnated set detecton s descrbed n followng steps:. For each ndvdual X n the populaton P compute the vector of the obectve functons f ( X ) = ( ( X ), ( X ), K, f m ( X )) (7) 2. Detect subset of nondomnated solutons P = X X P X P : X f X (8) { } 3. For each nondomnated soluton X from P compute ts strength value as s( X ) = { X X P X f X } P + The ftness for nondomnated solutons s equal to the reverse of the strength value f ( X ) = s( X ) (0) 4. For each domnated soluton X from P determne the ftness as f ( X = + ) s( X ), () X where X P X f X. In the orgnal approach [3] all ndvduals domnated by the same nondomnated ndvduals have equal ftness. We proposed an extenson by addng a term c. r( X ) ( P + ) nto the denomnator (), where r ( X ) s the number of ndvduals from P (not only from nondomnated solutons) whch domnate X and coeffcent c s set to very small number, for example Ths term s used to dstngush the mportance of ndvduals n the same nche (nche s an area n the obectve space domnated by the same part of Pareto front). Ths type of ftness evaluaton has the followng advantages: For all nondomnated ndvduals f ( X ), for domnated ndvduals holds f ( X ) <. If we use the replace-worst strategy, mplct Pareto eltsm s ncluded. Indvduals from Pareto front domnated smaller set of ndvduals receve hgher ftness, so the evoluton s guded towards the less-explored search space. Indvduals havng more neghbours n ther nche are more penalzed due to the hgher s ( X ) value of assocated nondomnated soluton. Indvduals domnated by smaller number of nondomnated ndvduals are more preferred. 5 Expermental results 5. Test benchmars In order to be able to compare the performance of our algorthm and other evolutonary algorthms wth nown results we used two napsac benchmars specfed by 00 (Kn00) and 250 tems (Kn250) publshed on the web ste [ Let us note that the uncorrelated profts p, and weghts w, were chosen, where p, and w, are random ntegers n the nterval <0,00>. We have compared our results wth presented results obtaned by two evolutonary algorthms SPEA [3] and NSGA[4]. These two algorthms represent the well worng evolutonary multobectve algorthms. (9)

4 5.2 Experments and results All experments were run on Sun Enterprse 450 machne (4 CPUs, 4 GB RAM), n future we consder the utlzng the cluster of Sun Ultra 5 worstatons. In Fg. the hstory of evoluton process for the benchmar Kn00 s depcted. The st, 25 th, 50 th and 00 th generaton of one run of Pareto optmal BOA s shown. In experment we set the populaton sze N=, but only 500 of the randomly chosen ndvduals/solutons are shown n the graph. The X-coordnate of each pont equals to the functon value (x) and the Y-coordnate equals to the functon (x). Ths experment shows the dynamsm of the evoluton process and the creaton of Pareto front. a) b) c ) d) Fg.. Dstrbuton of the solutons for Kn00 benchmar plotted n the a) st generaton, b) 25 th generaton, c) 50 th generaton, d) 00 th generaton. In Fg.2a there s the comparson of the fnal Pareto front produced by our Pareto BOA algorthm and by the SPEA [3] and NSGA [4] algorthms for the case of Kn00 benchmar and n Fg. 2b for the case of Kn250 benchmar. We performed 5 ndependent runs and constructed the fnal Pareto front from the 5 partcular Pareto fronts. From Fg. 2a t s evdent that for Kn00 the Pareto solutons produced by our Pareto BOA n the mddle part of Pareto front are slghtly better than the Pareto solutons produced by SPEA and NSGA. What s more mportant our Pareto BOA produces more solutons n the Pareto front margns. In Fg. 2b we see that for Kn250 the dfference between Pareto fronts s more expressve our Pareto BOA outperforms the SPEA and NSGA. We used the followng settng for our algorthm: for the Kn00 we set the populaton sze N to, for the case of the Kn250 the populaton sze N equals to The number of generatons used s about 300. The computaton tme s about 5 mnutes for the Kn00. In case of the Kn250 the tme was 2 hours for N=5000 and about 40 mnutes for N=. To reduce the computaton tme we consder the mplementaton of parallel verson of the Pareto BOA algorthm. It would be also possble to shorten the computatonal tme by decreasng the sze of populaton, but we wanted to eep those predefned values used n our prevous experments (e.g. n the case of hypergraph parttonng [7],[8]). In the context of algorthm comparson an mportant queston arses: What measure should be used to express the qualty of the results so that the varous evolutonary algorthms can be compared n a meanngful way. In [3] two measures are descrbed. One of them denoted as S represents the sze of the obectve space covered, the second measure denoted C represents the coverage of two sets accordng to ther domnance. We preferred n our comparson the topology/shape of the Pareto fronts.

5 NSGA SPEA BOA Fg. 2a Fnal Pareto fronts for Kn00, populaton sze N= NSGA SPEA BOA Fg. 2b Fnal Pareto fronts for Kn250, populaton sze N= Parallel Pareto BOA In [9], [0] we proposed the Dstrbuted Bayesan Optmzaton Algorthm. It uses a cluster of worstatons as a computng platform to speedup the evoluton process. Let s note that n the dstrbuted envronment the whole populaton P s splt nto several parts, each part P s beng generated and evaluated by dfferent processor. Ths approach can be extended to the Pareto BOA. We propose the followng modfcaton of the procedure for Pareto detecton and ftness assgnment: Frst, each processor wll compute the vector of obectve functons for all ndvduals from ts part P of populaton P. Then, each processor detects ts local set of nondomnated solutons P as P = { X X P X P : X f X } (2) and the master processor creates the global nondomnated set P from the unon of local nondomnated sets P as P = X X U P X U P : X f X (3)

6 The strength values for nondomnated solutons from P can be obtaned as the sum of local strength values computed n parallel by all processors: s( X ) = { X X P X f X } P + After that all nondomnated solutons and ther strength values are nown, so each processor s able to compute the Pareto ftness for all ndvduals from ts part of populaton accordng to equatons (0) and (). 7 Conclusons We have mplemented b-obectve Pareto BOA algorthm for two multple 0/ napsac problems Kn00 and Kn250. For ths purpose we have modfed the evaluaton phase of the sngle BOA algorthm [5], [7], [8] usng the concept of a strength crteron publshed n the SPEA algorthm [3]. Let us note that the SPEA s a modern multobectve optmzaton algorthm whch outperforms a wde range of classcal methods on many problems. That s why we compare the performance of our Pareto BOA algorthm manly wth the performance of the SPEA algorthm. The Pareto solutons produced by our algorthm are unformly dstrbuted along the Pareto front whch s more global than the Pareto fronts obtaned by NSGA and SPEA algorthms. But many problems reman to be solved, namely the greater computatonal complexty. The next possble mprovement les also n more sophstcated nchng technque, modfcaton of replacement phase of the algorthm and the ntroducton of problem nowledge nto optmzaton process. To reduce the computatonal complexty we proposed the dea of the parallelzaton of Pareto BOA ncludng the decomposton and detecton of the Pareto front. Ths approach s an extenson of Dstrbuted Bayesan Optmzaton Algorthm [9], [0] based on the parallelzaton of Bayesan networ constructon. The future wor wll be focused on the mplementaton of Parallel Pareto BOA algorthm for the cluster of SUN worstatons. Acnowledgement Ths research has been carred out under the fnancal support of the Czech Mnstry of Educaton FRVŠ No. 07/200 The parallelzaton of Bayesan Optmzaton Algorthm and the Research ntenton No. CEZ: J22/98: 2622 Research n nformaton and control systems. References [] Horn, J., Nafplots, N., Goldberg, D.E.: A Nched Pareto Genetc Algorthm for Multobectve Optmzaton, In Proceedngs of the Frst IEEE Conference on Evolutonary Computaton, IEEE World Congress on Computatonal Intellgence, Vol., pages 82-87, Pscataway, New Jersey, June 994. IEEE Servce Center. [2] Coello, C. A.: An Emprcal Study of Evolutonary Technques for Multobectve Optmzaton n Engneerng Desgn. PhD thess, Department of Computer Scence, Tulane Unversty, New Orleans, LA, Aprl 996. [3] Ztzler, E.: Evolutonary Algorthms for Multobectve Optmzaton: Methods and Applcatons PhD thess, Swss Federal Insttute of Technology (ETH), Zurch, Swtzerland, November 999. [4] Srnvas, N., Deb, K.: Multobectve Optmzaton usng Nondomnated Sortng n Genetc Algorthm. Evolutonary Computaton, Vol.2, No. 3, 994, pp [5] Pelan, M.: A Smple Implementaton of Bayesan Optmzaton Algorthm n C++(Verson.0). Illgal Report 990, February 999, pp. -6. [6] Pelan, M., Goldberg, D. E., & Cantú-Paz, E.: Lnage Problem, Dstrbuton Estmaton, and Bayesan Networs. IllGal Report No. 9803, November 998, pp [7] Schwarz, J., Očenáše, J..: Expermental Study: Hypergraph Parttonng Based on the Smple and Advanced Genetc Algorthm BMDA and BOA, Proceedngs of the Mendel'99 Conference, Brno Unversty of Technology, Faculty of Mechancal Engneerng, Brno, 999, pp , ISBN [8] Schwarz, J., Očenáše, J.: Evolutonary Multobectve Bayesan Optmzaton Algorthm: Expermental Study. Proceedngs of the MOSIS 0, 200, pp. 0-08, ISBN [9] Očenáše, J., Schwarz, J.: The Dstrbuted Bayesan Optmzaton Algorthm for Combnatoral Optmzaton, EUROGEN Evolutonary Methods for Desgn, Optmsaton and Control wth Applcatons to Industral Problems, Athens, Greece, September 9-2st, 200, accepted paper to be publshed. [0] Očenáše, J., Schwarz, J.: The Parallel Bayesan Optmzaton Algorthm, Proceedngs of the European Symposum on Computatonal Intellgence, Physca-Verlag, Košce, Slova Republc,, pp. 6-67, ISBN (4)