A compaative study of multiple-obective metaheuistics on the bi-obective set coveing poblem and the Paeto memetic algoithm Andze Jaszkiewicz * Woking pape RA-003/0 Septembe 200 * Institute of Computing Science Poznań Univesity of echnology ul. Piotowo 3a, 60-965 Poznań, Poland Jaszkiewicz@cs.put.poznan.pl www.cs.put.poznan.pl/~and_
Abstact he pape descibes a compaative study of multiple-obective metaheuistics on the bi-obective set coveing poblem. en epesentative methods based on genetic algoithms, multiple stat local seach, hybid genetic algoithms and simulated annealing ae evaluated in the computational expeiment. Nine of the methods ae well known fom the liteatue. he pape intoduces also a new hybid genetic algoithm called Paeto memetic algoithm. he esults of the expeiment indicate vey good pefomance of hybid genetic algoithms, howeve, no algoithm was able to outpefom all othe methods on all instances. Futhemoe, the esults indicate that the pefomance of multiple-obective metaheuistics may diffe adically even if the methods ae based on the same single obective algoithm and use exactly the same poblem-specific opeatos. Keywods: multiple-obective optimization, metaheuistics, set coveing poblem.. Intoduction In ecent yeas one could obseve a gowing inteest in multiple-obective metaheuistics. he goal of such methods is to geneate in a single un a set of solutions appoximating whole o a pat of the nondominated set. Methods of this kind ae applied to vaious computationally had multiple-obective poblems, e.g. combinatoial optimization poblems and nonlinea pogamming poblems. Although, authos of multiple-obective metaheuistics often claim that thei methods may effectively solve lage-scale poblems, a elatively low numbe of compaative expeiments, in paticula involving multiple-obective combinatoial optimization (MOCO) poblems, has been published in the liteatue. In this pape, we compae ten multiple-obective metaheuistics on the bi-obective set coveing poblem. Nine of the methods ae well known fom the liteatue. he methods ae: the multiple-obective genetic local seach algoithm (MOGLS) poposed by us [4], [6], Ishibuchi's and Muata's multiple-obective genetic local seach (IMMOGLS) [2], Seafini's multiple-obective simulated annealing (SMOSA) [23], multiple-obective simulated annealing poposed by Ulungu et al. (MOSA) [28], Paeto simulated annealing (PSA) poposed by us [2], nondominated soting genetic algoithm (NSGA) [25], contolled elitist non-dominated soting genetic algoithm (CENSGA) [3], stength Paeto evolutionay algoithm (SPEA) [30], and a simple multiple-obective multiple stat local seach (MOMSLS) with andom weight vectos [5], [6]. In addition, we intoduce a new multiple-obective metaheuistic called Paeto memetic algoithm (PMA). All methods wee implemented in C++ with the use of MOMHLib++ libay []. hei implementations shaed significant factions of common code. he code of all methods used in this expeiment is feely available at http://www-idss.cs.put.poznan.pl/~aszkiewicz/moscp. Set coveing poblem is an NP-complete combinatoial optimization poblem [6] often used to test single obective metaheuistics. It finds many pactical applications especially in cew scheduling [24]. In pactice, solutions of set coveing poblem often cannot be evaluated with a single obective only. Fo example, in the case of cew scheduling it may be necessay to
take into account both the cost and the quality of wok. Othe obectives impotant in the case of cew scheduling ae discussed in [20]. he pape is oganized in the following way. In the next section, some basic definitions ae given. he Paeto memetic algoithm is descibed in the thid section. In the fouth section, othe methods used in the expeiment ae biefly chaacteized. Adaptation of the methods to multiple-obective set coveing poblem is pesented in the fifth section. In the sixth section, the expeiment design is descibed. he appoach used to evaluate quality of obtained esults is pesented in the seventh section. In the eighth section, esults of the expeiments ae pesented and discussed. Concluding emaks ae given in the ninth section. 2. Basic definitions he geneal multiple-obective optimization (MOO) poblem is fomulated as: maximize s.t. { f ( ) z f ( x) } x D, x,..., whee: solution x [ x,..., ] J z J x I is a vecto of decision vaiables, D is the set of feasible solutions. If the vaiables ae discete the MOO poblem is called multiple-obective combinatoial optimization (MOCO) poblem. x x x he image of a solution x in the obective space is a point z [ z,..., z ] f ( x) x z f (x),,..,j. J (P),suchthat Point z dominates z 2, z fz 2 2,if 2 z z and z > z fo at least one. Solutionx dominates x 2 if the image of x dominates the image of x 2. A solution x D is Paeto-optimal (efficient) if thee is no x' D that dominates x. Point being an image of a Paeto-optimal solution is called nondominated. he set of all Paeto-optimal solutions is called the Paeto-optimal set. he image of the Paeto-optimal set in the obective space is called the nondominated set o Paeto font. An appoximation of the nondominated set is a set A of feasible points (and coesponding 2 2 solutions) such that z, z A such that z f z, i.e. set A is composed of mutually nondominated points. he point z * composed of the best attainable obective function values is called the ideal point: { f ( x) D},..., J. * z max x Range equalization factos ([26], ch. 8.4.2) ae defined in the following way: π R,,, J whee R is the (appoximate) ange of obective z in the nondominated set, o D o A. Obective function values multiplied by ange equalization factos ae called nomalized obective function values. 2
Weighted linea scalaizing functions ae defined in the following way: whee [ ] s l J J ( z, Λ) λ z λ f ( x) Λ λ,...,λ J is a weight vecto such that λ 0. Weighted chebycheff scalaizing functions ae defined in the following way: s 0 0 0 ( z, z, Λ) max{ ( z z )} max{ λ ( z f ( x) )} whee z 0 is a efeence point, [ ] λ, Λ λ,...,λ J is a weight vecto such that λ 0. Each weighted chebycheff scalaizing function has at least one global optimum (minimum) belonging to the set of Paeto-optimal solutions. Note, howeve, that if the optimum is not unique then some of the optima may be dominated, but must have at least one obective component equal to a Paeto-optimal solution. Fo each Paeto-optimal solution x thee exists a weighted chebycheff scalaizing function s such that x is a global optimum (minimum) of s ([26], ch. 4.8). Weight vectos that meet the following conditions: λ 0, λ, ae called nomalized weight vectos. Minimization of the weighted chebycheff scalaizing function coesponds to a min-max poblem. he poblem can be, howeve, tansfomed to the following one: minimize α (P2) s.t. J 0 0 ( z z ) λ ( z f ( x) ), J α λ,,..., x D. he bi-obective set coveing poblem (BOSCP) is the poblem of coveing the ows of a L-ow, I-column, zeo-one matix (a l ) by a subset of the columns minimizing two cost-type obectives. Defining x i if column i (with costs c i, c x i 0, othewise the BOSCP is: I I minimize {z c x i i, z 2 c 2 x i i } s.t. I i i a li x i, l,...,l i 2 i > 0) is selected in the solution and x i {0, },,...,J In BOSCP both obectives ae minimized, howeve, the poblem can be easily tansfomed to the fom of poblem (P) by changing signs of the obectives. hus, all definition intoduced above emain valid fo BOSCP. 3
3. Paeto memetic algoithm Hybid genetic algoithms (HGAs) ae elatively new metaheuistics that hybidize ecombination opeatos with local seach, o moe geneally, with othe local heuistics. Othe fequently used names ae memetic algoithms o genetic local seach. Methods of this type often pefom vey well on combinatoial optimization poblems, e.g., on tavelling salespeson poblem [22], gaph coloing poblem [7], and quadatic assignment poblem [27]. hey pefom also vey well on single obective set coveing poblem []. hus, the development of multiple-obective vesions of HGAs is one of the most pomising diections of eseach. Paeto memetic algoithm intoduced in this pape has been developed on the basis of peviously poposed multiple-obective genetic local seach (MOGLS) algoithm [4], [6]. he geneal scheme of PMA, MOGLS, as well as of IMMOGLS [2], may be summaized by the algoithm pesented in Figue. he thee algoithms diffe by the way in which solutions ae dawn fo ecombination. In the case of MOGLS, a elatively small tempoay elite population E composed of a numbe (e.g. 0) of solutions being the best on the cuent scalaizing function is selected fom the cuent set of solutions. hen, two solutions ae dawn fo ecombination fom the tempoay elite population with unifom pobability. In this way the solutions ecombined ae assued to be vey good on the cuent scalaizing function. his pocess in epeated in each iteation. Geneate the initial set of solutions epeat Daw at andom a weight vecto Λ defining the cuent scalaizing function Select pobabilistically two peviously geneated solutions being good on the cuent scalaizing function Recombine the two solutions Apply a local heuistic to the offsping until the stopping citeion is met Figue. Geneal scheme of the multiple-obective genetic local seach algoithm he selection pocess used in MOGLS, is howeve, time consuming. MOGLS was oiginally applied to multiple-obective taveling salespeson poblem using 2-opt local seach [6]. In this case the time needed fo selection of ecombined solutions is meaningless with espect to local seach time. In geneal, howeve, MOGLS could also be used with elatively simple and fast heuistics (see e.g. [3]). In this case, solutions selection time significantly influences oveall unning time. PMA uses anothe faste appoach based on tounament selection to select good solutions fo ecombination. Let denote cuent set of solutions. In each iteation, a sample is dawn at andom with epetitions fom. hen, two best solutions in ae winnes of the tounament and ae selected fo ecombination. he size of is set in ode to obtain expected ank induced by the cuent scalaizing function of ecombined solutions simila as in the case of MOGLS. Cuent scalaizing function s induces a ank in set with the best solution on s having ank. Fo simplicity assume that no two solutions in have equal value of s. Inthe 4
5 case of MOGLS, the expected ank of ecombined solutions is obviously E /2+0.5. Below we analyze expected ank in the case of tounament selection. Conside a solution x having ank and selected fo the tounament. hee ae - bette solutions and -+ not bette solutions (including x). he pobability P that x is the best solution in unde condition that x is selected fo the tounament is equal to the pobability of dawing - times a solution not bette than x: + P. he pobability P 2 that x is dawn fo the tounament is equal to: P 2, thus, the pobability P 3 that x is the best solution in the tounament is: 2 3 P P P, and the expected ank of the best solution in the tounament E is: E. In a simila way one may calculate expected ank of the second best solution. In this case, the pobability P 4 that a solution x having ank and selected fo the tounament is the second best solution is calculated fom Benoulli distibution and is equal to: ( ) 2 4 P. hus, the expected ank of the second best solution E2 isequalto: ( ) E 2 2 In esult, the expected ank of ecombined solutions E is equal to: ( ) + + E E E 2 2 2 2. he above fomula, although quite complicated, is within typical anges of ande, i.e. >>E 5, well appoximated by the following fomula:
3 E. 2 In PMA, the expected ank of ecombined solutions is used as a paamete of the method and the size of the tounament sample is set in the following way: 3. 2E In geneal, low values of E esult in vey high selection pessue but may cause a pematue convegence. he algoithm of PMA is summaized in Figue 2. he set of potentially Paeto-optimal solutions PP is the outcome of the method and is an appoximation of the nondominated set. It is empty at the beginning and is updated wheneve a new solution is geneated by the local heuistic. Updating the set of potentially Paeto-optimal solutions with solution x consists of: adding f(x) topp if no point in PP dominates f(x), emoving fom PP all points dominated by f(x). In this expeiment we used weighted echbycheff scalaizing functions in PMA and othe hybid genetic algoithms. he scalaizing functions used a efeence point based on the set of potentially Paeto-optimal solutions defined in the following way (note that in the case of BOSCP signs of the obectives wee changed, see section 2): z ** ( PP) max { z PP} + 0.( max { z z PP} min { z z PP} ),..., J z. Set is oganized as a queue of size N S, whees is the numbe of initial solutions and N is a paamete of the method typically equal to 2E- (a value assuing compatibility with appoach used in MOGLS). New solutions ae added to the beginning of the queue. If the size of the queue is bigge than N S then the last solution fom the queue is emoved. In ou expeiment, howeve, a value of N 2E- allowed to stoe all geneated solution in until the end of the algoithm un. 6
Paametes: E expected ank of ecombined solutions, S numbe of initial solutions o K - paamete used in condition (), stopping citeion Initialization: he set of potentially Paeto-optimal solutions PP: he cuent set of solutions : Geneation of the fist appoximation of the ideal point: fo each obective f Constuct a new feasible solution x by a andomized algoithm Apply a local heuistic optimizing obective f to solution x obtaining x Add x to the cuent set of solutions Update set PP with x Geneation of the initial set of solutions: epeat Daw at andom a weight vecto Λ Constuct a new feasible solution x by a andomized algoithm Apply a local heuistic optimizing s( z,...,λ) to solution x obtaining x Add x to the cuent set of solutions Update set PP with x until S initial solutions ae geneated o condition () is fulfilled Main loop: epeat Daw at andom a weight vecto Λ Fom daw at andom with unifom pobability a sample of Recombine the best and the second best solution on s( z,...,λ) obtaining x Apply a local heuistic optimizing s( z,...,λ) to solution x obtaining x if x isbetteon s( z,...,λ) than the second best solution in then Add x to the cuent set of solutions Update set PP with x until the stopping citeion is met Figue 2. he algoithm of Paeto memetic algoithm. s( z,...,λ) s z, z * ( PP) Λ o s ( z,λ) ( ), l. he fome vesion is used in this expeiment 3 2E solutions stands fo eithe he initial solutions ae geneated by local optimization of andomly selected scalaizing functions. he numbe of the initial solutions S is an additional paamete of the method. We use also anothe appoach that allows automatically stopping geneation of initial solutions. 7
Single obective hybid genetic algoithms often stat by geneation of a numbe of solution by local optimization. In the multiple-obective case, we popose to stop geneating the initial solutions when aveage quality of K best solutions in the set of initial solutions ove all scalaizing functions is the same as the aveage quality of solutions geneated by the local heuistic used fo optimization of these functions. Pecisely, the geneation of initial solutions is stopped when the following condition is met (minimization of the scalaizing function is assumed): x ( s x ( B( K,,, s ) s ( x) ) 0 x, () x whee x is an initial solution being obtained by local optimization of a scalaizing function s x (note that x does not need to be the best solution on s x in the set because solutions obtained by local optimization of othe functions may be bette on s x ), B( K,, x, s x ) is the set of K best solutions of function s x diffeent fom x and s x ( B( K,, s x )) of s x in B ( K,, x, ): s x s x ( B( K,,, s )) x ( y) sx y B( K,, x, s x ) x x. K, x is the aveage value In ode to daw at andom a weight vecto the algoithm pesented in Figue 3 is used. he algoithm unifomly samples the whole set of nomalized weight vectos [7]. Note that by changing the algoithm fo dawing the weight vectos one can concentate the method on a subegion of the nondominated set [3]. In eal-life applications the anges of obectives usually diffe a lot, often by many levels of magnitude. So, the nomalized weight vectos should be applied to nomalized obective values (see section 2). he ange equalization factos ae updated on-line fom the anges of paticula obectives in the set of potentially Paeto-optimal solutions. J λ,..., λ ( and() ) J and() λ l,..., l l J λ J λ l Figue 3. Algoithm fo geneating andom nomalized weight vectos. Function and() etuns a andom value fom the ange <0,> with unifom pobability 4. Othe methods tested in the expeiment Because of limited space only basic infomation about othe algoithms tested in the expeiment is given. Futhe details may be found in the liteatue. As it was mentioned in the pevious section, PMA nomalizes the obective values with ange equalization factos calculated on-line fom the anges of paticula obectives in the set of potentially Paeto-optimal solutions. Although desciptions of some algoithms do not efe to this issue, we use obective values nomalization in all algoithms that use scalaizing functions o distance measues in the obective space. 8
4.. Hybid genetic algoithms hee hybid genetic algoithms have been tested in the expeiment. he Paeto memetic algoithm (PMA) has been descibed in section 3. Multiple-obective genetic local seach algoithm (MOGLS) has also been biefly descibed in that section. Ishibuchi's and Muata's multiple-obective genetic local seach (IMMOGLS) [2] was the fist multiple-obective hybid genetic algoithm descibed in the liteatue. As it was aleady mentioned in section 3, its geneal scheme is the same as of PMA and MOGLS and is pesented in figue Figue. he method stoes elatively small numbe of solutions in a standad genetic population. New population eplaces the old one but some elite solutions ae peseved. he solutions ae selected fo ecombination accoding to the oulette wheel scheme, with fitness depending on the cuent scalaizing function. his esults in some pessue towads ecombination of solutions good on the cuent scalaizing function but, in geneal, the solutions ecombined ae wose on this function that in the case of MOGLS and PMA. 4.2. Multiple-obective multiple stat local seach (MOMSLS) he method coesponds to the Geneation of the initial set of solutions phase of PMA and othe hybid genetic algoithms. he method epeatedly daws a scalaizing function and uses a local heuistic to optimize the function stating fom a andom solution. 4.3. Simulated annealing algoithms he fist multiple-obective vesion of simulated annealing has been poposed by Seafini [23]. he method is called Seafini's multiple-obective simulated annealing (SMOSA). he algoithm of the method is almost the same as the algoithm of single obective SA. he method uses a single cuent solution. Seafini consideed a numbe of multiple-obective ules fo acceptance of new solutions. In this expeiment, a new neighbohood solution is accepted with an acceptance ule that may be intepeted as the local use of the weighted linea scalaizing function. In each iteation, the weight vecto used in the acceptance ule is modified andomly. Each weight is allowed to change by no moe than 0%. Ulungu et al. [28] poposed a method called multiple-obective simulated annealing (MOSA). he method uses a numbe of geneating solutions exploing diffeent egions of the nondominated set. With each of the solutions a pedefined weight vecto is associated. he weight vectos do not change duing the un of the method. he acceptance ule of new solutions is the same as in the case of SMOSA. Paeto simulated annealing (PSA) [2] also uses a sample of geneating solutions. Weight vectos associated with the geneating solutions ae modified in each iteation in ode to induce a epulsion mechanism assuing dispesion of the solutions ove all egions of the nondominated set. he method uses the same acceptance ule as the two above algoithms. Summaizing, all simulated annealing algoithms use locally scalaizing functions in acceptance ules, and use diffeent weight vectos, eithe pedefined o updated duing the un of the algoithm, in ode to assue dispesion of the seach ove all egions of the nondominated set. 9
4.4. Genetic algoithms Nondominated soting genetic algoithm (NSGA) [25] is a elatively simple Paeto-anking based multiple-obective GA. Its scheme is vey simila to the single obective GA with geneations eplacements. he only diffeence is in fitness assignment that is based on a anking induced by the dominance elation. In each iteation, NSGA assigns ank to all solutions nondominated in the cuent population. hen, the nondominated solutions ae tempoaily emoved fom the population and the next ank is assigned to the solutions nondominated in the emaining pat of the population. he pocess is continued until all solutions in the population ae anked. he method uses fitness shaing in ode to avoid convegence of all solutions to a small egion of the nondominated set caused by the genetic dift. Contolled elitist non-dominated soting genetic algoithm (CENSGA) [3] is a modification of NSGA that peseves good solutions fom the cuent population. In ode to assue divesification of the population it peseves not only solutions nondominated in the cuent population but also some dominated solutions. CENSGA uses also moe effective way fo ank assignment. In stength Paeto evolutionay algoithm (SPEA) [30] both solutions fom the cuent population and fom the set of potentially Paeto-optimal solutions paticipate in the selection which is also based on Paeto anking. In addition, the method uses a new Paeto-based niching method to peseve divesity of the population. Oiginal vesions of NSGA and CENSGA do not use the extenal set of potentially Paeto-optimal solutions and outcomes of the methods ae the final populations. In ou case, we used, howeve, such set in all methods. he set was updated with each newly geneated solution. 5. Adaptation of the methods to multiple-obective set coveing poblem Metaheuistics ae often used to solve single obective set coveing poblem. Vaious adaptations of metaheuistics to this poblem ae descibed fo example in [], [4], [9] and [2]. hus, we base ou adaptation of multiple-obective metaheuistics to BOSCP on these woks. Note, that although we use bi-obective instances in this expeiment, the adaptation may be used in the case of geneal multiple-obective set coveing poblem. Solutions encoding In ou case, a solution x is encoded as a set of columns, i.e. x C, wheec is the set of all columnsinmatix(a l ) (see section 2). Of couse, this encoding is equivalent to a binay epesentation using I binay vaiables o genes (see e.g. []), but this binay epesentation is not used diectly in ou implementation. Geneation of initial solutions Initial solutions ae constucted in a way simila to that poposed in [4]. At fist, we conside each ow l,,l, stating fom a andomly selected ow. If l is not coveed yet, a column i that coves l is dawn at andom. A column i coves ow l if a l. Solutions obtained in this way may include many edundant columns, i.e. columns that can be emoved without violating feasibility. hus, in the next step the edundant columns ae emoved iteatively until no edundant column emains in the solution. In the case of methods that use scalaizing 0
functions (simulated annealing algoithms, hybid genetic algoithms and MOMSLS), in each iteation, the edundant column with the highest atio of: scalaizing function value impovement caused be emoval of the column the numbe of non zeo elements in the column is emoved. In the case of genetic algoithms, in each iteation, a andomly selected edundant column is emoved. Neighbohood opeato We use a neighbohood opeato that is guided by a scalaizing function. At fist, a andomly selected column is emoved fom the solution. In esult an unfeasible solution is obtained. hen, the solution is epaied in geedy manne by inseting columns with the lowest atio of: scalaizing value decline caused by insetion of the column the numbe of uncoveed ows coveed by the column he column emoved in the fist step is not consideed by the geedy pocedue, thus, the neighbohood opeato always poduces a new solution. Local seach Local seach is guided by a scalaizing function. We use the steepest vesion of local seach, i.e. the whole neighbohood of the cuent solution is tested and the best local move is pefomed. Local seach ends when no impoving move may be found in the neighbohood of the cuent solution. In ode to test the whole neighbohood emoval of each column is consideed in the neighbohood opeato. Recombination opeato We use a ecombination opeato that poduces a single offsping fom two ecombined solutions (paents). he opeato is based on the idea of distance peseving cossove [22], i.e. it peseves elements common to both paents. At fist, we place in the offsping all columns that ae included in both ecombined solutions. he columns that appea in one of the paents only ae placed in the offsping with 50% pobability. his, in geneal, does not guaantees coveing of all ows. hen, all uncoveed ows with andomly selected columns. Afte that, ows ae selected fo mutation with 5% pobability. he mutation consists in dawing a andom column coveing the mutated ow and adding it to the solution. In the case of genetic algoithms, edundant columns ae then emoved in andom ode. In the case of hybid genetic algoithms, local seach assues emoval of edundant columns. 6. Computational expeiment design Evaluation of metaheuistics should involve at least two citeia the quality of obtained esults and unning time. he way we evaluate the quality of esults is descibed in section 7. In ode to measue unning time two main appoaches ae usually used: the numbe of geneated solutions/numbe of function evaluations (see e.g. [30]) o CPU time (see e.g. []). he advantage of CPU time is that it takes into account diffeences in time needed to geneate new solutions by diffeent opeatos and any CPU time oveload intoduced by a given method (e.g. time needed fo selection of solutions fo ecombination). CPU time has also the highest pactical impotance among unning time measues. he disadvantage of CPU time is that it is influenced by compute, pogamming language and implementation style chaacteistics. In ou case, howeve, we wee able to test all methods on the same compute, implemented in the same language and with the use of the same libay. In esult, most lines
of code used by each method wee shaed with othe methods. hus, we decided to use CPU time measue in this expeiment. We used 44 instances of BOSCP geneated by Gandibleux and available at http://www.univ-valenciennes.f/road/mcdm/listmoscp.html. he instances diffe not only by size but also by othe chaacteistics. he instances have fom 0 to 200 ows and fom 00 to 000 columns. hee ae fou sets of the instances. In the case of instances fom set A, costs wee geneated andomly and independently. In the case of instances fom set B, only coefficients of the fist obective wee geneated andomly while the coefficients of the second obectives wee set in the following way: 2 i c I i+ c, i,...,i. In the case of instances fom set C, columns ae divided into seveal subsets. All columns fom one subset have the same coefficients on both obectives. Set D of instances combines chaacteistics of sets B and C. Because of limited space we do not epot all esults, but only esults obtained fo 2 epesentative instances. All esults, as well as all geneated appoximations, ae available at http://www-idss.cs.put.poznan.pl/~aszkiewicz/moscp. able. Chaacteistics of the test instances Instance Numbe of ows Numbe of columns Set 2scp4A A 2scp4B 40 200 B 2scp4C C 2scp4D D 2scp8A A 2scp8B 80 800 B 2scp8C C 2scp8D D 2scp20A A 2scp20B 200 000 B 2scp20C C 2scp20D D On each instance, each method was un 0 times. Each method was allowed to un fo the same time. Since the instances diffe significantly it would not be a good appoach to use the same CPU time fo all instances. hus, at fist, PMA was un ten times with paametes descibed below and aveage unning time of PMA was used as the stopping citeion fo all othe methods. he paametes of all methods wee selected expeimentally on the basis of the best choice pinciple. We did not tune, howeve, the paametes to paticula instances, but set the paametes in ode to obtain a elatively good pefomance on all instances. he numbe of initial solutions was set in the same way fo all hybid genetic algoithms. We used the automatic appoach descibed in section 3. Since, the numbe of initial solutions obtained in this way is not deteministic (although usually is chaacteized by a low dispesion [6]) we decided to use the same numbe of initial solutions in all uns of the thee HGAs. he condition () was tested only once and then the same numbe of initial solutions was used in all othe uns of PMA, MOGLS and IMMOGLS on the same instance. he paamete K 2
was set to 0. he size of tempoay elite population used by MOGLS was set to 5. In the case of PMA, the expected ank E was set to 7.5. PMA was allowed to pefom ten times moe ecombinations than the numbe of initial solutions. In the case of IMOMGLS, the size of genetic population was set equal to the numbe of initial solutions and the elite size was set equal to 0% of the population size. In all multiple-obective genetic algoithms we used populations of size 00 which is quite typical value used in such methods [29]. he neighbohood distance used in NSGA was set to 0.2. he eduction ate in CENSGA was set to 0.8. he nondominated population size of SPEA was set to 00, and clusteing was pefomed when the size of the population exceeded 50. In all simulated annealing algoithms the stating tempeatue was set to 0.05 and final tempeatue to 0.0005. Afte each tempeatue plateau the tempeatue was multiplied 0.9. In esult, 44 tempeatue plateaus wee obtained. o each tempeatue plateau /44 of the aveage unning time of PMA was allocated. he numbe of geneating solutions used in PSA and the numbe of pedefined weight vectos used in MOSA was set to 8. he weights change coefficient used in PSA was set to 0.0003. 7. Quality evaluation As the quality measue of appoximations of the nondominated set we use aveage best value of the weighted chebycheff scalaizing function ove a set of systematically geneated nomalized weight vectos [8], [5], [4]. We use all nomalized weight vectos in which each individual weight takes on one of the following values: { l, l 0 k} k,...,,wheek is a sampling paamete. he set of such weight vectos is denoted by Ψ s and defined mathematically as: Ψ s { Λ [,..., λ ] Ψ λ { 0,, 2,..., k, } λ, J k k k whee Ψ is the set of all nomalized weight vectos. he measue is calculated as:, z A R ( A) (, A Λ) s * z 0, Λ Ψs * 0 0 0 whee s ( z, A, Λ) min{ s ( z, z Λ) } is the best value achieved by function s ( z, z ) appoximation A. Ψ s,,λ In ou expeiment we used sampling paamete k equal to 00, i.e. 0 weight vectos wee used. In the case of 6 instances with up to 40 ows and 400 columns we found exact ideal points optimizing each obective individually using Lingo [9] and Fontline Pemium [0] IP solves. he ideal points wee used as efeence points z 0 in the quality measue. In the case of othe instances, we used uppe appoximations of the ideal points obtained by optimization of individual obectives on elaxed vesion of BOSCP. he elaxed vesion was obtained by allowing vaiables x i to take continuous values fom ange <0,>. Befoe applying the quality measue all obective values wee nomalized using obective anges obseved duing individual optimization of the obectives, i.e. the ange one of the obectives obective was equal to the diffeence between its best value found by optimization of this obective and the value of this obective found by optimization of the othe obective. on 3
he measue achieves the best (minimum) value on the nondominated set. Howeve, the best value may be found without geneating the whole nondominated set if one is able to find the optimum value of the weighted chebycheff scalaizing function fo each weight vecto in set Ψ s, which equies solving poblem (P2). We used to this end the mentioned above two solves. Because of time equiements we wee able to find the best values of the measue only fo 6 instances with up to 40 ows and 400 columns. An inteesting obsevation was that instances fom sets C and D wee moe difficult fo the solves than instances fom sets A and B. he aveage unning time of the solves was about 4 times highe on instances fom sets C and D. We have also noticed that instances fom sets C and D contain much fewe nondominated points (cf. Figue 5 and Figue 6). In the case of othe 28 instances, we have found lowe bounds on the quality measue by solving elaxed vesion of poblem (P2) fo each weight vecto. he elaxed vesion was obtained by allowing vaiables x i to take continuous values fom ange <0,>. 8. Results and discussion able 2 pesents unning times and numbes of solutions geneated by paticula algoithms on the 2 epesentative instances. All calculations wee un on a 400 MHz PC. Note that in the case of HGAs and MOMSLS we count only local optima (we do not count intemediate solutions geneated duing local seach), so, the numbes of geneated solutions ae natually much lowe than in the case of othe algoithms. In the case of simulated annealing, we count accepted solutions only. hus, the esults should athe be used to compae algoithms fom the same class. Results of the expeiment on the 2 epesentative instances ae pesented in able 3 and Figue 4. Detailed numeical esults fo all instances ae available at http://www-idss.cs.put.poznan.pl/~aszkiewicz/moscp. Each chat in Figue 4 coesponds to a diffeent instance. Each chat contains ten esults epesenting the distibution of R fo (fom left to ight) the best possible value o lowe bound, PMA, MOGLS, IMMOGLS, MOMSLS, SPEA, NSGA, CENSGA, PSA and MOSA. Fo each method we pesent aveage value and ± standad deviation ange. Fo each chat the scale is fom the best possible value o lowe bound at the bottom to the highest obseved value at the top. We did not include esults of SMOSA in Figue 4 because aveage esults of SMOSA wee much wose than of othe algoithms with vey high standad deviations. Moe detailed analysis indicated that in most uns SMOSA gave esults compaable to esults of othe simulated annealing algoithm (PSA and MOSA). Howeve, in about 5% of uns of SMOSA the esults wee enomously poo. It indicates that sometimes the single solution used by the algoithm gets tapped in poo egions of the solution space. Othe simulated annealing algoithms ae less vulneable to such situations due to the use of seveal geneating solutions. he esults indicate that instances fom sets C and D ae moe difficult also fo multiple-obective metaheuistics (see section 7). In geneal, the gap between esults of the metaheuistics and the best possible value o lowe bound is highe in case of these instances. 4
able 2. Numbes of geneated solutions and unning times on the test instances Instance 2scp4A 2scp4B 2scp4C 2scp4D Numbe of initial solutions in HGAs 0 0 60 50 Running time [s] 27.5 26.4 3.9 3.4 PMA 20 20 660 550 MOGLS 39 067 566 496 IMMOGLS 695 662 422 402 Numbe of geneated solutions MOMSLS 426 426 30 284 SPEA 39580 39690 2430 20320 NSGA 3630 30000 2270 9820 CENSGA 27220 23340 800 0600 PSA 88749 89307 29774 3894 MOSA 80792 8643 22745 332 SMOSA 66252 2652 38488 46837 Instance 2scp8A 2scp8B 2scp8C 2scp8D Numbe of initial solutions in HGAs 40 50 20 30 Running time [s] 285.4 306.4 25.6 46.2 PMA 540 650 220 330 MOGLS 535 643 78 274 IMMOGLS 842 902 224 278 Numbe of geneated solutions MOMSLS 43 49 46 88 SPEA 89330 209020 5780 34080 NSGA 6070 92660 740 8290 CENSGA 3350 45550 9280 6730 PSA 63388 684540 35223 7235 MOSA 587273 637394 2926 39385 SMOSA 82799 850063 50898 77904 Instance 2scp20A 2scp20B 2scp20C 2scp20D Numbe of initial solutions in HGAs 40 30 70 40 Running time [s] 060.9 937.0 487.8 26.4 PMA 540 430 770 440 MOGLS 57 43 773 429 IMMOGLS 206 72 639 386 Numbe of geneated solutions MOMSLS 677 604 39 20 SPEA 87240 7240 29930 6760 NSGA 7390 70890 32600 24980 CENSGA 3320 7080 2890 2360 PSA 8938 87970 27459 53073 MOSA 799390 73029 28638 40693 SMOSA 3530 079046 327267 95030 aking into account all esults hybid genetic algoithms, especially PMA and MOGLS, ae the best pefomes in the expeiment. he pefomance of the two methods is vey simila. hus, the new faste selection mechanism intoduced in PMA does not deteioate quality with espect to MOGLS. PMA usually geneates slightly moe solutions than MOGLS but in some case the opposite situation is obseved. Since, PMA uses a faste selection mechanism one could expect that PMA should be able to geneate moe solutions than MOGLS. he eduction of selection time has, howeve, vey low influence on the oveall unning time in the case of BOSCP because most time is spent on local optimization. he advantage of PMA 5
selection mechanism should be moe evident if a simple and faste heuistic is used. IMMOGLS pefoms wose than PMA and MOGLS on all instances but 2scp8C and 2scp8D. In most cases, IMMOGLS was able to geneate fewe solutions than PMA and MOGLS in the same time. In ou opinion, this is because IMOMGLS ecombines, in geneal, wose solutions, and thus the offsping ae futhe fom local optima than in the case of PMA and MOMGLS. he esults indicate that selection of vey good solutions fo ecombination is cucial fo the pefomance of the two latte algoithms on BOSCP. MOMSLS pefoms wose than all HGAs. he method was able to geneate much fewe solutions than HGAs, because local seach needs moe time to achieve a local optimum when stated fom an initial solution. he esults show that hybidization of ecombination opeato with local seach is vey beneficial in the case of BOSCP. he pefomance of PSA and MOSA is vey simila and significantly depends on instance size. he methods gave the best esults on 2scp20C and 2scp20D instances, but pefomed pooly on 2scp4C and 2scp4D instances. his is pobably due to the difficulty in finding paametes appopiate fo all instances. We expect that pefomance on smalle instances could be significantly impoved if the paametes, especially stating and final tempeatue, wee tuned fo each instance. hus, although PSA and MOSA ae capable of giving high quality esults, they ae less obust than HGAs. Note that since we use nomalized obective values the above phenomenon cannot be explained by diffeent anges of obectives in diffeent instances. SMOSA always accepted in the same time many moe solutions than PSA and MOSA, and PSA, in most cases, accepted moes solutions than MOSA. his is because, SMOSA and PSA modify the weights used in scalaizing function in each iteation. Futhemoe, SMOSA makes geate changes to the weights in a single iteation. Excluding SMOSA, multiple-obective genetic algoithms ae the wost pefomes in the expeiment. Only on 2scp4C and 2scp4D instances CENSGA gives esults compaable to PMA and MOGLS. In most cases, howeve, the thee GAs give the wost esults. In addition, dispesion of the quality of esults is highe than in the case of othe methods. he simplest algoithm NSGA gives the wost esults among the thee GAs. hus, the extensions to the basic idea of Paeto-anking esulted in impoved pefomance of the newe algoithms, but did not make the new algoithm competitive to othe multiple-obective metaheuistics on BOSCP. Futhemoe, one may note that SPEA pefoms elatively well on instances fom sets A and B, while CENSGA pefoms bette on instances fom sets C and D. Note that CENSGA puts moe emphasis on divesification (see section 4.4) which appaently pays off on moe difficult instances. In most cases, SPEA geneates the highest numbe of solutions among GAs. 6
able 3. Aveage values and ± standad deviation of the quality of esults Instance 2scp4A 2scp4B 2scp4C 2scp4D PMA 0.097 ± 0.000 0.082 ± 0.0008 0.269 ± 0.0089 0.0888 ± 0.0033 MOGLS 0.08 ± 0.009 0.0830 ± 0.000 0.346 ± 0.0070 0.095 ± 0.004 IMMOGLS 0.202 ± 0.0027 0.0906 ± 0.0020 0.50 ± 0.0069 0.067 ± 0.004 MOMSLS 0.273 ± 0.005 0.0975 ± 0.00 0.957 ± 0.0099 0.303 ± 0.02 SPEA 0.35 ± 0.003 0.202 ± 0.00 0.659 ± 0.035 0.0977 ± 0.0098 NSGA 0.737 ± 0.0230 0.369 ± 0.0076 0.849 ± 0.0525 0.28 ± 0.0235 CENSGA 0.342 ± 0.008 0.208 ± 0.0085 0.353 ± 0.006 0.0926 ± 0.0075 PSA 0.96 ± 0.0022 0.0879 ± 0.008 0.4 ± 0.0068 0.040 ± 0.004 MOSA 0.9 ± 0.008 0.0884 ± 0.002 0.529 ± 0.075 0.9 ± 0.0039 SMOSA 0.2457 ± 0.205 0.64 ± 0.0902 0.697 ± 0.44 0.487 ± 0.206 Best possible value 0.05 0.0779 0.073 0.0668 Instance 2scp8A 2scp8B 2scp8C 2scp8D PMA 0.0844 ± 0.0003 0.085 ± 0.0003 0.0205 ± 0.002 0.0400 ± 0.0044 MOGLS 0.0844 ± 0.0002 0.086 ± 0.0006 0.0208 ± 0.0023 0.0486 ± 0.0075 IMMOGLS 0.0903 ± 0.0007 0.093 ± 0.003 0.0209 ± 0.0020 0.0402 ± 0.0052 MOMSLS 0.0958 ± 0.003 0.0977 ± 0.006 0.0264 ± 0.002 0.075 ± 0.0030 SPEA 0.260 ± 0.02 0.74 ± 0.050 0.44 ± 0.0726 0.2050 ± 0.070 NSGA 0.880 ± 0.0243 0.733 ± 0.043 0.2406 ± 0.0666 0.236 ± 0.29 CENSGA 0.525 ± 0.093 0.258 ± 0.004 0.69 ± 0.0696 0.908 ± 0.0334 PSA 0.0844 ± 0.0003 0.0833 ± 0.000 0.0203 ± 0.0006 0.0488 ± 0.0022 MOSA 0.0852 ± 0.0006 0.0860 ± 0.003 0.0207 ± 0.000 0.049 ± 0.0029 SMOSA 0.25 ± 0.2665 0.345 ± 0.399 0.6923 ±.0885 0.6369 ±.2544 Lowe bound 0.0800 0.077 0.00 0.037 Instance 2scp20A 2scp20B 2scp20C 2scp20D PMA 0.083 ± 0.000 0.0659 ± 0.0006 0.649 ± 0.0086 0.4967 ± 0.035 MOGLS 0.082 ± 0.0009 0.0662 ± 0.0006 0.659 ± 0.0047 0.4705 ± 0.058 IMMOGLS 0.0900 ± 0.003 0.0727 ± 0.0008 0.906 ± 0.0047 0.5296 ± 0.0249 MOMSLS 0.094 ± 0.000 0.0764 ± 0.002 0.2245 ± 0.00 0.6944 ± 0.0205 SPEA 0.338 ± 0.0080 0.089 ± 0.022 0.2526 ± 0.068 0.7228 ± 0.0843 NSGA 0.2024 ± 0.025 0.534 ± 0.079 0.27 ± 0.030 0.7590 ± 0.263 CENSGA 0.754 ± 0.036 0.559 ± 0.0209 0.2384 ± 0.062 0.6666 ± 0.0504 PSA 0.083 ± 0.0009 0.0665 ± 0.0004 0.506 ± 0.0044 0.446 ± 0.005 MOSA 0.0833 ± 0.0007 0.0677 ± 0.0007 0.504 ± 0.004 0.4226 ± 0.036 SMOSA 0.399 ± 0.3934 0.22 ± 0.2988 0.249 ± 0.2584.0697 ± 0.8540 Lowe bound 0.0590 0.053 0.0552 0.0789 7
0.8 0.6 0.4 0.2 0.4 0.3 0.2 0. 0.0 0.09 0.23 0.2 0.9 0.7 0.5 0.3 0.5 0.3 0. 0.09 0.0 0.08 0. 0.07 2scp4A 2scp4B 2scp4C 2scp4D 0.20 0.8 0.6 0.4 0.2 0.0 0.8 0.6 0.4 0.2 0.0 0.3 0.26 0.2 0.6 0. 0.06 0.36 0.3 0.26 0.2 0.6 0. 0.06 0.08 0.08 0.0 0.0 2scp8A 2scp8B 2scp8C 2scp8D 0.2 0.7 0.5 0.25 0.6 0.3 0.20 0. 0.5 0. 0.09 0.07 0.0 0.06 0.05 0.05 2scp20A 2scp20B 2scp20C Figue 4. Results of the expeiment in gaphical fom 0.88 0.68 0.48 0.28 0.08 2scp20D In Figues 5-8 some exemplay appoximations fo fou diffeent instances ae pesented (each of them was geneated in the fist un of a given algoithm). We did not include all appoximations in ode to peseve claity of the figues. he points efeed to as exact ae obtained by local optimization of weighted chebycheff scalaizing functions with IP solves (see section 7). Note, that the points may be dominated (see section 2), which appaently happens in many cases. he points efeed to as elaxed ae points obtained by local optimization of weighted chebycheff scalaizing functions on the elaxed vesion of BOSCP with continuous vaiables. One may note, that thee is a vey lage gap between the elaxed points and any of the appoximations in the case of 2scp20D instance. It is not clea whethe this gap is mainly due to the poo pefomance of metaheuistics on this instance o due to the gap between Paeto fonts of the oiginal and elaxed poblems. We have noted, howeve, that in the case of this instance solutions of the elaxed poblem contain many factional values. hus, it is possible that the solutions ae fa fom feasible solutions with binay values. 8
2850 2350 Exact PMA PSA CENSGA 850 350 850 850 050 250 450 650 850 2050 2250 2450 2650 Figue 5. Exemplay appoximations obtained fo 2scp4A instance 750 700 650 Exact PMA PSA CENSGA 600 550 500 450 400 700 900 00 300 500 700 900 Figue 6. Exemplay appoximations obtained fo 2scp4C instance 9
9000 7000 5000 Relaxed PMA PSA CENSGA 3000 000 9000 7000 5000 3000 000 000 3000 5000 7000 9000 000 3000 5000 7000 9000 Figue 7. Exemplay appoximations obtained fo 2scp20B instance 0500 9500 8500 Relaxed PMA PSA CENSGA 7500 6500 5500 4500 4400 4900 5400 5900 6400 6900 7400 7900 8400 Figue 8. Exemplay appoximations obtained fo 2scp20D instance 9. Conclusions en multiple-obective metaheuistics have been compaed on the bi-obective set coveing poblem. he main conclusions of the expeiment ae: instances with epeating values of obectives coefficients ae moe difficult fo both IP solves and metaheuistics, 20
PMA and MOGLS ae the best pefomes in the expeiment, although, they wee not able to outpefom all othe methods on all instances. compaison of PMA and MOGLS with IMMOGLS and MOMSLS indicates that the use of ecombination opeato and ecombination of good solutions is cucial fo the pefomance of the fome two algoithms, MOSA and PSA ae capable of giving high quality esults but ae vey sensitive to paametes setting, the use of a single geneating solution in SMOSA esults in vey high dispesion of the quality of esults, Paeto-anking based genetic algoithms pefom, in geneal, wose than othe algoithms, the new mechanisms intoduced in SPEA and CENSGA impove the esults of basic Paeto-anking based evaluation mechanism used in NSGA, CENSGA pefoms elatively well on small, difficult instances with epeating values of obectives coefficients. Futhemoe, the esults indicate that the pefomance of multiple-obective metaheuistics may diffe adically even if the methods ae based on the same single obective algoithm and use exactly the same poblem-specific opeatos. hus, the elements of the methods that ae specifictomultiple-obectivecasehavesubstantialinfluenceontheipefomance. In the futue we would like to conside moe advanced adaptations of the metaheuistics to the set coveing poblem taking into account the most effective appoaches poposed in single obective case, e.g. [2]. We plan also to extend the expeiments to instances with moe than two obectives and to include othe multiple-obective metaheuistics, e.g. methods based on tabu seach [5] o Paeto-anking based memetic algoithms [8]. 0. Refeences [] Beasly, J.E., Chu, P.C. (996), A Genetic Algoithm fo the Set Coveing Poblem. Euopean Jounal of Opeational Reseach, 94, 392-404. [2] Czyżak P., Jaszkiewicz A. (998), Paeto simulated annealing a metaheuistic technique fo multiple-obective combinatoial optimisation, Jounal of Multi-Citeia Decision Analysis, 7, 34-47. [3] Deb K., Goel. (200), Contolled elitist non-dominated soting genetic algoithms fo bette convegence, in: E. Zitzle, K. Deb, L. hiele, C.A. Coello Coello, D. Cone (eds.) Poceedings of the Fist Intenational Confeence on Evolutionay Multi-Citeion Optimization, Lectue Notes in Compute Science, 993, Spinge, Belin, 67-8. [4] Eemeev, A.V. (999), A genetic algoithm with a non-binay epesentation fo the set coveing poblem, in: Poc. of Opeations Reseach (OR'98). Spinge-Velag, 75-8. [5] Gandibleux X., Féville A. (2000), abu seach based pocedue fo solving the 0- multiobective knapsack poblem: the two obectives case, JounalofHeuistics, 6, 36-383. [6] Gaey M.R., Johnson D.S. (979), Computes and Intactability: A Guide to the heoy of NP-Completeness, W.F. Feeman and Co., San Fancisco. 2
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