ISRN Operatios Research Volume 2013, Article ID 203032, 10 pages http://dx.doi.org/10.1155/2013/203032 Research Article A New Formulatio of the Set Coverig Problem for Metaheuristic Approaches Nehme Bilal, Philippe Galiier, ad Fracois Guibault École Polytechique de Motréal, C.P. 6079, Succ. Cetre-Ville, Motréal, QC, Caada H3C 3A7 Correspodece should be addressed to Nehme Bilal; ehmebilal@gmail.com Received 18 February 2013; Accepted 4 April 2013 Academic Editors: L. Buza, P. Ekel, C. Moha, ad M. Wag Copyright 2013 Nehme Bilal et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Two difficulties arise whe solvig the set coverig problem (SCP) with metaheuristic approaches: solutio ifeasibility ad set redudacy.ithispaper,wefirstpresetareviewadaalysisoftheheuristicapproachesthathavebeeuseditheliteratureto address these difficulties. We the preset a ew formulatio that ca be used to solve the SCP as a ucostraied optimizatio problem ad that elimiates the eed to address the ifeasibility ad set redudacy issues. We show that all local optimums with respect to the ew formulatio ad a 1-flip eighbourhood structure are feasible ad free of redudat sets. I additio, we adapt a existig greedy heuristic for the SCP to the ew formulatio ad compare the adapted heuristic to the origial heuristic usig 88 kow test problems for the SCP. Computatioal results show that the adapted heuristic fids better results tha the origial heuristic o most of the test problems i shorter computatio times. 1. Itroductio The set coverig problem (SCP) is a popular optimizatio problemthathasbeeappliedtoawiderageofidustrial applicatios, icludig schedulig, maufacturig, service plaig, ad locatio problems [1 4]. The SCP is NP hard i the strog sese [5]. The mathematical formulatio of the SCPisasfollows.LetE = {e 1,...,e m } be a uiverse of elemets, ad let S = {s 1,...,s } be a collectio of subsets s j E,where s j =E.Eachsets j covers at least oe elemet of E ad has a associated cost c j >0.Theobjectiveistofid a subcollectio of sets X Sthat covers all of the elemets i E at a miimal cost. The mathematical programmig model of the SCP is usually formulated as follows. (i) Let A m be a zero-oe matrix where a ij =1if elemet i is covered by set j ad a ij =0otherwise. (ii) Let X={x 1,x 2,...,x } where x j =1if set j (with cost c j >0) is part of the solutio ad x j =0otherwise. Miimize c j x j (1) subject to 1 a ij x j, i=1,...,m (2) x j {0, 1}. (3) The objective fuctio (1) drives the search toward solutios at miimal cost. Costrait (2) (full coverage costrait) imposes the requiremet that all the elemets of the uiverse E must be covered. If costrait (2) is ot satisfied, the solutio is ifeasible. If costrait (2) is satisfied ad the objective fuctio is miimized, the solutio will cover all of the elemets at the miimal cost (optimal solutio). If costrait (2) is relaxed, the objective fuctiowill drive the search toward a empty solutio because the empty solutio has the lowest cost (0). These observatios show that the objective fuctio ad the full coverage costrait of the SCP guide the search i two opposite directios. Whe solvig the model with metaheuristic algorithms, two issues arise: solutio ifeasibility ad set redudacy. A solutio to the SCP is cosidered to be ifeasible if oe or more of the elemets of the uiverse E are ucovered. A set is cosidered to be redudat if all the elemets covered by the set are also covered by other sets i the solutio.
2 ISRN Operatios Research I this paper, we first review ad aalyze the literature to highlight the difficulties i dealig with solutio ifeasibility ad set redudacy whe solvig the SCP with metaheuristic algorithms (Sectio 2). We the preset a ew formulatio thatcabeusedtosolvethescpasaucostraiedoptimizatio problem ad that elimiates the eed for addressig the ifeasibility ad redudacy issues (Sectio 3). The ew formulatio uses a maximizatio objective that ca replace both the cost miimizatio objective ad the full coverage costrait of the classical formulatio. The ew formulatio ca also be see as a ew pealty approach that has may advatages over the existig pealty approaches (Sectio 3.2) for the SCP. Third, we preset a simple descet heuristic that is based o the ew formulatio ad that uses a simple 1-flip eighbourhood structure. The proposed descet heuristic is a adaptatio of a existig greedy heuristic for thescp.weshowthatalllocaloptimumswithrespectto the ew formulatio ad the 1-flip eighbourhood structure are feasible ad free of redudat sets. Fially, the proposed descet heuristic is compared to the origial greedy heuristic usig 88 kow set coverig problems (Sectio 5). 2. Literature Review I geeral, metaheuristic algorithms ca be divided ito three categories. (i) Costructive metaheuristics: i each iteratio, a ew local optimum is foud by costructig a ew solutio from scratch. A level of radomess is added to the costructio step i order to avoid costructig the same solutio over ad over. (ii) Evolutioary algorithms: i each iteratio, two or more solutios are combied to create a ew solutio. (iii) Local search: i each iteratio, the curret solutio is replaced by oe of its immediate eighbors (the solutio is usually modified slightly). I the followig sectios, we review the literature of solvig the SCP with metaheuristic approaches ad aalyze how each category of metaheuristics addresses solutio ifeasibility ad setredudacy. 2.1. Costructive Metaheuristics. Whe the SCP is solved with costructive metaheuristics, the local optimums foud at the ed of each costructive iteratio are usually feasible. I fact, the costructive iteratio eds whe all of the elemets are covered. For this reaso, these metaheuristics do ot have to deal with the ifeasibility issue. However, the local optimums are ot ecessarily free of redudat sets, ad a redudacy removal heuristic is eeded. Costructive metaheuristics for the SCP icludes at coloy optimizatio [6 10], Meta-RaPS [11], ad GRASP [12]. All of these metaheuristics use a dedicated redudacy removal operator that removes redudat sets at the ed of each iteratio. 2.2. Evolutioary Algorithms. Evolutioary algorithms for the SCP eed to address both ifeasibility ad set redudacy issues. Most evolutioary algorithms that are used to solve the SCP are based o the geetic algorithm (GA). Most of the GAs use a biary strig solutio represetatio where x j = 1 if the set s j is part of the solutio ad x j = 0 otherwise. The ifeasibility issue arises whe the crossover or mutatio operator of the GA produces a child (solutio) that does ot cover all of the elemets. I fact, a simple bit flip from 1 to 0 durig crossover or mutatio ca produce a ifeasible solutio. If a cost miimizatio objective fuctio is used, ifeasible solutios will be preferred over feasible oes because ifeasible solutios are usually cheaper. Two mai approaches have bee used i the literature to address the ifeasibility issue. The first approach uses a repair heuristic to trasform ifeasible solutios to feasible solutios before the evaluatio step of the GA. A greedy-like repair heuristic is usually used [13 15]. I each iteratio, the repair operator covers a ucovered elemet by selectig a ew set that covers theelemetadaddigittothesolutio.i[15], all of the solutios are repaired for evaluatio, but oly 5% of them are replaced with the correspodig repaired versios. The aim is to allow the search to explore ifeasible regios of the search space, which ted to be more effective tha limitig the search to oly feasible regios. A simpler repair heuristic is used i [16]. Durig the evaluatio of a solutio, a set is added to the solutio if it covers a ucovered elemet(s) ad is ot already part of the solutio. By addig ew sets, repair heuristics may itroduce redudat sets ito the solutios. Forthisreaso,geeticalgorithmsthatusearepairoperator also use a redudacy removal procedure that is applied after the repair ad just before evaluatio. The secod approach ivolves pealizig the objective valueofifeasiblesolutiostodrivethesearchtowardthe feasible regio. A pealty term that makes ifeasible solutios less attractive tha feasible oes is added to the objective fuctio. I [17], the same pealty M is added to the objective value of all ifeasible solutios. M is high eough to guaratee that all feasible solutios have lower objective values tha all ifeasible solutios (M =c j ). A drawback of usig such a objective fuctio is that ifeasible solutios caot be compared to each other because the objective fuctio does ot reflect the degree of ifeasibility. Objective fuctiosthatpealizeifeasiblesolutioswhilereflectig thedegreeofifeasibilityareproposedi[16, 18]. I [18], the pealty attributed to a ifeasible solutio is proportioal to the umber of elemets that are ot covered i the solutio. I [16], the pealty is proportioal to the miimum cost it wouldtaketocoveralloftheucoveredelemets.iall discussed pealty approaches, the pealties are high eough to esure that all ifeasible solutios have higher objective values tha all feasible oes. A immediate disadvatage of usig such high pealties is that feasible solutios will always be preferred over ifeasible oes. As a result, ifeasible solutioswillhavelowchacesofsurvivigithepopulatio, ad the ifeasible regio of the search space will ot be effectively explored. 2.3. Local Search. The feasibility costrait makes desigig a effective local search metaheuristic for the SCP a difficult
ISRN Operatios Research 3 task. For this reaso, few-local-search oly heuristics have bee developed for the SCP [8, 19]. Istead, most of the local search algorithms have combied local search with other techiques such as Lagragia relaxatio, subgradiet optimizatio, group theory, ad liear programmig [1, 20 24]. I [25], after otig the difficulty of defiig a good eighbourhood to solve the uicost set coverig problem with local search, the authors proposed that the problem could be trasformed to a equivalet satisfiability problem (SAT) that cabesolvedmoreadequatelywithlocalsearch. Most local search algorithms for the SCP use a simple 1-flip eighbourhood structure defied by moves that oly add (remove) oe set at a time to (from) the solutio. Whe a local optimum is reached, which is usually a feasible solutio, itisdifficulttodecideiwhichdirectiotocotiuethe search. Two cases arise. (i) If the search space is restricted to the feasible regio, oly redudat sets are allowed to be removed. If o redudat sets exist i the solutio, at least oe redudat set must be added before a remove move isallowedtobeperformed.asaresult,theifeasible regio of the search space will ot be explored ad the search will ted to fall ito local optimums ad cycles. A more complex eighbourhood called 3-flip is used i [8] tomakethesearchithefeasible regio more effective. The 3-flip eighbourhood of a give solutio cosists of all of the solutios that ca be obtaied by addig (removig) at most 3 sets to (from) the solutio. Evethough the proposed heuristicismoreeffectivethaasimple1-flipheuristic,itis ot sufficiet to avoid local optimums ad cycles ad it is sigificatly slower tha the 1-flip heuristics. (ii) If the search space is ot restricted to the feasible regio, the cost miimizatio objective drives the search toward the ifeasible regio, by removig sets from the curret cofiguratio (to miimize the cost), ad it is uclear whe to restore feasibility. I such situatios, pealty approaches are usually used to pealize ifeasible solutios. If the pealty weights are too high, eighbors i the feasibleregiowillbepreferredovereighborsitheifeasible regio, makig the ifeasible regio ureachable. Lower or dyamicpealtyweightsareusuallyusedtomakethesearch more effective by allowig it to reach ifeasible regios. If the pealty weights are too low, the fial solutio foud is ot guarateed to be feasible. A tabu search heuristic that uses such low pealties is proposed i [19]fortheuicostset coverig problem. A simple 1-flip eighbourhood structure is used. The objective is to miimize (C + E) where C is the umber of sets used i the solutio ad E is the umber of ucovered elemets. If a set covers oly oe ucovered elemet, addig (removig) it to (from) the solutio will ot have ay effect o the objective fuctio. As a result, this set might be left out of the solutio, makig it ifeasible. To overcome the fact that this objective fuctio does ot guaratee feasibility, the eighbourhood is restricted such that if a set is removed durig oe iteratio, oe or more sets must be added i the ext iteratio to restore feasibility. Evethoughsuchalowpealtyapproachallowsthesearch to reach the ifeasible regio, additioal eighbourhood restrictios are used to restore feasibility, ad the ifeasible regio is oly scratched. Dyamic pealty approaches, i which the pealty weights are repeatedly adjusted, are used to balace the search betwee the feasible ad ifeasible regios without usig a repair operator or eighbourhood restrictios [1, 20 22, 26]. The most frequet dyamic pealty approaches that have bee used i the literature are based o Lagragia relaxatio [27] ad subgradiet optimizatio [28]. Dyamic pealty approaches ca be very effective but are difficult to be desiged ad implemeted. 3. Proposed Formulatio I this work, we propose a ew formulatio of the SCP with a maximizatio objective. The aim of the proposed formulatio is to express the real objective of the SCP i the objective fuctio which is to cover all elemets at a miimal cost. We view coverig a elemet as collectig a gai at a give cost. I this perspective, we attribute a gai to each elemet. Because all elemets must be covered, the gai attributed to eachelemetmustbehigherthathecostofatleastoe of the sets that covers the elemet; otherwise, there is o beefit of coverig that elemet. Let c mi (e i ) be the cost of the cheapest set amog the sets that cover the elemet e i.agai g i =c mi (e i )+εis attributed to each elemet e i where ε is a small positive costat. (i) Let A m beazero-oematrixwherea ij =1if elemet e i is covered by set j ad a ij =0otherwise. (ii) Let X={x 1,x 2,...,x } where x j =1if set s j (with cost c j > 0) is part of the solutio ad x j = 0 otherwise. (iii) Let Y={y 1,y 2,...,y m } where y i =1if elemet e i (with gai g i >0) is covered i the solutio ad y i =0 otherwise. Maximize subject to y i m i=1 g i y i c j x j (4) a ij x j, i=1,...,m (5) x j,y i {0, 1}. (6) Costrait (5) is a relaxatio of costrait (2) because it does ot impose coverage of all the elemets; its oly purpose is to keep track of which elemets of E are part of the cover. Costrait (6) is the itegrity costrait i mathematical programmig. Costraits (5) ad(6) do ot eed to be addressed as costraits i heuristic approaches but are preseted for completeess of the mathematical programmig formulatio.
4 ISRN Operatios Research Claim 1. The optimal solutio of the proposed formulatio is a feasible solutio (covers all elemets). Proof. Suppose that the optimal solutio does ot cover all of the elemets ad has a objective value P. Lete i be a ucovered elemet. By the defiitio of the gai g i,wekow that there is at least a set s j that covers elemet e i ad has acostc j = c mi (e i ) = g i ε.ifthesets j is added to the cover,theewobjectivevalueisp = P + (g i c j ) = P+(g i (g i +ε))=p+ε>p.thus,pis ot optimal. By cotradictio, we coclude that the optimal solutio covers all of the elemets. Claim 2. The optimal solutio of the proposed formulatio covers all elemets at a miimal cost. Proof. We proved i Claim 1 that the optimal solutio covers all of the elemets. Hece, the first term of the objective fuctio (4) is a costat i the optimal solutio ( m i=1 g iy i = K). The objective fuctio becomes Maximize (K c j x j ) Maximize ( Miimize c j x j. c j x j ) Thus, the optimal solutio of the proposed formulatio is the cheapest feasible solutio, which is the objective of the SCP. From heuristic algorithms perspective, we replaced a costraied optimizatio problem with a ucostraied optimizatio problem that has the same optimal solutios. Ucostraied optimizatio problems are kow to be much easier to solve with heuristic algorithms tha costraied optimizatio problems. 3.1. Compariso to Pealty Approaches. Evethough the proposed formulatio is a full mathematical programmig formulatio for the SCP, it is similar to the existig pealty approaches but with some importat differeces. The objective fuctio preseted i (4)caberewritteas Miimize c j x j + m i=1 (7) g i y i, (8) where y i =1if elemet e i is ucovered ad y i =0otherwise. The value of the gai g i cabeseeasthepealtyassociated with ot coverig the elemet e i. The proposed approach is differet from high-pealty approaches because some ifeasible solutios might have a better objective value tha some feasible oes. For istace, let U={s 1,s 2,s 3 }, s 1 ={e 1,e 2,e 3 }, s 2 ={e 1,e 2 },ads 3 ={e 3 }. The costs of the sets are c 1 = 10, c 2 = 2,adc 3 = 1.The cheapest set that covers the elemet e 1 is s 2 with a cost c 2 =2. Thus, by defiitio of the gai, g 1 is equal to c 2 +ε=2+ε. Similarly, we fid that g 2 =2+εad g 3 =1+ε.LetX 1 ={s 1 } beafeasiblesolutio,adletx 2 ={s 2 } be a ifeasible oe. Usig the objective fuctio (8),theobjectivevalueofX 1 is 10 ad the objective value of X 2 is (c 2 +e 3 )=(3+ε). Thus, theifeasiblesolutiox 2 has a lower (better) objective value tha the feasible solutio X 1, which does ot occur with highpealty approaches. The proposed approach is differet from low-pealty approaches because the pealties are high eough to drive the search toward the feasible regio. We showed that the optimal solutios with respect to the ew formulatio are guarateed to be feasible. The proof of feasibility of the optimal solutio also shows that ay ifeasible solutio ca be trasformed to a feasible oe with a better objective value. For istace, i the previous example, the ifeasible solutio X 2 ca be trasformed to a feasible solutio X 3 = {s 2,s 3 } (by addig the set s 3 to X 2 )withaobjectivevalueof3, which is lower (better) tha the objective value of X 2 (3+ε). The proposed pealty approach is differet from dyamic pealty approaches because the pealty weights are static ad o adjustmet is eeded. Whe high-pealty approaches are used, the search process of a heuristic algorithm is disturbed by the high pealties ad drive immediately to the feasible regio. O the other had, low pealties do ot disturb the search but caot esure feasibility. The aim of our approach is to choose the lowest possible pealties that avoid disturbig the search process while esurig feasibility. Esurig feasibility meas that ay ifeasible solutio ca be trasformed to a feasible oewithabetterobjectivevalue. 3.2. Beefits of the New Formulatio with respect to Metaheuristics. The ew formulatio elimiates all issues related to solutio ifeasibility ad set redudacy that were discussed i the literature review (Sectio 2). Because the objective fuctio aturally pealizes redudat sets, the use of a redudacy removal operator is ot eeded. The objective fuctio also pealizes ifeasible solutios. As a result, the use of a repair or pealty approaches i evolutioary algorithms ad the use of eighbourhood restrictios i local search algorithms are ot eeded. Fially, because o costraits are ivolved ad the oly driver of the search is the objective fuctio proposed with the ew formulatio, desigig a good eighbourhood ad local search algorithm is quite simple. Such a simple eighbourhood is preseted i Sectio 4. 4. Proposed Descet Heuristic (DH) I this sectio, we preset a simple descet heuristic that is based o the ew formulatio ad that uses a 1-flip eighbourhoodstructure.wealsoshowthatalllocaloptimumswithrespecttotheewformulatioadthe1-flip eighbourhood are feasible ad free of redudat sets. Theproposeddescetheuristic(DH)isaadaptatio of the classical greedy heuristic that has bee used i the literature for the SCP [29]. I this greedy heuristic, the set s j with the miimum ratio η j =c j /card j (X) is added to the solutio i each iteratio. The term card j (X) is the umber
ISRN Operatios Research 5 sol empty solutio; loop fid the set s j with the maximum ratio R j ; if (R j >0) the flip bit x j ; else stop; ed if ed loop Algorithm 1: DH(). of elemets that are covered by s j ad are ot covered by the curret cofiguratio X.Ocealloftheelemetsarecovered, redudat sets are removed i decreasig order of cost. I DH, the term card j (X) of the classical greedy heuristic is replaced with δ j,whereδ j is the variatio i the objective fuctio associated with addig (removig) the set s j. DH starts from a give cofiguratio ad performs a sequeceofmovesoitutilthesolutioislocallyoptimal. It uses a simple 1-flip eighbourhood structure with two types of moves: add ad remove moves. add(j) addsthe set s j to the cofiguratio (flips x j from 0 to 1), while remove(j) removes the set s j from the cofiguratio. I each iteratio, the set s j with the maximum ratio R j = δ j /c j is added (removed) to (from) the solutio. The algorithm stops whe the curret cofiguratio is better tha all of its eighbors (R j 0for all j). The outlie of DH is preseted i Algorithm 1. 4.1. Redudacy Removal. I cotrast to the classical greedy heuristic, DH automatically removes the redudat sets from the solutio. Let X be a cofiguratio where the set s j is redudat. The ratio R j associated with removig s j from X is equal to (c j 0)/c j =1.BecauseR j >0, the move remove(j) will be performed ad the redudat sets will be removed. As a result, ay solutio that is improved with DH is ecessarily free of redudat sets. The redudat sets are removed at ay time durig the progress of DH ad ot oly at the ed. 4.2. Feasibility. Cosider X to be a cofiguratio where e i is ot covered. Let s i mi be the cheapest set that covers e i,ad let c i mi be its associated cost. The gai g i associated with e i is equal to c i mi +ε.ife i is the oly ucovered elemet covered by s i mi (worst case sceario), the ratio Ri mi associated with addig the set s i mi to X is equal to (ci mi +ε ci mi )/ci mi = ε/c i mi.becauseri mi >0(for all ε>0), the move add(si mi ) will be performed ad the solutio will be feasible. As a result, aysolutiothatisimprovedwithdhisfeasible. 4.3. Discussio. We showed that all of the solutios that are foud with DH are feasible ad free of redudat sets. With respect to the ew formulatio ad the 1-flip eighbourhood structure,thesesolutiosarelocaloptimums.thisisalso true for all solutios obtaied with ay descet heuristic that is based o the ew formulatio ad that uses the same eighbourhood structure. As a result, all local optimums with respect to the ew formulatio ad the 1-flip eighbourhood structure are feasible ad free of redudat sets. 5. Experimetal Aalysis I this sectio, we preset computatioal experimets with theproposeddescetheuristicthatisbasedotheewformulatio. Although we showed i the previous sectios that the ew formulatio provides may advatages over the classical formulatio, the fial performace of ay metaheuristic algorithm depeds o the implemetatio, the tuig of the parameters, ad the sophisticatio of the approach. We do ot assume that ay metaheuristic approach that is based o the ew formulatio will outperform all metaheuristic approaches that are based o the classical formulatio. I additio, experimetig with all classes of metaheuristics will ot prove (or disprove) the superiority of the proposed formulatio. Istead, we compare our descet heuristic to the origial greedy heuristic that is based o the classical formulatio.theaimistocomparethetwoformulatios usig similar algorithms. Sice greedy heuristics are used foritesificatioimostofthemetaheuristicapproaches for the SCP, evaluatig the effectiveess of a ew descet heuristic that ca replace these greedy heuristics provides a good idicatio of how suitable is the ew formulatio to metaheuristic approaches. We compare DH to the classical greedy heuristic (GH) [29] o three classes of the kow set coverig problems. (i) OR-Library bechmarks: this class icludes 65 small ad medium size radomly geerated problems that were frequetly used i the literature. Most metaheuristic approaches for the SCP have bee tested o these problems. They are available i OR-Library [30] ad are described i Table 1. (ii) Airlie ad bus schedulig problems: this class icludes fourtee real-world airlie schedulig problems (AA istaces) ad two bus driver schedulig problems (bus istaces). These problems were obtaied from [31] ad are described i Table 2. (iii) Railway schedulig problems: this class icludes seve large-scale railway crew schedulig problems from Italia railways ad are available i OR-Library [30]. These problems are described i Table 3. Most metaheuristic approaches for the SCP have bee exclusively tested o OR-Library bechmarks. Because these bechmarks are relatively small, we experimeted with larger problems that have bee less frequetly used i the literature. I all preseted tables, the ame of each istace is give i the first colum, the size of each istace is give i the secod colum (umber of elemets umber of sets), ad the desity of each istace is give i the third colum. The desity is the percetage of oes i the A m matrix described i Sectio 1). The optimal or best-kow solutio of each istace is give i the fourth colum. The solutios obtaied with each heuristic are preseted i colums 5 ad 6. The last two colums cotai the umber of iteratios performed by
6 ISRN Operatios Research Table 1: OR-Library bechmarks. Characteristics Cost Number of moves Istace Size Desity Best kow DH GH DH GH 4.1 200 1000 2% 429 433 434 77 93 4.2 200 1000 2% 512 523 552 75 94 4.3 200 1000 2% 516 531 546 79 96 4.4 200 1000 2% 494 503 507 70 93 4.5 200 1000 2% 512 515 518 72 95 4.6 200 1000 2% 560 575 597 78 83 4.7 200 1000 2% 430 444 449 74 77 4.8 200 1000 2% 492 493 525 70 77 4.9 200 1000 2% 641 672 672 82 99 4.10 200 1000 2% 514 519 528 71 86 5.1 200 2000 2% 253 265 273 76 88 5.2 200 2000 2% 302 314 335 71 82 5.3 200 2000 2% 226 230 230 66 82 5.4 200 2000 2% 242 246 254 69 86 5.5 200 2000 2% 211 214 215 73 87 5.6 200 2000 2% 213 216 227 69 85 5.7 200 2000 2% 293 297 305 76 84 5.8 200 2000 2% 288 297 304 77 85 5.9 200 2000 2% 279 281 290 68 84 5.10 200 2000 2% 265 271 274 74 81 6.1 200 1000 5% 138 149 143 39 56 6.2 200 1000 5% 146 156 154 44 53 6.3 200 1000 5% 145 149 157 43 46 6.4 200 1000 5% 131 134 140 46 51 6.5 200 1000 5% 161 180 182 47 50 A.1 300 3000 2% 253 258 269 82 97 A.2 300 3000 2% 252 262 268 78 93 A.3 300 3000 2% 232 243 248 80 105 A.4 300 3000 2% 234 240 243 84 107 A.5 300 3000 2% 236 240 246 79 107 B.1 300 3000 5% 69 72 71 41 45 B.2 300 3000 5% 76 79 78 44 50 B.3 300 3000 5% 80 84 84 47 46 B.4 300 3000 5% 79 84 88 44 50 B.5 300 3000 5% 72 72 75 46 48 C.1 400 4000 2% 227 237 252 102 110 C.2 400 4000 2% 219 230 225 93 128 C.3 400 4000 2% 243 249 258 89 102 C.4 400 4000 2% 219 229 239 94 115 C.5 400 4000 2% 215 222 222 93 106 D.1 400 4000 5% 60 64 66 49 54 D.2 400 4000 5% 66 68 69 52 50 D.3 400 4000 5% 72 77 80 54 59 D.4 400 4000 5% 62 62 66 52 54 D.5 400 4000 5% 61 65 67 49 61
ISRN Operatios Research 7 Table 1: Cotiued. Characteristics Cost Number of moves Istace Size Desity Best kow DH GH DH GH E.1 500 5000 10% 29 30 30 30 35 E.2 500 5000 10% 30 33 35 31 37 E.3 500 5000 10% 27 29 31 29 31 E.4 500 5000 10% 28 32 31 32 33 E.5 500 5000 10% 28 30 30 30 32 F.1 500 5000 20% 14 16 17 16 17 F.2 500 5000 20% 15 16 16 15 16 F.3 500 5000 20% 14 17 15 17 17 F.4 500 5000 20% 14 17 15 17 14 F.5 500 5000 20% 13 15 15 17 15 G.1 1000 10000 2% 176 186 191 132 146 G.2 1000 10000 2% 154 166 176 115 139 G.3 1000 10000 2% 166 178 182 126 147 G.4 1000 10000 2% 168 178 179 128 138 G.5 1000 10000 2% 168 179 182 127 131 H.1 1000 10000 5% 63 69 69 68 65 H.2 1000 10000 5% 63 70 72 62 67 H.3 1000 10000 5% 59 63 66 62 62 H.4 1000 10000 5% 58 65 64 65 61 H.5 1000 10000 5% 55 60 61 61 60 Table 2: Airlie ad bus driver crew schedulig problems. Characteristics Cost Number of moves Istace Size Desity Best kow DH GH DH GH AA03 106 8661 4.05% 33155 34637 35642 48 61 AA04 106 8002 4.05% 34573 36153 36749 45 62 AA05 105 7435 4.05% 31623 32249 32995 45 65 AA06 105 6951 4.11% 37464 38043 39422 43 70 AA11 271 4413 2.53% 35478 36965 39054 76 90 AA12 272 4208 2.52% 30815 33663 34044 77 85 AA13 265 4025 2.60% 33211 36337 37345 77 91 AA14 266 3868 2.50% 33219 36048 36530 77 95 AA15 267 3701 2.58% 34409 36269 37996 73 94 AA16 265 3558 2.63% 32752 36185 37160 79 85 AA17 264 3425 2.61% 31612 34326 36484 69 91 AA18 271 3314 2.55% 36782 39594 40603 84 101 AA19 263 3202 2.63% 32317 34749 36093 71 92 AA20 269 3095 2.58% 34912 37047 37744 82 86 BUS1 454 2241 1.89% 27947 28871 29673 88 100 BUS2 681 9524 0.51% 67760 69685 70606 282 280 each heuristic for each istace. The percetage deviatios from the best-kow solutios are preseted i Figures 1, 2, 3 ad 4. I both DH ad GH, each iteratio ivolves fidig the best set to be added (removed) to (from) the solutio ad updatig the uderlyig data structure after a move is performed. Thus, the algorithmic complexity of each iteratio is similar i both heuristics. I practice, the computatio times are highly depedet o the implemetatio ad the characteristics of the problem solved (size ad desity). For istace, fidig the best move to be performed i each iteratio ca be implemeted usig a loop that iterates over all sets
8 ISRN Operatios Research Table 3: Railway crew schedulig problems. Characteristics Cost Number of moves Istace Size Desity Best kow DH GH DH GH RAIL507 507 63009 1.2% 174 205 212 150 169 RAIL516 516 47311 1.3% 182 202 202 181 186 RAIL582 582 55515 1.2% 211 243 251 191 212 RAIL2586 2586 920683 0.4% 948 1102 1185 770 917 RAIL2536 2536 1081841 0.4% 691 828 891 581 660 RAIL4284 4284 1092610 0.2% 1065 1303 1385 997 1091 RAIL4872 4872 968672 0.2% 1534 1802 1900 1339 1521 σ (%) 14 12 10 8 6 4 2 0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 DH GH 4.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.1 6.1 6.2 6.3 6.4 6.5 Figure 1: Percetage deviatio from the best-kow solutio: OR- Library bechmarks 4.1 to 6.5. performed fewer iteratios tha GH for 56 istaces, equal umber of iteratios for seve istaces, ad more iteratios for oly two istaces. For the airlie, bus, ad railway schedulig problems, DH performed fewer iteratios tha GH for all problems except oe (more iteratios for BUS2). The average umber of iteratios performed by DH ad GH is preseted i Table 4. The average umber of iteratios shows that the umber of iteratios performed by DH is sigificatly smaller tha the umber of iteratios performed by GH. Thus, DH is theoretically faster tha GH. As a result, the proposed descet heuristic that is based o the ew formulatio performs better tha the correspodig greedyheuristicthatisbasedotheclassicalformulatio by fidig better results for most of the test problems usig fewer iteratios, which ca lead to shorter computatio times. 6. Coclusios ad Future Work or usig a priority-queue-based data structure. Prelimiary testig showed that choosig oe way or aother greatly affects the speed compariso of the discussed heuristics. To avoid a implemetatio-depedet compariso, ad because these aspects of the implemetatio are out of the scopeofthiswork,werecordedtheumberofiteratios istead. Both heuristics are determiistic, ad oly oe ru is required. The value of ε used i all DH rus is equal to 1 e 5. Smaller values of epsilo have caused umerical problems for some istaces. Our descet heuristic performed better tha GH by fidig better solutios for most of the test problems. For OR-Library bechmarks, DH foud better solutios tha GH for 47 istaces, equal solutios for 10 istaces, ad worse solutios for 9 istaces. For the airlie, bus, ad railway schedulig problems, DH foud better solutios tha GH for all problems except oe (equal solutios for RAIL516). The percetage deviatios preseted i Figures 1, 2, 3 ad 4 ad the average percetage deviatio preseted i Table 4 show that the solutios foud by DH are also sigificatly better i quality tha those foud by GH (up to 7.41% better for OR- Library, up to 6.83% better for airlie ad bus problems, ad up to 9.12% better for railway problems). DH also performed fewer iteratios tha GH for most of the test problems. For OR-Library bechmarks, DH I this paper, we idetified two issues that arise whe solvig the SCP with metaheuristic approaches: solutio ifeasibility ad set redudacy. We highlighted the difficulties of addressig these issues whe solvig the SCP with the differet classes of metaheuristics ad proposed a ew formulatio that overcomes these difficulties. We showed that this formulatio is, i fact, a ew pealty approach that uses static pealty weights that are low eough to avoid disturbig the search but high eough to esure the feasibility of the fial solutio. We also showed that all local optimums with respect to the ew formulatio ad the 1-flip eighbourhood structure are feasible ad free of redudat sets. As a result, buildig metaheuristic approaches for the SCP usig the ew formulatio is straightforward. To provide a first computatioal experiece usig the ew formulatio, we adapted a kow greedy heuristic for the SCP to the ew formulatio ad compared the adapted versio to the origial versio usig 88 set coverig problems. The adapted versio that is based o the ew formulatio foud better solutios tha the origial versio that is based o the classical formulatio for 69 tests problems, equal solutios for te problems, ad worse solutios for ie problems. I additio, the adapted versio performed fewer iteratios tha the origial versio for 78 test problems, equal umber of iteratios for two problems, ad more iteratios for eight problems. Thus the adapted versio fids better
ISRN Operatios Research 9 σ (%) 12 10 8 6 4 2 σ (%) 25 20 15 10 5 0 A.1 A.2 A.3 A.4 A.5 B.1 B.2 B.3 B.4 B.5 C.1 C.2 C.3 C.4 C.5 D.1 D.2 D.3 D.4 D.5 0 E.1 E.2 E.3 E.4 E.5 F.1 F.2 F.3 F.4 F.5 G.1 G.2 G.3 G.4 G.5 H.1 H.2 H.3 H.4 H.5 DH GH (a) DH GH (b) Figure 2: Percetage deviatio from the best-kow solutio: OR-Library bechmarks A.1 to H.5. Table 4: Average umber of iteratios ad percetage deviatios. Problems Average umber of iteratios Average percetage deviatio DH GH DH GH OR-Library bechmarks 64.89 74.51 5.46 7.31 Airlie ad bus problems 82.25 96.75 5.99 9.14 Railway problems 601.29 679.43 17.12 22.81 σ (%) 18 16 14 12 10 8 6 4 2 0 AA03 AA04 AA05 AA06 AA11 AA12 AA13 AA14 AA15 AA16 AA17 AA18 AA19 AA20 BUS1 BUS2 DH GH Figure 3: Percetage deviatio from the best-kow solutio: airlie ad bus schedulig problems. σ (%) 35 30 25 20 15 10 5 0 RAIL507 DH GH RAIL516 RAIL582 Figure 4: Percetage deviatio from the best-kow solutio: railway schedulig problems. RAIL2586 RAIL2536 RAIL4284 RAIL4827 solutios tha the origial versio i potetially shorter computatio times. Moreover, the adapted versio was easier to implemet because we did ot eed to hadle feasibility ad set redudacy. Most curret metaheuristic approaches for the SCP icorporate a descet or greedy heuristic that is resposible for the itesificatio part of the search. Thus, havig a more effective descet heuristic ca lead to better metaheuristic approaches. Refereces [1] A. Caprara, M. Fischetti, ad P. Toth, Heuristic method for the set coverig problem, Operatios Research, vol. 47, o. 5, pp. 730 743, 1999. [2] G.La,G.W.DePuy,adG.E.Whitehouse, Aeffectivead simple heuristic for the set coverig problem, Europea Joural of Operatioal Research,vol.176,o.3,pp.1387 1403,2007.
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