AN IMPROVED GENETIC ALGORITHM FOR RECTANGLES CUTTING & PACKING PROBLEM. Wang Shoukun, Wang Jingchun, Jin Yihui

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1 Copyrght 2002 IFAC 5th Trennal World Congress, Barcelona, Span A IPROVED GEETIC ALGORITH FOR RECTAGLES CUTTIG & PACKIG PROBLE Wang Shouun, Wang Jngchun, Jn Yhu Tsnghua Unversty, Beng 00084, P. R. Chna Department of Automaton, Tsnghua Unversty, Beng 00084, P. R. Chna Abstract: After revewng the bacground and descrpton of cuttng & pacng problem and pacng algorthms, ths paper proposed an mproved genetc algorthm for non-gullotne rectangles pacng problem. Frst, self-adaptve mutaton probablty s adopted to avod pre-maturty, some commonly used crossover operators are compared, and a stochastc hll-clmbng operator s desgned to mprove the local searchng ablty. Second, the greedy mechansm s ntroduced to acqure good pacng pattern after pacng order s determned by the GA. Computatonal results show the effectveness of these mprovements. Copyrght 2002 IFAC Keywords: Genetc algorthm, Random search, Heurstc search, Global optmzaton, Pacng Problem.. ITRODUCTIO The pacng problems have been wdely studed durng the last three decades, as they are often faced n ndustres such as metal, cloth, and papermang. The rectangular peces pacng problem, cut also from rectangular stoc, s one partcular case of ths set of problems. The am s often to acheve the mnmum trm loss. Pacng problems have been proven as P complete problems n many lteratures. Snce the geometrcal characterstcs of pacng obects should also be taen nto consderaton, ther computatonal algorthms are more complex to desgn. Based on ther descrptons and solutons, Dychoff (990) classfed the general stocng cuttng and pacng problems nto fve categores n hs classcal lterature: - Cuttng stoc and trm loss; - Bn pacng, strp pacng, and napsac; - Vehcle loadng, pallet loadng, and contaner loadng; - Assortment, depleton, desgn, dvdng, layout; - Captal budgetng, memory allocaton, and mult-processor schedulng. Generally, the stoc cuttng problems can be dvded nto regular pacng problems and rregular pacng problems by ther shape, also nto one-dmensonal, two-dmensonal, three-dmensonal and mult-dmensonal pacng problems by ther range of the soluton space. The earlest lterature on stoc cuttng problems appeared n 939, but actually the academc research wors have boomed snce 970s'. Gyson and Gregory (974) advanced a heurstc pacng algorthm n the frst tme. And then, tradtonal optmzaton algorthms such as rule based pacng algorthm (adsen, 979), branchng and boundng (Golden, 976), dynamc programmng (Golden, 976), and ntegral programmng (Farley, 988) were practced on ths subect comprehensvely. Coffman and Shor (990) brought forward a Benchmar to evaluate the effectveness of pacng algorthms, whch was descrbed as pacng a number of squares wth borders randomly generated between (0, ) nto a unt wde nfnte long rectangular stoc. odern optmzaton technologes have made a remarable progress snce 990s'. Genetc algorthms, smulated annealng, fuzzy searchng, neural networ and other meta-heurstc searchng algorthms

2 showed a strong advantage especally n combnatoral optmzaton problems. Jaobs (996) desgned a BL (bottom-left) -algorthm based genetc algorthm for pacng of rectangles; Hopper and Turton (999), Lu and Teng (999) brought forward ther mproved solutons on that algorthm respectvely; Fana (999) dscussed the applcaton of smulated annealng algorthm on pacng problems n detal. Based on the four papers above, an mproved genetc algorthm s presented for rectangular pacng problems wth a new set of algorthm evaluaton ndcators n ths paper. Smulaton results are ncluded to show ts effcency. Ths paper s organzed as follows. In Secton 2, the genetc algorthm, model of pacng problems and ther evaluaton ndcators are ntroduced. In Secton 3, the mplementaton of non-gullotne cut algorthms s dscussed n detal. In Secton 4, the smulaton results of numercal examples are showed and the advantages of the algorthms are explaned. Fnally, concluson and problems for further study are proposed n Secton GEETIC ALGORITHS AD DESCRIPTIO OF PACKIG PROBLES Genetc Algorthms (GAs) s a powerful global optmum strategy based on the smulaton of natural genetcs and evoluton (Holland, 975). Its applcatons n cuttng & pacng problems are scarce, and roughly dvded nto two methods. One s based on the coordnates of the small tems and the bnary or decmal codng [2]. The other s based on the order of the placement or cuttng and the nteger codng [7, 8, 9]. Through the exhaustve smulatons and GAs desgn experence, t s dscovered that the frst method calls for hgh desgnng arts on the penalty tem of the ftness functon, and s easly trapped nto the n-feasble soluton and local optmum; the second method should be combned wth other algorthm to generate pacng patterns, but t can somehow overturn the shortcomngs of the frst method. Based on the second method, a new pacng algorthm and genetc operators are desgned to mprove the structure and control parameters of GAs, and they also enhance the exactness and stablty of the algorthm. The general structure of cuttng & pacng problems s as follows:. One or more stocs of certan fgures, forms and szes; 2. A number of small tems to be cut or paced wth dfferent fgures, forms and szes; 3. Some cuttng & pacng restrctons; 4.Evaluatng crtera based on one or more obectves. For the rectangular cuttng & pacng problem, the thrd part mentoned above s very mportant. As to some ndustres such as glass and polystyrene, t s requred that the cut should be orthogonal form one edge to the other, whch s called gullotne cut. The cut wthout that requrement are called non-gullotne Fg.. on-gullotne cut and gullotne cut. cut. As shown n Fg.., (a) s non-gullotne cut, and (b) s gullotne cut. Generally, non-gullotne cut s more effcent than gullotne cut, but t requres larger searchng space and more complex cuttng and pacng algorthm. Consderng that non-gullotne cut s more general than gullotne cut, an mproved GAs for non-gullotne cut s presented and t can be easly modfed to ft the gullotne cut stuaton. The general obectve of pacng problems s to mnmze the waste area rate, and the most wdely used waste area rate s as follows: let S t be the area of the stoc, the area of rectangles are respectvely S and the number of them s, and then the waste area rate W s St S = W = 00% () S t Based on the area rate ndex above, the permeter ndex s used to mprove the qualty of the waste stoc. Let C w be the permeter of the waste stoc, expected waste stoc permeter s C E, and then the permeter ndex s Cw CE C = 00% (2) C C E w = 4 S S (3) t = C E represents the permeter of the square whose area s equal to the waste stoc s. nmzng the permeter ndex C can mae the shape of the waste stoc to be a square as close as possble, and mprove the reuse qualty especally for expensve stocs. 3. ALGORITH DESCRIPTIO The most wdely used smple heurstc pacng algorthm s BL algorthm. There are two steps n the pacng process of BL algorthm. Frst, randomly generate the pacng order of the rectangles. Second, accordng to the pacng order, place each rectangle nto the stoc and move t downward and leftward to the bottom-left pont of the stoc untl mpossble. Hopper and Turton (999) uses SGA (Smple Genetc Algorthm) as the frst step of the BL algorthm to generate and optmze the pacng order, however, the BL pacng process easly causes gaps among the paced rectangles and these gaps cannot be used agan. Fana (999) repeats the BL pacng process to fnd the gaps, and Lu and Teng (999) tres to fnd

3 the lowest pont of the unpaced stoc the mae up the gaps. Both of them mprove the qualty of the solutons but don t solve the gap problem completely. In ths paper, a greedy pacng algorthm s presented based on all the unpaced potental optmal pont of the stoc and the computatonal results show that t can completely avod gaps. Furthermore, the unform codng schema s used to dversfy the chromosome populaton, adaptve crossover and mutaton probabltes to prevent the pre-maturaton, and a Stochastc Hll-Clmbng operator s also desgned to mprove the searchng qualty. Step. Let generaton g=0, ntalze ndvduals accordng to unform codng schema. For any ndvduals and, defne the Hammng dstance H of the two ndvduals as: H = = 0 f = = otherwse (4) (5) represents the th bt of the th ndvdual. If H <T, then remove one ndvdual of and randomly, and regenerate a new ndvdual. T s a pre-decded threshold, and a larger T wll help to generate a more dversfed populaton. Step 2. Pac and evaluate ndvduals of the gth populaton accordng to the followng greedy algorthm:. Intalze the sngly lned lst P of optonal unpaced ponts. The frst element of P s the up-left pont of the stoc p Accordng to the pacng order, place the up-left pont of the next rectangle on the optonal pont p of the lst P, and calculate ts ftness f. After all the optonal ponts n the lst P are calculated, fnd the smallest ftness f =mn f and remar the relatve pont p. 3. Place the up-left pont of the rectangle on the pont p, and delete p, at the same tme nsert the bottom-left pont p, bottom-rght pont p +, and up-rght pont p +2 nto the lst P, then change the ponters. 4. If all the rectangles are place, turn to next step. Else go to 2. The ftness functon s f =αw +βc. For any ndvdual, W s the area ndex of the waste stoc, and C s the permeter ndex. The α and β are ndexes related to the generaton g. At the begnnng α>0.9 and β<0., after a certan number of generatons when the waste area ndex s close to the satsfed soluton, ncrease the value of β to mprove the qualty of the waste stoc. Step 3. Do select operaton on the current populaton by the roulette and eltsm schema. The probablty P of ndvdual to be chosen nto the next generaton s: P = = f (6) ( f ) Preserve 5%~0% ndvduals wth best ftness and put them drectly nto the next generaton. Step 4. Do crossover operaton on the chosen populaton. Four crossover operators ncludng Sngle-pont crossover, partal mappng crossover (PX), order crossover (OX) and on-abel crossover are tred and compared for ther effcency through the computatonal results. Step 5. Do mutaton operaton on the populaton. The swap and nverse operators are used, and the probablty of mutaton P m s: P m = P0 µ ( f f mn ) + ν H (7) P 0 s the ntal value, µ and ν are control parameters, f s the average ftness value, f mn s the best ftness value, and H s the average Hammng dstance of the populaton. Furthermore, a new stochastc-hll-clmbng (SHC) mutaton operator s proposed. For the 5% ndvduals wth the best ftness value and ther canddate chldren generated by the mutaton operaton, f f < f ', accept wth probablty ; otherwse accept ( f ' f ) / T g wth probablty e. Tg s dependent on the generaton g. At the begnnng T g s larger, helpng to enlarge the search space, and decreases as the generaton g ncreasng. Step 6. If g > g max or the best ftness value does not change n 0 generatons, turn to the next step; otherwse let g=g+, and go to Step 2. Step 7. Output the optmal soluton. 4. COPUTATIOAL RESULTS The benchmar desgned by Coffman and Shor (990) s used to compare the effcency of dfferent crossover operators and algorthm of ths paper and Fana s (999). Frst, ntalze a stoc wth wdth w and nfnte length; second, gve a certan number of rectangles (between 8 and 64) wth wdth and length randomly generated between (0, w). Each smulaton runs 00 tmes. Table compares the four crossover operators, two mutaton operators and the stochastc-hll-clmbng operator. When comparng crossover operators, the stochastc-hll-clmbng operator s used, and when comparng mutaton operators, PX crossover operators are used. Table 2 compares the effcency of the algorthm used n ths paper and n Fana s (999). Computatonal results n Table- show that except the on-abel operator, there s not much dfference

4 Table Comparson of Operators umber of rectangles: 32 Crossover Average % n/ax % utaton Average % n/ax % Sngle-pont / 96.9 Swap / PX / 97.7 Inverse / 96.3 OX / SHC / 97.7 on-abel / umber of rectangles Table 2 Comparsons of Algorthms Ths paper Fana s algorthm Average axmum Average n/ax (%) (%) Devaton (%) (%) n/ax (%) / / / / / / / / axmum Devaton (%) of the effcency among the rest of other three operators. Snce the on-abel operator s more le random search, t s not good to preserve the good schema, whch s the ey technque of the GAs, and the performance s not stable ether. As for the mutaton operators, t s obvous that the SHC operator has better performance. The SHC operator not only chooses offsprng wth hgher ftness value, but also accepts ndvduals wth lower ftness value by certan probablty, whch s a good example on the combnaton of the GAs and neghborhood local-search technques to mprove the performance. In Table-2, the performance of the algorthm used n ths paper s better than that of Fana s, and one pont to be mentoned s that the total evaluaton of ftness value n Fana s algorthm s 00 tmes of the number of the rectangles, whereas 40 tmes n ths paper, whch s much more tme effcent. After pacng order s generated by the GAs, greedy algorthm s used to pac the rectangles nto the stoc, and get a better pacng pattern each tme. Whereas Fana s algorthm uses a random pacng algorthm, whch does not guarantee better pacng pattern even though the pacng order s good. Furthermore, the mn/max stoc usage rate of ths paper s obvously smaller than that of Fana s algorthm. It s because of the hdden parallel and mass evoluton of GAs, whereas the smulatng annealng algorthm used n Fana s algorthm s more senstve on the ntal value when appled to combnatoral optmzaton problems. Jaobs (996), Hopper and Turton (999), and Lu and Teng (999) also use GA, but the performance cannot be compared because they do not gve the form and szes of the stoc and rectangles they used. Snce they only try on smple genetc algorthms and bottom-left pacng algorthm, our refned crossover and SHC operator wth greedy search algorthm s more effcent. 5. COCLUSIO In ths paper, an mproved GA combned wth greedy search algorthm s proposed for non- gullotne rectangles pacng problem. Frst, self-adaptve mutaton probablty s adopted to avod pre-maturty, compare and refne some common used crossover operators, and desgn a stochastc hll-clmbng operator s to mprove the local search ablty. Second, a greedy searchng mechansm s ntroduced to acqure good pacng pattern after pacng order determned by the GA. Fnally, computatonal results are presented to show the effectveness of these mprovements. Ths algorthm s also appled on gullotne rectangles pacng problem and got satsfed results. ow research on polygon pacng problems are undergong. REFERECE Coffman, E.G. and Jr. P. W. Shor (990). Average-case analyss of cuttng and pacng n two dmensons, Eur. J. Operatonal Research, 44, Dychoff, H. (990). A typology of cuttng and pacng problrms. Eur. J. Operatonal Research, 44, Fana, L. (999). An applcaton of smulated annealng to the cuttng stoc problem. Eur. J. Operatonal Research, 4, Farley, A. A. (988). athematcal Programmng odels for Cuttng-Stoc Problems n the Clothng Industry. J. Ope. Res. Soc., 39, Vol., 4-53 Golden, B.L. (976). Approaches to the cuttng stoc problem. AIIE Trans, 8, Gyson, R. G. and A. S. Gregory (974). The Cuttng Stoc Problem n the Flat Glass Industry, Operatonal Research Quarterly, 25, 4-53 Holland J H. (975). Adaptaton n natural and artfcal systems. I: Unversty of chgan

5 Press. Hopper, E. and B. Turton (999). A Genetc Algorthm for a 2D Industral Pacng Problem. Computers & Industral Engneerng, 37, Jaobs, S. (996). On genetc algorthms for the pacng of polygons. Eur. J. Operatonal Research, 88, Vol., 65-8 Kazarls, S. (994). Varyng qualty functon n genetc algorthms and the cuttng problem, n Proc. st IEEE Conf. Evolutonary Computaton, IEEE Servce Center, Vol., Lu, D. and H. Teng (999). An mproved BL-algorthm for genetc algorthm of the orthogonal pacng of rectangles. Eur. J. Operatonal Research, 2, adsen, O.B.G. (979). Glass cuttng n a small frm. ath. Prog. 7, 85-90