Approximate Packing Circles in a Rectangular Container: Valid Inequalities and Nesting

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1 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng I. Ltvnchev, and E. L. Ozuna* Complex Systems Department Computng Center of Russan Academy of Scences Moscow, Russa * luceroozuna@gmal.com Facultad de Ingenería Mecánca y Eléctrca Unversdad Autónoma de Nuevo León Monterrey, Nuevo León, Méxco ABSTRACT A problem of pacng a lmted number of unequal crcles n a fxed sze rectangular contaner s consdered. The am s to maxmze the (weghted) number of crcles placed nto the contaner or mnmze the waste. Ths problem has numerous applcatons n logstcs, ncludng producton and pacng for the textle, apparel, naval, automoble, aerospace and food ndustres. Frequently the problem s formulated as a nonconvex contnuous optmzaton problem whch s solved by heurstc technques combned wth the local search procedures. A new formulaton s proposed based on usng a regular grd approxmated the contaner and consderng the nodes of the grd as potental postons for assgnng centers of the crcles. The pacng problem s then stated as a large scale lnear 0- optmzaton problem. The bnary varables represent the assgnment of centers to the nodes of the grd. The resultng bnary problem s then solved by the commercal software. Two famles of vald nequaltes are proposed to strengthenng the formulaton. Nestng crcles nsde one another s also consdered. Numercal results are presented to demonstrate the effcency of the proposed approach. Keywords: Crcle Pacng, Integer Programmng, Large Scale Optmzaton. RESUMEN Se consdera el problema de empaquetar un número lmtado de círculos de rados dferentes en un contenedor rectangular de dmensones fas. El obetvo es maxmzar el número (ponderado) de círculos dentro del contenedor o mnmzar el desperdco de espaco dentro del msmo. Este problema tene numerosas aplcacones dentro de la logístca, ncluyendo la produccón y empaquetado para la ndustra textl, naval, automotrz, aeroespacal y la ndustra de almentos. Frecuentemente, el problema es formulado como un problema de optmzacón contnua no convexo que es resuelto con técncas heurístcas combnadas con procedmentos de búsqueda local. Se propone una nueva formulacón basada en el uso de una malla regular que cubre el contenedor y donde se consdera a los nodos de la malla como poscones potencales para la asgnacón de centros de los círculos. El problema de empaquetamento se escrbe entonces, como un problema de optmzacón 0- a gran escala y es resuelto con software comercal. Resultados numércos son presentados para demostrar la efcenca del enfoque propuesto y realzar una comparacón con los resultados conocdos.. Introducton Pacng problems consttute a famly of natural combnatoral optmzaton problems, whch occur n many felds of study such as computer scence, ndustral engneerng, logstcs and manufacturng and producton processes. For nstance, several real lfe ndustral applcatons requre the allocaton of a set of peces to a larger standardzed rectangular stoc unt. They generally consst of pacng a set of tems of nown dmensons nto one or more large obects n order to mnmze a certan obectve (e.g. the unused part of the obects or waste). The crcle pacng problem s a well studed problem [] whose am s the pacng of a certan number of crcles, each one wth a fxed nown 76 Vol., August 0

2 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 radus (not necessary the same for each crcle) nsde a contaner. The shape of the contaner may vary from a crcle, a square, a rectangular, etc. Ths problem has been appled n dfferent areas, such as the coverage of a geographcal area wth cell transmtters, storage of a cylndrcal drums nto contaners o stocng them nto an open area, pacagng bottles or cans nto the smallest box, plantng trees n a gven regon as to maxmze the forest densty and the dstance between the trees, and so forth [,,8]. Other applcatons one can fnd n the motor cycle ndustry, crcular cuttng, communcaton networs, faclty locaton and dashboard layout [,0,]. In ths paper we address the problem of pacng a set of crcular tems n a rectangular contaner. There are two prncpal types of obectves that have been used n the lterature: a) regard the crcles (not necessary equal) as beng of fxed sze and the contaner as beng of varable sze and b) regard the crcles and the contaner as beng of fxed sze and mnmze waste. Examples of the frst approach nclude [7]: For the square contaner mnmze the length of the sde and hence mnmze the permeter and area of the square; Mnmze the permeter of the rectangle; Mnmze the area of the rectangle; Consderng one dmenson of the rectangle as fxed, mnmze the other dmenson. Problems of ths type are often referred to as strp pacng problems (or as crcular open dmenson problems). For the second approach varous defntons of the waste can be used. The waste can be defned n relaton to crcles not paced (e.g. the number of unpaced crcles or the permeter/area of unpaced crcles), or ntroducng a value assocated wth each crcle that s paced (e.g. area of the crcles paced), etc. Many varants of pacng crcular obects n the plane have been formulated as nonconvex (contnuous) optmzaton problems wth decson varables beng coordnates of the centres. The nonconvexty s manly provded by no overlappng condtons between crcles. These condtons typcally state that the Eucldean dstance separatng the centres of the crcles s greater than a sum of ther rad. The nonconvex problems can be tacled by avalable nonlnear programmng (NLP) solvers, however most NLP solvers fal to dentfy global optma. Thus, the nonconvex formulaton of crcular pacng problem requres algorthms whch mx local searches wth heurstc procedures n order to wdely explore the search space. It s mpossble to gve a detaled overvew on the exstng soluton strateges and numercal results wthn the framewor of a sngle short paper. We wll refer the reader to revew papers presentng the scope of technques and applcatons for the crcle pacng problem (see, e.g. [,,7,8] and the references theren). In ths paper we propose a new formulaton for approxmate soluton of crcular pacng problems usng a regular grd to approxmate the contaner. The nodes of the grd are consdered as potental postons for assgnng centers of the crcles. The pacng problem s then stated as a large scale lnear 0- optmzaton problem. Two classes of vald nequaltes are proposed to strengthenng the formulaton. Nestng crcles nsde one another s also consdered. Numercal results are presented to demonstrate effcency of the proposed approach. To the best of our nowledge, the dea to use a grd was frst mplemented by Beasley [] n the context of cuttng problems. Ths approach was recently appled n [9,,6] for pacng problems. Ths wor s a contnuaton of [].. The model Suppose we have non-dentcal crcles C of nown radus R, K {,,... K}. Let at most M crcles C are avalable for pacng and at least m of them have to be paced. Denote by I {,..., n} the node ponts of a regular grd coverng the rectangular contaner. Let F I be Journal of Appled Research and Technology 77

3 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 the grd ponts lyng on the boundary of the contaner. Denote by d the Eucldean dstance between ponts and of the grd. Defne bnary varables x f centre of a crcle C s assgned to the pont ; x 0 otherwse. In what follows we wll dstngush two cases of crcle pacng, dependng on whether nestng crcles nsde one another s permtted o not. To the best of our nowledge, nestng problem was frst mentoned n [0] n the context of pacng ppes of dfferent dameters nto a shppng contaner. Comparng to the standard pacng, pacng wth nestng s much less nvestgated. Consder frst the problem wthout nestng. In order to the crcle C assgned to the pont be nonoverlappng wth other crcles beng paced, t s l necessary that x 0 for I, l K, such that d R Rl. For any fxed, let N {,: l, d R Rl }. Let n be the cardnalty of N : n N. Then the problem of maxmzng the area covered by the crcles can be stated as follows: Then the problem of maxmzng the area covered by the crcles s as follows: max Rx () I K m x M, K, I K x, I \ F R x mn d, I, K, F () () () l x x, for I, K, (, l) N () x {0,}, I, K (6) Constrants () ensure that the number of crcles paced s between m and M ; constrants () that at most one centre s assgned to any grd pont; constrants () that the pont can not be a centre of the crcle C f the dstance from to the boundary s less than R ; par-wse constrants () guarantee that there s no overlappng between the crcles; constrants (6) represent the bnary nature of varables. Note that for the partcular case of pacng equal crcles of radus we may smply reduce the dmensons of the contaner by and then apply the above model to a smaller contaner droppng the boundary condtons (). We may expect that the lnear programmng relaxaton of the problem ()-(6) provdes a poor upper bound for the optmal obectve. For example, for K and sutably chosen M, m, set x 0. for all I. Ths soluton s feasble to no overlappng constrants () and correspondng obectve value grows lnearly wth respect to the number of grd ponts. To tghtenng the LP-relaxaton of ()-(6) we propose two famles of vald nequaltes. The frst ensure that no grd pont s covered by two crcles, whle the second guarantee that there s at most one centre assgned to the area covered by a crcle. To present the frst famly, defne matrx as follows. Let for d R ; 0 otherwse. By ths defnton, f the crcle C centered at covers pont. The followng constrants ensure that no ponts of the grd can be covered by two crcles: x, I (7) K I Note that (7) s not equvalent to non-overlappng constrants (). Constrants (7) ensure that there s no overlappng n grd ponts, whle () guarantee that there s no overlappng at all. Smlar to setcoverng formulatons t s natural to refer to (7) as pont-coverng constrants. The second famly of nequaltes s stated as follows: 78 Vol., August 0

4 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 x x, for I, K. (8) To demonstrate that (8) s vald for the problem ()- (6) assume that x n (8). That s the centre of the crcle C s assgned at. By (8) we have x 0, for I, K and then t follows that d : R x 0for : d R. That s there are no other centres assgned to ponts nsde the crcle centred at. For x 0 we have x. Ths means d : R that among all grd ponts covered by the (magnary) crcle centred at, at most one pont can be assgned as a centre. Ths s true snce the dstance between any par of these ponts s less than R and assgnng the centres of C volates non overlappng constrants. To consder nestng crcles nsde one another, we only need to modfy the non-overlappng constrants. In order to the crcle C assgned to the pont be non-overlappng wth other crcles beng paced (ncludng crcles places nsde ths l crcle), t s necessary that x 0 for I, l K, such that R Rl d R Rl. Note that the later condton s always fulflled for R R l ( d 0 ), such that only smaller crcles can be placed nsde a gven crcle. For fxed, let {,: l, R Rl d R Rl }. Then the problem of pacng crcles wth nestng can be stated as follows: max d : R subect to I K wx m x M, K, I K x, I \ F Rx mn d, I, K, F l x x, for I, K, ( l, ) (9) In the problem (9) the weghtng coeffcents w may be assocated wth the area of crcles and/or represent the relatve mportance of subsets of the contaner. Note that nequaltes (7), (8) n general are not vald for the problem (9).. Computatonal results A rectangular unform grd was used n numercal experments, such that all grd ponts are defned by the grd ponts on ts edges. Let L be a horzontal dmenson (length) and W be a vertcal dmenson (wdth) of the contaner; M be a number of the equdstant grd ponts on the horzontal edge of the contaner, whle N be a number of the equdstant grd ponts on ts vertcal edge. Hence the grd has M N nm N. node ponts All optmzaton problems were solved by the system CPLEX.. The runs were executed on a PC Toshba Satellte L7, Intel Core I-,. Ghz and 8Gb RAM. In the frst part of our numercal experment we add vald nequaltes (7) or (8) or both to the problem ()-(6) and compare correspondng LP-relaxatons. Fve dfferent relaxatons were studed correspondng to constrants used/droped: LP: only orgnal constrants ()-(); LP: constrants ()-() and (7); LP: constrants ()-() and (8); LP: constrants ()-(), (7) and (8); LP: constrants ()-(), (7) and (8) Ten nstances wth equal crcles were used to compare relaxatons. The frst nstances were the same as n [9, Table ]: L, W 6 and raduses 0., 0.6, 0.6, 0.7 and 0. correspondngly. The second nstances were from [6] wth L W, R defned as follows: 00x00, ; 00x00, ; 00x00, 8; 00x00, ; 0x80,. In all nstances the obectve () was to maxmze the number of crcles paced. Journal of Appled Research and Technology 79

5 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 IP LP LP LP LP LP Table. LP bounds for pacng equal crcles. The results of the numercal experment are presented n Table. Here the second column presents the nteger soluton of the problem ()-(6) whle all the next columns gve the optmal obectves of the correspondng relaxatons. As we can see form Table, vald nequaltes mprove sgnfcantly LP, contnuous relaxaton of the orgnal problem, and provde a very tght bound for the optmal obectve of the orgnal nteger problem IP. In many cases roundng below the correspondng ratonal bound results n the optmal obectve value. Note that f we drop the par-wse non-overlappng constrants () and use both famles of vald nequaltes (relaxaton LP), the bound s stll good. We see that the values of LP-LP are very close to each over. From computatonal pont of vew the relaxaton LP s less expensve snce the par-wse nonoverlappng constrants () are relaxed. # m M crcles paced CPU (sec.) Table. Problem nstances and numercal results for pacng wth nestng. In the second part of the experment pacng of dfferent crcles wth nestng was studed. We fxed the dmenson of the contaner ( LW 60 ), raduses of the crcles ( R, R, R, R 0.7 ) and vary the bounds m, M for the crcles to be paced. The data for 6 nstances of the problem (9) consdered n the experment are presented n Table together wth the number of the crcles paced and correspondng CPU tme. Empty cells n ths table correspond to the case wth no lower/upper lmts for the crcles to be paced. All 6 problem nstances were solved usng the grd x and wth mpgap = % for runnng CPLEX. Fgure. Pacng pattern for nstance. The obectve was to maxmze the total area of the crcles paced. 70 Vol., August 0

6 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / Conclusons The plane crcle pacng problem was approxmated usng nteger formulaton based on a grd approxmaton of a contaner. Fgure. Pacng pattern for nstance. Fgure. Pacng pattern for nstance. Fgure. Pacng pattern for nstance. As we can see from Table the nstances wth lower bounds for the number of crcles to be paced are mostly expensve computatonally. Fgures -6 present pacng confguratons for the correspondng nstances. We can see that varyng the lmts for the number of crcles to be paced changes sgnfcantly the pacng confguraton. Fgure. Pacng pattern for nstance. The case of nestng crcles nsde one another was consdered. Ths problem was mentoned n [0] n Journal of Appled Research and Technology 7

7 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 the context of pacng ppes of dfferent dameters nto a shppng contaner and has not receved much attenton so far. References [] H. Aeb and M. Hf, Solvng the crcular open dmenson problem usng separate beams and looahead strateges, Computers & Operatons Research, vol.0, pp. -, 0. [] E. Baltacoglu and J.T. Moore, Hll R.R., The dstrbutor s three-dmensonal pallet-pacng problem: a human-based heurstcal approach, Internatonal Journal of Operatons Research, vol., pp. 9-66, 006. [].J.E. Beasey, An exact two-dmensonal nongullotne cuttng tree search procedure, Operatons Research, vol., pp. 9-6, 98. [].E.G. Brgn and J.M. Gentl, New and mproved results for pacng dentcal untary radus crcles wthn trangles, rectangles and strps, Computers & Operatons Research, vol. 7, pp. 8-7, 00. Fgure 6. Pacng pattern for nstance 6. The presented approach can be easly generalzed to three (and more) dmensonal case and to dfferent shapes of the contaner, ncludng rregulars. For the case wthout nestng two famles of vald nequaltes were ntroduced to strengthenng the formulaton. Numercal experment was presented to demonstrate the effcency of the proposed approach. An nterestng topc for the future research s to study the use of Lagrangan relaxaton [] or decomposton technques [7] to cope wth large dmenson of the problem formulaton. The other drecton for future research s usng metaheurstc approaches []. [].I. Castllo, F.J. Kampas and J.D. Pnter, Solvng crcle pacng problems by global optmzaton: Numercal results and ndustral applcatons, European Journal of Operatonal Research, vol. 9, pp , 008. [6].M.H. Correa, J.F. Olvera and J.S. Ferrera, Cylnder pacng by smulated annealng. Pesqusa Operaconal, vol. 0, pp , 000. [7] M. Elzondo-Cortes and R. Aceves-Garca, Strategy of soluton for the nventory-routng problem based on separable cross decomposton, Journal of Appled Research and Technology, vol., pp. 9-9, 00. [8] H.J. Frazer and J.A. George, Integrated contaner loadng software for pulp and paper ndustry, European Journal of Operatonal Research, vol.77, pp. 66-7, 99. [9] S.I. Galev and M.S. Lsafna. Lnear models for the approxmate soluton of the problem of pacng equal crcles nto a gven doman, European Journal of Operatonal Research, vol. 0, pp. 0-, 0. [0] J.A. George, J.M. George, Lamar B.W., Pacng dfferent szed crcles nto a rectangular contaner, European Journal of Operatonal Research, vol.8, pp. 69-7, 99. [] J.A. George, Multple contaner pacng: a case study of ppe pacng, Journal of the Operatonal Research Socety, vol. 7, pp , 996. [] M. Hf and R. M Hallah, A lterature revew on crcle and sphere pacng problems: models and methodologes, Advances n Operatons Research, vol. 009, pages, do:0./009/06. 7 Vol., August 0

8 Approxmate Pacng Crcles n a Rectangular Contaner: Vald Inequaltes and Nestng, I. Ltvnchev / 76 7 [] Y.C. Ln, Mxed-nteger constraned optmzaton based on memetc algorthm, Journal of Appled Research and Technology, vol., pp. -0, 0. [] I. Ltvnchev, S. Rangel, J. Saucedo, A Lagrangan bound for many-to-many assgnment problem, Journal of Combnatoral Optmzaton, vol.9, pp. -7, 00. [] I. Ltvnchev, E. L. Ozuna, Pacng crcles n a rectangular contaner, Proc. Intl. Congr. on Logstcs and Supply Chan, Queretaro, Mexco, October -, 0. [6] I. Ltvnchev, E. L. Ozuna, Integer programmng formulatons for approxmate pacng crcles n a rectangular contaner, Mathematcal Problems n Engneerng, 0 (to appear). [7] C.O. Lopez, J.E. Beasly, Pacng unequal crcles usng formulaton space search, Computers & Operatons Research, vol. 0, pp , 0. [8] Y.G. Stoyan and G.N. Yasov, Pacng congruent spheres nto a mult-connected polyhedral doman, Internatonal Transactons n Operatonal Research, vol. 0, pp , 0. Journal of Appled Research and Technology 7