XLVII SIMPÓSIO BRASILEIRO DE PESQUISA OPERACIONAL

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1 LP-BASED HEURISTIC FOR PACKING CIRCULAR-LIKE OBJECTS IN A RECTANGULAR CONTAINER Igor Ltvnchev Computng Center of Russan,Academy of Scences Moscow , Vavlov 40, Russa gorltvnchev@gmal.com Lus Alfonso Infante Rvera Faculty of Mechancal and Electrcal Engneerng, UANL Pedro de Alba S/N, San Ncolas de los Garza, NL 66450, Méxco lusnfantervera@gmal.com Lucero Ozuna Faculty of Mechancal and Electrcal Engneerng, UANL Pedro de Alba S/N, San Ncolas de los Garza, NL 66450, Méxco luceroozuna@gmal.com ABSTRACT A problem of pacng unequal crcles n a fxed sze rectangular contaner s consdered. An heurstc s proposed, based n a lnear relaxaton of a nteger mathematcal model as approxmaton of the optmal soluton of the problem. The crcle s consdered n a general sense, as a set of ponts that are all the same dstance (not necessary Eucldean) from a gven pont. The am s to maxmze the (weghted) number of crcles placed nto the contaner or mnmze the waste. An nteger formulaton s proposed usng a grd approxmaton of the contaner and consderng the grd nodes as potental postons for centers of the crcles. The bnary varables represent the assgnment of centers to the nodes of the grd. The pacng problem s then stated as a large scale lnear 0-1 optmzaton problem. Vald nequaltes are used to strengthenng the formulaton. Numercal results on pacng crcles and octagons are presented to demonstrate the effcency of the proposed approach. Keywords. Crcle pacng. Integer programmng. Large scale optmzaton. Man area. OC Combnatoral Optmzaton 2452

2 1. Introducton Pacng problems generally consst of pacng a set of tems of nown dmensons nto one or more large objects or contaners to mnmze a certan objectve (e.g. the unused part of the objects or waste). Pacng problems consttute a famly of natural combnatoral optmzaton problems appled n computer scence, ndustral engneerng, logstcs, manufacturng and producton processes (E. Baltacoglu, J.T. Moore and Hll R.R (2013), I. Castllo, F.J. Kampas and J.D. Pnter (2008), H.J. Frazer and J.A. George (1996)) The crcle pacng problem s a well studed problem (E.G. Brgn and J.M. Gentl (2010), J.A. George (1996), (2009)) whose am s pacng a certan number of crcles, each one wth a fxed nown radus nsde a contaner. The crcles must be totally placed n the contaner wthout overlappng. The shape of the contaner may vary from a crcle, a square, a rectangular, etc. Along wth aforementoned applcatons, crcle (sphere) pacng problems arse n automated radosurgcal treatment plannng for treatng bran and snus tumours (J. Wang (1999))]. Radosurgery uses t target tumor area. For large target regons multple shots of dfferent ntensty are used to cover dfferent parts of the tumor. However, ths procedure may result n large doses due to overlap of the dfferent shots. Optmzng the number, postons and ndvdual szes of the shots can reduce the dose to normal tssue and acheve the requre coverage. Many varants of pacng crcular objects 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, H. Aeb and M. Hf (2013), E.G. Brgn and J.M. Gentl (2010), C.O. Lopez and J.E. Beasley (2011 and 2013), Y.G. Stoyan and G.N. Yasov (2013) and the references theren). In ths paper we study pacng crcular-le objects usng a regular grd to approxmate the contaner. The crcular-le object s consdered n a general sense, as a set of ponts that are all the same dstance (not necessary Eucldean) from a gven pont. Thus dfferent shapes, such as ellpses, rhombuses, rectangles, octagons can be treated the same way by smply changng the norm used to defne the dstance. 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 0-1 optmzaton problem. Vald nequaltes are used to strengthenng the orgnal formulaton and mprove LP-bounds. Reduced costs of the LP-soluton are used to fx some varables n the orgnal problem to get an approxmate nteger soluton. Numercal results on pacng crcles and regular octagons 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 (1985) n the context of cuttng problems. Recently ths approach was appled for S.I. Galev and M.S. Lsafna 2453

3 (2013), I. Ltvnchev and L. Ozuna (2013 and 2014), Toledo et al (2013) for pacng problems. Ths wor s a contnuaton of I. Ltvnchev and L. Ozuna (2014). 2. The Model Suppose we have non-dentcal crcles C of nown radus R, K {1,2,... K}. Here we consder the crcle as a set of ponts that are all the same dstance R (not necessary Eucldean) from a gven pont. In what follows we wll use the same notaton C for the fgure bounded by the crcle assumng that t s easy to understand from the context whether we mean the curve or the fgure. Denote by S the area of C. Let at most M crcles C are avalable for pacng and at least m of them have to be paced. Denote by I {1,2..., n} the node ponts of a regular grd coverng the rectangular contaner. Let F I be the grd ponts lyng on the boundary of the contaner. Denote by d j the dstance (n the sense of norm used to defne the crcle) between ponts and j of the grd. Defne bnary varables x 1 f centre of a crcle C s assgned to the pont ; x 0 otherwse. In order to the crcle C assgned to the pont be non-overlappng wth other crcles beng l paced, t s necessary that x 0 for j I, l K, such that d j R R l. For fxed, let j N {,: j l j, dj 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: subject to max Sx (1) I K m x M, K, (2) I x 1, I \ F, (3) K Rx mn d, I, K, (4) j j F l x xj 1, for I, K, (,) jl N (5) x {0,1}, I, K (6) Constrants (2) ensure that the number of crcles paced s between m and M ; constrants (3) that at most one centre s assgned to any grd pont; constrants (4) 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 (5) guarantee that there s no overlappng between the crcles; constrants (6) represent the bnary nature of varables. Smlar to plant locaton problems [21] we can state non-overlappng condtons n a more compact form. Summng up par-wse constrants (5) over (,) jl N we get l j jl, N n x x n for I, K (7) Note that constrants smlar to (7) were used n [8] for pacng equal crcles. Proposton 1 [13, 15 ]. Constrants (5), (6) are equvalent to constrants (6), (7). 2454

4 Thus the problem (1)-(6) s equvalent to the problem (1)-(4), (6), (7). To compare two equvalent formulatons, let l P1 { x 0: x xj 1, for I, K, (,) jl N }, P2 { x 0: n x x n for I, K}. l j jl, N Proposton 2 [13, 15 ]. P1 P2. As follows from Proposton 2, the par-wse formulaton (1)-(6) s stronger (n the sense of [17]) than the compact one. Numercal experments presented n [14, 15] demonstrate that the par-wse formulaton s also computatonally more attractve snce t provdes a tghter LP-bound. Bearng n mnd these reasons we restrct ourselves by consderng below only par-wse formulatons. By the defnton, N {,: j l j, dj R Rl } and hence f ( jl,) N, then (, ) Njl. Thus a half of the constrants n (5) are redundant: and x l j l xj 1, for I, K, (,) jl N x x 1, for j I,l K, (, ) Njl. The redundant constrants can be elmnated wthout changng the qualty of LP-bound gvng a reduced par-wse non overlappng formulaton. In what follows we wll assume that the redundant constrants are elmnated from (5). Note that all constructons proposed above, ncludng Propostons 1,2, reman vald for any norm used to defne a crcular-le object. In fact, changng the norm affects only the dstance d j used n the defntons of the sets N, n the non-overlappng constrants (5). That s, by smple pre-processng we can use the basc model (1)-(6) for pacng dfferent geometrcal objects of the same shape. It s mportant to note that the non-overlappng condton has the form dj R Rl no matter whch norm s used. For example, a crcular object n the maxmum norm z : max{ z} s represented by a square, taxcab norm z : z 1 yelds a rhombus. In a smlar way we may manage rectangles, ellpses, etc. Usng a superposton of norms we can consder more complex crcular objects. For z : max{ z, z} (8) and a sutable we get an octagon, an ntersecton of a square and a rhombus. In partcular for 1/ 2 we get a regular octagon. 3. LP-based heurstc We may expect that the lnear programmng relaxaton of the problem (1)-(6) provdes a poor upper bound for the optmal objectve. For example, for K 1and sutable M, mthe pont x 0.5 for all I may be feasble to the relaxed problem wth the correspondng objectve growng lnearly wth respect to the number of grd ponts. 2455

5 To tghtenng the LP-relaxaton for (1)-(6) we consder vald nequaltes amed to ensure that no grd pont s covered by two crcles. Defne matrx j as follows. Let j 1 for dj R, j 0 otherwse. By ths defnton, j 1 f the crcle C centred at covers pont j. The followng constrants ensure that no ponts of the grd can be covered by two crcles: j xj 1, I. (9) K j I Note that (9) s not equvalent to non-overlappng constrants (5). Constrants (9) ensure that there s no overlappng n grd ponts, whle (5) guarantee that there s no overlappng at all. We can treat (9) as a relaxed non-overlappng condtons and expect that refnng the grd reduces overlappng. The vald nequaltes (9) hold for any norm used to defne the crcular object. Numercal experments presented n [14, 15 ] demonstrate that aggregatng vald nequaltes (9) to the orgnal problem (1)-(6) mproves sgnfcantly the value of the correspondng LP-bound. Moreover, vald nequaltes change the structure of the optmal LP-soluton. Below, we wll use the same term LP relaxaton (LP-bound) for the problem (1)-(5) as well as for problems (1)-(5), (9) and (1)- (4), (9). That s along wth relaxng ntegralty constrants (6) we may substtute non-overlappng condtons (5) for vald nequaltes (9). Let G be a set of the nodes of an orgnal grd and FG be a set of the nodes of the refned grd constructed such, that G FG,.e. all nodes of the orgnal grd reman the nodes of the refned one. Let z and G zfg be the optmal values of the nteger problems (1)-(6) or (1)-(6), (9) obtaned for correspondng grds GFG., Then we have zg zfg lpfg, (10) where lpfg s the value of the LP-bound correspondng to the grd FG. Here the frst nequalty holds snce we may construct a feasble soluton to the problem correspondng to FG by settng x 0 for FG Gand leavng all other components equal to G - optmal soluton. Thus we can construct LPbounds for the orgnal objectve usng grds dfferent from the orgnal one. Suppose that a relaxed problem for the grd FG s solved and correspondng reduced costs are nown. The heurstc algorthm below amed to reduce the number of varables n nteger problem (1)- (6), (9) by fxng x 0 for the nodes of G wth suffcently negatve reduced costs. LP-based heurstc. Step 1. For the orgnal grd G defne a refned grd FG, G FG, and solve LP-relaxaton for the grd FG. Let d, FGbe correspondng reduced costs. Step 2. Defne the set of non-postve reduced costs, FG { FG : d 0}. Step 3. For FG defne scaled reduced costs d [0,1] as follows: d d /(max FG d ) Step 4. For a fxed parameter (0,1) FG { FG : d } Step 5. Solve the nteger problem (1)-(6), (9) correspondng to the grd G fxng x 0 FG G. for 2456

6 4. Computatonal results In ths secton we numercally compare LP-relaxatons obtaned for dfferent grds and study the mpact of ntroducng vald nequaltes for the case of pacng equal crcular objects wthout the lmts (2) for the avalablty of the objects. A rectangular unform grd of sze ( s the space ) along both sdes of the contaner was used to form an ntal grd. The test bed set of 9 nstances from [8, Table 3] was used for pacng maxmal number of objects nto a rectangular contaner of wdth 3 and heght 6. All optmzaton problems were solved by the system CPLEX The runs were executed on a destop computer wth CPU AMD FX core processor 4 Ghz and 32Gb RAM. Frst, we compare lnear programmng bounds obtaned by dfferent grds for crcular objects defned by the Eucldan norm (crcles). The LP-bound was calculated for the problem (1)-(4), (9), that s non-overlappng constrants (5) were substtuted for vald nequaltes (9). The followng three grds were used: orgnal grd of sze generated the same way as n [8, Table 3] wth n nodes and two refned grds wth n /2 and n /3 nodes obtaned by reducng the orgnal grd sze to /2and /3, respectvely. The results of the numercal experment are gven n Table 1. Here the frst three columns show the characterstcs of the nstances, number of nstance, radus R used to defne the crcular object and orgnal sze of the grd, the four column show the optmal nteger soluton z I obtaned for the grd. The rest of the columns gve the number of grd ponts ( n ), value of the correspondng LPrelaxaton ( ) and CPU tme (n seconds). # R zi n CPU n /2 CPU n / Table 1. LP-bounds for crcles In the second part of the expermentaton, we compare the best bounds obtaned usng the orgnal model nteger wth the bounds obtaned wth the heurstc, the objectve s prove f the bounds obtaned wth the heurstc are of qualty smlar. Table 2 provdes results obtaned by the heurstc proposed n the prevous secton for 0.1 and the grd wth n /2nodes used to form the relaxed problem. Here the frst four columns present nstance number, number of nteger varables n correspondng to the orgnal grd, optmal nteger soluton zi and the correspondng CPU tme. For all nteger problems n Table 2 mpgap = 0 was set for runnng CPLEX. Computatons to get nteger soluton zi were nterrupted after the computatonal tme exceeded 12 hours CPU tme and the value n parenthess gves the correspondng mpgap. More detals on gettng z I one can fnd n [15]. The last CPU 2457

7 three columns gve the number of nteger varables n the reduced problem ( n reduced ), correspondng nteger soluton ( z H ) and CPU tme. For the heurstc the tme lmt was set to 1 hour CPU. # n z I CPU nreduced zh CPU (5%) > 12 h (5%) > 12 h (1.3%) 3600 Table 2. Heurstc solutons for crcles Tables 3, 4 present LP-bounds and heurstc solutons obtaned for pacng regular octagons correspondng to 1/ 2 n (7). In both Tables CPU tme was lmted to 1 hour. # R zi n CPU n /2 CPU n /3 CPU Table 3. LP-bounds for octagons As we can see from Tables 1, 3 refnng the grd typcally (but not always) results n mprovng the LP-bound. However, solvng LP-relaxaton for fne grds may be computatonally too expensve. Concernng the qualty of the nteger soluton obtaned by the heurstc, we may conclude that n most cases (except for the nstance 9 for pacng crcles) the optmal soluton was obtaned. The use of heurstc reduces CPU tme sgnfcantly. 2458

8 # n z I CPU nreduced zh CPU (4.8%) > 12 h (5.7%) 3600 Table 4. Heurstc solutons for octagons Fgures 1-4 present optmal pacng and grd ponts left after heurstc node-reducton based on reduced costs. Fg.1. Crcles, nstance 2 Fg.2. Crcles, nstance

9 Fg.3. Octagons, nstance 2 Fg.4. Octagons, nstance 6 4. Conclusons An nteger formulaton and LP-based heurstc were proposed for approxmated pacng crcular-le objects n a rectangular contaner. Dfferent shapes of the objects, such as crcles, ellpses, rhombuses, rectangles, octagons can be consdered by smply changng the norm used to defne the dstance. The presented approach can be easly generalzed to the three (and more) dmensonal case and to dfferent shapes of the contaner, ncludng rregulars. Vald nequaltes are proposed to strengthenng the orgnal formulaton. A heurstc approach s proposed based on analyss of the reduced costs obtaned by LP-relaxaton. An nterestng drecton for the future research s to study the use of Lagrangan relaxaton and correspondng heurstcs (see I. Ltvnchev, S. Rangel and J. Saucedo (2010)) to cope wth large dmenson of arsng problems. References Aeb, H. and Hf, M. (2013), Solvng the crcular open dmenson problem usng separate beams and loo-ahead strateges, Computers & Operatons Research, 40, Baltacoglu, E., Moore, J.T. and Hll, R.R. (2006), The dstrbutor s three-dmensonal pallet-pacng problem: a human-based heurstcal approach, Internatonal Journal of Operatons Research, 1, Beasley, J.E. (1985), An exact two-dmensonal non-gullotne cuttng tree search procedure, Operatons Research, 33, Brgn, E.G. and Gentl, J.M. (2010), New and mproved results for pacng dentcal untary radus crcles wthn trangles, rectangles and strps, Computers & Operatons Research, 37,

10 Castllo, I., Kampas, F.J. and Pnter, J.D. (2008), Solvng crcle pacng problems by global optmzaton: Numercal results and ndustral applcatons, European Journal of Operatonal Research, 191, Fasano, G, Solvng Non-standard Pacng Problems by Global Optmzaton and Heurstcs. Sprnger- Verlag, Frazer, H.J. and George, J.A. (1994), Integrated contaner loadng software for pulp and paper ndustry, European Journal of Operatonal Research, 77, Galev, S.I. and Lsafna, M.S (2013), Lnear models for the approxmate soluton of the problem of pacng equal crcles nto a gven doman, European Journal of Operatonal Research, 230, George, J.A. (1996), Multple contaner pacng: a case study of ppe pacng, Journal of the Operatonal Research Socety, 47, Hf, M. and, R. (2009), A lterature revew on crcle and sphere pacng problems: models and methodologes, Advances n Operatons Research, vol. 2009, 22 pages, do: /2009/ Ltvnchev, I., Rangel, S. and Saucedo, J. (2010), A Lagrangan bound for many-to-many assgnment problem, Journal of Combnatoral Optmzaton, 19, Ltvnchev, I. and Ozuna, L., Pacng crcles n a rectangular contaner, Proc. Intl. Congr. on Logstcs and Supply Chan, Queretaro, Mexco, October 24-25, Ltvnchev, I. and Ozuna, L. (2014), Integer programmng formulatons for approxmate pacng crcles n a rectangular contaner, Mathematcal Problems n Engneerng. Artcle ID , do: /2014/ Ltvnchev, I. and Ozuna, L. (2014), Approxmate pacng crcles n a rectangular contaner: vald nequaltes and nestng, Journal of Appled Research and Technologes, 12, Ltvnchev, I., Infante, L. and Ozuna, L. (2014), Approxmate crcle pacng n a rectangular contaner: nteger programmng formulatons and vald nequaltes, Lecture Notes n Computer Scence, 8760, Lopez, C.O. and Beasley, J.E. (2011), A heurstc for the crcle pacng problem wth a varety of contaners, European Journal of Operatonal Research, 214, Lopez, C.O. and Beasley, J.E. (2013), Pacng unequal crcles usng formulaton space search, Computers & Operatons Research, 40, Stoyan, Y.G. and Yasov, G.N. (2013), Pacng congruent spheres nto a mult-connected polyhedral doman, Internatonal Transactons n Operatonal Research, 20, Toledo, F.M.B., Carravlla, M.A., Rbero, C., Olvera, J.F. and Gomes, A.M. (2013), The dottedboard model: a new MIP model for nestng rregular shapes, Internatonal Journal of Producton Economcs, 145, Wang, J. (1999), Pacng of unequal spheres and automated radosurgcal treatment plannng, Journal of Combnatoral Optmzaton, 3, Wolsey, L.A., Integer Programmng. Wley, New Yor,