Simulated annealing for weighted polygon packing

Otob er 26, 2021 23:46 Engineering Optimization submission Engine ering Optimization V ol. 00, No. 00, Mon th 200x, 113 RESEAR CH AR TICLE Sim ulated Annealing for W eigh ted P olygon P a king Yi-Ch un Xu a , Ren-Bin Xiao b and Mart yn Amos c, † a Institute of Intel ligent Vision & Image Information, China Thr e e Gor ges University, China b Institute of System Engine ering, Huazhong University of Sien e & T e hnolo gy, China c Dep artment of Computing & Mathematis, Manhester Metr op olitan University, UK ( R e  eive d 00 Month 200x; nal version r e  eive d 00 Month 200x ) In this pap er w e presen t a new algorithm for a la y out optimization problem: this onerns the plaemen t of w eigh ted p olygons inside a irular on tainer, the t w o ob jetiv es b eing to minimize im balane of mass and to minimize the radius of the on tainer. This problem arries real pratial signiane in industrial appliations (su h as the design of satellites), as w ell as b eing of signian t theoretial in terest. Previous w ork has dealt with irular or retangular ob jets, but here w e deal with the more realisti ase where ob jets ma y b e represen ted as p olygons and the p olygons are allo w ed to rotate. W e presen t a solution based on sim ulated annealing and rst test it on instanes with kno wn optima. Our results sho w that the algorithm obtains on tainer radii that are lose to optimal. W e also ompare our metho d with existing algorithms for the (sp eial) retangular ase. Exp erimen tal results sho w that our approa h out-p erforms these metho ds in terms of solution qualit y . 1. In tro dution The L ayout Optimization Pr oblem (LOP) onerns the ph ysial plaemen t of in- strumen ts or piees of equipmen t in a spaeraft or satellite. Beause these ob jets ha v e mass, the system is sub jet to additional onstrain ts (b ey ond simple Cartesian pa king) that aet our solution. The t w o main onstrain ts that w e handle in this pap er are (1) the spae o upied b y a giv en olletion of ob jets (en v elopmen t), and (2) the non-equilibrium (i.e. im balane) of the system. The rest of the pap er is organized as follo ws: In Setion 2 w e rst presen t a detailed desription of the prob- lem, and desrib e previous related resear h. In Setion 3 w e desrib e our algorithm, and in Setion 4 w e giv e the results of n umerial exp erimen ts. 2. The La y out Optimization Problem in Satellites The La y out Optimization Problem (LOP) w as prop osed b y F eng et al. (5) in 1999, and has signian t impliations for the ost and p erformane of devies su h as satellites and spaeraft. It onerns the two dimensional ph ysial plaemen t of a olletion of obje ts (instrumen ts or other piees of equipmen t) within a spae- raft/satellite abinet", or  ontainer . The LOP is demonstrably NP-hard (6). Early w ork on this problem (11, 13, 15 , 17 ) almost alw a ys mo deled ob jets as irles in or- der to simplify the pa king pro ess. Ho w ev er, in real-w orld appliations, ob jets are generally retangular or p olygonal in shap e, and mo deling them as irles leads to † Email: M.Amosmm u.a.uk ISSN: 0305-215X prin t/ISSN 1029-0273 online   200x T a ylor & F ranis DOI: 10.1080/0305215YY xxxxxxx h ttp://www.informa w orld.om Otob er 26, 2021 23:46 Engineering Optimization submission 2 exp ensiv e w astage of spae. W e ha v e reen tly rep orted w ork on solving the retan- gular ase (14 ), and here w e rep ort a new algorithm (based on a dieren t approa h) to solv e the p olygonal ase. W e no w briey in tro due related w ork on the pa king of irregular items. Do wsland et al. use a so-alled "Bottom-Left" strategy to plae p olygonal items in a bin (3 ), with items ha ving xed orien tations. P osh y anonda et al. (12 ) om bine geneti algorithms with artiial neural net w orks to obtain reasonable pa king densities. In other related w ork, Bergin et al. study the pa king of iden tial retangles in an irregular on tainer (1, 7). Burk e and Kendall ha v e applied sim ulated annealing to translational p olygon pa king (i.e., without rotations) (2). Other authors ha v e applied sim ulated annealing to solv e the problem of rotational p olygon pa king on a on tin uous spae (8, 9 ). Giv en the additional onstrain t that im balane of mass m ust b e minimised, it is diult to see ho w these existing metho ds ma y b e diretly applied to the urren t problem. In what follo ws w e desrib e a new nonlinear optimization mo del for the LOP , and then sho w ho w it ma y b e solv ed using sim ulated annealing. 2.1. Notation and Denitions Here w e desrib e the formal optimization mo del, b y rst explaining our notation for the represen tation of p olygons. W e then sho w ho w to quan tify relations b et w een p olygons (su h as distane and degree of o v erlap), whi h are en tral to the problem of assessing the o v erall qualit y of a la y out. 2.1.1. Strutur e of a p olygon Supp ose there are k p olygons (1 , 2 , . . . , k ) to b e pa k ed. The strutur e of a p olygon inludes b oth its shap e and its mass . W e use str ( i ) to denote the initial struture of a p olygon, i : str ( i ) = ( n i , m i , ( r 1 , r 2 , ..., r n i ) , ( θ 1 , θ 2 , ..., θ n i )) (1) where m i is the mass of p olygon i , and n i is the n um b er of v erties in the graph represen tation of p olygon i . The p ositions of the n i v erties are dened b y t w o lists of p olar o ordinates. List ( r 1 , r 2 , ..., r n i ) denes the Eulide an distan e from ea h of the n i v erties to the p olygon's en tre of mass, and list ( θ 1 , θ 2 , ..., θ n i ) denes the orientation of ea h of the n i v erties relativ e to the en tre of mass. Figure 1 sho ws ho w to dene the initial struture of a square with edge length 1; in Figure 1 (a), the shap e's en tre of mass is lo ated at the shap e en tre, whereas in Figure 1 (b), the en tre of mass is lo ated at one v ertex. W e dene the p oin t of referene of ea h p olygon as its en ter of mass, in order to simplify the notation. 2.1.2. R adius of a p olygon The radius of p olygon i is dened as the maxim um of ( r 1 , r 2 , ..., r n i ) : r ( i ) = max { r 1 , r 2 , ..., r n i } (2) With the p olygon's en tre of mass at its o wn en tre, the irle with radius r ( i ) denes the minim um-sized irle that an ompletely o v er the p olygon. Otob er 26, 2021 23:46 Engineering Optimization submission 3 Figure 1. Illustration of the struture denition of p olygons Figure 2. Illustration of the state of a p olygon 2.1.3. State of a p olygon W e use Cartesian o ordinates to reord the p ositions of the p olygons, and set the en ter of the on tainer (that is, the irle) as the original p oin t. W e use sta ( i ) to denote the state of a p olygon i : sta ( i ) = ( x i , y i , α i ) (3) where x i , y i is the p osition of the en tre of mass, and α denes a rotation angle. Then with str ( i ) and sta ( i ) , w e an dra w a p olygon, i , as in Figure 2 . 2.1.4. Distan e b etwe en two p olygons The distane b et w een t w o p olygons is dened as the Eulidean distane b et w een their en tres of mass: dis ( i, j ) = q ( x i − x j ) 2 + ( y i − y j ) 2 ) (4) 2.1.5. Overlap b etwe en two p olygons If t w o p olygons do not o v erlap, this measure is zero. If t w o p olygons i and j o v erlap at all, w e measure this as: Otob er 26, 2021 23:46 Engineering Optimization submission 4 Figure 3. The o v erlap funtion is not on tin uous ov e ( i, j ) = max { 0 , r ( i ) + r ( j ) − d is ( i, j ) } (5) This measuremen t of o v erlap has ertain  harateristis: • In Equation (5), r ( i ) and r ( j ) are t w o onstan ts, therefore dis ( i, j ) an b e diretly obtained b y the p ositions of the t w o p olygons. • It is lear that ov e ( i, j ) ≥ 0 . T o satisfy the non-overlapping onstrain t, w e should minimize ov e ( i, j ) to zero. • ov e ( i, j ) is not a on tin uous funtion of the p ositions. As sho wn in Figure 3, when t w o squares with edge length 2 are adjaen t on one side, their o v erlap is zero, but when the left square is mo v ed a little to the righ t, the o v erlap jumps" to 2 √ 2 − 2 . Asertaining o v erlap b et w een t w o p olygons is not a diult problem in ompu- tational geometry or omputer graphis. In this pap er, w e lo ok at ea h edge of one p olygon in turn; if it is in terseted b y an y edge of another p olygon, then an o v erlap exists; otherwise, if one p olygon is on tained within another, then learly an o v erlap exists. So the asertaining of o v erlap has omplexit y O ( mn ) , where m, n are the n um b er of edges of the t w o p olygons. 2.1.6. State of a layout A la y out X is dened as the om bination of the states of k p olygons: X = ( x 1 , y 1 , α 1 , x 2 , y 2 , α 2 , ..., x k , y k , α k ) (6) 2.1.7. R adius of a layout If ( X x , X y ) denotes the p osition of the en tre of mass of a la y out X , then X x = P k i =1 m i x i P k i =1 m i , X y = P k i =1 m i y i P k i =1 m i . (7) W e dene the radius r ( X ) of a la y out X as the longest Eulidean distane from its en tre of mass to an y of the v erties of the p olygons. Beause of the imb alan e onstrain t, w e plae the en tre of mass of the la y out at the on tainer en ter, so r ( X ) denes the minim um-sized on tainer. 2.1.8. Overlap of a layout The o v erlap of a la y out is the sum of all o v erlaps b et w een its p olygons: Otob er 26, 2021 23:46 Engineering Optimization submission 5 ov e ( X ) = k X i =1 k X j =1 , j 6 = i ov e ( i, j ) (8) 2.1.9. Pr oblem denition F rom the denitions ab o v e, w e obtain an unonstrained optimization problem: minimize f ( X ) = λ 1 ov e ( X ) + λ 2 r ( X ) (9) where λ 1 , λ 2 are t w o onstan ts. Beause the o v erlap funtion ov e ( X ) is not on- tin uous, f ( X ) is not on tin uous. In general, the o v erlaps are extremely deleterious, so λ 1 should b e set large enough to prev en t their in tro dution. Ho w ev er, w e note that the omputation do es not in tro due o v erlaps when attempting to derease the radius of the la y out at the nal stage of optimization, b eause of the dison tin uous of the ov e funtion. 3. Sim ulated annealing algorithm Simulate d anne aling (SA) is a probabilisti meta-heuristi that is w ell-suited to global optimization problems (10 ). A nne aling refers to the pro ess of heating then slo wly o oling a material un til it rea hes a stable state. The heating enables the material to a hiev e higher in ternal energy states, while the slo w o oling allo ws the material more opp ortunit y to nd an in ternal energy state lower than the initial state. SA mo dels this pro ess for the purp oses of optimization. A p oin t in the sear h spae is regarded as a system state, and the ob jetiv e funtion is regarded as the in ternal energy . Starting from an initial state, the system is p erturb ed at random, mo ving to a new state in the neigh b ourho o d, and a  hange of energy ∆ E tak es plae. If ∆ E < 0, the new state is aepted (a downhil l mo v e), otherwise the new state is aepted with a probabilit y exp( − ∆ E K b T ) (an uphil l mo v e), where T is the temp erature at that time and K b is Boltzmann's onstan t. When the system rea hes equilibrium, T is dereased. When the temp erature approa hes zero, the probabilit y of an uphill" mo v e b eomes v ery small, and SA terminates. Let t 0 denote the initial temp erature, imax denote the maxim um n um b er of itera- tions, E ( x ) denote the energy funtion, and emax denote the "stopping" energy .The general pseudo-o de for sim ulated annealing ma y b e written as follo ws: Algorithm 1 : Standard SA set the initial state x and initial temp erature t = t 0 , let i = 0 while i < imax and E ( x ) > emax p erturb x in its neigh b ourho o d and get x ′ if E ( x ′ ) < E ( x ) then x = x ′ else x = x ′ with probabilit y exp( E ( x ) − E ( x ′ ) t ) derease the temp erature i = i + 1 return x The p erformane of SA ma y b e aeted b y sev eral parameters: the initial temp er- Otob er 26, 2021 23:46 Engineering Optimization submission 6 ature t 0 , the maxim um n um b er of iterations, imax , the stopping" energy , emax , the struture of the neigh b ourho o d, and the s hedule of o oling. F or a giv en prob- lem, the v alues of these parameters should b e arefully seleted. 3.1. SA for p olygon p aking 3.1.1. Neighb ourho o d strutur e In ea h iteration of SA, w e p erturb one p olygon, th us obtaining a new la y out, and then deide whether to aept or rejet the new la y out b y means of an ev aluation. In the i th iteration, the ( i mod k ) th p olygon will b e p erturb ed. Giv en an initial radius R 0 , whi h is large enough to on tain the p olygons, the neigh b ourho o d for p olygon j is dened as: x j , y j ∈ ( imax i − 2 × imax + 1 . 05 ) × R 0 × r andom ( − 1 , 1) (10) α j ∈ ( imax i − 2 × i max + 1 . 05 ) × π × r andom ( − 1 , 1) (11) Equations (10) and (11) sho w that, at the b eginning of the algorithm's exeution, the p osition of a p olygon ma y v ary b y ( − 0 . 55 R 0 , 0 . 55 R 0 ) , and its orien tation ma y b e p erturb ed b y ( − 0 . 55 π, 0 . 55 π ). This neigh b ourho o d is large, and and the p olygon an th us explore" more spae. As the algorithm pro eeds, the neigh b ourho o d b eomes inreasingly smaller. A t the end of the algorithm's exeution, the neigh b ourho o d shrinks to 0.05 times its original size, then SA  ho oses the b est solution in the neigh b ourho o d. 3.1.2. T emp er atur e de r e asing W e use a simple rule to derease the temp erature: ev ery cmax iterations, w e let t = d × t , where d < 1 . 3.1.3. Desription of the algorithm The detailed SA algorithm for p olygon pa king is therefore desrib ed as follo ws: Algorithm 2 : SA for the pa king problem randomly generate an initial la y out X . Let t = t 0 , i = 0 while i < imax and f ( X ) > emax let j = ( i mod k ) randomly selet x j , y j , α j b y (10) and (11 ) and get new X ′ if f ( X ′ ) < f ( X ) then X = X ′ else X = X ′ with probabilit y exp( f ( X ) − f ( X ′ ) t ) if i mod cmax = cmax − 1 then t = d × t i = i + 1 return x Otob er 26, 2021 23:46 Engineering Optimization submission 7 T able 1. F our instanes with kno wn optima Instane k R 0 Struture 1 5 2.3 str ( i ) = (4 , 30 , ( √ 10 2 , √ 10 2 , √ 10 2 , √ 10 2 ) , ( atan ( 1 3 ) , π − atan ( 1 3 ) , π + atan ( 1 3 ) , 2 π − atan ( 1 3 )) , i = 1 , 2 str ( i ) = (4 , 10 , ( √ 2 2 , √ 2 2 , √ 2 2 , √ 2 2 ) , ( π 4 , 3 4 π , 5 4 π , 7 4 π ) , i = 3 , 4 , 5 2 5 2.8 str ( i ) = (5 , 100 , (2 , 2 √ 2 − 2 , 2 , √ 2 , √ 2) , (0 , 1 4 π , 1 2 π , 5 4 π , 7 4 π )) , i = 1 , 2 , 3 , 4 , str ( i ) = (4 , 100 , ( √ 2 , √ 2 , √ 2 , √ 2) , (0 , 1 4 π , 1 2 π , 3 2 π )) , i = 5 3 6 3.4 str ( i ) = (3 , 100 , (2 , 2 , 2) , (0 , 2 3 π , 4 3 π ) , i = 1 , 2 , 3 , 4 , 5 , 6 4 12 5.0 str ( i ) = (3 , 10 , (1 , 1 , 1) , (0 , π , 3 2 π )) , i = 1 , 2 , 3 , 4 , str ( i ) = (4 , 20 , (2 , 2 , √ 2 , √ 2) , (0 , π , 5 4 π , 7 4 π )) , i = 5 , 6 , 7 , 8 , str ( i ) = (4 , 20 , (2 , 2 , √ 2 , √ 2) , (0 , π , 5 4 π , 7 4 π ) , i = 9 , 10 , 11 , 12 5 3 8.0 str ( i ) = (4 , 40 , (2 √ 2 , , 2 √ 2 , √ 2 , 2 √ 2) , ( 1 4 π , 3 4 π , 5 4 π , 7 4 π )) , i = 1 , str ( i ) = (8 , 60 , (2 √ 2 , 2 √ 5 , 2 √ 5 , 2 √ 5 , 2 √ 5 , 2 √ 2 , 2 , 2) , ( 1 4 π , atan (2) , π − atan (2) , π + atan (2) , − atan (2) , − 1 4 π , − 1 2 π , 1 2 π )) , i = 2 , 3 6 5 5.0 str ( i ) = (4 , 60 , ( √ 2 , √ 2 , √ 2 , √ 2) , ( 1 4 π , 3 4 π , 5 4 π , 7 4 π )) , i = 1 , 2 , 3 , 4 str ( i ) = (12 , 500 , ( √ 10 , √ 10 , √ 2 , √ 10 , √ 10 , √ 2 , √ 10 , √ 10 , √ 2 , √ 10 , √ 10 , √ 2) , ( − at an ( 1 3 ) , atan ( 1 3 ) , 1 4 π , 1 2 π − atan ( 1 3 ) , 1 2 π + atan ( 1 3 ) , 3 4 π , π − atan ( 1 3 ) , π + atan ( 1 3 ) , 5 4 π , 3 2 π − atan ( 1 3 ) , 3 2 π + atan ( 1 3 ) , 7 4 π )) , i = 5 4. Numerial Results W e are not a w are of an y standard library of b en hmark instanes for this p arti- ular problem, although su h libraries do exist for other related problems ( 4). W e therefore tak e a t w o-stage approa h to testing our algorithm; w e rst design six in- stanes with known optima, against whi h w e ma y initially v alidate our metho d. W e note that these instanes inlude b oth on v ex and nonon v ex p olygons. After es- tablishing the eetiv eness of our SA algorithm, w e then test our algorithm against other reen tly-desrib ed metho ds for r e tangle pa king (retangles, of ourse, b eing mem b ers of the p olygon lass), using b oth existing instanes from the literature and new, larger instanes. 4.1. Known Optima The instanes with kno wn optima are desrib ed in T able 1, with graphial repre- sen tations giv en in Figure 4 . F or ea h instane, w e use SA to try to nd the optimal la y out. The v alue of imax is set to 20000 × k , the v alue of cmax is set to 100 × k , and the initial temp erature is set to 100. The onstan ts λ 1 and λ 2 are set to 100 (to indue a large f ( x ) and adjust the probabilit y of uphill mo v emen t). In ea h ase, the algorithm is exeuted 40 times. V alues for the b est radius found, r best , mean radius, ¯ r , and v ariane v are pre- sen ted in T able 2 . Represen tations of the b est results obtained are giv en in Figure 5 . F rom T able 2 and Figure 5, w e observ e that the SA algorithm an nd la y outs that are v ery lose to the the optimal onguration for instanes 1, 2, and 3, where the n um b er of p olygons is relativ ely small and the o v erall strutures are simple. The optimal radius for instane 1 w as originally alulated as 3 2 √ 2 = 2 . 121 , but the our results yielded a smaller v alue. This prompted a re-estimation, giving a new optim um of q 4 9 64 ). In the rst three instanes, the errors to the optim um are Otob er 26, 2021 23:46 Engineering Optimization submission 8 Figure 4. Instanes with kno wn optima. ab out r best − r optimum r optimum = 2% . The algorithm p erforms less w ell on instane 4, with 12 p olygons in a relativ ely omplex onguration. In this ase, our algorithm annot nd the b est onguration, and the error to the optim um is 15%. Instanes 5 and 6 feature nono v ex p olygons. Beause of the omplexit y of the shap es, the algorithm is unable to solv e these instanes to optimalit y . The errors to the optimal radius are 11% and 4% for instanes 5 and 6 resp etiv ely . T able 2. Numerial results for SA on four instanes with kno wn optima Instane Est. r optimum r best r v 1 q 4 9 64 = 2 . 034 2.080 2.167 0.027 2 2 √ 2 = 2 . 828 2.861 3.209 0.010 3 2 √ 3 = 3 . 464 3.522 4.065 0.157 4 3 √ 2 = 4 . 242 4.887 5.149 0.024 5 4 √ 2 = 5 . 656 6.295 9.266 59.25 6 3 √ 2 = 4 . 242 4.423 7.034 119.44 4.2. R e tangular Instan es W e no w test our algorithm on four instanes of the LOP on taining only retangu- lar shap es. The rst three instanes w ere rst desrib ed in ( 16 ), and the fourth in (14 ), where all four instanes w ere used to b en hmark three dieren t approa hes: a geneti algorithm (GA), partile sw arm optimisation (PSO) and a h ybrid om- pation algorithm follo w ed b y partile sw arm lo al sear h (CA-PSLS). Depitions of ea h instane are depited in Figure 6 , and full desriptions are giv en in T able 3 . Sine b oth partile-based algorithms out-p erformed the GA, w e do not onsider this last metho d here. W e run ea h algorithm 50 times on ea h instane, reording the b est radius found, r best , a v erage radius, r , standard deviation of radii, r σ and a v erage run time in seonds, t . Ea h algorithm is o ded in C, ompiled with g++ 4.1.0 , and run Otob er 26, 2021 23:46 Engineering Optimization submission 9 Figure 5. Best results obtained for instanes with kno wn optima Figure 6. F our instanes of the LOP using retangles under SUSE Lin ux 10.1 (k ernel v ersion 2.6.16.54-0.2.5-smp) on a omputer with dual In tel Harp erto wn E5462 2.80GHz pro essors, 4GB of RAM and an 80GB hard driv e. The three algorithms (SA, CA-PSLS and PSO) ea h run o v er a n um b er of it- erations, whi h is ditated b y the v alue of the onstan t CYCLE. In this set of exp erimen ts, w e set CYCLE=3000 for ea h algorithm. The SA parameter v alues for cmax , initial temp erature, λ 1 , and λ 2 are set as b efore, and the imax v alues set as to 100000, 120000, 108000 and 100000 for instanes 1-4 resp etiv ely . The results obtained are depited in T able 4. On the rst three (small) instanes, b oth partile-based algorithms sligh tly out- p erform the SA metho d in terms of solution quality ; on a v erage, b y 10%. Ho w ev er, this omes at a signian t ost disadv an tage in terms of run time; o v er the rst three instanes, the partile-based metho ds require four times the exeution time Otob er 26, 2021 23:46 Engineering Optimization submission 10 T able 3. F our instanes from the literature Instane k R 0 Struture 1 5 20 str (1) = (4 , 12 , ( √ 25 , √ 25 , √ 25 , √ 25) , ( atan ( 3 4 )) , π − at an ( 3 4 ) , π + atan ( 3 4 ) , 2 π − a tan ( 3 4 ))) str (2) = (4 , 16 , ( √ 32 , √ 32 , √ 32 , √ 32) , ( atan ( 4 4 )) , π − at an ( 4 4 ) , π + atan ( 4 4 ) , 2 π − a tan ( 4 4 ))) str (3) = (4 , 15 , ( √ 34 , √ 34 , √ 34 , √ 34) , ( atan ( 3 5 )) , π − at an ( 3 5 ) , π + atan ( 3 5 ) , 2 π − a tan ( 3 5 ))) str (4) = (4 , 12 , ( √ 40 , √ 40 , √ 40 , √ 40) , ( atan ( 2 6 )) , π − at an ( 2 6 ) , π + atan ( 2 6 ) , 2 π − a tan ( 2 6 ))) str (5) = (4 , 9 , ( √ 18 , √ 18 , √ 18 , √ 18) , ( atan ( 3 3 )) , π − at an ( 3 3 ) , π + atan ( 3 3 ) , 2 π − ata n ( 3 3 ))) 2 6 40 str (1) = (4 , 12 , ( √ 25 , √ 25 , √ 25 , √ 25) , ( atan ( 3 4 )) , π − at an ( 3 4 ) , π + atan ( 3 4 ) , 2 π − a tan ( 3 4 ))) str (2) = (4 , 16 , ( √ 32 , √ 32 , √ 32 , √ 32) , ( atan ( 4 4 )) , π − at an ( 4 4 ) , π + atan ( 4 4 ) , 2 π − a tan ( 4 4 ))) str (3) = (4 , 15 , ( √ 34 , √ 34 , √ 34 , √ 34) , ( atan ( 3 5 )) , π − at an ( 3 5 ) , π + atan ( 3 5 ) , 2 π − a tan ( 3 5 ))) str (4) = (4 , 20 , ( √ 41 , √ 41 , √ 41 , √ 41) , ( atan ( 4 5 )) , π − at an ( 4 5 ) , π + atan ( 4 5 ) , 2 π − a tan ( 4 5 ))) str (5) = (4 , 25 , ( √ 50 , √ 50 , √ 50 , √ 50) , ( atan ( 5 5 )) , π − at an ( 5 5 ) , π + atan ( 5 5 ) , 2 π − a tan ( 5 5 ))) str (6) = (4 , 18 , ( √ 45 , √ 45 , √ 45 , √ 45) , ( atan ( 3 6 )) , π − at an ( 3 6 ) , π + atan ( 3 6 ) , 2 π − a tan ( 3 6 ))) 3 9 40 str (1) = (4 , 12 , ( √ 25 , √ 25 , √ 25 , √ 25) , ( atan ( 3 4 )) , π − at an ( 3 4 ) , π + atan ( 3 4 ) , 2 π − a tan ( 3 4 ))) str (2) = (4 , 16 , ( √ 32 , √ 32 , √ 32 , √ 32) , ( atan ( 4 4 )) , π − at an ( 4 4 ) , π + atan ( 4 4 ) , 2 π − a tan ( 4 4 ))) str (3) = (4 , 15 , ( √ 34 , √ 34 , √ 34 , √ 34) , ( atan ( 3 5 )) , π − at an ( 3 5 ) , π + atan ( 3 5 ) , 2 π − a tan ( 3 5 ))) str (4) = (4 , 20 , ( √ 41 , √ 41 , √ 41 , √ 41) , ( atan ( 4 5 )) , π − at an ( 4 5 ) , π + atan ( 4 5 ) , 2 π − a tan ( 4 5 ))) str (5) = (4 , 25 , ( √ 50 , √ 50 , √ 50 , √ 50) , ( atan ( 5 5 )) , π − at an ( 5 5 ) , π + atan ( 5 5 ) , 2 π − a tan ( 5 5 ))) str (6) = (4 , 12 , ( √ 40 , √ 40 , √ 40 , √ 40) , ( atan ( 2 6 )) , π − at an ( 2 6 ) , π + atan ( 2 6 ) , 2 π − a tan ( 2 6 ))) str (7) = (4 , 18 , ( √ 45 , √ 45 , √ 45 , √ 45) , ( atan ( 3 6 )) , π − at an ( 3 6 ) , π + atan ( 3 6 ) , 2 π − a tan ( 3 6 ))) str (8) = (4 , 24 , ( √ 52 , √ 52 , √ 52 , √ 52) , ( atan ( 4 6 )) , π − at an ( 4 6 ) , π + atan ( 4 6 ) , 2 π − a tan ( 4 6 ))) str (9) = (4 , 30 , ( √ 61 , √ 61 , √ 61 , √ 61) , ( atan ( 5 6 )) , π − at an ( 5 6 ) , π + atan ( 5 6 ) , 2 π − a tan ( 5 6 ))) 4 20 100 str (1) = (4 , 10 , ( √ 22 . 25 , √ 22 . 25 , √ 22 . 25 , √ 22 . 25) , ( atan ( 2 . 5 4 )) , π − atan ( 2 . 5 4 ) , π + a tan ( 2 . 5 4 ) , 2 π − at a n ( 2 . 5 4 ))) str (2) = (4 , 8 , ( √ 20 , √ 20 , √ 20 , √ 20) , ( atan ( 4 2 )) , π − at an ( 4 2 ) , π + atan ( 4 2 ) , 2 π − ata n ( 4 2 ))) str (3) = (4 , 15 , ( √ 34 , √ 34 , √ 34 , √ 34) , ( atan ( 3 5 )) , π − at an ( 3 5 ) , π + atan ( 3 5 ) , 2 π − a tan ( 3 5 ))) str (4) = (4 , 14 , ( √ 28 . 25 , √ 28 . 25 , √ 28 . 25 , √ 28 . 25) , ( atan ( 4 3 . 5 )) , π − atan ( 4 3 . 5 ) , π + a tan ( 4 3 . 5 ) , 2 π − at a n ( 4 3 . 5 ))) str (5) = (4 , 7 . 50 , ( √ 27 . 25 , √ 27 . 25 , √ 27 . 25 , √ 27 . 25) , ( atan ( 1 . 5 5 )) , π − at an ( 1 . 5 5 ) , π + atan ( 1 . 5 5 ) , 2 π − at a n ( 1 . 5 5 ))) str (6) = (4 , 18 , ( √ 45 , √ 45 , √ 45 , √ 45) , ( atan ( 3 6 )) , π − at an ( 3 6 ) , π + atan ( 3 6 ) , 2 π − a tan ( 3 6 ))) str (7) = (4 , 12 , ( √ 40 , √ 40 , √ 40 , √ 40) , ( atan ( 2 6 )) , π − at an ( 2 6 ) , π + atan ( 2 6 ) , 2 π − a tan ( 2 6 ))) str (8) = (4 , 18 , ( √ 45 , √ 45 , √ 45 , √ 45) , ( atan ( 3 6 )) , π − at an ( 3 6 ) , π + atan ( 3 6 ) , 2 π − a tan ( 3 6 ))) str (9) = (4 , 20 , ( √ 41 , √ 41 , √ 41 , √ 41) , ( atan ( 5 4 )) , π − at an ( 5 4 ) , π + atan ( 5 4 ) , 2 π − a tan ( 5 4 ))) str (10) = (4 , 5 . 25 , ( √ 14 . 50 , √ 14 . 50 , √ 14 . 50 , √ 14 . 50) , ( atan ( 1 . 5 3 . 5 )) , π − atan ( 1 . 5 3 . 5 ) , π + atan ( 1 . 5 3 . 5 ) , 2 π − at a n ( 1 . 5 3 . 5 ))) str (11) = (4 , 12 , ( √ 25 , √ 25 , √ 25 , √ 25) , ( atan ( 3 4 )) , π − atan ( 3 4 ) , π + atan ( 3 4 ) , 2 π − ata n ( 3 4 ))) str (12) = (4 , 6 , ( √ 18 . 25 , √ 18 . 25 , √ 18 . 25 , √ 18 . 25) , ( atan ( 1 . 5 4 )) , π − atan ( 1 . 5 4 ) , π + a tan ( 1 . 5 4 ) , 2 π − at a n ( 1 . 5 4 ))) str (13) = (4 , 15 , ( √ 34 , √ 34 , √ 34 , √ 34) , ( atan ( 3 5 )) , π − atan ( 3 5 ) , π + atan ( 3 5 ) , 2 π − ata n ( 3 5 ))) str (14) = (4 , 20 , ( √ 41 , √ 41 , √ 41 , √ 41) , ( atan ( 4 5 )) , π − atan ( 4 5 ) , π + atan ( 4 5 ) , 2 π − ata n ( 4 5 ))) str (15) = (4 , 17 . 50 , ( √ 37 . 25 , √ 37 . 25 , √ 37 . 25 , √ 37 . 25) , ( atan ( 3 . 5 5 )) , π − at an ( 3 . 5 5 ) , π + a tan ( 3 . 5 5 ) , 2 π − at a n ( 3 . 5 5 ))) str (16) = (4 , 15 , ( √ 42 . 25 , √ 42 . 25 , √ 42 . 25 , √ 42 . 25) , ( atan ( 2 . 5 6 )) , π − at an ( 2 . 5 6 ) , π + atan ( 2 . 5 6 ) , 2 π − at a n ( 2 . 5 6 ))) str (17) = (4 , 12 , ( √ 40 , √ 40 , √ 40 , √ 40) , ( atan ( 2 6 )) , π − atan ( 2 6 ) , π + atan ( 2 6 ) , 2 π − ata n ( 2 6 ))) str (18) = (4 , 20 , ( √ 41 , √ 41 , √ 41 , √ 41) , ( atan ( 4 5 )) , π − atan ( 4 5 ) , π + atan ( 4 5 ) , 2 π − ata n ( 4 5 ))) str (19) = (4 , 30 , ( √ 61 , √ 61 , √ 61 , √ 61) , ( atan ( 5 6 )) , π − atan ( 5 6 ) , π + atan ( 5 6 ) , 2 π − ata n ( 5 6 ))) str (20) = (4 , 9 , ( √ 18 , √ 18 , √ 18 , √ 18) , ( atan ( 3 3 )) , π − at an ( 3 3 ) , π + atan ( 3 3 ) , 2 π − a tan ( 3 3 ))) of the SA algorithm to terminate. When the problem size is inreased to 20, the b enets of the SA algorithm b egin to b eome apparen t, as it out-p erforms the other t w o algorithms in terms of b oth solution qualit y and run time. In order to establish the signiane of this, w e no w test all three metho ds on m u h larger instanes. 4.3. L ar ge R e tangular Instan es W e designed instanes with 40, 60, 80 and 100 retangles. Spae preludes a detailed desription of these, but the full problem set is a v ailable from the orresp onding author. As b efore, ea h metho d w as run 50 times on ea h instane. Beause of the omputational ost inurred, w e redued CYCLE to 1000 for ea h algorithm. The results are depited in T able 5, with an example solution for the 40 retangle instane depited in Figure 7. The SA metho d signian tly out-p erforms the other t w o metho ds in terms of solution qualit y , but with an asso iated ost in terms of run time. Ho w ev er, as sho wn b y the gures for standard deviation, SA oers a onsisten tly high-qualit y solution metho d (at a prie), whereas the other t w o algorithms oer solutions of more v ariable qualit y , but more qui kly . Otob er 26, 2021 23:46 Engineering Optimization submission 11 T able 4. Results for SA, CA-PSLS and PSO on four retangular instanes from the literature (CYCLE=3000) Instane Size Algorithm r best r r σ t (s) 1 5 SA 12.776 13.693 0.62 0.82 CA-PSLS 10.942 11.704 0.49 4.31 PSO 11.046 11.716 0.49 4.89 2 6 SA 16.004 17.377 0.69 1.66 CA-PSLS 14.686 15.590 0.67 7.76 PSO 14.320 15.349 0.56 8.28 3 9 SA 20.849 22.328 0.87 6.84 CA-PSLS 18.157 19.797 1.03 21.07 PSO 18.579 19.205 0.49 22.84 4 20 SA 29.969 31.680 0.98 92.01 CA-PSLS 27.927 33.129 5.48 125.52 PSO 32.596 34.426 2.23 138.00 T able 5. Results for SA, CA-PSLS and PSO on large instanes (CYCLE=1000) Instane Size Algorithm r best r r σ t (s) 1 40 SA 164.061 174.586 4.81 263.24 CA-PSLS 179.508 253.627 60.42 219.84 PSO 242.471 276.939 24.44 197.03 2 60 SA 170.284 187.312 6.32 905.75 CA-PSLS 184.984 288.642 124.26 579.31 PSO 272.282 317.739 26.17 451.72 3 80 SA 265.654 281.087 8.54 2178.31 CA-PSLS 298.524 544.421 162.81 1016.70 PSO 432.347 490.862 35.75 813.59 4 100 SA 406.991 423.087 7.87 4260.54 CA-PSLS 658.352 880.537 108.65 1611.52 PSO 598.265 688.785 48.06 1277.43 5. Conlusions In this pap er w e desrib e a no v el algorithm based on sim ulated annealing for the problem of pa king w eigh ted p olygons inside a irular on tainer. As w ell as b eing of signian t theoretial in terest, this problem has real signiane in domains su h as satellite design in the aerospae industry . Our algorithm onsisten tly generates high-qualit y solutions that oer a signian t impro v emen t o v er those generated b y other metho ds. Ho w ev er, this sup eriorit y omes with an asso iated omputational o v erhead, so the  hoie of metho d should largely b e driv en b y the an tiipated appli- ation. F uture w ork will in v olv e impro ving the metho d's p erformane on problems on taining nonon v ex p olygons, as w ell as its extension in to three dimensions. A  kno wledgemen ts This w ork w as partially supp orted b y the Dalton Resear h Institute, Man hester Metrop olitan Univ ersit y . Otob er 26, 2021 23:46 Engineering Optimization submission 12 REFERENCES Figure 7. Best 40 retangle solution generated b y SA ( r = 134 . 07 ) Referenes [1℄ E. G. Birgin, J. M. Martínez, F.H. Nishihara, and D. P . Rononi. Orthogonal pa king of retangular items within arbitrary on v ex regions b y nonlinear optimization. Computers and op er ations r ese ar h , 33(12):35353548, 2006. [2℄ E. Burk e and G. Kendall. Applying sim ulated annealing and the no t p olygon to the nesting problem. In Pr o  e e dings of the W orld Manufaturing Confer- en e , pages 2730, 1997. [3℄ K.A. Do wsland, V aid S., and W.B. Do wsland. An algorithm for p olygon plaemen t using a b ottom-left strategy . Eur op e an Journal of Op er ational R ese ar h , 141:371381, 2002. [4℄ S.P . F ek ete and J.C. v an der V een. P a kLib2: An in tegrated library of m ulti- dimensional pa king problems. Eur op e an Journal of Op er ational R ese ar h , 183(3):11311135, 2007. [5℄ E. F eng, X.L. W ang, X.M. W ang, and H. T eng. An algorithm of global optimization for solving la y out problems. Eur op e an Journal of Op er ational R ese ar h , 114:430436, 1999. [6℄ R.J. F o wler, M.S. P aterson, and S.L. T animoto. Optimal pa king and o v ering in the plane are NP-omplete. Information Pr o  essing L etters , 12(3):133137, 1981. [7℄ Birgin E. G., J. M. Martínez, W. F. Masarenhas, and D. P . Rononi. Metho d of sen tinels for pa king items within arbitrary on v ex regions. The Journal of the Op er ational R ese ar h So iety , 6(57):735746, 2006. [8℄ R. He kmann and T. Lengauer. A sim ulated annealing approa h to the nest- ing problem in the textile man ufaturing industry . A nnals of Op er ations R ese ar h , 57(1):103133, 1995. [9℄ P . Jain, P . F eynes, and R. Ri h ter. Optimal blank nesting using sim ulated annealing. T r ansations of the A meri an So iety of Me hani al Engine ers , 114:160165, 1992. [10℄ S. Kirkpatri k, C. D. Gelatt, and M. P . V e hi. Optimization b y sim ulated annealing. Sien e , 220(4598):671680, 1983. [11℄ N. Li, F. Liu, and D. Sun. A study on the partile sw arm optimization with m utation op erator onstrained la y out optimization. Chinese Journal of Computers , 27(7):897903, 2004. [12℄ P osh y anonda P . and C.H. Dagli. Geneti neuro-nester. Journal of Intel ligent Manufaturing , 15(2):201218, 2004. Otob er 26, 2021 23:46 Engineering Optimization submission REFERENCES 13 [13℄ F. T ang and H. T eng. A mo died geneti algorithm and its appliation to la y out optimization. Journal of Softwar e , 10:10961102, 1999. [14℄ Yi-Ch un Xu, Ren-Bin Xiao, and Mart yn Amos. P artile sw arm algorithm for w eigh ted retangle plaemen t. In Pr o  e e dings of the 3r d International Confer en e on Natur al Computation (ICNC'07) , pages 728732. IEEE Press, August 24-27, 2007. [15℄ Y. Y u, J. Cha, and X. T ang. Learning based GA and appliation in pa king. Chinese Journal of Computers , 24(12):12421249, 2001. [16℄ J. Zhai, E. F eng, Z. Li, and H. Yin. Non-o v erlapp ed geneti algorithm for la y out problem with b eha vioral onstrain ts. Journal of Dalian University of T e hnolo gy , 39(3):352357, 1999. [17℄ C. Zhou, L. Gao, and H. Gao. P artile sw arm optimization based algorithm for onstrained la y out optimization. Contr ol and De ision , 20(1):3640, 2005.