Ant Colony Algorithm for the Weighted Item Layout Optimization Problem

Transcription

1 Ant Colony Algorthm for the Weghted Item Layout Optmzaton Problem Y-Chun Xu 1, Fang-Mn Dong 1, Yong Lu 1, Ren-Bn Xao 2, Martyn Amos 3,* 1 Insttute of Intellgent Vson and Image Informaton, Chna Three Gorges Unversty, Chna. 2 Insttute of Systems Engneerng, Huazhong Unversty of Scence and Technology, Chna. 3 Department of Computng and Mathematcs, Manchester Metropoltan Unversty, Unted Kngdom. * Correspondng author. Emal: m.amos@mmu.ac.uk. Abstract Ths paper dscusses the problem of placng weghted tems n a crcular contaner n two-dmensonal space. Ths problem s of great practcal sgnfcance n varous mechancal engneerng domans, such as the desgn of communcaton satelltes. Two constructve heurstcs are proposed, one for packng crcular tems and the other for packng rectangular tems. These work by frst optmzng object placement order, and then optmzng object postonng. Based on these heurstcs, an ant colony optmzaton (ACO) algorthm s descrbed to search frst for optmal postonng order, and then for the optmal layout. We descrbe the results of numercal experments, n whch we test two versons of our ACO algorthm alongsde local search methods prevously descrbed n the lterature. Our results show that the constructve heurstc-based ACO performs better than exstng methods on larger problem nstances. Keywords - Layout optmzaton problem; Heurstc; Ant colony optmzaton; I. INTRODUCTION Cuttng and Packng (C&P) problems are optmzaton problems wth practcal sgnfcance to manufacturng ndustres (e.g. when cuttng glass, wood, and leather) or transportaton (Dyckhoff, 1990). Layout optmzaton problems, such as the placement of cells n VLSI desgn (Tang and Yao, 2007) or the layout of artcles n a newspaper (Gonzalez et al., 2002), are related to C&P problems. In ths doman, small tems must be placed n a contaner wthout overlap, accordng to some certan objectves. In general, these optmzaton problems are NP-hard (Dyckhoff, 1990). When constrants are attached to them, they usually become even more complex. 1

2 Exstng approaches for solvng C&P problems nclude local search heurstcs and constructve heurstcs. Local search heurstcs (e.g., genetc algorthms (Tang and Yao, 2007), smulated annealng (Gonzalez et. al, 2002; Cagan et al., 1998), and gradent descent method (Wang et al., 2002)) search the neghborhood of a gven startng soluton and mprove t contnuously, untl a near-optmal soluton s found. Constructve heurstcs place the tems one-by-one n a certan sequence (order); when the fnal tem s placed, a good soluton s thus constructed (Dowsland et. al, 2002; Wu et al, 2002). Several papers have successfully combned these two approaches; they frst use a constructve heurstc to place tems n a gven sequence, and then apply a local search to mprove the layout (Harwg, 2003; Sokea & Bng, 2006). In ths paper, we study the problem of packng weghted tems n a crcular contaner; the so-called Weghted Item Layout problem (WIL). The WIL was frst proposed n (Teng et al., 1994); as descrbed, tems may not overlap, and the whole system must be kept balanced n terms of ts dstrbuton of mass. One sgnfcant real-world applcaton of WIL may be found n the desgn of communcaton satelltes (Teng et al, 2001; Zhang et al., 2008), where balancng the dstrbuton of mass s crtcal to the stablty of a large rotatng cylnder. In order to reduce complexty, the problems we consder here are lmted to twodmensonal space. Two varants of the problem are consdered here; the frst places weghted crcles (Fg1-a), and the second places weghted rectangles (Fg1-b). We name these varants the Weghted-Crcle Layout problem (WCL) and Weghted- Rectangle Layout problem (WRL) respectvely. Several approaches to the WIL have already been proposed, and most of these are local search heurstcs (Teng et al., 1994; Teng et al., 2001; Zhang et al., 2008; Xao et al.,2007; Huang & Chen, 2006; Xu et al., 2007a). The man strategy of these local search heurstcs s to frst randomly place objects n the plane, and then gradually move them to new postons n order to decrease overlap and mbalance of mass wthn the system. When the tems are crcular, the overlap between them can be modeled as potental energy, and thus tems may be moved accordng to the vrtual force requred to decrease energy (Wang et al., 2002). Local search heurstcs show reasonable performance for the WCL (Xao, et al., 2007; Huang & Chen, 2006), but when the same heurstcs are appled to the WRL (or more complex problems n three-dmensonal space), ther performance quckly suffers (Xu et al., 2007). The man reason for ths degradaton s that the noton of a good layout s related not only to the postons of rectangles, but also to ther orentatons (whch are not consdered by local methods). We therefore focus on constructve heurstcs for both the WCL and WRL. For C&P problems, the most mportant components of a constructve heurstc are the postonng rules,.e., how and where to place each tem. By the postonng rules developed n (Xu et al., 2007; Xu & Xao, 2008; Xu et al., 2010), we may place the weghted tems one-by-one n a certan order, and then a good layout s generated. Ths method was named the Order-based 2

3 Postonng Technque (OPT). However, dfferent orders wll yeld layouts of dfferng qualty qualty; the key problem s how to fnd the best order to mprove the qualty of a layout. In the present paper, we develop ant colony optmzaton (ACO) algorthms to search for the optmal postonng order. ACO s a meta-heurstc for combnatoral optmzaton problems, and has ganed ncreasng use n the last decade. The frst verson of the ACO algorthm, named AS (ant system), was proposed by Dorgo as a soluton to the Travelng Salesman Problem (TSP) (Dorgo, 1992; Dorgo et al., 1996; Blum & Dorgo, 2004; Dorgo & Stützle, 2004). The authors based ther algorthm on the observaton that real ant colones can quckly fnd a shortest path from ther nest to a gven food source, usng chemcal sgnallng based on pheromones. In the AS, Dorgo desgned a colony of artfcal ants, where each ant can buld solutons by contrbutng to an artfcal pheromone tral. The tral can also be externally renforced accordng to the qualty of the soluton t represents. Ths mechansm forms a postve feedback loop, drvng the system towards optmalty. Snce the nventon of the AS, many mprovements have been proposed to the orgnal algorthm, and ACO has now been appled to many optmzaton problems (Dorgo & Stützle, 2004; Stützle & Hoos, 1997; Juang & Hsu, 2009). Some versons of ACO have already been appled to C&P problems. The frst such applcaton was to the one-dmensonal bnpackng problem (1-BPP) (Levn & Ducatelle, 2004). The authors used ACO to learn whch tems are lkely to be grouped together by ther length. They reported that ther ACO method outperformed other meta-heurstcs such as genetc algorthms. )In (Brugger et al. 2004) the authors also reported a verson of ACO for the 1-BPP, but they related the length of tem to the space left n the bn. On the benchmarks, ther ACO algorthm gave better results than (Levn & Ducatelle, 2004). Two-dmensonal packng problems have also been solved by ACO; (Thruvady et al., 2008) reports an ACO algorthm for the strp packng problem. Ther method was based on a constructve heurstc ("Bottom-Left" method), and ther method was used to learn the packng order. In (Burke & Kendall, 1999) the authors apply ant systems to the nestng problem, when the tems are rregular. They used the "No-Ft-Polygon" to compose the constructve heurstc, and the ACO was also appled to learn the packng order. Based on a survey of prevous work, t seems that when applyng ACO on C&P problems, t s necessary to frst develop a constructve heurstc. In ths paper, after ntroducton of the constructon heurstc OPT, we develop two verson of the ACO and compare ther performance on both the WCL and the WRL. The rest of the paper s organzed as follows: n Secton II, we frst defne the mathematcal foundatons of the problem. We then ntroduce, n Secton III, the constructve heurstc OPT for the WCL and WRL. The ACO algorthm s presented n Secton IV, and we report the results of numercal experments n Secton V. We conclude n Secton VI wth a dscusson and suggestons for future work. 3

4 II. MODELS FOR WEIGHTED CIRCLE LAYOUT AND WEIGHTED RECTANGLE LAYOUT Both problems are concerned wth the placement of objects wthn some contanng crcle, the dea beng to mnmse the contaner's radus (whlst mnmsng mass mbalance, whch we consder later). For both problems, we use R to denote the radus of the contanng crcle. Suppose there are n tems to be placed and ther masses are m 1, m 2,, m n. For the WCL, the szes of the crcular tems are denoted by ther rad, r 1, r 2,, r n. For the WRL, the sze of each rectangular tem s denoted by the lengths of ts two adjacent edges, such as (a 1, b 1 ), (a 2, b 2 ),, (a n, b n ). We also use r 1, r 2,, r n to denote the rad of the envelopng crcle of each rectangular tem, such that r = a + b. 2 In two-dmensonal space, we use (x, y ) to denote the poston of tem. For a crcular tem, (x, y ) s suffcent to defne the poston of tem, but for a rectangular tem, we requre addtonal nformaton; the orentaton z s used together wth (x, y ). (a) (b) Fgure 1. Illustraton of WCL and WRL. (a) WCL (b) WRL. By summarzng the models provded n the lterature, we descrbe the three man objectves for each problem n the followng subsectons. A. Requrements of Weghted Crcle Layout 1. Non-overlappng tems Ths s the fundamental requrement, n that no two crcles may overlap. Checkng the condton for zero overlap between two crcles s straghtforward, n that we only need to check whether (1) s satsfed for each par, j. ( x x ) + ( y y ) r + r (1) 2 2 j j j 2. Compactness of the layout We requre that tems are placed n a compact layout. The compactness of the layout s measured by the radus of the envelopng crcle, as llustrated n Fg1-a and Fg1-b. One objectve of our packng s to obtan the smallest possble envelopng 4

5 crcle. The radus of the envelopng crcle can be calculated for WCL by (2). R envelope = r + x + y (2) 2 2 _ max( ) 1 n 3. Balance of the system Another objectve s to mnmze the mbalance generated by weghted tems, defned by (3): (3) n 2 n 2 = 1 = 1 mbalance = ( m x ) + ( m y ) In recent work (Xao et al., 2007; Xu et al., 2007a), we showed that the optmzaton process can actually beneft from the zero-mbalance requrement. Under ths strcter constrant, the mass center of the system s arranged at the center of the contanng crcle. Then the radus of the envelopng crcle (to thus measure the compactness) s replaced by (4): m x R envelope r x y m y n n = 1 2 = 1 2 _ = max ( + ( ) + ( ) ) 1 n n n = 1 m = 1 m (4) It should be noted that we only consder statc mbalance. Dynamc balance s not consdered, as n (Teng et al., 1994), because n the 2-dmensonal case, when the statc mbalance becomes zero, the dynamc mbalance wll also get to zero. B. Requrements of Weghted Rectangle Layout 1. Non-overlappng tems Although t s more dffcult to judge the overlap of two rectangles compared wth crcles, t s stll a straghtforward ssue n computatonal geometry. The suffcent and necessary condtons for no overlap between two rectangles and j are: 1. Any edge of does not ntersect wth any edge of j, and 2. Any vertex of s not contaned n j, and vce versa. Condton 1 should be tested for every edge par (, j). After condton 1 s satsfed, only one vertex of rectangle and one vertex of rectangle j should be checked to consder condton 2. In general, the orentaton of a rectangle s not a fxed value. But when we lmt the orentaton to be ether 0 or 90 degrees, the condtons for no overlap become easer to check. Usng (xmn, ymn), (xmax, ymax) to denote the left-bottom vertex and toprght vertex, we only need to check whether the followng s satsfed (as llustrated by Fg 2): xmn >xmax j or xmax <xmn j or ymn >ymax j or ymax <ymn j 5

6 (xmax j, ymax j) j ymax (xmn, ymn ) ymn xmn xmax Fgure 2. Relatve poston of two rectangles wth fxed orentaton. 2. Compactness of the layout The envelopng crcle llustrated n Fg1-b s agan used to measure compactness. The calculaton of the radus of the envelopng crcle for WRL s a lttle more complex than n WCL, but t s stll relatvely easy. We calculate the dstances from each vertex of the rectangles to the weght center of the contaner, and select the longest dstance as the radus of the envelopng crcle. We denote the set of four vertces of rectangle to be {v,j j=1,2,3,4}, =1,2,..n, and d(a, b) to be the Eucldean dstance between pont a and b. Suppose the center of the contaner s o, then the radus of the envelopng crcle s defned as (5): R _, j = 1,2,..., n j= 1,2,3,4 envelope = max max ( d( v, o)) (5) 3. Balance of the system Snce the defnton of mbalance of WRL s the same as (3), and the mbalance s also requred to be zero, then the center of the envelopng crcle should also be placed at the mass center of the system. Accordngly, the radus of the envelopng crcle can also be found wth reference to the mass center, where we replace the o n equaton (5) wth a term descrbng the mass center of the system. C. Optmzaton for Weghted Item Layout Based on what we have descrbed, WIL becomes a sngle objectve optmzaton problem: Under the constrant of no overlap between any tems, the envelopng crcle centered at the mass center of the system should be mnmzed. 6

7 III. ORDER-BASED POSITIONING TECHNIQUE The Order-based Postonng Technque (OPT) OPT s a constructve method for WIL. Accordng to the defned postonng rules, objects are placed one-by-one n a gven order, and the compactness and balance of the layout are assessed durng ths postonng. In general, f the szes and masses of the tems are dfferent, the postonng order wll affect the qualty of the fnal layout. Wthout loss of generalty, we consder a postonng order of (1, 2,, n). The man dea of OPT s that when we add a new tem to a partal layout (the tems already placed), we should fnd a poston whch wll lead to a new partal layout wth a mnmally-szed envelopng crcle. Ths s a greedy polcy. The greedy polcy can not guarantee a globally optmal layout, but can always provde a reasonable layout good. Durng OPT, we address two questons: 1. How and where to place the frst tem(s); 2. How and where to place the -th tem after the prevous -1 tems. We now descrbe OPT for WCL and WRL n two separate subsectons. A. OPT for Weghted Crcle Layout The rules to decde the poston of a new crcle n a partal layout are related to the compactness defned by (4). It s obvous that the tangent of two crcles makes for the most compact layout f we have only two tems, so t s reasonable to nsst that a crcle should be tangental to other crcles when t s placed. Snce there exst too many postons for a crcle to be tangental to only one other crcle (one crcle can "run around" the other crcle), we requre that the new crcle be tangental to two exstng crcles, such that there are only two avalable postons. The packng rules are descrbed as follows: 1. Place crcle 1 and crcle 2 at poston (-r 1, 0), (r 2, 0). From ths rule, Crcle 1 s placed tangental to crcle From crcle >=3, we requre crcle to be tangental to two exstng crcles p and q, where p< and q<. p q Fgure 3. Postonng of a crcle. 7

8 There are two avalable postons for crcle to be tangental to p and q. (Fg 3). Although these two postons can guarantee no overlap between and p or q, overlap may stll exst between and other exstng crcles, so the valdty of the two postons s checked. Gven that there are (-1)(-2)/2 pars of p and q altogether, then we may have, at most, (-1)(-2) postons. From these postons, the one yeldng the mnmal R_envelope s chosen as the poston of. If the calculaton (wth valdaton check) for one poston s assumed to take unt tme, and we assume n>=3, the tme complexty of the OPT for WCL may be calculated as (6): n 3 ( ) ( 1)( 2) = ( ) T n O n (6) = 1 B. OPT for Weghted Rectangle Layout In the WCL, the postons of the all crcles defne the layout. In WRL, however, the orentaton of each rectangle must also be provded along wth ts poston. The less vacant space there s n a layout, the "better" the layout. So, when placng a rectangle, we try to fnd a poston and orentaton whch mnmse vacant space. The packng rules for WRL are descrbed as follows: 1. The poston and orentaton of the frst rectangle s drectly set to (x 1, y 1, z 1 )=(0, 0, 0). 2. When postonng the rectangle, we consult one exstng rectangle j as llustrated n Fg 4. At frst, we requre one edge of rectangle to be n contact wth one edge of rectangle j. By ths requrement and rule 1, we can nfer that the orentaton for any rectangle should be 0 or 90 degrees. Second, along the touchng edges, we only consder two postons, as llustrated n Fg4-a, where the left or rght vertces of the two touchng edges concde. We beleve these packng rules generate good layouts, snce fewer stles are generated. A stle ntroduces vacant space nto a layout. In Fg 4-a, the partal layout has only one stle, but n Fg 4-b t has two. j j (a) (b) Fgure 4. Illustraton of the postonng rules for WRL (a) Item s placed as defned, and generates one stle, (b) Item generates two stles. 8

9 As llustrated n Fg 5, rectangle has 16 avalable postons around rectangle j. We also check the valdty of the 16 postons, snce may overlap wth other rectangles. Because there are -1 exstng rectangles, rectangle has 16(-1) avalable postons at most. Snce rectangle needs only one poston, we also adopt a greedy strategy to choose the poston whch yelds a mnmal envelopng crcle. j Fgure postons for a rectangle. Takng one poston calculaton as a unt tme expense, we use (7) to estmate the tme complexty of OPT. 2 T ( n) O( ( n 1)) = O( n ) (7) C. Dsadvantages of OPT OPT can generate a layout usng a placng order. However, accordng to the constrants of the OPT, the layout can not always be optmal, especally when there are only a few tems. Ths s easly demonstrated by Fg 6; layouts (c) and (d) are better than (a) and (b), but OPT s only capable of generatng (a) and (b). Fortunately, as the problem sze ncreases, the advantages of OPT become apparent. In practce, we often need a satsfactory near-optmal soluton, f we cannot fnd the optmal ones. (a) (b) (c) (d) Fgure 6. Illustraton of OPT not able to provde the optmal layouts.(a) and (b) are the layout by OPT for WCL and WRL, where there are 4 tems wth same szes and weghts. (c) and (d) are two better layout than (a) and (b). 9

10 IV. ANT COLONY OPTIMIZATION OPT generates a layout accordng to a gven postonng order. But for n tems wth dfferent szes and dfferent masses, there exst n! postonng orders. Snce dfferent postonng orders can lead to dfferent layouts, we should search for a postonng order whch yelds an optmal layout. Ths knd of problem s sometmes called a permutaton constraned problem, where an optmal permutaton of (1, 2,, n) s requred. One famous permutaton constraned problem s the TSP (Travellng Salesman Problem), whereby a salesman should vst n ctes only once, along the shortest path. Such problems quckly become ntractable for even small values of n. ACO s a type of meta-heurstc well-suted to permutaton constraned problems. In ths Secton, we provde an ACO algorthm to search for the optmal postonng order for WIL. A. Standard ACO for TSP ACO s nspred by the behavor of real ants. As a knd of socal nsect, a sngle ant has very lmted ablty, but, through cooperaton, the colony can perform very complex tasks. Ethnologsts observe that, although the ant s almost blnd, the colony as a whole can fnd a shortest path from ther nest to the food source. When one ant moves on a path, t deposts pheromone (a chemcal sgnallng molecule) on the ground. The other ants follow the (perhaps many) paths defned by the pheromone, where the path wth the strongest concentraton of pheromone s more lkely to be chosen. Because the ants on the shorter path return more quckly, then the pheromone on the shorter path s renforced and as a result t wll attract more ants agan. Ths s a postve feedback loop, whch quckly leads to the emergence of a shortest path. Dorgo formalsed the above mechancs and nvented the ant colony optmzaton method for TSP (Dorgo et al., 1996). Gven n ctes (1, 2, n) wth the dstances d j between any two cty and j, the salesman should fnd the shortest closed tour, such that each cty s vsted once. Suppose L ants walk on the graph of n ctes. In ACO, each ant k n cty wll move to cty j wth probablty p k (, j), p (, j) k = α β τ (, j) η(, j),f j Allowed α β k τ (, s) η(, s) (8) s Allowed k 0, otherwse We defne τ (, j) as the pheromone level on the edge (, j); η(,j) s the heurstc nformaton gudng the ant, whch n TSP s often set to 1/ d j, snce shorter edges are preferred. Allowed k s a set ncludes the ctes not vsted by ant k. The parameters α 10

11 and β are used to defne the relatve mportance of the pheromone and the heurstc nformaton, whch may be adjusted by the user. After each ant performs a complete tour, the pheromone trals are updated. The pheromone on all edges "decays" at some rate ρ(0<ρ<1), and the pheromone on the edges traversed by ants s then renforced. m τ (, j) ρ τ (, j) + τ k (, j) (9) k = 1 Where Q, f (, j) walked by ant k τ k (, j) = Lk (10) 0, otherwse Q s a const, and L k s the length of the tour by ant k. After the update of the pheromone trals, the next teraton begns. The ACO algorthm s llustrated n Fg 7. ACO algorthm for TSP { Intalze the parameters and the pheromone tral. For (teraton = 1 to MAX) { For (ant k=1 to m) Whle (any cty s not vsted by k) Determne next cty wth probablty p k (, j) as defned by (8) Update the pheromone on the walked paths by (9) and (10) } Output the best tour and other data } Fgure 7. ACO pseudo-code. 11

12 B. Applcaton of ACO to optmzaton problems As a robust and versatle meta-heurstc, ACO may be appled to solve many other combnatoral optmzaton problems, such as the Quadratc Assgnment Problem and Job-shop Schedulng Problem (Dorgo & Stützle, 2004). The man assumptons, when applyng ACO to a problem are that: 1. Each path followed by an ant may be mapped to a canddate soluton to the gven problem; 2. After a canddate soluton s generated, the amount of pheromone to depost on the path s proportonal to the qualty of the soluton; 3. The path wth relatvely more pheromone has more chance of beng chosen by the ants. C. The ACO algorthm for WIL Because WIL s also a permutaton constraned problem, ACO s drectly applcable. In TSP, the ant fnd a sequence n whch to vst ctes, whle n WIL the ant fnd an order n whch to place tems. The man ssues concerned n our ACO are lsted as follows: 1. The frst mportant step s to defne the pheromone tral for the ACO applcaton. In WIL, The pheromone tral τ(, j) s defned as the "favorablty" of placng tem j after tem ; 2. Another mportant feature of ACO s the heurstc. In our ACO, the heurstc nformaton η(, j) n (8) s defned as m j *r j, because heaver and lager tems should be placed wth hgher prorty; 3. In our ACO, L k n (10) s defned as the radus of the envelopng crcle of the layout generated by the OPT, so compact layouts are preferred. 4. The update of the pheromone follows the ACO verson of Max-Mn Ant System (MMAS) (Stützle & Hoos, 1997), where only the ant wth the best soluton s allowed to depost pheromone on ts tral n each cycle of the teraton. To prevent the ACO convergng too quckly, the pheromone s lmted n [τ mn, τ max ]. 12

13 V. EXPERIMENTS AND RESULTS We now descrbe the results of numercal experments to test our method. The experments are carred on an Intel 1.83GHZ/512Mb Computer. All algorthms are mplemented n the C language and compled by MS VC++. Each program s run on each benchmark nstance 10 tmes. The mnmal radus of the envelopng crcle (r_best), the average radus of the envelope crcle (r_average), and the average computaton tme for each nstance (t_average) are used to evaluate the performance of the algorthms tested. The computatonal results are lsted n table I and table II. In what follows, two versons of ACO, the standard Ant System (AS) and the Mn-Max AS (MMAS), are mplemented and tested. The parameter settngs are derved from (Dorgo & Gambardella, 1997). The number of ants s set to 20, and the number of teratons to 100. The mportance factors of the pheromone tral α and heurstc β are both set to 1. The pheromone decayng rate ρ s set to 0.9. The value of Q n (10) s set to the mnmal radus of the envelopng crcle found. Before each teraton, pheromone τ(, j) s ntalzed to 1/n. The [τ mn, τ max ] s set to [0.1/n, 10/n]. A. Tests on Weghted Crcle Layout We frst use 10 problem nstances taken from (Xao et al., 2007a; Xu et al., 2007) to test the ACO for WCL. The number of crcles vares from 10 to 55. Two other algorthms are also run on the same benchmark nstances, n order to yeld meanngful comparsons. One s a local search heurstc from (Xao et al., 2007a), whch s a hybrd of the gradent descent method and Partcle Swarm Optmzaton (PSO), named CA-PSLS (compact algorthm wth partcle swarm local search). The other one s a GA based on OPT from (Xu et al., 2007). Both algorthms were the best known-methods for ths problem at the tme of publcaton. In CA-PSLS, we frst randomly select 100 layouts, and then use the compacton algorthm to get an output layout from each of them. The best layout from the 100 outputs s chosen as the start pont, and the PSO algorthm then performs a local search to mprove the layout. We use 20 partcles and the teraton number of the PSO s set to In the GA, the populaton sze s set to 20, and the program run for100 generatons. One-pont crossover s used and the mutaton probablty set to 12.5%. The computatonal results for the AS, MMAS, GA, CA-PSLS are lsted n Table I. We derve three fndngs from these results: 1. MMAS performs better than AS on WCL problems. Because only the best ant updates the pheromone tral n MMAS, whle every ant n AS updates ts own pheromone tral, MMAS runs a lttle faster than AS on all 10 nstances. Only the ant who fnds the best soluton s permtted to update the pheromone tral, whch suggests that MMAS spends 13

14 more on explotng than on exploraton. So t can be predcted that n larger soluton spaces, MMAS wll outperform AS. The results also show that on the frst two small nstances, AS fnds better layouts than MMAS, and on the followng 8 larger nstances, MMAS outperforms AS. 2. MMAS search s nearly as effectve as the GA on average. In Table I, we see that, on average, on nstance 1,2,3,7,8, the GA fnds better layouts than MMAS, but on the other fve nstances, MMAS outperforms the GA. Ths fndng demonstrates that, although ACO and GA are based on dfferent knds of search dea, f both of them apply heurstc nformaton they can have broadly smlar search abltes. 3. OPT s a powerful constructve method. Although the CA-PSLS method yelds very successful results on WCL problems compared to other local search methods (Xao et al., 2007), n Table 1, we fnd that the three algorthms based on OPT, (AS, MMAS, and GA) yeld smlar results (n terms of qualty) n about 20% of the computatonal run-tme. TABLE I: COMPUTATIONAL RESULTS FOR THE WCL nstance algorthm r_best r_average t_average (second) nstance algorthm r_best r_average t_average (second) 1 AS AS (10 tems) MMAS (35 tems) MMAS GA GA CA-PSLS CA-PSLS AS AS (15 tems) MMAS (40 tems) MMAS GA GA CA-PSLS CA-PSLS AS AS (20 tems) MMAS (45 tems) MMAS GA GA CA-PSLS CA-PSLS AS AS (25 tems) MMAS (50 tems) MMAS GA GA CA-PSLS CA-PSLS AS AS (30 tems) MMAS (55 tems) MMAS GA GA CA-PSLS CA-PSLS

15 B. Tests on Weghted Rectangle Layout We use the 4 nstances n (Xu et al., 2007a) to test the ACO for WRL, wth 5, 6, 9, 20 rectangles respectvely. To test the algorthms on a larger problem, we compose a new nstance wth 40 rectangles. A local search method, also named CA-PSLS, from (Xu et al., 2007a) s tested as the comparson algorthm. In CA_PSLS, we select one layout randomly and compact t, and use the output as the start pont. We run the PSO to perform a local search n order to get a refned layout. The number of partcles of PSO s set to 20 and the teraton number to The results are dsplayed n Table II. The fndngs are as follows: 1. We agan fnd that MMAS outperforms AS on the larger nstances. On the smaller nstances 1, 2, and 3, where the numbers of tems are less than 10, AS outperforms MMAS. However, on nstances 4 and 5, where the numbers of tems grow to 20 and 40 respectvely, MMAS wns. Ths fndng shows that MMAS s sutable for searchng larger soluton spaces. 2. OPT-based MMAS shows clear superorty over the local search method CA-PSLS on large nstances. On nstances 1 and 2, MMAS fnds nferor layouts compared to CA-PSLS, but the dfferences are very small. When the number of rectangles s rased to 9, CA-PSLS begn to lose ground, and when the number s rased to 40, the radus of the envelopng crcle found by MMAS s better than that found by CA-PSLS by a factor of 50% n terms of r_best, and by a factor of 100% n terms of r_average. Consderng computatonal resources, we fnd that MMAS was about 3 tmes faster than CA-PSLS on all nstances. Fg 8 presents the a graphcal llustraton of the results obtaned by both algorthms on nstance 5, whch can help us to understand the superorty of OPT. We see n Fg 8-b that all the rectangles are adjacent along ther edges, accordng to OPT rules, whch seems to be a hard task for CA-PSLS to acheve. 15

16 Table II: Computatonal Results for the WRL. nstance algorthm r_best r_average t_average (second) 1 (5 tems) AS MMAS CA-PSLS (6 tems) AS MMAS CA-PSLS (9 tems) AS MMAS CA-PSLS (20 tems) AS MMAS CA-PSLS (40 tems) AS MMAS CA-PSLS (a) (b) Fgure 8: The layouts generated from nstance 5 n the experment for WRL. (a) layout by CA-PSLS, (b) layout by MMAS. VI. CONCLUSIONS Ths paper proposes a constructve heurstc to pack weghted tems n a crcular contaner, where the compactness and balance should be consdered. By carefully desgnng packng rules, the heurstcs generate good layouts. An ant-based algorthm based on ths heurstc s then descrbed to optmze the packng order. In the ant algorthm, the pheromone matrx encodes the 16

17 favorablty of choosng an tem. Heurstc nformaton s also consdered, whch s the product of the sze and the weght of the next packed tem. Ths means that large and heavy tems have hgher prorty. Two versons of the ant-based algorthm, AS and Mn-Max AS, are compared wth exstng approaches, such as the genetc algorthm for Weghted Crcle L, and the hybrd partcle swarm algorthm for both Weghted Crcle and Weghted Rectangle Layout. The expermental results showed that: (1) Mn-Max AS performs better than AS on large-scaled nstances for both WCL and WRL, (2) The OPT based approaches, ncludng the ACO and GA, perform better than the local search based approach CA-PSLS. However, the crcle and rectangle are two specal shapes. Further work s requred to study packng of weghted and rregularshaped tems, whch wll be the subject of future research. REFERENCES C. Blum and M. Dorgo (2004). The hyper-cube framework for ant colony optmzaton. IEEE Transactons on System, Man, and Cybernetcs, Part B 34(2), pp B. Brugger, K.F. Doerner, R.F. Hartl and M. Remann (2004). AntPackng An ant colony optmzaton approach for the one-dmensonal bn packng problem. Lecture Notes n Computer Scence, Volume 3004, pp E. Burke and G. Kendall (1999). Applyng ant algorthms and the no ft polygon to the nestng problem. Lecture Notes n Computer Scence, Volume 1747, pp J. Cagan, D.Degentesh, and S.Yn (1998). A smulated annealng-based algorthm usng herarchcal models for general three-dmensonal component layout. Computer-Aded Desgn 30, pp M. Dorgo (1992). Optmzaton, learnng and natural algorthms, PhD thess, Poltecnco d Mlano, Italy. M. Dorgo, V. Manezzo and A. Colorn (1996). Ant system: optmzaton by a colony of cooperatng agents. IEEE Transactons on System, Man, and Cybernetcs, Part B 26(1), pp M. Dorgo and L.M. Gambardella (1997). Ant colony system: a cooperatve learnng approach to the Travellng Salesman Problem. IEEE Transactons on Evolutonary Computaton 1(1), pp M. Dorgo and T. Stützle (2004). Ant Colony Optmzaton, MIT Press. K A. Dowsland, S. Vad, and W. B. Dowsland (2002). An algorthm for polygon placement usng bottom-left strategy. European Journal of Operatonal Research 141(2), pp H. Dyckhoff (1990). A typology of cuttng and packng problems. European Journal of Operatonal Research 44(1), pp J. Gonzalez, I. Rojas, H. Pomares, M. Salmeron and J.J. Merelo (2002). Web newspaper layout optmzaton usng smulated annealng. IEEE Transactons on System, Man, and Cybernetcs, Part B 32(5), pp J. M. Harwg (2003). An adaptve tabu search approach to cuttng and packng problems. PhD thess, Unversty of Texas at Austn. W. Huang and M. Chen (2006). Note on: An mproved algorthm for the packng of unequal crcles wthn a larger contanng crcle. Computers & Industral Engneerng 50(3), pp

18 C.-F. Juang and C.-H. Hsu (2009). Renforcement nterval type-2 fuzzy controller desgn by onlne rule generaton and Q- Value-Aded ant colony optmzaton. IEEE Transactons on System, Man, and Cybernetcs, Part B, DOI: /TSMCB J. Levne and F. Ducatelle (2004). Ant colony optmzaton and local search for bn packng and cuttng stock problems. Journal of the Operatonal Research Socety 55(7), pp A. Sokea and Z. Bng (2006). Hybrd genetc algorthm and smulated annealng for two-dmensonal non-gullotne rectangular packng problems. Engneerng Applcatons of Artfcal Intellgence 19(5), pp T. Stützle and H. Hoos (1997). MAX-MIN ant system and local search for the Travelng Salesman Problem. Proceedngs of the 1999 IEEE Congress on Evolutonary Computaton, IEEE Press, pp M. Tang and X. Yao (2007). A memetc algorthm for VLSI floorplannng. IEEE Transactons on System, Man, and Cybernetcs, Part B 37(1), pp H. Teng, S. Sun, and W. Ge, and W. Zhong (1994). Layout optmzaton for the dshes nstalled on a rotatng table- the packng problem wth equlbrum behavoral constrants. Scence n Chna (Seres A) 37(10), pp H. Teng, S. Sun, D. Lu,and Y. L (2001). Layout optmzaton for the objects located wthn a rotatng vessel - a threedmensonal packng problem wth behavoral constrants. Computers & Operatons Research 28(6), pp D.R. Thruvady, B. Meyer and A.T. Ernst (2008). Strp packng wth hybrd ACO: placement order s learnable. Proc. IEEE Congress on Evolutonary Computaton, IEEE Press, pp H. Wang, W. Huang, and Q. Zhang, and D. Xu (2002). An mproved algorthm for the packng of unequal crcles wthn a larger contanng crcle. European Journal of Operatonal Research 141(2), pp Y. L. Wu, W. Huang, S. Lau, C.K. Wong and J.H. Yung (2002). An effectve quas-human based heurstc for solvng the rectangle packng problem. European Journal of Operatonal Research 141(2), pp R.-B. Xao, Y.-C. Xu, and M. Amos (2007). Two hybrd compacton algorthms for the layout optmzaton problem. BoSystems 90(2), pp Y.C. Xu, R. B. Xao, and M. Amos (2007a). Partcle swarm algorthm for weghted rectangle placement. Proceedngs of the 3rd Internatonal Conference on Natural Computaton (ICNC07), IEEE Press, pp Y.C. Xu, R. B. Xao, and M. Amos (2007b). A novel genetc algorthm for the layout optmzaton problem. Proceedngs of the 2007 IEEE Congress on Evolutonary Computaton (CEC07), IEEE Press, pp Y.C. Xu and R. B. Xao (2008). An ant colony algorthm for the layout optmzaton wth equlbrum constrants. Control and Decson 23(1), pp (n Chnese). Y.C. Xu, F. M. Dong, Y. Lu, and R.B. Xao (2010) A new genetc algorthm for layng out rectangles wth equlbrum constrants. Submtted to Pattern Recognton and Artfcal Intellgence (In Chnese). B. Zhang, H.F. Teng and Y.J. Sh (2008). A layout optmzaton of satellte module usng soft computng technques. Appled Soft Computng Journal 8(1), pp