From Tme Representaton n Schedulng to the Soluton of Strp Packng Problems Pedro M. Castro * and gnaco E. Grossmann Undade de Modelação e Optmzação de Sstemas Energétcos Laboratóro Naconal de Energa e Geologa 1649-038 Lsboa Portugal Department of Chemcal Engneerng Carnege Mellon Unversty Pttsburgh PA 15213 USA Abstract We propose two med-nteger lnear programmng based approaches for the 2-D orthogonal strp packng problem. Usng knowledge from the alternatve forms of tme representaton n schedulng formulatons we show how to effcently combne three dfferent concepts nto the - and y- dmensons. One model features a dscrete representaton on the -as (strp wdth) and a contnuous representaton wth general precedence varables on the y-as (strp heght). The other features a full contnuous-space representaton wth the same approach for the y-as and a sngle non-unform grd made up of slots for the -as. Through the soluton of a set of twenty nne nstances from the lterature we show that the former s a better approach even when compared to three alternatve MLP models rangng from a pure dscrete-space to a pure contnuous-space wth precedence varables n both dmensons. All models are avalable n www.mnlp.org. Keywords: Optmzaton; nteger programmng; Search algorthm; Event ponts. * Correspondng author. Tel.: +351-210924643. E-mal: pedro.castro@lneg.pt. 1
1. ntroducton Tme representaton s perhaps the most mportant classfcaton crteron for schedulng models. Mathematcal formulatons can ether be classfed as dscrete- or contnuous-tme and several dfferent alternatves have been proposed. Overall there are essentally four man concepts beng used whch are llustrated n Fgure 1. Dscrete tme grd U2 T5 T6 Contnuous tme wth sngle grd U2 T5 T6 U1 T1 T2 T3 T4 U1 T1 T2 T3 T4 2 3 4 5 6 7 8 9 10 11 12 1 13 2 3 4 1 5 mmedate Precedence (through sequencng varables) General 1 Contnuous tme wth multple grds 2 U2 T5 T6 U2 T5 T6 U1 T1 T2 T3 T4 U1 T1 T2 T3 T4 1 2 3 4 Fgure 1. Tme representaton alternatves n schedulng formulatons. Table 1. Classfcaton of schedulng approaches accordng to tme representaton Contnuous-tme Dscrete-tme Sngle grd Multple grds Precedence Kondl et al. (1993) Zhang & Sargent Pnto & Grossmann Méndez et al. (2000) (1996) (1995) Shah et al. (1993) Schllng & Panteldes earapetrtou & Méndez et al. (2001) (1996) Floudas (1998) Panteldes (1994) Castro et al. (2001) Gannelos & Harjunkosk & Georgads (2002) Grossmann (2002) Glsmann & Gruhn Maravelas & Castro & Gupta & Karm (2003) (2001) Grossmann (2003b) Grossmann (2005) Maravelas & Castro et al. (2004) Lu & Karm Prasad & Maravelas Grossmann (2003a) (2007) (2008) Castro et al. (2008) Sundaramoorthy & Castro & Novas Sundaramoorthy & Karm (2005) (2008) Maravelas (2008) Sundaramoorthy & Gménez et al. (2009) Shak & Floudas Ferrer-Nadal et al. Maravelas (2011a) (2009) (2008) Wassck & Ferro (2011) Castro et al. (2009) Susarla & Karm Capón-Garca et al. (2010) (2009) Table 1 provdes a lst of some of the most mportant schedulng references n the Process Systems Engneerng lterature n the last 20 years classfed accordng to the tme representaton concept 2
used. t s apparent that no partcular opton s becomng more common clearly reflectng that the best opton for a problem s very much dependent on ts features. Whle predctng the best performer s often dffcult some gudelnes can be gven. The dscrete-tme approach s perhaps the most powerful and has been shown capable of handlng problems of ndustral relevance (Wassck 2009; Wassck & Ferro 2011). The tme horzon of nterest s dvded nto a fed number of tme slots of predetermned duraton wth one knowng a pror the locaton of all tme ponts. Ths makes t straghtforward to handle holdng and backlog costs (Sundaramoorthy & Maravelas 2011b) thus allowng for easy ntegraton wth the hgher level plannng model (Maravelas & Sung 2009) ntermedate events such as release/due dates equpment mantenance as well as tme-dependent utlty prcng and avalablty (Castro et al. 2009 2011). On the downsde fed processng tmes need to be assumed and appromated to a multple of the nterval length. Contnuous-tme models on the other hand are more accurate and senstve to small changes n the duraton of processng and changeover tasks whch can be of a dfferent order of magntude. They are thus more approprate for ntegraton wth the lower level control layer (Capón-Garca et al. 2011). Decdng for a contnuous-tme model needs to be followed by the choce of the concept used to keep track of events takng place. Precedence based models (Méndez et al. 2000 2001) were the frst to appear and are known for ther ablty to provde hgh qualty solutons wth lmted computatonal resources even though t may be dffcult to prove optmalty. The concept of general precedence s used more frequently when compared to mmedate precedence snce t gves rse to smaller models that typcally perform better. Precedence based models tend to be less general than ther tme grd counterparts and are thus more commonly found for multstage plants where they are more effcent. n facltes wth a network structure nvolvng resource constrants other than equpment and unt avalablty.e. multpurpose plants tme grd based models become the only opton. Due to process complety they are lnked to unfed frameworks for process representaton the State-Task (Kondl et al. 1993) and Resource-Task (Panteldes 1994) Networks. When compared to the dscrete-tme representaton the tme horzon s also dvded nto a fed number of slots but now the grd(s) s/are 3
non-unform wth the duraton of the slots beng determned through a set of contnuous varables. Fewer slots are requred to represent the soluton more so f multple tme grds are employed. However the number of prespecfed slots has also a stronger nfluence on both soluton qualty and computatonal effort wth one typcally requrng an teratve search procedure to fnd the global optmal soluton (Méndez et al. 2006). Whle t s easer to rely on a sngle grd such opton forces allowng batch tasks to spread across multple slots whch severely compromses ther performance. Wth multple tme grds more effcent unt-specfc models can be used even though there s stll no general model of such type that can handle all dfferent types of resources and storage polces. 1.1. Gong multdmensonal n the contet of N-dmensonal allocaton problems (Westerlund 2005) schedulng problems can be vewed as one dmensonal wth the relevant dmenson beng tme. Strp packng problems are a class of 2-dmensonal allocaton problems (Amossen & Psnger 2010) that are open dmensonal (Wascher et al. 2007) meanng that all tems need to be packed nto a strp of a gven wdth so as to mnmze ts heght. n the paper ndustry for eample one goal s cuttng jumbo reels of paper nto smaller reels so as to mnmze trm losses. Beng a smple to defne albet challengng problem of ndustral relevance s probably the reason t has been wdely studed by the research communty and several soluton approaches have been proposed. Dependng mostly on problem sze and the tme avalable to generate a soluton one may rely on heurstcs (We et al. 2009; Ortmann et al. 2010) eact algorthms (Kenmoch et al. 2009; Martello et al. 2003; Bekrar et al. 2007; Alvarez- Valdes et al. 2009; Grancolas & Pnto 2010) or mathematcal programmng models (Castro & Olvera 2011). The latter two have the advantage of establshng f the best found soluton s ndeed optmal whle nformng of the mamum possble dstance to such optmum whch can be qute relevant. The drawback s ther sharp decrease n performance wth respect to problem sze. Mathematcal programmng approaches are more adaptable to changes n the problem constrants or objectve functon an essental feature consderng that ndustral problems rarely fall wthn the eact boundares of a standard problem defnton. For nstance Dow Chemcal has recently reported (Wassck & Ferro 2011) the soluton of a non-classcal packng problem consstng on loadng a 4
sem-traler wth packages of dfferent szes and weghts so that the traler payload weght and also the rear ale weght are kept below hghway allowable lmts. The soluton method was a mednteger lnear programmng (MLP) model that etended a dscrete-tme schedulng model (Wassck 2009) from one to two dmensons. The recently proposed MLP-based approaches by Castro and Olvera (2011) for 2-D packng problems were also nspred by schedulng models. The man novelty has been the combnaton of two dfferent concepts a dscrete-space representaton wth respect to the -as and a contnuousspace representaton for the y-as. Combnng dscrete wth contnuous representatons s known to be an effectve approach for plannng and schedulng wth a rollng-horzon strategy (Dmtrads et al. 1997; Castro et al. 2011). Lke n schedulng contnuous-space representatons for 2-D packng are more accurate snce the rectangles wdths and heghts do not need to be multples of the characterstc dmensons of the mesh used to dscretze the strp see Fgure 2 on the left. Classcal MLP models for 2-D packng problems feature contnuous varables to determne the (y) coordnates of a gven rectangle pont n the strp (see Fgure 2 on the rght) a concept that can naturally be etended to three (Wu et al. 2010) and hgher dmensonal allocaton problems (Westerlund et al. 2007). They can be derved from a Generalzed Dsjunctve Programmng formulaton featurng four sets of Boolean varables to dentfy the locaton of a rectangle wth respect to another followed by ether a conve-hull or bg-m reformulaton (Sawaya & Grossmann 2005). A closely related model usng the centrod coordnate nstead of the upper-left corner as the reference pont was proposed by Castllo et al. (2005) and reported to have a better computatonal performance so t s the one consdered n ths artcle for comparatve purposes. y Dscrete space n y as Contnuous space n y as H wth postonng varables H... 7 6 5 4 3 2 1 1 2 3 4 5... W W Fgure 2. Classcal representaton alternatves for 2-D allocaton problems. 5
The bnary postonng varables n the contnuous-space models are the N-dmensonal equvalents to the general precedence sequencng varables n schedulng. Models relyng on ths concept can be used as standalone procedures whereas the soluton from those relyng on one or multple spatal grds s dependent on the number of slots specfed and thus requres an teratve search procedure to fnd the global optmal soluton whch mght termnate wth a suboptmal soluton due to a temporary plateau n the value of the objectve functon. To overcome ths lmtaton we now propose a new hybrd dscrete/contnuous-space model featurng sequencng varables n the y-as whch can not only guarantee global optmalty but s also able to mprove the performance of the recent hybrd model of Castro & Olvera (2011). We also present a full contnuous-space model that reles on the same set of varables for the y-as and on a sngle grd for the -as. We have therefore combned all four major concepts for spatal representaton gven n Fgure 1 and effcently as wll be shown later on. The rest of the paper s structured as follows. Gven the problem defnton n secton 2 we dscuss the new 2-D representaton alternatves n secton 3. The contnuous-space model of Castllo et al. (2005) that s part of the computatonal studes s brefly descrbed n secton 4 whle the new MLP models are presented net. Secton 5 concerns the new hybrd dscrete/contnuous-space approach wth secton 6 dealng wth the new contnuous-space approach. The computatonal results are then the subject of secton 7. n secton 8 we summarze the advantages and lmtatons of the dfferent spatal representaton concepts studed wth the conclusons beng left for secton 9. 2. Problem Defnton We consder the two-dmensonal orthogonal strp packng problem. Gven are a set of rectangles wth wdth w and heght h. The objectve s to place the rectangles wthout overlap nto a strp of a gven wdth W so as to mnmze the heght H. The focus s set on the soluton of problems wth nteger data and no rotatons are allowed. Nevertheless the contnuous-space models that are presented net naturally handle real values for the wdths and heghts. 6
3. Prevous and Newly Proposed Spatal Representaton Alternatves n the hybrd model of Castro & Olvera (2011) the strp s dvded along ts wdth W (-as) nto vertcal strps of untary wdth. Each elementary strp has ts own vertcal grd made of allocaton ponts that can be lnked to the left-bottom corner of a partcular rectangle. All W grds have the same number of allocaton ponts but ther y-coordnate wll typcally be dfferent unless adjacent elementary strps share the same rectangle. Ths s llustrated on the left of Fgure 3. y Dscrete space n as Dscrete space n as Contnuous space n as Contnuous space n y as Contnuous space n y as wth sngle spatal grd wth multple spatal grds wth sequencng varables Contnuous space n y as H H H wth sequencng varables 1 2 3 4 5... W 1 2 3 4 5... W W Fgure 3. New representaton alternatves for 2-D packng problems. The newly proposed hybrd dscrete/contnuous-space model uses sequencng varables nstead of spatal grds to properly place rectangles wth respect to the y-as. Contnuous varables gve the y- coordnate of the bottom-edge of the dfferent rectangles wth the possblty of the same coordnate beng shared by two rectangles provded there s enough horzontal space among them. Fgure 3 on the mddle llustrates ths aspect showng a drect correspondence to the placement of the allocaton ponts from the other hybrd model (Castro & Olvera 2011) that are shown on the left (horzontal segments). Ths novel approach s not cursed by the uncertanty n the number of allocaton ponts meanng that the soluton of a sngle MLP wll gve the global optmal soluton to the problem. We also propose a somewhat related hybrd contnuous-space model that keeps the postonng varables n the y-as but uses a sngle horzontal contnuous grd nstead of a dscrete spatal grd. We thus return to the use of allocaton ponts and the need for an teratve search procedure to fnd the global optmal soluton. Ths alternatve s llustrated on the rght of Fgure 3. Note that 7
8 rectangles can span over multple slots so the vertcal lnes do not necessarly correspond to gullotne cuts. 4. A Contnuous-Space Model wth Postonng Varables (CS) Two-dmensonal contnuous-space models typcally employ ( y ) varables to defne the rectangle coordnates on the and y-as. n order to provde a comparson wth the newly proposed models we consder the adaptaton of the BLDP1 block layout desgn model by Castllo et al. (2005) for the strp packng problem. The bg-m formulaton by Sawaya and Grossmann (2005) s smlar but nvolves four sets of bnary varables n the no-overlap constrants where at most one can be actve. The objectve s the mnmzaton of the strp heght eq. (1). Contnuous varables ( y ) denote the coordnates of the center of rectangle whle bnary varables P and Q are sets of bnary varables (Westerlund et al. 2007) used n the overlap preventon constrants eqs. (2-5). Specfcally (2) and (3) are concerned wth vald placement of rectangle to the rght and left of rectangle whle (4) and (5) ensure vald placement of rectangle on top or below rectangle. Eqs. (6-9) ensure that the edges of every rectangle are wthn the boundares of the strp whle the doman of the model varables s gven by (10-13). mn H (1) Q P W w w ) ( ) ( 2 1 (2) Q P W w w ) (1 ) ( 2 1 (3) Q P M y y h h ) (1 ) ( 2 1 (4) Q P M y y h h ) (2 ) ( 2 1 (5) w 2 1 (6) w W 2 1 (7)
1 y h 2 (8) y H 1 2 h (9) P {01} > (10) Q {01} > (11) 0 (12) y 0 (13) 5. New Hybrd Dscrete-Space Approach (NDCS) We now propose a new model that s dscrete n the -as doman and contnuous n the y-as. The y dmenson s preferred for contnuous representaton to avod the teratve procedure nvolved n the effcent determnaton of the strp heght (Castro & Olvera 2011) whch can now be defned as a contnuous varable H. Let X={1 W} be the set of vertcal slots n the -as. Two sets of bnary varables are used: N to dentfy the assgnment of the left-edge of rectangle to the start of slot ; Z to specfy f the top-edge of rectangle s below the bottom-edge of rectangle ( >) provded that one bo s at least partly above the other see Fgure 4 and Fgure 5. The y- coordnate of the bottom-edge of rectangle s gven by varable Y. Z ={01} Z =1 f > Z =0 f > y y y y Fgure 4. Value of y-as sequencng varables Z for dfferent arrangements. 9
10 Fgure 5. Values of the varables for the new med dscrete-space approach (NDCS). Consder the eample n Fgure 5 that dentfes whch varables Z need to be fed for the arrangement shown. Rectangle 1 s beneath the others and so Z 12 = Z 13 = Z 14 =1. Rectangle 4 s below 2 whch s the same as sayng that 2 s above 4.e. Z 24 =0. Wth respect to the nteracton between (23) and (34) they do not occupy the same horzontal slots and so the values of the correspondng varables are rrelevant. Ths s apparent from bg-m constrants (14-15) whch are wrtten for every par of rectangles and every slot. f the left edge assgnments of rectangles and make them occupy slot then both summatons nsde the brackets on the rght-hand sde wll be equal to 1. Then there are two possbltes: Z =1 or 0. Wth the former the bg-m term n eq. (14) dsappears and the constrant Y Y +h s enforced.e. the bottom-edge of rectangle s above the top-edge of rectangle as ntended by the defnton of the bnary varables. For the latter opton t s the bg-m term n eq. (15) that s equal to zero leadng to Y Y +h.e. rectangle s above. For all other cases the constrants are relaed. X N N Z M h Y Y w X w X ) (3 1 1 (14) X N N Z M h Y Y w X w X ) (2 1 1 (15) Eq. (16) ensures that the top edge of every rectangle s below the strp heght. The same apples to the sum of the heghts of all rectangles occupyng a gven slot eq. (17). Then any gven rectangle can be assgned to a sngle slot eq. (18). Eqs. (19-22) gve the doman of the model varables whle the objectve functon n eq. (1) also apples. H 2 Y 2=7 N 23=1 Z 24=0 3 4 Y 4=3 N 42=1 Y 3=2 Z 14=1 N 35=1 1 Z 12=1 Y 1=0 N 11=1 Z 13=1 1 2 3 4 5... W
Y h H (16) X w 1 h N H X (17) N (18) X 1 ={:+w W+1} X (19) N {01} X (20) Z {01} > (21) Y 0 (22) 6. New Contnuous-Space Approach (NCS) Usng the same set of sequencng varables Z we now propose a new contnuous-space model that uses a sngle spatal grd n the -as. Let E={e 1 e E } be the set of allocaton ponts n the - as that mplctly fes the mamum number of rectangles that can be placed on a gven horzontal slce of the strp to E -1 see Fgure 6. Rectangles wll occupy one or more slots wth bnary varables N dentfyng the assgnment of the left-edge of rectangle to allocaton pont e and the e e rght-edge to pont e (e >e). The -coordnate of allocaton pont e wll be gven by contnuous varables X e. Usng the same smple eample we llustrate the value of the model varables n Fgure 7. Notce that rectangle 2 occupes a sngle slot startng at allocaton pont e 3 and endng at allocaton pont e 4 thus N 1. The same goes for 3 whle 4 occupes 2 slots and rectangle 1 4. t s mportant 234 to hghlght that the rght edge of the rectangle may not concde wth the -coordnate of the endng allocaton pont (e.g. e 5 ). Ths avods the need for further allocaton ponts n cases where the empty space cannot be occuped and s reflected n eqs. (23-24). t states that the dfference n the - coordnates of allocaton ponts e (replaced by the strp wdth W f e = E eq. 24) and e must be greater than the wdth of the rectangle provded that ts left and rght edges are assgned to such ponts. Notce that the coordnate of the frst allocaton pont s equal to zero eq (25). 11
slot1 slot 2 slot E 2 slot E 1 2 3 E 2 E 1 1 E allocaton X 1 X 2 ponts X 3 X E 2 X E 1 W Fgure 6. Sngle non-unform spatal grd n the -as used by new contnuous-space approach H Y 2=7 2 N 234=1 Z24=0 3 4 Y 4=3 N 424=1 Y 3=2 Z 14=1 N 345=1 1 Z12=1 Y1=0 N115=1 Z13=1 e 1 e 2 e 3 e 4 e 5 X 1=0 X 3=2 X 4=4 W X 2=1 Fgure 7. Values of the varables for the new contnuous-space approach (NCS). X e W X X w N e E e E e e e E (23) e e e e w N e e e E e E e e (24) X 1 =0 (25) The other sets of constrants are conceptually smlar to those of the hybrd dscrete/contnuousspace model consderng that nde has been replaced by nde e. n Eq. (26) the bg-m term s equal to zero whenever the top-edge of rectangle s below the bottom-edge of rectangle and both and occupy slot e of the -doman grd meanng that ther left-edges are allocated to some allocaton pont e matchng or located to the left of allocaton pont e and that ther rght-edges are assgned to some pont e that s located to the rght of both allocaton ponts e and e. f on the other-hand rectangle s above then t s eq. (27) that s enforced. Y Y h M [3 Z ( N e e N e e )] e E e E (26) e E e E ee e e e e Y Y h M [2 Z ( N e e N e e )] e E e E (27) e e e E e E e e e e 12
Eqs. (1) (16) (21-22) are shared wth NDCS whle eq. (28) states that the total heght of all rectangles occupyng a gven slot e must be lower than the strp heght. Eq. (29) states that the left and rght edges of every rectangle must be assgned to eactly one par of allocaton ponts (ee ) wth e >e. The doman of the new set of bnary varables s gven by eq. (30). Notce that contrary to NDCS there can be no doman reducton for the left-edge allocaton pont. e e e E e e e E e e h N e e H e E e E (28) ee e e E e N e e 1 (29) N e e {01} e E e E e e (30) 6.1. Heurstc search for the global optmal soluton The new contnuous-space model can fnd the optmal soluton for a gven number of allocaton ponts. Wth a mnmum of two ponts.e. one slot a feasble soluton wth H h can be obtaned by packng all rectangles on top of each other provded that the bg-m value n eqs. (26-27) s greater than H. n contrast the hybrd dscrete/contnuous-space model n (Castro & Olvera 2011) normally requres a few allocaton ponts to ensure feasblty. n general however two allocaton ponts wll be nsuffcent to fnd the real optmal soluton to the problem and a hgher value wll need to be specfed. The dffculty s that there s no eact method to predct the value to use even for one-dmensonal schedulng problems and despte recent efforts (L & Floudas 2010). Ths s a serous ssue consderng that one typcally gets a one order of magntude ncrease n computatonal effort followng a sngle ncrease n the number of allocaton ponts.e. events (see Castro & co-workers 2004 2008 2011). Snce global optmalty can only be ensured n the lmt of E = +1 whch wll almost certanly compromse tractablty a proper search procedure s needed. We adapt the teratve search procedure from (Castro & Olvera 2011) that s n turn borrowed from event-based schedulng models (Méndez et al. 2006). Startng wth E =2 we keep ncreasng the number of allocaton ponts and solvng the optmzaton problem untl the objectve functon 13
stops mprovng. Snce the soluton space for E +1 ponts ncludes the feasble space for E and to enhance computatonal performance we remove the current best soluton from the feasble space through the use of a cutoff value see eq. (31). Wth the purpose of solely searchng for a better soluton we make the problem nfeasble f the global optmal soluton has already been found. One can also specfy an absolute optmalty tolerance for the MLP solver ε whch s partcularly useful n the case of nteger data for the rectangles dmensons lke n the problems consdered n ths artcle.e. ε=0.999. The detaled search algorthm for the contnuous-space model NCS s gven n Fgure 8. Hcutoff (31) ntalzaton E =2 absolute optmalty tolerance=ε Solve MLP E = E +1 mn H (1) s.t. eqs. (3 21 31) store soluton cutoff=h ε NO problem nfeasble? YES Output Strp heght H Placements nto strp Fgure 8. Search algorthm for new slot-based contnuous-space approach (NCS). 6.2. The Temporary Plateau Lmtaton Although the teratve search procedure works qute well n practce (the frst teratons can be solved rather rapdly) t s mportant to hghlght that the plateau n the objectve functon may only be temporary meanng that there s no guarantee that the global optmal soluton wll result even f all teratons are solved to optmalty. n fact ths phenomenon was observed n 2 out of the 29 test problems solved contrary to what happened wth the hybrd dscrete/contnuous-space model from (Castro & Olvera 2011) where a smple eample had to be used for llustratve purposes. 14
Takng the well-known ngcut08 problem as an eample the search for the optmal strp heght starts wth H=149 for 2 allocaton ponts. For a sngle ncrement n the number of allocaton ponts there s a substantal decrease n heght H=75. The soluton contnues to mprove for 4 and 5 ponts wth the optmal arrangements gven n Fgure 9. Notce that the soluton for E =4 stll generates a consderable amount of waste and that the locaton of e 2 and e 3 defne gullotne cuts. The same can be sad for e 2 and e 3 concernng the soluton on the rght but not for e 4 snce rectangle 2 s located between e 3 and e 5. The search contnues for E =6 that features an optmal strp heght equal to 36. The same optmal soluton s returned for E =7 leadng to the termnaton of the search wth a best soluton that s n fact suboptmal. More specfcally f model NCS s solved for E =8 the global optmal soluton can be found (H=33). These two solutons are shown n Fgure 10. n the one on the rght there are already seven rectangles (2-4 7 8 10 12) for whch the actve bnary varable N features e e e >e+1. n partcular rectangle 8 spreads across three slots wth the left-edge assgned to allocaton pont e 5 and the rght-edge to pont e 8. 15
H=50 8 2 3 44 42 4 H=40 38 6 38 4 2 34 9 31 31 11 8 23 5 26 11 1 5 21 20 12 18 18 1 12 14 6 7 10 10 9 8 8 7 13 3 13 6 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 e1 e2 e3 e4 e1 e2 e3 e4 e5 Fgure 9. nfluence on the number of allocaton ponts on the optmal soluton for E16 part 1. On the left E =4 and H=50. On the rght E =5 and H=40. 16
H=36 4 H=33 30 5 7 13 9 10 12 23 6 22 18 2 25 3 21 20 11 5 16 1 15 9 10 6 11 8 8 7 1 2 12 6 8 13 4 3 8 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 e1 e2 e3 e4 e5 e6 e1 e2 e3 e4 e5 e6 e7 e8 Fgure 10. nfluence on the number of allocaton ponts on the optmal soluton for E16 part 2. On the left E =6 and H=36. On the rght E =8 and H=33 whch s a global optmal soluton. 7. Computatonal Results The performance of the gven soluton approaches (NDCS NCS and CS) s now llustrated through the soluton of 29 strp packng problems and compared to that of two recent soluton strateges (DS and DCS see Castro & Olvera 2011) that have been evaluated under the same software and hardware. More specfcally the models and search algorthms were mplemented n GAMS 23.2 usng CPLEX 12.1 as the MLP solver wth a sngle thread an absolute optmalty tolerance equal to 0.999 (recall that all data are nteger) and a mamum computatonal tme per run equal to 7200 CPU s. n the case of the search algorthms a new teraton started only f the accumulated computatonal tme was below 7200 CPU s makng t possble for the total computatonal tme to go up to 4 hours. n eqs. (4-5 14-15 26-27) we have used M=300 ecept for cgcut02 (M=3000 so that the soluton for E =2 s feasble). The hardware conssted of a laptop wth 17
an ntel Core2 Duo T9300 2.5 GHz processor wth 4 GB of RAM runnng Wndows Vsta Enterprse. The GAMS nput fles of all fve mathematcal programmng models and assocated search algorthms are avalable through the CMU-BM Cybernfrastructure for MNLP collaboratve ste www.mnlp.org n the MNLP lbrary of problems under the ttle: 2-D Orthogonal Strp Packng. 7.1. Key performance ndcators The ultmate goal of any optmzaton approach s to determne the global optmal soluton whch frst nvolves fndng such soluton and then provng t cannot be mproved further. For two of the approaches under evaluaton (CS and NDCS) ths nvolves solvng a smple MLP problem per eample whle for the others a few teratons may be nvolved. n partcular t s possble for DCS and NCS to termnate wth a suboptmal soluton snce the search strategy for the global optmal soluton s not rgorous but heurstc. n fact ngcut08 and ht02 led to temporary plateaus for NCS (see secton 6.2) whle the phenomena was not observed for DCS. When falng to fnd the optmal soluton most methods are capable of fndng near optmal solutons the ecepton beng DS for whch the frst feasble soluton s also optmal. Table 2 provdes the values for a comparatve analyss of these 4 performance ndcators. Table 2. Key performance ndcators on the soluton of 29 test problems Optmal solutons Suboptmal solutons No solutons Best performer Approach Reference Found Proven Dscrete-space (DS) (Castro & 19 19-10 8 Olvera 2011) Hybrd (Castro & 21 13 8-7 dscrete/contnuous (DCS) Olvera 2011) Contnuous-space (CS) (Castllo et 16 6 13-5 al. 2005) New hybrd Ths work 21 19 8-7 dscrete/contnuous (NDCS) New contnuous-space Ths work 17 15 12-2 (NCS) Total 29 18
Table 3. Best soluton found and total computatonal effort for dfferent approaches (best performer n bold suboptmal solutons n talc) Prevous work New work Prevous work New work Approach DS DCS CS NDCS NCS DS DCS CS NDCS NCS Problem W H CPU s E1 9 18 18 18 18 18 18 4.86 3.21 0.39 2.10 4.23 E2 9 18 18 18 18 18 18 3.45 2.99 0.39 0.75 4.31 E3 10 20 23 23 23 23 23 6.21 9578 a 7200 a 244 117 E4 11 20-22 22 22 22 7205 a 10762 a 7200 a 340 3230 E5 21 10 24 24 24 24 24 4.61 2.93 46.9 41.2 50.1 SCP16 14 6 33 33 33 33 33 5.30 2.87 4335 b 2.87 505 cgcut01 16 10 23 23 23 23 23 2.37 4.75 7200 a 57.2 8027 a cgcut02 23 70-67 68 72 74 7463 a 7201 a 7200 a 7200 a 8720 a ngcut01 10 10 23 23 23 23 23 81.6 461 293 2.71 3.46 ngcut02 17 10-30 30 30 30 7202 a 12368 a 5182 b 26.2 329 ngcut03 21 10 28 28 29 29 29 12.6 2.87 7200 a 7200 a 13682 a ngcut04 7 10 20 20 20 20 20 1.22 5.90 0.96 0.56 9.93 ngcut05 14 10 36 36 36 36 36 3.18 0.62 7200 a 0.81 2036 ngcut06 15 10-31 31 31 31 3742 c 8194 a 6235 b 87.5 159 ngcut07 8 20 20 20 20 20 20 0.98 3.37 0.23 0.4 1.24 ngcut08 13 20 33 33 34 33 36 29.5 7363 a 7200 a 118 1280 ngcut09 18 20-53 51 53 52 7210 a 7200 a 7200 a 7200 a 10616 a ngcut10 13 30-80 80 80 80 8760 a 7200 a 4198 c 5118 b 3665 ngcut11 15 30-52 52 52 54 7215 a 10028 a 7200 a 7200 a 12153 a ngcut12 22 30-87 87 87 87 10339 a 7202 a 5235 b 987 449 ht01 16 20 20 20 21 20 20 2.08 2.01 7200 a 26.6 9128 a ht02 17 20 20 21 21 20 21 5.28 7200 a 7200 a 6952 4093 ht03 16 20 20 20 21 20 20 2.51 9.34 7200 a 20.4 1226 ht04 25 40 15 16 17 17 17 91.8 7202 a 7200 a 7200 a 7203 a ht05 25 40 15 16 17 17 17 20.4 7200 a 7200 a 7200 a 12737 a ht06 25 40 15 15 16 15 16 18.3 11.5 7200 a 773 13428 a ht07 28 60 30 31 34 38 35 3771 7201 a 7200 a 7200 a 7203 a ht08 29 60-31 33 35 37 7200 a 7201 a 7200 a 7204 a 7200 a ht09 28 60-33 32 39 34 7200 a 7201 a 7200 a 7204 a 12752 a a Mamum computatonal tme termnaton. b Out of memory termnaton. c Abnormal termnaton wth suboptmal soluton. The new hybrd dscrete/contnuous-space model (NDCS) s the best overall performer. t s able to fnd the same 21 optmal solutons as ts DCS counterpart and was able to prove optmalty n 6 more cases. When added to the fact that no teratve procedure for the global optmal soluton s needed t shows that t s preferable to use sequencng varables than multple spatal grds for the y- as. The same number of proven optmal solutons (19) was obtaned by the dscrete-space approach whose man drawback s ether fndng an optmum or no soluton at all. t thus has to be consdered the thrd best performer. Further down the lne come the two contnuous-space approaches that ehbted a lower success rate n terms of fndng and provng optmalty wth the 19
new one havng the edge due to the hgher number of proven optmal solutons (15 vs. 6). t s thus better to employ a sngle tme grd on the -as than postonng varables despte the dsadvantage of the heurstc search procedure. The last column n Table 2 gves the number of problems n whch a gven formulaton was the best performer. To make the decson we frst consdered soluton qualty and then computatonal tme (see Table 3). Whenever two approaches had the same tme (e.g. DCS and NDCS n SCP16) we chose the one not requrng an teratve search procedure (NDCS). n ngcut11 we broke the te between CS and NDCS by pckng the one wth the hghest relaaton (best possble soluton) at the tme of termnaton (43.0 vs. 50.8). The most nterestng result s that a partcular formulaton s the best performer n at least two problems meanng that they all can be potentally useful. n the days of parallel computng where even a relatve nepensve computer features a few threads t s already possble to rely on multple approaches for decson-makng. Nevertheless a few recommendatons can be made that are the subject of secton 8. But frst let us dscuss the computatonal statstcs related to problem sze. 7.2. Computatonal Statstcs n Table 4 we show the number of enttes related to problem sze together wth the root node relaaton for the models that do not requre an teratve search procedure CS and NDCS (note that the dscrete-space model DS uses a dfferent objectve functon and that the relaaton for the slot based models DCS and NCS changes wth E ). The values n columns 2 and 3 show that the new hybrd dscrete/contnuous-space MLP model s consderably tghter than the contnuous-space model of Castllo et al. (2005). n fact the relaaton of the former s equal to the well-known contnuous lower bound (Martello et al. 2003) gven by eq. (32). The same apples to hybrd model DCS whle the relaaton of the new contnuous-space approach (NCS) les somewhere n between CS and NDCS. As an eample the largest ntegralty gap was for ngcut12 wth an optmal soluton H=87 and a root node relaaton equal to 24 (CS) 41.6 (NCS) and 83.5 (DCS and NDCS). w h /W (32) 20
Wth respect to problem sze the opton of dscretzng both dmensons leads to the largest number of bnary and total varables whch can easly go past the tens of thousands lsted n Table 4. Ths was hardly unepected consderng the use of three-nde bnary varables (y) n DS where X =W and Y =H. DCS also features three-nde bnares (e) but now the number of allocaton ponts n the y-as requred to fnd the optmal soluton s consderably lower than the strp heght. n terms of the models gven n ths paper NCS also features three ndces (ee ) but two of them are event based resultng n typcally fewer bnary varables than the two-nde () model NCS. The lowest number of bnares often results from the two-nde ( ) model CS. A smlar trend s observed n terms of total varables whle for total constrants the results are not as conclusve wth DS DCS and CS sharng top spot. The number of constrants for the new hybrd model (NDCS) s typcally one order of magntude larger wth NCS lyng n fourth place. Notce that the large majorty of the constrants for NDCS arse from the y-as no overlap constrants whch feature three-ndces ( ) whereas n the correspondng NCS constrants the frst s an allocaton pont nde (e ). The other three approaches have just two ndces n the no overlap constrants: one set of (y) equatons n DS; 3 sets of (e) constrants n DCS; and 2 sets of ( ) equatons n CS. Wth respect to the comparson between the two hybrd dscrete/contnuous-space models movng from DCS to NDCS can be vewed as swtchng the complety from the bnary varables to the constrants sde. Overall t s clear that an analyss based solely on problem sze s hardly sutable to predct the best performer. 21
Table 4. Statstcs for last problem solved (RMP=root node relaaton DV=dscrete varables TV=total varables SE=sngle equatons) Approach CS NDCS DS DCS CS NDCS NCS DS DCS CS NDCS NCS DS DCS CS NDCS NCS Problem RMP DV TV SE E1 15 15.4 1342 580 72 152 225 1667 671 91 162 241 334 359 180 1332 645 E2 15 15.4 1323 585 72 153 225 1648 676 91 163 241 334 359 180 1332 645 E3 11 20 2485 652 90 208 255 2946 733 111 219 272 471 322 220 1840 776 E4 13 20 2428 720 110 235 363 2849 801 133 247 382 432 323 264 2242 1107 E5 9 22.5 3729 1336 420 377 525 3970 1417 463 399 552 262 325 924 4252 2462 SCP16 11 31.8 1829 441 182 154 301 2028 484 211 169 321 213 168 420 1126 1153 cgcut01 8 22.5 2409 868 240 244 456 2640 939 273 261 479 247 282 544 2442 1814 cgcut02 31 62.1 63438 6425 506 1538 598 67849 6776 553 1562 627 4434 1413 1104 35536 2926 ngcut01 10 20 1069 420 90 105 145 1300 491 111 116 160 241 276 220 930 484 ngcut02 9 29 2582 1080 272 244 391 2873 1181 307 262 414 308 397 612 2764 1654 ngcut03 9 27.7 3820 1296 420 372 651 4101 1377 463 394 679 302 325 924 4252 3009 ngcut04 15 16.2 606 186 42 83 168 807 217 57 91 182 208 121 112 444 419 ngcut05 12 35.3 2954 560 182 205 385 3315 621 211 220 406 375 204 420 1858 1420 ngcut06 11 29 2391 792 210 204 330 2692 873 241 220 351 316 319 480 2140 1310 ngcut07 20 20 1603 393 56 159 148 2004 454 73 168 162 409 242 144 1156 421 ngcut08 18 31.7 4780 856 156 292 351 5441 937 183 306 371 674 325 364 3166 1241 ngcut09 20 48.7 10428 1578 306 416 532 11409 1699 343 435 556 999 486 684 6176 2256 ngcut10 30 57.5 12285 1512 156 330 351 14536 1693 183 344 371 2264 721 364 4736 1241 ngcut11 29 49.4 11076 1920 210 425 525 12577 2101 241 441 548 1516 723 480 6360 1927 ngcut12 24 83.5 23813 4277 462 560 561 26334 4668 507 583 589 2543 1556 1012 13934 2689 ht01 12 20 3907 1305 240 381 568 4308 1406 273 398 592 417 406 544 4852 2167 ht02 13 20 4222 1305 272 397 493 4623 1406 307 415 517 418 407 612 5494 2029 ht03 14 20 4054 1488 204 368 456 4455 1609 273 385 479 417 484 544 4852 1814 ht04 5 15 10691 4140 600 1128 675 11292 4341 651 1154 706 626 815 1300 24090 3430 ht05 7 15 10938 4230 600 1146 825 11539 4431 651 1172 857 626 815 1300 24090 4181 ht06 7 15 11131 5010 600 1135 825 11732 5251 651 1161 857 626 973 1300 24090 4181 ht07 13 30 35531 8394 756 1777 798 37332 8755 813 1806 832 1829 1456 1624 45476 4261 ht08 11 30 37249 7400 812 1886 841 39050 7701 871 1916 876 1830 1219 1740 48838 4558 ht09 14 30 35415 7055 756 1789 966 37216 7356 813 1819 1001 1829 1218 1624 45476 5186 22
Table 5. Man Characterstcs of Tested Approaches for 2-D Strp Packng Problems Spatal Approach Full Dscrete Hybrd dscrete/contnuous Full Contnuous Name DS DCS NDCS CS NCS -as Sngle unform grd Sngle unform grd Sngle unform grd General precedence Sngle slot-based grd y-as Sngle unform grd Multple slot-based grd General precedence General precedence General precedence A pror Sze of squares (rectangles) n the mesh Wdth of unform tme slots Wdth of unform tme None Number of allocaton decsons that (-as); number of slots (-as) ponts and number of can affect allocaton ponts of dfferent slots a rectangle can span soluton grds (y-as) (-as) Need for teratve Yes Yes No No Yes search procedure Strengths Perfect packng problems; possblty of usng a Tghtest formulaton (same Best overall performer; Ablty to fnd good Problem sze ndependent courser grd to keep problem tractable (data rounded relaaton as NDCS) tghtest formulaton solutons fast; leads to of strp dmensons. to multples of slot wdth and heght). (same relaaton as the smallest problem DCS) szes Drawbacks Optmal soluton s the frst feasble soluton from Soluton dependence on Strp wdth affects Not partcularly tght teratve search procedure search procedure; can lead to prohbtvely large number of allocaton ponts; problem sze meanng that global may end wth suboptmal problem szes when consderng large strp areas and Strp wdth affects problem optmalty may be soluton accurate data; soluton dependence on mesh sze sze dffcult to prove 23
8. Overvew of the Man Features of Alternatve Approaches The characterstcs strengths and drawbacks of the fve alternatve approaches tested are summarzed n Table 5. The frst aspect to hghlght s related to the motvaton behnd ths paper and concerns the type of spatal representaton employed. The deal formulaton should be fully contnuous-space to be ndependent of problem data and allow consderng real and not just nteger data for the rectangles wdths and heghts. f not we may need to employ the fnest dscretzaton possble (e.g. strp dvded n 11 squares) to consder the real accurate problem whch may lead to grds wth too many slots n other words to an ntractable problem. Ths property can also be consdered an advantage snce t gves an obvous way to reduce the problem sze and hence complety. By ncreasng the slot sze (coarser grd) and roundng up problem data an appromaton verson of the problem can stll be solved. Whle t s beyond the scope of ths paper to evaluate ths opton t s relevant to hghlght that ths strategy s frequently employed for the effectve soluton of ndustral szed schedulng problems (Méndez et al. 2006). A full dscrete-space approach (DS) has been shown partcularly effcent n zero-waste perfect packng problems where addng the constrant of no empty elementary squares has had a major mpact on effcency (Castro & Olvera 2011). Dscrete approaches are known to be consderably tghter than ther contnuous counterparts thus compensatng the larger sze of ther resultng mathematcal programmng problems. By keepng one of the dmensons dscrete (-as) we have kept the relaaton as tght as possble equal to the contnuous lower bound. Ths s somewhat related to a hgher lkelhood of fndng and provng optmalty as can be seen n Table 2. The advantage of usng general precedence sequencng varables nstead of multple grds n the y-as s that t avods the heurstc search procedure for the global optmal soluton whch can n theory lead to termnaton wth a suboptmal soluton (Castro & Olvera 2011). Such a drawback was ndeed observed for the new contnuous-space formulaton (NCS) suggestng that t may be more frequent when usng sngle rather than multple grds. Nevertheless NCS was tghter than ts general precedence counterpart (CS) whch can eplan why more problems were solved to global optmalty. However both are consderably less tght than DCS and 24
NDCS. The worst performance of CS n terms of provng optmalty s compensated by the generaton of the smallest MLPs of the lot whch s translated nto the ablty of fndng very good solutons n the early nodes of the search tree. 9. Conclusons Ths paper has presented two new med-nteger lnear programmng approaches for the 2- dmensonal strp packng problem the NDCS and NCS models. Contrary to the approach of relyng on two sets of bnary postonng varables to locate a partcular rectangle wth respect to another we keep the set of postonng varables for the y-as but use a dfferent approach for the -as. Whle n one model (NDCS) the -as s dscretzed through the use of a unform spatal grd consstng of a few slots n the other (NCS) a non-unform contnuous tme grd wth fewer slots s employed. Both can be vewed as hybrd models n the sense that dfferent concepts for spatal representaton are beng combned. The performance of the new models has been tested n several problems taken from the lterature and compared to three other MLP-based approaches both quanttatvely by usng the same hardware and software and qualtatvely by hghlghtng the man advantages and lmtatons of each partcular approach. The new hybrd dscrete/contnuous-space model was shown to be the best performer n key performance ndcators lke number of optmal solutons found and proven whle the new contnuous-space approach more than doubled the number of problems solved to optmalty wth respect to a prevously publshed contnuous-space model wth two sets of postonng varables. Overall t was nterestng to fnd out that all tested methods can be potentally useful snce each approach was the best performer n at least 2 out of 29 problems. Acknowledgments The authors gratefully acknowledge fnancal support from Luso-Amercan Foundaton and the Natonal Scence Foundaton under the 2011 Portugal U.S. Research Networks Program. 25
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