ANTZONE LAYOUT METAHEURISTIC: COUPLING ZONE-BASED LAYOUT OPTIMIZATION, ANT COLONY SYSTEM AND DOMAIN KNOWLEDGE

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1 NTZON LYOUT MTURST: OUPLN ZON-S LYOUT OPTMZTON, NT OLONY SYSTM N OMN KNOWL noit Montruil 1,2,3, Nabil Ouazzani 2,3, dith rothrton 2,3 and Mustapha Nourlfath 2,4 1. anada Rsarh hair in ntrpris nginring 2. NTOR Ntwork Organization Thnology Rsarh ntr 3. Laval Univrsity, Québ, anada 4. Univrsité du Québ n bitibi-témisamingu, Québ, anada noit.montruil@ntor.ulaval.a bstrat This paprs proposs a nw mtahuristi for solving th blok layout problm. Th mtahuristi xploits th ant olony systm approah, ombind with rnt advans in mathmatial modlling of th problm as wll as domain knowldg about blok layout. Th ovrall problm is dividd in two parts: layout dsign and layout nginring. Th ant olony approah dals with th layout dsign whil a linar optimization modl dals with layout nginring. Th optimization modl is a linar vrsion of th zon-basd layout optimization modl, whih optimizs a layout givn a ntr-to-zon layout od. layout od is th output of th paths gnratd by ah ant in th olony in its travl through th solution spa. Th papr dsribs th fundamntal lmnts dfining th mtahuristi. Thn it prsnts an intrativ java-basd softwar implmntation of th mtahuristi. inally it prsnts prliminary mpirial rsults. 1. ntrodution This paprs addrsss th fundamntal failitis layout optimization problm, onsisting of dfining th loation and shap of ntrs within a building so as to rspt ntr shap spifiations and loation onstraints and to optimiz proximity rlationships btwn ntrs and with th xtrior, oftn known as th blok layout problm. Th vrsion of this problm whr th availabl building spa is disrtizd as a st of fixd loations, whr a singl ntr an b laid out in ah loation, is th first to hav bn mathmatially modlld, indd as th quadrati assignmnt problm (illir & onnors [7]). Thn Montruil [11] has introdud a omprhnsiv sris of modls allowing to rprsnt layout optimization in ontinuous spa rathr than disrt spa, and dsribing how to xpliitly intgrat in th modling th input/output stations for ah ntr, th flow ntworks and aisl ntworks. Ths modls ar mixd intgr linar programs. n th last dad, svral xtnsions and improvmnts hav bn ahivd on ths optimization modls (.g. Mllr t al. [10], Shrali t al. [15]). owvr, vn taking advantag of th inrasd apabilitis of ommrial solvrs suh as plx, industrial siz ass annot b

2 solvd rapidly and rliably. This situation has alratd th urrnt dominan of huristi optimization, startd dads ago with th introdution of suh huristis as RT (rmour [1]) and rinford by rnt advans in mtahuristi optimization (s Tompkins t al. [18] for basis). Th ontribution of this papr is to invstigat th potntial of intgrating rnt advans in (1) mathmatial optimization modling of th layout problm, through th zon-basd layout optimization modl (Montruil t al. [13]), and (2) mtahuristi optimization, spially xploiting th ant olony mtahuristi framwork (.g. Manizzo & olorni [8]), oupld with domain knowldg about blok layout. and 1 and 2 and 3 and-basd struturation Z1 Z1 Z2 Z2 Z3 Z3 Z4 Z4 Z13 Z5 Z5 Z6 Z6 Z7 Z7 Z8 Z8 Z14 Z9 Z9 Z10 Z10 Z11 Z11 Z12 Z12 Z15 Zon-basd struturation Z1 Z2 Z3 Z4 Z13 Z17 Z5 Z6 Z7 Z8 Z14 Z18 Z9 Z10 Z11 - Z12 - Z15 and-basd layout oding ntrs zons Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Zon-basd layout oding Layout orrsponding to both odings igur 1: xampl for band-basd and zon-basd oding.

3 Th zon-basd layout optimization modl has bn introdud in an attmpt to mrg th bst aspts of both th quadrati assignmnt modl and th ontinuous spa layout modl, by xploiting th spa struturing stratgis undrlying most ontmporary layout huristis. ndd many authors (.g. Tat & Smith [17]) hav strongly onstraind th strutur of spa in th dsign of thir huristis. On of th most ommon ways to strutur th spa is with bands of flxibl aras to adapt to th ntrs plad within. Th rprsntation of this kind of spa struturing an b ahivd by a numbr of diffrnt oding shms. or a tn-ntr as, igur 1 prsnts th band-basd oding and th zon-basd oding of a strutur with thr horizontal bands. Th band-basd oding onsists of assigning ah ntr to a band and thn of spifying a linar ordr within its assignd band, whil th zon-basd oding onsists of assigning ah ntr to a zon. s shown in igur 2, th zon-basd layout optimization modl an support any band basd spa struturs suh as th thr horizontal bands abov, a struturing around four horizontal bands, as wll as mor omplx ons that ould bttr rprsnt spifi industrial ass, suh as thr horizontal bands on th wst sid and on vrtial bands on th ast sid of th building. n fat, any kind of spa strutur that rspt th following rul an b usd in th zon-basd modl: th rlativ positioning of any two zons is suh that ithr on of th two zons ontains th othr or th two zons annot intrst, with on zon stritly north, south, ast or wst of th othr. Z1 Z2 Z3 Z5 Z6 Z7 Z9 Z10 Z11 Z13 Z14 Z15 Z1 Z2 Z3 Z4 Z17 Z5 Z6 Z7 Z8 Z18 Z13 Z14 Z15 Z1 Z2 Z3 Z5 Z6 Z7 Z18 Z14 Z15 Z8 Z9 Z10 Z19 Z16 Z4 Z17 Z8 Z18 Z12 Z19 Z16 Z20 Z9 Z10 Z11 Z12 Z19 Z16 Z20 Z4 Z21 Z11 Z12 Z13 Z20 Z17 Z22 4V 31V 1V31V igur 2: llustrativ zon-basd spa struturs. fining th layout od an b thought as th layout dsign phas. Th sond phas is th layout nginring whih orrsponds to th gnration of th layout rspting th od, th shap and loation onstraints, and optimizing th intr-ntr proximity rlationships. Th zon modl globally optimizs both th dsign and nginring phass. Th dsign phas is dalt with by assigning ntrs to zons, whr ths zons ar spatially struturd rlativ to ah othr. Th portion of th zon modl dvotd to th dsign phas involvs 0-1 variabls and linar onstraints. Th nginring phas is modld using th ontinuous spa modling framwork introdud in Montruil [11], spifying th loation and shap of ah zon givn its ntr assignmnt and th imposd spatial strutur, th loation and shap of ah ntr within its assignd zon givn its shap and loation onstraints, and th

4 loation of th input/output stations of ah ntr. Th portion of th zon modl dvotd to th nginring phas involvs linar variabls and onstraints. Th objtiv funtion of th zon modl is linar. Th urrnt mpirial rsults whn solving th zon modl ar promising for small ass, providing th bst rsults vr on a sris of wll known ass. owvr it is lar that th modl, stritly solvd using ommrial solvrs, is not in a position to optimally solvd larg industrial ass (Montruil t al. [13]). Th basi ida at th sour of this papr is to us a mtahuristi to driv th layout dsign phas and to us th layout nginring portion of th zon layout modl, a linar programming modl, to optimiz th layout givn a dsign od. Th bauty of using th linar layout-from-od modl is (1) that for ah od th approah guarants that th optimal layout is xplord for that od and (2) that th modl asily allows to inorporat through linar variabls and onstraints suh notions as optimally loating th input/output station of ah ntr and laying out ntrs rstritd to spifi aras in th building. ndd th ida applis to any mtahuristi working from a oding of th solution. This papr xplors th potntial of ant olony basd huristi optimization. Th rmaindr of th papr is struturd as follows. Stion 2 dsribs th mixdintgr zon-basd layout optimization modl and th linar od-basd layout optimization modl. Stion 3 introdus th ntzon Layout mtahuristi. Stion 4 dsribs th intrativ softwar implmntation. Stion 5 prsnts prliminary mpirial rsults. Th final stion 6 provids onlusiv rmarks. 2. Layout optimization modls This stion first dfins formally th zon-basd layout optimization modl, whih is usd as a basis for dfining th od basd layout optimization modl. Th formr orrsponds to a mixd intgr program whil th lattr orrsponds to a rapidly solvabl linar program. 2.1 Zon-basd layout optimization modl s statd in th introdution, th zon-basd layout modl introdud by Montruil t al. (13) rquirs first to prdfin a st of zons whos rlativ positioning is to b fixd, or at last pr-struturd, yt ltting thm hav flxibl ontinuous-spa shaps and loations within this rlativ positioning onstraint st, and sond to assign ntrs to zons lik in th lassial Quadrati ssignmnt Problm (illir & onnors [7]). Th zon-basd ontinuous-spa failitis layout optimization modl is introdud formally blow. irst ar dfind th indis, sts, paramtrs and variabls usd in th modl, thn th objtiv funtion and onstraint sts ar statd, ah with a brif xplanation. ndis : : building, : ntr : : ntity (ithr a ntr, a zon z or building ) z, z : zon

5 Sts : : St of positiv intrations (flow or rlationship) : St of ntrs P x : St of pairs of zons (z,z ) suh that zon z is to b loatd to th lft of zon z along th X-axis P y : St of pairs of zons (z,z ) suh that zon z is to b loatd undr zon z along th y-axis S : St of nput/output stations for ntr Z : St of zons Z : St of zons in whih ntr is allowd to b laid out Paramtrs : a : Targt ara for ntity b xs, b xs, b ys, b ys : Min distan btwn lowr/uppr oordinat of /O station s of ntr on th X & Y axs d : Numbr of approximation dlta points for th ara of ntity (st to 10 in th xprimnt) i s s : ntration btwn /O station s of ntr and /O station s of ntr lx, lx, ly, ly : Minimal/maximal allowd lngth for ntity along th X & Y axs M : larg numbr p, p : Minimal/maximal allowd primtr for ntr r : Maximal allowd lngth-to-width ratio for ntity x, x, y, y : Minimal/maximal allowd oordinat for ntity along th X and Y axs Variabls : z : 0-1 assignmnt of ntr to b laid out in zon z X s, Y s : oordinat of /O station s of ntr along th X and Y axs X, X, Y, Y : Lowr/uppr oordinats of ntity along th X and Y axs + X s' s', X s' s' : Positiv/ngativ valu of X-distan btwn /O station s of ntr and /O station s of ntr + Y s' s', Y s' s' : Positiv/ngativ valu of Y-distan btwn /O station s of ntr and /O station s of ntr Objtiv funtion : minimizing th sum of intration tims rtilinar distan btwn th /O stations of ntrs + + i X + X + Y + Y (1.1) MN s' s' ( s' s' s' s' s' s' s' s' ) s' s' uilding and ntr shap ontrol : rspting th bounds on ntr and building lngth, width, primtr and ara 0 lx X - X lx U (2.1) 0 ly 0.5 a ( X - Y - Y X ) ( x i )² ( Y ly U (2.2) Y ) a x i U, i =1,, d (2.3)

6 whr : x i = lx + ly = min { d i 1 lx lx ) ubx = min { a r ; x } ( a r ; y } lx = a ly ly = ntr and zon loation ontrol: bounding ah ntr and zon to b within its allowd bounds in th building x x X X x x U Z (3.1) y y Y Y y y U Z (3.2) uilding loation ontrol : bounding th oordinats of th rtangular building x X X x (4.1) y Y Y y nput/output station loation in its ntr : ontrolling its loation to nsur adquat rprsntation Option a : Rstritd to b within a rtangl in th ntr X + b xs X s X - b xs ; s S (5.1a) Y + b ys Y s Y - a lx (4.2) b ys ; s S (5.2a) Option b : Rstritd to b at th ntr ntroid X s = ( X + X )/2 ; s S (5.1b) Y s = ( Y + Y )/2 ; s S (5.2b) omputation of th rtilinar distan btwn /O stations having intrations X s - X + s = X s' s' - X s' s' [(, s)-(, s )] (6.1) Y s - Y s = Y + s' s' - Y s' s' [(, s)-(, s )] (6.2) X + s' s', X s' s', Y + s' s', Y s' s' 0 [(, s)-(, s )] (6.3) ntrzon rlativ position ontrol : foring th dsird zon basd spa strutur X X z' (z, z ) P x (7.1) z Yz Y z' (z, z ) P y (7.2) ntr-to-zon assignmnt : nsuring that ah ntr is assignd to a zon, with at most on ntr par zon z Z z = 1 (8.1) z 1 z Z (8.2) z = 0 or 1 ; z Ζ (8.3)

7 ntr layout in its assignd zon : foring th zon to mbd th ntr whn th ntr is assignd to th zon X z X + M(1 - z ) ; z Ζ (9.1) Yz Y + M(1 - z ) ; z Ζ (9.2) X X z + M(1 - z ) ; z Ζ (9.3) Y Y z + M(1 - z ) ; z Ζ (9.4) s in Montruil [11], a layout is dfind through th valus of th lowr and uppr X and Y oordinats of ah ntr and of th building, as wll as th loation of th input/output station(s) of ah ntr. Th ntrs ar ford to hav a rtangular shap. onstraint sts 2.1 to 2.3 insur that th lowr and uppr bounds on th lngth and width of ah ntr and th building ar within spifiations, as wll as thir lngth-to-width ratio and thir ara (s Montruil [11] and Shrali t al. [15]), onstraints sts 3.1 and 3.2 for th ntrs to b within th building, or mor rstritivly within a givn ara of th building. onstraint 4.1 and 4.2 rstrit th loation of th building. ah ntr may hav any numbr of /O stations, ah rstritd through onstraint sts 5.1 and 5.2 to b laid out within som rlativly spifid rtangular spa within th ntr. t th simplst lvl, a ntr may hav a singl /O station, whih is rstritd to b loatd at th ntr s ntroid. positiv intration an b spifid btwn vry /O stations of any ntrs. This intration may orrspond, for xampl, to a flow from th output station of a ntr to th input station of a ntr. t may also orrspond to a qualitativ assssmnt of th importan that two ntrs b laid out in proximity of ah othr, in whih as th intration may b dfind btwn /O stations rstritd to b at th ntroid of thir rsptiv ntrs. Th objtiv funtion 1.1 minimizs th sum ovr all pairs of /O stations btwn whih an intration is dfind, of th produt of this intration and th rtilinar distan btwn th /O stations. This rtilinar distan is omputd through its positiv and ngativ omponnts through th us of onstraint sts 6.1.to 6.3 Th modl rquirs th dfinition of a st of zons within whih ntrs ar to b laid out, and a rlativ positioning of th zons whih forbids ll ovrlap. This rlativ positioning is nford through onstraint sts 7.1 and 7.2. ah ntr has to b assignd to a singl zon and ah zon may mbd at most a singl ntr, as nford through onstraint sts 8.1 through 8.3. This involvs 0-1 variabls z. Whn a ntr is assignd to a zon thn this rtangular zon must physially ontain th ntr, as imposd through onstraint sts 9.1 to od basd layout optimization modl n th zon layout optimization modl, layout dsign orrsponds to stting th z 0-1 variabls whil th layout nginring orrsponds to th rmaindr onstraint sts. Whn th layout dsign is prst and th z ar spifid a priori, thn th layout nginring orrsponds to a linar program. n this papr, assigning ah ntr to a uniqu zon is

8 trmd a layout od. n lin with this dfinition, th od basd layout optimization modl is thrfor th linarizd vrsion of th zon-basd layout optimization modl providd a prst layout od. Th od basd layout modl is to b solvd ah tim an ant oms up with a od in th ntzon mtahuristi, Sin only th valus of th assignmnts ar to vary from on rsolution to th nxt, th simplx algorithm dos not nd to b rstartd from srath. t may b solvd in th gnrally muh fastr snsitivity mod. 3. ntzon Layout mtahuristi This stion starts by prsnting th ant olony systm prinipls and pursus with a dtaild prsntation of th ky lmnts of th ntzon Layout mtahuristi. 3.1 nt olony systm prinipl n natur, ants lay down in som quantity an aromati substan, known as phromon, in thir way from food. Th phromon quantity that an ant dposits is dpndnt on th lngth of th path and th quality of th disovrd food sour. narby rih sour lads to strong dposit at ky loations along th path from it. Othr ants an obsrv th phromon trail and ar attratd to follow it. n ant hooss a spifi path in orrlation with th intnsity of th phromon dpositd by ants having travlld in th ara. Th phromon trail vaporats ovr tim if no mor phromon is laid down. Thus, th paths lading to rih food sours los to th nst ar mor frquntd and thir phromon lvl thrfor kps highr. n that way, paths to th bst food sours hav mor phromon and highr probability to b hosn by ants sking food. Th soial larning involvd prmits dynami huristi optimization of th food gathring pross by th ant olony. Th dsribd bhaviour of ral ant olonis an b usd to huristially solv ombinatorial optimization problms by simulation (.g. origo t al. [3]). rtifiial ants sarhing th solution spa for bst solutions simulat ral ants sarhing thir nvironmnt for bst food sours. path in this ant olony systm approah orrsponds to a od from whih a solution an b onstrutd. rtifiial phromon wights ar dpositd at dision points along th path. Ths wights ar usually proportional to th solution quality or to som guiding faturs of th solution. Th path of an artifiial ant, orrsponding to a solution od, is onstrutd in a randomizd huristi fashion biasd by th phromon trails lft by th prvious ants. Th phromon trails ar updatd aftr th onstrution of a solution, nforing that th bst faturs will hav a mor intnsiv phromon. pnding on th implmntation, all ants influn th phromon trail or only th ants providing th bttr solutions influn th phromon trail. lso phromon adjustmnt an b prformd as ah ant omplts a path or whn a givn st of ants hav all ompltd thir paths. n th huristi solution of a ombinatorial problm, th phromon wights an b sn as a kind of adaptiv mmory of th prvious solutions guiding th olltiv larning and optimization.

9 nt olony systm approahs ar gnrally atgorizd as mtahuristis as thy mbd a numbr of huristis. or xampl, a huristi is oftn usd to gnrat a solution from th od orrsponding to an ant path. lso, huristis ar ndd to gnrat th path of an ant and to updat th phromon wights. nt olony systms approahs hav bn usd to solv a varity of ombinatorial problms in th last dads (.g. origo t al. [3]). Suh approahs hav also bn oupld with othr huristi approahs suh as loal sarh and with mtahuristi approahs suh as simulatd annaling and tabu sarh (origo t al. [3], land [2]). Narst to th failitis layout problm addrssd in this papr is th appliation of ant olony systms approahs to th Quadrati ssignmnt Problm. numbr of rsarhrs hav publishd rsarh on th mattr in th last fw yars (.g. land [2], ambardlla t al. [6], Manizzo [8], Manizzo & olorni [9], and Stützl & oos [16]). owvr to our knowldg no rsarh has yt bn publishd to dat on applying ant olony systms approahs to th mor omplx faility layout problm. 3.2 sription of th ntzon Layout mtahuristi n gnral, dfining an ant olony systm mtahuristi rquirs to dsrib th following : 1. oding of an ant path in th solution spa 2. ration of an ant path givn phromon wights; 3. nration and valuation of a solution givn an ant path; 4. nitialization and updat of phromon wights; 5. trmination of th numbr of ants and thir travl disiplin. Using this dfining shm, th following stions dsrib th ntzon Layout mtahuristi oding of a solution as an ant path n th ntzon Layout mtahuristi th oding of an ant path xploits th notion of layout od dsribd in stion 2, whr a layout od spifis th ntr-to-zon assignmnts. Thus an ant path is xprssd as a path in a graph whos nods orrspond to ntr-to-zon assignmnts, with th onstraints that (1) thr is on nod assoiatd with ah ntr and (2) thr is at most on nod assoiatd to a givn zon. link from nod i to nod j in th path indiats that nod i has bn visitd just bfor nod j. s an illustrativ xampl, igur 3 prsnts an ant path for a as with tn ntrs and twlv zons. t shows that th ant first visits nod Z7 orrsponding to th assignmnt of ntr to zon 7, thn nod Z2 orrsponding to th assignmnt of ntr to zon 2, and so on. s an b sn in igur 3, an ant path is built not only on ntr-to-zon assignmnt nods, but also on intr-ntr proximity nods. or xampl, th link btwn nods Z9 and Z3 is highlightd as a dision in th path, xpliitly stating that th nxt nod to b visitd must b suh that ntr is assignd to a zon as nar as possibl to zon 2 to whih is alrady assignd ntr. ndd th sltion of nod Z3 is

10 stohastially indud by th link. Th following ruls ar usd for slting th nod indud by a sltd intr-ntr proximity link. irst, only availabl zons that ar narst to th zon of th alrady sltd ntr an b sltd. Sond, a zon is randomly sltd among th ligibl zons. ntr is to b assignd to zon 7 ntr is to b as nar as possibl to ntr ntr is to b assignd to zon 3 as stohastially indud by th dision to hav ntr as nar as possibl of ntr Z7 Z2 Z9 Z3 Z5 Z6 Z11 Z4 Z1 Z8 igur 3: llustrativ ant path. Th rasoning for adopting a doubl lvl oding involving both ntr-to-zon assignmnt nods and th intr-ntr proximity links is to rflt th quadrati natur of th assignmnt subproblm. Th ntr-to-zon assignmnt nods prmit to tak into onsidration th larning that bttr layouts ar xptd to b gnratd whn a ntr is assignd to som zons rathr than othrs. Th intr-ntr proximity links prmit to tak advantag of larning rlativ to th rlativ assignmnt of ntrs to zons, that bttr layouts ar gnratd whn a ntr i is laid out nar to som ntrs j rathr than othrs ration of an ant path givn phromon wights Th algorithm blow dsribs how ant path is ratd at som itration within th ntzon Layout mtahuristi, givn th urrnt phromon wights assoiatd to ah ntr-to-zon assignmnt nod and ah intr-ntr proximity link. 1. Prst th bst-sltion fator β btwn 0 and St all fasibl assignmnt nods as availabl for sltion and all proximity links as forbiddn (tabu). 3. f a numbr randomly sltd btwn 0 and 1 is gratr than β Thn 3.1 omput a sltion probability for ah availabl assignmnt nod and proximity link, proportional to its phromon wight, with th sum of probabilitis qual to on; 3.2 Randomly slt an assignmnt nod or proximity link among th availabl ons, basd on thir rsptiv probabilitis; Othrwis 3.3 Slt th assignmnt nod or proximity link having th highst phromon wight among th availabl ons, braking tis randomly. 4. f an assignmnt nod z is sltd, Thn

11 4.1 ssign ntr to zon z; 4.2 dd nod z to th ant path by linking it to th prvious visitd nod if any; Othrwis a proximity link is sltd, with ntr urrntly unassignd; 4.3 ind th narst zon(s) to th zon in whih is assignd ntr ; 4.4 Randomly assign ntr to a zon z among th narst zons, with qual probability of zon sltion; 4.5 dd assignmnt nod z and proximity link to th ant path by attahing link to th prviously visitd nod and rahing it to nod z; 5. St as forbiddn all othr assignmnts of ntr to som zon and all othr assignmnts of som ntr to zon z; thn st as availabl all proximity links btwn ntr and unassignd ntrs 6. f all ntrs ar assignd to a zon, thn stop, othrwis go bak to stp 3. Whn th bst-sltion fator β is st nar to on, thn th huristi slts mor oftn th availabl assignmnt or proximity with highst phromon wight rathr than rlying on th probabilisti draw basd on probabilitis basd on th phromon wights. high β thus promots nighbourhood xploration around th nods and links with highst phromon wights. low β onvrsly fors to rly havily on th phromon-wightd probabilisti draw. f β = 1, thn th highst-drivn st is always sltd and thr is no sarh as all ants follow th sam path, xpt whn som nods or links hav th sam phromon wight in th highst phromon st. f β = 0.8, thn th probability that th urrntly highst drivn st is sltd drops to or roughly 8,6%, allowing mor xploration of th solution whil still allowing nighbourhood xploration of th st of nods and links with urrntly top phromon wight. igur 4 graphially illustrats how an ant path is itrativly ratd givn th urrnt phromon wights. Th figur is struturd in four olumns and a row for ah itration of th path ration pross. Th first olumn draws th urrnt ant path at th nd of th atual itration. Th sond olumn dpits in spatial form th urrnt layout od assoiatd with th ant path by showing th ntrs alrady assignd to spifi zons. Th sltd or indud ntr-to-zon assignmnt is highlightd. Th third and fourth olumns show in matrix form th urrnt ntr-to-zon assignmnt matrix and th intr-ntr proximity matrix, not showing th atual phromon wights but rathr highlighting th status of ah assignmnt nod and proximity links. Th possibl status ar: impossibl (by usr input), alrady in ant path, availabl for sltion, xpliitly sltd, impliitly sltd (indud), and onsidrd for sltion (but not sltd). n th xampl, first an assignmnt nod, Z7, is sltd for starting th ant path. Thus ntr is to b assignd to zon 7. So in th sond itration th assignmnt matrix shows as forbiddn all othr assignmnts involving ntr and zon 7. Th proximity matrix shows that all intr-ntr proximitis involving ntr hav bom availabl. n this sond itration, again an assignmnt nod (Z2) is sltd. ntr is to b assignd to zon 2. Thus a link is drawn on th ant path from nod Z7 and nod Z2. n itration 3, nod Z9 is addd to th path. n itration 4, intr-ntr proximity link is sltd. This mans that ntr is to b assignd to a zon as nar as possibl to ntr alrady assignd to zon 2. ivn th thr horizontal band strutur of th zons,

12 URRNT NT PT URRNT LYOUT O N TROU N TROU NTR-ZON SSNMNTS N NTR-ZON SSNMNTS NTR-NTR PROXMTY SSNMNTS Z1 Z2 Z3 Z4 Z7 Z7 Z5 Z6 Z8 Z9 Z10 Z11 Z12 Z1 Z2 Z3 Z4 Z7 Z2 Z7 Z5 Z6 Z8 Z9 Z10 Z11 Z12 NTR NTR URRNT NTOR-ZON SSNMNT MTRX ZON URRNT NTR-NTOR PROXMTY SSNMNT MTRX NTR Z7 Z2 Z9 Z7 Z2 Z9 Z3 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR NTR Z7 Z2 Z9 Z5 Z3 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR Z7 Z2 Z9 Z6 Z5 Z3 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR Z11 Z11 Z4 Z7 Z2 Z9 Z6 Z5 Z3 Z7 Z2 Z9 Z6 Z5 Z3 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR NTR vailabl lrady in path onsidrd xpliitly sltd mpliitly sltd orbiddn, tabu mpossibl Z11 Z4 Z7 Z2 Z9 Z6 Z5 Z3 Z1 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR Z11 Z4 Z7 Z2 Z9 Z6 Z5 Z3 Z1 Z8 Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 NTR igur 4: llustrating th ant path ration pross.

13 th narst availabl zons to zon 2 ar zons 1, 3 and 6. Th huristi randomly slts zon 3 for assigning ntr. This indus nod Z3 to b impliitly sltd to ntr th ant path, linkd by to th prvious nod Z9. Th huristi ontinus until th last stp in whih th inlusion of nod Z8 omplts th ant path. igur 4 allows to study th volution of th assignmnt and proximity matris through th itrations. Th assignmnt matrix is gradually transformd from fully availabl to mostly forbiddn, with th hoi of sltions rduing at ah itration. Th proximity matrix starts ntirly forbiddn. Thn th numbr of availabl proximity links augmnts as th numbr of nods in th path inrasd, until a point at whih th inrasing numbr of forbiddn links btwn two alrady assignd ntrs gradually transforms th matrix again toward a mostly forbiddn stat nration and valuation of a solution givn an ant path s mntiond in th introdution, an ant path provids a layout od for whih th od-basd layout optimization modl is solvd. Th solution of th modl provids both th rsulting optimal layout givn th layout od, th objtiv funtion and th dsign onstraints. Tabl 1 provids th ky dsign information for a tn-ntr as introdud by ranis t al. [5]: th ara rquirmnts for ah ntr, th flows btwn ntrs, as wll as th fixd dimnsions for th building. Whn solving th od-basd layout modl for this as, givn th layout od of igur 3, th lft layout of igur 5 is gnratd assuming intrntroid travl. ts flow*distan sor of 16,345 is optimal for th providd layout od. Similarly th right layout of igur 5 rsults from solving th od-basd layout optimization modl assuming a singl nput/output station pr ntr, whos loation may b optimizd within th ntr. gain its sor of 6,183 is optimal for th providd od. oth layouts diffr from ah othr by som modifiation of shap and loation of a fw ntrs, and th loation of th /O stations. Thr ar two typs of ass whr th od basd layout modl dos not nd to b solvd as th rsulting optimal dsign an b found in a mor ffiint way. oth rquir that th shap of ah ntr b fixd and that th sum of th ntr aras qual th ara of a fixd building. n suh ass, ithr thr is a singl fasibl layout of th ntrs orrsponding to a layout od or thr is no fasibl layout orrsponding to th layout od. n th first as typ, th rlativ loation of ah /O station within its ntr is prst. This implis that on th layout of ntrs is drawn, thn th /O stations ar all loatd and th objtiv funtion of th solution an b alulatd. This as typ inluds as a subtyp th widly rsarhd vrsion of th quadrati assignmnt problm whr ntrs hav to b assignd to fixd zons. n th sond as typ, th rlativ loations of th /O station ar not prst, so th layout nginring must b ompltd by optimizing thir loations givn th now fixd layout of ntrs. This an b ahivd ffiintly through th polynomial algorithm introdud by Montruil and Ratliff [14].

14 Tabl 1: sign information for th tn-ntr ranis t al. as. ntrs ra ntr-ntr flows (ft²) uilding 90*95 On ntroid-loatd /O station pr ntr On optimally-loatd /O station pr ntr Sor = Sor = 6183 igur 5: Optimal layouts for th od of igur 3 assuming on /O station pr ntr, loatd (1) at th ntroid of thir ntr and (2) optimally loatd within thir ntr nitialization and updat of phromon wights Th spifi phromon wight assoiatd to ah ntr-to-zon assignmnt nod and ah intr-ntr proximity link plays an important rol in th mtahuristi as it influns signifiantly th ant path ration as dsribd in stion Thrfor th huristis for phromon wight initialization and updat ar ky fats of th ntzon Layout mtahuristi.

15 vn though svral implmntations of ant olony systms lt th ants starts with no initial wight, in ntzon Layout a domain knowldg basd huristi initialization of th phromon wights is prformd to fostr fastr larning. Th initialization of th phromon wight for a ntr-to-zon assignmnt nod and intr-ntr proximity links ar rsptivly ahivd through th following formula: W 0 = z i d za, z 10.1 W 0 i = 1+, ' 10.2 ' ' i Whr 0 W z d z = nitial wight of nod assigning ntr to zon z = vrag distan btwn zon z and all othr zons = i i s' s', 10.3 i s' s' = ntration btwn a ntr and all othr ntrs (summd ovr thir /O stations) = vrag ovr all ntrs i = i' i s' s',,' 10.4 ' s' s' = ntration btwn a ntr and a ntr (summd ovr thir /O stations) ' i = vrag ovr all ntrs i' Th intration dnsity ratio i /a divids th total intration assoiatd with a ntr by its ara. So a small ntr with a lot of intrations with othr ntrs has a muh highr intration dnsity ratio than a big ntr with limitd intration. Th initial phromon wight for a ntr-to-zon assignmnt nod divids th intration dnsity ratio of ntr by th avrag distan btwn zon z and all othr zons. r th intr-zon distans an b ithr omputd in trms of numbr of zons to ross to rah ah othr or by dvising an qualizd layout of th zons within th building and omputing th atual distan btwn zons (intr-ntroid or intr-boundary dpnding on th assumptions rlativ to /O station loation). n assignmnt of a ntr having a high intration dnsity ratio to a ntral zon has a high initial phromon wight whil th assignmnt of th sam ntr to a rmot zon has a lowr initial phromon wight. Th initial phromon wight for an intr-ntr proximity link is omputd as th sum of a onstant (qual to on) and a, intration importan ratio. Th ratio is simply omputd by dividing th total intration btwn th ntrs by th avrag intr-ntr

16 intration. y adding on to this ratio, this mans that a pair of ntrs having no dirt intration rsults in thir intr-ntr proximity link having an initial phromon wight of on rathr than zro. ntrs with high intration with ah othrs hav a havy wight assoiatd with thir proximity link. ftr a first omputation of th initial phromon wights, thy ar normalizd so that th avrag nod wight is qual to on and th avrag link wight quals th initial assignmnt-proximity ratio γ, whr γ 0 is prst by th usr. Whn γ = 1, thn th avrag assignmnt nod wight quals th avrag proximity link wight. Whn γ < 1, mor wight is rlativly givn to th assignmnt nods, and vi-vrsa whn γ > 1. Tabl 2 and 3 rsptivly provid th initial phromon wights for ah ntr-tozon assignmnt nod and intr-ntr proximity link for th 10-ntr as of ranis t al. [5] for a zon strutur basd on thr horizontal bands, ah allowd four horizontally alignd zons as shown in igur 1. Tabl 2: nitial phromon wights in th ntr-zon assignmnt matrix. ntr\zon ,18 0,22 0,22 0,18 0,20 0,26 0,26 0,20 0,18 0,22 0,22 0,18 2,11 2,64 2,64 2,11 2,44 3,17 3,17 2,44 2,11 2,64 2,64 2,11 0,78 0,98 0,98 0,78 0,90 1,17 1,17 0,90 0,78 0,98 0,98 0,78 0,21 0,26 0,26 0,21 0,24 0,31 0,31 0,24 0,21 0,26 0,26 0,21 1,33 1,66 1,66 1,33 1,53 1,99 1,99 1,53 1,33 1,66 1,66 1,33 0,98 1,22 1,22 0,98 1,13 1,47 1,47 1,13 0,98 1,22 1,22 0,98 0,47 0,59 0,59 0,47 0,54 0,70 0,70 0,54 0,47 0,59 0,59 0,47 1,27 1,59 1,59 1,27 1,47 1,91 1,91 1,47 1,27 1,59 1,59 1,27 0,47 0,59 0,59 0,47 0,54 0,70 0,70 0,54 0,47 0,59 0,59 0,47 0,59 0,74 0,74 0,59 0,69 0,89 0,89 0,69 0,59 0,74 0,74 0,59 Tabl 3: nitial phromon wights in th ntr proximity assignmnt matrix. ntrs ,598 2,072 0,598 0,5 0,5 0,5 0,5 0,5 0,5 2 0,893 0,598 0,893 0,893 0,893 0,598 0,893 0, ,5 0,5 0,5 0,5 0,5 0,5 0,5 4 0,5 0,598 0,5 0,5 0,5 0, ,5 0,5 0,5 0,5 6, ,072 0,5 0,893 0,5 7 0,5 2,072 0,5 8 0,5 6, ,072 Th ntzon Layout mtahuristi ontinuously maintains a list of th ant paths that hav laid to th urrntly N bst layouts, whr N is spifid by th usr. n ant is allowd to updat th phromon wights only if it oms bak with a solution that ntrs th list of

17 th top-n paths. asd on larning by th tam through xtnsiv xprimntation, this has bn nford to avoid having th multitud of bad layouts influning in quasi-random ways th volution of phromon wights whil th usr is indd sarhing for th ram of all layouts in th solution spa. Suh an litist stratgy has bn prvious usd with suss in th litratur (.g. origo t al. [3] and ambardlla t al. [6]). Whn an ant is allowd to updat th phromon wights on ntr-to-zon assignmnt nods and intr-ntr proximity links, it is prformd using th following formula: W W u u z Whr : a ( ) u 1 a = + 1 W z α, u 1 N, u 1 S d i α, z) u u P u S d i ( 10.5 ', u 1 N, u 1 p u 1 p = ( ) + S d i' ' 1 α W ' α u u ', ' 10.6 S d ' i W u z d ( ) W u 1 = 1 α z P u (, z) 10.7 α a α p α d = Smoothing fator [0,1] for phromon wights of assignmnt nods = Smoothing fator [0,1] for phromon wights of proximity links = Smoothing fator [0,1] for phromon day of assignmnt nods W u z = Phromon wight of nod assigning ntr to zon z aftr uth updating W u ' = Phromon wight of - intr-ntr proximity link aftr th u th updating P u S N, u S u d u, d u = St of ntr-to-zon assignmnt nods and intr-ntr proximity links dfining th ant path for th u th updating = varag layout sor of th N bst ant paths aftr th u th updating = Layout sor of th ant path for th u th updating = s' s' i s' s' + + ( X + X + Y + Y ) s' s' s' s' s' s' s' s' 10.8 i s' s' s' s' = vrag distan for intrations involving ntr in th layout of th ant path for th u th updating = vrag d u in th layout of th ant path for th u th updating

18 d u ' = s' s' i s' s' ' + + ( X + X + Y + Y ) s' s' s' s' s' s' i s' s' ' s' s' s' s' 10.9 ', d u = vrag distan for intrations involving ntr in th layout of th ant path for th u th updating = vrag d u ' in th layout of th ant path for th u th updating asd on quation 10.5, th phromon updat for th ntr-to-zon assignmnt nods solly for nods usd in th ant path is prformd as follows. Th nw wight is st qual to a wightd avrag btwn th old wight and an attributd wight, whr th avrag is wightd aording to th usr sltd smoothing fator. Th attributd wight is th produt of a path quality ratio and a nod ontribution ratio. Th path quality ratio simply divids th avrag layout sor for th N urrnt bst ant paths by th layout sor for th onsidrd ant path. Th nod ontribution ratio is basd on th fat that vn though a givn layout may hav an xllnt sor, som parts of th layout ontribut muh mor to this quality than othrs. or xampl, a ntr with high lvls of intration with othr ntrs that nds up with an avrag intration distan lowr than th avrag distan of all ntrs is indd ontributing highly to th solution quality. ntr with lowr intration intnsity has lss potntial for influning signifiantly th layout sor. Th ntr intration ratio rflts that fat by dividing th total intration of ntr by th avrag ovr all ntrs. ntr whos avrag intration distan is highr than th avrag ovr all ntrs is dtriorating th layout sor whil thos blow th avrag ontribut to minimiz th layout sor. Th ntr distan ratio rflts that fat by dividing th avrag ntr intration distan by that of ntr. Th nod ontribution ratio is th produt of its ntr intration ratio by its ntr distan ratio. quation 10.7 prmits to modl phromon day through vaporation whn nods ar not usd, as is ommon in natur. Th phromon wight of a nod not usd in an ant path sltd for phromon updating is simply rdud by a small phromon day smoothing fator. Whil an assignmnt is zro-on in natur, proximity is rlativ in natur and an b masurd gradually. This allows ntzon Layout to updat th phromon wights of all proximity links at ah ant path. Th updat omputation is prformd vry similarly to th nod updating. Th only diffrns ar th us of an intr-ntr distan ratio instad of a ntr distan ratio, and of an intr-ntr intration ratio instad of a ntr intration ratio, in th omputation of th link ontribution ratio. n summary, th updating pays in gnral mor whn th solution is bttr as ompard with th urrnt avrag of th N bst paths, yt maks disrimination among th

19 assignmnt nods and proximity links in trms of thir ontribution to th quality of th ant path trmination of th numbr of ants and thir travl disiplin Th numbr of ants prmits th onurrnt xploration of multipl paths within th solution spa. n paralll implmntation, ah ant may b dployd on a distint prossor, prmitting fast xploration. n usual singl-prossor implmntations, th ants ar typially launhd squntially, all of thm starting thir qust basd on th phromon updats from th prvious itration. n appliations suh as ntzon Layout whr th solution quality is valuatd on a path has bn ntirly ratd, thn th ntir path ration prods basd on th phromon wights from th prvious itration. Thus whn all ants hav ompltd thir path, thn th ants ar sltd for updating th phromons and th phromon updating pross is launhd, so as to guid th nxt round of ant launhing. Using fwr ants implis that a vry limitd numbr of ants xplor th solution spa nighbourhood around th phromon wights at th givn itration. Largr numbrs of ants slow down th phromon updating and thus th larning, hoping that ah xploration round is to allow th disovry of high quality solutions whih ar to influn signifiantly th dirtion of th sarh. n th ntzon Layout mtahuristi, th usr slts th numbr of ants to b usd through th solution pross. 4. Softwar implmntation and prliminary mpirial invstigation s dsribd in th prvious stion, a mtahuristi suh as ntzon Layout inluds multipl faturs influning its prforman. Partiularly, suh an ant olony systm mtahuristi mbds a rlian on probabilisti disions and on phromon basd distributd larning. This maks th bhaviour of th mtahuristi omplx to undrstand and prdit. This is why an intrativ softwar implmntation has bn ralizd, mphasizing th apability to dynamially xamin th volution of th phromon wights, th rsolution pross prforman, th atually bst layouts, and so on. igur 6 dpits a srnshot of th softwar implmntation of ntzon Layout. On th right sid lis th pan prmitting to kp trak of th top soring layouts. y liking on any on layout idntifiation, a ntral pan prmits to xamin th layout itslf. n th lowr ntral pan, a dynamially updatd graph prmits to kp trak of th layout sors as th mtahuristi progrsss. t shows at ah round th sor of th layout gnratd by solving th layout-from-od modl for th urrnt ant path, as wll as th avrag sor up to dat and th two-standard-dviation rgion around th man. Th uppr lft pans show th ntr-to-zon assignmnt matrix and th intr-ntr proximity matrix. olor shm from hot rd to nutral whit to dark blu prmits to monitor th volution of th phromon wights btwn high, avrag and low valus.

20 igur 6: Srnshot of th intrativ softwar implmntation of ntzon Layout. Whn ths graphial faturs ar nabld, it slows th solution pross. owvr it nabls to assss th dynami bhaviour of th mtahuristi. rom a rsarh prsptiv this has prmittd th dsign tam to prft th mtahuristi faturs, suh as th phromon wight updating pross and formula. rom a faility dsignr prsptiv, it prmits to avoid th blak box phnomnon and hlps larning th impat of th huristi faturs so as to bttr fin tun th mtahuristi. rom an aadmi prsptiv, th ava-odd th ntzon Layout implmntation is ntirly ompatibl and assibl through th intrnt-nabld WbLayout dsign platform (Montruil t al. [12]), prmitting profssors and studnts to xplor its potntial both as a layout dsign mtahuristi as an nabling tool within a largr sop fatory dsign nvironmnt. 5. Prliminary mpirial rsults This stion prsnts th rsults of a prliminary mpirial xprimnt aimd to assss th validity of th ntzon Layout mtahuristi. Th xprimnt has purposfully fousd on a wll known small as from ranis t al. [5] in ordr to both b abl to thoroughly study

21 th bhaviour of th mtahuristi and to prmit omparison with optimal rsults, both with a singl /O station pr ntr, a vrsion in whih it is rstritd to th ntroid of th ntr and a vrsion whos loation an b optimizd within its ntr. urthr xprimntations will aim to tst th mtahuristi on larg sal ass and to ompar its prforman to othr faility layout huristis and mtahuristis. 5.1 Solving th mixd-intgr zon-basd layout optimization modl Modlld using th zon basd layout optimization modl, th as has first bn solvd with th Plx 8.0 mixd intgr programming solvr on a M omputational srvr. s shown in Tabl 4, th ntroid /O vrsion was solvd in about 7.3 hours whil th optimally loatd /O vrsion was ountr-intuitivly solvd in a shortr tim of 5.7 hours. Ths solution tims ar aptabl whn th staks ar high and th absolut optimal is dsird. igur 7 dpits th 3-horizontal-band optimal solutions for both vrsions. t should b notd that th vrsions do not hav th sam optimal layout, dmonstrating by xampl that th hypothsis rlativ to /O loation has signifiant impat on th optimal solution. Optimal solutions suh as ths annot yt b gnrally found for ass with mor than 12 to 15 ntrs, vn whn ltting th solvr runs for many days. n ordr to analys th potntial of using an optimal modl, stopping its solution pross at a givn tim, and using its bst solution up to that tim, th solvr was stoppd aftr thr minuts and on hour. Th thr-minut markr orrsponds for that as to lt th ntzon Layout mtahuristi for about 10,000 ant path rations and valuations. Th on-hour markr is a surrogat to stopping th solvr aftr a vry long tim, yt bfor optimally, for largr ass. Whn stoppd at th thr-minut markr, Tabl 4 shows that th ntroid /O vrsion optimization providd a solution with 27% of optimality whil th optimally loatd /O vrsion had a gap of 32%. ftr on hour th gaps wr rsptivly down to 5% and 8%. Tabl 4: Rsults from solving th zon-basd layout optimization modl with Plx 8.0. Solution tim (s.) ntroid /O Optimally loatd /O Sor Optimality gap (%) Sor % % 3, % % 26,300 20, Optimality gap (%)

22 ntroid /O Optimally loatd /O Sor = 8567 Sor = 1404 igur 7: Optimal layout assuming on /O station pr ntr, loatd (1) at th ntroid of thir ntr and (2) optimally within thir ntr. 5.2 Solving through random sampling with random ants t th othr xtrm of solution produrs lis th wll known random sampling approah. Th ida is to us th spd of th omputr for gnrating and valuating solutions, and rmmbring th bst solution up to now, so as to lt th shr numbr of gnratd solutions hlp gnrat optimizd solutions. r this has bn implmntd by using fixd phromon wights qual to on for all ntr-to-zon assignmnt nods and all intr-ntr proximity links. So ant paths ar gnratd randomly thn ah is transformd into a layout and valuatd using th layout-from-od optimization modl. igur 8 dpits for th ntroid /O vrsion a typial rsult obtaind using this approah. Ovr 10,000 itrations of ant path ration and valuation, th graph dynamially shows th minimum, avrag and maximum sors obtaind ovr a group of 100 ants. t also shows a lin rprsnting th appliation of a simpl tim-phasd linar rgrssion modl. Th avrag 100-ant sor hovrs around th 14,500 lvl, with a slight dgradation by inrasing valu through tim. Th bst sor is tabl at around 11,000 until 4000 ants, thn drops to about 10,000 and stays thr until th nd. Th avrag random ant sor is thus 69% abov th optimal sor whil th bst random ant sor is 17% abov th optimal. Tabl 5 summarizs similar rsults avr 10 runs of 10,000 random ants for both vrsions.

23 22000 Sor volution by groups of 100 random ants Sor y = 1,8845x MN MX MN Linéair Linar rgrssion (MN) trations igur 8. : llustrating th prforman of random sampling with random ants using th ntroid /O vrsion of th ranis t al. (1992) as. Tabl 5: omparing random sampling through random ants to th optimal solution. Rsults ovr 10 runs with ntroid /O Optimally loatd /O 10,000 random ants /run Sor gap (%) Sor gap (%) st solution % % vrag minimum pr run % % vrag man pr run % % vrag maximum pr run % %

24 ntroid /O Optimally loatd /O Sor = 9175 Sor = 1594 Optimality gap = 7% Optimality gap = 14% igur 9: st solution ovr 10 runs of 10,000 random ants assuming on /O station pr ntr, loatd (1) at th ntroid of thir ntr and (2) optimally within thir ntr. 5.3 Solving with th ntzon Layout mtahuristi aving st th prforman arna btwn using an optimal solvr and random sampling in stions 5.1 and 5.2, this stion provids th prliminary mpirial rsults for th ntzon Layout mtahuristi. ll optimization runs hav usd a olony of 100 ants and hav stoppd aftr 10,000 ant paths, whih wr solvd in uniformly about thr minuts on th M ntllistation omputrs usd for th xprimntation. Not th mtahuristi implmntation in ava ass its wb portability at th widly aknowldgd xpns of omputational spd prforman rlativ to and ++ implmntations. So th thr-minut run ould surly b prformd in shortr tim through a spd-optimizd oding. Th first qustion to b answrd is whthr th mtahuristi has th apability of rahing nar optimal rsults. igur 10 larly illustrats th answr to this qustion by dpiting th bst rsults obtaind for both vrsions of th ranis t al. as. Th layout sor for th ntroid /O is optimal sin ntzon Layout gnrats th sam layout as th optimal solution shown in igur 7. or th optimally loatd /O vrsion, th mtahuristi gnrats an optimizd layout within 0,03% of optimality. Th layout od from whih th layout has bn gnratd, ( ), diffrs signifiantly from th

25 od of th optimal solution, ( ), whil having multipl strutural similaritis. Th sond qustion is whthr th mtahuristi is robust in trms of prforman givn that it taks probabilisti disions through its solution pross. Tabl 6 provids vidn rlativ to this qustion by dpiting th rsults for tn runs of 10,000 ants, at thr minuts ah run, solving th ntroid /O vrsion. ll runs usd a bst-sltion fator β = 0,9, smoothing fators α a = α a = 0,3 for phromon wights of assignmnt nods and proximity links, and a day smoothing fator st to zro. t indiats that th optimal solution of 8567 was gnratd by th ninth run. Th avrag ant path in this run sord 10,680. Th avrag bst sor of a run is 8633, whih orrsponds to an xllnt optimality gap of 0.77%. singl run rrd at 5, 54% of optimality, xmplifying th pitfalls of probabilisti dision making, whil ight runs wr within 0,27% of optimality, nin out of tn wr within 0,77%. ntroid /O Optimally loatd /O Sor = 8567 Sor = 1404 Optimality gap = Zro Optimality gap = 0.03% igur 10: st solution found by ntzon mtahuristi assuming on /O station pr ntr, loatd (1) at th ntroid of thir ntr and (2) optimally within thir ntr.

26 Tabl 6: Prforman of ntzon Layout mtahuristi for th ntroid /O vrsion of th ranis t al. [5]as using α=0.3 and β=0.9. Run of Sor Optimality ap (%) 10,000 ants Minimum vrag Maximum Minimum vrag Maximum ,27% 26,57% 84,34% ,54% 26,72% 84,28% ,27% 25,38% 103,68% ,27% 26,13% 82,79% ,27% 28,54% 99,42% ,27% 25,61% 78,22% ,27% 25,76% 112,83% ,27% 26,45% 98,26% ,00% 24,66% 83,32% ,27% 25,58% 85,97% st ,00% 24,66% 78,22% vrag ,77% 26,14% 91,31% Worst ,54% 28,54% 112,83% Sin th whol onpt of ant olony systms is about olltiv larning, thn th third qustion is whthr thr is vidn that th ants ar olltivly larning through th dynami phromon dposit pross. Th olltiv larning apability of ants in th ntzon Layout mtahuristi is wll illustratd by ontrasting igurs 8 and 11. Th formr shows th dynami prforman of random sampling whil th lattr xposs th th quivalnt for ntzon Layout using α=0.3 and β=0.9, t is asy to s that in igur 8 th random ants, with onstant phromon wights st to on, ar not larning sin thir prforman dos not improv through tim. n igur 11, th rlativ prforman of groups of 100 ants is gradually improving through tim until ar found optimal or nar optimal solutions. Thn it stabilizs to a lvl way bttr than random ants both in trms of avrag sor of ah ant paths and in trms of th bst sor rahd by an ant in thir group. ndd th avrag 100-ant sor gradually improvs to about 11,000 instad of a stabl 14,500 for random ants. Ovr a sampl of tn runs, th avrag prforman of random sampling run with 10,000 non-larning random ants is an optimality gap of 17% whil that of an ntzon Layout run with its 10,000 larning ants rahs an avrag optimality gap of 0.77%. This larly shows that th ants ar olltivly larning and that this larning prmits thm to ahiv suprior prforman. Th fourth qustion is about th alibration of th mtahuristi and its rlativ impat on prforman. Th rsults prviously shown up in th papr hav bn obtaind with 100 ants, a bst-sltion fator β = 0.9, smoothing fators α a = α a = 0,3 for phromon wights of assignmnt nods and proximity links, and a day smoothing fator st to zro. o ths sttings influn th prforman of th mtahuristi? s with most mtahuristis, th answr to this qustion is positiv. n this papr th xprimnt is fousd on th bst-

27 sltion fator and th smoothing fators for assignmnt nods and proximity links to dmonstrat th signifiant influn of th mtahuristi alibration Sor volution by groups of 100 ants Sor y = -12,207x trations MN MX MN Linéair Linar rgrssion (MN) igur 11: llustrating th prforman th ant larning apability of th ntzon Layout mtahuristi using th ntroid vrsion of th ranis t al. [5]as with a=0.3 and b=0.9. igur 12 and Tabl 7 provid th rsults of an xprimnt involving tn runs for ah ombination of bst-sltion fator and smoothing fator, ah with th following options. Th bst-sltion fator β was st to 0, 0.25, 0.5, 0.75, 0.9, 0.95 or 0,99. Th phromon updat smoothing fators wr st qual to ontrol th numbr of ombinations in this prliminary xprimnt, and allowd to tak valus multipls of 0.1 up to 0.9, as wll as Th numbr of ants was stationary at 100 as wll as th day fator at zro. Th xprimnt was run for th ntroid /O vrsion of th ranis t al. [5]as. or ah ombination of α and β, igur 12 plots th optimality gap of th avrag ovr th tn runs of bst sor obtaind at ah run. t larly shows that th bst-sltion fator is th most influntial. ts bst rang is around 0.75 and 0.9, with signifiant