An AGV-Routing Algorithm in the Mesh Topology with Random Partial Permutation

Transcription

1 A AGV-Rou Alorhm he Mesh Topoloy wh Radom aral ermuao Ze Jaya, Hsu We-J ad Vee Voo Yee ere for Advaed Iformao Sysems, Shool of ompuer eer Naya Teholoal Uversy, Sapore {p8589, hsu, Absra I hs paper, we model a reals AGV sysem by a mul robos sysem wh mesh layou Based o era reasoable assumpos, we propose a mproved rou alorhm, ad prove ha has a ood me performae wh hh probably Iroduo Auomaed Guded Vehles (or AGVs for shor) have beome a mpora opo maeral hadl [-7, 9-, 6 I may applaos, suh as oaer ermals[, 9-, he serve area s ofe arraed o reaular bloks, whh leads o a mesh-lke pah opoloy Therefore, develop effe alorhms for AGV rou o hs opoloy has beome a mpora researh op There are may exs resuls abou AGV [5 However, relavely lle s kow abou rou o he mesh opoloy [- ave he aalyss of me ad spae omplexes for some bas AGV rou operaos o -mesh opoloy The per bouds of me ad spae omplexes for AGV rou are Θ( ) ad Θ( ) respevely, where deoes he umber of odes he pah opoloy However, he paper does o ve he deals of he rou alorhms ad ehques o avod oeso, ofls, deadloks, e [6-7 preseed dffere mehods o shedule ad roue smulaeously a mesh-lke pah opoloy I hese papers, he rou proess s formulaed as a sor problem Alhouh here are o ofls dur he permuao, requres seps of well-defed physal moves, whh requres AGVs o ravel exra dsae ad osume exra eery o fsh he asks Aually a AGV sysem s also a mul robo sysem There has bee researh doe o he rou sraey he mul robo sysem[-5, bu hese soluos assume a small umber of robos o he mesh layou o more ha O ( N ) or O ( ) for a N mesh layou However, se here are odes he mesh layou, should be able o aommodae more AGVs/ robos I [5, O( ) umber of robos s osdered, ad a lo ood rou alorhm s preseed o fsh all asks O ( ) seps wh hh probably I hs paper, we mprove he rou alorhm of [5 ad we show ha us our rou alorhm, he permuao asks a be fshed O ( ) seps wh hher probably ha ha [5 The remader of he paper s orazed as follows Seo desrbes he rou model Seo ves he rou alorhm I Seo, we aalyze he me performae of he rou alorhm Fally, Seo 5 dsusses possbles of relax era osras ad pos ou dreos of fuure sudy Rou model I our AGV sysem, here are oal bloks, amely bloks eah olum ad bloks eah row ah blok has he same sze ah blok has oe

2 / k -rop off sao (or / sao for shor), loaed a he per rh ad per op orer of he blok O he per-lef sde, here s a vehle park where all AGVs are saoed ally ad o whh hey wll reur o ompleo of all asks We oraze he AGV movemes o hree phases I he frs phase, le AGVs se ou from he park o her pk saos I he seod phase, le AGVs pk loads ad ravel o her desaos ad drop-off loads I he hrd phase, le AGVs reur o he park from her drop-off saos Beause s easy for us o dspah he AGV mov whou ay ofl he frs phase ad he hrd phase, we wll fous oly o he seod phase whe he loaded AGVs move o he mesh layou I he follow, a sep of a AGV meas ha moves from oe ode o oe of s ehbor odes I he mesh opoloy, we assume ha he umber of!!" AGVs, m, s bouded by O( ) Thus, he lo Fure Reals mesh layou Alhouh here are some mpora deals for AGV rou, suh as he sze of he juo, he radus of urs, he leh of he AGV, e[-7, s reasoable ad reals for us o smplfy he mesh model for oveee of aalyss ad dsusso I he smplfed mesh layou model, as show Fure, here are juos of pahways A juo ad he assoaed ehbor sao are ollevely rearded as a ode ah ode s assed wh he oordaes as s address or I, where x ad y represe respevely he row ad olum Is Ths mesh layou s modeled by a raph The veres of he raph represe juo odes, ad he b-dreoal edes represe wo pahs bewee wo adjae juo odes, ad he leh of eah ede s a osa #%$%& ')( * +!, - follow, we spose ha m m lo The moveme paer s a - paral permuao, whh s defed as follows { σ σ : Z Z Z Z, σs, σ m }, where m O( ) lo A he same me, we assume ha he ommuao mehasm amo all AGVs allows eah AGV o dee he AGVs whh are oe u dsae aroud As show Fure, he AGV eer a dee he AGVs he "do" pos Fure ommuao level As [5, he mesh layou for rou s paroed o maary squares ah square osss of lo lo odes of he rds, as show Fure Fure Smplfed mesh rou model (a) There are lo rows of squares ad lo

3 olums of squares ah square s marked by he oordaes (,j) as s address or I, where ad j represe respevely he row ad olum A he same me, we assume all AGVs eah square a oly ravel he pre-spefed yle dreo show Fure (b) The dreos of ay wo ehbor yles are dffere The yles are represeed by L, L, L,, L k, where L k represes he boudary, ad L represes he ex eral yle,, L k represes he ermos yle he square lo lo efo (Job): A job s defed by a ordered par J((,Y),(,Y)), where (,Y) represes he address of he pk sao, (,Y) represes he address of he drop-off sao, ad (,Y ) (,Y ) efo (Or square job se): A or square job se S (, j) deo a job se whh eah job s oraed from he square (,j), e S (, j) { J((, Y),(, Y))(, Y) square(, j)} L L L L Noe ha, by our assumpo, S lo (, j) efo (esao square job se): A desao square job se S (, j) deo a job se whh eah job s desed o he square (,j), e (a)the paro of mesh layou (b) Imaary yles eah square maary squares Fure re-spealzao of he mesh layou A he same me, we follow he formal defo of ood paral permuaos defed by [5 efo A: For a permuao σ : Z Z Z Z, σ, f a mos lo AGVs are oraed from (or desed o) every square, we all σ a ood paral permuao, where m max{,6 } Se mesh layou, a radom σ s a ood paral permuao wh hh probably, for lare, s reasoable for us o assume ha our rou sysem, he permuao s a ood paral permuao Our rou alorhm s based o hs assumpo I Seo 5, we wll show how o deal wh he rou problem f hs assumpo s relaxed Based o he pre-spefed squares he mesh layou ad he ood paral permuao, we formally defe he follow oaos S (, j) { J((, Y),(, Y))(, Y) square(, j)} Noe ha, by our assumpo, S lo (, j) efo (Square yle): A square yle L deo a yle a AGV s job se whh eah job s desed o he square (,j), e S (, j) { J((,Y),(,Y)) (,Y) square(, j)} Noe ha, by our assumpo, S lo (, j) efo (AGV s saus): A AGV s saus deo he poso of he AGV he mesh layou s defed by A(( S,S ),L,( x,y )), where ( S,S ) s he AGV s square I, ad ( x,y) s he AGV s poso I wh he square ( S,S ) L s he AGV s yle poso he square efo (rory)[5: The prory s ha AGVs whh oue rl o he same square boudary are preferred over AGVs ha ry o o o a ehbor square boudary For example, he rh sde of Fure 5, f he AGV o ode 7 was o o o he boudary of he square o s rh had sde, he AGV o ode 5 has hher prory

4 Rou alorhm [5 proposed Square Alorhm, as llusraed Fure 5, whh osss of hree phases: I he frs phase, every robo res o move from s or o he boudary of he square, us he maary eral Hamloa yle he square oa s or I he seod phase, us oly odes o he boudary of he square ha oas s desao I he hrd phase, every robo moves from he square s boudary o he desao sde he square, also walk hrouh he Hamloa yle 8 7 Fure 5 The square alorhm [5 Our AGV rou alorhm also osss of hree phases The dfferee s ha he frs ad las phases, we use square yles sead of Hamloa yles I he mddle phase, we have more ha oe pah o o, o oly oe spefed way as he square alorhm [5 Based o he same assumpo of ood paral permuao, our rou alorhm s ve as follows Spose ha a AGV s saus s A(( S,S ),L,( x, y )) 5 6, ' ' ad s job s J((, Y),(, Y)) Square ( S,S ) s he ehbor square of S,S ), he we kow ha S ' S or S S ' ± x x ± y y ( The alorhm, dvded o hree phases, s ve as follows hase Move he AGVs from her ors o he square s boudary Repea for lo ha he wors ase, afer lo seps ( Seo, we wll prove seps, all AGVs wll fsh her frs phase) If he AGV s o he boudary L The advae o he boudary lokwse dreo lse f he AGV s o he yle L ad here s o AGV wh hher prory o he yle L -, The moves o he L - lse advaes o he yle L hase Move he AGVs from her or square boudares o her desao square boudares Repea a eah sep If S,S ) (,Y ) ( The sars las phase ' lse If S S + s( ) ad here s o x x ' ' AGV wh hher prory ( S,S ) for AGV A (where s(-), f >; oherwse, s(-)) ' ' The moves o he square ( S,S ) lse advaes o he boudary of square S,S ) ( hase Move he AGVs from he boudary of he desao s square o her desao Repea for lo ha wors ase, afer lo x seps ( Seo, we wll prove y seps, all AGVs wll fsh her las phase) If he AGV reahes s desao (,Y ) The eers he buffer ad leaves he mesh lse f he AGV s o he yle L ad here s o AGV wh hher prory o he yle L +, The jumps o he L + lse advaes o he yle L

5 ( lo ) seps o reah he yle L, he would ake aoher ( lo ) seps, he wors ase We (a) a aalyze he smlar ases for he oher yles Therefore, we e he ru me of he frs phase he wors ase T ( lo ) + [ ( lo ) + ( lo ) + + [ ( lo ) + ( lo ) + lo lam : I he wors ase, he hrd phase our rou alorhm akes O(lo ) seps (b) Fure 6 Our rou alorhm Aalyss of me omplexy We aalyze he me performae of eah phase our rou alorhm lam : I he wors ase, he frs phase our rou alorhm akes O(lo ) seps for all AGVs o omplee her permuao operaos [roof: Se he dreos of wo ehbor yles are dffere, he AGVs o oe yle a oly dsurb eah AGV o aoher yle oe ( dsurb meas a AGV--A bloks aoher oe o ome o he same yle beause of A has hher prory) Beause lo > lo, for he seod yle L, whh s lose o he square boudary ad he sze of whh s ( lo ) ( lo ), he wors ase, a AGV o wll ake ( lo ) seps o reah he boudary Smlarly, for he AGV o he ex yle L, wll ake [roof: The proof s very smlar o ha of lam ad s herefore omed lam : I he seod phase of our rou alorhm, wh hh probably ( ) lo, all AGVs wll reah her desao square s boudary O ( ) seps [roof: The proof uses a arume smlar o ha of [5 The follow verso of heroff boud [8 s used our proof heroff Boud[8 Le p, p,, p R wh p, for,,, Le ad p p + p + + m p, ad le,,, be depede Beroull radom varables wh p r ob[ p, for,,,, S The for r 6m, r ob[ S r r We also eed eed he follow lemma 5

6 Lemma ( ) lo, Afer he frs phase, wh probably dur eah of he frs lo rouds, every AGV moves o he ex square s pah, ad dur hese rouds, a eah sep, every square has o more ha lo AGVs, where 5 + ad he A also ours [roof of lam : The proof s he same as ha of [5 lam : For every r ob[ B [roof of lam : See Appedx A, lo [roof of Lemma : Frsly, le s rodue he defos of era eves also defed [5 {a mos lo AGVs are ouboud every or square}, {a mos lo AGVs are boud o arrve a des every square}, ad, or des where max{6, } m For > A {a roud all ouboud AGVs move o he ex square her pah }, B {a ed of roud here are a mos lo every square }, AGVs ad { A B }, where 5 + ad Se we assume he ood paral permuao, r ob[ I order o prove he lemma, we rodue he follow lams lam : For every, f ours, lo Based o lam ad lam, we olude ha r ob[ r ob[ B r ob[ A B Therefore, ( lo we have r ob[ r ob[ r ob[ r ob[ r ob[ Subsu lo ) ( r ob[ ) o he equaly, we have lo lo r ob[ ( ) Thus, we e he proof of Lemma Aord o Lemma, a eah oe of he frs lo rouds, all AGVs move o he ex square dur 6

7 her pahs ah pah oa a mos lo squares, lam : I our rou alorhm, wh hh probably, all AGVs wll reah her desaos O ( ) seps ad eah roud eeds wh probably lo ( ) lo seps, so we kow ha lo lo O( ) seps Therefore, we e he proof of lam, he seod phase akes [roof: Based o lam, lam ad lam, ad se O(lo ) O( ), we a easly e he proof 5 sussos ad olusos I hs paper, we have aalyzed a reals AGV sysem wh a mesh layou, ad osdered he ase where he umber of AGVs s bouded by O( ) Based o lo (k,l) some pre-spefed pah of he mesh layou ad he ood paral permuao, we prese a mproved rou alorhm, ad prove ha wh hh probably, a be doe O( ) seps (k,l) (,j) (h,) (a) I he square alorhm (,j) (h,) dow lef Our alorhm s a mproveme over he resuls [5 I he seod phase of he rou alorhm [5, eah robo a oly ravel oe speal pah o reah s desao I our rou alorhm, every AGV has more pahs o hoose from ha he square alorhm, whe res o move owards s desao Iuvely, beause we allow AGVs o move o ay square ha dereases he square dsae o her desaos, should have more haes o avod poeal ofls, so s easer o reah s desao From he probably aalyss, has also bee ofrmed We assume ha he AGVs have ood paral permuao However, whe hs assumpo s o sasfed, we a use a b Hamloa yle he whole mesh layou, he he wors ase, he permuaos whh are o rh ood paral oes a be fshed O( ) seps (b) I our rou alorhm Fure 7 The squares of (, j ) The pah marked by dashed le s he oe for he job J((k,l),(h,)) For a (, ) square (, j ) j, k + l j Our rou alorhm reles o he mmal loal ommuao mehasm However, he ommuao level a be exeded The here should exs a more effe rou alorhm for fsh he permuao operao We have assumed ha he permuaos are -, ad eah AGV s oly assed o oe job These assumpos a 7

8 also be relaxed Whe a AGV jus fshes dropp off a box (or oaer, e) ad pks a ew oe, we a reard a ew AGV ora a ha me mome (spose ha he assumpo of ood paral permuao s sll sasfed Therefore, remov hs assumpo would o add muh dffuly o our aalyss I hs paper, we oly osder he me performae he rou alorhm Bu our mesh rou alorhm, all AGVs should make may urs before hey reah her desaos, hus, hey osume more eery ha some oher reedy rou alorhms [5 Therefore, s mpora for us o osder he eery effey he rou alorhm There are sll may ope ssues for fuure researh Frsly, how o exed he smplfed rou model whh eah blok s o a square, bu sead, a reale Seodly, we assumed ha he buffer of eah ode a oly aommodae oe AGV, ad here s o queue he rou model How o deerme he sze of he buffer ad he queue, f he assumpo s relaxed? Thrdly, our sudy, we dd o osder he ase whe some AGVs break dow, or whe he ommuao sysem s broke These falures ould lead o a serous problem of he whole sysem Therefore, s esseal o osder faul-olera alorhms Akowledme We akowlede he Marme ad or Auhory, A*STAR ad Naya Teholoal Uversy, all of Sapore, for her spor of he researh proje Referees [ vers, J J M ad S A J Koppers Auoma uded vehle raff orol a a oaer ermal Trasporao Researh ar A, ():-,996 [ HSU, W-J ad HUANG, S-Y, 99, Roue pla of auomaed uded vehles roeeds of Ielle Vehles, ars, pp79-85 [ Hua, S-Y ad W-J Hsu Rou auomaed uded vehles o mesh lke opoloes I roeeds of Ieraoal oferee o Auomao, Robos ad ompuer Vso, 99 [ Qu, L ad W-J Hsu, A b-dreoal pah layou for ofl-free rou of AGVs Ieraoal Joural of roduo Researh, 9(): 77-95, [5 Qu, L, W J Hsu, Shell-Y Hua, ad Ha Wa, "Shedul ad Rou Alorhms for AGVs: a Survey" Ieraoal Joural of roduo Researh, Vol, No, pp 75-76, [6 Qu, L, W J Hsu, "Rou AGVs o a Mesh-lke ah Topoloy" I roeeds of he I Ielle Vehles Symposum (IVS ), pp 9-97, earbor, Mha, USA, O -5, [7 Qu, L, W J Hsu, "Alorhms for Rou AGVs o a Mesh Topoloy" I roeeds of he uropea oferee o arallel ompu (uro-par ), pp , Tehal Uversy of Muh, Muh, Germay, Au 9-Sep, [8 T Haer ad Rub, A uded our of heroff bouds Iformao roess Leers, 5-8, [9 Ye, R, W-J Hsu, ad V-Y Vee srbued rou ad smulao of auomaed uded vehles I roeeds of TNON, volume II, paes 5-, Kuala Lumpur, Malaysa, Sepember -7, [ Ye, R, V-Y Vee, W-J Hsu ad SN Shah arallel smulao of AGVs oaer por operaos I roeeds of h Ieraoal oferee/xhbo o Hh erformae ompu Asa-paf Reo (H-ASIA ), volume I, paes 58-6, Bej, ha, May -7, [5 Yu, ad S-Y Hua, A eralzed Rou Alorhm for AGVS oaer ors I roeeds of he h Ieraoal oferee o ompuer Ieraed Maufaur, Sapore, paes 589-6, 997 [ Y Moses ad M Teeholz, O ompuaoal aspes of arfal soal sysems I roeeds of AI-9, 99 8

9 [ Y Shoham ad M Teeholz, O raff laws for moble robos I Frs oferee o AI la Sysems, 99 [ Y Shoham ad M Teeholz, O soal laws for arfal ae soeew: Off-le des Arfal Iellee, vol 7, 995 [5 remer, S omplexy aalyss of moveme mul robo sysem Maser s hess, eparme of Appled Mahemas, he Wezma Isue of See, Rehovo, Israel, 995 [6 Ze, J, Hsu W J ad Qu L A ery-ffe Alorhm For ofl-free AGV Rou O A Lear ah Layou I roeeds of he Ieraoal ompuer Symposum (IS ), Huala, Tawa, e 8-, Appedx A: roof of lam [roof: From lam, f, he A ours, amely, all ouboud AGVs whh are o he boudary of he square a he be of roud, wll leave he square dur he roud So we oly eed o osder he AGVs eer he square a he -h roud For hs purpose, we osder he follow eves (, j ) B {a mos lo AGVs are arrv o square (,j) a roud }, he we kow ha (, j ) B (, j ) B { he se of all squares ha are a a dsae of squares from (,j)}, The squares of (, j ) are show Fure 7 (b) We use he follow Beroull varables f he m-h AGV ora orae (k, l) ), (, j) where (k, l), ad m lo (by he assumpo of ood paral permuaos) I order o use he heroff boud, eah varable mus be depede However, our formula, depede of (, j ) ( k',l' ),m' s o (, j ), for ( k, l ), m ( k', l' ), m' ah of he four ses marked by dffere paers Fure 7 (b) s depede of he ohers, so we rodue he follow depede eves aord o Fure 7 (b) ( {, j ) (, ) ( k,l ), k,m lo } j < ( {, j ) dow (, ) ( k,l ), k,m lo } j > ( {, j ) rh (, ) ( k,l ),l j, k,m lo } j > ( {, j ) lef (, ) ( k,l ),l j, k,m lo } j < Now we wll alulae r ob[, r ob[ dow, r ob[ lef, ad r ob[ rh respevely, where deoes he ompleme of r ob[ r ob[ r ob[ r ob[ r ob[ r ob[ ( ) r ob[ r ob[ ( r ob[ ) Aord o Fure 7 (b), we kow ha here are a leas from (k, l) has (, j) o s pah ( lo ) (for ) odes ha a be he (, j ) desaos of AGVs ha orae ( k,l ) for he m-h oherwse (lud he ase whh less ha m AGVs AGV(all he odes mus he odes of he se) Wha eres us are he odes ha a be possble 9

10 desao odes Aord o Lemma A, he lares umber of squares mos lo here are a mos (, j ) s 5, ad here are a lo desao odes every square So 5 5 lo lo odes lo Beause here are a mos eah square, ad > r ob{ (, ( k,l ) r ob{ (, ( k,l ) j ),l > j, k,m j ),l > j, k,m lo AGVs ora, we have (, j ) (, j ) lo } lo } ha a be possble desao odes Therefore, we have 5 lo (, j ) [ 5lo lo Nex, order o use he heroff boud, we arue ha (, j ) (, ) ( k,l ), < k, m s sohasally domaed by he j sum Y, where j j wh suess probably here are oally se) Thus we have [ lo Y are depede Beroull rals j 5 lo (we sum o se lo odes he 5lo Y j (, j ) (, j ) ( k,l ), k,m [ j < By heroff boud we e for r ob{ (, ( k,l ) r ob{ Y Therefore, we have r ob[ j (, j ) lo j ),< k,m } lo lo } Se he dow se s symmeral o he se, we have r ob[ dow ( 5 lo prob[ B So r ob[ Smlarly we e r ob[ rh lef Now we oue o prove lam ( Se B, j ) ( ), we e r ob[ B (, j ) r ob[ Thus, we have r ob[ B (, j ) dow r ob[ B (, j ) (, j ) prob[ B lo dow rh rh r ob[ (, j ), j ) (, j ) lef lef dow r ob[ B (, j ) Therefore we omplee he proof of lam rh Lemma A: osder a mesh wh umber of squares For a ve squares (,j), here are a mos 5 possble squares ha are he desaos of he AGVs ha orae from square whh s have he square (,j) o her pahs lef ad (, j )

11 [roof: Whe (,j) s he eer of he mesh, we have he maxmum of he possble desao, where For oveee, we se (,j) o be he (,) po of he (, j ) oordaes For a ve square (k,l), ad ay square (h,) s a square ha has he AGVs ha orae from square (k,l) ad have square (,j) o her pahs, as ( k + l )! show Fure 7 (b), here are oally S k!l! square pahs from (k,l) o (,j), ad oally ( h + )! S square pahs from (,j) o (h,) A he same h!! ( k + l + h + )! me, here are oally S pahs from ( h + k )!( l + )! (,j) o (h,) The probably, of whh (h,) a be he square ora ad hav he square (,j) o s pah, s ve as follows ( h, ) S S r S ( k + h )( k + h )( k + ) ( l + )( l + )(l + ) ( k + l + h + )( k + l + h + )( h + + ) where k + l ad h, Whe h or reases, r dereases Spose ha here are a mos S possble squares ha sasfes he requreme, we have # S r h ( h, ) For h x( x > ), we have x ( x,x ) r < x Therefore, we have # S ( + r h ( h, ) ) + dx + x ( ( + ) x x 5 )