Conflict-Free Routing of AGVs on the Mesh Topology Based on a Discrete-Time Model

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1 Conflc-Free Roung of AGVs on he Mesh Topolog Based on a Dscree-Tme Model Zeng Janang zengj@pmalnuedusg Wen-Jng Hsu hsu@nuedusg Cener for Advanced Informaon Ssems School of Compuer Engneerng Nanang Technologcal Unvers, Sngapore Absrac: Auomaed Guded Vehcles or AGVs for shor have become an mporan opon n maeral handlng In man applcaons, such as conaner ermnals, he servce area s ofen arranged no recangular blocks, whch leads o a mesh-lke pah opolog Therefore, developng effcen algorhms for AGV roung on he mesh opolog has become an mporan research opc In hs paper, we presen a dscree me model, based on whch a smple roung algorhm on he mesh opolog s presened The algorhm works b carefull choosng suable parameers such ha he vehcles usng a same juncon wll arrve a dfferen pons n me, and hence no conflcs wll occur durng he roung; meanwhle, hgh roung performance can be acheved Analses of he ask compleon me and he requremens on mng conrol durng he AGV roung are also presened Inroducon: Auomaed Guded Vehcles or AGVs for shor have become an mporan opon n maeral handlng [-7, 9- ] In man applcaons, such as conaner ermnals[, 9-], he servce area s ofen arranged no recangular blocks, whch leads o a mesh-lke pah opolog Therefore, developng effcen algorhms for AGV roung on hs opolog has become an mporan research opc There are man esng resuls abou AGV [5] However, relavel lle has been known abou roung on he mesh opolog [-] gave he analss of me and space complees for some basc AGV roung operaons on D-mesh opolog The upper bounds of me and space complees for AGV roung are Θ n and Θ n respecvel, where n denoes he number of nodes n he pah opolog However, he paper does no gve he deals of he roung algorhms and echnques o avod congeson, conflcs, deadlocks, ec [6-7] presened dfferen mehods o schedule and roue smulaneousl n an n n mesh-lke pah opolog The algorhms can schedule and roue smulaneousl up o 4n AGVs concurrenl a one me In hese papers, he roung process s formulaed as a sorng problem Alhough here are no conflcs durng he permuaon, requres n seps of well-defned phscal moves, whch requres AGVs o ravel era dsance and consume era energ o fnsh he asks In hs paper, we presen a dscree me model on mesh opolog for AGV roung Based on hs model, he roung algorhm s presened and me conrol requremen s analzed The ke dea les n makng use of he regular of he mesh, and hence he regular of pons of me when AGVs arrve a he nersecons B choosng a suable speed for he AGVs along dfferen drecons, we can ensure ha no conflcs among an AGVs wll occur We also analze he algorhm n erms of bounds on he ask compleon me and requremens on he roung precson conrols B our desgn, he AGVs can advance drecl o her desnaons, unlke n [6-7], and hence hgh performance can also be ensured The remaner of he paper s organzed as follows Secon descrbes he mesh laou and he roung model Secon gves he roung algorhm and he me conrol crera o avod conflcs In Secon 4, we analze he performance of he roung algorhm Secon 5 gves an eplc mehod o derve he mng conrols Fnall, Secon 6 dscusses possbles of relang ceran consrans and pons ou drecons of fuure sud Descrpon of mesh laou model In order o descrbe he marrow of our roung process clearl, we sar wh a smple bu general model n whch one ard block has onl one saon near an nersecon of pahwas refer o Fgure In hs mesh

2 +, laou, here are n oal N N blocks, namel N blocks n each column and N blocks n each row Each block has he same sze There are wo pahs wh dfferen drecons beween wo adjacen blocks Each Block has one Pck up-drop off saonor P/D saon for shor, locaed a he upper rgh and upper op corner of he block On he lef-op sde, here s a vehcle park where all AGVs are saoned nall and o whch he wll reurn upon compleon of all asks Fgure Realsc mesh laou Alhough here are some mporan deals for AGV roung, such as he sze of he juncon, he radus of 90 urn, he lengh of he AGV, ec[4-7], s reasonable and realsc for us o smplf he mesh model for convenence of analss and dscusson In he smplfed mesh laou model, shown as Fgure, here are N juncons of pahwas A juncon and he assocaed neghborng saon are collecvel regarded as a node Each node s o assgn wh he coordnaes, as s address or ID, where and represen respecvel he row and column IDs Ths mesh laou s modeled b a graph G V,E The N N verces of he graph represen juncon nodes, and he bdreconal edges represen wo pahs beween wo adjacen juncon nodes, and he lengh of each edge s a consan "!$#&% $* Fgure Smplfed mesh roung model We dvde he AGV movemens no hree phases In he frs phase, le AGVs se ou from he park o her pck up saons In he second phase, le AGVs pck up loads and ravel o her desnaons and drop-off loads In he hrd phase, le AGVs reurn o he park from her dropoff saons Because s eas for us o dspach he AGV movng whou an conflc n he frs phase and he hrd phase, we can focus onl on he second phase when he loaded AGVs move on he mesh laou We assume ha he me can be dvded no dscree uns of me, and ha each AGV alwas reaches ever juncon node a some dscree pon on me I s reasonable for us o make hs assumpon because he dsance beween wo adjacen nodes s a consan, and we can adjus he speed of he AGVs o le hem arrve a he juncons a mulples of he un me We assume ha when an AGV reaches s desnaon, eners he buffer and leaves he mesh grd Based on he mesh laou model, we formall defne he followng Defnon:,, f and onl f and Defnon Job: A job s denfed b an ordered par PX,PY,DX,DY, wherepx,py represens he address of he pckup saon, DX,DY represens he address of he drop-off saon, and PX,PY DX,DY Assume ha each job has a dsnc orgn and also a dsncbu dfferen desnaon, and an AGV s gven onl one job and an job s assgned o onl one AGV Defnon Job Se: A job se M denong a se of k N jobs, where k, s defned as follows: M{PX, PY, DX, DY PX, PY, DX, DY N, for,,, k Accordng o he poson of he orgns and desnaons of jobs, an gven job se M can be dvded no four subses, denoed b M, M, M respecvel, such ha M {PX, PY,DX, DY DX PX,DY PY for,,,k M {PX, PY,DX, DY DY PY,DX for,,,k M PX {PX, PY,DX, DY DY PY,DX PX, for,,,k,

3 We also dvde M no wo subses, denoed b M and M, such ha M + + {PX, PY,DX, DY DY > PY,DX PX, for,,,k M {PX, PY,DX, DY DY < PY,DX PX, for,,,k Accordngl, we have he followng noaons: A : he se of AGVs ha carr ou jobs n M ; A : he se of AGVs ha carr ou jobs n M ; A + : he se of AGVs ha carr ou jobs n + M ; A : he se of AGVs ha carr ou jobs n M ; Defnon Drecon of AGVs: v s a un vecor whch represens he drecon of a gven AGV, where v { +,, +, v v when v s n he same drecon as v v v when v s n he oppose drecon of v v v 0 when v s n a vercal drecon of v Defnon Sae of AGVs:,,,v s he sae of an AGV, where, represens he locaon n he mesh laou, and represens he dscree me pons of he AGV, and v { +,, +, Defnon Collson:,,,v s he sae of AGV, and,,,v s he saus of AGV When,,, and { +,, +, { v In hs case, we sa ha v AGV and AGV have a collson a, or, when on he mesh laou Conflc-free roung algorhm Based on he smplfed mesh laou and he dscree me, he roung algorhm s presened as follows Le all AGVs se ou from her pck up saons a he same me, when 0 0 Case a In he job se M In hs case, le he AGV ravel along he row PY from PX,PY saon o DX,DY Case b In he job se M In hs case, le he AGV ravel along he column PX from PX,PY saon o DX,DY Case c In he job se M + In hs case, le he AGV frsl ravel along he column PX from PX,PY saon o DX,DY saon Then le ravel along he row DY from DX,PY saon o DX,DY saon Case d In he job se M In hs case, le he AGV frsl ravel along he column PX from PX,PY saon o DX,DY Then le ravel along he row DX,PY DY from saon o DX,DY saon The roung algorhm looks smple, and f we le AGVs ravel n hs rule a an arbrar speed, s ver lkel o have collsons on he mesh laou However, as we wll show shorl, f we conrol he me when each AGV reaches ever juncon node we can do so b conrollng he AGV s speed, AGVs can run on he mesh laou wh he freedom of conflcs We le T + denoe he me requred for an AGV o ravel hrough one edge of he mesh along he + drecon Le T + T +, T be defned smlarl We assume ha AGVs ravel a he speed v +, v, v +, v n hese four cases respecvel Accordng o he precedng roung, we have he followng conclusons Clam : Accordng o our roung algorhm, here s no conflc beween an wo AGVs belongng o he same se A or A Accordng o he defnon of collson, and he assumpon ha each AGV has a dsnc orgn, s que clear ha here s no conflc n he AGVs belongng o A or A Clam : Based on he roung algorhm, an AGV wll no run no conflc wh oher AGVs on he mesh laou, f he followng relaon s sasfed lcm T N - ma T where T and T are an wo permuaons from { T +, T, T +, T gcd T+ + T - gcd T+ T - gcd T + T + -4 gcd T T + -5 gcd T, T N -6 ± ±

4 here gcd he Greaes Common Dvsor, and lcm he Leas Common Mulplecf Defnon A and Defnon A Frsl le us recall some defnons and heorems of number heor[8] whch wll be used n he dscussons laer Defnon A Dvsbl: If a and b are negers, we sa ha a dvdes b f here s an neger c such ha bac If a dvdes b, we also sa ha a s a dvsor or facor of b Wre a b f a dvde b; oherwse, wre a b f a does no dvde b Defnon A The Greaes Common Dvsor: The greaes common dvsor of wo negers a and b, no boh zero, s he larges posve neger ha dvdes boh a and b; s denoed b gcda,b Defnon A The Leas Common Mulple: The leas common mulple gcd of wo negers a and b, s he leas posve neger dvsble b boh a and b; s denoed b lcma,b Defnon A4 Lnear combnaon: If a and b are negers, hen a lnear combnaon of a and b s a sum of he form ma+nb, where boh m and n are negers Defnon A5 Lnear dophanne equaon: A lnear dophanne equaon n wo varables and s a dophanne equaon of he form a+bc, where a, b and c are negers Theorem A: The greaes common dvsor of he negers a and b, ha are no boh zero, s he leas posve neger ha s a lnear combnaon of a and b Theorem A: Le a and b be posve negers Then ab lcm a,b gcd a,b Theorem A: Le a and b be posve negers and dgcda,b The equaon a+bc has no neger soluons f d c If d c, hen here are nfnel man neger soluons Moreover, f 0, 0 s a parcular soluon of he equaon, hen all soluons are gven b + 0 b / d n 0 a / d n Now connue wh he proof of he clam All cases of possble conflcs are shown n Fgure We omed a few smlar cases, whch are smmercal o some of he followng cases -a -b -c -d -e M M or M M -a -b -c -d -e -f -g -h M M Fgure All cases of possble conflcs Assume he nal saes of AGV and AGV respecvel as follows AGV:,, 0,v ; AGV:,, 0,v When,,,, he saes of AGV and AGV are respecvel, AGV:,,,v ; AGV:,,,v Now le us prove ha n all cases of poenal conflcs Case Ths case covers -a, -b, -c, -d, - g We have he followng relaons + T T, + j T j T, where 0, j N, and T and T are an wo permuaons from { T +, T, T +, T Accordng o he defnon of lcm, we have lcm T, T mn T j mn lcm T, T T Accordng o Inequal -, we know mn, jmn N, whch conflc wh he condon such ha 0, j N Therefore n all hese cases, for an, j, where 0, j N 4

5 Case Ths case covers -e, -a, -b, -c, - e, -f, -h We have he followng relaons T+ + jt+, k T or or, T+ + jt k T or,, T + jt+ k T+,, T + j T k T+,, where 0, j,k N These four relaons are smlar o each oher, so we can focus on he frs one Frsl we ake a look a he followng equaon T + T kt where and are negers From Eq -6 and Eq -, we have gcd T, T k T + + Accordng o Theorem A, we know ha Eq -7 has no neger soluons, so for an, j [0,N ], Case relaon Ths case covers -d We have he followng T+ + jt+, k T+, T+ + jt+ kt+ j T+ k j T+ In order o prove ha namel,,, we need o show ha We know ha 0 k j N, hen hs suaon can be convered o he one n case ha we have proved Therefore, we conclude ha n all cases, here s no an conflc usng our roung algorhm and he me lm 4 Tme requremen Clam 4: The me requremen Tr for all AGVs o ranspor all jobs s upper-bounded b N ma{ T + + Snce all jobs are carred ou n parallel, he me requremen for a job se M s deermned b he mos me-consumng job n he se Formall, for an gven job se, we have ma{tpx,py,dx,dy,tpx,py, T r DX,DY,,TPX k,pyk,dx k,dyk, where TPX,PY,DX,DY s he me requremen for AGV o complee s job Assume ha here ess job,,n, N whch uses he mos me and use ma{ T, T me o + + go hrough one edge on he mesh laou, hen we oban he followng relaon T,, N,N N - ma{ T Thus, he me requremen for a job se s upperbounded b T r T,,N,N + N - ma{ T+ + Alhough our roung algorhm guaranees collsonfreedom under some specal crera, he conrol ssem needs o know he me pon when each AGV goes hrough ever juncon node So s necessar for us o consder he relaon beween dfferen me pons when each AGV goes hrough ever juncon node Defnon Tme dfference: The me dfference s he mnus of wo me pons when wo AGVs reach one specal juncon node The mnmum me dfference s he mnmum me dfference for all AGVs a ever juncon node on he mesh laou Clam 4: The mnmum me dfference on he mesh laou s lower-bounded b mn{gcd T,gcd T,gcd T, T, gcd T namel, ,gcd T mn{gcd, S, where S { T + +,gcd T + To ge he mnmum me dfference, we can fnd he mnmum of he followng equaon: T + j T where, j are negers, and T, T are an wo numbers from {,,, T + T Accordng o Theorem A, he leas posve neger of T + jt s gcd T Thus for an T, T from { T +, T, T +, T, he mnmum of he me dfference s mn{gcd T,gcd T,gcd T, T, T + T

6 gcd T namel +,gcd T mn{gcd,, S where S { T + +,gcd T + 5 The mehod o consruc T, T + + We nroduce he followng mehod o consruc T, T, whch sasf he crera of + + conflc-free roung a+ b g c+ d g We le T P P, T P P, T P + and T P + P P 5 a c f 4 Where P, P, P and P 4 are prmes; a,b log N ; P c,d log N ; P e log N ; P f log N ; P 4 g log N P 5 5 P P a c e Clam 5: The values of T, T consruced b hs mehod sasf + + he crera of conflc-free roung Accordng o Theorem A, we have lcm T+ T+ T T gcd T T + gcd T + c+ d g T P P5 g + P5 c + d P N N Smlarl we can prove ha for an wo permuaons T and T from { T +, T, T +, T, he followng relaons are sasfed lcm T N T + Accordng o he consrucon mehod, we have a+ b g a c f a gcd T+ + gcd P P5,P P P4 P N a Because P c d g P + P5, we have gcd T+ + T Smlarl we can prove ha he nequales ---6 are sasfed Therefore he values of T, T consruced b hs mehod sasf + + all he crera of conflc-free roung The dfferences n T+ + have mplcaons on he roung conrol The smaller value of hs dfference generall means he more accurae mng when vehcles arrve a he juncons Clam 5: The mnmum me dfference of T, T consruced b hs mehod + + mn{ P,P,whch s lower-bounded b Ω N Accordng o Clam 4, he mnmum me dfference on he mesh laou s lower-bounded b mn{gcd T,gcd T,gcd T, T, gcd T ,gcd T Subsung no he values of,gcd T + T+ +, we have mn{gcd T+,gcd T+ +,gcd T+, gcd T,gcd T,gcd T + a c a c mn{ P P,P,P,P,P a mn{ P,P Therefore, he mnmum me dfference of T, T consruced b hs mehod s + + lower-bounded b mn{ P,P Because P,P N, he mnmum me dfference s lowerbounded b Ω N From clam 4, he longes value of T, T generall means a hgher ask + + compleon me for he gven choce of me un The followng resul bounds hs value Clam 5: The values of bounded b Θ N + c,p g 5 T, T, T, T are + + Accordng o he consrucon mehod, we have a+ b g T + P P5 N N Smlarl, we can prove ha T N, T+ N, T N If we keep he values of T, T as small N + + as possble, we can make hem be bounded b Θ N Therefore, he values of T+ + bounded b Θ N We gve a smple eample o llusrae he consrucon + Le N7, we can choose T, + s 6

7 + T, T 7, T The mnmum me dfference of hs case s 8, and he rao beween he mamum and he mnmum of T, T s abou Dscussons and conclusons We have presened a dscree me model for AGV roung on he mesh opolog In hs model, each AGV s assumed o reach ever juncon node a dscree pons n me Based on hs model, we proposed a roung algorhm whch allows AGVs o ravel a dfferen mulples of he un me along dfferen drecons, whch guaranees he freedom of conflcs The mng conrol requremen was analzed, and he mehod o consruc he mulples of he un me was also nroduced Wh our roung algorhm, all he AGVs can move drecl owards her desnaons whou conflcs Therefore, he overall roung performance s ensured Moreover, snce each AGV makes a mos one urn durng he enre roung process, he speed of each AGVs s changed no more han once Therefore, he energ requremen b our roung algorhm s also relavel low In our roung model, each AGV on he mesh opolog s assumed o be one pon However, here are some deals ha we mus consder n acual mplemenaons, such as he sze of juncon, he lengh of he AGV, ec These consderaons have a requremen of he mnmum me dfference, whch can be adjused b he conrol ssem Accordng o Clam 5, he mnmum me dfference s decded b mn{ P,P Thus we can choose he value of { P,P o ncrease he mnmum me dfference Therefore, he dscree me model and he roung algorhm can be used n real mesh-lke laou Smlarl, he ask compleon me can be conrolled b choce of he uns of me, he dsance beween nersecons and/or he speeds of AGVs For nsance, b Clam 5, he mamum value of T, T s bounded b Θ N As N + + ncreases, he me requremen o fnsh he jobs seems o ncrease quckl However, we can choose a small un of me o keep he acual me requremen low, as long as he mnmum me dfference for avodng conflcs s sasfed We assumed ha when an AGV reaches s desnaon, eners he buffer and leaves he mesh grd Ths assumpon can be also relaed Usuall, when an AGV eners he buffer of he P/D saon, akes some me for he vehcle o compleel leave he mesh grd The suaon s smlar when an AGV goes hrough he juncon However, as long as he me requred for an AGV o ener he buffer of he saon s less han he + mnmum me dfference, here are sll no conflcs durng he AGV roung We assume ha each AGV has dsnc orgn and also a dsnc bu dfferen desnaon, namel, he paern of AGV movemen s permuaon Ths assumpon can also be relaed such ha each AGV has mulple orgns and mulple desnaons As long as we conrol he me pons for AGVs o se ou from he saons and le hem ener he buffers a proper me pon, we can sll guaranee he freedom of conflcs There are man open ssues for fuure research Frsl, how o deal wh he falure of AGVs? In our algorhm as well as ohers, a sngle blockage wll cause he falure of he whole ssem Therefore, s essenal o consder faul-oleran sraeges Secondl, f we allow each AGV o make more han one urn before reaches s desnaon, we need more complcaed scheme o avod conflcs Thrdl, we need o devse a mehod o decde he number of AGVs for he gven jobs and o deal wh dle AGVs Acknowledgemen We acknowledge he Marme and Por Auhor, A*STAR and Nanang Technologcal Unvers, all of Sngapore, for her suppor of he research projec References [] Evers, J J M and S A J Koppers Auomac guded vehcle raffc conrol a a conaner ermnal Transporaon Research Par A, 0:-4,996 [] HSU, W-J and HUANG, S-Y, 994, Roue plannng of auomaed guded vehcles Proceedngs of Inellgen Vehcles, Pars, pp [] Huang, S-Y and W-J Hsu Roung auomaed guded vehcles on mesh lke opologes In Proceedngs of Inernaonal Conference on Auomaon, Robocs and Compuer Vson, 994 [4] Qu, L and W-J Hsu, A b-dreconal pah laou for conflc-free roung of AGVs Inernaonal Journal of Producon Research, 90: 77-95, 00 [5] Qu, Lng, Wen-Jng Hsu, Shell-Yng Huang, and Han Wang, "Schedulng and Roung Algorhms for AGVs: a Surve" Inernaonal Journal of Producon Research, Vol 40, No, pp , 00 [6] Qu, Lng and Wen-Jng Hsu, "Roung AGVs on a Mesh-lke Pah Topolog" In Proceedngs of he IEEE Inellgen Vehcles Smposum 000 IVS 7

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