MODELLING DISTRIBUTED MANUFACTURING SYSTEMS VIA FIRST ORDER HYBRID PETRI NETS. Mariagrazia Dotoli, Maria Pia Fanti, Agostino Marcello Mangini

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1 MODELLING DISTRIBUTED MANUFACTURING SYSTEMS VIA FIRST ORDER HYBRID PETRI NETS Mariagrazia Dooli, Maria Pia Fani, Agosino Marcello Mangini Diarimeno di Eleroecnica ed Eleronica, Poliecnico di Bari, Ialy Absrac: A Disribued Manufacuring Sysem (DMS) is a collecion of indeenden comanies ossessing comlemenary sills and inegraed wih ransoraion and sorage sysems. This aer rooses a new model for DMS emloying firs order hybrid Peri nes, i.e., Peri nes based on firs order fluid aroximaion. More recisely, ransorers and manufacurers are described by coninuous ransiions, buffers are coninuous laces and roducs are reresened by coninuous flows rouing from manufacurers, buffers and ransorers. Moreover, discree evens occurring sochasically in he sysem are considered o ae ino accoun he sar of he reailer requess and he blocing of ransors and raw maerial suly. Wih he aim of showing he model effeciveness, a DMS examle is modelled and simulaed under wo differen oeraive condiions. Coyrigh 2006 IFAC Keywords: manufacuring sysems, Peri-nes, dynamic models, simulaion, erformance indices. 1. INTRODUCTION A Disribued Manufacuring Sysem (DMS) is a collecion of indeenden comanies ossessing comlemenary sills and inegraed wih ransoraion and sorage sysems (Viswanadham and Raghavan 2000). Aroriae modelling and analysis of hese highly comlex sysems are crucial for erformance evaluaion and comarison of comeing DMS. However, few conribuions face he roblem of modelling he DMS in order o analyze he sysem erformance measures and o oimize is funcional obecives. Viswanadham and Raghavan (2000) model he sysem as a Discree Even Dynamical Sysem (DEDS), in which he evoluion deends on he ineracion of discree evens. Generalized Sochasic Peri Nes (GSPN) model a aricular examle of SC and deermine he decouling oin locaion, i.e., he faciliy from which all finished goods are assembled afer cusomer order confirmaion. Moreover, in (Desrochers e al. 2005) a wo roduc SC is modelled by comlex-valued oen Peri nes and he erformance measures are deermined by simulaion. In addiion, Dooli and Fani (2005) roose a GSPN model in order o describe in a modular way a generic DMS. However, he limi of his formalism is he modelling of roducs by means of discree quaniies (i.e., oens). This assumion is no realisic in large sysems wih a huge amoun of maerial flow. Since DMS are DEDS whose number of reachable saes is very large, aroximaing fluid models can be used in his conex as in manufacuring sysems. The aim of he aer is o roose a new model for DMS emloying Firs Order Hybrid Peri Nes (FOHPN, Balduzzi e al. 2000), ha include coninuous laces holding fluid and discree laces conaining a non-negaive ineger number of oens and ransiions, where he laer are eiher discree or coninuous. Such a hybrid Peri ne model is based on he framewor roosed by Alla and David (1998) and resens he main ey feaure of allowing he Insananeous Firing Seeds (IFS) of he coninuous ransiions o be chosen in a given range by a conrol agen. Moreover, he se of all admissible IFS vecors is exlicily characerized by he feasible soluions of a linear consrain se. Furhermore, an oimal IFS vecor can be chosen according o a given obecive funcion. Using such a modelling aroach, his aer develos an FOHPN model of DMS by means of firs-order fluid aroximaions. In aricular, ransorers and manufacurers are described by coninuous ransiions, buffers are coninuous laces and roducs are reresened by coninuous flows

2 (fluids) rouing from manufacurers, buffers and ransorers. The model is buil by using a modular aroach based on he idea of he boom-u mehodology (Zhou and Venaesh, 1998). A reresenaive examle, including he yical DMS elemens, shows he effeciveness of he modelling echnique ha allows us o evaluae he sysem erformance indices by he simulaion. The aer is srucured as follows. Secion 2 describes he srucure and he dynamics of a generic DMS. Secion 3 reors a brief overview of he FOHPN modelling formalism and Secion 4 resens he modular DMS model. Secion 5 describes and analyzes an examle of DMS and a conclusion secion closes he aer. 2. THE SYSTEM DESCRIPTION A DMS may be described as a se of faciliies wih maerials ha flow from he sources of raw maerials o subassembly roducers and onwards o manufacurers and consumers of finished roducs. The DMS faciliies are conneced by ransorers of maerials, semi-finished goods and finished roducs. More recisely, he eniies of a DMS can be summarized as follows. 1- Suliers: a sulier is a faciliy ha rovides raw maerials, comonens and semi-finished roducs o manufacurers ha mae use of hem. 2- Manufacurers and assemblers: manufacurers and assemblers are faciliies ha ransform inu raw maerials/comonens ino desired ouu roducs. 3- Logisics and ransorers: sorage sysems and ransorers lay a criical role in disribued manufacuring. The aribues of logisics faciliies are sorage and handling caaciies, ransoraion imes, oeraion and invenory coss. 4- Reailers or cusomers: reailers or cusomers are sin nodes of maerial flows. Here, ar of he logisics, such as sorage buffers, is considered eraining o manufacurers, suliers and cusomers. Moreover, ransorers connec he differen sages of he roducion rocess. The dynamics of he disribued roducion sysem is raced by he flow of roducs beween faciliies and ransorers. Because of he large amoun of maerial flowing in he sysem, we model a DMS as a hybrid sysem: he coninuous dynamics models he flow of roducs in he DMS, he manufacuring and he assembling of differen roducs and is sorage in aroriae buffers. Hence, resources wih limied caaciies are reresened by coninuous saes describing he amoun of fluid maerial ha he resource sores. Moreover, we consider also discree evens occurring sochasically in he sysem, such as: a) he blocing of he raw maerial suly, e.g. modelling he occurrence of labour sries, accidens or sos due o he shifs; b) he blocing of he ransor oeraions due o he shifs or o unredicable evens such as amming of ransoraion roues, accidens, sries of ransorers ec.; c) he sar of a reques from he reailers. 3. FIRST-ORDER HYBRID PETRI NETS 3.1 The ne srucure and maring. This secion recalls he Firs Order Hybrid Peri Nes (FOHPN) formalism used in he following (Balduzzi e al. 2000). A FOHPN is a biarie digrah described by he six-ule PN=(P, T, Pre, Pos, D, F). The se of laces P=P d P c is ariioned ino a se of discree laces P d (reresened by circles) and a se of coninuous laces (reresened by double circles). The se of ransiions T=T d T c is ariioned ino a se of discree ransiions T d and a se of coninuous ransiions T c (reresened by double boxes). Moreover, he se of discree ransiions T d =T I T E is furher ariioned ino a se of immediae ransiions T I (reresened by bars) and a se of exonenially disribued ransiions T E (reresened by boxes). The marices Pre and Pos are he re-incidence and he os-incidence marices, resecively, of dimension P T. Noe ha he symbol A denoes he cardinaliy of se A. Such marices secify he ne digrah arcs and are defined as follows: + Pc T Pre,Pos :. Pd T We require ha for all T c and for all P d i holds Pre(,)=Pos(,) (well-formed nes). The funcion D: T R + secifies he iming associaed o exonenially disribued imed ransiion T E. More recisely, we associae o each T E he average firing delay δ =D( ). Moreover, he funcion F : T c R + R + secifies he firing seeds associaed o coninuous ransiions (we denoe R + =R + { }). For any coninuous ransiion T c we le F( )=(V m,v M ), wih V m V M. Here, V m reresens he minimum firing seed (mfs) and V M he Maximum Firing Seed (MFS) of he generic coninuous ransiion. Given a FOHPN and a ransiion T, he following ses of laces may be defined: ={ P: Pre(,)>0}, named re-se of ; ={ P: Pos(,)>0}, named os-se of. Moreover, he corresonding resricions o coninuous or discree laces are defined as (d) = P d or (c) = P c. Similar noaions may be used for re-ses and os-ses of laces. The incidence marix of he ne is defined as C(,)=Pos(,)-Pre(,). The resricion of C o P X and T X (wih X,Y {c, d}) is denoed by C XY. A maring Pd m : + Pc

3 is a funcion ha assigns o each discree lace a nonnegaive number of oens, reresened by blac dos, and o each coninuous lace a fluid volume; m i denoes he maring of lace i. The value of a maring a ime τ is denoed by m(τ). The resricion of m o P d and P c are denoed by m d and m c, resecively. A FOHPN sysem <PN,m(τ 0 )> is a FOHPN wih iniial maring m(τ 0 ). The following saemens rule he firing of he coninuous and discree ransiions: 1- a discree ransiion T d is enabled a m if for all i, m i >Pre( i,); 2- a coninuous ransiion T c is enabled a m if for all i (d), m i >Pre( i,). Moreover, we say ha an enabled ransiion T c is srongly enabled a m if for all laces i (c), m i >0; we say ha ransiion T C is wealy enabled a m if for some i (c), m i =0. In addiion, if <PN,m> is an FOHPN sysem and T c wih insananeous firing seed (IFS) v, i holds: 1- if is no enabled hen v i =0; 2- if i is srongly enabled, hen i may fire wih any firing seed v [V m,v M ]; 3- if is wealy enabled, hen i may fire wih any firing seed v [V m,v ], where V V M deends on he amoun of fluid enering he emy inu coninuous lace of i. We denoe by v(τ) [v 1 (τ) v 2 (τ) v Tc (τ)] T he IFS vecor a ime τ. Any admissible IFS vecor v a m is a feasible soluion of he following linear se: V v 0 ( m ) M m T ε v V 0 ( m ) (1) T ε v = 0 T υ ( m ) C(, ) v 0 P ε ( m ), Tε where Tε ( m ) Tc ( Tυ ( m ) Tc ) is he subse of coninuous ransiions ha are enabled (no enabled) a m and Pε ( m ) = { i Pc mi = 0} is he subse of emy coninuous laces. 3.2 The ne dynamics. The hybrid dynamics of he ne combines boh ime-driven and even-driven dynamics. We define macro evens he evens ha occur when: i) a discree ransiion fires or he enabling/disabling of a coninuous ransiion aes lace; ii) a coninuous lace becomes emy. The equaion ha governs he ime-driven evoluion of he maring of a lace i P C is: m () τ = C(, ) v () τ. (2) i i T C Now, if τ and τ +1 are he occurrence imes of wo macro-evens, we assume ha wihin he ime inerval [τ,τ +1 ) (macro eriod) he IFS vecor v(τ ) is consan. Then he coninuous behaviour of an FOHPN for τ [τ,τ +1 ) is described by: c c m ( τ ) = m ( τ ) + Ccc v( τ)( τ τ). (3) d d m () τ = m ( τ ) The evoluion of he ne a he occurrence of he macro-evens is described by: c c m ( τ ) = m ( τ ) + Ccdσ( τ), (4) d d m ( τ ) = m ( τ ) + Cddσ( τ) where σ(τ) is he firing coun vecor associaed o he firing of he discree ransiion. 4. THE MODULAR DMS MODEL Peri ne modelling and synhesis is a very imoran research area ha araced much aenion in he as (Zhou and Venaesh, 1998). Based on he idea of he boom-u aroach, his secion rooses a modular FOHPN model o describe a DMS. Such a mehod can be summarized in wo ses: decomosiion and comosiion. Decomosiion involves dividing a sysem ino several subsysems. In DMS his division can be erformed based on he deerminaion of disribued sysem faciliies (i.e., suliers, manufacurers, ransorers and cusomers). All hese subsysems are modelled by FOHPN. On he oher hand, comosiion involves he ineracing of hese sub-models ino a comlee model, reresening he whole DMS. The following FOHPN modules model each individual subsysem comosing he DMS. 1- The sulier module. The sulier is modelled as a coninuous ransiion and wo coninuous laces (see Fig 1). The coninuous lace S reresens he row maerial buffer of finie caaciy C S and S reresens he corresonding available caaciy such ha m S +m S =C S. Moreover, he coninuous ransiion S models he arrival of raw maerial in he sysem a a bounded rae v S wih F( S )=(V Smin,V Smax ). In addiion, we consider he ossibiliy ha he roviding of raw maerial is bloced. This siuaion is reresened by a discree even modelled by wo exonenially disribued ransiions and wo discree laces. In aricular, lace P d models he oeraive sae of he sulier and P d is he non-oeraive sae (see Fig 1). The blocing and he resoraion of he raw maerial suly corresonds o he firing of ransiions and, resecively. For he sae of clariy, Fig. 1 deics he ransiion T modelling he ransor oeraion. ' 0 s s Cs Fig. 1. The FOHPN modelling he sulier. T s

4 I C I I 0 h T ' O max max med C O 0 ' T Fig. 2. The FOHPN modelling he manufacurer and he assembler. ' med O Fig. 3. The FOHPN modelling logisics. R 0 0 R ' Fig. 4. The FOHPN modelling he reailer. 2- The manufacurer and assembler module. The manufacurers and assemblers are modelled by he FOHPN shown by Fig. 2. More recisely, lace I is he inu buffer of finie caaciy C I soring he inu goods of a aricular ye. The corresonding lace I models he available buffer sace so ha m I +m I =C I. Analogously, he ouu buffer of caaciy C O soring he ouu roduc of a aricular ye is modelled by he lace O reresening he occuied buffer level and by lace O modelling he corresonding available caaciy, wih m O +m O =C O. To manage he roducion in funcion of he ouu invenory of he roduc, we assume ha he roducion rae of he faciliy changes in funcion of he available sace in he ouu buffer. Indeed, wo differen raes are considered for he roducion: a nominal rae F( n )=(0,V mn ) associaed wih he coninuous ransiion n and a high rae F( h )=(V mn,v Mh ) associaed wih he coninuous ransiion h. If i holds m O (τ)<max (i.e., he invenory is high enough and he buffer free sace is no oo high) hen ransiion is disabled and he manufacurer wors a he nominal rae wih m =1 and m =0 wih, P d as deiced in Fig. 2. On he conrary, when he buffer level of ouu ars decreases and hence he available buffer sace reaches he maximum value m O (τ)=max, he F T n immediae ransiion is enabled and can fire. Afer he firing of, i holds m =0 and m =1, so ha he faciliy wors a he higher rae associaed wih he coninuous ransiion h and he ouu buffer O is relenished more raidly. Analogously, if he level of his buffer reaches he medium value m O (τ)=med, hen he ransiion is enabled and can fire, leading he faciliy rae a he nominal value. Noe ha ransiions T and T of Fig. 2 deics he ransors faciliies. 3- The logisics module. The ransorers connecing he differen faciliies are modelled by a coninuous ransiion T ha describes he flow of maerial from a faciliy o a subsequen one a a bounded rae 0 v T V MT. Moreover, he random so of he maerial ransor is reresened by wo laces, P d and wo exonenially disribued ransiions, T d. If lace P d is mared, hen he ransor is oeraive. On he conrary, if ransiion fires, hen he ransorers are no oeraive and he lace P d becomes mared. When ransiion fires, he ransorers are esablished again. 4- The reailer module. Finally, we consider he model of he reailers, reresened by a coninuous buffer lace R of infinie caaciy associaed wih each final roduc ye. Moreover, a coninuous ransiion R models he acquisiion of final roducs by he reailer. However, we consider ha he requess of he roducs are managed by discree random evens exressed by wo discree imed ransiions and and wo laces and. If m =1 hen he flow of maerial is enabled. On he conrary, if ransiion fires, hen a oen in means ha he reailer does no require any roduc and R is inhibied. The coninuous lace F models he sysem ouu and collecs all he roducs requesed by he reailer. Figure 4 deics he reailer module. 5. AN EXAMPLE OF DMS We consider a sysem roducing wo roduc yes E and F by he four sages deiced in Fig. 5. The firs sage includes wo comonen suliers S1 and S2, he second sage is comosed by wo subassembly manufacurers M1 and M2, he hird sage is comosed by wo assemblers A1 and A2 and he las sage is consiued by he reailers R1 and R2. Moreover, six logisics service roviders T1 o T6 suiably connec he DMS faciliies ha are locaed in differen geograhical sies. The buyers order wo brands of roducs (E and F). Such roducs are obained by wo assemblers ha assemble wo yes of roducs (C and D) obained from wo manufacurers. The subassemblies C and D are in urn roduced by he manufacurers of he second sage, which receives he comonens of ye A and B by he firs sulier sage.

5 Sulier S1 Sage 1 Sage 4 Sulier S2 A A B B Transorer Transorer T1 T2 Logisics 1 A A B B Manufacurer Manufacurer M1 M2 Sage 2 C C D D Transorer T3 Transorer T4 Logisics 2 C C D D Assembler Assembler A1 A2 Sage 3 E E F F Transorer Transorer T5 T6 Logisics 3 E E F F Reailer R1 Reailer R2 Fig. 5: The DMS configuraion S T maxm maxa meda maxa maxa meda m 30 m 32 m m 27 m 29 m m 4 3 m 3 m 5 5 m 6 M1 T A1 T5 R medm maxm medm 8 m 8 m m 1 1 m m78 2 m m 16 m 15 m 17 m m m m 10 m m 20 m 19 Fig. 6: The DMS model maxm medm maxm medm 13 m 10 m m 10 9 m 9 m m m 25 meda maxa m 26 meda m 23 m S2 T2 M2 A2 R T4 68 T6 Table 1: Firing seeds and average firing delay of coninuous and discree ransiions for Cases 1 and 2. Coninuous ransiions [V min,v max] (Case 1) [V min,v max] (Case 2) Discree Average firing ransiions delay (hours) 1 [0, 8] [0, 4] [0, 10] [0, 5] [0, 10] [0, 5] [0,8] [0, 3] [0,9] [0, 4] [8,20] [0, 6] [0, 12] [0, 2] [12,30] [0, 6] [0, 11] [0, 3] [0, 6] [4, 15] [6, 15] [0, 4] [0, 8] [4, 12] [8, 20] [0, 3] [0, 3] [0, 4] [0, 4] [4, 10] [0, 5] [0, 6] We model he DMS in Fig. 5 by using in a modular way he elemenary modules described in Secion 3. Figure 6 shows he merged FOHPN model and deics each faciliy module in dashed line squares. To analyze he DMS behaviour, we simulae he FOHPN in wo differen cases ha corresond o wo differen oeraive condiions. The daa relaive o Case 1 and 2 are shown in Table 1. In aricular, Case 1 corresonds o a sysem wih high roducion rae of each manufacurer and assembler. On he oher hand, in Case 2 manufacurers and assemblers exhibi lower roducion raes han in Case 1 bu he same ransoraion seed and buffer caaciies. Moreover, Table 1 reors for each T c he minimum and maximum firing seeds and for each T E he average firing delay. In addiion, Table 2 shows for Case 1 and Case 2 furher daa necessary o fully describe and simulae he sysem: he buffer caaciies for he invenories of each sage, he iniial marings of coninuous laces and he values of he edge weighs. To analyze he sysem dynamics, we define some erformance indices assumed as relevan measures for he DMS analysis: i) he hroughu T i wih i=1,2 of reailer Ri wih i=1,2 resecively, i.e., he average number of roducs obained by each reailer in a ime uni; ii) he sysem hroughu T=T 1 +T 2 ; iii) he average inu socs in manufacurers Mi wih i=1,2 (I Mi i=1,2) and in assemblers Ai wih i=1,2 (I Ai i=1,2) during he run ime TP; iv) he average ouu invenory in Mi wih i=1,2 (O Mi i=1,2) and in Ai wih

6 i=1,2 (O Ai i=1,2) during he run ime TP; v) he average sysem invenory SI, i.e., he amoun of roduc sorage in each buffer during he run ime TP; vi) he average lead ime LT ha is as follows: LT= SI/T (5) The resuls are obained by simulaing he FOHPN in he Malab environmen for a simulaion run of TP=480 hours. In aricular, Fig. 7 shows he average invenories in manufacurers and in he assemblers in Cases 1 and 2. The figure shows ha, also hans o he deendence of manufacurers and assemblers roducion raes wih heir ouu invenories, he DMS is able o ee socs a a saisfacorily high level, so ha he demand is efficienly saisfied and invenory is no excessive. Moreover, as execed he DMS inu socs are always higher han he corresonding ouu invenories. However, he invenories of Case 1 are usually higher han he invenories in Case 2 because of he higher roducion raes. In addiion, Table 3 reors he average lead imes and he sysem invenories for Cases 1 and 2. As execed, hese wo erformance indices are higher in Case 1, exhibiing a higher level of congesion caused by he faser sysem roducion. Moreover, Fig. 8 reors he hroughu of reailers R1 and R2 and he sysem hroughu T. We remar ha he hroughu values in Case 1 are higher han he corresonding values in Case 2: his is execed, because manufacurers wor in Case 1 wih higher roducion raes. CONCLUSIONS The aer considers Firs-Order Hybrid Peri Nes (FOHPN) o model Disribued Manufacuring Sysems (DMS), which are new emerging comany newors, very comlex o describe and manage. Combining coninuous and discree dynamics, FOHPN aears a romising formalism, able o caure he differen roeries of such discree even sysems, characerized by a large number of saes. Fuure research will aly he FOHPN conflic resoluion olicy in order o oimize a given DMS obecive funcion under differen managemen olicies. REFERENCES Alla, H., and R. David, (1998). Coninuous and Hybrid Peri Nes. J. of Circuis, Sysems and Comuers, 8 (1), Balduzzi, F, A. Giua, and G. Menga, (2000). Modelling and Conrol wih Firs order Hybrid Peri Nes. IEEE Trans. on Roboics and Auomaion, 4 (16), Desrochers, A., T.J. Deal and M.P. Fani, (2005) Comlex-Valued Toen Peri nes. IEEE Transacions on Auomaion Science and Engineering, 2(4), Dooli, M. and M.P. Fani, (2005). A Generalized Sochasic Peri Ne Model for Disribued Manufacuring Sysem Managemen. 44h IEEE Conf. on Decision and Conrol and Euroean Conrol Conference, Seville, (Sain), Viswanadham, N. and S. Raghavan, (2000). Performance analysis and design of suly chain: a Peri ne aroach. Journal of he Oeraional Research Sociey, 51 (10), Zhou, M.C. and K. Venaesh (1998). Modeling, Simulaion and Conrol of Flexible Manufacuring Sysems. A Peri Ne Aroach. World Scienific, Singaore. Table 2: Iniial maring of coninuous laces, caaciies and edge weighs for Cases 1 and 2. Iniial maring Produc unis Caaciies m 1 m 2 20 C 1,C 2 =200 m 3 m 5 m 9 m C 4,C 6,C 10,C 12 =100 m 7 m C 8,C 14 =80 m 15 m 21 m 17 m C 16,C 22, C 18,C 24 =100 m 19 m C 20,C 26 =80 m 27 m 28 m 29 0 m 30 m 31 m 32 0 Edge weighs max M =190 max A =190 med M =50 med A =50 Table 3: Overall lead ime and sysem invenory for Cases 1 and 2. Case 1 Case 2 LT (hours) SI (roduc unis) unis I M 1 I O O I I O O Case1 Case2 M 2 M 1 M 2 A 1 A 2 A 1 A 2 Fig. 7. The average invenory in manufacurers and in assemblers in Cases 1 and 2. unis/h Case 1 Case 2 T1 T2 T Fig. 8. Throughus (unis/hours) in Cases 1 and 2.