A discrete event model for the control and analysis of complex automated warehouse systems

Transcription

1 2011 IEEE Conference on Automation Science and Engineering Trieste, Italy - Augus4-27, 2011 ThB2.1 A discrete event model for the control and analysis of comlex automated warehouse systems F. Basile and P. Chiacchio and J. Coola Abstract In the last twenty years, a big effort has been made to find otimal strategies for lanning and control of automated warehouse systems. An automated warehouse system has two main comonents: an automated storage and retrieval subsystem consisting of a number of aisles, each one served by a crane, and a icking area which is formed by bays where stock units coming from the aisles are artially emtied by human oerators. These two comonents are connected via an interface area consisting of carousels, conveyors and buffers. The role of this interface area is usually neglected, while the overall system erformance deends on how this subsystem is controlled. A model focusing on the interface area is roosed in this aer to imrove the control and the analysis of comlex automated warehouse systems. A real case study is considered to show the effectiveness of the aroach. I. INTRODUCTION Since 1990 a big effort has been sent to find otimal strategies for lanning and control of warehouse systems. Planning involves long-term otimization: it usually has a day or week time horizon and it is based on simlified models of warehouse systems and on statistical characterization of the system erformance [1]. Conversely, a detailed model is used for the control which erforms the shortterm otimization of handling sequences: it usually has the objective to minimize the time to comlete a little number of icking or storage oerations and it is based on the current state of the system. We refer to a real warehouse architecture consisting of a number of aisles, each one served by a crane, a carousel, icking ositions and inut buffers (see Fig. 1 for a real layout). On both sides of each aisle there is a storage rack comosed of n r rows and n c columns; moreover, as said, each aisle is served by a crane, caable of moving both vertically and horizontally at the same time, which erforms the following oerations: i) icking of the Stock Unit (SU) at the inut buffer/bay of the aisle to be stored in a rack location S; ii) storage of the SU into the assigned location S of the rack; iii) movement to location R where a retrieval has been requested; iv) retrieval of the SU stored in R; v) movement to the outut buffer/bay of the aisle to deosit the SU. This set of oerations is called, in the warehousing system context, a Dual Command (DC) machine cycle [2], [3], [4], [5]. The DC cycle can be generalized to the case of multile storages and retrievals for cycle. As to the carousel, it consists of shuttles which can move a number of SUs. The shuttles move along a mono- The authors are with Di. Ing. Elettronica e Ing. Informatica, Università di Salerno, Italy ( {fbasile,jcoola,}@unisa.it). dimensional ath laced orthogonally with resect to the aisle axis. They erform icking actions (from the aisles outut bays and from the icking area outut bays) and deosit actions (into the aisles inut bays and into the icking area inut bays). The icking area reresents the outut oint of warehouse system. A icking bay consists of a icking location connected via conveyors to the carousel inut and outut interfaces, so that a SU can be artially emtied by a human oerator and then carried back to an aisle rack location. The inut buffer reresents the interface of the warehouse with the incoming area. It is used to load full SUs in the warehouse. A set of missions is given as inut to this kind of systems. Each mission requires that a certain quantity of an article, which can be stored in more than one aisle, is moved to a icking bay. Hence, the execution of a mission requires the choice of the SU to move among those containing the desired article (this choice includes also the choice of the crane since there is one crane in each aisle), the choice of a shuttle to transfer the SU to the icking area, the choice of the icking bay, again the choice of a shuttle to move again the SU in the storage area, the choice of the location where the SU must be stored among those available. The control roblem is to assign each available resource (a location, a icking bay, a crane or a shuttle) to a mission. When one resource is available for a set of missions, a Conflict occurs. When two or more resources are simultaneously available, several conflicts can be mutually deendent, in the sense that the solution of a conflict can influence the others. The outut of the control roblem consists in determining Who has to do What and in Which Order in a manner that a certain objective is reached over a certain time horizon. In other words, the control must solve these conflicts. A detailed model is needed since it is imortant to detect in which order these conflicts occur. This aer focuses on how to obtain a model oriented to the control and to the erformance analysis of these systems. In articular, the comlexity of modern warehouse systems, like the real one considered in the case study, requires big interfaces (e.g. a carousel) between cranes and icking area and so many shuttles must be used. Furthermore, when each crane cycle involves more than one icking and deosit, the number of SUs moved by shuttles at a time in the interface area grows, and then a significant time is required to cover the interface ath. From now on, the term Interface System (IS) is used to denote the set of devices between the aisles and the icking bays /11/$ IEEE 201

2 Fig. 1. Layout of a real warehouse and its zones: crane bays (on to), icking bays (on bottom), direct link between aisle1 and bay1 (on left), IS formed from carousel routes and branch oints (in center). Indeed, the challenging roblem is the control of the IS since the control of the cranes has been studied a lot in the literature. In a certain sense, the control of the whole warehouse reduces to that of the cranes if the time to cover the IS is negligible. In [6] a key oint in the develoment of the warehouse otimization is that the crane otimization can be considered indeendent of the shuttle otimization if a shuttle requires a negligible time with resect to the crane mean cycle time to reach the crane bay from a icking bay. In words, as soon as a cycle ends, a crane can start another cycle. This is why automated warehouse systems are usually sized according to FEM (see [7]). More in detail, the erformance of each aisle is estimated by comuting the time needed to erform a dual command cycle, named FEM cycle, consisting of storing a SU and retrieve another one in secific oints of storage rack. The coordinates of these oints are obtained from rack geometry. A similar aroach could be used also in general case, where the cranes, in a single cycle, can store and retrieve more than one SU (in our case study, u to 2 SUs). Once that a crane cycle has been created (i.e. the list of locations to visit in a single travel to store and to ick SUs), the cycle time is deterministic and can be analytically comuted. In this aer it is assumed that the cranes work according to an extended version of the algorithm resented by the authors in [6] to otimize DC cycles, but the effectiveness of the aroach here resented is indeendent of the crane algorithm. Note that cranes have not a discrete event behavior. The discrete event behavior of an automated warehouse is caused by the IS. The activity of the IS is more relevant when the number of shuttles and the number of interface bays grows, and consequently the sto and go state of the shuttles related to event occurrences grows (e.g. a collision of two shuttles must be avoided, a shuttle stos when it reaches a certain interface bay, etc.). This increases the time to move a SU from icking area to the aisle inut bay and reduces the crane erformances indeendently from the crane otimization algorithm. A. Related works and aer contribution The roblem of shuttle otimization in warehouse systems of the tye in Fig. 1 has been reviously discussed in the literature in [6] where an algorithm to otimize the sequence of icking and deosit oerations erformed by a single shuttle, so as to minimize the time required by each cycle, is roosed. A similar roblem has been treated in [8] where a unique model for automated storage and retrieval systems, comrising rail guided vehicles and narrow aisle cranes, is roosed. The contribution of this aer is the develoment of a control oriented model of an automated warehouse, as shown in Fig. 1, to take into account the role of the IS. Preliminary results on the control of IS are also resented. Using commercial simulation tools like Arena c as in [3] or Automod c is already ossible to model systems like the one resented in this aer. However, they are not based on a formal model, and so, they are not general enough to be alied to every IS. Moreover, they can be used only 202

3 to simulation uroses, while a formal model allows one to check system roerties (e.g. deadlock avoidance) by formal analysis. A discrete event model based on the standard Colored Timed Petri Net (CTPN) [9], [10] formalism is roosed in this aer, while in the aroach resented in [8] inhibitory arcs are used. Moreover, differently from [8], the activity of automated storage and retrieval subsystem reduces to a timed transition modeling the time to erform a given cycle: the aisles and their locations are not exlicitly modeled in our model. This allows one to obtain a model of a reasonable size for real warehouses like the one considered in the case study. A real case study is considered to show the effectiveness of the aroach. The CTPN model of the warehouse can be used both for the erformance evaluation and for online imlementation of control algorithms. II. NOTATIONS AND DEFINITIONS In this section at first Petri Nets (PNs) are introduced. Then, a brief overview on Colored PNs (CPNs) is furnished. For further details on PNs and on simulation of PNs, the reader can refer to [11] and to [12]. A Place/Transition (P/T) net is a 4-tule N = (P,T,Pre,Post), where P is a set of m laces (reresented by circles), T is a set of n transitions (reresented by black bars), Pre : P T N (Post : P T N) is the re (ost-) incidence matrix. C = Post Pre is the incidence matrix. The net marking is reresented as a vector m N m. The marking of a lace is a scalar value m i N. A transition t is enabled at m iff m Pre(,t) and this is denoted as m[t. An enabled transition t may fire yielding the marking m = m + C(,t) and this is denoted as m[t m. The symbols ( t) and (t ) are used for the re-set and ost-set of a lace P (transition t T ), resectively, e.g. t = { P Pre(,t) 0 }. As shown in Fig. 2a), there is a structural conflict when t i t j. There is a behavioral conflict between t i and t j at marking m iff there exist k,k N such that m k Pre(,t i ) m k Pre(,t j ), but m k Pre(,t i )+k Pre(,t j ). All the formal definitions given for PNs can be naturally extended to Colored PNs (CPNs). Formally, a CPN is a 6- tule N = (P,T,Pre,Post, Cl, Co). As in PNs, P is a set of m laces (reresented by circles), T is a set of n transitions (reresented by bars). Cl is the set of colors. Co: P T Cl is a color function that associates to each element in P T a non-emty ordered set of colors in the set of ossible colors Cl. P,Co( i ) = {a i,1,a i,2,...,a i,ui } Cl is the ordered set of ossible colors of tokens in i, andu i is their number. t T,Co(t j ) = { b j,1,b j,2,...,b j,vj } Cl is the ordered set of ossible occurrence colors in t j, and v j is their number. For each lace i P, the marking m i is defined as a non-negative multi-set over Co(P i ). The maing m i : Co(P i ) N associates to each ossible token color in P i a non-negative integer reresenting the number of tokens of that color that is contained in the lace i. Fig. 2. Conflict in Petri Nets: structural conflict ; behavioral conflict 1 0 c ( 1 c, 2 ) c (,1 ) ( c,1 2 c,,2 ) c (,1 ) e 1 (c) 1 0 c ( 2 ) c ( 1 c, 2 ) c (,1 ) ( c ( 2 ) c,1 2 c,,2 ) Fig. 3. Unmarked CPN; Marked CPN ; (c) Marked CPN in simlified by using notation of section III. The column vector of u i non-negative integers, whose hth comonent m i (h) is equal to the number of tokens of color a i,h that are contained in i, is denoted as m i. The marking of a CPN is an m-dimensional column vector of multisets: m = [m 1...m m ] T. For the sake of simlicity, a token of color c1 contained in a lace i will be indicated with the symbol (c1). In literature, more than one formal definition for CPNs exists, deending on how the incidence matrix and transition colors are defined. In the formalism chosen in this work, matrix entries are reresented by matrices. Pre and Post are the re-incidence and ost-incidence m n-sized matrices, resectively. Pre( i,t j ) is a maing from the set of occurrence colors of t j to a non-negative multiset over the set of colors of i, namely, Pre( i,t j ) : Co(t j ) N(Co( i )), for i = 1,...,m and j = 1,...,n. Pre( i,t j ) reresents a matrix of u i v j non-negative integers whose generic element Pre( i,t j )(h,k) is equal to the weight of the arc from lace i w.r.t color a i,h to transition t j w.r.t color b j,k. Post( i,t j ) : Co(t j ) N(Co( i )), for i = 1,...,m and j = 1,...,n. Post( i,t j ) reresents a matrix of u i v j non-negative integers whose generic element Post( i,t j )(h,k) is equal to the weight of the arc from transition t j w.r.t color b j,k to lace i w.r.t color a i,h. The incidence matrix C is a m n matrix, whose generic element C( i,t j ) : Co(t j ) Z(Co( i )), for i = 1,...,m and j = 1,...,n, is the u i v j matrix of integer numbers C( i,t j ) = Post( i,t j ) Pre( i,t j ). The concets of reset and ost-set of a lace P or a transition t T are naturally inherited from PNs, but colors must be also considered: t i c j = { t i c j T Pre( h c k,t i c j ) 0 }. In Fig. 3a) a CPN with a structural conflict is shown. It 203

4 is made u of a lace, having C o () = {c 1,c 2 }, and of two transitions, and with C o ( ) = c 1,1 and C o ( ) = {c 2,1,c 2,2 }. When fires, one token, corresonding to color c 1,1, is removed from lace ; can fire both under color c 2,1 and c 2,2 and when it fires, one c 2,1 or c 2,2 token, resectively, is removed from. In Fig. 3b) a token is added to the CPN in Fig. 3a); notice that the conflict is still structural (not behavioral), since no c 1,1 token is resent in and, consequently, transition cannot fire. When time is added to PNs and CPNs a time function is defined, which associates to each transition t i in the case of PNs, or to each transition color t i c j in the case of CPNs, a time duration from enabling to firing. In this case the PNs and CPNs become TPNs and CTPNs. Notice that timed and untimed transitions will be reresented with emty filled boxes and black bars, resectively. III. MODEL OF CAROUSEL Among the ISs commonly used to move SUs from icking area to cranes and viceversa, a very general and comlex one is the carousel. This is why we focus on the carousel modeling in this aer. However, the aroach can be immediately extended to other handling systems. A carousel consists of a ath along which shuttles move. This ath, without loss of generality, can be divided in these basic comonents: 1) Elementary zone, where a unique route is ossible and there are not interfaces with other subsystems. 2) Switching zone, where more than one route is ossible since the ath can branch off in different lines of travel. 3) Interface zone, where a shuttle stos to load (unload) a SU from (to) another subsystem bay. In this section it is shown how these comonents can be modeled each one by a single CTPN module, so the whole carousel can be modeled roerly connecting a set of CTPN modules. Moreover, it is shown that this formalism allows an easy conflicts detection. This is very imortant for the control of such systems since the control action consists in solving conflicts. These conflicts are usually solved by disatching rules, rather than by the design of comlex algorithms. This is the more reasonable and usual way to face the ossible combinatorial exlosion of this mission-to-resource assignment roblem. Although the solution of the control roblem via disatching rules roduces only locally otimal solutions, finding an analytical closed form solution for the whole set of missions does not make sense since the sudden arrival of a new mission, the breakdown of cranes or shuttles, the sto of a human oerator could require a comlete reschedule of the work. The roosed model allows also an easy imlementation of the disatching rules. Disatching rules are alied to each conflict detected by means of the model. The outut of disatching rules action is to disable all the events excet one. This is ossible since the structure of the model allows one to detect the events that can be involved in a conflict. These events are modeled by controllable transitions, i.e. they are associated to a boolean Fig. 4. Model of an elementary zone. control variable. If this variable is false, the transition cannot fires, even if it is enabled by the net marking. For the sake of simlicity, with reference to Section II, some secial notations are used to draw CTPNs in this aer: When transition occurrence colors can fire under any inut lace color, no color is indicated at transition side (see Fig. 3c) for transition ). If a transition occurrence color can fire only under some secific inut lace colors, they will be indicated near the transition (see Fig. 3c) for transition ). If no matrix is indicated near an arc, an identity matrix I is intended to be associated to that arc (see Fig. 3c); this is the case of the arc from lace to transition ). e h near an arc from lace i to transition t j or viceversa denotes a column vector of sizeu i, with the hth element equal to one and the other elements equal to zero (see Fig. 3c); this is the case of the arc from lace to transition ). In real automated warehouses each zone can be occuied by just one shuttle at time: a shuttle has to halt while the next zone is busy. Firing time of timed transitions deends on the length of the zone and reresents the time necessary to the shuttle to cross the zone. For the sake of simlicity, in this aer acceleration and deceleration have been neglected. Hence, shuttles can have only two seeds, v = 0 when they are stoed and v = V max when they are moving. With these simlifications, a carousel elementary zone can be modeled like a simle belt conveyor (see Fig. 4 wheref i denotes the zone end oint). This technique simlifies also the modeling of interface zones since cranes or icking bays consist of belt conveyors. Switching zones can be modeled as shown in Fig. 5. Notice that a behavioral conflict occurs every time a token is resent in the lace shuttle in Z i+1 of Fig. 5a), or in the lace shuttle in Z j+1 of Fig. 5b). Colors can be used to discriminate a free shuttle from a busy one, to identify where busy shuttles are directed and to identify a SU. SU identification (number) is not essential in the IS since SUs are moved according to their destination along the IS ath, but it is used to associate the storage location once that a SU reaches a crane. For this reasons each net lace has as many colors as the ossible shuttles 204

5 Fig. 5. Switching zone: outut branch oint; inut branch oint. Fig. 6. Interface zone: assage of SUs from bay to shuttle and from shuttle to bay. destinations lus one color that identifies an emty shuttle, and a SU number is associated to each token. This number allows one to associate a rack location to each SU when it reaches a crane bay. Interface zone between shuttles and crane (or icking) bays can be modeled as shown in Fig. 6. The assage of a SU from a bay to a shuttle can be modeled as shown in Fig. 6a). Assumec e is the color used for lace shuttle inz i+1 to reresent an emty shuttle. When transition SU transfer on Z i+1 fires, a token of color c e is removed from the lace shuttle in Z i+1 : it means that the emty shuttle resent in the zone, cannot continue to move along carousel since it has to wait for a SU. It is worth to notice that the firing of the transition SU transfer on Z i+1 does not ut any token in lace M, reresenting a mutual exclusion between Z i+1 - Z i+1 and Z i, i.e. if a shuttle is resent in the revious zone, it cannot move forward since the next zone is still occuied. When transfer between a bay conveyor named B h,j+1 and Z i+1 is comleted, a token of color c h (that identifies the SU destination) is ut in the lace shuttle in Z i+1. Thus, the emty shuttle is relaced with a busy one that now can kee moving along carousel. Even if a token is resent in the lace SU on B h,j+1, the exchange between bays and shuttles can occur only if in the interface lace shuttle in Z i+1 a token of color c e is resent; in this case a behavioral conflict occurs between transition SU transfer on Z i+1 and transition advancing on Z i+1. The assage of a SU from a shuttle to the bay of destination is modeled in Fig. 6b). Assume c h is the color used for lace shuttle in Z i+1 to model a SU having bay B h as destination. Transition SU transfer on B h,j can fired only if a token of colorc h, equals to the inut color of the bay, is in the lace shuttle in Z i+1. When such transition fires, it removes the token from the lace shuttle in Z i+1 (emtying of the shuttle). Just as the revious case, the firing of the transition SU transfer on B h,j stos the shuttle in the interface osition and does not ut any token in lace M, avoiding another shuttle enters in the zone. When transfer between Z i+1 and B h,j is comleted, a token of color c e (emty shuttle) is ut in the lace shuttle in Z i+1 and the free shuttle can kee 205

6 moving along the carousel. When in the lace shuttle in Z i+1 a token of color c h equal to the inut color of the bay is resent, a behavioral conflict occurs between transition SU transfer on B h,j and transition advancing on Z i+1. Modularity is an advantage of the CTPN model resented: it can be adated at several layouts just adding or removing elementary zones; as more, if a new bay or a new route is added, it can be connected at carousel just introducing the CTPN module modeling an interface zone or a switching zone resectively. A. Plant descrition IV. CASE STUDY The warehouse is divided in 10 aisles, each served by one crane. On both sides of each aisle there is a storage rack comosed of 17 rows and 47 columns. The whole systems is made u of locations. Each location can store one or more SUs, deending of their dimensions (mm x mm): EURO1, 800x1200; EURO3, 1200x1200; MINI- JUMBO, 1600x1200; JUMBO, 2500x1200. In the warehouse EURO 1 SUs can be stored. The crane moves along horizontal (vertical) axis with a maximum seed 4m/s (0.8m/s). The icking area consists of 9 bays where SUs are emtied by human oerators. Aisle1 and bay1 are directly connected by a system of conveyors and they can move only Jumbo SUs. The other 9 aisles are connected with the remaining 8 bays by a carousel (Fig. 1) where u to 23 shuttles can be moved. Carousel resents several arts: an outer ring (blue ath in Fig. 1) where the carousel interfaces with cranes or icking bays; an inner ring, connected with the outer ring by 6 switching oints (named NW, N, NE, SE, S, and SW in Fig. 1), with 4 alternative routes (from NW to N, from N to NE, from SE to S, from S to SW) that shuttles can use to arrive at a secific set of bays; a rail of arking (brown ath in Fig. 1) where shuttles can be stoed when they are not useful. In the warehouse, full SUs are moved in and out from the aisles, to be (artially) emtied at the icking bays. The relenishment of the articles in the structure reresents the 10% of the warehouse activity, and it is erformed during the night, in the manner that it does not influence warehouse erformance. For this reason, at the start of any working-day, we can assume that all the articles necessary to satisfy the orders are already resent in the warehouse. Several control laws, also very comlex, can be devised on the basis of the carousel model obtained according to the Section III. Deadlock avoidance is not the goal of this aer. For the sake of brevity, in the following we omit the discussion about deadlock analysis that can be done using the model resented in Section III, and about deadlock avoidance olicy used in the real case study. The focus is on the system erformance. As for the warehouse otimization, in this reliminary aer the following control strategy actually used in the real lant has been adoted in the simulations: a storage queue is associated to each crane; such a queue contains all the SUs that the crane has to store. When a new SU is available at a icking bay, it is assigned to the crane having a free location of the same SU size (the controller has access to warehouse ma) and with the lowest storage queue. Such a location is reserved, to avoid to assign two SUs to the same location. The aroach resented in Section III has been used to model the carousel. To build the CTPN model of the carousel 19 colors are used: 9 colors are used to identify shuttles transorting SUs directed to the aisles, 8 colors are used to identify shuttles carrying SUs directed to the icking bays, 1 color is used to identify emty shuttles and 1 color is used to identify shuttle designed for the arking zone. The overall net model is deterministic. This is realistic, since the timing duration of each transition associated to conveyors and shuttles is fixed, while for the human oerator at the interface bay a mean time has been used. B. Simulation analysis According to FEM document, the estimated average cycle time for each crane of the considered warehouse is 50 SUs/h (500 SUs/h for the overall system), but, as it will be shown later, this is not realistic. Indeed, this could be accetable when the carousel effects can be neglected. As said in the introduction, the activity of each crane reduces to a single timed transition whose firing time is fixed after the cycle has been created. When two SUs to be stored in a certain aisle reach the crane bay, a crane cycle is built by the controller, roerly adding two ickings so that the crane erformance is otimized (see [6]). Details on this are omitted here, since this is not the aer focus and it is a well known roblem in the literature. Starting from an initial warehouse occuation of about 80%, the behavior of the real warehouse during 5 workingdays, each one including 6000 missions, have been simulated (PNetLab has been used as simulation tool - see [12]). In Fig. 7, daily average system erformances in terms of makesan (time required to the system to comlete all the missions) are reorted. It results that the erformances are quite stable but the makesan is minor than the exected one (abou2h): the time to execute a FEM cycle is an uer bound of the average cycle time for the overall system. Thus, when the carousel travel time is comarable with the crane cycle time, its dynamic must be taken into account. This is frequent in automated warehouse involving heavy SUs where it is very exensive to increase the carousel seed. However, increasing the carousel seed is not always a good idea as it will be shown in the next. In a second set of simulations, a single working-day has been simulated varying shuttle seed in a range from 0.9 m/s to 3 m/s, actually used in real world. In Fig. 8a) it can be seen how the varying of shuttle seed influences the crane subsystem erformances. As for the crane occuation time, we mean the ercentage of time a crane remains busy during a simulation. This time deends on how much time the carousel sends to deosit on crane bays SUs that must be stored in the aisles. Indeed, if there are not SUs on the bay, a crane must wait. 206

7 Fig. 7. System erformance with 20 shuttles and a shuttle seed equal to 1,5 m/s: a) Daily makesan Average SUs/h. Fig. 8. Variation of the system erformances with 20 shuttles but increasing shuttles seed: ercentage of cranes utilization; makesan. Variation of makesan deending to shuttle seed is reorted in Fig. 8b). It could seem surrising that an increment of shuttle seed does not roduce an increment of the makesan. The nonlinear behavior is due to the interface zones where shuttles halt to load (unload) SUs: an increment of shuttle seed cause more stos and then it does not imly an increment of system erformance. In the last set of simulations, a single working-day has been simulated increasing the number of shuttles in the carousel: after a threshold (that is equal to 20 shuttles, in Fig. 9), adding other shuttles chokes u the carousel and gets worse the erformances. This result confirms again the nonlinear behavior due to the interface zones where the discrete event dynamic is relevant. V. CONCLUSIONS The imortance of having an accurate model of the interface zone between cranes and the icking area in automated warehouses has been treated in this aer. A Colored Timed Petri Net model has been roosed to take into account the discrete event nature of this subsystem. REFERENCES [1] J. Van den Berg, A literature survey on lanning and control of warehousing systems, IIE Transactions, vol. 31,. 1 13, [2] S. C. Graves, W. H. Hausman, and L. B. Schwarz, Storage-retrieval interleaving in automatic warehousing systems, Management Science, vol. 23, no. 9, , [3] Y. A. Bozer and J. A. White, Travel-time models for automated storage/retrieval systems, IEE Transactions, vol. 16, no. 4, , [4] M. H. Han, L. McGinnis, J. S. Shieh, and J. A. White, On sequencing retrievals in an automated storage/retrieval system, IEE Transactions, vol. 19, no. 3, , [5] S. Lee, R. de Souza, and E. Ong, Simulation modelling of a narrow aisle automated storage and retrieval system (AS/RS) serviced by railguided vehicles, Com. in Ind., vol. 30, , Fig. 9. Variation of makesan increasing the number of shuttles with a shuttle seed equal to 1,5 m/s. [6] F. Amato, F. Basile, C. Carbone, and P. Chiacchio, An aroach to control automated warehouse systems, Control Eng. Practice, vol. 13, , [7] Euroean Federation of Material Handling, FEM Performance data of storage and retrieval machines: cycle times, [8] M. Dotoli and M. Fanti, A coloured Petri net model for automated storage and retrieval systems serviced by rail-guided vehicles: a control ersective, International Journal of Comuter Integrated Manufacturing, vol. 18, , [9] K. Jensen, Colored Petri nets. Basic concets, analysis methods and ratical use. Volume 1, ser. Monograhs on Theoretical Comuter Science. New York: Sringer Verlag, [10] S. Hsieh, J. Hwang, and H. Chou, A Petri net based structure for AS/RS oeration modeling, Int. Journal of Production Research, vol. 36, no. 12, , [11] T. Murata, Petri nets: Proerties, analysis and alications, Proceedings of IEEE, vol. 77, no. 4, , Aril [12] F. Basile, C. Carbone, and P. Chiacchio, Simulation and analysis of discrete-event control systems based on Petri nets using PNetlab, Control Eng. Practice, vol. 15, , February