Multimodal Processes Rescheduling: Cyclic Steady States Space Approach

The paper concerns cyclic scheduling problems arising in aMultimodal TransportationNetwork (MTN) in which several unimodal networks (AGVs, hoists, lifts, etc.) interact with each other via common shared workstations as to provide a variety of demandresponsive material handling operations. The material handling transport modes provide movement of work-pieces between workstations along their manufacturing routes in the MTN. The goal is to provide a declarative framework enabling to state a constraint satisfaction problem aimed at AGVs fleet match-up scheduling taking into consideration assumed itineraries of concurrently manufactured product types. In that context, treating the different product types as a set of cyclic multimodal processes is the main objective to discuss the conditions sufficient for FMS rescheduling imposed by production orders changes. To conclude, the conditions sufficient for an FMS rescheduling imposed by changes of production orders treated as cyclic multimodal processes are stated as the paper’s main contribution.


Introduction
The design and operational issues arising in an Automated Guided Vehicle (AGV) served Flexible Manufacturing System (FMS) when deciding on how to achieve a desired system performance are usually hard to determine and evaluate.This is because productivity of an AGV-served flow shop producing a set of different kinds of products, repetitively, depends on both the job flow sequencing (aimed at maximizing throughput or minimizing cycle time) and the material handling system required to achieve a prespecified throughput, that is, AGVs fleet sizing, assignment and scheduling [1][2][3][4][5].So, assuming a deterministic system where demand is known in advance and the processing times of each job on each machine is a known constant, in order to improve its productivity, the AGV fleet size and the scheduling/dispatching influence on the system throughput are usually examined [1,[6][7][8][9].
In opposite to conventionally applied approach assuming AGVs fleet sizing, routing, and timetabling while matching up prescheduled multiproduct manufacturing flow (i.e., providing detailed transportation route and materials load/ unload schedule between workstations in FMS), we propose AGVs following a Multimodal Transportation Network (MTN) concept.MTN can be seen as composition of several unimodal networks (AGVs, hoists, lifts, etc.) interacting with each other via common shared workstations as to provide a variety of demand-responsive material handling operations.The material handling transport modes provide movement of work-pieces between workstations along their manufacturing routes in the MTN.The MTN encompasses a well-known multimodal approach to freight transportation supply chains and passenger transportation systems.So, similarly to such systems having ability to use the appropriate mode of transport for movement of goods and people, the assumed material handling transport modes provide movement of work-pieces between workstations along their manufacturing routes in the MTN.In that context, assuming that the modes can be treated as cyclic local transportation processes, and material (e.g., work-pieces/tools) flows supported by them can be treated as multimodal processes, the problems considered can be seen either as aimed at designing of a multimodal network infrastructure guaranteeing assumed materials flow or focused on work-pieces routing ensuring lag-free workstations service.
The problems arising concerning material transportation routing and scheduling belong to NP-hard ones [2].Since the steady state of production flows has a cyclic character, hence servicing them AGV-served transportation processes (usually executed along loop-like routes) encompasses also cyclic behavior.That means that the periodicity of FMS [10] depends on both the periodicity of production flow cyclic schedule and following this schedule the AGVs periodicity.Of course, the Multimodal Transportation Network (MTN) throughput is maximized by the minimization of its cycle time.
It seems to be obvious that not all the behaviors (including cyclic ones) are reachable under constraints imposed by the system's structure.The similar observation concerns the system's behavior that can be achieved in systems possessing specific structural constraints.Since system constraints determine its behavior, both the system structure and the desired cyclic schedule have to be considered simultaneously.So, the problem solution requires that the system structure must be determined for the purpose of processes scheduling, yet scheduling must be done to devise the system configuration.In that context, our contribution provides a discussion of some solvability issues concerning cyclic processes dispatching problems, especially the conditions guaranteeing solvability of the cyclic processes scheduling as well as direct and/or indirect transitions between assumed cyclic steady states.Their examination may replace exhaustive searches for solution satisfying required system capabilities.
Many models and methods have been considered to date [11].Among them, the mathematical programming approach [1,12], max-plus algebra [13], constraint logic programming [14][15][16] evolutionary algorithms, Petri nets [17,18], and heuristic frameworks [19] belong to the more frequently used.Most of these are oriented at finding a minimal cycle or maximal throughput while assuming deadlock-free processes flow.Note that processes' operations are blocking if they must stay on a resource (e.g., the station, the machine) after finishing when the next resource is occupied by a job from another process.During this stay the resource is blocked for other processes.The approaches trying to estimate the cycle time from cyclic processes structure and the synchronization mechanism employed (i.e., mutual exclusion instances) while taking into account deadlock phenomena are quite unique.
In our approach a declarative framework aimed at refinement and prototyping of the cyclic steady states for concurrently executed cyclic processes modelling material handling systems is employed.These systems are of the type of AGVs fleets in the flexible manufacturing systems (FMS) and are frequently encountered in industry.The following questions are the main focus of the research.Can the assumed material handling system, for example, AGVs, meet load/unload deadlines imposed by flow of scheduled work-pieces processing?Does there exist sufficient AGVS enabling to schedule the AGVs fleet as to ensure lag-free service of scheduled work pieces processing?So, the main question is Can the assumed AGVs fleet assignment reach its goal subject to constraints assumed on concurrent multiproduct manufacturing at hand?
Moreover the declarative framework enables to state a problem of multimodal processes rescheduling which boils down to searching for transient periods allowing the mutual reachability of MTN cyclic behaviors.In the case of MTNs, distinguishing many different cyclic steady states (cyclic behaviors), the following questions play a pivotal role.Does there exist the smooth (direct) transition between two assumed cyclic behaviors?What conditions guarantee the reachability of a given cyclic behavior from any other ones?Is it possible to come back to a given cyclic behavior from any disturbance-born one, for instance, caused by operation time delays, damage of devices, route modifications, and so forth?
In other words, the paper's objective concerns the MTN infrastructure assessment from the perspective of possible FMS oriented requirements imposed by AGVs fleet assignment, sizing, and scheduling (problems of cyclic processes scheduling and rescheduling).Due to the complexity implied in answering the above questions, the combined problem remains unsolved for all practical purposes.This is especially problematic as many manufacturing companies would stand to reap considerable rewards through better fleet assignment and scheduling.The presented approach addresses this through solving the combined rather than the separate problems individually.
The rest of the paper is organized as follows.Section 2 introduces the Automated Guided Vehicles System (AGVS) modelled in terms of concurrently flowing cyclic processes (SCCPs) and a cyclic steady state space concept.Section 3 regarding multimodal processes rescheduling provides a problem statement concerning AGVs fleet match-up scheduling with an assumed multiproduct manufacturing flow schedule.Sufficient conditions allowing one to search for states with mutual allocation as well as illustrative example of implementing them algorithm usage are discussed.The related work and concluding remarks are presented in Sections 4 and 5, respectively.

Multimodal Network Modeling
Considered case of multimodal processes rescheduling problem is presented on the example of Automated Guided Vehicle Systems (AGVS) modeled by Systems of Concurrently Flowing Cyclic Processes (SCCPs).To solve this kind of problems a nonempty state space, that is, containing nonempty set of possible cyclic behaviors, is required.For that reason the conditions guaranteeing cyclic behavior are presented, primarily.They are specified in terms of a declarative model formalism distinguishing the structure and behavior as the main components of an SCCP.Relationship linking the structure and reachable behaviors is the main subject of this section.

Model of AGVS.
Automated Guided Vehicle Systems are used for material handling within a Flexible Manufacturing System (FMS) and provide asynchronous movement pallets of products through a network of guide paths between the workstations by the AGVs.Each workstation is connected to the guide path network by a pick-up/delivery station where pallets are transferred from/to the AGVs.
In AGVS literature, most are related to AGVS design issues, which include determination of the number of vehicles required, flow path design, and route planning as well as vehicle dispatching and traffic management.Recently, an integrated problem of dispatching and conflict free routing of AGVs, that is, integrating the simultaneous assignment, scheduling, and conflict free routing of the vehicles, is receiving increasing attention.The above mentioned problems, in the general case, belong to the class of NP-hard problems.Since most processes observed in steady state manufacturing are periodic, cyclic schedules and following them, cyclic scheduling methods can be considered.Since cyclic schedules encompass repetitive character of manufacturing processes, the cyclic processes modeling approach seems to be a reasonable perspective [13][14][15].
To present some of the design and operational issues that arise in repetitive manufacturing systems served by AGVs in network of loops layout with unidirectional material flow, a case example of a simple FMS is shown in Figure 1.The concurrent multiproduct flows are depicted by bold green, blue, and orange color lines, while the transportation loop-like networks are distinguished by double solid line.Both kinds of material (jobs) and transportation (AGVs) flows shown in Figure 1 can be modeled in terms of Systems of Concurrently Flowing Cyclic Processes [14,15] as shown in Figure 2. In turn, the SCCP framework provides a formal model enabling to state and resolve problems of AGV fleet size minimization as well as steady state cycle time minimization.The cycle time minimization is required to obtain maximum throughput rate.
For example, the production route depicted by the orange line corresponds to the multimodal process  1 supported by AGVs (arbitrarily given), which in turn encompass local transportation streams  1  1 ,  1 2 ,  3 2 , and  2  4 .This means that the production route specifying how a multimodal process is executed can be considered to be composed of parts of the routes of local cyclic processes.
In the system considered each multimodal process consists of four streams: 3, which means that along each production route four work-pieces are processed serially (four balls on the one production lines in Figure 1).
Processes can interact with each other through shared resources, that is, the transportation sectors.The routes of the considered local processes (streams) are as follows (see Figure 2): In the general case, the route    is the sequence of resources used in order to execute the operations of the stream    .Note that the streams  1 2 ,  2 2 , and  3 2 which belong to  2 and the streams of processes  4 ,  5 , and  6 follow the same route but start from different resources (these streams correspond to vehicles moving along the same route).

Input resources
Output resources Similarly the cyclic multimodal processes  1 ,  2 , and  3 follow the routes (see Figure 2) as follows: Let us assume that   1 ,   2 , and   3 can be seen as follows (see Figure 2): where one has the following: ( 3 ,   (i) a new subsequent operation of the local process may start on a required resource only if the current operation has been completed and the resource has been released; (ii) each new consecutive operation in the multimodal process may start its execution on an assigned resource only if the current operation has been completed and the resource has been released and an appropriate local process begins its consecutive operation on this resource; (iii) local/multimodal processes share common resources in the mutual exclusion mode, the operation of a local/multimodal process can only be suspended if the necessary resource is occupied, suspended processes cannot be released, and processes are nonperceptible; that is, a resource may not be taken from a process as long as it is used by it; (iv) the SCCP resources can be shared by local and multimodal processes as well as by both of them; (v) multimodal processes encompassing production flow conveyed by AGVs follow local transportation routes; (vi) different multimodal processes can be executed simultaneously along the same local process; (vii) local and multimodal processes execute cyclically with periods  and , respectively; resources occur uniquely in each transportation route; (viii) in a cyclic steady state, each th stream must cover its local route the same number of times.
A resource conflict (caused by the application of the mutual exclusion protocol) is resolved with the aid of a priority dispatching rule [14][15][16], which determines the order in which streams access shared resources.For instance, in the case of the resource  5 , the priority dispatching rule 2 ) determines the order in which streams of local processes can access the shared resource  5 ; in the case considered the stream  2  2 is allowed to access as first, then the stream  3  2 as next, then streams  1 5 ,  2 5 , and  1 2 , and then once again  2  2 , and so on.The stream    occurs the same number of times (in the considered system once) in each dispatching rule associated with the resources featuring in its route (in the case considered each stream    occurs uniquely).The SCCP shown in Figure 2 is specified by the following set of dispatching rules: Θ = {Θ 0 , Θ 1 }, where Θ 0 = { 0 1 , . . .,  0 33 } is set of rules determining the orders of local processes and Θ 1 = { 1  1 , . . .,  1 33 } is set of rules determining the orders of multimodal processes.
In general, the following notation is used.
(i) A sequence    = (  ,1 ,   ,2 , . . .,   , , . . .,   ,() ) specifies the route of the stream of a local process    (the th stream of the th local process   ).Its components define the resources used in the execution of operations, where   , ∈  (the set of resources  = { 1 ,  2 , . . .,   , . . .,   }) denotes the resources used by the th stream of the th local process in the th operation; in the rest of the paper, the th operation executed on the resource   , in the stream    will be denoted by   , ; () is the length of the cyclic process route (all streams of   are of the same length).For example, the route  1  1 = ( 16 ,  3 ,  13 ,  6 ) of the stream of process  1 (Figure 2) is the sequence is equal to  16 , whereas the second,  1 1,2 =  3 , and so on,  1  1,3 =  13 , is the subsequence of the route In other words, the transportation route   is a sequence of parts of routes of local processes.For instance, the route followed by process  1 (see Figure 2) is as follows:   1 = (( 3 ,  and () is the length of the cyclic process route is the set of priority dispatching rules for local ( = 0)/multimodal ( = 1) processes where    = (  ,1 , . . .,   , , . . .,   ,() ) are sequence components which determine the order in which the processes can be executed on the resource   ,  0 , ∈  (where  is the set of local streams, e.g., in the case presented in Figure 2 ,  1 7 }), and  1 , ∈  (where  is the set of multimodal streams, e.g., in case from Figure 2 Using the above notation, a SCCP can be defined as a tuple: where one has the following:  = { 1 ,  2 , . . .,   , . . .,   }the set of resources, -the number of resources, SL = (, , Θ 0 )-the structure of local processes, that is,  = { 1 1 , . . .,  (1) 1 , . . .,  1  , . . .,  ()  }-the set of routes of local process, ()-the number of streams belonging to the process   , -the number of local processes,  = { 1  1 , . . .,  (1) 1 , . . .,  1  , . . .,  ()  }-the set of sequences of operation times in local processes, Θ 0 = { 0 1 ,  0 2 , . . .,  0  , . . .,  0  }-the set of priority dispatching rules for local processes, SM = (, , Θ 1 )-the structure of multimodal processes, that is,  = { 1  1 , . . .,  (1) 1 , . . .,  1  , . . .,  ()  }-the set of routes of a multimodal process, ()-the number of streams belonging to the process   -the number of multimodal processes,  = { 1  1 , . . .,  (1)  , . . .,  1  , . . .,  ()  }-the set of sequences of operation times in multimodal processes, and The SCCP model ( 5) can be seen as a multilevel model, cf. Figure 3, that is, a model composed of an " level" (resources), an "SL level" (local cyclic processes), and an "SM 1 level" (multimodal cyclic processes), as well as an "SM  level" (the th metamultimodal process).The SL level is defined by the transportation routes structure following the set  of local processes and the set of parameters Θ determining the required system behavior.In turn, the SM 1 level takes into account multimodal processes, as well as metamultimodal processes (SM 2 level) composed of multimodal processes from the SM 1 level.In other words, it is assumed that the variables describing SM  are the same as in the case of SM, whereas the routes of the multimodal process of the th level remain composed of the processes from the ( − 1)th level.The presented model is an extended version of a simplified model limited to  and SL levels, which is introduced in [15].
Therefore, in general, the SC = ((, SL), SM) model can be seen as composed of  levels as follows: The SC  model emphasizes structural characteristics of the SCCP modeled.In turn, the behavioral characteristics can be specified in terms of the admissible states reachability concept, that is, either taking into account the state space concept including both cyclic steady states and leading to them transient states [16] or the cyclic steady state space only [14,15].The second way, which does not take into account initial states leading through the transient states to the cyclic steady states, seems to be quite close to the cyclic scheduling methods widely used in many real-life cases [9,11].
In that context the relevant multilevel cyclic schedule   encompassing the SCCP behavior on each of its processes level, that is, local SL and multimodal SM  processes, can be defined as follows: = (((((, ) , ( 1 ,  1 )) , . . ., ) , . . ., (  ,   )) , . . ., (  ,   )) , (7) where one has the following: , /(  ,   )-sequence of commencement of operations and periodicity of local/multimodal (th level) processes executions.It should also be noted that the schedule   can be defined as a sequence of ordered pairs describing behaviors of local (, ) and multimodal (  ,   ) processes, where  is a sequence of timings   , (for  = 0th cycle) of commencement of operations   , from streams    executed along local processes  = ( 1  1,1 ,  Since values of   , /    , follow from system structure, hence the cyclic behavior   of SCCP is determined by its structure SM  .Moreover, the multimodal processes behavior (  ,   ) also depends on the local cyclic processes behavior (, ).
In the general case, besides depending on the structural characteristics of SCCP the values of the considered variables   , /    , depend on constraints from the mutual exclusion protocol, that is, the set of priority dispatching rules Θ, operation times, and so forth [14,15] as well as on the way the local and multimodal processes interacting with each other.For example, in case of the two levels structure model, that is, including levels SL and SM as shown in Figure 2  (ii) for multimodal processes: the timing of commencement of operation where one has the following: following the set of local processes , (X  , -the set of moments when operation   , of multimodal process may use required local process) and ⌈⌉ the smallest integer greater than or equal to  in terms of the set : ⌈⌉  = min{ ∈  :  ≥ }.
In other words, the timing of commencement of operation  , belongs to the set X  , and being coincident with operation processes by relevant local transportation process   .
The constraints determining (due to introduced rules ( 8) and ( 9)) the timing of commencement of operations for the SCCP from Figure 2 are gathered in Tables 1 and 2.

Cyclic Steady State Space.
According to the previous section, a set of cyclic process achieved in the structure SC  (6) can be represented as a cyclic schedule   (7) that meets the constraints (8) and (9). Figure 4 presents an example of a cyclic schedule which can be achieved in the system illustrated in Figure 1.It shows that for a set of production plan (represented by multimodal processes  1 ,  2 , and  3 ) it is possible to organize the work of AGVs responsible for transportation of elements between workstations (local processes  1 -P 8 ) in such a way that no deadlocks appear in the system or there is no process waiting at the resources.An  assumption was made that the transportation of work-pieces (streams of multimodal processes) takes one unit of time and their loading to/unloading from AGVs (local processes) takes place in the first and the last unit of each operation (see Figure 4).The work [20] shows that schedules of this kind (ensuring such an organization of AGVs that guarantees the accomplishment of the set of production routes without any stoppages) can be obtained by means of solving the following constraint satisfaction problem (10): where one has the following:   ,   -decision variables,   -cyclic schedule (7),   -sequence of operation times   = (((,  The sequence of operations times   and the sequence of their beginning moments   both of them follow constraints   ,   which are the sufficient conditions for the SCCP cyclic steady state behavior, that is, the state following the problem (10) solution.In case of two level systems, that is, SL and SM (see Figure 3), the constraints   ,   can be specified in terms of constraints (8) and their extension (9) [15].
Presenting cyclic processes as   schedules is commonly used both in SCCP [11,12,14,15] and in various problems of cyclic scheduling.Instead of cyclic schedules, the so-called behavior digraphs [16], that create the state space P, can also be applied.Constraints for stream 1 2,5 = ⌈max {(   The digraph   (denoted by  index, in case of cyclic processes representation) is a digraph whose vertexes represent the admissible states   of the system and arcs represent transitions between the states (while depicting a given transition function  [16]).
In the considered case, the state of SCCP describes the present (valid in a given  period) allocation on the resources of local (AGVs) and multimodal (technological routes) processes and describes the current access rights to the resources (determined by dispatching priority rules Θ).Formally, in the state definition, three elements are distinguished that characterize processes at the particular behavior levels (see Figure 3).
In this approach, the state of SCCP takes the following form: where one has the following: (i) -the number of levels of structure SC  (6) and (ii)   -the th state of local processes executed on the SL level (5) as follows: where one has the following: where one has the following:     = (   ,1 1 , . . .,    ,ℎ  , . . .,    ,((),)

𝑙𝑤(𝑙)
)-allocation of the process streams    ℎ  ; that is, where one has the following:    ℎ  -the ℎth stream of the th multimodal process from -level of SC  (6), -the number of resources ,  In the context of the introduced notions (allocation, semaphore, indexes) the   schedule presented in Figure 4 illustrates only the allocation of processes in time.The particular allocations are denoted by a frame and symbol " ⃝ " of the state   connected with the given allocation.Therefore, the cyclic process represented by   schedule can be described with use of 42 states (S 0 -S 41 ).
The digraph   = (  ,   ) corresponding to   schedule from Figure 4 was illustrated in Figure 5.The set of vertexes   = { 0 , . . .,  41 } includes all the states that can be achieved by the system in the course of implementing processes according to   schedule.The set of arcs   ⊆   ×  defines the transitions between the states.Transition of the system from the state  0 to the state  1 (denoted by  0 →  1 , and in Figure 5 as ⃝ → ⃝ ) occurs according to the transition function , which was described in [16].
According to this approach, the digraph   = (  ,   ) depicting the behavior from Figure 4 is a cycle including 42 vertexes; see Figure 5. Apart from   states, Figure 5 illustrates also local states   (12) which only represent the behavior of local processes.In the discussed case, the number of local states is the same as the number of   states.Generally, it does not have to be so, as there are situations when one local state is an element of numerous   states [14,15].The local states   as well as the digraph related to them should be perceived as a projection of   states and   digraph onto the level of local processes.
It should be noted that in the discussed example only one cyclic steady state is reachable, that is, as a result of the solution (10); that is, one schedule   was obtained along with one digraph   corresponding to it.Generally, in the given structure SC  a lot of cyclic steady states can be reachable (which depends on the initial phases of dispatching rules) and therefore, we can consider the cyclic steady state space P as a set of digraphs that can be reachable in a given structure of   digraphs.
The cyclic steady state space can be illustrated as a graph being the sum of all reachable digraphs: P = ( P ,  P ) =  ,1 ∪  ,2 ∪ ⋅ ⋅ ⋅ ∪  , (where  , ∪  , = (  ∪   ,   ∪   )) are reachable in a given structure.There are also situations where in a given structure no cyclic steady states are reachable; then P = (0, 0).The main reason for the lack of cyclic behaviors is the possible deadlocks.
This situation can be interpreted as introducing a new production order into the system, which requires a new method of processing elements.The presence of cyclic schedule   (providing a solution to the problem (10)) and   digraph (Figure 7) related to it mean that such an order can be implemented.
The presence of two different cyclic steady states leads to the question of their mutual reachability: Is it possible to smoothly change the system behavior from  ,1 to  ,2 (Figure 7)?In other words, is it possible to smoothly (with no need to stop the system) proceed from implementing the first production order  ,1 to the other  ,2 ?

Multimodal Processes Rescheduling
3.1.Problem Formulation.The question of the possibility of smoothly changing the cyclic behaviors of the system is related to the problem of mutual reachability of the two behavior digraphs  ,1 ,  ,2 .The digraphs presented so far (Figure 7) are cycles.In general, a digraph may take the form of the so-called vortex   , that is, a digraph including a cycle representing a cyclic steady states   and a tree  , representing transient states leading to cyclic steady states.An example of two vortexes  ,1 ,  ,2 is shown in Figure 8. Formally, the digraph of the vortex type is defined as follows: where In this approach, the problem of mutual reachability of the behavior digraphs of mutual space of states is defined as follows: there is a given structure SC  and the cyclic steady state space P = ( P ,  P ) resulting from it, which includes two vortexes  ,1 ,  ,2 that represent two cyclic steady processes.It is therefore necessary to find answer to the following question: Are the digraphs  ,1 ,  ,2 (being subdigraphs of  ,1 ,  ,2 ) mutually reachable (Figure 8)?In the case they are mutually reachable the next question is how to make a transition between the digraphs?
In other words, the question of mutual reachability of cyclic behaviors can be treated as the question of the possibility of direct or indirect transition between the digraphs  ,1 ,  ,2 .An example of such transitions is shown in Figure 8.An indirect transition should be understood as changing  ,1 into  ,2 , resulting from the transition of the state   into transitory path (a path of digraph  , ) leading to  ,2 .The direct transition means a transition from the state   right into the cycle  ,1 .

Sufficient Conditions.
The possibility of transitions between cyclic steady states depends on the state   (  ), which is an element of two digraphs:  ,1 and  ,2 .In other words, it is necessary that in case of direct transition the following transitions between states occur: In SCCP, transitions of this kind are not acceptable.It results from the property saying that each state has at least  one descendant [16] (a descendant is the consecutive state after   ).The presence of state   included simultaneously into two digraphs ( ,1 ,  ,2 ) would require the existence of at least two descendants of the state   .
Although there are no mutual states at the level of the space of states P, the different projections of states upon lower levels of behavior as well as their components (space of allocation, semaphores, etc.) may be elements belonging to numerous digraphs at the same time.
An example of such a situation is shown in Figure 9, where the idea of a multilevel model of the cyclic steady states space P has been applied.The figure shows the projection of the space P upon the level of local processes behavior P SL (the second level in Figure 9) and upon the space of allocation A (the lowest level in Figure 9).The elements of the space A are the allocations   of the local states   = (  ,   ,   ) occurring in the space P SL = ( SL ,  SL ) (where   ∈  SL ), which at the same time is the projection of the space of states P.
It is worth mentioning that some allocations of the space A are simultaneously mutual for several states.For example, the allocation  2 ∈ A is at the same time an element of the state  3 = ( 2 ,  3 ,  3 ) and the state  4 = ( 2 ,  4 ,  4 ).In practice, it means that in both states the local processes (representing, e.g., means of transport such as AGVs) are allocated on the same resources.However, since there are various access rights (determined by  3 ,  3 and  4 ,  4 ) their further implantation will be different.The existence of states of common allocation can be used for the transitions between various behavior digraphs.The states  3 and  4 differ only with the sequence of semaphores and indexes.Thus, if in the state  3 with allocation  2 , the form of  3 ,  3 changes into the form of  4 ,  4 , we will attain the state  4 leading to another cycle.
The modification of the state  3 into  4 by changing the form of semaphores and indexes allows for an indirect transition between cyclic processes of the space P 0 .The direct transition is possible as a result of modifying the semaphores and indexes of the states  1 = ( 1 ,  1 ,  1 ) and  2 = ( 1 ,  2 ,  2 ) mutually sharing the allocation  1 .Therefore, the presence of states of mutual allocation implies the possibility of changing the set of cyclic steady states.Sl j Sl 1

Evaluation
-The rth local state of SCCP: Sl r = (A r , Z r , Q r ) ation a a at a a The considered transitions between  3 and  4 as well as  1 and  2 refer mainly to digraphs occurring only at the local level (i.e., space P SL ).In case of digraphs  ,1 ,  ,2 of the space P a similar procedure should be followed.These states are the projections of the corresponding states from the space P. The state  1 is the projection of the state which is an element of the digraph  ,1 , and the state  2 = ( 2 ,  1  2 ) is the projection of the digraph  ,2 .
If, among states projecting upon  1 and  2 , there are states (e.g., states  1 and  2 ) with mutual allocation  1   of the multimodal processes, the transition between  ,1 and  ,2 (and therefore also between  ,1 and  ,2 ) is possible as a result of modifications of corresponding semaphores and indexes (from the form of The states  1 and  2 are characterized by mutual allocation of both local and multimodal processes.By means of modifying semaphores and indexes in the state  1 , a direct transition from digraph  ,1 to digraph  ,2 is possible. In practice, the change of semaphores and indexes related to them means simply the change of the control rules (change of signaling) of the system, the moment the processes are allocated properly (compatible with  1 ).The solution of this kind neither stops the implementation of the process (work of AGVs, technological routes) nor creates the need for changing their allocation (shift).In case of the modification of the state  6 the transition is immediate and noninvasive; it only requires the change of control.
To sum up the above considerations, we can say that the transition between the two digraphs  ,1 and  ,2 is possible if they include states characterized by mutual allocation at every behavior level.This observation leads to the following two properties.In this approach, the problem of digraphs reachability calls for an answer to the following question: Are there two states   ∈  ,1 and   ∈  ,2 , sharing the same allocations , for  = 1 ⋅ ⋅ ⋅ , among states included into  ,1 and  ,2 ?If such states exist, the transition between the states is possible as a result of modifications of semaphores and indexes.In this context the direct transition is defined as and the indirect transition as where one has the following:   1 , . . .,   (−1) , are known), determining the states with mutual allocation is not a complex task.Due to a small number of states (in practice no more than a few hundred states per vortex), all that has to be done is to successively compare all potential variants.An algorithm corresponding to such an approach looks as Algorithm 1.
The result of Algorithm 1 is the set AC including pairs of states (  ,   ) with mutual allocations.The states can be used to determine the direct and indirect transitions between digraphs  ,1 and  ,2 .Therefore, the existence of a nonempty set AC means a positive answer to the posed question.
To make things simple, an assumption can be made that we take into consideration that direct transitions and the digraphs have the same number of states (denoted by ); thus computational complexity of Algorithm 1 is expressed by the function () =  2 .
Algorithm 1 makes it possible to evaluate the mutual reachability of only two digraphs  ,1 and  ,2 ∈ DC (set of all cyclic digraphs existing in the space P).In case of evaluating the mutual reachability of all digraphs from the set DC, it is necessary to make a search of this kind for a pair of processes.The computational complexity in such case amounts to Owing to polynomial character of the function of computational complexity, the problem of evaluating the mutual reachability of behavior digraphs is not a difficult one.All the computational effort is focused on the stage of determining cyclic digraphs (set DC), that is, solving the problem (10).In order to do so, methods described in [11,12,[14][15][16]] can be applied.
Among the available methods, however, there is a deficit of those which can help determine transient states leading to cyclic digraphs (i.e., digraphs   ).Their usability is therefore limited to evaluating direct transitions (where no information on transient states (processes) is required).
In order to determine the transient states, the method of vortex generation can be used.Vortexes determined with use of this method include cyclic processes as well as all transient processes leading to them.The example below illustrates the use of this method for evaluating mutual reachability of behavior digraphs.

Numerical Example.
Consider two AGVs supporting multiproduct manufacturing flows and their SCCP models shown in Figures 2 and 7(b).Let us assume that the two series manufacturing of three different products follow the routes distinguished by different colored (green, orange, and blue) bold lines.
Each workstation can process only unique work-piece from a unique product kind at a given instance.Successive work-pieces following the product route of a given product kind while taking into account local material handling routes are treated as subsequent streams of manufacturing processes (in a given kind of product manufacturing flow) and are modeled as streams of multimodal processes.For instance, see streams of  1 ,  2 , and  3 from Figure 2 as follows:   approach to AGVs fleet dispatching and scheduling, for example, following from the SCCP concept [1,9,25] can be considered as well.Many models and methods have been proposed to solve the cyclic scheduling problem so far.Most of these are oriented at finding a minimal cycle or maximal throughput while assuming deadlock-free processes flow.The approaches trying to estimate the cycle time from cyclic processes structure and the synchronization mechanism employed (i.e., mutual exclusion instances) are quite unique.
In that context our main contribution is to propose a new modeling framework enabling to state and answer the following questions: Does there exist a control procedure (e.g., a set of dispatching rules and an initial state) guaranteeing an assumed steady cyclic state subject to SCCP's structure constraints?Does an SCCP's structure exist such that a steady cyclic state can be achieved?
Is the cyclic steady sate space empty?In the case the space is not empty the next question is do the cyclic steady states are mutually reachable?
The last question refers to the problem of multimodal processes rescheduling.Most of so far proposed approaches to the rescheduling of processes executed within multimodal transportation networks environment [17,[26][27][28] are only limited to the deadlock-free cases.It should be noted that, however, the evolutionary algorithms driven approaches frequently used [26][27][28] are not effective in cases allowing deadlocks occurrence.The approach proposed allows finding deadlock-free transitions between the reachable cyclic steady states while avoiding the deadlocks in the course of the transient state generation.

Conclusions
The complicated problem which integrates AGVs fleet assignment, routing, and scheduling, generating a suboptimal cyclic schedule for multiproduct manufacturing, is novelty of this research.Such integrated AGVs dispatching problems can be seen as a special case of the cyclic blocking flowshop one, where the jobs might block either the machine or the AGV at the processing time.Therefore the main class of problems of AGVs fleet match-up scheduling with reference schedule of assumed production flow belong to the NP-hard ones.
Besides the above mentioned AGVs dispatching/planning issues the research objective regards a quite large class of digital and/or logistics networks that share common properties even though they have huge intrinsic differences.In other words, besides AGVs that can be treated as a kind of internal transport, other emerging trends concerning the logistics (e.g., supply chains management) and city traffic  (e.g., infrastructure of the public transport) issues can be modeled.For instance, the relevant multimodal processes of passengers traveling between destination points in an environment of local processes encompassing the subway lines network might be considered as well.In other words, the passenger's itinerary including different metro lines can be considered as a plan of a multimodal process routing within a metro network.The proposed declarative approach aimed at AGVs fleet scheduling stated in terms of constraint satisfaction problem representation provides a unified method for performance evaluation of local as well as supported by them multimodal processes.
In general case, however, there is not any guarantee that the considered SCCP model of AGVS has a nonempty cyclic steady states space as well as that in case this space is not empty, any direct or indirect mutual reachability can be observed.That means that the decidability of the problem of the space P reachability and mutual reachability among its cyclic steady states plays a pivotal role in multimodal processes rescheduling.

Figure 2 :
Figure 2: Example of FMS-SCCP model of an AGVS.

Figure 6 .
Figure 6.It is a cyclic schedule   reachable in SC structure (Figure 2) in which the manner of implementing production routes was changed (routes   2

- 1 -Local process P 2 - 3 -Local process P 4 - 5 -Local process P 6 - 7 --
Execution of process's P j i operation Load time = 1 u.t.Unload time = 1 u.t.-Local process P Local process P Local process P Local process P Local process P 8 S 42 S a S b S c -Transition: S a → S b → S c

Figure 6 :
Figure 6: AGVs fleet cyclic schedule matching the multiproduct manufacturing for system with changed multimodal routes.

Figure 7 : 1 Figure 8 :
Figure 7: The state space for structures with different multimodal processes routes.

- 1 -
Transition Sl a → Sl b -Cyclic steady state projection of G W,1 onto the space  SL -Cyclic steady state represented by cyclic digraph G W,The rth state of SCCP S r = (Sl r , m 1

Figure 9 :
Figure 9: Illustration of cyclic steady state space projections, P SL and A, for behavior digraphs from Figure 8.

Figure 10 :
Figure 10: The indirect transition between schedules from Figures 4 and 6.

Figure 11 :
Figure 11: The indirect transition between schedules from Figures 6 and 4.

Figure 13 :
Figure 13: Illustration of SCCP composed of four local cyclic processes (a) and cyclic steady state space P including transient states depicting mutual reachability  ,1 →  ,2 (b).
11,2 , ..., (),() ), and by analogy, the sequence    = (   ) consists of the timing of commencement of multimodal processes operations (from the th level, where one has the following: ()-the number of multimodal processes at the th level, (, )-the number of streams of the th process at the th level, and (, )-the number of operation of the th process at the th level).Variables   , / , ∈ Z describe the timing of commencement of operations in the th cycle of the SCCP cyclic steady state behavior:   , () =   , +  ⋅ /    , () =     , +  ⋅   .

Table 1 :
Constraints determining the local processes behavior of the SCCP from Figure2.
:     , ,   , ∈ N, {  ,   }-the set of constraints   and   describing SCCP behavior,   -constraints determining cyclic steady state of local processes, that is, their cyclic schedule, and  constraints determining multimodal processes behavior.

Table 2 :
Constraints (9)determining the multimodal processes behavior of the SCCP from Figure2.

Table 3 :
Times (units) of operations executions of local and multimodal processes from Figure2.