2011 UKSim 5th European Symposium on Computer Modeling and Simulation
Simulation-Based Optimization for the Design of Discrete Event Systems Modeled by Parametric Petri Nets
Juan Ignacio Latorre Biel
Emilio Jiménez Macías
Mercedes Pérez de la Parte
Dept. of Mechanical Engineering, Energetics and Materials. Public University of Navarre 31500 Tudela, Spain
[email protected]
Department of Electrical Engineering. University of La Rioja 26004 Logroño, Spain
[email protected]
Department of Mechanical Engineering. University of La Rioja 26004 Logroño, Spain
[email protected]
synchronization and resource sharing. Moreover, they can be used to represent a DES in every stage of its life cycle [2]. There is a very active field of research that focuses on methodologies that solve this kind of problems by means of three steps [3], [4]: • Search in the solution space of the most promising solutions. • Simulation of the behavior of the chosen solutions in order to measure their performance. • Choice of the best solution available regarding the performance measurements. For the second step there exist a large number of tools for modeling and simulation of discrete event systems. On the other hand, the evaluation of the performance of a discrete event system requires the introduction of time in the Petri net formalism to measure the duration of the different activities performed by the system. The first step, which consists of searching in the solution space, may present important difficulties to be afforded by limited computer resources, since its size is usually extremely large for real systems due to the combinatorial explosion. This situation arises as a consequence of the existence in a system of a set of freedom degrees that can take different values leading to diverse configurations for the system. The number of feasible combinations of the values for these parameters usually is not a linear function of the number of freedom degrees. On the contrary, they increase very rapidly, especially when conflicts, a kind of freedom degrees, appear during the evolution of the system. As a consequence of the previous considerations, the search process in the solution space cannot be usually afforded by means of an exhaustive approach. Some kind of guided search should be considered to solve the decision problem using a reasonable amount of computer resources. What is reasonable in this context depends on the application field of the decision-making process, in particular in how much time should the decision be made and what the computer-based system that will support the decision process is. For example, in real-time applications the time constraints are important; hence the development of efficient methodologies for the decision making is an active field of research.
Abstract—Many technological, industrial or economical systems are described by discrete event system (DES) models. The decision making processes that arise in the design and operation of this kind of systems can be afforded by means of algorithmic methodologies. A large range of approaches based on the simulation of the behavior of the system have been reported to answer this problem. Their main advantage consists of being applicable to most of the systems. An important drawback is the significant computational resources required to perform an exhaustive exploration of the state space due to the combinatorial explosion. A manual choice of a reduced set of configurations to be simulated can be improved by the use of parametric Petri nets and a metaheuristic search of the most promising ones. In this paper, a review of some definitions of parametric Petri net found in the literature is presented, as well as a definition for the general framework of stating optimization problems of both, the operation and the design of the model of the DES. Moreover, a methodology to obtain such a parametric Petri net, called compound Petri net, from an easier-to-obtain set of alternative Petri nets is proposed and an application example is given. Keywords-parametric Petri net; parameterized Petri net; compound Petri net; alternative Petri nets; optimization; modeling & simulation; decision making.
I.
INTRODUCTION
Many technological systems where computer-based controllers are used to constrain their behavior can be modeled as discrete event systems. Moreover, a significant number of manufacturing systems and workflow systems, as well as supply chains and communication networks can also be classified in this category. The design and operation of these systems require affording some decision making processes. The decisions can be translated into several types of formal problems, such as constraint satisfaction problems if a feasible solution is wanted or optimization problems, when an optimal or quasi-optimal solution is required [1]. An important phase in the formalization of the decision problem consists of the modeling of the discrete event system. There are several formalisms and families of formalisms developed for the modeling of DES. In particular, Petri nets (PNs) are especially suited for modeling systems whose behavior may show parallelism, concurrency, 978-0-7695-4619-3/11 $26.00 © 2011 IEEE DOI 10.1109/EMS.2011.63
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set of simulations to a set of simulation-based optimization problems where the unknowns are non-structural parameters. There are so many different problems as structural configurations are planned to be tested. Up the stream, a completely automated search process is associated to a single problem, which due to the properties of the sets of exclusive entities, shows a complexity that does not increase exponentially but linearly with the number of the optimization problems that are merged into a single one [9]. The advantages of this fully automated search process are not only the ones described for the incomplete automated search but also the possibility of developing a compact model that does not include the redundant information associated to the shared subnets that may exist in a set of alternative Petri nets. On the other hand, the search of a single solution space allow to focus the search in the most promising areas of a complete solution space and not wasting computer resources in the solution spaces associated to some alternative Petri nets [8]. The rest of this paper has the following structure. The section II reviews some definitions of parametric Petri nets that can be found in the scientific literature. The section III will present the optimization problem resulting from an incomplete automation of the search in the solution space. The fourth section defines the compound Petri nets and the subsequent optimization problem with fully automated search in the solution space. The section V defines the alternative Petri nets, whereas the following section discusses the transformation algorithms from this formalism to a compound Petri net. This process leads from a set of optimization problems to a single one by the automation of the search in the solution space. In this same section, an example is briefly described to illustrate the theoretical concepts. The seventh section presents some conclusions and future research lines, while the final section is devoted to the references.
A manual choice of the configurations of the DES to be simulated is a classical methodology that has been described in the literature, such as in [4]. This approach has the advantage of not needing to program a search algorithm to choose the most promising solutions to test and the possibility of using the intuition of an experienced human decision-maker. However, the drawbacks are important, such as the difficulty of implementing a systematic search of the most promising configurations and the task of identifying them, as well as the possibility of developing a process that is non-dependent on the experience, skills or luck of a human decision-maker. In fact, the manual choice of promising configurations may lead to a too small set of tested solutions, due to a limited computation capacity, where the probability of finding a good solution may be low. In order to overcome these limitations, a significant research effort has been devoted during the last decades to the search algorithms applied to the solution space. This approach leads to an automatic search guided by some criteria, sometimes inspired in successful natural processes, such as the simulated annealing, genetic algorithms, ant colony optimization, etc. See, for example [3], [5]. However, the research developed so far has focused almost entirely in the automation of the search process applied to certain freedom degrees of the discrete event system, in particular to these that are related to the behavior of the system but not to its structure. An example of this approach can be found in [3]. The different structural configurations of a DES in process of being designed or controlled are usually chosen manually as in [3], [4], [6] or [7]. It is possible to interpret this way of stating an optimization problem as an incomplete automation of the search process, since the part of the solution that refers to structural freedom degrees is still chosen manually. This approach of considering a set of different structural configurations to be simulated can lead naturally to an intuitive model of the discrete event system given by a set of alternative Petri nets [8]. Every one of the alternative Petri nets that contains a set of non-structural freedom degrees are usually described by means of parametric Petri nets [3]. In these models the parameters represent the unknowns in the behavior of the DES that can take diverse values as a consequence of different decisions in the design or the operation of the DES. The incomplete automation of the search process overcomes some of the drawbacks of the manual choice of all the configurations to be tested, whereas keeps some of its own, associated to the structural freedom degrees whose optimal values are searched manually. The complete automation of the search in the solution space, including the structural freedom degrees, leads to the need of the definition of an appropriate formalism. One of these formalisms will be described in this paper; it is called compound Petri net. The automation of the search in the solution space can be understood as a movement against the stream produced by the paradigm of “divide and conquer” since it leads from a
II.
PARAMETRIC PETRI NETS
The parametric, parameterised or parameterized Petri nets are the names of formalisms based on the Petri net paradigm that contain parameters and have been developed by diverse authors to be used in different fields of the theory and applications of discrete event systems. Some of them are discussed in this section. [10] introduces the concept of model parameterization, in particular parameterization of P/T nets with respect to the initial marking is formally defined and studied. [11] defines the parameterized Petri net (PPN) for modelling DES whose structure can dynamically change. The definition of PPN considers the parameterization of transitions and places. [12] defines a Petri net model that consists of a parametric Petri nets whose parameters can be found in the initial marking. Predicates allow the specification of restrictions over the set of admissible parameter values, and/or the relations among the model parameters. [13] proposes a parameterization of Coloured Petri Nets as a component of a flexible and modular modelling
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single feasible solution or configuration. As a consequence this particular case is reduced to the simulation of the single existing configuration and the evaluation of the performance measurements. Each optimization problem can be stated in the following way. Definition 1. Optimization problem based on an alternative Petri net. Maximize f(x) subject to: i) Ri , alternative Petri net. ii) Sc = { g1(x) = 0, …, g i1 (x) = 0, g i 2 (x) < 0, … g i3 (x) < 0},
methodology. Three classes of parameters are identified: value, type and net structure parameters, inspired in the same classification existing for the module feature of standard ML, basis of the CPN ML language that describes coloured Petri nets. Net structure parameters are considered transitions that can be substituted by a module, with places used as module interface. The reference does not consider places and arcs as possible net structure parameters; however they are included in the open research fields. [14] defines the “parameterized coloured Petri nets”, PCPN, by the consideration of three kinds of parameters: type, expression and net parameters. This formalism is based in the definition given in [13] [15] provides with a definition of parametric Petri net, where the parameters are present in the weight of the arcs. [16] describes a modular, automated and systematic approach to complex systems modelling. This task is afforded by the integration of PN submodels described as parameterized building blocks for fast modelling of FMSs and re-design. The parameterized submodels can be reused since they represent common devices in FMS such as machines and buffers. [17] presents an extension of the high level Petri nets with parametric and dynamic net structures. The parameterization of the high level Petri net allows reusability of components. Components are defined as subnets with inputs and outputs and can contain parameters. In [3] a parametric Petri net is presented as model of a system in process of being designed. An optimization process led by the methodology of adaptive simulated annealing is applied to find the optimal values to the parameters of the Petri net. The parametric Petri net does not include any unknown in the structure of the model. [18] describes the abstract notion of Petri net types given by parameterized net classes. In order to allow a uniform approach to different kinds of Petri net classes, the concept of parameterized net classes was introduced. Two parameters are identified for describing the characteristics of a Petri net variant: the net structure and markings on the one side and the data type specifying the internal structure of tokens. The instantiation of these parameters leads to different Petri net types. [19] gives a definition of the PTPN (parametric time Petri nets), where they are described as model of Time Petri Nets (TPNs) extended with time parameters and its use to perform on-line diagnosis of distributed systems. A description of a parametric coloured Petri net is given in [20] and applied to the modeling of switched networks. III.
set of additional constraints. iii) xi ∈ Di, ∀ 1 ≤ i ≤ n , x ∈ D. Parameters of the Petri net, which define the solution of the problem. n
where f: D → R and a generic feasible solution can be represented by x = (x1, x2, … , xn) ∀i∈
N * such that 1 ≤ i ≤ n.
□ The search process consists of finding a feasible combination of values for the set of non-structural parameters of the alternative Petri net. The evaluation of the objective function will provide with the performance measurement that characterizes the quality of the solution, that is to say its proximity to the optimum. IV.
COMPOUND PETRI NET
The definition of the appropriate Petri net model to represent all the freedom degrees of the original discrete event system allows the complete automation of the search process of promising solutions in the solution space. There are several formalisms that have been developed for this purpose [8]. Every one of them presents its advantages and drawbacks. In this paper it will be described the compound Petri net, a parametric Petri net able to cope with a significant range of parameters of different types. The concept behind a compound Petri net consists in the resulting model of the translation into the formal language of the Petri nets of what is called an undefined discrete event system, characterized for having some freedom degrees in its structure that allow making decisions. The freedom degrees of the undefined discrete event system, also called undefined characteristics, can be modeled by means of undefined parameters. An undefined parameter can be defined as follows: Definition 2. Undefined parameter. Any numerical variable of a Petri net model or its evolution that has not a known value but it has to be assigned as a consequence of a decision from a set of at least two different feasible values. The value assigned to the undefined parameter must be unique. □ The parameters of a generalized Petri net can be classified regarding their role in the model itself. For example it can be defined the structural, marking, transitionfiring or interpretation parameters as it can be seen in [8].
OPTIMIZATION WITH INCOMPLETE AUTOMATION OF THE SEARCH IN THE SOLUTION SPACE
The problem of choosing the best configuration for a discrete event system according to its freedom degrees is stated as a collection of independent optimization problems. Every optimization problem is associated to a different alternative Petri net. If the alternative Petri net associated to a given problem does not include any parameter, this particular optimization problem does not require any search process, since there is a
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structural configuration of the discrete event system. This is the topic of the next section.
An important property of the set of feasible values of the undefined Parameters of a Petri net is the exclusiveness, which means that only one of them can be chosen as the result of a decision; hence discarding the others. In fact, only the exclusiveness of the sets of feasible values for the undefined structural parameters of a Petri net are interesting in the definition of a compound Petri net, since they introduce the possibility of modeling different structural configurations for a DES. In particular it can be stated that. Definition 3. Exclusiveness of the feasible combination of values for the undefined structural parameters. Let Sstrα = { α1,…, αn } be the set if undefined structural parameters of a Petri net.
V.
n
Let Svalstrα = { cvk = (v(1,k),…, v(n,k)) ∈ R | v(1,k) is a feasible value for αi ∀ i, k ∈ N such that 1 ≤ i ≤ n and 1 ≤ k ≤ m } be the set of feasible values for the undefined structural parameters of the Petri net. Notice that card(Svalstrα) =m. If given any cvk ∈ Svalstrα the assignment of (α1,…, αn) = cvk is the result of a decision and (α1,…, αn) ≠ cvj ∀ cvj ∈ Svalstrα such that cvk ≠ cvj ⇒ cvk and cvj are said to be exclusive in Svalstrα. □ A compound Petri net can be defined as a generalized Petri net where all the numerical values of the model can be known or belong to sets of feasible values where one has to be chosen. As a consequence, the following definition presents a compound Petri net as a collection of parameters that are ordered according to their type and role in the model. Definition 4. Compound Petri net. A (generalized) compound Petri net is a triple Rc = 〈np, Sγ, Svalγ〉 where *
np ∈ N is the number of places. Sγ = { γ1, γ2, … , γn } is a set of parameters. Svalγ = { cv1, cv2, … , cvm } is the set of feasible combinations of values for the parameters of Sγ. The elements of Svalstrα are exclusive.
ALTERNATIVE PETRI NETS
A set of alternative Petri nets can be used to represent the same discrete event system than an equivalent compound Petri net. As a consequence, the exclusiveness associated to the feasible combination of values for the undefined structural parameters of the latter should be also included in the former. The exclusiveness of the alternative Petri nets is presented in terms of mutually exclusive evolution. Definition 5. Exclusiveness of the alternative Petri nets. Let SR = { R1,…, Rm } be a set of alternative Petri nets. If given any Rk ∈ SR the choice of Rk as a configuration for the original DES is the result of a decision and m(Rj) = m0(Rj) ∀ Rj ∈ SR when m0(Rk) ≠ m0(Rj) ⇒ Rk and Rj are said to have exclusive evolutions in SR. □ Given a discrete event system with an undefined structure, it is possible to model every one of the structural configuration by means of an alternative Petri net. Some or all the alternative Petri nets may be parametric ones, having undefined non-structural parameters. Once the DES is represented by a set of alternative Petri net, an associated decision problem can be solved by means of an optimization problem with incomplete automation of the search process in the solution space. However, another possibility arises when a transformation algorithm is applied to the set of alternative Petri nets and an equivalent compound Petri net is obtained. In this case it is possible to state an optimization problem with complete automation of the search process or, in other words, it is possible to obtain a solution for the problem in a single optimization problem. VI.
TRANSFORMATION ALGORITHMS
The transformation of a set of alternative Petri nets into a compound Petri net requires the development of two main stages. On the first hand, it is necessary to adapt the size of the incidence matrices of all the alternative Petri nets, as well as the number of non-structural parameters associated to the models. This process is performed by adding to the Petri net models places and transitions that are isolated to the rest of the net, since their input and output arcs have weight zero. This operation does not alter the behavior of the system [21]. As a consequence it will be verified that: Sγ( Rim ) = Sγ( R mj ) ∀ Rim , R mj ∈ S Rm , where
□ *
Notice: it is verified that n = (k+1) · np, where k ∈ N (transitions) and ∀ cvi ∈ Svalγ, cvi = (v1, v2, … , vn); hence it is possible to construct the matrix-based and the graphical representation of the Petri net by means of this description. The concept of compound Petri net allows the modeling of a discrete event system with structural freedom degrees in a single model, since the unknowns in the structure can be represented by means of undefined structural parameters. It is true that the immediate construction of a compound Petri net as the model of a DES with undefined structure is not always an easy task. An exception may be a set of similar machines produced by the same manufacturer with a small number of differences that can be modeled by different values of the weight of some arcs. In fact, the natural way of modeling such systems, which can also be found in the literature, is the use of a set of alternative Petri nets, one alternative PN for every different
S Rm = { R1m ,…, Rmm } is the resulting set of the so called matching alternative Petri nets. The second step consists in transforming a set of matching alternative Petri net into a compound Petri net by means of the merging of the sets of feasible combination of values for the parameters of every alternative Petri net.
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R1
A. Algorithm 1. Obtaining a set of matching alternative Petri nets. Step 1. Define the size of the incidence matrices of the matching simple alternative Petri nets. ∀ Ri ∈ SR = { R1,…, Rm } perform the following steps: Step 2. Obtain W( Rim ) by the addition of columns and rows of the incidence matrix W(Ri). Step 3. Obtain Sβ(Ri> Rim ), set of defined parameters
R2
p1 α 5
p1
t1
t1
2
p1 t1
p2
p2 p3
t2
included in Rim as a consequence of the addition of rows and/or columns of zeros to W( Rim ).
R3 2
t3
2
p2 p3
t2
t2
Figure 1. Set of alternative Petri nets.
Step 4. Obtain Sγ( Rim ), set of all the parameters of Rim . Sγ( Rim ) = Sα( Rim ) ∪ Sβ( Rim ) = Sβ(Ri) ∪ Sβ(Ri> Rim ) ∪ Sα(Ri)
R2
R1
□ Notice that the subindices α, β and γ refer respectively to undefined parameters, defined ones and both types of them. On the other hand, it is convenient to mention that some authors such as [22] exclude explicitly the nets with isolated places or transitions from the definition Petri nets. For the applicability of this algorithm, a less constrained definition of Petri net should be considered, such as the ones given in [12] or [23].
p1 α 5 t1 p2 t2
B. Algorithm 2. Obtaining a compound Petir net from a set of matching alternative Petri nets.
R3
p1
t3
t1
2
p1
t3
t1
p2 p3
p3
2
t3
2
p2 p3 t2
t2
Figure 2. Set of matching alternative Petri nets.
Step 1. Obtain Svalγ( Ric ), set of feasible combinations of values for the parameters of the resulting compound PN. Svalγ( R ) = c i
m
∪S i =1
c
valγ
R1c
m i
(R )
p1 α19
c
Step 2. Obtain Sγ( R1 ) and Sα( R1 ) from Svalγ( Ric ). c
t1
c
Step 3. Obtain Svalα( R1 ) form Svalγ( Ric ) and Sα( R1 ).
α4
□ This last algorithm performs a merging of all the sets of feasible combinations of values for the parameters of every alternative Petri net.
p2
t3 α7
t2
C. Example In order to illustrate the previous transformation algorithm an example is given. The graphical representation of a set of non-matching alternative Petri nets is shown in the Fig. 1. The largest number of rows and columns corresponds to R3, whose incidence matrix is 3 x 3. This will be the size of the incidence matrices of the matching alternative Petri nets that will result from the application of the Algorithm 1. It will be added an isolated transition to R1 and R2, as well as an isolated place to R1 as application of the step 2 of this Algorithm 1. The resulting matching alternative PNs can be seen in the Fig. 2. The application of the Algorithm 2 leads to the compound Petri net whose graphical representation is given in the Fig. 3. The set of undefined parameters and the set of feasible values for the undefined parameters of the compound Petri net are represented under Fig. 3.
α12 α9
p3α21
α17
Figure 3. Resulting compound Petri net. c
Sα( R1 ) = { α4 , α7 , α9 , α12 , α17 , α19, α21} c
Svalstrα( R1 ) = { (2,0,0,0,0,1,0,0), (2,0,0,0,0,2,0,0), (1,1,0,0,1,2,0,1), (2,0,2,1,1,2,0,0)} VII. CONCLUSIONS AND FUTURE RESEARCH Some decision problems associated to a DES can be solved by means of searching the best configuration of the system in the solution space. Significant effort research has been invested in manual or semi-manual search. This second approach, that performs a manual search of the structural configurations, is usually applied by means of parametric Petri nets, whose parameters are associated to the freedom
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degrees of the system and constitute the unknowns that define the solutions of the problem. In this paper a fully automated search process is proposed and for its application an appropriate parametric Petri net, called compound Petri net, is presented. The compound Petri net provides with a broad framework for modeling DES in optimization problems that can be associated to design and operational decisions. Due to the fact that it is not always easy to obtain directly a compound Petri net model from a discrete event system with structural freedom degrees, a transformation algorithm from a set of models of the different structural configurations is given. The mentioned set of models is called set of alternative Petri nets. Both models, the compound Petri net and the set of alternative Petri nets, are equivalent and share the exclusiveness of some components that formalize the decision process associated to the specification of a realizable system from the original DES with freedom degrees. This approach in the use of parametric Petri nets is very promising since the application by the authors of another formalism also suitable for the modeling of undefined DES has given notable results when comparing the required computer resources with the classical methodology based on a set of alternative Petri nets. As future research in the context of the compound Petri nets it is planned to apply this methodology to a number of practical applications in different sectors such as manufacturing facilities, supply chain, computer-based systems, etc. It is expected that these applications will provide a deeper insight in the underlying theory and a better understanding of it applicability.
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