Complexity

Volume 2018 (2018), Article ID 5341346, 14 pages

https://doi.org/10.1155/2018/5341346

## Equivalent and Efficient Optimization Models for an Industrial Discrete Event System with Alternative Structural Configurations

Correspondence should be addressed to Emilio Jiménez-Macías

Received 21 August 2017; Revised 5 December 2017; Accepted 10 January 2018; Published 4 March 2018

Academic Editor: Miguel Romance

Copyright © 2018 Juan-Ignacio Latorre-Biel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Discrete event systems in applications, such as industry and supply chain, may show a very complex behavior. For this reason, their design and operation may be carried out by the application of optimization techniques for decision making in order to obtain their highest performance. In a general approach, it is possible to implement these optimization techniques by means of the simulation of a Petri net model, which may require an intensive use of computational resources. One key factor in the computational cost of simulation-based optimization is the size of the model of the system; hence, it may be useful to apply techniques to reduce it. This paper analyzes the relationship between two Petri net formalisms, currently used in the design of discrete event systems, where it is usual to count on a set of alternative structural configurations. These formalisms are a particular type of parametric Petri nets, called compound Petri nets, and a set of alternative Petri nets. The development of equivalent models under these formalisms and the formal proof of this equivalence are the main topics of the paper. The basis for this formal approach is the graph of reachable markings, a powerful tool able to represent the behavior of a discrete event system and, hence, to show the equivalence between two different Petri net models. One immediate application of this equivalence is the substitution of a large model of a system by a more compact one, whose simulation may be less demanding in the use of computational resources.

#### 1. Introduction

A significant number of systems of technological, social, or financial interest may present a high degree of complexity in their composition, in the interrelation among their components, in their behavior, or in all of these features [1, 2]. Many of these systems can be considered as discrete event systems (DES) [3, 4], such as manufacturing facilities, food industry, supply chains, airports, or traffic networks [5–8].

The design of such systems can be a challenging task, involving experts from a variety of knowledge fields [9]. In this context, an effective communication of the partial and final results of the design process may be achieved, when using a formal language to represent a model of the system of interest. Additionally, the formal verification of some specifications of the designed system, such as checking the compliance of certain structural properties or assessing the performance of the system, can be carried out before the system itself has been built up [10, 11]. One effective strategy for achieving this purpose with a certain degree of accuracy consists of using a formal model of the system [10, 12].

A very popular paradigm to represent the model of a DES is Petri nets [13]. This formalism has been applied to a wide range of different fields [14, 15]. The Petri nets present many favourable features, such as an important body of knowledge referred to as subclasses of formalisms, structural analysis techniques and properties, or methodologies for implementing performance evaluation and simulation [16]. Petri nets are the paradigm chosen in this research to represent models of DES.

The research presented in this paper deals with Petri net models applied to the design process of DES. In this design process, the DES modeled by the Petri net has not been completely specified yet; hence, the Petri net should model the lack of concretion in some of the features of the original DES. These nonspecified features of the DES in process of being designed constitute a set of degrees of freedom, which should be solved by decision making as long as the design process is being completed. The mentioned decision-making process can be carried out by means of different strategies, such as “what-if” analysis or the statement of an optimization problem. The formalisms presented in this paper, belonging to the paradigm of the Petri nets, are particularly suited for developing optimization processes. Nevertheless, they are also appropriate for other strategies of decision making [17].

Usually, in the design process of a DES, some of the degrees of freedom that should be solved by the designers refer to structural features of the system [18], while some others can be related to the system’s behavior. For example, the layout of the components of the system is a feature related to its structure. On the contrary, the dynamics or evolution of some of these components, such as raw materials supply, human resources, or communication packets, is associated with its behavior [19]. In a Petri net model, the structural features of a DES are explicitly represented in the so-called incidence matrices. These structural features can be clearly distinguished from the behavior of the system, which is described by successive markings of the net. An introduction of Petri nets can be found in [16, 20].

A large number of scientific reports on the application of decision-making methodologies for solving certain stages of the design process of a DES can be found in the literature. However, most of them refer to the management or control of the operation of such systems; hence, the methodologies they present are aimed at solving behavioral degrees of freedom of the DES.

The references presenting Petri net models of discrete event systems with structural degrees of freedom usually fall under one of the two following approaches:(a)Each alternative configuration for the structural degrees of freedom, in brief alternative structural configuration, is represented by a different Petri net model [7, 8, 21, 22]. These Petri nets can be called alternative Petri nets.(b)The incidence matrix representing the structure of the Petri net model contains a number of parameters. Giving diverse feasible values to these parameters, it is possible to specify different alternative structural configurations of the DES. These Petri nets can be called parametric, parameterized, parameterized, or compound Petri nets [6].

Many of these works, dealing with structural degrees of freedom, lack formal and systematic approach to the definition and implementation of the structural degrees of freedom. In fact, they mainly focus on the decision-making process to select one alternative structural configuration of the DES [6, 8, 21, 22].

Some previous works have advanced an equivalence relation between both formalisms, a set of alternative Petri nets and a compound Petri net, without providing a rigorous proof.

The main contribution of this paper consists of formally proving that both formalisms, a set of alternative Petri net and a compound Petri net, can be equivalent and describing the conditions for achieving this equivalence. This result leads to some consequences of interest, since from this result it is possible to do the following:(a)Apply appropriate algorithms to transform a compound Petri net into a set of alternative Petri nets, detailed in this paper, and vice versa.(b)Use any of both formalisms for modeling in different stages of the design process of a DES, profiting from the advantages of each one of them. For example, depending on the particular case, the development of the original Petri net model of a discrete event system with structural degrees of freedom may be easier with one of the two formalisms.(c)Reduce the amount of data required to describe a Petri net model with an associated set of feasible alternative structural configurations by its transformation into a compact compound Petri net. It has to be considered that there are virtually infinite compound Petri nets, of different sizes, that are equivalent to a given set of alternative Petri nets. Moreover, in the design of a discrete event system, the alternative structural configurations may share a large amount of data. If this redundant data is removed, significant reductions in the size of the simulation model of the system might be achieved.(d)Accelerate the simulation-based optimization of a Petri net model with an associated set of feasible alternative structural configurations by using an appropriate implementation of distributed computing profiting from the advantages of using compact models.

The rest of the paper is organized as follows. Section 2 is devoted to formally define a set of alternative Petri nets and a compound Petri net. Section 3 focuses on proving that a certain transformation on a compound Petri net, based on the concept of partition of a set, leads to a set of alternative Petri nets. Section 4 presents some definitions related to the graphs of reachability and the marking of a Petri net. Section 5 describes a methodology to construct the reachability graph of a set of alternative Petri nets and of a compound Petri net. Section 6 uses the concept of reachability graph to prove the equivalence between a compound Petri net and the set of alternative Petri nets that results from the application of the transformation algorithm presented in Section 3. Section 7 illustrates the concepts, definitions, proposition, and theorems from previous sections by means of an example of application. Last section details the conclusions and future research lines.

#### 2. Alternative Petri Nets and Compound Petri Net

One possible definition of a Petri net system is based on a weighted flow relation through two incidence functions [16, 20]. A Petri net system is also called marked Petri net or just Petri net for simplicity.

*Definition 1 (Petri net). *A (generalized)* Petri net* is a five-tuple:where and are disjoint, finite, nonempty sets of places and transitions, respectively. Pre: is the preincidence or input function. Post: is the postincidence or output function. is a marking of the set of places , where , whose th component is the marking of place .

The first four elements of the Petri net define its structure of a Petri net, which is a static feature, while the fifth one, the marking, represents the behavior of the system, that is, the system state and its changes.

It is possible to describe the structure of a Petri net by using the incidence matrices and . These matrices represent the incidence functions given in Definition 1 and can be called pre- and postincidence matrices, respectively.

A pair of place and transition is called a self-loop if is both an input and output place of . A Petri net is said to be pure if it has not any self-loop. Moreover, pure nets are completely characterized by a single incidence matrix .

As it has been stated in the Introduction, a Petri net associated with a set of alternative structural configurations can be presented as a set of alternative Petri nets. Every Petri net of the mentioned set consists of the complete Petri net model of the original DES, particularized with one of the alternative structural configurations.

Given a set of alternative Petri nets, any pair of these Petri nets verifies a property called mutually exclusive evolution, meaning that only one of the alternative Petri nets can be active at a given time, since all of them are exclusive models of the same DES. This property allows characterizing a set of alternative Petri nets as shown in the following.

*Definition 2 (mutually exclusive evolution). *Given two Petri nets and , they are said to have mutually exclusive evolutions if the following is verified:

(i) if ,

(ii) if .

Once this property has been stated, it is possible to carry out the definition of a couple of Petri nets and a set of alternative Petri nets, as it is stated below.

*Definition 3 (pair of alternative Petri nets). *Given two Petri nets and , they are said to be alternative Petri nets if it is verified that

(i)* R* and have mutually exclusive evolution;

(ii) , where are the incidence matrices of and , respectively.

*Definition 4 (set of alternative Petri nets). *Given a set of Petri nets , is said to be a set of alternative Petri nets if it verifies that

(i) ;

(ii) , such that and , ; then and are a pair of alternative Petri nets.

is called the th alternative Petri net of .

The second formalism, belonging to the paradigm of the Petri nets, which will be considered in this research document, is the compound Petri nets. It is a particular case of parametric Petri net, containing parameters in any of the two incidence matrices. It is associated with a set containing all the feasible combinations of values that can be assigned to these parameters.

*Definition 5 (parameter of a Petri net). *Any variable in a Petri net model, associated with a set of feasible values, its cardinality is greater than 1. A value can be assigned to this variable as a consequence of a decision. The outcome of this decision is a choice from a set of feasible values. Once a decision has been made, the value assigned to the parameter is unique.

*Definition 6 (structural parameter of a Petri net). *Any parameter of an incidence matrix of a Petri net.

*Definition 7 (compound Petri net). *A compound Petri net is a 7-tuple , where

(i) is the set of parameters of ;

(ii) is the feasible combination of values for the parameters, meaning that not all the combinations of values for the parameters of the Petri net lead to a valid structural configuration;

(iii) additionally, , set of structural parameters of , such that , meaning that a compound Petri net should contain at least one structural parameter among all its parameters.

In the previous definitions, two different Petri-net-based formalisms able to represent a discrete event system with alternative structural configurations have been formally presented. Two numerical examples of these two formalisms can be found in Figures 1 and 2. In the following section, an algorithm will be developed to transform a compound Petri net into a set of alternative Petri nets.