Keywords

1 Introduction

Currently, the competitiveness of industrial enterprises is closely related to the level of its digitalization and automation [1]. Digitalization affects the entire process from the conversion of measurements and documents into digital form to the creation of new business processes that are completely based on the digital model of units, the whole enterprise and software that ensures its relevance.

The process of transitioning an enterprise to digital production should be gradual and based on a long-term strategy. The optimization strategy for both the manufacturing processes themselves and their management is the key point on which the final result of the enterprise transformation depends. The development of such a strategy is the first step in the digitalization of production. It requires a comprehensive analysis of current processes, knowledge in the field of modern digital technologies, production automation and production management technologies, as well as taking into account the current position of the enterprise in the market, its capabilities, forecasts for the development of the market and the economy in general and for the enterprise in particular. Obviously, the strategy of development process itself also requires digitalization and automation.

Simulation requires the collection of a significant amount of heterogeneous information about the ongoing production process, its analysis and presentation in the required form, which can be done using methods of Process Mining [2] and Data Mining [3]. Logs of the functioning of information systems or data analysis from enterprise process control systems can be used depending on the possibilities to examine the production process. However, in this article we will touch upon only the issues of process simulation itself. The methods of simulation and analysis of processes using the mathematical apparatus of extended Petri nets with semantic relations are proposed.

2 Background

A lot of papers have been devoted to the search for approaches to modernizing enterprises, but most of these works simply represent theoretical assumptions without taking into account the specifics of specified entities. Among them, it should be noted paper [4], where solving problems of production modernization were systematized, work [5], which presents and formalizes the main ways of digitalization of production, and article [6], which shows the modernization methods defined by the survey of Russian enterprises, and which still has not lost its relevance. The work [7] is a point of special interest, where its authors propose models for presenting knowledge about the structure, parameters and functioning of the system for monitoring, diagnostics and modernization of technical systems at various stages of the life cycle, which makes it possible to analyze the current state of the technical system, predict subsequent states, and conduct fault diagnosis and offer options for modernization taking into account the goals of the enterprise. However, taking into account the limits of the enterprise when issuing recommendations for modernization is not provided. It is necessary to note the foreign works, e.g. [8] devoted to the problem of decision-making in the manufacturing process, works [9,10,11], in which the analysis and the effective implementation of modern information technologies in the production process is presented. Unfortunately, not all the ideas of foreign researchers can be applied in the Russian economy. Almost all researchers on modernization methods agree on the need to simulate processes to obtain information about its strengths and weaknesses.

The most detailed systematization of the process simulation methods is presented in [12], where the advantages and disadvantages of various methods are briefly presented, but there are no recommendations for their application at various levels of process consideration and for various stages of the life cycle. Simulation of manufacturing processes using various extensions of the classical theory of Petri nets is proposed by many scientists (for example, [13,14,15,16,17] and others) due to the simplicity of implementation, obviousness and the possibility of obtaining additional characteristics by adding temporal and conditional parameters to the network. Petri nets allow to consider in dynamics various aspects of the process, to study the temporal and structural characteristics of the process, however, the approaches to structural changes of the nets that can improve the production process are not proposed. In this work, the new approach to simulation using adaptive models based on extended Petri nets with semantic relations [18, 19] and its capabilities for simulation of manufacturing processes and increasing their efficiency is proposed.

3 Extended Petri Nets with Semantic Relations

An extended Petri net with semantic relations (ExPNSR) is a hierarchical color temporal Petri net with additional semantic relations and is a structural-parametric model defined by a following set:

$$ \varPsi = \{ \varPi ,{M}\} , $$
(1)

where \(\varPi = \{ A,Z^{C} ,Z^{S} ,Tr\}\) – a set that presents the structure of tripartite oriented graph that is a Petri net; \(A\) – a finite set of places of size \(J(a)\); \(Z^{C}\) – a finite set of transition by control relations of size \(J(z^{C} )\); \(Z^{S}\) – a finite set of transition by semantic relations of size \(J(z^{S} )\); \(Tr = \{ Tr_{i(col)} \}\) – a set of arcs of the net, \(Tr_{i(col)} = \{ \{ \tilde{R}_{i(col)}^{C} ,\hat{R}_{i(col)}^{C} \} ,\{ \tilde{R}_{i(col)}^{S} ,\hat{R}_{i(col)}^{S} \} \}\) – a set of arcs of the net of specified color; \(\tilde{R}_{i(col)}^{C}\)– an incidence matrix of size \(J(a) \times J(z^{C} )\) that maps the set of places to a set of transitions by control relations for token of the color \(i(col)\); \(\hat{R}_{i(col)}^{C}\) – an incidence matrix of size \(J(z^{C} ) \times J(a)\) that maps the set of transitions by control relations to a set of places for token of the color \(i(col)\); \(\tilde{R}_{i(col)}^{S}\) – an incidence matrix of size \(J(a) \times J(z^{S} )\) that maps the set of places to a set of transitions by semantic relations for token of the color \(i(col)\); \(\hat{R}_{i(col)}^{S}\) – an incidence matrix of size \(J(z^{S} ) \times J(a)\) that maps the set of transitions by semantic relations to a set of places for token of the color \(i(col)\); \(M = \{ h^{C} (t),h^{S} (t),\varLambda^{C} ,\varLambda^{S} \}\) – parameters for the net structure Π that define temporal and logical characteristics of ExPNSR; \(h^{C} (t)\) – a vector of size \(J(z^{C} )\) of the time of delay of control component of the token in the place before firing the corresponding transition by control relation; \(h^{S} (t)\) – a vector of size \(J(z^{S} )\) of the time of delay of semantic component of the token in the place before firing the corresponding transition by semantic relation; \(t\) – time; \(\varLambda^{C} = (\lambda_{i(z\_C)}^{C} )\) – a vector of logical conditions of size J(zC) that defines the possibility of firing the corresponding transition by control relation; \(\varLambda^{S} = (\lambda_{i(z\_S)}^{S} )\) – a vector of logical conditions of size \(J(z^{S} )\), that defines the possibility of firing the corresponding transition by control relation; \(I_{A} (Z^{C} )\) and \(O_{A} (Z^{C} )\) – respectively input and output functions of transitions by control relations; \(I_{A} (Z^{S} )\) and \(O_{A} (Z^{S} )\) – respectively input and output functions of transitions by semantic relations.

The place of the net represents individual operation or process step. At the same time, the net is built on the principle of decomposition, and one place of ExPNSR can be a subnet itself that represents the process in more details.

The separation of transitions into two types by control and semantic relations is caused by the need to analyze the weaknesses of manufacturing processes from the position of time costs and synthesis new model of more effective process. The transition by control relations is a change in activity, i.e. the transition from performing one operation to another. The transition by semantic relations characterizes the possibility of performing a new operation based on the completeness of the previous ones and providing access to information or physical access to objects. Transitions by control relations are represented by thickened bars; transitions by semantic relations are indicated by triangles.

There are data about the time associated with the transition by control relation, which shows the execution time of the operation preceding it, as well as data about the time associated with the transition by semantic relations, which shows the time to provide data or the time to transport objects or to transmit information to other participants of the process. With insufficient automation of the processes, the transition time by semantic relations can be comparable to or even more than the corresponding transition by control relations. Thus it indicates the need for modernization of these operations. In addition, there may be situations where the same information may be required by several participants in the process, and the possibility of using it from the point of view of the organization of the manufacturing process can be both parallel and sequential, that is represented by transitions by control relations. The logical conditions of firing transitions are required in case of branches, cycles or in special cases of information use.

All transitions by control relations can be divided into:

  • Ordinary (or primitive), the cardinal numbers of input and output functions of the transition are equal to one, i.e. the transition from one operation to other

  • Fork, the cardinal number of input function of the transition is equal to one and for output function it is more than one, i.e. the action, as a result of which one or several additional (parallel) processes are started in the system

  • Join, the cardinal number of input function of the transition is more than one and for output function it is equal to one, i.e. the action, as a result of which one or several parallel processes are combined in one

  • Synchro, the cardinal numbers of input and output functions of the transition are equal to numbers more than one, the action, as a result of which two or more parallel processes in the system are waiting for each other to complete, after which their independent execution continues. In this case, additional logical conditions imposed on such a transition may allow or prohibit the start of one or more processes until one or more processes complete the previous operations

Actually, the transition by semantic relations is the organization of information and material flows. All transitions by semantic relations in the net can be divided into:

  • Ordinary (or primitive), the cardinal numbers of input and output functions of the transition are equal to one, i.e. transferring information from one process to only one other

  • S-join, the cardinal number of input function of the transition is more than one and for output function it is equal to one, i.e. the operation of obtaining by the process the data or materials from all necessary sources

  • S-share, the cardinal numbers of input and output functions of the transition are equal to numbers more than one, i.e. the operation in which the presented data or material flows become available simultaneously to many users. Such operations are possible when information flows is organized using a common corporate environment with the separation of access rights and is best suited to the idea of parallel design

The transition by control relations and the transition by semantic relations are called joint if they connect the same places.

Tokens in the net are requests for the execution of a particular operation and can have a different type (color) that characterizes these requests in the considered processes. Tokens of any color are composite and they have a control and semantic part of the same color. In fact, the semantic component of the token represents the object or information over which or on the basis of which actions are taken, and the control component of the token indicates the availability of the resource for the operation and possibility to finish one and start another. Arcs in the net can also have color and are used the conditions for the functioning of transitions are set.

A transition by control relations may fire if for each of the colors the number of composite tokens in the input places is greater than or equal to the number of arcs of the corresponding color connecting this place with the transition (unless otherwise specified by a separate condition). The transition by control relations fires with a delay specified in the vector \(h^{C} (t)\). When the transition by control relations fires the control components of the tokens are removed from the input places of the transition in the amount equal to the number of input arcs of a given color that go from a given place. The control components of the token are transferred to the output places in the amount equal to the number of output arcs of a given color to a given place. It is important that all transitions by semantic relations associated with the place can fire only after the transition by control relations.

A transition by semantic relations can fire if for each of the colors the number of semantic component of the united tokens in the input places that have taken part in firing of the transitions by control relations is greater than or equal to the number of arcs of the corresponding color connecting this place with the transition (unless otherwise specified by a separate condition). The transition by semantic relations fires with a delay specified in the vector \(h^{S} (t)\). When a transition by semantic relations fires the semantic component of the token are removed from the input places of the transition in the amount equal to the number of input arcs of a given color that go from a given place. The semantic components of the token are transferred to the output places in the amount equal to the number of output arcs of a given color to a given place.

In the output places, the control and semantic components of the token are combined. In this case, situations are possible when the semantic first, and only then the controlling component of the chip appears in the position, and when, first, the controlling and then semantic components appear. The first case may indicate an ineffective organization of the process, if the next transition could work if a chip was added to the position of the control part. Improving the efficiency in such cases can be achieved by parallel execution of work. In the second case, it is necessary to analyze the delay in the arrival of the required information/objects in this position, and identify the causes of unproductive time and ways to eliminate them.

The advantage of the proposed approach is the ability to automate simulation and analysis of processes. In a computer, the net structure can only contain incidence matrices which are presented in a sparse form since they are predominantly filled with zero elements. Such form of matrix representation reduces the computer memory requirements. Any of the matrices can be represented as the following structure: Matrix {set A; set B; integer n; integer m} where n and m determine the number of rows and columns, respectively, set is some structure suitable for storing multisets. A reference to the given structure by indices will return ‘1’, if the corresponding pair is found in A and B, otherwise ‘0’ will return. Setting ‘1’ in the matrix creates a pair of values in A and B in accordance with the indices, setting ‘0’ deletes the corresponding pair. All temporal and logical characteristic can be represented by arrays or lists of time values or lists of sets that contains pairs for representation of logical conditions.

In addition, it is possible to automatically construct a graphical representation of the net based on the matrices. This process can be considered as the analysis of incidence matrices and the construction of an image with the corresponding transitions (bar or triangle) and relations (arc of the specified color) depending on the type of incidence matrices.

Based on additional logical and temporal data about the net it is possible to organize automatic or automated rebuilding of the net with a change in its structure in accordance with predetermined criteria for the effectiveness of the processes and given constraints. Consider the approaches to the analysis of the manufacturing processes based on the model ExPNSR in more detail.

4 Analysis of EXPNSR for Increasing the Effectiveness of Manufacturing Processes

To increase the efficiency of manufacturing processes, it is necessary to create the ExPNSR of the existing process. For this, it is necessary to associate each stage or operation for the selected level of model abstraction with the place of the net (in the case of sufficiently automated production some of these data can be taken from the automatic control system or from the ERP system [20]). Then, the arrangement of the transitions by control relations is carried out, representing the sequence of operations within the process. After that, the flows of information and objects in the net are simulated by adding transitions by semantic relations. Time and logical characteristics are added to the network. After that, the initial marking that shows the availability of information and objects at different stages of the net is set.

The ExPNSR should be analyzed as follows (Fig. 1). First of all, it is necessary to analyze isolated sources of information and material objects. It means searching for a cumulus of information sources/objects having output relations with a small number of the following places. If there are many such sources, the net can be rebuilt as follows: the generation places are combined into one/several places depending on the type of source, a transition of s-share type is added connected to all places - consumers of data and material objects. In a real system, this means applying a common information system for grant access the necessary data or automated warehouse storage. After that the analysis of possibilities of parallel execution of process is performed (Fig. 2). The cases of the arrival the semantic token to the place first are tracked. In such cases, the subnet associated with the transitions by control relations is rebuilt.

The next step is an analysis of the temporal characteristics of processes. The total execution time of the process is calculated as the maximum of the sums of times of transition by control and semantic relations that includes into the path from the initial position to the final (or given) place. Search for maximum time values and comparison with possible time values for these operations with use of up-to-date technology for definition of the not-effective ones. Perhaps the process should be studied in more detail at a different level of abstraction to identify the most time-consuming operations. The times of joint transitions by control and semantic relations should compares. If the time of the transition by semantic relations exceeds the time of the transition by control relations, it may be necessary to adopt an automated system for transmitting information, or to change the location of individual stages in space if constant transportation of objects is required. However, such decisions should be made taking into account the material and resource capabilities of the enterprise. So far, we will not touch on this issue in detail in the framework of this article.

Fig. 1.
figure 1

General algorithm of ExPNSR rebuild.

Full size image

5 Example of Simulation and Analysis of the Process Planning Based on ExPNSR

Consider the simulation and structural analysis of processes using the mathematical apparatus of the ExPNSR on the example of the process planning. Due to the limitation of the presentation of the results in the form of an article, we consider here a generalized principal model. For this representation, a single-color net is enough. Since optimization issues are not discussed here in detail, we consider only the part related to the structure of the net and the possibilities of rebuilding it in the presence of all the required resources. The ExPNSR of the initial (Fig. 3) and rebuilt (Fig. 4) process is presented.

Fig. 2.
figure 2

Parallelization algorithm: (a) general (b) linear.

Full size image

As it can be seen on Fig. 3, places 1 and 2 are not related by semantics and, therefore, do not require the transfer of information or material objects to each other and can be performed in parallel. We see a similar situation between places 8 and 9 and places 10 and 11. In addition, information from places 6 and 7, as well as from places 8 and 9 can be accumulated in one system which can be represented in the net using the s-share transition. The s-share transition always requires a condition for its execution, in this case \(\lambda_{6} = (\{ a_{3} ,a_{6} \} ,\{ a_{8} \} ) \vee (\{ a_{3} ,a_{7} \} ,\{ a_{9} \} )\) and \(\lambda_{7} = (\{ a_{3} ,a_{7} ,a_{8} \} ,\{ a_{11} \} ) \vee (\{ a_{3} ,a_{9} \} ,\{ a_{10} \} )\).

The increase in process efficiency is associated with a reduction in time presented in the model by transitions by control relations. Transitions 1 and 2, 8 and 9, 9 and 10, 11 and 12 of the original network are combined in pairs and only the maximum of their time remains in the net. The presence of one system that transmits information reduces the total time for information transfer from one system to another, for re-saving it in the required formats, and also for elimination its duplication.

Fig. 3.
figure 3

ExPNSR of process planning.

Full size image

Note that the existing approaches often represent only a part of the ExPNSR (most often only a semantic subnet), which shows only the possibilities for the development of the process, but not its real implementation represented here by the control subnet.

Fig. 4.
figure 4

Modified ExPNSR of process planning.

Full size image

6 Conclusion

The methods of simulation and analysis of manufacturing processes proposed in the article can be introduced into the industry to develop a modernization strategy in accordance with modern achievements of science and technology, which will allow you to plan and organize the process of step-by-step digitalization of the enterprise to increase the competitiveness of the Russian economy by making more efficient use of the resources available at the enterprise and the adoption of new information technologies. The proposed approach can be used in practice at the planning stage of production modernization or when planning new industries.