LOGICAL-STATIC PLANNING COMPLEX TECHNICAL OBJECTS OPERATIONS AND FUNCTIONING MODES

A definition of operating mode of complex technical object (CTO) concept is introduced. A logical-static interpretation of CTO operating modes is given. A formal grammar of transformations of logical functions for compatibility (incompatibility) of operations in constraints of a static planning model is presented. Transformation rules are introduced for specific cases. An algorithm for formalizing logical multi-mode structures of CTO functioning in static planning terms of operation flows and modes is developed. Results of computational experiments are presented.

1 Introduction

The main elements and subsystems of Industry 4.0 are complex technical objects (CTOs) [1]. Individual computers, computer networks, industrial lines, modern agricultural enterprises, supply chains and networks, etc. are CTOs, which perform many operations [2]. By operation, we mean an action performed by a single functional element (FE) of a CTO [3]. We assume that operations can be performed in parallel or series, with intensity control and the ability to reconfigure the composition of their elements and subsystems (multi-mode).

Based on the studies [4,5], we describe CTO functioning according to the basic provisions of the conceptual model of the functioning of an active mobile object. We suppose that for CTO control and management, it is important not to know its design features but to understand the manifestation of these features. At the same time, the goals of CTO functioning can be achieved in various ways (implementation of aggregated operations (AOs)). In this case, the functioning of the CTO can be described at one of four levels: the level of goals, the level of subgoals, the AO level, or the level of operating modes of subsystems. The CTO operating mode we consider as AO logical interrelations, which performed by various FE to achieve the specified goals. In this paper the aspect of compatibility and incompatibility of the CTO operating modes (described using logical functions), is considered [6].

1.1 Literature review

There is a large body of research focused on using linear programming (LP) methods for planning and scheduling CTO functioning [7,8]. Across a number of decades, this approach has served as the basis for solving specific planning and scheduling tasks in various fields [9].

There is evidence, however, that some CTO operations are linked by logical connections. To illustrate this, researchers have proposed using AND/OR graphs [10,11] to describe the logic of CTO functioning. Drawing on the logical-dynamic description of CTO functioning, researchers have demonstrated that it is possible to create an integrated operational and flow model [12]. The authors have shown that the resulting plans and schedules possess the properties of completeness, closure, and consistency, as they are based on fundamental results from optimal control theory.

The analysis showed that researchers use the principle of decomposition to address the problems of high dimensionality, nonstationary, and nonlinearity [4,13]. Studies have also shown that creating a specific model is a time-consuming process. Firstly, introducing restrictions in optimization planning tasks related to specific operating procedures is always individual in nature, requiring a deep understanding of the subject area. Secondly, for instance, the procedure for forming conjugate variables to solve optimal control problems is quite       complicated [14]. At the same time, a number of practical logical constraints cannot be formally described using logical-dynamic models. This means that the creator of the planning model must describe in detail all alternative ways to achieve the goal (technology) [15,16].

As an approach to choosing the composition of operations to ensure the CTO operation modes, the formalization and solution of the optimization problem are proposed. This is because ensuring the specified modes of operation of the CTO involves performing a set of operations that, as a rule, have functional redundancy (variability in the choice of operations to ensure the required mode of operation of the CTO). In particular, the optimization problem can be formalized and solved in terms of a class of LP models. However, at the moment, there is no single procedure for translating logical constraints into the constraints of an LP problem.

To address this gap, we take a rather novel approach. We propose implementing a formal representation of logical functions reflecting the compatibility or incompatibility of the implementation of operating modes in the form of a formal grammar for their transformation into constraints of a static planning model.

1.2 Conceptual statement of the problem

We propose formalizing the logic of implementing operating modes in the form of additional constraints on the time variables for activating the modes under consideration. In this case, the sought variables reflecting the integral time of mode activation must not only satisfy the specified logical conditions of compatibility/incompatibility in mode implementation but also ensure compliance with arbitrarily complex logical relationships between the operating modes and the operations supporting them [17,18]. These relationships are presented in this article in the form of a tree of logical functions.

Let us introduce the following notations:

	total operating time of a multi-mode CTO – ensuring the implementation of AO;
	the operating time of the CTO in one of the modes (submodes) at the n-th nesting level of the logical function tree:
	the operating time of the CTO in the mode (submode) that is ensured by the joint (parallel) execution   of specified operations;
	the operating time of the CTO in the mode (submode) that is ensured by the joint (parallel-sequential) execution   of specified operations;
	the operating time of the CTO, during which the execution of specified operations is prohibited, where s is the number   of modes (submodes) and operations at  the n-th nesting level of the logical function tree.

2     Materials and methods

2.1 Formal grammar of logical functions transformation of aggregate operation interconnections into constraints of linear programming problem

The process of transforming logical functions into constraints of the LP problem can be represented in the form of the following formal grammar:

where:

S - the CTO operating mode, ensuring the implementation of AO;

N - set of logical operations that characterize the relations between the modes and operations of the CTO functioning;

T - restrictions of the LP problem;

F - rules for transforming logical operations into constraints of the LP problem.

To specify the process of transforming logical functions into LP problem constraints, we define the elements of the sets described above.

As is known, the description of any logical function can be carried out with the presence of the operations "AND" (AND, &), "OR" (OR, |), "NOT" (NOT, ~) [19]. Let us introduce these logical relationships between the operating modes and the operations that provide them in the form of the following LP tasks:

-        the task of parallel execution of operations, the total time of such an operating mode $R_{s\mathrm{AND}}^n$ can be introduced into the system of constraints as the lower bound of the total time of joint (simultaneous or parallel) execution of operations x_i [20]:

	(2)

where I is a set of indices of submodes and operations that ensure the execution of the mode (submode) $R_{s\mathrm{AND}}^n$;

-        the task of non-execution (prohibition of execution) of operations and submodes, in which incompatibility with specified operations and submodes must be ensured over time:

	(3)

where  is the time horizon for planning the operation of the CTO as a whole;

-        the problem of parallel-sequential execution of operations and submodes, the total time of activation of $R_{s\mathrm{OR}}^n$  which can be introduced into the system of constraints both as the sum of the execution time of incompatible operations and submodes x_i, and as the lower bound of the execution time of joint operations and submodes x_i:

	(4)

In this case, partial compatibility/incompatibility of the execution of operations and submodes x_i is allowed.

Here it is necessary to explain a number of features of the presented models. Firstly, in all three models the inequality sign is deliberately used because when formalizing the general LP task there may be other restrictions interconnected with the modes (submodes) or the operations that provide them, and therefore part of the time resource allocated to the operations and susbmodes x_i can be used not only to ensure the execution of a given mode of operation of the CTO, and therefore, the equality sign should be used only when such allocation of resources is prohibited. Secondly, it is recommended to use the mode $R_{s\mathrm{NOT}}^n$ with caution in the case of several operations and submodes x_i, as practice has shown, clearly expected results are observed either when using a single such operation, or when using an OR mode as an operation. In the constraints of model (3), the sum of x_i can only be used for incompatible operations [21-23]. Thirdly, in subsystem (4), the first constraint is obvious (indeed, the mode $R_{s\mathrm{OR}}^n$ cannot be executed for more than the time of incompatible operations and submodes), while the lower group of constraints is not redundant. For example, when formalizing a complex mode $R_{s\mathrm{NOT}}^n \left(R_{s\mathrm{OR}}^{n+1}\right) \lor$, it would be necessary to minimize and rewrite the constraints of subsystem (4), then the model $R_{s\mathrm{OR}}^{n+1} \rightarrow \min R_{s\mathrm{OR}}^{n+1}$ becomes the upper limit of the durations of operations and submodes x_i.

To formalize an arbitrary mode of operation of the CTO (in the form of an arbitrary logical function) in terms of LP, one must first determine the criterion for selecting the optimal solution, and then construct a system of constraints in the form of equations and inequalities.

Since this paper focuses on the formalization of a single mode of CTO operation, the maximization of the execution time of the mode specified by a certain logical function is proposed as the criterion for selecting a solution. In other words, it is proposed to find a plan that considering the possibility of introducing additional restrictions into the LP problem, ensures the execution of this mode for as long as possible. In this case, it is possible to use the restrictions of models (2)-(4) as restrictions for some submodes (intermediate modes) that ensure the implementation of the mode $R_1^0$ with the sequential addition of constraints of the form (2)-(4) to the structure of the general planning model constraints, and the maximization of the mode $R_1^0$ should indirectly maximize these modes as well, since an arbitrary logical function can be transformed into a hierarchical tree of operations [24]. Such a tree of logical operations can be illustrated using the example of the operation "exclusive OR" (XOR). A variant of such an implementation and the order of tree traversal (classical depth-first traversal) (Fig. 1).

Fig. 1. A variant of a hierarchical tree for the logical operation XOR

In other words, when formalizing the algorithm for parsing the logical function of the mode  into a hierarchical tree, an analysis of its components – logical functions and operands – was performed in order to automatically add constraints to the LP model. For the basic logical functions "AND", "OR", "NOT" such constraints are formed (see models (2)-(4), while possible operands require formalization: "SYMBOL" – a symbolic variable (logical variable); constants "TRUE" (true) and "FALSE" (false).

When a symbolic logical variable is found as an operand, the corresponding intermediate mode can be added to the constraint structure of the LP problem as the following expression:

(5)

In cases where constants act as operands of a logical function, one of the expressions below should be used:

	(6)
	(7)

It should also be noted that it is impossible to use the introduced transformations directly when implementing complex logical functions of the type $R_{\mathrm{AND}}^n \left( R_{\mathrm{NOT}}^{n+1},\ R_{\mathrm{NOT}}^{n+1} \right)$ and $R_{\mathrm{NOT}}^n \left( R_{\mathrm{AND}}^{n+1} \right)$, since all variables of operations in nested modes will be automatically zeroed. Therefore, according to de Morgan's rules [25], additional transformations of the type (8)-(9) should be implemented:

	(8)
	(9)

Thus, if we take constraints (2)-(4) as a basis, take into account additional transformations (5)-(7), (8)-(9) and use submodes instead of operations x_i as operands of logical function $R_j^n$s, then the formal grammar of transformations of logical operations of an arbitrary logical function into constraints of the LP problem will finally take the following form [26], were

S is a set of mode (sub-modes) numbers, formed dynamically when traversing the operation tree;

J - a set of submode numbers included in the set of operands of logical functions;

I - a set of numbers of logical variables;

n - the number of the current nesting level of the operations tree.

	(10)
	(11)
	(12)
	(13)

2.2 Algorithm for transforming logical operations of an arbitrary logical function into constraints of a linear programming problem

Based on the transformation model presented above, an algorithm was developed, the central link of which is a recursive traversal of the tree of logical operations, at each step of which the following occurs:

-        adding a new constraint to the structure of the LP task depending on the type of logical operation and operands;

-        updating the set by adding the next index number.

After completing the traversal of the logical operation tree, it is necessary to add a group of restrictions on the permissible activation time:

(14)

The general scheme of the algorithm is presented in (Fig. 2 – 3).

Fig. 2. Scheme of the algorithm for recursive transformation of a logical function into a structure of constraints of a linear programming problem

Fig. 3. Scheme of the algorithm for recursive transformation of a logical function into the structure of constraints of a linear programming problem

3 Results and discussion

Based on the developed algorithm, computational experiments were conducted, including checking the implementation of the main properties of logical operations, three variants of implementing XOR and several other examples of logical operations [27,28]. For all the experiments, the time resource. T_k=1. Let us analyze the simulation results presented in Table 1. Here, the variables  are operations, and are modes R_i (submodes) at the corresponding nesting levels.

Table 1. Simulation results

Rows 1–3 in Table 1 contain three implementations of the XOR operation. Rows 4–8 are checks for basic logical constructions. Rows 9–16 contain calculations related to various classical logical functions of several arguments. For example, in row 12, the execution of a given mode must be ensured by two simultaneous operations, one of which must be chosen (for any instant of time) as either p4 or p5, and the other as one of p1, p2, p3.

Table 2 shows the verification calculations of three XOR implementations, to which additional constraints of the following type have been added.

Table 2. Verification calculations

3.1         Discussion

The calculations presented in Tables 1 and 2 indicate the reliability of the formalization of the proposed grammar for transforming logical constraints on the compatibility/incompatibility of operations and modes into constraints of the static model of operations planning. In particular, in all three cases given in Table 2, the variables p1, p2, p3 have the same values, such that both the requirements for the limited total time of operations and the additional requirements for the relationship between these variables are satisfied [29,30]. The results confirm the adequacy of the proposed formal grammar of transformations of logical functions.

4     Conclusions

In the course of the research, the authors introduced a generalized definition of the concept of "operating mode" of the CTO, which served as the basis for the development of a formal grammar for transforming logical constraints on the compatibility/incompatibility of operations and submodes into constraints of the static planning model (LP). An algorithm for formalizing logical multi-mode structures of the CTO functioning in terms of LP was developed and used to verify the correctness of the research results.

It is advisable to devote further research on this topic to the integration of the resulting constraints: into planning tasks involving dynamically changing structures of information, material and energy flows processing, transmission and storage systems; into planning tasks when making decisions on using resources to perform operations in time slices; into tasks for synthesizing the technical appearance of multimode objects (including agro-biological objects) during their design and optimization of their structures and business processes.

The study was funded by the Russian Science Foundation (project No. 24-19-00823), https://rscf.ru/project/24-19-00823/.

The authors declare that there are no conflicts of interest that could have influenced the research or its outcomes.

Alexander N. Pavlov: Conceptualization, methodology, formal analysis, writing original draft. Dmitry A. Pavlov: Software development, computational experiments, data analysis, writing review & editing. Anna Yu. Ivanyu: Analysis of the results, validation, visualization, writing original draft. Valerii V. Zakharov: Methodology, supervision, project administration, funding acquisition, writing review & editing. Nikolay A. Barashenkov: Resources, data curation, formal analysis, writing review & editing.

No external datasets were generated or analyzed in this study. The simulation code and model parameters not publicly available.

There are no supplementary materials to include.