Operational Scheduling of Continuous Production in an Automated Process Control System Environment
Abstract Generated abstract
This paper addresses operational scheduling for continuous production complexes under automated process control, where conventional coordination based on industry average norms fails to reflect the dependence of section efficiency on load. It formulates the daily allocation problem using constraints on input and output flows, static subsystem models, and an efficiency criterion depending on workloads and raw material expenditures. To make the resulting large scale mathematical programming problem tractable within computer memory and time limits, the authors propose a decomposition approach that separates subsystem planning, identifies coordinating loads, derives efficiency dependencies for subsystems, and optimizes the overall criterion over these linking parameters. The procedure is intended to yield coordinated daily subsystem loads, total complex productivity, and associated resource expenditures.
Full Text
OPERATIONAL SCHEDULING OF CONTINUOUS PRODUCTION IN AN AUTOMATED PROCESS CONTROL SYSTEM ENVIRONMENT
E. G. KLEIBORT, R. N. LYUBLINSKII
(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)
Operational scheduling of continuous production in existing practice consists of coordinating the throughput capacity of various units on the basis of industry-average normative information [1]. Such planning does not take into account the dependence of the operating efficiency of each section on the load and, consequently, is not optimal.
In an automated process control system environment, thanks to the use of computers capable of solving complex mathematical problems and storing a large amount of data, it becomes possible both to obtain dependencies of the operating efficiency of each production section on the load and to solve formalized planning problems using mathematical optimization methods.
In the present work, the problem of distributing the daily plan among production sections according to given static models of the sections and load characteristics of the sections is considered. In formalized form this problem can be represented as follows.
Suppose that the annual plan of the continuous-technology complex has already been divided in time, and that daily constraints are specified for the complex on the input and output product flows, beyond which it must not go:
\[ \underline{Y_i} \leq Y_i \leq \overline{Y_i}; \tag{1} \]
\[ \underline{X_i} \leq X_i \leq \overline{X_i}, \quad i=\overline{1,n}, \tag{2} \]
where \(Y_i\) is the vector of parameters of the output product of the \(i\)-th subdivision of the complex;
\(X_i\) is the analogous vector of the input flow;
\(n\) is the number of subsystems of the complex.
Suppose also that mathematical models have been constructed for the functioning of the sections of the complex:
\[ y_i = f_i(X_i, U_i), \tag{3} \]
where \(U_i\) is the vector of raw-material expenditures.
Under these conditions, for clear and rhythmic production operation it is necessary to draw up such daily plans for the subdivisions of the complex that, when implemented, would achieve the greatest effect from the standpoint of the selected planning criterion.
Given models (3), the efficiency criterion in general form depends on the workload of the departments and on raw-material costs:
\[ \Phi=\Phi(X_i, U_i),\quad i=\overline{1,n} \tag{4} \]
Modern large production complexes are complex techno-economic systems, with a large number of internal links and interactions. Therefore, the mathematical programming problem (1)–(4), formulated for finding the optimal operating plan of such a complex, is a large-scale problem requiring large expenditures of machine time when solved on a computer. Moreover, under APCS conditions, when the computer must process large arrays of information for solving problems of control, analysis, planning, and management, there is an acute shortage of operative memory.
All these circumstances make it difficult to apply optimal programming methods to the solution of the problem mentioned and compel one to seek another approach to its solution. An effective method for solving large-scale problems is the method of decomposing the original problem into a number of subproblems of smaller dimension [2]. The formation of subproblems proceeds from the structural scheme of the complex, on the basis of which the general model of the complex breaks down into submodels of departments. It should be noted that the decomposition of planning problems and of the corresponding mathematical programming problems arose not only in connection with the need to find ways of solving large-scale problems. Decomposition methods are closely connected with the idea of dividing planning functions between the center managing the economy of the complex and the subsystems making up the complex [3], i.e., with the problem of the optimal combination of centralized planning with the independent action of subsystems.
When the ideas of decomposition are used, the problem of optimal operational scheduling can be solved according to the following scheme.
I. Formation of efficiency criteria for the operation of the entire complex and of the subsystems included in it.
The stage of forming the efficiency criterion is extremely important, since the choice of it determines the goal we seek to achieve in optimizing the operation of the system. This problem has independent significance and is based on the principles set forth in [4].
II. Selection of the “linking” workloads of each subsystem from the entire group of input, output, and control parameters of the subsystems.
By the “linking” workloads of each subsystem are understood those of its parameters that connect the given subsystem with at least one other subsystem of the complex.
III. Breaking the existing links between the subsystems of the complex.
From this point until a certain moment in time, each of the subsystems is considered to exist independently, with its own sales market and supply.
IV. Obtaining dependencies of the efficiency of subsystem operation on the magnitude of the “linking” workloads.
These dependencies may be obtained in two ways.
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Construction of averaged dependencies of the efficiency criterion of subsystem operation on the magnitude of the “linking” workload from statistical data.
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Construction of dependencies of the efficiency criterion on the magnitude of the “linking” workload under optimal operation of the subsystem.
The first way of obtaining dependencies does not require a large volume of computational work, but is significantly inferior in accuracy to the second.
The second path is distinguished by its laboriousness, which is explained by the presence in it of tasks for optimizing the operation of subsystems.
The indicated dependencies may be represented in various forms.
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Values of the “linking” loads and the corresponding values of the “unlinked” parameters, stored in the computer memory in the form of nodes of a spatial lattice.
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Graphs of analogous dependencies.
V. Finding analytical formulas describing the dependencies of the operating efficiency of subsystems on the magnitudes of the “linking” loads.
VI. Substitution of the analytical dependencies of the operating efficiency of subsystems on the “linking” loads into the criterion of the operating efficiency of the complex.
As a result, the criterion of the operating efficiency of the complex proves to be a function only of the “linking” (coordinating) loads.
VII. Finding the optimal values of the coordinating parameters.
Optimization of the criterion of the efficiency of the complex with respect to the values of the coordinating parameters corresponds to optimal coordination of the productivities of the subsystems.
As a result of solving the problem, the optimal daily loads of all subsystems, the optimal daily load of the complex, the daily productivity of the complex, and the daily expenditures of resources for maintaining the calculated productivities are found.
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R. N. Lyublinskii, E. G. Kleibort, B.-Zh. Zhambalnimbuev. On an economic criterion for the efficiency of controlling a chemical apparatus. Article in the present collection.