On the Selection of Efficiency Criteria for Solving Operational Planning Problems at Enterprises with a Testing Nature of Production
Abstract Generated abstract
The paper addresses the choice of efficiency criteria for automated operational planning at enterprises whose production process consists chiefly of testing incoming products from many customer enterprises. It argues that neither expert assessment alone nor criteria derived directly from a mathematical planning model are sufficient, because daily testing schedules are affected by random order arrivals and directive targets are usually set for longer planning periods. A combined procedure is proposed: expert screening of relevant directive indicators, construction of a mathematical model with technological and resource constraints, formation of normalized local criteria from model inequalities, and aggregation through a global criterion using loss-of-optimality and utility coefficients. The paper also suggests reducing the number of local criteria by excluding less significant inequalities or by analyzing gradient components near an extremum, with practical validation of any simplification.
Full Text
ON THE CHOICE OF EFFICIENCY CRITERIA FOR SOLVING OPERATIONAL PLANNING PROBLEMS AT ENTERPRISES WITH A TEST-ORIENTED PRODUCTION PROCESS
A. M. MALYSHENKO
(Presented by the scientific and technical seminar of the Department of Automation and Telemechanics)
A specific feature of the class of enterprises under consideration is the dependence of their operation on the regularity and intensity of the flow, arriving from other enterprises, of orders for testing the products manufactured by them. With a large number of customer enterprises and a broad range of products being tested, operational planning of the work of production testing enterprises (centers) becomes especially difficult because of the large number of deviations from the plan for the arrival of products for testing. Under these conditions, because of the large scale of the problem being solved, the preparation of daily work plans for a testing center requires a high degree of responsiveness and the consideration of a considerable number of factors affecting the efficiency of the center’s operation. In this connection, a substantial technical and economic effect should be expected from the transition to automated preparation of daily plan-schedules for the enterprise’s work.
Naturally, the automation of test planning must provide not simply for the automatic execution of individual procedures for compiling plan-schedules, but for the selection, from among a large number of different algorithms for compiling test plan-schedules, of the most rational one, which, with little time spent on preparing the input data and on computer calculations, makes it possible to obtain a plan-schedule that ensures the best technical and economic indicators.
Both in the manual and in the automated variant of operational planning, there arises the need for a quantitative evaluation of the various possible variants of plans and, consequently, for a quantitative measure for such an evaluation—the efficiency criterion.
At present there is no generally accepted methodology for choosing an efficiency criterion. As a rule, for this purpose either the method of expert evaluations is used, or inequalities from the mathematical model of the planning problem being solved are used.
The first approach is usually easier to implement; however, because of the subjectivity of expert evaluations, the choice of local criteria and the formation on their basis of a global criterion may be carried out with deviation from the true goals of the system being optimized [1]. Therefore, rather often a system optimized according to criteria formulated by experts proves in fact to be inefficient.
In this connection, methods for forming global criteria based on analysis of the domains of ...
feasible solutions, taking into account the directive assignments and resources of the system being optimized [2]. In this case, the global criterion must maximize efficiency and minimize costs. To this end, in [2] it is proposed that the global criterion be formed from all inequalities included in the mathematical model of the optimization problem under consideration.
Such an approach to the choice of an efficiency criterion, as applied to problems of operational planning of the work of enterprises with a testing-oriented nature of production, cannot be used in its pure form, since directive assignments, as a rule, apply to fairly long intervals of time (year, quarter, month), and not to individual working days or shifts. In addition, the planning of the work of enterprises of the type under consideration is significantly affected by the essentially random, for a shift or daily cycle, nature of incoming orders for testing.
Under these conditions, it seems advisable, for the purpose of forming an efficiency criterion, to use a combination of the expert-assessment method and a method based on the use of a mathematical model.
To this end, at the first stage it is necessary to study the goals and nature of production, as well as the annual and quarterly directive assignments for the enterprise. At the second stage, from among the directive assignments, using the expert method, it is necessary to exclude indicators whose magnitude does not depend on changes (variation) in the operational plans for the enterprise’s work.
At the next stage, a mathematical model is compiled for the problem of operational planning of the enterprise’s work, taking into account the selected directive assignments and technological and resource constraints. In this case we obtain a system of equations and inequalities:
\[ f_i(\overline{X}) \geq B_i \quad (i=1,2,\ldots,m), \]
\[ f_i(\overline{X}) = B_i \quad (i=m+1,\ldots,n), \tag{1} \]
where \(B_i(i=1,2,\ldots,n)\) are quantities characterizing the directive assignments, resources, and technological constraints of the enterprise; \(\overline{X}\) is the vector of parameters characterizing variants of the operational production plan.
The system of equations (1) describes the conditions of operational production planning. The efficiency of the selected plan variant will evidently be determined by the extent to which the directive indicators are improved and the extent to which the resources provided to the enterprise are saved. In this connection, the fourth stage in choosing an efficiency criterion reduces to forming a global efficiency criterion on the basis of the inequalities of the system of equations (1). For this purpose, the methodology proposed by B. E. Kukharev in [2] may be recommended.
According to this methodology, the local criteria used are functions defined by the inequalities of the system of equations (1). The local criteria are formed in normalized form as
\[ L_i(\overline{X}) = \frac{f_i(\overline{X}) - B_i} {f_i(\overline{X}_0) - B_i}, \tag{2} \]
where
\[ f_i(\overline{X}_0)=\operatorname{ext}_{\overline{X}} f_i(\overline{X}) - \tag{3} \]
is the extremal value of the function \(f_i(\overline{X})\).
In this case
\[ 0 \leq L_i(\overline{X}) \leq 1. \]
The global criterion, according to [1], can be reduced to the form
\[ Q(\overline{X}, \beta)=\min_{\overline{X}}\left[\frac{\sum_{i=1}^{m} R_i^\beta(\overline{X})}{m}\right]^{\frac{1}{\beta}}, \tag{4} \]
where
\[ 1 \leq \beta \leq \infty, \tag{5} \]
\[ R_i(\overline{X})=\left[\frac{1-L_i(\overline{X})}{1+L_i(\overline{X})}\right]^{v_i} \tag{6} \]
is the loss-of-optimality function.
In the last expression, \(v_i\) is the utility coefficient of the \(i\)-th local criterion, with \(v_i \geq 1\). More important local criteria correspond to smaller \(v_i\), and for the most important of them \(v_i=1\). To choose the values of the utility coefficients, the expert method should again be used.
The value of the coefficient \(\beta\) must be chosen in accordance with (5), where \(\beta=1\) corresponds to the trend of development under the adopted form of the efficiency criterion, while as \(\beta \to \infty\) the stabilization trend is realized. In the latter case, criterion (4) imposes increasingly stringent requirements on the equality of losses across all local criteria.
Under the adopted method for choosing the efficiency criterion, its calculation—and still more, optimization according to the chosen criterion—may prove difficult because of the large number of local criteria used. In this connection, it is highly desirable to reduce the number \(m\) of local criteria used. To this end, if possible, when forming the criterion in the form (4), the least essential of the inequalities (2) should not be taken into account. In that case the latter are subsequently used as constraints on the feasible-solution region.
If, however, such an operation cannot be carried out with confidence that no substantial deterioration will occur in the technical-and-economic indicators of the plans optimized according to the chosen criterion, then an attempt to reduce the number of local criteria used may be undertaken after forming criterion (4) and the stage of estimating the values of the components
\[ \operatorname{grad} Q[R_i(\overline{X})] \]
in a neighborhood of the extremum point. Obviously, in the subsequent constraints one may first include those inequalities from (2) for which the components of the gradient of the function \(Q\) with respect to the corresponding \(R_i(\overline{X})\) have the smallest values. The validity of such simplifications of the efficiency criterion should be checked in the practical solution of the operational-planning problem.
It may be hoped that the method set forth may prove acceptable in choosing efficiency criteria for operational planning of production at enterprises of other profiles.
REFERENCES
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V. G. Babchuk. Choice of a global criterion. In: Complex Control Systems, Issue 2, Kiev, 1969.
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B. E. Kukharev. Development and investigation of decision-making methods in multicriteria systems. Dissertation for the degree of Candidate of Technical Sciences. Kharkov Institute of Radio Electronics, 1972.