# Optimality Criterion

## Optimality Criterion

a characteristic on the basis of which a comparison of possible solutions (alternatives) is made and the best is selected. The content of the optimality criterion is objectively determined by many factors: the nature of the social order, economic laws, the scale of the decisions (national economy, branch of production, individual enterprise), and the goals toward which actions are to be directed. The principle of optimality is borrowed from mathematical programming and control theory. The methodological basis of the theory of optimization of the economy is the principle of national economic optimality, that is, the study of economic phenomena from the point of view of the whole, from the point of view of the entire national economy.

A criterion of optimality is designed to help corroborate a decision. The practical problems of corroboration may be tentatively subdivided into three types. The essence of problems of the first type is the need to select the best variant of actions to ensure achievement of a completely defined, that is, assigned, result with minimum expenditure of resources. In problems of the second type the volume of existing resources is fixed and what must be found is the best variant for using them to obtain a maximum result. In problems of the third type, the search for the best variant is conducted without rigid constraints on either the volume of resources used or the final results. When substantiating decisions the concept of “degree of goal achievement,” which is described by a definite indicator, is used.

The resources available to a society, branch of production, or enterprise are limited; therefore, the volume of resources al-located to a single goal depends to some degree on how the resources are allocated to other goals. Therefore, any variant of resource distribution directly or indirectly affects several goals simultaneously and is thus characterized by several indicators.

The solution to a problem of any type can be reduced in principle to consideration of a set of alternatives, a comparative evaluation of them, and the selection of the best one. The “transportation problem” can serve as an example of the first type of problem. A country has n places where coal is extracted, and from these places the coal is delivered to m consumers located in different cities of the country. The cost of delivering a ton of coal from point of extraction i (i = 1, 2, . . ., n) to consumption pointy j(j = 1, 2, . . . ,m) is known.

The amount of coal Xj needed by each consumer is also known. The plan must be determined for delivering to consumers the required amounts of coal with minimum expenditures. The solution to such a problem is methodologically simple be-cause the values of all indicators describing the results of actions —xj—are fixed (they are constraints in the form of equalities). Each plan variant for providing coal to consumers is evaluated by one variable indicator, expenditures, and this is the criterion of optimality. It is significantly more complex to solve a problem of this type when, in addition to monetary expenditures, consideration must be given to the expenditure of physical resources, labor, and other resources that sometimes cannot be expressed in monetary form.

Similar difficulties arise in problems of the second type be-cause the results of distributing resources are described by several indicators having variable values. The case that involves a comparison of different variants of capital investments in the development of a branch, production association, or individual enterprise and the final work results corresponding to them is an example of a problem of the third type. These are the problems most often encountered in the planning process when it is necessary to decide which is better, to raise production capacities by increasing capital investments or, let us suppose, to keep both at their former level. The results of each decision are described by combining the values of several indicators. To establish which of the possible decisions is best they must be compared for several indicators. In this case it may be necessary to shape an optimality criterion that makes it easier to evaluate alternatives.

As is the case with individual indicators, the quantity used as the criterion of optimality may be measured on a continuous or discrete scale. Discrete evaluations may, in turn, be ordinal or metric. An ordinal scale is a sequence of different combinations of indicator values drawn up on the basis of the correspondence of these combinations to the defined goals. When such a scale is used to compare two variants it cannot be determined how much better one result is than another, but only which variant is better than the other. Unlike the ordinal scales, the metric scale permits an evaluation of the “distance” between two neighboring orders (ranks); that is, it makes it possible to determine how much better one alternative is than another. An example of an ordinal scale for one index could be word (qualitative) definitions of the degree of achievement of the set goal: complete satisfaction of a certain need, partial satisfaction of the need, and so on. An indicator represented in the metric scale may be volume of output for a certain purpose.

In practice the alternatives usually compared differ in final results and expenditures and are of the type “better and more expensive” and “worse and cheaper.” When this is the case, the results are described by several indicators. Problems of this type are sometimes called vector optimization problems. In this case the components of the vector are the indicators describing the degree of achievement of particular goals. Among the variants being compared the most efficient ones are usually singled out, including the variants that ensure achievement of a definite result with minimum expenditures or achievement of the maximum result for certain determined expenditures. Selection of the best (optimal) variant from the most efficient ones may be done by means of an appropriate criterion of optimality.

The objective need to compare variants on the basis of several incommensurable indicators is the primary cause of the difficul-ties that must be overcome in shaping the criterion of optimality. A variant for which one indicator cannot be increased further without decreasing the values of at least one of the others (the Pareto optimum or maximum) cannot be considered best. The criterion of optimality should be such that in the general case it is possible to compare variants when one of the indicators (one of the vector components) increases while another decreases. It is evident that the most that can be expected in comparing vectors (combinations of the values of several indicators that describe the degree of achievement of various goals) is establishing preferences among them, that is, evaluating vectors by means of an ordinal scale. It should be noted that evaluations of vectors according to an ordinal scale are entirely adequate for comparing variants and selecting the best ones.

In a socialist society all decisions made at different levels in the system of planning and control should correspond as much as possible to the highest goal: fullest possible satisfaction of society’s needs. This goal can be achieved provided that a definite set of socioeconomic goals that foresee the satisfaction of all society’s needs is formulated and then achieved. Society must produce different products to satisfy its needs. The need for these products depends on the level of satisfaction of personal and other nonproduction needs today and in the future. Thus, the level of development of production may be viewed as the variable in a function describing the degree of satisfaction of society’s nonproduction needs.

One of the problems of planning is to determine the most efficient proportions in production of different products. During the planning process, variants of distribution of labor and other resources at the disposal of society are reviewed, and the variant that best corresponds to society’s needs is selected. Marx wrote, “The social need, that is, the use value on societal scale appears here as a determining factor for the amount of total social labor time which is expended in various special spheres of production” (K. Marx and F. Engels, Soch., 2nd ed., vol. 25, part 2, p. 186). Thus, a comparative evaluation of variants of the national economic plan must be made according to a criterion that reflects the degree to which the plan corresponds to society’s needs.

Plans are realized in time and space; therefore, in the general case the values of particular indicators should characterize changes in the degree of satisfaction of needs in different years of the planning period and in different regions of the country. The comparison of plan variants for a large number of indicators presents significant difficulties. Generalization of data reduces the number of indicators. The higher the level of the planning agency the greater the degree of generalization. Thus, for decision-making at the highest level the degree of satisfaction of a certain public need can evidently be represented as a ratio between the planned volume of production of products of a certain type and the amount of goods and services that will provide for the given need in accordance with the effective demand of the population and social funds. In this the degree of satisfaction of the need will be characterized by a single indicator W. To avoid the necessity of working with values of this indicator in different years its value at the end of the planning period may be considered. This is permissible if a steady increase in the value of the indicator by years is assumed. If we begin from the necessity of satisfying n needs of society, then each variant of the national economic plan will be characterized, at a minimum, by a combi-nation of the values of n indices W1, W2, . . . ,Wn.

Comparative evaluation of variants of a plan being developed at any level can be done either directly, by combining the values of the indicators, or according to a specially shaped criterion of optimality. The chief requirement that the criterion of optimality used at any level should meet is that it should make possible the evaluation of variants on the basis of the assigned goal. One of the methods of reflecting how different combinations of the values of several indicators correspond to a higher goal is the ordered sequence of these combinations.

Selecting or shaping a criterion of optimality is the main question in comparative evaluation of alternatives. In this the funda-mental methodological principle is a systems approach to evaluating possible decisions. The essence of the systems approach is that the advisability of particular changes in the object is determined with due regard for its mutual relations based on the interests of the system of which the object under consideration is a part. No recommendations concerning the concrete content of the criterion of optimality can be given beforehand. They can only be made after consideration of the general goals and the establishment of the degree of correspondence of different combinations of indicator values describing the object to the goals that face the system.

Consideration of uncertainty, for example, descriptions of technology under development, its cost, the conditions under which it will be used, and so on, are especially important in corroborating decisions.

There is a formal decision theory that considers different methods for shaping the criterion for evaluating alternatives under uncertainty, such as the maximin criterion and the minimax regret criterion. Alternatives must always be compared by a single criterion, but this statement does not exclude the possibility of evaluating variants in turn by one criterion and then another.

The literature on systems analysis has devoted significant attention to questions of quantitative corroboration of decisions under uncertainty. Analysis of systems is a method of evaluating alternatives under uncertainty with several contradictory goals. The application of this method makes it easier to corroborate the goals of action and to identify the advantages and disadvantages of alternative variants of action. However, the final selection is made by the leader responsible for making the decision.

### REFERENCES

Luce, R. D., and H. Raiffa. Igry i resheniia. Moscow, 1961. (Translated from English.)
Pugachev, V. F. Optimizatsiia planirovaniia (teoreticheskie problemy). Moscow, 1968.
Fedorenko, N. P. O razrabotke sistemy optimal’nogo funktsionirovaniia ekonomiki. Moscow, 1968.
Solnyshkov, lu. S. Kak obosnovat’ reshenie. Moscow, 1972.

IU. S. SOLNYSHKOV

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