Linear Models of Production and Economic Optimization
As we have seen in Ch. I, the general problem of production planning can be expressed as that of selecting from among the feasible alternatives one which is economically optimal. From a formal point of view this is a problem of finding a maximum (or a min
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Linear Models of Production and Economic Optimization A. The Linear Production Model 1. As we have seen in Ch. I, the general problem of production planning can be expressed as that of selecting from among the feasible alternatives one which is econcmically optimal. From a formal point of view this is a problem of finding a maximum (or a minimum) of a function subject to a number of side conditi~ns which are the mathematical expressions of the technical and economic restrictions on the company's freedom of choice, and the feasible planning alternatives appear as alternative non-negative solutions to this system of relations. The problem will often be such that the side conditions have the form of linear equations or inequalities and we have a linear model of production. We shall now look into the question under what circumstances this will be the case, and compare with other types of production models known from the traditional economic theory of production.
2. The variables of the production model-i.e., the variables of the restrictions subject to which an optimum is sought--can be classified under two headings: (i) the quantities of the products (or outputs) to be produced during the planning period, and (ii) the quantities consumed of the various factors of production, or inputs. The latter group comprises such factors as labor input (measured in man-hours), raw materials (in physical units), energy (for example, kwhs. consumed), and the services of fixed equipment (machine hours). We will denote outputs by Xl, X2, ••• , Xn whereas VI, V2, ••• , Vm will stand for input quantities consumed. Now these variables will of course be interrelated. In the first place, there will exist a number of technological relations in the x's and the v's which are the mathematical expressions of the production processes by which inputs are transformed into outputs. The simplest example is the case of proportionality between inputs and outputs; the amounts of labor, raw materia.l, energy, machine services etc. consumed per unit of each output are constant, vCi=aCixi
(i=l, 2, .. , m; j=l, 2, .. , n),
(I)
where the constant a'i is the "technical coelficient of production" expressing input of factor no. i per unit of product no. j (or the reciprocal productivity
S. Danø, Linear Programming in Industry Theory and Applications © Springer-Verlag Wien 1963
The Linear Production Model
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of the factor with respect to this particular product). In the theory of production this is known as the case of limitational factors of production. However, it is often possible to produce the same output by several alternative combinations of the inputs; this is the case of factor substitution. In addition to these technological relationships there will usually be a number of economic restrictions on the problem. For example, in short-term production planning the company's capital equipment is given and the resulting capacity limitati0n8 for the fixed factors must be ta:ken into account. They will be inequalities of the type
(2) since total util
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