Linear programming emerged in the 1930s from the research of Soviet economist and mathematician Leonid Vitalievitch Kantorovitch (1912–1986), who remains the only Soviet researcher ever to have received the "Nobel Prize" in Economics, in 1975.
A decision-making process often involves choosing, from a range of possible solutions, the one that optimizes a particular function f of variables subject to constraints. For example, the aim may be to maximize production output with limited resources, to minimize the cost of a purchase that meets a given quality standard, or to produce a high-quality blend at the lowest possible cost.
In practice, particularly in economic applications, concrete problems are therefore expressed mathematically as the optimization—maximization or minimization—of a function f whose variables, denoted by x1, x2… xn , must satisfy constraints defining a set E in n-dimensional real space (see Mathematics and Economics. Tangente Library 62, 2018).
Effects proportional to their causes
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One special case, common in practice, is that in which the function under study is linear—that is, linear in all its variables. In this situation, the effects are proportional to their causes and additive. This is true of the quantity of a product manufactured: it is proportional to the quantities of raw materials used, while the output of two (or more) workshops making the same item is added together. Mathematically, the function f under consideration then has the form:
