# Industrial Applications

## Abstract

One of the first practical problems to be formulated and solved by linear programming methods was the so-called *diet problem*, which is concerned with planning a diet from a given set of foods which will satisfy certain nutritive requirements while keeping the cost at a minimum. For each food the nutritional values in terms of vitamins, calories, etc. per unit of food are known constants and these are the *a*’s of the problem, *a*_{ ij } being the amount of the *i*th nutritional factor contained in a unit of the *j*th food. If it is required that there shall be at least *b*_{ i } units of the *i*th nutrient in the diet the nutritional requirements will take the form of a set of linear inequalities1 in the variables *x*_{ j }, which represent the amounts of the respective foods which shall be present in the diet. These restrictions will in general be satisfied by a large number of combinations of ingredients (foods) and we want to select a combination which minimizes the total cost of ingredients, i. e., a linear function in the *x*_{ j } where the coefficients *c*_{ j } are the prices per unit of the respective foods.

## Keywords

Setup Time Linear Programming Problem Transportation Problem Slack Variable Total Profit## Preview

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## References

- 1.Cf. I. Katzman (1956).Google Scholar
- 3.An example of such a problem (only with additional restrictions on the quantities of crudes available and with two alternate processes for one of the crudes) is given in G. H. Symonds (1955), Ch. 3.Google Scholar
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- 1.Borrowed from A. Bordin (1954), Ch. III : Un esempio di programmazione lineare nell’industria, by A. Bargoni, B. Giardina and S. Ricossa. To preserve company confidentiality, and partly for didactic reasons, the authors hav e adjusted some of the technological and economic data used in the following.Google Scholar
- 1.This assumption of additivity—in chemical engineering known as the “Mixture Law”, cf. J. H. Perry (1941), p. 616—is not always satisfied even in cases of physical blending. For example, if alcohol and water are mixed the volume of the mixture will not be the sum of the volumes of the ingredients although there is no chemical reaction.Google Scholar
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- 1.Cf. Ch. IV, A. 31.Google Scholar
- 2.Cf. pp. 36–43 above.Google Scholar
- 4.Cf. Ch. IV, p. 27.Google Scholar
- 5.Cf. Ch. IV, pp. 28 ff.Google Scholar
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- 2.Cf. Ch. IV, p. 31.Google Scholar
- 5.See A. Henderson and R. Schlaifer, op. cit., pp. 86–89, where the procedure is illustrated by a numerical example.Google Scholar
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