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 ith nutritional factor contained in a unit of the jth food. If it is required that there shall be at least b i units of the ith 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.


Setup Time Linear Programming Problem Transportation Problem Slack Variable Total Profit 
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  1. 1.
    Cf. I. Katzman (1956).Google Scholar
  2. 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
  3. 4.
    Cf. N. V. Reinfeld and W. R. Vogel (1958), pp. 122–5.Google Scholar
  4. 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
  5. 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
  6. 1.
    See A. Charnes, W. W. Cooper and B. Mellon (1952) .Google Scholar
  7. 5.
    The example is borrowed from A. Henderson and R. Schlaifer (1954), pp. 83 f.Google Scholar
  8. 1.
    Cf. Ch. IV, A. 31.Google Scholar
  9. 2.
    Cf. pp. 36–43 above.Google Scholar
  10. 4.
    Cf. Ch. IV, p. 27.Google Scholar
  11. 5.
    Cf. Ch. IV, pp. 28 ff.Google Scholar
  12. 6.
    Cf. A. Henderson and R. Schlaifer, op. cit., p. 87.Google Scholar
  13. 7.
    Borrowed from A. Charnes, W. W. Cooper, D. Farr, and Staff (1953).Google Scholar
  14. 1.
    Cf. A. Henderson and R. Schlaifer, op. cit., p. 86.Google Scholar
  15. 2.
    Cf. Ch. IV, p. 31.Google Scholar
  16. 5.
    See A. Henderson and R. Schlaifer, op. cit., pp. 86–89, where the procedure is illustrated by a numerical example.Google Scholar
  17. 3.
    For an example, see S. Danø and E. L. Jensen (1958).Google Scholar
  18. 1.
    Cf. A. Charnes and W. W. Cooper (1955).Google Scholar
  19. 2.
    Cf. A. Henderson and R. Schlaifer, op. cit., pp. 77 f.Google Scholar
  20. 3.
    See, for example, A. Henderson and R. Schlaifer, op. cit., pp. 79 ff.Google Scholar

Copyright information

© Springer-Verlag Wien 1963

Authors and Affiliations

  • Sven Danø
    • 1
  1. 1.University of CopenhagenDänemark

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