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 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.