By Ola Bratteli
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35 Blending Problems Two of the ingredients, sweet cream and curd cheese, contribute to the fat content of the mix, containing 69 and 34% fat respectively. It is a legal requirement that the product shall have a fat content of not less than 20% and that the curd cheese content shall be not less than 51 %; furthermore, for technical reasons the maximum content of sweet cream is fixed at 5%. For total output equal to 100 Ibs. 69xl Xl where mix 1 . 34x2:2 20 < (1) 5 are the quantities of sweet cream and curd cheese in the In addition to these problems of human "diet," linear programming has been applied extensively to solving feed-mix problems in agriculture and in the feedstuff industry 2, as well as to calculating the optimal blend of fertilizers.
No blending proportions are specified for each product (such as in the previous examples), nor are the proportions in which the products are to be made specified in advance (as they were in the diet problem). Each output can be made from anyone of the inputs, one unit of the product Xj requiring one unit of the raw material Vi, or by blending them in arbitrary proportions, except that the blending proportions must be such that certain quality specifications for the products are satisfied. When these specifications can be expressed as linear restrictions on the variables we have a linear optimization model.
3. Since one unit of the final product x} requires a constant amount of the raw material v,--directly or by way of intermediate products-the technology of the problem is characterized by a linear model, each final output defining a linear process and the capacity limits for the raw materials giving rise to six linear inequalities. The coefficients of the model are given in the table on p. 40 1 . The problem with which the company is concerned is to determine the quantities to be produced per period (day) of the respective products such as to maximize total gross profit subject to the six side conditions.