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# Linear Proramming: Problem with three products and three additives plus a base

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Incredible indelible ink company mixes three additives, A1,A2, A3 to a base in different proportions to obtain different colors of ink.
Red ink is obtained by mixing A1, A2, and A3 in the ration of 3:1:2, blue ink in the ratio of 2:3:4, and green in the ratio of 1:2:3.
After these additives are mixed, an equal amount of base is added for each color. The company currently has 1000 gallons of A1, 1500 of A2, 2000 of A3, and 4000 of base. Given that the selling price per gallon for each type on ink is the same, formulate a LP model to determine how these resources should be used to obtain the maximum revenue.

#### Solution Preview

Let After mixing additives the three inks red, blue and green are x1, x2 and x3 gallons.
First of all change the scale of three ratios of additives as follows:
Red = 3:1:2 => sum = 3+1+2 = 6
Blue = 2:3:4=> sum = 2+3+4 = 9
Green = 1:2:3=> sum= 1+2+3 = 6
LCM of 6,9,6 = 18
Therefore multiply the ratios by 18 and divide by the sum of ratios:
Red= (3:1:2) *18/6 = 9:3:6 => sum = ...

#### Solution Summary

A linear programming problem is solved for maximum product.

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