class_ch7_minimize_1.doc 1/15/04 5:51 AM Page 1 of 2
7.106 The Fido Dog Food Company wishes to introduce a new brand of dog biscuits (composed of chicken and liver flavored biscuits) that meets certain nutritional requirements. The liver flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B, while the chicken flavored ones contain 1 unit of nutrient A and 4 units of nutrient B. According to federal requirements, there must be at least 40 units of nutrient A and 60 units of nutrient B in a package of the new mix. In addition, the company has decided that there can be no more than 15 liver flavored biscuits in a package. If it costs 1 cent to make a liver flavored biscuit and 2 cents to make a chicken flavored one, what is the optimal product mix for a package of the biscuits in order to minimize the firm's cost?
(a) Formulate this as a linear programming problem.
(b) Solve this problem corner point method, giving the optimal values of all variables.
(c) Are any constraints redundant? If so, which one or ones?
(d) What is the total cost of a package of dog biscuits using the optimal mix?
ANSWER:
(a) Let X1 = number of liver flavored biscuits in a package
X2 = number of chicken flavored biscuits in a package
Minimize X1 + 2X2
Subject to: X1 + X2 ³ 40
2X1 + 4X2 ³ 60
X1 £ 15
X1, X2 ³ 0
(b) Corner points (0,40) and (15,25)
Optimal solution is (15,25)
with cost of 65.
(c) 2X1 + 4X2 ³ 60 is redundant