Question

.. B.27 Doug Turner Food Processors wishes to introduce a new brand of dog biscuits composed of chicken- and liver-flavored biscuits that meet certain nutritional requirements. The liver-flavored biscuits contain 1 unit of nutrient A and 2 units of nutrient B; the chicken-flavored biscuits 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 to make 1 liver-flavored biscuit and 2c to make 1 chicken- flavored, what is the optimal product mix for a package of the biscuits to minimize the firm's cost? a. Formulate this as a linear programming problem. b. Solve this problem graphically, giving the optimal values of all variables c. What is the total cost of a package of dog biscuits using the optimal mix? Px

Answer 1

Assume that x units of liver flavoured biscuits and y of chicken flavoured biscuits are used in new composition

cost is to be minimised such that

Objective function - Min { x+2y}

such that

x+y=>40

2x+4y =>60

x<=15,

x, y =>0

Graph

The feasible region is shown as the unbound region between the lines x+y<=40 and x<=15 . points of optimality are (15,40)(0,40)  and ( 15,25)

Value of OF

at (15,40) = 15+80 =95

at (0,40) = 0+2x40 =80

at ( 15,25) =15+2x25 =65

Since the cost is minimum at ( 15,25), it is the most feasible option.

Hence the company should make 15 liver biscuits and 25 chicken biscuits to minimise the cost ( 65)