Pet Supplies Company produces 16-ounce cans of dog food by combining meat by-products, which cost $0.60 per pound, and chicken by-products, which cost $0.35 per pound. Meat by-products are 55% protein and 30% fat by weight, while chicken by-products are 40% protein and 10% fat by weight. To meet customer expectations, the final product should contain at least 50% protein and between 15% and 25% fat by weight. Pet Supplies advertises that each can of dog food has no more than 30% by weight of chicken by-products.
a) Formulate a linear programming solution to be used to determine what the composition of a can of dog food should be to meet the various requirements at the minimum cost. Hint: you can express the amount of each ingredient in a 16-ounce can of dog food as a fraction of a 16-ounce can rather than in ounces. Provide the objective function and the constraints for your linear programming formulation.
b) Using linear programming, find the optimal solution for the objective function and the optimal values of each of the variables. Provide a copy of your linear programming output, identifying the value of the objective function and the values of the variables at optimality.
c) Suppose the cost of the meat increased to 0.80 per pound. Don't resolve the problem but use the sensitivity analysis output from part b). Explain if this would change the optimal value of the objective function and the optimal value of the variables. Provide the information from part b) that you used in reaching your conclusion.
d) Suppose the maximum chicken by-product that is accepted increases by 4% to 34%. Don't resolve the problem but use the sensitivity analysis output from part b). Explain if this would change the optimal value of the objective function and the optimal value of the variables. Provide the information from those results that you used in reaching your conclusion.
e) Now suppose that each can of dog food sales for $1.50. If the required profit per can is at least $0.65, write the constraint satisfying this objective function. You do not need to resolve for the optimal answer.