To provide additional practice in formulating and interpreting the computer solution for linear programs involving more than two decision variables, we consider a minimization problem involving three decision variables. Bluegrass Farms, located in Lexington, Kentucky, has been experimenting with a special diet for its racehorses. The feed components available for the diet are a standard horse feed product, a vitamin enriched oat product, and a new vitamin and mineral feed additive. The nutritional values in units per pound and the costs for the three feed components are summarized in the table below. For example, each pound of the standard feed component contains 0.8 unit of ingredient A, 1 unit of ingredient B, and 0.1 unit of ingredient C. The minimum daily diet requirements for each horse are 3 units of ingredient A, 6 units of ingredient B, and 4 units of ingredient C. In addition, to control the weight of the horses, the total daily feed for a horse should not exceed 6 pounds. Bluegrass Farms would like to determine the minimum cost mix that will satisfy the daily diet requirements.
NUTRITIONAL VALUE AND COST DATA FOR THE BLUEGRASS FARMS PROBLEM:
Feed Component Standard Enriched Oat Additive
Ingredient A 0.8 0.2 0
Ingredient B 1.0 1.5 3
Ingredient C 0.1 0.6 2
Cost per pound $0.25 $0.5 $3
Combining all the constraints with the nonnegativity requirements enables us to write the complete linear programming model for the Bluegrass Farms problem as follows:
See attached file for the data
Answer the following questions based on the computer output provided:
a) What is the solution in terms of total cost, decision, slack and surplus variables?
b) If the minimum requirement for ingredient A is lowered to 1.6 units, how would this change impact the current solution? What will be the revised total cost?
c) If the minimum requirement for ingredient B is increased by 2 units, how would this change impact the current optimal solution? What will be the revised total cost?
d) If the total weight requirement of 6 pounds is increased to 8 pounds, how would this change impact the current optimal solution? What will be the revised total cost?
e) If the unit cost of enriched oat increases to $0.94, how would this change impact the current optimal solution? What will be the revised total cost?
f) If the unit price for the standard feed is decreased by 5 cents, how would this change impact the current optimal solution? What will be the revised total cost?
g) If the changes given in parts (e) and (f) occur at the same time, how would these simultaneous changes impact the current optimal solution? What will be the revised total cost?
h) Interpret the dual price of $-1.959 associated with the third constraint within the context of this business case.
i) Although this is not the case, for arguments sake, assume that the optimal value of "E" happens to be zero pounds in the solution and the associated reduced cost is $-0.30. For some reason, if one is forced to include 0.4 pounds of "E" in the mix, what will be the revised total cost?
Discussion is in attachment.