The Beef-up ranch feeds cattle for midwestern farmers and delivers them to proce
ID: 447754 • Letter: T
Question
The Beef-up ranch feeds cattle for midwestern farmers and delivers them to processing plants in Topeka,Kansas and Tulsa, Oklahoma. The ranch must determine the amounts of catlle feed to buy so that variouis nutritional requirements are met while minimizing total feed costs. The mixture fed to the cows must contain different levels of four key nutrients and can be made by blending three different feeds. The amount of each nutrient (in ounces) found in each pound of feed is summarized as follows:
Nutrient a- feed 1 (3) feed 2 (2) feed 3 (4)
b- feed 1 (3) feed 2 (1) feed 3 (3)
c- feed 1 ( 1) feed 2 (0) feed 3 (2)
d- feed 1 (6) feed 2 (8) feed 3 (4)
The cost per pound of feeds 1,2, and 3 are $2.00, $2.50, and $3.00, respectively. The minimum requirement per cow each month is 4 pounds of nutrient A, 5 pounds of nutrient B, 1 pound of nutrient C, and 8 pounds of nutrient D. However, cows should not be fed more than twice the minimum requirement for any nutrient each month. Additionally, the ranch can only obtain 1,500 pounds of each type of feed each month. Because there aree usually 100 cows at the beef-up ranch at any given time, this means that no more than 15 pounds of each type of feed can be used per cow each month.
Use Solver to create a Sensitivity Report for question 15 at the end of Chapter 3 and answer the following questions:
a. Is the solution degenerate?
b. Is the solution unique?
c. Explain the signs of the reduced costs for each of the decision variables. That is, considering the optimal value of each decison vaiable, why does the sign of its associated reduced costs make economic sense?
d. Suppose the cost per pound for feed 3 is increased by $3. Would the optimal solution change? Would the optimal objective function value change?
e. If the company could reduce any of the nutrient requirements, which one should it choose and why?
f. If the company could increase any of the nutrient requirements, which one should it choose and why?
Explanation / Answer
Decision variables: x,y,z for pounds of feed/ cow for feed 1, 2 and 3 respectively Objective function: Minimize (2x+2.5y+3z) {Minimizing the total cost} Constraints: x,y,z>=0 {non negativity constraint} Feed 1: 3x+2y+4z>=64 ounce (minimum requirement constraint) , 3x+2y+4z=80 (minimum requirement constraint), 3x+y+3z=16 (minimum requirement constraint) , x+2z=128 (minimum requirement constraint) , 6x+8y+4zRelated Questions
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