Please help me solve this problem. The Beef-Up Ranch feeds catte for Midwestern
ID: 447915 • Letter: P
Question
Please help me solve this problem.
The Beef-Up Ranch feeds catte for Midwestern farmers and delivers them to processing plants in Topeka, Kansas, and Tulsa, Oklahoma. The ranch must determine the amounts of cattle feed to buy so that various nutritinal 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:
Nutrients (in ounces) per Pound of Feed
Nutrient Feed 1 Feed 2 Feed 3
A 3 2 4
B 3 1 3
C 1 0 2
D 6 8 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 nutreint A, 5 pounds of nutreint B, 1 pound of nutrient C, and 8 pounds of nutrient D. However, cows should not ve fed more than twice the minimum requirement for any nutrient each month. (Note that there are 16 ounces in a pound.) Additionally, the ranch can only obtain 1,500 pounds of each type of feed each month. Because there are 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.
A. Use solver to solve this problem and create a sensivity report.
b. Is the solution degenrate?
c. Is the solution unique?
d. Explain the signs of the reduced costs for each of the decision variables. That is, considering the optimal value of each decision variable, why does the sign of its associated reduced cost make economic sense.
e. Suppose the cost per pound for Feed 3 increased by $3. Would the optimal soulution change? Would the optimal objective function value change?
f. If the company could reduce any of the nutrient requirements, which one would it choose and why?
g. If the company would increase any of the nutrient requirements which one would 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<=128 (maximum requirement constraint)
Feed 2: 3x+y+3z>=80 (minimum requirement constraint), 3x+y+3z<=160 (maximum requirement constraint)
Feed 3: x+2z>=16 (minimum requirement constraint) , x+2z<=32 (maximum requirement constraint)
Feed 4: 6x+8y+4z>=128 (minimum requirement constraint) , 6x+8y+4z<=256 (maximum requirement constraint)
x,y,z<=15 pounds (Maximum pounds available/cow)
A) Solver Solution:
Sensitivity report:
b) Note: A basic feasible solution is called degenerate if one of its RHS coefficients is 0.
Since, here no RHS coefficient is 0, hence the solution is not degenerate.
c) In linear programming, an LP can have multiple optimal solutions if it contains degenerate vertices.
Since it does not contain degenrate vertices, this LP solution is unique.
d) The reduced costs tell us how much the objective coefficients can be increased or decreased before the optimal solution changes.
In this case the reduced cost for Feed 1 is -3.25, i.e. if we increase the cost/pound of Feed 1 by more than $3.25, the optimal solution changes.
For example if add $3.5 the cost of feed 1:
e) cost per pound for Feed 3 increased by $3
The optimal solution does not change. Still
But, the objective function will change to Minimize (2x+2.5y+6z)
Feed 1 Feed 2 Feed 3 lbs needed/ cow 15.00 9.50 8.50 Nutrients (in ounces) per Pound of Feed (ounces) Nutrient feed 1 feed 2 feed 3 total Min Req Max Req A 3 2 4 98 64 128 B 3 1 3 80 80 160 C 1 0 2 32 16 32 D 6 8 4 200 128 256 Feed 1 Feed 2 Feed 3 Cost($/lb) 2 2.5 3 Max lbs /cow 15 15 15 total cost 79.25