Question: A boarding stable feeds and houses work horses used to pull tourist-fi
ID: 3553370 • Letter: Q
Question
Question: A boarding stable feeds and houses work horses used to pull tourist-filled carriages through the streets of a historic city. The stable owner wishes to strike a balance between a healthy nutritional standard for the horses and the daily cost of feed. This type of horse must consume exactly 5 pounds of feed per day. The feed mixes available are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a predictable amount of five ingredients needed daily to keep the average horse healthy. The table below shows these minimum requirements, units of each nutrient per pound of feed mix, and costs for the three mixes.
FEED MIX
OAT
GRAIN
MINERAL
NEEDED
NUTRIENT
(UNITS/LB)
(UNITS/LB)
(UNITS/LB)
(UNITS/LB)
A
2.0
3.0
1.0
6
B
0.5
1.0
0.5
2
C
3.0
5.0
6.0
9
D
1.0
1.5
2.0
8
E
0.5
0.5
1.5
5
Cost/lb
$ 0.33
$ 0.44
$ 0.57
10-2
Using the Sensitivity Report from 10-1, answer the following question:
If the price of grain decreases by $0.01 per pound, will the optimal solution change?
10-3
Using the Sensitivity Report from 10-1, answer the following question:
Which constraints are binding?
10-4
Using the Sensitivity Report from 10-1, answer the following question:
What would happen to the total cost if the price of mineral decreased by 20% from its current value?
10-5
Using the Sensitivity Report from 10-1, answer the following question:
For what price range of oats is the current solution optimal?
FEED MIX
OAT
GRAIN
MINERAL
NEEDED
NUTRIENT
(UNITS/LB)
(UNITS/LB)
(UNITS/LB)
(UNITS/LB)
A
2.0
3.0
1.0
6
B
0.5
1.0
0.5
2
C
3.0
5.0
6.0
9
D
1.0
1.5
2.0
8
E
0.5
0.5
1.5
5
Cost/lb
$ 0.33
$ 0.44
$ 0.57
Explanation / Answer
. a. If the price of grain decreases by $0.01 per pound, will the optimal solution change?
b. Which constraints are binding? Interpret the dual price for the binding constraints
d. For what price range of oats is the current solution optimal? Solutions to Question 2:
. c. What would happen to the total cost if the price of mineral decreased by 20% from its current value?
=========================================================
a. The decrease of $0.01 per pound is within the allowable decrease (infinity). Therefore the optimal solution will not change but the objective value will decrease by 0.04 (=0.01*4) to 0.356.
=========================================================
b. The constraints prescribing the minimum daily requirement for ingredient E and the max feed per day are binding (i.e. LHS 0.5*4+1.5*2=5, RHS=5, i.e LHS=RHS), (i.e LHS=0+2+4=6, RHS=6). For each additional unit of ingredient E required in the mix, the cost will increase by $ 0.156. For each additional unit of max feed per day, the cost will decrease by $ 0.064.
=========================================================
c. A 20% decrease in the cost of mineral implies the cost will decrease by $0.034 (=0.17*0.2), which is less the allowable decrease (0.128). The optimal solutions remain unchanged. The revised cost will be decrease by 0.068 (=2*0.034) to $0.328 (=0.396-0.068)
=========================================================
d. The price of oats can fluctuate between $0.014 (0.09-0.076) and infinity (0.09+infinity) per pound for the current solution to remain optimal.
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