Chapter 4 - Sensitivity Analysis & The Simplex Method : S-1

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Chapter 4

Sensitivity Analysis & The Simplex Method

3.See file: PRB4_3.XLS

Microsoft Excel Sensitivity Report

Adjustable Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$C$4 / Value: X1 / 12 / 0 / 2 / 1E+30 / 0
$D$4 / Value: X2 / 0 / 0 / 4 / 0 / 1E+30
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$E$10 / Used: / 12 / 0 / 2 / 10 / 1E+30
$E$8 / Used: / -12 / 0 / 8 / 1E+30 / 20
$E$9 / Used: / 12 / 2 / 12 / 1E+30 / 10

a.Constraint 2 is binding.

b.There is an alternate optimal solution. Variable X2 is at its lower bound (of zero) but also has a reduced cost of zero. This indicates that the value of X2 could be increased while having zero impact on the optimal objective function value.

c.There is no way to answer this question directly from the sensitivity report. We know that the optimal solution would change since the allowable decrease in the objective function value of X1 is zero, but we cannot tell what the new optimal solution would be. If we re-solve the revised model the solution is X1=2, X2=5.

d.It can decrease by any amount without changing the solution.

e.Constraint 2 is the only binding constraint. Therefore, we would want to increase its RHS value before any other.

24.See file: PRB4_24.XLS

Microsoft Excel Sensitivity Report
Adjustable Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$E$5 / Regular / 0.00 / 0.00 / 3.75 / 0 / 1E+30
$F$5 / Supreme / 150.00 / 0.00 / 7.75 / 1E+30 / 0
$E$6 / Regular / 160.87 / 0.00 / 5.25 / 0 / 0
$F$6 / Supreme / 189.13 / 0.00 / 9.25 / 0 / 0
$E$7 / Regular / 139.13 / 0.00 / 3.25 / 1.15 / 0
$F$7 / Supreme / 110.87 / 0.00 / 7.25 / 0 / 1E+30
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$G$5 / Used / 150 / 0.5 / 150 / 196.1538461 / 50
$G$6 / Used / 350 / 2 / 350 / 100 / 50
$G$7 / Used / 250 / 0 / 300 / 1E+30 / 50
$E$8 / Available Regular / 300 / 3.25 / 300 / 50 / 20.90909091
$F$8 / Available Supreme / 450 / 7.25 / 450 / 29.31034482 / 20.90909091
$E$11 / Actual Octane Level Regular / 97.6667 / 0.0000 / 90 / 7.666666667 / 1E+30
$F$11 / Actual Octane Level Supreme / 97.0000 / 0.0000 / 97 / 5.111111111 / 5.666666666

a. Yes, as reflected by the allowable increases and decreases of zero on the objective coefficients.

b. Note that the sensitivity information cannot be used to answer this (and the following) questions. Add a constraint holding the objective at its optimal value and then maximize E11 and F11 separately. Regular octane rating = 97.67, supreme octane rating = 97.

c. Regular octane rating = 90.0, supreme octane rating = 102.11.

d. The solution in part c should be implemented. The company would receive negative publicity if the answer in part b were implemented and the public learned that the inexpensive regular gas had a higher octane rating than the more expensive supreme gas.

e. This cannot be answered from the given information. They should be willing to buy an additional 100 barrels at this price, but beyond that the shadow price no longer holds.

11.See file: PRB4_11.XLS

Microsoft Excel Sensitivity Report
Adjustable Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$B$4 / Quantity Made HyperLink / 500 / -43.66666667 / 53 / 43.66666667 / 1E+30
$C$4 / Quantity Made FastLink / 1000 / 0 / 48 / 16 / 1E+30
$D$4 / Quantity Made SpeedLink / 1500 / 0 / 33 / 15 / 1
$E$4 / Quantity Made MicroLink / 2250 / 0 / 32 / 1 / 5.272727273
$F$4 / Quantity Made EtherLink / 500 / -9.666666667 / 38 / 9.666666667 / 1E+30
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$G$13 / PC Board (sq in) Used / 60,500 / 0 / 80000 / 1E+30 / 19500
$G$14 / Resistors Used / 100,000 / 0 / 100000 / 10500 / 6000
$G$15 / Memory chips Used / 30,000 / 7 / 30000 / 2000 / 2333.333333
$G$16 / Assembly Hours Used / 3,688 / 0 / 5000 / 1E+30 / 1312.5
$G$17 / Fast/Hyper>2 Used / 0 / 16 / 0 / 500 / 875

a. The constraints on the number of resistors and memory chips are binding. Also the constraint requiring twice as many Fastlink cards as Hyperlink cards is binding.

b. Hyperlinks. The company would earn $43.67 for each Hyperlink it could avoid producing.

c. Yes. Profits would increase by $7×1,000=$7,000.

d. The solution would change if the objective coefficients on the Speedlink decreased by $1 or if the objective coefficient on the Microlink increased by $1. Thus, the objective coefficients on these products would be of greatest concern.

e.Microlinks.

5.a. MIN: 260X13 + 220X14 + 290X15 + 230X23 + 240X24 + 310X25

S.T.:

X13 + X14 + X15  20

X23 + X24 + X25  20

X13 + X23 10

X14 + X24 15

X15 + X25 10

Xij 0

b. See file: PRB4_5.XLS

c. See below

Microsoft Excel Sensitivity Report
Adjustable Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$C$10 / Eustis Miami / 0 / 50 / 260 / 1E+30 / 50
$D$10 / Eustis Orlando / 10 / 0 / 220 / 20 / 0
$E$10 / Eustis Tallahassee / 10 / 0 / 290 / 0 / 310
$C$11 / Clermont Miami / 10 / 0 / 230 / 50 / 230
$D$11 / Clermont Orlando / 5 / 0 / 240 / 0 / 20
$E$11 / Clermont Tallahassee / 0 / 0 / 310 / 1E+30 / 0
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$C$12 / Shipped Miami / 10 / 230 / 10 / 5 / 10
$D$12 / Shipped Orlando / 15 / 240 / 15 / 5 / 5
$E$12 / Shipped Tallahassee / 10 / 310 / 10 / 5 / 5
$F$10 / Eustis Used / 20 / -20 / 20 / 5 / 5
$F$11 / Clermont Used / 15 / 0 / 20 / 1E+30 / 5

d. No.

e.No.

Capacity
Miami / Orlando / Tallahassee / Used / Available
Eustis / 0 / 15 / 5 / 20 / 20
Clermont / 10 / 0 / 5 / 15 / 20
Shipped / 10 / 15 / 10
Demand / 10 / 15 / 10
Total Cost / $8,600

f. The solution would not change. The current solution uses only 15 of the 20 tons of capacity available at Clermont.

g. Reducing the capacity in Eustis would increase costs by $20 per unit decrease yielding an objective function value of $8,600+20*5 = $8,700

h. Every additional ton of concentrate unit shipped from Eustis to Miami would increase costs by $50.