Class Exercises on Sensitivity Analysis of LP
BSNS2120, J. Wang
Name _____________________
The product mix problem of High Note Sound Company is given on page 278. The
linear program for solving that problem is shown below (and also on p.278), where X1=number
of CD players to produce, X2=number of receivers to produce.
Max. 50X1 + 120X2
(Total profit)
s.t.
2X1 + 4X2  80
(Electrician hours)
3X1 +
X2  60
(Audio tech hours)
X1, X2  0
The sensitivity ‘Ranging’ table of QM for Windows is shown in Program 7.5B, page
280. Answer the following questions by referring that ‘Ranging’ table.
(1) The optimal solution:
X1 (CD players) = _____________.
X2 (Receivers) = _____________.
(2) The optimal objective function value = __________________.
(Hint: Plug the optimal solution into the objective function.)
(3) With the optimal solution as in (1), is the resource of 80 electrician hours fully used?
(4) With the optimal solution as in (1), is the resource of 60 audio tech hours fully used?
(5) The shadow price of the resource of ‘electrician hours’ is __________ .
(Hint: Looking for the dual price of ‘Electrician hrs’).
(6) The shadow price of ‘electrician hours’, as you found in (5), remains unchanged as far as the
RHS of the first constraint is within the range ______________________.
(7) The shadow price (or dual value) of the resource of ‘audio tech hours’ is ___________.
(8) The shadow price of ‘audio tech hours’, as found in (7), remains unchanged as far as the RHS
of the second constraint is within the range ________________.
(9) If you could get one more electrician hour, in addition to the current 80, then how much
would the total profit increase?
(10) If you could get one more audio tech hour, in addition to the current 60, then how much
would the total profit increase?
(11) If you reduced the total available electrician hours from 80 to 79 hours, would the total
profit increase and decrease? By how much?
(12) If RHS of the first constraint (electrician hours) changes from 80 to 90, how much would
the total profit increase?
(13) If RHS of the second constraint (audio tech hours) changes from 60 to 65, how much would
the total profit increase?
(14) If you can increase either ‘electrician hours’ or ‘audio tech hours’ by one hour, but not both,
which resource would you select to increase?
(15) Between ‘electrician hours’ and ‘audio tech hours’, which is the more valuable resource?
(16) The current unit profit of a CD player is $50. How much could that unit profit increase or
decrease without changing the optimal solution as in (1)?
(17) Suppose $50/unit is not a good estimate of the unit profit of a CD player. Instead, $42/unit
is a better estimate. Is the optimal solution still the same as in (1) with the new profit $42 per
CD player? Comparing the new objective function value with the old one in (2).
(18) The current unit profit of a receiver is $120. How much could that unit profit increase or
decrease without changing the optimal solution as in (1)?
(19) Suppose $120/unit is not a good estimate of the unit profit of a receiver. Instead, $110/unit
is a better estimate. Is the optimal solution still the same as in (1) with the new profit $110 per
receiver? Comparing the new objective function value with the old one in (2).
(20) What would be the dual values if RHS of the first constraint changes from 80 to 145?
(21) What would be the dual values if RHS of the second constraint changes from 60 to54?