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This examination has 12 pages including this cover
The University of British Columbia Final Examination – 11 Dec 2009 Mathematics 340 Linear Programming
Closed book examination Time: 150 minutes
Name Signature
UBC Student Number
Special Instructions:
- Students are invited to write on both sides of each sheet.
- To receive full credit, all answers must be supported with clear and correct derivations.
- No calculators, notes, or other aids are allowed.
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CAUTION - Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action.
(a) Making use of any books, papers or memoranda, other than those authorized by the examiners.
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Total 100
[12] 1. Consider the following problem:
maximize ζ = − 5 x 1 + 6x 2 − 4 x 3 subject to 2 x 1 + 3x 2 − x 3 ≤ − 2 − x 1 + 2x 2 + 2x 3 ≤ 3 x 1 , x 2 , x 3 ≥ 0
(a) Write the dual problem. (b) Show that the dual problem is unbounded, by presenting a sequence of feasible inputs for the dual problem whose objective values diverge to −∞. (One reasonable approach starts with a sketch.) (c) What does the result in part (b) tell us about the problem stated above?
(Blank page for extra calculations.)
[12] 3. The matrix below shows the rewards to the column player in a standard zero-sum matrix game.
G =
[
]
(a) Set up a linear program to find the optimal mixed strategy for the column player. (b) Solve the LP in part (a). (Do at most four pivots.) (c) Find the optimal mixed strategy for the row player.
[6] 5. This problem involves an LP where some cost coefficients are not specified:
maximize ζ = c 1 x 1 + c 2 x 2 subject to x 1 + x 2 ≤ 4 x 2 + x 3 ≤ 2 x 1 − x 3 ≤ 3 x 1 , x 2 , x 3 ≥ 0
Suppose a dual minimizer has the form y∗^ = (3, 1 , γ) for some γ. Find the maximum value of ζ.
[15] 6. Rowena plays the rows and Callum plays the columns in a standard zero-sum matrix game in which the rewards to Callum are displayed in the following matrix:
G =
(a) Write a short, clear definition of “Nash equilibrium” applicable to zero-sum games. Use only words: no mathematical symbols or variables are allowed.
(b) Consider the mixed strategies x˜ =
for Callum and y∗^ =
for Rowena. Are
these strategies in Nash equilibrium? Explain, making reference to your definition in part (a). (c) Find all strategies x for Callum (if any) that can participate in a Nash equilibrium with Rowena’s choice of y∗^ from (b).
(Blank page for extra calculations.)
[15] 8. Our factory makes portable computer memory sticks in two styles. The Hello Kitty model, aimed at Facebook lovers, requires one logic circuit, one crypto chip, and two memory chips; each assembled unit earns a profit of 4 dollars. The Black Lightning model, styled for paranoid hackers, requires one logic circuit, two crypto chips, and one memory chip; each unit earns a profit of 5 dollars. We just got a new shipment of parts (the factory was empty!): 5000 logic circuits, 8000 crypto chips, and 8000 memory chips. Suppose we manufacture 1000x 1 Hello Kitty units and 1000x 2 Black Lightning units. (a) Write the linear program we can use to plan our production to maximize total profit. The resulting LP should look familiar. If it doesn’t, re-read all the questions on this exam. (b) The delivery driver offers to sell us a few extra logic circuits for $2.50 each. Should we buy some? Why? What if the asking price is $3.50 each? (c) The retailer who buys our products telephones with a nasty surprise: paranoid hackers are scaring away respectable customers. The retailer insists that x 2 ≤ x 1. Show how to use the Dual Simplex Method to determine our revised production plan when this constraint is added to our problem. What happens to our profit? (d) Part (c) was just a bad dream. Forget it. This morning we have good news. Our co-op student has figured out how to make a pocket music player using 2 logic circuits, 1 crypto chip, and 2 memory chips. What value(s) of profit-per-unit on music players would motivate us to change our current plan and make some? How will our overall production strategy change in this case?
