MATH3202 Operations Research and Mathematical Planning 2018
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Semester One Final Examinations, 2018
MATH3202 Operations Research and Mathematical Planning
Question 1 – Integer Programming
12 marks total
A severe storm is approaching Moreton Bay and there are currently 18 boats still out on the water. Each boat can seek shelter in the ports at Manly, Cleveland or Dunwich. The data in the Python stub gives the time (in minutes) that it will take each boat to make it to each port. Manly and Cleveland each have capacity remaining for 8 boats while the smaller Dunwich port can only accommodate 6 boats.
a) Which port should each boat be sent to in order to minimise the total travel time for the 18 boats? Formulate this as an integer programming problem. Write the formulation in the space below. Implement your formulation in Python. [8 marks]
b) Minimising the total travel time may still leave some boats out when the storm arrives. Suppose instead we want to minimise the maximum travel time for all 18 boats. Revise your formulation and show the new variables, objective and constraints in the space below. Implement your revised formulation in Python. [4 marks]
Question 2 – Dynamic Programming
8 marks
You are about to play 10 rounds of the Iterated Prisoner’s Dilemma with an opponent who initially has a 0.6 probability of cooperating. Every round you cooperate, their probability of cooperating increases by 0.1 (to a maximum of 1). However, every round you defect, their probability of cooperating decreases by 0.2 (to a minimum of 0).
The payoffs for you are given by the following table:
|
|
Opponent Cooperate Defect |
|
|
Cooperate You Defect |
3 |
0 |
|
5 |
1 |
|
What strategy should you pursue to maximise your expected payoff from the 10 plays? Implement a dynamic programming formulation of this problem in Python. Include comments in your code that describe the stages, state, actions and value function. Write the optimal strategy in the space below.
2022-06-11