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Unformatted text preview: chance he will receive a ¿100 ticket on route A and only a 30% chance he will receive a ¿100 ticket on route B. The police force has three options; namely, full police force on route A, full police force on route B and a 5050 split of the force on routes A and B. Determine the police payo matrix and determine the optimal mixed strategy for Alan and the police. (13 marks) AMA405 1 Turn Over AMA405 2 (i) Given the Minimax Theorem, min i n X j =1 a ij q j = max j m X i =1 a ij p i = ! ; prove the following statements: (a) There exists at least one pure strategy for each player which, if used against his opponent's optimal mixed strategy (M), yields the value of the game ( ! ) . (b) Against an opponent's optimal mixed strategy (M) a pure strategy cannot yield a higher expected payo than M. (c) A player's optimal mixed strategy (M) contains no pure strategy which yields less than the value of the game, when that pure strat egy is used against an opponent's optimal mixed strategy. (11 marks) (ii) Consider the IP problem max z = 3 x 1 + x 2 subject to 2 x 1 x 2 6 3 x 1 + 9 x 2 45 where x 1 and x 2 are nonnegative integers. The optimal tableau for the relaxed continuous LP problem is x 1 x 2 y 1 y 2 Solution z 8 = 7 5 = 21 123 = 7 x 1 1 6 = 14 1 = 21 33 = 7 x 2 11/7 2 = 21 24 = 7 where y 1 and y 2 are the slack variables of the rst and second constraints....
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 Spring '13
 405
 Operations Research, Statistics, Linear Programming, Optimization, Simplex algorithm

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