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strong duality and complementary slackness theorems
1) 1. A linear programming problem has three variables, x1, x2, and x3, and three constraints. The first constraint is 3x1 + 2x2 – 5x3 ≤ 150, but the other two constraints and the objective function are unknown. It is known that the optimal solution has an objective function value of 1025 and that x1=0, x2=75 and x3=25.
The objective function for the dual problem is min 150y1 + 250y2 - 200y3. The first constraint is
3y1 + y2 + 2y3 ≥ 3 and the second constraint is 2y1 + 3y2 – 3y3 ≥ 12. The third constraint is unkown.
Use the strong duality and complementary slackness theorems to determine the optimal solution (y1, y2, y3) to the dual problem. [Note, you do not need to find any of the missing constraints, there is enough information given here to solve the problem]
2. 1) Player A and B each choose a number between 1 and 5 and reveal them simultaneously.
a. Suppose that the player who chooses the larger number wins. If the row player chooses the larger number, she wins the sum of the two numbers from the columns player. If the column player wins, he wins the difference of the two numbers from the row players. If each player chooses the same number the game is a draw. What are the optimal strategies for each player, and what is the value of the game?
Next, suppose that we have a different set of payoffs. If the sum of the numbers is odd, then row player wins the sum of the numbers from the column player. If the sum of the numbers is even, then the column player wins the sum of the numbers from the row player. What are the optimal strategies for each player, and what is the value of the game?