Math340/Week 1

Question 1 (Tuesday, May 12)

You are in charge of ordering sushi for an event. The order can include a combination of Yam, Avocado and Salmon rolls.

The prices are: Yam roll- $4, Avocado roll- $3, Salmon roll $5.5.

Due to the guests preferences, there should be at least as many salmon rolls as yam rolls. There should also be at least 4 non-salmon rolls for vegetarians.

a. Express the problem as a linear programming problem.

b. Guess one solution, and show that it is feasible.

Question 2 (Wednesday, May 13)

What is the value of "Alice plays pure" for the game with the following payoff matrix (representing Alice's payoffs):

$A={\begin{pmatrix}1&5\\4&-2\end{pmatrix}},$

And what is the value of Betty plays pure?

Question 3 (Thursday, May 14)

a. Using your answers for yesterday's question, what is the duality gap of $A$?

b. What is the value of "Alice plays mixed", and what is Alice's optimal strategy?

c. Express the problem of finding Alice's optimal strategy as a linear programming problem.

Question 4 (Friday, May 15)

Define the slack variables and write out the first dictionary (corresponding to the initial solution $x_{1}=x_{2}=0$) for the LP problem:

${\begin{array}{ll}maximize&4x_{1}+5x_{2}\\subject\ to&x_{1}+x_{2}\leq 5,\\&x_{1}+2x_{2}\leq 8,\\&2x_{1}+x_{2}\leq 8,\\&x_{1},x_{2}\geq 0.\end{array}}$