### COMP 567B
Integer Programming Assignment 3

** Due: March 26, 2007 (give to Conor (MC232) by 5pm)**

Additional exercises to follow.

1. This is a variation of the Farmer's problem. Note that here the
selling/buying price of wheat and corn also varies according to the
scenario.

Consider a farmer who has 500 acres and has to decide how much

land to devote to each of the three crops: wheat, corn and sugar beets. He

knows that he needs 200T of wheat and 240T of corn to feed his cattle.

These amounts can be raised or purchased. Any production in excess can be

sold at a price 170$/T for wheat and 150$/T for corn. The purchase price

is 40% higher than the selling price. The farmer can sell sugar beet at a

price 36$/T. However, for sugar beet that is in excess of the quota he can

only sell for 10$/T. His quota is 6000. The planting cost is 150$, 230$

and 260$ per acre for wheat, corn and sugar beets respectively. On

average, the per acre yields are 2.5T, 3T and 20T for wheat, corn and sugar beets.

With probability 30%, it will be a good weather next year, which means all

yields are 120% of the normal ones and the selling/purchasing price of

wheat/corn is 90% of the normal ones. With probability 30%, it will be

bad weather, which means a 80% yield for all crops, and 110% of the

selling/purchasing price of wheat/corn. With 40% of probability, the

weather is normal and all coefficients are equal to those given above.

(Note that the price for sugar beet remains the same for all three scenarios)

Formulate this problem into a stochastic linear problem, in both extensive

form and concise form.Solve the LP using CPLEX.

2. You own a depanneur and want to sell La Presse. They will sell them
to you at 50 cents each , you can sell them for $1, and you will
receive 25 cents for any unsold newspapers. The demand for newspapers
is normally distributed with mean 100 and standard deviation 20. (Check
wiki if you forget how this distribution is defined.) You can order
between zero and 200 newspapers. How many do you
order to maximize your expected profit?

Solve this problem by discretizing the normal distribution, then
formulating the extended integer
linear program including all scenarios. Solve the problem using CPLEX.
In discretizing, choose seven scenarios given by the mean and the 6
points +/- 1,2,3 standard deviations from the mean. Compare the answer
you get from the LP to the exact solution given by Xu in class.

3. Please see pdf file.