stochastic project

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1. Consider the second-stage problem defined by
min
y
2y1 + y2
s.t. y1 + y2 ≥ 1 − x1,
y1 ≥ ξ − x1 − x2,
y1, y2 ≥ 0.
Show that this program has a finite recourse if E[ξ] is finite.
2. Consider the second-stage problem defined by
min
y
2y1 + y2
t.q. y1 − y2 ≤ 2 − ξx1,
y2 ≤ x2,
y1, y2 ≥ 0.
Determine K2(ξ) and K2 for
(a) ξ ∼ U[0, 1] ;
(b) ξ ∼ Poisson(λ), λ > 0. (P[X = k] = λke−λ/k!)
What properties can we reasonably expect for K2 ?

3. We consider a baker who decides the morning how many breads he will
cook, and can sell his/her production during the day with profit. In the
evening, the unsold items can be sold at a reduced price as the breads are
less fresh. Two types of bread can be produced : white bread and whole
wheat bread. A unit of white bread cost 1.5$ to produce, while the whole
wheat bread costs 1.8$. The are sold at 3$ and 4$ per unit respectively.
The unsold breads are sold the evening at 1$ et 1.2$. 200g of flour are
required to produce one unit of whole wheat bread, while for one unit of
white bread, 150g are sufficient. The baker has a total of 12kg of flour.
We assume that the demand for white bread follows a normal distribution
with mean 50 and standard devision 5, and the demand for whole wheat
bread follows a normal distribution with mean 30 and standard deviation
2. The two demands are correlated, with a covariance between of 0.4.