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STOCHASTIC PROGRAMMING

1. Solve the following problem
min 3z1 + 2z2 (1)
s.t.
P (2z1 + 3z2 ≥ ξ1) ≥ 0.90
P (3z1 + 5z2 ≥ ξ2) ≥ 0.95
z1 ≥ 0, z2 ≥ 0,
where ξ1, ξ2 are independent random variables uniformly distributed
on [0,100].
2. A news vendor goes to publisher every morning and buys x newspapers
at a price c > 0 per paper. The demand for newspapers varies over
days and is described by a random variable D. If the morning purchase
is less than the demand, the news vendor has to pay a backorder cost
b > c per paper to match the demand. Otherwise, if the morning
purchase exceeds the demand the news vendor has to pay a holding
cost h > 0. The problem can be formulated as 1
min ED[cx + b(D − x)+ + h(x − D)+] (2)
subject to
x ≥ 0
Assume that the demand is uniformly distributed on the interval [0,100],
c = 2, b = 4 and h = 1.
(a) Find the solution to problem (2).
(b) Find the cost of the optimal solution to problem (2).
1For a number a ∈ R, (a)+ denotes the maximum max{a, 0}
1
(c) Find the cost of the expected value solution to problem (2).
(d) Find the expected value of perfect information.
(e) Find the value of the stochastic solution.
3. Consider the following stochastic problems:
min z1 + z2 (3)
s.t.
P(z1 ≥ ξ1, z2 ≥ ξ2) ≥ 0.9
z1 ≥ 0, z2 ≥ 0,
and
min z1 + z2 (4)
s.t.
P(z1 ≥ ξ1) ≥ 0.9
P(z2 ≥ ξ2) ≥ 0.9
z1 ≥ 0, z2 ≥ 0,
where ξ1, ξ2 are independent standard normal random variables.
(a) Find the optimal solutions for both problems and compare the
optimal objective values.
(b) Choose the probabilities α1, α2 replacing 0.9 in the second problem
so that the optimum objective value would become equal to the
optimum value of the first problem.

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