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DECIS 209
corresponding equivalent deterministic linear program:
min z = cx + p 1 fy 1 + p 2 fy 2 + · · · + p W fy W
s/t Ax = b
−B 1 x + Dy 1 = d 1
−B 2 x + Dy 2 = d 2
.
.
.. .
−B W x + Dy W = d W
x, y 1 , y 2 , . . . , y W ≥ 0,
which contains all possible outcomes ω ∈ Ω. Note that for practical problems W is very large, e.g., a typical
number could be 10 20 , and the resulting equivalent deterministic linear problem is too large to be solved directly.
In order to see the two-stage nature of the underlying decision making process the folowing representation is also
often used:
min cx + E z ω (x)
Ax = b
x ≥ 0
where
z ω (x) = min f ω y ω
D ω y ω = d ω + B ω x
y ω ≥ 0, ω ∈ Ω = {1, 2, . . . , W }.
DECIS employs different strategies to solve two-stage stochastic linear programs. It computes an exact optimal
solution to the problem or approximates the true optimal solution very closely and gives a confidence interval
within which the true optimal objective lies with, say, 95% confidence.
1.3 Representing Uncertainty
It is favorable to represent the uncertain second-stage parameters in a structure. Using V = (V 1 , . . . , V h ) an
h-dimensional independent random vector parameter that assumes outcomes v ω = (v 1 , . . . , v h ) ω with probability
p ω = p(v ω ), we represent the uncertain second-stage parameters of the problem as functions of the independent
random parameter V :
f ω = f(v ω ), B ω = B(v ω ), D ω = D(v ω ), d ω = d(v ω ).
Each component V i has outcomes v ωi
i , ω i ∈ Ω i , where ω i labels a possible outcome of component i, and Ω i
represents the set of all possible outcomes of component i. An outcome of the random vector
consists of h independent component outcomes. The set
v ω = (v ω1
1 , . . . , vω h
h )
Ω = Ω 1 × Ω 2 × . . . × Ω h
represents the crossing of sets Ω i . Assuming each set Ω i contains W i possible outcomes, |Ω i | = W i , the set Ω
contains W = ∏ W i elements, where |Ω| = W represents the number of all possible outcomes of the random
vector V . Based on independence, the joint probability is the product
p ω = p ω1
1 pω2 2 · · · pω h
h .
Let η denote the vector of all second-stage random parameters, e.g., η = vec(f, B, D, d). The outcomes of η may
be represented by the following general linear dependency model:
η ω = vec(f ω , B ω , d ω , d ω ) = Hv ω ,
ω ∈ Ω
where H is a matrix of suitable dimensions. DECIS can solve problems with such general linear dependency
models.