Optimization Modeling

17.5. A worked example 195
This equality turns out to be a special instance of the following theorem.
Let the symbol ¯l denote any particular level of the event tree. Then the following
equality holds for each ¯l.
Theorem
¯l∑
∑
l=0 s∈L(l)
p s v s =
∑
p s w s
s∈L(¯l)
Note that when ¯l is equal to ˆl, then the term on the left of the equal sign is
nothing else but ∑ s p s v s , and the alternative formulation follows directly from
the theorem.
Corollary
The proof will be carried out by induction on the number of levels ¯l. For ¯l = 0,
the theorem holds trivially. Consider any particular ¯l >0, and assume that
the theorem holds for ¯l − 1. Then to prove that the theorem also holds for
¯l, you need to rewrite summations, use the above recursive definitions of ps
and w s , use the fact that ∑ k | α k=s π k = 1, and, of course, use the statement
of the theorem for ¯l − 1 during the last step. All this is done in the following
statements.
∑
p s w s =
∑
p s (v s + w αs )
s∈L(¯l)
s∈L(¯l)
= ∑
s∈L(¯l)
= ∑
s∈L(¯l)
= ∑
s∈L(¯l)
= ∑
s∈L(¯l)
= ∑
=
s∈L(¯l)
¯l∑
∑
l=0 s∈L(l)
p s v s +
p s v s +
p s v s +
p s v s +
p s v s +
p s v s
∑
s∈L(¯l)
∑
p s w αs
∑
s∈L(¯l−1) k | α k=s
∑
s∈L(¯l−1)
∑
s∈L(¯l−1)
¯l−1 ∑ ∑
l=0 s∈L(l)
p s w s
p s w s
p s v s
π k p αk w αk
∑
k | α k=s
π k
Proof
□
17.5 A worked example
In this section an input data set is provided together with an overview of the results
based on the multi-stage programming model developed in Section 17.3.
The initial probabilities are the same as in Figures 17.2 and 17.3. The revenues
This section