solution of homework 9
Problem 1
a.
Let
f
n
(
x
) denote the minimal cost from n to the end of the planning hori
zon given that x is the cumulative requisitions after the requisitions of the nth
period have arrived. Because you could choose either processing the requisitions
or not. The dynamic program should be as the following:
f
n
(
x
) = min
{
K
+
X
j
P
j
f
n
+1
(
j
)
,cx
+
X
j
P
j
f
n
+1
(
x
+
j
)
}
The boundary condition is:
f
N
+1
(
x
)
≡
K
b.
From the dynamic program, we can see that in the nth period:
if
K
+
∑
j
P
j
f
n
+1
(
j
)
> cx
+
∑
j
P
j
f
n
+1
(
x
+
j
), we will decide not processing
the requisitions
if
K
+
∑
j
P
j
f
n
+1
(
j
)
< cx
+
∑
j
P
j
f
n
+1
(
x
+
j
), we will decide processing the
requisitions.
K
+
∑
j
P
j
f
n
+1
(
j
) is a constant value we do not know.
cx
+
∑
j
P
j
f
n
+1
(
x
+
j
)
is nondecreasing in x, this could be explained in two ways: 1) By intuition. More
requisitions will cause the cost nondecreasing 2) By induction. Since
f
N
+1
(
x
)
≡
K
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 Spring '12
 Richard
 Boundary value problem, pj, dynamic program

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