Agriculture Reference
In-Depth Information
0 and the
feasible region is bounded by linear constraints. This implies that standard convex-
ity theorems can be used to prove that an optimal solution
The objective function
G
(
χ
) of Problem (
8.22
) is strictly convex if
χ>
χ
∗
always exists (see
Kokan and Khan
1967
). Additionally, Bethel (
1989
) used the Kuhn
Tucker theo-
rem to show that there are dual variables
ʻ
v
0, such that the optimal solution to
Problem (
8.22
)is
8
<
p
k
h
if
X
g
v
¼1
ʱ
v
a
vh
>
s
X
!
0
;
1
h H
q
g
v
¼1
ʱ
v
a
vh
X
k
h
X
v
ʱ
v
a
vh
H
ˇ
h
¼
ð
8
:
23
Þ
:
h
¼1
1
otherwise
X
v
¼1
ʻ
v
and therefore
X
g
g
v
¼1
ʱ
v
¼
ʻ
v
=
v
¼ 1.
where,
ʱ
The solution in Eq. (
8.23
) can only be used operationally if the normalized
Lagrange multipliers (
v
) are known. In the next sub-section, we describe an
algorithm for determining the optimal values,
ʱ
ʱ
v
.
If a solution is too expensive, it can be rescaled to suit the available budget using
a new optimal allocation that is constrained to being proportional to the original
solution. In this way,
ˇ
h
and
the precision of the sample estimates can be directly
determined.
Moreover, from Eq. (
8.23
) we can easily calculate the shadow prices (the partial
derivatives of the cost function with respect to the right hand side of the variance
constraints
χð =∂
k
h
). Therefore, we can use classic sensitivity analysis
methods to determine the cost reduction if one constraint is relaxed.
∂
G
8.4.1 Computational Aspects
The multipurpose allocation problem outlined in the previous section can be solved
using either the Bethel (Bethel
1989
) or Chromy (Chromy
1987
) algorithms. These
algorithms are iterative procedures that converge to the optimal solution of Problem
(
8.23
). We briefly summarize the two different algorithms in the following.
First, consider the Bethel algorithm. Let
ʴ
vz
¼ 1if
v
¼
z
, and
ʴ
vz
¼ 0 otherwise.
χ ðÞ
be the vector of variables that has an
h
-th entry
Let
ˇ
h
ðÞ
. This is calculated
t
. For optimal
* and
x
*
, we must
according to Eq. (
8.23
) for fixed
α
¼
ʱ
1
... ʱ
g
α
χ
α
ðÞ
¼
χ
∗
. The following steps are used to find
χ
∗
.
have that
t
,
so the algorithm has an initial solution that is the well-known optimal solution of
the univariate case (Neyman
1934
). Theoretically, every variable of interest in
ðÞ
v
1. For
s
¼1,
ʱ
¼
ʴ
v
1
,
v
¼ 1,
...
,
g
. In practice,
ʱ
¼ 10
ð
...
0
...
0
Þ
Search WWH ::
Custom Search