4. Optimal solutions
Sun Woong Kim, Steven G. Heeringa and Peter W. Solenberger
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Given the set of
possible arrays in
,
consider the subset
where
A solution
set to a controlled selection problem
denoted as
is
the set of the arrays that have the required positive selection probabilities
.
This solution set, or simply a “solution” to the controlled selection problem,
is usually obtained by an algorithm to control the constraints in (3.1) through
(3.6). As described in the introduction, since Goodman and Kish (1950), many
algorithms for obtaining solutions to controlled selection problems have been
developed.
Until Groves and Hess (1975)
suggested a computer algorithm, most solutions were manually obtained in a
process that resembled solving a mathematical puzzle. Furthermore, for most
problems it is possible that there is more than one solution set that meets the
constraints. Since the 1980s, the computer-intensive controlled selection
algorithms using transportation theory, network flow, integer programming, and
LP have been developed. These algorithms may depend on highly specialized
software or may be programmed to run in major software systems.
However, previous
solutions ranging from manual to computer-intensive algorithms have rarely been
compared empirically using a standard set of performance criteria. Therefore,
we begin with the description of a concept called optimal solution sets, or more simply, optimal solutions.
The controlled selection
problem
is only one array, but there may be many
possible arrays in
.
Also, only one array
from any solution to
is randomly chosen by
as the basis for choosing the stratified
sample. In general then, we might define an optimal solution as that satisfying
the following requirements (R1 and
R2):
R1.
The solution is obtained based on appropriate and objective measurements of the closeness between
and every single array
in
.
R2.
The solution, as much as possible, maximizes the probabilities of selection
over the arrays nearest to
under such measurements as referenced in R1.
The remainder of this section will address how to
specify R1 and R2 for optimal solutions. First, in order to define closeness in R1, a real number
representing the distance between
and
,
can be considered, where
is a distance function that satisfies the following
axioms:
(i)
(ii)
;
(iii)
Axiom (iii) is termed the triangle inequality axiom. Distance
functions satisfying (i), (ii), and (iii) can be defined by using the two-ordered
and
for
and
.
We first define the ordinary or Euclidean
distance (2-norm distance):
This
function is probably the most familiar measure to define the distance between
and
.
Also, we can define the
function called the Chebyshev distance (infinite
norm distance):
These distance functions give rise to distinct distance
spaces. Owing to (3.2), for any
,
the following holds.
and
For instance, for the
array in Problem 2.1 and the
array in Problem 2.3,
and
,
respectively.
Second, as mentioned in
R2, regarding the arrays nearest to
under such measurements described in R1,
consider the set of arrays in
having the minimum
or
value from
. Let
be the set of the arrays having the minimum
value from
and
be the set of the arrays having the minimum
value from
.
Assuming that all
possible arrays in
are known, we define optimum arrays as follows.
Definition. The arrays in
are called the optimum arrays.
Note that in the new
algorithm for controlled selection to be described in Section 6,
or
are chosen based on preference. We avoid
defining the intersection of
and
as the optimum arrays because this may exclude
the other arrays not in
with the same minimum
(
) value. We illustrate below that there may
exist a very small number of optimum arrays relative to the number of all
possible arrays in
for any
.
The details of how to find all possible arrays will be described in Section 6
and Section 7.
Illustrations.
For Problem 2.1 through
Problem 2.4, it is noted that
.
Thus, it is possible to use
only in illustrating the optimum arrays.
For Problem 2.1,
there are six possible arrays satisfying (3.1), (3.2), (3.3), and (3.4). That
is,
, as
given in Table 4.1. There exists only one optimum array,
,
with the minimum value of
.
Table 4.1
Controlled selection problem, optimum array with
and the other arrays
Table summary
This table displays the results of
Controlled selection problem. The information is grouped by
, , , , , , (appearing as column headers).
|
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0.8 |
0.5 |
0.7 |
0.8 |
0.8 |
0.8 |
Problem 2.2 has
30 possible arrays, and there are three optimum arrays, shown in Table 4.2
Table 4.2
Optimum arrays with
Problem 2.3 has 141
possible arrays. There are six optimum arrays, where each array has the same
.
One of them is given in Table 4.3.
Table 4.3
One of six optimum arrays with
There are 159
possible arrays for Problem 2.4, and there is only one optimum array given in
Table 4.4.
Table 4.4
Optimum array with
Accordingly, based on the
definition of optimum arrays as well as
and
satisfying the axioms (i), (ii), and (iii), we
suggest the following specifications (S1 and S2) of R1 and R2 of optimal solutions:
S1.
The solution is based on the values of the distance
between
and every single array
in
.
S2.
The solution maximizes the probabilities of selection of optimum arrays.
S1 and S2 will be the
rudiments of a new algorithm presented in Section 6, and in the next section we
turn into the discussion on the previous algorithms from the viewpoint of
optimal solutions.
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