Background
Methods
A framework for cost-efficient exposure assessment
The objective function - precision of the mean
The cost function
The general optimization problem
The two-variable reduction
The one-variable substitution approach
Results
Case A: α = 1, β = 1
Case A; substitution and objective function
Case A; boundaries of the choice set
Case A; optimization inside the choice set
Combination of αand β
| ||||
---|---|---|---|---|
A: α= 1;β= 1 | B: α= 1;β≠1 | C: α≠1;β= 1 | D: α≠1;β≠1 | |
Budget restriction | 5 | 12 | 16 | NA |
Objective variance function; independent variable | 6; n
s
| 13; n
d
| 17; n
s
| NA |
Derivative of objective function | 7 and 7a | 14 | 18 and 18a | NA |
Condition for choosing lower choice set boundary | 8 | 15 | 19 | NA |
Condition for choosing upper choice set boundary | 9 | NA | NA | NA |
Internal n
s
| 10 | NA | NA | NA |
Internal n
d
| 11 | NA | NA | NA |
Case B: α = 1, β≠1
Case B; substitution and objective function
Case B; boundaries of the choice set
Case B; optimization inside the choice set
Case C: α≠1, β = 1
Case C; substitution and objective function
Case C; boundaries of the choice set
Case C; optimization inside the choice set
Case D: α≠1, β≠1
Numerical examples
a. (s
2
BS
, s
2
BD
, s
2
μWD
) = (2, 10, 10) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
α: | 0.50 | 0.75 | 1.00 | 1.25 | 1.50 | ||||||
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
| ||
(π
s
, π
d
, c
q
) |
β
| ||||||||||
(2, 10, 10) | 0.50 | (14, 2) | 0.86 | (14, 2) | 0.86 | (10, 3) | 0.87 | (13, 2) | 0.92 | (9, 3) | 0.96 |
0.75 | (24, 1) | 0.92 | (13, 2) | 0.92 | (9, 3) | 0.96 | (12, 2) | 1.00 | (8, 3) | 1.08 | |
1.00 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (11, 2) | 1.09 | (10, 2) | 1.20 | |
1.25 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.50 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
(11, 5.5, 5.5) | 0.50 | (17, 3) | 0.51 | (15, 3) | 0.58 | (11, 4) | 0.64 | (7, 7) | 0.69 | (5, 10) | 0.80 |
0.75 | (22, 2) | 0.55 | (14, 3) | 0.62 | (10, 4) | 0.70 | (6, 7) | 0.81 | (6, 6) | 0.89 | |
1.00 | (39, 1) | 0.56 | (18, 2) | 0.67 | (9, 4) | 0.78 | (8, 4) | 0.88 | (6, 5) | 1.00 | |
1.25 | (39, 1) | 0.56 | (32, 1) | 0.69 | (14, 2) | 0.86 | (7, 4) | 1.00 | (6, 4) | 1.17 | |
1.50 | (39, 1) | 0.56 | (32, 1) | 0.69 | (13, 2) | 0.92 | (6, 4) | 1.17 | (5, 4) | 1.40 | |
(20, 1, 1) | 0.50 | (49, 5) | 0.12 | (27, 7) | 0.18 | (14, 12) | 0.26 | (8, 23) | 0.36 | (6, 28) | 0.45 |
0.75 | (52, 4) | 0.13 | (21, 9) | 0.20 | (13, 12) | 0.28 | (8, 19) | 0.38 | (6, 23) | 0.48 | |
1.00 | (80, 2) | 0.15 | (26, 5) | 0.23 | (13, 9) | 0.32 | (8, 14) | 0.43 | (6, 17) | 0.53 | |
1.25 | (74, 2) | 0.16 | (27, 4) | 0.26 | (13, 7) | 0.37 | (8, 10) | 0.50 | (6, 12) | 0.61 | |
1.50 | (134, 1) | 0.16 | (29, 3) | 0.30 | (12, 6) | 0.44 | (9, 6) | 0.59 |
(5, 12)
|
0.73
| |
b. (
s
2
BS
, s
2
BD
, s
2
μWD
) = (11, 5.5, 5.5)
| |||||||||||
α: |
0.50
|
0.75
|
1.00
|
1.25
|
1.50
| ||||||
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
| ||
(2, 10, 10) | 0.50 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 |
0.75 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.00 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.25 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.50 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
(11, 5.5, 5.5) | 0.50 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (12, 2) | 1.38 | (9, 2) | 1.83 |
0.75 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (12, 2) | 1.38 | (9, 2) | 1.83 | |
1.00 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (9, 2) | 1.83 | |
1.25 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (8, 2) | 2.06 | |
1.50 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (8, 2) | 2.06 | |
(20, 1, 1) | 0.50 | (134, 1) | 0.16 | (44, 2) | 0.38 | (19, 4) | 0.72 | (11, 6) | 1.17 | (7, 14) | 1.68 |
0.75 | (134, 1) | 0.16 | (43, 2) | 0.38 | (18, 4) | 0.76 | (11, 5) | 1.20 | (7, 12) | 1.70 | |
1.00 | (134, 1) | 0.16 | (42, 2) | 0.39 | (19, 3) | 0.77 | (11, 4) | 1.25 | (7, 9) | 1.75 | |
1.25 | (134, 1) | 0.16 | (40, 2) | 0.41 | (18, 3) | 0.81 | (10, 5) | 1.32 | (7, 7) | 1.80 | |
1.50 | (134, 1) | 0.16 | (53, 1) | 0.42 | (20, 2) | 0.83 | (11, 3) | 1.33 | (7, 5) | 1.89 | |
c. (
s
2
BS
, s
2
BD
, s
2
μWD
) = (20, 1, 1)
| |||||||||||
α: |
0.50
|
0.75
|
1.00
|
1.25
|
1.50
| ||||||
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
|
(
n
s
, n
d
)
|
s
2
μ
| ||
(2, 10, 10) | 0.50 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 |
0.75 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.00 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.25 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
1.50 | (24, 1) | 0.92 | (23, 1) | 0.96 | (22, 1) | 1.00 | (20, 1) | 1.10 | (17, 1) | 1.29 | |
(11, 5.5, 5.5) | 0.50 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (10, 1) | 2.20 |
0.75 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (10, 1) | 2.20 | |
1.00 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (10, 1) | 2.20 | |
1.25 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (10, 1) | 2.20 | |
1.50 | (39, 1) | 0.56 | (32, 1) | 0.69 | (22, 1) | 1.00 | (15, 1) | 1.47 | (10, 1) | 2.20 | |
(20, 1, 1) | 0.50 | (134, 1) | 0.16 | (53, 1) | 0.42 | (22, 1) | 1.00 | (12, 2) | 1.75 | (8, 3) | 2.58 |
0.75 | (134, 1) | 0.16 | (53, 1) | 0.42 | (22, 1) | 1.00 | (12, 2) | 1.75 | (8, 3) | 2.58 | |
1.00 | (134, 1) | 0.16 | (53, 1) | 0.42 | (22, 1) | 1.00 | (12, 2) | 1.75 | (8, 2) | 2.63 | |
1.25 | (134, 1) | 0.16 | (53, 1) | 0.42 | (22, 1) | 1.00 | (12, 2) | 1.75 | (8, 2) | 2.63 | |
1.50 | (134, 1) | 0.16 | (53, 1) | 0.42 | (22, 1) | 1.00 | (12, 1) | 1.83 | (8, 2) | 2.63 |