stratallo (version 3.0.1)

dopt: Multi-Domain Optimum Sample Allocation with Controlled-Precision under Upper-Bound Constraints

Description

Computes the optimum allocation for the following multi-domain optimum allocation problem, formulated in mathematical optimization terms:

Minimize $$f(T,\, \boldsymbol x) = T$$ over \(\mathbb R \times \mathbb R_+^{\lvert \mathcal H \rvert}\), subject to $$\sum_{(d,h) \in \mathcal H} x_{d,h} = n,$$ $$\sum_{h \in \mathcal H_d} (\frac{1}{x_{d,h}} - \frac{1}{N_{d,h}}) \frac{N_{d,h}^2 S_{d,h}^2}{\rho_d^2} = T, \qquad d \in \mathcal D,$$ $$x_{d,h} \leq N_{d,h}, \qquad (d,h) \in \mathcal H,$$ where:

\((T,\, \boldsymbol x) = (T,\, (x_{d,h},\, (d,h) \in \mathcal H))\)

the optimization variable,

\(\mathcal H \subset \mathbb N^2\)

the set of domain-stratum indices,

\(\mathcal D := \{d \in \mathbb N \colon\; \exists h,\, (d,h) \in \mathcal H\}\)

the set of domain indices,

\(\mathcal H_d := \{h \in \mathbb N \colon\; (d,h) \in \mathcal H\}\)

the set of strata indices in domain \(d\),

\(N_{d,h} > 0\)

size of stratum \((d,h)\),

\(S_{d,h} > 0\)

standard deviation of the study variable in stratum \((d,h)\),

\(\rho_d := t_d\, \sqrt{\kappa_d}\)

where \(t_d\) denotes the total in domain \(d\), i.e., the sum of the values of the study variable for population elements in domain \(d\), and \(\kappa_d\) is a priority weight for domain \(d\),

\(n \in (0,\, \sum_{(d,h) \in \mathcal H} N_{d,h}]\)

total sample size.

Usage

dopt(n, H_counts, N, S, total, kappa, return_T = FALSE)

Value

If return_T = FALSE (default), a numeric vector containing the optimal sample allocations \(x_{d,h}\) for each stratum \((d,h) \in \mathcal H\).

If return_T = TRUE, a list with components:

xopt

numeric vector of optimal sample allocations.

Topt

optimal value of the objective function \(T\).

Arguments

n

(integerish(1))
total sample size \(n\). Must satisfy 0 < n <= sum(N).

H_counts

(integerish)
strata counts in each domain.

N

(integerish)
strata sizes \((N_{d,h},\, (d,h) \in \mathcal H)\).

S

(numeric)
standard deviations \((S_{d,h},\, (d,h) \in \mathcal H)\) of surveyed variable in strata.

total

(numeric)
vector of domain totals, \(t_d,\, d \in \mathcal D\), i.e., the sum of the study variable over all population elements in each domain.

kappa

(numeric)
vector of priority weights for the domains, \(\kappa_d,\, d \in \mathcal D\).

return_T

(logical(1))
If TRUE, the function returns a list containing the optimal allocation and the optimal value of the objective function \(T\). If FALSE (default), only the optimal allocation vector is returned.

Details

The dopt() function uses the RDCA algorithm implemented in rdca().

References

WojciakPhDstratallo

See Also

rdca(), dca(), dca_nmax(), opt(), optcost().

Examples

# Three domains with 2, 2, and 3 strata, respectively,
# that is, H = {(1,1), (1,2), (2,1), (2,2), (3,1), (3,2), (3,3)}.
H_counts <- c(2, 2, 3)
# (N_{1,1}, N_{1,2}, N_{2,1}, N_{2,2}, N_{3,1}, N_{3,2}, N_{3,3})
N <- c(140, 110, 135, 190, 200, 40, 70)
# (S_{1,1}, S_{1,2}, S_{2,1}, S_{2,2}, S_{3,1}, S_{3,2}, S_{3,3})
S <- c(180, 20, 5, 4, 35, 9, 40)
total <- c(2, 3, 5)
kappa <- c(0.5, 0.2, 0.3)
n <- 828

# Optimum allocation.
dopt(n, H_counts, N, S, total, kappa)

# Example population with 9 domains and 278 strata
p <- pop9d278s
sum(p$N)
n <- 5000
x <- dopt(n, p$H_counts, p$N, p$S, p$total, p$kappa, return_T = TRUE)
x
all(x$xopt <= p$N)
sum(x$xopt)