opt_1sided  R Documentation 
Functions that implement selected optimal allocation algorithms that compute a solution to the optimal allocation problem defined in the language of mathematical optimization as follows.
Minimize
f(x_1,\ldots,x_H) = \sum_{h=1}^H \frac{A^2_h}{x_h}
subject to
\sum_{h=1}^H c_h x_h = c
and either
x_h \leq M_h, \quad h = 1,\ldots,H
or
x_h \geq m_h, \quad h = 1,\ldots,H,
where
c > 0,\, c_h > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0,\, h = 1,\ldots,H
,
are given numbers. The minimization is on \mathbb R_+^H
.
The inequality constraints are optional and user can choose whether and how
they are to be added to the optimization problem.
If onesided lower bounds m_h,\, h = 1,\ldots,H
, must be imposed, it
is then required that c \geq \sum_{h=1}^H c_h m_h
.
If onesided upper bounds M_h,\, h = 1,\ldots,H
, must be imposed, it
is then required that 0 < c \leq \sum_{h=1}^H c_h M_h
.
Lower bounds can be specified instead of the upper bounds only in case of the
LRNA algorithm. All other algorithms allow only for specification of
the upper bounds. For the sake of clarity, we emphasize that in the
optimization problem consider here, the lower and upper bounds cannot be
imposed jointly.
Costs c_h,\, h = 1,\ldots,H
, of surveying one element in stratum, can
be specified by the user only in case of the RNA and LRNA
algorithms. For remaining algorithms, these costs are fixed at 1, i.e.
c_h = 1,\, h = 1,\ldots,H
.
The following is the list of all the algorithms available to use along with the name of the function that implements a given algorithm. See the description of a specific function to find out more about the corresponding algorithm.
RNA  rna()
LRNA rna()
SGA sga()
SGAPLUS  sgaplus()
COMA  coma()
Functions in this family should not be called directly by the user. Use
opt()
or optcost()
instead.
rna(
total_cost,
A,
bounds = NULL,
unit_costs = 1,
check_violations = .Primitive(">="),
details = FALSE
)
sga(total_cost, A, M)
sgaplus(total_cost, A, M)
coma(total_cost, A, M)
total_cost 
( 
A 
( 
bounds 
(

unit_costs 
( 
check_violations 
( 
details 
( 
M 
( 
Numeric vector with optimal sample allocations in strata. In case
of the rna()
only, it can also be a list
with optimal sample
allocations and strata assignments (either to takeNeyman or takebound).
rna()
: Recursive Neyman Algorithm (RNA) and its
twin version, Lower Recursive Neyman Algorithm (LRNA)
dedicated to the allocation problem with onesided lowerbounds constraints.
The RNA is described in Wesołowski et al. (2021), while LRNA is
introduced in Wójciak (2023).
sga()
: StengerGabler type algorithm SGA, described in
Wesołowski et al. (2021) and in Stenger and Gabler (2005).
This algorithm solves the problem with onesided upperbounds constraints.
It also assumes unit costs are constant and equal to 1, i.e.
c_h = 1,\, h = 1,\ldots,H
.
sgaplus()
: modified StengerGabler type algorithm, described in
Wójciak (2019) as Sequential Allocation (version 1) algorithm.
This algorithm solves the problem with onesided upperbounds constraints.
It also assumes unit costs are constant and equal to 1, i.e.
c_h = 1,\, h = 1,\ldots,H
.
coma()
: Change of Monotonicity Algorithm (COMA),
described in Wesołowski et al. (2021).
This algorithm solves the problem with onesided upperbounds constraints.
It also assumes unit costs are constant and equal to 1, i.e.
c_h = 1,\, h = 1,\ldots,H
.
If no inequality constraints are added, the allocation is given by the Neyman allocation as:
x_h = \frac{A_h}{\sqrt{c_h}} \frac{n}{\sum_{i=1}^H A_i \sqrt{c_i}},
\quad h = 1,\ldots,H.
For stratified \pi
estimator of the population total with
stratified simple random sampling without replacement design in use,
the parameters of the objective function f
are:
A_h = N_h S_h, \quad h = 1,\ldots,H,
where N_h
is the size of stratum h
and S_h
denotes
standard deviation of a given study variable in stratum h
.
Wójciak, W. (2023).
Another Solution of Some Optimum Allocation Problem.
Statistics in Transition new series, 24(5) (in press).
https://arxiv.org/abs/2204.04035
Wesołowski, J., Wieczorkowski, R., Wójciak, W. (2021).
Optimality of the Recursive Neyman Allocation.
Journal of Survey Statistics and Methodology, 10(5), pp. 1263–1275.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/jssam/smab018")},
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.48550/arXiv.2105.14486")}
Wójciak, W. (2019). Optimal Allocation in Stratified Sampling Schemes.
MSc Thesis, Warsaw University of Technology, Warsaw, Poland.
http://home.elka.pw.edu.pl/~wwojciak/msc_optimal_allocation.pdf
Stenger, H., Gabler, S. (2005).
Combining random sampling and census strategies 
Justification of inclusion probabilities equal to 1.
Metrika, 61(2), pp. 137–156.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s001840400328")}
Särndal, C.E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling, Springer, New York.
opt()
, optcost()
, rnabox()
.
A < c(3000, 4000, 5000, 2000)
m < c(50, 40, 10, 30) # lower bounds
M < c(100, 90, 70, 80) # upper bounds
rna(total_cost = 190, A = A, bounds = M)
rna(total_cost = 190, A = A, bounds = m, check_violations = .Primitive("<="))
sga(total_cost = 190, A = A, M = M)
sgaplus(total_cost = 190, A = A, M = M)
coma(total_cost = 190, A = A, M = M)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.