MOSAlloc (version 1.2.3)

mosallocStepwiseFirst: Multiobjective sample allocation for constraint multivariate and multidomain optimal allocation in survey sampling (a stepwise optimality procedure is processed first to force Pareto optimality of the solution)

Description

Computes solutions to standard sample allocation problems under various precision and cost restrictions. The input data is transformed and parsed to the Embedded COnic Solver (ECOS) from the 'ECOSolveR' package. Multiple survey purposes are optimized simultaneously through a stepwise weighted Chebyshev minimization which forces Pareto optimality of solutions (cf. mosalloc()).

Usage

mosallocStepwiseFirst(
  D,
  d,
  A = NULL,
  a = NULL,
  C = NULL,
  c = NULL,
  l = 2,
  u = NULL,
  opts = list(sense = "max_precision", init_w = 1, mc_cores = 1L, max_iters = 100L)
)

Value

The function mosallocStepwiseFirst() returns a list containing the following components:

$w The initial preference weighting opts$init_w.

$n The vector of optimal sample sizes.

$J The optimal objective vector.

$Objective The objective value with respect to decision funtional f. NULL if opts$f = NULL.

$Utopian Always NULL (consitency to mosalloc()

output). NULL if opts$f = NULL.

$Normal The vector normal to the Pareto frontier at $J.

$dfJ Always NULL (consitency to mosalloc()

output).

$Sensitivity The dual variables of the objectives and constraints.

$Qbounds The Quality bounds of the Lorentz cones.

$Dbounds The weighted objective constraints ($w * $J).

$Scalepar An internal scaling parameter.

$Ecosolver A list of ECOSolveR returns including:

...$Ecoinfostring The info string of ECOSolveR::ECOS_csolve().

...$Ecoredcodes The redcodes of ECOSolveR::ECOS_csolve().

...$Ecosummary Problem summary of ECOSolveR::ECOS_csolve().

$Timing Run time info.

$Iteration Always NULL (consitency to mosalloc() output).

Arguments

D

(type: matrix) The objective matrix. A matrix of either precision or cost units.

d

(type: vector) The objective vector. A vector of either fixed precision components (e.g. finite population corrections) or fixed costs.

A

(type: matrix) A matrix of precision units for precision constraints.

a

(type: vector) The right-hand side vector ofthe precision constraints.

C

(type: matrix) A matrix of cost coefficients for cost constraints

c

(type: vector) The right-hand side vector of the cost constraints.

l

(type: vector) A vector of lower box constraints.

u

(type: vector) A vector of upper box constraints.

opts

(type: list) The options used by the algorithms:
$sense (type: character) Sense of optimization (default = "max_precision", alternative "min_cost").
$init_w (type numeric or matrix) Preference weightings (default = 1; The weight for first objective component must be 1).
$mc_cores (type: integer) The number of cores for parallelizing multiple input weightings stacked rowwise (default = 1L).
max_iters (type: integer) The maximum number of iterations (default = 100L).

References

See:

Willems, F. (2025). A Framework for Multiobjective and Uncertain Resource Allocation Problems in Survey Sampling based on Conic Optimization (Doctoral dissertation). Trier University. tools:::Rd_expr_doi("10.25353/ubtr-9200-484c-5c89").

Examples

# Artificial population of 50 568 business establishments and 5 business
# sectors (data from Valliant, R., Dever, J. A., & Kreuter, F. (2013).
# Practical tools for designing and weighting survey samples. Springer.
# https://doi.org/10.1007/978-1-4614-6449-5, Example 5.2 pages 133-9)

# See also https://umd.app.box.com/s/9yvvibu4nz4q6rlw98ac/file/297813512360
# file: Code 5.3 constrOptim.example.R

Nh <- c(6221, 11738, 4333, 22809, 5467) # stratum sizes
ch <- c(120, 80, 80, 90, 150) # stratum-specific cost of surveying

# Revenues
mh.rev <- c(85, 11, 23, 17, 126) # mean revenue
Sh.rev <- c(170.0, 8.8, 23.0, 25.5, 315.0) # standard deviation revenue

# Employees
mh.emp <- c(511, 21, 70, 32, 157) # mean number of employees
Sh.emp <- c(255.50, 5.25, 35.00, 32.00, 471.00) # std. dev. employees

# Proportion of estabs claiming research credit
ph.rsch <- c(0.8, 0.2, 0.5, 0.3, 0.9)

# Proportion of estabs with offshore affiliates
ph.offsh <- c(0.06, 0.03, 0.03, 0.21, 0.77)

budget <- 300000 # overall available budget
n.min  <- 100 # minimum stratum-specific sample size

#----------------------------------------------------------------------------
# Problem: Minimization of the maximum relative variation of estimates for
# the total revenue, the number of employee, the number of businesses claimed
# research credit and the number of businesses with offshore affiliates
# subject to cost restrictions

l <- rep(n.min, 5) # minimum sample size ber stratum
u <- Nh            # maximum sample size per stratum
C <- rbind(ch, ch * c(-1, -1, -1, 0, 0))
c <- c(budget, - 0.5 * budget)
A <- NULL # no precision constraint
a <- NULL # no precision constraint

# Variance components for multidimensional objective
D <- rbind(Sh.rev**2 * Nh**2/sum(Nh * mh.rev)**2,
           Sh.emp**2 * Nh**2/sum(Nh * mh.emp)**2,
           ph.rsch * (1 - ph.rsch) * Nh**3/(Nh - 1)/sum(Nh * ph.rsch)**2,
           ph.offsh * (1 - ph.offsh) * Nh**3/(Nh - 1)/sum(Nh * ph.offsh)**2)

d <- as.vector(D %*% (1 / Nh)) # finite population correction

opts = list(sense = "max_precision",
            init_w = 1,
            mc_cores = 1L,
            max_iters = 100L)

res1 <- mosallocStepwiseFirst(D = D, d = d, C = C, c = c, l = l, u = u,
                              opts = opts)
w <- res1$J[1] / res1$J
w # [1] 1.000000 3.879692 2.653655 1.000000

opts = list(sense = "max_precision",
            init_w = w,
            mc_cores = 1L,
            max_iters = 100L)

res2 <- mosallocStepwiseFirst(D = D, d = d, C = C, c = c, l = l, u = u,
                              opts = opts)
res2$w # [1] 1.000000 3.879692 2.653655 1.000000

# Compare to function mosalloc (without stepwise procedure)
opts = list(sense = "max_precision",
            f = NULL, df = NULL, Hf = NULL,
            init_w = w,
            mc_cores = 1L, pm_tol = 1e-05,
            max_iters = 100L, print_pm = FALSE)
res3 <- mosalloc(D = D, d = d, C = C, c = c, l = l, u = u, opts = opts)

# Compare objectives
rbind(res1$J, res2$J, res3$J)
#            [,1]         [,2]         [,3]       [,4]
#[1,] 0.00170589 0.0004396972 0.0006428453 0.00170589
#[2,] 0.00170589 0.0004396971 0.0006428420 0.00170589
#[3,] 0.00170589 0.0004396971 0.0006428440 0.00170589

# Compare optimal sample sizes
rbind(res1$n, res2$n, res3$n)
#          [,1]     [,2]     [,3]     [,4]     [,5]
# [1,] 958.0510 290.7446 147.1789 602.8856 638.2686
# [2,] 958.0455 290.7447 147.1871 602.8847 638.2692
# [3,] 958.0488 290.7446 147.1822 602.8853 638.2688