two_stage_design: Two-stage allocation with the primary-unit cost taken from...

In fieldopt: Area-Frame and Dual-Frame Survey Design under Real Field Cost

Two-stage allocation with the primary-unit cost taken from the routing

Description

two_stage_allocation() needs the cost c1 of visiting a primary unit, which is not a constant: a tour shares its travel among the units it visits, so the routed cost G(n) of visiting n units grows like ⁠c0 + a n + b sqrt(n)⁠ (Beardwood, Halton and Hammersley). This function measures G(n) by routing samples at a few sizes around the current allocation, fits that curve, and re-allocates with the marginal cost ⁠G'(n)⁠ in place of c1; with a non-linear cost the optimal number of secondary units is Cochran's formula with the marginal, not the average, cost per primary unit, as in Hansen, Hurwitz and Madow's travel-cost model. Under a budget the number of primary units then comes from the fitted curve, so the design spends the budget at its real routed cost. It stops when n no longer changes (or after max_iter rounds; a two-cycle stops with the better of the two designs).

Usage

two_stage_design(
  frame,
  depot,
  cost_model,
  m_secondary,
  s2_between,
  s2_within,
  target_variance = NULL,
  target_cv = NULL,
  budget = NULL,
  mean = NULL,
  c1_start = NULL,
  max_iter = 8,
  size = NULL,
  strata = NULL,
  selection = c("lpm", "systematic", "srs"),
  matrix = NULL,
  method = c("haversine", "euclidean", "osrm"),
  max_length = Inf,
  max_stops = Inf,
  n_rep = 5,
  iterations = 100,
  seed = 1
)

Arguments

frame	Data frame of units as in select_units(), including the depot row named in depot.
depot	Name of the depot unit in frame$unit.
cost_model	A field_cost_model().
m_secondary	Secondary units per primary unit in the population.
s2_between, s2_within	Variance among primary-unit means and within primary units.
target_variance, target_cv, budget	Exactly one: target variance of the estimated total, target coefficient of variation (needs mean), or budget.
mean	Population mean per secondary unit, for target_cv.
c1_start	Starting value of the cost per primary unit (default: the per-visit cost plus the travel cost of the median trip from the depot).
max_iter	Maximum number of rounds.
size	Optional size column for probability-proportional-to-size selection.
strata	Optional stratum column, passed to select_units().
selection	Selection method of select_units(): "lpm", "systematic" or "srs".
matrix	Optional travel_matrix() of the frame (computed from the coordinates otherwise).
method	Distance method of travel_matrix() when matrix is NULL ("osrm" queries the road network once for the whole frame).
max_length, max_stops	Route limits passed to route_fieldwork().
n_rep	Routed samples per sample size of the cost curve.
iterations	GRASP iterations per routing.
seed	Seed.

Details

Using the average cost G(n)/n instead takes too few primary units: in the package's experiments the marginal rule was within 0.2 percent of the cost of the best design found by exhaustive search, the average rule up to 3 percent above it, and a constant c1 guessed from depot distances 7 to 19 percent above.

Value

The final fieldopt_two_stage allocation, with cost the routed cost ⁠G(n) + c2 n m⁠, and the extra elements c1 (the marginal cost used), c1_average (G(n)/n), cost_curve (the routed points: n, cost_mean, cost_sd), cost_coef (c0, a, b), history (one row per round: round, c1, c1_average, n, m, cost, variance_total) and converged.

References

Hansen, M. H., Hurwitz, W. N. and Madow, W. G. (1953). Sample Survey Methods and Theory, Vol. II, Chapter 6, Section 11 (two-stage optimum with a travel cost proportional to sqrt(m)). Cochran, W. G. (1977). Sampling Techniques, 3rd ed., Section 10.6. Beardwood, J., Halton, J. H. and Hammersley, J. M. (1959). The shortest path through many points. Proc. Cambridge Phil. Soc., 55, 299-327.

Examples

set.seed(2)
cells <- expand.grid(x = 1:12, y = 1:12)
cells$unit <- paste0("c", seq_len(nrow(cells)))
frame <- rbind(data.frame(unit = "depot", x = 6, y = 6), cells)
model <- field_cost_model(
  per_travel = 3, per_unit = 40, per_interview = 20,
  interviews_per_unit = 1
)
two_stage_design(frame, "depot", model,
  m_secondary = 12, s2_between = 9, s2_within = 40,
  target_cv = 0.05, mean = 10, method = "euclidean", n_rep = 2,
  iterations = 20
)

fieldopt documentation built on Oct. 22, 2026, 1:07 a.m.