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

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.

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.

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.

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
)
#> 
#> ── Two-stage allocation ────────────────────────────────────────────────────────
#> n = 61 primary units with m = 3 secondary units each (optimal m 3.83): cost
#> 6353, variance of the total 743400 (SE 862.2, CV 4.99%).