Routed cost curve of a design: the travel constant of Hansen, Hurwitz and Madow

Selects samples of each size in n_grid, routes each one from the depot (or bases), prices it with the cost model, and fits the expected cost as G(n) = c0 + a n + b sqrt(n). This is the cost function of Hansen, Hurwitz and Madow (1953, Vol. I, Chapter 6, Section 12), C0 sqrt(m) + C1 m, with b in the role of their travel constant C0 and a of the cost per primary unit C1. They derived C0 from points on a regular grid and called it a rough approximation; here it is measured on the frame, with its roads (when the matrix comes from OSRM), its depots and its route limits. The travel of a tour through n scattered points grows like sqrt(n) (Beardwood, Halton and Hammersley 1959), and fixed costs per visit and per route add the linear term.

Usage

routed_cost_curve(
  frame,
  depot,
  cost_model,
  n_grid,
  size = NULL,
  strata = NULL,
  selection = c("lpm", "systematic", "srs"),
  replicates = 1,
  matrix = NULL,
  method = c("haversine", "euclidean", "osrm"),
  max_length = Inf,
  max_stops = Inf,
  n_rep = 10,
  iterations = 100,
  seed = 1
)

# S3 method for class 'fieldopt_cost_curve'
predict(object, newdata, type = c("total", "average", "marginal"), ...)

# S3 method for class 'fieldopt_cost_curve'
tidy(x, ...)

# S3 method for class 'fieldopt_cost_curve'
glance(x, ...)

# S3 method for class 'fieldopt_cost_curve'
autoplot(object, ...)

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().

n_grid

Sample sizes to route (at least three distinct values for the three coefficients).

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".

replicates

Replicates of a "systematic" selection.

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.

iterations

GRASP iterations per routing.

seed

Seed.

newdata

Sample sizes (a numeric vector) for predict().

type

"total" for G(n), "average" for G(n)/n, "marginal" for G'(n).

...

Unused.

x, object

A fieldopt_cost_curve.

Value

An object of class fieldopt_cost_curve: a list with points (a tibble with n, cost_mean, cost_sd, travel_mean, routes_mean per routed size), coef (c0, a, b), r_squared, currency and the settings. predict() gives the total, average or marginal cost at any n; tidy(), glance() and autoplot() are available.

Details

The fit is constrained to be increasing and concave (a >= 0, b >= 0), so the marginal cost G'(n) = a + b / (2 sqrt(n)) is positive and falls with n. The marginal cost is the one the classical allocation formulas need (see two_stage_design()).

References

Hansen, M. H., Hurwitz, W. N. and Madow, W. G. (1953). Sample Survey Methods and Theory, Vol. I, Chapter 6, Sections 11-15, and Vol. II, Chapter 6, Section 11. Wiley. 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 = 1:15, y = 1:15)
cells$unit <- paste0("c", seq_len(nrow(cells)))
frame <- rbind(data.frame(unit = "depot", x = 8, y = 8), cells)
model <- field_cost_model(per_travel = 3, per_unit = 40)
curve <- routed_cost_curve(frame, "depot", model,
  n_grid = c(10, 20, 40, 80), method = "euclidean", n_rep = 2,
  iterations = 20
)
curve
#> 
#> ── Routed cost curve ───────────────────────────────────────────────────────────
#> G(n) = 3.9 + 39.17 n + 46.46 sqrt(n) (cost; 4 sample sizes from 10 to 80, 2
#> routed samples each, lpm selection; R-squared 1).
#> In Hansen, Hurwitz and Madow's notation: C0 = 46.46 (travel), C1 = 39.17 (per
#> primary unit).
#> At n = 30: average cost 47.79, marginal cost 43.41 per unit.
predict(curve, c(30, 60), type = "marginal")
#> [1] 43.41366 42.17136