The variance and the standard deviation over the scenarios

How much a quantity varies from scenario to scenario, as opposed to what it is on average. expectation(cost) + 2 * std_dev(cost) is an objective that trades a low average cost against a steady one, and add(m, variance(cost) <= 100) limits the variation.

Usage

variance(x, sample = FALSE)

std_dev(x, sample = FALSE)

Arguments

x

A random expression (see is_random()).

sample

FALSE, the default, divides by the number of scenarios; TRUE divides by one less, as var() and sd() do.

Value

An expression of the same length as x, no longer random.

Details

By default the scenarios count as the whole distribution, each with weight 1 / n_scenarios, so variance(x) is expectation(x^2) - expectation(x)^2. R's var() and sd() divide by n_scenarios - 1 instead, because they estimate the variance of a population from a sample of it; sample = TRUE does the same. The two differ by a factor n_scenarios / (n_scenarios - 1), which changes the value reported but not which decision is best.

Unlike cvar(), these measure variation in both directions: a scenario far better than average increases them as much as one far worse.

Examples

m <- model()
stock  <- num_var(m, "stock", lower = 0, upper = 200)
demand <- rand_var(m, "demand", normal(100, 15))
set_scenarios(m, 512, seed = 42)
cost <- 3 * stock + 10 * max(demand - stock, 0)
minimize(m, expectation(cost) + 2 * std_dev(cost))