The average over the worst scenarios

cvar(x, alpha) is the average of x over the worst 1 - alpha share of the scenarios, the ones in which x is largest. With alpha = 0.95, it is the average over the worst 5%. The measure is known as the conditional value at risk. Minimizing it asks for a decision that keeps the bad scenarios as good as possible, rather than the average one.

Usage

cvar(x, alpha)

Arguments

x

A random expression (see is_random()).

alpha

A number strictly between 0 and 1. It cannot depend on a decision.

Value

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

Details

x is read as a cost: large values are bad. For a profit, use the negative, cvar(-profit, 0.95).

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, cvar(cost, 0.95))              # the average cost of the worst 5% of scenarios