The probability that a comparison holds

prob(a <= b) is the share of the scenarios in which a <= b holds: an estimate of its probability. It is a single number, so it can be used in a constraint. A requirement on a probability is called a chance constraint:

Usage

prob(rel)

Arguments

rel

A comparison of expressions, written with <= or >=.

Value

An expression with one probability, between 0 and 1, per element of the comparison.

Details

add(m, prob(demand <= stock) >= 0.9)

reads as "demand is met in at least 90% of the scenarios". The line holds two comparisons, which do different jobs: the inner one, demand <= stock, is the event checked in each scenario, and the outer one, >= 0.9, is the requirement on how often it happens. The margin argument of add() allows for the sampling error of the estimate.

The comparison is written with <= or >=, and at least one side must contain a random variable. == is not accepted, since the probability that a quantity which can take any value equals one particular value is 0. < and > are not accepted either, since for such a quantity they mean the same as <= and >=. For vector expressions, each element gets its own probability.

holds() turns the same comparison into a 0/1 value in each scenario, and expectation(holds(a <= b)) is the same number as prob(a <= b).

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)
add(m, prob(demand <= stock) >= 0.9)       # demand met in at least 90% of scenarios