# Maximising a minimum over continuous shares
#
# Divides a divisible resource so the worst-off share is as large as
# possible. Because the quantities are continuous the optimum equalises the
# shares and is generally fractional, which an integer formulation of the
# same idea would round away.
#
# Technique: t ≤ e_i for every i with t continuous, maximised — the
# continuous form of the maximin pattern.
#
# On this instance the optimum is 17.142857142857142 (maximisation), confirmed
# independently by two phase simplex before publication.
#
# Reference: Rawls, J. (1971). A Theory of Justice. Harvard University
# Press.
model m
set A = {1, 2}
const u = {6, 1}, forall a in A
const T = 20
var 0 <= x <= infty, forall a in A
var 0 <= t <= infty
max t
# t sits below every agent's utility, so maximising it lifts the worst off
constr t <= u_{a} * x_{a}, forall a in A
constr sum_{a in A}{x_{a}} <= T
end
import elytica
Read this through MCP: read with path: "formulations/maximin-continuous", or resource hlpl://formulations/maximin-continuous.