    -----------------------------------------------------------------------------
    -- |
    -- Module      :  Documentation.SBV.Examples.Optimization.Production
    -- Copyright   :  (c) Levent Erkok
    -- License     :  BSD3
    -- Maintainer  :  erkokl@gmail.com
    -- Stability   :  experimental
    --
    -- Solves a simple linear optimization problem
    -----------------------------------------------------------------------------

    module Documentation.SBV.Examples.Optimization.Production where

    import Data.SBV

    -- | Taken from <http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.html>
    --
    -- A company makes two products (X and Y) using two machines (A and B).
    --
    --   - Each unit of X that is produced requires 50 minutes processing time on machine
    --     A and 30 minutes processing time on machine B.
    --
    --   - Each unit of Y that is produced requires 24 minutes processing time on machine
    --     A and 33 minutes processing time on machine B.
    --
    --   - At the start of the current week there are 30 units of X and 90 units of Y in stock.
    --     Available processing time on machine A is forecast to be 40 hours and on machine B is
    --     forecast to be 35 hours.
    --
    --   - The demand for X in the current week is forecast to be 75 units and for Y is forecast
    --     to be 95 units.
    --
    --   - Company policy is to maximise the combined sum of the units of X and the units of Y
    --     in stock at the end of the week.
    --
    -- How much of each product should we make in the current week?
    --
    -- We have:
    --
    -- >>> optimize Lexicographic production
    -- Optimal model:
    --   X     = 45 :: Integer
    --   Y     =  6 :: Integer
    --   stock =  1 :: Integer
    --
    -- That is, we should produce 45 X's and 6 Y's, with the final maximum stock of just 1 expected!
    production :: Goal
    production = do x <- sInteger "X" -- Units of X produced
                    y <- sInteger "Y" -- Units of X produced

                    -- Amount of time on machine A and B
                    let timeA = 50 * x + 24 * y
                        timeB = 30 * x + 33 * y

                    constrain $ timeA .<= 40 * 60
                    constrain $ timeB .<= 35 * 60

                    -- Amount of product we'll end up with
                    let finalX = x + 30
                        finalY = y + 90

                    -- Make sure the demands are met:
                    constrain $ finalX .>= 75
                    constrain $ finalY .>= 95

                    -- Policy: Maximize the final stock
                    maximize "stock" $ (finalX - 75) + (finalY - 95)