Air conditioning

This tutorial was generated using Literate.jl. Download the source as a .jl file. Download the source as a .ipynb file.

Taken from Anthony Papavasiliou's notes on SDDP This is a variation of the problem that first appears in the book Introduction to Stochastic Programming by Birge and Louveaux, 1997, Springer-Verlag, New York, on page 237, Example 1. For a rescaled problem, they reported an optimal value of 6.25 with a first-stage solution of x1 = 2 (production)and y1 = 1 (store production). On this variation, without rescaling, it would be equivalent to 62500, 200 and 100, respectively.

Consider the following problem

• Produce air conditioners for 3 months
• 200 units/month at 100 $/unit
• Overtime costs 300 $/unit
• Known demand of 100 units for period 1
• Equally likely demand, 100 or 300 units, for periods 2, 3
• Storage cost is 50 $/unit
• All demand must be met

The known optimal solution is $62,500

using SDDP, HiGHS, Test

function air_conditioning_model(duality_handler)
    model = SDDP.LinearPolicyGraph(;
        stages = 3,
        lower_bound = 0.0,
        optimizer = HiGHS.Optimizer,
    ) do sp, stage
        @variable(
            sp,
            0 <= stored_production <= 100,
            Int,
            SDDP.State,
            initial_value = 0
        )
        @variable(sp, 0 <= production <= 200, Int)
        @variable(sp, overtime >= 0, Int)
        @variable(sp, demand)
        DEMAND = [[100.0], [100.0, 300.0], [100.0, 300.0]]
        SDDP.parameterize(ω -> JuMP.fix(demand, ω), sp, DEMAND[stage])
        @constraint(
            sp,
            stored_production.out ==
            stored_production.in + production + overtime - demand
        )
        @stageobjective(
            sp,
            100 * production + 300 * overtime + 50 * stored_production.out
        )
    end
    SDDP.train(model; duality_handler = duality_handler)
    lb = SDDP.calculate_bound(model)
    println("Lower bound is: $lb")
    @test isapprox(lb, 62_500.0, atol = 0.1)
    sims = SDDP.simulate(model, 1, [:production, :stored_production])
    x1 = sims[1][1][:production]
    y1 = sims[1][1][:stored_production].out
    @test isapprox(x1, 200, atol = 0.1)
    @test isapprox(y1, 100, atol = 0.1)
    println(
        "With first stage solutions $(x1) (production) and $(y1) (stored_production).",
    )
    return
end

for duality_handler in [SDDP.LagrangianDuality(), SDDP.ContinuousConicDuality()]
    air_conditioning_model(duality_handler)
end

-------------------------------------------------------------------
         SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
  nodes           : 3
  state variables : 1
  scenarios       : 4.00000e+00
  existing cuts   : false
options
  solver          : serial mode
  risk measure    : SDDP.Expectation()
  sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
  VariableRef                             : [6, 6]
  AffExpr in MOI.EqualTo{Float64}         : [1, 1]
  VariableRef in MOI.GreaterThan{Float64} : [4, 4]
  VariableRef in MOI.Integer              : [3, 3]
  VariableRef in MOI.LessThan{Float64}    : [2, 3]
numerical stability report
  matrix range     [1e+00, 1e+00]
  objective range  [1e+00, 3e+02]
  bounds range     [1e+02, 2e+02]
  rhs range        [0e+00, 0e+00]
-------------------------------------------------------------------
 iteration    simulation      bound        time (s)     solves  pid
-------------------------------------------------------------------
         1L  7.000000e+04  6.166667e+04  5.691469e-01         8   1
        40L  5.500000e+04  6.250000e+04  8.121698e-01       344   1
-------------------------------------------------------------------
status         : simulation_stopping
total time (s) : 8.121698e-01
total solves   : 344
best bound     :  6.250000e+04
simulation ci  :  6.091250e+04 ± 6.325667e+03
numeric issues : 0
-------------------------------------------------------------------

Lower bound is: 62500.0
With first stage solutions 200.0 (production) and 100.0 (stored_production).
-------------------------------------------------------------------
         SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
  nodes           : 3
  state variables : 1
  scenarios       : 4.00000e+00
  existing cuts   : false
options
  solver          : serial mode
  risk measure    : SDDP.Expectation()
  sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
  VariableRef                             : [6, 6]
  AffExpr in MOI.EqualTo{Float64}         : [1, 1]
  VariableRef in MOI.GreaterThan{Float64} : [4, 4]
  VariableRef in MOI.Integer              : [3, 3]
  VariableRef in MOI.LessThan{Float64}    : [2, 3]
numerical stability report
  matrix range     [1e+00, 1e+00]
  objective range  [1e+00, 3e+02]
  bounds range     [1e+02, 2e+02]
  rhs range        [0e+00, 0e+00]
-------------------------------------------------------------------
 iteration    simulation      bound        time (s)     solves  pid
-------------------------------------------------------------------
         1   3.000000e+04  6.250000e+04  4.096985e-03         8   1
        20   6.000000e+04  6.250000e+04  4.773283e-02       172   1
-------------------------------------------------------------------
status         : simulation_stopping
total time (s) : 4.773283e-02
total solves   : 172
best bound     :  6.250000e+04
simulation ci  :  5.675000e+04 ± 6.792430e+03
numeric issues : 0
-------------------------------------------------------------------

Lower bound is: 62500.0
With first stage solutions 200.0 (production) and 100.0 (stored_production).