Newsvendor
This tutorial was generated using Literate.jl. Download the source as a .jl file. Download the source as a .ipynb file.
This example is based on the classical newsvendor problem, but features an AR(1) spot-price.
V(x[t-1], ω[t]) = max p[t] × u[t]
subject to x[t] = x[t-1] - u[t] + ω[t]
u[t] ∈ [0, 1]
x[t] ≥ 0
p[t] = p[t-1] + ϕ[t]The initial conditions are
x[0] = 2.0
p[0] = 1.5
ω[t] ~ {0, 0.05, 0.10, ..., 0.45, 0.5} with uniform probability.
ϕ[t] ~ {-0.25, -0.125, 0.125, 0.25} with uniform probability.using SDDP, HiGHS, Statistics, Test
function joint_distribution(; kwargs...)
names = tuple([first(kw) for kw in kwargs]...)
values = tuple([last(kw) for kw in kwargs]...)
output_type = NamedTuple{names,Tuple{eltype.(values)...}}
distribution = map(output_type, Base.product(values...))
return distribution[:]
end
function newsvendor_example(; cut_type)
model = SDDP.PolicyGraph(
SDDP.LinearGraph(3),
sense = :Max,
upper_bound = 50.0,
optimizer = HiGHS.Optimizer,
) do subproblem, stage
@variables(subproblem, begin
x >= 0, (SDDP.State, initial_value = 2)
0 <= u <= 1
w
end)
@constraint(subproblem, x.out == x.in - u + w)
SDDP.add_objective_state(
subproblem,
initial_value = 1.5,
lower_bound = 0.75,
upper_bound = 2.25,
lipschitz = 100.0,
) do y, ω
return y + ω.price_noise
end
noise_terms = joint_distribution(
demand = 0:0.05:0.5,
price_noise = [-0.25, -0.125, 0.125, 0.25],
)
SDDP.parameterize(subproblem, noise_terms) do ω
JuMP.fix(w, ω.demand)
price = SDDP.objective_state(subproblem)
@stageobjective(subproblem, price * u)
end
end
SDDP.train(
model;
log_frequency = 10,
time_limit = 20.0,
cut_type = cut_type,
)
@test SDDP.calculate_bound(model) ≈ 4.04 atol = 0.05
results = SDDP.simulate(model, 500)
objectives =
[sum(s[:stage_objective] for s in simulation) for simulation in results]
@test round(Statistics.mean(objectives); digits = 2) ≈ 4.04 atol = 0.1
return
end
newsvendor_example(cut_type = SDDP.SINGLE_CUT)
newsvendor_example(cut_type = SDDP.MULTI_CUT)-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-23
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 8.51840e+04
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, 3]
AffExpr in MOI.LessThan{Float64} : [2, 2]
VariableRef in MOI.GreaterThan{Float64} : [3, 4]
VariableRef in MOI.LessThan{Float64} : [3, 3]
numerical stability report
matrix range [8e-01, 2e+00]
objective range [1e+00, 2e+00]
bounds range [1e+00, 1e+02]
rhs range [5e+01, 5e+01]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
10 5.250000e+00 4.888669e+00 1.688788e-01 1350 1
20 4.350000e+00 4.105556e+00 2.576878e-01 2700 1
30 5.000000e+00 4.099889e+00 3.567400e-01 4050 1
40 3.500000e+00 4.096596e+00 4.656050e-01 5400 1
50 5.250000e+00 4.095112e+00 6.099160e-01 6750 1
60 3.643750e+00 4.092827e+00 7.274399e-01 8100 1
70 2.643750e+00 4.091514e+00 8.448379e-01 9450 1
80 5.087500e+00 4.091301e+00 9.641309e-01 10800 1
90 5.062500e+00 4.090986e+00 1.082005e+00 12150 1
100 4.843750e+00 4.086917e+00 1.210888e+00 13500 1
110 3.400000e+00 4.085985e+00 1.338425e+00 14850 1
120 3.375000e+00 4.085819e+00 1.469565e+00 16200 1
130 5.025000e+00 4.085765e+00 1.606970e+00 17550 1
140 5.000000e+00 4.085640e+00 1.741043e+00 18900 1
150 3.500000e+00 4.085560e+00 1.880842e+00 20250 1
160 4.281250e+00 4.085411e+00 2.008721e+00 21600 1
170 4.562500e+00 4.085379e+00 2.137781e+00 22950 1
180 5.768750e+00 4.085379e+00 2.266474e+00 24300 1
190 3.468750e+00 4.085337e+00 2.401833e+00 25650 1
200 4.131250e+00 4.085202e+00 2.536483e+00 27000 1
210 4.512500e+00 4.085132e+00 2.671912e+00 28350 1
220 4.900000e+00 4.085130e+00 2.813945e+00 29700 1
230 4.025000e+00 4.085111e+00 2.957188e+00 31050 1
240 4.468750e+00 4.085094e+00 3.130660e+00 32400 1
250 4.062500e+00 4.085056e+00 3.268546e+00 33750 1
260 4.875000e+00 4.085019e+00 3.408702e+00 35100 1
270 3.850000e+00 4.084990e+00 3.549527e+00 36450 1
280 4.912500e+00 4.084971e+00 3.691307e+00 37800 1
290 2.987500e+00 4.084965e+00 3.838426e+00 39150 1
300 3.825000e+00 4.084940e+00 3.988740e+00 40500 1
310 3.250000e+00 4.084900e+00 4.136380e+00 41850 1
320 3.537500e+00 4.084889e+00 4.284208e+00 43200 1
330 3.950000e+00 4.084889e+00 4.422086e+00 44550 1
340 4.500000e+00 4.084886e+00 4.567498e+00 45900 1
350 5.000000e+00 4.084886e+00 4.712956e+00 47250 1
360 3.075000e+00 4.084864e+00 4.861357e+00 48600 1
370 3.500000e+00 4.084859e+00 5.018950e+00 49950 1
380 3.356250e+00 4.084854e+00 5.172823e+00 51300 1
390 5.500000e+00 4.084839e+00 5.330371e+00 52650 1
400 4.475000e+00 4.084834e+00 5.477461e+00 54000 1
410 3.750000e+00 4.084832e+00 5.631426e+00 55350 1
420 3.687500e+00 4.084832e+00 5.807130e+00 56700 1
430 4.337500e+00 4.084811e+00 5.966261e+00 58050 1
440 5.750000e+00 4.084811e+00 6.107514e+00 59400 1
450 4.937500e+00 4.084776e+00 6.266573e+00 60750 1
460 3.600000e+00 4.084776e+00 6.418093e+00 62100 1
470 4.387500e+00 4.084776e+00 6.565460e+00 63450 1
480 4.000000e+00 4.084771e+00 6.724265e+00 64800 1
490 2.975000e+00 4.084769e+00 6.882646e+00 66150 1
500 3.125000e+00 4.084769e+00 7.039539e+00 67500 1
510 4.250000e+00 4.084769e+00 7.203773e+00 68850 1
520 4.525000e+00 4.084767e+00 7.355491e+00 70200 1
530 3.875000e+00 4.084767e+00 7.520227e+00 71550 1
540 4.387500e+00 4.084762e+00 7.689125e+00 72900 1
550 5.287500e+00 4.084762e+00 7.854840e+00 74250 1
560 4.650000e+00 4.084762e+00 8.014209e+00 75600 1
570 3.062500e+00 4.084762e+00 8.174000e+00 76950 1
580 3.187500e+00 4.084758e+00 8.327413e+00 78300 1
590 3.812500e+00 4.084758e+00 8.502972e+00 79650 1
600 3.637500e+00 4.084746e+00 8.663835e+00 81000 1
610 3.925000e+00 4.084746e+00 8.824592e+00 82350 1
620 4.625000e+00 4.084746e+00 8.991285e+00 83700 1
630 4.218750e+00 4.084746e+00 9.159709e+00 85050 1
640 3.025000e+00 4.084746e+00 9.330123e+00 86400 1
650 2.993750e+00 4.084746e+00 9.491837e+00 87750 1
660 3.262500e+00 4.084746e+00 9.658766e+00 89100 1
670 3.575000e+00 4.084746e+00 9.831266e+00 90450 1
680 2.981250e+00 4.084746e+00 1.000096e+01 91800 1
690 4.187500e+00 4.084746e+00 1.017483e+01 93150 1
700 4.500000e+00 4.084746e+00 1.034269e+01 94500 1
710 3.225000e+00 4.084746e+00 1.051238e+01 95850 1
720 4.375000e+00 4.084746e+00 1.068776e+01 97200 1
730 2.650000e+00 4.084746e+00 1.086368e+01 98550 1
740 3.250000e+00 4.084746e+00 1.103796e+01 99900 1
750 4.725000e+00 4.084746e+00 1.122269e+01 101250 1
760 3.375000e+00 4.084746e+00 1.142509e+01 102600 1
770 5.375000e+00 4.084746e+00 1.160292e+01 103950 1
780 4.068750e+00 4.084746e+00 1.178571e+01 105300 1
790 4.412500e+00 4.084746e+00 1.197213e+01 106650 1
800 4.350000e+00 4.084746e+00 1.216172e+01 108000 1
810 5.900000e+00 4.084746e+00 1.235018e+01 109350 1
820 4.912500e+00 4.084746e+00 1.253547e+01 110700 1
830 4.387500e+00 4.084746e+00 1.271462e+01 112050 1
840 3.712500e+00 4.084746e+00 1.290220e+01 113400 1
850 5.375000e+00 4.084746e+00 1.308662e+01 114750 1
860 3.562500e+00 4.084746e+00 1.327653e+01 116100 1
870 3.075000e+00 4.084746e+00 1.346845e+01 117450 1
880 3.625000e+00 4.084746e+00 1.365377e+01 118800 1
890 2.937500e+00 4.084746e+00 1.383608e+01 120150 1
900 4.450000e+00 4.084746e+00 1.402761e+01 121500 1
910 4.200000e+00 4.084746e+00 1.422250e+01 122850 1
920 3.687500e+00 4.084746e+00 1.443890e+01 124200 1
930 4.687500e+00 4.084746e+00 1.462490e+01 125550 1
940 4.018750e+00 4.084746e+00 1.480239e+01 126900 1
950 4.625000e+00 4.084746e+00 1.497961e+01 128250 1
960 3.375000e+00 4.084746e+00 1.516323e+01 129600 1
970 3.812500e+00 4.084746e+00 1.534149e+01 130950 1
980 3.112500e+00 4.084746e+00 1.552260e+01 132300 1
990 3.600000e+00 4.084746e+00 1.570999e+01 133650 1
1000 5.500000e+00 4.084746e+00 1.589507e+01 135000 1
1010 3.187500e+00 4.084746e+00 1.607787e+01 136350 1
1020 4.900000e+00 4.084746e+00 1.627245e+01 137700 1
1030 3.637500e+00 4.084746e+00 1.646828e+01 139050 1
1040 3.975000e+00 4.084746e+00 1.665285e+01 140400 1
1050 4.750000e+00 4.084746e+00 1.683889e+01 141750 1
1060 4.437500e+00 4.084746e+00 1.704065e+01 143100 1
1070 5.000000e+00 4.084746e+00 1.723419e+01 144450 1
1080 4.143750e+00 4.084746e+00 1.744039e+01 145800 1
1090 5.625000e+00 4.084746e+00 1.761734e+01 147150 1
1100 3.475000e+00 4.084746e+00 1.780302e+01 148500 1
1110 4.156250e+00 4.084746e+00 1.799724e+01 149850 1
1120 4.450000e+00 4.084746e+00 1.818356e+01 151200 1
1130 3.225000e+00 4.084741e+00 1.837689e+01 152550 1
1140 5.375000e+00 4.084741e+00 1.855362e+01 153900 1
1150 4.800000e+00 4.084737e+00 1.874424e+01 155250 1
1160 3.300000e+00 4.084737e+00 1.892441e+01 156600 1
1170 4.356250e+00 4.084737e+00 1.910317e+01 157950 1
1180 3.900000e+00 4.084737e+00 1.929518e+01 159300 1
1190 4.450000e+00 4.084737e+00 1.948819e+01 160650 1
1200 5.156250e+00 4.084737e+00 1.968089e+01 162000 1
1210 4.512500e+00 4.084737e+00 1.986183e+01 163350 1
1218 3.287500e+00 4.084737e+00 2.001831e+01 164430 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.001831e+01
total solves : 164430
best bound : 4.084737e+00
simulation ci : 4.067790e+00 ± 4.080342e-02
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-23
-------------------------------------------------------------------
problem
nodes : 3
state variables : 1
scenarios : 8.51840e+04
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, 3]
AffExpr in MOI.LessThan{Float64} : [2, 2]
VariableRef in MOI.GreaterThan{Float64} : [3, 4]
VariableRef in MOI.LessThan{Float64} : [3, 3]
numerical stability report
matrix range [8e-01, 2e+00]
objective range [1e+00, 2e+00]
bounds range [1e+00, 1e+02]
rhs range [5e+01, 5e+01]
-------------------------------------------------------------------
iteration simulation bound time (s) solves pid
-------------------------------------------------------------------
10 4.312500e+00 6.850260e+00 1.805770e-01 1350 1
20 4.143750e+00 5.311682e+00 5.348649e-01 2700 1
30 4.018750e+00 4.053763e+00 9.704909e-01 4050 1
40 4.825000e+00 4.044604e+00 1.498003e+00 5400 1
50 4.400000e+00 4.040324e+00 2.118543e+00 6750 1
60 4.306250e+00 4.039161e+00 2.923838e+00 8100 1
70 4.781250e+00 4.039019e+00 3.886630e+00 9450 1
80 4.662500e+00 4.038969e+00 5.027150e+00 10800 1
90 3.750000e+00 4.038882e+00 6.210525e+00 12150 1
100 3.000000e+00 4.038855e+00 7.560289e+00 13500 1
110 4.350000e+00 4.038829e+00 8.956580e+00 14850 1
120 3.700000e+00 4.038810e+00 1.055849e+01 16200 1
130 3.075000e+00 4.038778e+00 1.210008e+01 17550 1
140 3.375000e+00 4.038777e+00 1.386517e+01 18900 1
150 4.625000e+00 4.038777e+00 1.575253e+01 20250 1
160 3.468750e+00 4.038770e+00 1.764538e+01 21600 1
170 3.750000e+00 4.038770e+00 1.987796e+01 22950 1
171 2.893750e+00 4.038770e+00 2.007726e+01 23085 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.007726e+01
total solves : 23085
best bound : 4.038770e+00
simulation ci : 4.107675e+00 ± 1.133143e-01
numeric issues : 0
-------------------------------------------------------------------