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, 2017-25
-------------------------------------------------------------------
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.EqualTo{Float64} : [1, 1]
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 4.888859e+00 1.775420e-01 1350 1
20 4.350000e+00 4.105855e+00 2.724020e-01 2700 1
30 5.000000e+00 4.100490e+00 3.775969e-01 4050 1
40 3.475000e+00 4.097376e+00 5.134380e-01 5400 1
50 5.250000e+00 4.095862e+00 6.369178e-01 6750 1
60 3.643750e+00 4.093345e+00 7.618129e-01 8100 1
70 2.643750e+00 4.091824e+00 8.839879e-01 9450 1
80 5.087500e+00 4.091594e+00 1.012147e+00 10800 1
90 5.062500e+00 4.091314e+00 1.139704e+00 12150 1
100 4.843750e+00 4.086999e+00 1.278727e+00 13500 1
110 3.437500e+00 4.086095e+00 1.415127e+00 14850 1
120 3.375000e+00 4.085928e+00 1.553908e+00 16200 1
130 5.025000e+00 4.085864e+00 1.702627e+00 17550 1
140 5.000000e+00 4.085733e+00 1.844867e+00 18900 1
150 3.500000e+00 4.085659e+00 1.989906e+00 20250 1
160 4.281250e+00 4.085459e+00 2.154009e+00 21600 1
170 4.500000e+00 4.085433e+00 2.295878e+00 22950 1
180 5.768750e+00 4.085431e+00 2.439945e+00 24300 1
190 3.468750e+00 4.085369e+00 2.592200e+00 25650 1
200 4.131250e+00 4.085231e+00 2.744016e+00 27000 1
210 4.506250e+00 4.085165e+00 2.891733e+00 28350 1
220 4.900000e+00 4.085162e+00 3.040874e+00 29700 1
230 4.025000e+00 4.085143e+00 3.189551e+00 31050 1
240 4.468750e+00 4.085124e+00 3.345205e+00 32400 1
250 4.062500e+00 4.085084e+00 3.499794e+00 33750 1
260 4.875000e+00 4.085047e+00 3.656076e+00 35100 1
270 3.850000e+00 4.085019e+00 3.815183e+00 36450 1
280 4.912500e+00 4.085000e+00 3.971057e+00 37800 1
290 2.987500e+00 4.084993e+00 4.134289e+00 39150 1
300 3.825000e+00 4.084961e+00 4.309523e+00 40500 1
310 3.250000e+00 4.084914e+00 4.467330e+00 41850 1
320 3.537500e+00 4.084897e+00 4.627977e+00 43200 1
330 3.950000e+00 4.084897e+00 4.781134e+00 44550 1
340 4.500000e+00 4.084895e+00 4.939983e+00 45900 1
350 5.000000e+00 4.084893e+00 5.099168e+00 47250 1
360 3.075000e+00 4.084867e+00 5.258696e+00 48600 1
370 3.500000e+00 4.084862e+00 5.428827e+00 49950 1
380 3.356250e+00 4.084857e+00 5.597045e+00 51300 1
390 5.500000e+00 4.084841e+00 5.775202e+00 52650 1
400 4.475000e+00 4.084837e+00 5.940873e+00 54000 1
410 3.750000e+00 4.084834e+00 6.106600e+00 55350 1
420 3.687500e+00 4.084834e+00 6.275382e+00 56700 1
430 4.337500e+00 4.084813e+00 6.447895e+00 58050 1
440 5.750000e+00 4.084813e+00 6.614158e+00 59400 1
450 4.937500e+00 4.084776e+00 6.786614e+00 60750 1
460 3.600000e+00 4.084776e+00 6.955675e+00 62100 1
470 4.387500e+00 4.084776e+00 7.116969e+00 63450 1
480 4.000000e+00 4.084771e+00 7.288643e+00 64800 1
490 2.975000e+00 4.084769e+00 7.456001e+00 66150 1
500 3.125000e+00 4.084769e+00 7.622070e+00 67500 1
510 4.250000e+00 4.084769e+00 7.797441e+00 68850 1
520 4.512500e+00 4.084767e+00 7.961719e+00 70200 1
530 3.875000e+00 4.084767e+00 8.135526e+00 71550 1
540 4.387500e+00 4.084762e+00 8.310143e+00 72900 1
550 5.287500e+00 4.084762e+00 8.484481e+00 74250 1
560 4.650000e+00 4.084762e+00 8.649041e+00 75600 1
570 3.062500e+00 4.084762e+00 8.837071e+00 76950 1
580 3.187500e+00 4.084758e+00 8.999417e+00 78300 1
590 3.812500e+00 4.084758e+00 9.155328e+00 79650 1
600 3.637500e+00 4.084746e+00 9.319168e+00 81000 1
610 3.925000e+00 4.084746e+00 9.485489e+00 82350 1
620 4.625000e+00 4.084746e+00 9.655270e+00 83700 1
630 4.218750e+00 4.084746e+00 9.829597e+00 85050 1
640 3.025000e+00 4.084746e+00 1.000312e+01 86400 1
650 2.993750e+00 4.084746e+00 1.016699e+01 87750 1
660 3.262500e+00 4.084746e+00 1.033747e+01 89100 1
670 3.575000e+00 4.084746e+00 1.051339e+01 90450 1
680 2.981250e+00 4.084746e+00 1.069047e+01 91800 1
690 4.187500e+00 4.084746e+00 1.086519e+01 93150 1
700 4.500000e+00 4.084746e+00 1.105563e+01 94500 1
710 3.225000e+00 4.084746e+00 1.122993e+01 95850 1
720 4.375000e+00 4.084746e+00 1.140574e+01 97200 1
730 2.650000e+00 4.084746e+00 1.158624e+01 98550 1
740 3.312500e+00 4.084746e+00 1.175916e+01 99900 1
750 4.725000e+00 4.084746e+00 1.195022e+01 101250 1
760 3.375000e+00 4.084746e+00 1.213821e+01 102600 1
770 5.375000e+00 4.084746e+00 1.231718e+01 103950 1
780 4.068750e+00 4.084746e+00 1.250136e+01 105300 1
790 4.412500e+00 4.084746e+00 1.268508e+01 106650 1
800 4.350000e+00 4.084746e+00 1.287110e+01 108000 1
810 5.887500e+00 4.084746e+00 1.305782e+01 109350 1
820 4.912500e+00 4.084746e+00 1.324217e+01 110700 1
830 4.400000e+00 4.084746e+00 1.343794e+01 112050 1
840 3.675000e+00 4.084746e+00 1.362423e+01 113400 1
850 5.375000e+00 4.084746e+00 1.380121e+01 114750 1
860 3.562500e+00 4.084746e+00 1.398982e+01 116100 1
870 3.075000e+00 4.084746e+00 1.418317e+01 117450 1
880 3.625000e+00 4.084746e+00 1.437202e+01 118800 1
890 2.937500e+00 4.084746e+00 1.456107e+01 120150 1
900 4.450000e+00 4.084746e+00 1.475167e+01 121500 1
910 4.200000e+00 4.084746e+00 1.493991e+01 122850 1
920 3.687500e+00 4.084746e+00 1.513438e+01 124200 1
930 4.725000e+00 4.084746e+00 1.532086e+01 125550 1
940 4.018750e+00 4.084746e+00 1.551524e+01 126900 1
950 4.675000e+00 4.084746e+00 1.569792e+01 128250 1
960 3.375000e+00 4.084746e+00 1.587552e+01 129600 1
970 3.812500e+00 4.084746e+00 1.605301e+01 130950 1
980 3.112500e+00 4.084746e+00 1.623633e+01 132300 1
990 3.600000e+00 4.084746e+00 1.642531e+01 133650 1
1000 5.500000e+00 4.084746e+00 1.661773e+01 135000 1
1010 3.187500e+00 4.084746e+00 1.680518e+01 136350 1
1020 4.900000e+00 4.084746e+00 1.699504e+01 137700 1
1030 3.637500e+00 4.084746e+00 1.719784e+01 139050 1
1040 3.975000e+00 4.084746e+00 1.738607e+01 140400 1
1050 4.750000e+00 4.084746e+00 1.757457e+01 141750 1
1060 4.437500e+00 4.084746e+00 1.779747e+01 143100 1
1070 5.000000e+00 4.084746e+00 1.799033e+01 144450 1
1080 4.143750e+00 4.084746e+00 1.818948e+01 145800 1
1090 5.625000e+00 4.084746e+00 1.837086e+01 147150 1
1100 3.475000e+00 4.084746e+00 1.856671e+01 148500 1
1110 4.156250e+00 4.084746e+00 1.876417e+01 149850 1
1120 4.450000e+00 4.084746e+00 1.895509e+01 151200 1
1130 3.225000e+00 4.084741e+00 1.916055e+01 152550 1
1140 5.375000e+00 4.084741e+00 1.934867e+01 153900 1
1150 4.800000e+00 4.084737e+00 1.956194e+01 155250 1
1160 3.300000e+00 4.084737e+00 1.975167e+01 156600 1
1170 4.356250e+00 4.084737e+00 1.993517e+01 157950 1
1174 3.600000e+00 4.084737e+00 2.001428e+01 158490 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.001428e+01
total solves : 158490
best bound : 4.084737e+00
simulation ci : 4.070355e+00 ± 4.161086e-02
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-25
-------------------------------------------------------------------
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.EqualTo{Float64} : [1, 1]
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 3.607143e+00 5.356537e+00 2.181170e-01 1350 1
20 4.750000e+00 4.047869e+00 5.996900e-01 2700 1
30 4.006250e+00 4.044163e+00 1.107378e+00 4050 1
40 3.662500e+00 4.043502e+00 1.719154e+00 5400 1
50 5.250000e+00 4.041806e+00 2.527084e+00 6750 1
60 4.750000e+00 4.041672e+00 3.499503e+00 8100 1
70 4.125000e+00 4.041319e+00 4.607560e+00 9450 1
80 5.062500e+00 4.040145e+00 5.788184e+00 10800 1
90 3.750000e+00 4.040055e+00 7.098457e+00 12150 1
100 4.462500e+00 4.039112e+00 8.496645e+00 13500 1
110 4.637500e+00 4.038966e+00 1.001234e+01 14850 1
120 4.625000e+00 4.038832e+00 1.175426e+01 16200 1
130 3.200000e+00 4.038829e+00 1.348181e+01 17550 1
140 3.250000e+00 4.038829e+00 1.542656e+01 18900 1
150 4.556250e+00 4.038779e+00 1.746624e+01 20250 1
160 3.425000e+00 4.038779e+00 1.957712e+01 21600 1
162 4.250000e+00 4.038775e+00 2.003992e+01 21870 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.003992e+01
total solves : 21870
best bound : 4.038775e+00
simulation ci : 4.065978e+00 ± 1.118561e-01
numeric issues : 0
-------------------------------------------------------------------