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 1.648998e-01 1350 1
20 4.350000e+00 4.105556e+00 2.509530e-01 2700 1
30 5.000000e+00 4.099889e+00 3.487120e-01 4050 1
40 3.500000e+00 4.096596e+00 4.545598e-01 5400 1
50 5.250000e+00 4.095112e+00 5.657809e-01 6750 1
60 3.643750e+00 4.092827e+00 6.840398e-01 8100 1
70 2.643750e+00 4.091514e+00 7.981379e-01 9450 1
80 5.087500e+00 4.091301e+00 9.148009e-01 10800 1
90 5.062500e+00 4.090986e+00 1.030168e+00 12150 1
100 4.843750e+00 4.086917e+00 1.160880e+00 13500 1
110 3.400000e+00 4.085985e+00 1.286889e+00 14850 1
120 3.375000e+00 4.085819e+00 1.414850e+00 16200 1
130 5.025000e+00 4.085765e+00 1.547287e+00 17550 1
140 5.000000e+00 4.085640e+00 1.675500e+00 18900 1
150 3.500000e+00 4.085560e+00 1.803753e+00 20250 1
160 4.281250e+00 4.085411e+00 1.955412e+00 21600 1
170 4.562500e+00 4.085379e+00 2.087663e+00 22950 1
180 5.768750e+00 4.085379e+00 2.217077e+00 24300 1
190 3.468750e+00 4.085337e+00 2.352901e+00 25650 1
200 4.131250e+00 4.085202e+00 2.488050e+00 27000 1
210 4.512500e+00 4.085132e+00 2.631313e+00 28350 1
220 4.900000e+00 4.085130e+00 2.767367e+00 29700 1
230 4.025000e+00 4.085111e+00 2.902879e+00 31050 1
240 4.468750e+00 4.085094e+00 3.044212e+00 32400 1
250 4.062500e+00 4.085056e+00 3.182452e+00 33750 1
260 4.875000e+00 4.085019e+00 3.328798e+00 35100 1
270 3.850000e+00 4.084990e+00 3.470359e+00 36450 1
280 4.912500e+00 4.084971e+00 3.611864e+00 37800 1
290 2.987500e+00 4.084965e+00 3.759992e+00 39150 1
300 3.825000e+00 4.084940e+00 3.913076e+00 40500 1
310 3.250000e+00 4.084900e+00 4.065368e+00 41850 1
320 3.537500e+00 4.084889e+00 4.213393e+00 43200 1
330 3.950000e+00 4.084889e+00 4.379249e+00 44550 1
340 4.500000e+00 4.084886e+00 4.524798e+00 45900 1
350 5.000000e+00 4.084886e+00 4.667817e+00 47250 1
360 3.075000e+00 4.084864e+00 4.813132e+00 48600 1
370 3.500000e+00 4.084859e+00 4.966779e+00 49950 1
380 3.356250e+00 4.084854e+00 5.120142e+00 51300 1
390 5.500000e+00 4.084839e+00 5.275948e+00 52650 1
400 4.475000e+00 4.084834e+00 5.422108e+00 54000 1
410 3.750000e+00 4.084832e+00 5.570609e+00 55350 1
420 3.687500e+00 4.084832e+00 5.724658e+00 56700 1
430 4.337500e+00 4.084811e+00 5.882289e+00 58050 1
440 5.750000e+00 4.084811e+00 6.022937e+00 59400 1
450 4.937500e+00 4.084776e+00 6.183115e+00 60750 1
460 3.600000e+00 4.084776e+00 6.341070e+00 62100 1
470 4.387500e+00 4.084776e+00 6.488384e+00 63450 1
480 4.000000e+00 4.084771e+00 6.646559e+00 64800 1
490 2.975000e+00 4.084769e+00 6.809901e+00 66150 1
500 3.125000e+00 4.084769e+00 6.986242e+00 67500 1
510 4.250000e+00 4.084769e+00 7.158323e+00 68850 1
520 4.525000e+00 4.084767e+00 7.313242e+00 70200 1
530 3.875000e+00 4.084767e+00 7.487013e+00 71550 1
540 4.387500e+00 4.084762e+00 7.660255e+00 72900 1
550 5.287500e+00 4.084762e+00 7.838036e+00 74250 1
560 4.650000e+00 4.084762e+00 7.994974e+00 75600 1
570 3.062500e+00 4.084762e+00 8.161216e+00 76950 1
580 3.187500e+00 4.084758e+00 8.325656e+00 78300 1
590 3.812500e+00 4.084758e+00 8.478542e+00 79650 1
600 3.637500e+00 4.084746e+00 8.639452e+00 81000 1
610 3.925000e+00 4.084746e+00 8.802190e+00 82350 1
620 4.625000e+00 4.084746e+00 8.967913e+00 83700 1
630 4.218750e+00 4.084746e+00 9.134595e+00 85050 1
640 3.025000e+00 4.084746e+00 9.306375e+00 86400 1
650 2.993750e+00 4.084746e+00 9.463239e+00 87750 1
660 3.262500e+00 4.084746e+00 9.645423e+00 89100 1
670 3.575000e+00 4.084746e+00 9.817763e+00 90450 1
680 2.981250e+00 4.084746e+00 9.985817e+00 91800 1
690 4.187500e+00 4.084746e+00 1.015689e+01 93150 1
700 4.500000e+00 4.084746e+00 1.032539e+01 94500 1
710 3.225000e+00 4.084746e+00 1.049425e+01 95850 1
720 4.375000e+00 4.084746e+00 1.067993e+01 97200 1
730 2.650000e+00 4.084746e+00 1.085818e+01 98550 1
740 3.250000e+00 4.084746e+00 1.104094e+01 99900 1
750 4.725000e+00 4.084746e+00 1.123251e+01 101250 1
760 3.375000e+00 4.084746e+00 1.142026e+01 102600 1
770 5.375000e+00 4.084746e+00 1.160450e+01 103950 1
780 4.068750e+00 4.084746e+00 1.178450e+01 105300 1
790 4.412500e+00 4.084746e+00 1.198229e+01 106650 1
800 4.350000e+00 4.084746e+00 1.216782e+01 108000 1
810 5.900000e+00 4.084746e+00 1.236381e+01 109350 1
820 4.912500e+00 4.084746e+00 1.257733e+01 110700 1
830 4.387500e+00 4.084746e+00 1.275904e+01 112050 1
840 3.712500e+00 4.084746e+00 1.294703e+01 113400 1
850 5.375000e+00 4.084746e+00 1.312981e+01 114750 1
860 3.562500e+00 4.084746e+00 1.333230e+01 116100 1
870 3.075000e+00 4.084746e+00 1.352700e+01 117450 1
880 3.625000e+00 4.084746e+00 1.371528e+01 118800 1
890 2.937500e+00 4.084746e+00 1.389716e+01 120150 1
900 4.450000e+00 4.084746e+00 1.408592e+01 121500 1
910 4.200000e+00 4.084746e+00 1.429070e+01 122850 1
920 3.687500e+00 4.084746e+00 1.449292e+01 124200 1
930 4.687500e+00 4.084746e+00 1.467831e+01 125550 1
940 4.018750e+00 4.084746e+00 1.486387e+01 126900 1
950 4.625000e+00 4.084746e+00 1.504508e+01 128250 1
960 3.375000e+00 4.084746e+00 1.522294e+01 129600 1
970 3.812500e+00 4.084746e+00 1.541984e+01 130950 1
980 3.112500e+00 4.084746e+00 1.560230e+01 132300 1
990 3.600000e+00 4.084746e+00 1.579720e+01 133650 1
1000 5.500000e+00 4.084746e+00 1.598597e+01 135000 1
1010 3.187500e+00 4.084746e+00 1.618315e+01 136350 1
1020 4.900000e+00 4.084746e+00 1.637049e+01 137700 1
1030 3.637500e+00 4.084746e+00 1.658073e+01 139050 1
1040 3.975000e+00 4.084746e+00 1.676955e+01 140400 1
1050 4.750000e+00 4.084746e+00 1.696201e+01 141750 1
1060 4.437500e+00 4.084746e+00 1.717360e+01 143100 1
1070 5.000000e+00 4.084746e+00 1.736055e+01 144450 1
1080 4.143750e+00 4.084746e+00 1.756261e+01 145800 1
1090 5.625000e+00 4.084746e+00 1.775128e+01 147150 1
1100 3.475000e+00 4.084746e+00 1.793851e+01 148500 1
1110 4.156250e+00 4.084746e+00 1.813441e+01 149850 1
1120 4.450000e+00 4.084746e+00 1.835230e+01 151200 1
1130 3.225000e+00 4.084741e+00 1.854592e+01 152550 1
1140 5.375000e+00 4.084741e+00 1.872128e+01 153900 1
1150 4.800000e+00 4.084737e+00 1.891283e+01 155250 1
1160 3.300000e+00 4.084737e+00 1.909349e+01 156600 1
1170 4.356250e+00 4.084737e+00 1.928687e+01 157950 1
1180 3.900000e+00 4.084737e+00 1.948563e+01 159300 1
1190 4.450000e+00 4.084737e+00 1.967977e+01 160650 1
1200 5.156250e+00 4.084737e+00 1.987337e+01 162000 1
1207 4.750000e+00 4.084737e+00 2.001382e+01 162945 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.001382e+01
total solves : 162945
best bound : 4.084737e+00
simulation ci : 4.068465e+00 ± 4.099361e-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 4.800000e+00 4.409619e+00 1.790011e-01 1350 1
20 5.250000e+00 4.399397e+00 5.437090e-01 2700 1
30 4.125000e+00 4.336935e+00 1.048731e+00 4050 1
40 2.550000e+00 4.047548e+00 1.669310e+00 5400 1
50 3.637500e+00 4.042638e+00 2.382298e+00 6750 1
60 3.875000e+00 4.040121e+00 3.108443e+00 8100 1
70 2.981250e+00 4.039179e+00 4.010416e+00 9450 1
80 4.125000e+00 4.039034e+00 5.039032e+00 10800 1
90 4.875000e+00 4.038962e+00 6.261034e+00 12150 1
100 4.275000e+00 4.038879e+00 7.517578e+00 13500 1
110 4.843750e+00 4.038879e+00 8.929122e+00 14850 1
120 5.625000e+00 4.038825e+00 1.041227e+01 16200 1
130 4.375000e+00 4.038808e+00 1.207345e+01 17550 1
140 4.962500e+00 4.038801e+00 1.373721e+01 18900 1
150 2.893750e+00 4.038777e+00 1.559038e+01 20250 1
160 5.287500e+00 4.038777e+00 1.757976e+01 21600 1
170 4.250000e+00 4.038770e+00 1.966844e+01 22950 1
172 4.037500e+00 4.038770e+00 2.010557e+01 23220 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.010557e+01
total solves : 23220
best bound : 4.038770e+00
simulation ci : 4.097582e+00 ± 1.128328e-01
numeric issues : 0
-------------------------------------------------------------------