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 5.500000e+00 5.473308e+00 1.802280e-01 1350 1
20 4.062500e+00 4.451907e+00 2.729421e-01 2700 1
30 2.993750e+00 4.101372e+00 3.747702e-01 4050 1
40 5.750000e+00 4.095757e+00 4.832520e-01 5400 1
50 5.125000e+00 4.093536e+00 5.949891e-01 6750 1
60 3.737500e+00 4.089225e+00 7.386382e-01 8100 1
70 4.500000e+00 4.088152e+00 8.584051e-01 9450 1
80 4.950000e+00 4.087904e+00 9.830451e-01 10800 1
90 4.125000e+00 4.087756e+00 1.116382e+00 12150 1
100 3.275000e+00 4.086300e+00 1.254984e+00 13500 1
110 5.250000e+00 4.086125e+00 1.389918e+00 14850 1
120 4.975000e+00 4.086054e+00 1.533996e+00 16200 1
130 3.500000e+00 4.085422e+00 1.673236e+00 17550 1
140 4.593750e+00 4.085327e+00 1.814593e+00 18900 1
150 3.000000e+00 4.085258e+00 1.954536e+00 20250 1
160 2.700000e+00 4.085247e+00 2.101151e+00 21600 1
170 3.812500e+00 4.085151e+00 2.254985e+00 22950 1
180 3.875000e+00 4.085121e+00 2.406813e+00 24300 1
190 4.737500e+00 4.085102e+00 2.552333e+00 25650 1
200 2.906250e+00 4.085073e+00 2.718421e+00 27000 1
210 3.750000e+00 4.085050e+00 2.869789e+00 28350 1
220 5.050000e+00 4.085037e+00 3.018799e+00 29700 1
230 2.925000e+00 4.085012e+00 3.162940e+00 31050 1
240 4.500000e+00 4.084970e+00 3.313683e+00 32400 1
250 4.875000e+00 4.084908e+00 3.470978e+00 33750 1
260 3.675000e+00 4.084905e+00 3.616996e+00 35100 1
270 4.725000e+00 4.084903e+00 3.774278e+00 36450 1
280 3.437500e+00 4.084900e+00 3.927018e+00 37800 1
290 3.750000e+00 4.084879e+00 4.084138e+00 39150 1
300 4.125000e+00 4.084879e+00 4.245228e+00 40500 1
310 4.875000e+00 4.084803e+00 4.402763e+00 41850 1
320 5.625000e+00 4.084803e+00 4.557153e+00 43200 1
330 3.825000e+00 4.084800e+00 4.732231e+00 44550 1
340 3.300000e+00 4.084796e+00 4.890705e+00 45900 1
350 5.887500e+00 4.084796e+00 5.055271e+00 47250 1
360 3.600000e+00 4.084786e+00 5.222467e+00 48600 1
370 3.618750e+00 4.084786e+00 5.382099e+00 49950 1
380 4.968750e+00 4.084782e+00 5.538792e+00 51300 1
390 3.300000e+00 4.084780e+00 5.695110e+00 52650 1
400 4.006250e+00 4.084779e+00 5.862309e+00 54000 1
410 4.050000e+00 4.084779e+00 6.030380e+00 55350 1
420 5.125000e+00 4.084776e+00 6.196335e+00 56700 1
430 4.000000e+00 4.084776e+00 6.362340e+00 58050 1
440 4.125000e+00 4.084776e+00 6.532712e+00 59400 1
450 3.112500e+00 4.084776e+00 6.700407e+00 60750 1
460 3.750000e+00 4.084771e+00 6.881885e+00 62100 1
470 3.187500e+00 4.084767e+00 7.037245e+00 63450 1
480 3.031250e+00 4.084757e+00 7.212752e+00 64800 1
490 4.181250e+00 4.084753e+00 7.380738e+00 66150 1
500 3.187500e+00 4.084746e+00 7.558795e+00 67500 1
510 4.875000e+00 4.084741e+00 7.726933e+00 68850 1
520 3.900000e+00 4.084737e+00 7.892418e+00 70200 1
530 4.350000e+00 4.084737e+00 8.059904e+00 71550 1
540 4.212500e+00 4.084737e+00 8.227000e+00 72900 1
550 4.250000e+00 4.084734e+00 8.406149e+00 74250 1
560 3.750000e+00 4.084734e+00 8.576610e+00 75600 1
570 5.306250e+00 4.084730e+00 8.748308e+00 76950 1
580 4.725000e+00 4.084730e+00 8.920720e+00 78300 1
590 4.250000e+00 4.084730e+00 9.106618e+00 79650 1
600 4.000000e+00 4.084730e+00 9.281422e+00 81000 1
610 4.600000e+00 4.084730e+00 9.449826e+00 82350 1
620 3.375000e+00 4.084730e+00 9.621901e+00 83700 1
630 3.981250e+00 4.084725e+00 9.799755e+00 85050 1
640 3.250000e+00 4.084725e+00 9.972482e+00 86400 1
650 3.625000e+00 4.084725e+00 1.014121e+01 87750 1
660 4.781250e+00 4.084725e+00 1.031430e+01 89100 1
670 4.275000e+00 4.084725e+00 1.049861e+01 90450 1
680 2.731250e+00 4.084725e+00 1.068273e+01 91800 1
690 5.237500e+00 4.084725e+00 1.086570e+01 93150 1
700 3.325000e+00 4.084725e+00 1.104192e+01 94500 1
710 4.750000e+00 4.084725e+00 1.123174e+01 95850 1
720 4.537500e+00 4.084725e+00 1.143757e+01 97200 1
730 4.725000e+00 4.084725e+00 1.163137e+01 98550 1
740 4.475000e+00 4.084725e+00 1.181422e+01 99900 1
750 2.893750e+00 4.084725e+00 1.199743e+01 101250 1
760 3.525000e+00 4.084725e+00 1.218509e+01 102600 1
770 3.525000e+00 4.084725e+00 1.237030e+01 103950 1
780 3.262500e+00 4.084725e+00 1.255919e+01 105300 1
790 4.918750e+00 4.084725e+00 1.275127e+01 106650 1
800 3.750000e+00 4.084725e+00 1.293358e+01 108000 1
810 4.687500e+00 4.084725e+00 1.312416e+01 109350 1
820 4.018750e+00 4.084725e+00 1.331697e+01 110700 1
830 4.725000e+00 4.084725e+00 1.351528e+01 112050 1
840 4.268750e+00 4.084725e+00 1.372358e+01 113400 1
850 5.175000e+00 4.084725e+00 1.390885e+01 114750 1
860 3.125000e+00 4.084725e+00 1.410734e+01 116100 1
870 2.762500e+00 4.084725e+00 1.428764e+01 117450 1
880 3.375000e+00 4.084725e+00 1.446668e+01 118800 1
890 3.875000e+00 4.084725e+00 1.466341e+01 120150 1
900 3.093750e+00 4.084725e+00 1.485074e+01 121500 1
910 4.125000e+00 4.084724e+00 1.505326e+01 122850 1
920 4.750000e+00 4.084724e+00 1.524667e+01 124200 1
930 5.431250e+00 4.084724e+00 1.543615e+01 125550 1
940 3.000000e+00 4.084724e+00 1.562966e+01 126900 1
950 4.243750e+00 4.084724e+00 1.582959e+01 128250 1
960 3.756250e+00 4.084724e+00 1.601514e+01 129600 1
970 3.750000e+00 4.084724e+00 1.620816e+01 130950 1
980 4.350000e+00 4.084724e+00 1.639128e+01 132300 1
990 3.375000e+00 4.084722e+00 1.658343e+01 133650 1
1000 4.750000e+00 4.084722e+00 1.677371e+01 135000 1
1010 3.825000e+00 4.084722e+00 1.695977e+01 136350 1
1020 4.000000e+00 4.084722e+00 1.713576e+01 137700 1
1030 4.500000e+00 4.084722e+00 1.732446e+01 139050 1
1040 3.637500e+00 4.084722e+00 1.751349e+01 140400 1
1050 4.050000e+00 4.084722e+00 1.769978e+01 141750 1
1060 4.925000e+00 4.084722e+00 1.790373e+01 143100 1
1070 4.500000e+00 4.084722e+00 1.810115e+01 144450 1
1080 5.000000e+00 4.084722e+00 1.830009e+01 145800 1
1090 4.393750e+00 4.084722e+00 1.849242e+01 147150 1
1100 3.750000e+00 4.084722e+00 1.870430e+01 148500 1
1110 4.306250e+00 4.084722e+00 1.889172e+01 149850 1
1120 4.200000e+00 4.084722e+00 1.908700e+01 151200 1
1130 4.343750e+00 4.084722e+00 1.928747e+01 152550 1
1140 4.650000e+00 4.084722e+00 1.948628e+01 153900 1
1150 4.537500e+00 4.084722e+00 1.970448e+01 155250 1
1160 4.250000e+00 4.084722e+00 1.991142e+01 156600 1
1165 5.025000e+00 4.084722e+00 2.001092e+01 157275 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.001092e+01
total solves : 157275
best bound : 4.084722e+00
simulation ci : 4.099569e+00 ± 4.216437e-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 4.012500e+00 4.753864e+00 2.050312e-01 1350 1
20 4.468750e+00 4.048691e+00 5.993659e-01 2700 1
30 3.562500e+00 4.043180e+00 1.105992e+00 4050 1
40 2.775000e+00 4.040344e+00 1.784556e+00 5400 1
50 5.000000e+00 4.039945e+00 2.591044e+00 6750 1
60 3.625000e+00 4.039145e+00 3.555521e+00 8100 1
70 5.343750e+00 4.039024e+00 4.622418e+00 9450 1
80 3.037500e+00 4.038992e+00 5.776634e+00 10800 1
90 3.825000e+00 4.038941e+00 7.155560e+00 12150 1
100 3.737500e+00 4.038828e+00 8.590671e+00 13500 1
110 4.125000e+00 4.038209e+00 1.012499e+01 14850 1
120 4.775000e+00 4.038203e+00 1.172979e+01 16200 1
130 3.600000e+00 4.038124e+00 1.341944e+01 17550 1
140 4.350000e+00 4.038110e+00 1.530374e+01 18900 1
150 4.275000e+00 4.038109e+00 1.723750e+01 20250 1
160 4.068750e+00 4.038102e+00 1.929068e+01 21600 1
164 4.200000e+00 4.038096e+00 2.015432e+01 22140 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.015432e+01
total solves : 22140
best bound : 4.038096e+00
simulation ci : 4.101393e+00 ± 1.141672e-01
numeric issues : 0
-------------------------------------------------------------------