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, Testfunction 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[:]endfunction 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 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 returnendnewsvendor_example(; cut_type = SDDP.SINGLE_CUT)newsvendor_example(; cut_type = SDDP.MULTI_CUT)-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-26
-------------------------------------------------------------------
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 5.500000e+00 5.473308e+00 1.932809e-01 1350 1
20 4.062500e+00 4.451907e+00 2.880671e-01 2700 1
30 2.993750e+00 4.101372e+00 3.924031e-01 4050 1
40 5.750000e+00 4.095757e+00 5.021641e-01 5400 1
50 5.125000e+00 4.093536e+00 6.159291e-01 6750 1
60 3.737500e+00 4.089225e+00 7.367611e-01 8100 1
70 4.500000e+00 4.088152e+00 8.568540e-01 9450 1
80 4.950000e+00 4.087904e+00 9.808161e-01 10800 1
90 4.125000e+00 4.087756e+00 1.105386e+00 12150 1
100 3.275000e+00 4.086300e+00 1.235460e+00 13500 1
110 5.250000e+00 4.086125e+00 1.368009e+00 14850 1
120 4.975000e+00 4.086054e+00 1.507555e+00 16200 1
130 3.500000e+00 4.085422e+00 1.646099e+00 17550 1
140 4.593750e+00 4.085327e+00 1.782902e+00 18900 1
150 3.000000e+00 4.085258e+00 1.950576e+00 20250 1
160 2.700000e+00 4.085247e+00 2.093822e+00 21600 1
170 3.812500e+00 4.085151e+00 2.236728e+00 22950 1
180 3.875000e+00 4.085121e+00 2.384791e+00 24300 1
190 4.737500e+00 4.085102e+00 2.527818e+00 25650 1
200 2.906250e+00 4.085073e+00 2.674457e+00 27000 1
210 3.750000e+00 4.085050e+00 2.825798e+00 28350 1
220 5.050000e+00 4.085037e+00 2.974187e+00 29700 1
230 2.925000e+00 4.085012e+00 3.117716e+00 31050 1
240 4.500000e+00 4.084970e+00 3.267206e+00 32400 1
250 4.875000e+00 4.084908e+00 3.423229e+00 33750 1
260 3.675000e+00 4.084905e+00 3.567961e+00 35100 1
270 4.725000e+00 4.084903e+00 3.723559e+00 36450 1
280 3.437500e+00 4.084900e+00 3.873526e+00 37800 1
290 3.750000e+00 4.084879e+00 4.031406e+00 39150 1
300 4.125000e+00 4.084879e+00 4.191570e+00 40500 1
310 4.875000e+00 4.084803e+00 4.348417e+00 41850 1
320 5.625000e+00 4.084803e+00 4.501798e+00 43200 1
330 3.825000e+00 4.084800e+00 4.659718e+00 44550 1
340 3.300000e+00 4.084796e+00 4.816911e+00 45900 1
350 5.887500e+00 4.084796e+00 4.980453e+00 47250 1
360 3.600000e+00 4.084786e+00 5.141091e+00 48600 1
370 3.618750e+00 4.084786e+00 5.298140e+00 49950 1
380 4.968750e+00 4.084782e+00 5.454209e+00 51300 1
390 3.300000e+00 4.084780e+00 5.633222e+00 52650 1
400 4.006250e+00 4.084779e+00 5.799307e+00 54000 1
410 4.050000e+00 4.084779e+00 5.961954e+00 55350 1
420 5.125000e+00 4.084776e+00 6.124999e+00 56700 1
430 4.000000e+00 4.084776e+00 6.287156e+00 58050 1
440 4.125000e+00 4.084776e+00 6.458108e+00 59400 1
450 3.112500e+00 4.084776e+00 6.623329e+00 60750 1
460 3.750000e+00 4.084771e+00 6.791167e+00 62100 1
470 3.187500e+00 4.084767e+00 6.947603e+00 63450 1
480 3.031250e+00 4.084757e+00 7.122863e+00 64800 1
490 4.181250e+00 4.084753e+00 7.289353e+00 66150 1
500 3.187500e+00 4.084746e+00 7.464522e+00 67500 1
510 4.875000e+00 4.084741e+00 7.630750e+00 68850 1
520 3.900000e+00 4.084737e+00 7.797462e+00 70200 1
530 4.350000e+00 4.084737e+00 7.964040e+00 71550 1
540 4.212500e+00 4.084737e+00 8.131009e+00 72900 1
550 4.250000e+00 4.084734e+00 8.303925e+00 74250 1
560 3.750000e+00 4.084734e+00 8.473961e+00 75600 1
570 5.306250e+00 4.084730e+00 8.643466e+00 76950 1
580 4.725000e+00 4.084730e+00 8.812862e+00 78300 1
590 4.250000e+00 4.084730e+00 8.980910e+00 79650 1
600 4.000000e+00 4.084730e+00 9.152898e+00 81000 1
610 4.600000e+00 4.084730e+00 9.315739e+00 82350 1
620 3.375000e+00 4.084730e+00 9.486081e+00 83700 1
630 3.981250e+00 4.084725e+00 9.661452e+00 85050 1
640 3.250000e+00 4.084725e+00 9.831698e+00 86400 1
650 3.625000e+00 4.084725e+00 1.001873e+01 87750 1
660 4.781250e+00 4.084725e+00 1.018699e+01 89100 1
670 4.275000e+00 4.084725e+00 1.036162e+01 90450 1
680 2.731250e+00 4.084725e+00 1.054212e+01 91800 1
690 5.237500e+00 4.084725e+00 1.072047e+01 93150 1
700 3.325000e+00 4.084725e+00 1.089116e+01 94500 1
710 4.750000e+00 4.084725e+00 1.107363e+01 95850 1
720 4.537500e+00 4.084725e+00 1.125055e+01 97200 1
730 4.725000e+00 4.084725e+00 1.142809e+01 98550 1
740 4.475000e+00 4.084725e+00 1.160147e+01 99900 1
750 2.893750e+00 4.084725e+00 1.177588e+01 101250 1
760 3.525000e+00 4.084725e+00 1.195400e+01 102600 1
770 3.525000e+00 4.084725e+00 1.213525e+01 103950 1
780 3.262500e+00 4.084725e+00 1.231392e+01 105300 1
790 4.918750e+00 4.084725e+00 1.249809e+01 106650 1
800 3.750000e+00 4.084725e+00 1.267627e+01 108000 1
810 4.687500e+00 4.084725e+00 1.285920e+01 109350 1
820 4.018750e+00 4.084725e+00 1.304511e+01 110700 1
830 4.725000e+00 4.084725e+00 1.323083e+01 112050 1
840 4.268750e+00 4.084725e+00 1.341490e+01 113400 1
850 5.175000e+00 4.084725e+00 1.358916e+01 114750 1
860 3.125000e+00 4.084725e+00 1.377959e+01 116100 1
870 2.762500e+00 4.084725e+00 1.395316e+01 117450 1
880 3.375000e+00 4.084725e+00 1.412610e+01 118800 1
890 3.875000e+00 4.084725e+00 1.431030e+01 120150 1
900 3.093750e+00 4.084725e+00 1.449064e+01 121500 1
910 4.125000e+00 4.084724e+00 1.470761e+01 122850 1
920 4.750000e+00 4.084724e+00 1.489122e+01 124200 1
930 5.431250e+00 4.084724e+00 1.507356e+01 125550 1
940 3.000000e+00 4.084724e+00 1.526156e+01 126900 1
950 4.243750e+00 4.084724e+00 1.544205e+01 128250 1
960 3.756250e+00 4.084724e+00 1.561929e+01 129600 1
970 3.750000e+00 4.084724e+00 1.581024e+01 130950 1
980 4.350000e+00 4.084724e+00 1.599370e+01 132300 1
990 3.375000e+00 4.084722e+00 1.618192e+01 133650 1
1000 4.750000e+00 4.084722e+00 1.636635e+01 135000 1
1010 3.825000e+00 4.084722e+00 1.654873e+01 136350 1
1020 4.000000e+00 4.084722e+00 1.672293e+01 137700 1
1030 4.500000e+00 4.084722e+00 1.691041e+01 139050 1
1040 3.637500e+00 4.084722e+00 1.709862e+01 140400 1
1050 4.050000e+00 4.084722e+00 1.728210e+01 141750 1
1060 4.925000e+00 4.084722e+00 1.747070e+01 143100 1
1070 4.500000e+00 4.084722e+00 1.766589e+01 144450 1
1080 5.000000e+00 4.084722e+00 1.786320e+01 145800 1
1090 4.393750e+00 4.084722e+00 1.805362e+01 147150 1
1100 3.750000e+00 4.084722e+00 1.825823e+01 148500 1
1110 4.306250e+00 4.084722e+00 1.844230e+01 149850 1
1120 4.200000e+00 4.084722e+00 1.863123e+01 151200 1
1130 4.343750e+00 4.084722e+00 1.882882e+01 152550 1
1140 4.650000e+00 4.084722e+00 1.902608e+01 153900 1
1150 4.537500e+00 4.084722e+00 1.922634e+01 155250 1
1160 4.250000e+00 4.084722e+00 1.942918e+01 156600 1
1170 3.312500e+00 4.084722e+00 1.962406e+01 157950 1
1180 4.968750e+00 4.084722e+00 1.984129e+01 159300 1
1188 3.825000e+00 4.084718e+00 2.000176e+01 160380 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.000176e+01
total solves : 160380
best bound : 4.084718e+00
numeric issues : 0
-------------------------------------------------------------------
-------------------------------------------------------------------
SDDP.jl (c) Oscar Dowson and contributors, 2017-26
-------------------------------------------------------------------
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 5.500000e+00 4.081493e+00 2.180440e-01 1350 1
20 4.375000e+00 4.048517e+00 6.047029e-01 2700 1
30 3.062500e+00 4.039763e+00 1.104559e+00 4050 1
40 4.062500e+00 4.039387e+00 1.737323e+00 5400 1
50 4.537500e+00 4.039129e+00 2.462221e+00 6750 1
60 3.218750e+00 4.039031e+00 3.309478e+00 8100 1
70 4.000000e+00 4.038778e+00 4.309766e+00 9450 1
80 3.875000e+00 4.038716e+00 5.383879e+00 10800 1
90 5.150000e+00 4.038129e+00 6.572116e+00 12150 1
100 3.162500e+00 4.038125e+00 7.825053e+00 13500 1
110 4.125000e+00 4.038045e+00 9.156189e+00 14850 1
120 4.875000e+00 4.038031e+00 1.060765e+01 16200 1
130 3.843750e+00 4.038031e+00 1.212711e+01 17550 1
140 4.125000e+00 4.038024e+00 1.376174e+01 18900 1
150 4.368750e+00 4.037526e+00 1.545144e+01 20250 1
160 3.612500e+00 4.037335e+00 1.720894e+01 21600 1
170 4.356250e+00 4.037322e+00 1.916161e+01 22950 1
175 4.312500e+00 4.037319e+00 2.008561e+01 23625 1
-------------------------------------------------------------------
status : time_limit
total time (s) : 2.008561e+01
total solves : 23625
best bound : 4.037319e+00
numeric issues : 0
-------------------------------------------------------------------