Example: two-stage newsvendor

This tutorial was generated using Literate.jl. Download the source as a .jl file. Download the source as a .ipynb file.

The purpose of this tutorial is to demonstrate how to model and solve a two-stage stochastic program.

It is based on the Two stage stochastic programs tutorial in JuMP.

This tutorial uses the following packages

using JuMP
using SDDP
import Distributions
import ForwardDiff
import HiGHS
import Plots
import StatsPlots
import Statistics

Background

The data for this problem is:

D = Distributions.TriangularDist(150.0, 250.0, 200.0)
N = 100
d = sort!(rand(D, N));
Ω = 1:N
P = fill(1 / N, N);
StatsPlots.histogram(d; bins = 20, label = "", xlabel = "Demand")

Kelley's cutting plane algorithm

Kelley's cutting plane algorithm is an iterative method for maximizing concave functions. Given a concave function $f(x)$, Kelley's constructs an outer-approximation of the function at the minimum by a set of first-order Taylor series approximations (called cuts) constructed at a set of points $k = 1,\ldots,K$:

\[\begin{aligned} f^K = \max\limits_{\theta \in \mathbb{R}, x \in \mathbb{R}^N} \;\; & \theta\\ & \theta \le f(x_k) + \nabla f(x_k)^\top (x - x_k),\quad k=1,\ldots,K\\ & \theta \le M, \end{aligned}\]

where $M$ is a sufficiently large number that is an upper bound for $f$ over the domain of $x$.

Kelley's cutting plane algorithm is a structured way of choosing points $x_k$ to visit, so that as more cuts are added:

\[\lim_{K \rightarrow \infty} f^K = \max\limits_{x \in \mathbb{R}^N} f(x)\]

However, before we introduce the algorithm, we need to introduce some bounds.

Bounds

By convexity, $f(x) \le f^K$ for all $x$. Thus, if $x^*$ is a maximizer of $f$, then at any point in time we can construct an upper bound for $f(x^*)$ by solving $f^K$.

Moreover, we can use the primal solutions $x_k^*$ returned by solving $f^k$ to evaluate $f(x_k^*)$ to generate a lower bound.

Therefore, $\max\limits_{k=1,\ldots,K} f(x_k^*) \le f(x^*) \le f^K$.

When the lower bound is sufficiently close to the upper bound, we can terminate the algorithm and declare that we have found an solution that is close to optimal.

Implementation

Here is pseudo-code fo the Kelley algorithm:

Take as input a convex function $f(x)$ and a iteration limit $K_{max}$. Set $K = 1$, and initialize $f^{K-1}$. Set $lb = -\infty$ and $ub = \infty$.
Solve $f^{K-1}$ to obtain a candidate solution $x_{K}$.
Update $ub = f^{K-1}$ and $lb = \max\{lb, f(x_{K})\}$.
Add a cut $\theta \ge f(x_{K}) + \nabla f\left(x_{K}\right)^\top (x - x_{K})$ to form $f^{K}$.
Increment $K$.
If $K > K_{max}$ or $|ub - lb| < \epsilon$, STOP, otherwise, go to step 2.

And here's a complete implementation:

function kelleys_cutting_plane(
    # The function to be minimized.
    f::Function,
    # The gradient of `f`. By default, we use automatic differentiation to
    # compute the gradient of f so the user doesn't have to!
    ∇f::Function = x -> ForwardDiff.gradient(f, x);
    # The number of arguments to `f`.
    input_dimension::Int,
    # An upper bound for the function `f` over its domain.
    upper_bound::Float64,
    # The number of iterations to run Kelley's algorithm for before stopping.
    iteration_limit::Int,
    # The absolute tolerance ϵ to use for convergence.
    tolerance::Float64 = 1e-6,
)
    # Step (1):
    K = 1
    model = JuMP.Model(HiGHS.Optimizer)
    JuMP.set_silent(model)
    JuMP.@variable(model, θ <= upper_bound)
    JuMP.@variable(model, x[1:input_dimension])
    JuMP.@objective(model, Max, θ)
    x_k = fill(NaN, input_dimension)
    lower_bound, upper_bound = -Inf, Inf
    while true
        # Step (2):
        JuMP.optimize!(model)
        x_k .= JuMP.value.(x)
        # Step (3):
        upper_bound = JuMP.objective_value(model)
        lower_bound = min(upper_bound, f(x_k))
        println("K = $K : $(lower_bound) <= f(x*) <= $(upper_bound)")
        # Step (4):
        JuMP.@constraint(model, θ <= f(x_k) + ∇f(x_k)' * (x .- x_k))
        # Step (5):
        K = K + 1
        # Step (6):
        if K > iteration_limit
            println("-- Termination status: iteration limit --")
            break
        elseif abs(upper_bound - lower_bound) < tolerance
            println("-- Termination status: converged --")
            break
        end
    end
    println("Found solution: x_K = ", x_k)
    return
end

kelleys_cutting_plane (generic function with 2 methods)

Let's run our algorithm to see what happens:

kelleys_cutting_plane(;
    input_dimension = 2,
    upper_bound = 10.0,
    iteration_limit = 20,
) do x
    return -(x[1] - 1)^2 + -(x[2] + 2)^2 + 1.0
end

K = 1 : -4.0 <= f(x*) <= 10.0
K = 2 : -2.25 <= f(x*) <= 10.0
K = 3 : -5.3125 <= f(x*) <= 10.0
K = 4 : 0.83984375 <= f(x*) <= 5.625
K = 5 : -1.3438585069444455 <= f(x*) <= 1.9791666666666667
K = 6 : 0.4532453748914933 <= f(x*) <= 1.7513020833333333
K = 7 : -2.794810401068801 <= f(x*) <= 1.3444010416666663
K = 8 : 0.19507712328139326 <= f(x*) <= 1.3179100884331594
K = 9 : 0.9073862122310157 <= f(x*) <= 1.3022015061077878
K = 10 : 0.7292616273896162 <= f(x*) <= 1.2835882279084943
K = 11 : 0.9856775767620292 <= f(x*) <= 1.1542808575464905
K = 12 : 0.9521967150117504 <= f(x*) <= 1.0538679846579115
K = 13 : 0.9907765147617908 <= f(x*) <= 1.0341945633777465
K = 14 : 0.990619313815891 <= f(x*) <= 1.0168012962055821
K = 15 : 0.9997569528573889 <= f(x*) <= 1.010937796651451
K = 16 : 0.9955736574995747 <= f(x*) <= 1.0023159378334365
K = 17 : 0.9981907645826057 <= f(x*) <= 1.001070011161672
K = 18 : 0.999293284088297 <= f(x*) <= 1.0010295293971427
K = 19 : 0.9997619192401398 <= f(x*) <= 1.0005033714074143
K = 20 : 0.9999234387181322 <= f(x*) <= 1.0003705497285347
-- Termination status: iteration limit --
Found solution: x_K = [1.0074056501552666, -1.9953397824465389]

L-Shaped theory

The L-Shaped method is a way of solving two-stage stochastic programs by Benders' decomposition. It takes the problem:

\[\begin{aligned} V = \max\limits_{x,y_\omega} \;\; & -2x + \mathbb{E}_\omega[5y_\omega - 0.1(x - y_\omega)] \\ & y_\omega \le x & \quad \forall \omega \in \Omega \\ & 0 \le y_\omega \le d_\omega & \quad \forall \omega \in \Omega \\ & x \ge 0. \end{aligned}\]

and decomposes it into a second-stage problem:

\[\begin{aligned} V_2(\bar{x}, d_\omega) = \max\limits_{x,x^\prime,y_\omega} \;\; & 5y_\omega - x^\prime \\ & y_\omega \le x \\ & x^\prime = x - y_\omega \\ & 0 \le y_\omega \le d_\omega \\ & x = \bar{x} & [\lambda] \end{aligned}\]

and a first-stage problem:

\[\begin{aligned} V = \max\limits_{x,\theta} \;\; & -2x + \theta \\ & \theta \le \mathbb{E}_\omega[V_2(x, \omega)] \\ & x \ge 0 \end{aligned}\]

Then, because $V_2$ is convex with respect to $\bar{x}$ for fixed $\omega$, we can use a set of feasible points $\{x^k\}$ construct an outer approximation:

\[\begin{aligned} V^K = \max\limits_{x,\theta} \;\; & -2x + \theta \\ & \theta \le \mathbb{E}_\omega[V_2(x^k, \omega) + \nabla V_2(x^k, \omega)^\top(x - x^k)] & \quad k = 1,\ldots,K\\ & x \ge 0 \\ & \theta \le M \end{aligned}\]

where $M$ is an upper bound on possible values of $V_2$ so that the problem has a bounded solution.

It is also useful to see that because $\bar{x}$ appears only on the right-hand side of a linear program, $\nabla V_2(x^k, \omega) = \lambda^k$.

Ignoring how we choose $x^k$ for now, we can construct a lower and upper bound on the optimal solution:

\[-2x^K + \mathbb{E}_\omega[V_2(x^K, \omega)] = \underbar{V} \le V \le \overline{V} = V^K\]

Thus, we need some way of cleverly choosing a sequence of $x^k$ so that the lower bound converges to the upper bound.

Start with $K=1$
Solve $V^{K-1}$ to get $x^K$
Set $\overline{V} = V^k$
Solve $V_2(x^K, \omega)$ for all $\omega$ and store the optimal objective value and dual solution $\lambda^K$
Set $\underbar{V} = -2x^K + \mathbb{E}_\omega[V_2(x^k, \omega)]$
If $\underbar{V} \approx \overline{V}$, STOP
Add new constraint $\theta \le \mathbb{E}_\omega[V_2(x^K, \omega) +\lambda^K (x - x^K)]$
Increment $K$, GOTO 2

The next section implements this algorithm in Julia.

L-Shaped implementation

Here's a function to compute the second-stage problem;

function solve_second_stage(x̅, d_ω)
    model = Model(HiGHS.Optimizer)
    set_silent(model)
    @variable(model, x_in)
    @variable(model, x_out >= 0)
    fix(x_in, x̅)
    @variable(model, 0 <= u_sell <= d_ω)
    @constraint(model, x_out == x_in - u_sell)
    @constraint(model, u_sell <= x_in)
    @objective(model, Max, 5 * u_sell - 0.1 * x_out)
    optimize!(model)
    return (
        V = objective_value(model),
        λ = reduced_cost(x_in),
        x = value(x_out),
        u = value(u_sell),
    )
end

solve_second_stage(200, 170)

(V = 847.0, λ = -0.1, x = 30.0, u = 170.0)

Here's the first-stage subproblem:

model = Model(HiGHS.Optimizer)
set_silent(model)
@variable(model, x_in == 0)
@variable(model, x_out >= 0)
@variable(model, u_make >= 0)
@constraint(model, x_out == x_in + u_make)
M = 5 * maximum(d)
@variable(model, θ <= M)
@objective(model, Max, -2 * u_make + θ)

\[ -2 u\_make + θ \]

Importantly, to ensure we have a bounded solution, we need to add an upper bound to the variable θ.

kIterationLimit = 100
for k in 1:kIterationLimit
    println("Solving iteration k = $k")
    # Step 2
    optimize!(model)
    xᵏ = value(x_out)
    println("  xᵏ = $xᵏ")
    # Step 3
    ub = objective_value(model)
    println("  V̅ = $ub")
    # Step 4
    ret = [solve_second_stage(xᵏ, d[ω]) for ω in Ω]
    # Step 5
    lb = value(-2 * u_make) + sum(p * r.V for (p, r) in zip(P, ret))
    println("  V̲ = $lb")
    # Step 6
    if ub - lb < 1e-6
        println("Terminating with near-optimal solution")
        break
    end
    # Step 7
    c = @constraint(
        model,
        θ <= sum(p * (r.V + r.λ * (x_out - xᵏ)) for (p, r) in zip(P, ret)),
    )
    println("  Added cut: $c")
end

Solving iteration k = 1
  xᵏ = -0.0
  V̅ = 1213.4729158321243
  V̲ = 0.0
  Added cut: -4.99999999999999 x_out + θ ≤ 0
Solving iteration k = 2
  xᵏ = 242.69458316642533
  V̅ = 728.0837494992736
  V̲ = 501.40140899824405
  Added cut: 0.10000000000000007 x_out + θ ≤ 1011.0600336477376
Solving iteration k = 3
  xᵏ = 198.2470654211254
  V̅ = 594.7411962633743
  V̲ = 552.1957352618741
  Added cut: -2.399 x_out + θ ≤ 473.09515615884374
Solving iteration k = 4
  xᵏ = 215.27205981948535
  V̅ = 558.9887080268181
  V̲ = 546.9708383130628
  Added cut: -1.1240000000000003 x_out + θ ≤ 735.549162714932
Solving iteration k = 5
  xᵏ = 205.84627965183392
  V̅ = 555.2278217399255
  V̲ = 552.8920162128899
  Added cut: -1.685000000000001 x_out + θ ≤ 617.7335943032163
Solving iteration k = 6
  xᵏ = 202.57484333945757
  V̅ = 553.9225186512873
  V̲ = 553.2571572724692
  Added cut: -1.940000000000001 x_out + θ ≤ 565.4116478728369
Solving iteration k = 7
  xᵏ = 201.12525427885248
  V̅ = 553.3441326161059
  V̲ = 553.1035749581341
  Added cut: -2.2460000000000004 x_out + θ ≤ 503.6267624055374
Solving iteration k = 8
  xᵏ = 201.91139041601193
  V̅ = 553.2969644478762
  V̲ = 553.236178705781
  Added cut: -2.093000000000001 x_out + θ ≤ 534.4584193970913
Solving iteration k = 9
  xᵏ = 202.3086828480091
  V̅ = 553.2731269019566
  V̲ = 553.2628952792838
  Added cut: -1.9910000000000012 x_out + θ ≤ 555.0836734249172
Solving iteration k = 10
  xᵏ = 202.20837282181816
  V̅ = 553.2637980695206
  V̲ = 553.2627418311824
  Added cut: -2.042000000000001 x_out + θ ≤ 544.769990172668
Solving iteration k = 11
  xᵏ = 202.22908337742217
  V̅ = 553.2636116745206
  V̲ = 553.2636116745196
Terminating with near-optimal solution

To get the first-stage solution, we do:

optimize!(model)
xᵏ = value(x_out)

202.22908337742217

To compute a second-stage solution, we do:

solve_second_stage(xᵏ, 170.0)

(V = 846.7770916622578, λ = -0.1, x = 32.22908337742217, u = 170.0)

Policy Graph

Now let's see how we can formulate and train a policy for the two-stage newsvendor problem using SDDP.jl. Under the hood, SDDP.jl implements the exact algorithm that we just wrote by hand.

model = SDDP.LinearPolicyGraph(;
    stages = 2,
    sense = :Max,
    upper_bound = 5 * maximum(d),  # The `M` in θ <= M
    optimizer = HiGHS.Optimizer,
) do subproblem::JuMP.Model, stage::Int
    @variable(subproblem, x >= 0, SDDP.State, initial_value = 0)
    if stage == 1
        @variable(subproblem, u_make >= 0)
        @constraint(subproblem, x.out == x.in + u_make)
        @stageobjective(subproblem, -2 * u_make)
    else
        @variable(subproblem, u_sell >= 0)
        @constraint(subproblem, u_sell <= x.in)
        @constraint(subproblem, x.out == x.in - u_sell)
        SDDP.parameterize(subproblem, d, P) do ω
            set_upper_bound(u_sell, ω)
            return
        end
        @stageobjective(subproblem, 5 * u_sell - 0.1 * x.out)
    end
    return
end

SDDP.train(model; log_every_iteration = true)

-------------------------------------------------------------------
         SDDP.jl (c) Oscar Dowson and contributors, 2017-24
-------------------------------------------------------------------
problem
  nodes           : 2
  state variables : 1
  scenarios       : 1.00000e+02
  existing cuts   : false
options
  solver          : serial mode
  risk measure    : SDDP.Expectation()
  sampling scheme : SDDP.InSampleMonteCarlo
subproblem structure
  VariableRef                             : [4, 4]
  AffExpr in MOI.EqualTo{Float64}         : [1, 1]
  AffExpr in MOI.LessThan{Float64}        : [1, 1]
  VariableRef in MOI.GreaterThan{Float64} : [2, 3]
  VariableRef in MOI.LessThan{Float64}    : [1, 1]
numerical stability report
  matrix range     [1e+00, 1e+00]
  objective range  [1e-01, 5e+00]
  bounds range     [2e+02, 1e+03]
  rhs range        [0e+00, 0e+00]
-------------------------------------------------------------------
 iteration    simulation      bound        time (s)     solves  pid
-------------------------------------------------------------------
         1   0.000000e+00  7.280837e+02  6.415129e-03       103   1
         2   4.055470e+02  5.947412e+02  2.316213e-02       406   1
         3   5.947412e+02  5.589887e+02  2.788210e-02       509   1
         4   5.807261e+02  5.552278e+02  3.246403e-02       612   1
         5   6.175388e+02  5.539225e+02  3.706813e-02       715   1
         6   6.077245e+02  5.533441e+02  4.158902e-02       818   1
         7   5.067479e+02  5.532970e+02  4.625010e-02       921   1
         8   6.037803e+02  5.532731e+02  5.087113e-02      1024   1
         9   4.929001e+02  5.532638e+02  5.554605e-02      1127   1
        10   6.066251e+02  5.532636e+02  6.011009e-02      1230   1
        11   4.217689e+02  5.532636e+02  6.490111e-02      1333   1
        12   6.066873e+02  5.532636e+02  6.966019e-02      1436   1
        13   3.992063e+02  5.532636e+02  7.448816e-02      1539   1
        14   5.750462e+02  5.532636e+02  7.921100e-02      1642   1
        15   4.274792e+02  5.532636e+02  8.403802e-02      1745   1
        16   6.066873e+02  5.532636e+02  8.881712e-02      1848   1
        17   5.841125e+02  5.532636e+02  9.355903e-02      1951   1
        18   5.468577e+02  5.532636e+02  9.833217e-02      2054   1
        19   5.742471e+02  5.532636e+02  1.030741e-01      2157   1
        20   3.917188e+02  5.532636e+02  1.078582e-01      2260   1
        21   4.605557e+02  5.532636e+02  1.265640e-01      2563   1
        22   5.514938e+02  5.532636e+02  1.313231e-01      2666   1
        23   5.249043e+02  5.532636e+02  1.360321e-01      2769   1
        24   6.066873e+02  5.532636e+02  1.407812e-01      2872   1
        25   4.488704e+02  5.532636e+02  1.455350e-01      2975   1
        26   6.066873e+02  5.532636e+02  1.504030e-01      3078   1
        27   6.066873e+02  5.532636e+02  1.551602e-01      3181   1
        28   6.066873e+02  5.532636e+02  1.599011e-01      3284   1
        29   6.066873e+02  5.532636e+02  1.646991e-01      3387   1
        30   6.066873e+02  5.532636e+02  1.694591e-01      3490   1
        31   6.066873e+02  5.532636e+02  1.742442e-01      3593   1
        32   6.066873e+02  5.532636e+02  1.790462e-01      3696   1
        33   6.066873e+02  5.532636e+02  1.838071e-01      3799   1
        34   6.066873e+02  5.532636e+02  1.887372e-01      3902   1
        35   5.468577e+02  5.532636e+02  1.935492e-01      4005   1
        36   6.066873e+02  5.532636e+02  2.909710e-01      4108   1
        37   6.066873e+02  5.532636e+02  2.960660e-01      4211   1
        38   6.066873e+02  5.532636e+02  3.010802e-01      4314   1
        39   5.586560e+02  5.532636e+02  3.061452e-01      4417   1
        40   5.586560e+02  5.532636e+02  3.111892e-01      4520   1
-------------------------------------------------------------------
status         : simulation_stopping
total time (s) : 3.111892e-01
total solves   : 4520
best bound     :  5.532636e+02
simulation ci  :  5.421768e+02 ± 3.480225e+01
numeric issues : 0
-------------------------------------------------------------------

One way to query the optimal policy is with SDDP.DecisionRule:

first_stage_rule = SDDP.DecisionRule(model; node = 1)

A decision rule for node 1

solution_1 = SDDP.evaluate(first_stage_rule; incoming_state = Dict(:x => 0.0))

(stage_objective = -404.45816675481, outgoing_state = Dict(:x => 202.229083377405), controls = Dict{Any, Any}())

Here's the second stage:

second_stage_rule = SDDP.DecisionRule(model; node = 2)
solution = SDDP.evaluate(
    second_stage_rule;
    incoming_state = Dict(:x => solution_1.outgoing_state[:x]),
    noise = 170.0,  # A value of d[ω], can be out-of-sample.
    controls_to_record = [:u_sell],
)

(stage_objective = 846.7770916622595, outgoing_state = Dict(:x => 32.229083377405004), controls = Dict(:u_sell => 170.0))

Simulation

Querying the decision rules is tedious. It's often more useful to simulate the policy:

simulations = SDDP.simulate(
    model,
    10,  #= number of replications =#
    [:x, :u_sell, :u_make];  #= variables to record =#
    skip_undefined_variables = true,
);

simulations is a vector with 10 elements

length(simulations)

and each element is a vector with two elements (one for each stage)

length(simulations[1])

The first stage contains:

simulations[1][1]

Dict{Symbol, Any} with 9 entries:
  :u_make          => 202.229
  :bellman_term    => 957.722
  :noise_term      => nothing
  :node_index      => 1
  :stage_objective => -404.458
  :objective_state => nothing
  :u_sell          => NaN
  :belief          => Dict(1=>1.0)
  :x               => State{Float64}(0.0, 202.229)

The second stage contains:

simulations[1][2]

Dict{Symbol, Any} with 9 entries:
  :u_make          => NaN
  :bellman_term    => 0.0
  :noise_term      => 201.67
  :node_index      => 2
  :stage_objective => 1008.29
  :objective_state => nothing
  :u_sell          => 201.67
  :belief          => Dict(2=>1.0)
  :x               => State{Float64}(202.229, 0.559534)

We can compute aggregated statistics across the simulations:

objectives = map(simulations) do simulation
    return sum(data[:stage_objective] for data in simulation)
end
μ, t = SDDP.confidence_interval(objectives)
println("Simulation ci : $μ ± $t")

Simulation ci : 549.8905807929784 ± 40.30426836713124

Risk aversion revisited

SDDP.jl contains a number of risk measures. One example is:

0.5 * SDDP.Expectation() + 0.5 * SDDP.WorstCase()

A convex combination of 0.5 * SDDP.Expectation() + 0.5 * SDDP.WorstCase()

You can construct a risk-averse policy by passing a risk measure to the risk_measure keyword argument of SDDP.train.

We can explore how the optimal decision changes with risk by creating a function:

function solve_newsvendor(risk_measure::SDDP.AbstractRiskMeasure)
    model = SDDP.LinearPolicyGraph(;
        stages = 2,
        sense = :Max,
        upper_bound = 5 * maximum(d),
        optimizer = HiGHS.Optimizer,
    ) do subproblem, node
        @variable(subproblem, x >= 0, SDDP.State, initial_value = 0)
        if node == 1
            @stageobjective(subproblem, -2 * x.out)
        else
            @variable(subproblem, u_sell >= 0)
            @constraint(subproblem, u_sell <= x.in)
            @constraint(subproblem, x.out == x.in - u_sell)
            SDDP.parameterize(subproblem, d, P) do ω
                set_upper_bound(u_sell, ω)
                return
            end
            @stageobjective(subproblem, 5 * u_sell - 0.1 * x.out)
        end
        return
    end
    SDDP.train(model; risk_measure = risk_measure, print_level = 0)
    first_stage_rule = SDDP.DecisionRule(model; node = 1)
    solution = SDDP.evaluate(first_stage_rule; incoming_state = Dict(:x => 0.0))
    return solution.outgoing_state[:x]
end

solve_newsvendor (generic function with 1 method)

Now we can see how many units a decision maker would order using CVaR:

solve_newsvendor(SDDP.CVaR(0.4))

182.17860665923672

as well as a decision-maker who cares only about the worst-case outcome:

solve_newsvendor(SDDP.WorstCase())

156.78434244660437

In general, the decision-maker will be somewhere between the two extremes. The SDDP.Entropic risk measure is a risk measure that has a single parameter that lets us explore the space of policies between the two extremes. When the parameter is small, the measure acts like SDDP.Expectation, and when it is large, it acts like SDDP.WorstCase.

Here is what we get if we solve our problem multiple times for different values of the risk aversion parameter $\gamma$:

Γ = [10^i for i in -4:0.5:1]
buy = [solve_newsvendor(SDDP.Entropic(γ)) for γ in Γ]
Plots.plot(
    Γ,
    buy;
    xaxis = :log,
    xlabel = "Risk aversion parameter γ",
    ylabel = "Number of pies to make",
    legend = false,
)

Things to try

There are a number of things you can try next:

Experiment with different buy and sales prices
Experiment with different distributions of demand
Explore how the optimal policy changes if you use a different risk measure
What happens if you can only buy and sell integer numbers of newspapers? Try this by adding Int to the variable definitions: @variable(subproblem, buy >= 0, Int)
What happens if you use a different upper bound? Try an invalid one like -100, and a very large one like 1e12.