@bernardokp I don't really understand how this formulation matches the description.

• Where does the y.in^2 come from?
• Is the buy decision Decision-Hazard?

Can we do something more explicit like this?

model = SDDP.LinearPolicyGraph(
    stages = T + 1,
    sense = :Min,
    lower_bound = 0.0,
    optimizer = Ipopt.Optimizer,
) do sp, t
    @variable(sp, x_inventory >= 0, SDDP.State, initial_value = x0)
    @variable(sp, x_unsatisfied_demand >= 0, SDDP.State, initial_value = 0)
    @variable(sp, u_buy >= 0)
    @variable(sp, u_sell >= 0)
    @variable(sp, w_demand == 0)
    @constraint(sp, x_inventory.out - x_inventory.in == u_buy - u_sell)
    @constraint(
        sp,
        x_unsatisfied_demand.out - x_unsatisfied_demand.in == w_demand - u_sell,
    )
    if t == 1
        @stageobjective(sp, 0)
    elseif t == T + 1
        @stageobjective(
            sp,
            α^(t - 1) * c * (x_unsatisfied_demand.in - x_inventory.in),
        )
    else
        @stageobjective(
            sp,
            α^(t - 1) * (
                c * u_buy +
                h * x_inventory.out +
                p * x_unsatisfied_demand.out,
            )
        )
        SDDP.parameterize(ω -> JuMP.fix(w_demand, ω), sp, Ω)
    end
    return
end

Hello Oscar,

thanks for adding this example.
1 - Looking at the cost expression, we have linear integrands in $y$. If the density function is constant, it will give rise to quadratic terms on $y$.
2 - Yes, it is decision-hazard.

What you wrote is correct, as it correctly captures what the loss function is. However, it adds another layer of approximation at the stage objective cost. The way I wrote makes this cost explicit and deterministic, without the need to resort to sampling.

One thing that bothers me is the plots you added to the example. I obtained completely different ones. For the finite case, the policy is to order up to 741, and then some end-of-horizon effects take place. For the infinite horizon case, the policy is a horizontal line at the number 741.

Best,
Bernardo

But then the stage objective for stage t includes the expected cost of meeting demand in stage t+1? That isn't how one would typically formulate a problem. It's cheating because you're assuming a uniform distribution and ignoring the samples. If we simulated the policy out-of-sample (with a non-uniform distribution) it wouldn't work?

I see your point. My claim is that it is not cheating because you know the distribution and you can integrate it in the present, so there is no need for samples for that. However, you cannot incorporate it explicitly into the future cost function because the functional form is too complicated. In that case, you must use samples.

In summary, you remove one layer of approximation by having an exact cost at each stage simply because you can integrate a function.

Where did you get the discount factor of α = 0.995 from? That's an interest rate of 0.5%

Standard, textbook parameter choices. Please take a look at Hillier and Lieberman, chapter 19.