Mathematical formulation

We are given the set of gases G={lin,lox} and the set of time periods T={1,2,...,N}, the maximum production level P_lin and P_lox for each period and the inventory capacity S_lin and S_lox for each gas. The demand is given by the parameter dem_tg for t∈ T,g∈ G.

The uncertainty set is defined by every vector of interruptions (χ)_{i∈ T} such that χ_i ∈ [0,1] and not more than K of its elements are positive:

∑_{i ∈ T} χ_i ≤ K.

Let us define the variables prod_tg and inv_tg representing the production and inventory level, respectively, at period t for gas g. Without uncertainty in the input data, the model would have simple of constraints: bounds on the variables and fixing of the initial inventory level:

0≤ prod_tg ≤ P_g ∀ g, ∀ t∈ {1,2,...,N}

0≤ inv_tg ≤ S_g ∀ g, ∀ t∈ {1,2,...,N}

0≤ inv_0g = I_g ∀ g;

and the production planning constraint would consist of a mass conservation constraint for every gas g and time period t:

inv_t-1,g + prod_tg = inv_tg + dem_tg ∀ t ∈ T, g∈ G.

The modeling is more involved for this case. A first attempt would be to write the robust constraint by multiplying the production variable by (1-χ_t), so that the actual production at time period t is zero if χ_t=1:

inv_t-1,g + (1-χ_t)prod_tg = inv_tg + dem_tg ∀ t ∈ T, g∈ G.

However, this would yield no feasible solution: for each robust constraint related to production at time period t the uncertain χ_t would be set to one, and no production would occur. We need to aggregate the production constraints in such a way that the balance between production and demand yields an inventory that is nonnegative at every time period. The above constraint is replaced by the following one:

inv_0,g + ∑_{τ∈ T: τ ≤ t} ((1-χ_τ)prod_{τ g} - dem_{τ g}) ≥ 0 ∀ t ∈ T, g∈ G.

This robust constraint considers uncertainty in the interruptions but allows for a production plan that satisfies all demands even with K interruptions. In order to deal with contour conditions and take into account the initial inventory, we add a slight modification to the uncertainty set and require that no interruption happens at time period 1:

χ₁ = 0.

Finally, the robust value of the inventory must take into account the situation in which, after planning the production, no interruption actually occurs: this is simply enforced by bounding the inventory at every time period as the sum of all accumulated production discounted by the accumulated demand:

inv_tg ≥ inv_0,g + ∑_{τ∈ T: τ ≤ t} (prod_{τ g} - dem_{τ g}) ∀ t ∈ T, g∈ G.

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