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.

© 2001-2020 Fair Isaac Corporation. All rights reserved. This documentation is the property of Fair Isaac Corporation (“FICO”). Receipt or possession of this documentation does not convey rights to disclose, reproduce, make derivative works, use, or allow others to use it except solely for internal evaluation purposes to determine whether to purchase a license to the software described in this documentation, or as otherwise set forth in a written software license agreement between you and FICO (or a FICO affiliate). Use of this documentation and the software described in it must conform strictly to the foregoing permitted uses, and no other use is permitted.

Contents

Index

Glossary

Search Results

Mathematical formulation