Inventory-eliminated formulation

Inventory-eliminated formulation

Class: ULSAlgorithms.Exact.Formulations.InventoryEliminatedFormulationSolver

Family: Solver-backed mathematical formulation
Time: Solver-dependent
Memory: O(T) variables + O(T²) coefficients/constraints
Applicability: General classical ULS

Description

Inventory-eliminated formulation is an exact solver-backed ULS strategy. It builds a mathematical formulation and delegates the optimization step to the selected external engine while keeping the common IUlsSolver result contract.

Mathematical model

After eliminating inventory, define

\[\bar p_t=p_t+\sum_{r=t}^{T-1}h_r, \qquad C=-\sum_{t=1}^{T-1}h_t\sum_{i=1}^{t}d_i. \]

The model is

\[\min \sum_{t=1}^{T}f_t y_t+\sum_{t=1}^{T}\bar p_t x_t+C \]

subject to

\[\sum_{i=1}^{t}x_i\ge\sum_{i=1}^{t}d_i, \qquad t=1,\ldots,T-1, \]

\[\sum_{i=1}^{T}x_i=\sum_{i=1}^{T}d_i, \]

\[x_t\le D_{t,T}y_t,\qquad D_{t,T}=\sum_{k=t}^{T}d_k, \qquad x_t\ge0,\ y_t\in\{0,1\}. \]

For the formulation taxonomy and historical context, see Mathematical Programming Formulations.

Minimal API

IUlsSolver solver = new InventoryEliminatedFormulationSolver();
UlsSolveResult result = solver.Solve(problem);

Scientific source

Brahimi, Dauzere-Peres, Najid & Nordli (2006), Single Item Lot Sizing Problems, European Journal of Operational Research 168(1), 1-16

Full class reference

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