🤖 AI Summary
This work proposes the first polynomial-time approximation scheme with provable performance guarantees for the economic lot-sizing problem, a classic inventory management challenge that has lacked such algorithms since its formulation in the 1950s. By integrating dynamic programming encodings, approximation techniques, and combinatorial optimization, the authors establish an efficient representation and optimization framework for dynamic replenishment policies, resolving the long-standing open issue that such policies inherently require exponential space. When the number of item types is constant, the method constructs an ε-optimal dynamic policy in polynomial time, overcoming prior limitations that relied on restrictive structural assumptions or offered no performance guarantees. This breakthrough substantially advances the algorithmic tractability of this fundamental problem.
📝 Abstract
In this focused technical paper, we present long-awaited algorithmic advances toward the efficient construction of near-optimal replenishment policies for a true inventory management classic, the economic warehouse lot scheduling problem. While this paradigm has accumulated a massive body of surrounding literature since its inception in the late'50s, we are still very much in the dark as far as basic computational questions are concerned, perhaps due to the intrinsic complexity of dynamic policies in this context. The latter feature forced earlier attempts to either study highly-structured classes of policies or to forgo provably-good performance guarantees altogether; to this day, rigorously analyzable results have been few and far between. The current paper develops novel analytical foundations for directly competing against dynamic policies. Combined with further algorithmic progress and newly-gained insights, these ideas culminate in a polynomial-time approximation scheme for constantly-many commodities. In this regard, the efficient design of $\epsilon$-optimal dynamic policies appeared to have been out of reach, since beyond their inherent algorithmic challenges, even the polynomial-space representation of such policies has been a fundamental open question.