Warehouse Problem with Bounds, Fixed Costs and Complementarity Constraints

TL;DR

This paper introduces polynomial-time algorithms for a complex warehouse trading problem involving fixed costs, bounds, and complementarity constraints, using network flow techniques and extended linear formulations.

Contribution

It provides the first polynomial-time algorithms for this warehouse problem with fixed costs and complementarity constraints, and extends to approximation schemes without fixed costs.

Findings

Polynomial-time algorithms for fixed-cost warehouse trading problem.

Exact characterization of feasible region's extreme points.

Polynomial extended linear formulations for the problem.

Abstract

This paper studies an open question in the warehouse problem where a merchant trading a commodity tries to find an optimal inventory-trading policy to decide on purchase and sale quantities during a fixed time horizon in order to maximize their total pay-off, making use of fluctuations in sale and cost prices. We provide the first known polynomial-time algorithms for the case when there are fixed costs for purchases and sales, optional complementarity constraints that prohibit purchasing and selling during the same time period, and bounds on purchase and sales quantities. We do so by providing an exact characterization of the extreme points of the feasible region and using this to construct a suitable network where a min-cost flow computation provides an optimal solution. We are also able to provide polynomial extended linear formulations for the original feasible regions. Our methods build on the work by Wolsey and Yaman (Discrete Optimization 2018). We also consider the problem without fixed costs and provide a fully polynomial time approximation scheme in a setting with time-dependent bounds.