Economic lot sizing problem with inventory dependent demand

We consider an economic lot sizing problem where the demand in a period is a piecewise linear and concave function of the amount of the available inventory after production in that period. We show that the problem is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {N}}}{{\mathcal {P}}}$$\end{document}-hard even when the production capacities are time invariant, and propose a polynomial time algorithm to the case where there are no capacity restrictions on production.