Uncapacitated lot sizing with backlogging: the convex hull (Q1016115)

From the abstract: ``An explicit description of the convex hull of solutions to the uncapacitated lot-sizing problem with backlogging in its natural space of production, setup, inventory and backlogging variables, has been an open question for many years.'' Uncapacitated lot-sizing problem with backlogging can be formulated as \[ \begin{gathered} z= \min\sum^n_{t=1} f_t x_t+ c_t y_t+ g_t r_t+ h_t s_t),\\ s_{t-1}+ y_t- r_{t-1}= d_t+ s_t- r_t,\quad t= 1,\dots, n,\\ y_t\leq dx_t,\quad t= 1,\dots, n,\quad d= \sum^n_{j=t} d_j,\\ r_0= s_0= r_n= s_n= 0,\\ y\in \mathbb{R}^n_+,\quad s\in\mathbb{R}^{n+1}_+,\quad r\in\mathbb{R}^{n+1}_+,\quad x\in \{0,1\}^n.\end{gathered} \] In this paper, the authors identify valid inequalities that subsume all previously known valid inequalities for this problem. They show that these inequalities are enough to describe the convex hull of solutions. The authors give polynomial separation algorithms for some special cases. Finally, they report a summary of computational experiments with our inequalities that illustrates their effectiveness.