Model formulations for the capacitated lot-sizing problem with service-level constraints

We present deterministic model formulations for the capacitated lot-sizing problem, including service-level constraints that address both the periodic and cyclic α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} service levels, as well as the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}, and η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} service levels. We assume that service levels for individual products are given and controlled over a given reporting period (i.e., one year). These deterministic model formulations may be used when all data are deterministic and decision makers intend to exploit given service levels to minimize setup and holding costs. A further application is to provide lower bounds for capacitated lot-sizing problems with stochastic demand and given service-level constraints. In contrast to well-known stochastic or robust optimization approaches, there are new proposals in the literature that do not require scenario modeling and thus have much greater potential for use in solving (large-scale) real-world production planning problems. However, an evaluation of the quality of solutions resulting from these new approaches is difficult. For this purpose, lower bounds showing the best possible (ideal) solution should be of great help. In a computational study, we provide insight into the computational efforts associated with deterministic model formulations with service-level constraints. Finally, lower bounds generated by the deterministic model with β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} service-level constraints are compared with the results of a rolling schedule strategy addressing a stochastic lot-sizing problem with given β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} service-level constraints. The resultant difference in the objective function values (costs) defines the uncertainty gap. We demonstrate its increase with forecast inaccuracy as well as machine utilization.