An approximation algorithm for stochastic multi-level facility location problem with soft capacities

Facility location problem is one of the most important problems in the combinatorial optimization. The multi-level facility location problem and the facility location with capacities are important variants for the classical facility location problem. In this work, we consider the multilevel facility location problem with soft capacities in the uncertain scenario. The uncertainty setting means the location process is stochastic. We consider a two-stage model. The soft-capacities setting means each facility has multiple capacities by paying multiple opening cost. The multi-level setting means the client needs to connect to a path. We propose a bifactor (1/α,6/(1-2α))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (1/\alpha ,6/(1-2\alpha ) )$$\end{document}-approximation algorithm for the stochastic multi-level facility location problem (SMLFLP), where α∈(0,0.5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha \in (0,0.5) $$\end{document} is a given constant. Then, we reduce the stochastic multi-level facility location problem with soft capacities to SMLFLP. The reduction implies a 1/α+6/(1-2α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left( 1/\alpha + 6/(1-2\alpha ) \right) $$\end{document}-approximation algorithm. The ratio is 14.9282 when setting α=0.183\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha = 0.183 $$\end{document}.