Design of the cost allocation rule for joint replenishment to an overseas warehouse with a piecewise linear holding cost rate

To obtain higher profits, a third-party overseas warehouse increases the delivery frequency of goods by setting a holding cost rate that increases with the holding time. To reduce the cost incurred from using the third-party overseas warehouse, small cross-border e-commerce retailers form a coalition to jointly replenish the overseas warehouse. Based on cooperative game theory, this paper designs a cost allocation rule to allocate the total cost of a cooperative coalition. The cost allocation rule is proved to be in the core of the corresponding cooperative game, indicating that it satisfies individual rationality and coalition rationality. An (1+ϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\epsilon )$$\end{document}-approximate algorithm is presented to search for the optimal replenishment cycle under joint replenishment. Numerical experiments show that joint replenishment can help small cross-border e-commerce retailers to reduce costs, shorten replenishment cycles, and expedite commodity turnover. Having a greater number of retailers involved in joint replenishment would help to save more costs. The average saving cost rate after cooperation can be 58% when the fixed costs are very high.