Multi-period empty container repositioning with stochastic demand and lost sales

Abstract

This study is concerned with empty container repositioning between multi-ports with stochastic demand and lost sales over multi-periods. Unmet demands due to the unavailability of empty containers will be lost forever and will incur a stockout cost in view of the fact that maritime container shipping is a highly competitive industry. The objective is to find an effective empty container repositioning policy by minimizing the total operating cost including container holding cost, stockout cost, importing cost and exporting cost. First, this study focuses on the empty container repositioning problem in a single port. This problem is mathematically formulated as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (An,Sn), i.e., importing empty containers up to An when the number of empty containers in the port is fewer than An; exporting empty containers down to Sn when the number of empty containers in the port is more than Sn; and doing nothing, otherwise. A polynomial time algorithm is developed to determine the two thresholds, i.e., An and Sn for each period. Two numerical examples are provided to illustrate the solution procedures based on the normal distribution and uniform distribution, respectively. The results show that the proposed algorithm performs highly effectively and efficiently. Next, this study extends the single-port results to the multi-port case. The multi-port problem is also mathematically formulated and a tight lower bound on the cost function is determined. The concept of relative error with respect to the tight lower bound is introduced, which is used to measure the performance of the algorithm. Based on the two-threshold optimal policy for a single port, a polynomial time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively as the calculated average relative error with respect to the tight lower bound is within 5 per cent for the normal distribution and uniform distribution, respectively. Furthermore, the algorithm performs very efficiently due to its polynomial running time. The stability of the algorithm improves as the number of ports increases. More importantly, the approximate repositioning policy is easy to understand and implement from a practical perspective as a result of its simplicity.