Mathematical expressions for the transit time of merchandise through a liner shipping network

Recent publications on the design of liner shipping networks are limited in their treatment of the level of service (LoS) experienced by shippers. We propose the use of inventory holding costs—a function of merchandise transit time—as a proxy for LoS. We assume the existence of a two-tier optimization model, where fleet deployment, vessel routing, and vessel speed are determined in the higher tier. Merchandise flows and transshipment quantities are determined in the lower tier. We partition the total merchandise transit time into time spent in open waters, time spent during port calls, and time spent dwelling in the terminal yard. Using the notions of service frequency and service phase, we develop mathematical expressions for the three aforementioned quantities within the lower tier of the optimization model. We arrive at a bilinear expression for overall inventory holding costs that is suitable for liner shipping network design.