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

Abstract

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.

Appendix

Theorem 1

• For two relatively prime service inter-arrival frequencies ζ _r , ζ _s ∈ℤ⁺, and for every dwell interval k∈{0, 1, 2, …(ζ _s −1)}, there exists an arrival q∈{0, 1, 2, …(ζ _s −1)} of a vessel in service r at time t=q·ζ _r such that the earliest subsequent arrival in service s occurs k units of time later.

Proof

• We denote by k _q the dwell interval of the merchandise arriving in the qth call of service r. We have established that k _q =⌈q·ζ _r /ζ _s ⌉·ζ _s −q·ζ _r . Given that a vessel in service s arrives every ζ _s units of time, we must have 0⩽k _q ⩽ζ _s −1 for any q. Given that the k _q can take only ζ _s integer values, we only need to show that the k _q are different for each of the ζ _s values of q in {0, 1, 2…(ζ _s −1)}.

In order to derive a contradiction, assume k _q =k _q′, with q≠q′. We would then have

However, ζ _s and ζ _r are relatively prime, and ζ _s does not divide (q−q′). We then have a contradiction, as the right-hand side of (22) is not integral.

This demonstrates that all the k _q are different for q∈{0, 1, 2…(ζ _s −1)}. □

Corollary 1

• For any two service inter-arrival frequencies ζ _r , ζ _s ∈ℤ⁺, and for every dwell interval k∈{0, 1·μ _rs , 2·μ _rs …(ζ _s /μ _rs −1)·μ _rs }, there exists an arrival q∈{0, 1, 2…γ _rs /ζ _r −1} of a vessel in service r at time t=q·ζ _r such that the earliest subsequent arrival in service s occurs k units of time later.

Proof

• We can compress the time dimension by μ _rs =gcd (ζ _r , ζ _s ), which does not alter the correspondence between vessels. We then apply Theorem 1 with ζ _r /μ _rs and ζ _s /μ _rs . □