The economic ship speed under time charter contract—A cash flow approach

Various deterministic models for economic ship speed optimisation exist in the literature, but none considered the time charter contract, and in particular the influence of the redelivery time. This paper studies the economic optimal speed of a ship on a (time) charter contract through the development of an Operational Research (OR) optimisation model. The ship charterer’s objective is to maximise the Net Present Value (NPV) of a cash-flow function of the ship’s activities over a relevant future horizon H, where H can be interpreted as any possible day within the redelivery time window as specified in the time charter clause. We develop a general time charter contract model PM(H), and three special cases P1, P∞ and PM(H→∞), each model mapping onto different contractual contexts, and present algorithms to each of these models for finding optimal ship speeds for any journey structure. While ships on time charter contracts may travel to any series of ports during the charter contract, examining the models’ behaviour when the ship repeatedly executes a roundtrip journey allows us to reach some important general insights about the impact of contract type for any journey structure. In particular, economic speeds in PM(H) follow a very different pattern than those in the classic models from the literature, as well as in the recent class of NPV models P (n,m,Go) from Ge et al., (2021). We prove that P1 and P∞ map quite generally to the classes P (1,n,0) and P (∞,n,−), respectively, while P (n,m,0) shows behaviour in approximation equal to the special case PM(H→∞). We prove that two main strands of speed optimisation models from the literature, which did not consider the contract type nor used the NPV approach, show equivalence under mild conditions to P1 and P∞, respectively. These results facilitate matching models to contract types. None of these models, however, matches the general time charter contract model PM(H) introduced in this paper. In general, the paper demonstrates how optimal economic speed is dependent on the (time) charter contract type, and that this should thus be reflected in the speed optimisation model developed.