Optimum speed economics
The cost-minimising speed derivation, the (n-1) coefficient, the cube-root damping against market swings, and the cargo inventory term most models leave out.
Optimum speed economics is the body of models that trade a ship’s fuel cost, which rises steeply with speed, against the opportunity cost of its time, which falls with speed. The result is a cost-minimising speed that is neither the design speed nor the slowest the ship can run.
With daily consumption modelled as a coefficient multiplied by speed to the power n, and the ship’s time valued at a daily opportunity cost, the optimum is the nth root of that cost divided by (n minus 1) times the coefficient times the bunker price. Two features are commonly got wrong. The denominator carries n minus 1, not n, because the exponent governing a fixed-distance voyage is one less than the one governing the instantaneous fuel rate. And distance cancels out entirely, so leg length does not affect the optimum.
The robust practical result is the damping: because the optimum varies as the cube root of the earnings-to-bunker ratio, doubling the daily time value raises it by about 26 percent while doubling the bunker price lowers it by only about 21 percent. Ronen (1982) set out three models under three different revenue schedules, and Psaraftis and Kontovas added the point that the cargo’s inventory carrying cost belongs in the objective function alongside the owner’s fuel bill. See slow steaming and time charter equivalent .