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Cargo time windows have been discussed in the shipping literature in several ways and within several contexts. One way considers the  hard  time windows, where a fixed date is assigned to both the open and close laycan for the cargo loading and discharging ports. According to this way, the cargo is rejected if the ship cannot meet these time windows. Another way develops  soft  time windows, where the ship for an additional cost or a penalty may violate the cargo laycan. The context in the previous ways includes a tramp deterministic model. In this paper, the soft time windows are considered within a context of a tramp stochastic model, where both the cargo transport demand and the laycan dates are random variables. The objective is a gross-profit-per-day accompanied with realistic shipping elements embedded into decision support systems known as shipping optimization systems (SOS). The chance-constraint programming and the modified version of the Dantzig–Wolfe decomposition principle known as block-angular linear ratio programming solve the problem. A case study demonstrates that using stochastic soft time windows with stochastic cargo transport demand has considerably improved the gross profit-per-day better than trying to arrive on time by varying ship speed or being within a non-stochastic shipping context.

Shipping optimization systems (SOS) for tramp: stochastic cargo soft time windows