Integrated optimization of feeder routing and stowage planning for containerships

Abstract

The sailing safety constraints of containerships were ignored in the previous studies on feeder containership routing problems; however, they are especially critical for small-sized feeder containerships. In this study, we optimize feeder routes while incorporating stowage plans to address the sailing safety of containerships. Firstly, a mixed integer nonlinear programming model for integrated optimization is formulated. Next, a heuristic algorithm is designed, by which feeder routes can be updated through a variable neighborhood search, and stowage plans are obtained using a genetic algorithm. Finally, through the computational study, we confirm that the integrated optimization can meet the sailing safety requirements of containerships and effectively reduce the total cost of the feeder service. Moreover, through the sensitivity analysis, we discuss the performance robustness of the proposed algorithm and further demonstrate the significance of this study.

Appendices

Appendix A

Formulations for GM, T, and Ms

a. GM can be calculated as $GM=KM-KG$. KM is the distance between the keel and metacenter of the containership, it is related to the draft, and can be accurately determined using the hydrostatic curve provided by the containership; KG is the distance between the keel and center of gravity of the containership, which can be calculated as $KG=\frac{\sum _{s\in S_k}W_b^{ks}\times H_s}{D}$, where D is the loaded displacement of the containership and $H_s$ is the height of the center of slot s.

b.  Ms can be calculated as $Ms=\frac{1}{2}\times (\gamma \times DL +\sum _{s\in S_k}W_b^{ks}\times I_s-D\times f\times L)$. $\gamma$ is the equivalent arm of the total moment of the deadweight of the first and the latter half to the center cross section of the containership; DL is the light displacement of the containership; $I_s$ is the horizontal distance between slot s and the vertical center of the containership; f is the equivalent arm of the buoyancy of the containership; and L is the length of the containership.

c.  The trim can be calculated as $T=\frac{\sum _{s\in S_k}I_s\times W_b^{ks}-LCB\times D}{MCT\times 100}$. LCB is the distance between the center of buoyancy and the vertical center of the containership and can be obtained from the trim diagram of the containership; and MCT is the moment required to change the trim of the containership by 1 cm, which can also be obtained from the trim diagram.

Appendix B

The framework of Hierarchical optimization-I and hierarchical optimization-II is shown in Tables 13 and 14, respectively.