# Erratum to: Shipping Optimisation Systems (SOS): tramp optimisation perspective

The Original Article was published on 10 July 2017

## Erratum

Unfortunately, the original PDF version of this article  contained some errors. The equation 8, 11, 13, 14, 15, 16, 17, 19, 22 all displayed incorrectly in the section of Appendix 1. The correct equations can be found below and the original article of PDF version has been updated to reflect this change.

$${V}_j^k\ge {V}_i^k-{v}_i{\sum}_{z\in Z}{x}_{ijz}^k,\kern0.75em i\in L,j\in E,\kern0.5em \mathrm{and}\kern0.5em k\in S,\kern0.5em \mathrm{where}\kern0.5em {\sum}_{z\in Z}{x}_{ijz}^k=1,$$
(8)
$${V}_j^k\ge {V}_i^k+{v}_i{\sum}_{z\in Z}{x}_{ijz}^k,i\in D,j\in E,\mathrm{and}\kern0.75em k\in S,\kern0.5em \mathrm{where}\ {\sum}_{z\in Z}{x}_{ijz}^k=1,$$
(11)
$${W}_i^k\ge {W}^k,i\in D,\kern0.5em \mathrm{and}\ k\in S,\kern0.5em \mathrm{where}\kern0.75em {\sum}_{z=3,4}{x}_{igz}^k=1,g\in G,$$
(13)
$${V}_i^k\ge {V}^k,i\in D,\mathrm{and}\ k\in S,\mathrm{where}\kern0.75em {\sum}_{z=3,4}{x}_{igz}^k=1,g\in G,$$
(14)
$${N}_i^k\ge {N}^k,i\in D,\mathrm{and}\ k\in S,\mathrm{where}\ {\sum}_{z=3,4}{x}_{igz}^k=1,g\in G,$$
(15)
$${W}_i^k\ge {W}_{0,}^k\ i\in D, and\ k\in S, where\ {\sum}_{z=1,2}{x}_{ijz}^k=1,j\in E,$$
(16)
$${T}_j^k\ge {T}_i^k+{t}_i+{\sum}_{z\in Z}{t}_{ijz}^k{x}_{ijz}^k,i\in {E}_f,j\in {E}_g,\mathrm{and}\ k\in S,\mathrm{where}\ {t}_{ifz}^k=0,\mathrm{and}{\sum}_{z\in Z}{x}_{ijz}^k=1,$$
(17)
$${T}_j^k\ge {a}_i,j\in E,\mathrm{and}\ k\in S,\mathrm{where}{\sum}_{i\in Ef}{\sum}_{z\in Z}{x}_{ijz}^k=1,$$
(19)
$${T}_g^k\le {T}^k,g\in G,k\in S,\mathrm{where}\ {\sum}_{z=3,4}{x}_{igz}^k=1\kern0.5em \mathrm{and}\kern0.5em i\in D,$$
(22)

## Reference

1. Said El Noshokaty (2017) Shipping Optimisation Systems (SOS): tramp optimisation perspective. J Shipp Trade 2:3. https://doi.org/10.1186/s41072-017-0021-y

## Author information

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Correspondence to Said El Noshokaty. 