There exists *m* supply stations *s*_{i}, each capable of producing *sup*(*s*_{i}) units of a commodity. There are *n* demand stations *t*_{j}, each demanding *dem*(*t*_{j}) units of the commodity. There are *w* warehouse stations *w*_{k}, each capable of receiving and reshipping (known as "transshipping") a maximum *max*_{k} units of the commodity at the fixed warehouse processing cost of *w**p*_{k} per unit. There is a fixed shipping cost of *d*(*i*, *j*) for each unit shipping from supply station *s*_{i} to demand stations *t*_{j}, a fixed transshipping cost of *ts*(*i*, *k*) for each unit shipped from supply station *s*_{i} to warehouse station *w*_{k}, and a fixed transshipping cost of *ts*(*k*, *j*) for each unit shipped from warehouse station *w*_{k} to demand station *t*_{j}. The goal is to determine the flow *f*(*i*, *j*) of units from supply station *s*_{i} to demand station *t*_{j} that minimizes the overall total cost, which can be concisely defined as:

Total Cost (TC) = Total Shipping Cost (TSC) + Total Transshipping Cost (TTC) |

TSC = Σ _{i} Σ _{j} d(i, j)*f(i, j) |

TTC = Σ _{i} Σ _{k} ts(i, k)*f(i, k)+ Σ _{j} Σ _{k} ts(j, k)*f(j, k) |

The goal is to find integer values for *f*(*i*, *j*)≥0 that ensure that *TC* is a minimum while meeting all of the supply and demand constraints. Finally, the net flow of units through a warehouse must be zero, to ensure that no units are lost (or added!). The supplies *sup*(*s*_{i}) and demands *dem*(*t*_{i}) are all greater than 0. The shipping costs *d*(*i*, *j*), *ts*(*i*, *k*), and *ts*(*k*, *j*) may be greater than or equal to zero.

**Solution**

We convert the Transshipment problem instance into ...

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