## The Unbalanced Transportation Problem

In practice supply rarely equals demand exactly, at least in the short term. The difference may be stored or taken from the supply chain, or from some warehouse. We can model the situation where supply does not equal demand by introducing dummy suppliers or customers. The table below shows demand for and supply of bread, with costs highlighted.

Warehouse 1 | Warehouse 2 | Warehouse 3 | Supply | |
---|---|---|---|---|

Bakery 1 | 6 | 6 | 7 | 7 |

Bakery 2 | 2 | 5 | 4 | 10 |

Demand | 6 | 8 | 9 |

Total demand is 6+8+9=23 and total supply is 7+10=17 so total demand exceeds total supply and not all warehouses will see their demand met.

We can balance supply and demand for the purpose of solving the transportation problem by introducing a dummy bakery. The dummy can meet the excess demand of 6 units of bread.

Warehouse 1 | Warehouse 2 | Warehouse 3 | Supply | |
---|---|---|---|---|

Bakery 1 | 6 | 6 | 7 | 7 |

Bakery 2 | 2 | 5 | 4 | 10 |

Dummy | 0 | 0 | 0 | 6 |

Demand | 6 | 8 | 9 |

The dummy has no costs, indicated by the zeros, because using the dummy does not involve transporting anything anywhere.

Similarly if supply exceeds demand, we can introduce a dummy supplier.

Warehouse 1 | Warehouse 2 | Warehouse 3 | Supply | |
---|---|---|---|---|

Bakery 1 | 6 | 6 | 7 | 16 |

Bakery 2 | 2 | 5 | 4 | 10 |

Demand | 6 | 8 | 9 |

Warehouse 1 | Warehouse 2 | Warehouse 3 | Dummy | Supply | |
---|---|---|---|---|---|

Bakery 1 | 6 | 6 | 7 | 0 | 16 |

Bakery 2 | 2 | 5 | 4 | 0 | 10 |

Demand | 6 | 8 | 9 | 3 |

The dummy can take the excess supply of 3.