Boris Milkem owns six assets that he must sell over the next three years. For example, if he sells asset 3 in year 2, he receives $30 million. Each asset can be sold at most once, of course. He must sell at least $20 million of assets in year 1, $30 million in year 2, and 35 million in year 3. How can he maximize his total revenue from the sales? He expects that the sales will generate the following revenue, in millions of $

## Expert Answer

Let Yij be a binary integer so that

Yij = 1 when the i-th Asset is sold in j-th year

=0, otherwise ∀i = {1,2,…,6} and ∀j = {1,2,3}

Yij | Year 1 | Year 2 | Year 3 |

Asset 1 | Y11 | Y12 | Y13 |

Asset 2 | Y21 | Y22 | Y23 |

Asset 3 | Y31 | Y32 | Y33 |

Asset 4 | Y41 | Y42 | Y43 |

Asset 5 | Y51 | Y52 | Y53 |

Asset 6 | Y61 | Y62 | Y63 |

Objective Function: max.Z = Total Revenue = ∑_{i,j} Rij x Yij where Rij is the revenue matrix given

Subject to,

1. All assets can be invested once only

- Yi1 + Yi2 + Yi3 = 1 ∀i = {1,2,…,6}

2. Minimum year-wise revenue

- ∑
_{i}(Yi1 x Ri1) ≥ 20 ∀i = {1,2,…,6} - ∑
_{i}(Yi2 x Ri2) ≥ 30 ∀i = {1,2,…,6} - ∑
_{i}(Yi3 x Ri3) ≥ 35 ∀i = {1,2,…,6}

3. Binary Integer constraints: Yij ∈ {0,1}

Implementation in excel:

Yij | Year 1 | Year 2 | Year 3 | Total | Rij x Yij | Year 1 | Year 2 | Year 3 | |||

Asset 1 | 0 | 0 | 1 | 1 | = | 1 | Asset 1 | 0 | 0 | 24 | |

Asset 2 | 1 | 0 | 0 | 1 | = | 1 | Asset 2 | 16 | 0 | 0 | |

Asset 3 | 0 | 1 | 0 | 1 | = | 1 | Asset 3 | 0 | 30 | 0 | |

Asset 4 | 0 | 0 | 1 | 1 | = | 1 | Asset 4 | 0 | 0 | 30 | |

Asset 5 | 1 | 0 | 0 | 1 | = | 1 | Asset 5 | 17 | 0 | 0 | |

Asset 6 | 0 | 0 | 1 | 1 | = | 1 | Asset 6 | 0 | 0 | 29 | |

Total | 33 | 30 | 83 | ||||||||

Revenue (Rij) |
Year 1 | Year 2 | Year 3 | ≥ | ≥ | ≥ | |||||

Asset 1 | 15 | 20 | 24 | 20 | 30 | 35 | |||||

Asset 2 | 16 | 18 | 21 | Total Revenue |
146 |
||||||

Asset 3 | 22 | 30 | 26 | ||||||||

Asset 4 | 10 | 20 | 30 | ||||||||

Asset 5 | 17 | 19 | 22 | ||||||||

Asset 6 | 19 | 25 | 29 |