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I've tried hours and hours to model this problem but i didn't succeed. Can you help me ?

We want to realize n projects in the next T years. For each project, a profitability index pi is known, which expresses the expected final gain (in Euro) and a cost profile ai = (ai1, ai2, ..., aiT) for each year of the period considered. Moreover, in each year j of the period a budget of bj € is available. Which projects should be selected to maximize the expected profit while respecting budget constraints?

Assume that the projects have a duration of T '< T years. For each selected project, we also want to identify the start year.

Sometime ago i succeeded in modeling the non-bold part of the problem, however if i try to add also the last requirement i dont understand what to change of my original formulation, that is :

$max \sum_{i=1}^{n} p_{i}x_{i}$

$ \sum_{i=1}^{n} a_{ij}x_{i}=b_{j}$ , $j=1..T$

$x_{i}\in \left \{ 0,1 \right \}$ , $i=1..n$

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Koinos
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  • I do not see how time affects the problem. How does $a_i$ interact with the start year and duration? – LinAlg Oct 08 '18 at 12:34
  • If i start the project at year 10 and the project last for 4 years, then the cost in the 10-th year is a1, the cost in the 11-th year is a2 and so on until year 13 – Koinos Oct 08 '18 at 16:42
  • use a binary variable $x_{it}$ to indicate that project $i$ starts at time $t$ – LinAlg Oct 08 '18 at 16:45
  • Nice ! Now what i dont understand is how to handle the budget every year (if i starta project start at year 2 , all the ai are shifted by 2 from the initial year ( that is 0)) – Koinos Oct 08 '18 at 18:38

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