Project Selection Model¶

Definition¶

The project selection model chooses complete projects under resource budgets. Let $y_j=1$ if project $j$ is selected, $r_j$ be its return, $a_{tj}$ its expenditure in year $t$ , and $B_t$ that year's budget. Then

\begin{gathered} \max\sum_jr_jy_j,\\ \sum_ja_{tj}y_j\le B_t\ \text{for every }t,\\ y_j\in\{0,1\}. \end{gathered}

Interpretation¶

The model chooses which complete projects to fund so that the total return is as large as possible without exceeding any year's budget. A project uses resources in each relevant year and contributes its full return only when selected; fractional project funding is excluded.

A table of five projects with returns and yearly budget use, with the selected projects 1, 3 and 4 highlighted.

The green columns are the selected projects 1, 3 and 4, with $y_j=1$ $1$ . Each year's budget row adds only their expenditures, $a_{t1}+a_{t3}+a_{t4}\le B_t$ $B_{t}$ , because the unselected projects 2 and 5 have $y_j=0$ and contribute nothing.

The product $a_{tj}y_j$ equals $a_{tj}$ when project $j$ is selected and 0 otherwise, so $\displaystyle \sum_ja_{tj}y_j$ is the spending on the selected projects in year $t$ . The same $y_j$ appears in every year's constraint: a selected project must be paid for in every year it needs money, and the projects compete for the budget of each year. The model is a binary integer program with one binary variable per project.

Ranking projects by return, or by return per unit of cost, and funding them in that order is a heuristic that can miss the optimum. With a single budget of 10, a project costing 6 with return 7 has the best ratio, but after funding it neither of two projects costing 5 with return 5 fits; funding those two instead gives return 10.

Key Results¶

A separate budget constraint is needed for each year.
The later development-project example uses one capital budget.
Relaxing a binary variable to a fraction supplies a bound but not an implementable partial project.

Examples¶

The green columns in the figure represent selected projects, $y_j=1$ $1$ ; each budget row sums only their expenditures.

The week 2 exercise has five projects and a budget of 25 in each of three years.
The development-project example has capital 20 and an optimal return of 3.4 from projects 1, 3, and 4.

Prerequisites¶

Binary Integer Program

Sources¶

OR Modelling · Week 2

Aliases: project selection problem Symbols: element of, sum

Last updated: 20 October 2026 03:02