When you face an analytics scenario that requires optimal allocation of resources—such as distributing a fixed budget across multiple projects—the first step is to translate the real-world problem into a linear programming (LP) model. This process involves three main steps: identifying decision variables, writing the objective function, and expressing constraints.

Start by pinpointing the decision variables—the quantities you control and need to determine. In a budget allocation scenario, these might be the amounts of money assigned to each project.

Next, define the objective function. This is a mathematical expression that represents what you want to optimize, such as maximizing the total expected return from all projects.

Symbolic LP formulation for budget allocation across projects:

Let $x_1, x_2, \ldots, x_n$ be the amounts allocated to projects $1, 2, \ldots, n$.

Objective: maximize total expected return $\max: \quad r_1 x_1 + r_2 x_2 + \ldots + r_n x_n$

Subject to:

• Budget constraint: $x_1 + x_2 + \ldots + x_n \leq \text{TotalBudget}$;
• Minimum allocation: $x_1 \geq \text{Min}_1,\ x_2 \geq \text{Min}_2,\ \ldots,\ x_n \geq \text{Min}_n$;
• Maximum allocation: $x_1 \leq \text{Max}_1,\ x_2 \leq \text{Max}_2,\ \ldots,\ x_n \leq \text{Max}_n$.

Decision variables: $x_j \geq 0$ for each project $j$

Finally, state your constraints. These are the limits or requirements that must be respected, such as not exceeding the total available budget or ensuring minimum investment in certain projects.