What is the optimal solution and maximum net return for the four-project selection example?

Conditions

• The model includes four projects with specific returns and monthly costs.
• The objective is to maximize total net return subject to monthly budget constraints.
• Variables are binary ($X_i \in \{0, 1\}$).

Reasoning, step by step

1. Solve the integer linear programming model using software.
2. Identify the values of the decision variables in the optimal solution.
3. Calculate the total net return by substituting the optimal variable values into the objective function.
4. Verify that the solution satisfies all budget constraints.

Example

The video states the optimal solution is $X_1=1, X_2=0, X_3=1, X_4=1$. The calculated optimal return is 414. Checking the January constraint: $58(1) + 44(0) + 26(1) + 23(1) = 58 + 26 + 23 = 107 \le 120$, which is feasible.

Common misconceptions

• Assuming all projects with positive returns should be selected.
• Believing the optimal return is the sum of all individual project returns without considering budget limits.
• Confusing the cost coefficients with the return coefficients when calculating the final value.

Watch the explanation

Explore next

• How do you formulate a project selection problem with binary variables and monthly budget constraints?
• Why is project 2 excluded in the optimal solution?
• What happens if the January budget is reduced to 100?