Maximizing Profit with Resource Constraints?

📌 CONCEPT: Linear programming is a method to find the optimal solution for a problem with multiple variables and constraints, often used to maximize profit with resource constraints.

📐 RULE / FORMULA: The general form of a linear programming problem is Maximize (or Minimize) Z = ax + by, subject to constraints such as cx + dy ≤ k, where x and y are decision variables, a, b, c, d, and k are coefficients, and Z is the objective function.

💡 WORKED EXAMPLE: Suppose we want to maximize profit by producing two products, A and B. The profit per unit of A and B is ₹ 150 and ₹ 100, respectively. The labor union has imposed a constraint of 180 hours per day for both products combined. Let x be the number of units of A produced and y be the number of units of B produced. The linear programming model would be Maximize Z = 150x + 100y, subject to 3x + 2y ≤ 180. The optimal production levels of A and B can be found by solving this model.

⚠️ COMMON MISTAKE: Students often forget to formulate the constraints correctly, such as not including all the given constraints or not considering the direction of the inequality signs.