A Small Scale Manufacturer produces two types of Products A and B which gives a profit margin of AED 70 and AED 50 respectively. There are two plants I and II, each having a capacity of 240 and 100 hours. Cycle time of A and B are 4 hours and 3 hours in plant I and 2 hour and 1 hour in Plant II. How many products of A and B to be manufactured? Formulate the problem.

To formulate the problem of maximizing profit for the small-scale manufacturer producing products A and B, we can define the variables, constraints, and the objective function. ### Variables: Let: - \( x_A \) = number of products A produced - \( x_B \) = number of products B produced ### Objective Function: The objective is to maximize the total profit, which can be expressed as: \[ \text{Maximize } Z = 70x_A + 50x_B \] ### Constraints: We need to consider the capacity constraints for both plants. #### Plant I: - The cycle time for product A is 4 hours, and for product B is 3 hours. - The total available hours in Plant I is 240 hours. The constraint for Plant I can be expressed as: \[ 4x_A + 3x_B \leq 240 \] #### Plant II: - The cycle time for product A is 2 hours, and for product B is 1 hour. - The total available hours in Plant II is 100 hours. The constraint for Plant II can be expressed as: \[ 2x_A + 1x_B \leq 100 \] ### Non-negativity Constraints: Since the number of products cannot be negative, we have: \[ x_A \geq 0 \] \[ x_B \geq 0 \] ### Summary of the Formulation: Putting it all together, the linear programming problem can be formulated as follows: **Objective:** \[ \text{Maximize } Z = 70x_A + 50x_B \] **Subject to:** \[ 4x_A + 3x_B \leq 240 \quad \text{(Plant I constraint)} \] \[ 2x_A + 1x_B \leq 100 \quad \text{(Plant II constraint)} \] \[ x_A \geq 0 \] \[ x_B \geq 0 \] This formulation can now be solved using linear programming techniques (such as the Simplex method or graphical method) to find the optimal values of \( x_A \) and \( x_B \) that maximize the profit \( Z \).