<p>A manufacturing company makes two types of television sets; one is black and white and the other is in coloured. The company has resources to make at most 300 sets a week. It takes ₹1800 to make a black and white set and ₹2700 to make a coloured set. The company can spend not more than ₹648000 a week to make television sets. If it makes a profit of ₹510 per black and white set and ₹675 per coloured set, how many sets of each type should be produced so that the company has maximum profit? Formulate this problem as a LPP given that the objective is to maximise the profit.</p>
<p>\(x + y \leq 600,\, 2x + 3y \leq 720,\, x \geq 0,\, y \geq 0\) and \(z = 225x + 675y\)</p>
<p>\(x + y \leq 300,\, 2x + 3y \leq 360,\, x \geq 0,\, y \geq 0\) and \(z = 510x + 325y\)</p>
<p>\(x + y \leq 600,\, 2x + 3y \leq 360,\, x \geq 0,\, y \geq 0\) and \(z = 255x + 325y\)</p>
<p>\(x + y \leq 300,\, 2x + 3y \leq 720,\, x \geq 0,\, y \geq 0\) and \(z = 510x + 675y\)</p>
Step-by-Step Solution
Key Concept: Identify decision variables, formulate linear constraints from resource limitations, and set up an objective function to maximize profit. The profit per unit must be multiplied by the quantity produced to get the total profit expression.
<p><strong>Step 1: Define Decision Variables</strong></p><p>Let x = number of black and white TV sets produced per week<br>Let y = number of coloured TV sets produced per week</p><p><strong>Step 2: Formulate the Objective Function</strong></p><p>Profit from B&W sets = 510x<br>Profit from coloured sets = 675y<br><strong>Maximize Z = 510x + 675y</strong></p><p><strong>Step 3: Identify and Formulate Constraints</strong></p><p><strong>Production capacity constraint:</strong> Total sets ≤ 300<br>x + y ≤ 300</p><p><strong>Budget constraint:</strong> Total cost ≤ ₹648000<br>1800x + 2700y ≤ 648000<br>Dividing by 900: 2x + 3y ≤ 720</p><p><strong>Non-negativity constraints:</strong><br>x ≥ 0, y ≥ 0</p><p><strong>Step 4: Complete LPP Formulation</strong></p><p><strong>Maximize:</strong> Z = 510x + 675y<br><strong>Subject to:</strong><br>x + y ≤ 300<br>2x + 3y ≤ 720<br>x ≥ 0, y ≥ 0</p><p>∴ Answer: D</p>
Correct Answer: D