Question 5

Formulation of the Linear Programming Problem

Let:

• x = Number of deluxe machines produced
• y = Number of standard machines produced

Objective Function:

The objective is to maximize the total profit (Z). The profit function is:

Maximize Z = 450x + 250y

Constraints:

• Labor Constraint: 18x + 3y ≤ 800 (Total labor hours available)
• Testing Constraint: 8x + 4y ≤ 600 (Total testing hours available)
• Demand Constraint: y ≥ 150 (Minimum demand for standard machines)
• Non-negativity Constraints: x ≥ 0, y ≥ 0 (Production cannot be negative)

Solving the Linear Programming Problem (Graphical Method)

To solve this LPP graphically, we need to plot the constraints on a graph and identify the feasible region. The optimal solution will lie at one of the corner points of the feasible region.

Step 1: Plotting the Constraints

• 18x + 3y = 800: When x=0, y = 800/3 ≈ 266.67. When y=0, x = 800/18 ≈ 44.44.
• 8x + 4y = 600: When x=0, y = 600/4 = 150. When y=0, x = 600/8 = 75.
• y = 150: A horizontal line at y = 150.
• x ≥ 0, y ≥ 0: Restricts the solution to the first quadrant.

Step 2: Identifying the Feasible Region

The feasible region is the area that satisfies all the constraints simultaneously. It is bounded by the lines representing the constraints.

Step 3: Finding the Corner Points of the Feasible Region

The corner points are the intersection points of the constraint lines. We need to find the coordinates of these points.

• A: (0, 150) - Intersection of y = 150 and x = 0
• B: (0, 266.67) - Intersection of 18x + 3y = 800 and x = 0
• C: Intersection of 18x + 3y = 800 and 8x + 4y = 600

  Solving these two equations simultaneously:

  Multiply the first equation by 4 and the second by 3:

  72x + 12y = 3200

  24x + 12y = 1800

  Subtracting the second equation from the first:

  48x = 1400

  x = 1400/48 ≈ 29.17

  Substituting x in 8x + 4y = 600:

  8(29.17) + 4y = 600

  233.36 + 4y = 600

  4y = 366.64

  y = 91.66

  However, this point does not satisfy y ≥ 150. Therefore, we need to find the intersection of 8x + 4y = 600 and y = 150.

  8x + 4(150) = 600

  8x + 600 = 600

  8x = 0

  x = 0

  This gives us point A again. Let's find the intersection of 18x + 3y = 800 and y = 150.

  18x + 3(150) = 800

  18x + 450 = 800

  18x = 350

  x = 350/18 ≈ 19.44

  So, C: (19.44, 150)

• D: Intersection of 8x + 4y = 600 and x = 0 - This is point B.

Step 4: Evaluating the Objective Function at the Corner Points

We evaluate the objective function Z = 450x + 250y at each corner point:

• A (0, 150): Z = 450(0) + 250(150) = 37500
• C (19.44, 150): Z = 450(19.44) + 250(150) = 8748 + 37500 = 46248

Since we are dealing with production quantities, we need to consider integer solutions. We can round the values of x and y to the nearest integers and check if they satisfy the constraints. Let's try x = 19 and y = 150.

18(19) + 3(150) = 342 + 450 = 792 ≤ 800

8(19) + 4(150) = 152 + 600 = 752 > 600. This doesn't satisfy the testing constraint.

Let's try x = 18 and y = 150.

18(18) + 3(150) = 324 + 450 = 774 ≤ 800

8(18) + 4(150) = 144 + 600 = 744 > 600. This doesn't satisfy the testing constraint.

Let's consider the intersection of 8x + 4y = 600 and y = 150, which gives x = 0. Then Z = 37500.

Let's try to maximize x while satisfying the testing constraint. If y = 150, then 8x = 0, so x = 0. If we reduce y, we can increase x. Let's try y = 151. Then 8x + 4(151) = 600, so 8x = -4, which is not possible.

Therefore, the optimal solution is approximately x = 19.44 and y = 150. Since we need integer solutions, we can consider x = 19 and y = 150, but it violates the testing constraint. The best integer solution is likely x = 0 and y = 150, giving a profit of 37500.