Question:

(a) A company sells three different products X, Y and Z. The company makes a profit of Rs. 35, Rs. 50 and Rs. 60 per unit on products X, Y and Z respectively. The three products are produced in a common production process and are sold in three different markets. The total production process has a total capacity of 20000 man-hours. It takes two hours to produce a unit of X, three hours to produce a unit of Y and two hours to produce a unit of Z.

The market has been surveyed and it is found that the maximum units of X, Y and Z that can be sold are 12000, 8000 and 6000 respectively. Explain fully how you would formulate a linear programming model for this problem.

(b) A product mix problem can be formulated mathematically as follows:

Max Z = 100x₁ + 75x₂

subject to the constraints

5x₁ + 10 x₂ ≤ 1500
8x₁ + 5x₂ ≤ 1200
x₁ ≤ 300 and x₂ ≤ 400
x₁ ≥ 0, x₂ ≥ 0

Solve the above product mix problem using graphical method.

(c) Explain briefly the term ‘sensitivity analysis'.