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A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for ₹ 48 per unit and product B is sold for ₹ 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum gross income? Formulate the above as a Linear Programming Problem and solve it, indicating clearly the feasible region in the graph.

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Let x units of product A and y units of product B be produced and sold by the carpenter. Then information given in the statement is tabulated as:

Then the LPP is maximise P = 48x + 40y 

Subject to the constraints: 

2x + y ≤ 90 …..(i) 

x + 2y ≤ 80 ….. (ii) 

x + y ≤ 50 ….. (iii) 

x ≥ 0, y ≥ 0

Draw the graphs of equations (i), (ii), (iii) 

Then shaded region is the required feasible region which is bounded with comer points 

O(0, 0), A(45, 0), B(40, 0), C(20, 30) and D(0, 40). 

At O(0, 0), the value of P = 0 + 0 = 0 

At A(45, 0), the value of P = 48 × 45 + 0 = 2160 

At B(40,10), the value of P = 48 × 40 + 40 × 10 = 2320 → Maximum 

At C(20, 30), the value of P = 48 × 20 + 40 × 30 = 2160 

At D(0, 40), the value of P = 0 + 40 × 40 = 1600 

We have the maximum value of P as ₹ 2320 and it is obtained at the vertex B (40, 10).

Hence, the maximum gross income of the carpenter should make 40 units of product A and 10 units of product B.

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