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Formulation of LP

It is simple linear equation representation of the constraints given in the question in terms of limited supply of legs and man-hour labour.
 
Assume that ‘x’ table and ‘y’ chairs are to be made.
 
Equation for legs ⇒
 
2x + 4y ≤ 1200 (we are taking ‘≤’ and not ‘=’ sign because we may not be using all the units supplied)
 
Equation for man-hour labour ⇒
 
4x + 3y ≤ 1000 (we are taking ‘≤’ and not ‘=’ sign because we may not be using all the units supplied)
 
Total profit = 1000 x + 1500 y
 
Objective is to maximize total profit ⇒ maximize (1000 x + 1500 y)

Process to Solve

Step 1: Solve all the pairs of equations (convert the inequations into equation by putting “=” sign instead of inequality sign) formed from the constraints (the way we solve simultaneous equations).

Step 2: Put all the point of intersection obtained into total profit one by one.

Step 3: The point which gives maximum total profit will give the units to be made.
 
In this question, we have only one pair of equation formed from the constraints viz., of legs and man-hour labour.
 
2x + 4y = 1200----------------------------(1)
 
4x + 3y = 1000----------------------------(2)
 
Solving these two equations, we get y = 280, x = 40
 
Hence carpenter should make 40 tables and 280 chairs to maximize his profit.
 
Total profit = 1000x + 1500y = 1000 × 40 + 1500 × 280 = 460,000
 
Note that if at all a solution can be found in case of LP, the solution has to be in the 1st quadrant (x ≥ 0, y≥ 0).





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