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Basics of Linear Programming (LP)

At the onset, I would like to clarify that we are not getting into the depth of formulation of Linear programming. Our focus is to learn the methods of solving the questions using the concepts of LP.
 
Example

There is a carpenter who sells only tables (with two legs only) or chairs (with four legs only). The top part of table and top part of chair (the wooden plank) are same and one wooden plank is required to configure one chair or one table. Carpenter gets the legs and top wooden plank separately and then assembles it to make either chair or table.

 

Its also given that Profit per table = Rs 1000/unit and profit per chair = Rs 1500/unit.
 

 

Question-1
If there is limited supply of 1200 legs and top wooden plank, which is more profitable to produce—table or chair to maximize the overall profit?
Solution
In this situation, for any limited supply of legs, it is more profitable to make table (if at all production to be made). We can simply see it through the concepts of average that in case of table, profit per leg = Rs 500, and for chair profit per leg = Rs 375/leg. We are not considering the top wooden plank because number of unit required for each of table and chair is same = 1 top wooden plank.
 
Now we are adding another aspect to this situation:
Carpenter takes 4 hours to assemble a table and 3 hours to assemble a chair.
 
 
Question-2
If there is limited supply of 1000 man-hour labor, which is more profitable to produce—table or chair to maximize the overall profit?
Solution
Calculating again the per unit time profit of table and chair,
 
Profit per hour for table = Rs 250 and
 
Profit per hour for chair = Rs 500. Hence it is more profitable to produce chair.
 
 

Question 3.
If we club the information given in Q.1 and Q.2, then which is more profitable to produce—table or chair to maximize the overall profit?
Solution
It is difficult to tell by only looking at the average profit per leg or per unit of man-hour labor because we are getting contradictory results. It shows that there will be some midway solution, possibly some units of table and some units of chair.
 
Concept of Linear Programming comes handy in this situation. As I have stated earlier too, we will be concerned only with our primary purpose of getting the solution of our question, and not the concepts of LP in depth.
 





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