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How do you Find the Constraints of an Objective Function?

TopWhen a word problem is given such that we can derive many constraints or Relations from it, information must be collected completely. This information is basically in the form of variables and constants that lead to formulation of expressions. So, how do you find the constraints of an objective function?

For this let us suppose an example of a word problem: There is factory that produces 3 different kinds of products namely A, B and C in its two workshops P and Q. Workshop 'P' produces 16 number of A, 12 B and 30 C products, while workshop Q produces 15 A, 18 B, and 20 C products in one day. Factory collects an order for 30 A, 40 B, and 50 C products. Cost of production for workshop P is $ 1200 a day and that for the workshop Q, it is $ 3000 a day. To minimize the cost of production, the factory head chooses the total number of days that are needed to work in each workshop. In our word problem of Linear Programming, variables (known and unknowns) are:
1. X = the total cost for working in workshop P.
2. Y = the total cost for working in workshop Q.
3. Also we know the information related to the number of A, B and C products manufactured.
4. The unknown variable are the total number of days required to be worked in worked in the two workshops to get the three products A, B and C.

So, in such situation with these constraints we can write the objective function as follows:
1. 3000X + 1200Y,
2. 1200X + 3000Y,
3. 58X + 53Y,
4. 30X + 40Y + 50Z,
5. None of the above
The correct answer is 1.

Which of the inequalities are possible to be formulated in this situation?
1. 30X +20Y < 50
2. 16X + 15Y > 30
3. X + Y + Z > 120
4. X + Y > 120
5. None of the above
Correct answer is 1 because we have to minimize the cost.