# Profit Maximization problem in linear programming where orderring new raw materials is allowed

My problem is this:

A factory manufactures two types of jewels.

Each one of them consists of gold and two
kinds of
gems- diamonds and pearls.

Every unit of jewel No. 1 consists of 3
diamonds, 5 pearls and 0.3 grams of gold. A
unit of Jewel NO. 1 is sold by the price of
200$, and its demand is 10. Every unit of jewel No. 2 consists of 1 diamond, 1 pearl and 0.1 grams of gold. A unit of Jewel NO. 2 is sold by the price of 70$, and its demand is 17.

It takes 15 minutes to set one gem ( Diamond
or pearl ) on a jewel( 1 or 2 ), and it
takes 20 minutes to polish a jewel.

The factory has a total of 80 diamonds, 120
pearls, 30 grams of gold, 400 Working
hours, and has an option of
purchasing extra gems in the price of 30$$per a diamond, and 20$$ per a pearl.

Model this problem in terms of linear
programming


Now, without extra gems, this would have been my solution( Where x1 and x2 are the amounts of jewel 1 and jewel 2 to be manufactured ):

Max{200x1+70x2}

s.t

3x1+x2<=80

5x1+x2<=120

[15*(3+5)+20]x1<=400*60
[15*(1+1)+20]x2<=400*60

0.3x1+0.1x2<=30

x1<=10
x<=17


My issue is I can't really figure out how to combine the extra gems part into the constraints. Can sombody give a piece of advice?

Even without the extra gems, your formulation needs some correction. Where you have a pair of constraints (one for $$x_1$$ and one for $$x_2$$), you should instead have a single constraint (involving both $$x_1$$ and $$x_2$$) for the shared resource. Also, the demands should impose upper bounds (rather than lower bounds) on $$x_i$$, because you cannot sell more than the demand.
For the extra gems, introduce two more decision variables, say $$d$$ and $$p$$, for the number of extra diamonds and pearls, respectively. Then include $$-30d-20p$$ in the objective function. And modify the corrected resource constraints. For example, the corrected diamond constraint $$3x_1+x_2 \le 80$$ becomes instead $$3x_1+x_2 \le 80+d$$.