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 $3x_1+x_2 \le 80+d$.