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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?

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

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  • $\begingroup$ Ok, and I guess you don't include the working hours, as two unit of the the two types gems can be processed at the same time. But still, do you have any idea for the extra gems issue? $\endgroup$ – Ori Lev Nov 20 at 18:22
  • $\begingroup$ Read your editted reply, Thanks a lot! $\endgroup$ – Ori Lev Nov 20 at 18:35

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