I'm studying linear programing for Operations Research. I'm at a complete lost in the first example in my textbook. The problem is about a company that will try to maximize profits by creating 2 new products.
The constraints for it are the following ones: Constraints
Where Z is the profit and X1 and X2 are the 2 new product units that are made each week. 3 and 5 correspond to the amount in 1000 dollars that each unit can be sold for. So a unit of product 1 is sold at 3000 dollars and product 2 at 5000.
Next they present the graphical solution step by step. First step they add a couple of lines. I understand why there's a line in 4 in the X1, because X1 can't be greater than 4. However, I don't understand why there's a line in 6 in X2. The constraint for X2 is 2X2 = 12, so why is it being placed at 6? I understand they're multiplying it by 2 at the intersection, but why is this done? It seems to me like they're arbitrarily multiplying 6 by the 2 in the X1, why is this necessary?
Lastly for this image. How is the line 3X1 + 2x2 = 18 supposed to represent 18? How were the intersections calculated? Which values are being used to multiply there?
Last image. 3 new lines. Once again I don't see how these lines represent what they're supposed to be representing. How does that lower line means 10? If I take the points which it seems to be touching, 3 at X1 and 2 at X2, then the equation would be 3(3) + 5(2) = 19, not 10, so where's the 10 coming from? I really feel completely lost with these type of problems.
[quote]However, I don't understand why there's a line in 6 in X2. The constraint for X2 is 2X2 = 12, so why is it being placed at 6?[/quote] Oh, dear. There may be a serious problem here. Have you ever taken and "algebra" class? If not you drop this course and not take it until after you have had the chance to learn algebra. Algebra is heavily used in "Operations Research"!
You do understand that 12/2= 6, don't you? If 2 times x is less than 12, x itself must be less than 6. They are not "arbritrarily" multiplying x by 2, they are doing it because that is what "$2x_2$" means- multiply %x_2$ by 2!
The (x1, x2) points satisfying "$x_1\ge 0$", "$x_2\ge 0$", "$x_1\le 4$", and "$x_2\le 6$" (which is the same as "$2x_2\le 12$") lie inside the rectangle formed by those four lines.
[quote]How is the line 3X1 + 2x2 = 18 supposed to represent 18?[/quote] I don't know what you mean by "represent 18"! First, do you understand that when we say something like "$3x_1+ 3x_2= 18$" corresponds to a given line in a coordinate system, every $(x_1, x_2)$ point satisfies that equation. You should also know, from basic geometry, that "a line is determined by two points". Two draw a straight line, we can just mark two points and draw the line through them. Because "0" is an easy number, I like to take $x_1= 0$, put it into the equation $3x_1+ 2x_2= 3(0)+ 2x_2= 2x_2= 18$ so, dividing both sides by 2,$x_2= 9$. Similarly if we take $x_2= 0$ we have $3x_1+ 2x_2= 3x_1+ 2(0)= 3x_1= 18$ so $x_1= 6$. Another point on the line is (6, 0). Draw the line through (0, 9) and (6, 0).
[quote]How were the intersections calculated?[/quote] We already know that the intesections of $3x_1+ 2x_2= 18$ with x= 0 and y= 0 are (0, 9) and (6, 0). At the intersection of $3x_1+ 2x_2= 18$ and $x_1= 4$ the same values of "$x_1$" and "$x_2$" must satisfy both equations. The second equation tells us that $x_1= 4$ so the first equation is $3(4)+ 2x_2= 12+ 2x_2= 18$. $2x_2= 18- 12= 6$ so $x_2= 3$. Those two lines intersect at (4, 3).
And $3x_1+ 2x_2= 18$ and $x_2= 6$ intersect where $x_2= 6$ and $3x_1+ 2(6)= 3x_1+ 12= 18$. $3x_1= 18- 12= 6$ so $x_1= 6/3= 2$. Those two lines intersect at (2,6).