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1.26. Use penalty method or big `M' method to solve Min z =4x1+3xz Subject to 2xi+xzzlO -3x1+2x2<6 X1+xzz6 Xl, X2z0...

Question

1.26. Use penalty method or big `M' method to solve Min z =4x1+3xz Subject to 2xi+xzzlO -3x1+2x2<6 X1+xzz6 Xl, X2z0

1.26. Use penalty method or big `M' method to solve Min z =4x1+3xz Subject to 2xi+xzzlO -3x1+2x2<6 X1+xzz6 Xl, X2z0



Answers

In Problems 9–18, solve each linear programming problem.
$$
\text { Minimize } z=2 x+3 y \text { subject to } x \geq 0, \quad y \geq 0, \quad x+y \geq 3, \quad x+y \leq 9, \quad x+3 y \geq 6
$$

We're trying to find the max or They is able to 2.5 x -3.1. Why? You're trying to find a maximum for this And we're told that X. It's great and equal to one. Why is less than or equal to 7? X. is also less than recalled to three and negative X. Plus. Why? He's great at night. You called to to also eggs plus. Why? It's great to be equal to 66 All right. So I can rewrite it for the constraints. X has to be greater than or equal to one and less than or equal to three. Let's just put it together like this. So that's one of the requirements. And uh having Z is equal to 2.5 X -3.1. Why? And um in order to do this to get the max, we need to have the biggest possible um X. Value and the smallest possible Y. Value. So what is the maximum value? X can be. Well, before we look into that, let's look into the values for Y. Because it has a bigger factor here. So one thing we can do over here is a look at the requirements that we have and try to find the constraint for why? What I mean is we have negative X. Us? Why? It's great gender equal to two. We also have eggs. That's why It's great. Yeah. You called to six. If we combined it to inequalities, you get to I He's created an import to eight or minimum value for why is four? It's a minimal value for why it's four. And um maximum value for X. Is three. But there's also this problem here to let me just put him over here. So I have one. His left hand is equal to X less than equal to three. And also have four. Is the senate equal to y. Let's enter equal to seven. Mhm. And what I need to do is to maximize X. And minimize why? But if I do that then X plus Y will be equal to seven. But I need to have X plus why? Um To be Last time a great Article 2 6 which it meets that requirement next I need negative eggs plus why? To be great and equal to two. My wife value has decided it's going to be negative three bus. My exile is gonna be -3. Uh It's gonna be three and negative X. Is gonna be negative three. Excuse me And my wife value is going to be four. That is gonna be one and that is not Great. Daniel Close to two. So what can we do? Uh The most logical thing to do here is two. Decrease the value of X. To something that would allow the requirement, meaning let X be too why not do that? But why why not increase increase? Why? Because it has a higher factor as we can see here? Okay, So let's plug it back in. My c. for max is going to be 2.5 Times 2 -3.1 times. For It's gonna be five minus 12.4, Giving us -7.4 forever max.

We're trying to find the minimum for the equal to 2.5 x Plus 3.1 Why? And Were given the constraints that X is created and recalled to zero. Same goes for why And X. This is the center equal to four and negative X. Plus why is less than or equal to two An X. That's why he's also less than or equal to six. All right. So if you have a positive number multiplied by eggs plus a puzzle number, Michael, Pablo Y. So we need individually the minimum possible value for X. And the minimum possible value for why? A minimum for X that we can have is zero. And I want for why? It's also zero and we cannot get them any lower than that. So let's see. Does that um comply with the other requirements? X less than four. Yeah, that is negative X plus Y. Less than or equal to two. Which would be zero in this case. Which is also correct. And zero plus zero. we're assuming right now that x and Y is zero, we'll also give us zero, which also meet this requirement. So our minimum would be zero is equal to 2.5. Uh I'm sorry, Z is equal to 2.5 times zero, Us 3.1 times zero For our minimum is zero.

We're trying to find the minimum for functions E is you call to 1/3 X -2/5 by And we have the following constraints we have six. Is the center will go to X plus Y. The center it looks wait and um for the last time equal to negative X plus Y. The central quarter to six. Now it becomes okay, add up to two constraints together. 6-plus 4 gives us 10 plus X plus Y minus X plus Y. Gives us Just to why? And 8-plus 6 gives us 14. Um Now though I did the numbers over by two gives us why? Let's listen our April two. Why the center april to seven? No, you can see that. The limits for X and Y. In our constraints, The concerns for them are one but in our function the constant for X is one third Now, for why is 2/5? Go over five is larger than 1/3. So why should be the priority first of all? So, let's try to Because you have a negative maximize why to get the minimum value for Z. All right. So what is maximum for why? It's seven By equals to 7? No. Considering dad. Let's try to find our minimum value possible for X. So for that, we need to subtract the first, the second equation from the first one that would give us Uh 6 -4 to It's less than or equal to X. Two X. A centre april 22 Oh yeah. So we have only one option for X. X. is equal to one. All right. So now let's find sleep genes equal to 1/3 X. You should just one third minus 2/5. Y. Because they have seven for radios 14/5 and the common denominator. We get 15 here, five here and -42 here and adding it all together, we get Neither. over 15. And that's how we're finally answered.

To begin solving this linear programming problem. We have our objective function up here written out from the problem, and we also have our constraints here that we need to graph. So we know that this will be in the first Quadrant because X and wire both greater than or equal to zero. And we want to graft these two functions down here. So I want to start by rewriting thes into my y equals MX plus B form just to make it easier to graph. So this 1st 1 we can subtract two extra. Both sides get three y is greater than or equal to negative two X plus six. Divide both sides by three, and we end up with why is greater than or equal to negative 2/3 X plus two. No, I don't want to graft that Mahdi Wind yourself is at two and going with the negative 2/3 slope. My ex intercept would be at three. And to graph expose wise less than or equal to eight. We can do something similar or we take why is less than or equal to negative X plus eight. So our Y intercept being AIDS and with a slope of negative one or X intercept also occurs at eight and we know that are feasible. Points occurred between our constraints and what we want to do next is find the coordinates of each of the corner points. So starting here, I know that corner point is zero to appeared. This corner point would be 08 Over here. This would be 30 and this would be 80 Now that we know with corner place of the graph of are feasible points, you want to plug them back in into our objective function. So we want a soft busy at each of these points. And when X is equal to zero and y is equal to at this point, we know that Z is equal to zero plus eight, which is eight at 08 z is equal to 32 at 30 z is equal to nine and at 80 z is equal to 24. Since we're asked to minimize, we want to find the lowest value, which would be Z is equal to eight. So the minimum value of Z is eight, and it occurs at the 0.2


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