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Questions 13 and 14 building materials merchant has Lo deliver [welve orders of bricks t0 LOwn several miles away. The orders have the following weights. in tonnes_...

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Questions 13 and 14 building materials merchant has Lo deliver [welve orders of bricks t0 LOwn several miles away. The orders have the following weights. in tonnes_Order WeightE F G H [ ] 6The orders are to be dispatched in lorries_ each of which can carry nO more than 19 tonnes and with each order packaged at source (80 that no order can be split between vehicles) . It is required to find the mninit nuber o vehicles neexled_Q13 What auswer is given by the first-fit packing method?Q14 What answ

Questions 13 and 14 building materials merchant has Lo deliver [welve orders of bricks t0 LOwn several miles away. The orders have the following weights. in tonnes_ Order Weight E F G H [ ] 6 The orders are to be dispatched in lorries_ each of which can carry nO more than 19 tonnes and with each order packaged at source (80 that no order can be split between vehicles) . It is required to find the mninit nuber o vehicles neexled_ Q13 What auswer is given by the first-fit packing method? Q14 What answer is given by the first-fit decreasing method? Oplions for QQuestions 13 and 14



Answers

Use the two steps for solving a linear programming problem, given in the box on page $837,$ to solve the problems in Exercises $17-23 .$
Food and clothing are shipped to survivors of a natural disaster. Each carton of food will feed 12 people, while each carton of clothing will help 5 people. Fach 20 -cubic-
foot box of food weighs 50 pounds and each 10 -cubic-foot box of clothing weighs 20 pounds. The commercial carriers transporting food and clothing are bound by the following aconstraints:
$\cdot$ The total weight per carrier cannot exceed $19,000$ pounds.
$\cdot$ The total volume must be less than 8000 cubic feet. How many cartons of food and clothing should be sent with each plane shipment to maximize the number of people who can be helped?

Hello on this problem. We have a warehouse selling Cem ends on. We have the description off the cost function, being satisfaction being C Oh, cute. See you again. Is a through Q plus make you we're being nbr. Some positive, constant so they can be are some positive bus is that what we're told is win when q increases thank you increases then in the text. Ah, it says cheaper on average to place large orders because this reduces the ordering costs. Ordering, wondering cause when q increases run, ordering costs the grace and also in q increases the story. Sorry is a question about storage, storage costs, UH, increases. This takes gives us a connection between ordering Costas Surgical's part. They asks in this equation here defining the third class. Ah, which term is through here or begins? Which term represents ordering, ordering which one is order? Let's see what this ordering costs. Ah, a through Q. Analyzing it through kid. When the Q rises when he rises, we have a larger number here, so the total for a Q will decrease when this decreases with C order and costs decrease. These are the ordering ghosts. Ah, because Bye, Degrees. Ah, que increases. All right. So big. You, when cube is on the rise says big here. So big. Q is represents these three cool run that was questioning before. Mrs. This one is, uh, ordering on. This one is 35 now. Part B. Oh, what is the cube for which the costs Q is at a mineral All right to file on extreme point. We need to count Lloyd Work are the critical points fan for that? We should When is the derivative VC by the U equals there? So we re run. See you cured as a times cute to the minus. First I'll speak you simply a different trades easier and we different shit. So this is going to minus one time saying which is minus C time Askew to the modern signal pearl us The earth this is big equals zero. So we transfer this to the other side B was a cues. My second, which divided by a be through a equals Q to the power of ministers one through Peter's Square, which means the Q squared is equal a through being on cue Eagles square of a through B. Now this is a critical point. Question is, is it a minimum point for humans? So we will use the second route test. We found that bigger to see these minus a cue to the second was being the second. See like you will be monster. Dunn's minus A. We will be to a Q to the money's 3rd 0 Huh? They were interested in here. What's the science? Second route. The side of secondary would do squared away through being silence. Second, we can always still that for any queue here. This is positive. This is positive and this is positive. This is always going to be great. Zero. So see, off the square root of intervene is going to be two times a time's a through B. The square root side is going to be more than three to embarrass you. Anyway, you look at it. We were given that both and be a positive. So, um, the power of a three B is also going to be positive. This is positive to positive. This is reading zero. So we are talking about a minute on there. We have it for this value. We will have the construction at a minimum. We went today. We're done here. I can hope it helps. Have a good day by

Let X b the number off cartons off food while y equals theme number off cartons off loads. Let's it be the number of people who can be helped with these cartons of food and clothes. So that will be said, which is equal to 12 x plus five i, which is subjected to the constraints. Start excess greater than or equal to zero. Why is greater than or equal to zero 20 x plus 10 by is less than or equal toe 8000 on 50 X plus 20 vie is less than or equal to 19,000. Now let's plot the graph to find the corner points. Now. From the graph, we can see that the corner points are at 00 0 800 302 100 Andi at the point for 380 0 Now let's find the value of Zed at these points on. Maximize that so said at the 0.0 800 will be equal to 12 multiplied with zero plus five multiplied with 800 which is equal to 4000 said at 00 will be equal to 12 multiplied with zero plus five multiplied by zero which is equal to zero said at 300 on 200 will be equal to trails multiplied with 300 plus five multiplied with 200 which is equal door 4600 Aun said. At 380 0 will be equal to 12 multiplied with 380 plus five multiplied with zero, which is equal to 4560. Therefore, to maximize the number of people who can be helped 300 cartons or food Andi 200 cartons off Bloods uploads have Toby sent so help a maximum off 4600 people.

Forgetting the fates of some articles here, and we have to. We have to rearrange them in, ah, in the sending order. So we have to foot the heaviest way in the phone and the lighter square in the back. So Alousi was We invite them in the order of their way, so we see that the heaviest one is 91. That's the obvious one. No later than that is 911 later than that is for 13 lower than that is 3 78 lighter than that is 2 31 like to them that it's to 21 learned, and that is 208. We'll just continue. This learned in 208 is one in 88 on Lived in 1 88 is 109 Richard Leitess. Now let's just see that we'll check if we have missed doubtfully or anything, or put a lighter, every Peyton the wrong place. So 90 modesty of this we can see that the 911 and for one thing, that slide in others and free summit that started in all those 1231 that started in almost 10 to 21 that started and all those to wait, which is lower than all those 1 88 and 19 So we have made it correctly and legis call. The number says. 123456789 and 123456789 So we have a raised them correctly.

There is a manufacturing company that is trying to sell two types of treadmills. Okay, there's the standard model and the deluxe they want to sell Were shipped 15 standard treadmills and 16 deluxe treadmills. Now they have two trucks that are able to ship orders out the large truck. The large truck can carry six standard treadmills, three deluxe and the medium truck Can carry four standard and six deluxe. I don't know how big that deluxe is, but must not be that big. You can carry six of them for the medium truck. It's kind of weird. Okay, so, um yes, So you basically want to know how many large trucks and how many medium trucks are needed in order to carry out the order of 15 standard and 16 deluxe treadmills. All right. So, if we're looking for the number of large and medium trucks, we just figured out what our variable is going to be. So let acts represent the number of large trucks and why represent the number of medium trucks. Medium is not a good name, but we'll go with it. All right. So, um, if the large truck can carry six standard three deluxe, the medium can carry four standard six deluxe. Our limitations are that has to carry at least 15 standard and 16 deluxe. So in order to carry 15 standard, I looked at this column right here. So however many large trucks we have X. Each one of those can carry six standard treadmills. So that represents the total number of standard treadmills that we have. Okay, And the medium truck is represented by why? And each of those can carry four standards. So, for Y gives us the number of standard Treadmills that the medium trucks can handle. So this total has to be at least 15. Right again. This is the number of standard treadmills coming from the large truck. So this is from the large truck and this is the number of standard treadmills coming from the medium truck. That has to be at least 15. Okay. By the same token, if we look over here we can write a similar inequality for the deluxe treadmills and we get that three acts plus six Y. Has to at least be 16 because we want to ship out 16 treadmills. Okay, so this looks like a system of two inequalities with of course the additional assumptions that X and Y both have to be positive, So we're only dealing with quadrant one. Okay, this is a bit of a weird graph and it's only the way that I'm interpreting the word problem, but what it basically says is we're going to have to graft these two lines. Okay? So the first one which I'll do in green um I'm going to move the six Y. Over. So we get four Y. Greater than or equal to 15 minus six X. And if you divide by four um This is what you get. Okay so 15/4 is 3.75. It doesn't really matter where it is to be honest but okay and then it's gonna decrease at a rate of three halves. What's the X. Intercept? Well if you plug in Y. Equals zero. Um That's gonna be that's going to be 2 1/2. Okay so that's the slide right? We're assuming we're only working in the first squadron and because it has to be greater than or equal to uh we want everything kind of in this shaded region right here. Okay now the blue curve and I'm gonna erase all this work that I just did for the um green curve. But the blue curve if you graph that I'm going to move the three X over And divide by six. So 16 6 is the same as 8/3 and then this will just be a half X. Okay so 8/3 is um 2.7 give or take. So the Y intercept the point is the y intercept is a little lower. It's about where 8/3 is and then it also decreases by one half X. So the Y intercept the X intercept would be 16 3rd which is About 5.3. It's almost double what this is. You grab this line look something like this and then you want everything greater than that. So I'm gonna go and she ate everything over here. Now it looks like there's a lot of common ground. Right? I mean obviously this this problem is not the best because it didn't allow us it didn't give us any restrictions on the number of trucks that were allowed to use. So from what I understand, I can use a million trucks if I want. I can use however many trucks that I want. But I think the goal of the exercise is to come up with the least number of trucks possible. And that's going to happen at this point of intersection right here. So let's go ahead and just find that just for fun. Um So that's going to take place whenever these curves are equal to each other. Okay, So would be six X plus four Y. Um six x plus four y equals 15 and three X plus six White sixteen's. We're gonna just try to solve that. So the best way to do it is just to solve for y um on one of these and plug it in. Um Maybe elimination would work better anyway. Okay, so I just took this and multiply it by -2. Take the entire thing times -2. And then you can just add these equations together together to solve for why. Okay, so it looks like why is going to be 17 8. I have no clue what that is. Um That's what it is and you plug it into X. And then you would find exactly what this equilibrium point is. And then basically any trucks kind of surrounding the area would give you a pretty good indication. Okay, so I think it's it's going to be somewhere around one comma two. Okay, so probably two comma three would be a safe bet. So I would use um maybe too large trucks and three medium trucks just to be on the safe side.


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