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Question 3) (25 points) Consider a machine in a manufacturing center that can process multiple types of products. During day; the machine is available for 8 hours. ...

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Question 3) (25 points) Consider a machine in a manufacturing center that can process multiple types of products. During day; the machine is available for 8 hours. We have a set of 'potential items that can be processed on this machine and after the process, each item can be sold to a customer: Assume that we only can sell one copy of each item_ Table below shows the processing times and the potential profit of each item. MI B c D TE Processing time 65 |25/100 90 /120 MInMAT Revenue (S) 5

Question 3) (25 points) Consider a machine in a manufacturing center that can process multiple types of products. During day; the machine is available for 8 hours. We have a set of 'potential items that can be processed on this machine and after the process, each item can be sold to a customer: Assume that we only can sell one copy of each item_ Table below shows the processing times and the potential profit of each item. MI B c D TE Processing time 65 |25/100 90 /120 MInMAT Revenue (S) 52919901748/805 1- Which items the manufacturer should select to process and sell to the customer in order to maximize his profit? Write a mathematical optimization model for this problem. Clearly define decision variables, objective function, and constraints for the optimization model. Use Excel t0 solve the optimization model. 2- How long (in minutes) the machine remains idle during day? 3 - What is the solution of the problem if we assume that we can sell multiple copies of each item? How the objective function value changes if we consider this new assumption? Use excel to solve the problem with this new assumption:



Answers

Use the two steps for solving a linear programming problem, given in the box on page $888,$ to solve the problems. A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$\begin{array}{lll} {} & {\text { Model } A} & {\text { Model } B} \\ {\text { Assembling }} & {5} & {4} \\ {\text { Painting }} & {2} & {3} \end{array}$$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are 25 for model A and 15 for model B . How many of each type should be produced to maximize profit?

As we have given hell. The post equation three X minus 2.5 is called to 7.125 So, first of all, we have to simplify this equation so into small dipping by the numbers 1000. Okay, so after simply flying by this number, the question then becomes s 2000 eggs minus 25,000. Why? With equal toe 71 to fight. Now we can more simplify it. So again, after simplification, it will be as 24 x minus 20. Why will be equal to 57? Let's say this is a question first. Now coming to the second question, which is giving us two point five X minus three y is equal to seven point T 1 to 5. Okay, now again for simplifying. I'm smartly playing by the number 10,000. Okay, After a simplification, it will. It will convert into the step of form and again more simplifying so on the side in the final simplifying forms 40 x minus 48 by will be called 2117 Okay, let's say this is a question. Second, now we have to solve basic two questions by additional method. So I'm just making the seeing coefficient off eggs. So in a question, First time while dipping by the number five and a cushion. Second Marie playing by the number minus 60. So after multiplication, that question will become s one drink tea, eggs minus 100 white. Well, the equal to 285 Okay. And said integration will be s minus one green tea, eggs and plus 1 44 Right. This accord do minus 3 51 Now, just additional. We have to use additional matters. So I'm just adding this equations. So this stumble begins allowed, and we have the value U S 44 by will be equal toe minus 66. Okay, so after simplification viable, be equal to hear minus tree by two. Okay. And again, when people are value off by any question for us to a question first will be s 24. X bless 30 is equal to 57. And after simplifying here. So I will get us 24. Eggs is equal to 24 X will be equal to 27 here. So excellent vehicle to 27 by 24. That means it will be equal toe nine by it. Okay, so we can say that here. The solutions that for these two giving equations are X is equal to nine by eight and rising call to my nasty right now. And when we put these two will lose any questions like in here. So we will find out Elytis physical are just That means the solutions that disconnect for these two equations.

For this problem. We're looking at the ball company, which manufactures three different types of lamps, and we're hoping to maximize the profit that they're making for the manufacturing. These lamps. So three things we need to try to find from what we're told and from the chart given in this problem the first what is our objective function? Well, our objective function in this case is going to be the equation that we're trying to maximize, which is the profit. So if I pick an equation Z to show the prophet looking at my chart, I can see that my profit per unit it's $5 per each of the lamp type A that we make $4 for each of lamp B and $3 for each of lamps. See? So the coefficients in this objective function are 54 and three, and that corresponds to choice number three. Now, what about our constraints? Well, the constraints are what limits us. Um, now the limits air not shown on our chart. The charges shows us our profit and how many hours we use. But reading through the problem, we're told about our constraints. There's Onley, so many work hours per day. Department one on Lee has 400 work hours per day available in department two has 600. So those there are constraints those air what gonna limit the number of lamps that we can make. So the constraints correspond to choice number four. And our last question is, let's look at the constraints in department one. Well, if I look at that top line of my chart, that tells me the number of work hours I need in department one two hours for every lamp of type A that I make three hours for every lamp of type B and one hour for every lamp of type C. And we're told that our work hours for department one on Lee have we only have 400 a day. Remember, we saw that I'm circling that in blue. That was one of our constraints that we just talked about. So these hours in department one have to be less than or equal to 400. Yeah. Now, I used a B and C because those were the letters here. If we rewrote this with X sub one, except to an ex up three, we can see that this is a match for number number three. So those are the correct options out of these lists

For this problem. We are looking at a machine shop that makes bolts. If you read through the problem, it's always good to read through the whole problem. Before you start trying to do something with this many numbers, you read through the problem. You can see that we're trying to maximize the revenue of this company. So this is a maximization problem. And what is my revenue? Well, my revenue is I make 15 cents for each bolt of type one, so I'm going to mark, except one. This is type one for my bolts, and I make a second bolt That gives me 20 cents in revenue. So we let, except to be type two. Okay, So if I knew how many of each type of bold I made, I could find my revenue again. This is incense. So at the very end will probably have to convert it to dollars. But this is my revenue. My constraints happen at my three machines. For machine one, I need to spend 0.2 minutes for each of bolt type one. And for bolt type two, I only have 300 minutes available. So all the time that I spend on. These has to be less than or equal to 300. Machine to I have a little more time. Actually have 720 minutes for machine too. Everyone of type one bolt takes 10.6 minutes. My type. Choose each take point to again. Now type three is the fastest. I only need 4/10 of a minute for type one bolts and 8/10 of a minute for type two bolts. And for that machine, I have 100 minutes each day that I can use. So these are my constraints on. I want to use them to find my maximum revenue. Now, I'm gonna be using my desk. Most graphing calculator to see what this looks like. You can use this graphing application, your your calculator, anything that you're comfortable with. And if I put these on here, these are three constraints. You can see I have a nice area where they overlap. Five boundary points are there. Now, My fifth boundary, 0.0 I am not even going to consider that. That means I have made no bolts, so I have no profit. That's my minimum profit. So certainly don't wanna look at that for maximizing profit. So I'm gonna look at the other four points. So if I take them and I'm not gonna write them no particular order 0, 1250 501,000, 1050 and 450 and 1200 0 those are the four boundary points I care about again. I'm not even looking at 00 So if I plug those points into my revenue function, I confined this with my first point. The revenue is 25,000 pennies or $250. My second point gives me a revenue of 27,500 the 3rd 1. 24,750. And the last one is 18,000 sets. If I'm looking to maximize, I'm going to find that right here. That maximum is going to be $275 and I'll need 500 bolts of type one and 1000 volts of type two. Now, what if I make some changes specifically, I'm going to change this number right here. My revenue for type one bolt. Currently it's 15 cents. But what if I start to increase it At what point will I increase it? Enough to change my answer. To change what? Uh, allocation of type one and type two bowls. I want to dio. Well, let's go back and look at our equation here right now. Going to remove thes. Okay, Right now that is our maximum profit at 501,000 if I increase the the costs are that the revenue for type one? That's going to mean that my slope is going to get larger and larger, and it's a negative number, So it's gonna be a larger and larger negative slope that's going to put me at that point. So that's gonna be my next If I do make a shift to a higher um uh, make a shift to a different allocation based on a higher revenue for part one, that's going to be my new allocation. So let's take a look at what I would have to have is my new price point. Well, I'm going to introduce two new variables here. I'm going to say, let me use why is just to keep them separate from my exes? Why one is going to be the revenue for type one. And why to is the revenue and the new for type two right now, those air 15 and 20. But I'm gonna be changing them. So right now, from my maximum profit, I have simply and I don't wanna lose that. We're just gonna keep that. That's for the first part. Right now, I have 500 units of type one, so my revenue is going to be 500 times the revenue for that type. That's my overall revenue. And I have 1000 of type two. That's my revenue. Currently, if I were to go to my next, uh, allocation point, that would mean I'd have a revenue off 1050 of type one and 450 of type two. Right now, that is true work, because this is this is my highest revenue at the moment, 500,000 that split. But I want to know at what point does it change? At what point will this side be higher Now, if I solve this, I'm going to subtract 450 from both sides are 450 wise up to, and I'm going to subtract 500 wise up one so I can simplify that it will happen when the revenue for type one is higher than the revenue for type two. Well, we know that the revenue for type two is 20. That is not changing. So I will change my my allocation point if my revenue for type one is higher than 20. And let's just kind of show that I'm gonna put a little bit. I'm only gonna look at these two points right here. I'm not gonna look at the top or the bottom. At my current 500 1000. Let's say I have a price point. Are the revenue point for that type one? Let's make it be 1999 20 which should be where they're equal and 20.1 Now I know these air pennies. I can't really have 1/100 of a penny, but I just want to see that little bit of a difference on either side. So where I currently am at 500 of Type one in 1000 of type two at 19.99 cents apiece, 29995 That is the max. That's my total revenue. At 20 it is 30,000, and at 20.1 it is 30,000 and five. What about this second point 1050 of type 1 450 of type two? If I look at that when it's 20 they are indeed equal. At 19.99 it has a revenue of 29 not 89.5. It is less and a 20.1 It is 30,000 10.5 I have not by much, but it is Mawr. So right there, that's where I'm going to change. If the revenue for Type one goes above 20 that's when we're going to switch to a new allocation.

All right. So we've got a machine shop manufacturing two different types of screws. We have sheet metal screws, which we are going to denote by the variable X, and we have good screws, which we are going to denote by the variable. Why? And we are going to have three different places or machines making these. We have Moe, Larry and Curly and Mo. It takes 20 seconds to make the sheet metal screw and takes five seconds to produce the wood screw on Larry's. It's the reverse. Larry takes five seconds to produce the sheet metal screw and 20 seconds to produce the wood screw. And then there's Curly, which takes 15 seconds to produce each type. So we need to come up with some systems of inequalities to solve this. And ultimately what we're trying to do is we are trying to maximize the revenue so our revenue formula is going to be that we earn 10 cents for each sheet metal screw, and we earn 12 cents for each would screw. So we need to come up with a system of equations we need to graph that system of equations, and then we have to look at the corners or the Vertex is of the enclosed region. So let's come up with our system of equations. Uh, we cannot produce a negative number of either type of screw. So therefore, we are going to have X has to be greater than or equal to zero as well as why has to be greater than or equal to zero. Now let's focus on the information about mo. It takes 20 seconds to produce the sheet metal screw. Plus, it takes five seconds for each of the wood screws. And as we read through this, each machine can only operate for three hours each day. So we have to turn three hours into seconds. So three hours is equivalent to 60 times three minutes. And then to turn them into seconds, we're going to have to have that each minute is worth 60 seconds. So we're gonna multiply that by 60 as well. So we know that we have. I guess it's going to be 10,800. So we know that the machine has to be running less than or equal to 10,800 seconds. Now let's focus on Larry. Larry uses five seconds for each sheet metal screw and 20 seconds for each would screw and machine. Larry has to run no more than three hours or 10 0 800 seconds. And then let's focus on Curly. We've got 15 seconds for each of the sheet metal screws and 15 seconds for each of the wood screws and machine curly can only run up to three hours as well. Now, in order to graph these, I would highly recommend putting them into our slope intercept form so we would have X is greater than or equal to zero. Why is greater than or equal to zero? This first one will turn out to be that why is less than or equal to negative for X Plus 2000 160 the next one will be y is less than or equal to negative 1/4 X plus 540 and the final one can be rewritten as y equals negative X plus 720. So now what you're gonna want to do is graph the five different inequality equations. So I'm going to bring in my graphing calculator and in why equals you can see that I have the four major ones. And the reason I have left out this one right here is that it's a vertical line and it happens to coincide with the Y axis. So when I hit graph, I'm going to get this type of shape. So if I try to duplicate that type of shape, I've got the vertical axis. I've got the horizontal axis and I have a line coming this way. Then I have another line coming this way, and then I have a third line coming kind of more steep, and it's coming this way. And if I were to follow all those inequalities, I would be shading this region right here. So I have to identify all of these vortices, and this one is easy to identify. This is 00 and if I were to come over and put my pen right there, I could see that this one right here is six 540. I want to find this one right there. That one is going to be 240 480 and the next one would be right here. That's going to be 480 240 and then the final one, and I'm gonna scoop this up a little bit, is down here and that happens to be 5 40 comma, zero. So now, when it comes time to linear programming, any maximum or minimum is going to occur at one of those five advertises, so we're going to bring our revenue equation down. So our revenue equation again was 0.10 x plus 0.12 y And what I want to do is I want to take each of those advertises and substituted in and see what the revenue would be. So I'm going to do 0.10 times zero plus 00.12 times zero. So that means if I produced neither type of screws, I'm going to have a revenue of $0 and then I want to look at 0 540. So that means I'm going to do 0.10 times zero plus 00.12 times 540 and that is going to yield $64.80. And then I want to try another verte, asi or corner. So I'm going to try 244 180 so that's gonna give me 0.10 times 240 plus 12 times 480 that is going to yield $81.60. Keep working my way around the figure. So now I'm gonna try the corner of 4. 82. 40. So I get 0.10 times for 80 plus 800.12 times 240 and that is going to yield $76 and 80 cents. And the final Vertex of this closed region was 540 comma zero. So I'm going to substitute it into my revenue equation. Mhm, mhm. And that's going to yield $54. So in this particular problem, we were trying to maximize revenue. So our maximum revenue was $81.60 and that was when X was 240. And why was 480? So, therefore, to maximize revenue, I'm going to have to produce 240 of the sheet metal screws and 480 wood screws to generate a maximum revenue of $81.60. And that concludes your solution


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