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Q5) ABC company in Pune has about 120 employees and produces two types of belts_ B1 and B2 Product Bl is superior to B2 and the profits on selling them in the mark...

Question

Q5) ABC company in Pune has about 120 employees and produces two types of belts_ B1 and B2 Product Bl is superior to B2 and the profits on selling them in the market are Rs 30 and Rs_ 22.5 per piece respectively. Produet Bl requires twice the processing time compared t0 B2 I the company were [0 produce B2 alone_ they could do about 1200 pieces day. But because of some restrictions in tle internal processes_ they are able to produce only HAXHII of 1000 pieces and B2 together, day. Also_ tlie mar

Q5) ABC company in Pune has about 120 employees and produces two types of belts_ B1 and B2 Product Bl is superior to B2 and the profits on selling them in the market are Rs 30 and Rs_ 22.5 per piece respectively. Produet Bl requires twice the processing time compared t0 B2 I the company were [0 produce B2 alone_ they could do about 1200 pieces day. But because of some restrictions in tle internal processes_ they are able to produce only HAXHII of 1000 pieces and B2 together, day. Also_ tlie market trend shows that at least 250 pieces of B2 be produced. Assuming all of the produets can be sold, formulate the LPP: Using proper scaling: solve the probler graphically: What are the changes permitted in the prolits of BL and B2 such that the present solution remains Optimal? Would the solution change il we introduce minit quantity of Bl to be produced? List Lie Tarns scenarios.



Answers

A lawn mower company produces a push mower and a riding mower. Company analysts predict that, for next spring, the company will need to produce at least 150 push mowers and 100 riding mowers per day, but they can produce at most 250 push mowers and 200 riding mowers per day. To satisfy demand, they will have to ship a total of at least 300 mowers per day. Let $p=$ number of push mowers produced per day $r=$ number of riding mowers produced per day a) Write the linear inequalities that describe the constraints on the number of mowers that can be produced per day. b) Graph the feasible region that describes how production can be distributed between the riding mowers and the push mowers. Let $p$ be the horizontal axis and $r$ be the vertical axis. c) What does the point ( 175,110 ) represent? Will this level of production meet the needs of the company? d) Find three points in the feasible region and discuss their meanings. e) Find one point outside the feasible region and discuss its meaning.

So in this problem were the Acme Steel Company. And what do we see? Well, we see that on one day. If I were to make a list here of say the capacity of containers that are made and the count or number have fun. Well we made 500 of 10 gallon. They're all in gallons. So I can use to do this way 10 gallons capacity here's and gallons that's kind of ugly. Let me fix that up Ellen. Okay. We made 350 five gallon ones and 400 one gallon capacities. Okay. And then on another day this is day one. The two count. We made 700 of the 10 gallon ones, 500 of the five gallon ones And 850 of the one gallon ones. All right. And so the first thing we're asked to do is find it too by three matrix representing these data. And then we're asked to find a three by two matrix to represent the same day of All right, so have the data here. So the first 1 to 2 by three could look like this, couldn't it? Okay, two columns and three rows and then the other one. Well, we would just switch this around, wouldn't we? We would go 500 3 50 400 700 500 8 50. Do him like that. Okay, The next question says that the amount of material used in 10 volunteers is £15. So we'll put the pounds here. The 10 Galaxy Tears is £15. And the amount used, the five Gallon containers is £8. The amount used in the one gallon containers is £3. Find the three x 1 matrix representing the amount of material used. Well, we just write those across in the road. Don't wait 15 eight three like so All right. So now we want to multiply the two x 3. I don't know. Hang on, I got the wrong one, two by three And the three x 1. Mhm. Okay, so we need to go back here and fix this for a minute. So I did this wrong. Right, I need to make a couple of notes here. First of all, This is the two x 3. This is the three x 2. Does the first number tells you how many rows you have? All right. So if I have a 3x1 column uh matrix of this data, that means I just do this, don't I? Three rows and one column. Okay, so now now yeah, part C. Well we're gonna take and multiply this one by this one to get A two x 1. Matrix. Right? So we'll have 500 times 15 Plus 350 times eight plus 400 Times three is 1 entry. And we have 700 times 15 Plus 500 times eight Plus 8. 50 times three. Okay, so what do I get? 500 times 15 Plus 350 times eight Plus 400 times three. That's 11, And the other entry 700 times 15 Plus 500 times eight Plus 8. 50 times three At 1705 0. There's our two x 1 matrix. Okay. To show the day's usage of materials. Right. All right. So then party, so the state that still costs act me 10 cents. Uh huh. Pound And aluminum. This is still An aluminum is five cents. Uh huh. Pound for aluminum. Find a one x 2 matrix representing the cost. Okay, If I'm doing a one by two, that's one row in two columns. So put 0.1 and 0.05 like that. Okay. And then e says multiply. The matrix is found in these last two pieces to find the total cost of the day's production. So I take 11 500 In 1705 Oh. Um me switch that up for a second because I got to do it this way, don't I? You got to the one x 2 times the to buy one to get one x 1. Right, Okay, so I need 0.10.05 times 11. 517. 05. Oh. And So I get 11,500 times 0.1 plus 1705 times 0.05 is 2000 $2.50. And there's our total cost

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.

Mhm. All right, here we have question number 80. It's a lot of simple maths. I haven't written out to make things a little bit Ask Mother to go through. But this question is, is they have, Ah, small neighborhood bakery that sells four items Sweet rolls, which are here, just wrote Britain's rolls, bread cakes and pies. And so that is one giant matrix that has three amount of each ingredient. Eggs, flour, sugar, shortening and milk. In those four, which makes up matrix A and then matrix speed is the cost of making a large lot and a small lot, um, over each ingredient. So if you buy in bulk, um, nearly all the times a little bit cheaper, but now attentive, I think all the time. Maybe Anyway, eso starting with the question part egg says to use matrix multiplication to find a matrix giving the cost per item of the to purchase orders. So I just did the math here what you would do. So for the first one, you have eggs and rules, and you have one egg parole. So for a small lot, that'll be $5. And for a large lot, it'll be $5 because there's no difference between those two. But then, if you move one over for flour for rules there for cops off our parole and the cost of a cup in a large lot is $8. So four times eight is 32. But the cost of flour in a small lot is $10 so four times 10 is 40. So you have to go ahead and do all of that math. Just easy. Simple math for all five ingredients across all of the different, um, recipes, rolls, bread cakes and pies. And so that's what you would do to make you two matrices so you could do a number of ways that you could have. One major is here for the large cost, and then one major is here for the small cost, and you could put, um, the individual ingredients into it. But really, um, that would be kind of a big matrix so easy. Anything that I have done here is that I have just added up all of the costs of different ingredients somebody the total large cost for roles and the total small cost for roles. Same for all of these other. Where's the pieces that when you make the Matrix, you would have your recipes see over here so open are here and then you would have large cost, small cost. So your first one would have that 47 that dot dot and in the small cost of 57. Yeah, and then you would just continue on down and put all of the other recipes and there to make that matrix easy peasy. So that is part a a lot of, ah, math, but very easy maths, which shouldn't take you too long to finish that one now, Part B Par P says suppose a day's order consist of 20 dozen sweet rolls, 20 loaves of bread, 50 cakes and six pies. 65 which I have up here as 2252 16. And that's the 1 to 4 matrix. So now you have to use matrix multiplication in order to write the matrix amount for the ingredients. So you're gonna go back up to this matrix that you just made here, you're gonna multiply the cost of a large and the cost of a small by that exact number. So the 1 to 4 matrix multiplying by the 4 to 5 matrix, which is gonna give you a 1 to 5. So he's just gonna have, um e would just go this way. And I have the, um, multiplication than here. So you would just put into the matrix 2. 20 8 90 and so on, all the way down until you have all of those written. So then finally, glass part is C and then see, it says use matrix multiplication to find a matrix giving the costs under the to purchase options to fill the days order. So you're going to take that, um 52 form a tax from before and multiply the four to I'm just going to result in a 4 to 2 matrix. And so once again, you're going to take, um, the cost of the large costs in the small costs. Um, multiply them at them all together to get the total large cost in the total small cost, which is a total large costs 11,120 small cost is 7550. So that is the cost under the two different purchase options. The large cost a small cost for the whole day supporters. And that's how you answer number 80

All right. So you're looking at a professional father here is put up a special colours are riding mowers have a look. At least 300. I can't have Make at least 150 push more at 22 15 Every day. And I have to leave 100 card lowers and no one has to wonder. So this creates three inequality. Uh huh. No, I am I right. It is that you This investment and the key is fascinated by 2:30. I have 100 bucks Are less than two people. 200. And then I know that my little push and writing Lord has to be greater than or equal to. Mhm. This creates wrap. Well, I hope in me that are going to intersect and create and that was the color. So for example, if I look at my diagnosis on PNR Yeah. Yeah. Mm hmm. Yeah. So if I didn't have to worry about anything else, I just had my one boundary life. Yeah. That would be my promise and the white people like our And another 100 300. But because I also am saying that I have learned it from what I come into. Not I have to put those values in there as well. So so those oh so I to me, no less than your infrastructure. It's only yeah. Mhm mm. Mhm. Yeah. Mhm. Well I can have I have to have a lead 150 go right away this point right here. The lowest kind of animal that line is it. I have to have I can't take more than Yeah. Uh huh. So that right there or the master to make others before. So that means right now Yeah. Cut with mine. You've been waiting says that is my boundaries with the gentle love to have. Now I have to look at my riding more about these as well. So mm there is the heat amount arise. I have so primary that to me today. Well there's my boundary line right there and I can see that because of all this type. It's actually going to chop this bottom section off. So this section of the line that I actually have running around down there can't work. Okay it landed another value. Uh huh. Family line is now all the way back. Right. And then the most ultimate already Mama with 200. No, they're my boundaries. What happened? So scrapping this boat, my family's country there and there there. Mhm. But we only inspection that actually work resolution with this. Yeah. What is actually within ordinary fashion. But I have and I know you dotted line for my boundaries even though but they are a lot of people to because I am allowed to make that exactly what I just started to show different Valerie I propose through what God does that mean? Right. Mhm. So there are more questions here. He asked, what was the point 1 75/min turn represent? 175 that is your peace with that. Police officials. We've made a hunting from the inside foot floors which is within your boundaries That we've made more than 150. All you need is the word mm hmm buying going also within your boundary and shipping line that's too high or that will not For me because it's less than three. Uh huh. Mhm Right. Well maybe I was required to circulate and then questions we find three points. The people has been discussed where me. So I can actually use my boundary for me here because they totally in there. Really. So I can use the point You just never hired. I can use the point Between 40 and 60 minutes and then as he is the point one thing. Yes you have it. All that means is like shipping those numbers. They only support from all over $300 to 50 50 Shift by at least 300 and they're all within the ranges of crime alive make in a day. And that says find one point of I just used relation of respective screens anywhere that's not shaded. What what is the point? And in the point they gave up 175 and 110 he was not feeling the feasible region As a not should be happening 300. So real interesting is the point Earth in this the questions as Yeah.


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