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Q5) ABC company Pune has about 120 emplovees prodluces TnE belt > BL ad B2 Product cuperior B2 ad prolit: OHL selling thet the market are Rs. 30 ;d Rs. 22.5 per ...

Question

Q5) ABC company Pune has about 120 emplovees prodluces TnE belt > BL ad B2 Product cuperior B2 ad prolit: OHL selling thet the market are Rs. 30 ;d Rs. 22.5 per piece respectively: Product BI requires twice the proce-sing comnared Ifthe company Were|o produce B2 alone; they couldl do about 1200 pieces day; But because SM restrictions Le inleral urucuse twy able produce ouly MAXImu ofIQ00 pieces B2 together, day; Al-o, the market Tene shows that least 250 pieces B2 be produced; A-Suming all of

Q5) ABC company Pune has about 120 emplovees prodluces TnE belt > BL ad B2 Product cuperior B2 ad prolit: OHL selling thet the market are Rs. 30 ;d Rs. 22.5 per piece respectively: Product BI requires twice the proce-sing comnared Ifthe company Were|o produce B2 alone; they couldl do about 1200 pieces day; But because SM restrictions Le inleral urucuse twy able produce ouly MAXImu ofIQ00 pieces B2 together, day; Al-o, the market Tene shows that least 250 pieces B2 be produced; A-Suming all of the products can sold; formulate the LPF b) Using proper scaling solve the problem graphically. Wlat the changes permitted the prolits o Bl and B2 >uch thal the present solution remaic optimal? Would the jolution change introduce Minimu quantity of BI be produced? List the various ECCMTIUS



Answers

Set up and solve Exercises 23–29 by the simplex method.

Profit The Muro Manufacturing Company makes two kinds of plasma screen television sets. It produces the Flexscan set that sells for $\$ 350$ profit and the Panoramic I that sells for $\$ 500$ profit. On the assembly line, the Flexscan requires 5 hours, and the Panoramic I takes 7 hours. The cabinet shop spends 1 hour on the cabinet for the Flexscan and 2 hours on the cabinet for the Panoramic I. Both sets require 4 hours for testing and packing. On a particular production run, the Muro Company has available 3600 work-hours on the assembly line, 900 work- hours in the cabinet shop, and 2600 work-hours in the testing and packing department. (See Exercise 10 in Section 3.3.)
a. How many sets of each type should it produce to make a maximum profit? What is the maximum profit?
b. Find the values of any nonzero slack variables and describe what they tell you about any unused time.

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.

Okay, so your cells three types off yogurt, non fat, regular and super green, super creamy. And it's select three locations. That first location non fat, is 50 gallons per day. 100 per day had no killings. Prevent particulars Body similarly for location to location three and income is for non 30 $12 regular $10. And this is 15. So first will represent need to represent three by three metrics off this information, and we need to write rose as location. So we from the data we already know let these metrics B A. So this is 12 and three third location so we can end. This is non fat, regular and super creamy. So we just have to put the data 50 10 60 and read 91 20 in here 30 50 and forties. So that is done. So question is dumped. Now he's asking for metrics product for daily income for each the so first will represent income. So let that metrics B I. And that will be $12.10 dollars, $15. Four. This is non fat, regular and super creamy. Okay, so this was part B, right? So sorry for mistaking and this is CND, so we're done with question number Be not to find. See, that is metrics product. So we have to multiply a and I get daily income right at each location. So for C, it will be eight times I transports. Right? So we have 50 130 10 90 50 61 2040 and we'll multiply this with a transfer that will make it column metric sites up to, well, 10 15 late. So if you multiply this, this rule will be multiplied with this column. So we'll have 50 times 12 plus 100 times 10 plus 30 times 15 Next will be 10 times 12 plus 90 times 10 plus 15 times 15. And the last one was 60 times 12 plus 1 20 times 10 plus 40 times 15 so on. Solving this we get and this can be represented at income metrics. Okay, so let's try to this I two metrics or we can write it As I see metrics, there's no issue. You can name it any variable. So in solving this, we get to 0501770 and lasted 25 to 0. Okay, so that's a total income at each location. So what will be the total? So this is done. See? Now we need to find total income. Total income will be some off all these three. So adding 2050 plus 1770 plus 25 20 On ordering this, we get adding this we get $6340. Thank you.

Through Texas are a huge medics multiplication toe. Find a Matics giving the cost per item for a chase options. So there s more to play. And so I multiple in group it column. So flushed on the questions again with the second column element hard rock element with particle argument Spend a year ago So one into five for in tow a 32 one way for into Then I do one way for in tow and a warning to fight. What of in this? Into this So one into five. So in this whole world by this. So this is the productive now? No, I had him. So this is the product? No, to be officer suppose a day artist Kansas 20 Dajun suit tools. So this 40 billion suit girls 200 laws of bread No wonder loss of bread Could be gig and 60 fight. Now multiply with given my picks. Notice will deploy with disputing medics, which is Quito. It's it'll into column. So this is a product of the dramatics? No hiding them 2200 20 plus 18 60 on one to be on this list. I would add this all. So this is the productive off to antics. This is a moment for its ingredient. This word. Big floor. So good sort ending and milk now supporting the use. Multiplication. No, see, this is cool. 20 8 71 05 1 25 and 70. Normal to play with my picks. This is a large lord. And this morning juices off it, then. Well, and this is five, then? Well, 15 and six normal to play this. So I'm going to play this. This is a total cost, Large lord. And this morning, large lot and this small knocked.

For this problem. We're being asked to maximize the profit for the mural manufacturing company, which is making plasma screen television sets. So let's see what our equation is for the profit that this company is going to make. Well, there are two types of television sets that it makes. One is a flex scan, which makes $350 of profit. So I'm just gonna mark here that my ex up one is going to be my flex scan model. And the second kind that it makes is the panoramic. And there's $500 of profit for that one. That's my panoramic in the ceramic. Okay, so this is my profit equation. $350 of profit for each flex scan. $500 of profit for each panoramic. Now, what are my constraints? Well, one constraint we have is the assembly line. Okay. Now, on the assembly line, we're told that the Flex scan requires five hours and the panoramic takes seven hours and at most I've got 3600 hours available on the assembly line. Our second constraint comes at the cabinet shop and there were told that the flex can only takes one hour in the cabinet shop, the panoramic takes too. Okay. And adding all of those up, I have 900 hours total available in the cabinet shop. My third constraint happens in testing and packing like that red. So in the testing and packing area, there were told that the flex scan takes four hours, as does the panoramic. So each one of those takes four hours altogether as 2600 hours. Now, I'm going to simplify this one slightly. I'm going to just divide everything by four just to keep my number's a little smaller. And this becomes 650. I do need to remember, though, that if something comes back with the slack variable for that equation, I need to remember that I made an adjustment. Will have toe, um, work on that at the end if something comes back for that one. Okay, Let's look at how I can add my slack variables so I could enter these into my grid. My first equation, my assembly. I'm just gonna write these in blue, so they kind of stand out a little bit. Five x up one plus seven x up to plus my first slack variable is going to equal 3600. My second equation. I'm going to add my second slack variable, and that will equal 900. And then for my third equation, I have my third slack variable. And that equals 650. Okay, Knowing all of those, I can now put together my grid. I have two X variables. I have three slack variables and I have ze. So let's enter our blue constraint equations in here. If I pull out all of my coefficients, I have 57 and one I don't have any of those variables and it equals 3600. My second equation is 12 It has the second slack variable in that equals 900. And my third one is one and one, and it has my third slack variable equaling 6 50 and my indicator row at the bottom I'm going to take and I'm gonna market with a blue arrow appear at the top, my equation for Z and I'm going to set everything equal to zero. So when I pull all of the terms over to the left hand side, I have negative 350 x of one minus 500 except to no slack variables and dizzy. Hey, so here's my grid. Okay, Now, I know that I'm going to be redoing my grid down here, So I'm going to going to set this up really quick. All of my same variables. Now let's take a look at where our pivot point's going to be. If I look at my indicator row, negative 500 is the biggest negative number I have. So we'll be looking at the ratio of my Constance to the X Up two column. And when I do that, 3 36 100 divided by seven is about 514. Yes, that's got it. With a decimal 900 divided by two is 450 and 6 50 divided by one is obviously 6 50. So the smallest one is the two. There's my pivot point. So the row with the pivot point my second row is not going to change. That stays the same to get rid of my other rose. Well, to get rid of the top row, I've kind of don't have a whole lot of room right there, so I'm just gonna put this off to the side. I'm going to take negative seven times the second row, plus twice the first row. And I'm going to put that back into the first row all the way across in my new grid. And when I do that, I end up with three 02 negative. 700 900. Remember, the goal was to make every number in the same row or same column with the pivot number all equal to zero. Okay, well, what about my third row? Well, in order to get that one that I'm circling here to get that to become a zero, what I'm gonna do is take negative the opposite of the second row, plus twice the third row, and that will go into the third row position. Doing that across the row gives me 100 negative. 1 to 0 and 400. Okay, finally, we need to dio are indicator row at the bottom. To get rid of that, I'll take 500 times the second row plus twice, the fourth wrote. And that's what I'll put in the fourth row all the way across. And when I dio, my output is negative. 200 00 500 02 and then 450,000. Hey, we are not done. We still have a negative in our indicator, wrote negative 200. So to find our pivot point, we're gonna look at the values in the x of one column or Exubera column and compare them to our constants. So no. 1 900 divided by three is 300 900 divided by one is 904 100 divided by one is 400. So this three right here gives me my smallest ratio. That's my new pivot point so and bring that down a bit. So I've got some room toe work and we make our next grid. So the road with the pivot point does not change. So I'm just going to copy Row one all the way across, as is. Hey, I want to get rid of every other number in that except one collar. I want them all to equal zero, so let's look at the second row to get rid of that. I will take the opposite of the first row, plus three times the second row, and that will go into the second row space across the board. When I do that, I end up with 06 Negative to 10. 00 1800 Right. Third row. Well, again, I have the same number in the second row and the third row. So my equation is gonna look the same. Negative are sub one plus three times the third row, and that will go into the third row position. Doing that gives me 00 negative. 2460 and 300. Now we just have to get rid of that negative 200 in the bottom row. To do that, I'm going to take 500 times the first row. I'm sorry. Timmy did write 500. I met, right, 200. Try that again. So 200 times the first row, plus three times the fourth row. And I'm going to put that into the fourth row position. Doing that all the way across that row gives me 00 401 100 06 and one million, 530,000. There are no more negatives in my indicator row, so we are complete so I can start reading the answer off of this grid. Uh huh. Except one just has one non zero. So I could say that three X up one equals 900 or X up one equals 300 except to also has only one non zero. So six X up to equals 1800 or accept to also equals 300. So without looking at any of the other variables, I know that the optimal number of units to make our 300 of each type of television set Now, I can also see that from my slack variables. S sub one and s up to are both going to be zero but s up. Three is not s up. Three. I could say six s up. Three equals 300 or as sub three equals 50. Now, I'm just gonna come back up to the top and remind you s up. Three. If you remember, I made those numbers a little bit smaller. I divided it by four. Everything in my first one divided by four. My second one. So I really need to multiply this by four to go back to the original equation. So s up three for my original equation is 200. So what does that mean? Well, that means that in that third constraint, which was my testing and packing facility, I have 200 unused hours. So I've got some, You know, everything else. I maxed out to capacity, but I still had room available here in that particular department. And what is? We come back so you can see this little bit better. What is my maximum profit? Well, six z equals 1,530,000 or Z equals 200 55,000. That is the maximum profit for selling 300 of each type of television set.


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