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Damian takes carc of a machine. At the end of the ath day; € {1,2,3,4}, preventive maintenance performed. During the days, faliures can happen to the machin...

Question

Damian takes carc of a machine. At the end of the ath day; € {1,2,3,4}, preventive maintenance performed. During the days, faliures can happen to the machine. Each faliure results in a cost of C1. Denote by Z(a) the number of faliures during days; and suppose Z(a) Poi(A(a)) , Where A(a) J A(-)dz. Assume A(a) In(a + 1). Denote by Y(a) the total cost per day (given and by C the preventive maintenance cost incurred at the end of the ath day: Derive E[Y (a)]: Assume that C1 =2 and C2 3. SolveEL

Damian takes carc of a machine. At the end of the ath day; € {1,2,3,4}, preventive maintenance performed. During the days, faliures can happen to the machine. Each faliure results in a cost of C1. Denote by Z(a) the number of faliures during days; and suppose Z(a) Poi(A(a)) , Where A(a) J A(-)dz. Assume A(a) In(a + 1). Denote by Y(a) the total cost per day (given and by C the preventive maintenance cost incurred at the end of the ath day: Derive E[Y (a)]: Assume that C1 =2 and C2 3. Solve ELY (a)] min ae{123}



Answers

A manufacturing company owns a major piece of equipment that depreciates at the (continuous) rate $ f = f(t) $, where $ t $ is the time measured in months since its last overhaul. Because a fixed cost $ A $ is incurred each time the machine is overhauled, the company wants to determine the optimal time $ T $ (in months) between overhauls.

(a) Explain why $ \displaystyle \int^t_0 f(s) ds $ represents the loss in value of the machine over the period of time $ t $ since the last overhaul.
(b) Let $ C = C(t) $ be given by $$ C(t) = \frac{1}{t} \Biggl[ A + \int^t_0 f(s) ds \Biggr] $$
What does $ C $ represent and why would the company want to minimize $ C $?
(c) Show that $ C $ has a minimum value at the numbers $ t = T $ where $ C(T) = f(T) $.

So we saw part a leg fft be the value of Okay, the equipment at the time t mhm mhm. So we know that from the given information that the rate of depiction of the equipment is fft So because the delivery will give us the right off change of a function So we can say the derivative of capital f of T equal to FFT by the attacking entry angle to both sides and applying the fundamental theorem of calculus, we find that okay Capital f of T equal to and regal F of s Niaz zero to t because we define capital fft as the value of equipment. The equation tell us that the intrigue will give us the total loss of the value at Time T So it is all for Alonso now. We saw hard B here C of t equal to one upon t e plus in real f of S Diaz zero to t equal to a plus Capital f of T upon d represent the average amount of just expenditure per unit of Time T on the interval, zero to t. So we assume that there has only been on our all during that raid. The company wants to minimize the average expenditure. So our final answer is average among golf. Okay. Expenditure. Yeah. Mhm. Oh, Interval zero to t minimize Mhm average expenditure. Yeah. So it is a fire answer? No, we sold part c. Yeah, we know that c of t equal to one upon t bracket eight plus and trigger f of S d s zero to t. So at this point, the world maximum should be ringing a bell. How do we find minimum and maximum? So we take the derivative of our equation and set it equal to zero the capital T from the kitchen. We will take discuss at the end. So let's take the Davidoff t of CFT. So here, derivative of C of t equal to D one upon t a plus in trickle f of s d s zero to t upon DT So we use product rule. So here we get mhm negative one up on the square time e plus in triangle f of s Lee s zero to t plus zoo plus fft time one of on t. So here no devotee of T equal to zero so you can see that here? Negative one upon d dying one upon t time A plus Intrigue Aled F of s Yes. Zero to t negative f off D thought Okay. Yeah, equal to zero. So here we get one of bounty. A bloods intrigue ALS app of s time? Yes, CEO duty. Yeah, negative fft equal to zero. So we get fft equal to one upon t a plus Intrigue f of S d s zero to t So here you can see that it is the value of C f T therefore f of t equal to C of t. So it is all final answer. No, please wait no, We have to put equal to Capital T So here we get c of t equal to half of capital t So here f of capital t equal to one upon capital t time a plus and wriggled f of s time. Diaz zero to capital t equal to Sea of Capital T So it is a one final answer

Did anybody off sea of X equal to 0.0 three times X. Choir negative. See your 0.12 time X plus 20. So here a dozen of Dayton Industries manufactures a dealer's roster. Own management has determined that daily marginal cost function associated with producing these lost toaster Owen is given by delivery of sea of X equal to 0.3 times X. Choir negative 0.12 times X plus 20 were delivery of C of X is measured in dollar per unit and X denotes the number of unit produced. And here management has also determined that the daily fixed coast incurred in the production is $800. So in part E we have to find the total coast incurred by Dayton is producing the 4th 300 units of these tossed her own party. So here we know that the daily coast in good is sea of ceo equal to 800. So we have to find see off 300. so here net change in the total cost function sea of X or interval 0 to 300 to the fundamental to your um see off 300 negative of SEO equal to definite intrigue all of delivery of sea of X D X from 0 to 300 so here we know that the value of delivery of sea of X so we put here and we get definitely and real of cedar point cyril cyril 03 times X choir 90 0.12 time X plus 20 D X from SEO 2 300 so here and everybody of 0.3 time ex choir negative 0.12 time explosives you is 0.1 time actually powerful now you do 0.6 time ex choir plus 20 times X from 0 to 300 so now you can see that here. Opera value is 300. So I put here at school 23 300 sorry so we put here 300 so here we get zero point seal. Seal one time 300 to the power fool. Now I do see your 0.6 time 300 to the power do plus 20 times 300. Now you can see that here lower value is zero. So I put here at school zero so here we get zero now we simplify this and here we get 2700 negative. 5400 flows 6000 So here again approximate value is 300 sorry 3300. That is the value of C. Of 300 negative. See off seal. So we know that here we have to aid to the both sides by sea of zero. So here we get sea of 300 equal to 3300 positive. See off ceo. So now you can see that sea of zero value is 800. So we put here 3300 plus 800 equal to 4100. That is the value of C. Of 300 doll. So fine laundries. Sea of 300 equal to $4100. It is all for an answer for part A. Now we sold part B. What is the cost of incurred by Dayton? Is soy bean for easing the 200 one S. T. True 300 unit per day True the fundamental theorem here we know that see all 300 now I do see off 200 equal to definite intrigue. Als oh delivery of sea of X. B. X. From 200 to 300. So here you can see that definitely not real of 0.3 times X. Choir negative. 0.12 time X plus 20 time D. X 202 300. So now here we know that the and did anybody wolf 0.3 time ex choir negative. See your 0.12 time explodes 20 is 0.0 01 time xQ 90 0.6 time X square plus 20 times X. From 200 to 300. So now here I probably used 300 so we put here X equal to 300. So here we get 2700 now you do 5400 plus 6000. Now you do know you can see that here laurel you is 200 so I put your X cold too. So we get after simplify this 8000. Now you do 2400 plus 400. That is approximate value. So here we get 3000 300. Now you do 2400 equal to 900. So here we can say the total cost in cured by data is producing the 2000, sorry, 200 through 300 unit per day is 900 dollar. So here we get $900. So it is a final answer.

Probably 47. The delivery time of a circuit board to replace a failed one is represented by white where y is a random variable follows in from distribution over the interval from 1 to 5 days. The cost associated with this failure equals the fix it cost of a new board plus a cost proportional to the delay of the delivery. See one are deployed by y squared, it's proportional to the square of the delivery time for birthday. We want to find the probability that the delivery time exceeds two days, which means we want to calculate the probability for why to be greater than two. We can rewrite this probability as one minus the probability for a while to be smaller than equals two, then equals one minus the integration from minus infinity to to four ever. Why do you want? What is if a boy we can get a void because why follows an informal distribution, it equals one divided by two to to mind state on data to minus set on five minus one through the interval from 0 to 1 potato two. And it's defined as zero elsewhere. Then the probability for why to be greater than two equals one minus the integration from minus infinity 22 is the integration from 1 to 2. Because the function is defined as zero from minus infinity to one and from 1 to 2, if a boy is defined as one quarter then it's one minus. Why? Divided by four. We integrate from 1 to 2 equals one minus. We substitute by y equals two. First then minus y equals one. Whether by four it's just three quarters mm For Berkeley we want to calculate the expected value for the cost, selected the value for the cost, it equals the expected value. Four C note plus c one Y squared which means it acquires the expected well, for C note plus the expected value for C. One. Well I squid. The expected value of a constant is just the value of this contest. Plus we can get the constant out of expected. Then it's C. One multiplied by the expected value of Y squared. As long as boy is a random variable. We can rewrite it using the definition of the expected but C note, glossy one multiplied. The definition of the expected value is the integration from minus infinity to infinity. Four Y squared multiplied by F of Y do it? We know that well. The function is defined as zero elsewhere and it's defined as one quarter from one 2 to 5. Then it equals C. Note plus the one multiplied by the integration from 1 to 5. For why he squared multiplied by one quarter. Do you want then equals C. Note plus C one deployed by the integration of voice square is why cubed divided by three. Multiplied by four, gives 12. Integration from 1 to 5, then equals C. Note plus C. +11 by 12 blow it by. We substitute first by Y equals five gives 100 25 minus researched you by Y equals one gives one, then it's 124 divided by 12, gives 31 divided by three. Then the final answer is C note close 31. See one divided vital. This is an answer for being.

A company that manufactures travel clocks, has determined that the daily Marginal Coast function associated with producing T's clocks is derivative of C of X equals 0.9 times X square, minus 0.9 times X plus eight we're seeing Derivative of X is measure in dollars per unit, and X denotes the number of units produced. Management has also determined that the daily fixed coast incurring producing these clocks is $120. And with that information, we want to find the total cost incurred by the company in producing the first 500 trouble clocks per day. Right so way have the marginal cost function that is the derivative of the cost function. We want to find the glucose. So we got to find the until derivative or indefinite integral off. It's c prime. So we know that total cost see of X will be equal to the integral off the marginal coast. Uh huh. What? And that will be equal to the integral. The indefinite integral off the expression given to Thean marginal coast that is zero point 000 seriously or nine x square, then minus 0.9 x plus eight differential ipix. And we got to integrate this and this is a constant. So we get 0.9 times interval of X Square minus 0.9 which is a constant times the integral of X plus eight Internal of Differential of X And these are powers of acts and we know how to integrate that. It's 0.0 00 Sierra nine x to the third over three minus 0.9 times X squared over two plus eight eggs, plus a constant of integration. We put the continent's constant of integration at the end because even if each integral has a constant of integration, we know that some of them will be a unique constant of integration. So the tractor off being indefinite is the reason why we have constant of integration, which has to be determined with some initial condition. Let's simplify a little bit here, so we have 0.0 9/3 0 point 000003 X cube minus 0.9 Number two is 0.0 45 x square plus eight x plus c and this is the total cost function. With it, we can calculate it'll cost at any for any number of units. But before we get to determine this value, the value of this constant of integration. And for that we have the initial value given here. The management determined that daily fixed coast in current position disclose clocks is $120 the people. So because see of zero is 120 the fixed cost of position the unit, it's $120. I remember this function is given in dollars. It's important because the marginal coast is given in dollars per unit. So the coast, because it will be given in dollars so see of Siri's is 120. If we relate this expression at zero X equals zero, we get C. So so you gotta be 120. Okay, then The total cost function is 0.3 X cube minus 0.0 45 X square, plus a X plus 120 in dollars. And now we are able to calculate. What the problem is asking for is to find the total cost incurred by the company in producing the 1st 505 o'clock per day. So total cause mhm off producing to first. Okay, 500 clocks or traffic clock ace. See, at 500 that is 0.3 times 500 to the third, minus 0.0 45 500 square. Right. Plus eight times 500 plus 120. And using a calculator, find disease equal to 3000 370. So okay, during coast Uh huh. Incurred by the company? Yeah. In producing the first in producing the first 500 travel clocks per day is three 1003 $170. And that's do you final answer off given probe.


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