So, welcome to the review session for Module four.

It's the same format as in the past. We going to work through a couple of

questions. This time it's going to be five questions.

And we'll do this in the format that I'll present the question to you,

You pause the video, And then I'll present the answer to you

once you're ready to check your results. Okay.

So, the first question is about this firm my-law.com.

And the way my-law works is basically they have a lawyer, and he's answering e-mails.

and e-mails come in from eight to six:00 p.m.

And there are ten e-mails arriving per hour.

There's only one lawyer on call and it takes the lawyer five minutes to write a

response e-mail with a standard deviation of four minutes.

Alright, a couple of questions are coming up.

Pause me now and take a shot at these questions yourself.

Alright. This is obviously a waiting time problem.

The magic word here, as you see, wait. How long does the customer have to wait?

And we want to be careful with this question, because it is asking you to

consider not just the wait questions, the time, and the queue.

But you have to also, if you want to wait for the e-mail, the customer has to wait

for the agent or the lawyer to get to the e-mail and then the customer also has to

wait for the five minutes that it takes to respond to the e-mail.

So, we're also looking for the processing time.

Now, the hardest thing is really coming up with the wait time.

And so just remember, time in the q is P times U divided by one minus u times CVa

square plus CVp square divided by two. Let me emphasize it's one lawyer so m

equals one. For bigger m, as we will see in a moment,

for, on the next question, for bigger m, we need the other, the bigger formula.

So, this is a simple case for m equals one.

Alright, ten e-mails per hour that means there's an e-mail coming in every six

minutes so the inter-arrival time is six. The processing time, we notice, P is five

minutes and so if you look, then it's the utilization.

Utilization is flow rate divided by capacity.

Flow rate as e-mails flow in the system every six minutes.

One over five is our capacity, and so the utilization is really five over six.

That means here I have P equals to five times U five over six divided by one minus

five over six times, okay, CVa square coefficient variation of the inter-arrival

time is one and so that is simply a one, One squared stays a one,

Plus CVp squared Well, so standard deviation here is four minutes and the

mean is five minutes. And so, we are dealing here with 0.8, four

divided by five, 0.8 and we're going to square this.

And then we're going to divide both of these by two,

Alright? So, that is basically giving you here, we

have one minus five-sixths. That is one over six, five over six by one

over six. That is going to be five,

Five times five is 25. And the other ratio here,

This is 0.64 plus one is 1.64, divided by two is 0.82.

And that is giving you 20.5 minutes. Again, this is just the time in the, the

queue. If you want to add up to that,

The actual writing time of the e-mail then you have to add another five minutes and

that would give you then a total of 25.5 minutes.

That's the answer to the question. Now, B and C are actually,

I don't know what I did when I wrote these questions, but these are really friendly

question in many ways. Just over ten hour days.

Remember, ten e-mails are coming in per hour, and so really we have nothing but a

hundred e-mails here that are answered. And then finally, the last question asks,

is the time the lawyer can use to pursue these kind of large settlement cases, and

for that, we have to look at the idle time which is one minus the utilization, times

the ten hours in the day we know that the idle time is one-sixth because the

utilization is five-sixths and so that is simply ten over six hours per day.

Okay. The next question,

A small computer store here with five employees that is doing repairs for people

like Jim, who has his computer broken. And so, this computer store, PC Fixers, or

PF for short, is going to get customers walk in every ten minutes and they take 40

minutes to fix the computers. And there are five computer technicians to

do the work. Alright.

Same business as usual. I shut up for a while.

You work on the problem and then press on play again so that you hear my answers.

Okay. So, again, we are dealing with a waiting

time problem. But this time, because of the multiple

technicians, we are dealing with the more complicated waiting time formula that has

Tq as a function of m, multiple servers expressed.

So let's just start writing down the formula from the lecture.

So we're looking at the expected wait time, P divided m times u raised to the

square root 2m+1.1. Now, very careful, the square root ends

after the one, minus one. The minus one is outside the square root.

One minus u times CVa square plus CVp squared divided by two.

Alright, so now, let's look for what's easy and what is hot.

We clearly see that there's a customer walking in every ten minutes, so that's

the inter-arrival time a. We have a service time P of 40 minutes.

And so, we can first compute the utilization.

As of flow rate, which is one over a divided by the capacity, which is m over p

and so that simplifies to p divided by a times m, and that is, in this case where

we have a P of 40, divided by ten times five so that is 40 by 50 or 0.8

utilization. Now, note that in this example here, CVa

is equal to one, because both the mean inter-arrival time and the standard

deviation is equal to ten, so that cancels out.

And CVp is also equal to one, because 40 is both the mean and the standard

deviation. So, this whole thing here at the end is

all going to be equals to one. That simplifies the math because now we

have 40 minutes divided by five servers times 0.8, the utilization is raised to a

power of twelve, which is 2m+1,1. Five plus one is six,

Two times six is twelve. So, square root of twelve minus one

divided by 0.2 times one. And you plug that into your calculator,

and at least I get a 23.08. Now, the next question asks for the length

of the line or how many customers will be waiting for their computer to be fixed.

This is ultimately something that we can determine with Little's Law, I equals R

times T. Remember, already we know that there is a

customer coming in every ten minutes. So, one over ten customers per minute is

going to be our flow rate. And our flow time, well that's a 23.08

here, here in the queue plus the 40 minutes that they spent for the, waiting

for the actual service time. And so, this is then going to be 63.08

minutes. And that gives you simply 6.308 customers,

and that is the inventory of customers in the store.

Now, Recompute is a company that offers super computers to users via the internet.

And they get jobs every four minutes. Standard deviation of the inter-arrival

time is also four minutes. It takes ten minutes to execute these

jobs. And here's a trick that if the super

computers are busy, the job actually gets rerouted to another super computer of

another I mean, or a supplier of ours, or a partner of ours.

In which case, we have to pay these people actually $40 per job.

So, you see the question here test your self and then put me back on play.

Alright, the key to this question is that this is not a waiting time question but a

loss question. If you hear words in the questions, see

words in the questions, such that you'll redirect or the job you know, a

probability that the job can or cannot be executed.

We're not dealing with waiting time questions,

We're dealing with loss questions. For loss question, we have the LN loss

function, and so, remember, we start by looking at the ratio of P divided by a.

So, we have a processing time here that is ten minutes, and we have a inter-arrival

time that is four minutes. So, that ration here is 2.5.

We have m equals four. We then look at the table that I had

mentioned in the lecture, but it is posted in the, in the wiki for the current week.

Look at the table, and there you will find that the probability that we have a loss

where is R equals 2.5 and M equals four is equals to 14.99%,.

So 0.4, 1.499. So, notice a question here is asking

what's that probability which, with which an incoming job can be executed, and so we

have to take one minus that probability as the answer to the question.

And so, the answer to this question is that it is 85 or one%..

The second part of the question looks at the and sometimes, sometimes looks at the

overflow and it's really interesting to keep in mind or important to keep in mind

that we have fifteen jobs coming in per hour,

Right, That is just 60 minutes divided by the

excuse me, by the inter-arrival time of four minutes here.

So, fifteen jobs come per hour and we know from the previous question that jobs will

be turned down or turned away to the partner company on computer that happens

with the probability of 14.99%.. So, in that case, simply, we have to turn

2.2485 expected jobs away. And if we multiply that with the penalty

of $40 per job, We're going to get a hourly payment, an

expected hourly payment to our partner company of $89.94, so this is dollars per

hour. Okay, next question.

We have a contractor here that is doing lots of renovation and building work.

And the contractor has six projects lined up.

You'll see their processing times listed here.

And he wants to minimize the average wait time that projects are waiting for.

And so, that's how he sequences his work. And the question is what will he be doing

in 30 days from now. Alright ready?

You go first and then, I'll give you the solution.

Alright. The key thing here that these not working

first come first serve, but he is using the shortest processing time rule.

The shortest processing time rule will minimize the average wait time of these

projects. Now, notice that regardless in which

sequence first come first serve, shortest processing time rule, or whatever else,

the total amount of time it takes is simply just the sum of these various

tasks. But, by doing the shortest processing time

rule, he can reduce the average wait time. It simply makes more sense that we start

with the short jobs so that the long job will waits for the short job, as opposed

to the other way around. So, if he follows the shortest processing

time rule, he will do this job first. This job second, third,

Fourth, fifth, and, this one's here's going to be last.

And so, now I know the sequence in which he's going to do the work, and so the, the

first job is going to be over after two days.

The second job he's going to touch is going to be over of the two plus six

equals to eight days. The next job is going to be over of the

sixteen days. Then, he's going to take the job number

four that takes ten days, so that's hoing to get him to 26 days.

So, after 26 days, he has the first four jobs completed, and then he's going to

start the fifth job in his, in his sequence, and that is going to be the new

kitchen here at, at Rosemont. And so, with that in mind, it is the new

kitchen in Rosemont that answers a question on what he will be working 30

days from now. Alright.

The last question is about a call center. The call center, right now, has a constant

staff available all over the day, and there is some seasonality in the demand,

in particular, there is lots of demand in the morning hours, and very little demand

in the afternoon hours. So, utilization of the call center is high

in the morning and low in the afternoon. Now, which of the following action will

decrease the average waiting time in the call center?

Take some time for yourself, and then I share my thoughts with you.

Alright so, this is a qualitative question, there is really no number here

that you could crunch into the waiting time formula.

Nevertheless, I think it's helpful to remind ourselves of the waiting time

formula. Tq is equals to P divided by m.

Utilization raise to the 2m+1 minus one divided by one minus u times CVa squared

plus CVp squared divided by two. Alright, now let's look at the suggestions

that I put up here of adding more servers. So, adding more servers is a good idear,

right? You notice a larger m here will decrease

Tq. Now I want to be careful here in reminding you that the utilization, which

we defined as flow rate divided by capacity, and as one over the

inter-arrival time times m over P and then P divided by a times m, I want to just

emphasize that the utilization is also a function of m and so if you're taking the

sensitivity analysis derivative of Tq with respect to m, you cannot just look at this

term. Also u is changing as you're adding

servers, but, either way, adding servers is a good thing.

It will shorten the waiting time, Decreases service coefficient of

variation, That one is fairly straightforward.

You see the marginal impact here. Cvp shows up just linearly in here, and

everything that you do to CVp will be reducing the wait time, so this one also

makes sense. Decreasing the average service time.

You see this will help through P here in the formula.

That same comment is on mP, is also sitting in u, and will also have an x3

effect by allowing you to operate at a lower level of utilization.

And so, that one is certainly true as well.

Now, the last one, option D, is a little tricky here.

Leveling the demand. Why would that matter, and how do you

think about that? For that place, remember that if you plot

the time and the cube as a function of the utilization u, you have this very steep,

nonlinear effect. And so right now, in the morning hours, we

have a very high level of utilization. So, say, say, we are, we are here, high

level of utilization. And in the afternoon hours, we have a low

level of utilization. Now, conceptually, you see that if we

could now bring this together, and level the demand,

So we move some of the demand from the morning into the afternoon hours,

Well, the folks who had previously a super short wait time, well, they will lose a

little. They wait a little longer here, right?

So, you see, there's a little increase here in the waiting time, for these guys.

But the morning people with really, really benefit, right?

And so, you notice that, on average, it's nice to actually have a constant level of

utilization as opposed to having half of the customers get really high level of

utilization, and half of the customers having a really low level of utilization.

So, leveling the demand because of these convexities, of steep increase of the

waiting time as a function of u will be a good thing.

And thus, we can check option E, all, all of the above.

额外单元复习

运营管理概论（中文版）

与讲师见面

额外单元 复习

运营管理概论（中文版）

与讲师见面

额外单元复习