了解如何提升工作效率和提高质量标准，学会分析和改善服务业或制造业商务流程。主要概念包括流程分析、瓶颈、流程速率和库存量等。成功完成本课程后，您可以运用所学技能处理现实商务挑战，这也是沃顿商学院商务基础专项课程的组成部分。

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Из курса от партнера University of Pennsylvania

运营管理概论（中文版）

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了解如何提升工作效率和提高质量标准，学会分析和改善服务业或制造业商务流程。主要概念包括流程分析、瓶颈、流程速率和库存量等。成功完成本课程后，您可以运用所学技能处理现实商务挑战，这也是沃顿商学院商务基础专项课程的组成部分。

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第 4 单元 - 质量

质量并不是运营管理唯一的重点，但是质量对于企业长期发展和成功至关重要。本模块将介绍运营中与质量的几个主要方面，导致缺陷的常见原因、发现质量问题以及保障可靠性和标准的常用实践方法。本模块教学结束后，您将了解缺陷可能发生的原因，并且能针对质量和稳定性提出合理的方法。

- Christian TerwieschAndrew M. Heller Professor at the Wharton School, Senior Fellow Leonard Davis Institute for Health Economics Co-Director, Mack Institute of Innovation Management

The Wharton School

In the previous session, we saw that waiting time can occur even if the

resource utilization is below 100%.. This was driven by the variability in the

process flow. Now unfortunately, the type of tools that

we've introduced so far in this course, can really not explain that waiting time.

We, so far naively believed that if the utilization is less than 100%,, there

wouldn't be any waiting. The reason for this is our previous

analysis was always built on averages. We have simply ignored the concept of

variability, so far. The purpose of this session is to help us

describe the variability in this process, and start putting together a set of tools

that help us predict the amount of waiting, even if the utilization is less

than 100%.. Now, we first need to describe the

variability, we need some form of measuring it.

We'll stay with the example of the doctor's office, this is pretty

representative for most service operations.

I have some waiting area and I have a resource.

Notice that in the section now, we're going to look at the case where the flow

rate is constrained by demand. If we have more demand than capacity, our

implied utilization is more than a 100% and we don't need fancy variability models

to tell us that there's waiting. This was my first doctor's office in the

previous session. We further assume for the remainder of

this session that everybody who arrives to the practice will wait in line till they

are served and leave the patients after having seen the doctor.

Okay. Now, let's first describe variability in

demand. We notice that the demand process was

somewhat random. When we said random, what we meant is that

the customers were not lined up at Toyota Camrys at the end of an assembly line.

They came when they wanted. So, a formal way of capturing this is we

define the arrival time as a time when patients arrive to the practice.

And then we define the inter-arrival time, as the time between two subsequent

arrivals. When we say that the arrivals are random,

what we really mean is that these inter-arrival times are drawn from some

underlined statistical distribution. We will denote with a, the average

inter-arrival time. Like any distribution, the inter-arrival

time distribution not just has an average but also a standard deviation.

We denote by the coefficient of variation of the inter-arrival time, CVa, as the

ratio between the standard deviation and the mean.

We call the idea of the coefficient of variation from the module of customer

choice and variety. The idea here is that the standard

deviation by itself is not a good measure of variability.

Is a standard deviation of ten minutes a lot or a little?

Well, it really depends on the underlying demand process.

You might have heard concepts such as exponential distribution or Poisson

distributions. For those cases, the CVa is equal to one, and the assumption is

really that at any given moment in time, the likelihood of a new patient arriving

in the next unit of time is constant. I'll talk about seasonal arrival times a

little later on in this module. For now, I'll just assume that the

likelihood of a patient coming in is random, but is reasonably constant over

time. Variability however, is not limited to the

demand process. We also have variability in what we do

internally. In other words, we have variability in our

processing time. Let me use P to define the average

processing time, Keeping in mind that the processing time

will vary from one customer to the other. Just like we used the coefficient of

variation for the inter-arrival times, it also defines a coefficient of variation

for the processing time, as a standard deviation of the processing times divided

by the average processing time. The coefficient of variation of the

processing time really measures how standardized the work is.

This varies a lot by application. In an assembly line forever, you might see

situations where this coefficient of variation of the processing time might be

close to zero. In other settings, if you think about an

operating room, you think about developing a piece of software, there might be a fair

bit of variability in the processing time. Hence, CVps can be one or bigger.

Again, this is something that you would have to measure as you go into the

analysis that we're about to do. You just collect a bunch of processing

times, put them in Excel, And just compute the standard deviation

and the mean. Equipped with all of these definitions, we are now ready to compute

the time in the queue. Notice that the time in the queue will

vary for each individual customer. So, the best we can do, is we can compute

the expected or the average time in the queue.

Now, the following formula will allow us to compute this time in the queue.

The time in the queue is given as a product of the average processing time P

for the average activity time, I use those interchangeably here,

Times the utilization divided by one minus the utilization times the coefficients of

variations added up and then divided by two.

Now, let me comment on this and go to each of these three factors step by step.

The first one is simply the average processing time.

In other words, we see that the time in the que grows linearly here with the

processing time P. Whenever in math you see a formula that

has the shape u divided by one minus u, you know that it gets ugly as you're

approaching one, then we just plug in some numbers. At an 80% utilization,

This ratio is 0.8 divided by 0.2 which is simply four.

At a 90% utilization, The same ratio is 0.9 divided by 0.1 which

is already nine. You notice that a simple ten percent

increase in utilization can more than double the waiting time.

You see this on the graph here on the right that the utilization grows very

steeply as we approach 100% utilization. This is practically very important.

The reason for that is that managers and service operations are incurring very big,

fixed costs. So, it is in their interest to squeeze

more and more customers through the process.

However, you'll notice that those last customers, which from a profitability

perspective are really interesting because their revenue goes right into the bottom

line after all the fixed costs are already paid for.

They look very profitable, but they create havoc to the system.

Again, as these last customers get over to the process, our waiting times get up to

the roof. Finally, look at the variability here.

I'm squaring the coefficient of variation for both the arrival process as well as

for the processing times. The more variability there is in the

process, the longer is the time in the queue.

If you look at the case where you have a coefficient of variation of a exponential

distribution, you notice that these two fellows here are going to be equal to one,

and this whole loss factor will degenerate to one.

Let's practice our new equation with a quick example.

Again, let's look at a doctor's office. Let's assume that patients come every 30

minutes, or is with a standard deviation of 30 minutes, and that consultations last

fifteen minutes with a standard deviation of fifteen minutes.

The first thing that I encourage you to do for these sets of problems, is simply

write down the waiting time formula. The time in the queue is P divided by u

divide by one minus u times the coefficients of variations added up after

they got squared individually. Now, which one of these ingredients here

in the formula is easy? Well, we see that the processing time is

simply fifteen minutes. What else? The coefficient of variation of

the inter-arrival time is 30 minutes off the standard deviation divided by 30

minute average. So, this means it's one, which gets

squared, but it still stays one, same for the processing times, the standard

deviation here is fifteen minutes, the average is fifteen minutes, so I have

another one squared. And I divide those two by two.

So, this old fellow here at the end is simply going to be equals to one.

Now, utilization is not immediately visible in this question, and so we have

to remind ourselves that utilization is the flow rate divided by the capacity.

Flow rate in this practice is one patient every 30 minutes unconstrained by demand.

I divide this by capacity, which is one over fifteen, then the utilization of

50%.. In other words then, if I plug this in,

this middle factor here is simply 50% divided by one minus 50%, which is equals

to one. So, the total wait time does, is fifteen

times one times one equals to fifteen minutes.

Now, I want to be a little hairsplitting here, because the question actually

doesn't ask for the waiting time, but asks what's the time when our friend, Newt, can

walk out of the practice again. That includes the time in the queue,

Which we said was fifteen minutes, but also the time in the practice when he is

in service. And so, this is the time in the queue the

plus the processing time, Altogether, 30 minutes.

And so, at ten:30,30, he can expect to be out of the office again.

We can predict the average time in the queue, based on the processing time, the

utilization, and the amount of variability, in the system.

We saw on the waiting time formula that as the utilization goes up and it approaches

100%, the utilization will drive the waiting time through the roof.

So, once the system becomes more congested than a 90, 95%, utilization, it becomes

very sensitive to every additional customer walking in.

Please only use this waiting time formula for situations where the utilization is

less than 100%, or we have more capacity than we have demand.

If demand exceeds capacity, first of all, remember, we speak of an implied

utilization, utilization by itself at the maximum be 100%..

But if demand exceeds capacity we really don't have a variability in dues waiting

time. We just have a problem of a doctor who is

taking six patients in and can only serve three.

To predict that this waiting room will fill up over time, doesn't require some

fancy math. It's just a simple matter of understanding

that we are serving three fewer customers than we have demand.

Also notice that once we have computed the time in the queue, we can compute the

total flow time, the time in the system, by simply adding the time in the queue

plus the processing time P. We can also compute inventory using

Little's Law. So, once you have computed the time in the

queue, every other measure that we talked about in this class can be computed.

It is why this waiting time formula is so important.

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