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Part Three Planning and control
334
Variability
The concept of variability is central to understanding the behaviour of queues. If there were
no variability there would be no need for queues to occur because the capacity of a process
could be relatively easily adjusted to match demand. For example, suppose one member of
staff (a server) serves at a bank counter customers who always arrive exactly every five minutes
(i.e. 12 per hour). Also suppose that every customer takes exactly five minutes to be served,
then because,
(a) the arrival rate is ≤ processing rate, and
(b) there is no variation
no customer need ever wait because the next customer will arrive when, or before, the
previous customer. That is, WIP
q
= 0.
Also, in this case, the server is working all the time, again because exactly as one customer
leaves the next one is arriving. That is, u = 1.
Even with more than one server, the same may apply. For example, if the arrival time at
the counter is five minutes (12 per hour) and the processing time for each customer is now
always exactly 10 minutes, the counter would need two servers, and because,
(a) arrival rate is ≤ processing rate m, and
(b) there is no variation
again, WIP
q
= 0, and u = 1.
Of course, it is convenient (but unusual) if arrival rate/processing rate = a whole number.
When this is not the case (for this simple example with no variation),
Utilization = processing rate/(arrival rate multiplied by m)
For example, if arrival rate, r
a
= 5 minutes
processing rate, r
e
= 8 minutes
number of servers, m = 2
then, utilization, u = 8 / (5 × 2) = 0.8 or 80%
Incorporating variability
The previous examples were not realistic because the assumption of no variation in arrival or
processing times very rarely occurs. We can calculate the average or mean arrival and process
times but we also need to take into account the variation around these means. To do that we
need to use a probability distribution. Figure S11.1 contrasts two processes with different
arrival distributions. The units arriving are shown as people, but they could be jobs arriving
at a machine, trucks needing servicing, or any other uncertain event. The top example shows
low variation in arrival time where customers arrive in a relatively predictable manner. The
bottom example has the same average number of customer arriving but this time they arrive
unpredictably with sometimes long gaps between arrivals and at other times two or three
customers arriving close together. Of course, we could do a similar analysis to describe pro-
cessing times. Again, some would have low variation, some higher variation and others be
somewhere in between.
In Figure S11.1 high arrival variation has a distribution with a wider spread (called
‘dispersion’) than the distribution describing lower variability. Statistically the usual measure
for indicating the spread of a distribution is its standard deviation, σ. But variation does not
only depend on standard deviation. For example, a distribution of arrival times may have
a standard deviation of 2 minutes. This could indicate very little variation when the average
arrival time is 60 minutes. But it would mean a very high degree of variation when the
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Supplement to Chapter 11 Analytical queuing models
335
Figure S11.1 Low and high arrival variation
average arrival time is 3 minutes. Therefore to normalize standard deviation, it is divided
by the mean of its distribution. This measure is called the coefficient of variation of the
distribution. So,
c
a
= coefficient of variation of arrival times = σ
a
/t
a
c
e
= coefficient of variation of processing times = σ
e
/t
e
Incorporating Little’s law
In Chapter 4 we discussed on of the fundamental laws of processes that describes the relation-
ship between the cycle time of a process (how often something emerges from the process),
the working in progress in the process and the throughput time of the process (the total time
it takes for an item to move through the whole process including waiting time). It was called
Little’s law and it was denoted by the following simple relationship.
Work-in-progress = cycle time × throughput time
Or,
WIP = C × T
We can make use of Little’s law to help understand queuing behaviour. Consider the queue
in front of a station.
Work-in-progress in the queue = the arrival rate at the queue (equivalent to cycle time)
× waiting time in the queue (equivalent to throughput
time)
WIP
q
= r
a
× W
q
and
Waiting time in the whole system = the waiting time in the queue + the average process
time at the station
W = W
q
+ t
e
We will use this relationship later to investigate queuing behaviour.
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336
Types of queuing system
Conventionally queuing systems are characterized by four parameters.
A – the distribution of arrival times (or more properly interarrival times, the elapsed
times between arrivals)
B – the distribution of process times
m – the number of servers at each station
b – the maximum number of items allowed in the system.
The most common distributions used to describe A or B are either
(a) the exponential (or Markovian) distribution denoted by M; or
(b) the general (for example normal) distribution denoted by G.
So, for example, an M/G/1/5 queuing system would indicate a system with exponentially
distributed arrivals, process times described by a general distribution such as a normal dis-
tribution, with one server and a maximum number of items allowed in the system of 5. This
type of notation is called Kendall’s notation.
Queuing theory can help us investigate any type of queuing system, but in order to
simplify the mathematics, we shall here deal only with the two most common situations.
Namely,
● M/M/m – the exponential arrival and processing times with m servers and no maximum
limit to the queue.
● G/G/m – general arrival and processing distributions with m servers and no limit to the
queue.
And first we will start by looking at the simple case when m = 1.
For M/M/1 queuing systems
The formulae for this type of system are as follows.
WIP =
Using Little’s law,
WIP = cycle time × throughput time
Throughput time = WIP / cycle time
Then,
Throughput time =×=
and since, throughput time in the queue = total throughput time − average processing time,
W
q
= W − t
e
=−t
e
==
= t
e
u
(1 − u)
t
e
− t
e
− ut
e
1 − u
t
e
− t
e
(1 − u)
1 − u
t
e
1 − u
t
e
1 − u
1
r
a
u
1 − u
u
1 − u
M/M/m queues
G/G/m queues
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again, using Little’s law
WIP
q
= r
a
× W
q
= t
e
r
a
and since
u ==r
a
t
e
r
a
=
then,
WIP
q
=×t
e
×
=
For M/M/m systems
When there are m servers at a station the formula for waiting time in the queue (and there-
fore all other formulae) needs to be modified. Again, we will not derive these formulae but
just state them.
W
q
= t
e
From which the other formulae can be derived as before.
For G/G/1 systems
The assumption of exponential arrival and processing times is convenient as far as the
mathematical derivation of various formulae are concerned. However, in practice, process
times in particular are rarely truly exponential. This is why it is important to have some idea
of how a G/G/1 and G/G/m queue behaves. However, exact mathematical relationships are
not possible with such distributions. Therefore some kind of approximation is needed. The
one here is in common use, and although it is not always accurate, it is for practical purposes.
For G/G/1 systems the formula for waiting time in the queue is as follows.
W
q
= t
e
There are two points to make about this equation. The first is that it is exactly the same as the
equivalent equation for an M/M/1 system but with a factor to take account of the variability
of the arrival and process times. The second is that this formula is sometimes known as the
VUT formula because it describes the waiting time in a queue as a function of:
V – the variability in the queuing system
U – the utilization of the queuing system (that is demand versus capacity), and
T – the processing times at the station.
In other words, we can reach the intuitive conclusion that queuing time will increase as
variability, utilization or processing time increases.
D
F
u
(1 − u)
A
C
D
F
c
a
2
+ c
e
2
2
A
C
u
2(m+1)−1
m(1 − u)
u
2
(1 − u)
u
t
e
u
(1 − u)
u
t
e
r
a
r
e
u
(1 − u)
VUT formula
Supplement to Chapter 11 Analytical queuing models
337
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For G/G/m systems
The same modification applies to queuing systems using general equations and m servers.
The formula for waiting time in the queue is now as follows.
W
q
==
t
e
D
F
u
2(m+1)−1
m(1 − u)
A
C
D
F
c
a
2
+ c
e
2
2
A
C
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338
‘I can’t understand it. We have worked out our capacity figures and I am sure that one
member of staff should be able to cope with the demand. We know that customers arrive
at a rate of around 6 per hour and we also know that any trained member of staff can
process them at a rate of 8 per hour. So why is the queue so large and the wait so long?
Have at look at what is going on there please.’
Sarah knew that it was probably the variation, both in customers arriving and in how
long it took each of them to be processed, that was causing the problem. Over a two-day
period when she was told that demand was more or less normal, she timed the exact
arrival times and processing times of every customer. Her results were as follows.
The coefficient of variation, c
a
of customer arrivals = 1
The coefficient of variation, c
e
of processing time = 3.5
The average arrival rate of customers, r
a
= 6 per hour
therefore, the average inter-arrival time = 10 minutes
The average processing rate, r
e
= 8 per hour
therefore, the average processing time = 7.5 minutes
Therefore the utilization of the single server, u = 6/8 = 0.75
Using the waiting time formula for a G/G/1 queuing system
W
q
= 7.5
= 6.625 × 3 × 7.5 = 149.06 mins
= 2.48 hours
Also because,
WIP
q
= cycle time × throughput time
WIP
q
= 6 × 2.48 = 14.68
So, Sarah had found out that the average wait that customers could expect was 2.48 hours
and that there would be an average of 14.68 people in the queue.
‘Ok, so I see that it’s the very high variation in the processing time that is causing the queue
to build up. How about investing in a new computer system that would standardize
processing time to a greater degree? I have been talking with our technical people and
they reckon that, if we invested in a new system, we could cut the coefficient of variation
of processing time down to 1.5. What kind of a different would this make?’
Under these conditions with c
e
= 1.5
W
q
= 7.5
= 1.625 × 3 × 7.5 = 36.56 mins
= 0.61 hour
D
F
0.75
1 − 0.75
A
C
D
F
1 + 2.25
2
A
C
D
F
0.75
1 − 0.75
A
C
D
F
1 + 12.25
2
A
C
Worked example 1
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Supplement to Chapter 11 Analytical queuing models
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A bank wishes to decide how many staff to schedule during its lunch period. During
this period customers arrive at a rate of 9 per hour and the enquiries that customers
have (such as opening new accounts, arranging loans, etc.) take on average 15 minutes
to deal with. The bank manager feels that four staff should be on duty during this period
but wants to make sure that the customers do not wait more than 3 minutes on average
before they are served. The manager has been told by his small daughter that the dis-
tributions that describe both arrival and processing times are likely to be exponential.
Therefore,
r
a
= 9 per hour, therefore
t
a
= 6.67 minutes
r
e
= 4 per hour, therefore
t
e
= 15 minutes
The proposed number of servers, m = 4
therefore, the utilization of the system, u = 9/(4 × 4) = 0.5625.
From the formula for waiting time for a M/M/m system,
W
q
= t
e
W
q
=×0.25
=×0.25
= 0.042 hour
= 2.52 minutes
Therefore the average waiting time with 4 servers would be 2.52 minutes, which is well
within the manager’s acceptable waiting tolerance.
0.5625
2.162
1.75
0.5625
10−1
4(1 − 0.5625)
u
2(m+1)−1
m(1 − u)
Worked example 2
Therefore,
WIP
q
= 6 × 0.61 = 3.66
In other words, reducing the variation of the process time has reduced average queuing
time from 2.48 hours down to 0.61 hour and has reduced the expected number of
people in the queue from 14.68 down to 3.66.
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Introduction
Operations managers often have an ambivalent attitude towards
inventories. On the one hand, they are costly, sometimes tying
up considerable amounts of working capital. They are also risky
because items held in stock could deteriorate, become obsolete
or just get lost, and, furthermore, they take up valuable space in
the operation. On the other hand, they provide some security in
an uncertain environment that one can deliver items in stock,
should customers demand them. This is the dilemma of inventory
management: in spite of the cost and the other disadvantages
associated with holding stocks, they do facilitate the smoothing
of supply and demand. In fact they only exist because supply
and demand are not exactly in harmony with each other
(see Fig. 12.1).
Chapter 12
Inventory planning
and control
Key questions
➤ What is inventory?
➤ Why is inventory necessary?
➤ What are the disadvantages of
holding inventory?
➤ How much inventory should an
operation hold?
➤ When should an operation replenish
its inventory?
➤ How can inventory be controlled?
Figure 12.1 This chapter covers inventory planning and control
Check and improve your understanding of this chapter using self assessment
questions and a personalised study plan, audio and video downloads, and an
eBook – all at www.myomlab.com.
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No inventory manager likes to run out of stock. But for
blood services, such as the UK’s National Blood Service
(NBS) the consequences of running out of stock can
be particularly serious. Many people owe their lives to
transfusions that were made possible by the efficient
management of blood, stocked in a supply network
that stretches from donation centres through to hospital
blood banks. The NBS supply chain has three main
stages:
1 Collection, which involves recruiting and retaining
blood donors, encouraging them to attend donor
sessions (at mobile or fixed locations) and transporting
the donated blood to their local blood centre.
2 Processing, which breaks blood down into its
constituent parts (red cells, platelets and plasma)
as well over twenty other blood-based ‘products’.
3 Distribution, which transports blood from blood
centres to hospitals in response to both routine and
emergency requests. Of the Service’s 200,000
deliveries a year, about 2,500 are emergency
deliveries.
Inventory accumulates at all three stages, and in
individual hospitals’ blood banks. Within the supply
chain, around 11.5 per cent of donated red blood cells
donated are lost. Much of this is due to losses in
processing, but around 5 per cent is not used because
it has ‘become unavailable’, mainly because it has been
stored for too long. Part of the Service’s inventory control
task is to keep this ‘time-expired’ loss to a minimum.
In fact, only small losses occur within the NBS, most
blood being lost when it is stored in hospital blood banks
that are outside its direct control. However, it does
attempt to provide advice and support to hospitals to
enable them to use blood efficiently.
Blood components and products need to be stored
under a variety of conditions, but will deteriorate
over time. This varies depending on the component;
platelets have a shelf life of only five days and demand
can fluctuate significantly. This makes stock control
particularly difficult. Even red blood cells that have a
shelf life of 35 days may not be acceptable to hospitals
if they are close to their ‘use-by date’. Stock accuracy
is crucial. Giving a patient the wrong type of blood can
be fatal.
At a local level demand can be affected significantly
by accidents. One serious accident involving a cyclist
used 750 units of blood, which completely exhausted the
available supply (miraculously, he survived). Large-scale
accidents usually generate a surge of offers from donors
wishing to make immediate donations. There is also a
more predictable seasonality to the donating of blood,
however, with a low period during the summer vacation.
Yet there is always an unavoidable tension between
maintaining sufficient stocks to provide a very high level
of supply dependability to hospitals and minimizing
wastage. Unless blood stocks are controlled carefully,
they can easily go past the ‘use-by date’ and be wasted.
But avoiding outdated blood products is not the only
inventory objective at NBS. It also measures the
percentage of requests that it was able to meet in full,
the percentage emergency requests delivered within
two hours, the percentage of units banked to donors
bled, the number of new donors enrolled, and the
number of donors waiting longer than 30 minutes before
they are able to donate. The traceability of donated blood
is also increasingly important. Should any problems with
a blood product arise, its source can be traced back to
the original donor.
Chapter 12 Inventory planning and control
341
Operations in practice The UK’s National Blood Service
1
Source: Alamy/Van Hilversum
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What is inventory?
Inventory, or ‘stock’ as it is more commonly called in some countries, is defined here as
the stored accumulation of material resources in a transformation system. Sometimes the term
‘inventory’ is also used to describe any capital-transforming resource, such as rooms in a
hotel, or cars in a vehicle-hire firm, but we will not use that definition here. Usually the term
refers only to transformed resources. So a manufacturing company will hold stocks of materials,
a tax office will hold stocks of information, and a theme park will hold stocks of customers.
Note that when it is customers who are being processed we normally refer to the ‘stocks’ of
them as ‘queues’. This chapter will deal particularly with inventories of materials.
Revisiting operations objectives; the roles of inventory
Most of us are accustomed to keeping inventory for use in our personal lives, but often we
don’t think about it. For example, most families have some stocks of food and drinks, so
that they don’t have to go out to the shops before every meal. Holding a variety of food
ingredients in stock in the kitchen cupboard or freezer gives us the ability to respond quickly
(with speed) in preparing a meal whenever unexpected guests arrive. It also allows us the
flexibility to choose a range of menu options without having to go to the time and trouble
of purchasing further ingredients. We may purchase some items because we have found
something of exceptional quality, but intend to save it for a special occasion. Many people
buy multiple packs to achieve lower costs for a wide range of goods. In general, our inventory
planning protects us from critical stock-outs; so this approach gives a level of dependability
of supplies.
It is, however, entirely possible to manage our inventory planning differently. For example,
some people (students?) are short of available cash and/or space, and so cannot ‘invest’ in
large inventories of goods. They may shop locally for much smaller quantities. They forfeit
the cost benefits of bulk-buying, but do not have to transport heavy or bulky supplies.
They also reduce the risk of forgetting an item in the cupboard and letting it go out of date.
Essentially, they purchase against specific known requirements (the next meal). However, they
may find that the local shop is temporarily out of stock of a particular item, forcing them,
for example, to drink coffee without their usual milk. How we control our own supplies
is therefore a matter of choice which can affect their quality (e.g. freshness), availability or
speed of response, dependability of supply, flexibility of choice, and cost. It is the same for
most organizations. Significant levels of inventory can be held for a range of sensible and
pragmatic reasons but it must also be tightly controlled for other equally good reasons.
Why is inventory necessary?
No matter what is being stored as inventory, or where it is positioned in the operation, it
will be there because there is a difference in the timing or rate of supply and demand. If
the supply of any item occurred exactly when it was demanded, the item would never be
stored. A common analogy is the water tank shown in Figure 12.2. If, over time, the rate of
supply of water to the tank differs from the rate at which it is demanded, a tank of water
(inventory) will be needed if supply is to be maintained. When the rate of supply exceeds the
rate of demand, inventory increases; when the rate of demand exceeds the rate of supply,
inventory decreases. So if an operation can match supply and demand rates, it will also
succeed in reducing its inventory levels.
Inventory
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Types of inventory
The various reasons for an imbalance between the rates of supply and demand at differ-
ent points in any operation lead to the different types of inventory. There are five of these:
buffer inventory, cycle inventory, de-coupling inventory, anticipation inventory and pipeline
inventory.
Buffer inventory
Buffer inventory is also called safety inventory. Its purpose is to compensate for the
unexpected fluctuations in supply and demand. For example, a retail operation can never
forecast demand perfectly, even when it has a good idea of the most likely demand level.
It will order goods from its suppliers such that there is always a certain amount of most
items in stock. This minimum level of inventory is there to cover against the possibility
that demand will be greater than expected during the time taken to deliver the goods. This
is buffer, or safety inventory. It can also compensate for the uncertainties in the process of
the supply of goods into the store, perhaps because of the unreliability of certain suppliers
or transport firms.
Cycle inventory
Cycle inventory occurs because one or more stages in the process cannot supply all the
items it produces simultaneously. For example, suppose a baker makes three types of bread,
each of which is equally popular with its customers. Because of the nature of the mixing and
baking process, only one kind of bread can be produced at any time. The baker would have
to produce each type of bread in batches (batch processes were described in Chapter 4)
as shown in Figure 12.3. The batches must be large enough to satisfy the demand for each
kind of bread between the times when each batch is ready for sale. So even when demand
is steady and predictable, there will always be some inventory to compensate for the inter-
mittent supply of each type of bread. Cycle inventory only results from the need to produce
Chapter 12 Inventory planning and control
343
Figure 12.2 Inventory is created to compensate for the differences in timing between supply
and demand
Buffer inventory
Safety inventory
Cycle inventory
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