Transcript Why is Inventory Important?

Chapter 5: Service Processes
Service Businesses
A service business is the management of
organizations whose primary business
requires interaction with the customer to
produce the service
 Generally classified according to who the
customer is:
 Financial services
 Health care
 A contrast to manufacturing
Service-System Design Matrix
Degree of customer/server contact
High
Buffered
Permeable
core (none) system (some)
Reactive
system (much)
Low
Face-to-face
total
customization
Face-to-face
loose specs
Sales
Opportunity
Face-to-face
tight specs
Internet &
on-site
technology
Mail contact
Low
Production
Efficiency
Phone
Contact
High
Characteristics of Workers, Operations,
and Innovations Relative to the Degree of
Customer/Service Contact
Queuing Theory
Waiting occurs in
Service facility
 Fast-food restaurants
 post office
 grocery store
 bank
Manufacturing
Equipment awaiting repair
Phone or computer network
Product orders
Why is there waiting?
Customer Service Population Sources
Population Source
Finite
Example: Number
of machines
needing repair
when a company
only has three
machines.
Infinite
Example: The
number of people
who could wait in
a line for
gasoline.
Service Pattern
Service
Pattern
Constant
Example: Items
coming down an
automated
assembly line.
Variable
Example: People
spending time
shopping.
The Queuing System
Length
Queue Discipline
Queuing
System
Service Time
Distribution
Number of Lines &
Line Structures
Examples of Line Structures
Single
Phase
One-person
Single Channel
barber shop
Multichannel
Bank tellers’
windows
Multiphase
Car wash
Hospital
admissions
Measures of System Performance
 Average number of customers
waiting
In the queue
In the system
 Average time customers wait
In the queue
In the system
 System utilization
Number of Servers
Single Server
Multiple
Servers
Multiple Single
Servers
Some Assumptions
Relative Frequency
 Arrival Pattern: Poisson
.18
.16
.14
.12
.10
.08
.06
.04
.02
0
1
2
3
4
5
6
7
8
9
Cu st om e r s p e r t i me u ni t
 Service pattern:
exponential
Relative Frequency (%)
Service Time
 Queue Discipline: FIFO
10 11 12
13
Some Models
1. Single server, exponential service time (M/M/1)
2. Multiple servers, exponential service time (M/M/s)
A Taxonomy
Arrival
Distribution
where
M/M/s
Service
Distribution
Number of
Servers
M = exponential distribution (“Markovian”)
Given
l
m
s
=
=
=
customer arrival rate
service rate (1/m = average service time)
number of servers
=
=
=
=
=
=
average number of customers in the queue
average number of customers in the system
average waiting time in the queue
average waiting time (including service)
probability of having n customers in the system
system utilization
Calculate
Lq
L
Wq
W
Pn
r
Note regarding Little’s Law: L = l * W and Lq = l * Wq
Model 1: M/M/1 Example
The reference desk at a library receives request for assistance at an
average rate of 10 per hour (Poisson distribution). There is only one
librarian at the reference desk, and he can serve customers in an average
of 5 minutes (exponential distribution). What are the measures of
performance for this system? How much would the waiting time decrease
if another server were added?
M/M/s Queueing Model Template
l 
m 
s =
0
Data
10
12
1
Prob(W > t) =
when t =
0.135335
1
Prob(Wq > t) =
0.112779
1
when t =
(mean arrival rate)
(mean service rate)
(# servers)
Results
L =
5 Number of customers in the system
Lq =
4.166666667 Number of customers in the queue
W =
Wq =
r 
P0 =
0.5 Waiting time in the system
0.416666667 Waiting time in the queue
0.833333333 Utilization
0.166666667 Prob zero customers in the system
Application of Queuing Theory
We can use the results from
queuing theory to make the
following types of decisions: Cost
How many servers to employ
Whether to use one fast
server or a number of
slower servers
Whether to have general
purpose or faster specific
servers
Goal:
Total Cost
Cost of
Service Capacity
Cost of customers
waiting
Optimum
Service Capacity
Minimize total cost = cost of servers + cost of waiting
Example #1: How Many Servers?
In the service department of an auto repair shop, mechanics
requiring parts for auto repair present their request forms at the
parts department counter. A parts clerk fills a request while the
mechanics wait. Mechanics arrive at an average rate of 40 per hour
(Poisson). A clerk can fill requests in 3 minutes (exponential). If the
parts clerks are paid $6 per hour and the mechanics are paid $18 per
hour, what is the optimal number of clerks to staff the counter.
l
m
s=
Data
40
20
3
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 2.0383E-08
when t =
1
Results
L = 2.888888889
Lq = 0.888888889
Service Cost = s * Cs
Waiting Cost = l * W * Cw
W = 0.072222222
W q = 0.022222222
Prob(W q > t) = 1.8321E-09
0
when t =
1
r  0.666666667
P0 = 0.111111111
3 $
4 $
5 $
18.00
24.00
30.00
$
$
$
52.00
39.13
36.72
$
$
$
70.00
63.13
66.72
S = 4 IS THE SMALLEST
Example #2: How Many Servers?
Beefy Burgers is trying to decide how many registers to
have open during their busiest time, the lunch hour.
Customers arrive during the lunch hour at a rate of 98
customers per hour (Poisson distribution). Each service
takes an average of 3 minutes (exponential
distribution). Management would not like the average
customer to wait longer than five minutes in the
system. How many registers should they open?
 Need at least 5 (why?) Increment from there
For six servers
M/M/s Queueing Model Template
l
m
s=
Data
98
20
6
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 1.19E-08
when t =
1
0
Prob(Wq > t) = 2.77E-10
when t =
1
Results
L=
7.359291808 Number of customers in the system
Lq =
2.459291808 Number of customers in the queue
W=
Wq =
r
P0 =
0.075094814 Waiting time in the system
0.025094814 Waiting time in the queue
0.816666667 Utilization
0.00526507 Prob zero customers in the system
Choose s = 6 since W = 0.0751 hour is less than 5 minutes.
Example #3: One Fast Server or Many
Slow Servers?
Beefy Burgers is considering changing the way that they serve
customers. For most of the day (all but their lunch hour), they have
three registers open. Customers arrive at an average rate of 50
per hour. Each cashier takes the customer’s order, collects the
money, and then gets the burgers and pours the drinks. This takes
an average of 3 minutes per customer (exponential distribution).
They are considering having just one cash register. While one
person takes the order and collects the money, another will pour
the drinks and another will get the burgers. The three together
think they can serve a customer in an average of 1 minute. Should
they switch to one register?
3 Slow Servers
l
m
s=
Data
50
20
3
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 6.38E-05
when t =
1
W = 0.120224719 Waiting time in the system
Wq = 0.070224719 Waiting time in the queue
Prob(Wq > t) = 4.34E-05
0
when t =
1
r  0.833333333 Utilization
1 Fast Server
l
m
s=
Data
50
60
1
Prob(W > t) = 4.54E-05
when t =
1
Prob(Wq > t) = 3.78E-05
0
when t =
1
Results
L = 6.011235955 Number of customers in the system
Lq = 3.511235955 Number of customers in the queue
(mean arrival rate)
(mean service rate)
(# servers)
P0 =
0.04494382 Prob zero customers in the system
Results
L=
5 Number of customers in the system
Lq = 4.166666667 Number of customers in the queue
W=
0.1 Waiting time in the system
Wq = 0.083333333 Waiting time in the queue
r  0.833333333 Utilization
P0 = 0.166666667 Prob zero customers in the system
W is less for one fast server, so choose this option.
Example 4: Southern Railroad
The Southern Railroad Company has been subcontracting for painting of its
railroad cars as needed. Management has decided the company might
save money by doing the work itself. They are considering two
alternatives. Alternative 1 is to provide two paint shops, where painting
is to be done by hand (one car at a time in each shop) for a total hourly
cost of $70. The painting time for a car would be 6 hours on average
(assume an exponential painting distribution) to paint one car.
Alternative 2 is to provide one spray shop at a cost of $175 per hour.
Cars would be painted one at a time and it would take three hours on
average (assume an exponential painting distribution) to paint one car.
For each alternative, cars arrive randomly at a rate of one every 5
hours. The cost of idle time per car is $150 per hour.
 Estimate the average waiting time in the system saved by alternative 2.
 What is the expected total cost per hour for each alternative? Which
is the least expensive?
Answer: Alt 2 saves 1.87 hours. Cost of Alt 1 is: $421.25 / hour and cost of Alt 2 is
$400.00 /hour.
Example 5
A large furniture company has a warehouse
that serves multiple stores. In the
warehouse, a single crew with four
members is used to load/unload trucks.
Management currently is downsizing to
cut costs and wants to make a decision
about crew size.
Trucks arrive at the loading dock at a mean
rate of one per hour. The time required
by the crew to unload/and-or load a truck
has an exponential distribution
(regardless of crew size). The mean of
the distribution for a four member crew
is 15 minutes – i.e., 4 trucks per hour. If
the crew size is changed, the service
rate is proportional to its size – i.e., a
three member crew could do 3 per hour,
etc.
The cost of providing each member of the
crew is $20 per hour and the cost for a
truck waiting is $30 per hour. The
company has a service goal such that the
likelihood of a truck spending more than
one hour being served is 5% or less.
a) For the current configuration, what is
the average waiting time in the system?
What is the average number of trucks
waiting to be unloaded (not including the
truck currently being unloaded? What is
the probability that a truck waits more
than one hour to be unloaded? What is
the total cost of the four person crew?
b) Suppose the company is looking at
alternatives. One is a three member
crew. What is the cost of this crew?
Compare the statistics mentioned in part
a) with comparable statistics for the
three member crew. Would you select
the three member crew over the crew in
part a)? Why or why not?
c) One person suggested that rather than
have one four member crew, the firm
should use two, two member crews, where
each crew could load/unload two trucks
per hour. What is the cost of this
solution? What is the probability that a
truck waits longer than one hour for
loading/unloading? Would you recommend
that they implement this solution? Why
or why not?
Example 5 (Answer)
part a)
W:
Lq:
Pr(w>1 hour) =
Total Cost = $
0.333
hours
0.083
trucks
0.05
90.00
20 min
(L = .33)
part b)
W:
Lq:
Pr(w>1 hour) =
Total Cost =
0.5
0.167
0.14
$75.00
30 min
(L = .5)
hours
trucks
The cost is less even though the service is worse. Based on costs, select
the three person crew; o/w go with the 4 person crew
part c)
Assume that there is one waiting line for the two, two member crews
Total Cost =
$96.00
per unit so $100 total
Pr(w>1 hour) =
0.220
No; the cost is greater as is the probability that a truck waits longer
is over 20%
If assuming each crew has its own waiting line:
Cost for each: $
Total cost for 2: $
Pr(w>1 hour) =
50.00
100.00
0.220
l
m
s=
Cs = $
1
4
1
80.00
Cw = $
30.00
l
m
s=
Cs = $
1
3
1
Cw = $
30.00
l
m
s=
60.00
1
2
2
Cs = $
40.00
Cw = $
30.00