OMS552: Operations Management
Flow Time Analysis
Operations Management
¬ Operations Strategy (#1-#2)
- Process Analysis (#3-#5)
– #3 - #4: Process Flow Rate (Capacity) Analysis
– #4 - #5: Process Flow Time Analysis
»
»
»
»
»
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Theoretical Flow Time, Average Flow time & Waiting Time
Critical Path & Critical Activities
Variability and Waiting (Queueing)
Strategies for managing waiting time
Cases: Manazana Insurance
Supply Chain Management 101
Total Quality Management
World Class Operations
Wrap Up
Flow Time Analysis
Ravi Anupindi
Outline
u Process
Flow Time
– Value Added / Non-Value Added
– Effect of Variability & Utilization
– Key levers for improving process flow time
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Kristen’s Cookie: Process Flow Chart
Receive
Order
Computer (0 mins)
Spoon
to Tray
Wash &
Mix
You
(6 mins.; up to 3 doz.)
You; (2 min./doz.)
Deliver &
Accept
Payment
RM
(1 min. / order)
Pack
Load &
Set Timer
Bake
RM / Oven
(1 min./doz.)
Oven
(9 mins.)
Cool
Unload
(5 mins.)
RM (0 mins)
RM
(2 min./doz.)
Flow Time Analysis
Ravi Anupindi
Operational Measure: Flow Time
Driver: Activity Times, Critical Activity
u (Theoretical)
u Average
Flow Time
Flow Time
– Flow times of 2nd, 4th, 10th order??
u Flow
Time efficiency =
u Critical
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Theoretica l Flow Time
Average Flow Time
Path & Activity
Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Gantt Chart: Kristen Cookie
: 1st order
: 2nd order
Oven
RM
You
10
20
26
30
36
Time
Flow Time Analysis
Ravi Anupindi
Most time inefficiency comes from waiting:
E.g.: Flow Times in White Collar Processes
Industry
Process
Average
Flow Time
Theoretical
Flow Time
Flow Time
Efficiency
Life Insurance
New Policy
Application
72 hrs.
7 min.
0.16%
Consumer
Packaging
New Graphic
Design
18 days
2 hrs.
0.14%
Commercial Bank
Consumer
Loan
24 hrs.
34 min.
2.36%
Hospital
Patient Billing
10 days
3 hrs.
3.75%
Automobile
Manufacture
Financial
Closing
11 days
5 hrs
5.60%
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Critical Path & Critical Activities
u Critical
Path: A path with the longest total flow
time.
5
10
A
B
20
10
D
C
u Critical
Activity: An activity on the critical path.
Ravi Anupindi
Flow Time Analysis
Levers for Reducing Flow Time
u
Decrease the work content of critical activities
– work smarter
– work faster
– do it right the first time
– change product mix
u
Move work content from critical to non-critical activities
– to non-critical path or to ``outer loop’’
u
Reduce waiting time.
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Flow Time Analysis
4
OMS552: Operations Management
Flow Time Analysis
Key Ideas: Flow Time
u Flow
Time:
– (Theoretical) Flow time, Average Flow Time, Flow Time
Efficiency
» Key Observation: Most time is non-value added (or waiting)
– Focus on the Critical Path
– Levers for improving Flow time
u Next:
Key Drivers of waiting time & Primary Levers
to manage it …
Flow Time Analysis
Ravi Anupindi
Motivation
u Kristen’s
Cookie Company ... Can you quote a delivery
time of 26 minutes to an arriving customer? (Recall that the capacity of
the process is 6 doz. cookies per hour.)
– Suppose orders for 1 doz. Cookies come in exactly 11 minutes apart.
– Suppose orders for 1 doz. Cookies come in randomly but the average
time between orders is 11 minutes.
u Visit
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to a Doctor
Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Motivation
National Cranberry on Sept 23, 1970
Histogram of Truck Weights
40
35
35
Frequency (# of trucks)
Frequency (# of trucks)
Histogram of Truck inter-delivery times
40
30
25
20
15
10
5
30
25
20
15
10
5
0
0
0
2
4
6
8
10
12
14
16
18
20
0
4
Truck interarrival time (min)
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8
12
16
20
24
28
32
36
40
Truck Weight (Kpounds)
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Flow Time Analysis
Ravi Anupindi
Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Queuing Model of a Simple Process
(a call center)
Tp
Ti
Arrivals
K...
Throughput (R)
321
Queue
Service
The Firm
Flow Time: T
Flow Time Analysis
Ravi Anupindi
Tp
Arrivals
(Ri)
Attributes of a Queuing System
u
Service
… 21
K
Queue
Thru’put
[# of
servers (c)]
Arrivals (or demand for service)
» Specified as either a rate (Ri ) or time between subsequent arrivals (flow
units)
» Two parameters: mean & standard deviation
u
Service
» Specified as either a rate or time (Tp ) to service one arrival (flow unit)
» Two parameters: mean & standard deviation
u
u
u
u
Number of servers (c)
Maximum Queue Size (K)
Number of queues
Queue discipline and priorities
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Performance Characteristics
u Throughput
u Waiting
(R)
(Ri)
u Resource
K… 2 1
Queue
Service
[# of
servers (c)]
Thru’put
(R)
T
Time (Ti)
u Inventory/WIP
Tp
Ti
Arrivals
(Ii = R.Ti)
(Server) utilization (ρ)

Throughput
R
=
ρ =

R
Resource
Capacity
p 


» where resource capacity Rp = c/Tp, c is the # of servers,
and Tp is the average service time.
Ravi Anupindi
Flow Time Analysis
Example: MBPF Call Center
u
MBPF Inc.’s call center has one customer service
representative (CSR) taking calls. When the CSR is busy, the
caller is put on hold. Calls are taken in the order received.
Assume that calls arrive exponentially at the rate of one every
3 minutes. The CSR takes on average 2.5 minutes to complete
the reservation. The time for service is also assumed to be
exponentially distributed.
u
What is the average waiting time for a customer?
u
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
MBPF Call Center analysis …
Fallacy of Averages
u Identify
parameters
– (Avg.) Arrival Rate (Ri) = 1 call every 3 minutes = 0.33
calls per min.
– (Avg.) Service time (Tp) = 2.5 minutes per call
– # of servers (c) = 1
u Compute
processing rate
– Avg. Processing rate (Rp) = 1 every 2.5 mins. = 0.4 calls
per min.
u Observe:
– Average Arrival Rate < Average Processing Rate
Flow Time Analysis
Ravi Anupindi
Key Assumptions & Tools for
Performance Evaluation
u
Assumptions:
A. Time between arrivals and service time is exponentially distributed
B. There is ONE queue and customers are served first-in-first-out
u
Tools for Performance Evaluation
»
»
Spreadsheets can be used (under Assumptions A & B); see
performance.xls on CTools
Formula for Approximate Queue Length is available (under
Assumption B)*
*See an textbook, page 215, formula (8.9).
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Spreadsheet (performance.xls) Approach …
u
Recall: spreadsheet assumes that inter-arrival and service times follow
exponential distribution; so mean and S.D. are equal.
u Specify
–
–
–
–
input FOUR parameters:
Average Arrival rate Ri = 0.33 calls/min
Average Service Time Tp = 2.5 minutes / call
# of servers c = 1
Max. queue size K; if this is not a constraint pick some
large # for K, say 200.
u The
spreadsheet computes the rest for you.
Flow Time Analysis
Ravi Anupindi
MBPF Call Center analysis …
Ri
0.333
0.35
0.375
0.39
Tp c
2.5
2.5
2.5
2.5
1
1
1
1
ρ
Rp
0.4
0.4
0.4
0.4
0.83
0.88
0.94
0.98
Ti
11.79
17.50
37.13
76.28
90
80
Waiting Time
70
60
50
40
30
20
10
0
0.8325
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Flow Time Analysis
0.875
0.9375
0.975
Utilization
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OMS552: Operations Management
Flow Time Analysis
Effect of Utilization on Waiting Time
(when variability exists)
u Safety
Capacity: Capacity
carried in excess of expected
demand to cover for system
variability (arrivals or
service)
» Safety capacity provides a
buffer against system variability
(arrivals or service) and reduces
waiting time in queue
Avg. Flow Time
A
B
C
Theoretical Flow Time
100%
Utilization
Flow Time Analysis
Ravi Anupindi
Application to multi-stage systems
u The
analysis so far considered a single stage process
u Same analysis can be applied to multi-stage systems
(e.g., Manzana Insurance) by analyzing each stage
independently to estimate delays:
– For each stage, first estimate
» Arrival Rates
» Service Times
» # of servers
– Evaluate Performance
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Analysis using Queue Length Formula
Flow Time Analysis
Ravi Anupindi
MBPF Call Center analysis …using Queue
Length Formula
u Identify
parameters
– (Avg.) Service time (Tp) = 2.5 minutes per call
– (Avg.) Arrival Rate (Ri) = 1 call every 3 minutes = 0.33 calls per min.
– # of servers (c) = 1
u Compute
processing rate
– (Avg.) Processing rate (Rp) = 1 every 2.5 mins. = 0.4 calls per min.
u
Compute Throughput
– Throughput (R) = Arrival Rate (Ri)
u Compute
Utilization
– Utilization (ρ) = R/ Rp = 0.33/0.4 = 0.833 = 83.3%
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Effect of Variability on
Inventory &Waiting Time
Ø Queue length formula (with unlimited max. queue size): The
average queue length is (page 215 of MBPF text):
Ii =
2
2
r 2(c +1) (C i + C p )
´
1- r
2
variability
utilization
x effect
effect
(see next slide)
Little’s Law: Ti = Ii / R
Ravi Anupindi
Flow Time Analysis
Variability Effect Explained
2
u Ci
is the squared coefficient of variation of time
between arrivals; that is,
2
 S D o f tim e b e t w e e n arrivals 
C i2 = 

 M e a n tim e b e t w e e n arrivals 
– Specifically, when time between arrivals are exponential C i2 is equal
to 1.
u C p2
is the squared coefficient of service time; that is,
2
 S D o f s e r v i c e tim e 
C p2 = 

 M e a n s e r v i c e tim e 
– Specifically, when service time is exponential, C p2 is equal to 1.
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
MBPF Call Center analysis … using Queue
length formula
Recall: For exponential distributions, since S.D. = Mean.
This means variability effect = 1.
Ii =
2(1+1)
0.833
(0.833)2
´1 =
= 4.16
1 - 0.833
0.167
variability
utilization
x effect
effect
Recall:
•Tp = 2.5 mins.
• ρ = 0.833
Then,
• Ii = 4.16, and by Little’s law
•Ti = Ii/R = 4.16/0.33
= 12.6 mins.
(see next slide)
* with unlimited inventory buffer & exponential arrival time and service time distributions
Flow Time Analysis
Ravi Anupindi
Effect of Variability on
Inventory &Waiting Time
Ø Queue length formula (with unlimited buffer): The
average queue length is (page 166 of MBPF text):
2
2
r 2(c +1) (C i + C p )
Ii =
´
1- r
2
variability
utilization
x effect
effect
Avg. Flow Time
Variability
Theoretical Flow Time
(see next slide)
100%
Utilization
Little’s Law: Ti = Ii / R
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Other Examples …
Ravi Anupindi
Flow Time Analysis
MBPF Call Center with one server &
unlimited buffer
u
u
u
Consider MBPF Inc. that has a customer service representative
(CSR) taking calls. When the CSR is busy, the caller is put on
hold. The calls are taken in the order received.
Assume that calls arrive exponentially at the rate of one every
3 minutes. The CSR takes on average 2.5 minutes to complete
the reservation. The time for service is also assumed to be
exponentially distributed.
The CSR is paid $20 per hour. It has been estimated that each
minute that a customer spends in queue costs MBPF $2 due to
customer dissatisfaction and loss of future business.
– MBPF’s waiting cost =
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
MBPF Call Center Resource Pooling
u
2 phone numbers
50%
– MBPF hires a second CSR
who is assigned a new
telephone number. Customers
are now free to call either of
the two numbers. Once they
are put on hold customers tend
to stay on line since the other
may be worse..
u
Queue Server
50%
Queue Server
1 phone number: pooling
– both CSRs share the same
telephone number and the
customers on hold are in a
single queue
Queue
Servers
Flow Time Analysis
Ravi Anupindi
Process Structure & Resource
Capabilities
u Specialization
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vs. Flexibility
Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Psychology of Waiting Lines
The Principles of Waiting
ü
ü
ü
ü
ü
ü
ü
ü
by
David Maister
Unoccupied time feels longer than occupied time
Preprocess waits feel longer than in-process waits
Anxiety makes wait seem longer
Uncertain waits are longer than known, finite waits
Unexplained waits are longer than explained waits
Unfair waits are longer than equitable waits
The more valuable the service, the longer the
customer will wait
Solo waits feel longer than group waits
Flow Time Analysis
Ravi Anupindi
Motivation:Telemarketing at L.L.Bean
u $580m
in annual sales (1988).
u About 65% of sales through two telemarketing centers
in Maine.
T
u During some half hours, 80%
T
Thru’put
Service
Arrivals
of calls dialed received a busy
(R )
(R)
Queue
signal.
T
u Customers getting through had
Balking (R )
Abandoning (R )
to wait on average 10 minutes
for an available agent.
p
i
… 21
i
b
[# of
servers (c)]
a
In 1988, L.L. Bean conservatively estimated that it lost
$10 million of profit
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Example 3: MBPF Call Center
limited buffer size
u
u
In reality only a limited number of people can be put on hold
(this depends on the phone system in place) after which a
caller receives busy signal. Assume that at most 5 people can
be put on hold. Any caller receiving a busy signal simply calls
a competitor resulting in a loss of $100 in revenue.
– # of servers c =
– buffer size K =
What is the hourly loss because of callers not being able to get
through?
Flow Time Analysis
Ravi Anupindi
Key Ideas on Flow Time
u Components
of Flow time:
– (Theoretical) Flow time, Waiting Time, and Average Flow
Time
u Focus
on the Critical Path
u Metric: Flow Time Efficiency
u Most inefficiency is in waiting.
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Flow Time Analysis
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OMS552: Operations Management
Flow Time Analysis
Key Ideas: Waiting Time
u Queues
build up due to variability increasing flow
time
u Reducing variability improves performance
– Supply & Demand Variability
u If
service cannot be provided from stock, safety
capacity must be provided to cover for variability
– If response time is important, do not plan for 100% utilization.
u Tradeoff
is between cost of waiting, (lost sales,) and
cost of capacity
u Pooling servers improves performance
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Flow Time Analysis
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