Quantitative Review III

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Quantitative Review III
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Chapter 6 6S

• Managing Quality
• Statistical Process Control (SPC)

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How do we ensure quality of manufacturing and
service delivery processes? Using Statistical
Process Control (SPC) to help identify and
eliminate unwanted causes of variation in the
process To monitor the degree of product
conformance to a specification on continuous
scale of measurement such as length, weight, and
time (continuous metric), we use X-bar and
R-charts. To monitor the noncontinous scale of
measurement (discrete metric)such as from data
that are counted (e.g. no. of compete orders,
no. of good service experiences), we use p- or
c-charts
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Control Charts for Variable Data(length, width,
etc.)

• Mean (x-bar) charts
• Tracks the central tendency (the average or mean
  value observed) over time
• Range (R) charts
• Tracks the spread of the distribution over time
  (estimates the observed variation)

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Control Charts for Variable Data(length, width,
etc.)
K Number of samples n Sample size
The mean for each sample (sample average)
R the range for each sample
The overall mean of all the sample means
The average range of all the sample ranges
Upper Control Limit Lower Control Limit
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standard deviation of the sample means
standard deviation of the process
n of observations in each sample
k of samples
standard normal variable or the of std.
deviations desired to use to develop the control
limits
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Example X Bar-Chart
Assume the standard deviation of the process is
given as 1.13 ouncesManagement wants a 3-sigma
chart (only 0.26 chance of alpha error)Observed
values shown in the table are in ounces.
Calculate the UCL and LCL.
Sample 1 Sample 2 Sample 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample means 15.875 15.975 15.9
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We have 3 samples here
We have 4 observations in each sample
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Example X Bar-Chart
Inspectors want to develop process control charts
to measure the weight of crates of wood. Data
(in pounds) from three samples are Sample
Crate 1 Crate 2 Crate 3 Crate 4 1
18 38 22 26 2
26 30 28 20 3
26 26 26 26
What are the upper and lower control limits
for this process?
Z3
4.27
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We first need to calculate x-bar for each
sample

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26

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We have 3 samples
We have 4 observations in each sample
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Range or R Chart
k of sample ranges
Range Chart measures the dispersion or variance
of the process while The X-Bar chart measures
the central tendency of the process. When
selecting D4 and D3 from the table, use sample
size or number of observations for n.
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Control Chart Factors
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Example R-Chart
Sample 1 Sample 2 Sample 3
Observation 1 15.8 16.1 16.0
Observation 2 16.0 16.0 15.9
Observation 3 15.8 15.8 15.9
Observation 4 15.9 15.9 15.8
Sample means 15.875 15.975 15.9
Sample ranges 0.2 0.3 0.2
Sample ranges highest observation lowest
observation
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R-chart Computations(Use D3 D4 Factors Table
6-1)
D4 from control chart factors when n4
D3 from control chart factors when n4
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Example R-Chart
Ten samples of 5 observations each have been
taken form a Soft drink bottling plant in order
to test for volume dispersion in the bottling
process. The average sample range was found To be
.5 ounces. Develop control limits for the sample
range.
0.5, n 5
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P- Chart
Many quality characteristics assumes only two
values, such as good or bad, pass or fail. The
proportion of nonconforming items can be
monitored using a control chart called a
p-chart, where p is the proportion of
nonconforming items found in a sample.
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(P) Fraction Defective Chart

• Used for yes or no type judgments (acceptable/not
  acceptable, works/doesnt work, on time/late,
  etc.)
• p proportion of nonconforming items

average proportion of nonconforming items
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(P) Fraction Defective Chart
K of samples
n of observations in each sample
standard normal variable or the of std.
deviations desired to use to develop the control
limits
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P-Chart Example A Production manager for a tire
company has inspected the number of defective
tires in five random samples with 20 tires in
each sample. The table below shows the number of
defective tires in each sample of 20 tires. Z
3. Calculate the control limits.
Sample Number of Defective Tires Number of Tires in each Sample Proportion Defective
1 3 20 .15
2 2 20 .10
3 1 20 .05
4 2 20 .10
5 1 20 .05
Total 9 100 .09
Solution
Note Lower control limit cant be negative, if
LCLp is less than zero, a value of zero is used.
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C-Chart
A p-chart monitors the proportion of
nonconforming items, but a nonconforming item
may have more than one conformance. E.g. a
customers order may have several errors, such
as wrong item, wrong quantity, wrong price. To
monitor the number of nonconformance per unit, we
use a c-chart. It is used extensively in service
applications.
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Number-of-Defectives or C-Chart
Used when looking at of nonconformances c
of nonconformances
average of nonconformances per sample
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Number-of-Defectives or C Chart
standard normal variable or the of std.
deviations desired to use to develop the control
limits
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C-Chart Example The number of weekly customer
complaints are monitored in a large hotel using a
c-chart. Develop three sigma control limits
using the data table below. Z3.
Week Number of Complaints
1 3
2 2
3 3
4 1
5 3
6 3
7 2
8 1
9 3
10 1
Total 22

• Solution

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Process Capability The natural variation in a
process that results from common
causes Process Capability Index A measure
that quantify the relationship between the
natural variation and specifications
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Process Capability

• Product Specifications
• Preset product or service dimensions, tolerances
• e.g. bottle fill might be 16 oz. .2 oz.
  (15.8oz.-16.2oz.)
• Based on how product is to be used or what the
  customer expects
• Process Capability Cp and Cpk
• Assessing capability involves evaluating process
  variability relative to preset product or service
  specifications
• Cp assumes that the process is centered in the
  specification range
• Cpk helps to address a possible lack of centering
  of the process

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Process Capability
USL
LSL
Spec Width
-3d -2d -1d µ 1d 2d 3d
If Cp is 1 process capable of meeting
design specs
If Cpk is 1 process capable of meeting
design specs
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Process Capability Index

• Can a process or system meet its requirements?

Cp lt 1 process not capable of meeting design
specs Cp 1 process capable of meeting design
specs

• One shortcoming, Cp assumes that the process is
  centered on the specification range

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min minimum of the two
mu or mean of the process
If
is less than 1, then the process is not capable
or
is not centered.
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Capability Indexes

• Centered Process (Cp)
• Any Process (Cpk)

• CpCpk when process is centered

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Example

• Design specifications call for a target value of
  16.0 /-0.2 ounces (USL 16.2 LSL 15.8)
• Observed process output has a mean of 15.9 and a
  standard deviation of 0.1 ounces

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Computations

• Cp
• Cp lt 1 process not capable of meeting design
  specs
• Cpk

C pk lt1, then this measure also indicates that
process is not capable
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Chapter 3

• Project Management

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Project Planning, Scheduling and
Controlling Calculate Path Durations

• The longest path (ABDEGIJK) limits the projects
  duration (project cannot finish in less time than
  its longest path)
• ABDEGIJK is the projects critical path

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ES16 EF30 LS16 LF30
ES32 EF34 LS33 LF35
ES10 EF16 LS10 LF16
ES4 EF10 LS4 LF10
Project Network
E Buffer
ES39 EF41 LS39 LF41
ES35 EF39 LS35 LF39
D(6)
E(14)
H(2)
B(6)
A(4)
K(2)
G(2)
J(4)
ES0 EF4 LS0 LF4
C(3)
ES30 EF32 LS30 LF32
F(5)
I(3)
ES4 EF7 LS22 LF25
ES7 EF12 LS25 LF30
ES32 EF35 LS32 LF35
Critical Path the sequence of activities that
takes the longest time defines the total
project completion time
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Calculate Early Starts Finishes
ES32 EF34
ES16 EF30
ES10 EF16
ES4 EF10
D(6)
E(14)
H(2)
B(6)
Latest EF Next ES
ES35 EF39
ES39 EF41
A(4)
K(2)
G(2)
J(4)
ES0 EF4 LS0 LF4
ES30 EF32 LS30 LF32
C(3)
F(5)
I(3)
ES4 EF7F25
ES7 EF12 LS25 LF30
ES32 EF35 LS32 LF35
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Calculate Late Starts Finishes
ES32 EF34 LS33 LF35
ES10 EF16 LS10 LF16
ES16 EF30 LS16 LF30
ES4 EF10 LS4 LF10
ES39 EF41 LS39 LF41
ES35 EF39 LS35 LF39
D(6)
E(14)
H(2)
B(6)
A(4)
K(2)
G(2)
J(4)
ES0 EF4 LS0 LF4
ES30 EF32 LS30 LF32
C(3)
F(5)
I(3)
ES4 EF7 LS22 LF25
ES7 EF12 LS25 LF30
ES32 EF35 LS32 LF35
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What is the critical path?
E(5)
B(3)
G(4)
A(4)
H(3)
C(5)
D(2)
F(6)

1. ABEGH
2. ACEGH
3. ACFGH
4. ADFGH

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What is ES, EF for G?
E(5)
B(3)
G(4)
A(4)
H(3)
C(5)
D(2)
F(6)

1. 14, 18
2. 15, 19
3. 16, 20
4. 17, 21

Latest EF Next ES
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What is LS, LF for C?
E(5)
B(3)
G(4)
A(4)
H(3)
C(5)
D(2)
F(6)

1. 4, 9
2. 5, 10
3. 3, 8
4. 6, 11

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ES 9 LS 10 EF 14 LF 15
ES 4 LS 7 EF 7 LF 10
ES 19 LS 19 EF 22 LF 22
ES 15 LS 15 EF 19 LF 19
ES 4 LS 4 EF 9 LF 9
ES 0 EF 4
ES 4 LS 7 EF 6 LF 9
ES 9 LS 9 EF 15 LF 15
Latest EF Next ES
Earliest LS Previous LF
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Activity Slack Time
TES earliest start time for activity TLS
latest start time for activity TEF earliest
finish time for activity TLF latest finish time
for activity Activity Slack TLS - TES TLF
- TEF
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Calculate Activity Slack
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Path Slack
Duration of Critical Path (41) -
Path Duration Path Slack
Path Slack
1
0
19
18
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Consider the following project information.    Consider the following project information.    Consider the following project information.    Consider the following project information.   
Activity Activity Time (weeks) Immediate Predecessor(s)
A 3 none
B 5 A
C 2 B
D 5 B
E 12 C, D
F 3 E
G 6 E
H 5 G, F

What is the critical path? It is ABDEGH What is the critical path? It is ABDEGH What is the critical path? It is ABDEGH What is the critical path? It is ABDEGH

A (3) B (5) C (2) E (12) F (3) H(5) D (5) G (6) A (3) B (5) C (2) E (12) F (3) H(5) D (5) G (6) A (3) B (5) C (2) E (12) F (3) H(5) D (5) G (6) A (3) B (5) C (2) E (12) F (3) H(5) D (5) G (6)
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A (3) B (5) C (2)

E (12) F (3)
H(5)
D (5)
G (6)
What is ES and EF for E? ES for E 35513 EF
for E 1312 25 What is LS and LF for C? LF
for C ES for E 13 LS for C 13-211
What is the activity slack for C? ES for C
35 8 Activity slack for C LS-ES11-83 EF
for C 8210 If as a project manager you see
that Activity F in the project above is going to
take 3 extra weeks, what should you do? Watchful
waiting
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Waiting Line Models
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• How long you wait in line depends on a number
• of Factors
• the number of people served before you
• the number of servers working
• the amount of time it takes to serve each
  individual customer
• Wait time is affected by the design of the
  waiting
• line system. A waiting line system (or queuing
• system) is defined by two elements
• The customer population (people or objects to be
  processed)
• The process or service system

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Waiting Line System Performance Measures

• The average number of customers waiting in queue
• The average number of customers in the system
• The average waiting time in queue
• The average time in the system
• The system utilization rate ( of time servers
  are busy)

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Infinite Population, Single-Server, Single Line,
Single Phase Formulae
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Infinite Population, Single-Server, Single Line,
Single Phase Formulae
Do not confuse L from LQ or W from WQ
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Example

• A help desk in the computer lab serves students
  on a first-come, first served basis. On average,
  15 students need help every hour. The help desk
  can serve an average of 20 students per hour.
  Assess the performance measures in this case.
• Based on this description, we know
• µ 20 (exponential distribution)
• ? 15 (Poisson distribution)

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- Average System Utilization
- Average Number of Students in the System
- Average Number of Students Waiting in Line
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- Average Time a Student Spends in the System
.2 hours or 12 minutes
- Average Time a Student Spends Waiting (Before
Service)
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- Probability of n Students in the Line
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Example
Consider a single-server queuing system. If the
arrival rate is 40 units per hour, and each
customer takes 60 seconds on average to be
served, what is the average of customers in the
line? Answer to 2 decimal places. ? 40 units/
hr µ (60 60)/60 60 units / hr Average
number of customers in the line LQ ?L (? /µ)
(? /µ-?) (40/60)(40/20) 1.33
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Example
Consider a single-line, single-server waiting
line system. Suppose that customers arrive
according to a Poisson distribution at an
average rate of 60 per hour, and the average
(exponentially distributed) service time is 36
seconds per customer. What is the average
number of customers in the system? Should they
add servers?
? 60 customers/hour Service rate is 36
seconds for each customer, so µ 6060 / 36
100 customers / hour The average number of
customers in the system L ? / (µ-?) 60 /
(100-60) 1.5, no need to add servers
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Example
Consider a single-server queuing system. If
the arrival rate is 25 customers per hour and it
takes 3 minutes on average to serve a customer,
what is the average waiting time in the waiting
line in minutes? ? 25 customers/hr, µ 60/3
20 customers/hr WQ ?W(?/µ)(1/µ-?)
(25/20)(1/20-25) -.25 Answer the waiting time
will continuously increase
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Example
Consider a single-line, single-server waiting
line system. The arrival rate lambda is 80
people per hour, and the service rate µ is 120
people per hour. What is the probability of
having 3 units in the system?
? ?/µ 80/120 .66 Probability Pn (1- ?)?n
(1-.66) (.663) (.33)(.287) .0947
9.47