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Indian Institute of Health Management Research JAIPUR
December 2010
TABLE OF CONTENTS 1. Process, Defects, and Variation 2. Six Sigma Overview 3. Cross-Functional Teams 4. DMAIC Breakthrough Strategy 5. Define Phase
Develop Problem Statement
Map High Level Process Map (SIPOC)
Determine Critical to Quality Characteristics (CTQ’s)
Develop Project Charter
6. Measure Phase
Develop Detailed “AS IS” Process Map
Determine What to Measure (Process Y)
Validate Measurement System
Quantify Current Performance
7. Analyze Phase
Cause and Effect Diagram
Why? Why? Why? Analysis
Testing and Validation of Theories
Validating the Root Causes
Finalizing the Charter
8. Improve Phase
Determine Solutions to Counteract the Root Causes
Provide Statistical Evidence that Solutions Work
Prepare the “Should Be” Process Map
9. Control Phase
Prepare and Implement the Control Plan
Control Charts
Improvement Dashboards
Final Project Report
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Lean Six Sigma Overview
SIX SIGMA TRAINING
SIX SIGMA GREEN BELT 1
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Quality – The Definition Quality is a state in which value entitlement is realized by the customer and provider in every aspect of the business relationship.
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What is a Process?? ¾ Any sequence of activities that use a set of INPUTS to produce an OUTPUT is called a PROCESS ¾ A Process is a means for doing work ¾ Every Process has a CUSTOMER. A Customer is the immediate recipient of the Output from the Process
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Lean Six Sigma Overview
Components of a Process?? Supplier: The provider of inputs to your process Input:
Materials, resources or data required to execute your process
Process:
A collection of activities that takes one or more kinds of input and creates output that is of value to the customer
Output:
The products or services that result from the process
Customer: The recipient of the process output – may be internal or external 4
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Graphical Representation of a Process Process
Outputs
Input Variables
Process Variables
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What Can Go Wrong in a Process?? ¾ A Process may not produce the desired output leading to CUSTOMER DISSATISFACTION. ¾ The output from a process may have defects or errors in it and this leads to REWORK or REJECTION. This leads to the generation of WASTE WASTE. ¾ The produced output may be unpredictable in its ability to meet customer requirements and this is caused due to high VARIATION in a Process. ¾ The process may be unstable and this leads to generation of WASTE in the process itself © 2009, Qimpro
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Lean Six Sigma Overview
What are the Implications of Variation and Waste The key deficiencies of any Process include: ¾ VARIATION ¾ WASTE
Both these deficiencies have the following implications: ¾ CUSTOMER DISSATISFACTION ¾ INCREASE IN COSTS OF DELIVERING SERVICES © 2009, Qimpro
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How to Prevent Variation and Waste ¾ Identify Chronic Problems (diseases) in the Process ¾ Ensure that adequate Measurement Systems have been defined to accurately measure the damage i.e. Rework, Rejections, Variation, etc caused by these Chronic Problems ¾ Use structured Problem Solving Methodologies such as Six Sigma to permanently eliminate or minimize the Waste and Variation ¾ Improve the Capability of the Process to meet customer requirements Consistently at Optimized Costs © 2009, Qimpro
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How to Achieve Process Improvement ¾ Process Deficiencies are solved by a Project by Project approach. ¾ Each Project needs to address a specific PAIN (deficiency) in the process ¾ Each Project j is a structured approach pp to Problem Solving involving the five steps; 9 Defining the Problem – Define Phase 9 Measuring the Problem – Measure Phase 9 Analyzing the Root Causes – Analyze Phase 9 Implementing the Improvements – Improve Phase 9 Sustaining the Gains – Control Phase © 2009, Qimpro
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How to Achieve Process Improvement ¾ Each Project needs to have a specific GOAL for improvement in terms of either eliminating or minimizing the deficiency. ¾ Each Project needs to be conducted by a CROSSFUNCTIONAL TEAM consisting of members b ffrom the h ffunctions i most affected ff d by the pain. ¾ Each Project needs to be TIMEBOUND
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How to Achieve Process Improvement ¾ Each Project must have a goal to generate savings as ELIMINATING OR MINIMIZING DEFICIENCIES will always REDUCE COSTS. This reduction in costs translates to SAVINGS TO THE BOTTOMLINE
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What is Sigma? Sigma is used in statistics to denote standard deviation. A sigma value is used to relate the ability of a process to perform defect free work. The lower the value of Standard Deviation the better the process is performing and the lower the probability that a defect will occur.
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What is Six Sigma? ¾ The higher the Sigma Level i.e. 3σ, 4σ, etc the better the process is performing and the lower the probability that a defect will occur. ¾ At Six Sigma (6σ) level of performance the probability of a defect occurring is reduced to 3.4 out of 1 million and that is considered to be virtual perfection.
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What is Sigma? Standard Deviation: Metric that displays variation from it’s “target”.
1 Std. Dev. (“Sigma”)
One standard deviation around the mean is about 68% of the total “opportunities” for meeting customer requirements! 14
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What is Six Sigma? If we can squeeze six standard deviations in between our target and the customer’s requirements...
6 5 4 3 2 1
1 2 3 4 5 6
then: 99.99966% of “opportunities” to meet customer requirements are included! © 2009, Qimpro
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Lean Six Sigma Overview
Defect Rates and Sigma Levels Sigma Level
PPM/DPMO
1
691,462
2
308,538
3
66,807
4
6,210
5
233
6
3.4
3σ - Historical Standard 10 Times Improvement
4σ - Current Standard 1800 Times Improvement
6σ – New Standard
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Six Sigma in Practical Terms 99% GOOD (4σ)
99.99966% GOOD (6σ)
20,000 LOST ARTICLES OF MAIL PER HR.
SEVEN LOST ARTICLES OF MAIL PER HR.
UNSAFE DRINKING WATER 15 MIN. PER DAY
UNSAFE DRINKING WATER FOR ONE MINUTE EVERY SEVEN MONTHS
5,000 INCORRECT SURGICAL OPERATIONS PER WEEK
1.7 INCORRECT SURGICAL OPERATIONS PER WEEK
2 SHORT OR LONG LANDINGS AT MOST MAJOR AIRPORTS EACH DAY
ONE SHORT OR LONG LANDING EVERY FIVE YEARS
200,000 WRONG DRUG PRESCRIPTIONS EACH YEAR
68 WRONG DRUG PRESCRIPTIONS EACH YEAR
NO ELECTRICITY FOR ALMOST 7 HOURS PER MONTH
NO ELECTRICITY FOR ONE HOUR EVERY 34 YEARS
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Focus of Six Sigma Reduce Variation Reduce Waste Reduce Defects Delighting Patients Reduce Cost Reduce Delivery Time
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Lean Six Sigma Overview
Cross Functional Teams Cross Functional Teams are made up of individuals who represent the different functions or departments who are impacted by the problem. They are carefully selected Subject Matter Experts (SME) The key advantage of cross functional teams is that the representation form all the impacted departments promotes acceptance and implementation of change throughout the organization © 2009, Qimpro
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Team Meeting Structure 1. Develop an agenda and distribute the agenda in advance 2. Start and finish on Time 3. Appoint a recorder to record the minutes 4 Use visual aids liberally 4. 5. Summarize key points 6. Review assignments and completion dates and set deliverables for the next meeting 7. Distribute minutes promptly 8. Review meeting effectiveness periodically © 2009, Qimpro
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Team Facilitator / Leader Must Do: 1. Extract balanced participation from all members 2. Identify members who need coaching or training for effective participation and provide the same 3. Keep the team on track with the project 4 Provide 4. P id an outside t id neutral t l perspective ti 5. Help in securing resources that the team needs 6. Resolve conflicts and focus on progress towards achieving the team goal 7. Ensure that all members complete all assigned tasks between two meetings. © 2009, Qimpro
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Team Facilitator / Leader Must NOT Do: 1. Being judgmental of team members or their ideas and opinions 2. Taking sides or becoming caught-up in the subject matter 3. Solving a problem or giving an answer 4. Making suggestions on the task instead of the process 5. Steer the team towards pre-conceived solutions
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Black Belt Facilitator: Six Sigma implementation experts with the ability to develop, coach, and lead multiple cross-functional process improvement teams Use tools to quickly and efficiently drive improvement Facilitate to keep team focused on the project objective Ensure that the Six Sigma methods are followed Help teams learn and understand Six Sigma tools and techniques through regular project reviews Responsible for the ultimate success of the project Trains and develops Green Belts Spread Six Sigma awareness throughout the organization. © 2009, Qimpro
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Green Belt Project Team Leaders: Execute Six Sigma as part of their daily jobs Form Six Sigma project teams Process experts Part time on Six Sigma Projects Trained on Six Sigma methods and quality tools.
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Lean Six Sigma Overview
Team Roles – Team Member ¾ Team Members are the Process Experts and are vital for success in the Project. ¾ The key Roles and Responsibilities of the Team Member include: Good knowledge of product, process and customer t requirements i t 9 Willing to work in teams and dedicate time to work on projects 9 High Participation and Active in Data Collection 9 Responsible for Implementing the Changes and Improvements 9
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Team Stages Forming: Forming is the beginning of team life. Members typically start out by exploring the boundaries of acceptable group behaviour.
Storming: The second stage consists of conflicts and resistance to the group’s task and structure. This is the most difficult stage for any team to work through. Team members tend to cling on to their own opinions , based on personal experience and resist seeking the opinions of others. This can lead to hurt feelings and unnecessary disputes. The role of the team facilitator / leader is crucial in this stage. © 2009, Qimpro
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Team Stages Norming: During the third stage, a sense of group cohesion develops. Team members begin to focus more on collection and analysis of data rather than experiences. Norming takes place as the team keeps meeting routinely as per a fixed schedule and the team becomes more relaxed and steady. Norming is essential. A team cannot perform if it does not norm.
Performing: This is the payoff stage. The team begins to work effectively and cohesively towards achieving the common goal. © 2009, Qimpro
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Lean Six Sigma Overview
Team Stages PERFORMING: Members: Show maturity, focus on the process, achieve goals and operate smoothly
PERFROMA ANCE
NORMING: Members: Cooperate, talk things out, focus on objectives
STORMING: Members have: confrontations, divided loyalties and individual thinking,
FORMING: Members are: Inexperienced, excited, anxious and proud
TIME 28
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DMAIC Breakthrough Strategy Recognize
9
Establish focus to ensure improvements will make a strategic difference.
Define
9
Identify the product or process to be improved and top few critical to quality (CTQ) customer requirements.
9
Quantify how the process performs today and set improvement goal goal.
9
Identify the input variables that affect the CTQ’s the most.
9
Determine solutions for controlling the key process input variables, quantify their impact and compare to goal.
Measure
Process Characterization Analyze
Process Optimization
Improve
Control
Integrate into Daily Work
9 Implement process design modifications and standardization methods for maintaining the improved performance level over time.
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DEFINE PHASE Key Objectives are….. ¾
Identify the factors that are critical to Customer Satisfaction (CTS).
¾
Define Project Boundaries. Boundaries
¾
Define the project objective and impact.
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Lean Six Sigma Overview
MEASURE PHASE Key Objectives are….. ¾
Prepare the “AS IS” process map and determine outputs.
¾
Determine what to measure
Gage name: Date of study: Reported by: Tolerance: Misc:
Gage R&R (ANOVA) for NO. OF DAYS
and validate the
Components of Variation
Percent
measurement system.
%Contribution %Study Var
10
50
5
0
0 Gage R&R
Repeat
Reprod
PROJECT
Part-to-Part
1
2
3
R Chart by ENGINEER 1
Sample Range
¾
By PROJECT
100
Quantify current
3
UCL=1.198
6
7
8
9
10
5 R=0.3667 LCL=0
0.0
0
0
ENGINEER 1
Xbar Chart by ENGINEER 1
2
3
ENGINEER*PROJECT Interaction
2
3
5
10
Average
Sample Mean
5
10
0.5
10
performance and estimate
4
By ENGINEER
2
1.0
UCL=4.773 Mean=4.083 LCL=3.394
ENGINEER 1 2 3
5
0
0 0
PROJECT
1
2
3
4
5
6
7
8
9
10
improvement target. © 2009, Qimpro
ANALYZE PHASE R e g r e s s io n P lo t
Key Objectives are….. ¾
N o. o f D ay s = 0.201 735 + 0.19 153 9 No. O f E r ro r + 0 .0 0 3 1 6 0 6 N o . O f E r r o r * * 2 S = 0.66 489 1
R-S q = 96.4 %
R - S q ( a d j) = 9 5 .3 %
10 9
Identify causes of variation and defects.
8 7 6 5 4 3 2 1 0
¾
Provide statistical evidence that causes are real.
0
10
20
30
Fishbone Diagram for Internal Logistics
Men
Machine
Salary for the storekeeper required to manage Warehouse
¾
Commit to improvement target.
Cost of Internal Logistics of Imported Equipment is very High.
All inland transportation of imported equipment is done through Air Freight. Cost of Inventory of Material Stored in Warehouse.
Cost of Rent paid for the Warehouse
Cost of Insurance premium paid for the Insuring the Warehouse.
Method
Material
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IMPROVE PHASE Key Objectives are…..
Control Chart for the cause of high cost of Internal Logistics Current DPMO
¾
Determine solutions (ways to counteract causes) including operating levels and tolerances. Install solutions and provide statistical evidence that the solutions work.
1,000,000
Target DPMO
1,000,000
1,000,000
1,000,000
800,000 P ro ce ss D P M O
¾
1,200,000
600,000
400,000
200,000 142,800
0
6,210 Sep-01
6,210 Oct-01
6,210 Nov-01
6,210 Dec-01
142,800
Jan-02
6,210
6,210 0 Feb-02 Feb-02
Sep-01
Oct-01
Nov-01
Dec-01
Jan-02
Current DPMO
1,000,000
1,000,000
1,000,000
142,800
142,800
0
Target DPMO
6,210
6,210
6,210
6,210
6,210
6,210
Period in Months
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Lean Six Sigma Overview
CONTROL PHASE Key Objectives are….. Put controls in place to maintain improvement over time.
Control Chart for the sustained process control 5
Individual Value
¾
UCL=4.450
4 3 2
Mean=1.65
1 0 1 -1
LCL= 1 150 LCL=-1.150
-2 Subgroup
Provide statistical evidence that the improvement is sustained (3 months of data).
10
20
1
1
4
Moving Range
¾
0
No. Of Days
UCL=3.439 3 2 1
R=1.053
0
LCL=0
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Effort Over a Project Life Cycle Black Belt/Green Belt and Team Team and Process Owner
Level of E Effort
Champion
Define Project
M
A
I
C
Integrate into Daily Work
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SIX SIGMA DEFINE PHASE
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Define Phase Contents 1. Develop Problem Statement 2. Map High Level Process (SIPOC) 3. Determine Critical to Quality Characteristics (CTQ’s) 4 Develop Project Charter 4.
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Problem Statement Description of the “pain”
What is wrong or not meeting our customer’s needs?
When and where does the problem occur?
How big is the problem?
What’s the impact of the problem?
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Problem Statement Key Consideration/Potential Pitfalls
Is the problem based on observation (fact)
Does the problem statement prejudge a root cause?
Can data be collected by the team to verify and analyze the problem?
Is the problem statement too narrowly or broadly defined?
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Problem Statement Key Consideration/Potential Pitfalls (conti...)
Is a solution included in the statement?
Is the statement blaming any person or function?
Would customers be happy if they knew we were working on this?
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Problem Statement – Examples Example 1 Poor Statement
Because our customers are dissatisfied with our service, they are late paying their bills.
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Problem Statement – Examples Example 1 (conti...) Improved Statement
In the last 6 months (when) 20% of our repeat customers – not first timers (where) – were over 60 days late (what) paying our invoices. When surveyed all of these customers reported extreme surveyed, dissatisfaction with our service (what). The current rate of late payments is up from 10% in 1990 and represents 30% of our outstanding receivables (how big). This negatively affects our operating cash flow (impact)
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Lean Six Sigma Overview
Problem Statement – Examples Example 2 Poor Statement Customers are unable to access the call centre half the time leading to high revenue losses.
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Problem Statement – Examples Example 2 (conti...) Improved Statement During the year 2003, (when) 40% of our customers (extent) were unable to access the call centre at the first attempt (what). This causes dissatisfaction to our customers and a loss of revenue opportunities to the organization (impact).
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High Level Process Mapping CTP
CTQ
S
I
P
O
Suppliers pp
Inputs
Process
Outputs p
Measures
C Customers
Measures Process Map
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Lean Six Sigma Overview
Process Boundaries - Example Credit Decision Cycle Time
Customer’s Perspective
I
O
S Stop
Internal View Initial Metric Completed Application Received From Customer
Customer Sends Application
Decision Sent To The Customer
Customer Receives Decision
New Metric
View Process From The Customer’s Perspective Not The Internal Perspective 46
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SIPOC Worksheet S Supplier
I Input (Use nouns)
P
O
Process (Use verbs)
Output (Use nouns)
C Customer
1
2
3
4
5
6
7
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Gather VOC Data Surveys Project Team 1
Personal Visits Questionnaires
Project Team 2
Interviews Phone Calls
Customer
Project Team 3
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Lean Six Sigma Overview
Gather VOC Data Key Considerations In Collecting Customer Data:
Collector’s bias may affect what is heard What contact/relationship do you have with the customer? What are your time constraints? What budget is available? How much certainty to do you need to move forward with the project? Ensure customer expectations are aligned with our intentions/actions
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Managing Customer Expectations
Manage customer expectations throughout VOC data collection
Select customers carefully
Explain your intent for gathering the information
Clarify your ability to act on information gathered
Communicate next steps to the customer Asking For Information Does Not Translate To A Promise To Act 50
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Steps To Determining CTQs A Process To Identify Customers And Understand Their CTQs Identify Customers
• • •
List customers Define customer segments Narrow list
Determine CTQs
Voice Of The Customer (VOC)
• •
•
Review existing VOC data Decide what to collect/ select VOC tools Collect data
• • • • •
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Organize all customer data Translate VOC to specific needs Define CTQs for needs Prioritize CTQs Contain problem if necessary 51
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Lean Six Sigma Overview
Effective Brainstorming
Brainstorming is used to establish common method for a team to creatively and efficiently generate high volume of ideas on any topic
Brainstorming encourages open thinking
Gets the involvement of all the team members without the dominance of anyone team member
Allows team members to build on each others creativity while staying focus on the joint mission
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Affinity Diagram
Record each VOC on a post it note in bold letters
Without talking sort the ideas simultaneously as a team into 5 –10 related groupings
For each grouping create summary or header cards using consensuses
Draw the final affinity diagram connecting all finalized header cards with their grouping
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Affinity Diagram - Example FEEDBACK & RESPONSIVENESS
MATERIAL
MATERIAL
DELIVERY
FULFILLMENT
SYSTEMS MANAGEMENT
Action plan for pending STO
Quality of packing
Stores to punch entire STO in SAP
Communicating new part codes and part numbers to users
Advance intimation regarding availability of material
Cycle time for 100% fulfillment of STO requirements
Common tracking no. for all material issues
Streamlining of part codes in all database systems
Dispatch intimation as per STO
Weekly dispatch schedule
Material balancing across all lines
List of obsolete or discontinued parts
Stores and Production should have part lists for equivalent components
Reconciliation with balance STO’s
All accessories for new models should be bundled with 1st dispatch
Response time for acknowledgement of requirement
Alternate process to be standardized for urgent requirements Availability of obsolete or discontinued parts for repeat orders
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Lean Six Sigma Overview
Kano Model
To identify & prioritize the full range of the customers needs
Kano model helps to describe which needs, if fulfilled contribute to customer dissatisfaction neutrality or delight
Kano Model Identifies • Must be needs - Critical to customer expectation • More is better – Critical to customer satisfaction • Delighter – Converting wants to needs 55
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CTQ Prioritization Matrix Process: Invoicing Primary output (product or service): Invoices sent to customers’ accounts payable departments Strong relationship
Rs. NonReceivable
Customer Clarifications
Total Cycle Time
CTQ Output Characteristic
Errors Per Invoice
Weak relationship
Potential Project Y Metrics Deviation In Delivery
M d Moderate relationship l i hi
Cycle Time for Invoicing Accuracy of Invoices Legible Invoices
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What is a Pareto Diagram? A diagram that shows 20% of the inputs (Xs) cause 80% of the problems with dependent process outputs (Ys) A Pareto diagram allows a team to: Discover what type of categories relate to the problem Focus on the most important items
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Lean Six Sigma Overview
Pareto Diagram A Pareto diagram is a bar chart organized with the largest bar to the left and the smaller bars to the right in order of frequency. Pareto Chart for Type 100
50
80
40
60
30
40
20 10 0
Percent
Count
60
20 0
Defect © 2009, Qimpro
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Pareto Diagrams – Key Points!
Pareto diagrams are typically used to prioritize competing or conflicting problems and to distinguish the “vital few” from the “trivial many.”
Pareto diagrams determine which of several classifications have the most count or cost associated with them.
The base data gathered must be in terms of either counts or costs.
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Pareto Diagrams – Key Points!
Do not use terms that can't be added, such as percent yields or error rates.
Remember to use Pareto Diagrams creatively.
If the first one doesn’t show an 80-20 pattern, then reconsider the problem and try again.
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Lean Six Sigma Overview
CTQ Tree Why use it
Identifies critical to quality (CTQ) characteristics, features by which customers evaluate our product or service that is to be used as the measure for our project.
A useful CTQ characteristic has the following features: • It is critical to the customers perception of quality • It can be measured • A specification can be set to tell whether the CTQ characteristic has been achieved.
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CTQ Tree What does the CTQ tree do…
Links customer needs gathered from the voice of customer data with process drivers and with specific, measurable characteristics.
Enables the project team to transform general data into specific data.
Makes the measuring process easier for the team.
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Setting up a CTQ Tree
Gather the sorted customer needs from the VOC data
List the major customer needs on the left hand side of the tree structure
View each need from the customers point of view
For each need ask “what would that mean” from the customers stand point
Each answer becomes the driver for the CTQs
Keep asking “what would that mean” until you reach a level where it would be absurd to continue
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Lean Six Sigma Overview
Alignment of Project Y BUSSINESS Y Key output metrics that are aligned with strategic goals/objective of the business. Big Ys provide a direct measure of business performance
CORE PROCESS Y’S
PROCESS
Key output metrics that summarize process performance
MANAGEMENT
PROJECT Y Key project metric defined from the customer perspective
X1
X2
X3
INPUT PARAMETERS THAT INFLUENCE THE “Y”
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Select The Project Y Think Outside-In
For your key CTQ, how does the customer define process performance?
S I
Supplier
Input
P O C
Process
Output
Customer
CTQs
y=______ What is the exact definition the customer recognizes for this metric?_
How does this compare with existing performance metrics?
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Select The Project Y Key Questions To Address
If the Project Y changes, will my customer feel the impact?
Does the Project Y match with how the customer describes the process?
Does the Project Y link to one at the Big Ys for your business?
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Lean Six Sigma Overview
What is a Charter? One of the most important things necessary to get a team started on a footing is a charter A Charter:
Clarifies what is expected of the project
Keep the team focused
Keeps the team aligned with organizational priorities
Transfers the project from the Champion to the Improvement Team
Used as a tool by the Apex Council to review project progress
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SIX SIGMA MEASURE PHASE
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Measure Phase - Steps 1. Develop Detailed “AS IS” Process Map 2. Determine What to Measure (Process Y) 3. Validate Measurement System 4. Quantify Current Performance
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Lean Six Sigma Overview
What is a Process Map? Process flow diagram that visualizes how work is done. Start
Step 1
Step 2A
Step 2B
Step 2C
Step 3
Rework
No
Good?
Yes
End
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Process Map Symbols Symbol
Meaning Start or End of Process Activity or Process Step Decision or Inspection Point Connector Direction of Flow
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Why Create a Process Map?
A Cross Functional Process Map shows “AS IS” process.
Process start and end points are identified and made measurable.
Makes process steps visible and keeps the team from drifting outside the project boundaries. boundaries
Non-valued-added steps become clear and can be discarded or minimized.
Rework and repair is obvious. They are the hidden factories. Hidden Factories have to be shut down.
Is a consensus on how work is done. © 2009, Qimpro
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Lean Six Sigma Overview
Analyzing a Flow Diagram 1.Examine Each Decision Symbol Is this a checking activity? Is this a complete check, or do some types of errors go undetected Is this a redundant check?
2.Examine Each Rework Loop Would we need to perform these activities if we had no failure? How ‘long’ is this rework loop (steps, time lost, resources consumed, etc?) Does this rework loop prevent the problem from reoccurring?
4.Examine Each Document or Database Symbol Is this necessary? How is this kept up to date? Is there a single source for this information? How can we use this information to monitor and improve the process?
3.Examine Each Activity Symbol Is this a redundant activity? What is the value of this activity relative to its cost? How have we prevented errors in this activity?
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Learn to Recognize Waste
Waste of correction (rework)
Waste of waiting
Waste of inventory
Waste of over p production
Waste of transportation
Waste of motion
Waste of over processing
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Types Of Data Data Type Is An Important Consideration Discrete Data Binary (Yes/No, Defect/No Defect) Ordered categories (1-5) Counts E Examples l Number of incomplete applications Percent of responding with a “5” on survey Number of Green Belts trained
Continuous Data Can be broken down into increments Infinite number of possible values E Examples l Cycle time (measured in days, hours, minutes, etc.) Weight (measured in tons, pounds, etc.)
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Data Collection Plan
Establish Data Collection Goals
Develop Operational Definitions & Procedures
Ensure Data Consistency & Stability
Collect Data & Monitor Consistency
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Breaking Down the Overall Variation Overall Variation
Part-to-Part Variation
Which variation component do we want to be large?
Measurement System Variation Variation due to gageRepeatability
Operator
Variation due to operatorReproducibility
Operator by part Interaction
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Variable Gage R&R - Method To conduct a variable gage R&R study…. At least two operators (persons doing the measuring) should participate. Two or three operators are typical. At least 10 parts should be measured. These are 10 yp p product that represent p the full units of the same type range of manufacturing variation. Each operator will measure each part two or three times. Parts should be measured in random order. It is very important that an operator not be aware of his or her earlier measurement when doing a repeat measurement on the same part. © 2009, Qimpro
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Attribute Gage R&R
It is also important to have good repeatability and reproducibility when obtaining attribute data.
If one operator, for example, decides a unit has an “appearance” defect and another operator concludes the same unit has no defect, then there is a problem with the measurement system. system
Similarly, the measurement system is inadequate when the same person draws different conclusions on repeat evaluations of the same unit of product.
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Attribute Gage R&R
An attribute measurement system compares each part to a standard and accepts the part if the standard is met.
The screen effectiveness is the ability of the attribute measurement system to properly discriminate good from bad. bad
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Attribute Gage R&R - Method
Select a minimum of 30 parts from the process. These parts should represent the full spectrum of process variation (good parts, defective parts, borderline parts).
An “expert” inspector performs an evaluation of each part, part classifying it as “Good” Good or “Not Not Good Good.”
Independently and in a random order, each of 2 or 3 operators should assess the parts as “Good” or “Not Good.”
Enter the data into the Attribute Gage R&R.xls spreadsheet to quantify the effectiveness of the measurement system.
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Attribute Gage R&R - Example SCORING REPORT DATE: NAME: PRODUCT: SBU: TEST CONDITIONS:
Attribute Legend 1G 2 NG
Known Population Sample # Attribute 1 G 2 G 3 G 4 G 5 G 6 G 7 G 8 G 9 NG 10 NG 11 G 12 G 13 NG 14 G 15 G 16 G 17 NG 18 G 19 G 20 G
% APPRAISER SCORE
Operator #1 Try #1 Try #2 G G G G G G G G G G NG G G G G G G G NG NG G G G G NG NG G G G G G G NG NG G G G G G G (1)
% SCORE VS. ATTRIBUTE
-> (2)
->
Operator #2 Try #1 Try #2 G G G G G G G G G G G G G G G G NG NG G G G G G G NG NG G G G G G G NG NG G G G G G G
95.00%
100.00%
90.00%
95.00%
SCREEN % EFFECTIVE SCORE
Y/N Agree Y Y Y Y Y N Y Y N N Y Y Y Y Y Y Y Y Y Y
(3)
->
Y/N Agree Y Y Y Y Y N Y Y N N Y Y Y Y Y Y Y Y Y Y
This is the overall measure of consistency among operators and “expert.” 100% is best!
85.00%
SCREEN % EFFECTIVE SCORE vs. ATTRIBUTE
(4)
->
85.00%
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Interpreting the Results
% Appraiser Score is the consistency within one person.
% Score vs. Attribute is a measure of how well the operator’s evaluation agrees with that of the “expert”.
Screen % Effective Score is a measure of how well the operators agree with each other.
Screen % Effective Score vs. Attribute is an overall measure of consistency between operators and agreement with the “expert”.
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Role of Defect-Based Metrics When Using Attribute Data... These Metrics Quantify Process Capability: DPMO - Defects per Million Opportunities PPM - Parts per Million convert to... Sigma Level for the Process © 2009, Qimpro
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Defects versus Defective Items Out of these 12 Marble Slabs… there are...
3 Defective Slabs 6 Defects
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Estimate Process Capability A cutting operation cuts tubes to a target length. PPM represents the number of defective items per million items inspected. Variable Data: Length in mm LSL
9% Defective
USL
7%
or
2%
Attribute Data: Go/No Go Length Gage 45(NoGo ) = 0.09 500(Inspected)
90,000 PPM 9% Defective or 90,000 PPM
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Why Count Number of Defects
Whenever a number of things can go wrong, or things can go wrong at any of several steps, then…
Counting number of defects provides a more meaningful indicator of process capability than merely counting the number of defective items at the end of the process.
First, we must determine the opportunities for defect.
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Determine the Total Opportunities If we measured the outcomes of our processes (products, services and information) like volume... then, the combined volume... should equal???
Defects
Successes 88
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The Outcomes of our Process
Either a defect or a success:
? Defects
Successes 89
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Not Good!!!!
Defects
Successes
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Better Output
But good enough???
Defects Successes
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Defects Per Million Opportunities If... each glass can hold one million drops... and, our process generates one million drops... then the number of drops in the “defects” glass represents...
DPMO!
Defects
Successes
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DPMO Converts to Sigma 999,996.6
34 3.4 Defects
Successes
3.4 DPMO is equal to 6σ © 2009, Qimpro
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Measures of Central Tendency Mean or Average The sum of the values in a data set divided by the number of values.
Placement Time for Technical Positions (in days) 22, 26, 26, 31, 33, 37, 37, 42, 52, 52, 52, 57, 59
X = 40.5 days
Mode The most frequently occurring data value.
Mode = 52 days
Median The middle observation in the data set that has been arranged in a ascending or descending order.
Median = 37 days
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Sample vs. Population
x
Sample s s2
μ
Population σ σ2
Sample Statistics
Population Parameters
x = Mean
μ = Mean
s = Standard Deviation
σ = Standard Deviation
s2 =
σ2 = Variance
Variance
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Histograms A histogram is a frequency polygon in which data are grouped into classes. The height of each bar shows the frequency in each class. 20
20
12
10
10 4
3
1 0
10
15
20
25
30
35
40
Days
Does a histogram preserve the time order in which the data was collected? © 2009, Qimpro
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Shapes of Data Sets Bell Shape – The Normal Distribution Right Skewed (Positively Skewed) Left Skewed (Negatively Skewed) Uniform Distribution Bimodal Distribution 97
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Histograms to Evaluate Shape
When creating a histogram, the data must be properly grouped in order to understand the shape of the data distribution. For the given sample size, the right number of classes should be used.
Number of Data Points
Number of Classes
Under 50
5-7
50 – 100
6-10
100 – 250
7-12
Over 250
10-20
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The Normal Distribution
Since many process outputs have this shape, the properties of the normal curve can be used to make predictions about the process population.
Data that is non-normal can sometimes be transformed to the normal distribution, to use the properties of the normal curve to make predictions.
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Properties of the Normal Distribution
Standard Deviation
-2σ
-3σ
Average
-1σ
+1σ
68%
+2σ
+3σ
95% 99.73% 100
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What is Process Capability? Seat Track Rolling Mill
Hiring Process
Inner Dimension of Seat Track (mm)
Placement Time (in Days)
LSL
USL
USL
The capability of a process to meet customer requirements.
Placements that take too long!
Product exceeds Specification Limits!
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PPM (Parts per Million) Inner Dimension of Seat Track (mm) LSL
3%
Placement Time (in Days)
USL
USL
15%
3%
6% of the seat tracks do not meet specifications. This translates to… 60,000 PPM
15% of the time, we take too long to fill the open positions. This translates to… 150,000 PPM
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A Stable Process LSL
USL
Mean
Individual Measurements
6σ
A process is said to be stable when the distribution of all individual measurements are contained within ± 3σ from the mean. Stable processes provide the most reliable estimates of process capability. 103
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Four Stable Processes A
B
LSL
LSL
USL
C
USL
What can be said about the capability of these stable processes?
D
LSL
USL
LSL
USL
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Average & Standard Deviation For example: The average of all data = 178.6 and the average range = 8.4 from a stable control chart that used a sample size of 5. Population
Then, σ = R = 8.4 = 3.6 d2 2.326 X = 178.6
If the target = 171, is the process centered on target?
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LSL = 160
USL = 182
Estimating Process Capability
Process Variation We expect that 99.73% of the time, we will produce product that falls between 167.8 and 189.4
Specification Limits According to the specifications, we want all product to fall between 160 and 182.
Target
167.8
171
178.6
x - 3σ
189.4
x +3σ
x
If centered, would this process be capable of meeting specifications?
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Determine the Potential Capability (CP) USL
LSL
Cp = 1
Cp > 1
The Cp index reflects the potential of the process if the average were perfectly centered between the specification limits. The larger the Cp index, index the better!
USL Cp = Cp < 1
- LSL 6σˆ
For a Six Sigma Process, Cp = 2 107
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Estimate Percentage Beyond Specs
LSL = 160
USL = 182
To estimate the percentage of product (or PPM) that falls outside the specification limits, we must first compute Z upper and Z lower.
167.8
Z lower is the number of standard deviations between the Process Average and the Lower Specification Limit.
178.6
189.4
Z upper is the number of standard deviations between the Process Average and the Upper Specification Limit.
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Estimating % Beyond Specifications LSL = 160
USL = 182
From a Z table, we find that Z = 0.94 corresponds to proportion = 0.1736 This converts to 17.36% Defective or 173,600 PPM=2.4 σ (from Sigma
167.8
178.6
Z upper = 0.94
Table)
189.4
Zupper = USL – X σ Zlower = X – LSL σ 109
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Process Capability Metrics Attribute Process O tp t Output Y
PPM, DPU, DPO DPMO, RTY
Data Type
Sigma g Level
Variable
CP, CPK, PP, PPK PPM
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SIX SIGMA ANALYZE PHASE
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Analyze Phase - Steps 1. Cause and Effect Diagram 2. WHY? WHY? WHY? Analysis 3. Testing and Validation of Theories 4. Validating the Root Causes 5. Finalize the Charter
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Cause and Effect Diagrams A visual tool used by an improvement team to brainstorm and logically organize possible causes for a specific problem or effect Measurement
Methods
Potential
Mother Nature
Machinery
High Level
People
Causes (Xs)
Effect Y
Materials
Summarize potential high-level causes Provide visual display of potential causes Stimulate the identification of deeper potential causes 113
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Cause and Effect Diagram Let us Brainstorm???
Process Map Analysis Why Is There Difference In The V i i In Variation I Cycle C l Time Between Line 1 and Line 2
Data Analysis
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Cause and Effect Diagram Brainstorm the “major” cause categories and connect to the centerline of the Cause & Effect diagram Measurement
Methods
Why y Is There Difference In The Variation In Cycle Time Between Line 1 and Line 2
People
Machinery
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Cause and Effect Diagram Measurement
Methods
No tracking of in process rejection of components
No. of components inserted per operator on either line No. of boards reworked
Speed of individual conveyor belts not known
Why Is There Difference In The Variation In Cycle Time Between Line 1 and Line 2
Breakdown time are different for either line
Line 1 has more temporary operators
Untrained Inspectors in Final Inspection
People
Machinery 116
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Cause and Effect Diagram-Example 1 MACHINE Irregular prev. maintenance
MAN Time loss for Magazine setup
Dirty line & surrounding
Improper data in comp.library
No program protection No control of loading list
Unnecessary checks during model change
METHOD
Time loss for PCB inspection
Implementation of ECRs/Mod.notes
Mismatched cycle time
UNDERUTILIZATION OF SMT LINE Alternate Feeder Bank not used
Time loss for magazine setup
Insufficient buffer at input
Time loss for PCB setup
MATERIAL
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Cause and Effect Diagram–Example 2 Men
Machine Overloaded Branch Engineers
Interpretation of customer specs by Branch Engineer
Branch Engineer is not available most of the time
Information by Sales Engineer is Incorrect Submission of Offer is Hasty
Response Time to Query is very Large
Interpretaion of branch information by CoEE Branch does not fill DITO correctly
New Bunch of fresh Engineers in CoEE Lack of experience of CoEE engineer in executing applications at site
Resistance to Changes in way of working Frequent Revisions in the Standard Formats
Resource not there Flooding of Projects to CoEE Too many Fast Track jobs
There is lack of communication between branch and CoEE
No documented procedure to fill the DITO
Too many Assumptions are made by CoEE regarding the Design Too many questions required to be asked by CoEE to Branch
DITO has to be checked for errors
Branch Engineer wastes time in rework Transfer information from Branch formats to CoEE formats There are standard CoEE formats applicable to all branches
Start of Production by CoEE takes too long
Low confidence on the DITO provided by Branch
Incorrect and Incomplete information from Branches
Stagerred Information from the Branches
Information passes through many hands in various formats
Irregurality in despatch of Information from the Branches Information availability at the Branches is not organised
Method
CoEE Standard Formats not being used from the Sales stage itself
Material
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Sorting the Possible Causes Non-Controllable Causes: These are causes that the team unanimously conclude are beyond the control of the present process boundaries or outside the physical location of the process execution. Lack of Solution OR Direct Improvements: These are causes that are actually solutions that can be implemented directly and need no further analysis. They are usually stated as lack of resources, equipment, tools or training. Likely and Controllable Causes: These causes are the causes that have passed the above two filters and need further analysis. © 2009, Qimpro
Asking 5 Why’s?
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4.2.1
The immediate next step after the segregation is to attack the likely and controllable causes and ask at least 3 – 5 why’s?….. for each cause. This is called root cause drill down.
Only after we have asked why 3 - 5 times to each of the likely causes, causes we will be able to arrive at the possible root cause, also known as KPIV.
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5 Why’s? – Example WHY do we have poor and declining participation in improvement programs? 1. Because people resist change . Why do people resist change? 2. Because they fear making mistakes. Why do people fear making mistakes? 3. Because they are criticized for mistakes. Why are people criticized for mistakes? No ideas, let’s move on. Okay, then Why else do people fear making mistakes 4. Because they are penalized for mistakes. 121
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Prioritizing the Possible Root Causes There are two tools that are widely used for the prioritizing of the possible root causes that are obtained from the Cause and Effect diagram.
Cause and Effect Matrix (C&E Matrix)
Failure Mode Effect Analysis (FMEA)
Desired Output List of Prioritized Possible Root Causes 122
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What is a Cause & Effect Matrix? The Cause & Effect Matrix is a team tool used to prioritize the … key process input variables (KPIV) that affect key process output variables (KPOV)
Totals
8 4 5 1 4 0
Burns
8 3 5 3 5 3
3
Flash
2 0 2 6 5 0
9
Tears
2 0 1 8 9 0
7
Voids
IL D
% ISO Time Temperature Fill Time Pressure Mold Temperature
Pad Stiffness
Process Input
7
Pad Width
Blending Blending Blending Molding Molding Molding
2
Pad Thickness
Process Step
Process Outputs 9 6
2
2 7 6 7 6 5
2 9 2 1 6 9
3 0 0 6 9 2
3 0 0 7 3 6
198 162 136 182 267 150
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Anatomy of a Cause and Effect Matrix
3
Totals
3 0 0 7 3 6
198 162 136 182 267 150
Tears
9
Flash
2 0 2 6 5 0
7
Voids
2 0 1 8 9 0
Process Outputs 9 6
Pad Stiffne ess
% ISO Time Temperature Fill Time Pressure Mold Temperature
7
IL D
Process Input
2
Pad Thickn ness
Process Step Blending Blending Blending Molding Molding Molding
2
Pad Width
Proces s Steps
Burns
Customer Importan ce Rating
Process Input Variables
8 3 5 3 5 3
8 4 5 1 4 0
2 7 6 7 6 5
2 9 2 1 6 9
3 0 0 6 9 2
Strength of Correlations
Key Process Output V i bl Variables
Cross Multiply & Prioritize 124
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Cause and Effect Matrix - Matrix PCB breakage
5
wrong components
10
Incomplete kit
5
Delay in reloading of different layouts
Breakdown
10
Breakdown
Outputs Process Step Model changeover
10
high m/c PPM
3
Rating of Importance to Customer ==>
1
1
7
1
1
1
103
1 1 7 3 5
1 1 7 3 3
3 3 5 3 3
1 1 1 1 1
3 7 1 1 1
1 1 1 1 1
83 123 161 89 95 109
Inputs Un-necessary checks during model change No control of loading list No program protection Irregular Preventive Maintenance. Dirty line & surrounding Improper data in comp.library
Total
Component replenishment
Alternate Feeder Bank not used
3
1
7
1
1
1
Program change due to ECR/Mod notes
Implementation of ECRs/Mod.notes
1
1
3
1
3
1
83
Insufficient buffer at I/p Time loss for PCB inspection Time loss for PCB setup Time loss for magazine setup
1
1
10
10
1
1
178
3 1
1 1
7 5
1 3
1 1
1 1
109 93
Run time loss Other losses
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What is FMEA? A Failure Mode and Effects Analysis is a systemized group of activities intended to:
Recognize and evaluate potential failure and its effects
Identify actions which will reduce or eliminate the chance of failure
Document analysis findings
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Objectives of FMEA in Six Sigma Identify the high priority failure modes and causes of defects in a process. Identify
high priority input variables (Xs) that impact important output variables (Ys).
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When to use FMEA?
FMEA is designed to prevent failures from occurring or from getting to internal and external customers.
Therefore, FMEA is essential for situations where failures might occur and the effects of those failures occurring are potentially serious. serious
FMEA can be used on all Six Sigma projects. It serves as an overall control document for the process.
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FMEA – Worksheet Process Potential Potential Step/Part Failure Failure Number Mode Effects
S E V
Potential Causes
O C C
Current Controls
D E T
R P N
Actions Recom mended Resp.
Actions Taken
S E V
O C C
D E T
R P N
0
0
0
0
RPN = Severity X Occurrence X Detection
0
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FMEA – Severity Rating Scale Severity – The consequences of a failure should it occur. Rating Criteria – A Failure Could 10 9 8 7 6 5 4 3 2 1
Injure a customer or employee Be illegal / cause controllership issues Render the product or service unfit for use Cause extreme customer dissatisfaction Result in partial malfunction Cause a loss of performance which is likely to result in a complaint Cause minor performance loss Cause a minor nuisance, but be overcome with no performance loss Be unnoticed and have only minor effect on performance Be unnoticed and not affect the performance
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Hypothesis Testing
Refers to the use of statistical analysis to determine if observed differences between two or more data samples are due to random chance or to true differences in the samples
Used to test the theories established during the Cause and Effect analysis. analysis
Increases your confidence that probable Xs are statistically significant
Used when you need to be certain that a statistical difference exists
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Why Do Hypothesis Testing? Number Of Scrapped Prototype Seats
Is the observed difference real?
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Program B 132
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Kinds Of Differences Continuous data Differences in averages Differences in variation Differences in distribution shape” of values Discrete data Differences in proportions
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Hypothesis Testing Definition Of Terms Null Hypothesis – H0 The Null Hypothesis is the antithesis to our claim regarding the relationship of two or more data sets.
Alternate Hypothesis – H1 or HA The Alternate Hypothesis is our claim statement. This is the theory that we want to test.
The Null Hypothesis and Alternative Hypothesis Are Mutually Exclusive and Complimentary. © 2009, Qimpro
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Hypothesis Testing – Assumptions Sampling from a distribution must be representative or independent
Random sampling is the key assumption
Normality is not the key assumption
The random sampling assumption is also known as the statistical independence assumption
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Hypothesis Testing – The Decision As a result of the hypothesis test, we will either….
Reject the Null Hypothesis, or
Fail to Reject the Null Hypothesis
In Hypothesis Testing We Always Work With The Null Hypothesis. The Test Result Will Tell Us If We Can Reject Or Fail To Reject The Null Hypothesis
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Risk of Hypothesis Whenever a hypothesis test is run, there is a risk associated with the decision that is made. There are two types of errors (risks): Type I error (also known as alpha risk, denoted by α) – The probability of rejecting the null hypothesis when it is true. true Type II error (also known as beta risk, denoted by β) – The probability of accepting the null hypothesis when it is false.
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Type I and Type II Errors Reality
Accept Null Hypothesis
Null Hypothesis True
Null Hypothesis False
Correct decision
Type II error
Type I error
Correct decision
Decision Reject Null Hypothesis
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Type I and Type II Errors Reality
My test decides there is no difference Decision My test decides there is a difference
In fact there is no difference
In fact there is a difference
I have confidence in my test
I have taken a “β” risk, my test did not have enough power
I have taken an “α” risk, could not have enough confidence in my test
My test is powerful !
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Hypothesis Testing – Exercise Refer to the exercise A1 in the CSSGB workbook and state the Null and Alternate Hypothesis for the situations provided. For each of the situations, state … H0 = ???? And HA = ????
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Hypothesis Testing – Common Tests
1 sample z-test / t-test
2 sample t-test
1 way analysis of variance (ANOVA)
Chi square test for independence
1 proportion test
2 proportion test
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z – test Features
The z statistic or the t statistic is a ratio of difference in means and variation of the means
The z – statistic is used for samples with; •
Large sample size of more than 30 data points
•
Standard deviation of the population is known
Used to compare average performance of two groups.
Tests the null (H0) hypothesis of “no difference between means of two groups”
Draws critical values from standard normal table or z table and t distribution table respectively. © 2009, Qimpro
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t - test Features
The t – statistic is used for samples with; •
Small sample size of less than 30 data points
•
Standard deviation of population is not known.
Used to compare average performance of two groups groups.
Tests the null (H0) hypothesis of “no difference between means of two groups”
Draws critical values from standard normal table or t distribution table.
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Z Test – Example The 1-sample Z test is used when…
Testing the equality of a population mean to a specific value, and
Sample size is large (n > 30)
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Z Test – Example (conti...) You are attempting to assess the cycle time for packaging of goods when utilizing two different methods. The use of a metal strapping has traditionally been assumed to generate the best response, but that assumption is now going to be tested against a process of using self adhesive tape. Historically, when utilizing the metal staple sheets: Average cycle time = 6 minutes Standard deviation = 2 minutes A random sample of size 36 was collected from the self adhesive tape process, yielding: x = 4.7 minutes
s = 2.0 minutes 145
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Z Test – Example (conti...) Establish both the Alternative and Null Hypotheses. H0 : μ = 6 minutes HA : μ ≠ 6 minutes Determine the level of significance, significance α : α = 0.05 0 05
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Z Test – Example (conti...) Calculate the test statistic, Z : We use the sample standard deviation, s, as our estimate of σ population. Then… z=
x - μ = 4.7 - 6.0 = - 1.3 = 3.94 σ 2 0.33 T the same Try n 36 with y = 5.5 mins
For a = 0.05 (95% confidence) : z critical = 1.96 Since the computed Z value = -3.94 < -1.96; We REJECT the Null Hypothesis. Conclusion: Cycle Time for packaging with Self Adhesive Tape is not equal to 6 minutes. © 2009, Qimpro
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Scatter Diagrams A Scatter Diagram is an Important Graphical Tool For Exploring the Relationship Between Predictor Variables (X’s) And The Response Variable (Y’s) (i.e., Causes and the Effect)
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Analyzing Relationships Use Scatter Diagrams To Study The Relationship Between Two Variables 40 35 30 25 20 15 10 5 1K
2K
3K
4K
5K
6K
7K
8K
9K
10K
No. Of Components for Insertion (X) 149
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Correlation and Causation Warning! Correlation Does Not Imply Causation 100
200
300
80
80
Population (In Thousands))
Correlation Between
70
70
60
60
Number Of Storks A d And Human Population
50 100
200
300
50
Number Of Storks Source: Box, Hunter, Hunter. Statistics For Experimenters. New York, NY: John Wiley & Sons. 1978
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Interpreting A Scatter Diagram Look for Patterns 1
3
Strong Positive Correlation 2
5
Strong Negative Correlation
4
Positive Correlation
No Correlation
6
Negative Correlation
Other Pattern
For all charts: Y = Participant satisfaction (scale: 1 – worst to 100 – best) X = Trainer experience (# of hours) 151
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Regression Measures Of Correlation
The Correlation Coefficient r measures the strength of linear relationships
–1 ≤ r ≤ 1
When a relationship exists, the variables are said to be correlated
Perfect negative relationship No linear correlation Perfect positive relationship
r2 measures the percent of variation in Y explained by the linear relationship of X and Y and is called the Coefficient of Determination.
r2 value will always be smaller than r values
r = –1.0 r= 0 r = +1.0
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What is Regression?
In simple linear regression, you obtain the graph and the equation of the straight line that best represent the relationship between two variables. Given a sample p of p paired data, the regression equation y = β0 + β1x describes the relationship between two variables.
Y: C Cost $k
Best Fit Line 150
100
50
10
20
X:Time (Days)
30
The graph of the regression equation is called the regression line (or best fit line). © 2009, Qimpro
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The Regression Equation Dependent Variable
Y = β 0 + β 1x y-intercept
slope
Line of Best Fit
Independent Variable
Where β0 = Predicted Value Of Y When X1 = 0
Y
β1 = Slope Of Line Change In Y Per Unit Change In X X © 2009, Qimpro
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SIX SIGMA IMPROVE PHASE
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Improve Phase - Steps 1. Determine Solutions to Counteract the Root Causes 2. Provide Statistical Evidence that Solutions Work 3. Prepare “Should Be” Process Map
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What is a Counteraction? A counteraction is anything that is done to reduce or eliminate the effect of a cause. Reduce the Effect of a Cause A helmet (counteraction) reduces the injury (effect) from an impact (cause). R d Reduce the h Cause, C Thereby Th b Reducing R d i the h Effect. Eff A refrigerator (counteraction) reduces spoilage (effect) by reducing temperature (cause) Eliminate the Cause, Thereby Eliminating the Effect. Antibiotics (counteraction) eliminate some diseases (effect) by eliminating Bacteria (cause) © 2009, Qimpro
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Useful Counteractions Good counteractions are Well defined Actionable Examples of well defined, actionable counteractions: Create quarterly goals for the Six Sigma Implementation Plan Add cavity pressure transducer to cut off injection pressure at set point.
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158
Less Useful Counteractions Examples of less useful Counteractions (too vague, not actionable): Improve communication Develop process focus Improve injection molding quality Using such counteractions only leads to confusion, since people don’t understand the specific actions to take. When these kinds of counteractions arise in brainstorming, ask, “What specific actions do we need to take to accomplish this item?” © 2009, Qimpro
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Evaluation Matrix A Decision Matrix that helps the team to screen possible solutions against three criteria…..
Effectiveness in eliminating or reducing the verified root causes – X’s
Ease of Implementation
Cost of Implementation
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Evaluation Matrix – Example Counteractions – Reducing Cycle Time from Customer Order to Supplier Purchase Order.
Effectiveness
Ease to Implement
Cost
Fax from Master Contact List
{
{
~
Send Supplier E-Mail version of the PO file
{
Ì
~
Auto-fax: Modify Access Database to fax and e-mail confirm.
~
~
{
Request Call Return from Supplier
Ì
{
~
Legend: ~ Strong Relationship { Moderate Relationship Ì Weak Relationship
Quiz: Which idea should the team select?
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161
Objectives of FMEA in Six Sigma Identify the high priority failure modes and causes of defects in a process. Identify
high priority input variables (Xs) that impact important output variables (Ys).
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When to use FMEA?
FMEA is designed to prevent failures from occurring or from getting to internal and external customers.
Therefore, FMEA is essential for situations where failures might occur and the effects of those failures occurring are potentially serious. serious
FMEA can be used on all Six Sigma projects. It serves as an overall control document for the process.
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FMEA – Worksheet Process Potential Potential Step/Part Failure Failure Effects Number Mode
S E V
Potential Causes
O C C
Current Controls
D E T
R P N
Actions Recom mended Resp.
Actions Taken
S E V
O C C
D E T
R P N
0
0
0
0
RPN = Severity X Occurrence X Detection
0
164
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Cost/Benefits Analysis – Example Costs (First Year) Equipment Training
Benefits (Yearly) $3,000 500
Travel and living
250
Team labor
500
Total Cost
$4250
Increased capacity (reduced cycle time)
$750
Reduce rejects by 50%
4,000
Reduce labor hours for the job
500
Reduced interest expense
750
Total Benefit
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$6000
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Cost/Benefits Analysis Tips
Concentrate on tangible costs and benefits; use indirect costs that are generally acceptable to all stakeholders
Use the process map and personnel in associated departments to identify cost and benefit information
Keep the analysis simple; focus on cost of implementation and a few key benefits that clearly exceed the cost
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Cost/Benefits Analysis Tips (conti...)
Use standard methods and rates in your calculations
List all activities that contribute to either cost or benefit and identify as much as possible how these activities will be measured
Keep the presentation simple and easy to understand
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Outside – In Perspective Take One Final Look at Your Solution Before Implementation
Does it address the root cause?
Can the solution been verified through the data that it will drive your process toward successfully meeting the customer CTQs?
Will your customer be satisfied with the solution?
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Pilot Solutions A Test Of All Or Part Of A Proposed Solution On A Small Scale In Order To Better Understand Its Effects And To Learn About How To Make The Full Scale Implementation More Effective
169
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Benefits Of Piloting
Improved solution that meets customer CTQs
Refined implementation plans
Lower risk of failure by identifying and fixing problems
Confirmation expected results and relationships (of X and Y)
Increased opportunity for feedback and buy-in
Get early version of a solution out quickly to a particular segment
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Verification Of Pilot Results Statistically Verify The Pilot Results Before
After
Time To Process In Days
Time To Process In Days
Does The Output Data Show A Significant Difference That Can Be Attributed To The New Solution?
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Test Effectiveness of Pilot To Assess The Effectiveness Of A Solution:
Calculate the new Process Capability (sigma) and compare it with the improvement goal and the original process.
Compare before and after Visual Tools so you can analyze the data visually.
Use Hypothesis Testing to see if a significant statistical difference exists between the old versus the new process.
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Piloting Tips
The improvement team should be there as much as possible during the pilot process; what they learn and observe will be worth the time they invest
Collect data on process and external factors that may be influential
If possible, make sure that the full range of inputs and process conditions are tested in the pilot
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173
Piloting Tips (conti...)
Expect “scale-up” issues after even the most successful pilots
Identify critical differences between the pilot environment and the full-scale implementation environment; note potential issues/problems for full scale plan full-scale
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Need for Hypothesis Testing During the Measure Phase, we collected data on the process output (Y) to establish the baseline performance……x1 During the Improve Phase, we collected data on the process output (Y) after the process has been improved improved…… x1 Now we need to answer the following…..
Is there really a difference between x1 and x2
Is x2 better than x1
Is x2 worse than x1
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Hypothesis Testing Hypothesis Testing gives us the answer……. Case1 : Higher the better. H0 : x1 = x2 HA: x1 < x2 Case 2 : Lower the better. H0 : x1 = x2 HA: x1 > x2
Use t Test for Continuous Data and p Test for Attribute Data © 2009, Qimpro
176
SIX SIGMA CONTROL PHASE
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Control Phase - Steps 1. Prepare and Implement the Control Plan 2. Provide Statistical Evidence that the Improvements are Sustained. (3 months of data)
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What is a Control Plan?
A written summary description of the system for controlling a process
Describes actions required to maintain the “desired state” of the process and minimize process and p p product variation
A living document which evolves and changes with the process and product requirements
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Control Plan Strategy A good control plan strategy…
Minimizes process tampering.
Clearly states the reaction plan to out-of-control conditions.
Describes training g needs for standard operating p g procedures
Describes maintenance schedule requirements.
A good control plan clearly describes what actions to take, when to take them, and who should take them… thereby reducing “fire fighting” activities.
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What to Control Y = f ( x1, x2, x3…) KPOV
KPIVs
Monitor
Control
A control plan controls the X’s to ensure the desired state for Y. Merely monitoring the output, Y, is not an effective way to control a process. © 2009, Qimpro
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Why use a Control Plan?
Provides a single point of reference for understanding process characteristics, specifications, and Standard Operating Procedures (SOP)
Enables orderly transfer of responsibility for sustaining the gain gain” “sustaining
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Developing a Control Plan Questions to Get Started
What do you want to control? How often do you need to measure the process? Do you have an effective measurement system? Who needs to see the data? What type of tool/chart is necessary? Who will generate the data? Who will control the process? Have they been trained? What are the system requirements for auditing & maintenance? © 2009, Qimpro
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Control Charts Different control charts exist for different types of data. Common Control Charts Variable Data: → X Bar a and a d R Chart C at → Individuals and Moving Range Chart Attribute Data: → p Chart (for Binomial Data) → u Chart (for Poisson Data)
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The Value of Process Control When a process is in control: You can predict what it will do in the future in terms of its average performance and its variation. You can estimate the capability of the process to meet specifications specifications. It reduces process variation and process cost. CAUTION! When a process is not stable, we cannot draw valid conclusions about the process' ability to meet specifications! 185
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What Control Charts Say
Is the process stable? Should action be taken?
Should the process be left alone?
What types of causes are present?
Special Cause
Dimension in mm
6.4 6.3 6.2 6.1 6.0 5.9 5.8 5.7 5.6
UCL
= x
Common Causes LCL
0
5
10
What is the average process output?
What is the variation of the process?
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15
20
25
Subgroup Number
186
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Looking at the Data Statistically Statistical Control limits have been calculated from the data and are drawn as dashed lines on the graph. X Bar Control Chart—Average Daily Production Cost by Week $19,500
UCL Average Daily Productio on Cost
$19,000
$18,500
$18,000
$17,500
$17,000
$16,500
LCL $16,000 Week Number
Since all points are randomly distributed within the statistical bounds, this production system is stable. 187
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Do Not Try to Explain Differences
6.2.1
This means that the observed variation in average daily production cost is the natural fluctuation one would expect from a non-changing system X Bar Control Chart—Average Daily Production Cost by Week $19,500 Average Daily Produc ction Cost
UCL $19,000 $18,500 $18,000 $17,500 $17,000 $16,500
LCL $16,000 Week Number
It does not make sense to attempt to explain the difference between any of these points.
188
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Ineffective Actions
When we implemented p corrective actions, we wrongly assumed they were effective.
X Bar Control Chart—Average Daily Production Cost by Week $19,500
UCL Average Daily Production Cost A
Without looking at the data statistically, we wrongly concluded the process was changing cost was increasing!
6.2.1
$19,000 $18,500 $18,000 $17,500 $17,000 $16,500
LCL
$16,000 Week Number
The corrective actions could not produce sustainable improvement because they corrected causes that are not real. © 2009, Qimpro
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Understanding Types of Variation
6.2.1
Without proper understanding of the types of variation, over-reaction or incorrect actions are taken.
Special Cause
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Two Types of Variation Type of Variation
6.2.1
Definition
Characteristics
Common cause
No undue Influence by any of the 5M and 5Ms d 1P
• • • • •
Expected Predictable Normal Random Chance
Special cause
Undue influence by any of the 5Ms and 1P
• • • • •
Unexpected Unpredictable Not Normal Not Random Assignable
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Responding to Variation
6.2.1
MEASUREMEN TS Common Causes MEASURE IInvestigate ti t allll off the th variation by identifying the “vital few” process inputs - X’s
Common or Special
Special Causes MEASURE IInvestigate ti t th the specific ifi data points related to the special causes
MEASURE
ANALYZE Develop solutions for the “vital few” process inputs - X’s
Develop solutions for special causes and implement as appropriate
IMPROVE © 2009, Qimpro
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MEASURE 192
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Lean Six Sigma Overview
Special and Common Cause Variation
6.2.1
How you treat variation... Common Causes
What the variation really is...
Special Causes
Common Causes
Special Causes
Focus on fundamental process change
Mistake 1 Tampering (increases variation)
Mistake 2 Under reacting (missed prevention)
Focus on investigating special causes
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Summary of Variation
To improve any process, it is useful to understand its variation.
All variation is caused by common and/or special causes.
There are two major classifications of causes which help you select appropriate managerial actions:
9
If all variation is due to “common causes,” the result will be a predictable or stable system
9
If some variation is from “special causes,” the result is an unstable or unpredictable system.
Variation causes customer dissatisfaction.
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Control Limits Upper Control limit
Lower Control limit
Control Limits are statistical bounds used to determine process stability.
In the X Bar chart, if all the averages stay within these bounds (and fluctuate in a random manner with 2/3 of the points near the center line), then the process is stable. © 2009, Qimpro
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Specification Limits Specification Limits are applied to individual measurements.
LSL
USL
We want to know whether or not all foam pads made will fall within the given specification.
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Process Not in Statistical Control 1. Points beyond Control Limits: Points beyond control limits are isolated high or low points. Usually both the X chart and R chart will show the same point beyond control limits.
2. Points Hugging Control Limits Points are hugging control limits if a chart doesn't satisfy the rule of two-thirds of the points being within one-third of the centerline. 197
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Process Not in Statistical Control 3. Points Hugging the Centerline Almost all of the points are within one-third (of the distance between the control limits) of the centerline centerline. 4. Sudden Shift in Level Sudden shift in level occurs when the points seem to move to a new average over a short period of time. © 2009, Qimpro
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Process Not in Statistical Control 5. Trend Trends will continue up or down without a well defined end.
6. Cycle A cycle produces a pattern of up and down points, very much as if the values of the points were time dependent.
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X Bar & R Chart 10.0 UCL
8.0 The
6.0
Chart 4.0 2.0
LCL
0.0
16.0
UCL
12.0 The Range Chart
80 8.0
R
4.0 0.0
Sample # 1 2 3 4 5 Average Range
8:00 10.0 1.0 4.0 9.0 8.0 6.4 9.0
The X chart (averages) is accompanied by the range (R) chart (variation).
8:30 7.0 4.0 10.0 2.0 8.0 6.2 8.0
9:00 5.0 2.0 6.0 2.0 3.0 3.6 4.0
9:30 10:00 10:30 11:00 9.0 2.0 2.0 5.0 3.0 4.0 4.0 6.0 7.0 2.0 8.0 4.0 3.0 6.0 8.0 10.0 1.0 1.0 6.0 3.0 4.6 3.0 5.6 5.6 8.0 5.0 6.0 7.0
It is important to simultaneously monitor a process' average performance and its variation. 200
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Construction of X Bar & R Chart 1. Compute the average for each subgroup: Add the measurements together and divide by the number of measurements in the subgroup. For the 8:00 subgroup:
x =
Sample #
8:00
8:30
9:00
9:30 10:00
10:30
11:00
1
10.0
7.0
5.0
9.0
2.0
2.0
5.0
2
1.0
4.0
2.0
3.0
4.0
4.0
6.0
3
4.0
10.0
6.0
7.0
2.0
8.0
4.0
8.0
10.0
4 5
9.0
2.0
2.0
3.0
6.0
8.0
8.0
3.0
1.0
1.0
6.0
3.0
Average
6.4
6.2
3.6
4.6
3.0
5.6
5.6
Range
9.0
8.0
4.0
8.0
5.0
6.0
7.0
1 (10.0 +1.0 + 4.0 + 9.0 + 8.0) = 6.4 5
2. Compute the range for each subgroup: Subtract the smallest measurement from the largest measurement in the subgroup. For the 8:00 subgroup: R =10.0 - 1.0 = 9.0 © 2009, Qimpro
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Construction of X Bar & R Chart 3. Compute X double bar (Average of the Averages): Add all the averages of the subgroups together and divide by the number of subgroups.
Sample #
8:00
8:30
9:00
9:30 10:00 10:30 11:00
1
10.0
7.0
5.0
9.0
2.0
2.0
5.0
2
1.0
4.0
2.0
3.0
4.0
4.0
6.0
3
4.0
10.0
6.0
7.0
2.0
8.0
4.0
8.0
10.0
4
9.0
5
2.0
2.0
3.0
6.0
8.0
8.0
3.0
1.0
1.0
6.0
3.0
Average
6.4
6.2
3.6
4.6
3.0
5.6
5.6
Range
9.0
8.0
4.0
8.0
5.0
6.0
7.0
1 X = (6.4 + 6.2 + 3.6 + 3.6 + 3.0 + 5.6 + 5.6 ) = 5.0 7
4. Compute R bar (Average of the Ranges): Add all the ranges of the subgroups together and divide by the number of subgroups. R=
1 (9.0 + 8.0 + 4.0 + 8.0 + 5.0 + 6.0 + 7.0 ) = 6.7 7 202
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Construction of X Bar & R Chart 5. Calculate the control limits:
Sample #
For the x bar chart...
8:00
9:00
9:30 10:00 10:30 11:00
10.0
7.0
5.0
2.0
5.0
2
1.0
4.0
2.0
3.0
4.0
4.0
6.0
3
4.0
10.0
6.0
7.0
2.0
8.0
4.0
8.0
10.0
4
9.0
5
UCL x = X + A2 R UCL x = 5.0 +(0.577 × 6 7) = 8.9
8:30
1
2.0
2.0
9.0
3.0
2.0
6.0
8.0
8.0
3.0
1.0
1.0
6.0
3.0
Average
6.4
6.2
3.6
4.6
3.0
5.6
5.6
Range
9.0
8.0
4.0
8.0
5.0
6.0
7.0
LCL x = X - A2 R LCL x = 5.0 - (0.577 × 6.7) =1.1
Subgroup Size (n) 2 3 4 5
For the R chart... UCL R = D4 R UCL R = 2.114 × 6.7 =14.2 LCL R = D3 R LCL R = 0.000 × 6.7 = 0.0
A2 1.880 1.023 0.729 0.577
D3 0.000 0.000 0.000 0.000
D4 3.267 2.574 2.282 2.114
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Interpreting an X bar and R Chart Data is the dimension of a feature detail on a headliner.
Sam mple Mean
Xbar/R Chart for Dimension 0.756 0.755 0.754 0.753 0.752 0.751 0.750 0.749 0.748 0 747 0.747 Subgroup
UCL=0.7551
Mean=0.7512
LCL=0.7473 0
5
10
15
20
Sample Range
0.015
25
UCL=0.01434
0.010 R=0.00678 0.005 0.000
LCL=0
Is the Process in Statistical Control? © 2009, Qimpro
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Collecting Data for an X Bar & R Chart Rules of Thumb:
At least 20 subgroups of about n=5 data are required.
The data within a subgroup should be collected close together in time (for example, 5 consecutively i l produced d d parts). )
Longer time intervals are used between subgroups. (Depending on the process and purpose of the study, these time intervals could be 15 min., 30 min., 1hr., 2 hr., or longer).
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Process Capability Key metric to prove improvement from the baseline performance level – Process Capability
A1
LSL
A2
USL
Capability of A1 CPK, PPM, DPMO Sigma Level
LSL
< =
USL
Capability of A2 CPK, PPM, DPMO Sigma Level
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Process Capability Metrics Attribute
Process O t t Output Y
PPM, DPU, DPO DPMO, RTY
Data Type
Sigma Level
Variable
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CP, CPK, PP, PPK PPM
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Lean Six Sigma Overview
Control Dashboards
A dashboard is a template which gives the top management a snap shot view of the improved process.
A single graph that displays the current performance of the process against the target.
Each output parameter needs to have one dashboard.
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Dashboard When to use a Dashboard
For the BB from the exit of Improve Phase and to the end of Control Phase
For the process Owner from the hand over of project by BB to lifetime of the process or till the next improvement
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Dashboard - Examples
May
April
June
May
June
Target
April
Mar
Feb
Jan
Nov
Dec
Oct
Aug
July
Month
Sept
On ha nd stock ra tio
100 80 60 40 20 0 Baseline
Ja n Fe b M ar Ap ril M ay Ju ne
ct No v De c
pt Se
O
g
Ba
Au
e Ju ly
On Ha nd Stock Ra tio
Jan
On Hand stock Ratio CPT
On Hand Stock Ratio - Cabinet
100 80 60 40 20 0 se lin
Mar
Month
Target
On hand Stock Ratio - Carton
On Hand stock Ratio Carton
Feb
Nov
Oct
Month On Hand stock Ratio Cushions
Dec
Aug
July
Sept
100 80 60 40 20 0 Baseline
May
April
On Hand Stock Ratio
On Hand Stock Ratio - CPT
June
Jan
Mar
Feb
Nov
Dec
Oct
Aug
Sept
July
Baseline
On ha nd stock Ra tio
On Hand Stock Ratio - Cushions 80 60 40 20 0
Month Target
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On Hand stock Ratio Cabinet
Target
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Lean Six Sigma Overview
Final Report
At the end of Control Phase, each BB is required to make a report on the project.
The report should cover all the stages of the Six Sigma project DMAIC
The Report should have the SOP’s & formats attached as per the counter measure matrix
The report should end with the Dashboard updated till the end of the 3rd month of Control Phase
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Thank You!
[email protected] www.qimpro.com 91-22-6634 8701
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Z Table Values (One Tail) from Z = 0 to Z = 4.99 Z 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
5.00e-001
4.96e-001
4.92e-001
4.88e-001
4.84e-001
4.80e-001
4.76e-001
4.72e-001
4.68e-001
4.64e-001
4.60e-001
4.56e-001
4.52e-001
4.48e-001
4.44e-001
4.40e-001
4.36e-001
4.33e-001
4.29e-001
4.25e-001
4.21e-001
4.17e-001
4.13e-001
4.09e-001
4.05e-001
4.01e-001
3.97e-001
3.94e-001
3.90e-001
3.86e-001
3.82e-001
3.78e-001
3.74e-001
3.71e-001
3.67e-001
3.63e-001
3.59e-001
3.56e-001
3.52e-001
3.48e-001
3.45e-001
3.41e-001
3.37e-001
3.34e-001
3.30e-001
3.26e-001
3.23e-001
3.19e-001
3.16e-001
3.12e-001
3.09e-001
3.05e-001
3.02e-001
2.98e-001
2.95e-001
2.91e-001
2.88e-001
2.84e-001
2.81e-001
2.78e-001
2.74e-001
2.71e-001
2.68e-001
2.64e-001
2.61e-001
2.58e-001
2.55e-001
2.51e-001
2.48e-001
2.45e-001
2.42e-001
2.39e-001
2.36e-001
2.33e-001
2.30e-001
2.27e-001
2.24e-001
2.21e-001
2.18e-001
2.15e-001
2.12e-001
2.09e-001
2.06e-001
2.03e-001
2.00e-001
1.98e-001
1.95e-001
1.92e-001
1.89e-001
1.87e-001
1.84e-001
1.81e-001
1.79e-001
1.76e-001
1.74e-001
1.71e-001
1.69e-001
1.66e-001
1.64e-001
1.61e-001
1.59e-001
1.56e-001
1.54e-001
1.52e-001
1.49e-001
1.47e-001
1.45e-001
1.42e-001
1.40e-001
1.38e-001
1.36e-001
1.33e-001
1.31e-001
1.29e-001
1.27e-001
1.25e-001
1.23e-001
1.21e-001
1.19e-001
1.17e-001
1.15e-001
1.13e-001
1.11e-001
1.09e-001
1.07e-001
1.06e-001
1.04e-001
1.02e-001
1.00e-001
9.85e-002
9.68e-002
9.51e-002
9.34e-002
9.18e-002
9.01e-002
8.85e-002
8.69e-002
8.53e-002
8.38e-002
8.23e-002
8.08e-002
7.93e-002
7.78e-002
7.64e-002
7.49e-002
7.35e-002
7.21e-002
7.08e-002
6.94e-002
6.81e-002
6.68e-002
6.55e-002
6.43e-002
6.30e-002
6.18e-002
6.06e-002
5.94e-002
5.82e-002
5.71e-002
5.59e-002
5.48e-002
5.37e-002
5.26e-002
5.16e-002
5.05e-002
4.95e-002
4.85e-002
4.75e-002
4.65e-002
4.55e-002
4.46e-002
4.36e-002
4.27e-002
4.18e-002
4.09e-002
4.01e-002
3.92e-002
3.84e-002
3.75e-002
3.67e-002
3.59e-002
3.51e-002
3.44e-002
3.36e-002
3.29e-002
3.22e-002
3.14e-002
3.07e-002
3.01e-002
2.94e-002
2.87e-002
2.81e-002
2.74e-002
2.68e-002
2.62e-002
2.56e-002
2.50e-002
2.44e-002
2.39e-002
2.33e-002
2.28e-002
2.22e-002
2.17e-002
2.12e-002
2.07e-002
2.02e-002
1.97e-002
1.92e-002
1.88e-002
1.83e-002
1.79e-002
1.74e-002
1.70e-002
1.66e-002
1.62e-002
1.58e-002
1.54e-002
1.50e-002
1.46e-002
1.43e-002
1.39e-002
1.36e-002
1.32e-002
1.29e-002
1.25e-002
1.22e-002
1.19e-002
1.16e-002
1.13e-002
1.10e-002
1.07e-002
1.04e-002
1.02e-002
9.90e-003
9.64e-003
9.39e-003
9.14e-003
8.89e-003
8.66e-003
8.42e-003
8.20e-003
7.98e-003
7.76e-003
7.55e-003
7.34e-003
7.14e-003
6.95e-003
6.76e-003
6.57e-003
6.39e-003
6.21e-003
6.04e-003
5.87e-003
5.70e-003
5.54e-003
5.39e-003
5.23e-003
5.08e-003
4.94e-003
4.80e-003
4.66e-003
4.53e-003
4.40e-003
4.27e-003
4.15e-003
4.02e-003
3.91e-003
3.79e-003
3.68e-003
3.57e-003
3.47e-003
3.36e-003
3.26e-003
3.17e-003
3.07e-003
2.98e-003
2.89e-003
2.80e-003
2.72e-003
2.64e-003
2.56e-003
2.48e-003
2.40e-003
2.33e-003
2.26e-003
2.19e-003
2.12e-003
2.05e-003
1.99e-003
1.93e-003
1.87e-003
1.81e-003
1.75e-003
1.69e-003
1.64e-003
1.59e-003
1.54e-003
1.49e-003
1.44e-003
1.39e-003
1.35e-003
1.31e-003
1.26e-003
1.22e-003
1.18e-003
1.14e-003
1.11e-003
1.07e-003
1.04e-003
1.00e-003
9.68e-004
9.35e-004
9.04e-004
8.74e-004
8.45e-004
8.16e-004
7.89e-004
7.62e-004
7.36e-004
7.11e-004
6.87e-004
6.64e-004
6.41e-004
6.19e-004
5.98e-004
5.77e-004
5.57e-004
5.38e-004
5.19e-004
5.01e-004
4.83e-004
4.66e-004
4.50e-004
4.34e-004
4.19e-004
4.04e-004
3.90e-004
3.76e-004
3.62e-004
3.49e-004
3.37e-004
3.25e-004
3.13e-004
3.02e-004
2.91e-004
2.80e-004
2.70e-004
2.60e-004
2.51e-004
2.42e-004
2.33e-004
2.24e-004
2.16e-004
2.08e-004
2.00e-004
1.93e-004
1.85e-004
1.78e-004
1.72e-004
1.65e-004
1.59e-004
1.53e-004
1.47e-004
1.42e-004
1.36e-004
1.31e-004
1.26e-004
1.21e-004
1.17e-004
1.12e-004
1.08e-004
1.04e-004
9.96e-005
9.57e-005
9.20e-005
8.84e-005
8.50e-005
8.16e-005
7.84e-005
7.53e-005
7.23e-005
6.95e-005
6.67e-005
6.41e-005
6.15e-005
5.91e-005
5.67e-005
5.44e-005
5.22e-005
5.01e-005
4.81e-005
4.61e-005
4.43e-005
4.25e-005
4.07e-005
3.91e-005
3.75e-005
3.59e-005
3.45e-005
3.30e-005
3.17e-005
3.04e-005
2.91e-005
2.79e-005
2.67e-005
2.56e-005
2.45e-005
2.35e-005
2.25e-005
2.16e-005
2.07e-005
1.98e-005
1.89e-005
1.81e-005
1.74e-005
1.66e-005
1.59e-005
1.52e-005
1.46e-005
1.39e-005
1.33e-005
1.28e-005
1.22e-005
1.17e-005
1.12e-005
1.07e-005
1.02e-005
9.77e-006
9.34e-006
8.93e-006
8.54e-006
8.16e-006
7.80e-006
7.46e-006
7.12e-006
6.81e-006
6.50e-006
6.21e-006
5.93e-006
5.67e-006
5.41e-006
5.17e-006
4.94e-006
4.71e-006
4.50e-006
4.29e-006
4.10e-006
3.91e-006
3.73e-006
3.56e-006
3.40e-006
3.24e-006
3.09e-006
2.95e-006
2.81e-006
2.68e-006
2.56e-006
2.44e-006
2.32e-006
2.22e-006
2.11e-006
2.01e-006
1.92e-006
1.83e-006
1.74e-006
1.66e-006
1.58e-006
1.51e-006
1.43e-006
1.37e-006
1.30e-006
1.24e-006
1.18e-006
1.12e-006
1.07e-006
1.02e-006
9.68e-007
9.21e-007
8.76e-007
8.34e-007
7.93e-007
7.55e-007
7.18e-007
6.83e-007
6.49e-007
6.17e-007
5.87e-007
5.58e-007
5.30e-007
5.04e-007
4.79e-007
4.55e-007
4.33e-007
4.11e-007
3.91e-007
3.71e-007
3.52e-007
3.35e-007
3.18e-007
3.02e-007
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Page 1 of 2
Z Table Values (One Tail) from Z = 5 to Z = 9.99 Z 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70 9.80 9.90
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
2.87e-007
2.72e-007
2.58e-007
2.45e-007
2.33e-007
2.21e-007
2.10e-007
1.99e-007
1.89e-007
1.79e-007
1.70e-007
1.61e-007
1.53e-007
1.45e-007
1.37e-007
1.30e-007
1.23e-007
1.17e-007
1.11e-007
1.05e-007
9.96e-008
9.44e-008
8.95e-008
8.48e-008
8.03e-008
7.60e-008
7.20e-008
6.82e-008
6.46e-008
6.12e-008
5.79e-008
5.48e-008
5.19e-008
4.91e-008
4.65e-008
4.40e-008
4.16e-008
3.94e-008
3.72e-008
3.52e-008
3.33e-008
3.15e-008
2.98e-008
2.82e-008
2.66e-008
2.52e-008
2.38e-008
2.25e-008
2.13e-008
2.01e-008
1.90e-008
1.79e-008
1.69e-008
1.60e-008
1.51e-008
1.43e-008
1.35e-008
1.27e-008
1.20e-008
1.14e-008
1.07e-008
1.01e-008
9.55e-009
9.01e-009
8.50e-009
8.02e-009
7.57e-009
7.14e-009
6.73e-009
6.35e-009
5.99e-009
5.65e-009
5.33e-009
5.02e-009
4.73e-009
4.46e-009
4.21e-009
3.96e-009
3.74e-009
3.52e-009
3.32e-009
3.12e-009
2.94e-009
2.77e-009
2.61e-009
2.46e-009
2.31e-009
2.18e-009
2.05e-009
1.93e-009
1.82e-009
1.71e-009
1.61e-009
1.51e-009
1.43e-009
1.34e-009
1.26e-009
1.19e-009
1.12e-009
1.05e-009
9.87e-010
9.28e-010
8.72e-010
8.20e-010
7.71e-010
7.24e-010
6.81e-010
6.40e-010
6.01e-010
5.65e-010
5.30e-010
4.98e-010
4.68e-010
4.39e-010
4.13e-010
3.87e-010
3.64e-010
3.41e-010
3.21e-010
3.01e-010
2.82e-010
2.65e-010
2.49e-010
2.33e-010
2.19e-010
2.05e-010
1.92e-010
1.81e-010
1.69e-010
1.59e-010
1.49e-010
1.40e-010
1.31e-010
1.23e-010
1.15e-010
1.08e-010
1.01e-010
9.45e-011
8.85e-011
8.29e-011
7.77e-011
7.28e-011
6.81e-011
6.38e-011
5.97e-011
5.59e-011
5.24e-011
4.90e-011
4.59e-011
4.29e-011
4.02e-011
3.76e-011
3.52e-011
3.29e-011
3.08e-011
2.88e-011
2.69e-011
2.52e-011
2.35e-011
2.20e-011
2.06e-011
1.92e-011
1.80e-011
1.68e-011
1.57e-011
1.47e-011
1.37e-011
1.28e-011
1.19e-011
1.12e-011
1.04e-011
9.73e-012
9.09e-012
8.48e-012
7.92e-012
7.39e-012
6.90e-012
6.44e-012
6.01e-012
5.61e-012
5.23e-012
4.88e-012
4.55e-012
4.25e-012
3.96e-012
3.69e-012
3.44e-012
3.21e-012
2.99e-012
2.79e-012
2.60e-012
2.42e-012
2.26e-012
2.10e-012
1.96e-012
1.83e-012
1.70e-012
1.58e-012
1.48e-012
1.37e-012
1.28e-012
1.19e-012
1.11e-012
1.03e-012
9.61e-013
8.95e-013
8.33e-013
7.75e-013
7.21e-013
6.71e-013
6.24e-013
5.80e-013
5.40e-013
5.02e-013
4.67e-013
4.34e-013
4.03e-013
3.75e-013
3.49e-013
3.24e-013
3.01e-013
2.80e-013
2.60e-013
2.41e-013
2.24e-013
2.08e-013
1.94e-013
1.80e-013
1.67e-013
1.55e-013
1.44e-013
1.34e-013
1.24e-013
1.15e-013
1.07e-013
9.91e-014
9.20e-014
8.53e-014
7.91e-014
7.34e-014
6.81e-014
6.31e-014
5.86e-014
5.43e-014
5.03e-014
4.67e-014
4.33e-014
4.01e-014
3.72e-014
3.44e-014
3.19e-014
2.96e-014
2.74e-014
2.54e-014
2.35e-014
2.18e-014
2.02e-014
1.87e-014
1.73e-014
1.60e-014
1.48e-014
1.37e-014
1.27e-014
1.17e-014
1.09e-014
1.00e-014
9.30e-015
8.60e-015
7.95e-015
7.36e-015
6.80e-015
6.29e-015
5.82e-015
5.38e-015
4.97e-015
4.59e-015
4.25e-015
3.92e-015
3.63e-015
3.35e-015
3.10e-015
2.86e-015
2.64e-015
2.44e-015
2.25e-015
2.08e-015
1.92e-015
1.77e-015
1.64e-015
1.51e-015
1.39e-015
1.29e-015
1.19e-015
1.10e-015
1.01e-015
9.33e-016
8.60e-016
7.93e-016
7.32e-016
6.75e-016
6.22e-016
5.74e-016
5.29e-016
4.87e-016
4.49e-016
4.14e-016
3.81e-016
3.51e-016
3.24e-016
2.98e-016
2.75e-016
2.53e-016
2.33e-016
2.15e-016
1.98e-016
1.82e-016
1.68e-016
1.54e-016
1.42e-016
1.31e-016
1.20e-016
1.11e-016
1.02e-016
9.36e-017
8.61e-017
7.92e-017
7.28e-017
6.70e-017
6.16e-017
5.66e-017
5.21e-017
4.79e-017
4.40e-017
4.04e-017
3.71e-017
3.41e-017
3.14e-017
2.88e-017
2.65e-017
2.43e-017
2.23e-017
2.05e-017
1.88e-017
1.73e-017
1.59e-017
1.46e-017
1.34e-017
1.23e-017
1.13e-017
1.03e-017
9.48e-018
8.70e-018
7.98e-018
7.32e-018
6.71e-018
6.15e-018
5.64e-018
5.17e-018
4.74e-018
4.35e-018
3.99e-018
3.65e-018
3.35e-018
3.07e-018
2.81e-018
2.57e-018
2.36e-018
2.16e-018
1.98e-018
1.81e-018
1.66e-018
1.52e-018
1.39e-018
1.27e-018
1.17e-018
1.07e-018
9.76e-019
8.93e-019
8.17e-019
7.48e-019
6.84e-019
6.26e-019
5.72e-019
5.23e-019
4.79e-019
4.38e-019
4.00e-019
3.66e-019
3.34e-019
3.06e-019
2.79e-019
2.55e-019
2.33e-019
2.13e-019
1.95e-019
1.78e-019
1.62e-019
1.48e-019
1.35e-019
1.24e-019
1.13e-019
1.03e-019
9.40e-020
8.58e-020
7.83e-020
7.15e-020
6.52e-020
5.95e-020
5.43e-020
4.95e-020
4.52e-020
4.12e-020
3.76e-020
3.42e-020
3.12e-020
2.85e-020
2.59e-020
2.37e-020
2.16e-020
1.96e-020
1.79e-020
1.63e-020
1.49e-020
1.35e-020
1.23e-020
1.12e-020
1.02e-020
9.31e-021
8.47e-021
7.71e-021
7.02e-021
6.39e-021
5.82e-021
5.29e-021
4.82e-021
4.38e-021
3.99e-021
3.63e-021
3.30e-021
3.00e-021
2.73e-021
2.48e-021
2.26e-021
2.05e-021
1.86e-021
1.69e-021
1.54e-021
1.40e-021
1.27e-021
1.16e-021
1.05e-021
9.53e-022
8.66e-022
7.86e-022
7.14e-022
6.48e-022
5.89e-022
5.35e-022
4.85e-022
4.40e-022
4.00e-022
3.63e-022
3.29e-022
2.99e-022
2.71e-022
2.46e-022
2.23e-022
2.02e-022
1.83e-022
1.66e-022
1.51e-022
1.37e-022
1.24e-022
1.12e-022
1.02e-022
9.22e-023
8.36e-023
7.57e-023
6.86e-023
6.21e-023
5.63e-023
5.10e-023
4.62e-023
4.18e-023
3.79e-023
3.43e-023
3.10e-023
2.81e-023
2.54e-023
2.30e-023
2.08e-023
1.88e-023
1.70e-023
1.54e-023
1.39e-023
1.26e-023
1.14e-023
1.03e-023
9.32e-024
8.43e-024
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SIGMA TABLES
ZST
PPMLT (-1.5σ)
CpkLT
ZST
-6.0 -5.9 -5.8 -5.7 -5.6 -5.5 -5.4 -5.3 -5.2 -5.1 -5.0 -4.9 -4.8 -4.7 -4.6 -4.5 -4.4 -4.3 -4.2 -4.1 -4.0 -3.9 -3.8 -3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 999,999 999,999 999,998 999,997 999,995 999,991 999,987 999,979 999,968 999,952 999,928 999,892 999,841 999,767 999,663 999,517 999,313 999,032 998,650 998,134 997,445 996,533 995,339 993,790 991,802 989,276 986,097 982,136 977,250 971,284 964,070 955,435 945,201 933,193
-2.5 -2.5 -2.4 -2.4 -2.4 -2.3 -2.3 -2.3 -2.2 -2.2 -2.2 -2.1 -2.1 -2.1 -2.0 -2.0 -2.0 -1.9 -1.9 -1.9 -1.8 -1.8 -1.8 -1.7 -1.7 -1.7 -1.6 -1.6 -1.6 -1.5 -1.5 -1.5 -1.4 -1.4 -1.4 -1.3 -1.3 -1.3 -1.2 -1.2 -1.2 -1.1 -1.1 -1.1 -1.0 -1.0 -1.0 -0.9 -0.9 -0.9 -0.8 -0.8 -0.8 -0.7 -0.7 -0.7 -0.6 -0.6 -0.6 -0.5 -0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0
PPMLT (-1.5σ) CpkLT 933,193 919,243 903,199 884,930 864,334 841,345 815,940 788,145 758,036 725,747 691,462 655,422 617,911 579,260 539,828 500,000 460,172 420,740 382,089 344,578 308,538 274,253 241,964 211,855 184,060 158,655 135,666 115,070 96,801 80,757 66,807 54,799 44,565 35,930 28,716 22,750 17,864 13,903 10,724 8,198 6,210 4,661 3,467 2,555 1,866 1,350 968 687 483 337 233 159 108 72 48 32 21 13 8.5 5.4 3.4
-0.5 -0.5 -0.4 -0.4 -0.4 -0.3 -0.3 -0.3 -0.2 -0.2 -0.2 -0.1 -0.1 -0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 0.8 0.8 0.9 0.9 0.9 1.0 1.0 1.0 1.1 1.1 1.1 1.2 1.2 1.2 1.3 1.3 1.3 1.4 1.4 1.4 1.5 1.5
