Stats Notes

Stat 435/835 Statistical Methods for Process Improvement

Course Overview

Stefan Steiner, [email protected]

Background

Capstone statistics course No new statistical methods introduced But, we use what you have previously learnt

Numerical and graphical data summaries (Stat 231)

Linear regression (Stat 231 [+331]) Design of experiments (Stat 332 [+430]) Analysis of Variance (Stat 332)

Practice appropriate application Develop statistical thought process

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Course Learning Objectives

Learn about the Statistical Engineering algorithm, strategies and approaches for solving chronic excess variation problems think strategically about how to achieve cost-effective

variation reduction reduce variation by following a step-by-step algorithm learn how to use appropriate statistical plans and tools

to achieve the goal of Statistical Engineering

Understand sources of variation and their role in process improvement

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Course Learning Objectives (cont.)

Learn how to better use empirical methods; that is, learn effective and efficient ways to plan, execute and analyze the results of a process investigation

Apply the methodology to Watfactory, a virtual manufacturing process, to aid in learning Tell me, I will forget Show me, I may remember Involve me and I will understand

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Textbook

This course covers the material in textbook Statistical Engineering: An algorithm for reducing variation in manufacturing processes published by Quality Press 2005

You are expected to read the textbox on your own Download electronic version and/or Borrow text for the term from me

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Watfactory

The course (though not the textbook) is designed around a virtual process called Watfactory

Watfactory is a web based virtual process to produce camshafts demo later

Watfactory website www.student.math.uwaterloo.ca/~watfacto/login.htm

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Watfactory Project

You will be divided into teams and assigned different versions of Watfactory to improve

Watfactory projects involve: nine weekly written progress reports two class presentation on your progress two management reviews of another team

See course outline and the report and presentations guidelines for more information

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Videos

You are also expected to watch the series of the videos on your own time A suggested schedule is given in the written

course outline

The videos cover all the material from the textbook as well as the Watfactory virtual process

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MINITAB Statistical Software

General purpose statistical software Most commonly used package in the quality

improvement area Very easy to use

data window looks like a spreadsheet pull down menus to access numerical analysis and

graphs better than Excel for statistics/graphics

Used throughout these course notes and in the corresponding book

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Course Topics (Book Chapters in Brackets) Introduction Overview and Goals (1-3, 5) Watfactory The Statistical Engineering Algorithm (4) Problem Selection and Definition (6) Measurement System Analysis (7) Choosing a Variation Reduction Approach (3, 8) Finding a Dominant Cause using the Method of Elimination and

Families of Variation (9) Tools for Finding a Dominant Cause (10-12) Verification of the Dominant Cause (13) Revisiting the Choice of Variation Reduction Approach (14) Determining the Feasibility of an Approach (15-20) Implementation and Holding the Gains (21) Wrap Up and Conclusions

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Introduction

Problems are only opportunities in work clothes

Henry J. Kaiser

Variation Definition

Variation is both deviation of output from target changing value of output from part to part

V6 piston diameters target diameter = 101.591 mm measured diameters for 3 consecutive pistons:

101.593, 101.589, 101.597

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Consequences of Variation

Excess output variation leads to Poor performance Scrap and/or rework Low customer satisfaction Extra costs

sProcess improvement

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Reducing Variation

We can improve the process by Better centering to target Reducing variation among the parts

Reducing variation among parts is usually harder than moving the process center

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Truck Pull

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Pull is a critical alignment characteristic Target pull: 0.23 Newton-meters Almost all trucks in last 2 months were

within specs -0.12 to 0.58 Nm Goal: reduce pull variation about the target

Engine Block Leaks

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Cast iron engine blocks were tested for leaks

Current scrap rate was 2-3% Goal: reduce leak rate to less than 1%

Camshaft Lobe Runout

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Camshaft lobe geometry is critical Base circle run-out is a positive measure

of the max. deviation from an ideal circle Goal: reduce average run-out Issue: physical lower limit of zero

Sand Core Strength

Breakage of sand cores occurred in handling Goal: increase average core strength Issue: cores that were too strong led to

casting defects

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Crankshaft Main Diameter

Excessive main diameter variation Histogram suggests process off target Goal: move average diameter to target Issue: asymmetric costs

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Paint Film Build Vehicle paint appearance is critical Film build lower specification is 15 thou Goal: reduce film build variation Issue: reducing variation would allow decrease

in average film build and cost savings

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Refrigerator Frost Build

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Customer complaints about frost build up in frost free fridges

Goal: eliminate frost build up Issues:

difficult to measure frost except during customer usage

causes found to be in usage environment

Describing Processes

If I had to reduce my message to management to just a few words, Id say it all had to do with reducing variation

W. Edward Deming, 1900-1993

Process A series of actions which are carried out

in order to achieve a particular result. (Collins Dictionary ) Manufacturing processes

e.g. production of automobiles or automobile parts Service processes

e.g. credit applications, customer returns, Math faculty admissions

Measurement processes e.g. gauges, operators, etc. produce measurements

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Process Map

Each time the (e.g. exhaust manifold) process operates it creates a unit/part/realization

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Melting

Core Making

Molding

Pouring Shakeout Machining

Melting

Core Making

Molding

Pouring Shakeout Machining

Process Outputs and Inputs

Outputs: characteristics of the realizations of interest to the customers characteristics related to performance or ease

of assembly, e.g. strength of casting, dimensions, etc.

Inputs: features of the process itself e.g. operators, pouring temperature,

properties of the sand, etc. Inputs and Outputs can be

continuous, binary, ordinal, etc.

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Critical to Quality (CTQ)

Every manufactured product has 1+ critical to quality (CTQ) output characteristics e.g. piston head diameter, credit application

decision Often we can make the process better if

we reduce variation in the CTQ(s). CTQs typically have a target value and

specification limits e.g. 595 5 microns from nominal for piston

head diameter 5

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Output Distribution

We are interested in the distribution of output values from the process

We can summarize the output distribution graphically by histogram numerically by the center (average), standard

deviation, min, max, etc. A histogram shows the distribution of the

output values, the bar heights give the relative frequency for each range of output values

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Describing Variation Truck alignment (pull): target 0.23, specs -0.12 to 0.58, well centered good process

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Types of Problems Excessive variation Poor targeting

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3210-1-2-3

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Defect rate too high

Combination

Types of Problems

Chronic versus sporadic problems chronic problems are persistent and resist solution sporadic problems are urgent and short-lived

(firefighting) Problems with a continuous output

characteristic e.g. time, length, etc. excessive variation (high scrap and/or rework) poor targeting of the process center

Problems with a binary output characteristic, e.g. pass/fail, defective/not defective defect rate too high

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Measure of Variation (StDev)

We quantify variability (across units) as where is output for ith part and is the average

Stdev is expressed in the same measurement units as the process output

For bell shaped histograms almost all values will fall within

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iy 1,2,...,i n

Fixed and Varying Process Inputs A fixed input changes only when we deliberately

change it, e.g. control plan iron pouring temperature target value process or product design changes

A varying input naturally changes from part to part or time to time, e.g. core dimensions change from casting to casting operators change each shift raw material characteristics change each batch environmental conditions

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Causes (of Variation) Variation in the output(s) as the process

runs must have a cause! Only varying (not fixed) inputs can be

causes of this output variation Some causes have a large (or dominant)

effect others have little or no effect Denote output (y), fixed inputs (z) and

varying inputs (x), then we might model 12 1 2 1 2, ,..., , ,...Y f x x z z

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What is a Cause?

Can the product design (process design) be a large cause of output variation?

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cause

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Scatterplot of output vs cause

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Scatterplot of output vs cause

Dominant Cause of Variation We shall assume (to start) that for every

problem, there is a SINGLE DOMINANT CAUSE Pareto Principle applied to causes see the next page

Secondary causes can be identified, but the tools and strategies used in the search for a cause work best if there is a single dominant cause

The assumption is more likely to hold with a focused problem

e.g. one with a single failure mode

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Pareto Principle

First proposed by Vilfredo Pareto in 1906 80% of Italian land owned by 20% of the people 80/20 rule

Since then principle has been shown to be widely applicable

Here we apply it to causes of variation Most of the output variation can be explained

by just one or a few causes (varying inputs) Vital few, trivial many 15

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Model for Single Cause

Suppose where x represents a single cause

Then, assuming independence between the cause x and all other causes, we have

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Y f x R

2 2Rstdev Y stdev due to x V

Effect of Square Root Sum of Squares Formula

17 1.00.90.80.70.60.50.40.30.20.10.0

1009080706050403020100

standard deviation(due to cause) / standard deviation(total)

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Dominant Cause Continuous Output

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Continuous cause Discrete cause

Dominant Cause with Binary Output (Ggood, Bbad)

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Input value

G GGG B BBBCh

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Interaction and Correlation

There is an interaction between 2+ inputs if the effect on the output of changing either input depends on the level of the other input

Interaction is not to be confused with correlation between two inputs a correlation exists between two inputs if they

vary together in some way, e.g. when input1 is low, input2 also tends to be low

note two inputs can be correlated whether they have an effect on the output or not

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Cause and Output Relationship

In the search for a dominant cause we look for a strong correlation between a varying input and the output, such as

We assume that reducing the variation in a dominant cause will reduce variation in the output

However, correlation does not guarantee this! Verify assumption later

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Variation Reduction Steps

To reduce variation Juran suggests two steps Diagnostic journey find the cause(s) of

the variation Focus on varying inputs (xs)

Remedial journey find a solution To improve we must change something Focus on fixed inputs (zs)

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Solutions

To change the long term output variation (i.e. solve the problem) we will need to change one or more fixed inputs!

Change to a fixed input might help if it reduces the variation in the dominant cause changes the relationship between a dominant

cause and the output moves the process output center

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Seven Variation

Reduction Approaches

A fool can learn from his own experiences;

the wise learn from the experience of others

Democritus, 460-370 B.C.

The Seven Variation Reduction Approaches Fix the Obvious Based on Knowledge of a

Dominant Cause Desensitize the Process to Variation in a

Dominant Cause Feedforward Control Based on a Dominant Cause Feedback Control Make Process Robust to Noise 100% Inspection Change the Process Center

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Fix the Obvious Based on Dominant Cause Reduce variation in the dominant cause

Existing Process Improved Process

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Truck Pull In the early phases of improving the truck

alignment process, the team looked at right caster stratified by alignment gauge

As the trucks enter the gauges haphazardly the dominant cause is the gauges

An obvious solution was to recalibrate the gauges (and monitor them over time)

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Hubcap Damage

Customers complained of wheel trim and hubcap damage

A dominant cause of the broken retaining legs was found to be a combination of cold weather and contact with curbs.

An obvious solution was to replace the inherently brittle existing ABS hubcap with a new design made of mineral reinforced polypropylene

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Desensitization

Desensitize a process to variation in a dominant cause


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Engine Block Porosity Problem: cast iron engine block subsurface porosity Dominant cause: iron pouring temperature. Low

temp. occurred during (un)planned stoppages Using an experiment the team explored the effect of

a new core wash

Solution: change the block core wash to reduce the effect of the iron temperature variation

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alternateregular

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Feedforward Control Monitor the dominant cause and predict the

future behavior of the output If the prediction is far enough from the target,

adjust the process Existing Process Improved Process

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Truck Alignment (Pull) Pull is an important characteristic as it indicates

how well a truck will track on a standard highway Variation in truck frame geometry is a dom. cause

of variation in the key alignment characteristic left caster that affects pull

Solution: Adjustment for each alignment assembly measure geometry from bar coded label on each frame predict left caster and pull using a predictive equation drill cam to adjust predicted pull closer to target

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Feedback Control

Monitor the output characteristic and predict future behavior from current and past observations

If the prediction is far enough from the target, make an adjustment to the process


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Target

time

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V6 Piston Diameter Excess piston diameter variation was a problem Stratifying the process by streams found structural

variation in the diameters

Solution: Informal feedback controller (one on each stream) Every 15 minutes select and measure two pistons If their average is outside the range 2.7 to 10.7 (target is

6.7 microns) adjust the process center to compensate

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Make the Process Robust

Change fixed inputs to reduce the effects of unidentified causes.

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control input settings

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Paint Thickness Door paint thickness variation was a problem Dominate variability acted vehicle-to-vehicle An investigation to find the cause failed An investigation to search for more robust settings

was conducted An experiment involving five fixed inputs was conducted Each experimental run consisted of painting five

consecutive cars Performance measure was the log standard deviation of

thickness over the five cars Solution: Change the process settings

high Zone X voltage, high conductivity, low temperature

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100% Inspection

Reduce the variability by identifying and then scraping or reworking all parts that have values of the output beyond selected inspection limits

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output

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UpperInspectionLimit

LowerInspectionLimit

Blocked Exhaust Manifolds

Blocked exhaust manifold ports are very rare, but result in catastrophic failure of the engine

A blocked port is relatively difficult to detect since it is not visible

Search for a cause is difficult because blocked ports are so rare Ten year search was fruitless

Automatic 100% inspection of all manifolds using ultrasound was expensive, but outweighed the potential cost of a blocked port reaching the customer

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Change the Process Centre

Adjust process center to move it closer to the target


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Process Target

output

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Process Target

Battery Seal Leaky battery seals resulted in rework and

customer complaints Low tensile seal strength was the cause of leaks The problem was reformulated to increase the

tensile strength of the seal

An experiment looked at the effect of temp., melt time and elevator speed on the tensile strength

Solution: Low melt temp. increases seal strength

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speedtemptime

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The Seven Variation Reduction Approaches

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Process Output

control

Feedback Control

Process Output???

Making a Process Robust

Process

Output Inspection

Process

Change the Process Center

Process

control

Feed-forward ControlDominant

Cause Output

ProcessDominantCause

Desensitize Process

Output

Process Output

Fix the Obvious by ReducingVariation in a Dominant Cause

Statistical Engineering: An Algorithm for Reducing Variation in Manufacturing

Processes

Begin with the end in mind

Stephen Covey

Statistical Engineering

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A union of engineering and statistics applied to chronic manufacturing problems Statistical methods are needed to plan

investigations and to analyze the collected data

Engineering methods are needed to help plan the investigations, interpret the results and to act on the acquired information

INCREASED PROCESS KNOWLEDGE pp

OPPORTUNITIES FOR PROCESS IMPROVEMENTS

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The Key is Knowledge

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There is no substitute for knowledge W. Edwards Deming

The greatest obstacle to discovery is not ignorance it is the illusion of knowledge

Daniel Boorstin

By increasing knowledge of how and why a process behaves as it does, we will discover cost effective changes to the process that will reduce variation

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Goal of Algorithm

Quickly find a low cost solution to a chronic problem

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StatEng Algorithm Uses engineering knowledge and statistical

methods to reduce variation Statistical methods are needed to plan

investigations and to analyze the collected data

Engineering knowledge is needed to help plan the investigations, interpret the results and to act on the acquired information

Requirements for success A high volume manufacturing process A clearly defined chronic process problem A small team of dedicated problem solvers Management support and understanding

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StatEng Algorithm

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Define Focused Problem

Check the Measurement System

Find and Verify a DominantCause of Variation

Implement and Validate Solutionand Hold the Gains

Choose Working Variation Reduction Approach

Fix the ObviousDesensitize Process

Feedforward Control

Feedback ControlMake Process Robust

100% Inspection

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Assess Feasibility and PlanImplementation of Approach

Change Process (or Sub-process) Center

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Competing Algorithms

Shainin Red X Strategy Six Sigma (DMAIC, Breakthrough

Cookbook) Taguchis Parameter and Tolerance Design Demings PDSA Cycle

Statistical Engineering is more focused and prescriptive

Statistical Engineering reflects the iterative nature of real problem solving.

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Structured Problem Solving

A systematic approach to Problem Solving / Variation Reduction is good because it: Prevents jumping to incorrect solutions Is a good communication tool Encourages teamwork Is teachable Is manageable

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Define Problem and Check Measurement Stages

Part of other algorithms, but

Benefits of establishing a problem baseline can be enormous Allows us to know if the problem should be priority

and later whether we have solved problem Effects design of future investigations

The measurement system is critical Provides our only view of the process Checking the measurement system starts the search

for a dominant cause Often (in our experience) a source of trouble

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Third Stage

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Feedforward Control


100% Inspection

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Feedforward Control


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Choosing a Variation Reduction Approach Stage

Begin with the end in mind Stephen Covey 7 approaches to reduce variation

Fix the Obvious Based on a Dominant Cause Desensitize the Process to Dominant Cause Variation Feedforward Control Based on a Dominant Cause Feedback Control Make Process Robust to Noise 100% Inspection Change the Process Center

Choice of approach effects how we proceed

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Fourth Stage

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Feedforward Control


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Feedforward Control


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Finding Dominant Cause Stage

Focus on varying inputs Use families of causes and method of

elimination (more on this later) Based (mostly) on sequence of observational

studies Often the most time consuming stage Not always needed, but usually worth it

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Assessing Feasibility and Implementation Stages How to assess feasibility or implement is

different for each of the 7 approaches. e.g. not all solutions require knowledge of a

dominant cause Use designed experiments on fixed inputs

to assess possible process changes

Note: we delay the use of (expensive) designed experiments until the assessing feasibility stage of the algorithm

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StatEng Algorithm Keys Structured (Stage by Stage) Algorithm

prevents jumping to incorrect solutions is a good communication tool encourages teamwork is teachable and manageable

Selecting a working (tentative) solution approach early on to drive what we do next

Seven possible variation reduction approaches Fix the Obvious (or Reformulate) Using a Dominant Cause Desensitize the Process to Variation in a Dominant Cause Feedforward Control Feedback Control Make Process Robust 100% Inspection Change the Process Center

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StatEng Algorithm Keys (cont.)

An important consideration in the algorithm is whether or not to search for a dominant cause. looking for a dominant cause is strongly

recommend! Separating the search for a dominant cause

from the search for a solution Specific tools and strategies are associated

with the various stages in the algorithm A series of investigations is (normally) required

to find a solution

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Process Investigations

A series of investigations are required within the StatEng algorithm

Problem definition Measurement system analysis Searching for a dominant cause Verification of the dominant cause Determining if a proposed approach is

feasible Testing a proposed solution

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QPDAC (Question, Plan, Data,

Analysis and Conclusion) Framework

There is no substitute for knowledge

W. Edward Deming, 1900-1996

Observational/Experimental Plans Observational plan: observe the current process

in action does not interfere with existing process may measure inputs/outputs not usually measured usually low cost (relative to experimental plan)

Experimental plan: deliberately manipulate the values of one or more inputs (fixed or varying) usually high cost logistical challenges may need to contain produced parts as they may be of

suspect quality

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QPDAC Statistical Method

For each investigation, we propose the QPDAC (Chap. 5) framework

Specify a clear Question(s) that tells us what we want to know about the process

Develop a Plan that specifies how we will collect data to try to answer the question

Collect the Data according to the Plan Perform Analysis to summarize the data Draw Conclusions from the investigation to

(try to) answer the question

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Issues in Process Studies We want to infer how the process will

operate in the future from data collected over a short period of time

It's tough to make predictions, especially about the future Yogi Berra

How we collect the data and its quality are crucial

Process consistency is needed to make reasonable predictions

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Key Decisions in the Plan of an Empirical Investigation

What are the parts and population available for the investigation? i.e. over what time frame will we conduct the investigation? defines the study population

How will we select units to be included in the sample? includes the choice of the number of parts defines the sampling protocol

What inputs and outputs will we measure or deliberately change on the selected parts?

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study population

time0

sample

target populationstudy population

time0

sample

target population

5

Choosing the Problem

Our plans miscarry because they have no aim. When you dont know what harbor youre aiming for,

no wind is the right wind.Lucuis Annaeus Seneca, 5 BC-65 AD

First Stage of the Algorithm

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Feedforward Control


100% Inspection

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Feedforward Control


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Projects

Management should choose projects/problems based on customer and/or business requirements

(use Pareto Principle, 80/20 rule) greatest $ return lowest cost of problem solving likelihood of success availability of trained and knowledgeable people

Need management input/decisions to prioritize DO NOT start a large number of projects

simultaneously!

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Problem Definition Statistical Engineering requires a focused problem

general problems may not have a single dominant cause One project can generate several Statistical Engineering

problem solving efforts Example leaking engine blocks

Project: Reduce scrap rate due to casting defects in machined engine blocks

Problems: Eliminate three different failure modes (center, cylinder bore and rear intake wall) that caused leaks

Focusing may require studies, new measurement systems, redefinition of the problem(s).

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Link Between Projects, Problems and Investigations

Translate management defined projects into specific problems

Use StatEng algorithm to guide choice of investigation different at each stage

Use QPDAC framework to help plan, conduct and analyze each individual investigation

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Project

Problem A Problem B

Question A1Baseline

Apply StatEngAlgorithm Question A3

...Question A2Measurement

...

Connecting Rods Project to Problem

Managements goal was to reduce the rod scrap rate from 3.2% to less than 1.5% would be easier to address a more specific

problem defined in terms of a binary output (scrap or not),

we prefer a continuous output

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Rod Scrap by Day

Scrap rate fairly stable over time

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Connecting Rod Scrap Locations

Grinding (68%) was the dominant location for scrap detection looking more closely (not shown here), 90% of the scrap at grind

was due to undersized rods Rod thickness was selected to define the baseline

if thickness variation can be reduced so that undersized rods are eliminated, scrap reduction is approximately 3.2% x 0 .68 x 0.9 = 1.96%, so overall scrap rate will be approximately 1.25 % (Goal met)

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grind

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85 24 14 6 264.9 18.3 10.7 4.6 1.564.9 83.2 93.9 98.5 100.0

0

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Key Elements of Focusing a Project to One or More Problems

Identify and address the most important failure modes

Replace a binary or discrete output by a continuous one, if possible

Define the problem in terms of an output that can be measured locally and quickly (e.g. refrigerator frost buildup)

Choose the problem goal to meet the management project goal

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Process Certification Process certification is a desirable prerequisite to

Statistical Engineering FIX THE OBVIOUS!

ensure basic good process management follow standard operating procedures as written include safety, training, housekeeping, maintenance need to have a defined process before improvements

can be made Elements covered by Quality system standards

such as ISO 9000/QS 9000 Statistical control (i.e. a stable process as defined

by a control chart) is not required for Statistical Engineering to work

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Selecting an Output To define the problem we need to select an output

characteristic (or more than one) that can be used to summarize the size and nature of the problem

Select a critical process output continuous characteristic (dimension, time, ...) discrete characteristic (defect count, scrap, ...)

We can summarize the output using a performance measure, e.g. mean, standard deviation, histogram, run chart,

capability ratio, ... scrap/rework rate, run chart, cost, ...

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Quantifying the Baseline

If you know a thing only qualitatively, you know it no more than vaguely. If you know it quantitatively - grasping some numerical measure that distinguishes it from an infinite number of other possibilities you

are beginning to know it deeply. Carl Sagan, 1932-1996

First Stage of the Algorithm

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Determining the Problem Baseline To complete the first stage of the StatEng algorithm,

we must establish the problem baseline, i.e. quantify the size of the current problem

The baseline performance is used to set goals [when is the project completed?] track progress help in the search for a dominant cause!

used to plan investigations used to help in the analysis of the results of

investigations

validate success of a solution

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Problem Baseline Investigation We conduct a study (i.e. sample and measure parts from

the process) to determine the problem baseline The specific goals of this baseline investigation are to

estimate/determine the distribution of output values process center and

process standard deviation, etc. full extent of variation (FEoV) in the output nature of the process variation over time (time family

of output variation) The time family of the output variation provides strong

clues about the nature of the dominant cause (the dominant cause must act in the same time family as the output variation)

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Time Families of Variation Some outputs (causes) change a lot from one part to

the next, others change more slowly over time. e.g. raw material properties usually change slowly whereas piston dimension is different from part to

part What is slow and fast depends on your perspective

and specific process e.g. plant environment (daily/seasonal changes),

operators (change each shift) There are many time families

part to part, hour to hour, shift to shift, day to day, week to week, etc.

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Time Family Example Problem: Excessive scrap due to diameter variation in a

piston manufacturing process. To assess the time families part to part and hour to

hour suppose we measure diameter on three consecutive pistons once per hour for 12 hours output varies slowly output varies quickly

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Time Families Example

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output varies slowly output varies quickly

Uses of Time Family Knowledge

Knowing the output time family is extremely useful for planning, it helps us select an appropriate time frame (i.e. study

population) for future observational investigations

define a run for future experimental investigations

Output time family also allows us to eliminate varying inputs that act in other time families as suspect dominant causes

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Establishing the Baseline Goal: assess process performance (center and

variation), and output time families Investigation should

capture effect of all major sources of variation e.g. different machines, raw material, operators, etc.

consist of 100s (continuous output) or 1000s of parts (binary output)

use a systematic sampling plan designed to allow us to assess a variety of time families

Appropriate time frame for baseline data is key longer is better, but is more expensive how long is long enough?

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Connecting Rod Baseline Select 20 consecutive rods twice haphazardly

each day for five days, total of 200 rods Measure the thickness of each rod at each of the

four positions total of 800 thickness measurements

Questions are five days enough? How can we tell/check? are 800 measurements enough? why are the two batches of 20 consecutive rod

chosen haphazardly from within each days production?

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Row/Column Format Baseline Data

Each row represents a different rod and position

Each column gives the values for a different input

MINITAB worksheet Most convenient format

for data analysis Not the default way to

store data in Excel

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Rod Baseline Results Numerical Summaries, thickness = deviation (in thousands of an inch) from nominal (0.9 inches) mean: 34.6 standard deviation: 11, min and max: 2, 59

Histogram with specification limits 10 and 60

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thickness

Freq

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5648403224168

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0

10 60

Histogram of thickness

MINITAB Histogram

Graph , Adding reference lines for specification limits

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Setting the Problem Goal

Want to eliminate undersized rods process well centered already, so need to reduce

thickness variation Specification range is 10 to 60 thou Set goal to reduce thickness standard deviation to less than

Corresponds to a ~25% reduction from the baseline standard deviation of 11

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60 108.3

6

Stratifying the Output We can learn a lot about the process and the

nature of the dominant cause by stratifying the output in a number of ways by time family, e.g. by day, batch, etc. by location family, e.g. position

To graphically compare the distribution of output (or input) values stratified into subprocesses use an individual values plot with groups (plot on left

on next slide), or a box plot with groups (plot on right on next

slide) if the number of observations is large

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Rod Baseline Comparing Different Positions

Position 3 lower on average Would the undersized rods (scrap) problem be

solved if we could increase the average thickness at position 3?

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MINITAB Individual Values Plot Graph /sW

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MINITAB Boxplot (With Groups) Graph

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Box (and Whiskers) Plot shows a five number summary of the distribution

min, max, median, 25th and 75th percentiles a summary of a histogram turned on its side outlying observations are shown with a separate symbol

(rule for outlier vs. min or max varies with software)

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Boxplot of thickness

median

75th percentile

25th percentile

max

minoutliers

Rod Baseline Day to Day Pattern

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We see little output variation from day to day, i.e. the variation in thickness is large and roughly the same in each of the five days

Rod Baseline Time Pattern

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Little variation from batch to batch 20 consecutive parts give the FEoV

helps us choose time frame for future investigations tremendous clue about the dominant cause

Rod Baseline Time Series Plot

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Multivari Chart

The proposed sampling plan for a baseline investigation is systematic

As a result, the elapsed time between parts follows a consistent pattern but is not the same for all parts

The standard time series plot is not ideal. A multivari chart is designed for this sort of data We look at multivari investigations later when

searching for the dominant cause

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Rod Baseline Multivari Charts

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MINITAB Instructions Multivari

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Multivari Dialog Box For a multivari chart always using the option

Display individual data values Note that the factor used to define horizontal

axis is the last factor in the list

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Rod Baseline Conclusions An estimate of the long term rod thickness variation

(standard deviation, denoted ) is 11 To meet the goal we need to reduce the output

variation to around 8 Full extent of output variation (FEoV) is 2 to 59 Output varies in the part to part family Subsequent investigations conducted over a short

time interval should result in the output FEoV Dominant cause must act in the part to part and

position to position families Can almost solve the problem by increasing the

average thickness at position three by around 10 units

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yV

Baseline Over Too Short a Time Suppose we see the following hypothetical

pattern of output by day

Large day to day effect it is hard to tell what will happen tomorrow we need to collect data over many more days to

be sure that the baseline variation represents the long term behavior of the process

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54321

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Watfactory

An Online Virtual Manufacturing Environment

Tell me and I will forget. Show me and I may remember.

Involve me and I will understand. Chinese Proverb

Watfactory (Camshaft) Manufacturing Process Watfactory is designed to model a

manufacturing process that produces automotive camshafts

Consider a single output, denoted y300 The target for y300 is zero (measured from

nominal) and specification limits are -10 to 10

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Watfactory Process Map

There are three types of process characteristics one output (y), can be measured at y100, y200, y300 60 varying inputs (xs), change as the process runs 30 fixed inputs (zs), normally constant, but changeable

Machine 1

Machine 2

Machine 3

Stream 1Machine B

Stream 2Machine B

Step 200Step 300

Varying Inputsx16, ..., x25

Fixed Inputsz1, ..., z6

y300

Final Output

y200

Step 200Output

Assembly

Step 100

Varying InputsMachine #: x31

x32, ..., x45

y100

Step 100Output

Component A

Component E

Component D

Component C

Component B

Components

Varying Inputsx1, x2, x3





Varying InputsStream #: x46x47, ..., x53

Welding

Stream 2Machine A

Stream 1Machine A



Fixed InputsCan be Set by Machine

z13, ..., z18

Step 150

Assembly

Welding Heat Treatment

Step 250


z25, ..., z30

Fixed InputsCan be Set by Stream

z19, ..., z24

Machine 1

Machine 2

Machine 3

Stream 1Machine B

Stream 2Machine B

Step 200Step 300



y300

Final Output

y200

Step 200Output

Assembly

Step 100

Varying InputsMachine #: x31

x32, ..., x45

y100

Step 100Output

Component A

Component E

Component D

Component C

Component B

Components







Welding

Stream 2Machine A

Stream 1Machine A



Fixed InputsCan be Set by Machine

z13, ..., z18

Step 150

Assembly

Welding Heat Treatment

Step 250


z25, ..., z30

Fixed InputsCan be Set by Stream

z19, ..., z24

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Watfactory Process Game Management has determined that the final

output (y300 - straightness measured in microns from nominal) exhibits too much variation Your teams goal is to find a cost effective way to

reduce the variation in y300 so that (virtually) all camshafts are within the specification limits You have a budget of $10,000 to find a solution Your team will follow the Statistical Engineering

algorithm (covered in the textbook and associated videos) and conduct a series of process investigations looking for a way to reduce variation in y300

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More Process Information Process runs 3 shifts, 5 days a week, 1 part per minute i.e. 1440 camshafts are produced per day

Varying Inputs (x1, , x60) Type (continuous/categorical) Process step in which they act (assembly, welding,

heat treatment) History (pattern of variation over time) e.g. x25 is the operator in the assembly step, x42 the cooling temperature in the welding operation x50 the heating time in the heat treatment step

Fixed inputs (z1, , z30) Current level, possible range e.g. z22 is coil length in the heat treatment step

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Varying Inputs Information

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Varying Input

Description Type (# levels)

Observed Range

Varying Input

Description Type (# levels)

Observed Range

x1 dimension A continuous unknown x31 machine # categorical (3) 1, 2, 3 x2 diameter A continuous unknown x32 squeeze time continuous unknown x3 hardness A continuous -2.2, 7.2 x33 feed rate continuous unknown x4 dimension B continuous unknown x34 temperature continuous -17, 29.6 x5 diameter B continuous -17.1, 22.2 x35 dimension 1 continuous unknown x6 hardness B continuous -14.3, 18.5 x36 electrode force continuous unknown x7 dimension C continuous unknown x37 humidity continuous unknown x8 diameter C continuous -20, 26.6 x38 dimension 2 continuous 0.2, 10.4 x9 hardness C continuous unknown x39 mandrel position continuous -1.5, 12.1

x10 dimension D continuous unknown x40 weld time continuous unknown x11 diameter D continuous unknown x41 load time continuous -1.6, 9.4 x12 hardness D continuous -7.5, 19.6 x42 cooling temp. continuous unknown x13 dimension E continuous unknown x43 spacing continuous unknown x14 diameter E continuous unknown x44 operator categorical (5) 1, 2, , 5 x15 hardness E continuous -12.6, 21.2 x45 fixture categorical (12) 1, , 12 x16 temperature continuous unknown x46 stream # categorical (2) 1, 2 x17 fixture categorical (5) 1, 2, , 5 x47 power density continuous unknown x18 humidity continuous -3.0, 12.2 x48 induction level continuous -20, 26.2 x19 ball size continuous unknown x49 frequency continuous -14.5, 29.5 x20 orientation categorical (3) 1, 2, 3 x50 heating time continuous 0.9, 8.4 x21 position continuous unknown x51 operator categorical (4) 1, 2, 3, 4 x22 pressure continuous -10.3, 22.4 x52 depth continuous unknown x23 force continuous unknown x53 coupling degree continuous unknown x24 offset continuous -2.5, 6.9 x54 surface area continuous -14, 21 x25 operator categorical (3) 1, 2, 3 x55 coil categorical (8) 1, , 8 x26 temperature continuous -10.6, 15 x56 current continuous unknown x27 fixture categorical (5) 1, 2, , 5 x57 hold time continuous unknown x28 operator categorical (4) 1, 2, 3, 4 x58 air gap continuous unknown x29 power continuous unknown x59 inductance continuous -8.8, 13.4 x30 static continuous unknown x60 quench temp. continuous -4.3, 8.8

6

Input Time Family Information

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Watfactory Login Web site: www.student.math.uwaterloo.ca/~watfacto/login.htm Login ID and Password are given at registration (each

team has access to a different copy of the process) A guest login (to a different version of the process) is

also available

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Team Home Page Gives summary information on virtual date remaining funds y300 specification limits links to more information

You can request data from previous

studies change your password see investigation/solution

history

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Available Empirical Investigations Observational: prospective, retrospective Experiments: with varying inputs, fixed inputs or both Offline experiments: e.g. component swap Solutions: process changes

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Conducting Investigations For each investigation you need to specify Type of investigation

(observation/experimental,) What input(s): x1, , x60 (if any) and/or

output(s): y100, y200, y300 to measure How many parts and which parts (camshafts) to

measure For experimental plans you also need to specify

which fixed inputs (z1, z30) and/or varying inputs (x1, , x60) to control to which levels and when

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Investigation Costs There is a cost (in $ and time) for each investigation. Cost influences: Type of investigation Prospective/observational studies are cheaper Number of parts

Which inputs/outputs are selected. Cost/part measuring each input and output, e.g. $1/part for y300 tracing parts through the process, i.e. matching inputs

and/or outputs measured at different processing steps The cost for each investigation you conduct is recorded! Costs can be determined before running an investigation

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Prospective Measurement Costs

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Process Step

Varying Input

Measurement Costs Per

Part

Process Step

Varying Input

Measurement Costs Per

Part x1 3 x31 2 x2 2 x32 1 x3 5 x33 1 x4 2 x34 1 x5 3 x35 2 x6 5 x36 2 x7 1 x37 1 x8 1 x38 1 x9 5 x39 1

x10 1 x40 1 x11 3 x41 4 x12 5 x42 2 x13 3 x43 2 x14 2 x44 1

Components

x15 5

200

x45 1 x16 1 250, 300 x46 1 x17 1 x47 1 x18 2 x48 2 x19 1 x49 2 x20 1 x50 2 x21 1 x51 1 x22 2 x52 1 x23 6

250

x53 1 x24 1 x54 2

100

x25 1 x55 1 x26 1 x56 2 x27 1 x57 12 x28 1 x58 1 x29 5 x59 1

150

x30 2

300

x60 2

13

Tracing Costs (per part) Tracing costs are applicable when inputs and/or

outputs are measured at different process steps. This cost accounts for the additional expense of

tracing parts through the manufacturing process. Tracing costs are in addition to the standard

measurement costs for any input or

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Output Based Tracing Costs Per Part Upstream

Output Downstream

Output Cost

y100 y200 12 y100 y300 22 y200 y300 10

Input Based Tracing Costs Per Part Inputs Cost Links to Output

Components (x1, , x15) 6 y100 Step 100 (x16, , x25) 3 y100 Step 150 (x26, x30) 6 y200 Step 200 (x31, , x45) 3 y200 Step 250 (x46, , x53) 6 y300 Step 300 (x54, , x60) 3 y300

Investigation Time Virtual time elapses when you conduct

investigations in Watfactory Time elapsed depends on your choice of study

population time elapsed is rounded up to nearest shift minimum investigation time is 1 shift

Your team home page shows your current virtual time in terms of weeks, days and shifts since the start

15

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Other Investigations There are also special investigation costs and time

associated with the other types of investigations such as measurement assessment retrospective assembly versus components component swap experiments with varying inputs, fixed inputs or both

We cover these costs and time elapsed later when the investigation is needed 16

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Possible Solutions Goal is to reduce variation in y300 a solution requires a process change

Possible solutions include changing 1+ fixed input (z1,z30) adding 100% inspection reducing varying input variation (x1,,x60) adding a feedback controller adding a feedforward controller

Solution cost (per part) depends on the type of solution selected

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Hints and Suggestions Use the StatEng algorithm To find a solution look for a dominant cause(s) of output

variation use a series of studies focus on fixed inputs that act in the same

processing step as the dominant cause(s) Use only process knowledge obtained

inside Watfactory (realistic, but not real)

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Organization of Data and Results You will conduct a series of empirical investigation in

Watfactory Often the plan for the next investigation will best be

determined using knowledge gained from previous investigations As a result, it is helpful to stay organized Suggestions Create a new Minitab project with a sensible name

for each investigation Summarize the plan and conclusions from each

investigation in a single document

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Watfactory Project Reports

Summarize progress in 9 written reports Each report describes 1+ investigation plan investigation collect the data in Watfactory conduct an analysis to draw conclusions write a short report that summarizes your rationale

for choices made in the plan and gives a summary of your conclusions

Use the QPDAC (Question, Plan, Data, Analysis and Conclusion) framework to organize each report

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Available Watfactory Videos Baseline/prospective investigation Measurement system assessment

Assembly/disassembly and component swap investigations Retrospective investigations Experimental (with varying and/or fixed

inputs) investigations Possible solutions

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1st Watfactory Investigation Establishing the Baseline

, 4 BC AD 65

Prerequisites

Watched videos and read textbook for Chapters 1-6 Chapter 6 covers the baseline investigation

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Current Algorithm Stage

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Implement Approach



Feedforward Control


100% Inspection

Validate Solutionand Hold the Gains

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Assess Feasibility of Approach


Baseline Goals

Complete the first stage of the Statistical Engineering algorithm by Estimating the process variability, i.e. , and

center Determining the full extent of variation (FEoV) in

the output Determining the time pattern in the output

variation, e.g. does the output vary a lot from part to part, hour to hour, shift to shift, day to day,

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yVyP

Investigation Plan

Select a plan to address the baseline goals Decide what inputs/outputs to measure Choose the study population period of time when you collect data

Select a sampling protocol and sample size

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Investigation Cost and Time

Costs See prospective investigation costs in the

Watfactory introduction video or Watfactory diagnostic journey written summary Recall measurement and tracing costs

Elapsed Time Depends on study population Should not be more than 1 week (at least for first

baseline investigation)

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Baseline Investigation Selection

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Random Sampling Example

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Systematic Sampling Example

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Stats Notes

Documents

Transcript of Stats Notes