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Simulation and Optimizationin Six Sigma Projects

Crystal Ball Six Sigma Partner Program

Crystal Ball for Six Sigma (DMAIC) Module

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Crystal Ball for Six Sigma

What Is Included in Six Sigma Training Module?

The module includes:

• Training module notes (Six Sigma Training Module Notes.doc)

• PowerPoint slides for teaching the basics and the Simulation with DoE case study (Crystal Ball Module for Six Sigma.ppt)

• Simulation with DoE case study model without Crystal Ball enhancements (Simulation with DoE and Cost Exercise.xls)

• Simulation with DoE case study model with Crystal Ball enhancements (Simulation with DoE and Cost Sim Solution.xls)

• Simulation with DoE case study model with Crystal Ball and OptQuest enhancements

(Simulation with DoE and Cost Opt Solution.xls)

• Optimization settings file for Simulation with DoE case study (Simulation with DoE and Cost Sim Solution.opt)

• Step-by-step class exercise for Simulation with DoE case study(Simulation with DoE and Cost Solution.doc)

• Overview of Crystal Ball handout(Common Questions about Crystal Ball.doc)

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How Best to Start?

• The best way to begin is to review the slides to see what is covered. You are free to edit these slides as necessary. They feature basic Crystal Ball information, application details of Crystal Ball for Six Sigma (DMAIC), and a review of the Simulation with DoE case study.

• Next, we suggest that you open the Simulation with DoE and Cost Exercise.xls model and walk through the step-by-step final solution described in Simulation with DoE and Cost Solution.doc . You can use the Simulation with DoE and Cost Sim Solution.opt file to import the optimization settings into OptQuest, but it is not mandatory for the exercise.

• Once you are familiar with the model, you can apply your own teaching methods to this case study. In particular, you will need to determine whether or not you want the students to run through the exercise on their own or as a group.

• The handout Common Questions about Crystal Ball.doc is an additional reference meant to answer common questions concerning the software.

Crystal Ball for Six Sigma

Simulation and Optimizationin Six Sigma Projects

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Crystal Ball for Six Sigma

Topics Covered in This Module

• Defining Simulation Models

• Monte Carlo Simulation

• What Is Crystal Ball?

• Benefits of Simulation and Optimization for Six Sigma

• DoE Example – Simulation

• DoE Example – Optimization

• Additional Resources

Introductory Concepts

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Crystal Ball for Six Sigma

MODELS SIMULATION

Y = f (x)

Y = f (x)

NoiseVariables

Models and Simulation

• Models are an attempt to capture behavior and performance of business processes and products.

• Simulation is the application of models to predict future outcomes with known and uncertain inputs.

ControlInputs

1 2 3 LO HI

OutcomePredictions

amF

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Crystal Ball for Six Sigma

What is a Mathematical Model?

• Models come in many different forms

– Mathematical relationships based on established physical principles

– Regression equations derived from historical data

– Design of Experiments (DOE) response equations from measured observations

– General knowledge of business system or product

Y = β0 + β1x1 + β2x2 + β12x1x2 + β11x12 + β22x2

2

440

016

idd

Td

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Crystal Ball for Six Sigma

Models and Simulation

• Once the business process or product behavior is captured with mathematical and logical statements:

– Place the model into Excel

– Apply Crystal Ball probabilistic methods

A B C

1 $ $ $

2 $ $ $

Y = f (x)

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Crystal Ball for Six Sigma

What Is Monte Carlo Simulation?

• A system that uses random numbers to measure the effects of uncertainty.

• A computer simulation of N trials where

– Each trial samples input values from defined probability distribution functions (PDFs)

– Applies the input values to the model and records the output

– Sampling statistics then utilized to characterize output variation (mean, standard deviation, fitted probability distributions)

• Outputs:

– Prediction of Output Variation (DPU, Cpk, PPM, Z-score)

– Identification of Primary Variation Drivers (Sensitivity Analysis)

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Probability Distributions As Inputs Simulation requires probabilistic inputs.

Distributions use ranges of values and assign a likelihood of occurrence for values (e.g., a normal distribution could represent variation of the part dimensions).

Range

Probability

Parameters

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Crystal Ball for Six Sigma

Monte Carlo Simulation Results As Outputs

Number of simulation trials

performed

Lower Spec Limit (LSL)

Upper Spec Limit (USL)Certainty (probability)

that the forecast lies between LSL and USL

Parts within the spec limits are shown in blue, parts outside spec limits are shown red

Explore the range of possible outcomes AND the probability of their occurrence

Quality Metricssuch as Cpk, ZST, p(N/C), etc....

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Crystal Ball for Six Sigma

Sensitivity Analysis: A Critical Tool

• Examine which few critical factors (X’s) in your analysis cause the predominance of variation in the response variable of interest (Y)

• Operates during the simulation, calculating the relationships between all X’s and Y’s

• Similar to Pareto Chart in interpretation but is not a Main Effects plot

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Sensitivity Analysis: Using the Results

• Acts as communication tool to help team understand what’s driving defects

• Generally see a few factors having strongest impact on forecast variation

• Shows where to focus your energies (and where not to focus them)

• After reducing the variation for these few critical X’s, you can rerun the simulation and examine the effects on the output

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Next Step: Stochastic Optimization

Simulation can help you to understand and reduce variation but does not by itself offer the best solution.

The combination of simulation and optimization lets you make the best (optimal) decisions while accounting for the variability or uncertainty inherent within a process.

You will see this at work in the DoE with Simulation Example.

An optimization model answers the question "What's best?" rather than "What happened?" (statistics), "What if?" (simulation) or "What will happen?" (forecasting).

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Crystal Ball for Six Sigma

What is Stochastic Optimization?

Y1-max

X3X2X1

Y3-minY2-target

Y2 = f2(X1,X3,X4)

X4

Y1 = f1(X1,X2,X3) Y3 = f3(X2,X3,X4)

? ? ?• Stochastic optimization finds

the best solution while using the results of simulation.

• Goal: Determine a set of input values that will influence multiple outputs to target values.

• Example: Decrease Process Cost and Cycle Time while meeting quality requirements

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Crystal Ball for Six Sigma

Stochastic Optimization

• X contains natural variation (x)

• Y required to be between YR1 and YR2 (LSL & USL) with an acceptable defect rate of 3 sigma

Y

x

Yopt

Xgood including x

YR1

YR2

Y

x

Yopt

Xbad including x

YR1

YR2

GOOD BADUnacceptableDefect Rate

AcceptableDefect Rate

What Is Crystal Ball?

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Crystal Ball for Six Sigma

• Crystal Ball Excel-based Monte Carlo simulation tool, includes plug-in tools for setup and analysis (CB Tools), distribution fitting, sensitivity analysis, and output charts and reports

• OptQuest Global optimization for uncertain models

• CB Predictor Time-series forecasting and multiple linear regression

• Crystal Ball and CB Predictor Developer Kits VBA customization tools

is a suite of software for Microsoft® Excel

Professional Edition includes:

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Crystal Ball for Six Sigma

How Does Crystal Ball Appear in Excel?

Define Menu Run Menu Analyze Menu

Toolbar

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Crystal Ball for Six Sigma

How Does Crystal Ball Work (in Six Sigma terminology)?

Here’s another way to describe how simulation works:

• Describe the Effect (Y) as a function of the causal Factors (X’s)

• Describe Factors using probability distributions (e.g., Normal, Uniform, Binomial, etc.)

• Repeatedly Sample the Input Factors (X’s) and Compute the Effect (Y)

• Describe the Distribution of the Effect (Y) and plot in a histogram

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Crystal Ball for Six Sigma

DEFI

NE

MEA

SU

RE

AN

ALY

ZE

IMPR

OV

EC

ON

TR

OLTypical Crystal Ball Roles in Six Sigma

Projects

6 PHASES• Monte Carlo Simulation and

Optimization can be used in variety of Six Sigma phases

– DEFINE: Project Selection

– ALL PHASES: Service Process

– ANALYZE/DESIGN: Process Simulation and Optimization (Strongest Application)

• Crystal Ball does not replace other statistical packages (Minitab or JMP)

– It complements other codes by incorporating their outputs (input variable characterization and response models) into simulations and optimizations

Tra

nsa

ctio

nal S

erv

ice P

roce

ss S

imu

lati

on

Project Selection

DoE

MSA

FISHBONE

Process Simulation

andStochastic

Optimization

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Crystal Ball for Six Sigma

Benefits of Simulation in Six Sigma Projects

• Reduce the Uncertainty Around Project Success

– Account for uncertainty of costs and success in initial stages

– Understand impacts on customer satisfaction and profitability and prioritize opportunities

• Improve Your Understanding of the Critical X’s

– Discover and validate underlying causes of variation and waste

– Use simulation to predict variation where data is minimal or non-existent

• Evaluate Effects of Process Changes Prior to Implementation

– Save on expenses and resources by experimenting first

– Build team consensus and gain early approval of process owners

Case Study: DoE with Simulation

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Crystal Ball for Six Sigma

Problem Statement

• An Injection Mold Process has resulted in incomplete filling of the mold or different part lengths. A Six Sigma Project team has been assigned to reduce the variation not meeting length requirements.

• Customer: Part Buyers

• Approach:

– Perform 23 Full Factorial DoE (5 replicates) to determine Response Surface model of Part Length

– Use Crystal Ball Capability features to predict current quality metrics

– Use OptQuest Optimization techniques to determine process settings that minimize process cost while meeting minimum quality targets.

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Case Study Overview by Phase

Define

Control

- Measure current parameter capabilityMeasure

Analyze

- Perform Design of Experiments- Characterize current process state with simulation- Determine variation drivers w/ Sensitivity Analysis- Address drivers and reiterate simulation

- Review problem statement

Improve - Optimize design for cost and performance

- Run capability study on proposed process settings to confirm quality

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Crystal Ball for Six Sigma

Step 1: Measure Current Parameter Capability

• As part of the Measure Phase, the variation of the Control Parameters (Inputs, Factors) is characterized during Capability Studies

– Input Factors are Mold Temp, Cycle Time, and Hold Pressure

– 30 samples of each are made during the studies and Factors are assumed to behave normally

Each set of samples passes Normality Test

Means and Standard Deviations are recorded

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Step 2: Perform Design of Experiments

• 23 Full Factorial DOE with 5 replicates is performed (40 runs)

– RESPONSE: Part Length

– FACTORS : LO HI

Mold Temperature (x1) 100 200

Cycle Time (x2) 60 140

Hold Pressure (x3)120 140

• Response polynomial equation developed (R2

adj = 92.5%)

– 3 Main Effects

– 1 Interaction Term

Measure

Define

Analyze

Improve

Control Y = β0 + β1x1 + β2x2 + β3x3 + β23x2x3

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Crystal Ball for Six Sigma

Step 3: Characterize Current Process State

• Define the Inputs (Factors) as Normal Assumptions (Cells E5:E7)

– Cell Reference Assumption Name from Column B

– Cell Reference Assumption Mean from Column F

– Cell Reference Assumption StDev from Column G

• Define the Response (Length in Cell E9) as a Forecast

– Cell Reference the LSL from Cell F9

– Cell Reference the USL from Cell G9

• Run Simulation

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Monte Carlo Simulation to Predict Variation

Nominal Response of 64.59 mm close to target but 2% will fall out of the spec limits! → Sigma Level of ~ 2.0

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Step 4: Review Sensitivity Analysis

• Run Sensitivity Analysis to determine major driver of variation.

• Can anything be done to reduce standard deviation of Mold Temperature?

– Assume standard deviation can be reduced by 50% in Cell G5.

– Run simulation.

Measure

Define

Analyze

Improve

Control

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Step 5: Reiterate Monte Carlo Analysis

• Run Monte Carlo again → ~ 1% are out of specification → Sigma Level of ~ 2.5

• The Part Length quality has been improved

– Can it be improved even more while minimizing cost to run the process?

Measure

Define

Analyze

Improve

Control

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Step 6: Optimize Design for Cost & Performance

How can the process settings be configured so that a minimum quality goal is reached while reducing the process cost per part?

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Optimize Design for Cost & Performance

• Must consider relationship between process parameters and cost.

– Energy consumed by molding equipment is proportional to product of Cycle Time and Mold Temperature ($ ∞ Temp * Time)

– Labor Cost to run molding equipment proportional to Cycle Time ($ ∞ Time)

• Create Cost Response as a function of

– Cycle Time

– Mold Temperature

• Define Process Cost Forecast (Cell E10)

$PROCESS = K1*Temp*Time + K2*Time

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Exercise: Process DoE Optimization

• Characterize Current Quality Levels (Cpk & ZST)

– Enable Capability Metrics in Run Preferences

– In Define Forecast, use cell references for LSL & USL and auto-extract Capability Metrics

• Assuming you can control the nominal process settings but not the variation, use Optimization to determine the settings that results in the best quality (maximum Z-score)

• Process Parameters

– Mold Temp → LO (100) to HI (200), Step = 10

– Cycle Time → LO (60) to HI (140), Step = 1

– Hold Pressure → LO (120) to HI (140), Step = 2.5

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Helping You Optimize: Decision Variables

Decision variables are Crystal Ball model elements for quantities over which you have control (e.g., percentage of dollars to allocate in a project, amount of product to produce, man-hours required for a project, unit cost for a given product, go/no-go decision).

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Crystal Ball for Six Sigma

Define Decision Variables

• Define Decision Variable Lower and Upper Bounds of all Factor means (Cells E5:E7) by cell referencing corresponding adjacent cells:

Cell reference Name from Column B

Cell reference Upper Bound from Column C (LO)

Cell reference Lower Bound from Column E (HI)

• Ensure the correct Discrete Step Size is used within each Decision Variable as listed below

Decision Variables

Lower Bound Upper Bound Discrete Step Size

Mold Temp 100 200 10

Cycle Time 60 140 1

Hold Pressure 120 140 2

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

OptQuest: A Blend of Approaches

OptQuest excels at stochastic optimization because it:

• Uses several optimization techniques (Scatter Search and Advanced Tabu Search) vs. relying on a single method or genetic algorithm,

• Employs heuristics (problem solving techniques that use self-education to improve performance),

• Has both short-term and long-term Adaptive Memory,

• Can escape local optimal solutions to find global optimal solution,

• Uses neural network technology that predicts performance after only running 10% of simulation and typically reduces number of required simulations by 50%, and

• Features a wizard tool that makes setup easy.

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Crystal Ball for Six Sigma

Optimize Design for Cost & 4 Performance

• Run OptQuest and Define Forecast Selections

Optimization Goals:

– Primary is to Minimize Cost

– Requirement is to Reduce Variation of Part Length to 4 levels

Zst required to have a lower bound of 4

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Optimize Design for Cost & 4 Performance

New Design results in a Process Cost of $1.16 per part and increase to 4 quality!

Measure

Define

Analyze

Improve

Control

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Crystal Ball for Six Sigma

Comparison of Design Performance & Cost

Iter-ation

#

Mold Temp Mean

Mold Temp StDev

Cycle Time Mean

Cycle Time

StDdev

Hold Press Mean

Hold Press StDev

Sigma Level

of Flow Rate

Process Cost

1 160 10 100 10 130 5 1.94 $2.03

2 160 5 100 10 130 5 2.53 $2.03

3 150 5 61 10 140 5 4.01 $1.16

Where have we been, and where are we going?

Six Sigma team proceeds to run Capability Study on proposed process settings to confirm quality during Control phase.

Measure

Define

Analyze

Improve

Control

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Case Study Conclusions

• Quality Levels will be increased by decreasing variation on driving input variables.

– Monte Carlo analysis predicts quality levels.

– Sensitivity analysis identified Mold Temperature as most influential design variable.

• Process Cost decreases with decreasing Mold Temperature and Cycle Time.

– Simply reducing the Temp and Time to their lowest allowed value would result in unacceptable Part Length quality.

• Stochastic Optimization of input variable (Factor) means will increase Part Length quality levels while minimizing Process Cost impact.

Measure

Define

Analyze

Improve

Control

Summary

Crystal Ball for Six Sigma

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Crystal Ball for Six Sigma

Benefits of Simulation in Six Sigma

• Use insights into what drives variation to improve process or product

• Little or no customer exposure to a “bad” process, product, or service

• Easy to “change design” — can perform “what-if” analysis with only a mouse click — prior to implementation

• Virtual implementation of process changes means little or no waste of materials or staff resources

• Instant feedback of results

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Crystal Ball for Six Sigma

Additional Crystal Ball Resources

• Other Six Sigma Example Models with Crystal Ball 7.2

– In Excel/CB: Help > Crystal Ball > Examples Guide

• Process Capability Guide

– Start > All Programs > Crystal Ball 7 > Documentation > Process Capability Guide

• Crystal Ball Website (www.crystalball.com)

– Risk Resources > Case Studies

– Risk Resources > Example Models

– Training > Course List

– Six Sigma - Articles, Papers & Success Stories (www.crystalball.com/sixsigma/papers.html)

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