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**Crystal Ball Six Sigma Partner Program**

Crystal Ball for Six Sigma (DMAIC) Module Simulation and Optimization in Six Sigma Projects

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**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.

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**Crystal Ball for Six Sigma**

Simulation and Optimization in Six Sigma Projects

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**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

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**Introductory Concepts**

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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. MODELS SIMULATION Control Inputs 1 2 3 LO HI Noise Variables Y = f (x) Y = f (x) Outcome Predictions

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**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 + β22x22

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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 Y = f (x) A B C 1 $ 2

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**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). Probability Range Parameters

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**Monte Carlo Simulation Results As Outputs**

Explore the range of possible outcomes AND the probability of their occurrence Number of simulation trials performed Parts within the spec limits are shown in blue, parts outside spec limits are shown red Lower Spec Limit (LSL) Upper Spec Limit (USL) Quality Metrics such as Cpk, ZST, p(N/C), etc.... Certainty (probability) that the forecast lies between LSL and USL

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**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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**What is Stochastic Optimization?**

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 Y1-max X3 X2 X1 Y3-min Y2-target Y2 = f2(X1,X3,X4) X4 Y1 = f1(X1,X2,X3) Y3 = f3(X2,X3,X4) ?

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**Stochastic Optimization**

X contains natural variation (sx) Y required to be between YR1 and YR2 (LSL & USL) with an acceptable defect rate of 3 sigma GOOD BAD Acceptable Defect Rate Unacceptable Defect Rate Y Y YR2 YR2 Use animations to illustrate concepts of constraints and requirements. Yopt Yopt YR1 YR1 x x Xgood including sx Xbad including sx

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What Is Crystal Ball?

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**is a suite of software for Microsoft® Excel**

Professional Edition includes: 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

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**How Does Crystal Ball Appear in Excel?**

Toolbar Define Menu Run Menu Analyze Menu

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**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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**Typical Crystal Ball Roles in Six Sigma Projects**

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 6s PHASES DEFINE Project Selection FISHBONE MSA MEASURE DoE ANALYZE Process Simulation and Stochastic Optimization Transactional Service Process Simulation IMPROVE CONTROL

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**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

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**Case Study: DoE with Simulation**

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**Define Measure Analyze Improve Control 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. Define Measure Analyze Improve Control

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**Case Study Overview by Phase**

Define - Review problem statement Measure - Measure current parameter capability - Perform Design of Experiments - Characterize current process state with simulation Determine variation drivers w/ Sensitivity Analysis Address drivers and reiterate simulation Analyze Improve - Optimize design for cost and performance Control - Run capability study on proposed process settings to confirm quality

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**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 Define Measure Analyze Improve Control

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**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) Cycle Time (x2) Hold Pressure (x3) Response polynomial equation developed (R2adj = 92.5%) 3 Main Effects 1 Interaction Term Define Measure Analyze Improve Control Y = β0 + β1x1 + β2x2 + β3x3 + β23x2x3

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**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 Define Measure Analyze Improve Control

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**Monte Carlo Simulation to Predict Variation**

Nominal Response of mm close to target but 2% will fall out of the spec limits! → Sigma Level of ~ 2.0 Define Measure Analyze Improve Control

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**Step 4: Review Sensitivity Analysis**

Run Sensitivity Analysis to determine major driver of variation. Define Measure Analyze Can anything be done to reduce standard deviation of Mold Temperature? Assume standard deviation can be reduced by 50% in Cell G5. Run simulation. 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? Define Measure Analyze Improve Control

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**Step 6: Optimize Design for Cost & Performance**

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

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**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) Define Measure Analyze Improve $PROCESS = K1*Temp*Time + K2*Time Control

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**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 Define Measure Analyze Improve Control

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**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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**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 Define Measure Analyze Decision Variables Lower Bound Upper Bound Discrete Step Size Mold Temp 100 200 10 Cycle Time 60 140 1 Hold Pressure 120 2 Improve Control

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**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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**Optimize Design for Cost & 4s Performance**

Run OptQuest and Define Forecast Selections Optimization Goals: Primary is to Minimize Cost Requirement is to Reduce Variation of Part Length to 4s levels Zst required to have a lower bound of 4 Define Measure Analyze Improve Control

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**Optimize Design for Cost & 4s Performance**

Define New Design results in a Process Cost of $1.16 per part and increase to 4s quality! Measure Analyze Improve Control

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**Comparison of Design Performance & Cost**

Where have we been, and where are we going? Define 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 10 100 130 5 1.94 $2.03 2 2.53 3 150 61 140 4.01 $1.16 Measure Analyze Iteration #1) First cut at process settings resulted in ~2 sigma. Iteration #2) Second cut at process settings while halving the Mold Temperature standard deviation resulted in ~2.5 sigma. Iteration #3) Optimizing for Process Cost (minimization) with 4 sigma requirement on Part Length results in reduced cost (from iteration #2) and increased quality. Instructor should be prepared to answer the question: “Why is my best identified solution not exactly the same as the instructor’s best identified solution?” The reason lies in the nature of Monte Carlo analysis in that each and every Monte Carlo run for the same set of Decision Variable values may not yield the same Z-score (due to accuracies associated with Number of Trials). Sometimes Z-scores will be overestimated and sometimes underestimated. Thus, what could’ve been a “best” and feasible (meets Z-score requirement) solution may not be “best” or even feasible in a Monte Carlo simulation with the same set of Decision Variable values. Improve Six Sigma team proceeds to run Capability Study on proposed process settings to confirm quality during Control phase. 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. Define Measure Analyze Improve Control

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**Crystal Ball for Six Sigma**

Summary

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**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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**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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