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<h1>Six Sigma Component</h1>

<table>
	<th colspan=2>Six Sigma Analysis</th>
	<tr>
		<td>Description:</td>
		<td>This model demonstrates the use of the Six Sigma component for analyzing the quality -- <i>reliability and
			robustness</i> -- of a design.  The example system used in this model, a welded plate supporting a load, is
			illustrated below.  The following random variables and responses are defined in this example:
			<ul>
			    <li>Random Variables (8):
			    <ul>
					<li>L - the cantilevered length of the plate
					<li>s - the welded length of the plate
					<li>h - the height of the plate
					<li>b - the thickness of the plate
					<li>t - the thickness of the weld
					<li>E - Young's modulus
					<li>G - shear modulus
					<li>P - the applied load
			    </ul>
			    <li>Responses (5):
				<ul>
					<li>Bending Stress
					<li>Deflection
					<li>Ratio Pc - buckling strength, ratio of the critical load to the load
					<li>Ratio TB - a geometric constraint, the ratio of the size of the bead to the thickness of the plate
					<li>Shear Stress
				</ul>
			</ul>
			All random variables are normally distributed with a coefficient of variation of 1%.
			<p><br>
			The FORM (First Order Reliability Method) reliability analysis technique is employed to calculate response
			statistics, probabilities, and sigma levels.
			<p><br>
			The execution summary reported on the Summary tab of the Runtime Gateway and the Six Sigma and Reliability
			specific graphs and tables can be used to review and explore results.
			<p>
			<br>
			<img src="WeldedPlate.jpg">
		</td>
	</tr>
	<tr>
		<td>Model:</td>
		<td><a href="SS-WeldCostRobustAnalysis.zmf">SS-WeldCostRobustAnalysis.zmf</a></td>
	</tr>
	<tr>
		<td>Illustrated components:</td>
		<td>Six Sigma, Calculator</td>
	</tr>
	<tr>
		<td>Other illustrated features:</td>
		<td>Reliability technique plugin - FORM
			<p>
			<br>
			Six Sigma / Reliability Post Processing Results:
			<ul>
				<li>Execution Summary Results - a summary of the Six Sigma analysis problem formulation and results
				<li>Six Sigma Graph - a graph displaying sigma level, limits, mean, and standard deviation for all responses
				with limits defined
				<li>Pareto plot - a graph ranking the effects of the random variables on a response
				<li>Reliability Table - a table displaying the lower limit reliability, upper limit reliability, and total
				reliability for all selected responses with limits defined
				<li>Response Percentiles Table - a table displaying response values at multiple percentiles within a normal
				distribution based on the response mean and standard deviation
				<li>Six Sigma Table - a table showing the sigma level, probability of success (reliability), probability of
				failure, and defects per million for each selected response with limits defined
			</ul>
			NOTE:  Other graphs and tables are available depending on the six sigma analysis type chosen (for example, Monte
			Carlo graphs and tables are available when Monte Carlo analysis type is selected, DOE graphs and tables are
			available when DOE analysis type is selected.)
		</td>
	</tr>
	<tr>
		<td>Simcodes needed:</td>
		<td>None</td>
	</tr>
	<tr>
		<td>Support files needed:</td>
		<td>None</td>
	</tr>
</table>
<br>
<p>
<table>
	<th colspan=2>Six Sigma Robust Optimization</th>
	<tr>
		<td>Description:</td>
		<td>This model demonstrates the use of the Six Sigma component for optimizing the quality -- <i>reliability and
			robustness</i> -- of a design.  The Six Sigma Analysis example above is wrapped in an optimization component
			for this example.  The welded plate six sigma optimization problem is defined as follows:
			<ul>
			    <li>Design Variables (4):
			    <ul>
					<li>s - the welded length of the plate
					<li>h - the height of the plate
					<li>b - the thickness of the plate
					<li>t - the thickness of the weld
			    </ul>
			    <li>Constraints (5):
				<ul>
					<li>Sigma Level of Bending Stress >= 4.5
					<li>Sigma Level of Deflection >= 4.5
					<li>Sigma Level of Ratio Pc  >= 4.5 (Ratio Pc is buckling strength, ratio of the critical load to the load)
					<li>Sigma Level of Ratio TB  >= 4.5 (Ratio TB is a geometric constraint, the ratio of the size of the bead to the thickness of the plate)
					<li>Sigma Level of Shear Stress >= 4.5
				</ul>
			    <li>Objectives (7):
				<ul>
					<li>Minimize Mean Total Cost
					<li>Minimize Standard Deviation of Total Cost
					<li>Minimize Standard Deviation of Bending Stress
					<li>Minimize Standard Deviation of Deflection
					<li>Minimize Standard Deviation of Ratio Pc (Ratio Pc is buckling strength, ratio of the critical load to the load)
					<li>Minimize Standard Deviation of Ratio TB (Ratio TB is a geometric constraint, the ratio of the size of the bead to the thickness of the plate)
					<li>Minimize Standard Deviation of Shear Stress
				</ul>
			</ul>
			The NLPQL optimization technique is used to solve this six sigma optimization problem.
			Again FORM (First Order Reliability Method) reliability analysis technique is employed to calculate response
			statistics, probabilities, and sigma levels during the optimization process.
			<p><br>
			The execution summary reported on the Summary tab of the Runtime Gateway for the optimization component
			and the optimization, Six Sigma, and Reliability
			specific graphs and tables can be used to review and explore results.
			<p>
			<br>
			<img src="WeldedPlate.jpg">
		</td>
	</tr>
	<tr>
		<td>Model:</td>
		<td><a href="SS-WeldCostRobustOptimization.zmf">SS-WeldCostRobustOptimization.zmf</a></td>
	</tr>
	<tr>
		<td>Illustrated components:</td>
		<td>Six Sigma, Optimization, Calculator</td>
	</tr>
	<tr>
		<td>Other illustrated features:</td>
		<td>Optimization technique plugin - NLPQL
			<br>
			Reliability technique plugin - FORM
			<p>
			<br>
			Six Sigma / Reliability Post Processing Results:
			<ul>
				<li>Execution Summary Results - a summary of the Six Sigma analysis problem formulation and results
				<li>Six Sigma Graph - a graph displaying sigma level, limits, mean, and standard deviation for all responses
				with limits defined
				<li>Pareto plot - a graph ranking the effects of the random variables on a response
				<li>Reliability Table - a table displaying the lower limit reliability, upper limit reliability, and total
				reliability for all selected responses with limits defined
				<li>Response Percentiles Table - a table displaying response values at multiple percentiles within a normal
				distribution based on the response mean and standard deviation
				<li>Six Sigma Table - a table showing the sigma level, probability of success (reliability), probability of
				failure, and defects per million for each selected response with limits defined
			</ul>
			NOTE:  Six Sigma graphs and tables are available for each step during the optimization process using the
			slider at the bottom of the Runtime Gateway Graphs tab.
			<p><br>
			NOTE:  Other graphs and tables are available depending on the six sigma analysis type chosen (for example, Monte
			Carlo graphs and tables are available when Monte Carlo analysis type is selected, DOE graphs and tables are
			available when DOE analysis type is selected.)
		</td>
	</tr>
	<tr>
		<td>Simcodes needed:</td>
		<td>None</td>
	</tr>
	<tr>
		<td>Support files needed:</td>
		<td>None</td>
	</tr>
</table>
</br>
<br>
<p>
<table>
	<th colspan=2>Six Sigma Robust Optimization using Random Variable Standard Deviations</th>
	<tr>
		<td>Description:</td>
		<td>This model demonstrates the use of the Six Sigma component for optimizing the quality -- <i>robustness</i> -- of a design
			and looking at the trade off between the cost and robustness.  This simple example consists of a beam whose cost vs std. deviation of stress tradeoff is to be studied:
			<ul>
			    <li>Design Variables (2):
			    <ul>
					<li>The standard deviation of Height
					<li>The standard deviation of Length
			    </ul>
			    <li>Objectives (2):
				<ul>
					<li>Minimize Total Cost
					<li>Minimize Standard Deviation of Stress
				</ul>
			</ul>
			The multi-objective partcle swarm optimization technique is used to solve this six sigma optimization problem.
			Mean Value Method (MVM) reliability analysis technique is employed to calculate response
			statistics during the optimization process.
			<p><br>
			The execution summary reported on the Summary tab of the Runtime Gateway for the optimization component
			and the optimization, Six Sigma, and Reliability
			specific graphs and tables can be used to review and explore results.
			<p>
		</td>
	</tr>
	<tr>
		<td>Model:</td>
		<td><a href="Beam_cost_ss_optimization.zmf">Beam_cost_ss_optimization.zmf</a></td>
	</tr>
	<tr>
		<td>Illustrated components:</td>
		<td>Six Sigma, Optimization, Calculator</td>
	</tr>
	<tr>
		<td>Other illustrated features:</td>
		<td>Optimization technique plugin - Multi-Objective Particle Swarm
			<br>
			Reliability technique plugin - MVM
			<p>
			<br>
			Using Standard Deviations of random variables as design variables in optimization.
		</td>
	</tr>
	<tr>
		<td>Simcodes needed:</td>
		<td>None</td>
	</tr>
	<tr>
		<td>Support files needed:</td>
		<td>None</td>
	</tr>
</table>
</br>
<br>
<p>
<table>
	<th colspan=2>Six Sigma Robust Analysis using Importance Sampling</th>
	<tr>
		<td>Description:</td>
		<td>This model demonstrates the use of the Six Sigma component for analyzing the quality -- <i>reliability and
			robustness</i> -- of a design.  The example system used in this model is a cantelever beam. 
                  The following random variables and responses are defined in this example:
			<ul>
			    <li>Random Variables (10):
			    <ul>
					<li> Width (5 random variables)
					<li> Height (5 random variables)
			    </ul>
			    <li>Responses (5):
				<ul>
					<li>Stress (4 responses)
					<li>Tip Deflection
				</ul>
			</ul>
			All random variables are normally distributed with a coefficient of variation of 10%.
			<p><br>
			Simple mportance sampling is employed to calculate response	statistics, probabilities, and sigma levels. Simple Importance
			computes the probabilities within 5% of the actual values using a fraction (1/10 in this case) of sub flow evaluations
			needed for Monte Carlo.
			<p><br>
			The execution summary reported on the Summary tab of the Runtime Gateway and the Six Sigma and Reliability
			specific graphs and tables can be used to review and explore results.
			<p>
			<br>
		</td>
	</tr>
	<tr>
		<td>Model:</td>
		<td><a href="SS-ImportanceSampling.zmf">SS-ImportanceSampling.zmf</a></td>
	</tr>
	<tr>
		<td>Illustrated components:</td>
		<td>Six Sigma, Calculator</td>
	</tr>
	<tr>
		<td>Other illustrated features:</td>
		<td>Importance Sampling
		</td>
	</tr>
	<tr>
		<td>Simcodes needed:</td>
		<td>None</td>
	</tr>
	<tr>
		<td>Support files needed:</td>
		<td>None</td>
	</tr>
</table>
</br>

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