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| {{standard model overview gompz}}
| | #REDIRECT [[Gompertz_Models]] |
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| {{parameter estimation using least squares in nonlinear regression}}
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| {{cumulative reliability gumpz}}
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| {{modified gompertz model}}
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| ==General Examples==
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| ===Example 4===
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| <br>
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| A new design is put through a reliability growth test. The requirement is that after the ninth stage the design will exhibit an 85% reliability with a 90% confidence level. Given the data in Table 7.5, do the following:
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| <br>
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| :1) Estimate the parameters of the Standard Gompertz model.
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| :2) What is the initial reliability at <math>T=0</math> ?
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| :3) Determine the reliability at the end of the ninth stage and check to see if the goal has been met.
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| <br>
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| {|style= align="center" border="1"
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| |+'''Table 7.5 - Grouped per configuration data for Example 4'''
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| !Stage
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| !Number of Units
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| !Number of Failures
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| |-
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| |1|| 10|| 5
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| |-
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| |2|| 8|| 3
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| |-
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| |3|| 9|| 3
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| |-
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| |4|| 9|| 2
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| |-
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| |5|| 10|| 2
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| |-
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| |6|| 10|| 1
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| |-
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| |7|| 10|| 1
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| |-
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| |8|| 10|| 1
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| |-
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| |9|| 10|| 1
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| |}
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| <br>
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| ====Solution to Example 4====
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| :1) The data is entered in cumulative format and the estimated Standard Gompertz parameters are shown in Figure Gompex4a.
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| [[Image:rga7.6.png|thumb|center|400px|Entered data and the estimated Standard Gompertz parameters.]] | |
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| :2) The initial reliability at <math>T=0</math> is equal to:
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| ::<math>\begin{align}
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| & {{R}_{T=0}}= & a\cdot b \\
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| & = & 0.9497\cdot 0.5249 \\
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| & = & 0.4985
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| \end{align}</math>
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| :3) The reliability at the ninth stage can be calculated using the Quick Calculation Pad (QCP) as shown in Figure Gompex4b.
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| [[Image:rga7.7.png|thumb|center|400px|Calculate the reliability at the end of the ninth stage with 90% confidence bounds.]]
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| <br>
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| The estimated reliability at the end of the ninth stage is equal to 91.92%. However, the lower limit at the 90% 1-sided confidence bound is equal to 82.15%. Therefore, the required goal of 85% reliability at a 90% confidence level has not been met.
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| ===Example 5===
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| Using the data in Table 7.6, determine whether the Standard Gompertz or Modified Gompertz would be better suited for analyzing the given data.
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| <br>
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| {|style= align="center" border="1"
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| |+'''Table 7.6 - Reliability data for Example 5'''
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| !Stage
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| !Reliability (%)
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| |-
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| |0|| 36
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| |-
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| |1|| 38
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| |-
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| |2|| 46
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| |-
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| |3|| 58
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| |-
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| |4|| 71
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| |-
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| |5|| 80
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| |-
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| |6|| 86
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| |-
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| |7|| 88
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| |-
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| |8|| 90
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| |-
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| |9|| 91
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| |}
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| ====Solution to Example 5====
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| The Standard Gompertz Reliability vs. Time plot is shown in Figure Ex5Std.
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| The Standard Gompertz seems to do a fairly good job of modeling the data. However, it appears that it is having difficulty modeling the S-shape of the data. The Modified Gompertz Reliability vs. Time plot is shown in Figure Ex5Mod.
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| The Modified Gompertz, as expected, does a much better job of handling the S-shape presented by the data and provides a better fit for this data.
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|
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| [[Image:rga7.8.png|thumb|center|400px|Standard Gompertz Reliability vs. Time plot]]
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| <br>
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| <br>
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| [[Image:rga7.9.png|thumb|center|400px|Modified Gompertz Reliability vs. Time plot.]]
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