Logistic

The Logistic reliability growth model has an S-shaped curve and is given by Kececioglu [3]:


 * $$R = \frac{1}{1+be^{-kt}}, b > 0, k > 0, T \simeq 0$$

where $$b\,\!$$ and $$k\,\!$$ are parameters. Similar to the analysis given for the Gompertz curve, the following may be concluded:


 * 1)	The point of inflection is given by:


 * $${{T}_{i}}=\frac{\ln (b)}{k}\,\!$$


 * 2)	When $$b>1\,\!$$, then $${{T}_{i}}>0\,\!$$ and an S-shaped curve will be generated. However, when $$0<b\le 1\,\!$$, then $${{T}_{i}}\le 0\,\!$$ and the Logistic reliability growth model will not be described by an S-shaped curve.


 * 3)	The value of $$R\,\!$$ is equal to 0.5 at the inflection point.

=Parameter Estimation= In this section, we will demonstrate the parameter estimation method for the Logistic model using three examples for different types of data.

Example: Logistic for Reliability Data
Using the reliability growth data given in the table below, do the following:


 * 1)	Find a Gompertz curve that represents the data and plot it with the raw data.


 * 2)	Find a Logistic reliability growth curve that represents the data and plot it with the raw data.

Solution


 * 1) The figure below shows the entered data and the estimated parameters using the standard Gompertz model.




 * Therefore:


 * $$\begin{align}

& \widehat{a}= & 0.9999 \\ & \widehat{b}= & 0.2485 \\ & \widehat{c}= & 0.6858 \end{align}\,\!$$


 * $$R=(0.9999){{(0.2485)}^}\,\!$$


 * The values of the predicted reliabilities are plotted in the figure below.




 * Notice how the standard Gompertz model is not really capable of handling the S-shaped characteristics of this data.


 * 2. The least squares estimators of the Logistic growth curve parameters are given by Crow [9]:


 * where:


 * $$\begin{align}

{{{\hat{b}}}_{1}}= & \frac{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ \\ {{{\hat{b}}}_{0}}= & \bar{Y}-{{{\hat{b}}}_{1}}\bar{T} \\ \\ {{Y}_{i}}= & \ln \left( \frac{1}-1 \right) \\ \\ \bar{Y}= & \frac{1}{N}\underset{i=0}{\overset{N-1}{\mathop \sum }}\,{{Y}_{i}} \end{align}\,\!$$

In this example $$N=9\,\!$$, which gives:


 * $$\begin{align}

\overline{Y}&=\frac{1}{9}\sum_{i=0}^{8}ln\left (\frac{1}{R_{i}}-1 \right ) \\ &= -1.7355 \\ \\ \overline{T}&=\frac{1}{9}\sum_{i=0}^{8}T_{i} = 4 \\ \sum_{i=0}^{8}T_{i}^{2} &= 204 \\ \sum_{i=0}^{8}T_{i}Y_{i} &= -106.8630 \end{align}\,\!$$

From the equations for $$b_{i}\,\!$$ and $$\hat{b_{0}}\,\!$$:


 * $$\begin{align}

\hat{b_{1}} &= \frac{-106.8630 - 9(4)(-1.7355)}{204-9(4)_{2}} \\ & = 0.7398 \\ \hat{b_{0}} &= -1.7355 - (-0.7398)(4)\\ &= 1.2235 \end{align}\,\!$$

And from the least squares estimators for $$\hat{b}\,\!$$ and $$\hat{k}\,\!$$:


 * $$\begin{align}

\widehat{b}= & {{e}^{1.2235}} \\ = & 3.3991 \\  \widehat{k}= & -(-0.7398) \\ = & 0.7398 \end{align}\,\!$$

Therefore, the Logistic reliability growth curve that represents this data set is given by:


 * $$R=\frac{1}{1+3.3991\,{{e}^{-0.7398\,T}}}\,\!$$

The following figure shows the Reliability vs. Time plot. The plot shows that the observed data set is estimated well by the Logistic reliability growth curve, except in the region closely surrounding the inflection point of the observed reliability. This problem can be overcome by using the modified Gompertz model.



Example: Logistic for Sequential Success/Failure Data
A prototype was tested under a success/failure pattern. The test consisted of 15 runs. The following table presents the data from the test. Find the Logistic model that best fits the data set, and plot it along with the reliability observed from the raw data.

Solution

The first run is ignored because it was a success, and the reliability at that point was 100%. This failure will be ignored throughout the analysis because it is considered that the test starts when the reliability is not equal to zero or one. The test essentially begins at time 1, and is now considered as time 0 with $$N=14\,\!$$. The observed reliability is shown in the last column of the table. Keep in mind that the observed reliability values still account for the initial suspension.

Therefore:


 * $$\begin{align}

\bar{Y}= & \frac{1}{N}\underset{i=0}{\overset{N-1}{\mathop \sum }}\,{{Y}_{i}} \\ = & \frac{1}{14}\underset{i=0}{\overset{13}{\mathop \sum }}\,\ln \left( \frac{1}-1 \right) \\ = & -0.43163 \end{align}\,\!$$

and:


 * $$\begin{align}

\bar{T}= & \frac{1}{14}\underset{i=0}{\overset{13}{\mathop \sum }}\,{{T}_{i}} \\ = & 6.5 \\  \underset{i=0}{\overset{13}{\mathop \sum }}\,T_{i}^{2}= & 819.0 \\ \underset{i=0}{\overset{13}{\mathop \sum }}\,{{T}_{i}}{{Y}_{i}}= & -61.69 \end{align}\,\!$$

Now, from the least squares estimators, the values are:


 * $$\begin{align}

{{{\hat{b}}}_{1}}&= \frac{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{N-1}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ \\ &= \frac{-61.69-14 \cdot 6.5 \cdot (-.43163)}{819.0-14 \cdot 6.5^{2}} \\ &= -0.0985 \\ \\

\hat{b_{0}}&= \overline{Y} - \hat{b_{1}}\overline{T} \\ &= (-.043163)-(-0.0985) \cdot 6.5 \\ &= 0.2087 \end{align}\,\!$$

Therefore:


 * $$\begin{align}

\widehat{b}= & {{e}^{0.2087}} \\ = & 1.2321 \\  \widehat{k}= & -(-0.0985) \\ = & 0.0985 \end{align}\,\!$$

The Logistic reliability model that best fits the data is given by:


 * $$R=\frac{1}{1+1.2321\cdot \ \,{{e}^{-0.0985T}}}\,\!$$

The following figure shows the Reliability vs. Time plot.



Example: Logistic for Grouped per Configuration Data
Some equipment underwent testing in different stages. The testing may have been performed in subsequent days, weeks or months with an unequal number of units tested every day. Each group was tested and several failures occurred. The data set is given in columns 1 and 2 of the following table. Find the Logistic model that best fits the data, and plot it along with the reliability observed from the raw data.

Solution

The observed reliability is $$1-\tfrac{\#\text{ of failures}}{\#\text{ of units}}\,\!$$ and the last column of the table above shows the values for each group. With $$N=9\,\!$$, the least square estimator $$\overline{Y}$$ becomes:


 * $$\begin{align}

\bar{Y}= & \frac{1}{9}\underset{i=0}{\overset{8}{\mathop \sum }}\,\ln \left( \frac{1}-1 \right) \\ = & -1.4036 \end{align}\,\!$$

and:


 * $$\begin{align}

\bar{T}= & \frac{1}{9}\underset{i=0}{\overset{8}{\mathop \sum }}\,{{T}_{i}} \\ = & 4 \\  \underset{i=0}{\overset{8}{\mathop \sum }}\,T_{i}^{2}= & 204 \\ \underset{i=0}{\overset{8}{\mathop \sum }}\,{{T}_{i}}{{Y}_{i}}= & -68.33 \end{align}\,\!$$

Now from the least squares estimators, $$\hat{b_{i}}\,\!$$ and $$\hat{b_{0}}\,\!$$, we have:


 * $$\begin{align}

{{{\hat{b}}}_{1}}= & \frac{\underset{i=0}{\overset{8}{\mathop{\sum }}}\,{{T}_{i}}{{Y}_{i}}-N\cdot \bar{T}\cdot \bar{Y}}{\underset{i=0}{\overset{8}{\mathop{\sum }}}\,T_{i}^{2}-N\cdot {{{\bar{T}}}^{2}}} \\ = & \frac{-68.33-9\cdot 4\cdot \left( -1.4036 \right)}{204-9\cdot {{4}^{2}}} \\ = & -0.2967 \\   &  \\   {{{\hat{b}}}_{0}}= & \bar{Y}-{{{\hat{b}}}_{1}}\bar{T} \\ = & \left( -1.4036 \right)-\left( -0.2967 \right)\cdot 4.0 \\ = & -0.2168 \end{align}\,\!$$

Therefore:


 * $$\begin{align}

\widehat{b}= & {{e}^{-0.2168}} \\ = & 0.8051 \\  \widehat{k}= & -(-0.2967) \\ = & 0.2967 \end{align}\,\!$$

The Logistic reliability model that best fits the data is given by:


 * $$R=\frac{1}{1+0.8051\cdot \ \,{{e}^{-0.2967T}}}\,\!$$

The figure below shows the Reliability vs. Time plot.



=Confidence Bounds= Least squares is used to estimate the parameters of the following Logistic model.


 * $$\ln (\frac{1}-1)=\ln (b)-k{{T}_{i}}\,\!$$

Thus, the confidence bounds on the parameter $$b\,\!$$ are given by:


 * $$b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!$$

where:


 * $$\begin{align}

SE(\ln \hat{b})&=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{({{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}},\ \ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}} \\ \\ \sigma &=\sqrt{SSE/(n-2)} \end{align}\,\!$$

and the confidence bounds on the parameter $$k\,\!$$ are:


 * $$k=\hat{k}\pm {{t}_{n-2,\alpha /2}}SE(\hat{k})\,\!$$

where:


 * $$SE(\hat{k})=\frac{\sigma }{\sqrt},\ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}}\,\!$$

Since the reliability is always between 0 and 1, the logit transformation is used to obtain the confidence bounds on reliability, which is:


 * $$CB=\frac{{{{\hat{R}}}_{i}}+(1-{{{\hat{R}}}_{i}}){{e}^{\pm {{z}_{\alpha }}{{{\hat{\sigma }}}_{R}}/\left[ {{{\hat{R}}}_{i}}(1-{{{\hat{R}}}_{i}}) \right]}}}\,\!$$

Example: Logistic Confidence Bounds
For the data given above for the reliability data example, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:


 * 1)	The parameters $$b\,\!$$ and $$k\,\!$$.


 * 2)	Reliability at month 5.

Solution


 * 1)	The values of $$\hat{b}\,\!$$ and $$\hat{k}\,\!$$ that were estimated from the least squares analysis in the reliability data example are:


 * $$\begin{align}

\widehat{b}= & 3.3991 \\ \widehat{\alpha }= & 0.7398 \end{align}\,\!$$


 * Thus, the 2-sided 90% confidence bounds on parameter $$b\,\!$$ are:


 * $$\begin{align}

{{b}_{lower}}= & 2.5547 \\ {{b}_{upper}}= & 4.5225 \end{align}\,\!$$


 * The 2-sided 90% confidence bounds on parameter $$k\,\!$$ are:


 * $$\begin{align}

{{k}_{lower}}= & 0.6798 \\ {{k}_{upper}}= & 0.7997 \end{align}\,\!$$


 * 2) First, calculate the reliability estimation at month 5:


 * $$\begin{align}

{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\ = & 0.9224 	\end{align}\,\!$$
 * Thus, the 2-sided 90% confidence bounds on reliability at month 5 iare:


 * $$\begin{align}

{{[{{R}_{5}}]}_{lower}}= & 0.8493 \\ {{[{{R}_{5}}]}_{upper}}= & 0.9955 \end{align}\,\!$$

The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.



=More Examples=

Auto Transmission Reliability Data
The following table presents the reliabilities observed monthly for an automobile transmission that was tested for one year.


 * 1)	Find a Logistic reliability growth curve that best represents the data.
 * 2)	Plot it comparatively with the raw data.
 * 3)	If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99% be achieved?
 * 4)	If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of January the following year?

Solution 
 * 1)	The next figure shows the estimated parameters.


 * 2)	The next figure displays the Reliability vs. Time plot.


 * 3)	Using the QCP, the next figure displays when the reliability goal of 99% will be achieved.


 * 4)	The last figure shows the reliability at the end of January the following year (i.e., after 20 months of testing and development).



Sequential Data from Missile Launch Test
The next table presents the results for a missile launch test. The test consisted of 20 attempts. If the missile launched, it was recorded as a success. If not, it was recorded as a failure. Note that, at this development stage, the test did not consider whether or not the target was destroyed.


 * 1)	Find a Logistic reliability growth curve that best represents the data.
 * 2)	Plot it comparatively with the raw data.
 * 3)	If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99.5% with a 90% confidence level be achieved?
 * 4)	If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of the 35th launch?

Solution
 * 1)	The next figure shows the entered data and the estimated parameters.


 * 2)	The next figure displays the Reliability vs. Time plot.


 * 3)	The next figure displays when the reliability goal of 99.5% will be achieved with a 90% confidence level.


 * 4)	The next figure displays the reliability achieved after the 35th launch.



Sequential Data with Failure Modes
Consider the data given for the missile launch test in the previous example. Now suppose that the engineers assigned failure modes to each failure and that the appropriate corrective actions were taken.

The table below presents the data.
 * 1)	Find the Logistic reliability growth curve that best represents the data.
 * 2)	Plot it comparatively with the raw data.
 * 3)	If design changes continue to be incorporated and the testing continues, when will the reliability goal of 99.50% be achieved?
 * 4)	If design changes continue to be incorporated and the testing continues, what will be the attainable reliability at the end of the 35th launch?

Solution
 * 1)	The next figure shows the estimated parameters.


 * 2)	The next figure displays the Reliability vs. Time plot.


 * 3)	The next figure displays when the reliability goal of 99.5% will be achieved.


 * 4)	The last figure displays the reliability after the 35th launch.