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==Target Reliability Estimator==
#REDIRECT [[Target_Reliability_Tool]]
 
Product reliability affects total product costs in multiple ways. Increasing product reliability increases the initial cost of producing a product but decreases other costs incurred over the life of the product. For example, increased reliability results in lower warranty and replacement costs for defective products. Increased reliability also results in greater market share as satisfied customers typically become repeat customers and recommend reliable products to others. A minimal total product cost can be determined by calculating the optimum reliability for such a product. The Target Reliability Estimator does this by minimizing the sum of lost sales costs, warranty costs and manufacturing costs.
 
===Cost factors in Determing Target Reliability===
 
'''Lost Sales Cost'''
 
The lost sales cost is caused due to lost market share. It is caused by customers choosing to go elsewhere for goods and services. The lost sales cost depends on the total market value for a product and the actual sales revenue of a product.
 
\begin{equation} \text{Lost sales cost}=\text{Max}\{0, \text{Total Market Value}–\text{Sales Revenue}\} \end{equation}
 
In Weibull++, we assume the total potential market value is the product of maximum market potential (number of units of product) and the best unit sale price.
 
\begin{equation} \text{Total Market Value}=\text{Maximum Market Potential}\times\text{Best Market Unit Sale Price} \end{equation}
 
For example, if the maximum number of units of product demanded by the market were 100,000 and the best market unit sale price was $12.00, then the total market value would be:
 
\begin{equation} 100,000\times \$12.00=\$1,200,000.00 \end{equation}
 
Calculating sales revenue requires knowledge of market share and unit sale price. The function for market share <math>{{f}_{Market\_Share}}</math> is given by the equation:
 
<center><math>{{f}_{Market\_Share}}(R)=1-{{e}^{-{{\left( \frac{R}{a} \right)}^{b}}}}</math></center>
 
where ''a'' and ''b'' are parameters fit to market share data, and ''R'' is the product reliability.
 
 
The function for '''unit sale price''' is given by
 
<center><math> f_{Sale\_Price}\left(R\right)=b\times e^{a\cdot R} </math> </center>
 
where ''a'' and ''b'' are parameters fit to data, and ''R'' is the product reliability.
 
 
As a function of reliability ''R'', sales revenue is then calculated as:
 
:: \begin{equation}\text{Sales Revenue}\left(R\right)=\text{Total Market Value}\times\text{Market Share}\left(R\right)\times\text{Unit Price}\left(R\right) \end{equation}
 
Once the total market value and the sales revenue are obtainted, they can then be used to calculate the lost sales cost using the formula at the beginging of this section.
 
 
'''Production  Cost'''
 
Production cost is a function of total market value, market share, and manufacturing cost per unit. The function <math>(f_{Production\_Cost}(R))</math> for '''production cost per unit''' is given as:
 
<center> <math>f_{Production\_Cost} (R)=b\times e^{\frac{a}{\left(1-R\right)}}</math> </center>
 
where ''a'' and ''b'' are parameters fit to data, and ''R'' is the product reliability.
 
Using the substitution of variable <math> R'=\frac{1}{1-R} </math> results in the equation:
 
<center> <math> f_{Production\_Cost}\left(R'\right)=b\times e^{a\cdot R'} </math> </center>
 
for which the parameters ''a'' and ''b'' can be determined using simple regression tool such as functions in the [[Degradation Data Analysis]] in Weibull++.
 
'''Warranty Cost'''
 
Warranty cost is a function of total market value, market share, reliability, and cost per failure. The function of '''cost per failure''' <math>(f_{Failure\_Cost}(R))</math> is given by:
 
<center> <math>f_{Failure\_Cost}(R)=b\times e^{a\cdot R}</math> </center>
 
where ''a'' and ''b'' are parameters fit to data. For a given reliability value ''R'', the warranty cost is given by:
\begin{equation}\text{Warranty Cost}\left(R\right)=\text{Total Market Value}\times \text{Market Share}\left(R\right)\times\left(1-R\right)\times\text{Cost Per Failure}\left(R\right)
\end{equation}
 
 
'''Unreliability Cost'''
 
The sum of the '''Lost Sales Cost''' and '''Warranty Cost''' is called as '''Unreliability Cost'''.
 
 
'''Total Cost'''
 
For a given reliability ''R'', the expected total cost is given by:
 
\begin{equation} \text{Total Cost}\left(R\right) = \text{Lost Sales Cost}\left(R\right) + \text{Warranty Cost}\left(R\right) + \text{Production Cost}\left(R\right)=\text{Unreliability Cost}\left(R\right)+\text{Production Cost}\left(R\right) \end{equation}
 
The manufacturing cost is a pre-shipping cost whereas the warranty and lost sales costs are incurred after a product is shipped. These pre- and post-shipping costs can be seen in the figure below.
 
[[Image:Chart_totalcost.jpg|thumb|center|400px| ]]
 
The relaibility value resulting the lowest total cost will be the target reliaibility for the product.
 
===Profit at Target Reliability===
 
 
===The Weibull++ Target Reliability tool===
The purpose of this tool is to qualitatitivaly explore different options with regards to a target reliability for component, subsystem or system. All the costs are caculated using the equation given in the section before.
 
'''Inputs'''
 
There are five inputs for 3 specific cases. More specifcally:
 
<br>
 
{| border="1" cellspacing="1" cellpadding="1" align="center"
|-
! scope="col" | Input Title
! scope="col" | Input Value
|-
| Expected failures/returns per year as&nbsp;%&nbsp;of Sales
| valign="middle" align="center" | <math>Q% \text{ where } 0\le Q\le 100 \,\!</math>
|-
| &nbsp;% of market share you expect to capture
| valign="middle" align="center" | <math>S% \text{ where } 0\le S\le 100 \,\!</math>
|-
| Average unit sales price
| valign="middle" align="center" | <math>P \text{ where } 0< P \,\!</math>
|-
| Average cost per unit to produce
| valign="middle" align="center" | <math>C \text{ where } 0< C < P+O \,\!</math>
|-
| Other costs per failure
| valign="middle" align="center" | <math>O \text{ where } 0< O < C+O \,\!</math>
|}
 
<br>
 
These five inputs are then repeated for three specific cases, Best Case, Most Likely and Worst Case. <!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
 
<br>
 
{| border="1" cellspacing="1" cellpadding="1" align="center"
|-
! scope="col" | Input Title
! scope="col" | Best Case
! scope="col" | Most Likely
! scope="col" | Worst Case
|-
| valign="middle" align="left" | Expected failures/returns per year as&nbsp;%&nbsp;of Sales
| valign="middle" align="center" | <math>{{Q}_{1}}\,\!</math>
| valign="middle" align="center" | <math>{{Q}_{2}}\,\!</math>
| valign="middle" align="center" | <math>{{Q}_{3}}\,\!</math>
|-
| valign="middle" align="left" | &nbsp;% of market share you expect to capture
| valign="middle" align="center" | <math>{{S}_{1}}\,\!</math>
| valign="middle" align="center" | <math>{{S}_{2}}\,\!</math>
| valign="middle" align="center" | <math>{{S}_{3}}\,\!</math>
|-
| valign="middle" align="left" | Average unit sales price
| valign="middle" align="center" | <math>{{P}_{1}}\,\!</math>
| valign="middle" align="center" | <math>{{P}_{2}}\,\!</math>
| valign="middle" align="center" | <math>{{P}_{3}}\,\!</math>
|-
| valign="middle" align="left" | Average cost per unit to produce
| valign="middle" align="center" | <math>{{C}_{1}}\,\!</math>
| valign="middle" align="center" | <math>{{C}_{2}}\,\!</math>
| valign="middle" align="center" | <math>{{C}_{3}}\,\!</math>
|-
| valign="middle" align="left" | Other costs per failure
| valign="middle" align="center" | <math>{{O}_{1}}\,\!</math>
| valign="middle" align="center" | <math>{{O}_{2}}\,\!</math>
| valign="middle" align="center" | <math>{{O}_{3}}\,\!</math>
|}
 
&nbsp;
 
&nbsp; Based on the above inputs&nbsp;four models are then fitted as functions of reliability, <math>R=(1-Q)\,\!</math>, or
 
::&nbsp;<math>\begin{align}
  & f_{Market\_Share}(R)=1-{{e}^{-{{\left( \frac{R}{a} \right)}^{b}}}} \\
& f_{Sale\_Price}(R)=b\cdot {{e}^{\left( a\cdot R \right)}} \\
& f_{Production\_Cost}(R)=b\cdot {{e}^{\left( a\cdot \left( \frac{1}{1-R} \right) \right)}} \\
& f_{Failure\_Cost}(R)=b\cdot {{e}^{\left( a\cdot R \right)}} \\
\end{align}
</math>
 
&nbsp;
 
An additional variable needed then is maximum market potential, ''M''. It is defined by users in the following text box:
 
[[Image: Target Reliability Estimator Market Potential icon.png|thumb|center|400px]]
 
All the related costs then are defined as given in the previous section and calcualted as a function of reliability ''R''. The value giving the lowest total cost is the optimal or the target reliability.
 
'''Example 1:'''
{{Example: Target Reliability-Determine Target Reliability Based on Costs}}

Latest revision as of 06:47, 13 August 2012

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