Template:Target reliability estimator

Target Reliability Estimator
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 $${{f}_{Market\_Share}}$$ is given by the equation:

$${{f}_{Market\_Share}}(R)=1-{{e}^{-{{\left( \frac{R}{a} \right)}^{b}}}}$$

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

$$ f_{Sale\_Price}\left(R\right)=b\times e^{a\cdot R} $$

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 $$(f_{Production\_Cost}(R))$$ for production cost per unit is given as:

$$f_{Production\_Cost} (R)=b\times e^{\frac{a}{\left(1-R\right)}}$$

where a and b are parameters fit to data, and R is the product reliability.

Using the substitution of variable $$ R'=\frac{1}{1-R} $$ results in the equation:

$$ f_{Production\_Cost}\left(R'\right)=b\times e^{a\cdot R'} $$

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 $$(f_{Failure\_Cost}(R))$$ is given by:

$$f_{Failure\_Cost}(R)=b\times e^{a\cdot R}$$

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 production 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.



The relaibility value resulting the lowest total cost will be the target reliaibility for the product.

Profit and Return at Target Reliability
With all the above costs, the profit at a given reliability value can be calculated as:

\begin{equation} \text{Profit}\left(R\right) = \text{Sales Revenue}\left(R\right) - \text{Warranty Cost}\left(R\right) - \text{Production Cost}\left(R\right) \end{equation}

Traditional ROI

First, traditional Return On Investment (ROI) is a performance measure used to evaluate the efficiency of an investment, or to compare the efficiency of a number of different investments. In general to calculate ROI, the benefit (return) of an investment is divided by the cost of the investment; and the result is expressed as a percentage or a ratip. The following equation illustrates this. $$ROI=\frac{\text{Gain from Investment - Cost of Investment}}{\text{Cost of Investment}}$$

In this formula "gains from investment" refers to the revenue or proceeds obtained the investment of interest.

Return on investment is a very popular metric because of its versatility and simplicity. That is, if an investment does not have a positive ROI, or if there are other opportunities with a higher ROI, then the investment should be not be undertaken. Reliability ROI, is similarly computed by looking at the investment as the the investment in improving the reliability.

ReliaSoft's Reliability Return on Investment (R3OI)

R3OI considers the cost and return due to the product reliability. As we discussed before, high reliability will reduce the unreliability cost, but will increase the sales revenue and production cost. A balanced reliability target should be determined based on all the involved costs. For a given initial investment value, the R3OI is calcualted by: $$R3OI=\frac{\text{Profit(R)-Initial Investment }}{\text{Initial Investment}}$$

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:

These five inputs are then repeated for three specific cases, Best Case, Most Likely and Worst Case.

Based on the above inputs four models are then fitted as functions of reliability, $$R=(1-Q)\,\!$$, or


 * $$\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} $$

An additional variable needed then is maximum market potential, M. It is defined by users in the following text box:



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: