
Calculating Optimal Reliability Using Weibull++'s Target Reliability Tool
Product reliability affects total product costs in multiple ways. Improving the reliability increases the initial cost of production, 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 and goodwill. A minimal total product cost can be determined by calculating the optimum reliability for such a product. The Target Reliability tool in Weibull++ 8 does this by minimizing the sum of lost sales costs, warranty costs and manufacturing costs.
Cost Factors in Determining Target Reliability
Lost Sales Cost
The lost sales cost is 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.
In Weibull++, we assume that the total potential market value is the product of the maximum market potential (number of units that could be sold) and the best unit sale price, or:
For example, if the maximum number of units demanded by the market is 100,000 and the best market unit sale price is $16.00, then the total market value would be:
Calculating sales revenue requires knowledge of market share and unit sale price. The function for market share is given by the following equation:
where a and b are parameters fitted to market share data, and R is the product reliability.
The function for unit sales price is given by:
where a and b are parameters fitted to the data, and R is the product reliability.
As a function of reliability, R, the sales revenue is then calculated as:
Once the total market value and the sales revenue are obtained, they can then be used to calculate the lost sales cost using the following equation:
Production Cost
Production cost is a function of total market value, market share and manufacturing cost per unit. The function for production cost per unit is given by:
where a and b are parameters fitted to market share data, and R is the product reliability.
Using the substitution of variable results in the equation:
for which the parameters a and b can be determined using simple regression tools such as the 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 is given by:
where a and b are parameters fitted to market share data.
For a given reliability value, R, the warranty cost is given by:
Unreliability Cost
The unreliability cost is the sum of the lost sales cost and warranty cost:
Total Cost
For a given reliability, R, the expected total cost is given by:
Profit and Return at Target Reliability
With all of the costs described above, the profit at a given reliability can be calculated as:
First, consider that 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 ratio. The following equation illustrates this.
In this formula, Gains from Investment refers to the revenue or proceeds obtained due to the investment that is being considered.
Return on investment is a very popular metric because of its versatility and simplicity. If an investment does not have a positive ROI, or if there are other opportunities with a higher ROI, then the investment should not be undertaken. Reliability ROI is computed in a similar manner by looking specifically at 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 production cost. A balanced reliability target should be determined based on all of the costs involved. For a given initial investment value, the R3OI is calculated by:
Weibull++ Target Reliability Tool
The purpose of the Target Reliability tool is to qualitatively explore different options with regards to a target reliability for a component, subsystem or system. All the costs are calculated using the equations given above.
This tool requires five inputs (Q, S, P, C and O) for each of the three cases: Best Case, Most Likely and Worst Case, as shown in the following table.
Inputs  Best Case  Most Likely  Worst Case 
Expected failures/returns per period (as % of sales)  Q_{1}  Q_{2}  Q_{3} 
% of market share you expect to capture  S_{1}  S_{2}  S_{3} 
Average unit sales price  P_{1}  P_{2}  P_{3} 
Average cost per unit to produce  C_{1}  C_{2}  C_{3} 
Other costs per failure (in addition to replacement costs  O_{1}  O_{2}  O_{3} 
Given the value of the maximum market potential (M), Weibull++ uses the inputs in the table above to fit the following functions:
All the related costs are defined as given in the previous section and calculated as a function of reliability. The value that gives the lowest total cost is the optimal (target) reliability.
Example: Determining Reliability Based on Cost
The following table provides information for a particular product regarding market share, sales prices, cost of production and costs due to failure.
The first row in the table indicates the probability of failure during the warranty period. In the best case scenario, the expected probability of failure is 1% (i.e., the reliability will be 99%). Under this reliability, the expected market share is 90%, average unit sale price is $2.20, average cost per unit to produce is $1.50 and other costs per failure add up to $0.50. The rest of the table can be read in a similar manner.
The assumed maximum market potential is 1,000,000 units and the initial investment is $10,000.
Solution
To perform the analysis, open the Target Reliability tool in Weibull++ by choosing Insert > Tools > Target Reliability. This inserts a new folio into the project.
Enter the information from the given table into the Target Estimation Inputs area of the folio, and then enter 1,000,000 in the Max. Market Potential (Units) field in the control panel.
Click the Plot icon on the control panel. The following Cost vs. Reliability plot shows the cost models. The green vertical line on the plot represents the estimated reliability value that will minimize costs. Point your mouse cursor to the line to display the value, as shown next. In this example, the reliability that will minimize costs is estimated to be 98.1982%.
The following figure shows the Profit vs. Reliability plot.
For the R3OI plot, enter 10,000 in the Initial Investment field in the control panel and click the Plot icon again to redraw the plot. The following figure shows the result.
Click the Analysis Details button on the control panel to generate a report of the analysis. The following report shows the cost models.