By Issa Bass
 

Because it seeks to improve business performance, Six Sigma should whenever it is possible use traditional business metrics. But Six Sigma  is deeply rooted in Statistical analysis, that's why some of the techniques it uses are not commonly applied in business. The Defects Per Unit and the Rolled Throughput Yield are just a few examples examples.                                       

DPU and Yield
Let's consider a company manufacturing circuit boards. A circuit board is composed of multiple elements such as switches, resistors, capacitors, and so on; having every part of a board within specification is critical to the quality of each manufactured unit. Anytime an element of a unit (a board for this example) is outside its specified limits, it is considered as a defect, in other words a defect is a non conformance on a defective unit.

To measure the quality of his throughput, the manufacturer will want to know how many defects are found per unit. Since there are multiple parts per unit, it is conceivable to have more than one defect on one unit.

If we call the number of defects D, the number of units U and the defects per unit DPU,

Let's consider 15 units with defects spread as follow: 

The probability for finding defects on a unit follows a Poisson distribution.

Where P(x) is the probability for a unit to contain x defects and  is the mean defect per unit. This equation can be rearranged if DPU is known.

Example:

If the DPU is known to be 0.5, what is the probability of having 2 defects on a unit?

The probability for having 2 defects is 0.0758165

What is the probability of having 1 defect?

The probability for having 1 defect on a unit will be 0.303265

The objective of a manufacturer is to produce defect free products. The probability to produce defect free units will be:

Since

Manufacturing processes are made up of several operations, several linked steps. The probability for a unit to pass a step defect free will be

If we call Y (yield) the probability for a unit to pass a step the first time defect free, then

If Y is known, DPU can be found by simply rearranging the formula above.

Since ln e = 1

Example:

If a process has a first pass yield of 0.759,

Rolled Throughput Yield

A yield measures the probability for a unit to pass a step defect free, Rolled Throughput Yield (RTY) measures the probability for a unit to pass a set of processes defect free.

The Rolled Throughput Yield is obtained by multiplying the individual yields of the different processes.

What is the RTY for a product that goes through 4 processes with the following respective yields for each process? What is the DPU?

RTY = 0.78 * 0.86 * 0.88 * 0.83 = 0.489952

The probability for a unit to pass all the processes defect free is 0.489952. The probability for a defect will be 1 - 0.489952 = 0.510048.

DPU = - ln (0.489952) = 0.71345

About the author
Issa Bass is the managing editor of SixSigmaFirst. He can be reached at issa@sixsigmafirst.com

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