DMAIC or DMADV - Which Methodology To Use?

DMAIC
Define the project goals and customer (internal and external) deliverables
Measure the process to determine current performance
Analyze and determine the root cause(s) of the defects
Improve the process by eliminating defects
Control future process performance

DMADV
Define the project goals and customer (internal and external) deliverables
Measure and determine customer needs and specifications
Analyze the process options to meet the customer needs
Design (detailed) the process to meet the customer needs
Verify the design performance and ability to meet customer needs

The DMAIC methodology, instead of the DMADV methodology, should be used when a product or process is in existence at your company but is not meeting customer specification or is not performing adequately.
 
The DMADV methodology, instead of the DMAIC methodology, should be used when:
A product or process is not in existence at your company and one needs to be developed
The existing product or process exists and has been optimized (using either DMAIC or not) and still does not meet the level of customer specification or Six Sigma level
In summary, the DMAIC methodology is useful in improving an existing product or process to reduce defects, while the DMADV methodology is useful to create a new product or process design, or to optimize an existing product or process.

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