On the variance upper bound theorem and its applications

The exact upper bound of the variance of properties from multiple sources is attained from sampling not more than two sources. Important applications of this result referred to as variance upper bound theorem are discussed. We show that often, the variation of a property associated with the common pool of supplied/manufactured components is not the maximum possible variation that can be expected. Usually, there exists a particular combination of suppliers/machine centres which yields a higher variation. The variance upper bound theorem makes it possible to determine easily this upper bound and provide a conservative estimate of the variation of properties. This estimate is then used to produce a conservative estimate of the capability index characterizing the process which is of significant importance to statistical process control. The upper-bound variance theorem is particularly useful for processes where the best performance is obtained when the output parameter does not deviate too much from a specified target. The conservative estimate of the maximum variation of properties can be used for developing robust designs. If the design is capable of accommodating the maximum possible variance of properties, it is highly likely that it will be capable of accommodating variations produced by any particular combination of mixing proportions from the sources of variation. On the basis of the variance upper bound theorem, an algorithm is proposed for determining the source, whose removal reduces most significantly the maximum variance of properties, irrespective of the mixing proportions of the separate sources. The applications of the variance upper bound theorem are illustrated with a number of engineering examples.