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Mathematical Problems in Engineering
Volume 2014, Article ID 619639, 12 pages
Research Article

A Study on Overestimating a Given Fraction Defective by an Imperfect Inspector

Department of Industrial Engineering, Dankook University, 119 Dandae-ro, Dongnam-gu, Cheonan-si, Chungnam 330-714, Republic of Korea

Received 10 May 2014; Revised 28 July 2014; Accepted 4 August 2014; Published 15 September 2014

Academic Editor: Themistoklis P. Sapsis

Copyright © 2014 Moon Hee Yang and Kyung Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


It has been believed that even an imperfect inspector with nonzero inspection errors could either overestimate or underestimate a given FD (fraction defective) with a 50 : 50 chance. What happens to the existing inspection plans, if an imperfect inspector overestimates a known FD, when it is very low? We deal with this fundamental question, by constructing four mathematical models, under the assumptions that an infinite sequence of items with a known FD is given to an imperfect inspector with nonzero inspection errors, which can be constant and/or randomly distributed with a uniform distribution. We derive four analytical formulas for computing the probability of overestimation (POE) and prove that an imperfect inspector overestimates a given FD with more than 50%, if the FD is less than a value termed as a critical FD. Our mathematical proof indicates that the POE approaches one when FD approaches zero under our assumptions. Hence, if a given FD is very low, commercial inspection plans should be revised with the POE concept in the near future, for the fairness of commercial trades.