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Group testing procedures with quantitative features and incomplete identification

Author

Listed:
  • Shaul K. Bar‐Lev
  • Onno Boxma
  • Andreas Löpker
  • Wolfgang Stadje
  • Frank A. Van der Duyn Schouten

Abstract

We present a group testing model for items characterized by marker random variables. An item is defined to be good (defective) if its marker is below (above) a given threshold. The items can be tested in groups; the goal is to obtain a prespecified number of good items by testing them in optimally sized groups. Besides this group size, the controller has to select a threshold value for the group marker sums, and the target number of groups which by the tests are classified to consist only of good items. These decision variables have to be chosen so as to minimize a cost function, which is a linear combination of the expected number of group tests and an expected penalty for missing the desired number of good items, subject to constraints on the probabilities of misclassifications. We treat two models of this kind: the first one is based on an infinite population size, whereas the second one deals with the case of a finite number of available items. All performance measures are derived in closed form; approximations are also given. Furthermore, we prove monotonicity properties of the components of the objective function and of the constraints. In several examples, we study (i) the dependence of the cost function on the decision variables and (ii) the dependence of the optimal values of the decision variables (group size, group marker threshold, and stopping rule for groups classified as clean) and of the target functionals (optimal expected number of tests, optimal expected penalty, and minimal expected cost) on the system parameters.© 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

Suggested Citation

  • Shaul K. Bar‐Lev & Onno Boxma & Andreas Löpker & Wolfgang Stadje & Frank A. Van der Duyn Schouten, 2012. "Group testing procedures with quantitative features and incomplete identification," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(1), pages 39-51, February.
    • Handle: RePEc:wly:navres:v:59:y:2012:i:1:p:39-51
    DOI: 10.1002/nav.20489
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    References listed on IDEAS

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    1. Ming-Chin Hung & William H. Swallow, 2000. "Use of Binomial Group Testing in Tests of Hypotheses for Classification or Quantitative Covariables," Biometrics, The International Biometric Society, vol. 56(1), pages 204-212, March.
    2. Lawrence M. Wein & Stefanos A. Zenios, 1996. "Pooled Testing for HIV Screening: Capturing the Dilution Effect," Operations Research, INFORMS, vol. 44(4), pages 543-569, August.
    3. Graham Hepworth & Ray Watson, 2009. "Debiased estimation of proportions in group testing," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 58(1), pages 105-121, February.
    4. Shaul K. Bar‐Lev & Wolfgang Stadje & Frank A. Van der Duyn Schouten, 2006. "Group testing procedures with incomplete identification and unreliable testing results," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 22(3), pages 281-296, May.
    5. Shaul K. Bar-Lev & Wolfgang Stadje & Frank A. van der Duyn Schouten, 2004. "Optimal Group Testing with Processing Times and Incomplete Identification," Methodology and Computing in Applied Probability, Springer, vol. 6(1), pages 55-72, March.
    6. Shaul K. Bar‐Lev & Arnon Boneh & David Perry, 1990. "Incomplete identification models for group‐testable items," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(5), pages 647-659, October.
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