Approximately optimum two-stage sampling plans (Q1071461)

Two-stage sampling plans \((n_ 1,n_ 2,a_ 1,r_ 1,a_ 2)\) for sampling by attributes respectively by variables are considered, \(n_ i\) being the sample size, \(a_ i\) the acceptance number at the i-th stage \((i=1,2)\) and \(r_ 1\) the rejection number (at the first stage). The losses both for acceptance and for rejection are assumed to be linear in the fraction defective. The optimality criterion of minimizing the maximal expected avoidable loss is used. With the aid of many approximations - sometimes obtained by numerical comparisons - an approximately optimum sampling plan for sampling by attributes is obtained, the binomial distribution being approximated by the normal distribution. Some numerical examples show an improvement upon the optimum single sample plans of about 7\%. In the same manner an approximately optimum two-stage sampling plan by variables in the one-sided as well as in the two-sided case with normally distributed random variables is derived. The lot of approximations during the derivations of the optimum solutions seems to make it impossible to estimate the cumulative error arising from these approximations.