Geometric programming for optimal allocation of integrated samples in quality control

Abstract

We apply geometric programming, developed by Duffin, Peterson Zener (1967), to the optimal allocation of stratified samples. As an introduction, we show how geometric programming is used to allocate samples according to Neyman (1934), using the data of Cornell (1947) and following the exposition of Cochran (1953).

Then we use geometric programming to allocate an integrated sample introduced by Schwartz (1978) for more efficient sampling of three U. S. Federal welfare quality control systems, Aid to Families with Dependent Children, Food Stamps and Medicaid.

We develop methods for setting up the allocation problem, interpreting it as a geometric programming primal problem, transforming it to the corresponding dual problem, solving that, and finding the sample sizes required in the allocation problem. We show that the integrated sample saves sampling costs.

Keywords:

• Neyman allocation
• mathematical programming
• sampling costs
• variance constraints
• college enrollments
• integrated sampling
• Aid to Families with Dependent Children
• Food Stamps
• Medicaid
• primal and dual problems
• convexity
• maximization
• Lagrange multiplier
• Eureka: The Solver
• Davidon-Fletcher Powell method
• Rosen conjugate-gradient projection method