## DIMACS TR: 2008-07

## Optimal Sequential Inspection Policies

### Authors: Noam Goldberg, Jonathan Word, Endre Boros and Paul Kantor

**
ABSTRACT
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We consider the problem of combining a given set of diagnostic tests into an inspection system that can classify items of interest (cases) with maximum accuracy subject to budget constraints.
One motivating application is sequencing diagnostic tests for container inspection problems, where the diagnostic tests may correspond to radiation sensors, documents checks, or imaging systems. We consider mixtures of decision trees as inspection systems following the work of Boros, Fedzorah, Kantor, Saeger and Stroud.
We establish some properties of efficient inspection systems. Given an available set of elementary or compound tests (all are called "devices"), we characterize the optimal classification of cases, based on the the various "readings" or test scores given by the device. More generally, we consider the problem of optimally assigning a set of cases to a set of possible actions, based on the device ``readings'', while satisfying a budget constraint. The measure of performance is the fraction of all cases, in a specific class of interest, which are classified correctly. We find that this problem is a special case of a known variant of the knapsack problem, known as the Linear Multiple Choice Knapsack problem (LMCK). Exploiting the special features that we establish, we propose an algorithm that solves our special variant more efficiently than the existing LMCK algorithms.
Finally, we propose a dynamic programming algorithm that enumerates all of the efficient, undominated, inspection policies in a two dimensional cost-detection space. Our inspection policies may sequence any arbitrary number of tests and are not restricted in the branching factor (or number of channels). Our approach directly solves the bi-criterion optimization problem of maximizing detection and minimizing cost, and thus supports sensitivity analysis over different budget or detection requirements, and the optimization of any monotone (possibly nonlinear) utility function over the efficient set.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2008/2008-07.pdf

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