Integer Programming with Logistic Item Exposure Penalties for Automated Test Assembly

Abstract

One feature of e-testing for educational assessment is an automated test assembly of parallel test forms, for which each form has equivalent measurement accuracy but with a different set of items. An important task for automated test assembly is to assemble as many tests as possible. Although many automatic uniform test assembly methods exist, the maximum clique using the integer programming method is known to assemble the greatest number of uniform tests with the highest measurement accuracy. However, the automated test assembly often causes a bias of item exposure. This bias problem decreases the reliability of items and tests. To solve this problem, this study formulates the test assembly problem as the objective function of integer programming with two logistic item exposure penalties. The first penalty is a deterministic penalty of logistic item exposure. The second penalty is a stochastic penalty with logistic item exposure based on the Big-M method, a standard technique in mathematical programming. Numerical experiments demonstrate that the proposed methods reduce the bias of item exposure without decreasing the number of tests.