In educational practice, a test assembly problem is formulated as a system of inequalities induced by test specifications. Each solution to the system is a test, represented by a 0–1 vector, where each element corresponds to an item included (1) or not included (0) into the test. Therefore, the size of a 0–1 vector equals the number of items n in a given item pool. All solutions form a feasible set—a subset of 2n vertices of the unit cube in an n-dimensional vector space. Test assembly is uniform if each test from the feasible set has an equal probability of being assembled. This paper demonstrates several important applications of uniform test assembly for educational practice. Based on Slepian’s inequality, a binary program was analytically studied as a candidate for uniform test assembly. The results of this study establish a connection between combinatorial optimization and probability inequalities. They identify combinatorial properties of the feasible set that control the uniformity of the binary programming test assembly. Computer experiments illustrating the concepts of this paper are presented.

A set of linear conditions on item response functions is derived that guarantees identical observed-score distributions on two test forms. The conditions can be added as constraints to a linear programming model for test assembly that assembles a new test form to have an observed-score distribution optimally equated to the distribution on an old form. For a well-designed item pool and items fitting the IRT model, use of the model results into observed-score pre-equating and prevents the necessity of post hoc equating by a conventional observed-score equating method. An empirical example illustrates the use of the model for an item pool from the Law School Admission Test.