Lagrangian relaxation for constrained curve-fitting with binary variables: Applications in educational testing.

摘要

This dissertation offers a mathematical programming approach to curve fitting with binary variables. Various Lagrangian Relaxation (LR) techniques are applied to constrained curve fitting. Applications in educational testing with respect to test assembly are utilized. In particular, techniques are applied to both static exams (i.e. conventional paper-and-pencil (P&P)) and adaptive exams (i.e. a hybrid computerized adaptive test (CAT) called a multiple-forms structure (MFS)). This dissertation focuses on the development of mathematical models to represent these test assembly problems as constrained curve-fitting problems with binary variables and solution techniques for the test development.; Mathematical programming techniques are used to generate parallel test forms with item characteristics based on item response theory. A binary variable is used to represent whether or not an item is present on a form. The problem of creating a test form is modeled as a network flow problem with additional constraints. In order to meet the target information and the test characteristic curves, a Lagrangian relaxation heuristic is applied to the problem. The Lagrangian approach works by multiplying the constraint by a "Lagrange multiplier" and adding it to the objective. By systematically varying the multiplier, the test form curves approach the targets.; This dissertation explores modifications to Lagrangian Relaxation as it is applied to the classical paper-and-pencil exams. For the P&P exams, LR techniques are also utilized to include additional practical constraints to the network problem, which limit the item selection.; An MFS is a type of a computerized adaptive test. It is a hybrid of a standard CAT and a P&P exam. The concept of an MFS will be introduced in this dissertation, as well as, the application of LR as it is applied to constructing parallel MFSs.; The approach is applied to the Law School Admission Test for the assembly of the conventional P&P test as well as an experimental computerized test using MFSs.