Abstract
One-dimensional bin-packing problems require the assignment of a collection of items to bins with the goal of optimizing some criterion related to the number of bins used or the ‘weights’ of the items assigned to the bins. In many instances, the number of bins is fixed and the goal is to assign the items such that the sums of the item weights for each bin are approximately equal. Among the possible applications of one-dimensional bin-packing in the field of psychology are the assignment of subjects to treatments and the allocation of students to groups. An especially important application in the psychometric literature pertains to splitting of a set of test items to create distinct subtests, each containing the same number of items, such that the maximum sum of item weights across all bins is minimized. In this context, the weights typically correspond to item statistics derived from difficulty and discrimination indices. We present a mixed zero-one integer linear programming (MZOILP) formulation of this one-dimensional minimax bin-packing problem and develop an approximate procedure for its solution that is based on the simulated annealing algorithm. In two comparisons that focused on 34 practically-sized test problems (up to 6000 items and 300 bins), the simulated annealing heuristic generally provided better solutions than were obtained when using a commercial mathematical programming software package to solve the MZOILP formulation directly.
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Notes
Test items are the questions asked in a test; examinees are supposed to select appropriate answers and submit these to be scored and interpreted. Items are the building blocks of any test. They typically consist of an item stem, which contains the stimulus materials to which the examinees are to respond, and a system of response options—a means for the examinees to record their responses.
Certain testing situations might require us to measure a given attribute using two different tests (i.e., two different sets of items). Tests measuring the same attribute are commonly referred to as alternate forms of a test. If in addition they satisfy certain psychometric properties, then alternate test forms are called parallel. The classical true-score model represents an observed score, Y, as the sum of true-score and error-of-measurement components: Y=τ+ε. This leads naturally to the notion of parallel measurements, Y and Y ∗, scores on tests that yield a shared true score, τ, and errors of measurement, ε and ε ∗, assumed to be uncorrelated with the true score and with each other, and to have equal variances.
In classical test theory, the item difficulty score—or item difficulty for short—is defined as the proportion of examinees in a sample who master an item; hence, an item with a high difficulty is “easy” and one with a low difficulty is “difficult”. Item discrimination is defined as the correlation between the score on a single item and the total test score. Items with a high discrimination score discriminate well between examinees with high and low ability levels, whereas items with low discrimination scores do not. Item difficulty and discrimination are always estimated during item pretesting, and specify the typical order of magnitude of the estimation errors.
The precision of measurement of a test score is typically referred to as its reliability and is defined in classical test theory as the squared correlation between the observed score, Y, and the true score, τ. The split-half method is one among many techniques for estimating the reliability of a test: a single test of I items (suppose I an even number) is administered once only, and the items are split (in some way) into two subtests, each of I/2 items; the correlation between the scores of the two half-tests represents their reliability. The reliability of the total test score on I items can then be approximated by the Spearman-Brown prophecy (or correction) formula.
From an applied point of view, the differences in the objective function values observed between MZOILP/CPLEX and our SA implementation might seem of a rather academic nature, with negligible practical consequences. Within an academic perspective, however, they mark the difference when global optimality is set as a goal; and they emphasize the need for careful consideration of such a claim.
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Douglas Steinley was supported by a grant from NIAAA (5K25AA017456).
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Brusco, M.J., Köhn, H.F. & Steinley, D. Exact and approximate methods for a one-dimensional minimax bin-packing problem. Ann Oper Res 206, 611–626 (2013). https://doi.org/10.1007/s10479-012-1175-5
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DOI: https://doi.org/10.1007/s10479-012-1175-5