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.
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.
References
Adema, J. J. (1992). Methods and models for the construction of weakly parallel tests. Applied Psychological Measurement, 16, 53–63.
Anily, S., Bramel, J., & Simchi-Levi, D. (1984). Worst-case analysis of heuristics for the bin-packing problem with general cost structures. Operations Research, 42, 287–298.
Armstrong, R. D., & Jones, D. H. (1992). Polynomial algorithms for item matching. Applied Psychological Measurement, 16, 365–373.
Baker, K. R., & Powell, S. G. (2002). Methods for assigning students to groups: a study of alternate objective functions. Journal of the Operational Research Society, 53, 397–404.
Brusco, M. J., & Stahl, S. (2005). Branch-and-bound applications in combinatorial data analysis. New York: Springer.
Brusco, M. J., Thompson, G. M., & Jacobs, L. W. (1997). A morph-based simulated annealing heuristic for a modified bin-packing problem. Journal of the Operational Research Society, 48, 433–439.
Coffman, E. G., Garey, M. R., & Johnson, D. S. (1997). Approximation algorithms for bin packing: A survey. In D. Hochbaum (Ed.), Approximation algorithms for NP-hard problems (pp. 46–93), Boston: PWS Publishing.
Csirik, J., Johnson, D. S., Kenyon, C., Orlin, J. B., Shor, P. W., & Weber, R. R. (2006). On the sum-of-squares algorithm for bin packing. Journal of the ACM, 53, 1–65.
Dash Optimization, Ltd. (2001–2007). Modeling with Xpress-MP. http://www.dashoptimization.com/home/downloads/pdf/Modeling_with_Xpress-MP.pdf.
Dell’Amico, M., & Martello, S. (1995). Optimal scheduling of tasks on identical parallel processors. Informs Journal on Computing, 7, 191–200.
Fleszar, K., & Hindi, K. S. (2002). New heuristics for one-dimensional bin-packing. Computers & Operations Research, 29, 821–839.
Hall, N. G., Ghosh, S., Kankey, R. D., Narasimhan, S., & Rhee, W. T. (1988). Bin-packing problems in one dimension: heuristic solutions and confidence intervals. Computers & Operations Research, 15, 171–177.
ILOG Inc. (2009). CPLEX 12.1, http://www.ilog.com/products/cplex/news/whatsnew.cfm.
Johnson, D. S., Garey, M. R., Graham, R. L., Demers, A., & Ullman, D. (1974). Worst-case performance bounds for simple one dimensional packing algorithms. SIAM Journal on Computing, 3, 299–325.
Kämpke, T. (1988). Simulated annealing: use of a new tool in bin packing. Annals of Operations Research, 16, 327–332.
Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220, 671–680.
Klein, G., & Aronson, J. E. (1991). Optimal clustering: a model and method. Naval Research Logistics, 38, 447–461.
Rao, M. R. (1971). Cluster analysis and mathematical programming. Journal of the American Statistical Association, 66, 622–626.
Rao, R. L., & Iyengar, S. S. (1994). Bin-packing by simulated annealing. Computers and Mathematics with Applications, 27, 71–82.
Scholl, A., & Voss, S. (1996). Simple assembly line balancing—heuristic approaches. Journal of Heuristics, 2, 217–244.
van der Linden, W. J. (1981). Using aptitude measurements for the optimal assignment of subjects to treatments with and without mastery scores. Psychometrika, 46, 257–274.
van der Linden, W. J. (1998). Optimal assembly of psychological and educational tests. Applied Psychological Measurement, 22, 195–211.
van der Linden, W. J. (2005). Linear models for optimal test design. New York: Springer.
van der Linden, W. J., & Boekkooi-Timminga, E. (1988). A zero-one programming approach to Gulliksen’s matched random subtests method. Applied Psychological Measurement, 12, 201–209.
van der Linden, W. J., & Boekkooi-Timminga, E. (1989). A maximin model for test design with practical constraints. Psychometrika, 54, 237–247.
Veldkamp, B. P. (1999). Multiple objective test assembly problems. Journal of Educational Measurement, 36, 253–266.
Veldkamp, B. P. (2005). Optimal test construction. In K. Kempf-Leonard (Ed.), Encyclopedia of social measurement (Vol. 2, pp. 933–941). San Diego: Academic Press.
Author information
Authors and Affiliations
Corresponding author
Additional information
Douglas Steinley was supported by a grant from NIAAA (5K25AA017456).
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-012-1175-5