Exact and approximate methods for a one-dimensional minimax bin-packing problem

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.

Variable neighborhood search heuristics for a test assembly design problem

“…The results of the method clearly outperform those provided by previous algorithms, such as the SA method proposed in Brusco et al (2013). More specifically, the proposed method is able to routinely reach a BPP-based lower bound in instances with a large number of questions per questionnaire (i.e., over ten questions per questionnaire) with characteristics of real-life problems, and still manages to obtain small deviations from the theoretical bound in other cases.…”

“…A different approach, which also links test assembly design to a classical combinatorial optimization problem was proposed in Brusco et al (2013). In this case, the problem is formulated as a constrained Bin Packing Problem…”

“…For large real-life problems, the authors propose a Simulated Annealing (SA) algorithm that outperforms the quality of the solutions provided by CPLEX. While Brusco et al (2013) do not address the complexity of the problem, the analysis of the results showed that all of the algorithms proposed in the paper were able to optimally solve large problems with 2 questionnaires, leading the authors to hypothesize that the special case of 2 questionnaires may be solvable 30 in polynomial time (Brusco et al, 2013, p. 623). Furthermore, the quality of the solutions provided by the SA deteriorates as the number of questionnaires increases, which may identify possible improvements if different solution methods were used.…”

“…As pointed out by van der Linden & Boekkooi-Timminga (1988) and Brusco et al (2013), the division of items into groups has multiple advantages as it can be used to obtain tests with similar composition as well as to enforce constraints which may be difficult to specify. The objective of the problem is then to 200 obtain tests with similar discrimination index on the ability level of interest.…”

“…In this paper, we extend the work of Brusco et al (2013) by considering its complexity and by providing a new efficient solution method for the MIN-IMAX BSC. The computational experiment shows that we are able to solve instances with up to 60.000-item pools within less than a minute in a modern commodity computer, and the solutions obtained show very small optimality gaps when compared to a theoretical lower bound on the maximum discrimination index.…”