### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 611-626 |

Number of pages | 16 |

Journal | Annals of Operations Research |

Volume | 206 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2013 |

### Fingerprint

### Keywords

- Combinatorial optimization
- Integer programming
- One-dimensional bin-packing
- Simulated annealing
- Test splitting

### ASJC Scopus subject areas

- Decision Sciences(all)
- Management Science and Operations Research

### Cite this

*Annals of Operations Research*,

*206*(1), 611-626. https://doi.org/10.1007/s10479-012-1175-5

**Exact and approximate methods for a one-dimensional minimax bin-packing problem.** / Brusco, Michael J.; Koehn, Hans Friedrich; Steinley, Douglas.

Research output: Contribution to journal › Article

*Annals of Operations Research*, vol. 206, no. 1, pp. 611-626. https://doi.org/10.1007/s10479-012-1175-5

}

TY - JOUR

T1 - Exact and approximate methods for a one-dimensional minimax bin-packing problem

AU - Brusco, Michael J.

AU - Koehn, Hans Friedrich

AU - Steinley, Douglas

PY - 2013/7/1

Y1 - 2013/7/1

N2 - 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.

AB - 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.

KW - Combinatorial optimization

KW - Integer programming

KW - One-dimensional bin-packing

KW - Simulated annealing

KW - Test splitting

UR - http://www.scopus.com/inward/record.url?scp=84879410139&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879410139&partnerID=8YFLogxK

U2 - 10.1007/s10479-012-1175-5

DO - 10.1007/s10479-012-1175-5

M3 - Article

AN - SCOPUS:84879410139

VL - 206

SP - 611

EP - 626

JO - Annals of Operations Research

JF - Annals of Operations Research

SN - 0254-5330

IS - 1

ER -