Exact and approximate methods for a one-dimensional minimax bin-packing problem
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...
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| Vydáno v: | Annals of operations research Ročník 206; číslo 1; s. 611 - 626 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer US
01.07.2013
Springer Springer Nature B.V |
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| ISSN: | 0254-5330, 1572-9338 |
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| 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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| AbstractList | Issue Title: Including Special Section: Applications of Operations Research in Educational Measurement in Memory of Ronald D. Armstrong (1945-2011) 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.[PUBLICATION 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. 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. Keywords One-dimensional bin-packing * Combinatorial optimization * Integer programming * Simulated annealing * Test splitting |
| Audience | Academic |
| Author | Brusco, Michael J. Steinley, Douglas Köhn, Hans Friedrich |
| Author_xml | – sequence: 1 givenname: Michael J. surname: Brusco fullname: Brusco, Michael J. organization: Florida State University – sequence: 2 givenname: Hans Friedrich surname: Köhn fullname: Köhn, Hans Friedrich email: hkoehn@cyrus.psych.uiuc.edu organization: University of Illinois at Urbana-Champaign – sequence: 3 givenname: Douglas surname: Steinley fullname: Steinley, Douglas organization: University of Missouri-Columbia |
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| Cites_doi | 10.1007/BF02293734 10.1007/BF00127358 10.1016/S0305-0548(00)00082-4 10.1016/0898-1221(94)90077-9 10.1145/1120582.1120583 10.1287/ijoc.7.2.191 10.1137/0203025 10.1080/01621459.1971.10482319 10.1177/014662169201600106 10.1177/01466216980223001 10.1007/BF02294518 10.1111/j.1745-3984.1999.tb00557.x 10.1016/0305-0548(88)90009-3 10.1177/014662168801200210 10.1057/palgrave.jors.2601307 10.1287/opre.42.2.287 10.1016/B0-12-369398-5/00447-3 10.1177/014662169201600408 10.1002/1520-6750(199106)38:3<447::AID-NAV3220380312>3.0.CO;2-0 10.1126/science.220.4598.671 10.1007/BF02283751 10.1057/palgrave.jors.2600356 10.1007/0-387-29054-0 |
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| Keywords | Integer programming One-dimensional bin-packing Combinatorial optimization Test splitting Simulated annealing |
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| SubjectTerms | Algorithms Analysis Approximation Bins Business and Management Combinatorial optimization Combinatorics Integer programming Linear programming Mathematical analysis Mathematical programming Methods Minimax technique Operations research Operations Research/Decision Theory Optimization Packing problem Quantitative psychology Simulated annealing Simulated annealing (Mathematics) Software Software packages Statistics Studies Theory of Computation |
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