### Abstract

In this paper we consider the problem of decomposing a given integer matrix [Formula presented] into an integer conic combination of consecutive-ones matrices with a bound on the number of columns per matrix. This problem is of relevance in the realization stage of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf collimators with limited width. Constrained and unconstrained versions of the problem with the objectives of minimizing beam-on time and decomposition cardinality are considered. We introduce a new approach which can be used to find the minimum beam-on time for both constrained and unconstrained versions of the problem. The decomposition cardinality problem is shown to be [Formula presented]-hard and an approach is proposed to solve the lexicographic decomposition problem of minimizing the decomposition cardinality subject to optimal beam-on time.

Original language | English |
---|---|

Pages (from-to) | 87-103 |

Number of pages | 17 |

Journal | Discrete Applied Mathematics |

Volume | 250 |

DOIs | |

Publication status | Published - 11 Dec 2018 |

### Keywords

- Beam-on time
- Decomposition cardinality
- Field splitting
- Intensity modulated radiation therapy
- Multileaf collimator sequencing

### Cite this

*Discrete Applied Mathematics*,

*250*, 87-103. https://doi.org/10.1016/j.dam.2018.04.016

}

*Discrete Applied Mathematics*, vol. 250, pp. 87-103. https://doi.org/10.1016/j.dam.2018.04.016

**Minimizing the number of apertures in multileaf collimator sequencing with field splitting.** / Baatar, Davaatseren; Ehrgott, Matthias; Hamacher, Horst W.; Raschendorfer, Ines M.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Minimizing the number of apertures in multileaf collimator sequencing with field splitting

AU - Baatar, Davaatseren

AU - Ehrgott, Matthias

AU - Hamacher, Horst W.

AU - Raschendorfer, Ines M.

PY - 2018/12/11

Y1 - 2018/12/11

N2 - In this paper we consider the problem of decomposing a given integer matrix [Formula presented] into an integer conic combination of consecutive-ones matrices with a bound on the number of columns per matrix. This problem is of relevance in the realization stage of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf collimators with limited width. Constrained and unconstrained versions of the problem with the objectives of minimizing beam-on time and decomposition cardinality are considered. We introduce a new approach which can be used to find the minimum beam-on time for both constrained and unconstrained versions of the problem. The decomposition cardinality problem is shown to be [Formula presented]-hard and an approach is proposed to solve the lexicographic decomposition problem of minimizing the decomposition cardinality subject to optimal beam-on time.

AB - In this paper we consider the problem of decomposing a given integer matrix [Formula presented] into an integer conic combination of consecutive-ones matrices with a bound on the number of columns per matrix. This problem is of relevance in the realization stage of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf collimators with limited width. Constrained and unconstrained versions of the problem with the objectives of minimizing beam-on time and decomposition cardinality are considered. We introduce a new approach which can be used to find the minimum beam-on time for both constrained and unconstrained versions of the problem. The decomposition cardinality problem is shown to be [Formula presented]-hard and an approach is proposed to solve the lexicographic decomposition problem of minimizing the decomposition cardinality subject to optimal beam-on time.

KW - Beam-on time

KW - Decomposition cardinality

KW - Field splitting

KW - Intensity modulated radiation therapy

KW - Multileaf collimator sequencing

UR - http://www.scopus.com/inward/record.url?scp=85047207718&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2018.04.016

DO - 10.1016/j.dam.2018.04.016

M3 - Article

VL - 250

SP - 87

EP - 103

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -