### Abstract

Clustering involves partitioning a set of related objects into a set of mutually exclusive and completely exhaustive clusters. The objective is to form clusters which reflect minimum difference among objects as measured by the relevant clustering criterion. Most statements of clustering problems assume that the number of clusters, g, in the partition is known. In reality, a value for g may not be immediately obvious. It is known that as g increases, there is an improvement in the value of the clustering criterion function. However, for some values of g, this rate of improvement may be less than expected. Because there may be a cost factor involved, there is also interest in identifying those values of g that offer attractive rates of improvement. Partitions that are optimal for a given g, and for which the given g offer an attractive rate of improvement, are referred to as being w-efficient; other partitions, even if optimal for a given g, are referred to as being w-inefficient. We present a linear programming approach for generating the w-efficient partitions of the sequential clustering problem, and demonstrate the importance of w-efficient partitions to the efficient solution of the sequential clustering problem.

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

Pages (from-to) | 305-319 |

Number of pages | 15 |

Journal | Annals of Operations Research |

Volume | 74 |

State | Published - Dec 1 1997 |

### Fingerprint

### Keywords

- Bi-objective programming
- Clustering
- Parametric programming
- W-efficiency

### ASJC Scopus subject areas

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

### Cite this

*Annals of Operations Research*,

*74*, 305-319.

**W-efficient partitions and the solution of the sequential clustering problem.** / Joseph, Anito; Bryson, Noel.

Research output: Contribution to journal › Article

*Annals of Operations Research*, vol. 74, pp. 305-319.

}

TY - JOUR

T1 - W-efficient partitions and the solution of the sequential clustering problem

AU - Joseph, Anito

AU - Bryson, Noel

PY - 1997/12/1

Y1 - 1997/12/1

N2 - Clustering involves partitioning a set of related objects into a set of mutually exclusive and completely exhaustive clusters. The objective is to form clusters which reflect minimum difference among objects as measured by the relevant clustering criterion. Most statements of clustering problems assume that the number of clusters, g, in the partition is known. In reality, a value for g may not be immediately obvious. It is known that as g increases, there is an improvement in the value of the clustering criterion function. However, for some values of g, this rate of improvement may be less than expected. Because there may be a cost factor involved, there is also interest in identifying those values of g that offer attractive rates of improvement. Partitions that are optimal for a given g, and for which the given g offer an attractive rate of improvement, are referred to as being w-efficient; other partitions, even if optimal for a given g, are referred to as being w-inefficient. We present a linear programming approach for generating the w-efficient partitions of the sequential clustering problem, and demonstrate the importance of w-efficient partitions to the efficient solution of the sequential clustering problem.

AB - Clustering involves partitioning a set of related objects into a set of mutually exclusive and completely exhaustive clusters. The objective is to form clusters which reflect minimum difference among objects as measured by the relevant clustering criterion. Most statements of clustering problems assume that the number of clusters, g, in the partition is known. In reality, a value for g may not be immediately obvious. It is known that as g increases, there is an improvement in the value of the clustering criterion function. However, for some values of g, this rate of improvement may be less than expected. Because there may be a cost factor involved, there is also interest in identifying those values of g that offer attractive rates of improvement. Partitions that are optimal for a given g, and for which the given g offer an attractive rate of improvement, are referred to as being w-efficient; other partitions, even if optimal for a given g, are referred to as being w-inefficient. We present a linear programming approach for generating the w-efficient partitions of the sequential clustering problem, and demonstrate the importance of w-efficient partitions to the efficient solution of the sequential clustering problem.

KW - Bi-objective programming

KW - Clustering

KW - Parametric programming

KW - W-efficiency

UR - http://www.scopus.com/inward/record.url?scp=0042729539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042729539&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042729539

VL - 74

SP - 305

EP - 319

JO - Annals of Operations Research

JF - Annals of Operations Research

SN - 0254-5330

ER -