### Abstract

The aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty (1980). A multiplicative model is used for the paired comparisons data which are collected in a ratio scale in this set-up in any level of this hierarchy. An iterative scheme is found for the maximum likelihood estimation of the worth parameters in this multiplicative model. The iterative values are shown to be convergent monotonically to the estimates. We also obtain the asymptotic dispersion matrix of the maximum likelihood estimates of the relative worths of the alternatives according to a single criterion as well as those according to the over-all suitability when compared under several criteria. A numerical example is presented to illustrate the method developed in this paper. Simulation techniques are employed to find the average number of iterations required for the convergence of the above iterative scheme.

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

Pages (from-to) | 3719-3738 |

Number of pages | 20 |

Journal | Communications in Statistics - Theory and Methods |

Volume | 18 |

Issue number | 10 |

DOIs | |

State | Published - Jan 1 1989 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability

### Cite this

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**Estimation of the Multi-Criteria Worths of the Alternatives in a Hierarchical Structure of Comparisons.** / Basak, Indrani.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Estimation of the Multi-Criteria Worths of the Alternatives in a Hierarchical Structure of Comparisons

AU - Basak, Indrani

PY - 1989/1/1

Y1 - 1989/1/1

N2 - The aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty (1980). A multiplicative model is used for the paired comparisons data which are collected in a ratio scale in this set-up in any level of this hierarchy. An iterative scheme is found for the maximum likelihood estimation of the worth parameters in this multiplicative model. The iterative values are shown to be convergent monotonically to the estimates. We also obtain the asymptotic dispersion matrix of the maximum likelihood estimates of the relative worths of the alternatives according to a single criterion as well as those according to the over-all suitability when compared under several criteria. A numerical example is presented to illustrate the method developed in this paper. Simulation techniques are employed to find the average number of iterations required for the convergence of the above iterative scheme.

AB - The aggregated worths of the alternatives, when compared with respect to several criteria, are estimated in a hierarchical comparisons model introduced by Saaty (1980). A multiplicative model is used for the paired comparisons data which are collected in a ratio scale in this set-up in any level of this hierarchy. An iterative scheme is found for the maximum likelihood estimation of the worth parameters in this multiplicative model. The iterative values are shown to be convergent monotonically to the estimates. We also obtain the asymptotic dispersion matrix of the maximum likelihood estimates of the relative worths of the alternatives according to a single criterion as well as those according to the over-all suitability when compared under several criteria. A numerical example is presented to illustrate the method developed in this paper. Simulation techniques are employed to find the average number of iterations required for the convergence of the above iterative scheme.

UR - http://www.scopus.com/inward/record.url?scp=0001757701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001757701&partnerID=8YFLogxK

U2 - 10.1080/03610928908830119

DO - 10.1080/03610928908830119

M3 - Article

AN - SCOPUS:0001757701

VL - 18

SP - 3719

EP - 3738

JO - Communications in Statistics - Theory and Methods

JF - Communications in Statistics - Theory and Methods

SN - 0361-0926

IS - 10

ER -