Exact bounds for efficient consistent matrices obtained from a reciprocal matrix
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911209279717376 |
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| author | Furtado, Susana Johnson, Charles |
| author_facet | Furtado, Susana Johnson, Charles |
| contents | For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The maximal sets of entries in the lower and upper bound matrices of each interval that are attainable by some consistent matrix in the interval are described. This allows us to understand which subsets of the alternatives lie above which other subsets in all efficient orders for each interval. As a result, the partial order on the alternatives dictated by the efficient vectors follows. Then, we use the tools developed to also show that, when the n-by-n reciprocal matrices A,B are simple perturbed consistent matrices, or n=4, the sets of efficient vectors for A and B coincide only if A=B. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact bounds for efficient consistent matrices obtained from a reciprocal matrix Furtado, Susana Johnson, Charles Combinatorics 90B50, 91B06, 15B48 For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The maximal sets of entries in the lower and upper bound matrices of each interval that are attainable by some consistent matrix in the interval are described. This allows us to understand which subsets of the alternatives lie above which other subsets in all efficient orders for each interval. As a result, the partial order on the alternatives dictated by the efficient vectors follows. Then, we use the tools developed to also show that, when the n-by-n reciprocal matrices A,B are simple perturbed consistent matrices, or n=4, the sets of efficient vectors for A and B coincide only if A=B. |
| title | Exact bounds for efficient consistent matrices obtained from a reciprocal matrix |
| topic | Combinatorics 90B50, 91B06, 15B48 |
| url | https://arxiv.org/abs/2510.12358 |