Semiorders: Properties, Representations, Applications

Front Cover
Springer Science & Business Media, Jun 30, 1997 - Business & Economics - 190 pages
0 Reviews
Semiorder is probably one of the most frequently ordered structures in science. It naturally appears in fields like psychometrics, economics, decision sciences, linguistics and archaeology. It explicitly takes into account the inevitable imprecisions of scientific instruments by allowing the replacement of precise numbers by intervals. The purpose of this book is to dissect this structure and to study its fundamental properties. The main subjects treated are the numerical representations of semiorders, the generalizations of the concept to valued relations, the aggregation of semiorders and their basic role in a general theoretical framework for multicriteria decision-aid methods.
Audience: This volume is intended for students and researchers in the fields of decision analysis, management science, operations research, discrete mathematics, classification, social choice theory, and order theory, as well as for practitioners in the design of decision tools.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

FIRST PRESENTATION OF THE BASIC CONCEPTS
7
12 Introduction to the concept of semiorder
9
122 Strict complete order associated to a semiorder
12
124 Example
13
13 Introduction to the concept of interval order
14
132 Example
15
14 Interval graph and indifference graph
16
15 Valued semiorder
17
412 Semiorders as interval orders and the representation of interval orders
86
413 Ordered structures associated with a steptype matrix
88
414 Representations of all interval orders associated with a steptype matrix
89
415 Minimal representation of an interval order
91
416 Proofs of the theorems
93
VALUED SEMIORDERS
103
53 Graph and matrix representations of a valued relation
104
54 Semiordered valued relation
105

HISTORICAL REVIEW AND APPLICATIONS
19
22 Semiorders in genetics
22
23 Semiorders bandwidth and other difficult combinatorial problems
25
24 Semiorders in information storage
26
25 Semiorders in decisionaid
27
26 Semiorders in scheduling
28
28 Interval orders and semiorders in archaeology
29
29 Interval orders and semiorders in psychology
31
211 Semiorders in classification
32
212 Semiorders and rough sets
42
213 Semiorders and fuzzy sets
43
214 Semiorders and a theory of evolution of rationality
44
215 Appendix
45
BASIC CONCEPTS AND DEFINITIONS
49
32 Asymmetric and symmetric parts of a relation
50
35 Complement converse and dual of a relation
51
37 Semiorder
52
39 Dual of a semiorder strict semiorder
53
310 Complete preorder
54
313 Complete order
55
316 Complete preorders and strict complete order associated to a semiorder
56
317 Graph and matrix representations of a semi order
57
318 Example of semiorder
58
319 Interval order
60
321 Dual of an interval order strict interval order
61
323 Graph and matrix representations of an interval order
62
324 Example of interval order
63
325 Proofs of the theorems
65
MINIMAL REPRESENTATIONS
71
42 Alternative definition of a finite semiorder
72
44 An algorithm for finding the numerical representation of a semiorder
73
46 Minimal representation of a semiorder
74
47 Integrality of the minimal representation
75
48 Maximal contrast property
76
49 An example and the synthetic graph of a se miorder
77
410 Noses and Hollows
79
411 More about paths of SSG
82
56 Example of a semiordered valued relation
106
57 Minimal representation of a semiordered valued relation
108
58 Ipsodual semiordered valued relation
109
510 Example of an ipsodual semiordered valued relation
111
513 Numerical representation of a linear semiordered valued relation
112
514 Example of a linear semiordered valued relation
113
517 Numerical representation of an ipsodual linear semiordered valued relation
114
519 Minimal representation of an ipsodual semi ordered valued relation
115
522 Minimal representation
118
523 Proofs of the theorems
119
AGGREGATION OF SEMIORDERS
125
61 Arrows theorem for semiorders
126
62 Lexicographic aggregation of semiorders
127
622 First variant
128
623 Second variant
129
624 Third variant
130
625 A note on strict preference
131
626 A note on indifference
132
627 Fourth variant
133
628 Fifth variant
135
63 Aggregation of semiorders by Bordas method
136
632 Use of the concept of rank of an element in a graph without circuit
137
633 Using scores
139
64 The dominant aggregation paradigm applied to semiorders
141
65 Theoretical results related to the overall evaluation approach
144
66 The pairwise comparisons paradigm
148
67 A general framework
151
68 Aggregation of valued serniorders and families of semiorders
157
69 Proofs of the theorems Proof of theorem
160
MISCELLANEOUS
167
72 Semiordered mixture sets
168
74 Double threshold models
169
76 Enumerating semiorders
170
78 Indifference graphs and families of indifference graphs
171
CONCLUSION
173
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page ii - In particular, formal treatment of social phenomena, the analysis of decision making, information theory and problems o ("inference will be central themes of this part of the library. Besides theoretical results, empirical investigations and the testing of theoretical models of real world problems will be subjects of interest. In addition to emphasizing interdisciplinary communication, the series will seek to support the rapid dissemination of recent results.
Page 178 - Some applications of graph theory and related nonmetric techniques to problems of approximate seriation: The case of symmetric proximity measures. The British Journal of Mathematical and Statistical Psychology.
Page 179 - Nitzan, S. and Rubinstein, A. (1981). A further characterization of Borda ranking method, Public choice 36: 153-158.
Page 175 - Abbas, M., Pirlot, M. and Vincke, Ph. (1996). Preference structures and cocomparability graphs, Journal of Multicriteria Decision Analysis 5: 81-98.

References to this book

All Book Search results »

Bibliographic information