Table Of ContentLecture Notes in Economics and Mathematical
Systems 347
David Rios Insua
Sensitivity Analysis
in Multi-objective
Decision Making
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Author
Dr. David Rios Insua
School of Computer Studies
University of Leeds
Leeds LS2 9JT, UK
Opt. de Inteligencia Artificial, Facultad de Informatica
Universidad Politecnica de Madrid
Boadilla del Monte, 28660-Madrid, Spain
ISBN 978-3-540-52692-6 ISBN 978-3-642-51656-6 (eBook)
DOI 10.1007/978-3-642-51656-6
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1990
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Dedicated to my parents
Introduction
The axiomatic foundations of the Bayesian approach to decision making assurne
precision in the decision maker's judgements. In practicc, dccision makers often
provide only partial and/or doubtful information. We unify and expand results to
deal with those cases introducing a general framework for sensitivity analysis in
multi-objective decision making.
We study first decision making problems under partial information. We provide
axioms leading to modelling preferences by families of value functions, in problems
under certainty, and moJelling beliefs by families of probability distributions and
preferences by familics of utility functions, in problems under uncertainty. Both
problems are treated in parallel with the same parametric model. Alternatives
are ordered in a Pareto sense, the solution of the problem being the set of non
dominated alternatives. Potentially optimal solutions also seem acceptable, from
an intuitive point of view and due to their relation with the nondominated ones.
Algorithms are provided to compute these solutions in general problems and in
cases typical in practice: linear and bilinear problems. Other solution concepts
are criticised on the grounds of being ad hoc. In summary, we have a more ro
bust theory of decision making based on a weaker set ofaxioms, but embodying
coherence, since it essentially implies carrying out a family of coherent dccision
anitlyses.
These results are used to detect when a current optimal solution has competi
tors. In this case, sensitivity analyses are necessary as it means to help the DM
explore her thoughts to find which of her judgements are the most influential in de
termining choice. This includes a criticism of the Flat Maxima Principle. Several
concepts of adjacent potential optimality help us to reduce the set of competitars
of the current optimal alternative, detecting which alternatives may share opti
mality with it. A class of easily implcmcntable sensitivity tools based on gauge
functions is introduced, allowing us to identify critical judgements, competitors of
the current optimal alternative, and vitrious scnsitivity issues itccording to certain
scnsitivity indices. We then explore some questions on error modelling in judge
mental mcitsurement, providing some other lools, based on Baycsian hypüthcsis
testing, to deepen the analysis.
Most of the above ideas are implemented in SEl\'SATO, a prototype library für
a sensitivity analysis package far dccision aids. Some computational experience is
VI
described, proving the feasibility and success of our framework.
I wish to thank my supervisor, Professor Simon French, for his permanent en
couragement, discussions and criticisms, his patience with my sometimes imprecise
approaches and for guiding my interest to my research area and my thinking to
the Bayesian path.
My first steps in Decision Theory were given under the guidance of Professor
Sixto Rios, who has provided invaluable criticisms to my manuscript. I have
received many suggestions for my research from Professor Doug White, Dr. Roger
Cooke and Dr. Les Proll.
My gratitude to staff and fellow students in Manchester University, for taking
part in my experiments, and Leeds University, for helping me with the computers
and supplying an enjoyable environment. Mike Wiper belongs to the intersection
of both sets; in addition, he has discussed many concepts and read most of the
manuscript. Valeria Rios Insua helped me with the figures.
Computer facilities were provided by the University of Manchester Regional
Computer Center and the University of Leeds Computing Service. This research
was supported by a grant of the Bank of Spain.
Finally, I wish to thank my family for their support and understanding, and
Cornelia for showing me that even the impossible is beatable.
Contents
1 Partial Information and Sensitivity Analysis in Decision Making.
Introduction 1
1.1 Bayesian decision analysis 2
1.1.1 Problems under certainty . 3
1.1.2 Problems und er uncertainty 4
1.1.3 Comment ........ . 5
1.2 Some views on other decision aids 6
1.2.1 Fuzzy-set based decision analysis: Yager's method 6
1.2.2 The Analytic Hierarchy Process: a comment .. 9
1.2.3 Outranking methods: Comments on ELECTRE I 13
1.2.4 Interactive decision aids 16
1.2.5 A general comment 20
1.3 Our problem. . . . . . . 21
1.4 Some previous thoughts 22
1.4.1 The problem. . . 22
1.4.2 Foundations of decision making under partial information. 23
1.4.3 Ordering the alternatives. Decision making under partial
information . . . . 23
1.4.4 Additional criteria 23
1.4.5 Hierarchical approaches 24
1.4.6 Detection of sensitivity . 24
1.4.7 Sensitivity problems are not important 24
1.4.8 Sensitivity measures .. 25
1.4.9 Detection of competitors ... 25
1.4.10 What to do about sensitivity? 25
1.4.11 Error modelling . . . . . . . 25
1.4.12 Sensitivity analysis in commercial decision aids 25
1.5 Comments. . .. 27
VIII
2 Decision Making under Partial Information: Theory and Algo-
rithms 28
2.1 Some basic results . 29
2.2 Judgemental quasi orders. 29
2.2.1 General preference quasi orders under certainty 30
2.2.2 Additive models. . . . . . . .. . ... 32
2.2.3 Preference quasi orders under uncertainty 45
2.3 A parametric model .... 51
2.3.1 The certainty case 52
2.3.2 The uncertainty case 52
2.4 Orders and solution concepts . 53
2.5 General algorithms ..... . 59
2.5.1 Nondominated alternatives. 59
2.5.2 Potentially optimal alternatives 61
2.6 Some particular cases . 63
2.6.1 Linear models . 63
2.6.2 Bilinear models 67
2.7 Additional criteria .. 69
2.7.1 Some additional criteria 69
2.7.2 Hierarchical approaches 71
2.8 Comments ........... . 73
3 Sensitivity Analysis in Multi-objective Decision Making 74
3.1 Some basic results .......... . 75
3.1.1 The maximum ranking function 75
3.1.2 The optimality subsets 80
3.2 Do we need sensitivity analysis? (Yes) 83
3.2.1 Strang optimality ....... . 83
3.2.2 A Flat Maxima Principle? ., 86
3.2.3 Demands of a sensitivity analysis tool . 88
3.3 Adjacent potentially optimal solutions 89
3.3.1 Adjacent potential optimality .. 89
3.3.2 Adjacent potential optimality according to w . 92
3.3.3 Straight adjacent potential optimality . 92
3.3.4 Relations between the concepts 93
3.4 Some sensitivity tools . . . " 93
3.4.1 A dass of sensiti,ity tools 94
IX
3.4.2 Gauge function based sensitivity toels 96
3.5 Error modelling in decision analysis ... 103
3.5.1 Errors in judgemental modelling . 104
3.5.2 An error model ......... . 106
3.5.3 Some thoughts on the elements of the problem. 112
3.6 Comments ....... . 126
4 Computational experience 121
4.1 An introduction to SENSATO · 127
4.2 Linear models: Flood-plain management 132
4.2.1 The model ..... 133
4.2.2 Input to SENSLIN 134
4.2.3 Results ...... . · 134
4.3 Bilinear models: Portfollo selection · 143
4.3.1 The model ..... · 144
4.3.2 Input to SENSBIL · 145
4.3.3 Results....... · 146
4.4 General models: Technology-purchasing decision . · 147
4.4.1 The model ..... . · 149
4.4.2 Input to SENSGEN . · 150
4.4.3 Results...... 151
4.5 Dallas' problem revisited 151
4.5.1 The model .... 152
4.5.2 Results...... · 153
4.6 Questioning a model: Road selection · 153
4.6.1 Analysis 1 ....... . · 154
4.6.2 The model (Analysis 1) . 155
4.6.3 Results ......... . 156
4.6.4 Analysis 2 . . . . . . . . · 156
4.6.5 The model (Analysis 2) . 157
4.6.6 Results(Analysis 2) 158
4.7 Some counterexamples 158
4.7.1 The model. 159
4.7.2 Results. 160
4.8 Comments..... 161
x
5 Conclusions 164
5.1 Summary . 164
5.2 Topics for further research . 165
5.2.1 Decision making under partial information 165
5.2.2 Sensitivity analysis .. . 167
5.2.3 Computation ..... . 168
5.2.4 Group decision making . 170
5.2.5 Imprecision in expert systems 170
5.2.6 Restructuring ........ . 170
5.2.7 Integration with other methodologies 171
Bibliography 172
Appendix 186
