## Evaluating sets of multi-attribute alternatives with uncertain preferences

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##### Date

2020-09

##### Authors

Toffano, Federico

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University College Cork

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##### Abstract

In a decision-making problem, there can be uncertainty regarding the user preferences concerning the available alternatives. Thus, for a decision support system, it is essential to analyse the user preferences to make personalised recommendations. In this thesis we focus on Multiattribute Utility Theory (MAUT) which aims to define user preference models and elicitation procedures for alternatives evaluated with a vector of a fixed number of conflicting criteria.
In this context, a preference model is usually represented with a real value function over the criteria used to evaluate alternatives, and an elicitation procedure is a process of defining such value function. The most preferred alternative will then be the one that maximises the value function.
With MAUT models, it is common to represent the uncertainty of the user preferences with a parameterised value function. Each instantiation of this parameterisation then represents a user preference model compatible with the preference information collected so far. For example, a common linear value function is the weighted sum of the criteria evaluating an alternative, which is parameterised with respect to the set of weights. We focus on this type of preference models and in particular on value functions evaluating sets of alternatives rather single alternatives. These value functions can be used for example to define if a set of alternatives is preferred to another one, or which is the worst-case loss in terms of utility units of recommending a set of alternatives.
We define the concept of setwise minimal equivalent subset (SME) and algorithms for its computation. Briefly, SME is the subset of an input set of alternatives with equivalent value function and minimum cardinality. We generalise standard preference relations used to compare single alternatives with the purpose of comparing sets of alternatives. We provide computational procedures to compute SME and evaluate preference relations with particular focus on linear value functions.
We make extensive use of the Minimax Regret criterion, which is a common method to evaluate alternatives for potential questions and recommendations with uncertain value functions. It prescribes an outcome that minimises the worst-case loss with respect to all the possible parameterisation of the value function. In particular, we focus on its setwise generalisation, namely \textit{Setwise Minimax Regret} (SMR), which is the worst-case loss of recommending a set of alternatives. We provide a novel and efficient procedure for the computation of the SMR when supposing a linear value function.
We also present a novel incremental preference elicitation framework for a supplier selection process, where a realistic medium-size factory inspires constraints and objectives of the underlying optimization problem. This preference elicitation framework applies for generic multiattribute combinatorial problems based on a linear preference model, and it is particularly useful when the computation of the set of Pareto optimal alternatives is practically unfeasible.

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##### Keywords

Multiattribute utility theory , Uncertain preferences , Multiobjective optimisation , Interactive preference elicitation , Preference relations , Minimax regret

##### Citation

Toffano, F. 2020. Evaluating sets of multi-attribute alternatives with uncertain preferences. PhD Thesis, University College Cork.