Estimating criteria weight distributions in multiple criteria decision making: a Bayesian approach
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Estimating criteria weight distributions in multiple criteria decision making: a Bayesian approach Barbaros Yet1
1 · Ceren Tuncer Sakar ¸
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract A common way to model decision maker (DM) preferences in multiple criteria decision making problems is through the use of utility functions. The elicitation of the parameters of these functions is a major task that directly affects the validity and practical value of the decision making process. This paper proposes a novel Bayesian method that estimates the weights of criteria in linear additive utility functions by asking the DM to rank or select the best alternative in groups of decision alternatives. Our method computes the entire probability distribution of weights and utility predictions based on the DM’s answers. Therefore, it enables the DM to estimate the expected value of weights and predictions, and the uncertainty regarding these values. Additionally, the proposed method can estimate the weights by asking the DM to evaluate few groups of decision alternatives since it can incorporate various types of inputs from the DM in the form of rankings, constraints and prior distributions. Our method successfully estimates criteria weights in two case studies about financial investment and university ranking decisions. Increasing the variety of inputs, such as using both ranking of decision alternatives and constraints on the importance of criteria, enables our method to compute more accurate estimations with fewer inputs from the DM. Keywords Multiple criteria decision making · Bayesian models · Decision analysis · Criteria weights · Additive utility
1 Introduction While classical optimization theory deals with problems that aim to maximize or minimize a single criterion, most realistic decision making problems have multiple conflicting criteria. Since all criteria cannot be simultaneously optimized, special methods from multiple criteria decision making (MCDM) domain are used for these problems (Belton and Stewart 2002; Greco et al. 2016). Generally speaking, a criterion is a measure of effectiveness that needs to be considered for the problem at hand, for example, profit. When the direction of improvement
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Barbaros Yet [email protected] Department of Industrial Engineering, Hacettepe University, 06800 Ankara, Turkey
123
Annals of Operations Research
is added to the criterion, we obtain an objective, like maximizing profit. In general, a multiple objective problem with n maximization objectives can be formulated as in (1): Maximize f (x) ( f 1 (x), . . . , f n (x)), subject to x ∈ X
(1)
where x is the decision variable vector, X is the feasible region in decision space and f i is the ith objective function. Let us denote the feasible region in objective space by Z. This set is the image set of X. A vector z (z1 , …, zn ) ∈ Z is called nondominated if and only if there does not exist some y ∈ Z such that yi ≥ zi for all i 1, …, n and yi > zi for at least one i. Otherwi
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