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Environmental decision-making under uncertainty using intuitionistic fuzzy analytic hierarchy process (IF-AHP)

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Abstract

Analytic hierarchy process (AHP) is a utility theory based decision-making technique, which works on a premise that the decision-making of complex problems can be handled by structuring them into simple and comprehensible hierarchical structures. However, AHP involves human subjective evaluation, which introduces vagueness that necessitates the use of decision-making under uncertainty. The vagueness is commonly handled through fuzzy sets theory, by assigning degree of membership. But, the environmental decision-making problem becomes more involved if there is an uncertainty in assigning the membership function (or degree of belief) to fuzzy pairwise comparisons, which is referred to as ambiguity (non-specificity). In this paper, the concept of intuitionistic fuzzy set is applied to AHP, called IF-AHP to handle both vagueness and ambiguity related uncertainties in the environmental decision-making process. The proposed IF-AHP methodology is demonstrated with an illustrative example to select best drilling fluid (mud) for drilling operations under multiple environmental criteria.

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Abbreviations

A i (i = 1, 2,..., m):

Possible courses of actions or alternatives

A, B :

Intuitionistic fuzzy set (IFS)

\({\left\langle {{\left[ {{\left({{a}^{\prime}_{1}, {b}^{\prime}_{1}, {c}^{\prime}_{1}} \right)};\mu_{A}} \right]},\;{\left[ {{\left({a_{1}, b_{1}, c_{1}} \right)};\upsilon_{A}} \right]}} \right\rangle}\) :

Triangular intuitionistic fuzzy set

C j (j = 1, 2, ..., n):

Performance criteria or attributes

\(\bar{\bar{F}}_{Ai}\) :

Intuitionistic fuzzy AHP score

\(\bar{\bar{G}}_{k}\) :

Intuitionistic fuzzy global preference weights

\(\bar{\bar{J}}\) :

Intuitionistic fuzzy judgment matrix

\(\bar{\bar{j}}_{{ij}}\) :

Pairwise comparison index in intuitionistic fuzzy judgment matrix

K :

Number of levels in a hierarchical structure

\({\left[ {w^{{LI}}_{i}, w^{{UI}}_{i}} \right]}\) :

Lower and upper interval weights

\((\hat{w}_{i})^{{LI}}_{\alpha},(\hat{w}_{i})^{{UI}}_{\alpha}\) :

Lower and upper normalized interval weights

W = (w 1, w 2, ..., w n ):

Weight vector

\(\bar{\bar{W}}= (\bar{\bar{w}}_{i}, i = 1, 2, \ldots, n)\) :

Intuitionistic fuzzy weight vector

X :

Universe of discourse

x d :

Discrete points defined over the universe of discourse

x ij :

Performance rating of alternative A i for criterion C j

\(\bar{x}(A_{i})\) :

Generalized mean of an alternative A i (a reduced fuzzy set)

Δμ, ΔμL and ΔμU :

fuzzification factors

π x :

Degree of non-determinacy

υ x :

Non-membership function of x

υ A and υ B :

Non-membership function of IFS A and B

μ A and μ B :

Membership function of IFS A and B

μ L and μ U :

Lower and upper bound of membership function μ x

μ x :

Membership function of x

σ(A i ):

Standard deviation of an alternative A i (a reduced fuzzy set)

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Authors and Affiliations

  1. Urban Infrastructure Program, National Research Council Canada—Institute for Research in Construction, Ottawa, ON, Canada, K1A 0R6

    Rehan Sadiq & Solomon Tesfamariam

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  1. Rehan Sadiq
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  2. Solomon Tesfamariam
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Correspondence to Rehan Sadiq.

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Sadiq, R., Tesfamariam, S. Environmental decision-making under uncertainty using intuitionistic fuzzy analytic hierarchy process (IF-AHP). Stoch Environ Res Risk Assess 23, 75–91 (2009). https://doi.org/10.1007/s00477-007-0197-z

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