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A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process

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Abstract

The intuitionistic fuzzy analytic hierarchy process (IF-AHP), in which intuitionistic fuzzy numbers are utilized in defining decision makers’ linguistic judgment, has been used to solve various multi-criteria decision-making problems. Previous theories have suggested that interval-valued intuitionistic fuzzy numbers (IVIFN) with hesitation degree can act as alternative fuzzy numbers that can handle vagueness and uncertainty. This paper proposes a new preference scale in the framework of the interval-valued intuitionistic fuzzy analytic hierarchy process (IVIF-AHP). The comparison matrix judgment is expressed in IVIFN with degree of hesitation. The proposed new preference scale concurrently considers the membership function, the non-membership function and the degree of hesitation of IVIFN. To define the weight entropy of the aggregated matrix of IVIFN, a modified interval-valued intuitionistic fuzzy weighted averaging is proposed, by considering the interval number of the hesitation degree. Three multi-criteria decision-making problems are used to test the proposed method. A comparison of the results is also presented to check the feasibility of the proposed method. It is shown that the ranking order of the proposed method is slightly different from that of the other two methods because of the inclusion of the hesitation degree in defining the preference scale.

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Abbreviations

\(\mu _A ( x)\) :

Membership degree of \(x\) to the set \(A\)

\(v_A ( x)\) :

Non-membership degree of \(x\) to the set \(A\)

\(\pi _A ( x)\) :

Hesitation degree of \(x\) to the set \(A\)

\(\alpha =( {\mu _\alpha ,v_\alpha ,\pi _\alpha })\) :

Intuitionistic fuzzy number

\(\left( {\left[ {\mu _{\tilde{A}}^L ( x),\mu _{\tilde{A}}^U ( x)} \right] ,\left[ {\nu _{\tilde{A}}^L ( x),\nu _{\tilde{A}}^U (x)} \right] }\right) \) :

Interval-valued Intuitionistic fuzzy number notation

\(\mu _{\tilde{A}}^L ( x)\) and \(\mu _{\tilde{A}}^U ( x)\) :

Lower and upper membership degree of \(x\) to the set \(\tilde{A}\)

\(\nu _{\tilde{A}}^L ( x)\) and \(\nu _{\tilde{A}}^U ( x)\) :

Lower and upper non-membership degree of \(x\) to the set \(\tilde{A}\)

\(w=( {w_1 ,w_2 ,\ldots ,w_n })^T\) :

Weight vectors of criterion and alternatives

\(\lambda _k \) :

Weights of decision makers’ linguistic variables importance

\(D_k =( {\mu _k ,v_k ,\pi _k })\) :

Intuitionistic fuzzy variables for decision makers’ importance

\(\prod \limits _{j=1}^t {( {\nu _j })} \) :

Product of all values in range of series

\(\bar{\bar{w}}_i \) :

Entropy weight of aggregated interval-valued fuzzy comparison matrix judgement

\(w_i \) :

Normalized entropy weight of aggregated interval-valued fuzzy comparison matrix judgement

IFS:

Intuitionistic fuzzy sets

IVIFS:

Interval-valued intuitionistic fuzzy sets

IVIFN:

Interval-valued intuitionistic fuzzy number

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Authors and Affiliations

  1. School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu, 21030, Kuala Terengganu, Terengganu, Malaysia

    Lazim Abdullah & Liana Najib

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  1. Lazim Abdullah
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  2. Liana Najib
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Correspondence to Lazim Abdullah.

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Communicated by V. Loia.

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Abdullah, L., Najib, L. A new preference scale mcdm method based on interval-valued intuitionistic fuzzy sets and the analytic hierarchy process. Soft Comput 20, 511–523 (2016). https://doi.org/10.1007/s00500-014-1519-y

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