Abstract
We study the consistency property, and especially the acceptably consistent property, for incomplete interval-valued intuitionistic multiplicative preference relations. We propose a technique which first estimates the initial values for all missing entries in an incomplete interval-valued intuitionistic multiplicative preference relation and then improves them by a local optimization method. Two examples are presented in order to illustrate applications of the proposed method in group decision-making problems.
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Notes
In Saaty (1977), the MPRs assessed on the discrete \((1/9){-}9\) scale has values only in the set \(\{1/9, 1/8, 1/7,\ldots , 1/2, 1, 2, \ldots , 7, 8, 9\}.\)
Aczél and Saaty (1983) proved that to synthesize the group judgments, the geometric mean must be used in order to preserve the reciprocal property, that is, \(r_{ij}r_{ji}=1,\; \forall \; i,j\in N,\) must hold in the resultant aggregated matrix. Note that the weighted geometric mean preserves reciprocality on aggregation resulting in an aggregated MPR, while the same gets distorted if instead arithmetic mean is used for aggregating two or more MPRs.
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Acknowledgements
The authors are thankful to the esteemed referees for their valuable comments which help to improve the presentation of the paper substantially. The authors acknowledge the editor-in-chief for being considerate and supportive.
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Appendices
Appendix-1
We briefly include the operators of interval-valued intuitionistic fuzzy sets and numbers for the completeness of the ground-work theory. All notions and operators are described in Atanassov and Gargov (1989) and Xu and Cai (2012), while some of the interesting applications of I-MPRs and IVI-MPR are also included in (Xu et al. 2012).
Definition 13
Let X be a universal set. An intuitionistic fuzzy set (IFS) drawn from X is defined as \({\tilde{A}} = \{\langle x, \mu _{{\tilde{A}}}(x), \nu _{{\tilde{A}}}(x)\rangle \mid x \in X\},\) where \( \mu _{{\tilde{A}}},\; \nu _{{\tilde{A}}}: X \rightarrow [0,1]\) define, respectively, the degree of membership and the degree of non-membership of an element \(x\in X\) to the set \({\tilde{A}}\subseteq X\) such that \( \mu _{{\tilde{A}}}(x)+\nu _{{\tilde{A}}}(x) \le 1, \forall \;x \in X.\)
An intuitionistic fuzzy number (IFN) is defined by \(\alpha = (\mu _{\alpha }, \nu _{\alpha }),\) where \(\mu _{\alpha },\, \nu _{\alpha } \in [0,1],\;\mu _{\alpha }+\nu _{\alpha } \le 1.\)
Definition 14
Let X be a universal set. An IVIFS drawn from X is \( {\tilde{A}} = \{\langle x, \;{\tilde{\mu }}_{{\tilde{A}}}(x), \;{\tilde{\nu }}_{{\tilde{A}}}(x)\rangle \mid x \in X \}\,, \) where \({\tilde{\mu }}_{{\tilde{A}}},\; {\tilde{\nu }}_{{\tilde{A}}}: X\rightarrow 2^{[0,1]}\) are, respectively, the interval-valued membership and the interval-valued non-membership degrees of an element \(x\in X\) to the set \({\tilde{A}},\) such that, \(\sup _{x\in X} {\tilde{\mu }}_{{\tilde{A}}}(x) + \sup _{x\in X} {\tilde{\nu }}_{{\tilde{A}}}(x) \le 1.\)
Clearly if, \(\;\inf _{x\in X} {\tilde{\mu }}_{{\tilde{A}}}(x) = \sup _{x\in X} {\tilde{\mu }}_{{\tilde{A}}}(x)\;\) and \(\;\inf _{x\in X} {\tilde{\nu }}_{{\tilde{A}}}(x) = \sup _{x\in X} {\tilde{\nu }}_{{\tilde{A}}}(x),\;\) then the IVIFS \({\tilde{A}}\) reduces to the intuitionistic fuzzy set (IFS).
Definition 15
Let \({\tilde{A}} = \{\langle x, {\tilde{\mu }}_{{\tilde{A}}}(x), {\tilde{\nu }}_{{\tilde{A}}}(x)\rangle \mid x \in X \}, \; \tilde{A_1} = \{\langle x, {\tilde{\mu }}_{\tilde{A_1}}(x), {\tilde{\nu }}_{\tilde{A_1}}(x)\rangle \mid x \in X \}\) and \(\tilde{A_2} = \{\langle x, {\tilde{\mu }}_{\tilde{A_2}}(x), {\tilde{\nu }}_{\tilde{A_2}}(x)\rangle \mid x \in X \}\) be IVIFSs, and \(\lambda > 0\).
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1.
$$\begin{aligned} {\tilde{A}}^{\mathbf {c}} = \{\langle x, \;{\tilde{\nu }}_{{\tilde{A}}}(x), \;{\tilde{\mu }}_{{\tilde{A}}}(x)\rangle \mid x \in X \}; \end{aligned}$$
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2.
$$\begin{aligned}&\tilde{A_1} \cap \tilde{A_2} = \{\langle x, \;\left[ \min \{\inf {\tilde{\mu }}_{\tilde{A_1}}(x), \inf {\tilde{\mu }}_{\tilde{A_2}}(x)\}, \right. \\&\left. \quad \min \{\sup {\tilde{\mu }}_{\tilde{A_1}}(x), \sup {\tilde{\mu }}_{\tilde{A_2}}(x)\} \right] , \\&\quad \left[ \max \{\inf {\tilde{\nu }}_{\tilde{A_1}}(x), \inf {\tilde{\nu }}_{\tilde{A_2}}(x)\},\right. \\&\quad \max \{\sup {\tilde{\nu }}_{\tilde{A_1}}(x), \sup {\tilde{\nu }}_{\tilde{A_2}}(x)\} ]\rangle \mid x \in X \}\,; \end{aligned}$$
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3.
$$\begin{aligned}&\tilde{A_1} \cup \tilde{A_2} = \{\langle x, \;\left[ \max \{\inf {\tilde{\mu }}_{\tilde{A_1}}(x), \inf {\tilde{\mu }}_{\tilde{A_2}}(x)\},\right. \\&\left. \quad \max \{\sup {\tilde{\mu }}_{\tilde{A_1}}(x), \sup {\tilde{\mu }}_{\tilde{A_2}}(x)\} \right] , \\&\quad \left[ \min \{\inf {\tilde{\nu }}_{\tilde{A_1}}(x), \inf {\tilde{\nu }}_{\tilde{A_2}}(x)\},\right. \\&\left. \quad \min \{\sup {\tilde{\nu }}_{\tilde{A_1}}(x), \sup {\tilde{\nu }}_{\tilde{A_2}}(x)\} \right] \rangle \mid x \in X \}\,; \end{aligned}$$
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4.
$$\begin{aligned}&\tilde{A_1} + \tilde{A_2} = \{\langle x, \;\left[ \inf {\tilde{\mu }}_{\tilde{A_1}}(x) + \inf {\tilde{\mu }}_{\tilde{A_2}}(x) - \inf {\tilde{\mu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\mu }}_{\tilde{A_2}}(x),\right. \\&\quad \left. \sup {\tilde{\mu }}_{\tilde{A_1}}(x) + \sup {\tilde{\mu }}_{\tilde{A_2}}(x) - \sup {\tilde{\mu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\mu }}_{\tilde{A_2}}(x)\right] ,\\&\quad \left[ \inf {\tilde{\nu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\nu }}_{\tilde{A_2}}(x),\; \sup {\tilde{\nu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\nu }}_{\tilde{A_2}}(x)\right] \rangle \mid x \in X\}\,; \end{aligned}$$
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5.
$$\begin{aligned}&\tilde{A_1}\,\tilde{A_2} = \{\langle x, \;\left[ \inf {\tilde{\mu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\mu }}_{\tilde{A_2}}(x),\; \sup {\tilde{\mu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\mu }}_{\tilde{A_2}}(x)\right] , \\&\quad \left[ \inf {\tilde{\nu }}_{\tilde{A_1}}(x) + \inf {\tilde{\nu }}_{\tilde{A_2}}(x) - \inf {\tilde{\nu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\nu }}_{\tilde{A_2}}(x),\right. \\&\left. \quad \sup {\tilde{\nu }}_{\tilde{A_1}}(x) + \sup {\tilde{\nu }}_{\tilde{A_2}}(x) - \sup {\tilde{\nu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\nu }}_{\tilde{A_2}}(x)\right] \rangle \mid x \in X\}\,; \end{aligned}$$
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6.
$$\begin{aligned}&\lambda {\tilde{A}} = \{\langle x, \;\left[ 1- (1- \inf {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }, \;1- (1- \sup {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }\right] ,\\&\quad \left[ (\inf {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }, \;(\sup {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }\right] \rangle \mid x \in X\}\,; \end{aligned}$$
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7.
$$\begin{aligned}&{\tilde{A}}^{\lambda } = \{\langle x,\;\left[ (\inf {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }, \;(\sup {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }\right] , \\&\quad \left[ 1- (1- \inf {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }, \; 1- (1- \sup {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }\right] \rangle | x \in X\}. \end{aligned}$$
The basic component of an IVIFS is an ordered pair, characterized by an interval-valued membership degree and an interval-valued non-membership degree of x in \({\tilde{A}}\). This ordered pair is called an interval-valued intuitionistic fuzzy number (IVIFN).
An IVIFN is generally simplified as \(\left( [a, b],\; [c, d]\right) \), where \([a,b] \subseteq [0,1]\,,\; [c,d] \subseteq [0,1]\,, \;b+d \le 1\).
Obviously, when \(a=b\) and \(c=d\), then IVIFN reduces to an intuitionistic fuzzy number (IFN).
Definition 16
Let \({\tilde{\alpha }} = ([a, b],\; [c, d]), \;\tilde{\alpha _1} = ([a_1, b_1],\; [c_1, d_1])\) and \(\tilde{\alpha _2} = ([a_2, b_2], \;[c_2, d_2])\) be IVIFNs, and \(\lambda > 0\). Then
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1.
\({\tilde{\alpha }}^{\mathbf {c}} = ([c, d],\;[a, b])\)
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2.
$$\begin{aligned}&\tilde{\alpha _1} \wedge \tilde{\alpha _2} = (\left[ \min \{a_1,a_2\}, \min \{b_1,b_2\}\right] ,\;\\&\quad \left[ \max \{c_1,c_2\}, \max \{d_1,d_2\}\right] ); \end{aligned}$$
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3.
$$\begin{aligned}&\tilde{\alpha _1} \vee \tilde{\alpha _2} = (\left[ \max \{a_1,a_2\}, \max \{b_1,b_2\}\right] ,\\&\quad \left[ \min \{c_1,c_2\}, \min \{d_1,d_2\}\right] ); \end{aligned}$$
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4.
\(\tilde{\alpha _1} \oplus \tilde{\alpha _2} = (\left[ a_1+a_2-a_1a_2, b_1+c_2-b_1b_2\right] , \;\left[ c_1c_2, d_1d_2\right] )\,;\)
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5.
\(\tilde{\alpha _1} \otimes \tilde{\alpha _2} = (\left[ a_1a_2, b_1b_2\right] ,\; \left[ c_1+c_2-c_1c_2, d_1+d_2-d_1d_2\right] )\,;\)
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6.
\(\lambda {\tilde{\alpha }} = (\left[ 1-(1-a)^{\lambda }, 1-(1-b)^{\lambda }\right] ,\; \left[ c^{\lambda }, d^{\lambda }\right] )\,;\)
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7.
\({\tilde{\alpha }}^{\lambda } = (\left[ a^{\lambda }, b^{\lambda }\right] , \;\left[ 1-(1-c)^{\lambda }, 1-(1-d)^{\lambda }\right] ).\)
Appendix-2
Meng and Chen (2015) developed a deviation model for an MPR. In the same spirit, we formulate an equivalent deviation model for the model (P) proposed in Sect. 3.
where \(d_{ij,\; k}^{(q)+}=\left( \log a_{ij}^{(q)}-(\log a_{ik}^{(q)}+ \log a_{kj}^{(q)})\right) \vee 0,\; k \in \Omega (q),\) and \(d_{ij,\;k}^{(q)-}=\left( (\log a_{ik}^{(q)}+ \log a_{kj}^{(q)})-\log a_{ij}^{(q)}\right) \vee 0,\; k\in \Omega (q),\) and
It is observed that both model (P) and the above deviation model are equivalent and yield the same optimal outputs. Moreover, the model formulated by Meng and Chen (2015) is a particular case of the above model and hence of model (P).
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Sahu, M., Gupta, A. & Mehra, A. Acceptably consistent incomplete interval-valued intuitionistic multiplicative preference relations. Soft Comput 22, 7463–7477 (2018). https://doi.org/10.1007/s00500-018-3358-8
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DOI: https://doi.org/10.1007/s00500-018-3358-8