Acceptably consistent incomplete interval-valued intuitionistic multiplicative preference relations

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.

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 $\lambda > 0$.

• 1.
  $$\begin{aligned} {\tilde{A}}^{\mathbf {c}} = \{\langle x, \;{\tilde{\nu }}_{{\tilde{A}}}(x), \;{\tilde{\mu }}_{{\tilde{A}}}(x)\rangle \mid x \in X \}; \end{aligned}$$
• 2.
  $$\begin{aligned}&\tilde{A_1} \cap \tilde{A_2} = \{\langle x, \;\left[ \min \{\inf {\tilde{\mu }}_{\tilde{A_1}}(x), \inf {\tilde{\mu }}_{\tilde{A_2}}(x)\}, \right. \\&\left. \quad \min \{\sup {\tilde{\mu }}_{\tilde{A_1}}(x), \sup {\tilde{\mu }}_{\tilde{A_2}}(x)\} \right] , \\&\quad \left[ \max \{\inf {\tilde{\nu }}_{\tilde{A_1}}(x), \inf {\tilde{\nu }}_{\tilde{A_2}}(x)\},\right. \\&\quad \max \{\sup {\tilde{\nu }}_{\tilde{A_1}}(x), \sup {\tilde{\nu }}_{\tilde{A_2}}(x)\} ]\rangle \mid x \in X \}\,; \end{aligned}$$
• 3.
  $$\begin{aligned}&\tilde{A_1} \cup \tilde{A_2} = \{\langle x, \;\left[ \max \{\inf {\tilde{\mu }}_{\tilde{A_1}}(x), \inf {\tilde{\mu }}_{\tilde{A_2}}(x)\},\right. \\&\left. \quad \max \{\sup {\tilde{\mu }}_{\tilde{A_1}}(x), \sup {\tilde{\mu }}_{\tilde{A_2}}(x)\} \right] , \\&\quad \left[ \min \{\inf {\tilde{\nu }}_{\tilde{A_1}}(x), \inf {\tilde{\nu }}_{\tilde{A_2}}(x)\},\right. \\&\left. \quad \min \{\sup {\tilde{\nu }}_{\tilde{A_1}}(x), \sup {\tilde{\nu }}_{\tilde{A_2}}(x)\} \right] \rangle \mid x \in X \}\,; \end{aligned}$$
• 4.
  $$\begin{aligned}&\tilde{A_1} + \tilde{A_2} = \{\langle x, \;\left[ \inf {\tilde{\mu }}_{\tilde{A_1}}(x) + \inf {\tilde{\mu }}_{\tilde{A_2}}(x) - \inf {\tilde{\mu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\mu }}_{\tilde{A_2}}(x),\right. \\&\quad \left. \sup {\tilde{\mu }}_{\tilde{A_1}}(x) + \sup {\tilde{\mu }}_{\tilde{A_2}}(x) - \sup {\tilde{\mu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\mu }}_{\tilde{A_2}}(x)\right] ,\\&\quad \left[ \inf {\tilde{\nu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\nu }}_{\tilde{A_2}}(x),\; \sup {\tilde{\nu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\nu }}_{\tilde{A_2}}(x)\right] \rangle \mid x \in X\}\,; \end{aligned}$$
• 5.
  $$\begin{aligned}&\tilde{A_1}\,\tilde{A_2} = \{\langle x, \;\left[ \inf {\tilde{\mu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\mu }}_{\tilde{A_2}}(x),\; \sup {\tilde{\mu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\mu }}_{\tilde{A_2}}(x)\right] , \\&\quad \left[ \inf {\tilde{\nu }}_{\tilde{A_1}}(x) + \inf {\tilde{\nu }}_{\tilde{A_2}}(x) - \inf {\tilde{\nu }}_{\tilde{A_1}}(x)\,\inf {\tilde{\nu }}_{\tilde{A_2}}(x),\right. \\&\left. \quad \sup {\tilde{\nu }}_{\tilde{A_1}}(x) + \sup {\tilde{\nu }}_{\tilde{A_2}}(x) - \sup {\tilde{\nu }}_{\tilde{A_1}}(x)\,\sup {\tilde{\nu }}_{\tilde{A_2}}(x)\right] \rangle \mid x \in X\}\,; \end{aligned}$$
• 6.
  $$\begin{aligned}&\lambda {\tilde{A}} = \{\langle x, \;\left[ 1- (1- \inf {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }, \;1- (1- \sup {\tilde{\mu }}_{{\tilde{A}}}(x))^{\lambda }\right] ,\\&\quad \left[ (\inf {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }, \;(\sup {\tilde{\nu }}_{{\tilde{A}}}(x))^{\lambda }\right] \rangle \mid x \in X\}\,; \end{aligned}$$
• 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

• 1. ${\tilde{\alpha }}^{\mathbf {c}} = ([c, d],\;[a, b])$

• 2.
  $$\begin{aligned}&\tilde{\alpha _1} \wedge \tilde{\alpha _2} = (\left[ \min \{a_1,a_2\}, \min \{b_1,b_2\}\right] ,\;\\&\quad \left[ \max \{c_1,c_2\}, \max \{d_1,d_2\}\right] ); \end{aligned}$$
• 3.
  $$\begin{aligned}&\tilde{\alpha _1} \vee \tilde{\alpha _2} = (\left[ \max \{a_1,a_2\}, \max \{b_1,b_2\}\right] ,\\&\quad \left[ \min \{c_1,c_2\}, \min \{d_1,d_2\}\right] ); \end{aligned}$$

• 4. $\tilde{\alpha _1} \oplus \tilde{\alpha _2} = (\left[ a_1+a_2-a_1a_2, b_1+c_2-b_1b_2\right] , \;\left[ c_1c_2, d_1d_2\right] )\,;$

• 5. $\tilde{\alpha _1} \otimes \tilde{\alpha _2} = (\left[ a_1a_2, b_1b_2\right] ,\; \left[ c_1+c_2-c_1c_2, d_1+d_2-d_1d_2\right] )\,;$

• 6. $\lambda {\tilde{\alpha }} = (\left[ 1-(1-a)^{\lambda }, 1-(1-b)^{\lambda }\right] ,\; \left[ c^{\lambda }, d^{\lambda }\right] )\,;$

• 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.

$$\begin{aligned}&\qquad \min \;\;D=\sum _{q=1}^{4}\; \sum _{{i,j=1},{i<j},\;k\in \Omega (q)}^{n}(d_{ij,\; k}^{(q)+}+d_{ij,\;k}^{(q)-})\\&\quad \mathrm{subject\,to} \\&\quad \quad \begin{array}{lr} \delta _{ij}\left( \log a_{ij}^{(q)}-(\log a_{ik}^{(q)}+ \log a_{kj}^{(q)})\right) -d_{ij,\; k}^{(q)+}+d_{ij,\; k}^{(q)-}=0,\\ i<j,\; \forall \;i,\,j\in N, \;k\in \Omega (q),\; q=1,2,3,4\\ 1/9 \le a_{ij}^{(q)}, q = 1,2,\; \quad a_{ij}^{(q)}\in \Gamma (q)\\ a_{ij}^{(q)} \le 9,\; q = 3,4, \;\quad a_{ij}^{(q)}\in \Gamma (q)\\ a_{ij}^{(q)}a_{ij}^{(q+2)}\le 1,\quad q = 1,2, \;i,\,j,\,k \in N,\; \;i \ne j \ne k\\ a_{ij}^{(q+1)} \le a_{ij}^{(q)},\quad q = 1,3,\; i< j,\; \;i,j \in N\\ a_{ij}^{(q+1)} \ge a_{ij}^{(q)},\quad q = 1,3, \;i > j,\; \;i,j \in N\\ d_{ij,\;k}^{(q)+},\;\; d_{ij,\;k}^{(q)-} \ge 0, \quad i,\,j=1,\ldots , n, \;i<j\,,\\ \end{array} \end{aligned}$$

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

$$\begin{aligned} \delta _{ij}= \left\{ \begin{array}{lr} 1 &{}\quad k\in \Omega (q)\\ 0 &{}\quad \text {otherwise}\\ \end{array} \right. \end{aligned}$$

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).