1 Introduction

The original proof of Proposition 3.3 by Wu (2003) can only be correct when the extra assumption \(\mu _{\widetilde{y}_i}(0)= 1\) is added. A counterexample for the case of \(\mu _{\widetilde{y}_i}(0)\ne 1\) is given below.

Example

Let \(\widetilde{y}=(-4,-2,2,4)\) be trapezoidal fuzzy number and its membership function be

$$\begin{aligned} \mu _{\widetilde{y}}(r)=\left\{ \begin{array}{ll}(r+6)/2, &{} \quad \hbox {if}\;\; r\leqslant -4,\\ 1,&{}\quad \hbox {if}\;\; r\in [-4,-2],\\ (2-r)/4, &{}\quad \hbox {if}\;\; r\geqslant -2,\\ 0,&{}\quad \hbox {otherwise.}\end{array}\right. \end{aligned}$$

We see that trapezoidal fuzzy number \(\widetilde{y}\) is not a nonnegative or nonpositive fuzzy number. As shown in Fig. 1, \((\widetilde{y}^-)^U_\alpha =2-4\alpha <0=(\widetilde{y}^+)^L_\alpha \) and \((\widetilde{y}^+)^U_\alpha =0\ne 2-4\alpha =\widetilde{y}^U_\alpha \) if \(\alpha \in (0.5,1].\) It implies that, for all \(\alpha \in [0,1], (\widetilde{y}^+)^L_\alpha =0=(\widetilde{y}^-)^U_\alpha \) and \((\widetilde{y}^+)^U_\alpha =\widetilde{y}^U_\alpha , (\widetilde{y}^-)^L_\alpha =\widetilde{y}^L_\alpha \) is incorrect, if \(\widetilde{y}\) be not a nonnegative or nonpositive fuzzy number.

Fig. 1
figure 1

\(\widetilde{y}, \widetilde{y}^+\) and \(\widetilde{y}^-\) of Example

Full size image

2 The correct proof of Proposition 3.3

Now, we propose the correct proof of Proposition 3.3.

Proposition 3.1

Let \(\widetilde{\mathbf {x}}\) be in \(\digamma ^n{(\mathbf {R})}\). If  \(\widetilde{\mathbf {x}}\) is nonnegative or nonpositive, then

$$\begin{aligned} \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}\rangle \rangle = \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^+\rangle \rangle \oplus \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^-\rangle \rangle . \end{aligned}$$
(1)

Proof

Let \(\widetilde{\mathbf {x}}=(\widetilde{x}_1,\ldots ,\widetilde{x}_n),\widetilde{\mathbf {y}}=(\widetilde{y}_1,\ldots ,\widetilde{y}_n)\in \digamma ^n{(\mathbf {R})}\) and \(\widetilde{\mathbf {y}}\) be not a nonnegative or nonpositive fuzzy number vector. By Propositions 3.1 and 3.2 in  Wu (2003), we just need to show that, for any \(\alpha \in [0,1],\)

$$\begin{aligned} \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}\rangle \rangle _\alpha = (\langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^+\rangle \rangle \oplus \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^-\rangle \rangle )_\alpha =\langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^+ \rangle \rangle _\alpha + \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^-\rangle \rangle _\alpha , \end{aligned}$$

i.e., for any \(\alpha \in [0,1],\)

$$\begin{aligned} \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}\rangle \rangle ^L_\alpha =\langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^+ \rangle \rangle ^L_\alpha + \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^-\rangle \rangle ^L_\alpha , \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}\rangle \rangle ^U_\alpha =\langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^+ \rangle \rangle ^U_\alpha + \langle \langle \widetilde{\mathbf {x}},\widetilde{\mathbf {y}}^-\rangle \rangle ^U_\alpha , \end{aligned}$$

which also says that

$$\begin{aligned}&(\widetilde{x}_1\otimes \widetilde{y}_1)_\alpha ^L+\cdots +(\widetilde{x}_n\otimes \widetilde{y}_n)_\alpha ^L\\&\quad =\left[ \left( \widetilde{x}_1\otimes \widetilde{y}^+_1\right) _\alpha ^L+\cdots +\left( \widetilde{x}_n\otimes \widetilde{y}^+_n\right) _\alpha ^L\right] + \left[ \left( \widetilde{x}_1\otimes \widetilde{y}^-_1\right) _\alpha ^L+\cdots +\left( \widetilde{x}_n\otimes \widetilde{y}^-_n\right) _\alpha ^L\right] \\&\quad =\left[ \left( \widetilde{x}_1\otimes \widetilde{y}^+_1\right) _\alpha ^L+\left( \widetilde{x}_1\otimes \widetilde{y}^-_1\right) _\alpha ^L\right] +\cdots + \left[ \left( \widetilde{x}_n\otimes \widetilde{y}^+_n\right) _\alpha ^L+\left( \widetilde{x}_n\otimes \widetilde{y}^-_n\right) _\alpha ^L\right] , \end{aligned}$$

and

$$\begin{aligned}&(\widetilde{x}_1\otimes \widetilde{y}_1)_\alpha ^U+\cdots +(\widetilde{x}_n\otimes \widetilde{y}_n)_\alpha ^U\\&\quad =\left[ \left( \widetilde{x}_1\otimes \widetilde{y}^+_1\right) _\alpha ^U+\cdots +\left( \widetilde{x}_n\otimes \widetilde{y}^+_n\right) _\alpha ^U\right] + \left[ \left( \widetilde{x}_1\otimes \widetilde{y}^-_1\right) _\alpha ^U+\cdots +\left( \widetilde{x}_n\otimes \widetilde{y}^-_n\right) _\alpha ^U\right] \\&\quad =\left[ \left( \widetilde{x}_1\otimes \widetilde{y}^+_1\right) _\alpha ^U+\left( \widetilde{x}_1\otimes \widetilde{y}^-_1\right) _\alpha ^U\right] +\cdots + \left[ \left( \widetilde{x}_n\otimes \widetilde{y}^+_n\right) _\alpha ^U+\left( \widetilde{x}_n\otimes \widetilde{y}^-_n\right) _\alpha ^U\right] . \end{aligned}$$

In order to prove (1), it suffices to show that, for all \(\alpha \in [0,1]\) and \(i=1,2,\ldots ,n,\)

$$\begin{aligned} \left( \widetilde{x}_i\otimes \widetilde{y}_i\right) _\alpha ^L= & {} \left( \widetilde{x}_i\otimes \widetilde{y}^+_i\right) _\alpha ^L+\left( \widetilde{x}_i\otimes \widetilde{y}^-_i\right) _\alpha ^L,\nonumber \\ \left( \widetilde{x}_i\otimes \widetilde{y}_i\right) _\alpha ^U= & {} \left( \widetilde{x}_i\otimes \widetilde{y}^+_i\right) _\alpha ^U+\left( \widetilde{x}_i\otimes \widetilde{y}^-_i\right) _\alpha ^U. \end{aligned}$$
(2)

We shall prove (2) by considering the three cases below. Without loss of generality, let \(\widetilde{\mathbf {x}}\) be nonnegative. The proof is similar if \(\widetilde{\mathbf {x}}\) is nonpositive.

Case 1

For the case of \(\mu _{\widetilde{y}_i}(0)=1\), it implies that \((\widetilde{y}^+_i)^L_\alpha =0=(\widetilde{y}^-)^L_{i\alpha }\) and \((\widetilde{y}^+_i)^U_\alpha =\widetilde{y}^U_{i\alpha }, (\widetilde{y}^-_i)^L_\alpha =\widetilde{y}^L_{i\alpha }\) for all \(\alpha \in [0,1]\). The result follows immediately from Propositions 2.1 and 3.2 in Wu (2003).

Case 2

For the case of \(\mu _{\widetilde{y}_i}(0)\ne 1\) and \(a^U=\max \{r|\mu _{\widetilde{y}_i}(r)=1\}<0\), let \(a^L=\min \{r|\mu _{\widetilde{y}_i}(r)=1\}\). Then, there exists some \(\alpha _0\in [0,1]\) such that \(\widetilde{y}^U_\alpha <0\) if \(\alpha >\alpha _0\), otherwise, \(\widetilde{y}^U_\alpha \geqslant 0\). It is obvious that \(\widetilde{y}_i^L=(\widetilde{y}^-_i)^L\leqslant 0, (\widetilde{y}^+_i)^L=0.\) If \(\alpha \geqslant \alpha _0\), then \(\widetilde{y}_i^U=(\widetilde{y}^-_i)^U\) and \((\widetilde{y}^+_i)^U=0\), otherwise, \(\widetilde{y}_i^U=(\widetilde{y}^+_i)^U\) and \((\widetilde{y}^-_i)^U=0\). For all \(\alpha \in [0,1],\) we have \(\widetilde{x}_{i\alpha }^U\geqslant \widetilde{x}_{i\alpha }^L\geqslant 0\) since \(\widetilde{x}_i\) is a nonnegative fuzzy number. So, we can obtain

$$\begin{aligned}&\displaystyle \left( \widetilde{x}_i\otimes \widetilde{y}_i\right) _\alpha ^L=\widetilde{x}_{i\alpha }^U\widetilde{y}_{i\alpha }^L,\\&\displaystyle \left( \widetilde{x}_i\otimes \widetilde{y}^+_i\right) _\alpha ^L+(\widetilde{x}_i\otimes \widetilde{y}^-_i)_\alpha ^L=\widetilde{x}_{i\alpha }^L\left( \widetilde{y}^+_{i}\right) _\alpha ^L+ \widetilde{x}_{i\alpha }^U\widetilde{y}_{i\alpha }^L=\widetilde{x}_{i\alpha }^U\widetilde{y}_{i\alpha }^L,\\&\displaystyle \left( \widetilde{x}_i\otimes \widetilde{y}_i\right) _\alpha ^U=\left\{ \begin{array}{ll}\widetilde{x}_{i\alpha }^L\widetilde{y}_{i\alpha }^U,&{}\quad \text{ if } \alpha \geqslant \alpha _0,\\ \widetilde{x}_{i\alpha }^U\widetilde{y}_{i\alpha }^U,&{}\quad \text{ if } \alpha <\alpha _0,\end{array}\right. \\&\displaystyle \left( \widetilde{x}_i\otimes \widetilde{y}^+_i\right) _\alpha ^U=\left\{ \begin{array}{ll}0,&{}\quad \text{ if } \;\alpha \geqslant \alpha _0,\\ \widetilde{x}_{i\alpha }^U\widetilde{y}_{i\alpha }^U,&{}\quad \text{ if }\;\alpha <\alpha _0,\end{array}\right. \\&\displaystyle \left( \widetilde{x}_i\otimes \widetilde{y}^-_i\right) _\alpha ^U=\left\{ \begin{array}{ll}\widetilde{x}_{i\alpha }^L\widetilde{y}_{i\alpha }^U,&{}\quad \text{ if }\;\alpha \geqslant \alpha _0,\\ 0,&{}\quad \text{ if }\;\alpha <\alpha _0.\end{array}\right. \end{aligned}$$

This implies that (2) holds for all \(\alpha \in [0,1]\) and \(i=1,2,\ldots ,n.\)

Case 3

For the case of \(\mu _{\widetilde{y}_i}(0)\ne 1\) and \(a^L=\min \{r|\mu _{\widetilde{y}_i}(r)=1\}> 0\), the similar proof of case 2 is still valid.