Analyzing industrial robot selection based on a fuzzy neural network under triangular fuzzy numbers
This section covers the Triangular fuzzy number (TFN), its score function, the triangular fuzzy numbers’ operational rules, the triangular fuzzy Yager weighted averaging (TFYWA) aggregation operator which is based on Yager norms, and FFNNs.
Definition 1
Let \(\:\overline{\overline{\text{M}}}\) be a universal set and \(\:\overline{\overline \Gamma}\) be a fuzzy set16 on \(\:\overline{\overline{\text{M}}}\:\), which is defined as follows:
$$\:\overline{\overline \Gamma}=\left\{\left(\overline{\overline{\text{m}}},\:{\psi\:}_{\overline{\overline \Gamma}}\left(\overline{\overline{\text{m}}}\right)\right)|\overline{\overline{\text{m}}}\in\:\overline{\overline{\text{M}}}\right\},$$
(1)
where \(\:{\psi\:}_{\overline{\overline \Gamma}}\left(\overline{\overline{\text{m}}}\right)\):\(\:\:\overline{\overline{\text{M}}}\to\:\left[\text{0,1}\right]\) represents the membership degree.
Definition 2
Triangular fuzzy numbers (TFNs)19\(\:\:\mathcal{H}=\left({\mathcal{H}}^{\mathcal{l}},{\mathcal{H}}^{\mathcal{m}},{\mathcal{H}}^{\mathcal{u}}\right)\) are fuzzy sets defined on real numbers, with the following membership function:
$$\:{\psi\:}_{\overline{\overline \Gamma}}\left(\overline{\overline{\text{m}}}\right)=\left\{\begin{array}{c}0,\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:for\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overline{\overline{\text{m}}}<{\mathcal{H}}^{\mathcal{l}}\\\:\frac{\overline{\overline{\text{m}}}-{\mathcal{H}}^{\mathcal{l}}}{{\mathcal{H}}^{\mathcal{m}}-{\mathcal{H}}^{\mathcal{l}}},\:\:\:\:for\:{\mathcal{\:}\mathcal{\:}\mathcal{\:}\mathcal{H}}^{\mathcal{l}}\le\:\overline{\overline{\text{m}}}\le\:{\mathcal{H}}^{\mathcal{m}}\\\:\frac{{\mathcal{H}}^{\mathcal{u}}-\overline{\overline{\text{m}}}}{{\mathcal{H}}^{\mathcal{u}}-{\mathcal{H}}^{\mathcal{m}}},\:\:\:\:\:for\:\:\:{\mathcal{H}}^{\mathcal{m}}\le\:\overline{\overline{\text{m}}}\le\:{\mathcal{H}}^{\mathcal{u}}\\\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:for\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overline{\overline{\text{m}}}>{\mathcal{H}}^{\mathcal{u}}\end{array}\right.$$
(2)
where \(\:{\mathcal{H}}^{\mathcal{l}},\:{\mathcal{H}}^{\mathcal{m}},\:\)and \(\:{\mathcal{H}}^{\mathcal{u}}\:\)represent the lower bound, mode, and upper bound of the TFN, respectively. Figure 3 show the graphical representation of the TFN.
\(\:{\mathcal{H}}^{\mathcal{l}}\ge\:0\:\)and \(\:{\mathcal{H}}^{\mathcal{u}}\le\:0\:,\:\) are called positive and negative TFNs, respectively. A graphical representation of the TFN is displayed in Figure. The three points listed below are used to classify the membership function:
-
a.
\(\:{\mathcal{H}}^{\mathcal{l}}\) to\(\:{\mathcal{H}}^{\mathcal{m}}\).
-
b.
The function decreases from \(\:{\mathcal{H}}^{\mathcal{m}}\) to \(\:{\mathcal{H}}^{\mathcal{u}}\).
-
c.
\(\:{\mathcal{H}}^{\mathcal{l}}\le\:\:{\mathcal{H}}^{\mathcal{m}}\le\:{\mathcal{H}}^{\mathcal{u}}\)
Graphical representation of the TFN.
Definition 3
Let \(\:{\mathcal{H}}_{1}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}},{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)\) and \(\:{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{2}}^{\mathcal{l}},{{\mathcal{H}}_{2}}^{\mathcal{m}},{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)\) be the two TFNs, and \(\:\lambda\:\ge\:0,\:\)then we have:
1)
$$\:{\mathcal{H}}_{1}+{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}}+{{\mathcal{H}}_{2}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}}+{{\mathcal{H}}_{2}}^{\mathcal{m}},\:{{\mathcal{H}}_{1}}^{\mathcal{u}}+{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)$$
2)
$$\:{\mathcal{H}}_{1}\times\:{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}}\times\:{{\mathcal{H}}_{1}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}}\times\:{{\mathcal{H}}_{2}}^{\mathcal{m}},\:{{\mathcal{H}}_{1}}^{\mathcal{u}}\times\:{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)$$
3)
$$\:{\lambda\:\mathcal{H}}_{1}=\left(\lambda\:{{\mathcal{H}}_{1}}^{\mathcal{l}},\:\lambda\:{{\mathcal{H}}_{1}}^{\mathcal{m}},\:{{\lambda\:\mathcal{H}}_{1}}^{\mathcal{u}}\right)$$
4)
$$\:{{\mathcal{H}}_{1}}^{\lambda\:}=\left({{{\mathcal{H}}_{1}}^{\mathcal{l}}}^{\lambda\:},\:{{{\mathcal{H}}_{1}}^{\mathcal{m}}}^{\lambda\:},\:{{{\mathcal{H}}_{1}}^{\mathcal{u}}}^{\lambda\:}\right)$$
Definition 4
Let \(\:\mathcal{H}=\left({\mathcal{H}}^{\mathcal{l}},{\mathcal{H}}^{\mathcal{m}},{\mathcal{H}}^{\mathcal{u}}\right)\) be the TFN; then, the score function of the TFN is defined as:
$$\:\aleph\:\left(\mathcal{H}\right)=\frac{{\mathcal{H}}^{\mathcal{l}}+{\mathcal{H}}^{\mathcal{m}}+\:{\mathcal{H}}^{\mathcal{u}}}{3},\:\aleph\:\left(\mathcal{H}\right)\in\:\left[\text{0,1}\right].$$
(3)
Let \(\:{\mathcal{H}}_{1}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}},{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)\) and \(\:{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{2}}^{\mathcal{l}},{{\mathcal{H}}_{2}}^{\mathcal{m}},{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)\) be the two TFNs and \(\:\aleph\:\left(\mathcal{H}\right)\)be the score function of the TFN. Then, we have:
-
a.
If \(\:\aleph\:\left({\mathcal{H}}_{1}\right)<\aleph\:\left({\mathcal{H}}_{2}\right)\), then\(\:\:{\mathcal{H}}_{1}<{\mathcal{H}}_{2}\).
-
b.
If\(\:\:\aleph\:\left({\mathcal{H}}_{1}\right)>\aleph\:\left({\mathcal{H}}_{2}\right)\), then\(\:{\mathcal{H}}_{1}>{\mathcal{H}}_{2}\).
-
c.
If \(\:\aleph\:\left({\mathcal{H}}_{1}\right)=\aleph\:\left({\mathcal{H}}_{2}\right),\:\)then\(\:{\mathcal{H}}_{1}\:\sim\:{\mathcal{H}}_{2}\).
Definition 5
Let \(\:{\mathcal{H}}_{1}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}},{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)\) and \(\:{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{2}}^{\mathcal{l}},{{\mathcal{H}}_{2}}^{\mathcal{m}},{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)\) be the two TFNs. Then, the distance measure\(\:\varDelta\:\left({\mathcal{H}}_{1},{\mathcal{H}}_{2}\right)\) can be defined as:
$$\:\varDelta\:\left({\mathcal{H}}_{1},{\mathcal{H}}_{2}\right)=\sqrt{\frac{1}{3}\left[{\left({{\mathcal{H}}_{2}}^{\mathcal{l}}-{{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{2}+{\left({{\mathcal{H}}_{2}}^{\mathcal{m}}-{{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{2}+{\left({{\mathcal{H}}_{2}}^{\mathcal{u}}-{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{2}\right]}$$
(4)
Definition 6
Let \(\:\mathcal{T}\) be the Yager t-norm and \(\:\mathcal{S}\) be the Yager t-conorm. Then, the Yager t-norm and Yager t-conorm24 are defined as:
$$\:\mathcal{T}\left(\mathcal{x},\mathcal{y}\right)=1-\text{m}\text{i}\text{n}\left(1,{\left({\left(1-\mathcal{x}\right)}^{{\upgamma\:}}+{\left(1-\mathcal{y}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)$$
(5)
$$\:\mathcal{S}\left(\mathcal{x},\mathcal{y}\right)=\text{m}\text{i}\text{n}\left(1,{\left({\mathcal{x}}^{{\upgamma\:}}-{\mathcal{y}}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),$$
(6)
where\(\:\:{\upgamma\:}>0\).
Definition 7
Let \(\:{\mathcal{H}}_{1}=\left({{\mathcal{H}}_{1}}^{\mathcal{l}},{{\mathcal{H}}_{1}}^{\mathcal{m}},{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)\) and \(\:{\mathcal{H}}_{2}=\left({{\mathcal{H}}_{2}}^{\mathcal{l}},{{\mathcal{H}}_{2}}^{\mathcal{m}},{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)\) be the two TFNs. Then, the Yager operations laws based on the Yager t-norm and Yager t-conorm are defined as:
i.
$$\:{\mathcal{H}}_{1}{\oplus}_{\text{y}}{\mathcal{H}}_{2}=\left(\begin{array}{c}min\left(1,{\left({\left({{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}+{\left({{\mathcal{H}}_{2}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),min\left(1,{\left({\left({{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}+{\left({{\mathcal{H}}_{2}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\\\:min\left(1,{\left({\left({{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}+{\left({{\mathcal{H}}_{2}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
ii.
$$\:{\mathcal{H}}_{1}{\otimes\:}_{\text{y}}{\mathcal{H}}_{2}=\left(\begin{array}{c}1-min\left(1,{\left({\left(1-{{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}+{\left(1-{{\mathcal{H}}_{2}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\\\:1-min\left(1,{\left({\left(1-{{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}+{\left(1-{{\mathcal{H}}_{2}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\\\:1-min\left(1,{\left({\left(1-{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}+{\left(1-{{\mathcal{H}}_{2}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
iii.
$$\:{\upbeta\:}{\odot\:}_{\text{y}}{\mathcal{H}}_{1}=\left(\text{m}\text{i}\text{n}\left(1,{\left({\upbeta\:}{\left({{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\text{m}\text{i}\text{n}\left(1,{\left({\upbeta\:}{\left({{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\text{m}\text{i}\text{n}\left(1,{\left({\upbeta\:}{\left({{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\right)$$
iv.
$$\:{{\mathcal{H}}_{1}}^{{\upbeta\:}}=\left(\begin{array}{c}1-min\left(1,{\left({\upbeta\:}{\left(1-{{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),1-min\left(1,{\left({\upbeta\:}{\left(1-{{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right),\\\:1-min\left(1,{\left({\upbeta\:}{\left(1-{{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right),$$
where \(\:{\upbeta\:},{\upgamma\:}\ge\:1\)
Theorem 1
Let \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs and \(\:w={\left({w}_{1},{w}_{2},{w}_{3},\dots\:,{w}_{k}\right)\:}^{t}\) be the related weights of \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)\) such that hat \(\:{w}_{i}\) ∈ [0, 1] and \(\:\sum\:_{i=1}^{k}{w}_{i}\)=1. Then, the triangular fuzzy Yager weighted averaging aggregation operator maps TFYWAA:\(\:{\phi\:}^{k}\to\:\phi\:\)
$$\:\text{T}\text{F}\text{Y}\text{W}\text{A}\text{A}\left({\mathcal{H}}_{1},{\mathcal{H}}_{2},{\mathcal{H}}_{3},\dots\:,{\mathcal{H}}_{k}\right)={\oplus\:}_{i=1}^{k}{{w}_{i}{\odot\:}_{y}\mathcal{H}}_{i}=\:\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
(7)
Proof
For this purpose, we use the mathematical induction method.
For \(\:k=2\), we have
$$\:{{w}_{1}\mathcal{H}}_{1}{\oplus}_{y}{{w}_{2}\mathcal{H}}_{2}=\left(\begin{array}{c}min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right){\oplus}_{y}\left(\begin{array}{c}min\left(1,{\left({w}_{2}{\left({{\mathcal{H}}_{2}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{2}{\left({{\mathcal{H}}_{2}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{2}{\left({{\mathcal{H}}_{2}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
$$\:=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{2}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{2}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{2}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
Hence, for \(\:k=2\)
For \(\:k=n\), we have
$$\:{\oplus\:}_{i=1}^{n}{{w}_{i}\mathcal{H}}_{i}=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
For \(\:k=n+1\), we have
$$\:{\oplus\:}_{i=1}^{n}{{w}_{i}\mathcal{H}}_{i}{\oplus}_{y}{{w}_{1}\mathcal{H}}_{1}=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right){\oplus}_{y}\left(\begin{array}{c}min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left({w}_{1}{\left({{\mathcal{H}}_{1}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
$$\:{=\oplus\:}_{i=1}^{n+1}{{w}_{i}\mathcal{H}}_{i}=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{n+1}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n+1}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{n+1}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
Hence, \(\:k=n+1\)
Theorem 2
(Idempotency): Let \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs and \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)=\left({\mathcal{H}}^{\mathcal{l}},{\mathcal{H}}^{\mathcal{m}},{\mathcal{H}}^{\mathcal{u}}\right)=\mathcal{H}\).
Proof
Since \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs.
$$\:\text{T}\text{F}\text{Y}\text{W}\text{A}\text{A}\left({\mathcal{H}}_{1},{\mathcal{H}}_{2},{\mathcal{H}}_{3},\dots\:,{\mathcal{H}}_{k}\right)={\oplus\:}_{i=1}^{k}{{w}_{i}{\odot\:}_{y}\mathcal{H}}_{i}=\:\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
= \(\:\left({\mathcal{H}}^{\mathcal{l}},{\mathcal{H}}^{\mathcal{m}},{\mathcal{H}}^{\mathcal{u}}\right)\).
Theorem 3
(Boundedness); Let \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs. Consider \(\:{\mathcal{H}}^{-}=\text{m}\text{i}\text{n}\)(\(\:{\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\) and \(\:{\mathcal{H}}^{+}=\text{m}\text{a}\text{x}\)(\(\:{\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\). Then
$$\:{\mathcal{H}}^{-}\le\:\text{T}\text{F}\text{Y}\text{W}\text{A}\text{A}({\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\le\:\:{\mathcal{H}}^{+}$$
Proof
Since \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs. Consider \(\:{\mathcal{H}}^{-}=\text{m}\text{i}\text{n}\)(\(\:{\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\) and \(\:{\mathcal{H}}^{+}=\text{m}\text{a}\text{x}\:\)(\(\:{\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\).
$$\:=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{-}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{-}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{-}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)\le\:\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)\le\:\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{+}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{+}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{{\mathcal{H}}^{+}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
Therefore
$$\:{\mathcal{H}}^{-}\le\:\text{T}\text{F}\text{Y}\text{W}\text{A}\text{A}({\mathcal{H}}_{1},\:{\mathcal{H}}_{2},\dots\:,{\mathcal{H}}_{n})\le\:\:{\mathcal{H}}^{+}$$
Theorem 4
(Monotonicity); Let \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs and \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)\le\:\dddot{\mathcal{H}}_{\imath}=\left(\dddot{\mathcal{H}}_{\imath}^{\mathcal{l}},\dddot{\mathcal{H}}_{\imath}^{\mathcal{m}},\dddot{\mathcal{H}}_{\imath}^{\mathcal{u}}\right)\).
Proof
Since \(\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)|i=\text{1,2},3,\dots\:,k\) be the collection of the TFNs. \(\:{\mathcal{H}}_{i}=\le\:\overline{\overline{\mathcal{H}}_{\imath}}\), this implies that
$$\:=\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left({{\mathcal{H}}_{i}}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)\le\:\left(\begin{array}{c}min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left(\dddot{\mathcal{H}}_{\imath}^{\mathcal{l}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left(\dddot{\mathcal{H}}_{\imath}^{\mathcal{m}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\\\:min\left(1,{\left(\sum\:_{i=1}^{k}{w}_{i}{\left(\dddot{\mathcal{H}}_{\imath}^{\mathcal{u}}\right)}^{{\upgamma\:}}\right)}^{\frac{1}{{\upgamma\:}}}\right)\end{array}\right)$$
Therefore
$$\:{\mathcal{H}}_{i}=\left({{\mathcal{H}}_{i}}^{\mathcal{l}},{{\mathcal{H}}_{i}}^{\mathcal{m}},{{\mathcal{H}}_{i}}^{\mathcal{u}}\right)\le\:\overline{\overline{\mathcal{H}}_{\imath}}=\left(\dddot{\mathcal{H}}_{\imath}^{\mathcal{l}},\dddot{\mathcal{H}}_{\imath}^{\mathcal{m}},\dddot{\mathcal{H}}_{\imath}^{\mathcal{u}}\right)$$
Definition 8
When there is no cycle in the connections between nodes, the artificial neural network is called a feed forward neural network9,10,11. First and foremost, artificial neural networks1,2,3 are of feed forward variety. Data in this network flow only in one direction from the input nodes to the output nodes via the hidden node, and a cycle is not formed by it. Figure 4 shows the feed forward neural network.
Feed forward neural network.
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