Hybrid big data analytics and Industry 4.0 approach to projecting cycle time ranges

Abstract

This study proposes a hybrid big data analytics and Industry 4.0 (BD-I4) approach to enhancing the effectiveness of cycle time range projections for factory jobs. As a joint application of big data analytics and Industry 4.0, the BD-I4 approach is distinct from existing methods in this field. In this approach, each expert first constructs a fuzzy deep neural network to project the cycle time range of a job, an application of big data analytics (i.e., deep learning). Subsequently, the fuzzy weighted intersection operator is applied to aggregate the projected cycle times such that unequal authority levels can be considered, an application of Industry 4.0 (i.e., artificial intelligence). Applying the BD-I4 approach to a real case that the proposed methodology improved the projection precision by up to 72%, suggesting that instead of relying on a single expert, collaboration among multiple experts may be more effective and efficient.

Appendix

Proof theorem 1

Substituting Eq. (9) into Eq. (8) gives.

$$\tilde{o}_{j} (m) = \frac{1}{{1 + e^{{(\tilde{\theta }^{o} (m)( - )\tilde{I}_{j}^{o} (m))}} }}$$

(23)

Therefore,

$$o_{j3} (m) = \frac{1}{{1 + e^{{(\theta_{1}^{o} (m) - I_{j3}^{o} (m))}} }}$$

(24)

The other parameters are not fuzzified, so $I_{j3}^{o} (m)$ is equal to $I_{j2}^{o*} (m)$, which has a fixed value:

$$o_{j3} (m) = \frac{1}{{1 + e^{{(\theta_{1}^{o} (m) - I_{j2}^{o*} (m))}} }}$$

(25)

To minimize AR(m), $o_{j3} (m)$ should be as low as possible, which corresponds to maximization of $\theta_{1}^{o} (m)$. However, $\tilde{o}_{j} (m)$ should include $a_{j}$:

$$o_{j3} (m) \ge a_{j};\,j = {1} \sim n$$

(26)

In addition,

$$o_{j3} (m) \ge o_{j2}^{*} (m);j = {1} \sim n$$

(27)

Therefore,

$$o_{j3} (m) \ge \max (o_{j2}^{*} (m),\;a_{j} );\,j = {1} \sim n$$

(28)

Substituting Eq. (24) into Constraint (28) gives

$$\theta_{1}^{o} (m) \le I_{j2}^{o*} (m) + \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1);j = {1} \sim n$$

(29)

To maximize $\theta_{1}^{o} (m)$,

$$\theta_{1}^{o*} (m) = \mathop {\min }\limits_{j} \left( {I_{j2}^{o*} (m) + \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1)} \right);\,j = {1} \sim n$$

(30)

Theorem 1 is proved.

Proof theorem 2

The required proof is similar to that of Theorem 1.

Proof theorem 3

According to Eq. (7),

$$I_{j3}^{o} (m) = \max \left( {\sum\limits_{q = 1}^{{Q_{m} }} {(w_{q3}^{o} (m)h_{jq1}^{(2)} (m))} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {(w_{q3}^{o} (m)h_{jq3}^{(2)} (m))} } \right)$$

(31)

because $\tilde{h}_{jq}^{(2)} (m)$ is positive, whereas $\tilde{w}_{q}^{o} (m)$ may be negative. Substituting Eq. (31) into Eq. (24) gives

$$o_{j3} (m) = \frac{1}{{1 + e^{{\left( {\theta_{1}^{o} (m) - \max (\sum\limits_{q = 1}^{{Q_{m} }} {(w_{q3}^{o} (m)h_{jq1}^{(2)} (m)} ),\;\sum\limits_{q = 1}^{{Q_{m} }} {(w_{q3}^{o} (m)h_{jq3}^{(2)} (m))} )} \right)}} }}$$

(32)

Only $w_{{l_{f} }}^{o}$ is fuzzified; the other parameters are set to their optimized cores:

$$\begin{aligned} o_{j3} (m) & = \frac{1}{{1 + e^{{\left( {\theta_{2}^{o*} (m) - \max (w_{{q_{f} 3}}^{o} (m)h_{{jq_{f} 2}}^{(2)*} (m) + \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m} )),\;w_{{q_{f} 3}}^{o} (m)h_{{jq_{f} 2}}^{(2)*} (m) + \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m)} ))} \right)}} }} \\ & = \frac{1}{{1 + e^{{\left( {\theta_{2}^{o*} (m) - w_{{q_{f} 3}}^{o} (m)h_{{jq_{f} 2}}^{(2)*} (m) - \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m} ))} \right)}} }} \\ \end{aligned}$$

(33)

Substituting Eq. (33) into Constraint (28) gives

$$\frac{1}{{1 + e^{{\left( {\theta_{2}^{o*} (m) - w_{{q_{f} 3}}^{o} (m)h_{{jq_{f} 2}}^{(2)*} (m) - \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m} ))} \right)}} }} \ge \max (o_{j2}^{*} (m),\;a_{j} );\,j = {1} \sim n$$

(34)

Consequently,

$$w_{{q_{f} 3}}^{o} (m) \ge \frac{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m))} }}{{h_{{jq_{f} 2}}^{(2)*} (m)}};\,j = {1} \sim n$$

(35)

Therefore,

$$w_{{q_{f} 3}}^{o} (m) \ge \mathop {\max }\limits_{j} \left( {\frac{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m))} }}{{h_{{jq_{f} 2}}^{(2)*} (m)}}} \right)$$

(36)

Widening $\tilde{w}_{{q_{f} }}^{o} (m)$ increases the fuzziness of $\tilde{o}_{j} (m)$. Therefore, the following is reasonable:

$$w_{{q_{f} 3}}^{o*} (m) = \mathop {\max }\limits_{j} \left( {\frac{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {(w_{q2}^{o*} (m)h_{jq2}^{(2)*} (m))} }}{{h_{{jq_{f} 2}}^{(2)*} (m)}}} \right)$$

(37)

Theorem 3 is proved.

Proof theorem 4

The required proof is similar to that of Theorem 3.

Proof theorem 5

The required proof is straightforward because the other variables in Eqs. (13) and (14) are not affected by the fuzzification of $\theta^{o} (m)$.

Proof theorem 6

Substituting Eq. (5) into Eq. (6) gives.

$$\tilde{h}_{jq}^{(2)} (m) = \frac{1}{{1 + e^{{{(}\tilde{\theta }_{q}^{h(2)} (m)( - )\tilde{I}_{jq}^{h(2)} {\text{(m))}}}} }}$$

(38)

which is decomposed into

$$h_{jq1}^{(2)} (m) = \frac{1}{{1 + e^{{(\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m))}} }}$$

(39)

$$h_{jq3}^{(2)} (m) = \frac{1}{{1 + e^{{(\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m))}} }}$$

(40)

Substituting Eqs. (39) and (40) into Eq. (31) leads to

$$I_{j3}^{o} (m) = \max \left( {\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} } \right)$$

(41)

Similarly, we can derive

$$I_{j1}^{o} (m) = \min \left( {\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} } \right)$$

(42)

Substituting Eq. (41) into Eq. (25) gives

$$o_{j3} (m) = \frac{1}{{1 + e^{{\left( {\theta_{1}^{o} (m) - \max (\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} )} \right)}} }}$$

(43)

Similarly, we also have

$$o_{j1} (m) = \frac{1}{{1 + e^{{\left( {\theta_{3}^{o} (m) - \min (\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} )} \right)}} }}$$

(44)

The following requirements should be met:

$$\frac{1}{{1 + e^{{\left( {\theta_{1}^{o} (m) - \max (\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} )} \right)}} }} \ge \max (o_{j2}^{*} (m),\;a_{j} );\,j = {1} \sim n$$

(45)

$$\frac{1}{{1 + e^{{\left( {\theta_{3}^{o} (m) - \min (\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} )} \right)}} }} \le \min (o_{j2}^{*} (m),\;a_{j} );\,j = {1} \sim n$$

(46)

Equivalently,

$$\begin{aligned}\max \left( {\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q3}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} } \right) \\\ge \theta_{1}^{o} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(47)

$$\begin{aligned}\min \left( {\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q3}^{h(2)} (m) - I_{jq1}^{h(2)} (m){)}}} }}} ,\;\sum\limits_{q = 1}^{{Q_{m} }} {\frac{{w_{q1}^{o} (m)}}{{1 + e^{{{(}\theta_{q1}^{h(2)} (m) - I_{jq3}^{h(2)} (m){)}}} }}} } \right) \\\le \theta_{3}^{o} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(48)

Only $\theta_{{q_{f} }}^{h(2)}$ is fuzzified; the other fuzzy parameters are set to their optimized cores:

$$\begin{aligned}\max \left( \begin{gathered} \frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 3}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} , \hfill \\ \;\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 1}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} \hfill \\ \end{gathered} \right)\\ \ge \theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(49)

$$\begin{aligned}\min \left( \begin{gathered} \frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 3}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} , \hfill \\ \;\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 1}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} \hfill \\ \end{gathered} \right)\\ \le \theta_{2}^{o*} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1);j = {1} \sim n\end{aligned}$$

(50)

If $w_{{q_{f} 2}}^{o*} (m) \ge 0$,

$$\begin{aligned}\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 1}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} \ge \theta_{2}^{o*} (m)\\ - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(51)

$$\begin{aligned}\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 3}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} \le \theta_{2}^{o*} (m)\\ - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(52)

Therefore,

$$\begin{aligned}\theta_{{q_{f} 1}}^{h(2)} (m) &\le \ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1)\\ &+ I_{{jq_{f} 2}}^{h(2)*} (m);\,j = {1} \sim n\end{aligned}$$

(53)

$$\theta_{{q_{f} 3}}^{h(2)} (m) \ge \ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m);\,j = {1} \sim n$$

(54)

Thus,

$$\theta_{{q_{f} 1}}^{h(2)} (m) \le \mathop {\min }\limits_{j} \left( {\ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m)} \right)$$

(55)

$$\theta_{{q_{f} 3}}^{h(2)} (m) \ge \mathop {\max }\limits_{j} \left( {\ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m)} \right)$$

(56)

To reduce the fuzziness of $\tilde{\theta }_{{q_{f} }}^{h(2)} (m)$, the following is reasonable:

$$\theta_{{q_{f} 1}}^{h(2)*} (m) = \mathop {\min }\limits_{j} \left( {\ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m)} \right)$$

(57)

$$\theta_{{q_{f} 3}}^{h(2)*} (m) = \mathop {\max }\limits_{j} \left( {\ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m)} \right)$$

(58)

Otherwise (i.e., $w_{{q_{f} 2}}^{o*} (m) < 0$), Constraints (49) and (50) become

$$\begin{aligned}&\frac{w_{q_f2}^{o\ast}\left(m\right)}{1+e^{\left(\theta_{q_f3}^{h\left(2\right)}\left(m\right)-I_{jq_f2}^{h\left(2\right)\ast}\left(m\right)\right)}}+\sum_{q\neq q_f}\frac{w_{q_f2}^{o\ast}\left(m\right)}{1+e^{\left(\theta_{q_f3}^{h\left(2\right)}\left(m\right)-I_{jq_f2}^{h\left(2\right)\ast}\left(m\right)\right)}}\geq\theta_2^{o\ast}\left(m\right)\\&-\ln\left(\frac1{max\left(o_{i2}^\ast\left(m\right),\alpha_j\right)}-1\right);\;j=1\sim n\end{aligned}$$

(59)

$$\begin{aligned}\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{1 + e^{{{(}\theta_{{q_{f} 1}}^{h(2)} (m) - I_{{jq_{f} 2}}^{h(2)*} (m){)}}} }} + \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} \le \theta_{2}^{o*} (m)\\ - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1);\,j = {1} \sim n\end{aligned}$$

(60)

Therefore,

$$\begin{aligned}\theta_{{q_{f} 3}}^{h(2)} (m) &\ge \ln \left( {\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\max (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1} \right)\\& + I_{{jq_{f} 2}}^{h(2)*} (m);\,j = {1} \sim n\end{aligned}$$

(61)

$$\theta_{{q_{f} 1}}^{h(2)} (m) \le \ln (\frac{{w_{{q_{f} 2}}^{o*} (m)}}{{\theta_{2}^{o*} (m) - \ln (\frac{1}{{\min (o_{j2}^{*} (m),\;a_{j} )}} - 1) - \sum\limits_{{q \ne q_{f} }} {\frac{{w_{q2}^{o*} (m)}}{{1 + e^{{{(}\theta_{q2}^{h(2)*} (m) - I_{jq2}^{h(2)*} (m){)}}} }}} }} - 1) + I_{{jq_{f} 2}}^{h(2)*} (m);\,j = {1} \sim n$$

(62)

The same results are derived. Theorem 6 is proved.