C++ Neural Networks and Fuzzy Logic: Preface

Activations, Outputs, and Their Updating

We denote the activation of the neuron in the ith row and jth column by a

ij

, and the output is denoted by x

ij

. A

time constant Ä, and a gain » are used as well. A constant m is another parameter used. Also, ”t denotes the

increment in time, from one cycle to the next. Keep in mind that the index for the summation £ ranges from 1

to n, the number of cities. Excluded values of the index are shown by the use of the symbol `.

The change in the activation is then given by ”a

ij

, where:

”a

ij

 = ”t (Term

1

 + Term

2

 + Term

3

 + Term

4

 + Term

5

)

Term

1

 = − a

ij

/Ä

Term

2

 = − A

1 k`j

x

ik

Term

3

 = − A

2

£

k`i

x

kj

Term

4

 = − A

3

(£

i

 £

k

x

ik

 − m)

Term

5

 = − A

4

 £

k`i

d

ik

(x

k,j+1

 + x

k,j−1

)

To update the activation of the neuron in the ith row and jth column, you take:

a

ijnew = 

a

ijold + ”

a

ij

The output of a neuron in the ith row and jth column is calculated from:

x

in

 = (1 + tanh(»a

ij

))/2

NOTE: 

, which is the original sigmoid function

The function used here is the hyperbolic tangent function. The gain parameter mentioned earlier » is. The

output of each neuron is calculated after updating the activation. Ideally, you want to get the outputs as 0’s

and 1’s, preferably a single one for each row and each column, to represent a tour that satisfies the conditions

of the problem. But the hyperbolic tangent function gives a real number, and you have to settle for a close

enough value to 1 or 0. You may get, for example, 0.96 instead of 1, or 0.07 instead of 0. The solution is to be

obtained from such values by rounding up or down so that 1 or 0 will be used, as the case may be

C++ Neural Networks and Fuzzy Logic:Preface

Inputs

342

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C++ Neural Networks and Fuzzy Logic:Preface

Inputs

343

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