C++ Neural Networks and Fuzzy Logic: Preface

C++ Neural Networks and Fuzzy Logic

by Valluru B. Rao

MTBooks, IDG Books Worldwide, Inc.

ISBN: 1558515526   Pub Date: 06/01/95

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Other Approaches to Solve the Traveling Salesperson Problem

The following describes a few other methods for solving the traveling salesperson problem.

Anzai’s Presentation

Yuichiro Anzai describes the Hopfield network for the traveling salesperson problem in a slightly different

way. For one thing, a global inhibition term is not used. A threshold value is associated with each neuron,

added to the activation, and taken as the average of A

1

 and A

2

, using our earlier notation. The energy function

is formulated slightly differently, as follows:

E = A

1

 £

1

(£

k

 x

ik

−1)

2

 + A

2

 £

i

(£

k

 x

ki

−1)

2

 + A

4

 £

k

 £

j`k

 £

i`k

 d

kj

 x

ki

(x

j,i+1

 + x

j,i−1

)

The first term is 0 if the sum of the outputs is 1 in each column. The same is true for the second term with

regard to rows.

The output is calculated using a parameter », here called the reference activation level, as:

x

ij

 = (1 + tan tanh(a

ij

/»))/2

The parameters used are A

1

 = 1/2, A

2

 = 1/2, A

4

 = 1/2, ”t = 1, Ä = 1, and » = 1. An attempt is made to solve the

problem for a tour of 10 cities. The solution obtained is not crisp, in the sense that exactly one 1 occurs in

each row and each column, there are values of varying magnitude with one dominating value in each row and

column. The prominent value is considered to be part of the solution.