Differential Evolution for Permutation-Based Combinatorial Problems - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 30

Ta b l e 2 . 5 . Discrete/binary Solution Representation

j 1 2 3 4 5

n j 4 1 5 2 3

X π j 16 5 22 8 14

d π j π j +1 d 16 , 5 d 5 , 22 d 22 , 8 d 8 , 14 d 14 , 16

the third node is randomly chosen from the fifth cluster (here 22 is randomly chosen); the
fourth node is randomly chosen from the second cluster (here 8 is randomly chosen); and
the fifth node is randomly chosen from the third cluster (here 14 is randomly chosen).
As already illustrated, the objective function value implied by a solution x with m
nodes is the total tour length, which is given by:

m − 1
j =1 d π j π j +1 + d π m π 1

F ( π )= (2.18)

This leads to the total tour length being obtained as

m 1

j =1 d π j π j +1 + d π m π 1 = d 16 , 5 + d 5 , 22 + d 22 , 8 + d 8 , 14 + d 14 , 16

−

F ( π )=

2.3.7 Discrete Set Handling Approach

Discrete set handling is an algorithmic approach how to handle in a numerical way
objects from discrete set. Discrete set usually consist of various elements with non-
numerical nature. In its canonical form DE is only capable of handling continuous vari-
ables. However extending it for optimization of discrete variables is rather easy. Only a
couple of simple modifications are required. In evolution instead of the discrete value
x i itself, we may assign its index, i ,to x . Now the discrete variable can be handled as
an integer variable that is boundary constrained to range < 1 , 2 , 3 ,....., N > .Soasto
evaluate the objective function, the discrete value, x i , is used instead of its index i .In
other words, instead of optimizing the value of the discrete variable directly, we opti-
mize the value of its index i . Only during evaluation is the indicated discrete value used.
Once the discrete problem has been converted into an integer one, the methods for han-
dling integer variables can be applied. The principle of discrete parameter handling is
depicted in chapter 7.3.

2.3.8 Anatomy of Some Approaches

based combinatorial DE approaches
described in their topic (see Table 2.6). This exercise excludes smallest position value,
discrete/binary and discrete set handling approaches.

[10] carried out anatomy of the four permutation −