An Improved Discrete Artificial Bee Colony Algorithm for Hybrid Flow Shop Problems - Intelligent Computing for Sustainable Energy and Environment - page 297

S 1 ( i )= g ( S 2 ( i )) ( i =1,2,... n )meansthat S 1 ( i ) is the permutation of (1,2,... n )
based on the ascending order of S 2 ( i )( i =1,2,... n ); and argmi k {

IM k }

stands for
the argument of the minimum, i.e. the set of points of the given argument for
which the given function attains its minimum value.
In the above recursive equations, we firstly calculate the completion time of
the jobs at the stage one, then that of the stage two, until the last stage.

3 The Improved Discrete Artificial Bee Colony (IDABC)
Algorithm for the HFS Problem

3.1 Individual Representation and Initialization

Owing to the continuous nature of the ABC algorithm, it can not be directly used
for the HFS problem. So it is important to find a suitable mapping which can
conveniently convert a harmony to a solution. The model of the HFS problem
is formulated by using the vector representation in last section. As a result, we
adopt this representation in the proposed IDABC algorithm. The individual in
the IDABC algorithm is represented by a permutation of jobs at the stage one
=

.
To guarantee the initial population with a certain quality and diversity, it is
constructed randomly except that one is established by the aforementioned NEH
heuristic [7]. According to [6], the NEH heuristic, a typical constructive method
for the permutation flow shop scheduling problem, is also very robust and well
performing for the HFS problem.

{ π (1) ,π (2) ,...,π ( n ) }

3.2 Employed Bee Phase

In the original ABC algorithm, the employed bees exploit the given food sources
in their neighborhood. Here we propose a novel differential evolution scheme
for the employed bees to generate neighboring food sources. The differential
evolution scheme consists of three steps: mutation, crossover, and selection.
In the mutation part, two parameters mutation rate ( MR ), insert times ( IT )
are introduced. For each incumbent individual, a uniformly random number is
generated in the range of [0,1]. If it is less than MR , the mutant individual is
obtained by operating the insert operation on the best individual π best in the
population IT times; otherwise, the mutant individual is gained by operating
the insert operation on a randomly selected individual IT times.
Next, partially mapped crossover (PMX) [19], a widely used crossover operator
for permutation-based, is used in the crossover part. The incumbent individual
and the mutated individual will undergo the PMX operation with a crossover rate
( CR ) to obtain the two crossed individuals. On the other hand, the two crossed
individuals are the same as the mutated individual with about a probability of
(1- CR ).
Following the crossover operation, the selection is conducted. The one with
the lowest value of the objective function among the two crossed individuals and