International Journal of Computer Application (2250-1797)
Volume 6– No.5, September- October 2016
An Enhance Firefly Algorithm for Flexible Job Shop Scheduling
Amit kumarSrivastava1, Hari Mohan Singh2
Department of Computer Science & Engineering
SHAITS, Allahabad
DOI: 10.17632/wg46kj4jmw.1
Abstract: – The Flexible Job Shop scheduling Problem has
originated from the classical job shop scheduling problem. The
FJSSP finds its importance in many research fields and can be
applied to a large variety of real-world problems that can be
modeled as a FJSSP. In this paper we propose an Enhance Firefly
algorithm to solve the FJSSP. In this machine allocation and the
problem of sequencing the operation are solved by developing a
suitable conversion of the continuous functions. The functions
such as attractiveness, distance, and movement are transformed
into a form of new discrete functions. The computational results
show that the developed Enhance Firefly Algorithm gave better
results than the other author's algorithm
Keywords: Job Shop Scheduling, Firefly algorithm,
Flexible Job Shop Scheduling Problem.
1 Introduction
The scheduling job on the machines difficulties has
been very well known between the researchers. In
practical manufacturing environment, there are all
time a few uncertainties inducing unavailability of
machines, such as unavailability of staff, operational
errors of staff, machine breakdown, and so onward.
Several jobs intend to be done or it may be list of a
sequence of events to be completed. The term
scheduling mentions to the action of allocating
machine for a time to complete a collection of jobs.
The production systems are very complicated due to
the large number of combined entities and controlling
of available resources is always a crucial decision
with an aim to maximize or minimize certain
parameters like maximizing use of machines and
minimizing total make-span of the jobs.
1.1 Scheduling Problem, Classification
and Challenges
Scheduling in a manufacturing industry is required
for many altered aims. Li J. et al. suggested that the
aim of scheduling in manufacturing is to minimize
the manufacturing costs and time. There are some
problems in manufacturing, transportations and
engineering field; hence, there is a requirement of the
scheduling in real time system such as to be able to
change to the situation and reassign the jobs to
different machine in case of machine failure, making
of fresh schedules may be required in case of new job
arrival, in maintenance activity new schedules need
to be ready, change in objective function where
planner may need to change the objective function to
be optimized thereby new schedules need to be
prepared.
Scheduling is classified in various categories. First is
Single Machine Scheduling problem where a set of
independent, single operation jobs are to be schedule
on the machines with the same single operation on
each of the jobs. Next is Flow shop-scheduling
problem (FSSP) where there is N number of jobs, and
each job required to be process on M different
machines. All the jobs have the same process
sequence, but the processing times of different jobs
on a machine may be different. The Job-shop
Scheduling problem (JSSP) where each job needs to
follow a sequence of operations and exactly one
machine is required for each operation. The operation
and the sequence of operation are understood in
advance.It is which stated as non-deterministic
polynomial-time (NP) hard by Agarwal et al.
The latest is the Flexible Job-Shop Scheduling
problem (FJSSP) given by Agarwal et al. where each
operation processes on a machine chosen from in the
middle of a finite subset of capable machines. The
FJSSP is more difficult than the classical JSP since it
put a new decision level beside the sequencing one.
Thus, the flexibility shall considerably increase the
complexity of the problem, as it requires an extra
level of decisions.
The following notations and its description are given
below in Table 1.
Table 1: Symbol Used
Parameter
Description
i,h
index of jobs,
j,g
index of operation sequence,
k
index of machines,
n
total number of jobs,
m
total number of machines,
ni
total number of operation of job i,
Oij
the jth operation of job i,
Mij
set of available machines for the operation Oij
Pijk
processing time of operation Oij on machine k
tijk start time of operation Oij on machine k
Cij
completion time of operation Oij
Ck
the completion time of Mk
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International Journal of Computer Application (2250-1797)
Volume 6– No.5, September- October 2016
Formally, Flexible job-shop scheduling problem can
be state as follows: There are m number of machines
and n number of jobs. For each job Ji (1 ≤ i ≤n) there
are ni sequence of operations. From the set of
operations, an operation Oij (i=1,2,3,…..,n;
j=1,2,3,….,ni ) of job (Ji) may be processed by a
machine mij from the set of available capable
machines Mij. Pijk denotes the processing time of the
operation Oij on machine k ϵMij.
Minf1=
…
s.t.: Cij–Ci(j-1)≥PijkXijk,j= 2 to ni
(5)
The Attractiveness is a monotonically decreasing
function. The force with which each firefly attracts
other fireflies is determined by their unique
attractiveness (β). The attractiveness of the firefly can
be define with the distance r by Eq. (6).
(6)
(1)
(2)
(Chg-Cij-thjk )XhgkXijk≥0]v[(Cij-Chg-tijk) Xhji Xijk ≥ 0],∀
(i, j), (h, g),
(3)
∀
(4)
where, decision variables taken into account is
given as below:
Xijk = 1, if machine k
the operation Oij , otherwise 0
select
for
In above model the objective function is to minimize
the make span time. Equation (1) ensures minimum
make-span. Equation (2) maintains the operation
precedent constrained. Equation (3) checks that each
machine processes only one operation at a time when
all the stated elements satisfies by the first or the
second constraint. Equation (4) depicts that one
machine from the set of available capable machine
could be selected for each operation.
1.2 Firefly Algorithm
The Firefly Algorithm was developed by Xin She
yang in late 2007 and 2008 in Cambridge University.
It is inspired by fireflies behave and interact socially.
In its basic form, the firefly algorithm was designed
to solve continuous problem. However, a Firefly
algorithm was also adjusted for solving FJSSP. In the
firefly algorithm, we can idealize flashing
characteristics as the following three rules:
1.The fireflies are unisex so that all the fireflies are
attracted to each other regardless of their sex.
2.The attractiveness of the firefly is directly
proportional to the brightness of the firefly and them
both decreases with the distance.
3.The brightness of the firefly is determined by the
objective function.
The difficulties faced in implementing this algorithm
are to compute discrete distance between the fireflies
and to model their coordinated movement. The
Distance between any two flashing fireflies i and j,
located at positions xi and xj, respectively, may be
defined as Cartesian distance defined in Eq. (5).
β0 gives attractiveness at r=0 and ɣ is light absorption
coefficient. Movement of a firefly towards more
attractive (brighter) firefly j is determined by Eq. (7).
x j  xi  0e
rij 2
x
j

 xi    rand  1

2 (7)
The basic Firefly Algorithms describes as follows:
Algorithm 1: Firefly Algorithm
Objective function f(x),
x=(x1,x2,………xd)T
Generate initial population of fireflies xi
(i=1,2,…….,n)
Light Intensity Ii at xi is calculated by f(xi)
Define light absorption coefficient ɣ
While t < Max Generation do
for i=1 : n all n fireflies
for j= 1 : i all n fireflies
If Ij < Ii then
Move firefly i towards j in
d-dimension;
Endif
Attractiveness varies with distance r via exp [ɣr2]
Evaluate new solutions and update light
intensity
end for j
end for i
Rank the fireflies and find the current best
end while
Post process results and visualization.
Firefly algorithm is simple in terms of complexity
and it is easy to implement. Unfortunately, the firefly
algorithm performs a full pair-wise comparison i.e.
that is based on the comparison and computation of
attractiveness between each firefly in the swarm and
all the other fireflies, which can be time consuming
for large optimization problem.
Thus, following are the main contributions of
proposed work:
1).To enhance firefly algorithm for flexible job shop
scheduling problem.
2).To evaluate and compare performance of enhance
firefly algorithm with existing algorithm.
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International Journal of Computer Application (2250-1797)
Volume 6– No.5, September- October 2016
2. Related Background
Wang Y. et al., took up two major difficulties FJSSP
problems named Machine Assignment and Operation
Sequencing on machines into consideration. A hybrid
of PSO algorithm and Tabu search (TS) algorithm is
used to solve FJSSP for minimizing make-span.
Zhao C. et al., applied PSO to obtain the solution of
FJSSP problem with the objective to reduce the
criteria of maximum completion time. Meybodi M.
R. et al., proposed CLA-FA having modules Cellular
Learning Automata (CLA) and the Firefly Algorithm
(FA). Wong K. et al., stated that FJSSP is an NPhard problem in the vast field of optimization and
dynamic nature of real world adds many
complications to this problem. Author proposed a
mathematical model of the FJSSP problem with some
constraints to maximize profit of the total scheduling
job. Chandshekharan M. et al., also, reported Job
shop scheduling problem as non polynomial
deterministic (NP) hard problem. Using fuzzy logic.
Krasimira G. et al., suggested that many real lifescheduling problems might be formulated as FJSSPs
which simultaneously optimize several conflicting
criteria and pose high computational complexity.
Rahmati A. et al., suggested that one of the efficient
flexible manufacturing environments is flexible jobshop problem (FJSSP), which is an extended
derivation of classical job-shop scheduling problem
to meet the requirements of modern job-shop.
3. Proposed Methodology
Generate the initial population of fireflies or xi
(i=1,2,…..,n)
Determine the light intensity of Ii at xi via f(xi)
Initialize Global best
Whilet<MaxGet
{Solution Construction}
for i=1 to n (all n fireflies)
for j=1 to n (n fireflies)
if Ij>Ii then
move firefly i towards j by using eqn
Xit+1 =xit+β0e-ɣrij2(xjt-xit)+αtϵit
end if
attractiveness varies with distance r via exp [ɣr2];
evaluate new solution and update light intensity;
end for j;
end for i;
Rank the fireflies and find the current best;
{evaluation}
if current best < global best
then
global
best=current best
else
choose a random firefly from the population
replace firefly with global best
end if
end while;
Process results and visualization;
end procedure.
Here, for global optima firefly’s movement is used.
Global optimum can be a firefly that has the
maximum or minimum value. In addition, the global
optima will be update in any iteration.
Enhance Firefly Algorithm
4. Results and Discussion
In the firefly algorithm 1, firefly moves regardless of
the global optima and it can increase the number of
iterations to find global best scheduling. To
overcome the limitations of the firefly algorithm
especially
for
discrete
and
combinatorial
optimization, which eliminates drawback of original
firefly algorithm 1 and improve the movement of
fireflies, propose a new version of firefly algorithm
called Enhance Firefly Algorithm. The new enhance
firefly algorithm 2 depicts as follows:
Algorithm 2: Enhance Firefly Algorithm
Initialize algorithm parameters:
MaxGen: the maximal no of generations
γ=the light absorption coefficient
r=the particle distance from the light source
d=the domain space
Begin;
Define the objective function of f(x),
where x=(x1,.....xd)T
{Initialization}
The parameters of the algorithm are initialized as
follows: Attractiveness of the fireflies i.e.β0 =1.0,
Light absorption coefficient of the
environmenti.e.ɣ=0.1and randomization parameter
i.e. α=1.0.
4.1 Small scale instance problem (4x5)
Best Schedule
A= 4 2 3 1 5 1 3 2 4 4 1 2
B= 7 1 2 8 9 4 3 10 11 5 12 6
Best time=11
4.2 Medium scale instance problem (8x8)
Best Schedule
A= 2 5 6 3 4 7 5 7 4 1 2 6 3 1 4 6 7 3 8 2 3 8 4 1
285
B= 14 4 18 8 15 1 21 5 9 6 16 19 24 10 11 12 22
2 23 3 7 13 25 17 20 26 27
Best time=15
4.3 Large scale instance problem (10x10)
Best Schedule
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International Journal of Computer Application (2250-1797)
Volume 6– No.5, September- October 2016
A= 1 2 4 1 4 3 10 4 5 7 2 4 9 9 2 6 9 7 1 3 3 5 10
2676328
B= 19 25 16 26 28 29 7 1 8 20 22 2 21 23 9 10
27 30 3 11 13 14 15 4 17 5 24 12 18
Best time=8
Table 1 Comparative result based on the objective function
TD
CGA
AL
AL+CGA
HPSO
FFA
4x5
11
13
12
16
11
11
8x8
19
15
16
15
16
15
10x10
16
7
8
7
7
8
4.5 Comparative
execution time
result
based
on
Table 2 Comparative result based on execution time
Figure 1
Gantt chart for the optimized schedule of the
problem 4x5
TD
CGA
AL
AL+
CGA
HPSO
FFA
4x5
0.49
6.01
0.87
1.33
1.78
0.32
8x8
0.83
7.92
1.57
2.18
3.06
0.67
10x10
1.21
9.86
2.43
3.75
4.39
0.93
5. Conclusion and Future Work
Figure 2 Gantt chart for the optimized schedule of the
problem 8x8
Figure 3 Gantt chart for the optimized schedule of the
problem 10x10
4.4 Comparative result based on the
objective function
The Firefly algorithm is one of the best methods for
flexible job shop-scheduling problem. The
experimental results suggest that the developed
enhance firefly algorithm is good and better to solve
the flexible job shop-scheduling problem without any
hybridization. Moreover, the result proved the
efficiency and feasibility of the proposed firefly
algorithm. The results obtained and the time taken
demonstrates that enhance firefly algorithm is better.
Finally It is one of the easiest method and easy to put
in any NP hard problem.
The future work is to extend the merging capability
of the algorithm and to generalize the application of
the proposed firefly algorithm for other combinatorial
optimization problems. There is a lot of scope for
further research in bench marking example and as
well as solving multi objective flexible job shopscheduling problem. The progress of fast and better
algorithms solving multi-objective flexible job shop
scheduling problems remains a challenge for the
researchers from different areas of applied
mathematics.
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