The document describes a discrete firefly algorithm proposed to solve hybrid flowshop scheduling problems with two objectives: minimizing makespan and mean flow time. Hybrid flowshop scheduling problems involve scheduling jobs through multiple stages with parallel machines in some stages, and are known to be NP-hard. The proposed discrete firefly algorithm adapts the continuous firefly algorithm to the discrete problem by using a smallest position value rule to map continuous firefly positions to discrete job permutations. Computational experiments show the proposed algorithm outperforms other metaheuristics for hybrid flowshop scheduling problems.

1. IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 2, APRIL 2014 301
Letter
A Discrete Firefly Algorithm for the Multi-Objective
Hybrid Flowshop Scheduling Problems
Mariappan Kadarkarainadar Marichelvam,
Thirumoorthy Prabaharan, and Xin She Yang
Abstract—Hybrid flowshop scheduling problems include the general-
ization of flowshops with parallel machines in some stages. Hybrid flow-
shop scheduling problems are known to be NP-hard. Hence, researchers
have proposed many heuristics and metaheuristic algorithms to tackle
such challenging tasks. In this letter, a recently developed discrete firefly
algorithm is extended to solve hybrid flowshop scheduling problems with
two objectives. Makespan and mean flow time are the objective functions
considered. Computational experiments are carried out to evaluate the
performance of the proposed algorithm. The results show that the pro-
posed algorithm outperforms many other metaheuristics in the literature.
Index Terms—Discrete firefly algorithm (DFA), heuristics, hybrid
flowshop scheduling (HFS), makespan, mean flow time, metaheuristics.
I. Introduction
SCHEDULING may be considered a process of allocating
resources over time to perform a collection of tasks [1].
Hybrid flowshop scheduling (HFS) is one of the most impor-
tant scheduling problems. Many researchers have concentrated
on HFS problems since they were proposed by Arthanari and
Ramamurthy [2]. Many real industries, including the iron and
steel, textile, electronics, and chemical industires, resemble
the hybrid flowshop environment. Hybrid flowshop may be
considered the combination of flowshop and parallel machine
environments. But hybrid flowshop scheduling problems are
more complex than flowshop scheduling problems. Hybrid
flowshop scheduling problems were proved to be NP-hard
in [3] and [4]. Due to their complexity, hybrid flowshop
scheduling problems cannot be solved by exact algorithms.
Hence, researchers have developed many heuristics and meta-
heuristics. Rajendran and Chaudhuri [5] have proposed some
heuristics to minimize total flow time for multistage parallel
processor flowshop problems. Different heuristics have been
developed by researchers for different objective functions [6]–
[8]. Neron et al. [9] applied energetic reasoning and global
operations for enhancing the efficiency of branch and bound
algorithm to minimize the makespan for HFS. They tested the
algorithm on the benchmark problems in the literature. Agent-
based scheduling incorporated with game theory was proposed
Manuscript received March 28, 2012; revised August 17, 2012; accepted
December 29, 2012. Date of publication January 15, 2013; date of current
version March 28, 2014.
M. K. Marichelvam is with the Kamaraj College of Engineer-
ing and Technology, Virudhunagar, Tamil Nadu 626001, India (e-mail:
mkmarichelvamme@ gmail.com).
T. Prabaharan is with the Department of Mechanical Engineering, Mepco
Schlenk Engineering College, Sivakasi, Tamil Nadu 626001, India (e-mail:
prabaharan−369@yahoo.co.in).
X. S. Yang is with the School of Science and Technology, Middlesex
University, London NW4 4BT, U.K. (e-mail: xy227@cam.ac.uk).
Digital Object Identifier 10.1109/TEVC.2013.2240304
by Babayan and He [10] to minimize makespan for solving
n job 3 stage flexible flowshop scheduling problems. They
tested their methodology for randomly generated problems. An
immune algorithm was presented by Alisantoso et al. [11] to
solve the problem of scheduling a flexible printed circuit board
flowshop. Engin and Doyen [12] also proposed an artificial
immune system algorithm for the HFS problem to minimize
makespan. Different types of metaheuristics have been used to
solve hybrid flowshop scheduling problems. Yang et al. [13]
have proposed a tabu search simulation optimization to solve
flowshop scheduling problems with multiple processors. Tang
and Wang [14] also applied the tabu search algorithm to solve
the HFS problem. Genetic algorithm is a widely used meta-
heuristics algorithm to solve the HFS problem [15]–[21]. Re-
searchers have also applied the ant colony optimization (ACO)
algorithm [22], [23], the particle swarm optimization algorithm
[24]–[26], and the simulated annealing (SA) algorithm [27]
to solve the HFS problem. Jungwattanakit et al. [28] have
compared three different metaheuristics algorithms, namely,
genetic algorithm (GA), tabu search, and SA, to minimize the
convex sum of makespan and the number of tardy jobs for
flexible flowshop problems with unrelated parallel machines.
Recently, Ruiz and Vazquez-Rodriguez [29] provided a review
of the HFS problem. The different types of hybrid flowshop
scheduling, their complexity, the different algorithms, and the
objective function can be found in [29].
Firefly algorithm is one of the recently developed meta-
heuristic algorithms developed by Yang [30]. Yang [31] pro-
posed a firefly algorithm for multimodal optimization appli-
cations. Lukasik and Zak [32] presented a further study on
the firefly algorithm for constrained continuous optimization
problems. Yang [33] applied the firefly algorithm for the
optimization of pressure vessel design. He also presented a
few new test functions to validate the firefly optimization
algorithm. Sayadia et al. [34] presented a discrete firefly
algorithm to minimize makespan for flowshop scheduling
problems. Chai-ead et al. [35] have proposed bees and fire-
fly algorithms to find optimal solutions of noisy nonlinear
continuous mathematical models. Banati and Bajaj [36] have
presented a new feature selection approach by combining the
rough set theory with the firefly algorithm. Apostolopoulos
and Vlachos [37] have applied the firefly algorithm to solve
economic emission load dispatching problems to minimize
fuel cost and emission generating units. Basu and Mahanti [38]
proposed firefly algorithm and artificial bee colony algorithms
for the antenna design. Gandomi et al. [39] applied the
firefly algorithm to solve mixed continuous/discrete structural
optimization problems. Kazemzadeh Azad and Kazemzadeh
Azad [40] also proposed an improved firefly algorithm to solve
structural optimization problems. A discrete firefly algorithm
was proposed by Jati and Suyanto [41] to solve the travelling
salesman problem. Recently, Khadwilard et al. [42] have
solved the job shop scheduling problems using the firefly
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2. 302 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 2, APRIL 2014
algorithm. They have also investigated the different parameters
for the proposed algorithm and compared the performance
with different parameters. Yang et al. [43] have applied the
firefly algorithm to solve economic load dispatching problems.
Reviewing the literature, it may be easily concluded that
the applications of the firefly algorithm to the combinatorial
optimization problems are very limited. Hence, in this letter
a discrete firefly algorithm is proposed to minimize makespan
for M-stage hybrid flowshop scheduling problems with the
objective function of makespan and mean flow time. The
rest of the letter is organized as follows. Section II provides
the problem formulation. The firefly algorithm is presented
in Section III. The proposed discrete firefly algorithm is
discussed in Section IV. The computational experiments and
the result comparisons are provided in Section V. Finally,
Section VI concludes the letter.
II. Problem Formulation
Hybrid flowshop consists of a series of production stages.
Each stage has multiple machines operating in parallel. The
machines may be identical, uniform, or unrelated. Some stages
may have only one machine. But, at least one stage must have
multiple machines. Each job is processed by one machine
in each stage. The layout of the M-stage hybrid flowshop
scheduling environment is given in Fig. 1.
HFS problems are commonly found in many real-world
industries and they are the most important and difficult NP-
hard problems. Hence, in this letter we consider the M-stage
hybrid flowshop scheduling problem. The objective function
considered in this letter is the minimization of weighted sum
of makespan and mean flow time. Makespan is defined as the
completion time of the last job to leave the system. Makespan
is important for effective utilization of resources. Mean flow
time is the average time spent by the job in the system.
Mean flow time is important to minimize the work-in-process
inventories [1].
The problem is formulated mathematically as
Z = min(w1Cmax + w2f) (1)
subject to
Cmax ≥ Cjs, for all s = 1, 2, ..., M, j = 1, 2, ..., n, (2)
Cjs = Sjs + Psj (3)
for all
s = 1, 2, ..., M, j = 1, 2, ..., n (4)
Cjs ≤ Sj(s+1), for s = 1, 2, ..., M − 1 (5)
Shs ≥ Cjs − KWhjs, for all job pairs (h, j) (6)
Sjs ≥ Chs + K − 1, for all job pairs (h, j) (7)
Sj1 ≥ Rj, for allj = 1, 2, ..., n (8)
Yjis ∈ {0, 1}, Wjhj ∈ {0, 1}, for all
j = 1, ..., n, i = 1, 2, ..., ms, and s = 1, 2, ..., M.
(9)
Cjs ≥ 0, for all s = 1, 2, ..., M, j = 1, 2, ..., n (10)
f =
n
j=1 CjM
n
(11)
f ≥ 0. (12)
In this letter, the following assumptions are made.
1) All n jobs are available at the beginning of scheduling.
2) Each stage has infinite storage capacity.
3) One machine can process only one job at a time.
4) One job can be processed by only one machine at any
time.
5) For all the jobs, the processing times at each stage are
known in advance and deterministic.
6) Job set-up times are sequence-independent and are in-
cluded in the job processing time of the jobs at the
corresponding stage.
7) Travel time between consecutive stages is negligible.
8) Preemption is not allowed.
III. Firefly Algorithm
The Firefly algorithm is a recently developed nature-inspired
metaheuristic algorithm. The Firefly algorithm is inspired by
the social behavior of fireflies. Fireflies may also be called
lightning bugs. There are about 2000 firefly species in the
world. Most of the firefly species produce short and rhythmic
flashes. The pattern of flashes is unique for a particular species.
A firefly’s flash mainly acts as a signal to attract mating
partners and potential prey. Flashes also serve as a protective
warning mechanism. The following three idealized rules are
considered in [30] to describe the firefly algorithm.
1) All fireflies are unisex so that one firefly will be attracted
to other fireflies regardless of their sex.
2) Attractiveness is proportional to their brightness; thus,
for any two flashing fireflies, the less bright one will
move toward the brighter one. The attractiveness is
proportional to the brightness and they both decrease
as their distance increases. If there is no brighter one
than a particular firefly, it will move randomly.
3) The brightness of a firefly is affected or determined by
the landscape of the objective function. For a maximiza-
tion problem, the brightness may be proportional to the
objective function value. For the minimization problem,
the brightness may be the reciprocal of the objective
function value. The pseudocode of the firefly algorithm
was given by Yang [30]. The pseudocode of the firefly
algorithm is given in Algorithm 1.
A. Attractiveness
The attractiveness of a firefly is determined by its light
intensity. The attractiveness may be calculated by using the
equation
β(r) = β0e−γr2
. (13)

3. MARICHELVAM et al.: A DISCRETE FIREFLY ALGORITHM FOR THE MULTI-OBJECTIVE HYBRID FLOWSHOP SCHEDULING PROBLEMS 303
Algorithm 1 Pseudocode of the Firefly Algorithm
Objective function f(x), x = (x1, ..., xd)T
Generate initial population of fireflies xi(i = 1, 2, ..., n)
Light intensity Ii at xi is determined by f(xi)
Define light absorption coefficient γ
While (t < MaxGeneration)
for i = 1 : n all n fireflies
for j = 1 : i all n fireflies
if (Ij > Ii), Move firefly i toward j in d-dimension; end
if
Attractiveness varies with distance r via exp (−γr)
Evaluate new solutions and update light intensity
end for j
end for i
Rank the fireflies and find the current best
end while
Postprocess results and visualization
B. Distance
The distance between any two fireflies k and l at Xk and
Xl is the Cartesian distance as follows:
rkl = Xk − Xl =
d
k=1
Xk,o − Xl,o
2
. (14)
C. Movement
The movement of a firefly k that is attracted to another more
attractive firefly l is determined by
Xk = Xk + βoe−γr2
kl (Xl − Xk) + α rand −
1
2
. (15)
IV. Discrete Firefly Algorithm
The firefly algorithm has been originally developed for
solving continuous optimization problems. The firefly algo-
rithm cannot be applied directly to solve the discrete op-
timization problems. In this letter, we use and extend the
smallest position value (SPV) rule described by Bean [44]
to enable the continuous firefly algorithm to be applied to
discrete HFS scheduling problems. For this, a discrete firefly
algorithm (DFA) is proposed. The SPV rule has already been
applied by the researchers to solve the scheduling problems
[45].
A. Implementation of the DFA for HFS Problems
1) Solution Representation: Solution representation is one
of the most important issues in designing a DFA. The solution
search space consists of n dimensions as n number of jobs
are considered in this letter. Each dimension represents a job.
The vector Xt
i = (Xt
i1, Xt
i2,...,Xt
in) represents the continuous
position values of fireflies in the search space. The SPV rule is
used to convert the continuous position values of the fireflies
to the discrete job permutation. The solution representation of
a firefly with six jobs is illustrated in Table I.
Fig. 1. Layout of M-stage hybrid flowshop scheduling environment.
TABLE I
Solution Representation of a Firefly
Dimension j
1 2 3 4 5 6
xij 0.81 0.90 0.12 0.09 0.71 0.63
Jobs 5 6 2 1 4 3
The smallest position value is xt
i4 = 0.09, and the dimension
j = 4 is assigned to be the first job in the permutation
according to the SPV rule. The second smallest position value
is xt
i3 = 0.12, and the dimension j = 3 is assigned to be
the second job in the permutation. Similarly, all the jobs are
assigned in the permutation.
2) Population Initialization: In most of the metaheuristics,
the initial population is generated at random. In the DFA, the
initial population is also generated at random. The continuous
values of positions are generated randomly using a uniform
random number between 0 and 1.
3) Solution Updation: By using the permutation, each
firefly is evaluated to determine the objective function value.
The objective function value of each firefly is associated with
the light intensity of the corresponding firefly. A firefly with
less brightness is attracted and moved to a firefly with more
brightness. The attractiveness of the firefly is determined using
(13). The distance between each pair of fireflies is determined
by (14). The SPV rule is applied to obtain the job permutation.
The attractiveness is calculated for each firefly. Then, the
movement of the firefly is determined by (15) depending on
the attractiveness of the firefly. The above steps are repeated
until the termination criterion is met.
V. Computational Experiments
To test the performance of the proposed algorithm, computa-
tional experiments were carried out. The hybrid discrete firefly
algorithm was coded in C++ and run on a PC with an Intel
Core Duo 2.4 GHz CPU, 2 GB RAM, running Windows XP.
The performance of the proposed algorithm is evaluated by the
mean relative deviation index (MRDI), which is given below
MRDI =
R
b=1
(Cmax∗ − Cmax−mh)
Cmax∗
× 100/R. (16)
A. Test Instances
Two types of test instances are used in this letter. In the
first test instance, we consider a case study of a real industrial
scheduling problem within a steel furniture manufacturing
company. In the second test, we develop some random
instances.

4. 304 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 2, APRIL 2014
TABLE II
Processing Times of Jobs (in Seconds) at Different Stages
Stages → Power Drilling
Jobs ↓ Punching Bending Welding pressing stage
stage stage stage stage
1 40 30 0 0 0
2 60 70 110 0 0
3 10 20 50 0 0
4 20 0 0 0 0
5 30 40 0 0 0
6 30 70 70 0 0
7 40 0 0 0 0
8 40 0 0 50 0
9 30 250 40 0 0
10 10 20 25 0 0
11 0 0 0 10 0
12 0 0 0 20 0
13 50 90 0 0 0
14 0 0 0 20 0
15 0 0 0 40 0
16 10 20 0 0 30
17 0 0 0 10 0
18 0 0 0 10 0
19 0 0 0 10 0
20 0 0 0 6 0
Fig. 2. Pareto optimal solutions for the case study problem.
1) Case Study: In this letter, we use the data collected
from a leading steel furniture manufacturing company in
Chennai, Tamil Nadu, India. The company produces a variety
of steel furniture components. From among them, the two-
drawer vertical filing cabinet is considered in this letter. Each
two-drawer vertical filing cabinet consists of 20 parts. Each
part may be considered a different job. The jobs are processed
in five different stages: punching, bending, welding, power
pressing, and drilling. Each stage uses a different number
of machines. The number of machines in punching, bending,
welding, power pressing, and drilling stages are 5, 8, 3, 5, and
1, respectively. The processing times for the different jobs at
different stages are given in Table II. The jobs are produced
in lots and the lot size is 120. In the company, the scheduling
is done manually. For the case study, the values of w1 and
w2 are assumed to be 0.5 and 0.5, respectively. The proposed
DFA is tested with the data from the case study company. The
Pareto optimal solution obtained by the DFA is presented in
Fig. 2.
2) Random instances: In order to test the performance of
the proposed DFA, experiments were conducted on random
TABLE III
Factor Levels for Random Instances
Sl. No. Factors Levels
1 Number of stages 2, 5, and 10
2 Number of machines at each stage 2, 3, and 5
3 Number of jobs 10, 20, and 50
4 Processing time distribution Uniform [1–50]
TABLE IV
Parameters of the Discrete Firefly Algorithm
Sl. No. Factors Levels
1 Attractiveness of firefly β0
0.0 (low)
0.5 (medium)
1.0 (high)
2 Light Absorption coefficient γ
0.5 (low)
0.75
(medium)
1.0 (high)
3 Randomization parameter α
0.0 (low)
0.5 (medium)
1.0 (high)
TABLE V
MRDI Comparison of Different Algorithms for the
Test Problems
Sl. No. Algorithms
MRDI
Makespan Mean flow time
1 DFA 0.0 0.0
2 GA 5.26 5.38
3 PGA 4.68 4.82
4 ACO 7.26 7.42
5 SA 9.14 9.30
instances. Table III gives the factor levels for the design of
experiments to define the production systems: the number of
stages, number of machines, number of jobs, and processing
times of the jobs.
B. Parameter Setting
The proposed algorithm is tested with different types of
parameter settings. The attractiveness of fireflies, light absorp-
tion coefficient, and randomization parameter are the important
parameters. The parameters used in this letter are presented in
Table IV.
C. Computational Results
The performance of the DFA for the random problems is
compared with the parallel genetic algorithm (PGA) [18],
GA [20], ACO [22], and [27]. The results are presented in
Table V. From Table V, it is concluded that the proposed
DFA outperforms the earlier reported literature results. It
is observed that the DFA provides 5.26% improvement
with respect to GA and 4.68% with respect to PGA for
the makespan criterion. The DFA also provides 7.26%
improvement with respect to ACO and 9.14% with respect
to SA for the random problems for the makespan criterion.
In addition, the DFA provides better results than PGA, GA,
ACO, and SA for the mean flow time criterion.
VI. Conclusion
In this letter, we presented a multistage hybrid flowshop
scheduling problem with the objective of minimizing the sum

5. MARICHELVAM et al.: A DISCRETE FIREFLY ALGORITHM FOR THE MULTI-OBJECTIVE HYBRID FLOWSHOP SCHEDULING PROBLEMS 305
of makespan and mean flow time. A discrete firefly algorithm
was presented to solve these problems. The algorithm is tested
with a real-world case study and also with random problem
instances. The computational results show that the proposed
algorithm outperforms the GA, ACO, and SA.
Further studies on extensive parametric study and compar-
ison may be fruitful. It may also be an important application
to use the proposed algorithm to solve the HFS problems
with unrelated or nonuniform machines in each stage. The
algorithm may also be applied to HFS problems with due date-
related performance measures.
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