IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 2, APRIL 2014 301

Letter

A Discrete Firefly Algorithm for the Multi-ObjectiveHybrid 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, researchershave proposed many heuristics and metaheuristic algorithms to tacklesuch challenging tasks. In this letter, a recently developed discrete fireflyalgorithm is extended to solve hybrid flowshop scheduling problems withtwo objectives. Makespan and mean flow time are the objective functionsconsidered. Computational experiments are carried out to evaluate theperformance 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, hybridflowshop scheduling (HFS), makespan, mean flow time, metaheuristics.

I. Introduction

SCHEDULING may be considered a process of allocatingresources 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 concentratedon HFS problems since they were proposed by Arthanari andRamamurthy [2]. Many real industries, including the iron andsteel, textile, electronics, and chemical industires, resemblethe hybrid flowshop environment. Hybrid flowshop may beconsidered the combination of flowshop and parallel machineenvironments. But hybrid flowshop scheduling problems aremore complex than flowshop scheduling problems. Hybridflowshop scheduling problems were proved to be NP-hardin [3] and [4]. Due to their complexity, hybrid flowshopscheduling problems cannot be solved by exact algorithms.Hence, researchers have developed many heuristics and meta-heuristics. Rajendran and Chaudhuri [5] have proposed someheuristics to minimize total flow time for multistage parallelprocessor flowshop problems. Different heuristics have beendeveloped by researchers for different objective functions [6]–[8]. Neron et al. [9] applied energetic reasoning and globaloperations for enhancing the efficiency of branch and boundalgorithm to minimize the makespan for HFS. They tested thealgorithm 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; acceptedDecember 29, 2012. Date of publication January 15, 2013; date of currentversion 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, MepcoSchlenk 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, MiddlesexUniversity, 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 solvingn job 3 stage flexible flowshop scheduling problems. Theytested their methodology for randomly generated problems. Animmune algorithm was presented by Alisantoso et al. [11] tosolve the problem of scheduling a flexible printed circuit boardflowshop. Engin and Doyen [12] also proposed an artificialimmune system algorithm for the HFS problem to minimizemakespan. Different types of metaheuristics have been used tosolve hybrid flowshop scheduling problems. Yang et al. [13]have proposed a tabu search simulation optimization to solveflowshop scheduling problems with multiple processors. Tangand Wang [14] also applied the tabu search algorithm to solvethe 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] havecompared three different metaheuristics algorithms, namely,genetic algorithm (GA), tabu search, and SA, to minimize theconvex sum of makespan and the number of tardy jobs forflexible flowshop problems with unrelated parallel machines.Recently, Ruiz and Vazquez-Rodriguez [29] provided a reviewof the HFS problem. The different types of hybrid flowshopscheduling, their complexity, the different algorithms, and theobjective 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 onthe firefly algorithm for constrained continuous optimizationproblems. Yang [33] applied the firefly algorithm for theoptimization of pressure vessel design. He also presented afew new test functions to validate the firefly optimizationalgorithm. Sayadia et al. [34] presented a discrete fireflyalgorithm to minimize makespan for flowshop schedulingproblems. Chai-ead et al. [35] have proposed bees and fire-fly algorithms to find optimal solutions of noisy nonlinearcontinuous mathematical models. Banati and Bajaj [36] havepresented a new feature selection approach by combining therough set theory with the firefly algorithm. Apostolopoulosand Vlachos [37] have applied the firefly algorithm to solveeconomic emission load dispatching problems to minimizefuel cost and emission generating units. Basu and Mahanti [38]proposed firefly algorithm and artificial bee colony algorithmsfor the antenna design. Gandomi et al. [39] applied thefirefly algorithm to solve mixed continuous/discrete structuraloptimization problems. Kazemzadeh Azad and KazemzadehAzad [40] also proposed an improved firefly algorithm to solvestructural optimization problems. A discrete firefly algorithmwas proposed by Jati and Suyanto [41] to solve the travellingsalesman problem. Recently, Khadwilard et al. [42] havesolved the job shop scheduling problems using the firefly

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302 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 18, NO. 2, APRIL 2014

algorithm. They have also investigated the different parametersfor the proposed algorithm and compared the performancewith different parameters. Yang et al. [43] have applied thefirefly algorithm to solve economic load dispatching problems.

Reviewing the literature, it may be easily concluded thatthe applications of the firefly algorithm to the combinatorialoptimization problems are very limited. Hence, in this lettera discrete firefly algorithm is proposed to minimize makespanfor M-stage hybrid flowshop scheduling problems with theobjective function of makespan and mean flow time. Therest of the letter is organized as follows. Section II providesthe problem formulation. The firefly algorithm is presentedin Section III. The proposed discrete firefly algorithm isdiscussed in Section IV. The computational experiments andthe 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. Themachines may be identical, uniform, or unrelated. Some stagesmay have only one machine. But, at least one stage must havemultiple machines. Each job is processed by one machinein each stage. The layout of the M-stage hybrid flowshopscheduling environment is given in Fig. 1.

HFS problems are commonly found in many real-worldindustries and they are the most important and difficult NP-hard problems. Hence, in this letter we consider the M-stagehybrid flowshop scheduling problem. The objective functionconsidered in this letter is the minimization of weighted sumof makespan and mean flow time. Makespan is defined as thecompletion time of the last job to leave the system. Makespanis important for effective utilization of resources. Mean flowtime is the average time spent by the job in the system.Mean flow time is important to minimize the work-in-processinventories [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 allj = 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 =

∑nj=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 thecorresponding 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-inspiredmetaheuristic algorithm. The Firefly algorithm is inspired bythe social behavior of fireflies. Fireflies may also be calledlightning bugs. There are about 2000 firefly species in theworld. Most of the firefly species produce short and rhythmicflashes. The pattern of flashes is unique for a particular species.A firefly’s flash mainly acts as a signal to attract matingpartners and potential prey. Flashes also serve as a protectivewarning mechanism. The following three idealized rules areconsidered in [30] to describe the firefly algorithm.

1) All fireflies are unisex so that one firefly will be attractedto other fireflies regardless of their sex.

2) Attractiveness is proportional to their brightness; thus,for any two flashing fireflies, the less bright one willmove toward the brighter one. The attractiveness isproportional to the brightness and they both decreaseas their distance increases. If there is no brighter onethan a particular firefly, it will move randomly.

3) The brightness of a firefly is affected or determined bythe landscape of the objective function. For a maximiza-tion problem, the brightness may be proportional to theobjective function value. For the minimization problem,the brightness may be the reciprocal of the objectivefunction value. The pseudocode of the firefly algorithmwas given by Yang [30]. The pseudocode of the fireflyalgorithm is given in Algorithm 1.

A. Attractiveness

The attractiveness of a firefly is determined by its lightintensity. The attractiveness may be calculated by using theequation

β(r) = β0e−γr2

. (13)

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 firefliesfor j = 1 : i all n firefliesif (Ij > Ii), Move firefly i toward j in d-dimension; endifAttractiveness varies with distance r via exp (−γr)Evaluate new solutions and update light intensityend for j

end for i

Rank the fireflies and find the current bestend whilePostprocess results and visualization

B. Distance

The distance between any two fireflies k and l at Xk andXl 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 moreattractive 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 forsolving 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 thesmallest position value (SPV) rule described by Bean [44]to enable the continuous firefly algorithm to be applied todiscrete HFS scheduling problems. For this, a discrete fireflyalgorithm (DFA) is proposed. The SPV rule has already beenapplied by the researchers to solve the scheduling problems[45].

A. Implementation of the DFA for HFS Problems

1) Solution Representation: Solution representation is oneof the most important issues in designing a DFA. The solutionsearch space consists of n dimensions as n number of jobsare considered in this letter. Each dimension represents a job.The vector Xt

i = (Xti1, Xt

i2,. . . ,Xtin) represents the continuous

position values of fireflies in the search space. The SPV rule isused to convert the continuous position values of the firefliesto the discrete job permutation. The solution representation ofa 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 j1 2 3 4 5 6

xij 0.81 0.90 0.12 0.09 0.71 0.63Jobs 5 6 2 1 4 3

The smallest position value is xti4 = 0.09, and the dimension

j = 4 is assigned to be the first job in the permutationaccording to the SPV rule. The second smallest position valueis xt

i3 = 0.12, and the dimension j = 3 is assigned to bethe second job in the permutation. Similarly, all the jobs areassigned in the permutation.

2) Population Initialization: In most of the metaheuristics,the initial population is generated at random. In the DFA, theinitial population is also generated at random. The continuousvalues of positions are generated randomly using a uniformrandom number between 0 and 1.

3) Solution Updation: By using the permutation, eachfirefly is evaluated to determine the objective function value.The objective function value of each firefly is associated withthe light intensity of the corresponding firefly. A firefly withless brightness is attracted and moved to a firefly with morebrightness. The attractiveness of the firefly is determined using(13). The distance between each pair of fireflies is determinedby (14). The SPV rule is applied to obtain the job permutation.The attractiveness is calculated for each firefly. Then, themovement of the firefly is determined by (15) depending onthe attractiveness of the firefly. The above steps are repeateduntil 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 fireflyalgorithm was coded in C++ and run on a PC with an IntelCore Duo 2.4 GHz CPU, 2 GB RAM, running Windows XP.The performance of the proposed algorithm is evaluated by themean 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 thefirst test instance, we consider a case study of a real industrialscheduling problem within a steel furniture manufacturingcompany. In the second test, we develop some randominstances.

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 DrillingJobs ↓ Punching Bending Welding pressing stage

stage stage stage stage

1 40 30 0 0 02 60 70 110 0 03 10 20 50 0 04 20 0 0 0 05 30 40 0 0 06 30 70 70 0 07 40 0 0 0 08 40 0 0 50 09 30 250 40 0 010 10 20 25 0 011 0 0 0 10 012 0 0 0 20 013 50 90 0 0 014 0 0 0 20 015 0 0 0 40 016 10 20 0 0 3017 0 0 0 10 018 0 0 0 10 019 0 0 0 10 020 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 collectedfrom a leading steel furniture manufacturing company inChennai, Tamil Nadu, India. The company produces a varietyof steel furniture components. From among them, the two-drawer vertical filing cabinet is considered in this letter. Eachtwo-drawer vertical filing cabinet consists of 20 parts. Eachpart may be considered a different job. The jobs are processedin five different stages: punching, bending, welding, powerpressing, and drilling. Each stage uses a different numberof machines. The number of machines in punching, bending,welding, power pressing, and drilling stages are 5, 8, 3, 5, and1, respectively. The processing times for the different jobs atdifferent stages are given in Table II. The jobs are producedin lots and the lot size is 120. In the company, the schedulingis done manually. For the case study, the values of w1 andw2 are assumed to be 0.5 and 0.5, respectively. The proposedDFA is tested with the data from the case study company. ThePareto optimal solution obtained by the DFA is presented inFig. 2.

2) Random instances: In order to test the performance ofthe proposed DFA, experiments were conducted on random

TABLE III

Factor Levels for Random Instances

Sl. No. Factors Levels1 Number of stages 2, 5, and 102 Number of machines at each stage 2, 3, and 53 Number of jobs 10, 20, and 504 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 γ0.5 (low)0.75(medium)1.0 (high)

3 Randomization parameter α0.0 (low)0.5 (medium)1.0 (high)

TABLE V

MRDI Comparison of Different Algorithms for the

Test Problems

Sl. No. AlgorithmsMRDI

Makespan Mean flow time1 DFA 0.0 0.02 GA 5.26 5.383 PGA 4.68 4.824 ACO 7.26 7.425 SA 9.14 9.30

instances. Table III gives the factor levels for the design ofexperiments to define the production systems: the number ofstages, number of machines, number of jobs, and processingtimes of the jobs.

B. Parameter Setting

The proposed algorithm is tested with different types ofparameter settings. The attractiveness of fireflies, light absorp-tion coefficient, and randomization parameter are the importantparameters. The parameters used in this letter are presented inTable IV.

C. Computational Results

The performance of the DFA for the random problems iscompared with the parallel genetic algorithm (PGA) [18],GA [20], ACO [22], and [27]. The results are presented inTable V. From Table V, it is concluded that the proposedDFA outperforms the earlier reported literature results. Itis observed that the DFA provides 5.26% improvementwith respect to GA and 4.68% with respect to PGA forthe makespan criterion. The DFA also provides 7.26%improvement with respect to ACO and 9.14% with respectto 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 flowshopscheduling problem with the objective of minimizing the sum

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 algorithmwas presented to solve these problems. The algorithm is testedwith a real-world case study and also with random probleminstances. The computational results show that the proposedalgorithm outperforms the GA, ACO, and SA.

Further studies on extensive parametric study and compar-ison may be fruitful. It may also be an important applicationto use the proposed algorithm to solve the HFS problemswith unrelated or nonuniform machines in each stage. Thealgorithm may also be applied to HFS problems with due date-related performance measures.

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