Application of Firefly Algorithm in Job Shop Scheduling Problem for Minimization of Makespan

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Procedia Engineering 97 ( 2014 ) 1798 – 1807

Selection and peer-review under responsibility of the Organizing Committee of GCMM 2014

doi: 10.1016/j.proeng.2014.12.333

ScienceDirect

12th GLOBAL CONGRESS ON MANUFACTURING AND MANAGEMENT, GCMM 2014

Application of Firefly Algorithm in Job Shop Scheduling Problem

for Minimization of Makespan

K.C.Udaiyakumar

a

*, M.Chandrasekaran

b a_{Research Scholar, Sathyabama University,Chennai 600119, India}

b_{Director, Vels University, Chennai 600 117, India}

Abstract

Job shop scheduling problem is a well known scheduling problem in which most of them are categorised into non polynomial deterministic (NP) hard problem because of its complexity. Many researchers intended to solve the problem by applying various optimization techniques. While using traditional methods they observed huge difficulty in solving high complex problems. Later

90’s many researchers addressed JSSP by using intelligent technique such as fuzzy logic, simulated annealing etc. After that genetic algorithm (GA), Selective breeding algorithm (SBA), taboo search algorithm and Ant colony algorithm [ are popularly known as Meta heuristic algorithms were proved most efficient algorithms to solve various JSSP so far. The objective of this paper is as follows i) to make use of a recently developed meta heuristic called Firefly algorithm (FA) because of inspiration on Firefly and its characteristic.ii) to find the makespan minimization using1-25 Lawrance problems as a bench marking from a classical OR- library.iii) the analysis of the experimental resultson Firefly algorithm is compared with other algorithms.

Selection and peer-review under responsibility of the Organizing Committee of GCMM 2014. Keywords:Jobshop scheduling problem; fire-fly; makespan; benchmark.

1.Introduction

Scheduling is considered to be a major task for shop floor productivity improvement. Scheduling is the allocation of resources by applying the limiting factors of time and cost to perform a collection of tasks. Scheduling theory is concerned primarily with mathematical models that relate to the scheduling function and development of useful models and techniques. Two kinds of feasibility constraints are commonly found in scheduling problems, The first set of constraints is related to the amount of resources available(like number of machines available in each type).The second set of constraints is based on technological restrictions on the sequences in which tasks can be performed.

* Corresponding author. Tel.: 9381889996

E-mail address:kcudaiyakumar1967@gmail.com

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The objectives of the scheduling problem are listed below.

x Determining the sequence in which tasks are to be performed. x Determining the start time and finish time are to be assessed.

In other words , the essence of scheduling is to make allocation, decisions pertaining to start and finish times of tasks. Scheduling can be classified into: Single machine scheduling ,Flow Shop scheduling and Job shop scheduling[27].

2.Job Shop Scheduling Problems

Scheduling is the allocation of resources over time to perform a collection of tasks .The job shop scheduling problem(JSP) consists of a set m machines {M1,M2,...Mn}, and a collection of n jobs {J1,J2...Jn}

to be scheduled ,where each job must pass through each machine once only .Each job has its own processing order and it has no relation to the processing order of the any other job. Job Shop Scheduling problems are Non Polynomial-hard problem, so its complexity [16] is more.

The main purpose of JSSP is commonly used to find the best machine schedule for servicing all jobs in order to optimize either single criterion/objective or multi scheduling objectives. They are also known as job shop performance measures such as the makespan minimization (Cmax) or mean flow time or the mean tardiness or earliness etc.

Scheduling problem in their static and deterministic forms are simple to describe and formulate, but are difficult to solve as it involves complex combinatorial optimisation .For example, if there are m machines, each of which is required to perform n independent operations. The combination can be potentially exploded up to (n!)m _{operational sequences .Job Shop Scheduling is one of the most famous scheduling problems, most of}

which are categorised in to NP hard problem[15,16]. This means that due to the combinatorial explosion ,even a computer can take unacceptably large amount of time to seek a satisfied solution on even moderately large scheduling problem. Another potential issue of complexity is the assembly relationship [4, 5]. 2.1 ASSUMPTIONS

The following assumptions and constraints are to be considered in solving of job shop scheduling problem such as i) all jobs are independent. ii) Job setup time is included along with the machining time. iii) Job descriptions are known in advance. iv) Machines are continuously available. v) Jobs are processed without break. vi) Machine cannot process the parallel job at a time.vii) each machine will process a job.viii) Each job requires m machines to complete the required process. ix) No Pre-emptions are allowed. The order of processing is not the same and x) Operations cannot be interrupted.

2.2 CONSTRAINTS

In general, the constraints used in job shop scheduling are i) A job does not visit the same machine more than once. ii) Each machine can process only one job at a time iii) Due date for the job.

Many optimization techniques have been widely applied to solve the JSSP. Traditional methods based on mathematical model or numerical search such as branch and bound [6, 7] and Lagrangian relaxation [8, 9] which can assure the optimum solution. These methods have been effectively and efficiently used to solve JSSP. Even though these methods are used for moderating- large problem size (10 X 10) and to solve JSSP but it may consume high computational time resources and therefore there is a computational limitation exist [19, 20, 21].

Last few decades a larger size of JSSP have been solved by an approximation optimization methods or metaheuristics (for example Tabu search [10] and simulated annealing [11] these methods usually follow stochastic steps in their iterative or search process.However these methods do not guarantee the optimum solution.

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In this problem, each job requires three operations to be processed on a pre-defined machine sequence. The first job (J1) need to be initially operated on the machine M1 for 10 time units and then sequentially processed on M2 and M3 for 9 and 8 time units, respectively. The second job (J2) has to be initially performed on M3 for 9 time units and sequentially followed by M1and M2 for 8 and 7 time units, respectivelyLikewise the third job (J3) to be performed using three machines with scheduled processing time . Our taskis to search for the best schedule(s) for operating all pre-defined jobs in order to optimise either single or multiple scheduling objectives, which are used for identifying the goodness of schedule such as the minimisation of the makespan (Cmax).

Table 1. An Example of 3 –Jobs 3-Machines Scheduling Problems With Processing Times

Job Operation Time Machine (Mk)

(Ojk) (tjk) M1 M2 M3 J1 011 10 10 - - 012 9 - 9 - 013 8 - - 8 J2 023 9 - - 9 021 8 8 - - 022 7 - 7 - J3 033 10 - - 10 031 8 8 - - 032 11 - 11 -

3.Fire Fly Algorithm

3.1 INSPIRATION

Fireflies, which belong to the family of Lampyridae, are tiny winged beetles having capability of producing light with little or no heat and it is called a cold light. It flashes the light in order to attract mates[1,25].They are whispered to have a capacitor-like mechanism, that gradually charges until the definite threshold is reached, at which they discharge the energy in the form of light ,subsequent to which the cycle repeats.

Firefly algorithm was developed by Xin-She Yang (2008).It is enthused by the light dwindling over

the distance and fireflies’ communal attraction , even though by the occurrence of the fireflies’ light

flashing. Algorithm considers what every firefly observes at the point of its position , when trying to move to a better light-source , than is his own.

When Nature inspires algorithm such as particles & warm optimization(PSO) [15,23] asfirefly algorithm are the most powerful algorithm for optimization,

3.2 ALGORITHM

The Firefly Algorithm [1] is one of the newest meta-heuristics as it idealizes some of the characteristics of the firefly behaviour. They follow the following three rules: a) all the fireflies are unisex, b) each firefly is attracted only to the fireflies ,that are brighter than itself; Strength of the attractiveness is

proportional to the firefly’s brightness ,which attenuates over the distance ; the brightest firefly moves

randomly and, c) brightness of every firefly determines is quality of solution ; in most of the cases, it can be proportional to the objective function. Using three rules, as pseudo-code of the Firefly Algorithm may look as follows:

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Algorithm 1: Basic Firefly Algorithm Pseudo-Code

Input:f(x), x = (x1, x2 ..., xd); // Objective function

n, I0,ϒ, α; // User-defined constants

Output: x min_; _{// position of minimum in objective function}

for i Å 1 to n do

xi_Å_{Initial_Solution ( );}

end

While termination requirements are not met do

minÅarg min (f (xi ₎₎ i ε{1,..,n}

for i Å 1 to ndo

for j Å 1 to ndo

if f(xi_{) <}_f₍_xj₎_then

di,jÅ Distance (xi,xj); //move xi towards xj βÅ Attractiveness (I0,ϒ,di,j); Xi _Å_(1-_β₎_Xi ₊_β_Xj₊_α_{(Random ( )}_{1); //movement} 2 end end end

xmin_Åxmin + _α_{(Random ( )}_{1); //best firefly moves randomly}

2 end

In the above algorithm , n is the number of the fireflies, I0 is the light intensity at the source, ϒ is

the absorption coefficient and α is the size of the random step . 3.3 ALGORITHM

Night is filled with darkness and the only visible light is the light produced by fireflies. The light intensity of each firefly is proportional to the quality of the solution, based on its specific location . In order to improve its own solution ,the firefly needs to advance towards the fireflies that have brighter light emission than its own . Each firefly observes decreased light intensity than the one firefly which actually emit , due to the air absorption over the distance. Light intensity reduction abides the law:

3.4 APPLICATION OF FIREFLY ALGORITHM FOR JOB SHOP SCHEDULING

1) Introduction: The objectives of this paper are as follows i) to make use of a recently developed meta heuristic algorithm called Firefly algorithm (FA) based on the inspiration of Firefly and its characteristic.ii) to find the single objective of JSSP (i.e. makespan minimization using1-25 Lawrance problems as a bench marking from a classical OR- library).iii) the analysis of the experimental results on Firefly algorithm is compared with other algorithms with best known solution. The computational experiment was designed and conducted using twenty five benchmarking datasets of the JSSP instance from the OR-Library published by Beasley[13].

2) Firefly evaluation:The next stage is to measure the flashing light intensity of the firefly, which depends on the problem considered. In this work, the evaluation on the goodness of the schedules is measured by the makespan, which can be calculated using equation [1], where Ck is completed time of job k.

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3) Distance: The distance between any two fireflies i and j at Xi and Xj, respectively ,can be defined as

Cartesian distance (rij)) using equation(2), where Xi,kis the component of the spatial coordinate xi of the ith

firefly and d is the number of dimensions [12,22].

rij = xi-xj = d

Ʃ( Xi,k -Xj,k)2 ---(2) k=1

4) Attractiveness:The calculation of attractiveness function of a firefly are shown in equation (3) , where

r is the distance between any two fireflies,β0is the initial attractiveness r=0, and γ is an absorption

coefficient which controls the decrease of the light intensity [12,22]. β(r)=β0*exp(-γγm),with m >1 ---(3)

5) Movement: The movement of a firefly i which is attracted by a more attractive (i.e., brighter) firefly j is given by the following equation(4),where Xiis the current position of a firefly m the β(r) =β0*exp(-γγij 2) *(xj-xi) is attractiveness of a firefly is seen by adjacent fireflies .The α(rand-1/2) is a firefly’s random movement .The coefficient α is a interest with αɛ [0 to 1], while rand is random number obtained from the uniform distribution in the space[0,1]

These recently developed algorithms have been applied by few researchers for solving optimization problems, majority of them have been formulated into mathematical equations. In this work, the settings of FFA parameter such as number of fireflies (n), number of generations/iterations (G) the light absorption coefficient (γ), randomization parameter (α) and attractiveness value (β0) have to be chosen in an adhoc fashion. Generally the

combination factor (nG) determines the amount of search in the solution space conducted by this algorithm. This factor is directly related to the size of the problem considered. If high value of combination is considered then it helps the probability of achieving best solution however it involves longer computational time and resources. In this research, the acceptable computational limitations are practically implemented, therefore the combination factor was fixed at 1000 in order to accommodate computational search within the time limit. The light absorption coefficient

(γ) was varied from 0 to 10, the randomised parameter was usually set between 0 to 1 and the attractiveness function

was also chosen between 0 to 1 [12, 22].

Table 2.Comparison of FFA parameters’ setting used in previous Researchers.

Authors Problems FA parameters

nG Γ α β0

Apostolopoulos and Vlachos[33]

Economic emissions load dispatch _12*50 _1.0 _0.2 1.0

Xin-She Yang [31] Numerical studies 40*5 1.0 0.2 1.0

Lukasik and Zak [32] Continuous equation _40*250 _1.0 _0.01 _1.0 Aphirak Khadwilard1 et all [30] _{Job Shop Scheduling} _100*25 _0.1 _0.5 _1.0

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4.Experimental Design And Analysis

To formulate optimization problems we need to determine the parameters that can be decision/design variables. The objective function of minimization of make span (eqn-1) of any JSSP is not so easy because these are discrete optimization problems and there is no traditional method used to solve these problems and also are time consuming. To identify parameters of objective function we wish to use firefly algorithm. For this purpose in advance we have

to find out the parameters such as combination factor [nG], light absorption co efficient[γ], randomization parameter[α], attractiveness value [β0], m values by sensitive analysis [24] and by taking consideration above stated

authors mentioned in Table 2 for their problems. Firstly to test LA1-25 instances with best known solutions (Benchmark datasets) and the obtained datasets are compared with other researcher’s experimental research datasets

(obtained by using of other algorithms). SA is carried out to preset five FFA parameters the average of 25 iteration have been taken by considering different values of all parameter within the range mentioned. For that we need to run six lakhs time (5!*25*1000= 6 lakhs time, memory reset etc, are considered),in our program we can change number of flies and number of iteration as well to preset the parameters, if the FF and iteration are increased resulting in large computational and time consuming. We attempted to increase combination nG greater than 1000 but there is no appreciable change in the result. Many research works can be possible by varying parameter values. In order to solve the optimization problem we have to accomplish it in Matlab under Windows XP operating system,the following parameter used in solving JSSPare α = 0.05, βo= 0.02, γ =0.0001, m=1,number of fireflies is 10 and

maximum generation of fireflies is 100 hencetotal no of functional evolution is 1000. The results computational experiments for LA 01 to LA 25 are shown in Table 3 and Fig.1& 2.The results are also compared with other algorithm results [28, 29] for the same benchmark problem as shown in Table 4 and Fig 3.

Table 3.Comparisons of FEA Lawrance Bench Mark Data Base. Lawrance Problem (LA 01-25)

Law Problem n m _BenchmarkMakespan Makespan Actual _{Mean Error}Percentage

1 10 5 666 666 0 2 10 5 655 658 0.4580153 3 10 5 597 597 0 4 10 5 590 604 2.3728814 5 10 5 593 593 0 6 15 5 926 926 0 7 15 5 890 890 0 8 15 5 863 863 0 9 15 5 951 951 0 10 15 5 958 958 0 11 20 5 1222 1222 0 12 20 5 1039 1039 0 13 20 5 1150 1150 0 14 20 5 1292 1295 0.2321981 15 20 5 1207 1207 0 16 10 10 945 945 0 17 10 10 784 784 0 18 10 10 848 848 0 19 10 10 842 842 0 20 10 10 902 902 0 21 15 10 1046 1046 0 22 15 10 927 929 0.2157497 23 15 10 1032 1032 0 24 15 10 935 935 0 25 15 10 977 977 0

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Fig.1.Percentage Mean Error (La 01-25)

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Table 4.Comparisons of Lawrance Bench Mark with Other Algorithm

Law Problem

Makespan

BENCHMARK FIREFLY SFHM SBA

1 666 666 666 666 2 655 658 658 655 3 597 597 597 597 4 590 604 590 590 5 593 593 593 593 6 926 926 926 926 7 890 890 890 890 8 863 863 863 863 9 951 951 951 951 10 958 958 958 958 11 1222 1222 1222 1222 12 1039 1039 1039 1039 13 1150 1150 1150 1150 14 1292 1295 1292 1292 15 1207 1207 1207 1207 16 945 945 945 945 17 784 784 784 784 18 848 848 848 848 19 842 842 842 842 20 902 902 902 902 21 1046 1046 1046 1046 22 927 929 927 927 23 1032 1032 1032 1032 24 935 935 935 935 25 977 977 977 977

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Fig. 3. Comparison of Makespan with other Algorithm

5.Conclusion

The firefly algorithm is one of the best method for visualizing problems related to Job Shop Scheduling and provide best possible optimization. It is one of the simplest method and easy to apply any NP hard problem. This

algorithm was applied to find the minimization of makespan (Cmax) of 25 benchmarking JSSP datasets taken

from the OR Library. The parameters of FFA can be assigned as such as the absorption coefficient[γ] , the

population of fireflies[n],size of random step[α],attractiveness value[β0] and the number of iterations which

depends upon the optimized problem. Experimental design and sensitive analysis were carried out to find out the appropriate parameter settings of FFA. Out of 25 problems we found that 21problemswere exactly and successfully matched with the best known solution (actual bench mark) and three problems were nearly optimal with 96% confidence level. Our present goal is to test the tools using FFA. There is a vivid scope for further research in bench marking problems and as well as solving multi objective JSSP.

References

[1] X. S. Yang, Nature-Inspired Metaheuiristic Algorithms, Luniver Press, London, 2008.

[2] X. S. Yang, Firefly algorithms for multimodal optimization, Stochastic Algorithms: Foundations and Applications, SAGA, Lecture Notes in Computer Sciences5792 (2009), pp 169–178.

[3] M.K. Sayadi, R. Ramezanian, N. Ghaffari-Nasab, A discrete firefly meta-heuristic with local search for makespan minimization in permutation flow shop scheduling problems, Int.J. Industrial Eng. Computations1 (2010) 1–10.

[4] P. Pongcharoen, C. Hicks, P. M. Braiden, D. J. Stewardson, Determining optimum Genetic Algorithm parameters for scheduling the manufacturing and assembly of complex products, International Journal of Production Economics78 (2002) 311-322.

[5] P. Pongcharoen, C. Hicks,P. M. Braiden, The development of genetic algorithms for the finite capacity scheduling of complex products, with multiple levels of product structure, European Journal of Operational Research 152 ( 2004) 215-225.

[6] C. Artigues, D. Feillet, A branch and bound method for the job-shop problem with sequence dependent setup times, Annals of Operations Research 159 (2007) 135-159.

[7] C. Artigues, M.-J.Huguetand, P. Lopez, Generalized disjunctive constraint propagation for solving the job shop problem with time lags, Engineering Applications of Artificial Intelligence 24 ( 2007) 220-231.

[8] P. Baptiste, M. Flamini, F. Sourd, Lagrangian bounds for just-in-time job-shop scheduling, Computers & Operations Research, 35 (2008) 906-915.

[9] H. X. Chen, P. B. Luh, An alternative framework to Lagrangian relaxation approach for job shop scheduling, European Journal of Operational Research149 ( 2003) 499-512.

[10 H. Gröflin, A. Klinkert, A new neighbourhood and tabu search for the Blocking Job Shop, Discrete Applied Mathematics 157(2009) 3643-3655.

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[11] R. Zhang, C. Wu, A simulated annealing algorithm based on block properties for the job shop scheduling problem with total weighted tardiness objective, Computers & operations Research 38 ( 2011) 854-867.

[12] X.-S. Yang, Firefly Algorithms for multimodal optimization, Lecture Notes in Computer Science, 5792 (2009) pp. 169-178.

[13] J. E. Beasley, OR-library: distributing test problems by electronic mail, Journal of the Operational Research Society, 41 (1990) 1069-1072. [14] G. Moslehi, M. Mahnam, A Pareto approach to multi objective flexible job-shop scheduling problem using particle swarm optimization and

local search International Journal of Production Economics129 (2011) 14-22.

[15] X.-J. Wang, C.-Y. Zhang, L. Gao, P.-G. Li, A survey and future trend of study on multi-objective scheduling, Proceedings of the 4th International Conference on Natural Computation 2008, pp. 382-391.

[16] L. Asadzadeh, K. Zamanifar, An agent-based parallel approach for the job shop scheduling problem with genetic algorithms, Mathematical and Computer Modelling 52 (2010) 1957-1965.

[17] K. Ripon, C.-H. Tsang, S. Kwong, An Evolutionary approach for solving the multi-objective job-shop scheduling problem, Studies in Computational Intelligence 49 (2007) pp.165-195.

[18] A.K. Gupta, A.I.Sivakumar, Job shop scheduling techniques in semiconductor manufacturing, International, Journal of Advanced Manufacturing Technology 27 (2006) 1163-1169.

[19] S. Yang, D. Wang, T. Chai, G. Kendall, An improved constraint satisfaction adaptive neural network for job-shop scheduling, Journal of Scheduling 13 (2010) 17-38.

[20] S. X. Yang, D. W. Wang, A new adaptive neural network and heuristics hybrid approach for job-shop scheduling, Computers & Operations Research, 28 (2001) 955-971.

[21] O. Bilkay, O. Anlagan, S. E. Kilic, Job shop scheduling using fuzzy logic, International Journal of Advanced Manufacturing Technology 23 ( 2004) .

[22] X.S. Yang.Nature-Inspired Metaheuristic Algorithms, Luniver Press, 2008

[23] T.-L. Lin, S.-J. Horng, T.-W.Kao, Y.-H.Chen, R.-S.Run, R.-J.Chen, J.-L. Lai, I. H. Kuo, An efficient job-shop scheduling algorithm based on particle swarm optimization, Expert Systems with Applications37 (2010) 2629-2636.

[24] D. C. Montgomery, Design and Analysis of Experiments, 5 ed. New York, John Wiley & Sons, 2001.

[25] Encyclopedia Britannic Firefly, http://www.britannica.com/EBchecked/topic/207935/firefly, May 2011 [26] X.S.yang, Firefly algorithm-Matlabfiles, http://www.mathworks.com/matlabcentral/fileexchange/29693-firefly-algorithm, 2011 [27] R. Pannerselvam, Production and Operation Management, PHI Learning Private Limited, New Delhi, Second Edition, 2010, pp. 312-349. [28] M.Chandirasekaran,P.Ashokan,S.Kumanan, S. Umamaheswari, Multi Objective optimization of Job shop scheduling using sheep flocks

Heredity Model Algorithm,Internation Journal of Manufacturing Science and Technology, USA, 9(12) (2007) 47-54.

[29] M.Chandirasekaran, P.Ashokan, S.Kumanan, S.Arunachalam, Application of selective breeding algorithm for solving Job Shop Scheduling Problems,Internation Journal of Manufacturing Science and Technology, USA 9( 12) (2007) 103-118.

[30] Aphirak Khadwilard1 et all, Investigation of Firefly Algorithm Parameter Setting for Solving Job Shop Scheduling Problems, Operation Research Network, Thailand 2554 pp89-97.

[31] Xin-She Yang, Firefly Algorithm, L´evy Flights and Global Optimization, Research and Development in Intelligent Systems XXVI, © Springer-Verlag London Limited, 2010

[32] S. Lukasik, S. Zak, Firefly Algorithm for continuousconstrained optimization Tasks, Lecture Notes in Computer Science, 5796 (2009) pp. 97-106.

[33] T. Apostolopoulos, A. Vlachos, Application of the FireflyAlgorithm for solving the economic emissions load dispatchproblem, International Journal of Combinatorics, ( 2011).