Abstract
Nature inspired meta-heuristic algorithms are an integral part of modern optimization techniques. One such algorithm is bat algorithm which is inspired from echolocation behavior of bats and has been successfully applied to non-linear single-objective optimization problems. In this paper, a multi-objective extension of bat algorithm is proposed using the concepts of Pareto non-dominance and elitism. The novel algorithm is tested using thirty multi-objective test problems. The performance is measured using metrics namely, hyper-volume ratio, generational distance and spacing. The newly developed algorithm is then applied to a real-world multi-objective optimization problem of a phthalic anhydride reactor. It shows faster convergence for test problems as well as the industrial optimization problem than two popular nature inspired meta-heuristic algorithms, i.e. multi-objective non-dominated sorting particle swarm optimization and real-coded elitist non-dominated sorting genetic algorithm.
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Abbreviations
- A min :
-
Minimum loudness
- A max :
-
Maximum loudness
- A i :
-
Loudness of ith bat
- At :
-
Average loudness of all the bats at tth iteration
- BA:
-
Bat algorithm
- e :
-
Exponential
- f i :
-
Frequency of ith bat
- f min :
-
Minimum frequency
- f max :
-
Maximum frequency
- Ng :
-
Bats having rank 1
- Np :
-
Total number of solutions (bats) in the population
- NV :
-
Total number of decision variables
- NSPSO:
-
Non-dominated sorting PSO
- NSBAT-II:
-
Elitist non-dominated sorting BA
- r :
-
Pulse emission rate
- r i :
-
Pulse emission rate at ith bat
- r i,t :
-
Pulse emission rate at tth iteration
- RNSGA-II:
-
Real-coded NSGA-II
- t :
-
Number of iterations
- t max :
-
User-specified maximum number of iterations
- v i :
-
Current velocity of ith bat
- \(v_{i,j}^{0}\) :
-
Initial ith bat velocity
- \(v_{i,j}^{t}\) :
-
jth component of ith bat velocity at tth iteration
- w i :
-
Inertia weight for ith bat
- \(\varvec{x}_{i}\) :
-
Current position of ith bat
- \(x_{j}^{low}\) :
-
Lower value of decision variable
- \(x_{j}^{high}\) :
-
Upper of decision variable
- \(x_{g}^{best}\) :
-
Total number of solution having rank 1
- \(x_{i,j}^{0}\) :
-
jth component of ith position at 0th iteration
- \(x_{i,j}^{t}\) :
-
jth component of ith position at tth iteration
- \(\varepsilon\) :
-
Random number between −1 and 1
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Acknowledgments
The partial financial support from Science and Engineering Research Board, Department of Science and Technology, Government of India, New Delhi [through Grant SERB/F/1510/2014-2015, dated June 5, 2014] is gratefully acknowledged.
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Prakash, S., Trivedi, V. & Ramteke, M. An elitist non-dominated sorting bat algorithm NSBAT-II for multi-objective optimization of phthalic anhydride reactor. Int J Syst Assur Eng Manag 7, 299–315 (2016). https://doi.org/10.1007/s13198-016-0467-6
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DOI: https://doi.org/10.1007/s13198-016-0467-6