Abstract
The set-union knapsack problem (SUKP) is a variation of the 0–1 knapsack problem (KP) in which each item is a set of elements, each item has a nonnegative value, and each element has a nonnegative weight. The weight of one item is given by the total weight of the elements in the union of the items’ sets. The SUKP accommodates a number of real-life applications and is more complicated and computationally difficult than the 0–1 KP. In this paper, we propose a novel hybrid Jaya algorithm with double coding (DHJaya) to solve the SUKP. In the DHJaya, double coding is used to represent the individual, which includes the solutions for solving the SUKP by adopting a mapping function. The Jaya algorithm and differential evolution algorithm are combined to improve the exploration ability. To enhance the exploitation ability, the Cauchy mutation is performed on some individuals. Meanwhile, an improved repairing and optimization algorithm (MS-GROA) is proposed to repair the infeasible solutions and optimize the feasible solutions. We test the DHJaya using three sets of SUKP instances to demonstrate its efficiency, and the obtained results are compared with those in the previous study. Extensive experiments show a remarkable performance of the proposed approach.
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References
Ali M, Pant M (2011) Improving the performance of differential evolution algorithm using Cauchy mutation. Soft Comput 15(5):991–1007
Altman A (2004) Minimization of tool switches for a flexible manufacturing machine with slot assignment of different tool sizes. IIE Trans 36(2):95–110
Arulselvan A (2014) A note on the set union knapsack problem. Discrete Appl Math 169:214–218
Balakrishnan N, Nevzorov VB (2003) A primer on statistical distributions. Wiley, Hoboken
Cormen T, Charles E, Rivest R, Stein Clifford (2009) Introduction to algorithms, 3rd edn. The MIT Press, Cambridge
Crama Y (1997) Combinatorial optimization models for production scheduling in automated manufacturing systems. Eur J Oper Res 99(1):136–153
Engelbrecht AP, Pampara G (2008) Binary differential evolution strategies. In: IEEE congress on evolutionary computation, 2007. CEC’07, pp 1942–1947
Feller W (1971) An introduction to probability theory and its applications, vol 2. Wiley, Hoboken
Feng Y, Wang G, Li W, Li N (2017) Multi-strategy monarch butterfly optimization algorithm for discounted 0–1 knapsack problem. Neural Comput Appl 2017:1–18
Goldschmidt O, Nehme D, Yu G (1994) On the set-union knapsack problem. Naval Res Logist 41(6):833–842
Graaf JMD, Kok JN, Kosters WA (2001) Theory of genetic algorithms. Current Trends Theor Comput Sci 259:1–61
He YC, Xie H, Wong T, Wang X (2017) A novel binary artificial bee colony algorithm for the set-union knapsack problem. Future Gener Comput Syst 78:77. https://doi.org/10.1016/j.future.2017.05.044
He YC, Wang XZ, Zhao XL, Zhang XL (2018) The design and applications of discrete evolutionary algorithm based on encoding transformation. J Softw 28(9):2580–2594
Hirabayashi R, Suzuki H, Tsuchiya N (1984) Optimal tool module design problem for nc machine tools. J Oper Res Soc Jpn 27(3):205–229
Kellerer H, Pferschy U, Pisinger D (2004) Knapsack problems. Springer, Berlin
Khuller S, Moss A, Naor J (1999) The budgeted maximum coverage problem. Inf Process Lett 70(1):39–45
Lan K, Lan C (2008) Notes on the distinction of Gaussian and Cauchy mutations. In: Eighth international conference on intelligent systems design and applications, vol 1, pp 272–277
Li C, Mao Y, Zhou J et al (2017a) Design of a fuzzy-PID controller for a nonlinear hydraulic turbine governing system by using a novel gravitational search algorithm based on Cauchy mutation and mass weighting. Appl Soft Comput 52:290–305
Li C, Zhang N et al (2017b) Design of a fractional-order PID controller for a pumped storage unit using a gravitational search algorithm based on the Cauchy and Gaussian mutation. Inf Sci 396:162–181
Liu Y, Yao X (2002) How to control search step size in fast evolutionary programming. In Proceeding of the IEEE congress on evolutionary computation (CEC 2002), vol 1, pp 652–656
Moradi M, Foroutan V, Abedini M (2017) Power flow analysis in islanded micro-grids via modeling different operational modes of DGs: a review and a new approach. Renew Sustain Energy Rev 69:248–262
Navathe S, Ceri S, Wiederhold G, Dou J (1984) Vertical partitioning algorithms for database design. ACM Trans Database Syst 9(1984):680–710
Ozsoydan F B, Baykasoglu A (2018) A swarm intelligence-based algorithm for the set-union knapsack problem. Future Gener Comput Syst. https://doi.org/10.1016/j.future.2018.08.002
Rao RV (2016) Jaya: a simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Int J Ind Eng Comput 7(1):19–34
Rao R, Savsani V, Vakharia D (2011) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315
Rao R, Rai D, Balic J (eds) (2016) Surface grinding process optimization using Jaya Algorithm. Computational intelligence in data mining—Volume 2, vol 411. Advances in intelligent systems and computing. Springer, New Delhi
Sriyanyong P (2008) Solving economic dispatch using Particle Swarm Optimization combined with Gaussian mutation. In: International conference on electrical engineering/electronics, computer, telecommunications and information technology, Krabi, Thailand, 14–17 May 2008
Storn R, Price K (1997) Differential evolution: a simple and efficient heuristic for global optimization over continuous spaces. J Global Optim 11(4):341–359
Wang H, Li H, Liu Y, Li C (2007) Opposition-based particle swarm algorithm with Cauchy mutation. In: Proceedings of the IEEE congress on evolutionary computation (CEC 2007), pp 25–28, Singapore, September 2007
Warid W, Hizam H, Mariun N, Abdul-Wahab NI (2016) Optimal power flow using the Jaya algorithm. Energies 9(9):678
Wolpert DH, Macready WG (1997) No free Lunch Theorems for optimization. IEEE Trans Evol Comput 1(1):67–82
Wu Q, Law R (2011) Cauchy mutation based on objective variable of Gaussian particle swarm optimization for parameters selection of SVM. Expert Syst Appl 38(6):6405–6411
Yao X, Liu Y, Lin G (2002) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102
Zamli K, Din F, Kendall G, Ahmed BS (2017) An experimental study of hyper-heuristic selection and acceptance mechanism for combinatorial t-way test suite generation. Inf Sci 399:121–153
Zhang Y, Yang X, Cattani C, Rao V et al (2016) Tea category identification using a novel fractional Fourier entropy and Jaya Algorithm. Entropy 18(3):77
Acknowledgements
We are especially grateful to Dr. Xiangyun Gao and Dr. Bowen Sun for their help and support during article writing process. This research is supported by Scientific Research Project Program of Colleges and Universities in Hebei Province (Grant No. ZD2016005), Natural Science Foundation of Hebei Province (Grant No. F2016403055).
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Wu, C., He, Y. Solving the set-union knapsack problem by a novel hybrid Jaya algorithm. Soft Comput 24, 1883–1902 (2020). https://doi.org/10.1007/s00500-019-04021-3
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DOI: https://doi.org/10.1007/s00500-019-04021-3