Abstract
The discounted {0–1} knapsack problem (D{0–1}KP) is a kind of knapsack problem with group structure and discount relationships among items. It is more challenging than the classical 0–1 knapsack problem. A more effective hybrid algorithm, the discrete hybrid teaching-learning-based optimization algorithm (HTLBO), is proposed to solve D{0–1}KP in this paper. HTLBO is based on the framework of the teaching-learning-based optimization (TLBO) algorithm. A two-tuple consisting of a quaternary vector and a real vector is used to represent an individual in HTLBO and that allows TLBO to effectively solve discrete optimization problems. We enhanced the optimization ability of HTLBO from three aspects. The learning strategy in the Learner phase is modified to extend the exploration capability of HTLBO. Inspired by the human learning process, self-learning factors are incorporated into the Teacher and Learner phases, which balances the exploitation and exploration of the algorithm. Two types of crossover operators are designed to enhance the global search capability of HTLBO. Finally, we conducted extensive experiments on eight sets of 80 instances using our proposed approach. The experiment results show that the new algorithm has higher accuracy and better stability than do previous methods. Overall, HTLBO is an excellent approach for solving the D{0–1}KP.
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B G (2007) Heuristic and exact algorithms for discounted knapsack problems. University of Erlangen-Nurnberg, Germany, Master thesis
Rong A, Figueira JR, Klamroth K (2012) Dynamic programming based algorithms for the discounted {0–1} knapsack problem. Applied Mathematics & Computation 218(12):6921–6933
He YC, Wang XZ, He YL, Zhao SL, Li WB (2016) Exact and approximate algorithms for discounted {0-1} knapsack problem. Inf Sci 369:634–647
He YC, Wang XZ, Li WB et al (2016) Research on genetic algorithm for discounted {0-1} knapsack problem. Chinese Journal of Computers 39(12)
Feng Y, Wang GG, Li W, Li N (2017) Multi-strategy monarch butterfly optimization algorithm for discounted {0-1} knapsack problem. Neural Comput Applic:1–18
Feng YH, Wang GG (2018) Binary moth search algorithm for discounted {0-1} knapsack problem. IEEE Access 6(99):10708–10719
Zhu H, He Y, Wang X, Eric C.C. Tsang (2017) Discrete differential evolutions for the discounted {0-1} knapsack problem. International Journal of Bio-Inspired Computation 10(4):219
Rao RV, Savsani VJ, Vakharia DP (2011) Teaching-learning-based optimization: a novel method for constrained mechanical design optimization problems. Comput Aided Des 43(3):303–315
Rao RV, Savsani VJ, Balic J (2012) Teaching-learning-based optimization algorithm for unconstrained and constrained real-parameter optimization problems. Eng Optim 44(12):1447–1462
Rao RV (2012) Teaching-learning-based optimization:a novel optimization method for continuous non-linear large scale problems. Inf Sci 183(1):15
Rao RV, Patel V (2012) An elitist teaching-learning-based optimization algorithm for solving complex constrained optimization problems. Int J Ind Eng Comput 3(4):535–560
Tang Q, Li Z, Zhang LP et al (2017) Balancing stochastic two-sided assembly line with multiple constraints using hybrid teaching-learning-based optimization algorithm[J]. Comput Oper Res 82:102–113
Kumar Y, Singh P K (2018) A chaotic teaching learning based optimization algorithm for clustering problems[J]. Appl Intell, 2018
Rao RV, Savsani VJ (2012) Mechanical design optimization using advanced optimization techniques. Springer London
Rao RV, Patel V (2013) Multi-objective optimization of heat exchangers using a modified teaching-learning-based optimization algorithm. Appl Math Model 37(3):1147–1162
Rao RV, Patel V (2013) Multi-objective optimization of two stage thermoelectric cooler using a modified teaching–learning-based optimization algorithm. Eng Appl Artif Intell 26(1):430–445
Toğan V (2012) Design of planar steel frames using teaching–learning based optimization. Eng Struct 34(1):225–232
Ji X, Ye H, Zhou J, Yin Y, Shen X (2017) An improved teaching-learning-based optimization algorithm and its application to a combinatorial optimization problem in foundry industry. Appl Soft Comput 57:504–516
El Ghazi A (2017) Ahiod B (2017) energy efficient teaching-learning-based optimization for the discrete routing problem in wireless sensor networks[J]. Appl Intell
Li L, Weng W, Fujimura S (2017) An improved teaching-learning-based optimization algorithm to solve job shop scheduling problems. Ieee/acis International Conference on Computer and Information Science 2017:797–801
Gunji AB, Deepak BBBVL, Bahubalendruni CMVAR, Biswal DBB (2018) An optimal robotic assembly sequence planning by assembly subsets detection method using teaching learning-based optimization algorithm. IEEE Transactions on Automation Science & Engineering PP 99:1–17
Chen X, Mei C, Xu B, Yu K, Huang X (2018) Quadratic interpolation based teaching-learning-based optimization for chemical dynamic system optimization. Knowl-Based Syst
Rao RV, Rai DP (2016) Optimization of fused deposition modeling process using teaching-learning-based optimization algorithm. Engineering Science & Technology An International Journal 19(1):587–603
Yu K, Lyndon W, Reynolds M, Wang X, Liang JJ (2018) Multiobjective optimization of ethylene cracking furnace system using self-adaptive multiobjective teaching-learning-based optimization. Energy:148
Storn R, Price K (1997) Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359
Eberhart R (1995) A new optimizer using particle swarm theory. Procsixth Intlsympmicro Machine & Human Science:39–43
Karaboga D (2005) An Idea Based on Honey Bee Swarm for Numerical Optimization, Technical Report - TR06
Karaboga D, Basturk B (2007) A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm. J Glob Optim 39(3):459–471
Wang L, Zheng XL, Wang SY (2013) A novel binary fruit fly optimization algorithm for solving the multidimensional knapsack problem. Knowl-Based Syst 48(2):17–23
Yang XS (2010) Firefly algorithm, stochastic test functions and design optimisation. International Journal of Bio-Inspired Computation 2(2):78–84(77)
He YC, Wang XZ, Zhang SL (2016) The design and applications of discrete evolutionary algorithms based on encoding transformation. Journal of Software
Tasgetiren MF, Pan QK, Suganthan PN, Chen HL (2011) A discrete artificial bee colony algorithm for the total flowtime minimization in permutation flow shops. Information Sciences An International Journal 181(16):3459–3475
Gong M, Cai Q, Chen X, Ma L (2014) Complex network clustering by multiobjective discrete particle swarm optimization based on decomposition. IEEE Trans Evol Comput 18(1):82–97
Cai Q, Gong M, Ma L, Ruan S, Yuan F, Jiao L (2015) Greedy discrete particle swarm optimization for large-scale social network clustering. Information Sciences An International Journal 316(C):503–516
He Y, Xie H, Wong TL, Wang X (2017) A novel binary artificial bee colony algorithm for the set-union knapsack problem. Futur Gener Comput Syst
Feng Y, Yang J, Wu C et al (2016) Solving 0–1 knapsack problems by chaotic monarch butterfly optimization algorithm with Gaussian mutation[J]. Memetic Computing
Chen Y, Hao JK (2016) Memetic search for the generalized quadratic multiple knapsack problem. IEEE Trans Evol Comput 20(6):908–923
Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Xiii (7):2104–2116
Wu C, He Y, Chen Y et al (2017) Mutated bat algorithm for solving discounted {0-1} knapsack problem[J]. Journal of Computer Applications (China) 37(5):1292–1299
He Y, Zhang X, Li W, Li X, Wu W, Gao S (2016) Algorithms for randomized time-varying knapsack problems. J Comb Optim 31(1):95–117
Avci M, Topaloglu S (2017) A multi-start iterated local search algorithm for the generalized quadratic multiple knapsack problem[J]. Comput Oper Res 83:54–65
Acknowledgments
We are especially grateful to Dr. Xiangyun Gao, Dr. Qingru Sun and Dr. Bowen Sun for their language help and some valuable suggestions during writing process. This work was supported by National Natural Science Foundation of China (No. 61806069).
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Wu, C., Zhao, J., Feng, Y. et al. “Solving discounted {0-1} knapsack problems by a discrete hybrid teaching-learning-based optimization algorithm”. Appl Intell 50, 1872–1888 (2020). https://doi.org/10.1007/s10489-020-01652-0
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DOI: https://doi.org/10.1007/s10489-020-01652-0