{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T20:39:20Z","timestamp":1761597560365,"version":"build-2065373602"},"reference-count":54,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2014,11,28]],"date-time":"2014-11-28T00:00:00Z","timestamp":1417132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete differential evolution (TDDE) for solving 0-1 knapsack problem. In TDDE, an enhanced selection operator inspired by the principle of the minimal free energy in thermodynamics is employed, trying to balance the conflict between the selective pressure and the diversity of population to some degree. An experimental study is conducted on twenty 0-1 knapsack test instances. The comparison results show that TDDE can gain competitive performance on the majority of the test instances.<\/jats:p>","DOI":"10.3390\/e16126263","type":"journal-article","created":{"date-parts":[[2014,11,28]],"date-time":"2014-11-28T14:11:21Z","timestamp":1417183881000},"page":"6263-6285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem"],"prefix":"10.3390","volume":"16","author":[{"given":"Zhaolu","family":"Guo","sequence":"first","affiliation":[{"name":"Institute of Medical Informatics and Engineering, School of Science, JiangXi University of Science and Technology, Ganzhou 341000, China"}]},{"given":"Xuezhi","family":"Yue","sequence":"additional","affiliation":[{"name":"Institute of Medical Informatics and Engineering, School of Science, JiangXi University of Science and Technology, Ganzhou 341000, China"}]},{"given":"Kejun","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Computer Science and Technology, Zhejiang University, Hangzhou 310027, China"}]},{"given":"Shenwen","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information Engineering, Shijiazhuang University of Economics, Shijiazhuang 050031, China"}]},{"given":"Zhijian","family":"Wu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Software Engineering, Wuhan University, Wuhan 430072, China"}]}],"member":"1968","published-online":{"date-parts":[[2014,11,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/s10479-009-0577-5","article-title":"Upper bounds for the 0-1 stochastic knapsack problem and a B&B algorithm","volume":"176","author":"Kosuch","year":"2010","journal-title":"Ann. Oper. Res"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2405","DOI":"10.1016\/j.asoc.2012.11.050","article-title":"Incorporating \u03f5-dominance in AMOSA: Application to multiobjective 0\/1 knapsack problem and clustering gene expression data","volume":"13","author":"Bandyopadhyay","year":"2013","journal-title":"Appl. Soft Comput"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"711","DOI":"10.1016\/j.asoc.2009.08.037","article-title":"Assessing solution quality of biobjective 0-1 knapsack problem using evolutionary and heuristic algorithms","volume":"10","author":"Kumar","year":"2010","journal-title":"Appl. Soft Comput"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1556","DOI":"10.1016\/j.asoc.2010.07.019","article-title":"Solving 0-1 knapsack problem by a novel global harmony search algorithm","volume":"11","author":"Zou","year":"2011","journal-title":"Appl. Soft Comput"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"11042","DOI":"10.1016\/j.amc.2012.05.001","article-title":"A modified binary particle swarm optimization for knapsack problems","volume":"218","author":"Bansal","year":"2012","journal-title":"Appl. Math. Comput"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1774","DOI":"10.1016\/j.asoc.2012.11.048","article-title":"Chemical reaction optimization with greedy strategy for the 0-1 knapsack problem","volume":"13","author":"Truong","year":"2013","journal-title":"Appl. Soft Comput"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3419","DOI":"10.3390\/e15093419","article-title":"Determination of optimal water quality monitoring points in sewer systems using entropy theory","volume":"15","author":"Lee","year":"2013","journal-title":"Entropy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1109\/TSMCB.2012.2209115","article-title":"Multiple populations for multiple objectives: A coevolutionary technique for solving multiobjective optimization problems","volume":"43","author":"Zhan","year":"2013","journal-title":"IEEE Trans. Cybern"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1080\/0305215X.2013.832237","article-title":"Flower pollination algorithm: A novel approach for multiobjective optimization","volume":"46","author":"Yang","year":"2014","journal-title":"Eng. Optim"},{"key":"ref_10","first-page":"223","article-title":"Bat algorithm is better than intermittent search strategy","volume":"22","author":"Yang","year":"2014","journal-title":"J. Mult.-Valued Log. Soft Comput"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"854","DOI":"10.3390\/e16020854","article-title":"Fast feature selection in a GPU cluster using the Delta Test","volume":"16","author":"Sovilj","year":"2014","journal-title":"Entropy"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Shi, H. (2006, January 20\u201323). Solution to 0\/1 knapsack problem based on improved ant colony algorithm. Weihai, China.","DOI":"10.1109\/ICIA.2006.305887"},{"key":"ref_13","first-page":"3489","article-title":"A GA-PSO layered encoding evolutionary approach to 0\/1 knapsack optimization","volume":"6","author":"Neoh","year":"2010","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"010308","DOI":"10.1088\/0256-307X\/27\/1\/010308","article-title":"Quantum immune clonal selection algorithm for multi-objective 0\/1 knapsack problems","volume":"27","author":"Shang","year":"2010","journal-title":"Chin. Phys. Lett"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1504\/IJBIC.2011.042260","article-title":"A novel quantum inspired cuckoo search for knapsack problems","volume":"3","author":"Layeb","year":"2011","journal-title":"Int. J. Bio-Inspired Comput"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Chiu, C.H., Yang, Y.J., and Chou, Y.H. (2011, January 12\u201316). Quantum-inspired tabu search algorithm for solving 0\/1 knapsack problems. Dublin, Ireland.","DOI":"10.1145\/2001858.2001891"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Sabet, S., Farokhi, F., and Shokouhifar, M. (2012, January 2\u20134). A novel artificial bee colony algorithm for the knapsack problem. Trabzon, Turkey.","DOI":"10.1109\/INISTA.2012.6247029"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1504\/IJBIC.2012.048063","article-title":"Solving 0-1 knapsack problems by a discrete binary version of cuckoo search algorithm","volume":"4","author":"Gherboudj","year":"2012","journal-title":"Int. J. Bio-Inspired Comput"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.cam.2013.04.004","article-title":"A hybrid quantum inspired harmony search algorithm for 0-1 optimization problems","volume":"253","author":"Layeb","year":"2013","journal-title":"J. Comput. Appl. Math"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.ins.2013.04.018","article-title":"An adaptive population multi-objective quantum-inspired evolutionary algorithm for multi-objective 0\/1 knapsack problems","volume":"243","author":"Lu","year":"2013","journal-title":"Inf. Sci"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Deng, C., Zhao, B., Yang, Y., and Zhang, H. (2013, January 23\u201325). Binary encoding differential evolution with application to combinatorial optimization problem. Yangzhou, China.","DOI":"10.1007\/978-3-642-38460-8_9"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9959","DOI":"10.1016\/j.amc.2013.04.023","article-title":"Solving 0-1 knapsack problems based on amoeboid organism algorithm","volume":"219","author":"Zhang","year":"2013","journal-title":"Appl. Math. Comput"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential evolution\u2014A simple and efficient heuristic for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"646","DOI":"10.1109\/TEVC.2006.872133","article-title":"Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems","volume":"10","author":"Brest","year":"2006","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s10462-009-9137-2","article-title":"Recent advances in differential evolution: A survey and experimental analysis","volume":"33","author":"Neri","year":"2010","journal-title":"Artif. Intell. Rev"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/TEVC.2010.2059031","article-title":"Differential evolution: A survey of the state-of-the-art","volume":"15","author":"Das","year":"2011","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"398","DOI":"10.1109\/TEVC.2008.927706","article-title":"Differential evolution algorithm with strategy adaptation for global numerical optimization","volume":"13","author":"Qin","year":"2009","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1109\/TEVC.2009.2014613","article-title":"JADE: Adaptive differential evolution with optional external archive","volume":"13","author":"Zhang","year":"2009","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1016\/j.cor.2010.06.007","article-title":"A differential evolution algorithm with self-adapting strategy and control parameters","volume":"38","author":"Pan","year":"2011","journal-title":"Comput. Oper. Res"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/TEVC.2010.2087271","article-title":"Differential evolution with composite trial vector generation strategies and control parameters","volume":"15","author":"Wang","year":"2011","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1679","DOI":"10.1016\/j.asoc.2010.04.024","article-title":"Differential evolution algorithm with ensemble of parameters and mutation strategies","volume":"11","author":"Mallipeddi","year":"2011","journal-title":"Appl. Soft Comput"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1109\/TSMCB.2012.2213808","article-title":"Gaussian bare-bones differential evolution","volume":"43","author":"Wang","year":"2013","journal-title":"IEEE Trans. Cybern"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1109\/TSMCB.2010.2056367","article-title":"Enhanced differential evolution with adaptive strategies for numerical optimization","volume":"41","author":"Gong","year":"2011","journal-title":"IEEE Trans. Syst. Man. Cybern. B"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.cor.2008.12.004","article-title":"A novel hybrid discrete differential evolution algorithm for blocking flow shop scheduling problems","volume":"37","author":"Wang","year":"2010","journal-title":"Comput. Oper. Res"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1109\/TEVC.2010.2052054","article-title":"Orthogonal learning particle swarm optimization","volume":"15","author":"Zhan","year":"2011","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1166\/sl.2013.2714","article-title":"Social emotional optimization algorithm with Gaussian distribution for optimal coverage problem","volume":"11","author":"Cui","year":"2013","journal-title":"Sens. Lett"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"5475","DOI":"10.3390\/e15125475","article-title":"Entropy diversity in multi-objective particle swarm optimization","volume":"15","author":"Pires","year":"2013","journal-title":"Entropy"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1616","DOI":"10.1016\/j.cor.2011.09.026","article-title":"Multiobjective cuckoo search for design optimization","volume":"40","author":"Yang","year":"2013","journal-title":"Comput. Oper. Res"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Guo, Z., Wu, Z., Dong, X., Zhang, K., Wang, S., and Li, Y. (2014). Component thermodynamical selection based gene expression programming for function finding. Math. Probl. Eng, 2014.","DOI":"10.1155\/2014\/915058"},{"key":"ref_40","unstructured":"Mori, N., Yoshida, J., Tamaki, H., and Nishikawa, H. (December, January 29). A thermodynamical selection rule for the genetic algorithm. Perth, Australia."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ying, W., Li, Y., Peng, S., and Wang, W. (2007, January 27\u201330). A steep thermodynamical selection rule for evolutionary algorithms. Beijing, China.","DOI":"10.1007\/978-3-540-72590-9_152"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.3724\/SP.J.1001.2008.01613","article-title":"Improving the computational efficiency of thermodynamical genetic algorithms","volume":"19","author":"Ying","year":"2008","journal-title":"Chin. J. Softw"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Yu, F., Li, Y., and Ying, W. (2010, January 12\u201315). An improved thermodynamics evolutionary algorithm based on the minimal free energy. Beijing, China.","DOI":"10.1007\/978-3-642-13495-1_66"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"526","DOI":"10.1109\/TEVC.2008.2009457","article-title":"Differential evolution using a neighborhood-based mutation operator","volume":"13","author":"Das","year":"2009","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"2044","DOI":"10.1016\/j.ins.2009.12.010","article-title":"Advanced nonparametric tests for multiple comparisons in the design of experiments in computational intelligence and data mining: Experimental analysis of power","volume":"180","author":"Luengo","year":"2010","journal-title":"Inf. Sci"},{"key":"ref_46","first-page":"2677","article-title":"An extension on Statistical comparisons of classifiers over multiple data sets for all pairwise comparisons","volume":"9","author":"Herrera","year":"2008","journal-title":"J. Mach. Learn. Res"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/j.ins.2012.10.012","article-title":"Diversity enhanced particle swarm optimization with neighborhood search","volume":"223","author":"Wang","year":"2013","journal-title":"Inf. Sci"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"4699","DOI":"10.1016\/j.ins.2011.03.016","article-title":"Enhancing particle swarm optimization using generalized opposition-based learning","volume":"181","author":"Wang","year":"2011","journal-title":"Inf. Sci"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/j.ins.2011.09.001","article-title":"Enhancing the search ability of differential evolution through orthogonal crossover","volume":"185","author":"Wang","year":"2012","journal-title":"Inf. Sci"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/4235.771163","article-title":"Evolutionary programming made faster","volume":"3","author":"Yao","year":"1999","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/4235.585893","article-title":"No free lunch theorems for optimization","volume":"1","author":"Wolpert","year":"1997","journal-title":"IEEE Trans. Evol. Comput"},{"key":"ref_52","first-page":"1","article-title":"Statistical comparisons of classifiers over multiple data sets","volume":"7","year":"2006","journal-title":"J. Mach. Learn. Res"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s10732-008-9080-4","article-title":"A study on the use of non-parametric tests for analyzing the evolutionary algorithms behaviour: A case study on the CEC2005 special session on real parameter optimization","volume":"15","author":"Molina","year":"2009","journal-title":"J. Heuristics"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/j.ins.2014.04.013","article-title":"Multi-strategy ensemble artificial bee colony algorithm","volume":"279","author":"Wang","year":"2014","journal-title":"Inf. Sci"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/12\/6263\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:09Z","timestamp":1760217009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/12\/6263"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,11,28]]},"references-count":54,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2014,12]]}},"alternative-id":["e16126263"],"URL":"https:\/\/doi.org\/10.3390\/e16126263","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,11,28]]}}}