Abstract
To address indeterminism in the bilevel knapsack problem, an uncertain bilevel knapsack problem (UBKP) model is proposed. Then, an uncertain solution for UBKP is proposed by defining the \({\mathcal{P}_E}\) Nash equilibrium and \({\mathcal{P}_E}\) Stackelberg-Nash equilibrium. To improve the computational efficiency of the uncertain solution, an evolutionary algorithm, the improved binary wolf pack algorithm, is constructed with one rule (wolf leader regulation), two operators (invert operator and move operator), and three intelligent behaviors (scouting behavior, intelligent hunting behavior, and upgrading). The UBKP model and the \({\mathcal{P}_E}\) uncertain solution are applied to an armament transportation problem as a case study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai T, Wei J, Yang WW, et al., 2018. Multi-objective parameter estimation of improved Muskingum model by wolf pack algorithm and its application in upper Hanjiang River, China. Water, 10(10):1415. https://doi.org/10.3390/w10101415
Brotcorne L, Hanafi S, Mansi R, 2013. One-level reformulation of the bilevel knapsack problem using dynamic programming. Dis Optim, 10(1):1–10. https://doi.org/10.1016/j.disopt.2012.09.001
Caprara A, Carvalho M, Lodi A, et al., 2014. A study on the computational complexity of the bilevel knapsack problem. SIAM J Optim, 24(2):823–838. https://doi.org/10.1137/130906593
Carvalho M, Lodi A, Marcotte P, 2018. A polynomial algorithm for a continuous bilevel knapsack problem. Oper Res Lett, 46(2):185–188. https://doi.org/10.1016/j.orl.2017.12.009
Chih M, Lin CJ, Chern MS, et al., 2014. Particle swarm optimization with time-varying acceleration coefficients for the multidimensional knapsack problem. Appl Math Model, 38(4):1338–1350. https://doi.org/10.1016/j.apm.2013.08.009
Cohn AM, Barnhart C, 1998. The stochastic knapsack problem with random weights: a heuristic approach to robust transportation planning. Proc Triennial Symp on Transportation Analysis, p.1–13.
Dempe S, Richter K, 2000. Bilevel programming with knapsack constraints. Centr Eur J Oper Res, 8(2):93–107.
Gao F, Cui G, Wu ZB, et al., 2009. Virus-evolutionary particle swarm optimization algorithm for knapsack problem. J Harbin Inst Technol, 41(6):103–107 (in Chinese).
Gao YJ, Zhang FM, Zhao Y, et al., 2019. A novel quantum-inspired binary wolf pack algorithm for difficult knapsack problem. Int J Wirel Mob Comput, 16(3):222–232. https://doi.org/10.1504/IJWMC.2019.099861
Gherboudj A, Layeb A, Chikhi S, 2012. Solving 0-1 knapsack problems by a discrete binary version of cuckoo search algorithm. Int J Bio-Inspir Comput, 4(4):229–236. https://doi.org/10.1504/IJBIC.2012.048063
Han ZH, Tian XT, Ma XF, et al., 2018. Scheduling for re-entrant hybrid flowshop based on wolf pack algorithm. IOP Conf Ser Mater Sci Eng, 382(3):032007. https://doi.org/10.1088/1757-899X/382/3/032007
Li H, Zhang L, Jiao YC, 2014. Solution for integer linear bilevel programming problems using orthogonal genetic algorithm. J Syst Eng Electron, 25(3):443–451. https://doi.org/10.1109/JSEE.2014.00051
Li H, Xiao RB, Wu HS, 2016. Modelling for combat task allocation problem of aerial swarm and its solution using wolf pack algorithm. Int J Innov Comput Appl, 7(1): 50–59. https://doi.org/10.1504/IJICA.2016.075473
Liu B, 2009a. Some research problems in uncertainty theory. J Uncert Syst, 3(1):3–10.
Liu B, 2009b. Theory and Practice of Uncertain Programming. Springer Verlag, Berlin, Germany.
Liu B, 2010. Uncertainty Theory: a Branch of Mathematics for Modeling Human Uncertainty. Springer Verlag, Berlin, Germany.
Liu B, 2016. Uncertainty Theory (5th Ed.). Springer Verlag, Berlin, Germany.
Liu JQ, He YC, Gu QQ, 2007. Solving knapsack problem based on discrete particle swarm optimization. Comput Eng Des, 28(13):3189–3191, 3204 (in Chinese). https://doi.org/10.3969/j.issn.1000-7024.2007.13.052
Özaltın OY, Prokopyev OA, Schaefer AJ, 2010. The bilevel knapsack problem with stochastic right-hand sides. Oper Res Lett, 38(4):328–333. https://doi.org/10.1016/j.orl.2010.04.005
Qian J, Zheng JG, 2012. Greedy quantum-inspired evolutionary algorithm for quadratic knapsack problem. Comput Integr Manuf Syst, 18(9):2003–2011 (in Chinese).
Qiu X, Kern W, 2015. Improved approximation algorithms for a bilevel knapsack problem. Theor Comput Sci, 595(30): 120–129. https://doi.org/10.1016/j.tcs.2015.06.027
Wang R, Guo N, Xiang FH, et al., 2012. An improved quantum genetic algorithm with mutation and its application to 0-1 knapsack problem. Proc Int Conf on Measurement, Information and Control, p.484–488. https://doi.org/10.1109/MIC.2012.6273347
Wang ZT, Guo JS, Zheng MF, et al., 2015. A new approach for uncertain multiobjective programming problem based on \({\mathcal{P}_E}\) principle. J Ind Manag Optim, 11(1):13–26. https://doi.org/10.3934/jimo.2015.11.13
Wu HS, Zhang FM, Wu LS, 2013. New swarm intelligence algorithm—wolf pack algorithm. Syst Eng Electron, 35(11):2430–2438 (in Chinese).
Xue JJ, Wang Y, Li H, et al., 2016. A smart wolf pack algorithm and its convergence analysis. Contr Dec, 31(12): 2131–2139 (in Chinese). https://doi.org/10.13195/j.kzyjc.2015.1202
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Science and Technology Innovation 2030 Major Project of the Ministry of Science and Technology of China (No. 2018AAA0101200), the National Natural Science Foundation of China (No. 61502534), the Natural Science Foundation of Shaanxi Province, China (No. 2020JQ-493), and the Domain Foundation of China (No. 61400010304)
Contributors
Ren-bin XIAO designed the research. Jun-jie XUE and Jin-qiang HU processed the data. Hu-sheng WU drafted the manuscript. Ren-bin XIAO helped organize the manuscript. Hu-sheng WU revised and finalized the paper.
Compliance with ethics guidelines
Hu-sheng WU, Jun-jie XUE, Ren-bin XIAO, and Jin-qiang HU declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Wu, Hs., Xue, Jj., Xiao, Rb. et al. Uncertain bilevel knapsack problem based on an improved binary wolf pack algorithm. Front Inform Technol Electron Eng 21, 1356–1368 (2020). https://doi.org/10.1631/FITEE.1900437
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/FITEE.1900437