An enhanced Genetic Algorithm with an innovative encoding strategy for flexible job-shop scheduling with operation and processing flexibility

Xuewen Huang; Xiaotong Zhang; Sardar M. N. Islam; Carlos A. Vega-Mejía

doi:10.3934/jimo.2019088

Previous Article
Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks
JIMO Home
This Issue
Next Article
A zero-forcing beamforming based time switching protocol for wireless powered internet of things system

doi: 10.3934/jimo.2019088

An enhanced Genetic Algorithm with an innovative encoding strategy for flexible job-shop scheduling with operation and processing flexibility

Xuewen Huang ^1, , Xiaotong Zhang ^1, , Sardar M. N. Islam ^2,, and Carlos A. Vega-Mejía ^2,3,

1.	Faculty of Management and Economics, Dalian University of Technology, Dalian, Liaoning 116023, China
2.	ISILC, Victoria University, Melbourne, Australia
3.	Operations & Supply Chain Management Research Group, Universidad de La Sabana, Bogotá, Colombia

^* Corresponding author: Sardar M. N. Islam

Received November 2017 Revised March 2019 Published July 2019

Fund Project: Xuewen Huang is supported by the National Science and Technology Plan of China under Grant No. 2015BAF09B01 and the China Scholarship under Grant No. 201606060170

This paper considers the Flexible Job-shop Scheduling Problem with Operation and Processing flexibility (FJSP-OP) with the objective of minimizing the makespan. A Genetic Algorithm based approach is presented to solve the FJSP-OP. For the performance improvement, a new and concise Four-Tuple Scheme (FTS) is proposed for modeling a job with operation and processing flexibility. Then, with the FTS, an enhanced Genetic Algorithm employing a more efficient encoding strategy is developed. The use of this encoding strategy ensures that the classic genetic operators can be adopted to the utmost extent without generating infeasible offspring. Experiments have validated the proposed approach, and the results have shown the effectiveness and high performance of the proposed approach.

Keywords: IPPS, flexible job-shop scheduling, operation flexibility, processing flexibility, Genetic Algorithm.

Mathematics Subject Classification: Primary: 90B35, 90C59; Secondary: 68M20.

Citation: Xuewen Huang, Xiaotong Zhang, Sardar M. N. Islam, Carlos A. Vega-Mejía. An enhanced Genetic Algorithm with an innovative encoding strategy for flexible job-shop scheduling with operation and processing flexibility. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019088

References:

[1]	A. Azzouz, M. Ennigrou and L. B. Said, A hybrid algorithm for flexible job-shop scheduling problem with setup times, Int. J. of Prod. Mgmt. & Eng., 5 (2017), 23-30. doi: 10.4995/ijpme.2017.6618. Google Scholar
[2]	K. R. Baker, Introduction to Sequencing and Scheduling, Wiley, New York, 1974.Google Scholar
[3]	W. Banzhaf, The "molecula" traveling salesman, Bio. Cyber., 64 (1990), 7-14. Google Scholar
[4]	A. Baykasoglu, Linguistic-based meta-heuristic optimization model for flexible job shop scheduling, Int. J. of Prod. Research, 40 (2002), 4523-4543. doi: 10.1080/00207540210147043. Google Scholar
[5]	A. Baykasoglu, L. Ozbakir and A. Sonmez, Using multiple objective tabu search and grammars to model and solve multi-objective flexible job shop scheduling problems, J. of Intell. Manufacturing, 15 (2004), 777-785. doi: 10.1023/B:JIMS.0000042663.16199.84. Google Scholar
[6]	S. Benjaafar and R. S. Ramakrishnan, Modelling, measurement and evaluation of sequencing flexibility in manufacturing systems, Int. J. of Prod. Research, 34 (1996), 1195-1220. doi: 10.1080/00207549608904961. Google Scholar
[7]	C. Bierwirth, D. C. Mattfeld and H. Kopfer, On permutation representations for scheduling problems, in International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, 1141, Springer, Berlin, 1996,310–318. doi: 10.1007/3-540-61723-X_995. Google Scholar
[8]	C. Bierwirth and D. Mattfeld, Production scheduling and rescheduling with genetic algorithms, Evol. Computation, 7 (1999), 1-17. doi: 10.1162/evco.1999.7.1.1. Google Scholar
[9]	P. Brandimarte, Routing and scheduling in a flexible job shop by tabu search, Annals of Ops. Research, 41 (1993), 157-183. doi: 10.1007/BF02023073. Google Scholar
[10]	I. A. Chaudhry and M. Usman, Integrated process planning and scheduling using genetic algorithms, Tehnicki Vjesnik, 24 (2017), 1401-1409. doi: 10.17559/TV-20151121212910. Google Scholar
[11]	R. Cheng, M. Gen and Y. Tsujimura, A tutorial survey of job-shop scheduling problems using genetic algorithms-Ⅰ. Representation, Comp. & Indust. Engineering, 30 (1996), 983-997. doi: 10.1016/0360-8352(96)00047-2. Google Scholar
[12]	R. Cheng, M. Gen and Y. Tsujimura, A tutorial survey of job-shop scheduling problems using genetic algorithms: Part Ⅱ. Hybrid genetic search strategies, Comp. & Indust. Engineering, 37 (1999), 51-55. doi: 10.1016/s0360-8352(99)00022-4. Google Scholar
[13]	G. Chryssolouris, S. Chan and W. Cobb, Decision making on the factory floor: An integrated approach to process planning and scheduling, Robotics & Comp. Integ. Manufacturing, 1 (1984), 315-319. doi: 10.1016/0736-5845(84)90020-6. Google Scholar
[14]	G. Chryssolouris, S. Chan and N. P. Suh, An integrated approach to process planning and scheduling, CIRP Annals - Manufacturing Tech., 34 (1985), 413-417. doi: 10.1016/S0007-8506(07)61801-0. Google Scholar
[15]	D. A. Dabney, L. Green and V. Topalli, A priority-based heuristic algorithm (PBHA) for optimizing integrated process planning and scheduling problem, J. of Criminal Justice Edu., 2 (2015), 44-68. doi: 10.1080/23311916.2015.1070494. Google Scholar
[16]	L. Davis, Job shop scheduling with genetic algorithms, In: International Conference on Genetic Algorithms, L. Erlbaum Asssociates Inc., Hillsdale, 1985, 136–140.Google Scholar
[17]	H. H. Doh, J. M. Yu, J. S. Kim, D. H. Lee and S. H. Nam, A priority scheduling approach for flexible job shops with multiple process plans, Int. J. of Prod. Research, 51 (2013), 3748-3764. doi: 10.1080/00207543.2013.765074. Google Scholar
[18]	D. B. Fogel, An evolutionary approach to the traveling salesman problem, Bio. Cybernetics, 60 (1988), 139-144. doi: 10.1007/BF00202901. Google Scholar
[19]	J. Gao, M. Gen, L. Sun and X. Zhao, A hybrid of genetic algorithm and bottleneck shifting for multiobjective flexible job shop scheduling problems, Comp. & Indust. Engineering, 53 (2007), 149-162. doi: 10.1016/j.cie.2007.04.010. Google Scholar
[20]	K. Z. Gao, P. N. Suganthan, T. J. Chua, C. S. Chong, T. X. Cai and Q. K. Pan, A two-stage artificial bee colony algorithm scheduling flexible job-shop scheduling problem with new job insertion, Expert Syst. with Appl., 42 (2015), 7652-7663. doi: 10.1016/j.eswa.2015.06.004. Google Scholar
[21]	M. Gen, Y. Tsujimura and E. Kubota, Solving job-shop scheduling problems by genetic algorithm, in IEEE International Conference on Systems, (1994), 576–579. doi: 10.1109/ICSMC.1994.400072. Google Scholar
[22]	M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley & Sons, 2000. doi: 10.1002/9780470172261. Google Scholar
[23]	D. E. Goldberg and R. Lingle, Alleles, loci and the traveling salesman problem, Proc. of 1st Int. Conf. on Genetic Algorithms and Their Applications, 12 (1985), 154-159. Google Scholar
[24]	N. B. Ho, J. C. Tay and M. K. Lai, An effective architecture for learning and evolving flexible job-shop schedules, European J. of Oper. Research, 179 (2007), 316-333. doi: 10.1016/j.ejor.2006.04.007. Google Scholar
[25]	Y. C. Ho and C. L. Moodie, Solving cell formation problems in a manufacturing environment with flexible processing and routing capabilities, Int. J. of Prod. Research, 34 (1996), 2901-2923. doi: 10.1080/00207549608905065. Google Scholar
[26]	J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich., 1975. Google Scholar
[27]	K. Ida and K. Oka, Flexible job-shop scheduling problem by genetic algorithm, Electrical Engineering in Japan, 177 (2011), 28-35. doi: 10.1002/eej.21194. Google Scholar
[28]	L. Jin, Q. Tang, C. Zhang, X. Shao and G. Tian, More MILP models for integrated process planning and scheduling, Int. J. of Prod. Research, 54 (2016), 4387-4402. doi: 10.1080/00207543.2016.1140917. Google Scholar
[29]	B. Khoshnevis and Q. M. Chen, Integration of process planning and scheduling functions, J. of Intell. Manufacturing, 2 (1991), 165-175. doi: 10.1007/BF01471363. Google Scholar
[30]	Y. K. Kim, J. Y. Kim and K. S. Shin, An asymmetric multileveled symbiotic evolutionary algorithm for integrated FMS scheduling, J. of Intell. Manufacturing, 18 (2007), 631-645. doi: 10.1007/s10845-007-0037-5. Google Scholar
[31]	Y. K. Kim, K. Park and J. Ko, A symbiotic evolutionary algorithm for the integration of process planning and job shop scheduling, Comp. & Ops. Research, 30 (2003), 1151-1171. doi: 10.1016/S0305-0548(02)00063-1. Google Scholar
[32]	T. Kis, Job-shop scheduling with processing alternatives, European J. of Oper. Research, 151 (2003), 307-332. doi: 10.1016/S0377-2217(02)00828-7. Google Scholar
[33]	R. Kumar, M. K. Tiwari and R. Shankar, Scheduling of flexible manufacturing systems: An ant colony optimization approach, Proceedings of the Inst. of Mech. Engineers Part B: J. of Engineering Manufacture, 217 (2003), 1443-1453. doi: 10.1243/095440503322617216. Google Scholar
[34]	A. Ławrynowicz, Integration of production planning and scheduling using an expert system and a genetic algorithm, J. of the Oper. Research Society, 59 (2008), 455-463. doi: 10.1057/palgrave.jors.2602423. Google Scholar
[35]	H. Lee and S. S. Kim, Integration of process planning and scheduling using simulation based genetic algorithms, The Int. J. of Adv. Manufacturing Tech., 18 (2001), 586-590. doi: 10.1007/s001700170035. Google Scholar
[36]	K. M. Lee, T. Yamakawa and K. M. Lee, A genetic algorithm for general machine scheduling problems, Int. J. of Knowledge-Based Intell. Electronic Systems, 2 (1998), 60-66. doi: 10.1109/KES.1998.725893. Google Scholar
[37]	X. Y. Li and L. Gao, An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem, Int. J. of Prod. Economics, 174 (2016), 93-110. doi: 10.1016/j.ijpe.2016.01.016. Google Scholar
[38]	X. Y. Li, L. Gao, X. Shao, C. Zhang and C. Wang, Mathematical modeling and evolutionary algorithm-based approach for integrated process planning and scheduling, Comp. & Ops. Research, 37 (2010), 656-667. doi: 10.1016/j.cor.2009.06.008. Google Scholar
[39]	X. Y. Li, L. Gao and X. Y. Shao, An active learning genetic algorithm for integrated process planning and scheduling, Expert Syst. with Appl., 39 (2012), 6683-6691. doi: 10.1016/j.eswa.2011.11.074. Google Scholar
[40]	X. Y. Li, X. Y. Shao, L. Gao and W. Qian, An effective hybrid algorithm for integrated process planning and scheduling, Int. J. of Prod. Economics, 126 (2010), 289-298. doi: 10.1016/j.ijpe.2010.04.001. Google Scholar
[41]	X. Y. Li, C. Zhang, L. Gao, W. Li and X. Y. Shao, An agent-based approach for integrated process planning and scheduling, Expert Syst. with Appl., 37 (2010), 1256-1264. doi: 10.1016/j.eswa.2009.06.014. Google Scholar
[42]	Y. J. Lin and J. J. Solberg, Effectiveness of flexible routing control, Int. J. of Flexible Manufacturing Syst., 3 (1991), 189-211. doi: 10.1007/BF00170206. Google Scholar
[43]	M. Liu, S. Yi and P. Wen, Quantum-inspired hybrid algorithm for integrated process planning and scheduling, Proceedings of the Inst. of Mech. Engineers Part B: J. of Engineering Manufacture, 232 (2016), 1-18. doi: 10.1177/0954405416661006. Google Scholar
[44]	G. Luo, X. Wen, H. Li, W. Ming and G. Xie, An effective multi-objective genetic algorithm based on immune principle and external archive for multi-objective integrated process planning and scheduling, Int. J. of Adv. Manufacturing Tech., 91 (2017), 3145-3158. doi: 10.1007/s00170-017-0020-z. Google Scholar
[45]	V. K. Manupati, G. D. Putnik, M. K. Tiwari, P. Ávila and M. M. Cruz-Cunha, Integration of process planning and scheduling using mobile-agent based approach in a networked manufacturing environment, Comp. & Industrial Engineering, 94 (2016), 63-73. doi: 10.1016/j.cie.2016.01.017. Google Scholar
[46]	Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Second edition, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-02830-8. Google Scholar
[47]	C. Moon, Y. H. Lee, C. S. Jeong and Y. S. Yun, Integrated process planning and scheduling in a supply chain, Comp. & Indust. Engineering, 54 (2008), 1048-1061. doi: 10.1016/j.cie.2007.06.018. Google Scholar
[48]	N. Morad and A. Zalzala, Genetic algorithms in integrated process planning and scheduling, J. of Intell. Manufacturing, 10 (1999), 169-179. doi: 10.1023/a:1008976720878. Google Scholar
[49]	H. E. Nouri, O. B. Driss and K. Ghédira, Solving the flexible job shop problem by hybrid metaheuristics-based multiagent model, J. of Indust. Engineering Int., 14 (2018), 1-14. doi: 10.1007/s40092-017-0204-z. Google Scholar
[50]	M. Nouiri, A. Bekrar, A. Jemai, S. Niar and A. C. Ammari, An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem, J. of Intell. Manufacturing, 29 (2018), 603-615. doi: 10.1007/s10845-015-1039-3. Google Scholar
[51]	C. Özgüven, L. Özbakır and Y. Yavuz, Mathematical models for job-shop scheduling problems with routing and process plan flexibility, Appl. Math. Modelling, 34 (2010), 1539-1548. doi: 10.1016/j.apm.2009.09.002. Google Scholar
[52]	J. J. Palacios, M. A. González, C. R. Vela, I. González-Rodríguez and J. Puente, Genetic tabu search for the fuzzy flexible job shop problem, Comp. & Ops. Research, 54 (2015), 74-89. doi: 10.1016/j.cor.2014.08.023. Google Scholar
[53]	M. Petroviä, J. Petronijeviä, M. Mitiä, N. Vukoviä, Z. Miljkoviä and B. Babiä, The ant lion optimization algorithm for integrated process planning and scheduling, Appl. Mechanics & Materials, 834 (2016), 187-192. Google Scholar
[54]	R. K. Phanden, A. Jain and R. Verma, Integration of process planning and scheduling: A state-of-the-art review, Int. J. of Comp. Integrated Manufacturing, 24 (2011), 517-534. doi: 10.1080/0951192X.2011.562543. Google Scholar
[55]	L. Qiao and S. Lv, An improved genetic algorithm for integrated process planning and scheduling, The Int. J. of Adv. Manufacturing Tech., 58 (2012), 727-740. doi: 10.1007/s00170-011-3409-0. Google Scholar
[56]	A. Rajabinasab and S. Mansour, Dynamic flexible job shop scheduling with alternative process plans: An agent-based approach, The Int. J. of Adv. Manufacturing Tech., 54 (2011), 1091-1107. doi: 10.1007/s00170-010-2986-7. Google Scholar
[57]	C. Saygin and S. E. Kilic, Integrating flexible process plans with scheduling in flexible manufacturing systems, The Int. J. of Adv. Manufacturing Tech., 15 (1999), 268-280. doi: 10.1007/s001700050066. Google Scholar
[58]	X. Y. Shao, X. Y. Li, L. Gao and C. Y. Zhang, Integration of process planning and scheduling-a modified genetic algorithm-based approach, Comp. & Ops. Research, 36 (2009), 2082-2096. doi: 10.1016/j.cor.2008.07.006. Google Scholar
[59]	O. Sobeyko and L. Mã¶Nch, Integrated process planning and scheduling for large-scale flexible job shops using metaheuristics, Int. J. of Prod. Research, 55 (2017), 392-409. doi: 10.1080/00207543.2016.1182227. Google Scholar
[60]	D. Sreeramulu, Y. Sagar, P. Suman and A. S. Kumar, Integration of process planning and scheduling of a manufacturing systems using petri nets and genetic algorithm, Indian J. of Science & Tech., 9 (2016), 2-8. doi: 10.17485/ijst/2016/v9i41/100874. Google Scholar
[61]	G. Syswerda, Uniform Crossover in Genetic Algorithms, International Conference on Genetic Algorithms, (1989), 2–9.Google Scholar
[62]	Z. Wang and T. Ju, The research about integration of process planning and production scheduling based on genetic algorithm, International Conference on Computer Science and Software Engineering, (2008), 9–12. doi: 10.1109/CSSE.2008.845. Google Scholar
[63]	T. N. Wong, C. W. Leung, K. L. Mak and R. Y. K. Fung, An agent-based negotiation approach to integrate process planning and scheduling, Int. J. of Prod. Research, 44 (2006), 1331-1351. doi: 10.1080/00207540500409723. Google Scholar
[64]	T. N. Wong, C. W. Leung, K. L. Mak and R. Y. K. Fung, Integrated process planning and scheduling/rescheduling-an agent-based approach, Int. J. of Prod. Research, 44 (2006), 3627-3655. doi: 10.1080/00207540600675801. Google Scholar
[65]	T. N. Wong, S. C. Zhang, G. Wang and L. P. Zhang, Integrated process planning and scheduling – multi-agent system with two-stage ant colony optimisation algorithm, Int. J. of Prod. Research, 50 (2012), 6188-6201. doi: 10.1080/00207543.2012.720393. Google Scholar
[66]	H. Xia, X. Li and L. Gao, A hybrid genetic algorithm with variable neighborhood search for dynamic integrated process planning and scheduling, Comp. & Indust. Engineering, 102 (2016), 99-112. doi: 10.1016/j.cie.2016.10.015. Google Scholar
[67]	L. Yin, L. Gao, X. Li and H. Xia, An improved genetic algorithm with rolling window technology for dynamic integrated process planning and scheduling problem, IEEE International Conference on Computer Supported Cooperative Work in Design, (2017), 414–419. doi: 10.1109/CSCWD.2017.8066730. Google Scholar
[68]	Y. Yuan and H. Xu, Multiobjective flexible job shop scheduling using memetic algorithms, IEEE Transactions on Automation Science and Engineering, 12 (2015), 336-353. doi: 10.1109/TASE.2013.2274517. Google Scholar
[69]	G. Zhang, L. Zhang, X. Song, Y. Wang and C. Zhou, A variable neighborhood search based genetic algorithm for flexible job shop scheduling problem, Cluster Computing, (2018), 1–12. doi: 10.1007/s10586-017-1420-4. Google Scholar
[70]	S. Zhang and T. N. Wong, Integrated process planning and scheduling: An enhanced ant colony optimization heuristic with parameter tuning, J. of Intell. Manufacturing, 29 (2018), 585-601. doi: 10.1007/s10845-014-1023-3. Google Scholar
[71]	L. Zhang and T. N. Wong, An object-coding genetic algorithm for integrated process planning and scheduling, European J. of Oper. Research, 244 (2015), 434-444. doi: 10.1016/j.ejor.2015.01.032. Google Scholar
[72]	S. Zhang, Z. Yu, W. Zhang, D. Yu and Y. Xu, An extended genetic algorithm for distributed integration of fuzzy process planning and scheduling, Math. Problems in Engineering, 3 (2016), 1-13. doi: 10.1155/2016/3763512. Google Scholar

show all references

References:

[1]	A. Azzouz, M. Ennigrou and L. B. Said, A hybrid algorithm for flexible job-shop scheduling problem with setup times, Int. J. of Prod. Mgmt. & Eng., 5 (2017), 23-30. doi: 10.4995/ijpme.2017.6618. Google Scholar
[2]	K. R. Baker, Introduction to Sequencing and Scheduling, Wiley, New York, 1974.Google Scholar
[3]	W. Banzhaf, The "molecula" traveling salesman, Bio. Cyber., 64 (1990), 7-14. Google Scholar
[4]	A. Baykasoglu, Linguistic-based meta-heuristic optimization model for flexible job shop scheduling, Int. J. of Prod. Research, 40 (2002), 4523-4543. doi: 10.1080/00207540210147043. Google Scholar
[5]	A. Baykasoglu, L. Ozbakir and A. Sonmez, Using multiple objective tabu search and grammars to model and solve multi-objective flexible job shop scheduling problems, J. of Intell. Manufacturing, 15 (2004), 777-785. doi: 10.1023/B:JIMS.0000042663.16199.84. Google Scholar
[6]	S. Benjaafar and R. S. Ramakrishnan, Modelling, measurement and evaluation of sequencing flexibility in manufacturing systems, Int. J. of Prod. Research, 34 (1996), 1195-1220. doi: 10.1080/00207549608904961. Google Scholar
[7]	C. Bierwirth, D. C. Mattfeld and H. Kopfer, On permutation representations for scheduling problems, in International Conference on Parallel Problem Solving from Nature, Lecture Notes in Computer Science, 1141, Springer, Berlin, 1996,310–318. doi: 10.1007/3-540-61723-X_995. Google Scholar
[8]	C. Bierwirth and D. Mattfeld, Production scheduling and rescheduling with genetic algorithms, Evol. Computation, 7 (1999), 1-17. doi: 10.1162/evco.1999.7.1.1. Google Scholar
[9]	P. Brandimarte, Routing and scheduling in a flexible job shop by tabu search, Annals of Ops. Research, 41 (1993), 157-183. doi: 10.1007/BF02023073. Google Scholar
[10]	I. A. Chaudhry and M. Usman, Integrated process planning and scheduling using genetic algorithms, Tehnicki Vjesnik, 24 (2017), 1401-1409. doi: 10.17559/TV-20151121212910. Google Scholar
[11]	R. Cheng, M. Gen and Y. Tsujimura, A tutorial survey of job-shop scheduling problems using genetic algorithms-Ⅰ. Representation, Comp. & Indust. Engineering, 30 (1996), 983-997. doi: 10.1016/0360-8352(96)00047-2. Google Scholar
[12]	R. Cheng, M. Gen and Y. Tsujimura, A tutorial survey of job-shop scheduling problems using genetic algorithms: Part Ⅱ. Hybrid genetic search strategies, Comp. & Indust. Engineering, 37 (1999), 51-55. doi: 10.1016/s0360-8352(99)00022-4. Google Scholar
[13]	G. Chryssolouris, S. Chan and W. Cobb, Decision making on the factory floor: An integrated approach to process planning and scheduling, Robotics & Comp. Integ. Manufacturing, 1 (1984), 315-319. doi: 10.1016/0736-5845(84)90020-6. Google Scholar
[14]	G. Chryssolouris, S. Chan and N. P. Suh, An integrated approach to process planning and scheduling, CIRP Annals - Manufacturing Tech., 34 (1985), 413-417. doi: 10.1016/S0007-8506(07)61801-0. Google Scholar
[15]	D. A. Dabney, L. Green and V. Topalli, A priority-based heuristic algorithm (PBHA) for optimizing integrated process planning and scheduling problem, J. of Criminal Justice Edu., 2 (2015), 44-68. doi: 10.1080/23311916.2015.1070494. Google Scholar
[16]	L. Davis, Job shop scheduling with genetic algorithms, In: International Conference on Genetic Algorithms, L. Erlbaum Asssociates Inc., Hillsdale, 1985, 136–140.Google Scholar
[17]	H. H. Doh, J. M. Yu, J. S. Kim, D. H. Lee and S. H. Nam, A priority scheduling approach for flexible job shops with multiple process plans, Int. J. of Prod. Research, 51 (2013), 3748-3764. doi: 10.1080/00207543.2013.765074. Google Scholar
[18]	D. B. Fogel, An evolutionary approach to the traveling salesman problem, Bio. Cybernetics, 60 (1988), 139-144. doi: 10.1007/BF00202901. Google Scholar
[19]	J. Gao, M. Gen, L. Sun and X. Zhao, A hybrid of genetic algorithm and bottleneck shifting for multiobjective flexible job shop scheduling problems, Comp. & Indust. Engineering, 53 (2007), 149-162. doi: 10.1016/j.cie.2007.04.010. Google Scholar
[20]	K. Z. Gao, P. N. Suganthan, T. J. Chua, C. S. Chong, T. X. Cai and Q. K. Pan, A two-stage artificial bee colony algorithm scheduling flexible job-shop scheduling problem with new job insertion, Expert Syst. with Appl., 42 (2015), 7652-7663. doi: 10.1016/j.eswa.2015.06.004. Google Scholar
[21]	M. Gen, Y. Tsujimura and E. Kubota, Solving job-shop scheduling problems by genetic algorithm, in IEEE International Conference on Systems, (1994), 576–579. doi: 10.1109/ICSMC.1994.400072. Google Scholar
[22]	M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley & Sons, 2000. doi: 10.1002/9780470172261. Google Scholar
[23]	D. E. Goldberg and R. Lingle, Alleles, loci and the traveling salesman problem, Proc. of 1st Int. Conf. on Genetic Algorithms and Their Applications, 12 (1985), 154-159. Google Scholar
[24]	N. B. Ho, J. C. Tay and M. K. Lai, An effective architecture for learning and evolving flexible job-shop schedules, European J. of Oper. Research, 179 (2007), 316-333. doi: 10.1016/j.ejor.2006.04.007. Google Scholar
[25]	Y. C. Ho and C. L. Moodie, Solving cell formation problems in a manufacturing environment with flexible processing and routing capabilities, Int. J. of Prod. Research, 34 (1996), 2901-2923. doi: 10.1080/00207549608905065. Google Scholar
[26]	J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich., 1975. Google Scholar
[27]	K. Ida and K. Oka, Flexible job-shop scheduling problem by genetic algorithm, Electrical Engineering in Japan, 177 (2011), 28-35. doi: 10.1002/eej.21194. Google Scholar
[28]	L. Jin, Q. Tang, C. Zhang, X. Shao and G. Tian, More MILP models for integrated process planning and scheduling, Int. J. of Prod. Research, 54 (2016), 4387-4402. doi: 10.1080/00207543.2016.1140917. Google Scholar
[29]	B. Khoshnevis and Q. M. Chen, Integration of process planning and scheduling functions, J. of Intell. Manufacturing, 2 (1991), 165-175. doi: 10.1007/BF01471363. Google Scholar
[30]	Y. K. Kim, J. Y. Kim and K. S. Shin, An asymmetric multileveled symbiotic evolutionary algorithm for integrated FMS scheduling, J. of Intell. Manufacturing, 18 (2007), 631-645. doi: 10.1007/s10845-007-0037-5. Google Scholar
[31]	Y. K. Kim, K. Park and J. Ko, A symbiotic evolutionary algorithm for the integration of process planning and job shop scheduling, Comp. & Ops. Research, 30 (2003), 1151-1171. doi: 10.1016/S0305-0548(02)00063-1. Google Scholar
[32]	T. Kis, Job-shop scheduling with processing alternatives, European J. of Oper. Research, 151 (2003), 307-332. doi: 10.1016/S0377-2217(02)00828-7. Google Scholar
[33]	R. Kumar, M. K. Tiwari and R. Shankar, Scheduling of flexible manufacturing systems: An ant colony optimization approach, Proceedings of the Inst. of Mech. Engineers Part B: J. of Engineering Manufacture, 217 (2003), 1443-1453. doi: 10.1243/095440503322617216. Google Scholar
[34]	A. Ławrynowicz, Integration of production planning and scheduling using an expert system and a genetic algorithm, J. of the Oper. Research Society, 59 (2008), 455-463. doi: 10.1057/palgrave.jors.2602423. Google Scholar
[35]	H. Lee and S. S. Kim, Integration of process planning and scheduling using simulation based genetic algorithms, The Int. J. of Adv. Manufacturing Tech., 18 (2001), 586-590. doi: 10.1007/s001700170035. Google Scholar
[36]	K. M. Lee, T. Yamakawa and K. M. Lee, A genetic algorithm for general machine scheduling problems, Int. J. of Knowledge-Based Intell. Electronic Systems, 2 (1998), 60-66. doi: 10.1109/KES.1998.725893. Google Scholar
[37]	X. Y. Li and L. Gao, An effective hybrid genetic algorithm and tabu search for flexible job shop scheduling problem, Int. J. of Prod. Economics, 174 (2016), 93-110. doi: 10.1016/j.ijpe.2016.01.016. Google Scholar
[38]	X. Y. Li, L. Gao, X. Shao, C. Zhang and C. Wang, Mathematical modeling and evolutionary algorithm-based approach for integrated process planning and scheduling, Comp. & Ops. Research, 37 (2010), 656-667. doi: 10.1016/j.cor.2009.06.008. Google Scholar
[39]	X. Y. Li, L. Gao and X. Y. Shao, An active learning genetic algorithm for integrated process planning and scheduling, Expert Syst. with Appl., 39 (2012), 6683-6691. doi: 10.1016/j.eswa.2011.11.074. Google Scholar
[40]	X. Y. Li, X. Y. Shao, L. Gao and W. Qian, An effective hybrid algorithm for integrated process planning and scheduling, Int. J. of Prod. Economics, 126 (2010), 289-298. doi: 10.1016/j.ijpe.2010.04.001. Google Scholar
[41]	X. Y. Li, C. Zhang, L. Gao, W. Li and X. Y. Shao, An agent-based approach for integrated process planning and scheduling, Expert Syst. with Appl., 37 (2010), 1256-1264. doi: 10.1016/j.eswa.2009.06.014. Google Scholar
[42]	Y. J. Lin and J. J. Solberg, Effectiveness of flexible routing control, Int. J. of Flexible Manufacturing Syst., 3 (1991), 189-211. doi: 10.1007/BF00170206. Google Scholar
[43]	M. Liu, S. Yi and P. Wen, Quantum-inspired hybrid algorithm for integrated process planning and scheduling, Proceedings of the Inst. of Mech. Engineers Part B: J. of Engineering Manufacture, 232 (2016), 1-18. doi: 10.1177/0954405416661006. Google Scholar
[44]	G. Luo, X. Wen, H. Li, W. Ming and G. Xie, An effective multi-objective genetic algorithm based on immune principle and external archive for multi-objective integrated process planning and scheduling, Int. J. of Adv. Manufacturing Tech., 91 (2017), 3145-3158. doi: 10.1007/s00170-017-0020-z. Google Scholar
[45]	V. K. Manupati, G. D. Putnik, M. K. Tiwari, P. Ávila and M. M. Cruz-Cunha, Integration of process planning and scheduling using mobile-agent based approach in a networked manufacturing environment, Comp. & Industrial Engineering, 94 (2016), 63-73. doi: 10.1016/j.cie.2016.01.017. Google Scholar
[46]	Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Second edition, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-662-02830-8. Google Scholar
[47]	C. Moon, Y. H. Lee, C. S. Jeong and Y. S. Yun, Integrated process planning and scheduling in a supply chain, Comp. & Indust. Engineering, 54 (2008), 1048-1061. doi: 10.1016/j.cie.2007.06.018. Google Scholar
[48]	N. Morad and A. Zalzala, Genetic algorithms in integrated process planning and scheduling, J. of Intell. Manufacturing, 10 (1999), 169-179. doi: 10.1023/a:1008976720878. Google Scholar
[49]	H. E. Nouri, O. B. Driss and K. Ghédira, Solving the flexible job shop problem by hybrid metaheuristics-based multiagent model, J. of Indust. Engineering Int., 14 (2018), 1-14. doi: 10.1007/s40092-017-0204-z. Google Scholar
[50]	M. Nouiri, A. Bekrar, A. Jemai, S. Niar and A. C. Ammari, An effective and distributed particle swarm optimization algorithm for flexible job-shop scheduling problem, J. of Intell. Manufacturing, 29 (2018), 603-615. doi: 10.1007/s10845-015-1039-3. Google Scholar
[51]	C. Özgüven, L. Özbakır and Y. Yavuz, Mathematical models for job-shop scheduling problems with routing and process plan flexibility, Appl. Math. Modelling, 34 (2010), 1539-1548. doi: 10.1016/j.apm.2009.09.002. Google Scholar
[52]	J. J. Palacios, M. A. González, C. R. Vela, I. González-Rodríguez and J. Puente, Genetic tabu search for the fuzzy flexible job shop problem, Comp. & Ops. Research, 54 (2015), 74-89. doi: 10.1016/j.cor.2014.08.023. Google Scholar
[53]	M. Petroviä, J. Petronijeviä, M. Mitiä, N. Vukoviä, Z. Miljkoviä and B. Babiä, The ant lion optimization algorithm for integrated process planning and scheduling, Appl. Mechanics & Materials, 834 (2016), 187-192. Google Scholar
[54]	R. K. Phanden, A. Jain and R. Verma, Integration of process planning and scheduling: A state-of-the-art review, Int. J. of Comp. Integrated Manufacturing, 24 (2011), 517-534. doi: 10.1080/0951192X.2011.562543. Google Scholar
[55]	L. Qiao and S. Lv, An improved genetic algorithm for integrated process planning and scheduling, The Int. J. of Adv. Manufacturing Tech., 58 (2012), 727-740. doi: 10.1007/s00170-011-3409-0. Google Scholar
[56]	A. Rajabinasab and S. Mansour, Dynamic flexible job shop scheduling with alternative process plans: An agent-based approach, The Int. J. of Adv. Manufacturing Tech., 54 (2011), 1091-1107. doi: 10.1007/s00170-010-2986-7. Google Scholar
[57]	C. Saygin and S. E. Kilic, Integrating flexible process plans with scheduling in flexible manufacturing systems, The Int. J. of Adv. Manufacturing Tech., 15 (1999), 268-280. doi: 10.1007/s001700050066. Google Scholar
[58]	X. Y. Shao, X. Y. Li, L. Gao and C. Y. Zhang, Integration of process planning and scheduling-a modified genetic algorithm-based approach, Comp. & Ops. Research, 36 (2009), 2082-2096. doi: 10.1016/j.cor.2008.07.006. Google Scholar
[59]	O. Sobeyko and L. Mã¶Nch, Integrated process planning and scheduling for large-scale flexible job shops using metaheuristics, Int. J. of Prod. Research, 55 (2017), 392-409. doi: 10.1080/00207543.2016.1182227. Google Scholar
[60]	D. Sreeramulu, Y. Sagar, P. Suman and A. S. Kumar, Integration of process planning and scheduling of a manufacturing systems using petri nets and genetic algorithm, Indian J. of Science & Tech., 9 (2016), 2-8. doi: 10.17485/ijst/2016/v9i41/100874. Google Scholar
[61]	G. Syswerda, Uniform Crossover in Genetic Algorithms, International Conference on Genetic Algorithms, (1989), 2–9.Google Scholar
[62]	Z. Wang and T. Ju, The research about integration of process planning and production scheduling based on genetic algorithm, International Conference on Computer Science and Software Engineering, (2008), 9–12. doi: 10.1109/CSSE.2008.845. Google Scholar
[63]	T. N. Wong, C. W. Leung, K. L. Mak and R. Y. K. Fung, An agent-based negotiation approach to integrate process planning and scheduling, Int. J. of Prod. Research, 44 (2006), 1331-1351. doi: 10.1080/00207540500409723. Google Scholar
[64]	T. N. Wong, C. W. Leung, K. L. Mak and R. Y. K. Fung, Integrated process planning and scheduling/rescheduling-an agent-based approach, Int. J. of Prod. Research, 44 (2006), 3627-3655. doi: 10.1080/00207540600675801. Google Scholar
[65]	T. N. Wong, S. C. Zhang, G. Wang and L. P. Zhang, Integrated process planning and scheduling – multi-agent system with two-stage ant colony optimisation algorithm, Int. J. of Prod. Research, 50 (2012), 6188-6201. doi: 10.1080/00207543.2012.720393. Google Scholar
[66]	H. Xia, X. Li and L. Gao, A hybrid genetic algorithm with variable neighborhood search for dynamic integrated process planning and scheduling, Comp. & Indust. Engineering, 102 (2016), 99-112. doi: 10.1016/j.cie.2016.10.015. Google Scholar
[67]	L. Yin, L. Gao, X. Li and H. Xia, An improved genetic algorithm with rolling window technology for dynamic integrated process planning and scheduling problem, IEEE International Conference on Computer Supported Cooperative Work in Design, (2017), 414–419. doi: 10.1109/CSCWD.2017.8066730. Google Scholar
[68]	Y. Yuan and H. Xu, Multiobjective flexible job shop scheduling using memetic algorithms, IEEE Transactions on Automation Science and Engineering, 12 (2015), 336-353. doi: 10.1109/TASE.2013.2274517. Google Scholar
[69]	G. Zhang, L. Zhang, X. Song, Y. Wang and C. Zhou, A variable neighborhood search based genetic algorithm for flexible job shop scheduling problem, Cluster Computing, (2018), 1–12. doi: 10.1007/s10586-017-1420-4. Google Scholar
[70]	S. Zhang and T. N. Wong, Integrated process planning and scheduling: An enhanced ant colony optimization heuristic with parameter tuning, J. of Intell. Manufacturing, 29 (2018), 585-601. doi: 10.1007/s10845-014-1023-3. Google Scholar
[71]	L. Zhang and T. N. Wong, An object-coding genetic algorithm for integrated process planning and scheduling, European J. of Oper. Research, 244 (2015), 434-444. doi: 10.1016/j.ejor.2015.01.032. Google Scholar
[72]	S. Zhang, Z. Yu, W. Zhang, D. Yu and Y. Xu, An extended genetic algorithm for distributed integration of fuzzy process planning and scheduling, Math. Problems in Engineering, 3 (2016), 1-13. doi: 10.1155/2016/3763512. Google Scholar

Figure 1. Alternative process plans networks

Solution approach	EA	Modified GA	PBHA	GA-OP
Makespan	16	14	14	14
Mean CPU time (ms)	N/A	N/A	N/A	191
¹ N/A means the result was not given by the author.

Solution approach	ALGA	GA-OP
Parameters	$ PopSize=400 $	$ PopSize=200 $
Parameters	$ P_c=0.8 $, $ P_m=0.1 $, $ IterNo=100 $
Makespan	188	181
Mean CPU time (ms)	N/A	2,907

Problem	$ n \times m $	TABC	HGTS	MA2	HA	PSO	VNSGA	GA-OP
mk01	10x6	40	40	40	40	41	40	40
mk02	10x6	26	26	26	26	26	26	26
mk03	15x8	204	204	204	204	207	204	204
mk04	15x8	60	60	60	60	65	60	60
mk05	15x4	173	172	172	172	171	173	172
mk06	10x15	60	57	59	57	61	58	57
mk07	20x5	139	139	139	139	173	144	139
mk08	20x10	523	523	523	523	523	523	523
mk09	20x10	307	307	307	307	307	307	307
mk10	20x15	202	198	202	197	312	198	198

Problem	n x m	TABC	HGTS	MA2	HA	PSO	VNSGA	GA-OP
mk01	10x6	3.36	5	20.16	0.06	N/A	N/A	0.27
mk02	10x6	3.72	15	28.21	0.59	N/A	N/A	3.75
mk03	15x8	1.56	2	53.76	0.16	N/A	N/A	0.12
mk04	15x8	66.58	10	30.53	0.49	N/A	N/A	3.41
mk05	15x4	78.45	18	36.36	4.57	N/A	N/A	6.23
mk06	10x15	173.98	63	80.61	53.82	N/A	N/A	65.14
mk07	20x5	66.19	33	37.74	20.01	N/A	N/A	25.4
mk08	20x10	2.15	3	77.71	0.02	N/A	N/A	0.2
mk09	20x10	304.43	24	75.23	0.86	N/A	N/A	4.1
mk10	20x15	418.19	104	90.75	33.21	N/A	N/A	73.12

Solution approach	Experiment 6(a)		Experiment 6(b)
Solution approach	ACO	GA-OP	Enhanced ACO	GA-OP
Makespan	589	522	484	482
Mean CPU time (ms)	128,700	12,056	120534	12063

Operation	Job
Operation	1	2	3	4	5	6
1	2, 4	2, 4	3, 5	3, 5	2, 4	2, 4
2	7, 8	7, 8	4, 6	4, 6	3, 5	3, 5
3	1, 2	1, 2	2, 3, 5	2, 3, 5	1, 2, 6	1, 2, 6
4	8, 6	8, 6	6, 7	6, 7	6, 8	6, 8
5	3, 5	3, 5	1, 8	1, 8	1, 7	1, 7
6	1, 2, 4	1, 2, 4	1, 4	1, 4	1, 3	1, 3
7	5, 6	5, 6	6, 7	6, 7	6, 7	6, 7
8	1, 7	1, 7	5, 8	5, 8	5, 8	5, 8
9	3, 8	3, 8	4, 2	4, 2	4, 3	4, 3

Operation	Job
Operation	1	2	3	4	5	6
1	18, 22	18, 22	12, 15	12, 15	18, 22	18, 22
2	39, 36	39, 36	24, 23	24, 23	12, 15	12, 15
3	11, 10	37, 39	30, 31, 29	30, 31, 29	50, 52, 54	50, 52, 54
4	31, 34	20, 21	21, 22	21, 22	19, 21	53, 51
5	21, 23	21, 23	32, 30	32, 30	32, 31	22, 24
6	10, 12, 15	10, 12, 15	22, 25	22, 25	22, 25	22, 25
7	32, 30	36, 38	24, 22	42, 44	24, 22	24, 22
8	45, 44	45, 44	20, 19	41, 43	20, 18	20, 18
9	26, 24	26, 24	27, 22	27, 22	27, 22	27, 22

Machine number	1	2	3	4	5	6	7	8
1	0	3	7	10	3	5	8	12
2	3	0	4	7	5	3	5	8
3	7	4	0	3	8	5	3	5
4	10	7	3	0	10	8	5	3
5	3	5	8	10	0	3	7	10
6	5	3	5	8	3	0	4	7
7	8	5	3	5	7	4	0	3
8	12	8	5	3	10	7	3	0

[1]	Didem Cinar, José António Oliveira, Y. Ilker Topcu, Panos M. Pardalos. A priority-based genetic algorithm for a flexible job shop scheduling problem. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1391-1415. doi: 10.3934/jimo.2016.12.1391
[2]	Y. K. Lin, C. S. Chong. A tabu search algorithm to minimize total weighted tardiness for the job shop scheduling problem. Journal of Industrial & Management Optimization, 2016, 12 (2) : 703-717. doi: 10.3934/jimo.2016.12.703
[3]	Behrad Erfani, Sadoullah Ebrahimnejad, Amirhossein Moosavi. An integrated dynamic facility layout and job shop scheduling problem: A hybrid NSGA-II and local search algorithm. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-34. doi: 10.3934/jimo.2019030
[4]	Adel Dabah, Ahcene Bendjoudi, Abdelhakim AitZai. An efficient Tabu Search neighborhood based on reconstruction strategy to solve the blocking job shop scheduling problem. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2015-2031. doi: 10.3934/jimo.2017029
[5]	Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098
[6]	Ji-Bo Wang, Mengqi Liu, Na Yin, Ping Ji. Scheduling jobs with controllable processing time, truncated job-dependent learning and deterioration effects. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1025-1039. doi: 10.3934/jimo.2016060
[7]	F. Zeyenp Sargut, H. Edwin Romeijn. Capacitated requirements planning with pricing flexibility and general cost and revenue functions. Journal of Industrial & Management Optimization, 2007, 3 (1) : 87-98. doi: 10.3934/jimo.2007.3.87
[8]	Jiping Tao, Zhijun Chao, Yugeng Xi. A semi-online algorithm and its competitive analysis for a single machine scheduling problem with bounded processing times. Journal of Industrial & Management Optimization, 2010, 6 (2) : 269-282. doi: 10.3934/jimo.2010.6.269
[9]	Ethel Mokotoff. Algorithms for bicriteria minimization in the permutation flow shop scheduling problem. Journal of Industrial & Management Optimization, 2011, 7 (1) : 253-282. doi: 10.3934/jimo.2011.7.253
[10]	M. Ramasubramaniam, M. Mathirajan. A solution framework for scheduling a BPM with non-identical job dimensions. Journal of Industrial & Management Optimization, 2007, 3 (3) : 445-456. doi: 10.3934/jimo.2007.3.445
[11]	Min-Fan He, Li-Ning Xing, Wen Li, Shang Xiang, Xu Tan. Double layer programming model to the scheduling of remote sensing data processing tasks. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1515-1526. doi: 10.3934/dcdss.2019104
[12]	Yaw Chang, Lin Chen. Solve the vehicle routing problem with time windows via a genetic algorithm. Conference Publications, 2007, 2007 (Special) : 240-249. doi: 10.3934/proc.2007.2007.240
[13]	Abdel-Rahman Hedar, Alaa Fahim. Filter-based genetic algorithm for mixed variable programming. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 99-116. doi: 10.3934/naco.2011.1.99
[14]	Mostafa Abouei Ardakan, A. Kourank Beheshti, S. Hamid Mirmohammadi, Hamed Davari Ardakani. A hybrid meta-heuristic algorithm to minimize the number of tardy jobs in a dynamic two-machine flow shop problem. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 465-480. doi: 10.3934/naco.2017029
[15]	Jiping Tao, Ronghuan Huang, Tundong Liu. A $2.28$-competitive algorithm for online scheduling on identical machines. Journal of Industrial & Management Optimization, 2015, 11 (1) : 185-198. doi: 10.3934/jimo.2015.11.185
[16]	Yuanjia Ma. The optimization algorithm for blind processing of high frequency signal of capacitive sensor. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1399-1412. doi: 10.3934/dcdss.2019096
[17]	Bin Zheng, Min Fan, Mengqi Liu, Shang-Chia Liu, Yunqiang Yin. Parallel-machine scheduling with potential disruption and positional-dependent processing times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 697-711. doi: 10.3934/jimo.2016041
[18]	Chengxin Luo. Single machine batch scheduling problem to minimize makespan with controllable setup and jobs processing times. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 71-77. doi: 10.3934/naco.2015.5.71
[19]	Chuanli Zhao, Yunqiang Yin, T. C. E. Cheng, Chin-Chia Wu. Single-machine scheduling and due date assignment with rejection and position-dependent processing times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 691-700. doi: 10.3934/jimo.2014.10.691
[20]	Xianyu Yu, Dar-Li Yang, Dequn Zhou, Peng Zhou. Multi-machine scheduling with interval constrained position-dependent processing times. Journal of Industrial & Management Optimization, 2018, 14 (2) : 803-815. doi: 10.3934/jimo.2017076

American Institute of Mathematical Sciences

An enhanced Genetic Algorithm with an innovative encoding strategy for flexible job-shop scheduling with operation and processing flexibility

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors