Hybrid Genetic Simulated Annealing Algorithm for Improved Flow Shop Scheduling with Makespan Criterion
Abstract
:1. Introduction
 (1)
 We propose the novel HGSA algorithm to solve FSSPs. It is characterized by an operational coding in GA and a hormone regulation mechanism in the simulated annealing part of the algorithm. The MME [28] algorithm, combined with the NEH [5] and MinMax (MM) [29] heuristic algorithms, is used for initialization of the population. We use locationbased intersection and twopoint intersection for crossover operations with either one of these two being randomly selected. In the mutation process, twors mutation or inversion mutation [30] is randomly employed to mutate the population. After the crossover and mutation operation is completed, the best individuals are retained, and the simulated annealing operation is performed on these solutions.
 (2)
 Using the widely adopted Taillard benchmark FSSPs, we conducted extensive experiments and showed that our HGSA algorithm achieved better results than the baseline algorithms to which it was compared in our study. We showed that our HGSA algorithm’s high performance for FSSPs is based on its hybrid search strategy, twors/inversion mutation, locationbased/twopoint crossovers, and their combination with the MME heuristic algorithm for population initialization.
2. Flow Shop Scheduling Problem Description
3. Hybrid Genetic Simulated Annealing Algorithm
3.1. Overview of the HGSA Algorithm
HGSA {Hybrid Genetic Simulated Annealing Algorithm} Initialize population by MME while (not stop condition) do Step 1: Select the population Step 2: Crossover operation If(random==0) locationbased Crossover else twopoint Crossover Step 3: Mutation operation If(random==0) Insertion Mutation else Reverse order Mutation Step 4: Simulated annealing opearation end_while 
3.2. Encoding Representation
3.3. Initial Population
3.4. Crossover Operation
3.4.1. LocationBased Crossover
3.4.2. TwoPoint Crossover
3.5. Mutation Operation
3.5.1. Twors Mutation
3.5.2. Inversion Mutation
3.6. Simulated Annealing Operation
3.6.1. Neighborhood Structure
3.6.2. Initial Temperature
3.6.3. Annealing Rate
3.6.4. The Terminating Condition
3.7. Benchmark Selected
3.8. Computational Complexity
4. Experimental Results and Analysis
5. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
References
 Johnson, S.M. Optimal two and threestage production schedules with setup times included. Naval Res. Logist. Q. 2010, 1, 61–68. [Google Scholar] [CrossRef]
 Hong, Z.Y.; Pang, H.L. Study on a constructive heuristic algorithm based on compromise policy for Blocking flowshop scheduling. Syst. Eng. Theory Pract. 2008, 28, 114–118. [Google Scholar]
 Ignall, E.; Schrage, L. Application of the Branch and Bound Technique to Some FlowShop Scheduling Problems. Oper. Res. 1965, 13, 400–412. [Google Scholar] [CrossRef]
 Bansal, S.P. Minimizing the Sum of Completion Times of n Jobs over m Machines in a Flowshop A Branch and Bound Approach. AIIE Trans. 1977, 9, 306–311. [Google Scholar] [CrossRef]
 Nawaz, M.E.E.E., Jr.; Ham, I. A heuristic algorithm for the m machine, n job flowshop sequencing problem. Omega 1983, 11, 91–95. [Google Scholar] [CrossRef]
 Cui, Q.; Xiuli, W.U.; Jianjun, Y.U. Improved genetic algorithm variable neighborhood search for solving hybrid flow shop scheduling problem. Comput. Integr. Manuf. Syst. 2017, 23, 1917–1927. [Google Scholar]
 Marichelvam, M.K.; Prabaharan, T.; Yang, X.S. Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan. Appl. Soft Comput. 2014, 19, 93–101. [Google Scholar] [CrossRef]
 Burdett, R.L.; Kozan, E. A sequencing approach for creating new train timetables. OR Spectr. 2010, 32, 163–193. [Google Scholar] [CrossRef]
 Rathinam, B. Rule based heuristic approach for minimizing total flow time in permutation flow shop scheduling. Teh. Vjesn. 2015, 22, 25–32. [Google Scholar] [CrossRef] [Green Version]
 Govindan, K.; Balasundaram, R.; Baskar, N.; Asokan, P. A Hybrid Approach for Minimizing Makespan In Permutation Flowshop Scheduling. J. Syst. Sci. Syst. Eng. 2017, 26, 50–76. [Google Scholar] [CrossRef]
 Han, Y.Y.; Gong, D.; Sun, X. A discrete artificial bee colony algorithm incorporating differential evolution for the flowshop scheduling problem with blocking. Eng. Optim. 2015, 47, 927–946. [Google Scholar] [CrossRef]
 Pan, C.H.; Huang, H.C. A hybrid genetic algorithm for nowait job shop scheduling problems. Expert Syst. Appl. 2009, 36, 5800–5806. [Google Scholar] [CrossRef]
 Gao, K.; Pan, Q.; Suganthan, P.N.; Li, J. Effective heuristics for the nowait flow shop scheduling problem with;total flow time minimization. Int. J. Adv. Manuf. Technol. 2013, 66, 1563–1572. [Google Scholar] [CrossRef]
 Bertolissi, E. Heuristic algorithm for scheduling in the nowait flowshop. J. Mater. Process. Technol. 2000, 107, 459–465. [Google Scholar] [CrossRef]
 Nowicki, E.; Smutnicki, C. A fast tabu search algorithm for the permutation flowshop problem. Eur. J. Oper. Res. 1996, 91, 160–175. [Google Scholar] [CrossRef]
 Sayoti, F.; Ri, M.E. Golden Ball Algorithm for solving Flow Shop Scheduling Problem. Ijimai 2016, 4, 15–18. [Google Scholar] [CrossRef]
 Kasihmuddin, M.S.B.M.; Mansor, M.A.B.; Sathasivam, S. Genetic Algorithm for Restricted Maximum kSatisfiability in the Hopfield Network. Int. J. Interact. Multimedia Artif. Intell. 2016, 4, 52. [Google Scholar] [Green Version]
 Tseng, L.; Lin, Y. A hybrid genetic algorithm for nowait flowshop scheduling problem. Int. J. Prod. Econ. 2010, 128, 144–152. [Google Scholar] [CrossRef]
 Ding, J.Y.; Song, S.; Gupta, J.N.; Zhang, R.; Chiong, R.; Wu, C. An improved iterated greedy algorithm with a Tabubased reconstruction strategy for the nowait flowshop scheduling problem. Appl. Soft Comput. 2015, 30, 604–613. [Google Scholar] [CrossRef]
 Tasgetiren, M.F.; Kizilay, D.; Pan, Q.K.; Suganthan, P.N. Iterated greedy algorithms for the blocking flowshop scheduling problem with makespan criterion. Comput. Oper. Res. 2017, 77, 111–126. [Google Scholar] [CrossRef]
 Pan, Q.K.; Wang, L.; Sang, H.Y.; Li, J.Q.; Liu, M. A High Performing Memetic Algorithm for the Flowshop Scheduling Problem with Blocking. IEEE Trans. Autom. Sci. Eng. 2013, 10, 741–756. [Google Scholar]
 Davendra, D.; Bialicdavendra, M. Scheduling flow shops with blocking using a discrete selforganising migrating algorithm. Int. J. Prod. Res. 2013, 51, 2200–2218. [Google Scholar] [CrossRef]
 Eddaly, M.; Jarboui, B.; Siarry, P. Combinatorial particle swarm optimization for solving blocking flowshop scheduling problem. J. Comput. Des. Eng. 2016, 3, 295–311. [Google Scholar] [CrossRef]
 Burdett, R.L.; Kozan, E. Evolutionary algorithms for flowshop sequencing with nonunique jobs. Int. Trans. Oper. Res. 2000, 7, 401–418. [Google Scholar] [CrossRef]
 Yin, H.L. Genetic Algorithm Nested with Simulated Annealing for Big Job Shop Scheduling Problems. In Proceedings of the 2013 9th International Conference on Computational Intelligence and Security (CIS), Emei Moutain, China, 14–15 December 2013. [Google Scholar]
 Andresen, M.; BräSel, H.; MöRig, M.; Tusch, J.; Werner, F.; Willenius, P. Simulated annealing and genetic algorithms for minimizing mean flow time in an open shop. Math. Comput. Model. 2008, 48, 1279–1293. [Google Scholar] [CrossRef]
 Dai, M.; Tang, D.; Giret, A.; Salido, M.A.; Li, W.D. Energyefficient scheduling for a flexible flow shop using an improved geneticsimulated annealing algorithm. Robot. Comput.Integr. Manuf. 2013, 29, 418–429. [Google Scholar] [CrossRef] [Green Version]
 Ronconi, D.P. A note on constructive heuristics for the flowshop problem with blocking. Int. J. Prod. Econ. 2004, 87, 39–48. [Google Scholar] [CrossRef]
 Merz, P.; Freisleben, B. Memetic Algorithms for the Traveling Salesman Problem. Complex Syst. 1997, 13, 297–345. [Google Scholar]
 Abdoun, O.; Abouchabaka, J.; Tajani, C. Analyzing the Performance of Mutation Operators to Solve the Travelling Salesman Problem. Int. J. Emerg. Sci. 2012, 2, 61–77. [Google Scholar]
 Koulamas, C.; Kyparisis, G.J. The threestage assembly flowshop scheduling problem. Comput. Oper. Res. 2001, 28, 689–704. [Google Scholar] [CrossRef]
 Chang, P.; Hsieh, J.; Lin, S. The development of gradualpriority weighting approach for the multiobjective flowshop scheduling problem. Int. J. Prod. Econ. 2002, 79, 171–183. [Google Scholar] [CrossRef]
 Fink, A.; Vos, S. Solving the continuous flowshop scheduling problem by metaheuristics. Eur. J. Oper. Res. 2003, 151, 400–414. [Google Scholar] [CrossRef] [Green Version]
 Wang, J.; Xia, Z.Q. Flowshop scheduling with a learning effect. J. Oper. Res. Soc. 2005, 56, 1325–1330. [Google Scholar] [CrossRef]
 Agarwal, A.; Colak, S.; Eryarsoy, E. Improvement heuristic for the flowshop scheduling problem: An adaptivelearning approach. Eur. J. Oper. Res. 2006, 169, 801–815. [Google Scholar] [CrossRef]
 Rajendran, C.; Ziegler, H. Antcolony algorithms for permutation flowshop scheduling to minimize makespan/total flowtime of jobs. Eur. J. Oper. Res. 2007, 155, 426–438. [Google Scholar] [CrossRef]
 Yagmahan, B.; Yenisey, M.M. Ant colony optimization for multiobjective flow shop scheduling problem. Comput. Ind. Eng. 2008, 54, 411–420. [Google Scholar] [CrossRef]
 Zhang, G.; Shao, X.; Li, P.; Gao, L. An effective hybrid particle swarm optimization algorithm for multiobjective flexible jobshop scheduling problem. Comput. Ind. Eng. 2009, 56, 1309–1318. [Google Scholar] [CrossRef]
 Sayadi, M.K.; Ramezanian, R.; Ghaffarinasab, N. A discrete firefly metaheuristic with local search for makespan minimization in permutation flow shop scheduling problems. Int. J. Ind. Eng. Comput. 2010, 1, 1–10. [Google Scholar] [CrossRef]
 Pan, Q.K.; Tasgetiren, M.F.; Suganthan, P.N.; Chua, T.J. A discrete artificial bee colony algorithm for the lotstreaming flow shop scheduling problem. China Mech. Eng. 2011, 181, 2455–2468. [Google Scholar] [CrossRef]
 Deng, G.; Gu, X. A hybrid discrete differential evolution algorithm for the noidle permutation flow shop scheduling problem with makespan criterion. Comput. Oper. Res. 2012, 39, 2152–2160. [Google Scholar] [CrossRef]
 Li, X.; Yin, M. A hybrid cuckoo search via Lévy flights for the permutation flow shop scheduling problem. Int. J. Prod. Res. 2013, 51, 4732–4754. [Google Scholar] [CrossRef]
 Xie, Z.; Zhang, C.; Shao, X.; Lin, W.; Zhu, H. An effective hybrid teaching–learningbased optimization algorithm for permutation flow shop scheduling problem. Adv. Eng. Softw. 2014, 77, 35–47. [Google Scholar] [CrossRef]
 Lin, Q.; Gao, L.; Li, X.; Zhang, C. A hybrid backtracking search algorithm for permutation flowshop scheduling problem minimizing makespan and energy consumption. Comput. Ind. Eng. 2015, 85, 437–446. [Google Scholar] [CrossRef]
 Lin, J.; Zhang, S. An effective hybrid biogeographybased optimization algorithm for the distributed assembly permutation flowshop scheduling problem. Comput. Ind. Eng. 2016, 97, 128–136. [Google Scholar] [CrossRef]
 Deng, J.; Wang, L.; Wang, S.Y.; Zheng, X.L. A competitive memetic algorithm for the distributed twostage assembly flowshop scheduling problem. Int. J. Prod. Res. 2017, 54, 3561–3577. [Google Scholar] [CrossRef]
 Chen, P.; Wen, W.; Li, R.; Li, X. A hybrid backtracking search algorithm for permutation flowshop scheduling problem minimizing makespan and energy consumption. In Proceedings of the 2017 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Singapore, 10–13 December 2017. [Google Scholar]
 Bewoor, L.; Prakash, V.C.; Sapkal, S. Evolutionary Hybrid Particle Swarm Optimization Algorithm for Solving NPHard NoWait Flow Shop Scheduling Problems. Algorithms 2017, 10, 121. [Google Scholar] [CrossRef]
 Sun, Z.; Gu, X. Hybrid Algorithm Based on an Estimation of Distribution Algorithm and Cuckoo Search for the No Idle Permutation Flow Shop Scheduling Problem with the Total Tardiness Criterion Minimization. Sustainability 2017, 9, 953. [Google Scholar]
 Meng, T.; Pan, Q.K.; Li, J.Q.; Sang, H.Y. An improved migrating birds optimization for an integrated lotstreaming flow shop scheduling problem. Swarm Evol. Comput. 2018, 38, 64–78. [Google Scholar] [CrossRef]
 Yahyaoui, A.; Fnaiech, N.; Fnaiech, F. A Suitable Initialization Procedure for Speeding a Neural Network JobShop Scheduling. IEEE Trans. Ind. Electron. 2011, 58, 1052–1060. [Google Scholar] [CrossRef]
 Liu, S.Q.; Kozan, E. Scheduling a flow shop with combined buffer conditions. Int. J. Prod. Econ. 2009, 117, 371–380. [Google Scholar] [CrossRef]
 Tao, S.; Wang, S. An algorithm for weighted subgraph matching based on gradient flows. Inf. Sci. 2016, 340–341, 104–121. [Google Scholar] [CrossRef]
 Ku, L. An Adaptive Variable Neighbourhood Search Algorithm for the Hybrid Flowshop Scheduling Problem. Syst. Eng. 2015, 11, 121–129. [Google Scholar]
 Dai, M.; Tang, D.; Zheng, K.; Cai, Q. An Improved GeneticSimulated Annealing Algorithm Based on a Hormone Modulation Mechanism for a Flexible FlowShop Scheduling Problem. Adv. Mech. Eng. 2013, 5, 124903. [Google Scholar] [CrossRef]
 Taillard, E. Benchmarks for basic scheduling problems. Eur. J. Oper. Res. 1993, 64, 278–285. [Google Scholar] [CrossRef]
 Salido, M.A.; Escamilla, J.; Giret, A.; Barber, F. A genetic algorithm for energyefficiency in jobshop scheduling. Int. J. Adv. Manuf. Technol. 2016, 85, 1303–1314. [Google Scholar] [CrossRef]
 Rajkumar, R.; Shahabudeen, P. An improved genetic algorithm for the flowshop scheduling problem. Int. J. Prod. Res. 2009, 47, 233–249. [Google Scholar] [CrossRef]
Author, Year, and Reference  Algorithm  Optimization Criteria  Benchmarks  Summary 

Koulamas (2001) [31]  Heuristics with ratio performance guarantees (HRPG)  Makespan 
 
Chang et al. (2002) [32]  Gradualpriority weighting (GPW)  Makespan, total flowtime, total tardiness 
 
Andreas et al. (2003) [33]  Metaheuristics  Total flowtime  Taillard 

Wang et al. (2005) [34]  Polynomial algorithm  Makespan, total flowtime 
 
Agarwal (2006) [35]  Improvement heuristic approach (IHA)  Taillard, Carlier, Heller 
 
Rajendran (2007) [36]  Proposed ant colony algorithms  Makespan, total flowtime  Taillard 

Yagmahan et al. (2008) [37]  Ant colony optimization (ACO)  Makespan, total flow time, total machine idle time  Reeves 

Zhang et al. (2009) [38]  Hybrid particle swam optimization (PSO) algorithm  Makespan, maximal machine workload 
 
Sayadi et al. (2010) [39]  Firefly metaheuristic (FMH)  Makespan  Demirkol 

Pan et al. (2011) [40]  Discrete artificial bee colony (DABC) algorithm  Total weighted earliness and tardiness penalties 
 
Deng et al. (2012) [41]  Hybrid discrete differential evolution (HDDE) algorithm  Makespan  Ruiz 

Li et al. (2013) [42]  Cuckoo search (CS)based memetic algorithm (HCS)  Makespan, total flow time  Car, Rec 

Xie et al. (2014) [43]  Hybrid teaching–learningbased optimization (HTLBO) algorithm  Makespan, maximum lateness  Carlier’s, Reeves Yamada’s 

Lin et al. (2015) [44]  Hybrid backtracking search algorithm (HBSA)  Makespan  Civicioglu 

Lin et al. (2016) [45]  Hybrid biogeographybased optimization (HBBO) algorithm  Makespan  Hatami 

Deng et al. (2017) [46]  Competitive memetic algorithm (CMA)  Makespan, total tardiness  Taillard 

Peng et al. (2017) [47]  Hybrid backtracking search (HBSA)  Makespan, energy consumption  Car 

Bewoor et al. (2017) [48]  Hybrid particle swarm optimization (PHPSO) algorithm  Total flow time  Taillard 

Sun et al. (2017) [49]  Hybrid estimation of the distribution algorithm and cuckoo search (HEDA_CS)  Total Tardiness  Ruiz 

Meng et al. (2018) [50]  Improved migrating birds optimization (IMMBO)  Makespan  Randomly generated 

Parameter  Meaning 

J = {1, 2, …, n}  Set of n jobs 
M = {1_{1}, 2, …, m}  Set of m machines 
p_{ij}  The processing time when job i is processed on machine tool j 
S_{ij}  The starting time when job i is processed on machine tool j 
C_{ij}  The finishing time when job i is processed on machine tool j 
π = {π_{1}, π_{2}, …, π_{n}}  A sequence of jobs 
Π  Set of all jobs’ sequences 
C_{max}(π)  Makespan of one job’s sequence π 
Problem Size  Function Factor (n)  ARPD% 

200 × 20  1.0  9.21 
1.2  7.82  
1.5  5.16  
2.0  8.34  
2.5  11.06  
500 × 20  1.0  6.28 
1.2  5.49  
1.5  3.29  
2.0  7.13  
2.5  9.56 
Problem Size  Instance  Upper Bound  MA  IGRIS  HGA  IIGA  DSOMA  HGSA 

20 × 5  Ta001  1278      1449  1486  1374  1324 
Ta002  1359      1460  1528  1408  1442  
Ta003  1081      1386  1460  1280  1098  
Ta004  1293      1521  1588  1448  1469  
Ta005  1235      1403  1449  1341  1291  
Ta006  1195      1430  1481  1363  1391  
Ta007  1239      1461  1483  1381  1299  
Ta008  1206      1433  1482  1379  1292  
Ta009  1230      1398  1469  1373  1306  
Ta010  1108      1324  1377  1283  1233  
20 × 10  Ta011  1582      1955  2011  1698  1713 
Ta012  1659      2123  2166  1833  1718  
Ta013  1496      1912  1940  1676  1555  
Ta014  1377      1782  1811  1546  1516  
Ta015  1419      1933  1933  1617  1573  
Ta016  1397      1827  1892  1590  1457  
Ta017  1484      1944  1963  1622  1622  
Ta018  1538      2006  2057  1731  1749  
Ta019  1593      1908  1973  1747  1624  
Ta020  1591      2001  2051  1782  1722  
20 × 20  Ta021  2297      2912  2973  2436  2331 
Ta022  2099      2780  2582  2234  2280  
Ta023  2326      2922  3013  2479  2480  
Ta024  2223      2967  3001  2348  2362  
Ta025  2291      2953  3003  2435  2507  
Ta026  2226      2908  2988  2383  2375  
Ta027  2273      2970  3052  2390  2341  
Ta028  2200      2763  2839  2328  2279  
Ta029  2237      2972  3009  2363  2410  
Ta030  2178      2919  2979  2323  2401  
50 × 5  Ta031  2724  3000  3002  3127  3161  3033  2731 
Ta032  2834  3199  3201  3438  3432  3045  2934  
Ta033  2621  3011  3011  3182  3211  3036  2638  
Ta034  2751  3128  3128  3289  3339  3011  2785  
Ta035  2863  3162  3166  3315  3356  3128  2864  
Ta036  2829  3166  3169  3324  3347  3166  2907  
Ta037  2725  3013  3013  3183  3231  3021  2764  
Ta038  2683  3067  3073  3243  3235  3063  2706  
Ta039  2552  2908  2908  3059  3072  2908  2610  
Ta040  2782  3111  3120  3301  3317  3120  2784  
50 × 10  Ta041  2991  3638  3638  4251  4274  3638  3198 
Ta042  2867  3486  3507  4139  4177  3511  3020  
Ta043  2839  3483  3488  4083  4099  3492  3055  
Ta044  3063  3656  3656  4480  4399  3672  3124  
Ta045  2976  3629  3629  4316  4322  3633  3129  
Ta046  3006  3596  3621  4282  4289  3621  3293  
Ta047  3093  3692  3696  4376  4420  3704  3232  
Ta048  3037  3562  3572  4304  4318  3572  3390  
Ta049  2897  3527  3532  4162  4155  3541  3237  
Ta050  3065  3622  3624  4232  4283  3624  3251  
50 × 20  Ta051  3850  4479  4500  6138  6129  4511  4105 
Ta052  3704  4276  4276  5721  5725  4288  3992  
Ta053  3640  4261  4289  5847  5862  4289  3900  
Ta054  3720  4366  4377  5781  5788  4378  3921  
Ta055  3610  4261  4268  5891  5886  4271  4020  
Ta056  3681  4280  4280  5875  5863  4202  3971  
Ta057  3704  4304  4308  5937  5962  4315  4093  
Ta058  3691  4317  4326  5919  5926  4326  4090  
Ta059  3743  4315  4316  5839  5876  4329  4107  
Ta060  3756  4413  4428  5935  5958  4422  4113  
100 × 5  Ta061  5493  6143  6151  6492  6397  6151  5536 
Ta062  5268  6022  6022  6353  6234  6064  5302  
Ta063  5175  5927  5927  6148  6121  6003  5221  
Ta064  5014  5756  5772  6080  6026  5786  5044  
Ta065  5250  5957  5960  6254  6200  6021  5358  
Ta066  5135  5812  5852  6177  6074  5869  5197  
Ta067  5246  5989  6004  6257  6274  6004  5414  
Ta068  5094  5856  5915  6225  6130  5924  5130  
Ta069  5448  6066  6123  6443  6370  6154  5546  
Ta070  5322  6142  6159  6441  6381  6186  5480  
100 × 10  Ta071  5770  7016  7042  8115  8077  7042  5964 
Ta072  5349  6740  6791  7986  7880  6813  5596  
Ta073  5676  6878  6936  8057  8028  6943  5796  
Ta074  5781  7116  7187  8327  8348  7198  5928  
Ta075  5467  6810  6810  7991  7859  6815  5748  
Ta076  5303  6614  6666  7823  7801  6685  5446  
Ta077  5595  6783  6801  7915  7866  6827  5679  
Ta078  5617  6790  6874  7379  7913  6874  5723  
Ta079  5871  6981  7055  8226  8161  6092  5934  
Ta080  5845  6814  6965  8186  8114  6990  5998  
100 × 20  Ta081  6202  7796  7844  10,745  10,700  7854  6395 
Ta082  6183  7845  7894  10,655  10,594  7910  6433  
Ta083  6271  7794  7794  10,672  10,611  7825  6689  
Ta084  6269  7797  7899  10,630  10,607  7902  6419  
Ta085  6314  7817  7901  10,548  10,539  7901  6536  
Ta086  6364  7826  7888  10,700  10,690  7921  6527  
Ta087  6268  7923  7930  10,827  10,825  8051  6542  
Ta088  6401  7984  8022  10,863  10,839  8025  6712  
Ta089  6275  7877  7969  10,751  10,723  7969  6760  
Ta090  6434  7913  7933  10,794  10,798  8036  6621  
200 × 10  Ta091  10,862  13,348  13,406  15,739  15,319  13,507  11,120 
Ta092  10,480  13,242  13,313  15,534  15,085  16,458  10,658  
Ta093  10,922  13,318  13,416  15,755  15,376  13,521  11,224  
Ta094  10,889  13,290  13,344  15,842  15,200  13,686  11,075  
Ta095  10,524  13,247  13,360  15,692  15,209  13,547  10,793  
Ta096  10,326  13,079  13,192  15,622  15,109  13,247  10,467  
Ta097  10,854  13,517  13,598  15,877  15,395  13,910  11,394  
Ta098  10,730  13,483  13,504  15,733  15,237  13,830  11,011  
Ta099  10,438  13,277  13,310  15,573  15,100  13,410  10,725  
Ta100  10,657  13,325  13,439  15,803  15,340  13,744  10,786  
200 × 20  Ta101  11,195  14,912  14,912  20,148  19,681  15,027  11,642 
Ta102  11,203  14,876  15,002  20,539  20,096  15,211  11,683  
Ta103  11,281  15,057  15,186  20,511  19,913  15,247  11,930  
Ta104  11,275  14,975  15,082  20,461  19,928  15,174  11,791  
Ta105  11,259  14,733  14,970  20,339  19,843  15,047  11,728  
Ta106  11,176  14,861  15,101  20,501  19,942  15,212  11,690  
Ta107  11,360  14,988  15,099  20,680  20,112  15,168  11,958  
Ta108  11,334  14,926  15,141  20,614  20,056  15,247  11,730  
Ta109  11,192  14,885  15,034  20,300  19,918  15,136  12,138  
Ta110  11,288  14,921  15,122  20,437  19,935  15,243  12,084  
500 × 20  Ta111  26,059  35,677  35,372  49,095  46,689  37,064  26,859 
Ta112  26,520  35,953  35,743  49,461  47,275  37,419  27,220  
Ta113  26,371  35,732  35,452  48,777  46,544  37,059  27,511  
Ta114  26,456  36,084  35,687  49,283  46,899  37,014  26,912  
Ta115  26,334  35,774  35,417  48,950  46,741  36,894  26,930  
Ta116  26,477  35,948  35,747  49,533  46,941  37,372  27,354  
Ta117  26,389  35,631  35,395  48,943  46,509  36,698  26,888  
Ta118  26,560  35,943  35,568  49,277  46,873  36,944  27,229  
Ta119  26,005  35,658  35,304  49,207  46,743  36,862  28,103  
Ta120  26,457  36,016  35,643  49,092  46,847  37,098  27,290 
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Wei, H.; Li, S.; Jiang, H.; Hu, J.; Hu, J. Hybrid Genetic Simulated Annealing Algorithm for Improved Flow Shop Scheduling with Makespan Criterion. Appl. Sci. 2018, 8, 2621. https://doi.org/10.3390/app8122621
Wei H, Li S, Jiang H, Hu J, Hu J. Hybrid Genetic Simulated Annealing Algorithm for Improved Flow Shop Scheduling with Makespan Criterion. Applied Sciences. 2018; 8(12):2621. https://doi.org/10.3390/app8122621
Chicago/Turabian StyleWei, Hongjing, Shaobo Li, Houmin Jiang, Jie Hu, and Jianjun Hu. 2018. "Hybrid Genetic Simulated Annealing Algorithm for Improved Flow Shop Scheduling with Makespan Criterion" Applied Sciences 8, no. 12: 2621. https://doi.org/10.3390/app8122621