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July  2018, 14(3): 1219-1249. doi: 10.3934/jimo.2018007

## A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment

 1 Department of Industrial Engineering, Yalova University, Yalova, Turkey 2 Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

* Corresponding author: Omer Faruk Yilmaz

Received  August 2016 Revised  June 2017 Published  January 2018

This paper focuses on the batch scheduling problem in multi-hybrid cell manufacturing systems (MHCMS) in a dual-resource constrained (DRC) setting, considering skilled workforce assignment (SWA). This problem consists of finding the sequence of batches on each cell, the starting time of each batch, and assigning employees to the operations of batches in accordance with the desired objective. Because handling both the scheduling and assignment decisions simultaneously presents a challenging optimization problem, it is difficult to solve the formulated model, even for small-sized problem instances. Three metaheuristics are proposed to solve the batch scheduling problem, namely the genetic algorithm (GA), simulated annealing (SA) algorithm, and artificial bee colony (ABC) algorithm. A serial scheduling scheme (SSS) is introduced and employed in all metaheuristics to obtain a feasible schedule for each individual. The main aim of this study is to identify an effective metaheuristic for determining the scheduling and assignment decisions that minimize the average cell response time. Detailed computational experiments were conducted, based on real production data, to evaluate the performance of the metaheuristics. The experimental results show that the performance of the proposed ABC algorithm is superior to other metaheuristics for different levels of experimental factors determined for the number of batches and the employee flexibility.

Citation: Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007
##### References:

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##### References:
Batch-flow and one-piece flow
Average cell response time (Solution decoding)
Multi-Hybrid Cell Manufacturing System for the illustrative example
Crossover operator
Box-plot of RPD values with respect to the algorithms (for nine employees)
Box-plots of RPD values obtained by the algorithms for each level of NBEC (for nine employees)
Box-plots of RPD values obtained by the algorithms for each level of NESL (for nine employees)
Box-plots of RPD values obtained by the ABC algorithm for each level of NESL (for nine employees)
Illustrative example data for the problem
 $i$ $cx_i$ $cn_i$ $q_i$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $o_i$ $a_i$ 1 66 35 5 5 15 0 10 0 5 15 20 16 4 2 86 30 10 0 20 0 15 5 5 0 30 16 4 3 80 30 5 15 15 5 5 0 30 20 10 20 5 4 55 20 10 10 10 0 5 0 20 20 0 20 5
 $i$ $cx_i$ $cn_i$ $q_i$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $o_i$ $a_i$ 1 66 35 5 5 15 0 10 0 5 15 20 16 4 2 86 30 10 0 20 0 15 5 5 0 30 16 4 3 80 30 5 15 15 5 5 0 30 20 10 20 5 4 55 20 10 10 10 0 5 0 20 20 0 20 5
Maximum and minimum number of employees
 $i$ $W_i$ (Maximum) $W1_i$ (Minimum) 1 (Cell1) 2 1 2 (Cell1) 3 1 3 (Cell2) 3 1 4 (Cell2) 3 1
 $i$ $W_i$ (Maximum) $W1_i$ (Minimum) 1 (Cell1) 2 1 2 (Cell1) 3 1 3 (Cell2) 3 1 4 (Cell2) 3 1
The batch list representation scheme (Encoding scheme)
 position numbers 1 2 3 4 batch list 2 4 1 3 batch-employee assignment 1-3 1-2 2 1-3 employee-machine assignment (1-2) (3-4) (2-3) (1-4) (1-2-3-4) (3-4) (1-2)
 position numbers 1 2 3 4 batch list 2 4 1 3 batch-employee assignment 1-3 1-2 2 1-3 employee-machine assignment (1-2) (3-4) (2-3) (1-4) (1-2-3-4) (3-4) (1-2)
The cell cycle times and the first lead times
 $i$ $cn_i$ employee1 employee2 employee3 $cy_i$ $FT_i$ 1 (Cell1) 35 - 66 - 66 86 2 (Cell1) 30 43 43 - 43 91 3 (Cell2) 30 50 - 25 50 120 4 (Cell2) 20 35 20 - 35 85
 $i$ $cn_i$ employee1 employee2 employee3 $cy_i$ $FT_i$ 1 (Cell1) 35 - 66 - 66 86 2 (Cell1) 30 43 43 - 43 91 3 (Cell2) 30 50 - 25 50 120 4 (Cell2) 20 35 20 - 35 85
Cell cycle times, batch sizes, operation times and total walking times
 $i$ $cx_i$ $cn_i$ $q_i$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $o_i$ 1 957 450 5 16 150 MO MO 450 155 108 62 16 2 622 468 4 16 150 110 358 MO 160 108 62 16 3 915 432 10 15 144 MO MO 432 148 105 55 16 4 590 455 5 15 144 100 355 MO 152 105 58 16 5 887 420 15 15 140 MO MO 420 145 96 55 16 6 555 438 8 15 140 88 350 MO 145 96 55 16 7 741 319 15 12 135 MO MO 319 130 84 45 16 8 508 355 5 12 135 75 280 MO 138 84 48 16 9 626 250 10 9 120 MO MO 250 117 71 44 15 10 436 272 10 9 120 60 212 MO 117 71 44 15 11 472 155 15 8 108 MO MO 155 88 66 32 15 12 363 186 9 8 108 26 160 MO 102 66 38 15 13 525 182 12 8 115 MO MO 182 95 71 40 14 14 414 210 8 8 115 50 160 MO 112 71 44 14 15 617 226 14 11 131 MO MO 226 102 85 48 14 16 469 258 8 11 131 58 200 MO 120 85 50 14 17 755 318 10 14 145 MO MO 318 115 94 55 14 18 541 360 5 14 145 80 280 MO 134 94 60 14 19 940 431 5 18 153 MO MO 431 139 118 65 16 20 653 490 15 18 153 120 370 MO 155 118 73 16 21 1070 485 10 25 168 MO MO 485 163 135 78 16 22 739 545 8 25 168 135 410 MO 175 135 85 16 23 1202 528 10 34 185 MO MO 528 188 161 90 16 24 857 607 5 34 185 147 460 MO 211 161 103 16
 $i$ $cx_i$ $cn_i$ $q_i$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $r_{im}$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $e_{zim}$ $o_i$ 1 957 450 5 16 150 MO MO 450 155 108 62 16 2 622 468 4 16 150 110 358 MO 160 108 62 16 3 915 432 10 15 144 MO MO 432 148 105 55 16 4 590 455 5 15 144 100 355 MO 152 105 58 16 5 887 420 15 15 140 MO MO 420 145 96 55 16 6 555 438 8 15 140 88 350 MO 145 96 55 16 7 741 319 15 12 135 MO MO 319 130 84 45 16 8 508 355 5 12 135 75 280 MO 138 84 48 16 9 626 250 10 9 120 MO MO 250 117 71 44 15 10 436 272 10 9 120 60 212 MO 117 71 44 15 11 472 155 15 8 108 MO MO 155 88 66 32 15 12 363 186 9 8 108 26 160 MO 102 66 38 15 13 525 182 12 8 115 MO MO 182 95 71 40 14 14 414 210 8 8 115 50 160 MO 112 71 44 14 15 617 226 14 11 131 MO MO 226 102 85 48 14 16 469 258 8 11 131 58 200 MO 120 85 50 14 17 755 318 10 14 145 MO MO 318 115 94 55 14 18 541 360 5 14 145 80 280 MO 134 94 60 14 19 940 431 5 18 153 MO MO 431 139 118 65 16 20 653 490 15 18 153 120 370 MO 155 118 73 16 21 1070 485 10 25 168 MO MO 485 163 135 78 16 22 739 545 8 25 168 135 410 MO 175 135 85 16 23 1202 528 10 34 185 MO MO 528 188 161 90 16 24 857 607 5 34 185 147 460 MO 211 161 103 16
Experimental factors and their levels
 Factor Level Number of Batches on Each Cell (NBEC) 1 NB (Problem size factor) 2 [5$\times$ NB] 3 [10$\times$ NB] Number of Employees for each Skill Level Junior Normal Senior (NESL) 1 33% 33% 33% 2 66% 33% 00% 3 66% 00% 33% 4 100% 00% 00% 5 00% 66% 33% 6 33% 66% 00% 7 00% 100% 00% 8 33% 00% 66% 9 00% 33% 66% 10 00% 00% 100%
 Factor Level Number of Batches on Each Cell (NBEC) 1 NB (Problem size factor) 2 [5$\times$ NB] 3 [10$\times$ NB] Number of Employees for each Skill Level Junior Normal Senior (NESL) 1 33% 33% 33% 2 66% 33% 00% 3 66% 00% 33% 4 100% 00% 00% 5 00% 66% 33% 6 33% 66% 00% 7 00% 100% 00% 8 33% 00% 66% 9 00% 33% 66% 10 00% 00% 100%
The coefficient of skill levels
 Junior Normal Senior Skill level coefficients 0.63 1 1.29
 Junior Normal Senior Skill level coefficients 0.63 1 1.29
The promising values of the parameters for the metaheuristics
 Algorithm Notation Values Combination NBEC=1 NBEC=2 NBEC=3 GA $PS$ 20, 40, 60, 80,100 40 60 80 $pcross$ 0.2, 0.4, 0.6, 0.8 0.8 0.6 0.6 $pmutation$ 0.1, 0.2, 0.3, 0.4 0.3 0.2 0.2 ABC $NFS$ 20, 40, 60, 80,100 40 40 60 $\lambda$ 2, 4, 6, 8, 10 2 4 4 $limit$ 2, 4, 6, 8, 10 6 6 4 SA $initialtemp$ $10^3\times$(1, 3, 5, 7) 1 3 5 $coolingrate$ 0.9, 0.95, 0.99 0.99 0.99 0.99 $epoch$ 5, 10, 15, 20 10 15 15
 Algorithm Notation Values Combination NBEC=1 NBEC=2 NBEC=3 GA $PS$ 20, 40, 60, 80,100 40 60 80 $pcross$ 0.2, 0.4, 0.6, 0.8 0.8 0.6 0.6 $pmutation$ 0.1, 0.2, 0.3, 0.4 0.3 0.2 0.2 ABC $NFS$ 20, 40, 60, 80,100 40 40 60 $\lambda$ 2, 4, 6, 8, 10 2 4 4 $limit$ 2, 4, 6, 8, 10 6 6 4 SA $initialtemp$ $10^3\times$(1, 3, 5, 7) 1 3 5 $coolingrate$ 0.9, 0.95, 0.99 0.99 0.99 0.99 $epoch$ 5, 10, 15, 20 10 15 15
The tuned values of the parameters for the metaheuristics
 Algorithm Notation Combination NBEC=1 NBEC=2 NBEC=3 GA $PS$ 30, 40, 50 50, 60, 70 70, 80, 90 $pcross$ 0.7, 0.8, 0.9 0.5, 0.6, 0.7 0.5, 0.6, 0.7 $pmutation$ 0.25, 0.3, 0.35 0.15, 0.2, 0.25 0.15, 0.2, 0.25 ABC $NFS$ 30, 40, 50 30, 40, 50 50, 60, 70 $\lambda$ 1, 2, 3 3, 4, 5 3, 4, 5 $limit$ 5, 6, 7 5, 6, 7 3, 4, 5 SA $initialtemp$ 750, 1000,1250 2500,3000, 3500 4000,5000, 6000 $coolingrate$ 0.98, 0.99 0.98, 0.99 0.98, 0.99 $epoch$ 8, 10, 12 13, 15, 17 13, 15, 17
 Algorithm Notation Combination NBEC=1 NBEC=2 NBEC=3 GA $PS$ 30, 40, 50 50, 60, 70 70, 80, 90 $pcross$ 0.7, 0.8, 0.9 0.5, 0.6, 0.7 0.5, 0.6, 0.7 $pmutation$ 0.25, 0.3, 0.35 0.15, 0.2, 0.25 0.15, 0.2, 0.25 ABC $NFS$ 30, 40, 50 30, 40, 50 50, 60, 70 $\lambda$ 1, 2, 3 3, 4, 5 3, 4, 5 $limit$ 5, 6, 7 5, 6, 7 3, 4, 5 SA $initialtemp$ 750, 1000,1250 2500,3000, 3500 4000,5000, 6000 $coolingrate$ 0.98, 0.99 0.98, 0.99 0.98, 0.99 $epoch$ 8, 10, 12 13, 15, 17 13, 15, 17
The maximum computational times
 ABC GA SA NBEC=1 14.87 12.68 9.16 NBEC=2 53.40 46.07 38.73 NBEC=3 98.25 95.53 83.68
 ABC GA SA NBEC=1 14.87 12.68 9.16 NBEC=2 53.40 46.07 38.73 NBEC=3 98.25 95.53 83.68
Median of RPD values and computational time of algorithms for 9 employees
 NESL$\times$NBEC RPD LB SA GA ABC ABCWLS GAWLS CPU 1$\times$1 23580 17.66 16.96 18.82 18 17, 14 15 2$\times$1 18989 19.34 20.01 16.08 20.04 21.39 15 3$\times$1 22263 17.99 18.97 16.62 18.83 19.05 15 4$\times$1 23196 20.22 19.23 17.4 18.55 20.13 15 5$\times$1 22223 16.05 17.66 18.19 18.99 18.25 15 6$\times$1 22097 18.05 19.18 16.02 17.39 21.05 15 7$\times$1 20685 20.57 18.87 18.53 18.32 19.08 15 8$\times$1 24351 19.93 21.03 16.56 17.36 21.58 15 9$\times$1 19621 17.45 18.6 18.57 18.92 20.4 15 10$\times$1 23961 20.42 20.57 15.97 20.02 21.16 15 Medians 18.768 19.108 17.276 18.642 19.923 15 1$\times$2 123093 34.99 30.13 26.88 27.85 32.15 51.2 2$\times$2 116383 33.24 31.81 29.88 28.44 32.73 51.2 3$\times$2 125259 28.87 27.55 29.16 29.28 29.71 51.2 4$\times$2 114564 30.38 28.49 28.59 29.36 29.73 51.2 5$\times$2 108065 28.37 30.16 29.74 29.65 32.76 51.2 6$\times$2 121750 30.88 30.67 29.75 30.49 32.6 51.2 7$\times$2 119274 31.85 31.92 27.21 31.93 33.58 51.2 8$\times$2 118577 30.56 29.65 28.24 32.1 29.16 51.2 9$\times$2 104521 29.57 30.03 25.62 32.12 30.22 51.2 10$\times$2 111100 29.45 29.11 29.65 31.03 35.54 51.2 Medians 30.816 29.952 28.472 30.225 31.818 51.2 1$\times$3 242399 36.45 35.13 34.05 36.48 39.72 96.9 2$\times$3 249240 40.28 39.55 36.66 36 41.11 96.9 3$\times$3 238045 39.79 36.61 34.87 36.76 39.92 96.9 4$\times$3 246187 37.21 38.62 35.04 38.24 39.51 96.9 5$\times$3 229842 36.44 35.87 36.48 39.46 41.09 96.9 6$\times$3 228375 36.34 36.01 36.14 38.63 39.45 96.9 7$\times$3 211540 37.34 36.91 35.92 39.04 38.87 96.9 8$\times$3 225562 40.26 40.27 36.21 37.61 41.78 96.9 9$\times$3 232299 39.72 38.53 33.32 36.2 42.21 96.9 10$\times$3 237518 39.76 37.6 34.73 36.86 41.06 96.9 Medians 38.359 37.51 35.342 37.528 40.472 96.9
 NESL$\times$NBEC RPD LB SA GA ABC ABCWLS GAWLS CPU 1$\times$1 23580 17.66 16.96 18.82 18 17, 14 15 2$\times$1 18989 19.34 20.01 16.08 20.04 21.39 15 3$\times$1 22263 17.99 18.97 16.62 18.83 19.05 15 4$\times$1 23196 20.22 19.23 17.4 18.55 20.13 15 5$\times$1 22223 16.05 17.66 18.19 18.99 18.25 15 6$\times$1 22097 18.05 19.18 16.02 17.39 21.05 15 7$\times$1 20685 20.57 18.87 18.53 18.32 19.08 15 8$\times$1 24351 19.93 21.03 16.56 17.36 21.58 15 9$\times$1 19621 17.45 18.6 18.57 18.92 20.4 15 10$\times$1 23961 20.42 20.57 15.97 20.02 21.16 15 Medians 18.768 19.108 17.276 18.642 19.923 15 1$\times$2 123093 34.99 30.13 26.88 27.85 32.15 51.2 2$\times$2 116383 33.24 31.81 29.88 28.44 32.73 51.2 3$\times$2 125259 28.87 27.55 29.16 29.28 29.71 51.2 4$\times$2 114564 30.38 28.49 28.59 29.36 29.73 51.2 5$\times$2 108065 28.37 30.16 29.74 29.65 32.76 51.2 6$\times$2 121750 30.88 30.67 29.75 30.49 32.6 51.2 7$\times$2 119274 31.85 31.92 27.21 31.93 33.58 51.2 8$\times$2 118577 30.56 29.65 28.24 32.1 29.16 51.2 9$\times$2 104521 29.57 30.03 25.62 32.12 30.22 51.2 10$\times$2 111100 29.45 29.11 29.65 31.03 35.54 51.2 Medians 30.816 29.952 28.472 30.225 31.818 51.2 1$\times$3 242399 36.45 35.13 34.05 36.48 39.72 96.9 2$\times$3 249240 40.28 39.55 36.66 36 41.11 96.9 3$\times$3 238045 39.79 36.61 34.87 36.76 39.92 96.9 4$\times$3 246187 37.21 38.62 35.04 38.24 39.51 96.9 5$\times$3 229842 36.44 35.87 36.48 39.46 41.09 96.9 6$\times$3 228375 36.34 36.01 36.14 38.63 39.45 96.9 7$\times$3 211540 37.34 36.91 35.92 39.04 38.87 96.9 8$\times$3 225562 40.26 40.27 36.21 37.61 41.78 96.9 9$\times$3 232299 39.72 38.53 33.32 36.2 42.21 96.9 10$\times$3 237518 39.76 37.6 34.73 36.86 41.06 96.9 Medians 38.359 37.51 35.342 37.528 40.472 96.9
Median of RPD values and computational time of algorithms for 15 employees
 NESL$\times$ NBEC RPD LB SA GA ABC ABCWLS GAWLS CPU 7$\times$1 20685 3.68 3.45 3.11 3.53 3.65 15 7$\times$2 119274 5.11 4.9 4.65 5.07 5.15 51.2 7$\times$3 228375 6.32 5.98 5.6 5.92 6.44 96.9
 NESL$\times$ NBEC RPD LB SA GA ABC ABCWLS GAWLS CPU 7$\times$1 20685 3.68 3.45 3.11 3.53 3.65 15 7$\times$2 119274 5.11 4.9 4.65 5.07 5.15 51.2 7$\times$3 228375 6.32 5.98 5.6 5.92 6.44 96.9
Three-way ANOVA: RPD versus NBEC, NESL, and Algorithms
 Source F Sig. ($p$) Partial eta squared NBEC 132.802 0.000 0.708 NESL 1.186 0.395 0.155 Algorithms 22.228 0.002 0.626 Interactions NBEC*NESL 1.324 0.199 0.362 NBEC*Algorithms 1.284 0.266 0.226 NESL*Algorithms 0.814 0.748 0.337
 Source F Sig. ($p$) Partial eta squared NBEC 132.802 0.000 0.708 NESL 1.186 0.395 0.155 Algorithms 22.228 0.002 0.626 Interactions NBEC*NESL 1.324 0.199 0.362 NBEC*Algorithms 1.284 0.266 0.226 NESL*Algorithms 0.814 0.748 0.337
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