American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022009
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Dynamic virtual cellular reconfiguration for capacity planning of market-oriented production systems

 1 Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan city, Hubei province, China 2 School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou city, China

*Corresponding author: Lei Yue

Received  July 2021 Revised  October 2021 Early access January 2022

Fund Project: The authors are supported by the Guangdong Province Key Field R & D Program (2020B0101050001) and the National Natural Science Foundation of China (No. 51905196)

Market-oriented production systems generally have the characteristics of multi-product and small-batch production. Dynamic virtual cellular manufacturing systems create virtual manufacturing cells periodically in a planning horizon to respond to changing demands flexibly and quickly, and thus are suitable for production planning problems of market-oriented production systems. In the current research, we propose a dynamic virtual cell reconfiguration framework under a dynamic environment with unstable demands and multiple planning cycles. In this framework, we formulate a two-phase dynamic virtual cell formation (DVCF) model. In the first phase, the proposed model aims to maximize processing similarity and balance the workload in the system. In the second phase, we consider the objective of reconfiguration stability based on the first phase model. To address the proposed model, we design a hybrid metaheuristic named Lévy-NSGA-Ⅱ, and perform various computational experiments to examine the performance of the proposed algorithm. Results of experiments indicate that the proposed Lévy-NSGA-Ⅱ based algorithm outperforms multi-objective cuckoo search (MOCS) and NSGA-Ⅱ in solution quality and optimal solution size.

Citation: Zhengmin Zhang, Zailin Guan, Weikang Fang, Lei Yue. Dynamic virtual cellular reconfiguration for capacity planning of market-oriented production systems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022009
References:

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References:
The dynamic virtual cell reconfiguration framework under multiple planning cycles
Discrete Lévy flight search strategy
Flow chart of Lévy-NSGA-Ⅱ
Schematic structure of the chromosomes in Lévy-NSGA-II
Number of machine allocation for all cells
Crossover and mutation operations
Local random search
Global random search
The applications of Lévy-NSGA-II in the two-phase DVCF model
Pareto-optimal solutions for A1-A6
Pareto-optimal solutions for B1-B6
Comparisons of Lévy-NSGA-II, NSGA-II, and MOCS
The Pareto-optimal solutions for the first phase
The Pareto-optimal solutions for the second period
Notations used in the DVCF model
 Indices $j$ part types, $j=1, 2, \cdots, J$ $r$ process routings, $r =1, 2, \cdots, R_j$ $m$ machine types, $m=1, 2, \cdots, M$ $c$ virtual cells, $c=1, 2, \cdots, C$ $t$ formation periods Input parameters $J_t$ number of part types in period $t$ $R_j$ number of process routings for part type $j$ M number of machine types $N_m$ number of machines included in machine type $m$ $B_U$ upper bounds of virtual cells $B_L$ lower bounds of virtual cells $D_{j, t}$ demand for part type $j$ in period $t$ $A_m$ production capacity of each machine of type $m$ $\alpha_{j, r, m}$ 1, if $r$-th process routing of part type $j$ needs to use the machine type $m$, 0 otherwise $T_{j, r, m}$ processing time of $r$-th process routing of part type $j$ at machine type $m$ $S_{j, j', r, r'}$ the similarity coefficient between $r$-th process routing of part type $j$ and $r'$-th process routing of part type $j'$ $Z_{m, c, t-1}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t-1\ (t>1)$ Variables $C_t$ number of virtual cells in period $t$, $B_L\leq C \leq B_U$ $X_{j, r, c, t}$ 1, part type $j$ to be assigned to routing $r$ and to be assigned to virtual cell $c$ in period $t$, 0 otherwise $Y_{m, c, t}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t$ (real number) $Z_{m, c, t}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t$ (integer)
 Indices $j$ part types, $j=1, 2, \cdots, J$ $r$ process routings, $r =1, 2, \cdots, R_j$ $m$ machine types, $m=1, 2, \cdots, M$ $c$ virtual cells, $c=1, 2, \cdots, C$ $t$ formation periods Input parameters $J_t$ number of part types in period $t$ $R_j$ number of process routings for part type $j$ M number of machine types $N_m$ number of machines included in machine type $m$ $B_U$ upper bounds of virtual cells $B_L$ lower bounds of virtual cells $D_{j, t}$ demand for part type $j$ in period $t$ $A_m$ production capacity of each machine of type $m$ $\alpha_{j, r, m}$ 1, if $r$-th process routing of part type $j$ needs to use the machine type $m$, 0 otherwise $T_{j, r, m}$ processing time of $r$-th process routing of part type $j$ at machine type $m$ $S_{j, j', r, r'}$ the similarity coefficient between $r$-th process routing of part type $j$ and $r'$-th process routing of part type $j'$ $Z_{m, c, t-1}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t-1\ (t>1)$ Variables $C_t$ number of virtual cells in period $t$, $B_L\leq C \leq B_U$ $X_{j, r, c, t}$ 1, part type $j$ to be assigned to routing $r$ and to be assigned to virtual cell $c$ in period $t$, 0 otherwise $Y_{m, c, t}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t$ (real number) $Z_{m, c, t}$ number of machines of type $m$ assigned to virtual cell $c$ in period $t$ (integer)
Pattern of data generation
 Parameter Generation pattern Parameter Generation pattern $M$ 8 $R_j$ U [1,3] $\sum_{m=1}^{M}{N_m}$ U [2J, 3J] $\sum_{m=1}^{M}{\alpha_{j, r, m}}$ U [2,6] $B_U$ Random{3, 4} $T_{j, r, m}$ U 10 * [4,32] $B_L$ Random{2, 3} $A_m$ U 100 * [10,30] $D_j$ U [0, 10]
 Parameter Generation pattern Parameter Generation pattern $M$ 8 $R_j$ U [1,3] $\sum_{m=1}^{M}{N_m}$ U [2J, 3J] $\sum_{m=1}^{M}{\alpha_{j, r, m}}$ U [2,6] $B_U$ Random{3, 4} $T_{j, r, m}$ U 10 * [4,32] $B_L$ Random{2, 3} $A_m$ U 100 * [10,30] $D_j$ U [0, 10]
Type and dimension of test problems
 Size1 Size2 Size3 Size4 Size5 Size6 (15*40*28) (18*49*33) (21*55*39) (24*60*46) (27*69*54) (30*76*62) $T=1$ A1 A2 A3 A4 A5 A6 $T>1$ B1 B2 B3 B4 B5 B6
 Size1 Size2 Size3 Size4 Size5 Size6 (15*40*28) (18*49*33) (21*55*39) (24*60*46) (27*69*54) (30*76*62) $T=1$ A1 A2 A3 A4 A5 A6 $T>1$ B1 B2 B3 B4 B5 B6
The Pareto-optimal solutions for problems A3
 Solution number Lévy-NSGA-Ⅱ Solution number MOCS Solution number NSGA-Ⅱ Dissimilarity coefficient Workload balance Dissimilarity coefficient Workload balance Dissimilarity coefficient Workload balance 1 7.533333 2.022256 1 7.939286 1.976695 1 7.804762 2.074396 2 7.719048 1.921483 2 8.025000 1.830073 2 7.829762 1.976695 3 7.833333 1.855261 3 8.092063 1.802253 3 7.876190 1.918146 4 7.901190 1.851695 4 8.177778 1.629786 4 7.954762 1.830073 5 7.933730 1.835300 5 8.344444 1.503174 5 8.005159 1.694708 6 7.954762 1.830073 6 8.563492 1.457544 6 8.229762 1.661732 7 8.005159 1.694708 7 8.683730 1.391009 7 8.308333 1.653314 8 8.254762 1.629786 8 8.790476 1.266009 8 8.379762 1.559564 9 8.282143 1.490863 9 9.265476 1.224086 9 8.430159 1.503174 10 8.560714 1.457544 10 9.383333 1.167368 10 8.626190 1.470198 11 8.711111 1.391009 11 9.487302 1.131726 11 8.656349 1.391009 12 8.717063 1.297040 12 8.727778 1.266009 13 8.727778 1.266009 13 9.080556 1.224086 14 9.059524 1.224086 14 9.364286 1.131726 15 9.255952 1.167368 16 9.267857 1.131726
 Solution number Lévy-NSGA-Ⅱ Solution number MOCS Solution number NSGA-Ⅱ Dissimilarity coefficient Workload balance Dissimilarity coefficient Workload balance Dissimilarity coefficient Workload balance 1 7.533333 2.022256 1 7.939286 1.976695 1 7.804762 2.074396 2 7.719048 1.921483 2 8.025000 1.830073 2 7.829762 1.976695 3 7.833333 1.855261 3 8.092063 1.802253 3 7.876190 1.918146 4 7.901190 1.851695 4 8.177778 1.629786 4 7.954762 1.830073 5 7.933730 1.835300 5 8.344444 1.503174 5 8.005159 1.694708 6 7.954762 1.830073 6 8.563492 1.457544 6 8.229762 1.661732 7 8.005159 1.694708 7 8.683730 1.391009 7 8.308333 1.653314 8 8.254762 1.629786 8 8.790476 1.266009 8 8.379762 1.559564 9 8.282143 1.490863 9 9.265476 1.224086 9 8.430159 1.503174 10 8.560714 1.457544 10 9.383333 1.167368 10 8.626190 1.470198 11 8.711111 1.391009 11 9.487302 1.131726 11 8.656349 1.391009 12 8.717063 1.297040 12 8.727778 1.266009 13 8.727778 1.266009 13 9.080556 1.224086 14 9.059524 1.224086 14 9.364286 1.131726 15 9.255952 1.167368 16 9.267857 1.131726
Comparisons of 6 sets of test problems
 Problem No Pareto distance $V_{pd}$ Pareto distance $V_{np}$ Pareto distance $V_{rd}$ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ A1 0.1016 0.0484 0.0158 6 8 10 0.0367 0.0333 0.0333 A2 0.0754 0.0346 0.0286 2 6 10 0.0367 0.0400 0.0333 A3 0.1460 0.0644 0.0030 1 4 14 0.0367 0.0467 0.0533 A4 0.2443 0.5967 0.0014 0 1 10 0.0367 0.0267 0.0400 A5 2.0610 2.3134 0.0001 0 2 18 0.0367 0.0233 0.0633 A6 2.3025 2.7586 0.0000 0 0 16 0.0500 0.0500 0.0533 B1 0.4266 1.5974 0.0506 32 27 42 0.1333 0.1000 0.1467 B2 1.0217 2.0321 0.0123 18 18 51 0.1067 0.1167 0.1800 B3 0.3373 0.4506 0.2616 50 67 107 0.2600 0.3000 0.3900 B4 0.7522 1.6053 0.5043 48 20 67 0.3600 0.3633 0.2800 B5 1.5456 3.0284 0.4457 26 43 42 0.2267 0.2233 0.2600 B6 0.8931 2.0602 0.3225 29 16 100 0.2700 0.2400 0.4267
 Problem No Pareto distance $V_{pd}$ Pareto distance $V_{np}$ Pareto distance $V_{rd}$ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ MOCS NSGA-Ⅱ Lévy-NSGA-Ⅱ A1 0.1016 0.0484 0.0158 6 8 10 0.0367 0.0333 0.0333 A2 0.0754 0.0346 0.0286 2 6 10 0.0367 0.0400 0.0333 A3 0.1460 0.0644 0.0030 1 4 14 0.0367 0.0467 0.0533 A4 0.2443 0.5967 0.0014 0 1 10 0.0367 0.0267 0.0400 A5 2.0610 2.3134 0.0001 0 2 18 0.0367 0.0233 0.0633 A6 2.3025 2.7586 0.0000 0 0 16 0.0500 0.0500 0.0533 B1 0.4266 1.5974 0.0506 32 27 42 0.1333 0.1000 0.1467 B2 1.0217 2.0321 0.0123 18 18 51 0.1067 0.1167 0.1800 B3 0.3373 0.4506 0.2616 50 67 107 0.2600 0.3000 0.3900 B4 0.7522 1.6053 0.5043 48 20 67 0.3600 0.3633 0.2800 B5 1.5456 3.0284 0.4457 26 43 42 0.2267 0.2233 0.2600 B6 0.8931 2.0602 0.3225 29 16 100 0.2700 0.2400 0.4267
Parts information for the numerical example
 Parts Routes Operation Demand Time Parts Routes Operation Demand Time P1 R1 1 5, 7 360 P7 R3 2 90 8 90 8 120 6 160 5 160 2 180 2 50 R2 8 90 P8 R1 1 4, 7 360 5 180 7 90 1 360 2 160 6 160 4 180 2 180 6 90 P2 R1 2 10, 8 180 R2 2 180 6 80 8 360 4 120 5 160 7 90 2 340 R2 1 160 P9 R1 1 7, 5 360 5 90 5 160 3 120 2 180 7 90 3 190 R3 4 120 7 80 1 160 1 160 5 60 R2 1 360 3 90 8 100 P3 R1 8 0, 5 80 2 160 1 160 4 180 6 80 6 90 2 200 2 180 P4 R1 2 5, 0 120 P10 R1 4 0, 10 90 6 100 8 100 1 200 2 120 4 140 5 60 1 150 R2 6 60 P5 R1 2 6, 10 120 1 140 3 60 7 100 7 140 3 80 R2 1 120 R3 5 60 4 60 2 100 7 120 7 100 5 120 3 70 P6 R1 2 4, 0 120 P11 R1 2 10, 0 360 7 60 3 90 6 100 7 120 2 120 1 180 P7 R1 1 6, 4 90 4 90 5 160 7 40 2 50 5 360 7 120 P12 R1 4 9, 8 360 R2 2 120 6 90 8 60 2 300 6 80 8 180 2 120 5 90
 Parts Routes Operation Demand Time Parts Routes Operation Demand Time P1 R1 1 5, 7 360 P7 R3 2 90 8 90 8 120 6 160 5 160 2 180 2 50 R2 8 90 P8 R1 1 4, 7 360 5 180 7 90 1 360 2 160 6 160 4 180 2 180 6 90 P2 R1 2 10, 8 180 R2 2 180 6 80 8 360 4 120 5 160 7 90 2 340 R2 1 160 P9 R1 1 7, 5 360 5 90 5 160 3 120 2 180 7 90 3 190 R3 4 120 7 80 1 160 1 160 5 60 R2 1 360 3 90 8 100 P3 R1 8 0, 5 80 2 160 1 160 4 180 6 80 6 90 2 200 2 180 P4 R1 2 5, 0 120 P10 R1 4 0, 10 90 6 100 8 100 1 200 2 120 4 140 5 60 1 150 R2 6 60 P5 R1 2 6, 10 120 1 140 3 60 7 100 7 140 3 80 R2 1 120 R3 5 60 4 60 2 100 7 120 7 100 5 120 3 70 P6 R1 2 4, 0 120 P11 R1 2 10, 0 360 7 60 3 90 6 100 7 120 2 120 1 180 P7 R1 1 6, 4 90 4 90 5 160 7 40 2 50 5 360 7 120 P12 R1 4 9, 8 360 R2 2 120 6 90 8 60 2 300 6 80 8 180 2 120 5 90
Process-machine incidence matrix for the numerical example
 Operation Machine type Machine name Machine number Available processing time /min 1 M1 CNC lathes 5 3000 2 M2 Ordinary lathes 5 3000 3 M3 Slotting machines 2 1700 4 M4 CNC slotting machines 5 1600 5 M5 Grinders 6 1200 6 M6 Grinding machines 4 1200 7 M7 Gun Drill 3 1600 8 M8 Drilling machines 3 1200
 Operation Machine type Machine name Machine number Available processing time /min 1 M1 CNC lathes 5 3000 2 M2 Ordinary lathes 5 3000 3 M3 Slotting machines 2 1700 4 M4 CNC slotting machines 5 1600 5 M5 Grinders 6 1200 6 M6 Grinding machines 4 1200 7 M7 Gun Drill 3 1600 8 M8 Drilling machines 3 1200
One of the schemes for the first period of the numerical example
 Cell Part(routing) Number of each machine type in each cell f1 f2 M1 M2 M3 M4 M5 M6 M7 M8 1 1(2), 4(1), 8(1), 9(2), 12(1) 3 3 0 4 2 3 1 3 4.0190 0.5939 2 2(2), 5(2), 11(1) 2 2 2 1 5 0 3 0 3 6(1), 7(2) 0 1 0 0 0 1 1 1
 Cell Part(routing) Number of each machine type in each cell f1 f2 M1 M2 M3 M4 M5 M6 M7 M8 1 1(2), 4(1), 8(1), 9(2), 12(1) 3 3 0 4 2 3 1 3 4.0190 0.5939 2 2(2), 5(2), 11(1) 2 2 2 1 5 0 3 0 3 6(1), 7(2) 0 1 0 0 0 1 1 1
One of the schemes for the second period of the numerical example
 Number of each machine type in each cell f1 f2 f3 Cell Part(routing) M1 M2 M3 M4 M5 M6 M7 M8 1 1(2), 8(1), 9(2), 12(1) 3 3 0 4 2 3 1 3 3.3619 0.6384 6 2 2(2), 5(2), 7(1), 10(3) 1 1 1 1 3 0 3 0 3 3(1) 1 1 0 0 0 1 0 1
 Number of each machine type in each cell f1 f2 f3 Cell Part(routing) M1 M2 M3 M4 M5 M6 M7 M8 1 1(2), 8(1), 9(2), 12(1) 3 3 0 4 2 3 1 3 3.3619 0.6384 6 2 2(2), 5(2), 7(1), 10(3) 1 1 1 1 3 0 3 0 3 3(1) 1 1 0 0 0 1 0 1
Detailed machine changes between two formation periods
 M1 M2 M3 M4 M5 M6 M7 M8 Cell 1 0 0 0 0 0 0 0 0 Cell 2 -1 -1 -1 0 -2 0 0 0 Cell 3 +1 0 0 0 0 0 -1 0
 M1 M2 M3 M4 M5 M6 M7 M8 Cell 1 0 0 0 0 0 0 0 0 Cell 2 -1 -1 -1 0 -2 0 0 0 Cell 3 +1 0 0 0 0 0 -1 0
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