Dynamic virtual cellular reconfiguration for capacity planning of market-oriented production systems

Zhengmin Zhang; Zailin Guan; Weikang Fang; Lei Yue

doi:10.3934/jimo.2022009

Article Contents

2023, Volume 19, Issue 3: 1611-1635. 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
^*Corresponding author: Lei Yue

Received: June 30, 2021

Revised: September 30, 2021

Early access: January 2022

Published: March 2023

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 90B50, 90B30.

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Figure 1. The dynamic virtual cell reconfiguration framework under multiple planning cycles

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Figure 2. Discrete Lévy flight search strategy

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Figure 3. Flow chart of Lévy-NSGA-Ⅱ

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Figure 4. Schematic structure of the chromosomes in Lévy-NSGA-II

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Figure 5. Number of machine allocation for all cells

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Figure 6. Crossover and mutation operations

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Figure 7. Local random search

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Figure 8. Global random search

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Figure 9. The applications of Lévy-NSGA-II in the two-phase DVCF model

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Figure 10. Pareto-optimal solutions for A1-A6

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Figure 11. Pareto-optimal solutions for B1-B6

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Figure 12. Comparisons of Lévy-NSGA-II, NSGA-II, and MOCS

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Figure 13. The Pareto-optimal solutions for the first phase

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Figure 14. The Pareto-optimal solutions for the second period

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Table 1. 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)

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Table 2. 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]

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Table 3. Type and dimension of test problems

	Size1	Size2	Size3	Size4	Size5	Size6
	(154028)	(184933)	(215539)	(246046)	(276954)	(307662)
$ T=1 $	A1	A2	A3	A4	A5	A6
$ T>1 $	B1	B2	B3	B4	B5	B6

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Table 4. The Pareto-optimal solutions for problems A3

Solution number	Lévy-NSGA-Ⅱ		Solution number	MOCS		Solution number	NSGA-Ⅱ
Solution number	Dissimilarity coefficient	Workload balance	Solution number	Dissimilarity coefficient	Workload balance	Solution number	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

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Table 5. Comparisons of 6 sets of test problems

Problem No	Pareto distance $ V_{pd} $			Pareto distance $ V_{np} $			Pareto distance $ V_{rd} $
Problem No	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

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Table 6. 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

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Table 7. 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

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Table 8. 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
Cell	Part(routing)	M1	M2	M3	M4	M5	M6	M7	M8	f1	f2
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

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Table 9. 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	f1	f2	f3
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

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Table 10. 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

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