Clusters and filling-curve-based storage assignment in a circuit board assembly kitting area

Figures & data

Fig. 1 One-dimensional rack.

Fig. 2 A feasible filling curve of a square-in-time automatic storage and retrieval system.

Fig. 3 Picking route of a cluster stored along a filling curve.

Fig. 4 Two filling curves in an 8 × 8 square space.

Fig. 5 Multiple filling curves in an 8 × 8 square space.

Fig. 6 Multiple one-dimensional racks.

Fig. 7 Zoning in two one-dimensional racks.

Fig. 8 Zoning along two parallel one-dimensional racks.

Fig. 9 The dual filling curves of the 2p × 2q space.

Fig. 10 (a) A straight path; and (b) a straight-and-turn configuration.

Table 1 Estimated travel distance as a function of the number of extra clusters visited (m)

	Average m
	0	1	2	3	4	5	6	7
E(T)	65 283	75 241	83 636	91 033	97 720	103 869	109 593	114 968
Percentage reduction	46.5	38.3	31.5	25.4	19.9	14.9	10.2	5.8

Table 2 Curve parameters and travel distance using different filling curves

	Description	μω	σω	λω
ω 2	Non-decreasing rectilinear distance	1.89	0	2.30
ω 3	Variant of Moore's Hilbert space-filling curve	1	0	3.02/4.63
ω 4	Multiple filling curves	1	0	0
ω 5	Dual filling curve	1	0	1.75
ω 6	Multiple dual filling curves	1	0	0

Table 3 Estimated travel distance of different data sets

Curve	ω2	ω3	ω4	ω5	ω6	Random	Lower
Data set 1	80 321	64 314	64 775	57 775	55 324	122 044	36 129
Data set 2	116 783	91 324	94 599	88 455	79 337	174 938	52 275
Data set 3	66 569	54 053	53 991	50 146	44 845	105 163	31 913
Data set 4	104 638	85 337	86 357	79 412	71 988	163 775	50 854
Average percentage improvement	35	48	47	51	56

Table 4 Clustering result using different β values

	β
	0.5	0.6	0.7	0.8	0.9
Γ*	0.68	0.69	0.72	0.74	0.76
Average m	7.94	7.62	7.65	7.64	7.15
Percentage improvement	29.5	30.5	31.0	31.4	27.6

Fig. 11 Savings obtained using a common pick area with different sizes.

Fig. A1 Zoning result on three one-dimensional racks using Algorithm 1.

Table A1 The generated data sets

	Data set 1			Data set 2			Data set 3			Data set 4
	BOMs	r n	f n	BOMs	r n	f n	BOMs	r n	f n	BOMs	r n	f n
1	5	10	127	5	12	224	5	6	149	5	5	123
2	4	16	145	4	11	140	4	7	97	4	5	117
3	6	11	99	6	10	124	6	6	82	6	10	196
4	6	11	96	6	11	120	6	10	90	6	5	83
5	4	18	144	4	14	147	4	7	61	4	6	89
6	4	10	79	4	18	157	4	10	60	4	7	102
7	5	11	65	5	16	132	5	15	87	5	8	114
8	5	13	76	5	11	90	5	8	46	5	9	110
9	4	12	65	4	16	124	4	10	56	4	21	182
10	6	17	89	6	14	108	6	22	115	6	16	137
11	3	11	55	3	17	129	3	10	49	3	17	133
12	5	14	66	5	12	86	5	22	87	5	20	135
13	5	15	69	5	11	74	5	12	46	5	14	86
14	3	19	78	3	19	118	3	18	67	3	19	104
15	4	15	60	4	17	89	4	20	60	4	15	79
16	4	17	68	4	18	86	4	21	62	4	20	100
17	4	18	65	4	19	90	4	21	59	4	18	76
18	9	16	43	9	14	65	9	16	43	9	17	70
19	6	16	41	6	17	72	6	23	57	6	24	92
20	8	19	43	8	18	62	8	21	35	8	24	61
Total	100	289	1573	100	295	2237	100	285	1408	100	280	2189