Optimal two-class-based storage in a live-cube compact storage system

Figures & data

Figure 1. A live-cube compact storage system with a lift.

Figure 2. A virtual aisle in a live-cube system.

Figure 3. Instances of ABC curves with different values for the skewness parameter.

Figure 4. Sub-cases to be optimized individually. R₁ represents the first zone closer to I/O point and R₂ represents the second zone.

Figure 5. Fitted function b*_fit(s) versus optimal values b^*(s) for s ≤ s′ in sub-case A1.

Figure 6. Fitted function b*_fit(s) versus optimal values b^*(s) for s > s′ in sub-case C1.

Table 1. The optimal dimensions and zone boundary (in time units) for any s and V.

Subcase	Skewness parameter (s)	Optimal length (l^*), width (w^*), and height (h^*)	Estimated optimal boundary () b*fit(s) = (a ArcTan[c s + d] + e)V^1/3
A1. b/2 ≤ h & b ≤ l & h ≤ l	s ≤ 0.354 907
C1. b/2 ≤ h & l ≤ b ≤ 2l & h ≤ l	s > 0.354 907

Table 2. The optimal solutions for some selected skewness parameters (ABC curve).

s	ABC curve	Sub-case	l^*(= w^*) (s)	h^* (s)	b^* (s)	G1^*(%)	(s)
0.065	20–90%	A1	1.069 37	0.874 46	0.517	2.0	0.830
0.139	20–80%	A1	1.069 37	0.874 46	0.773	6.7	1.072
0.22	20–70%	A1	1.069 37	0.874 46	0.925	11.6	1.206
0.32	20–60%	A1	1.069 37	0.874 46	1.040	16.4	1.304
0.43	20–50%	C1	1.069 37	0.874 46	1.122	20.6	1.374
0.57	20–40%	C1	1.069 37	0.874 46	1.191	24.6	1.433
0.75	20–30%	C1	1.069 37	0.874 46	1.251	28.3	1.485
1	20–20%	/	1.069 37	0.874 46	/	/	1.531

“/”: for b^* is any value between 0 and 3.0132 (max{l^* + w^*, h^*} + h^*); “/” for G₁^* is any value between 0% and 100%.

Figure 7. Optimal dimensions and first zone boundary in time units for different values of s.

Table 3. Parameters describing storage system and storage policy derived from Hyundai Elevator (2013).

Fixed parameters	Data
Dimensions of a standard vehicle (mm)	Height = 1550, length = 5160, width = 2100^*
Dimensions of a location (mm)	Height = 2945, length = 6400, width = 3200
Shuttle speed in the x-direction (vx) (m/min)	100
Shuttle speed in the y-direction (vy) (m/min)	50
Lift speed (vz) (m/min)	90
Varying parameters	Base example data	Range for scenarios
Dimensions of system (s) ^**	Height = 15.70, length = 19.20, width = 19.20
Dimensions of system (m)	Height = 26.50, length = 38.40, width = 19.20
ABC curve (percentage)	20–80%	[20–20%, 20–90%]
First zone size (percentage of system size)	6.70%^***	[20%, 100%]
System size (no. of slots)^****	324	[96, 8748]
Length/width (no. of tiers)	1	[0.25, 4]

^* Length and width denote the size in the x- and y-directions, respectively.

^** Optimal dimensions of the system for the given system size.

^*** Optimal first zone size for the base example.

^**** Size of the system including empty slots = Length × Width × Height = 6 × 6 × 9 = 324 (in number of tiers).

Table 4. The optimal expected retrieval time for the base case.

ABC curve	Sub-case	Optimal first zone size (%)	Optimal boundary (b) (s)	E[TO] (s) ^*	E[TD] (s)	GapD (%)^**	E[TC] (s)	GapC (%)	E[TR] (s)	GapR (%)
20–80%	A1	6.70	13.86	19.22	18.55	3.51	19.61	2.03	27.48	30.05

^* E[T_x] is the expected retrieval time for method x (in seconds), where x = O (Optimal two-class-based), D (Discrete live-cube system), C (two-class-based system with Cuboid first zone), R (a system with Random storage).

^** Gap_x=((|E[T_x] − E[T_O]|)/E[T_O]) × 100%, represents the gap between the expected retrieval time of the optimal two-class-based live-cube system and the expected retrieval time of method x. For Gap_C, the first zone size and system dimensions of the cuboid two-class-based system are the same as the optimal two-class-based live-cube system.

Table 5. Results of sensitivity analysis when varying skewness parameter.

ABC curve	Sub-case	Optimal boundary (b) (s)	Optimal first zone size (%)	E[TO] (s)	E[TD] (s)	GapD (%)	E[TC] (s)	GapC (%)	E[TR] (s)	GapR (%)
20–90%	A1	9.33	2.0	14.94	14.25	4.62	15.52	3.88	27.48	45.63
20–70%	A1	16.65	11.6	21.68	21.05	2.92	22.01	1.52	27.48	21.11
20–50%	C1	20.14	20.6	24.67	24.47	0.80	24.92	1.01	27.48	10.23
20–30%	C1	22.44	28.3	26.64	26.78	0.53	26.75	0.41	27.48	3.06
20–20%	/	/	/	27.48	27.34	0.51	27.48	0.00	27.48	0.00

“/” means any sub-case, any value for b or any first zone size.

Table 6. Results of sensitivity analysis when varying system size.^*

System size (No. of slots)^**	Relative size change to base case (%)	Sub-case	Optimal boundary (b) (s)	Optimal first zone size (%)	E[TO] (s)	GapO (%)^***	E[TD] (s)	GapD (%)	E[TC] (s)	GapC (%)	E[TR] (s)	GapR (%)
96 (4 × 4 × 6)^****	−70.37^*****	A1	9.24	6.70	12.81	−33.35	12.19	4.84	13.07	2.03	18.31	30.05
768 (8 × 8 × 12)	137.04	A1	18.48	6.70	25.63	33.35	24.97	2.57	26.15	2.03	36.64	30.05
1500 (10 × 10 × 15)	362.96	A1	23.11	6.70	32.05	66.73	32.72	2.10	32.70	2.03	45.81	30.05
2592 (12 × 12 × 18)	700.00	A1	27.73	6.70	38.45	100.05	39.18	1.90	39.23	2.03	54.97	30.05
8748 (18 × 18 × 27)	2600.00	A1	41.59	6.70	57.68	200.10	58.62	1.63	58.85	2.03	82.46	30.05

^* For each instance, the dimensions and first zone boundary are optimal.

^** Including empty slots.

^*** Gap_O=((E[T_O] − E[T_base case])/E[T_base case]) × 100%, represents the gap between E[T] of base case (19.22 s) and E[T] of each instance.

^**** Length × Width × Height in number of tiers.

^***** ((96–324)/324) × 100%.

Table 7. Results of sensitivity analysis when varying the first zone size.

First zone size (%)	Sub-case	Boundary (b) (s)	E[T1st zone] (s)	Gap1st zone (%)^*	E[TD] (s)	GapD (%)	E[TC] (s)	GapC (%)	E[TR] (s)	GapR (%)
20	C1	19.94	20.04	4.09	19.20	4.19	20.70	3.29	27.48	27.07
40	C1	25.41	21.76	11.67	21.09	3.08	22.86	5.06	27.48	20.82
60	C1	29.83	23.48	18.14	22.86	2.62	24.44	4.09	27.48	14.56
80	O1	34.98	25.22	23.79	25.14	0.32	26.03	3.21	27.48	8.22
100	/	/	27.48	30.06	27.41	0.25	27.48	0.00	27.48	0.00

^* Gap_1st zone = ((E[T_1st zone] − E[T_base case])/E[T_base case]) × 100%, represents the gap between E[T] of base case (19.22 s) and E[T] of each instance.

Table 8. Results of sensitivity analysis when varying the shape factor (length/width in number of tiers).^*

Shape factor (L/W)	Sub-case	Optimal boundary (b) (s)	Optimal first zone size (%)	E[TO] (s)	E[TD] (s)	GapD (%)	E[TC] (s)	GapC (%)	E[TR] (s)	GapR (%)
4.00 (12/3)	D2	15.69	9.38	21.98	21.14	3.82	23.99	9.14	33.27	33.93
2.25 (9/4)	B2	14.59	8.07	20.14	19.45	3.43	21.05	4.52	29.42	31.54
1.00 (6/6)	A1	13.86	6.70	19.22	18.55	3.51	19.61	2.03	27.48	30.05
0.44 (4/9)	B2	14.59	8.07	20.14	19.45	3.43	21.05	4.52	29.42	31.54
0.25 (3/12)	D2	15.69	9.38	21.98	21.14	3.82	23.99	9.14	33.27	33.93

^* The system size (in number of slots) and height (in number of tiers) are fixed for all instances.

Table A1. The 36 sub-cases, each corresponding to a specific E[T] formula.

	Configuration
Case	1(h ⩽ w)	2 (w ⩽ h ⩽ l)	3 (l ⩽ h ⩽ l + w)	4 (l + w ⩽ h)
A (b/2 ≤ h and b ≤ w)	A1	A2	A3	A4
B (b/2 ≤ h and w ≤ b ≤ 2w and b ≤ 1)	B1	B2	B3	B4
C (b/2 ≤ h and l ≤ b ≤ 2w)	C1	C2	C3	C4
D (b/2 ≤ h 2 and w ≤ b ≤ l)	—	D2	D3	D4
E (b/2 ≤ h 2 and w ≤ b and l ≤ b ≤ l + w)	—	E2	E3	E4
F (b/2 ≤ h and l + w ≤ b ≤ 2l)	—	F2	F3	F4
G (b/2 ≤ h 2 and l ≤ b ≤ 2(l + w))	—	—	G3	G4
H (b/2 ≤ h 2 and (l + w) ≤ b ≤ 2h)	—	—	—	H4
I (b/2 ≥ h and b ≤ w)	I1	—	—	—
J (b/2 ≥ h and w ≤ b ≤ w + h and b ≤ l)	J1	—	—	—
K (b/2 ≥ h and l ≤ b ≤ w + h)	K1	—	—	—
L (b/2 ≥ h and w + h ≤ b ≤ l)	L1	L2	—	—
M (b/2 ≥ h and w + h≤ b and l ≤ b ≤ min{l + w,  l + h})	M1	M2	—	—
N (b/2 ≥ h and l + w ≤ b ≤ l + h)	—	N2	—	—
O (b/2 ≥ h and l + h ≤ b ≤ l + w)	O1	—	—	—
P (b/2 ≥ h and max{l + h,  l + w}≤ b≤ + w + h)	P1	P2	P3	—

Table A2. Optimal solutions of sub-cases I1, K1, O1, P1.

Subcases (conditions)	s	Optimal length (l^*)	Optimal boundary (b^*)
I1. b/2 ≥ h and b ≤ l and h ≤ l	s > 0.275732	(0.5(67 986.3 ArcTan[663 825.0s + 50 017.7] − 106 791.0))^− 1/2	67 986.3 ArcTan[663 825.0s + 50 017.7] − 106 791.0
	s ≤ 0.275732	1.259 92	1.259 92
K1.1. 2h ≤ b ≤ l + h and h ≤ l	s > 0.275732	(0.5(39 587.7 ArcTan[18 516.9s + 22 176.5] − 62 181.4))^− 1/2	39 587.7 ArcTan[18 516.9s + 22 176.5] − 62 181.4
	s ≤ 0.275732	1.259 92	1.259 92
K1.2. l ≤ b ≤ l + h and h ≤ l	s > 0.663535	1.259 92	1.259 92
	s ≤ 0.663535	1.259 92	19.2377 ArcTan[63.4077s + 19.408] − 28.6456
O1. b/2 ≥ h and l + h ≤ b ≤ 2l and h ≤ l	0 < s ≤ 1	0.833 ArcTan[ − 10.333s − 2.965] + 2.315
P1. 2l ≤ b ≤ 2l + h and h ≤ l	s > 0.278079	1	2
	s ≤ 0.278079	3932.9 ArcTan[32 245.0s + 17 556.2] − 6176.6	2(3932.9 ArcTan[32 245.0s + 17 556.2] − 6176.6)

Table A3. Set of possible sub-cases for any given system configuration: (a) configuration 1; (b) configuration 2; (c) configuration 3; and (d) configuration 4.

Config. 1	Sub-config.	Sub-config.	Boundary	Sub-case	Config. 2	Sub-config.	Sub-config.	Boundary	Sub-case
h ≤ w	w ≤ 2h	l ≤ 2h	b ≤ w	A1	w ≤ h ≤ l	l + w ≥ 2h	l ≤ 2w	b ≤ w	A2
			w ≤ b ≤ l	B1				w ≤ b ≤ l	B2
			l ≤ b ≤ 2h	C1				l ≤ b ≤ 2w	C2
			2h ≤ b ≤ w + h	K1				2w ≤ b ≤ 2h	E2
			w + h ≤ b ≤ l + h	M1				2h ≤ b ≤ l + w	M2
			l + h ≤ b ≤ l + w	O1				l + w ≤ b ≤ l + h	N2
			l + w ≤ b ≤ l + w + h	P1				l + h ≤ b ≤ l + w + h	P2
		2h ≤ l ≤ w + h	b ≤ w	A1			2w ≤ l ≤ 2h	b ≤ w	A2
			w ≤ b ≤ 2h	B1				w ≤ b ≤ 2w	B2
			2h ≤ b ≤ l	J1				2w ≤ b ≤ l	D2
			l ≤ b ≤ w + h	K1				l ≤ b ≤ 2h	E2
			w + h ≤ b ≤ l + h	M1				2h ≤ b ≤ l + w	M2
			l + h ≤ b ≤ l + w	O1				l + w ≤ b ≤ l + h	N2
			l + w ≤ b ≤ l + w + h	P1				l + h ≤ b ≤ l + w + h	P2
		w + h ≤ l	b ≤ w	A1			2h ≤ l	b ≤ w	A2
			w ≤ b ≤ 2h	B1				w ≤ b ≤ 2w	B2
			2h ≤ b ≤ w + h	J1				2w ≤ b ≤ 2h	D2
			w + h ≤ b ≤ l	L1				2h ≤ b ≤ l	L2
			l ≤ b ≤ l + h	M1				l ≤ b ≤ l + w	M2
			l + h ≤ b ≤ l + w	O1				l + w ≤ b ≤ l + h	N2
			l + w ≤ b ≤ l + w + h	P1				l + h ≤ b ≤ l + w + h	P2
	w ≥ 2h	2h ≤ l ≤ w + h	b ≤ 2h	A1		l + w ≤ 2h	l ≤ 2w	b ≤ w	A2
			2h ≤ b ≤ w	I1				w ≤ b ≤ l	B2
			w ≤ b ≤ l	J1				l ≤ b ≤ 2w	C2
			l ≤ b ≤ w + h	K1				2w ≤ b ≤ l + w	E2
			w + h ≤ b ≤ l + h	M1				l + w ≤ b ≤ 2h	F2
			l + h ≤ b ≤ l + w	O1				2h ≤ b ≤ l + h	N2
			l + w ≤ b ≤ l + w + h	P1				l + h ≤ b ≤ l + w + h	P2
		w + h ≤ l	b ≤ 2h	A1			2w ≤ l	b ≤ w	A2
			2h ≤ b ≤ w	I1				w ≤ b ≤ 2w	B2
			w ≤ b ≤ w + h	J1				2w ≤ b ≤ l	D2
			w + h ≤ b ≤ l	L1				l ≤ b ≤ l + w	E2
			l ≤ b ≤ l + h	M1				l + w ≤ b ≤ 2h	F2
			l + h ≤ b ≤ l + w	O1				2h ≤ b ≤ l + h	N2
			1 + w ≤ b ≤ l + w + h	P1				l + h ≤ b ≤ l + w + h	P2
(a)					(b)
Config. 3		Sub-config	Boundary	Sub-case	Config. 4		Sub-config	Boundary	Sub-case
l ≤ h ≤ l + w		l ≤ 2w	b ≤ w	A3	l + w ≤ h		l ≤ 2w	b ≤ w	A4
			w ≤ b ≤ l	B3				w ≤ b ≤ l	B4
			l ≤ b ≤ 2w	C3				l ≤ b ≤ 2w	C4
			2w ≤ b ≤ l + w	E3				2w ≤ b ≤ l + w	E4
			l + w ≤ b ≤ 2l	F3				l + w ≤ b ≤ 2l	F4
			2l ≤ b ≤ 2h	G3				2l ≤ b ≤ 2(l + w)	G4
			2h ≤ b ≤ l + w + h	P3				2(l + w) ≤ b ≤ 2h	H4
		l ≥ 2w	b ≤ w	A3			l ≥ 2w	b ≤ w	A4
			w ≤ b ≤ 2w	B3				w ≤ b ≤ 2w	B4
			2w ≤ b ≤ l	D3				2w ≤ b ≤ l	D4
			l ≤ b ≤ l + w	E3				l ≤ b ≤ l + w	E4
			l + w ≤ b ≤ 2l	F3				l + w ≤ b ≤ 2l	F4
			2l ≤ b ≤ 2h	G3				2l ≤ b ≤ 2(l + w)	G4
			2h ≤ b ≤ l + w + h	P3				2(l + w) ≤ b ≤ 2h	H4
(c)					(d)

Hyundai Elevator. (2013) Hyundai integrated parking system (HIP). http://hyundaielevator.co.kr/eng/parking/car/automobile.jsp. [13 March 2017]

 Google Scholar