Travel time model for multi-deep automated storage and retrieval systems with different storage strategies

Figures & data

Figure 1. View from the front and top on a rack in a triple-deep AS/RS (Lehmann and Hußmann 2022).

A triple-deep rack with descriptions. A top view and a front view are available. A S/R machine is placed in front of the rack in both views. The I/O point is located on the lower left corner of the rack.

Table 1. Notation for the travel time calculation in this work.

$a_x$, $a_y$	Acceleration and deceleration of the S/R machine
$a_n$	Acceleration and deceleration of the telescopic arm/satellite vehicle
$C_k$	Storage channel state with k storage goods
$l_x$, $l_y$, $l_n$	The length of a storage location in horizontal, vertical and depth directions
k	Number of storage goods in a storage channel
M	Transition matrix of a Markov chain
n	Depth of storage channels
$P\left(S_k\right)$	Probability that a storage operation is carried out into a storage channel of type $C_k$
$P\left(p_k\right|S)$	Conditional probability that a storage channel of type $C_k$ is chosen during a storage operation or the relocation-storage component
$P\left(S\right|p_k)$	Conditional probability that a storage operation or the relocation-storage component is carried out when regarding storage channels of type $C_k$
$P\left(R_k\right)$	Probability that a retrieval operation retrieves a storage good from a storage channel of type $C_k$
$P\left(p_k\right|R)$	Conditional probability that a storage channel of type $C_k$ is chosen during a retrieval operation
$P\left(R\right|p_k)$	Conditional probability that a relocation operation retrieves a storage good when regarding storage channels of type $C_k$
$p_k$	Probability that a random storage channel is in the channel state $C_k$ and has k storage goods inside
$p\left(\text{β}\right)$	Probability of relocations during a storage or retrieval operation in a n-deep AS/RS
S; R	Storage or retrieval operation
$S_k$; $R_k$	Storage or retrieval operation into or from a storage channel of type $C_k$
$t_A$	Average movement time of the S/R machine from the I/O-point to the storage/retrieval channel and vice versa
$t_{C,S}$	Channel time of the telescopic arm of the S/R machine during a storage operation S
$t_{C,R}$	Channel time of the telescopic arm of the S/R machine during the pure retrieval component of a retrieval operation R
$t_{C,\text{β},S}$	Channel time of the telescopic arm of the S/R machine during the relocation-storage component
$t_{C,\text{β},R}$	Channel time of the telescopic arm of the S/R machine during the relocation-retrieval component
$t_d$	Dead time – e.g. reaction time or sensor response time – which is constant for all command cycles (Arnold and Furmans 2005)
$t_{DC}$	Average travel time of a dual-command cycle
$t_E$	Average movement time of the S/R machine from the storage/retrieval channel to the relocation channel and to another storage/retrieval channel
$t_h$	Handling time of the S/R machine for the picking process of a storage good
$t_S$; $t_R$	Average travel time of a single-command storage/retrieval cycle in a n-deep AS/RS
$t_{step}$	Movement time the telescopic arm of the S/R machine needs to move one depth step into a channel
$v_x$; $v_y$	Velocity of the S/R machine in horizontal and vertical directions
$v_n$	Velocity of the telescopic arm or satellite vehicle in depth direction
X; Y	Number of storage channels horizontally/vertically
z	Stock filling level
α	Probability that storage channels of the same channel type $C_k$ are not affected during a storage or retrieval operation
β	Average number of expected relocations during a storage or retrieval operation in a n-deep AS/RS.

Table 2. Abbreviations used in this work.

AS/RS	Automated storage and retrieval system
AVS/RS	Autonomous vehicle storage and retrieval system
S/R machine	Storage and retrieval machine
RSCS	Random storage channel strategy
RSLS	Random storage location strategy
I/O point	Input and output point

Figure 2. The three possible storage operations in a triple-deep AS/RS (Lehmann and Hußmann 2022).

A scheme of a triple-deep rack. 16 storage channels are shown (two empty, three with one box, two with two boxes and nine with three boxes). An arrow symbolises a storage operation into three of these storage channels.

Figure 3. The six possible retrieval operations in a triple-deep AS/RS (Lehmann and Hußmann 2022).

A triple-deep rack with 16 storage channels. Two channels are empty, three storage channels have one box, two storage channels have two boxes inside and nine storage channels are full. Arrows from six storage channels symbolise retrieval operations. One arrow is for every possible storage location and possible storage channel.

Figure 4. Markov chain of the triple-deep AS/RS (Lehmann and Hußmann 2022).

A Markov chain with four states and nine transitions. The states are the four channel states empty, one box, two boxes and full of a triple-deep AS/RS. The transitions are represented by arrows and are the conditional probabilities that a channel state is changed during a storage or retrieval operation.

Figure 5. Example of the top view of a rack in a triple-deep AS/RS with a minimal variance strategy (Lehmann and Hußmann 2022).

A triple-deep rack with 16 storage channels. Five storage channels have one box inside, 11 storage channels have two boxes. A S/R machine is placed in front of the rack.

Table 3. Input parameters for the simulation. $v_x$, $v_n$, $a_x$, $a_n$, $t_d$ are obtained from Marolt, Kosanić, and Lerher (2022). The other parameters are based on the existing AS/RS and AVS/RS.

X	33	$v_y$	1 m/s
Y	11	$v_n$	1.5 m/s
$l_x$	0.5 m	$a_x$	2 m/s${}^2$
$l_y$	0.4 m	$a_y$	1.5 m/s${}^2$
$l_n$	0.6 m	$a_n$	1 m/s${}^2$
n	4	$t_h$	1 s
$v_x$	3 m/s	$t_d$	5 s

Figure 6. Relocation probabilities $p\left(\text{β}\right)$ for the example AS/RS for the four strategies and the relative error between the simulation and the analytic model (Lehmann and Hußmann 2022).

Two different diagrams. The left diagram shows the analytic relocation probability of all four strategies and the corresponding simulation results depending on the stock filling level. The right diagram shows the relative error that stays constantly low.

Figure 7. Relocation quantities β for the example AS/RS for the four strategies and the relative error between the simulation and the analytic model (Lehmann and Hußmann 2022).

Two different diagrams. The left diagram shows the analytic relocation quantity of all four strategies and the corresponding simulation results depending on the stock filling level. The right diagram shows the relative error which stays constantly low.

Figure 8. Travel time $t_{DC}$ for the example AS/RS for the four different strategies and the relative error between the simulation and the analytic model (Lehmann and Hußmann 2022).

Two different diagrams. The left diagram shows the analytic travel time of a dual command cycle of all four strategies and the corresponding simulation results depending on the stock filling level. The right diagram shows the relative error which stays constantly low.

Figure 9. Analytically derived channel state probabilities for a 4-deep AS/RS and the RSCS and RSLS strategies (Lehmann and Hußmann 2022).

Two different diagrams. The left diagram shows the analytical channel state probabilities of a 4-deep AS/RS and different stock filling levels compared with the corresponding simulation for the RSCS. The right diagram shows the analytical channel state probabilities of a 4-deep AS/RS and different stock filling levels compared with the corresponding simulation for the RSLS.

Table 4. Relocation probability $p\left(\text{β}\right)$ and quantity β for AS/RS with depth n = 2, n = 3, n = 4 and n = 5 and the RSCS and RSLS strategies for z from $5\mathrm{\%}$ to $95\mathrm{\%}$ and $z=99\mathrm{\%}$.

	RSCS								RSLS
	n = 2		n = 3		n = 4		n = 5		n = 2		n = 3		n = 4		n = 5
z	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β
0.05	0.05	0.05	0.07	0.08	0.10	0.10	0.12	0.13	0.03	0.03	0.05	0.05	0.08	0.08	0.10	0.10
0.10	0.09	0.09	0.14	0.15	0.18	0.21	0.22	0.27	0.05	0.05	0.10	0.11	0.15	0.16	0.19	0.22
0.15	0.13	0.13	0.20	0.23	0.26	0.32	0.31	0.41	0.08	0.08	0.15	0.16	0.22	0.25	0.28	0.33
0.20	0.17	0.17	0.26	0.30	0.33	0.43	0.39	0.55	0.11	0.11	0.20	0.22	0.29	0.34	0.36	0.45
0.25	0.20	0.20	0.31	0.37	0.39	0.53	0.45	0.69	0.14	0.14	0.25	0.28	0.34	0.42	0.42	0.57
0.30	0.23	0.23	0.35	0.44	0.44	0.63	0.51	0.83	0.17	0.17	0.30	0.34	0.40	0.51	0.48	0.69
0.35	0.26	0.26	0.39	0.50	0.48	0.73	0.55	0.95	0.20	0.20	0.34	0.40	0.45	0.60	0.53	0.81
0.40	0.29	0.29	0.43	0.55	0.52	0.82	0.59	1.07	0.22	0.22	0.38	0.45	0.49	0.69	0.57	0.92
0.45	0.31	0.31	0.46	0.61	0.55	0.90	0.62	1.19	0.25	0.25	0.42	0.51	0.53	0.77	0.60	1.03
0.50	0.33	0.33	0.49	0.66	0.58	0.97	0.65	1.29	0.28	0.28	0.45	0.57	0.56	0.85	0.63	1.14
0.55	0.35	0.35	0.51	0.70	0.61	1.04	0.67	1.39	0.31	0.31	0.48	0.62	0.59	0.93	0.66	1.24
0.60	0.38	0.38	0.54	0.74	0.63	1.11	0.70	1.47	0.33	0.33	0.51	0.67	0.62	1.01	0.68	1.34
0.65	0.39	0.39	0.56	0.78	0.65	1.17	0.71	1.56	0.36	0.36	0.54	0.72	0.64	1.08	0.71	1.44
0.70	0.41	0.41	0.58	0.82	0.67	1.23	0.73	1.63	0.38	0.38	0.56	0.76	0.66	1.15	0.72	1.53
0.75	0.43	0.43	0.60	0.86	0.69	1.28	0.75	1.71	0.40	0.40	0.58	0.81	0.68	1.21	0.74	1.62
0.80	0.44	0.44	0.61	0.89	0.70	1.33	0.76	1.77	0.43	0.43	0.60	0.85	0.70	1.28	0.75	1.70
0.85	0.46	0.46	0.63	0.92	0.72	1.38	0.77	1.84	0.45	0.45	0.62	0.89	0.71	1.34	0.77	1.78
0.90	0.47	0.47	0.64	0.95	0.73	1.42	0.78	1.89	0.46	0.46	0.64	0.93	0.73	1.39	0.78	1.86
0.95	0.49	0.49	0.65	0.97	0.74	1.46	0.79	1.95	0.48	0.48	0.65	0.97	0.74	1.45	0.79	1.93
0.99	0.50	0.50	0.66	0.99	0.75	1.49	0.80	1.99	0.50	0.50	0.66	0.99	0.75	1.49	0.80	1.99

Table 5. Relocation probability $p\left(\text{β}\right)$ and quantity β for AS/RS with depth n = 2, n = 3, n = 4 and n = 5 and the Minimal Variance and Maximal Variance strategies for z from $5\mathrm{\%}$ to $95\mathrm{\%}$ and $z=99\mathrm{\%}$.

	Minimal variance								Maximal variance
	n = 2		n = 3		n = 4		n = 5		n = 2		n = 3		n = 4		n = 5
z	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β	$p\left(\text{β}\right)$	β
0.05	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.10	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.15	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.20	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.25	0.00	0.00	0.00	0.00	0.00	0.00	0.20	0.20	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.30	0.00	0.00	0.00	0.00	0.17	0.17	0.33	0.33	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.35	0.00	0.00	0.05	0.05	0.29	0.29	0.43	0.43	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.40	0.00	0.00	0.17	0.17	0.38	0.38	0.50	0.50	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.45	0.00	0.00	0.26	0.26	0.44	0.44	0.56	0.67	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.50	0.00	0.00	0.33	0.33	0.50	0.50	0.60	0.80	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.55	0.09	0.09	0.39	0.39	0.55	0.64	0.64	0.91	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.60	0.17	0.17	0.44	0.44	0.58	0.75	0.67	1.00	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.65	0.23	0.23	0.49	0.49	0.62	0.85	0.69	1.15	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.70	0.29	0.29	0.52	0.57	0.64	0.93	0.71	1.29	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.75	0.33	0.33	0.56	0.67	0.67	1.00	0.73	1.40	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.80	0.38	0.38	0.58	0.75	0.69	1.13	0.75	1.50	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.85	0.41	0.41	0.61	0.82	0.71	1.24	0.76	1.65	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.90	0.44	0.44	0.63	0.89	0.72	1.33	0.78	1.78	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.95	0.47	0.47	0.65	0.95	0.74	1.42	0.79	1.89	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00
0.99	0.49	0.49	0.66	0.99	0.75	1.49	0.80	1.98	0.50	0.50	0.67	1.00	0.75	1.50	0.80	2.00

Lehmann, Timo, and Jakob Hußmann. 2022. “Travel Time Model for Multi-Deep Automated Storage and Retrieval Systems with different Storage Strategies.” doi:10.5445/IR/1000147666

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Arnold, Dieter, and Kai Furmans. 2005. Materialfluss in Logistiksystemen. Vol. 6. Berlin/Germany: Springer.

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Marolt, Jakob, Nenad Kosanić, and Tone Lerher. 2022. “Relocation and Storage Assignment Strategy Evaluation in a Multiple-deep Tier Captive Automated Vehicle Storage and Retrieval System with Undetermined Retrieval Sequence.” The International Journal of Advanced Manufacturing Technology118 (9): 3403–3420.

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Data availability statement

The data that support the findings of this study are available from the corresponding author, Timo Lehmann, upon reasonable request (Lehmann andHußmann 2022).