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Travel time analysis of the dual command cycle in the split-platform AS/RS with I/O dwell point policy

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Abstract

In conventional automated storage and retrieval systems (AS/RS), storage and retrieval (S/R) machine travels simultaneously in the horizontal and vertical directions. However, S/R machine cannot support overly heavy loads, such as sea containers, so a new AS/RS, called split-platform AS/RS (SP-AS/RS), was introduced and studied in recent years. The SP-AS/RS employs vertical and horizontal platforms, which move independently, and are capable of handling heavy loads. The vertical platform which represents an elevator (or lift) with the elevator’s lifting table carries the load up and down among different tiers and the horizontal platform which represents the shuttle carrier or the shuttle vehicle can access all cells of the tier in which it belongs to. Single command cycle (SC) and dual command cycle (DC) are two main operating modes in AS/RSs. However, travel time models in all previous articles related to the SP-AS/RS are only for the SC. In this study, we first present a continuous travel time model for the DC in the SP-AS/RS under input and output (I/O) dwell point policy and validate its accuracy by computer simulations. Our model and simulation results both show that the square-in-time rack incurs the smallest expected travel time. After comparing with the existing model for the SC, we find that the DC is better than the SC in terms of the expected travel time.

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Acknowledgments

This research was partially supported by the National Natural Science Foundation of China (Grant No. 71131004).

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Correspondence to Xianhao Xu.

Appendix

Appendix

Calculation of \(P\left( {\left| {x_{1} - x_{2} } \right| \le z} \right)\)

$$P\left( {\left| {x_{1} - x_{2} } \right| \le z} \right) = P\left( { - z \le x_{1} - x_{2} \le z} \right) = P\left( {x_{2} - z \le x_{1} \le x_{2} + z} \right) = \int\limits_{{Q_{z} }} {f_{{x_{1} x_{2} }} \left( {u,t} \right)} dudt$$

where Q z is the area bounded by the lines: \(u = t + z,\) \(u = t - z,\) \(0 \le u \le 1\) and \(0 \le t \le 1\).

Then, we have

$$P\left( {\left| {x_{1} - x_{2} } \right| \le z} \right) = \int\limits_{0}^{1 - z} {\int\limits_{0}^{t + z} {f_{{x_{1} x_{2} }} \left( {u,t} \right)} dudt} + \int\limits_{1 - z}^{z} {\int\limits_{0}^{1} {f_{{x_{1} x_{2} }} \left( {u,t} \right)} dudt} + \int\limits_{z}^{1} {\int\limits_{t - z}^{1} {f_{{x_{1} x_{2} }} \left( {u,t} \right)} dudt}$$

As x 1 and x 2 are independent and have the same probability distribution function, it is easy to get

$$f_{{x_{1} x_{2} }} \left( {u,t} \right) = f_{{x_{1} }} \left( u \right) \times f_{{x_{2} }} \left( t \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {0 \le u \le 1,0 \le t \le 1} \hfill \\ {0,} \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

Thus, we have

$$P\left( {\left| {x_{1} - x_{2} } \right| \le z} \right) = \left\{ {\begin{array}{*{20}l} {2z - z^{2} ,} \hfill & {0 \le z \le 1} \hfill \\ {1,} \hfill & {z \ge 1} \hfill \\ \end{array} } \right.$$

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Liu, T., Xu, X., Qin, H. et al. Travel time analysis of the dual command cycle in the split-platform AS/RS with I/O dwell point policy. Flex Serv Manuf J 28, 442–460 (2016). https://doi.org/10.1007/s10696-015-9221-7

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