Retrieval scheduling in crane-based 3D automated retrieval and storage systems with shuttles

Abstract

Retrieval task scheduling has been extensively studied for 2D automated retrieval and storage systems (AS/RS). A good schedule can significantly reduce the makespan for finishing a given group of retrieval tasks. However, the task scheduling problem has never been studied for crane-based 3D AS/RS with shuttle-based depth movement mechanisms (DMMs), which has become increasingly popular in practice. This study considered how to schedule a group of retrieval requests in a crane-based 3D AS/RS with shuttle-based DMMs with the objective to minimize the makespan. A mixed-integer programing model was developed to represent the problem, and the problem was proven to be NP-hard. Four heuristics were investigated for their computational performance. First-Come-First-Serve is the current practice while the Percentage Priority to Shuttle Reallocation with the Shortest Leg rule was developed based on the existing rule for scheduling storage and retrieval tasks in 3D AS/RS with conveyor-based DMMs. The Genetic Algorithm, which is popular for 2D systems, was adapted to deal with the 3D system. The Lowest-Waiting-Time-First heuristic was proposed based on the optimality condition of the scheduling problem and was demonstrated to outperform the other three algorithms in terms of solution quality and computational time. Further numerical results revealed insights for improving 3D AS/RS productivity. When the number of retrieval tasks is small (e.g., when a short planning horizon is adopted for high responsiveness), having more shuttles can improve the system performance. When there are many tasks to schedule, for example, in a situation with a long planning horizon, using a crane with higher speed rather than adding more shuttles can improve system efficiency more.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Abelwomack. (2018). Deep lane high density pallet storage: Unique storage technology that maximizes warehouse space. https://www.abelwomack.com/deep-lane-high-density-pallet-storage/. Retrieved July 6, 2018.

  2. Asokan, P., Jerald, J., Arunachalam, S., & Page, T. (2008). Application of adaptive genetic algorithm and particle swarm optimization in scheduling of jobs and AS/RS in FMS. International Journal of Manufacturing Research, 3(4), 393–405.

    Article  Google Scholar 

  3. Azadeh, K., De Koster, M., & Roy, D. (2017). Robotized warehouse systems: Developments and research opportunities. Social Science Research Network.

  4. Bastian Solutions. (2019). Options available with ASRS technology. https://www.bastiansolutions.com/options-available-with-asrs-technology/. Retrieved March 5, 2019.

  5. Bessenouci, H. N., Sari, Z., & Ghomri, L. (2012). Metaheuristic based control of a flow rack automated storage retrieval system. Journal of Intelligent Manufacturing, 23(4), 1157–1166.

    Article  Google Scholar 

  6. Boysen, N., Emde, S., Hoeck, M., & Kauderer, M. (2015). Part logistics in the automotive industry: Decision problems, literature review and research agenda. European Journal of Operational Research, 242(1), 107–120.

    Article  Google Scholar 

  7. Boysen, N., & Stephan, K. (2016). A survey on single crane scheduling in automated storage/retrieval systems. European Journal of Operational Research, 254(3), 691–704.

    Article  Google Scholar 

  8. Cai, X., Heragu, S. S., & Liu, Y. (2014). Modeling and evaluating the AVS/RS with tier-to-tier vehicles using a semi-open queueing network. IIE Transactions, 46(9), 905–927.

    Article  Google Scholar 

  9. Cao, W., & Zhang, M. (2017). The optimization and scheduling research of shuttle combined vehicles in automated automatic three-dimensional warehouse. Procedia Engineering, 174, 579–587.

    Article  Google Scholar 

  10. Carlo, H. J., & Vis, I. F. A. (2012). Sequencing dynamic storage systems with multiple lifts and shuttles. International Journal of Production Economics, 140(2), 844–853.

    Article  Google Scholar 

  11. Chang, S. H., & Egbelu, P. J. (1997). Relative pre-positioning of storage/retrieval machines in automated storage/retrieval systems to minimize maximum system response time. IISE Transactions, 29(4), 303–312.

    Google Scholar 

  12. Chetty, O. K., & Reddy, M. S. (2003). Genetic algorithms for studies on AS/RS integrated with machines. The International Journal of Advanced Manufacturing Technology, 22(11–12), 932–940.

    Article  Google Scholar 

  13. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to algorithms. Cambridge: MIT press.

    Google Scholar 

  14. Dooly, D. R., & Lee, H. F. (2008). A shift-based sequencing method for twin-shuttle automated storage and retrieval systems. IISE Transactions, 40(6), 586–594.

    Article  Google Scholar 

  15. Ekren, Banu Y., & Heragu, S. S. (2010). Simulation-based regression analysis for the rack configuration of an autonomous vehicle storage and retrieval system. International Journal of Production Research, 48(21), 6257–6274.

    Article  Google Scholar 

  16. Ekren, Banu Y., & Heragu, S. S. (2012). Performance comparison of two material handling systems: AVS/RS and CBAS/RS. International Journal of Production Research, 50(15), 4061–4074.

    Article  Google Scholar 

  17. Ekren, Banu Y., Heragu, S. S., Krishnamurthy, A., & Malmborg, C. J. (2013). An approximate solution for semi-open queueing network model of an autonomous vehicle storage and retrieval system. IEEE Transactions on Automation Science and Engineering, 10(1), 205–215.

    Article  Google Scholar 

  18. Federation Europeenne De La Manutention. (2001). FEM 9.851.

  19. Fukunari, M., & Malmborg, C. J. (2008). An efficient cycle time model for autonomous vehicle storage and retrieval systems. International Journal of Production Research, 46(12), 3167–3184.

    Article  Google Scholar 

  20. Gademann, A. N. (1999). Optimal routing in an automated storage/retrieval system with dedicated storage. IISE Transactions, 31(5), 407–415.

    Google Scholar 

  21. Gagliardi, J. P., Renaud, J., & Ruiz, A. (2012). Models for automated storage and retrieval systems: A literature review. International Journal of Production Research, 50(24), 7110–7125.

    Article  Google Scholar 

  22. Gaku, R., & Takakuwa, S. (2017). Simulation modeling of shuttle vehicle-type mini-load AS/RS systems for E-commerce industry of Japan. In Proceedings of the 2017 winter simulation conference (p. 259). IEEE Press.

  23. Gue, K. R. (2006). Very high-density storage systems. IISE Transactions, 38(1), 79–90.

    Article  Google Scholar 

  24. Ha, Y., & Chae, J. (2018). Free balancing for a shuttle-based storage and retrieval system. Simulation Modelling Practice and Theory, 82, 12–31.

    Article  Google Scholar 

  25. Ha, Y., & Chae, J. (2019). A decision model to determine the number of shuttles in a tier-to-tier SBS/RS. International Journal of Production Research, 57(4), 963–984.

    Article  Google Scholar 

  26. Hall, N. G., & Magazine, M. (2013). Scheduling and sequencing. In S. I. Gass & M. C. Fu (Eds.), Encyclopedia of operations research and management science (pp. 1356–1363).

  27. Han, M. H., McGinnis, L. F., Shieh, J. S., & White, J. A. (1987). On sequencing retrievals in an automated storage/retrieval system. IISE Transactions, 19(1), 56–66.

    Article  Google Scholar 

  28. Haq, A. N., Karthikeyan, T., & Dinesh, M. (2003). Scheduling decisions in FMS using a heuristic approach. The International Journal of Advanced Manufacturing Technology, 22(5–6), 374–379.

    Article  Google Scholar 

  29. Khojasteh-Ghamari, Y., & Son, J. D. (2008). Order picking problem in a multi-aisle automated warehouse served by a single storage/retrieval machine. International Journal of Information and Management Sciences, 19(4), 651–665.

    Google Scholar 

  30. Kim, B. I., Heragu, S. S., Graves, R. J., & Onge, A. (2003). Clustering-based order-picking sequence algorithm for an automated warehouse. International Journal of Production Research, 41(15), 3445–3460.

    Article  Google Scholar 

  31. Kumar, M., Husian, M., Upreti, N., & Gupta, D. (2010). Genetic algorithm: Review and application. International Journal of Information Technology and Knowledge Management, 2(2), 451–454.

    Google Scholar 

  32. Lee, H. F. (1997). Performance analysis for automated storage and retrieval systems. IISE Transactions, 29(1), 15–28.

    Article  Google Scholar 

  33. Lerher, T. (2016). Travel time model for double-deep shuttle-based storage and retrieval systems. International Journal of Production Research, 54(9), 2519–2540.

    Article  Google Scholar 

  34. Lerher, T., Ekren, Banu Y., Dukic, G., & Rosi, B. (2015). Travel time model for shuttle-based storage and retrieval systems. International Journal of Advanced Manufacturing Technology, 78(9–12), 1705–1725.

    Article  Google Scholar 

  35. Li, J., Huang, R., & Dai, J. B. (2017). Joint optimization of order batching and picker routing in the online retailer’s warehouse in China. International Journal of Production Research, 55(2), 447–461.

    Article  Google Scholar 

  36. Mahajan, S., Rao, B. V., & Peters, B. A. (1998). A retrieval sequencing heuristic for mini-load end-of-aisle automated storage/retrieval systems. International Journal of Production Research, 36(6), 1715–1731.

    Article  Google Scholar 

  37. Man, X., Zheng, F., Chu, F., Liu, M., & Xu, Y. (2019). Bi-objective optimization for a two-depot automated storage/retrieval system. Annals of Operations Research, 1–20.

  38. MHI: Strategic Solutions for Supply Chain. (2018). http://www.mhi.org/downloads/free/ACF6AE.pdf. Retrieved July 6, 2018.

  39. Popović, D., Vidović, M., & Bjelić, N. (2014). Application of genetic algorithms for sequencing of AS/RS with a triple-shuttle module in class-based storage. Flexible Services and manufacTuring Journal, 26(3), 432–453.

    Article  Google Scholar 

  40. Roodbergen, K. J., & Vis, I. F. (2009). A survey of literature on automated storage and retrieval systems. European Journal of Operational Research, 194(2), 343–362.

    Article  Google Scholar 

  41. Stadtler, H. (1996). An operational planning concept for deep lane storage systems. Production and Operations Management, 5(3), 266–282.

    Article  Google Scholar 

  42. Tappia, E., Roy, D., De Koster, M., & Melacini, M. (2017). Modeling, analysis, and design insights for shuttle-based compact storage systems. Transportation Science, 51(1), 269–295.

    Article  Google Scholar 

  43. Tutam, M., & White, J. A. (2019). A multi-dock, unit-load warehouse design. IISE Transactions, 51(3), 232–247.

    Article  Google Scholar 

  44. Van den Berg, J. P., & Gademann, A. J. R. M. (1999). Optimal routing in an automated storage/retrieval system with dedicated storage. IIE Transactions, 31(5), 407–415.

    Article  Google Scholar 

  45. Vose, M. D. (1999). The simple genetic algorithm: Foundations and theory. Cambridge: MIT Press.

    Google Scholar 

  46. Wang, Y., Mou, S., & Wu, Y. (2015). Task scheduling for multi-tier shuttle warehousing systems. International Journal of Production Research, 53(19), 5884–5895.

    Article  Google Scholar 

  47. Wen, U. P., Chang, D. T., & Chen, S. P. (2001). The impact of acceleration/deceleration on travel-time models in class-based automated S/R systems. IISE Transactions, 33(7), 599–608.

    Google Scholar 

  48. Westfalia. (2018). High density, multiple deep AS/RS. https://www.westfaliausa.com/products/automated-storage-retrieval-systems/storage-density. Retrieved July 6, 2018.

  49. Wu, K. Y., Xu, S. S. D., & Wu, T. C. (2013). Optimal scheduling for retrieval jobs in double deep AS/RS by evolutionary algorithms. In Abstract and applied analysis (Vol. 2013). Hindawi.

  50. Wu, Y., Zhou, C., Ma, W., & Kong, X. T. R. (2020). Modelling and design for a shuttle-based storage and retrieval system. International Journal of Production Research, 1–21.

  51. Xu, X., Gong, Y., Fan, X., Shen, G., & Zou, B. (2018). Travel-time model of dual-command cycles in a 3D compact AS/RS with lower mid-point I/O dwell point policy. International Journal of Production Research, 56(4), 1620–1641.

    Article  Google Scholar 

  52. Yang, P., Miao, L., Xue, Z., & Qin, L. (2015). An integrated optimization of location assignment and storage/retrieval scheduling in multi-shuttle automated storage/retrieval systems. Journal of Intelligent Manufacturing, 26(6), 1145–1159.

    Article  Google Scholar 

  53. Yang, P., Yang, K., Qi, M., Miao, L., & Ye, B. (2017). Designing the optimal multi-deep AS/RS storage rack under full turnover-based storage policy based on non-approximate speed model of S/R machine. Transportation Research Part E: Logistics and Transportation Review, 104, 113–130.

    Article  Google Scholar 

  54. Yu, Y., & De Koster, M. (2009a). Optimal zone boundaries for two-class-based compact three-dimensional automated storage and retrieval systems. IISE Transactions, 41(3), 194–208.

    Article  Google Scholar 

  55. Yu, Y., & De Koster, M. (2009b). Designing an optimal turnover-based storage rack for a 3D compact automated storage and retrieval system. International Journal of Production Research, 47(6), 1551–1571.

    Article  Google Scholar 

  56. Yu, Y., & De Koster, M. (2012). Sequencing heuristics for storing and retrieving unit loads in 3D compact automated warehousing systems. IISE Transactions, 44(2), 69–87.

    Article  Google Scholar 

  57. Zaerpour, N., Yu, Y., & Koster, M. (2015). Storing fresh produce for fast retrieval in an automated compact cross-dock system. Production and Operations Management, 24(8), 1266–1284.

    Article  Google Scholar 

  58. Zhao, X., Wang, Y., Wang, Y., and Huang, K. (2019). Integer programming scheduling model for tier-to-tier shuttle-based storage and retrieval systems. Processes, 7(4), 223.

  59. Zou, B., Xu, X., Gong, Y., & De Koster, M. (2016). Modeling parallel movement of lifts and vehicles in tier-captive vehicle-based warehousing systems. European Journal of Operational Research, 254(1), 51–67.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Mingzhou Jin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Appendices

Appendices

Appendix 1

In that study, only lanes with retrieval tasks are considered for the scheduling problem. However, in practice, a system may have more lanes than the SKU types and one type of goods is possibly stored in more than one lane. The lane-to-task assignment, which selects lanes to fulfill demand, can be modeled based on the MIP (1-21) proposed for the scheduling problem in this paper. In addition to the notations defined before, set \( A \) is defined as the set of SKU types, and \( N_{a} \), \( a \in A \), is the set of lanes used for storing SKU \( a \). The demand on SKU type \( a \) is \( D_{a} \). The lane-to-task assignment problem is formulated as follows based on the scheduling problem (1-21) by having parameter \( Q_{i} \) be a decision variable and adding an additional constraint set (22) to meet the demand.

$$ \begin{array}{*{20}l} {\text{Min}} \hfill & {(1)} \hfill & {} \hfill \\ {{\text{s}} . {\text{t}} .} \hfill & {\mathop \sum \limits_{{i \in N_{a} }} Q_{i} = D_{a} ,} \hfill & {a \in A,} \hfill \\ {} \hfill & {\left( {2 - 21} \right)} \hfill & {} \hfill \\ {} \hfill & {Q_{i} \ge 0.} \hfill & {} \hfill \\ \end{array} $$
(22)

Appendix 2

Parameters

\( N \): set of lanes with shuttles or having retrieval tasks, \( i \) and \( j \) are their indices;

\( N^{0} \): set of lanes having a shuttle at the beginning, \( N^{0} \subseteq N \) and \( \left| {N^{0} } \right| = \) number of shuttles;

\( N^{1} \): set of lanes without a shuttle at the beginning, \( N^{0} \cup N^{1} = N \)

\( Q_{i} \): number of retrieval tasks from lane \( i \);

\( Q^{\prime} \): largest number of tasks at a lane over all lanes, \( Q^{\prime} = \mathop {\hbox{max} }\limits_{i \in N} Q_{i} + 1 \);

\( M \): number of total SC tasks, \( M = \left| {N^{1} } \right| + \mathop \sum \limits_{i \in N} Q_{i} \);

\( p_{iq}^{1} \): time for a shuttle finishing the \( q \)th task in lane \( i \), \( p_{{i,Q_{i} + 1}}^{1} = 0 \);

\( p_{ij}^{2} \): time for the crane traveling from lane \( i \) to lane \( j \);

\( p_{ij}^{3} \): time for the crane traveling from lane \( i \) to the picking position, unloading a unit, and then traveling to lane \( j \) for the next task;

\( p_{i}^{4} \): time for the crane traveling from lane \( i \) to the picking position and unloading a unit;

\( P_{{i_{0} ,i}} \): time for the crane traveling from original position to lane \( i \); and

\( T \): planning horizon.

Variables

\( y_{mi} \) = 1 if the \( m \)th SC task is for retrieving a load or a shuttle from lane \( i \); = 0, otherwise;

\( t_{m} \): starting time for the crane to handle the \( m \)th task;

\( z_{mq} \) = 1 if the \( q \)th task at the lane is the \( m \)th SC task; = 0, otherwise;

\( x_{ij} \) = 1 if lane \( j \) is immediately handled by lane \( i \) with the same shuttle; = 0, otherwise;

\( \theta_{m} \) = 1 if task \( m \) is for moving a shuttle; = 0, otherwise;

\( r_{iq} \): moment when the \( q \)th task in lane \( i \) is ready for the crane to pick up, including the shuttle moving task; \( r_{i0} = 0 \)

\( l_{i} \) and \( \mu_{i} \): an artificial variable to avoid sub-tours and make sure that \( m \)th SC task associated with a lane which has a shuttle.

Appendix 3

Theorem 1

The 3D AS/RS retrieval task scheduling problem is NP-hard.

Proof

According to Han et al. (1987), the task scheduling problem in 2D AS/RS operating in the DC mode with multiple open locations is equivalent to the traveling salesman problem (TSP) in its simplified version in which there is only one open location. Since the TSP is NP-hard (Cormen et al. 2009), the task scheduling problem in 2D AS/RS is NP-hard. For the 3D AS/RS scheduling problem considered in our study, if the shuttle speed is assumed to be infinite, the crane can pick up an SKU/shuttle instantly when arrives at a target lane. In that case, after moving a shuttle to a lane, the crane will pick up an SKU from the lane immediately instead of going to serve another lane. If we consider all locations of shuttles as open locations, the crane operates in the DC mode when transferring shuttles: starts from the I/O point to an open location, picks up the shuttle there and transfers it to lane \( j \), hands over the shuttle and picks up an SKU form lane \( j \), and travels back to I/O point. It is obvious that the problem considered by Han et al. (1987) is a special case of our problem with infinite shuttle speed. As this special case is NP-hard, our scheduling problem for the 3D AS/RS is also NP-hard.

Theorem 2

(Optimality Condition) Consider the \( m{\rm th} \) and \( \left( {m + 1} \right){\rm th} \) tasks in an optimal task schedule. If the \( m{\rm th} \) task is for retrieving the \( \left( {q_{1} } \right){\rm th} \) SKU from lane \( j \), the \( \left( {m + 1} \right){\rm th} \) task is for retrieving the \( \left( {q_{1} } \right){\rm th} \) SKU from lane \( k \), and the \( \left( {m - 1} \right){\rm th} \) task is associated with lane \( i, \) we should always have \( p_{m} \ge s_{m} \) where \( p_{m} = max\left\{ {r_{{k,q_{2} }} - t_{m - 1} - p_{ik}^{3} ,0} \right\} \) and \( s_{m} = \hbox{max} \left\{ {r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} ,0} \right\} \) if the \( \left( {m - 1} \right){\rm th} \) task is a retrieval task, otherwise; \( p_{m} = \hbox{max} \{ r_{{k,q_{2} }} - t_{m - 1} - p_{i,l}^{2} - p_{l,k}^{2} ,0\} \) and \( s_{m} = \hbox{max} \left\{ {r_{{j,q_{1} }} - t_{m - 1} - p_{i,l}^{2} - p_{l,j}^{2} ,0} \right\} \) if the \( \left( {m - 1} \right){\rm th} \) task is to reallocate a shuttle from lane \( i \) to \( l \).

Proof

Suppose the \( \left( {m - 1} \right){\rm th} \) task is for retrieving an SKU, \( p_{m} = { \hbox{max} }\left\{ {r_{{k,q_{2} }} - t_{m - 1} - p_{ik}^{3} ,0} \right\} \) and \( s_{m} = \hbox{max} \left\{ {r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} ,0} \right\} \). In addition, Let’s define \( w_{m} \) and \( w_{m + 1} \) as the crane’s waiting time of the \( m{\rm th} \) and \( m + 1{\rm th} \) task. Clearly, \( w_{m} = s_{m} \) and \( w_{m + 1} = { \hbox{max} }\left\{ {r_{{k,q_{2} }} - t_{m} - p_{jk}^{3} ,0} \right\} \). To prove the optimality, we have to show that the total waiting time of these two tasks will increase if we switch the sequence. Let’s define \( w_{m}^{'} \) and \( w_{m + 1}^{'} \) as the crane’s waiting time of the new \( m{\rm th} \) and \( m + 1{\rm th} \) task after switching the sequence. It is clear that when \( s_{m} = w_{m} = 0 \), the solution is optimal, since we can always have \( w_{m} = w_{m + 1}^{'} = 0 \) and \( w_{m}^{'} \ge w_{m + 1} \). When \( s_{m} > 0 \), two cases can be considered.

Case 1 \( p_{m} = 0, r_{{k,q_{2} }} \le t_{m - 1} + p_{ik}^{3} \)

In that case, we can have

$$ w_{m}^{'} = \hbox{max} \left\{ {p_{m} ,0} \right\} = 0, $$

and

$$ w_{m}^{'} + w_{m + 1}^{'} = w_{m + 1}^{'} = \hbox{max} \left\{ {r_{{j,q_{1} }} - t_{m - 1} - p_{ik}^{3} - p_{jk}^{3} ,0} \right\} < r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} . $$

Clearly, when \( s_{m} > 0 \), we must have \( p_{m} > 0 \) to guarantee the optimality.

Case 2 \( p_{m} > 0 \)

When \( p_{m} > 0 \), \( w_{m}^{'} = r_{{k,q_{2} }} - t_{m - 1} - p_{ik}^{3} > 0 \), and \( w_{m + 1}^{'} = \hbox{max} \left\{ {r_{{j,q_{1} }} - r_{{k,q_{k} }} - p_{jk}^{3} ,0} \right\} \). If \( w_{m + 1}^{'} > 0 \), which is equivalent to \( r_{{k,q_{2} }} - r_{{j,q_{1} }} < - p_{jk}^{3} \), \( w_{m + 1} = \hbox{max} \left\{ {r_{{k,q_{2} }} - r_{{j,q_{1} }} - p_{jk}^{3} ,0} \right\} = 0 \). Therefore, after switching, the total waiting time of these two tasks will be

$$ w_{m}^{'} + w_{m + 1}^{'} = r_{{j,q_{1} }} - t_{m - 1} - p_{ik}^{3} - p_{jk}^{3} < r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} = w_{m} + w_{m + 1} = w_{m} . $$

To guarantee the optimality of the given solution, \( r_{{k,q_{2} }} - r_{{j,q_{1} }} \ge - p_{jk}^{3} \) and \( w_{m + 1}^{'} = 0 \). In that case, if \( w_{m + 1} = 0 \) and \( r_{{k,q_{2} }} - r_{{j,q_{1} }} \le p_{jk}^{3} \),

$$ \begin{aligned} & w_{m}^{'} + w_{m + 1}^{'} = w_{m}^{'} = r_{{k,q_{2} }} - t_{m - 1} - p_{ik}^{3} ,{\text{and}} \\ & w_{m} + w_{m + 1} = w_{m} = r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} . \\ \end{aligned} $$

To guarantee the optimally,

$$ r_{{k,q_{k} }} - t_{m - 1} - p_{ik}^{3} \ge r_{{j,q_{1} }} - t_{m - 1} - p_{ij}^{3} , $$

which is equivalent to

$$ p_{jk}^{3} \ge r_{{k,q_{2} }} - r_{{j,q_{1} }} \ge p_{ik}^{3} - p_{ij}^{3} . $$

And if \( w_{m + 1} > 0 \),

$$ w_{m}^{'} + w_{m + 1}^{'} = w_{m}^{'} = r_{{k,q_{2} }} - t_{m - 1} - p_{ik}^{3} > r_{{k,q_{2} }} - t_{m - 1} - p_{ij}^{3} - p_{jk}^{3} = w_{m} + w_{m + 1} . $$

According to the discussion, if \( s_{m} = 0 \), \( p_{m} \) can take any value. However, when \( s_{m} > 0 \), we need \( r_{{k,q_{2} }} - r_{{j,q_{1} }} \ge p_{ik}^{3} - p_{ij}^{3} \), which is equivalent to \( p_{m} \ge s_{m} . \) The same logic can be applied for the scenario when \( m - 1{\rm th} \) task is for reallocating a shuttle.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dong, W., Jin, M., Wang, Y. et al. Retrieval scheduling in crane-based 3D automated retrieval and storage systems with shuttles. Ann Oper Res (2021). https://doi.org/10.1007/s10479-021-03967-8

Download citation

Keywords

  • Material handling
  • Crane-based 3D AS/RS with shuttles
  • Retrieval task scheduling
  • Optimization