# Integer Programming Scheduling Model for Tier-to-Tier Shuttle-Based Storage and Retrieval Systems

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. System Description

- If shuttles and lifts reach the maximum velocity before deceleration, the operation includes three steps: accelerating from 0 to maximum velocity ${v}^{\mathrm{max}}$, traveling at ${v}^{\mathrm{max}}$, and decelerating from ${v}^{\mathrm{max}}$ to 0. The patterns of acceleration and velocity are shown in Figure 4a.
- If shuttles and lifts do not reach the maximum velocity before deceleration, the operation includes two steps: accelerating from 0 to v and decelerating from v to 0. The patterns of acceleration and velocity are shown in Figure 4b.

## 4. Integer Programming Scheduling Model

- The retrieval SKU can be found in its assigned aisles randomly within a single time window.
- At the beginning of retrieval tasks, all shuttles remain at rest in the buffer area on tier 1. When the first shuttle is in a task with new tasks emerging, the system schedules the second shuttle for the next task, and this pattern is followed for shuttles third through h.
- While a shuttle performs a task in a certain tier, it is forbidden to allow other shuttles to step in since it could lead to deadlock or conflict.
- For retrieval tasks, lifts firstly transport the shuttles to the tier where the tasks is located, then the starting times of the first lift task, first shuttle task, and first retrieval task are set equal to 0.

## 5. Case Study

## 6. Conclusions

- This paper only considers retrieval tasks, without considering storage tasks. In practice, SBS/RSs normally perform both storage and retrieval tasks. Besides, under dual cycle command (DCC), methods to model and schedule the SBS/RSs require further research.
- The integer programming scheduling model established here can obtain the optimized result by Gurobi. However, model-solving time will increase as the number of orders increases. Thus, determining the best heuristic algorithm for optimization requires further study.
- The developed automated computing program can be further packaged into software. Then, the decision maker would be able to control the inputs and read the best retrieval sequence. By sending information about the best sequence to the automation device, the automatic system’s efficiency could be improved, and the costs of the warehouse could be cut.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Operation of Lifts. (

**a**) The movement of an idle shuttle; (

**b**) The movement of a loading shuttle; (

**c**) The movement of a lift between two tiers; (

**d**) The movement of an idle lift; (

**e**) The movement of a loaded lift.

**Figure 4.**Operation Features of shuttles and lifts. (

**a**) Patterns of acceleration and velocity for situation 1; (

**b**) Patterns of acceleration and velocity for situation 2.

Parameter | Definition | Unit |
---|---|---|

${\mu}_{w}$ | Width of a single storage location | m |

h | Height of a single storage location | m |

${v}_{S}$ | The maximum horizontal velocity of the shuttle | m/s |

${v}_{L}$ | The maximum vertical velocity of the lift | m/s |

${a}_{S}$ | The horizontal acceleration/deceleration of the shuttle | m/s^{2} |

${a}_{L}$ | The vertical acceleration/deceleration of the lift | m/s^{2} |

t_{S} | The time for the shuttle to load or unload a SKU | s |

t_{L} | The time for the lift to load or unload a shuttle in | s |

T_{lift} | Time of the picking and delivery of lifts | s |

Parameter | Definition | Unit |
---|---|---|

$O$ | The number of outbound orders | number |

$H$ | The number of shuttles | number |

$M$ | Total number of missions of shuttles (i.e., 20) | number |

$C$ | The number of columns of storage racks | number |

$D$ | Scale constant | number |

$N$ | The number of rows of storage racks | number |

${S}_{i}$ | The number of retrieval tasks in tier i | number |

${Q}_{i}$ | Combination of retrieval tasks on tier i in the number of columns | number |

${T}_{h}$ | The number of tasks for shuttle h | number |

Variable | Definition | Unit |
---|---|---|

${t}_{m}$ | Starting time of performing lift task m | s |

${y}_{mi}^{f}$ | If task m is the retrieval task of tier i with conveying the loading shuttle, then the value is 1; otherwise, 0 | |

${y}_{mi}^{e}$ | If task m is the retrieval task of tier i with conveying the idle shuttle, then take value 1; otherwise, 0 | |

${x}_{isq}$ | If retrieval task s in tier i is also in column 1, then take value 1; otherwise, 0 | |

$S{S}_{is}$ | Start time of task s in tier i | s |

${F}_{is}$ | End time of task s in tier i | s |

${L}_{ms}$ | If lift task m is the retrieval task s for a particular tier, then take value 1; otherwise, 0 | |

${r}_{is}$ | Waiting time for lifts in column 1 after picking | s |

${B}_{ht}$ | Starting time of task t for shuttle h | s |

${\theta}_{ht}$ | If it is task t for shuttle h, then take value 1; otherwise, 0 | |

${\beta}_{tis}$ | If task t is task s in tier i for a shuttle, then take value 1; otherwise, 0 |

Scenario | Number of Positions | Number of Tiers | Number of Columns | Number of Tasks | Number of Shuttles |
---|---|---|---|---|---|

A | 1200 | 11 | 60 | 30 | 4 |

B | 1600 | 11 | 80 | 30 | 4 |

C | 2000 | 11 | 100 | 30 | 4 |

D | 960 | 9 | 60 | 30 | 3 |

E | 1280 | 9 | 80 | 30 | 3 |

F | 1600 | 9 | 100 | 30 | 3 |

Parameter | Value |
---|---|

Height of each rack | 0.6 m |

Width of each rack | 0.5 m |

Maximum horizontal velocity of shuttles | 2 m/s |

Maximum vertical velocity of lifts | 3 m/s |

Horizontal acceleration or deceleration of shuttles | 1 m/s^{2} |

Vertical acceleration or deceleration of lifts | 2 m/s^{2} |

Storage or retrieval time of shuttles | 4.5 s |

Taking or returning time of lifts | 3 s |

Task Cycle Time (s) | |||||||
---|---|---|---|---|---|---|---|

Random Sequence | Model Result | Optimized Percentage | |||||

1 | 2 | 3 | 4 | Average | |||

A | 559.99 | 542.96 | 560.17 | 553.00 | 554.03 | 490.79 | 12.89% |

B | 584.65 | 571.78 | 586.94 | 568.33 | 577.93 | 527.23 | 9.62% |

C | 602.90 | 551.72 | 592.92 | 578.78 | 581.58 | 532.06 | 9.31% |

D | 566.30 | 584.51 | 537.04 | 568.52 | 564.09 | 508.16 | 11.01% |

E | 539.70 | 541.81 | 559.93 | 559.93 | 550.34 | 513.35 | 7.21% |

F | 583.47 | 539.94 | 558.16 | 556.09 | 559.41 | 521.18 | 7.34% |

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**MDPI and ACS Style**

Zhao, X.; Wang, Y.; Wang, Y.; Huang, K. Integer Programming Scheduling Model for Tier-to-Tier Shuttle-Based Storage and Retrieval Systems. *Processes* **2019**, *7*, 223.
https://doi.org/10.3390/pr7040223

**AMA Style**

Zhao X, Wang Y, Wang Y, Huang K. Integer Programming Scheduling Model for Tier-to-Tier Shuttle-Based Storage and Retrieval Systems. *Processes*. 2019; 7(4):223.
https://doi.org/10.3390/pr7040223

**Chicago/Turabian Style**

Zhao, Xiaofeng, Yanyan Wang, Yunge Wang, and Ke Huang. 2019. "Integer Programming Scheduling Model for Tier-to-Tier Shuttle-Based Storage and Retrieval Systems" *Processes* 7, no. 4: 223.
https://doi.org/10.3390/pr7040223