# AMR-Assisted Order Picking: Models for Picker-to-Parts Systems in a Two-Blocks Warehouse

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. The Two-Blocks Warehouse Case Study

## 4. Implementations of an AMR-Assisted Picker-to-Parts System

#### 4.1. Separated Handover Locations for Sub-Aisles

#### 4.1.1. A Mixed Integer Linear Programming Formulation

_{b}. It always includes the depot, which is the starting and arrival point of the AMR on each of its tours. The set of customer orders and batches are indicated respectively with O and B. The composition of the batches is known through the binary indicator v

_{bo}, which takes value 1 if the order o is in batch b, 0 otherwise. The AMR fleet is modelled by set A = {1, 2, …, m}. The set of tours (for each AMR) to which batches can be assigned is represented by set K. The total number of batches to be processed is the upper bound on this set.

_{bi}is the time needed to walk and collect the items, l

_{bi}is the time taken to load the picked items on AMR. The parameter t

_{ij}represents the time needed by the AMR to travel from one location $i\in V$ to location $j\in V$, while u

_{b}is the time needed to download the articles of a batch b from the AMR at the end of the tour. Each customer order o is associated with a due date d

_{o}.

_{bi}denotes the moment the AMR handling batch b arrives at sub-aisle i; s

_{bi}represents the moment the picker of sub-aisle i starts collecting items of batch b; h

_{bi}is the moment the picker of sub-aisle i starts loading the items of batch b on the AMR. The completion time of each batch process by the AMR a with position k is denoted by ct

_{ak}. The assignment of each batch to a specific AMR tour is part of the optimization problem and is regulated by the binary indicator x

_{abk}; it takes value 1 if batch b is completed by AMR a with position k; 0 otherwise. Starting from position k = 0 for each AMR, k increases by one unit each time a batch is assigned to the AMR; the process ends when all batches have been assigned. For each sub-aisle i, the binary indicator z

_{bdi}takes values 1 if batch b is processed before batch d by the order picker; 0 otherwise. The tardiness of a customer order o is denoted by t

_{o}; the object of optimization is the minimization, for all orders, of the total tardiness. Note that the binary indicator z

_{bdi}only refers to a specific sub-aisle i because, given two different warehouse sub-aisles i$\in V$ and j$\in V$ and two different batches b $\in B$ and d $\in B$, it can happen that batch b$\in B$ is processed before batch d $\in B$ by the picker of sub-aisle i$\in V$ and, on the contrary, batch d $\in B$ is processed before batch $\in B$b by the picker of sub-aisle j$\in V$.

**Box 1.**Main notation used along the paper

**Sets**- B: batches
- O: customer orders
- A: AMRs
- K: sequencing numbers of a batch for each AMR
- V: composed by the copies of the depot and the sub-aisles (V = {0, 1, 2, …, n, …, 2n − 1, 2n, 2n + 1})
- C
_{b}: relevant picking sub-aisles for batch b ∈ B **Parameters**- d
_{o}: due date of customer order o ∈ O - M: a sufficiently large positive number
- v
_{bo}: has value 1 if the customer order o ∈ O is included in batch b ∈ B; 0 otherwise - p
_{bi}: time the order picker requires to walk and retrieve the items of batch b ∈ B which are stored in his picking sub-aisle i ∈ V - l
_{bi}: time the order picker requires to pass the items of batch b ∈ B stored in his picking sub-aisle i ∈ V to the associated AMR - u
_{b}: time required to unload the items of batch b ∈ B from the associated AMR - t
_{ij}: time an AMR requires to travel from the depot/handover location i ∈ V to the depot/handover location j ∈ V **Continuous decision variables**- t
_{o}: tardiness of customer order o ∈ O - a
_{bi}: arrival time of the AMR handling batch b B at handover location of picking sub-aisle i ∈ V - s
_{bi}: order picker’s start time of picking the items of batch b ∈ B stored in picking sub-aisle i ∈ V - h
_{bi}: order picker’s start time of passing the items of batch b ∈ B stored in picking sub-aisle i ∈ V to the associated AMR at the handover location - ct
_{ak}: completion time of the batch completed by the AMR a ∈ A with sequencing number k ∈ K **Binary decision variables**- x
_{abk}: takes value 1 if batch b ∈ B is completed by AMR a ∈ A with sequencing number k ∈ K; 0 otherwise - z
_{ibd}: takes value 1 if batch b ∈ B is handled before batch d ∈ B by the order picker of subaisle i ∈ V; 0 otherwise - q
_{bi}: takes value 1 if the items of batch b ∈ B stored in sub-aisle i ∈ V are collected in i ∈ V; 0 if if they are collected in y(i) ∈ V **Function**- y(i): function mapping each sub-aisle i ∈ V/{0, 2n + 1} to the sub-aisle y(i) ∈ V/{0, 2n + 1} that shares the same handover location

_{d}to the completion time of batch b ∈ B (plus the time the AMR requires to travel from the depot to handover location i ∈ C

_{d}) if the following holds: (i) batch b ∈ B is handled by the same AMR as batch d ∈ B, and (ii) batch b ∈ B is processed before batch d ∈ B by the AMR. Constraint (12) calculate the tardiness for each customer order. Finally, the continuous decision variables and the binary decision variables are defined in Constraints (13) and (14), respectively.

_{bi}, that contain for each batch b the (eventual) start of the picking process. It is straightforward to derive a user-friendly list for each picker starting from this information. Combining the optimal value of the set of variables x and ct it is finally possible to derive the sequence of operations carried out by the AMRs and to consequently program them.

#### 4.2. Separated Handover Locations for Sub-Aisles

#### A Mixed Integer Linear Programming Formulation

_{bi}takes different values, depending on whether the operator of a corridor i loads the items on the AMR in correspondence to the visit for its corridor or in that associated to corridor y(i).

_{b}when items of sub-aisle i are picked in i (see Constraint (24)) and when the items of i are picked in y(i) (see Constraint (25)). Constraints (26) and (27) determine the arrival time of the AMR handling batch b ∈ B at each couple of sub-aisle y(j) of j ∈ C

_{b}when items of sub-aisle i are picked in i (see Constraint (26)) and when the items of i are picked in y(i) (see Constraint (27)). Inequalities (28) and (29) determine the completion time of each batch processed by the AMR a ∈ A with sequencing number k ∈ K. Inequalities (30) and (31) determine the completion time of the first batch of each AMR. Constraint (32) link the arrival time of the AMR handling batch d ∈ B at the first relevant handover location i ∈ C

_{d}to the completion time of batch b ∈ B (plus the time the AMR requires to travel from the depot to handover location i ∈ C

_{d}) if the following holds: (i) batch b ∈ B is handled by the same AMR as batch d ∈ B, and (ii) batch b ∈ B is processed before batch d ∈ B by the AMR. Constraint (33) calculate the tardiness for each customer order. Finally, the continuous decision variables and the binary decision variables are defined in Constraints (34) and (35), respectively.

## 5. Numerical Experiments

#### 5.1. Details of the Case Study

- Distance between two close handover locations (meters): 10
- Distance between the depot and the leftmost handover location (meters): 2
- Distance between two adjacent storage locations (houses) (meters): 2
- Distance between each handover location and the first storage location of the corridor (meters): 3

- Unit picking time (seconds): 8
- Unitary loading/unloading time (time required to load/unload a unit from the cart, seconds): 4
- Walking time of the operator between two adjacent storage locations (seconds): 3
- Traveling time of the AMR between two consecutive handover locations (seconds): 15
- Traveling time of the AMR from the storage location to the first handover location (seconds): 3
- Traveling time of the operator from the handover location to the first storage location (seconds): 4.5

#### 5.2. Experiments on the AMS Fleet Size and Speed

#### 5.3. Comparison of the Implementations

#### 5.3.1. Tardiness Values for Different Instance Sizes

#### 5.3.2. Tardiness Values for Different AMR Fleet Sizes and Speeds

#### 5.3.3. Tardiness Percentage Improvements for Different Implementations and Conditions

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Small | Medium | Large | |||
---|---|---|---|---|---|

AMR Speed (km/h) | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 3 |

2.4 | 2634 | 5645 | 4131 | 16,954 | 1870 |

3.6 | 0 | 4863 | 2312 | 14,820 | 0 |

4.8 | 0 | 3559 | 2241 | 13,173 | 0 |

6.0 | 0 | 3513 | 2227 | 12,869 | 0 |

7.2 | 0 | 2953 | 2112 | 11,314 | 0 |

Small | Medium | Large | |||
---|---|---|---|---|---|

AMR Speed (km/h) | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 2 | Number of AMR = 3 |

2.4 | 1269 | 7341 | 2945 | 10,207 | 95 |

3.6 | 0 | 2981 | 2085 | 9885 | 0 |

4.8 | 0 | 2946 | 1656 | 9912 | 0 |

6.0 | 0 | 2857 | 1245 | 8603 | 0 |

7.2 | 0 | 1250 | 904 | 3199 | 0 |

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**MDPI and ACS Style**

Pugliese, G.; Chou, X.; Loske, D.; Klumpp, M.; Montemanni, R.
AMR-Assisted Order Picking: Models for Picker-to-Parts Systems in a Two-Blocks Warehouse. *Algorithms* **2022**, *15*, 413.
https://doi.org/10.3390/a15110413

**AMA Style**

Pugliese G, Chou X, Loske D, Klumpp M, Montemanni R.
AMR-Assisted Order Picking: Models for Picker-to-Parts Systems in a Two-Blocks Warehouse. *Algorithms*. 2022; 15(11):413.
https://doi.org/10.3390/a15110413

**Chicago/Turabian Style**

Pugliese, Giulia, Xiaochen Chou, Dominic Loske, Matthias Klumpp, and Roberto Montemanni.
2022. "AMR-Assisted Order Picking: Models for Picker-to-Parts Systems in a Two-Blocks Warehouse" *Algorithms* 15, no. 11: 413.
https://doi.org/10.3390/a15110413