# Effects of Multiple Depots on Total Travel Distance in Parallel-Aisle Manual Order Picking Systems

Thao Huong Tran-Vo, Thuy Mo Nguyen, Soondo Hong

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Thao Huong Tran-Vo, Thuy Mo Nguyen, Soondo Hong. Effects of Multiple Depots on Total Travel Distance in Parallel-Aisle Manual Order Picking Systems. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.310-318, 10.1007/978-3-031-16407-1_37. hal-05386060

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# Effects of Multiple Depots on Total Travel Distance in Parallel-Aisle Manual Order Picking Systems

Thao Huong Tran-Vo<fnref n="1" />, Thuy Mo Nguyen<fnref n="2" />, and Soondo Hong<fnref n="2" />(✉)

**Abstract.** This study introduces an order batching problem in a parallel-aisle warehouse system with multiple depots. In such a system, orders are organized into batches for picking and these batches are distributed across multiple depots, where pickers receive batch-specific picking information and unload all picked items after each trip. We formulate the problem using a mixed-integer linear programming model with the objective of minimizing the total travel distance while ensuring the workload balance among depots. Then, on small-scale instances, we also conduct several experiments to validate the model and to investigate how using multiple depots affects warehouse performance in terms of total travel distance. The results show that multi-depot order picking systems outperform single-depot systems in saving the total travel distance, and the performance can be enhanced by properly defining depot locations.

**Keywords:** Order picking system · Order batching · Multiple depots · MILP · Total travel distance

## 1 Introduction

The order-picking problem has been considered as one of the most important factors in improving warehouse productivity since this area can consume as much as 60% of a warehouse's total operating cost [1]. Pickers receive picking information from a depot, travel along pre-determined routes to retrieve items, then bring them back to the same depot before downstream workers inspect and package the orders. Pickers, in practical, often combine orders into batches to reduce the number of travel trips as well as to shorten the total travel distance, which is known as an NP-hard problem [2]. The target order picking system (OPS) in this study is a one-block parallel-aisle warehouse. We consider the one-way S-shape routing method which is the most popular in practice and may be helpful to avoid aisle congestion [3, 4].

Integrated picking operations have recently drawn high attention [5], in that warehouse layout design is also considered as a component [6]. Most previous studies were

<footnote n="1">¹ Major in Industrial Data Science and Engineering, Department of Industrial Engineering, Pusan National University, Busan 46241, Korea</footnote>

<footnote n="2">² Department of Industrial Engineering, Pusan National University, Busan 46241, Korea
soondo.hong@pusan.ac.kr</footnote>
conducted under the assumption that there is only one depot location, which may lead to congestion and become a bottleneck of the whole warehouse process of picking and its downstream operations. Recently the multi-depot model has been widely used in real manual warehouses to cope with this problem. Operating multiple depots may reduce the waiting time for loading containers and unloading picked items at depots, shorten the completion time, and thus have a high ability to enhance customer service. When using the multi-depot model, the workload balancing among depots must also be considered to achieve high performance in terms of time reduction.

For these aforementioned motivations, in this study, we *first* develop a new order batching problem with multiple depots (MDOB) under the objective of minimizing total travel distance. We present a formulation of MDOB using a mixed-integer linear programming model to find exact solutions for small-scale problems. *Second*, we run several experiments to assess the impact of using multiple depots in order picking systems in terms of total travel distance.

The remaining of this paper is structured as follows: Sect. 2 defines the multi-depot order batching problem and its assumptions. Section 3 proposes the mathematical model of the problem. An example of problem's solution, the small-instance experimental design, and computational results to analyze the impact of multiple depots on travel distance are described and discussed in Sect. 4. Finally, Sect. 5 concludes and makes recommendations for future research.

# 2 Problem Definition

## 2.1 Problem Statement

In this study, we consider an OPS with parallel aisles and two cross aisles at the front and the back of parallel aisles. Each parallel aisle has two aisle sides with the same number of item storage locations per side. Several depots are assumed to be in front of parallel aisles. Figure 1 displays a model with six one-way parallel aisles with ten storage locations per aisle side, and there are two depots in the middle of the front cross aisle. The depot workload balance is also necessary to be considered to ensure that all operated depots have a similar amount of work. As customers' shopping habits have changed from ordering large amounts with less variety of items to ordering smaller amounts with more variety [3], we focus on the OPS whose order size is 1~3 items.

A batch contains a set of orders picked in a single route. In this study, we assume that pickers apply the sort-while-pick strategy [3, 7] in the picking process, i.e., the capacity of each batch is measured by the number of orders that can be assigned into a batch. Pickers receive batch information from one depot, travel following the assigned route to collect all items of orders in that batch, then return to the same depot for conducting inspection and packaging. Pickers do not need to travel every aisle but if they enter an aisle, they must pass through all of it, following the one-way S-shape routing method. This study is conducted under the assumptions that travel time only depends on travel distance, as pickers' walking speed and picking time are unchanged; and blocking within aisles is also ignored.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Multi-depot order picking problem in a six-aisle OPS</figcaption></figure>

## 2.2 Route Length Calculation

We denote $L_a$ and $W_a$ as the length and width of an aisle, $n_{va}$ as the number of aisles visited by a picker; $s$ and $e$ be the starting aisle and ending aisle passed through by the picker with the route. The length $L$ of the whole route including assigned depot $d$ is:

$$L = L_a n_{va} + W_a(e - s) + W_a(|d - e| + |d - s|) \quad (1)$$

<figure><img src="image_4.png" /><figcaption>Fig. 2. Routing method in six-parallel-aisle OPS</figcaption></figure>

Considering the routing method for a six parallel-aisle system as shown in Fig. 2 with $L_a = 15$ and $W_a = 3$. The total number of combinations of 12 within-aisle routes and 6 depots is $12*6 = 72$. We consider the same route which enters aisle 3 ($s = 3$), travels through aisle 4, 5, and exits at the end of aisle 6 ($e = 6$). If it is combined with depot $d = 1$ (Fig. 2a), the length of the whole route will be $L = 15*4 + 3*(6-3) + 3*(|1-6|+|1-3|) = 90$, while combined with depot $d = 5$ (Fig. 2b) the length is $L = 15*4 + 3*(6-3) + 3*(|5-6|+|5-3|) = 78$ units of distance. As a result of this finding, selecting the appropriate depot for each picking trip has the potential to considerably shorten trip length.
## 3 Mathematical Formulation

We formulate MDOB problem as a mixed-integer linear programming model (MILP), where decision variables answer following questions: which orders are combined into a batch; and which route (including depot) is used to retrieve all the items of orders assigned into a batch. The initial number of batches is set equal to the number of orders to allow the worst case that every single order holds a separate batch. The set of routes and their travel distance when combined with each depot follows the definition given by Eq. (1).

The objective of this study is to minimize the total travel distance to retrieve all orders but still ensure the depot-workload balance. To do that, we set only “total travel distance minimizing” as the prime optimal objective, whereas “workload balancing” is considered as a constraint by setting a lower bound to the amount of workload at each depot. As the depot workload in this study is measured by the number of batches assigned to a depot, its lower bound (w) is the required minimum number of batches on each depot to make workload balance and is defined by the average integer number of batches per depot under the assumption that all batches reach the full capacity.

$$w = \left\lfloor \frac{\text{total number of orders}}{\text{cart capacity} \times \text{number of depots}} \right\rfloor \quad (2)$$

Indices and parameters:

A = the set of aisles a.

D = the set of operated depots d.

R = the set of routes r.

O = the set of orders o.

B = the set of batches b.

$L_{rd}$ = the length of route r with depot d.

C = the capacity of a cart.

$P_{oa} = \begin{cases} 1, & \text{if order } o \text{ has at least one pick on aisle } a \\ 0, & \text{otherwise} \end{cases}$

$V_{ra} = \begin{cases} 1, & \text{if route } r \text{ visits aisle } a \\ 0, & \text{otherwise} \end{cases}$

w = the minimum number of batches on each depot.

Decision variables:

$X_{ob} = \begin{cases} 1, & \text{if order } o \text{ is assigned into batch } b \\ 0, & \text{otherwise} \end{cases}$

$Y_{brd} = \begin{cases} 1, & \text{if batch } b \text{ takes route } r \text{ and depot } d \\ 0, & \text{otherwise} \end{cases}$

$$(MDOB) \min \sum_{b \in B} \sum_{d \in D} \sum_{r \in R} L_{rd} Y_{brd} \quad (3)$$

Subject to:

$$\sum_{b \in B} X_{ob} = 1, \quad \forall o \in O \quad (4)$$
$$X_{ob} \leq X_{bb}, \quad \forall o \in O, b \in B \qquad (5)$$

$$\sum_{o \in O} X_{ob} \leq C, \quad \forall b \in B \qquad (6)$$

$$\sum_{d \in D} \sum_{r \in R} Y_{brd} = X_{bb}, \quad \forall b \in B \qquad (7)$$

$$X_{ob}P_{oa} \leq \sum_{d \in D} \sum_{r \in R} V_{ra}Y_{brd}, \quad \forall a \in A, b \in B, o \in O \qquad (8)$$

$$\sum_{b \in B} \sum_{r \in R} Y_{brd} \geq w, \quad \forall d \in D \qquad (9)$$

$$X_{ob}, Y_{brd} \in \{0, 1\}, \quad \forall o \in O, b \in B, r \in R, d \in D \qquad (10)$$

The objective function (3) aims to minimize the total travel distance for picking all orders. Constraints (4) ensure that each order is assigned into one batch. Constraints (5) ensure that an order is assigned into a batch only when that batch is active (a batch is indexed by a specific order inside it). Constraints (6) limit the number of orders of each batch. Constraints (7) and (8) choose the route and depot for each batch so that all orders assigned to that batch can be picked. Constraints (9) ensure that the workloads among depots are balanced.

# 4 Experimental Results and Discussions

In this section, we present a small-scale solution example to help explain the considered problem and validate the proposed model. Then we conduct experiments on various small instances to investigate the effects of multi-depot application on the total travel distance. All experiments are set under some common parameters. We consider two configurations of an OPS with random storage policy, which are 6- and 8-parallel-aisle systems. The length and width of an aisle are 15 and 3 distance units. There are 30 picking locations on each aisle side. The number of one-way S-shape routes with the number of aisles equating to 6 and 8 are 12 and 33 routes, respectively [7]. The number of items per order is uniformly distributed (U[1, 3]). We implement our experiments using the Python API for IBM ILOG CPLEX Optimization Studio 20.1 on a Windows 10 Pro 64-bit server system with Intel(R) Core (TM) i5-10400 CPU @ 2.90 GHz and 16 GB memory.
## 4.1 A Small-Scale Solution

Table 1 shows the result of a small-size problem on the 6-aisle case in which depot 3 and depot 4 are operated (as illustrated in Fig. 1). The number of picking orders is 20 and the capacity is set at 3 orders per batch. An order picking information is represented by its relationship with aisle (e.g. [1, 1, 0, 0, 0, 0] depicts the order with picking locations belong to aisle 1 and aisle 2).

<table><caption>**Table 1.** An example of a small-size solution</caption><thead><tr><th>Batch</th><th>Route</th><th>Depot</th><th>Orders</th><th>Distance</th></tr></thead><tbody><tr><td>1</td><td>[1,1,1,0,0,1]</td><td>4</td><td>([0,1,1,0,0,1],[1,0,0,0,0],[0,0,1,0,1])</td><td>90</td></tr><tr><td>2</td><td>[0,0,1,1,0,0]</td><td>4</td><td>([0,0,0,1,0,0],[0,0,1,0,0],[0,0,0,1,0])</td><td>36</td></tr><tr><td>3</td><td>[0,0,1,1,0,0]</td><td>4</td><td>([0,0,1,0,0,0],[0,0,1,0,0],[0,0,1,0,0])</td><td>36</td></tr><tr><td>4</td><td>[0,0,0,0,1,1]</td><td>4</td><td>(([0,0,0,0,0,1],[0,0,0,0,1])</td><td>42</td></tr><tr><td>5</td><td>[1,1,0,0,0,0]</td><td>3</td><td>([1,1,0,0,0,0],[0,1,0,0,0],[0,1,0,0,0])</td><td>42</td></tr><tr><td>6</td><td>[1,1,1,1,0,0]</td><td>3</td><td>([0,1,0,1,0,0],[1,0,0,0,0],[0,1,1,0,0])</td><td>78</td></tr><tr><td>7</td><td>[1,0,0,1,1,1]</td><td>3</td><td>(([0,0,0,1,1,0],[0,0,0,1,1],[1,0,0,0,1,1])</td><td>90</td></tr><tr><td colspan="4">Total Travel Distance</td><td>414</td></tr></tbody></table>

As shown in the result, the optimal solution tries to fulfill a batch's capacity (6 full-capacity batches and only one two-order batch) and assign a best-fit route for all batches. The workloads between two depots are also balanced with 3 batches assigned to Depot 3 and 4 batches assigned to Depot 4. The optimal total travel distance of this problem is 414 units of distance.

## 4.2 Effects of Multiple Depots on Total Travel Distance

**Single-depot and Dual-depot Systems.** First, we aim to compare the performances between single-depot and dual-depot (two operated depots) situations by conducting experiments on several different configurations of MDOB with the capacity of each pick tour is equal to 3 orders. Each experiment consists of 20 randomly generated instances and the experiment result is represented by the average value of these instances. The average travel distance per order is used as the measurement for easily comparison. Figure 3 shows the results of four system configurations, in that D1, D2, ..., D8 denote the location of a depot by the index of the corresponding aisle. In each configuration, the diagonal cells (thick-border cells) are the results of the cases using only a single depot (e.g., D1-D1 is the situation of using only Depot 1), while the others are the results of operating two corresponding depots simultaneously. The index color demonstrates the performance comparison, in which the brighter color index indicates a better performance (i.e., the average travel distances are shorter) and vice versa.
<figure><img src="image_5.png" /><figcaption>a) Picking 20 orders in a 6-aisle OPS</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>b) Picking 30 orders in a 6-aisle OPS</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>c) Picking 20 orders in a 8-aisle OPS</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>d) Picking 30 orders in a 8-aisle OPS</figcaption></figure>

<caption>**Fig. 3.** Single-depot and dual-depot experimental results</caption>

For single-depot cases, the results show that travel distance is shortened when the depot location is closer the center of the front cross aisle, which is consistent with the conclusions of previous studies on single depot location [7]. The dual-depot application generally obtains better results than single-depot cases in all four configurations in terms of shortening travel distance. The best result of dual-depot cases can save 2.1~3.3% of the average travel distance on each order compared to the best result of single-depot cases. It leads to the opportunity of improving the problem performance by choosing the best proper locations of operated depots. However, these results also depict that combining the first and the last depot or combining two central depots is not a good choice for reducing travel distance.

**Multi-depot Systems.** We further investigate the effect of increasing the number of depots on the total travel distance by conducting experiments of picking 30 orders on the 6-aisle and 8-aisle systems with various cases of the number of depots. For each case, we investigate all the possible combinations of depots and the representing result for each combination is the average value of 20 randomly generated instances. The statistical results of these experiments are presented in Fig. 4 observably prove that using multiple depots significantly reduces the travel distance per order as well as the total travel distance compared to using only one depot. But for considered systems in this study in which depots are equipped in front of every aisle (i.e., the number of depots is equal to the number of aisles), the results show that if the number of depots is larger than a half of the number of aisles then operating more depots will not gain more advantages in saving the total travel distance. Besides, most of the outline points from multi-depot cases come from the combinations which include depots at farthest aisles. This emphasizes again that combining edge depots will not yield good performance.
<figure><img src="image_9.png" /><figcaption>Fig. 4. Effect of the number of depots on the total travel distance</figcaption></figure>

## 5 Conclusions

This study introduces a new order picking problem with multiple depots (MDOB) in that batching, routing, and depot-workload balancing processes are integrated. Orders are grouped into batches then each batch is assigned on a suitable combination of one route and one depot so that the total travel distance to retrieve all orders is minimized while the numbers of batches processed by selected depots are balanced. The concept of depots' workload balancing is expected to give good performance when combined with minimizing travel distances. The problem was formulated by a MILP model. Our experiments on small instances depict that operating a proper number of parallel depots does not only take advantage of reducing the completion time and bottleneck prevention but also gives positive effects on significantly shortening the total travel distance. These results are expected to lay the foundation for many interesting research topics on multi-depot warehouses in the future, particularly large-scale problems and the problem of defining the number of operated depots to achieve the best performance.

**Acknowledgment.** This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIT) (No. NRF-2020R1A2C2004320).

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