# Average Flow Time Estimation and Its Application for Storage Relocation in an Order Picking System

Jeongwon Park, Permata Joatiko, Chiwoo Park, Soondo Hong

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Jeongwon Park, Permata Joatiko, Chiwoo Park, Soondo Hong. Average Flow Time Estimation and Its Application for Storage Relocation in an Order Picking System. IFIP International Conference on Advances in Production Management Systems (APMS), Sep 2022, Gyeongju, South Korea. pp.60-66, 10.1007/978-3-031-16407-1_8. hal-05386007

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# Average Flow Time Estimation and Its Application for Storage Relocation in an Order Picking System

Jeongwon Park<fnref n="1" />, Permata Vallentino Eko Joatiko<fnref n="2" />, Chiwoo Park<fnref n="3" />, and Soondo Hong<fnref n="2" />(✉)

**Abstract.** This study proposes an average flow time estimation model based on Gaussian process regression that can be applied to adjust the storage locations of partial products and manages demand fluctuations and other dynamic order picking issues. We use the historical order picking data of a progressive zone picking system to extract features for the model. Subsequently, we train the estimation model and acquire the new storage location assignment by relocating part of the total products based on the estimated average flow time from the learning model. We test the proposed model using a simulation model based on a real cosmetic company's distribution center in South Korea. The simulation results indicate that the proposed model improves the performance by 9.61% with four relocation operations compared with the original storage location assignment before reassignment. The proposed model shows significant effectiveness when workloads are unbalanced, even in environments with high product diversity. We conclude that the proposed model could improve the productivity of real distribution centers with fewer reassignment operations.

**Keywords:** Facility logistics · Order picking · Flow time estimation · Storage location assignment · Warehouse

## 1 Introduction

The correct relocation of products assists in maintaining performance when warehouses and distribution centers (DCs hereafter) experience demand fluctuations and other problems that affect order picking operations. Balancing workloads minimizes the possibility of congestion in progressive zone picking systems [1]. Pan et al. [2] used a genetic algorithm for storage location assignments, and Chen et al. [3] formulated a mathematical model and used the tabu search algorithm to relocate items in a warehouse considering

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

<footnote n="2">² Department of Industrial Engineering, Pusan National University, Busan 46241, South Korea
soondo.hong@pusan.ac.kr</footnote>

<footnote n="3">³ Department of Industrial and Manufacturing Engineering, Florida State University, Tallahassee, FL 32306, USA</footnote>
the relocation effort. To avoid relocating an entire warehouse inventory, Accorsi et al. [4] proposed a bi-objective integer linear programming model for an adaptive storage allocation policy. Kim and Hong [5] developed a dynamic storage location assignment method in progressive zone picking with a bypass conveyor system and S/R crane. They regarded the dynamic storage assignment problem deterministically and formulated the problem using mixed-integer programming. In this study, we extend [5] by including Gaussian process regression in solving the dynamic storage location assignment problem. We develop an average flow time estimation model to evaluate the expected increase in performance from the relocation operation as a response to demand fluctuation. Then we optimize the storage location assignment by searching for the new storage locations that will yield the minimum estimated average flow time.

# 2 Problem Definition

## 2.1 Progressive Zone Picking System

We consider a progressive zone picking system with an S/R crane consisting of *z* zones with *r* racks and a picker per zone, as shown in Fig. 1. The pickers assigned to picking zones fill totes with an order's products and place the totes on a conveyor that transports them throughout the zones to an unloading station. If a zone's buffer is full, totes bypass the zone to avoid blocking and return after recirculation.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Progressive zone picking system with S/R crane and recirculation conveyor</figcaption></figure>

In this study, we assume that a product occupies one pick face. A picker loads product *p* into the tote from the assigned pick face *k*. When pickers start or finish order picking, they are at their depot stations located in the middle of each zone. The flow time of order *o* is the interval between the moment a tote enters the picking system and when the order is complete. The S/R crane transfers products to storage locations.

We obtain the average flow time $\bar{FT}$ from the sum of the flow times of the orders divided by the number of orders. An order list *l* is a group of orders processed within the same shift or time slot. We let $w_l^k$ be the number of picks at pick face *k* in order list *l*, and $K_z$ and $K_r$ be the set of pick faces allocated in zone *z*, and the set of pick faces
allocated in rack position *r*, respectively. We formulate the total number of picks in zone *z* of order list *l* as:

$$zw_{lz} = \sum_{k \in K_z} w_{lk}.$$

and the total number of picks in rack *r* of order list *l* as:

$$rw_{lr} = \sum_{k \in K_r} w_{lk}.$$

We use the following notations and parameters.

### Set and its indices

*O, o* Set of orders and its indices *o* ∈ *O*

*L, l* Set of order lists and its indices *l* ∈ *L*

*Z, z* Set of zones and its indices *z* ∈ *Z*

*R, r* Set of racks and its indices *r* ∈ *R*

*P, p* Set of products and its indices *p* ∈ *P*

*K, k* Set of pick faces and its indices *k* ∈ *K*

### Parameters

*$w_{lk}$* Number of picks at pick face *k* of order list *l*, $\forall l \in L$ and $k \in K$

*$zw_{lz}$* Number of picks in zone *z* of order list *l*, $\forall l \in L$ and $z \in Z$

*$rw_{lr}$* Number of picks in rack *r* of order list *l*, $\forall l \in L$ and $l \in L$

*$d_r$* Distance between depot station and rack *r* ∈ *R*

# 3 Model Formulation

## 3.1 Feature Preprocessing

To train the flow time estimation model, we extract the maximum and minimum numbers of workloads of the zones, and pickers' travel distances from historical order picking data. We calculate the maximum number and the minimum number of picks for order lists *l*, $x_l^1$ and $x_l^2$ as:

$$x_l^1 = \max_{\forall z \in Z} zw_{lz},$$

$$x_l^2 = \min_{\forall z \in Z} zw_{lz}.$$

and pickers' total travel distance of order list *l* as $x_l^3$ as:

$$x_l^3 = \sum_{r \in R} d_r \cdot rw_{lr}$$

Last, we merge $x_l^1$, $x_l^2$, and $x_l^3$ to form matrix *X* which is the input or feature.
## 3.2 Average Flow Time Estimation

Historical data from order picking system are noisy observations due to measurement errors and the influence of workload balance, rack storage policies, sequencing, etc., on order picking performance. Based on the definition of the Gaussian process adapted from Schulz et al. [6], Rasmussen and Williams [7], we formulate the joint probability of noisy observed data $(X, X_*, y)$ and posterior function $f_*$ as:

$$ \begin{bmatrix} y \\ f_* \end{bmatrix} \sim \mathcal{N} \left( 0, \begin{bmatrix} K(X, X) + \sigma_n^2 I & K(X, X_*) \\ K(X_*, X) & K(X, X_*) \end{bmatrix} \right), $$

where $\sigma_n$ controls the noise level. In this study, $X$ and $X_*$ denote the training input and testing input, respectively, $y$ denotes the training target (see Joatiko [8] for the details of the estimation model with the Gaussian process).

Prediction using the Gaussian process depends entirely on the kernel function Mackay [9]. The kernel function expresses the dependence of the two observations, $x_i$ and $x_j$. We use a Gaussian kernel combined with a linear kernel as the kernel function. We formulate test target $y_*$ the mean function of the posterior distribution as:

$$ p(f_*|X_*, X, y) = \mathcal{N}(\bar{f}_*, \text{cov}(f_*)). $$

To identify an optimal storage location, we use average flow time estimation. We measure the accuracy of the average flow time estimation model using the Pearson correlation coefficient, also called Pearson's r. Figure 2 shows the correlation along with Pearson's r value between the real value and the predicted value of 200 order lists using several experimental configurations as described in Sect. 4. We obtain real values using simulation and predicted values using the proposed flow time estimation model.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Flow time correlations between simulation and prediction values</figcaption></figure>
## 3.3 Storage Relocation Procedure

As mentioned, proper assignments of storage locations determine the best storage locations for the products in the order lists with minimal labor and cost. We assume a constant picking time per picker and that each picker picks each product independently. We assume identical zone size and rack size and that the termination time of the relocation operation is unknown because relocation times vary in practice.

We generate a candidate for the new storage assignment by switching the locations of two products. We consider only two products to switch at a time because we assume an unknown termination time. Thus, the total number of possible candidates is a combination of two of the total number of products ($\binom{|P|}{2}$). We evaluate the $\overline{FT}$ of each candidate using the proposed average flow time estimation and select the pair of products estimated to have the minimum $\overline{FT}$ if we switch them.

If more time is available to relocate the products again after the relocation operation, we use the same relocation procedure for the new storage assignment. Stage n denotes the number of switched pairs as determined by the experience of management, considering the number of storage locations that can be reassigned.

# 4 Experiment and Results

## 4.1 Experiment

Table 1 shows the order picking system's configuration based on the real DC. We base the size of each order on uniform distribution and base the picker's travel time to pick each product on its storage location. Classes A, B, and C differentiate the storage locations in each zone. We set each zone's buffer size at 3 and the constant work-in-process level (CONWIP) at 60. We train the estimation model on 300 order lists and test them with 200 different order lists. According to the workload distribution among zones, 40% of the order lists are classified as the balanced scenario and the remaining 60% as the unbalanced scenario. We use Python 3.7 and the scikit-learn toolbox to construct the model and Tecnomatix® Plant Simulation 12 to generate the synthetic historical data.

<table><caption>**Table 1.** Configuration of the order picking system experiment</caption><thead><tr><td>Parameter</td><td>Values</td></tr></thead><tbody><tr><td>Number of zones</td><td>8</td></tr><tr><td>Number of products</td><td>96, 144</td></tr><tr><td>Order size</td><td>Uniform (3,10)</td></tr><tr><td>Size of the rack in each zone</td><td>3 rows × 6 columns</td></tr><tr><td>Number of orders in each order list</td><td>100</td></tr><tr><td>Number of switched pairs (n)</td><td>1, 2, 3, 4</td></tr></tbody></table>
## 4.2 Experiment Results

We defined performance improvement as the $\overline{FT}$ reduction percentage relative to the initial storage assignment ($n = 0$). Table 2 shows the performance improvement of the proposed model from $n = 1$ to $n = 4$ depending on the number of products. To measure performance, we first consider the number of products to be 96. The proposed model yields an average improvement of 3.29% in one relocation operation and an average improvement of 8.44% in four relocation operations.

<table><caption>**Table 2.** Performance improvement of the proposed model</caption><thead><tr><th rowspan="3">Scenario</th><th colspan="8">Improvement percentage (%)</th></tr><tr><th colspan="4">|P| = 96</th><th colspan="4">|P| = 144</th></tr><tr><th>n = 1</th><th>n = 2</th><th>n = 3</th><th>n = 4</th><th>n = 1</th><th>n = 2</th><th>n = 3</th><th>n = 4</th></tr></thead><tbody><tr><td>Balance</td><td>1.30</td><td>1.85</td><td>2.38</td><td>2.63</td><td>1.76</td><td>2.98</td><td>3.65</td><td>4.16</td></tr><tr><td>Unbalanced</td><td>4.62</td><td>7.86</td><td>10.44</td><td>12.31</td><td>4.95</td><td>8.59</td><td>11.23</td><td>13.24</td></tr><tr><td>Average</td><td>3.29</td><td>5.46</td><td>7.22</td><td>8.44</td><td>3.67</td><td>6.34</td><td>8.20</td><td>9.61</td></tr></tbody></table>

Next, to analyze performance with high product diversity, we consider the number of products to be 144. The proposed model yields an average improvement of 3.67% in one relocation operation and an average improvement of 9.61% in four relocation operations. For both the unbalanced and the balanced scenarios, the improvement increases as the number of switched pairs increases, but the improvement in the unbalanced scenario is higher.

## 5 Conclusion

An average flow time estimation model was proposed to optimize storage location assignments in a progressive zone picking system considering the number of relocated products. Experiments based on a real DC showed that the proposed model successfully shortens the average flow time of order lists with a limited number of relocated products and that the proposed model significantly shortens the average flow time in an environment with high product diversity.

Having extracted features from historical data that were roughly related to the real workload balance of the real DC's picking system, future research will study the preprocessing step to confirm whether it could improve the accuracy of the proposed model. Further experiments will consider environments when stochastic picking times and pickers' capacities are not the same.

Despite the proposed model's performance, there is a risk that it will not achieve sufficient improvement considering relocation costs if there is not much room for improvement, or if the estimation model is not sufficiently trained. Future research will investigate all possible search procedures and approaches to guarantee estimates suitable for Gaussian process-based optimization.
**Acknowledgements.** This work was supported by 2022 BK21 FOUR Program of Pusan National University and was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. NRF-2020R1A2C2004320). This work was also supported by the Brain Pool Fellowship of the National Research Foundation of Korea (No. NRF-2019H1D3A2A01100649).

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