Performance Analysis of Picking Routing Strategies in the Leaf Layout Warehouse

Abstract

The routing strategy for order picking is an important factor in the efficiency of warehouse picking, and improvements to the warehouse layout provide more routing options for picking. The number of storage locations to be visited during the picking operation also has an impact on the selection of routing strategies. In order to achieve an effective improvement in the efficiency of picking operations in warehouse distribution centers, this paper focuses on the leaf warehouse layout based on the previous single-command operation strategy and extends it to study the multi-command operation strategy, in which three heuristic routing strategies, the S-shape, the return, and the composite, are introduced to solve the walking distance problem of picking operations, with the study of the selection of the routing strategy for different numbers of storage locations to be visited. Based on the distance equation between any two storage locations to be visited in the leaf layout warehouse, travel distance models corresponding to the three routing strategies in the picking operation are constructed, and the cuckoo search algorithm is introduced to solve and calculate the travel distance of the composite strategies for the experiments. In addition, the computational experiments of the three path strategies are carried out according to the different numbers of storage locations to be visited in the picking operation. By analyzing the numerical results, we find that the composite strategy has the best overall results among the three routing strategies, with the average values of optimization rates exceeding 30% (the S-shape) and 40% (the return), respectively. At the same time, the return strategy outperforms the S-shape strategy when the number of locations to be visited is less than seven. As the number of locations to be visited increases, the S-shape strategy gradually outperforms the return strategy. From a managerial and practical perspective, compared to the single-command operation strategy that is the focus of the current research, the multi-command operation strategy we studied is more relevant to the actual situation of order merging picking in warehouses and can effectively improve the efficiency of picking operations, the competitiveness of enterprises, and customer satisfaction of e-commerce enterprises.

1. Introduction

With the arrival of the “digital transformation era”, e-commerce has been able to develop rapidly, and customers’ demands for the efficiency of order item delivery have gradually increased [1]. Warehousing is the key to the efficiency of the supply chain [2], and the order picking operation is the main factor affecting the operational efficiency of warehouse distribution centers [3,4]. Among them, the cost of picking operations accounts for about 55% of the total cost of warehouse operations [5], while the walking time in picking operations accounts for more than 50% of the total operational time [6]. Therefore, in order to provide customers with a good experience, major e-commerce platforms have gradually started to realize the “after-order integration” of logistics and continuously improve the efficiency of picking and distribution. However, there are still some goods and orders that are difficult to pick and process efficiently due to their characteristics of multiple varieties and small batches [7,8]. The design of the storage layout and the routing strategy of the picking operation are important factors that affect the operational efficiency of the warehouse and distribution center [9,10]. Therefore, this paper will study the warehouse and distribution center from these two aspects in order to improve operational efficiency.

The current non-traditional design of the warehouse layout is mainly achieved by increasing the number of diagonal cross-aisles and changing the placement angles of the shelves. The flying-V layout and fishbone layout with two diagonal cross aisles were first proposed by Gue et al. [11,12]. Based on this, they further proposed the fishbone triangle layout [13] and the inverted-V layout [14]. They also achieved the chevron layout by varying the angle of shelf placement [15]. Öztürkoğlu [16] proposed the leaf layout and butterfly layout by changing the number of diagonal cross-aisles and the shelf placement angle simultaneously. Compared with the V-shape layout and fishbone layout, the design of the leaf layout changes the placement angle of the shelves in the area below the cross-aisle, further improving picking routing flexibility. Additionally, compared with the butterfly layout, the leaf layout has more warehouse storage area. Therefore, we perform the study of the routing strategy in the leaf layout warehouse in this paper. Travel time accounts for more than half of the total time of order picking operations [6], indicating that the routing strategy is critical to reducing picking operation time costs. Therefore, the main issues related to this study are the travel distance problem for picking operations in the leaf layout warehouse and the effect of the number of storage locations to be visited during the picking operation on the decision of the routing strategy. Further, based on Öztürkoğlu et al. [17] and Liu et al. [18], the performance analysis of routing strategies in the leaf layout warehouse is achieved by inputting different numbers of storage locations to be visited and outputting travel distances for different routing strategies under the condition of fixed parameters of the leaf warehouse layout design.

The motivations for this paper are:

• For non-traditional warehouse layouts, most of the current research is focused on the V-shaped layout [19,20,21] and fishbone layout [22,23,24,25,26,27], and less research is conducted on the leaf layout [28]. Therefore, we select the leaf layout as the research object in this paper to enrich the study of the routing strategy in the picking operation in non-traditional layout warehouses;
• There is no effective and efficient method to calculate the travel distance between any two storage locations to be visited in the leaf layout warehouse;
• The existing routing strategies in the leaf layout warehouse only have four applications: the S-shape, the return, the midpoint, and the largest gap, and more routing strategies need to be explored and extended.

Additionally, the main contributions of our work are:

• By focusing on the leaf warehouse layout, this paper develops the distance equation between any two storage locations to be visited in the picking operation of the leaf layout warehouse to meet the construction of walking distance models of heuristic routing strategies for different numbers of locations to be visited in order picking operations, which is one of the innovations of our work;
• We construct walking distance models corresponding to the three picking routing strategies of the leaf layout: the S-shape, the return, and the composite, in which the composite strategy of the leaf layout proposed by our work is also one of the innovative points of this paper.
• In this paper, the cuckoo search algorithm is introduced to solve and calculate the walking distance of the composite strategy in leaf layout warehouses for experiments, and the effectiveness of the cuckoo search algorithm is also proved based on the comparison of the numerical experimental results of the three routing strategies while demonstrating the selection of the routing strategy under different numbers of storage locations to be visited in the picking operation in leaf layout warehouses.

The structure of the rest of this paper is as follows: The next section reviews the related literature. Section 3 describes the problems studied in this paper and constructs the travel distance models of three routing strategies: the S-shape, the return, and the composite. In Section 4, the cuckoo search algorithm is introduced to solve and calculate the travel distance of the composite strategies for the experiments, and three path strategies are compared by the computational results that are carried out according to the different numbers of storage locations to be visited in the picking operation. Section 5 summarizes the research and points out the shortcomings of the paper and future research directions.

2. Literature Review

Our research focuses on the warehouse layout design, routing strategy, and S/R operations. Therefore, we mainly focus on these three aspects.

2.1. Warehouse Layouts

Warehouse layout design has a significant impact on the walking distance in picking operations and exceeds 60% [29]. The current literature on warehouse layout design also mostly focuses on travel cost or distance minimization as the optimization objective [6,12,14,30,31,32], with less consideration of operating costs [33], storage location assignment [34], storage space utilization [35,36], warehouse throughput [37,38], and operational strategies [39,40,41,42]. Warehouse layout problems can be divided into two categories according to their specific design: one is the layout of facilities and equipment, and the other is the design of the internal aisles of the warehouse [6]. The former is mainly achieved through planning the location of different functional areas within the warehouse (such as receiving and dispatching areas, storage areas, picking areas, etc.) to achieve effective collaboration of different functional areas and optimization of total costs; the latter focuses on the design of the angle, width, length, location, and the number of picking aisles within each functional area, as well as the number and location of workstations. The following discussion mainly focuses on non-traditional layout designs in the second category of warehouse layout problems.

In a traditional layout warehouse, shelves are placed straight and parallel to each other. The non-traditional layout design innovates by breaking the two traditional layout design principles: one is that the picking aisles must be straight and parallel to each other, and the other is that the cross-aisles must be straight and parallel to each other and perpendicular to the picking aisles. The design of non-traditional layouts first started with the design of radial picking aisles in non-rectangular warehouses [43]. After that, the field received no scholarly attention for nearly 30 years until the introduction of the fling-V layout and the fishbone layout [15]. The former has two non-planar diagonal cross-aisles, which can save about 10% travel distance compared with the traditional layout design, and the latter has two 45° and 135° diagonal cross-aisles, with the shelves above the diagonal cross-aisles placed vertically, and the shelves below placed horizontally, which can save about 20% travel distance for the traditional layout design. Further, the chevron layout [15], the fishbone triangle layout [13], the inverted-V layout [14], the leaf layout [17], the butterfly layout [17], and the discrete aisle layout [44] are derived, where the discrete aisle layout is equivalent to a variation of the traditional layout, similar to the multi-block traditional layout design. Compared with the fling-V layout and fishbone layout, the chevron layout, leaf layout, and butterfly layout break the constraint that shelves must be placed horizontally or vertically [17]. Moreover, some scholars have extended the study from a single I/O (input/output) point in non-traditional layouts to multiple I/O points [14,30,45,46]. In addition, other scholars have studied specific parameters in non-traditional layouts, such as the relationship between the optimal angles of the diagonal cross-aisles and the warehouse shape in the fishbone layout [23], the optimal angles of the diagonal cross-aisles in the V-shaped layout [31,47], and the 3D extension of the Flying-V and the fishbone layouts [48].

Through the existing literature, it can be found that the research on non-traditional layout designs is mostly focused on the fishbone layout and Flying-V layout, while the research on leaf layout is relatively lacking, and the existing literature on leaf layout is mainly shown in

Table 1. The existing literature on the leaf layout.

	Research Points	Storage Policy	Routing Strategy	Model/Method
Öztürkoğlu et al. [17]	Optimal layout	Random	Single-command	Expected walking distance model
Öztürkoğlu [49]	I/O points	Random	Single-command	Warehouse network model
Zhang et al. [50]	Twin leaf		Single-command	Expected walking distance model
Masae et al. [28]	Routing	Random, turnover-based	Optimal, midpoint, S-shape, largest gap, return	Eulerian graph, dynamic programming

.

2.2. Routing Strategies

The existing routing strategies mainly include the S-shape strategy, the return strategy, the midpoint strategy, the largest gap strategy, the composite strategy, and the optimal routing strategy [22,28,51,52]. Dijkstra et al. [53] comprehensively considered the picking routing problem and location allocation problem in the warehouse picking system, established the expected walking distance model of four routing strategies (the S-shape strategy, the return strategy, the midpoint strategy, and the largest gap strategy) under any location allocation strategy, and the expected picking walking distance model after dynamic planning of location allocation and made a comparative analysis. Using a dynamic programming algorithm to optimize the location allocation strategy, the picking walking distance is better than that under the four routing strategies. Rao et al. [54] designed an accurate analysis model to calculate the walking distance of the S-shape strategy based on the class-based storage strategy, and the accuracy was verified to be within 2%. Zhu et al. [55] improved the additional distance of the S-shape strategy order stochastic model based on class-based storage, which provides a reference for further optimizing the walking route of the picking operation.

More research focuses on optimizing the routing strategy combined with other warehousing system strategies or intelligent optimization algorithms. Hsu et al. [56] solved the shortest order batch picking routing problem through a genetic algorithm to shorten the walking route of the picking operation to the greatest extent. Theys et al. [57] used the TSP heuristic algorithm to shorten the walking distance of picking operations by 47% for the traditional multi-parallel warehouse system. Kulak et al. [58] improved the tabu search algorithm with a clustering algorithm to solve the problems of order batch picking and routing search in multi-picking-aisle warehouses and effectively improved the search quality and efficiency. Pan et al. [59] built a travel time estimation model for advanced storage systems and verified the accuracy of the model. Cheng et al. [60] combined the particle swarm optimization algorithm and the ant colony optimization algorithm to solve the problem of joint batch order picking routing optimization, which effectively improved the order picking efficiency. Chen et al. [61] combined the composite coding genetic algorithm and the ant colony algorithm to solve the optimization problem, integrating order batching, order sorting, and routing strategies in the warehouse. Experiments show that the optimization effect of the composite algorithm is better. Based on the perspective of a dynamic order picking routing, Lu et al. [62] applied the intrusive routing algorithm to optimize the walking route of a dynamically changing order goods picking operation and proved the feasibility and effectiveness of the algorithm. Gómez-Montoya et al. [63] applied a discrete particle swarm optimization algorithm and a genetic algorithm to optimize and solve the problems of cold storage equipment scheduling and picking operations. The experimental verification shows that the genetic algorithm has a higher degree of optimization for the walking distance of the picking operation. Hong et al. [64] established the order batch picking model using mixed integer programming and optimized it using the simulated return algorithm, which effectively improved the picking efficiency under the narrow aisle S-shape strategy. Scholz et al. [65] comprehensively considered order batching, sorting, and routing strategies, constructed a mathematical model, and obtained a highly efficient solution through a variable neighborhood descent algorithm. In order to improve the efficiency of warehouse picking, De Santis et al. [66] designed a meta-artery algorithm to minimize the walking distance of picking. Bódis et al. [67] optimized the routing strategy of order picking in the unit picking warehouse by a bacterial memory algorithm and a simulated annealing algorithm, respectively. The simulation results show that the optimization effect of the bacterial memory algorithm is more obvious. Zhou et al. [68] applied the genetic algorithm, the ant colony algorithm, and the cuckoo search algorithm to optimize the routing strategy optimization model of the picking operation under multiple instructions based on the fishbone storage layout under the random storage strategy. The results show that the cuckoo search algorithm has a better optimization effect under comprehensive conditions. Diefenbach et al. [69] sought the optimal storage allocation result through a mixed integer algorithm for a U-shaped layout so as to minimize the walking distance of the routing strategy in the picking operation. Yang et al. [70] designed a positioning interval distance algorithm, a location selection algorithm, a routing algorithm, and other algorithms to solve the shortest walking distance problem of picking operations in a multipoint storage system. Sebo et al. [71] compared and analyzed the effects of genetic algorithms, simple heuristic algorithms, and brute force algorithms in reducing the walking distance during picking operations. The results show that the brute force algorithm has more obvious advantages when picking personnel who are more experienced. Amorim-Lopes et al. [72] used the mixed integer optimization model to solve the shortest walking distance of a picking operation, which effectively improved the picking efficiency of a warehouse picking operation. Masae et al. [28,52] proposed an algorithm based on the graph theory and the dynamic programming method to calculate the walking distance of picking operations in the chevron layout and the leaf layout, respectively.

2.3. S/R Operations

The S/R operation strategy refers to the number of storage or retrieval operations performed during the travel of the picker or picking device from the I/O point and back to the I/O point to complete a picking operation. In warehouse picking systems, S/R operation strategies include single-command operation, dual-command operation, and multi-command operation. The single-command strategy refers to the picker’s travel from the I/O point and back to the I/O point to complete only one storage or retrieval operation, while the dual-command strategy is used to complete one storage and one retrieval operation. Since the multi-command strategy is similar to the routing strategy, the following is a brief description of the single-command and dual-command operation strategies in the context of warehouse picking.

The application of S/R operation strategies in non-traditional layouts is mainly the application of single-command and double-command strategies in picker-to-parts unit-load warehouses, and the main studies are shown in

Table 2. The studies of S/R operation strategies in non-traditional layouts.

	Warehouse Layout	S/R Operations
Gue et al. [12]	Fling-V, fishbone	Single
Pohl et al. [24]	Fishbone	Single
Pohl et al. [73]	Fishbone	Single, dual
Öztürkoğlu et al. [17]	Chevron, leaf, butterfly	Single
Gue et al. [14]	Fling-V	Single
Cardona et al. [23]	Fishbone	Single
Clark et al. [48]	Fling-V, fishbone	Single
Jiang et al. [74]	Fishbone	Single
Öztürkoğlu et al. [30]	Chevron	Single
Bortolini et al. [31]	V-shaped	Single
Mesa [75]	Diamond	Single
Öztürkoğlu et al. [49]	Leaf	Single
Accorsi et al. [76]	V-shaped	Single
Bortolini et al. [21]	V-shaped	Single
Zhang et al. [50]	Twin leaf	Single
Bortolini et al. [77]	V-shaped	Integration of single and dual
Kocaman et al. [45]	Chevron	Single

.

2.4. Research Gaps and Novelties

By summarizing and analyzing the existing literature, as shown in

Table 3. Comparison with the existing literature.

	Warehouse Layout	Routing Strategy, or S/R Operations	Model/Method
Zhou et al. [19]	V-shaped	Return, S-shape	Travel distance model, stochastic
Öztürkoğlu et al. [17]	Chevron, leaf, butterfly	Single-command	Expected walking distance model
Accorsi et al. [20]	V-shaped	Single-command	Access time model
Bortolini et al. [21]	V-shaped	Single-command	Analytic model
Masae et al. [52]	Chevron	Optimal, S-shape, midpoint, largest gap	Graph theory, dynamic programming
Masae et al. [28]	Leaf	Optimal, return, S-shape, midpoint, largest gap	Eulerian graph, dynamic programming
Zhou et al. [27]	Fishbone	S-shape, return	Travel distance model, stochastic
Liu et al. [18]	Chevron	S-shape, return, composite	Analytic model
Our work	Leaf	S-shape, return, composite	Analytic model

, it can be found that:

• Compared with the traditional layout, the non-traditional layouts mainly include the fishbone layout, the Flying-V layout, the chevron layout, the leaf layout, etc. Among them, there are relatively more studies on the fishbone layout and the Flying-V layout, and there is a lack of studies on the chevron layout and the leaf layout;
• The existing studies on S/R operation strategies in non-traditional layout warehouses in the literature mainly apply the single-command strategy to unit-load warehouses, focus mostly on the fishbone and Flying-V layouts, and fewer scholars have studied the multi-command strategy in the leaf layout warehouse;
• Few studies have examined the effect of the number of storage locations to be visited in picking operations in non-traditional layout warehouses on the selection of routing strategies.

Therefore, in this paper: (1) The leaf layout design is selected to enrich the theoretical basis for its practical application. (2) In the leaf layout warehouse, the travel distance models of three routing strategies, the S-shape, the return, and the composite, are constructed, and their performances are compared to meet the research needs of routing strategies in the non-traditional warehouse. (3) The selection of heuristic routing strategies under different numbers of storage locations to be visited in picking operations is investigated with the leaf layout as the research object.

Additionally, the novelties of our work are mainly focused on:

• Developing a new heuristic routing strategy in the leaf layout: the composite strategy;
• Exploring the distance equation between any two storage locations to be visited for the three heuristic path strategies, the S-shape, the return, and the composite, in the leaf layout warehouse, and based on the construction of walking distance models for the three routing strategies for different numbers of storage locations to be visited. We introduced the cuckoo search algorithm to numerically experiment and solve them;
• Evaluating three heuristic strategies by comparing them under different demand scales and storage assignments commonly used in warehouses.

3. Problem Description and Model Construction

The leaf warehouse layout is shown in Figure 1.

As the specific parameter settings in a warehouse picking system can have varying degrees of influence on the picking routing, the parameter settings for the leaf storage layout design are a prerequisite for the study of routing optimization under this layout. When the shelf angle of the lower right area in the leaf layout is approximately 32.33° (corresponding to the lower left area of 147.67°), and the angle of the right diagonal cross-aisle is 57.67° (corresponding to the angle of the left diagonal cross-aisle of 122.33°), the picking efficiency is the best in the picking operation, according to Öztürkoğlu et al. [17]. Therefore, the routing strategy of the picking operation in leaf layout is studied under the above parameter settings.

3.1. Notations and Assumptions

3.1.1. Notations

In order to facilitate problem description and model construction, as shown in

Table 4. Descriptions of the notations.

Notations	Descriptions
$L_R$$, W_R$	The length and width of the right-half warehouse (excluding the aisle width around the warehouse).
$l_S$$, l_A$$, l_U$	The widths of the shelf, the aisle, and the storage location.
${\alpha }_R$	The angle of the shelf and picking aisle in the lower right warehouse
${\beta }_R$	The angle of the diagonal cross-aisle in the right-half warehouse
$S_1,S_2,\cdots ,S_6$	Six picking areas divided by ${\beta }_R$$, {\alpha }_R$, and the central cross-aisle.
${\alpha }_L$	The parallel line of ${\alpha }_R$ passing through the top right vertex of the warehouse.
$D$	The total walking distance to complete a picking operation.
$i, j$	The $i$th and $j$th storage locations to be visited $1\le i\le n$$, 1\le j\le n$$, \mathrm{and} i\ne j$.
$d_{ij}$	The distance from the $i$th storage location to the $j$th storage location.
$x_{ij}$	Indicates whether to select the route from location $i$ to location $j$.
$(s,x,y,z)$	The coordinate of any location to be visited.
$s$$, s_i$$, s_j$	The serial number of the picking area.
$x$$, x_i$$, x_j$	The picking aisle where the location to be visited is located.
$y$$, y_i$	The side of the location to be visited in the picking aisle.
$z$$, z_i$	The location to be visited on the $y$ or $y_i$ side in the picking aisle.
$d_{01}$	The distance from the I/O point to the first visited location.
$d_{n0}$	The distance from the last visited location to the I/O point.
$x_{01}$	The route from the I/O point to the first visited location.
$x_{n0}$	The route from the last visited location to the I/O point.
$K$	The number of goods to be picked.
$n$	The number of storage locations to be visited.
$(s_1,x_1,y_1,z_1)$$, (s_i,x_i,y_i,z_i)$$, (s_n,x_n,y_n,z_n)$	The coordinates of the first, the $i$th, and the $n$th locations to be visited.
$d_{0i}$	The distance from the I/O point to $(s_i,x_i,y_i,z_i)$.
$d_{0i}{}^{\left(1\right)}$$, d_{0j}{}^{\left(1\right)}$	The walking distance from the I/O point to the diagonal cross-aisle or the front aisle.
$d_{0i}{}^{\left(2\right)}$$, d_{0j}{}^{\left(2\right)}$	The walking distance in the diagonal cross-aisle or front aisle.
$d_{0i}{}^{\left(3\right)}$$, d_{0j}{}^{\left(3\right)}$	The walking distance from the diagonal cross-aisle or front aisle to the $i$th or $j$th storage location.
$d_{0i}{}^{\left(4\right)}$$, d_{0j}{}^{\left(4\right)}$	The walking distance in the picking aisle to be visited.
${\varphi }_i$$, {\varphi }_j$	The length of picking aisle $x_i$ or $x_j$.
${\psi }_i$$, {\psi }_j$$, m_i$$, m_j$	The notations to judge the specific position of picking aisle $x_i$ or $x_j$.
${\phi }_i$$, {\phi }_j$	The remaining distance in picking aisle $x_i$ except $d_{0i}{}^{\left(4\right)}$ or $x_j$ except $d_{0j}{}^{\left(4\right)}$.
$W_i$$, W_j$	The mapping distance of the picking aisle $x_i$ or $x_j$ in the central cross-aisle when it is below ${\alpha }_L$.
$L_i$$, L_j$	The mapping distance of the picking aisle $x_i$ or $x_j$ in the front aisle when it is above ${\alpha }_L$.

, the notations used are defined as follows: $L_R$ and $W_R$ are the length and width of the right-half warehouse (excluding the aisle width around the warehouse), and $L_R=W_R=100$; $l_S$, $l_A$, and $l_U$, respectively, represent the widths of the shelf, the aisle, and the storage location, and $l_U=\frac{1}{2}{l}_S=\frac{1}{2}{l}_A=1$; ${\alpha }_R$ is the angle of the shelf and picking aisle in the lower right warehouse, and ${\alpha }_R={32.33}^{\circ }$; ${\beta }_R$ is the angle of the diagonal cross-aisle in the right-half warehouse, and ${\beta }_R={57.67}^{\circ }$; $S_1,S_2,\cdots ,S_6$ are six picking areas divided by ${\beta }_R$ and ${\alpha }_R$, and the central cross-aisle; ${\alpha }_L$ is the parallel line of ${\alpha }_R$ passing through the top right vertex of the warehouse; $D$ is the total walking distance to complete a picking operation; $d_{ij}\left(1\le i,j\le n,i\ne j\right)$ is the distance from location $i$ to location $j$; $x_{ij}\left(1\le i,j\le n,i\ne j\right)$ indicates whether to select the route from location $i$ to location $j$, specifically:

$$x_{ij}=\left\{\begin{array}{l}1, \mathrm{indicates} \mathrm{that} \mathrm{the} \mathrm{picking} \mathrm{operation} \mathrm{passes} \mathrm{through} \mathrm{path} x_{ij}\hfill \\ 0, \mathrm{indicates} \mathrm{that} \mathrm{the} \mathrm{picking} \mathrm{operation} \mathrm{does} \mathrm{not} \mathrm{pass} \mathrm{through} \mathrm{path} x_{ij}\hfill \end{array}\right.,i,j=1,2,3,\dots ,n.$$

At the same time, coordinate the location in the warehouse. Set the coordinate of any location to be visited as $(s,x,y,z)$, where $s$ is the serial number of the region, $s=1,2,3,4,5,6$; $x$ is the picking aisle where the location to be visited is located, $x=1,2,\cdots$; $y$ is the side of the location to be visited in the picking aisle. If the location to be visited is on the upper or right side of the picking aisle, $y=0$; otherwise, $y=1$; $z$ is the location to be visited on the $y$ side of the picking aisle $x$, and the numbering rule is that the I/O point is numbered successively from the I/O point to the far I/O point. For example, coordinates $(1,2,0,3)$ indicate that the location to be visited is located in the second picking aisle of area $S_1$, and the lower shelf is the third shelf from the lower right to the upper left. Here, set the I/O point coordinate to $(0,0,0,0)$ and the code to 0.

3.1.2. Assumptions

To ensure the accuracy of the model, referring to Öztürkoğlu et al. [17], Liu et al. [18], and Masae et al. [28], the following assumptions need to be made:

• The warehouse’s layout is symmetrical [17,18], and there is only one I/O point in the middle of the front aisle [17,28];
• The picker begins at the I/O point and returns to it at the end of the picking operation [18,28];
• The shelves in the warehouse are composed of unit storage locations with equal length and width, and the shelf height is not considered [28];
• There is no shortage of goods in the warehouse, the goods at each storage location can meet the picking demand, and the picker or equipment can complete each batch of picking list at one time [18];
• There is no order division, reorganization, or batching during picking [18,28];
• The picking operation runs along the picking aisle’s center line, and the picking operation of shelves on both sides can be completed from the picking aisle’s center line [28];
• Taking the right-half warehouse as an example, its length and width are the same [17], the shelf and the aisle are the same widths, and the shelf width is half of the shelf width [18].

3.2. Problem Description

According to the leaf layout settings and assumptions, the objective function of the walking distance in the leaf warehouse layout warehouse is:

$$D=\min (d_{01}{x}_{01}+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{d}_{ij}{x}_{ij}+d_{n0}{x}_{n0}}}),$$

(1)

subject to:

$${\displaystyle \sum _{i=1}^n{x}_{ij}}=1,j=1,2,3,\cdots ,n,$$

(2)

$${\displaystyle \sum _{j=1}^n{x}_{ij}}=1,i=1,2,3,\dots ,n,$$

(3)

$${\displaystyle \sum _{i,j\in K}{x}_{ij}}\le \left|K\right|-1,K\subset V,$$

(4)

$$x_{ij}=0 \mathrm{or} 1.$$

(5)

Equation (1) is the shortest walking distance required to complete a picking operation, which is essentially the practice of TSP problems in warehouse picking. Additionally, in Equation (1), $d_{01}$ is the distance from the I/O point to the first visited location and $d_{n0}$ is the distance from the last visited location to the I/O point. Equations (2) and (3) indicate that all the locations to be visited are picked only once in the picking process. Equation (4) avoids incomplete picking, where $K$ represents the number of goods to be picked. Additionally, $x_{01}=1$, $x_{n0}=1$, where $x_{01}$ indicates the route from the I/O point to the first visited location and $x_{n0}$ indicates the route from the last visited location to the I/O point.

Firstly, solve $d_{01}$ and $d_{n0}$ and set the coordinates of the first location to be visited and the $n$th location to be visited as $(s_1,x_1,y_1,z_1)$ and $(s_n,x_n,y_n,z_n)$, respectively. Assuming that the coordinates of any location to be visited are $(s_i,x_i,y_i,z_i)$, the distance $d_{0i}$ from the I/O point to $(s_i,x_i,y_i,z_i)$ can be divided into four parts: ① $d_{0i}{}^{\left(1\right)}$ is the walking distance from the I/O point to the diagonal cross-aisle or the front aisle; ② $d_{0i}{}^{\left(2\right)}$ is the walking distance in the diagonal cross-aisle or front aisle; ③ $d_{0i}{}^{\left(3\right)}$ is the walking distance from the diagonal cross-aisle or front aisle to the picking aisle to be visited; ④ $d_{0i}{}^{\left(4\right)}$ is the walking distance in the picking aisle to be visited. Namely:

$$d_{0i}=d_{0i}{}^{\left(1\right)}+d_{0i}{}^{\left(2\right)}+d_{0i}{}^{\left(3\right)}+d_{0i}{}^{\left(4\right)}.$$

(6)

Among them, $d_{0i}{}^{\left(1\right)}$, $d_{0i}{}^{\left(2\right)}$, $d_{0i}{}^{\left(3\right)}$, and $d_{0i}{}^{\left(4\right)}$ are as follows:

$$\begin{gathered} d_{0i}{}^{\left(1\right)}=\left\{\begin{array}{l}0.5\cdot l_A,s_i=1,6\\ l_A,s_i=2,3,4,5\end{array}\right., \\ {d}_{0i}{}^{\left(2\right)}=\left\{\begin{array}{l}\frac{\left(x_i-0.5\right)\left(l_A+l_S\right)}{\sin {\alpha }_R}-\frac{0.5l_A}{\tan {\alpha }_R},s_i=1 \mathrm{or} 6\\ \frac{\left(x_i-0.5\right)\left(l_A+l_S\right)}{\sin \left({\beta }_R-{\alpha }_R\right)}-\frac{0.5l_A}{\tan \left({\beta }_R-{\alpha }_R\right)},s_i=2 \mathrm{or} 5\\ \frac{\left(x_i-0.5\right)\left(l_A+l_S\right)}{\sin \left({90}^{°}-{\beta }_R\right)},s_i=3 \mathrm{or} 4\end{array}\right., \\ {d}_{0i}{}^{\left(3\right)}=\left\{\begin{array}{l}\frac{0.5l_A}{\sin {\alpha }_R},s_i=1\mathrm{or}6\\ \frac{0.5l_A}{\sin \left({\beta }_R-{\alpha }_R\right)},s_i=2\mathrm{or}5\\ \frac{0.5l_A}{\cos {\beta }_R},s_i=3\mathrm{or}4\end{array}\right., \end{gathered}$$

$$d_{0i}{}^{\left(4\right)}=\left\{\begin{array}{l}\left(z_i-0.5\right)\cdot l_U+\frac{\left(y_i-1\right)\cdot l_U}{\tan {\alpha }_R}+\frac{y_i\cdot 0.5\cdot \left(l_A+l_S\right)}{\tan {\alpha }_R},s_i=1 \text{or} 6\\ \left(z_i-0.5\right)\cdot l_U+\frac{\left(1-y_i\right)\cdot 0.5\cdot \left(l_A+l_S\right)}{\tan \left({\beta }_R-{\alpha }_R\right)}-\frac{y_i\cdot l_U}{\tan \left({\beta }_R-{\alpha }_R\right)},s_i=2 \text{or} 5\\ \left(z_i-0.5\right)\cdot l_U+\frac{y_i\cdot 0.5\cdot \left(l_A+l_S\right)}{\tan \left(90-{\beta }_R\right)}+\frac{\left(y_i-1\right)\cdot l_U}{\tan \left(90-{\beta }_R\right)},s_i=3\\ \left(z_i-0.5\right)\cdot l_U+\frac{\left(1-y_i\right)\cdot 0.5\cdot \left(l_A+l_S\right)}{\tan \left(90-{\beta }_R\right)}-\frac{y_i\cdot l_U}{\tan \left(90-{\beta }_R\right)},s_i=4\end{array}\right.$$

Meanwhile, when $s_i=$ 1 or 6, the length of the picking aisle $x_i$ is:

$${\varphi }_i=\left[L_R-\frac{0.5l_A}{\sin {\beta }_R}-\frac{(x_i-0.5)\cdot (l_A+l_S)}{\sin {\alpha }_R}\right]/\cos {\alpha }_R.$$

(7)

When $s_i=$ 2 or 5, it is necessary to judge the specific position of picking aisle $x_i$, including:

$${\psi }_i=\left\{\begin{array}{l}0,m_i=0\left(\mathrm{Picking} \mathrm{aisle} x_i \mathrm{is} \mathrm{located} \mathrm{below} {\alpha }_L\right)\\ 1,m_i>0\left(\mathrm{Picking} \mathrm{aisle} x_i \mathrm{is} \mathrm{above} {\alpha }_L\right)\end{array}\right.,$$

(8)

where $m_i$ is:

$$m_i=\max \left\{\left[L_R-\frac{0.5\cdot l_A}{\sin {\beta }_R}-\frac{(x_i-0.5)\cdot (l_A+l_S)}{\sin ({\beta }_R-{\alpha }_R)}\cdot \cos {\beta }_R\right]/\cos {\alpha }_R-{\varphi }_i,0\right\}.$$

When ${\psi }_i=0$, the length of the picking aisle $x_i$ is:

$${\varphi }_i=\left(L_R-\frac{0.5l_A}{\sin {\beta }_R}\right)/\cos {\alpha }_R-\frac{(x_i-0.5)\cdot (l_A+l_S)}{\tan \left({\beta }_R-{\alpha }_R\right)}+(x_i-0.5)(l_A+l_S)\tan {\alpha }_R.$$

(9)

When ${\psi }_i=1$, the length of the picking aisle $x_i$ is:

$${\varphi }_i=\frac{W_R}{\sin {\alpha }_R}-\frac{(x_i-0.5)\cdot (l_A+l_S)}{\tan \left({\beta }_R-{\alpha }_R\right)}-\frac{(x_i-0.5)\cdot (l_A+l_S)}{\tan {\alpha }_R}.$$

(10)

When $s_i=$ 3 or 4, the length of the picking aisle $x_i$ is:

$${\varphi }_i=W_R-\frac{0.5l_A}{\cos {\beta }_R}-(x_i-0.5)\cdot (l_A+l_S)\cdot \tan {\beta }_R.$$

(11)

The remaining distance in the picking aisle $x_i$ except $d_{0i}{}^{\left(4\right)}$ is:

$${\phi }_i={\varphi }_i-d_{0i}{}^{\left(4\right)}.$$

(12)

When the routing strategies are different, the travel routes between the two locations are different, so the expression forms of the objective function are different. The discussion is divided into the following situations.

3.3. Walking Distance Model of the Return Strategy

As shown in Figure 2, it is a return strategy. During the picking operation, the picker enters the goods to be picked aisle from the I/O point to pick the goods on one side. After completing the picking operation at the farthest storage location on one side of the aisle, the picker returns and completes the picking on the other side according to the original route until the picker exits the picking aisle and enters the next picking aisle. The above process is repeated until all picking operations in the warehouse are completed and the I/O point is returned.

As shown in Figure 2, Equation (1) of the return strategy is shown as follows:

$$D=d_{01}+{\displaystyle \sum _{i=1}^{n-1}{d}_{ij}}+d_{n0},j=i+1.$$

(13)

Among them, $d_{ij}$ is as follows:

(1) When two locations to be visited $i$ and $j$ are located in the same picking aisle, i.e., $s_i=s_j$, $x_i=x_j$,

$$d_{ij}=\left|d_{0i}{}^{\left(4\right)}-d_{0j}{}^{\left(4\right)}\right|.$$

(14)

(2) When two locations to be visited $i$ and $j$ are located in different picking aisles of the same area, i.e., $s_i=s_j$, $x_i\ne x_j$,

$$d_{ij}=\left|d_{0i}{}^{\left(2\right)}-d_{0j}{}^{\left(2\right)}\right|+d_{0i}{}^{\left(3\right)}+d_{0j}{}^{\left(3\right)}+d_{0i}{}^{\left(4\right)}+d_{0j}{}^{\left(4\right)}.$$

(15)

(3) When the two locations to be visited $i$ and $j$ are located in different picking areas, there are: when the two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_3$ or areas $S_4$ and $S_5$ respectively, $d_{ij}$ is the same as Equation (15). When the two locations to be visited $i$ and $j$ are located in different areas and in addition to the above situations,

$$d_{ij}=d_{0i}{}^{\left(2\right)}+d_{0j}{}^{\left(2\right)}+d_{0i}{}^{\left(3\right)}+d_{0j}{}^{\left(3\right)}+d_{0i}{}^{\left(4\right)}+d_{0j}{}^{\left(4\right)}.$$

(16)

3.4. Walking Distance Model of the S-Shape Strategy

As shown in Figure 3, it is an S-shape strategy. During the picking operation, the picker starts from the I/O point, enters the goods to be picked aisle from one end to complete the goods picking, and leaves from the other end to enter the next goods to be picked aisle. The aisle without goods for picking does not enter during the picking process, and the walking distance in the aisle with goods for picking is the length of the whole picking aisle. In the process of picking, if the total number of aisles to be visited is even and the last aisle to be visited is the last picking aisle of the warehouse, pick according to the S-shape strategy. If the total number of aisles to be visited is odd and the last aisle to be visited is the last picking aisle of the warehouse, the return strategy is used for picking.

As shown in Figure 3, Equation (1) of the S-shape strategy is shown as follows:

$$D=d_{01}+{\displaystyle \sum _{i=1}^{n-1}{d}_{ij}}+d_{n0},j=i+1.$$

(17)

If $n$ is odd, it will return to the I/O point directly after completing the picking operation of the $n$th location to be visited. The details of $d_{ij}$ are as follows:

(1) When two locations to be visited are located in the same picking aisle, that is $s_i=s_j$, $x_i=x_j$, $d_{ij}$ is the same as in Equation (14);

(2) When two locations to be visited $i$ and $j$ are located in the different picking aisles of the same area, i.e., $s_i=s_j$, $x_i\ne x_j$, $i$ is even and $d_{ij}$ is the same as in Equation (15). When $i$ is odd and $s_i=$ 1 or 6,

$$d_{ij}=\frac{\left|x_i-x_j\right|(l_A+l_S)}{\cos {\alpha }_R}+{\phi }_i+{\phi }_j+\frac{l_A}{\cos {\alpha }_R}.$$

(18)

When $i$ is odd and $s_i=$ 2 or 5, it is taken according to ${\psi }_i$ and ${\psi }_j$. When ${\psi }_i=0$, ${\psi }_j=0$, $d_{ij}$ is the same as in Equation (18). When ${\psi }_i=1$, ${\psi }_j=1$,

$$d_{ij}=\frac{\left|x_i-x_j\right|(l_A+l_S)}{\sin {\alpha }_R}+\frac{l_A}{\sin {\alpha }_R}+{\phi }_i+{\phi }_j.$$

(19)

When ${\psi }_i=0$, ${\psi }_j=1$,

$$d_{ij}=\frac{0.5l_A}{\cos {\alpha }_R}+W_i+l_A+L_j+\frac{0.5l_A}{\sin {\alpha }_R}+{\phi }_i+{\phi }_j.$$

(20)

When $s_i=$2 or 5, $W_i$ is the mapping distance of the aisle $x_i$ in the central cross-aisle. When picking aisle $x_i$ is below ${\alpha }_L$, and $L_j$ is the mapping distance of aisle $x_j$ in the front aisle when picking channel $x_j$ is above ${\alpha }_L$,

$$W_i=W_R-\frac{L_R-0.5l_A/\sin {\alpha }_R+0.5l_A}{\tan {\alpha }_R}-\frac{(x_i-0.5)(l_A+l_S)}{\cos {\alpha }_R},L_j=m_j\cdot \cos {\alpha }_R-\frac{0.5l_A}{\tan {\alpha }_R}.$$

When ${\psi }_i=1$ and ${\psi }_j=0$,

$$d_{ij}=\frac{0.5l_A}{\cos {\alpha }_R}+W_j+l_A+L_i+\frac{0.5l_A}{\sin {\alpha }_R}+{\phi }_i+{\phi }_j.$$

(21)

When $i$ is odd and $s_i=$3 or 4,

$$d_{ij}=\left|x_i-x_j\right|(l_A+l_S)+{\phi }_i+{\phi }_j+l_A.$$

(22)

(3) When the two locations to be visited $i$ and $j$ are located in different picking areas:

(1) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_6$, respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (16). When $i$ is odd,

$$d_{ij}=\frac{l_A}{\cos {\alpha }_R}+2(l_A+L_R+W_R)+{\displaystyle \sum _{i,j}^{}\left[\phi -(\varphi +\frac{0.5l_A}{\cos {\alpha }_R})\cdot \sin {\alpha }_R\right]}.$$

(23)

(2) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_5$, respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (16). When $i$ is odd,

$$d_{ij}=\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_i,{\psi }_i=0\\ L_R-L_i+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_i=1\end{array}\right.+{\phi }_i+{\phi }_j+\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_j,{\psi }_j=0\\ L_R-L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right..$$

(24)

(3) The two locations to be visited $i$ and $j$ are located in areas $S_3$ and $S_4$, respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (16). When $i$ is odd,

$$d_{ij}={\phi }_i+{\phi }_j+l_A+\left(x_i+x_j-1\right)(l_A+l_S).$$

(25)

(4) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_2$ or areas $S_5$ and $S_6$ respectively. When $i$ is even,

$$d_{ij}=d_{0i}{}^{\left(2\right)}+d_{0j}{}^{\left(2\right)}+d_{0i}{}^{\left(4\right)}+d_{0j}{}^{\left(4\right)}+d_{0i}{}^{\left(3\right)}+d_{0j}{}^{\left(3\right)}+0.5\cdot l_A.$$

(26)

When $i$ is odd,

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+{\phi }_j+\left\{\begin{array}{l}\frac{0.5l_A}{\cos {\alpha }_R}+\frac{(x_i+x_j-1)(l_A+l_S)}{\cos {\alpha }_R},{\psi }_j=0\\ W_R-({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R})\cdot \sin {\alpha }_R+L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right..$$

(27)

(5) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_3$ or areas $S_4$ and $S_5$ respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (15). When $i$ is odd,

$$d_{ij}={\phi }_i+{\phi }_j+0.5l_A+L_R-(x_j-0.5)(l_A+l_S)+\left\{\begin{array}{l}\frac{0.5l_A}{\cos {\alpha }_R}+W_i+l_A,{\psi }_i=0\\ -L_i+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_i=1\end{array}\right..$$

(28)

(6) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_3$ or areas $S_4$ and $S_6$ respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (26). When $i$ is odd,

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+W_R-({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R})\cdot \sin {\alpha }_R+l_A+L_R-(x_j-0.5)(l_A+l_S)+0.5l_A+{\phi }_j.$$

(29)

(7) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_4$ or areas $S_3$ and $S_5$ respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (16). When $i$ is odd,

$$d_{ij}=0.5l_A+\left(x_j-0.5\right)\left(l_A+l_S\right)+{\phi }_i+{\phi }_j+\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_i,{\psi }_i=0\\ L_R-L_i+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_i=1\end{array}\right..$$

(30)

(8) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_4$ or areas $S_3$ and $S_6$ respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (26). When $i$ is odd,

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+W_R-({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R})\cdot \sin {\alpha }_R+l_A+L_R+(x_j-0.5)(l_A+l_S)+0.5l_A+{\phi }_j.$$

(31)

(9) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_5$ or areas $S_2$ and $S_6$ respectively. When $i$ is even, $d_{ij}$ is the same as in Equation (26). When $i$ is odd,

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+W_R-\left({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R}\right)\cdot \sin {\alpha }_R+l_A+L_R+{\phi }_j+\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_j,{\psi }_j=0\\ L_R-L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right..$$

(32)

3.5. Walking Distance Model of the Composite Strategy

As shown in Figure 4, the composite strategy is mainly the combination of the S-shape strategy and the return strategy. At the same time, the design of the diagonal cross-aisle brings more possibilities to improve the picking efficiency.

As shown in Figure 4, Equation (1) of the composite strategy is shown as follows:

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+W_R-({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R})\cdot \sin {\alpha }_R+l_A+L_R+(x_j-0.5)(l_A+l_S)+0.5l_A+{\phi }_j.$$

(33)

Among them, $d_{ij}$ is as follows:

(1) When two locations to be visited $i$ and $j$ are located in the same picking aisle, that is, $s_i=s_j$, $x_i=x_j$, $d_{ij}$ is the same as in Equation (14);

(2) When two locations to be visited $i$ and $j$ are located in different picking aisles in the same area, i.e., $s_i=s_j$, $x_i\ne x_j$: (1) When $s_i=$1 or 6, $d_{ij}=\min \left(\mathrm{Equations} \left(15\right), \left(18\right)\right)$; (2) When $s_i=$2 or 5, $d_{ij}=\min \left(\mathrm{Equation} \left(15\right), \left[\mathrm{Equations} \left(18\right), \left(19\right), \left(20\right), \left(21\right)\right]\right)$; (3) When $s_i=$3 or 4, $d_{ij}=\min \left(\mathrm{Equations} \left(15\right), \left(22\right)\right)$.

(3) When the two locations to be visited $i$ and $j$ are located in different picking areas:

(1) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_6$, respectively, and mainly include the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (16);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (23).

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(16\right), \left(23\right)\right)$.

(2) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_5$, respectively, and mainly include the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (16);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (24);
• Via the corresponding picking aisles in areas $S_3$ and $S_4$, and from the diagonal cross-aisle to visit the picking aisle $x_j$ to complete the picking operation on the location $j$,

$$\begin{array}{cc}{d}_{ij}& =d_{0i}{{}^{\left(4\right)}}_{s_i=2}+d_{0i}{{}^{\left(3\right)}}_{s_i=2}+\left|d_{0i}{{}^{\left(2\right)}}_{s_i=2}-d_{0i}{{}^{\left(2\right)}}_{s_i=3}\right|+d_{0i}{{}^{\left(3\right)}}_{s_i=3}+{\varphi }_{i,s_i=3}\hfill \\ & +l_A+\left(x_i+x_j-1\right)\left(l_A+l_S\right)\hfill \\ & +{\varphi }_{j,s_j=4}+d_{0j}{{}^{\left(3\right)}}_{s_j=4}+\left|d_{0j}{{}^{\left(2\right)}}_{s_j=5}-d_{0j}{{}^{\left(2\right)}}_{s_j=4}\right|+d_{0j}{{}^{\left(3\right)}}_{s_j=5}+d_{0j}{{}^{\left(4\right)}}_{s_j=5}\hfill \end{array}$$

(34)

• Via the corresponding picking aisle in area $S_3$ and the back aisle, and from the back aisle or the right aisle to visit the picking aisle $x_j$ to complete the picking operation on the location $j$,

$$\begin{array}{cc}{d}_{ij}& =d_{0i}{{}^{\left(4\right)}}_{s_i=2}+d_{0i}{{}^{\left(3\right)}}_{s_i=2}+\left|d_{0i}{{}^{\left(2\right)}}_{s_i=2}-d_{0i}{{}^{\left(2\right)}}_{s_i=3}\right|+d_{0i}{{}^{\left(3\right)}}_{s_i=3}+{\varphi }_{i,s_i=3}+0.5l_A+\left(x_i-0.5\right)\left(l_A+l_S\right)+{\phi }_j\hfill \\ & +\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_j,{\psi }_j=0\\ L_R-L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right.\hfill \end{array}$$

(35)

• Via the back aisle and the corresponding picking aisle in area $S_4$, and from the diagonal cross-aisle to visit the picking aisle $x_j$ to complete the picking operation on the location $j$,

$$\begin{array}{cc}{d}_{ij}& =d_{0j}{{}^{\left(4\right)}}_{s_j=5}+d_{0j}{{}^{\left(3\right)}}_{s_j=5}+\left|d_{0j}{{}^{\left(2\right)}}_{s_j=5}-d_{0j}{{}^{\left(2\right)}}_{s_j=4}\right|+d_{0j}{{}^{\left(3\right)}}_{s_j=4}+{\varphi }_{j,s_j=4}+0.5l_A+\left(x_j-0.5\right)\left(l_A+l_S\right)+{\phi }_i\hfill \\ & +\left\{\begin{array}{l}{L}_R+l_A+\frac{0.5l_A}{\cos {\alpha }_R}+W_i,{\psi }_i=0\\ L_R-L_i+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_i=1\end{array}\right.\hfill \end{array}$$

(36)

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(16\right), \left(24\right), \left(34\right), \left(35\right), \left(36\right)\right)$.

(3) The two locations to be visited $i$ and $j$ are located in areas $S_3$ and $S_4$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (16);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (25).

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(16\right), \left(25\right)\right)$.

(4) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_2$ or areas $S_5$ and $S_6$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (27);
• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (26).

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(26\right), \left(27\right)\right)$.

(5) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_3$ or areas $S_4$ and $S_5$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (28);
• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (15).

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(15\right), \left(28\right)\right)$.

(6) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_3$ or areas $S_4$ and $S_6$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (29);
• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (26).

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(26\right), \left(29\right)\right)$.

(7) The two locations to be visited $i$ and $j$ are located in areas $S_2$ and $S_4$ or areas $S_3$ and $S_5$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (16);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (30);
• Via the diagonal cross-aisle, visit the picking aisle $x_1$ in area $S_3$, the back aisle, and from the back aisle to the picking aisle $x_j$ to complete the picking operation on the location $j$.

$$d_{ij}=d_{0i}{{}^{\left(4\right)}}_{s_i=2}+d_{0i}{{}^{\left(3\right)}}_{s_i=2}+\left|d_{0i}{{}^{\left(2\right)}}_{s_i=2}-d_{0i}{{}^{\left(2\right)}}_{s_i=3}\right|+d_{0i}{{}^{\left(3\right)}}_{s_i=3}+{\varphi }_{i,s_i=3}+l_A+\left(x_i+x_j-1\right)\left(l_S+l_A\right)+{\phi }_j.$$

(37)

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(16\right), \left(30\right), \left(37\right)\right)$.

(8) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_4$ or areas $S_3$ and $S_6$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (26);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (31);
• Via the left aisle, visit the picking aisle $x_1$ in area $S_2$, the diagonal cross-aisle, and from the diagonal cross-aisle to the picking aisle $x_j$ to complete the picking operation on the location $j$.

$$d_{ij}={\phi }_i+\frac{l_A}{\cos {\alpha }_R}+\frac{x_i(l_A+l_S)}{\cos {\alpha }_R}+{\varphi }_{1,s_i=2}+d_{0i}{{}^{\left(3\right)}}_{s_i=2}+d_{0i}{{}^{\left(2\right)}}_{s_i=2,x_i=1}+d_{0j}-d_{0j}{}^{\left(1\right)}.$$

(38)

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(26\right), \left(31\right), \left(38\right)\right)$.

(9) The two locations to be visited $i$ and $j$ are located in areas $S_1$ and $S_4$ or areas $S_3$ and $S_6$, respectively, and mainly have the following routing choices:

• It is the same as the S-shape strategy when $i$ is even and $d_{ij}$ is the same as in Equation (26);
• It is the same as the S-shape strategy when $i$ is odd and $d_{ij}$ is the same as in Equation (32);
• Via the left aisle, visit the picking aisle $x_1$ in area $S_2$, the diagonal cross-aisle, and from the diagonal cross-aisle to the picking aisle $x_j$ to complete the picking operation on the location $j$, and $d_{ij}$ is the same as in Equation (38).
• Via the bottom aisle, visit the picking aisle $x_1$ in area $S_6$ the right aisle, and from the right or back aisle to visit the picking aisle $x_j$ to complete the picking operation on the location $j$.

$$d_{ij}=d_{0i}-d_{0i}{}^{\left(1\right)}+d_{0i}{{}^{\left(2\right)}}_{s_i=6,x_i=1}+d_{0i}{{}^{\left(3\right)}}_{s_i=6}+{\varphi }_{1,s_i=6}+\frac{0.5l_A}{\cos {\alpha }_R}+{\phi }_j+\left\{\begin{array}{l}\frac{0.5l_A}{\cos {\alpha }_R}+\frac{x_j(l_A+l_S)}{\cos {\alpha }_R},{\psi }_j=0\\ W_R-\left({\varphi }_{1,s_i=6}+\frac{0.5l_A}{\cos {\alpha }_R}\right)\cdot \sin {\alpha }_R+l_A+L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right..$$

(39)

• Via the left aisle, visit the picking aisle $x_1$ in area $S_2$, the picking aisle $x_1$ in area $S_6$, the right aisle, and from the right or back aisle to the picking aisle $x_j$ to complete the picking operation on the location $j$,

$$\begin{array}{cc}{d}_{ij}& ={\phi }_i+\frac{l_A}{\cos {\alpha }_R}+\frac{x_i(l_A+l_S)}{\cos {\alpha }_R}+{\varphi }_{1,s_i=2}+d_{0i}{{}^{\left(3\right)}}_{s_i=2}+d_{0i}{{}^{\left(2\right)}}_{s_i=2,x_i=1}+0.5l_A+d_{0i}{{}^{\left(2\right)}}_{s_i=6,x_i=1}+d_{0i}{{}^{\left(3\right)}}_{s_i=6}+{\varphi }_{1,s_i=6}+\frac{0.5l_A}{\cos {\alpha }_R}+{\phi }_j\hfill \\ & +\left\{\begin{array}{l}\frac{0.5l_A}{\cos {\alpha }_R}+\frac{x_j(l_A+l_S)}{\cos {\alpha }_R},{\psi }_j=0\\ W_R-\left({\varphi }_{1,s_i=6}+\frac{0.5l_A}{\cos {\alpha }_R}\right)\cdot \sin {\alpha }_R+l_A+L_j+\frac{0.5l_A}{\sin {\alpha }_R},{\psi }_j=1\end{array}\right.\hfill \end{array}$$

(40)

• Via the left aisle, visit the back aisle, the diagonal cross-aisle, and from the diagonal cross-aisle to the picking aisle $x_j$ to complete the picking operation on the location $j$,

$$d_{ij}={\phi }_i+\frac{0.5l_A}{\cos {\alpha }_R}+W_R-\left({\varphi }_i+\frac{0.5l_A}{\cos {\alpha }_R}\right)\cdot \sin {\alpha }_R+l_A+L_R+\frac{W_R+0.5l_A}{\tan {\beta }_R}+\frac{W_R+0.5l_A}{\sin {\beta }_R}-d_{0j}{}^{\left(2\right)}+d_{0j}{}^{\left(3\right)}+d_{0j}{}^{\left(4\right)}.$$

(41)

Therefore, $d_{ij}=\min \left(\mathrm{Equations} \left(26\right), \left(32\right), \left(38\right), \left(39\right), \left(40\right), \left(41\right)\right)$.

Due to the randomness of order goods information and goods storage in the warehouse, the routing strategy selected in different picking operations will be different. Therefore, there will be different routing schemes between any two locations to be visited in the warehouse picking system. The above is the process of solving the walking distance between any two locations to be visited in the composite strategy of the leaf warehouse layout under the random storage strategy.

4. Results and Analysis

After the walking distance models of the three routing strategies (return, S-shape, and composite) were constructed, the return strategy and S-shape strategy results were easier to obtain, while the composite strategy needed to be solved using an optimization algorithm. According to Zhou et al. [68], the cuckoo search (CS) algorithm provides better results than the genetic algorithm and the ant colony algorithm under the fishbone layout, and the leaf layout is developed from the fishbone layout and the V-shaped layout, so the CS algorithm is applied here to solve the walking distance model of the composite strategy in the leaf layout warehouse, and the three routing strategies are compared and analyzed. The symbols used in the algorithm are as follows: $n$ is the number of host nests; $p_a$ is the probability of being discovered by the host (the probability of a new solution); $nd-1$ is the quantity of the location to be visited; $N\_iter$ is the maximum number of iterations in the solution process. The pseudo-code of the CS algorithm for solving the walking distance of the composite routing strategy is as Algorithm 1.

Algorithm 1: CS algorithm to solve the walking distance of composite routing strategy.

1:   begin 2:     Set up the nest: $n$, $X_i(i=1,2,\cdots ,n)$ 3:     Calculate the initial fitness value:$F_i(i=1,2,\cdots ,n)$ 4:     while termination conditions not met 5:       A new solution $X_i$ is generated by Lévy flight 6:       Calculate the fitness $F_i$ of the new solution 7:       Select the current optimal solution $X_j$ 8:       if $(F_i>F_j)$ 9:         Update current optimal solution 10:       end 11:       Find, and replace the inferior solution according to probability $p_a$ 12:       Save the optimal solution 13:     end 14:   end

4.1. Numeric Results

For the convenience of calculation, $l_A=l_S=2$ in the leaf layout set here, and the relevant parameters of the cuckoo search algorithm are: $n=100, Pa=0.25, N\_iter=500$. As the arrival of orders is random and unpredictable during order processing in actual e-commerce warehouse centers, and the results of optimizing the walking route of picking operations can vary between different goods in an e-commerce warehouse center, the coordinates of 100 randomly generated locations in a leaf layout e-commerce warehouse center were used, and each of the 10, 20, 30, and 40 locations containing them was randomly selected for 10 order picking operations.

Table 5. Coordinates and number when the number of locations to be visited is 10.

Coordinates	No.	Coordinates	No.	Coordinates	No.	Coordinates	No.
(0,0,0,0)	0	(2,6,0,75)	3	(3,2,0,28)	6	(6,2,0,83)	9
(5,8,1,29)	1	(3,7,0,16)	4	(3,14,0,1)	7	(2,1,1,110)	10
(5,2,1,71)	2	(4,4,0,23)	5	(4,8,0,38)	8	—	—

,

Table 6. Coordinates and number when the number of locations to be visited is 20.

Coordinates	No.	Coordinates	No.	Coordinates	No.	Coordinates	No.
(0,0,0,0)	0	(4,9,1,29)	6	(1,8,1,26)	12	(5,12,1,4)	18
(2,6,1,76)	1	(2,4,0,19)	7	(2,2,1,62)	13	(5,8,1,29)	19
(6,8,0,9)	2	(3,3,1,55)	8	(5,8,0,5)	14	(3,3,0,11)	20
(4,1,1,10)	3	(4,3,0,35)	9	(1,7,0,45)	15	—	—
(6,3,0,4)	4	(4,4,0,23)	10	(2,1,0,49)	16	—	—
(5,7,0,21)	5	(2,4,0,68)	11	(6,1,1,10)	17	—	—

,

Table 7. Coordinates and number when the number of locations to be visited is 30.

Coordinates	No.	Coordinates	No.	Coordinates	No.	Coordinates	No.
(0,0,0,0)	0	(4,4,0,23)	8	(4,3,1,42)	16	(6,8,1,11)	24
(3,7,0,55)	1	(5,7,0,53)	9	(4,9,0,19)	17	(6,7,0,32)	25
(5,8,1,29)	2	(1,8,1,26)	10	(2,4,0,68)	18	(6,5,1,60)	26
(5,12,1,4)	3	(5,7,0,9)	11	(6,8,1,14)	19	(6,3,0,4)	27
(5,2,1,3)	4	(2,6,1,76)	12	(4,5,1,48)	20	(2,2,1,91)	28
(1,3,0,46)	5	(2,10,0,32)	13	(6,11,0,16)	21	(6,4,1,73)	29
(1,9,1,5)	6	(1,1,1,92)	14	(4,10,0,8)	22	(3,7,1,19)	30
(4,3,1,16)	7	(3,1,1,82)	15	(5,7,0,21)	23	—	—

and

Table 8. Coordinates and number when the number of locations to be visited is 40.

Coordinates	No.	Coordinates	No.	Coordinates	No.	Coordinates	No.
(0,0,0,0)	0	(6,9,1,31)	11	(3,14,0,1)	22	(6,4,1,47)	33
(2,10,0,32)	1	(2,4,0,68)	12	(3,9,0,23)	23	(1,7,0,45)	34
(5,1,1,16)	2	(3,12,1,8)	13	(1,2,0,15)	24	(4,12,1,13)	35
(5,7,0,9)	3	(6,8,0,25)	14	(3,1,1,82)	25	(2,4,0,19)	36
(4,7,1,24)	4	(4,9,1,29)	15	(3,7,1,31)	26	(6,8,0,9)	37
(6,7,0,32)	5	(1,9,1,5)	16	(4,3,1,42)	27	(1,13,0,7)	38
(2,8,1,42)	6	(2,1,1,24)	17	(4,8,0,38)	28	(2,6,0,77)	39
(5,7,0,53)	7	(2,2,1,25)	18	(4,7,1,18)	29	(1,2,1,75)	40
(6,8,0,47)	8	(5,12,1,4)	19	(5,8,1,29)	30	—	—
(5,12,1,11)	9	(5,1,1,94)	20	(3,3,0,11)	31	—	—
(3,7,0,16)	10	(6,11,0,16)	21	(2,1,0,66)	32	—	—

show the coordinates and numbers of the individual orders with 10, 20, 30, and 40 picking positions, respectively, where the I/O points were considered as picking positions and the number was 0, as the walking route generated during the picking operation is a closed loop.

When the number of locations to be visited in a single picking order was 10, the walking distance generated by the return strategy and the S-shape strategy was 1310.9619 and 1180.4058, respectively. The walking distance of the composite strategy obtained by the cuckoo search algorithm was 630.2524, and the optimal visiting sequence in the picking operation was: 0→6→1→10→7→3→5→1→8→2→9→0. Figure 5 show the optimization convergence curve of the cuckoo search algorithm when the number of locations to be visited in a single picking order is 10.

When the number of locations to be visited in a single picking order was 20, the walking distance generated by the return strategy and the S-shape strategy was 1660.1804 and 1831.4547, respectively. The walking distance of the composite strategy obtained by the cuckoo search algorithm was 1202.7395, and the optimal visiting sequence in the picking operation was: 0→3→9→10→5→18→14→19→6→1→8→20→7→11→13→12→15→16→4→2→17→0. Figure 6 show the optimization convergence curve of the cuckoo search algorithm when the number of locations to be visited in a single picking order is 20.

When the number of locations to be visited in a single picking order was 30, the walking distance generated by the return strategy and the S-shape strategy was 2690.1585 and 2369.2273, respectively. The walking distance of the composite strategy obtained by the cuckoo search algorithm was 1435.9555, and the optimal visiting sequence in the picking operation was: 0→27→19→24→25→21→26→29→3→22→2→17→11→23→9→20→8→4→7→16→15→30→1→12→13→18→28→14→10→6→5→0. Figure 7 show the optimization convergence curve of the cuckoo search algorithm when the number of locations to be visited in a single picking order is 30.

When the number of locations to be visited in a single picking order is 40, the walking distance generated by the return strategy and the S-shape strategy was 2976.7253 and 2541.5162, respectively. The walking distance of the composite strategy obtained by the cuckoo search algorithm was 1618.9179, and the optimal visiting sequence in the picking operation was: 0→24→40→16→38→34→32→17→18→31→36→12→13→1→39→22→6→23→10→26→25→27→2→20→15→7→3→29→4→28→30→35→19→9→33→5→21→11→8→14→37→0. Figure 8 show the optimization convergence curve of the cuckoo search algorithm when the number of locations to be visited in a single picking order is 40.

4.2. Analysis of Numeric Results

The following is a summary of the calculation results of the travel distance of a single order and 10 orders for the S-shape strategy, return strategy, and composite strategy with 10, 20, 30, and 40 locations to be visited in the leaf layout warehouse, as shown in

Table 9. Results of the single order.

Number of Locations	Routing Strategy	CPU Running Time	Distance	Optimized Proportion
				to S-Shape	to Return	to Composite
10	S-shape	0.170682	1180.4058	0.00%	9.96%	−87.29%
	return	0.139839	1310.9619	−11.06%	0.00%	−108.01%
	composite	3.876069	630.2524	46.61%	51.92%	0.00%
20	S-shape	0.220125	1831.4547	0.00%	−10.32%	−52.27%
	return	0.164067	1660.1804	9.35%	0.00%	−38.03%
	composite	3.231893	1202.7395	34.33%	27.55%	0.00%
30	S-shape	0.239186	2369.2273	0.00%	11.93%	−62.72%
	return	0.175856	2690.1585	−13.55%	0.00%	−84.76%
	composite	4.420893	1456.0361	38.54%	45.88%	0.00%
40	S-shape	0.231844	2541.5162	0.00%	14.62%	−56.50%
	return	0.231118	2976.7253	−17.12%	0.00%	−83.30%
	composite	4.792556	1623.9958	36.10%	45.44%	0.00%

,

Table 10. Results of 10 orders.

Orders		1	2	3	4	5
10	S-shape	1180.4058	1150.3016	1109.7596	1056.5574	1072.8256
	return	1310.9619	1327.9820	1170.8092	1336.0670	943.3111
	composite	630.2524	797.9164	795.3686	585.9644	791.4205
20	S-shape	1831.4547	1590.7011	2012.4419	1783.7113	1892.3519
	return	1660.1804	1747.4264	1852.1397	1792.9413	2017.2153
	composite	1202.7395	977.0313	1183.2679	1056.9142	1137.5424
30	S-shape	2369.2273	1959.3428	2321.9254	2691.2847	2369.3568
	return	2690.1585	2163.4744	2716.2634	2885.6680	2886.8075
	composite	1456.0361	999.5249	1297.6445	1492.6086	1421.8737
40	S-shape	2541.5162	2524.2810	2570.8561	2754.9982	2920.5409
	return	2976.7253	2962.3912	2887.1379	3091.3559	2992.8084
	composite	1623.9958	1364.7088	1700.6997	1660.9445	1568.4643
Orders		6	7	8	9	10
10	S-shape	1086.0787	851.4957	1034.5959	1020.0817	1153.3554
	return	1459.9661	1016.7349	1428.8193	1074.5994	1095.0887
	composite	799.5363	649.7566	627.4810	682.9333	761.3741
20	S-shape	1801.2077	1825.5588	1662.3446	1804.1432	1761.4635
	return	1686.1954	1910.6276	1986.7992	2250.8594	1829.9145
	composite	1254.7663	1040.5726	1192.2926	1036.7237	1181.0978
30	S-shape	2289.9545	2364.2413	2320.6033	2118.5214	2059.3743
	return	2488.5332	2726.7524	2549.8209	2054.9456	2425.8369
	composite	1228.4882	1298.9774	1419.1072	1420.8772	1084.4516
40	S-shape	2728.6124	2594.4353	2592.3312	2937.5072	2678.2096
	return	2863.1707	3103.1463	3233.6603	3279.1435	3299.7715
	composite	1694.2977	1557.3505	1534.8450	1718.7293	1698.7772

and

Table 11. Summary of optimization proportion.

Group	Composite to S-Shape				Composite to Return
	10	20	30	40	10	20	30	40
<20%	0	0	0	0	1	0	0	0
20–30%	4	1	0	0	0	2	0	0
30–40%	4	5	4	6	5	3	1	0
40–50%	2	4	6	4	1	4	3	8
>50%	0	0	0	0	3	1	6	2
Total	10	10	10	10	10	10	10	10

.

By analyzing the results in Table 9, Table 10 and Table 11, in the leaf layout warehouse, it can be concluded that:

• The walking distance generated by the composite strategy is better than the S-shape strategy and the return strategy in the picking process;
• Compared with the S-shape strategy, the composite strategy provides better results than the return strategy, and the average proportion of optimization is higher than 40%;
• When the number of locations to be visited increases, the optimization effect of the composite strategy improves when compared to the S-shape strategy and the return strategy.

By calculating the walking distance generated when the number of locations to be visited in the picking operation is from 1 to 50, the results are shown in Figure 9. Additionally, according to Figure 9, it can be obtained that:

• In the leaf layout warehouse, regardless of the number of storage locations to be visited during the picking operation, the composite strategy outperforms both the S-shape and return strategies, which is consistent with the findings in Table 9, Table 10 and Table 11;
• When the order contains a small number of locations to be visited, the walking distance generated by the return strategy in the picking process is better than the S-shape strategy;
• With the increase in the number of locations to be visited in the order (the number of locations to be visited is greater than 7), the walking distance generated by the S-shape strategy is better than the return strategy.

To sum up, in the leaf layout storage center, among the three routing strategies of the return, S-shape, and composite, the composite strategy showed the best overall result. Among them, when the number of locations to be visited was less than seven, the return strategy outperformed the S-shape strategy, and the S-shape strategy was gradually better than the return strategy with the increase in the number of locations to be visited. In addition, the composite strategy had more obvious advantages in the leaf layout compared to the results of the chevron layout [18], but the design of more diagonal cross-aisles in the leaf layout may lead to a reduction in the number of storage locations.

4.3. Analysis of Storage Loss

When changing the warehouse layout design, the important thing related to the warehouse practitioner is the capacity of the warehouse. In the leaf layout warehouse, the design of the diagonal cross-aisle will lead to the change of storage area inside the warehouse. The loss of storage area in the leaf layout is analyzed below mainly by the variation of the number of storage locations that can be accommodated in the warehouse, and the results are shown in

Table 12. Storage loss of the leaf layout.

${\mathit{l}}_{\mathit{A}}$	${\mathit{l}}_{\mathit{S}}$	${\mathit{l}}_{\mathit{U}}$	Leaf Layout		Traditional Layout		Leaf Layout vs. Traditional Layout
			Number of Aisles	Number of Storage Locations	Number of Aisles	Number of Storage Locations	Storage Location Loss	Storage Area Loss	Loss Rate
1	1	0.5	166	38,918	100	40,000	1082	270.5	2.705%
1	2	1	112	12,778	68	13,400	622	622	4.642%
2	1	0.5	112	25,644	68	26,800	1156	289	4.313%
2	2	1	84	9460	50	10,000	540	540	5.400%
2	4	2	54	3056	34	3300	244	976	7.394%
4	2	1	56	6158	34	6600	442	442	6.697%
4	4	2	42	2238	26	2500	262	1048	10.480%

where $L_R=W_R=100$ and $l_U=\frac{1}{2}{l}_S$. In the traditional layout, the shelves are arranged longitudinally and parallel from the center to both sides, and there is no cross-aisle.

As can be seen from Table 12, compared with the traditional layout, the area loss of the leaf layout warehouse is directly related to the number of storage locations, which is mainly affected by the size of picking aisles, shelves, and storage locations, as follows:

• As shown in

  Table 13. Storage loss of the leaf layout at a certain width of the picking aisle.

  	${\mathit{l}}_{\mathit{A}}$	${\mathit{l}}_{\mathit{S}}$	${\mathit{l}}_{\mathit{U}}$	Loss Rate
  $l_A=1$
  	1	1	0.5	2.705%
  	1	2	1	4.642%
  $l_A=2$
  	2	1	0.5	4.313%
  	2	2	1	5.400%
  	2	4	2	7.394%
  $l_A=4$
  	4	2	1	6.697%
  	4	4	2	10.480%

  , at a certain width of the picking aisle, the wider the shelf, the larger the storage location, which leads to more loss of the storage area.
• As shown in

  Table 14. Storage loss of the leaf layout when the shelf width and the size of the storage location are certain.

  	${\mathit{l}}_{\mathit{A}}$	${\mathit{l}}_{\mathit{S}}$	${\mathit{l}}_{\mathit{U}}$	Loss Rate
  $l_S=1$$, l_U=0.5$
  	1	1	0.5	2.705%
  	2	1	0.5	4.313%
  $l_S=2$$, l_U=1$
  	1	2	1	4.642%
  	2	2	1	5.400%
  	4	2	1	6.697%
  $l_S=4$$, l_U=2$
  	2	4	2	7.394%
  	4	4	2	10.480%

  , when the shelf width and the size of the storage location are certain, the wider the picking aisle, the higher the storage area loss rate.
• As shown in

  Table 15. Storage loss of the leaf layout when the ratio of the picking aisle width to the shelf width is certain.

  	${\mathit{l}}_{\mathit{A}}$	${\mathit{l}}_{\mathit{S}}$	${\mathit{l}}_{\mathit{U}}$	Loss Rate
  $l_A:l_S=1:1$
  	1	1	0.5	2.705%
  	2	2	1	5.400%
  	4	4	2	10.480%
  $l_A:l_S=1:2$
  	1	2	1	4.642%
  	2	4	2	7.394%
  $l_A:l_S=2:1$
  	2	1	0.5	4.313%
  	4	2	1	6.697%

  , when the ratio of the picking aisle width to the shelf width is certain, the loss rate of the storage area increases gradually with the increase of both the picking aisle width and the shelf width.

In summary, from Table 12, Table 13, Table 14 and Table 15, it can be seen that, in a fixed-sized warehouse, the area loss caused by the leaf layout design is mainly related to the picking aisle width, shelf width, and the size of the storage location. In the seven scenarios provided above, changing the traditional layout to the leaf layout will, to some extent, lead to a loss of storage area (warehouse capacity), with a loss rate ranging from 2.705% to 10.480%, which is mainly caused by the design of the diagonal cross-aisle and the diagonal placement of shelves. Taking $l_A=l_S=2$ adopted in this paper as an example, the loss rate of storage area, in this case, is 5.400%, which is within the acceptable range. At the same time, the design of the diagonal cross-aisle in the leaf layout also brings more routing options for pickers during the picking operation, which can reduce the walking distance and improve the picking operation efficiency. Therefore, the performance analysis of routing strategies in the leaf layout warehouse is necessary and meaningful.

5. Conclusions

5.1. Theoretical Contributions

Based on Öztürkoğlu et al. [17], we selected the leaf warehouse layout among the three layouts they proposed as the object of our work and investigated the multi-command picking operation in it, which is also an extension of Öztürkoğlu et al. [17] on the single-command operation in the leaf layout. Additionally, the multi-command operation was more suitable for the combined picking of orders in picking operations, which in turn improved the efficiency of e-commerce warehousing. In the leaf layout warehouse, the shelf angle of the lower right area was about 32.33°, the angle of the right diagonal cross-aisle was 57.67°, and the left-half warehouse and the right-half warehouse were symmetrical. We constructed the travel distance models of the return strategy, the S-shape strategy, and the composite strategy and introduced the cuckoo search algorithm for the calculation and numerical experiment of the walking distance of the composite strategy. In the process of numerical experiment, there were mainly three scenarios: (1) comparison of the travel distances of the three strategies for a single order with 10, 20, 30, and 40 locations to be visited; (2) comparison of the travel distances of the three strategies for 10 orders with 10, 20, 30, and 40 locations to be visited; (3) comparison of the expected travel distances of the three strategies when the number of locations to be visited is from 1 to 50.

Based on the numerical experiments and comparative analysis of the results of the three strategies, our findings were mainly: (1) The cuckoo search algorithm can effectively achieve the optimization and calculation of the picking route in the leaf layout warehouse. (2) The composite strategy among the three strategies can achieve the shortest walking distance and the highest operational efficiency. (3) Compared with the S-shape strategy, the return strategy outperforms the S-shape strategy when the number of locations to be visited during the picking operation is less than seven. As the number of locations to be visited increases, the S-shape strategy gradually outperforms the return strategy. (4) Compared with the chevron layout, the composite strategy has a more obvious advantage in the leaf layout warehouse.

5.2. Managerial and Operational Implications

Based on the aforementioned computational results and analysis, from the managerial and operational perspective, our findings were mainly as follows:

• High customer satisfaction and low operational costs are the keys to gaining a competitive advantage for e-commerce companies, in which the non-conventional layout design of the warehouse and routing strategies both have an impact on the efficiency and cost of picking operations. We took the leaf warehouse layout design as the research object, of which three heuristic routing strategies, the S-shape, the return, and the composite, were studied and compared, and we found that the composite strategy had advantages over both the S-shape strategy and the return strategy, and the advantage over the return strategy was higher than 40%. Therefore, in the leaf layout warehouse, the use of the composite strategy was more helpful for companies to improve their competitiveness compared to the other two routing strategies.
• E-commerce sales were mainly affected by promotions and seasonal fluctuations, and consumer demand curves and order sizes were similarly subject to change, which can create significant challenges for warehouse picking operations. Our work shows that when the number of storage locations to be visited in picking operations is high, i.e., during the peak sales season, and for simplicity, the S-shape strategy can achieve better results in practice, while during the low sales season, when order sizes are small and fewer locations need to be visited, the return strategy is simpler and easier to implement.
• In addition, according to the results of Liu et al. [18], compared to the chevron layout, the composite strategy can achieve better results in the leaf layout, which also brings more options for the warehouse layout of e-commerce enterprises. However, at the same time, due to the design of the cross-aisle, the number of storage locations in the leaf layout was less than that of the chevron layout warehouse. Therefore, when choosing a warehouse layout for e-commerce companies, it is necessary to consider the balance between the number of storage locations and picking efficiency in the adopted layout design.

5.3. Future Research

There are many influencing factors in the research process of the actual operation and efficiency improvement of e-commerce warehousing centers. Based on the leaf layout design, this paper presented an in-depth study on the selection and routing strategy optimization of picking operations under the random storage policy, which can effectively enrich the theoretical content of picking operations and routing optimization and provide reliable ideas and methods for subsequent related research. However, at the same time, there are some limitations that cannot be ignored. In the future, we will consider further in-depth research on related problems from different warehouse layouts, different storage policies, different routing strategies, and different algorithms, such as the butterfly layout, which has received less attention in existing research, the classed-based storage policy, the midpoint and the largest gap routing strategies, and the optimal location and the optimal number of I/O points in the non-conventional layout warehouse. Furthermore, research can be conducted from the standpoints of picking equipment capacity limitations [78], multi-equipment picking operations [79], and energy consumption of facilities and equipment [80].