A Modified Brain Storm Optimization Algorithm for Solving Scheduling of Double-End Automated Storage and Retrieval Systems

Abstract

As a product of modern development, logistics plays a significant role in economic growth with its advantages of integrated management, unified operations, and speed. With the rapid advancement of technology and economy, traditional manual storage and retrieval methods can no longer meet industry demands. Achieving efficient storage and retrieval of goods on densely packed, symmetrically shaped logistics shelves has become a critical issue that needs urgent resolution. The brain storm optimization (BSO) algorithm, introduced in 2010, has found extensive applications across various fields. This paper presents a modified BSO algorithm (MBSO) aimed at addressing the scheduling challenges of double-end automated storage and retrieval systems (DE-AS/RSs). Traditional AS/RSs suffer from slow scheduling efficiency and the current heuristic algorithms exhibit low accuracy. To overcome these limitations, we propose a new scheduling strategy for the stacker to select I/O stations in DE-AS/RSs. The MBSO incorporates two key enhancements to the basic BSO algorithm. First, it employs an objective space clustering method in place of the standard k-means clustering to achieve more accurate solutions for AS/RS scheduling problems. Second, it utilizes a mutation operation based on a greedy strategy and an improved crossover operation for updating individuals. Extensive comparisons were made between the well-known heuristic algorithms NIGA and BSO in several specific enterprise warehouse scenarios. The experimental results show that the MBSO has significant accuracy, optimization speed, and robustness in solving scheduling of AS/RSs.

1. Introduction

With the rapid development of intelligent manufacturing and significant advancements in stacker transportation, the demands for storage capacity and stacker access efficiency in enterprises are increasing. As intelligent warehousing technology continues to progress, optimizing the efficiency of automated warehouses has become a critical trend in modern logistics. The traditional layout of single-end warehouse storage and retrieval platforms is no longer sufficient to meet the high-throughput requirements of contemporary enterprise warehouses. In this context, the operation mode and path optimization of stackers, also known as storage/retrieval (S/R) machines, play a pivotal role. As the most crucial transportation equipment in the warehouse, enhancing the efficiency of stacker operations directly impacts the overall storage and retrieval efficiency of the warehouse. Therefore, research on stacker optimization holds substantial theoretical and practical value.

Some scholars conducted extensive research on the layout structure optimization and storage strategy optimization of AS/RSs. Soyaslan et al. [1] studied a single-direction, high-density, deep-lane warehouse and developed a truck-based single-cycle order picking automatic storage and retrieval system (AS/RS), which can minimize travel time or energy consumption. Khojasteh et al. [2] solved the order-picking problem in a multi-channel unit load AS/RS served by one S/R machine. Lehmann et al. [3] proposed a multi-depth AS/RS travel time model, which includes four storage allocation strategies: two random processing strategies and strategies to minimize and maximize the variance of storage channel filling. Finally, the effectiveness of these models was tested and validated in detail through discrete event simulation. Xu et al. [4] integrated the input and output (I/O) points of AS/RS and considered separating and enhancing them by proposing four I/O point strategies. Finally, they established a travel time model to analyze these strategies and validated them using simulations.

In addition to the optimization strategies mentioned above, many researchers have also considered warehouse process constraints and physical space constraints when solving the optimization problem of location allocation, thus establishing a more realistic location allocation optimization model. Ma et al. [5] introduced the concept of conflicts in stacker loading/unloading sequences and warehouse collisions, establishing a mathematical programming model to minimize total completion time by eliminating conflict features. Tu et al. [6] added picking-load balancing and replenishment time constraints to the location allocation optimization model, proposed a heuristic multi-objective genetic algorithm to optimize the model, and verified the effectiveness of the algorithm through case studies. Liu et al. [7] introduced a continuous travel time model for the distribution center in a split platform AS/RS using an input and output (I/O) dwell point policy. Tang et al. [8] consider the principle of storing items in different lanes and propose a bacterial foraging algorithm to solve the multi-objective cargo space allocation model of a three-dimensional warehouse in the shipbuilding industry.

The scheduling optimization issue for AS/RSs is classified as NP-hard. This complexity poses significant challenges for conventional mathematical programming and heuristic methods to attain optimal outcomes. As a result, many researchers have investigated the use of various intelligent optimization algorithms to solve AS/RS scheduling problems. Yan et al. [9] proposed a non-dominated sorting artificial bee colony algorithm to address AS/RS scheduling problems, considering the sequencing constraints of multi-carrier S/R machine inbound and outbound tasks. Yu et al. [10] studied a TD-AS/RS with two storage areas and developed a search-based optimization algorithm to minimize the expected travel time of S/R machines. Gharehgozli et al. [11] developed a polynomial-time algorithm to sequence storage and retrieval tasks, aiming to minimize the total travel time of S/R machines in automated warehouses. Fandi et al. [12] improved and utilized a genetic algorithm to optimize AS/RS systems involving multi-channel AS/RSs and multi-shuttle S/R machines. Cai et al. [13] used a vortex search algorithm to optimize solutions, enhancing algorithm efficiency and storage efficiency in automated warehouses. Additionally, other researchers have explored the potential of combining multiple intelligent algorithms to address AS/RS scheduling problems. Wang et al. [14] introduced a two-stage path planning and real-time path trajectory control for mobile robots in AS/RSs. They combined tabu search with a genetic algorithm to optimize task scheduling time, improving system efficiency. Bessenoci et al. [15] developed tabu search and simulated annealing to achieve flow-rack AS/RS control, demonstrating their effectiveness in optimizing system performance. Li et al. [16] proposed a cost matrix-based greedy algorithm to address path planning issues for transportation equipment in AS/RSs. Kazemi et al. [17] designed an automated three-dimensional warehouse with a shared storage strategy, introducing an intelligent algorithm that combines ant colony optimization and variable neighborhood search to solve AS/RS scheduling problems.

The above intelligent optimization algorithms have achieved significant results in improving scheduling efficiency and optimizing storage strategies. However, these methods still have certain limitations when dealing with complex constraints or large-scale problems. Brain storm optimization (BSO), proposed by Shi in 2011 [18], is a promising swarm intelligence optimization method that simulates the human collective behavior known as brainstorming. Due to its flexible individual generation mechanism and powerful swarm intelligence optimization capability, BSO has been applied in various fields. For example, to address distributed flow shop and distribution scheduling problems, Hou et al. [19] proposed a multi-objective brain storm optimization algorithm to maximize processing quality and minimize the total weighted earliness and tardiness. Fu et al. [20] developed a multi-objective framework integrating a brain storm optimizer and a simulation system to solve stochastic multi-objective hybrid open shop scheduling problems, aiming to minimize total tardiness and energy consumption while meeting manufacturing requirements. Wu et al. [21] introduced an aggregated greedy BSO (AG-BSO) algorithm to tackle the classical traveling salesman problem (TSP), and simulations on standard TSP datasets showed that the AG-BSO algorithm excels in solution accuracy, optimization speed, and robustness. Cai et al. [22] proposed an alternative search mode BSO (ABSO) algorithm based on a search pattern to address the weak search capability and premature convergence of BSO, validating ABSO’s performance on 29 benchmark functions. These studies demonstrate that BSO is highly effective in solving various optimization problems. However, the implementation of BSO involves grouping the population into multiple clusters and generating new individuals using information from these clusters, which limits the algorithm’s exploration capability. Therefore, when optimizing problems with BSO, it is crucial to tailor the algorithm structure according to the specific characteristics of the problem to maximize BSO’s optimization potential.

In summary, existing research primarily focuses on optimizing strategies or improving existing heuristic algorithms to enhance stacker efficiency and optimize AS/RS inbound and outbound scheduling. Although these strategies have improved current AS/RSs, they often suffer from the limitations of a single approach. Therefore, building on previous research, this paper proposes a double-end automated storage and retrieval system (DE-AS/RS) structure to enhance stacker scheduling efficiency. Additionally, we have tailored BSO to the characteristics of DE-AS/RS to achieve better scheduling optimization results.

The main contributions of this paper on the AS/RS are as follows:

(1) For the DE-AS/RS problem, the I/O station that the stacker chooses to return is used as a scheduling strategy.

(2) The MBSO introduces two significant enhancements to the fundamental BSO algorithm. The first is the replacement of the original k-means clustering algorithm with an objective space clustering algorithm, which offers a more accurate and effective solution to the AS/RS scheduling problem. The second is the incorporation of a mutation operation based on a greedy strategy and an improved sequential crossover operation, which updates individuals within the population, increases solution space diversity, and enhances the effectiveness of information exchange between individuals.

The remainder of this paper is structured as follows: the problem description of DE-AS/RSs and basic assumptions are given in Section 2. In Section 3, the optimization model of the DE-AS/RS scheduling problem is presented. The MBSO used to solve the DE-AS/RS scheduling problem is introduced in detail in Section 4. The experiments and discussion are presented in Section 5. Finally, a summary of the paper with conclusions and directions for future improvement is presented in Section 6.

2. Problem Description and Assumptions

2.1. Problem Description of DE-AS/RS

The layout of the DE-AS/RS is similar to that of the regular AS/RS. The difference is that the DE-AS/RS has two I/O stations at both ends of the rack. The S/R machines can be started and parked from these I/O stations during work, which is convenient for the stacker to continue working. In addition, in the DE-AS/RS, it takes the same time for the tasks to be transported from the external starting point to the left and right I/O stations; the time for retrieving tasks to leave the warehouse from the two I/O stations and then transport to the designated delivery point is the same. The top view of the DE-AS/RS is shown in Figure 1.

There are two cycle operation modes of S/R machines: the single-command (SC) and the dual-command (DC) cycles. In the SC cycle, the stacker starts from an I/O station, completes a storage or retrieval task, and returns to any I/O station. In the DC cycle, the stacker begins at an I/O station, performs a storage task by placing items in designated storage units, then retrieves items, and finally returns to the I/O station. The specific SC and DC cycles of the DE-AS/RS can be seen in Figure 2a,b. The DC cycle can reduce the times of S/R machines to and from the I/O station and greatly improves the working efficiency of the stacker. However, in a batch of storage and retrieval task orders, maintaining consistency in the quantity of storage tasks and retrieval tasks is difficult, so it is inevitable for the stacker to perform the SC cycle.

For standard AS/RSs, when the I/O station is limited to being fixed at one end of the shelf, it can only optimize the distribution and combination of storage and retrieval tasks. In addition, in the DE-AS/RS, there is a scheduling strategy where the S/R machine can choose to return to either the left or right I/O station after each DC or SC operation. It can be seen from Figure 2b that when the stacker is performing a DC operation, the starting position at which the S/R machine picks up new order items depends on where it stops during the previous order. The I/O station that the S/R machine will return to depends on the station that the stacker takes less time to travel to from the current position. Similarly, when the stacker is performing an SC cycle, the principle for choosing which I/O station to return the stacker to is the same as mentioned above.

2.2. Basic Assumptions

In addition to the aforementioned description, further assumptions are put forward to simplify the DE-AS/RS model.

(1) Assume that there is only one S/R machine and that it moves at a consistent speed in both the horizontal and vertical directions. Effects of acceleration and deceleration are ignored.

(2) The S/R machine moves simultaneously in both horizontal and vertical directions. The times required for lifting, lowering, and other typical overhead tasks are disregarded.

(3) Each storage location can store only one task. During the operation of the stacker, only one task can be loaded at a time.

According to the fundamental assumptions provided, the time taken by the stacker to move from location (x1, y1) of task 1 to the position (x2, y2) of task 2 is determined by the longer duration of either the horizontal or vertical movement:

$$t_{ab}=max({\displaystyle \frac{|Xa-Xb|l}{Vx}},{\displaystyle \frac{|Ya-Yb|h}{Vy}})$$

(1)

where l and h are the width and height of the storage unit. Vx represents the moving velocity of the S/R machine in the horizontal direction, while Vy is the velocity in the vertical direction. Although these assumptions may not perfectly align with real-world conditions, they do not have a significant enough effect to greatly impact the optimization method being examined in our research.

3. Optimization Model of DE-AS/RS Scheduling Problem

According to the above content, the scheduling model established in this paper focuses on solving two problems: (1) Assigning appropriate DC and SC order sequences to a batch of tasks and (2) selecting the best I/O docking point for the stacker after each DC or SC operation. The key point to solve the above two problems is to minimize the travel time of the S/R machine to complete the retrieval process of the orders.

In the DE-AS/RS, the total duration for the S/R machine to finish this batch of warehouse tasks is:

$$T={\displaystyle \sum _{i=1}^m{T}_{D{C}_i}}+{\displaystyle \sum _{j=1}^n{T}_{S{C}_j}}$$

(2)

where T is the total time required to complete the order; T_DCi is the time taken to complete the i-th DC cycle; Tsc_j is the time taken for the j-th SC cycle to be completed; m and n represent the total number of DC and SC operations in storage and retrieval tasks orders, respectively.

If the time for loading and unloading of the stacker is ignored, the expression for the time taken to complete a DC cycle is as follows:

$$T_{DCi}=T_{I{O}_{w}I{P}_i}+T_{I{P}_{i}O{P}_i}+T_{O{P}_{i}I{O}_v},w,v\in \{0,1\}$$

(3)

where T_IOwIPi is the time for the S/R machine moving from the last docking I/O station IOw to the i-th storage position IPi during the i-th DC cycle; T_IPiOPi indicates the time when the stacker moves from IPi to the i-th retrieval position OPi; similarly, T_OPiIOv is the time for the machine to move from OPi to the optimal I/O station IOv during the i-th DC cycle. w, v are the variables selected between 0 and 1, according to the actual situation of optimization. And 0 represents the left I/O station, and 1 represents the right I/O station.

Therefore, when there are m DC tasks in a batch of storage and retrieval orders, the total time for the stacker to execute the DC cycles is as follows:

$$T_{DC}={\displaystyle \sum _{i=1}^m(T_{I{O}_{w}I{P}_i}+T_{I{P}_{i}O{P}_i}+T_{O{P}_{i}I{O}_v})}$$

(4)

Similarly, for the SC operation mode, the expression for the duration of an SC cycle is as follows:

$$T_{S{C}_j}=T_{I{O}_w{P}_j}+T_{P_{j}I{O}_v}$$

(5)

where T_IOwPj is the time for the S/R machine moving from the last docking I/O station to the storage or retrieval unit of task Pj; T_PjIOv is the time for the stacker to move from the position Pj to the nearest I/O station during the j-th SC cycle.

Therefore, when there are n SC tasks in a batch of storage and retrieval orders, the total time for the S/R machine to execute the SC cycles is as follows:

$$T_{SC}={\displaystyle \sum _{j=1}^n(T_{I{O}_w{P}_j}+T_{P_{j}I{O}_v})}$$

(6)

To sum up, Equation (2) is expressed as:

$$\begin{array}{l}T=T_{DC}+T_{SC}\\ ={\displaystyle \sum _{i=1}^{\mathrm{m}}(T_{I{O}_{w}I{P}_i}}+T_{I{P}_{i}O{P}_i}+T_{O{P}_{i}I{O}_v})+{\displaystyle \sum _{j=1}^n(T_{I{O}_w{P}_j}+T_{P_{j}I{O}_v})}\end{array}$$

(7)

In a batch of storage and retrieval tasks, the number of storage and retrieval tasks is generally not equal. Suppose there are n1 storage tasks and n2 retrieval tasks in a batch of orders. Take N1 = min (n1, n2) and N2 = max (n1, n2). Therefore, the batch retrieval storage orders are allocated into N1 DC cycles and N2-N1 SC cycles. In the DE-AS/RS, the total completion time that the S/R machine spends on finishing the order of tasks is:

$$T={\displaystyle \sum _{i=1}^{N1}(T_{I{O}_{w}I{P}_i}+T_{I{P}_{i}O{P}_i}+T_{O{P}_{i}I{O}_v})}+{\displaystyle \sum _{j=1}^{N2-N1}(T_{I{O}_w{P}_j}+T_{P_{j}I{O}_v})}$$

(8)

Finally, the optimization model of the DE-AS/RS scheduling problem is equivalent to minimize the travel time of the AS/RS machine to complete the retrieval process of the orders, given by Equation (9).

$$h(T)=min(T)$$

(9)

4. MBSO for DE-AS/RS Scheduling Optimization

4.1. The Principle of BSO

The issue of scheduling AS/RSs is well known and extensively researched in the fields of computer science and operations research. Similar to many other combinatorial optimization and routing problems, the scheduling optimization problem of AS/RSs is an NP-hard problem. Recently, various intelligent optimization algorithms, such as the GA, SA, PSO, and ACO have been proposed for solving and optimizing this kind problem. However, these algorithms, which need many iterations to find the approximate optimal solution, easily fall into local minima and have low convergence efficiency. Considering these shortcomings, this paper proposes an objective space clustering BSO algorithm with the greedy strategy to optimize the travel route of the S/R machine in the DE-AS/RS. The BSO algorithm, capable of gradually reducing the search space, has garnered attention from researchers due to its high convergence speed and solution accuracy [23,24]. BSO is usually combined with individual clustering and mutation mechanisms. The algorithm relies on two main operations: convergence operation and divergence operation. The core of the algorithm when solving a problem consists of the classification, selection, and individual updating of the solution scheme. The detailed steps of the BSO can be found in [25]. In this paper, we introduced a greedy strategy-based reverse mutation operation and an improved order crossover operation into the BSO algorithm for generating new individuals and named this new method the MBSO. These enhancements enable the MBSO to explore the solution space more effectively, thereby significantly accelerating the convergence speed and improving the quality of the results in solving AS/RS scheduling problems.

4.2. The Design of the MBSO

4.2.1. Multi-Layer Encoding Design

In view of the characteristics of the task scheduling of stackers in the DE-AS/RS and considering the selection of I/O station as a scheduling strategy, the original single encoding method in BSO cannot accurately process the storage and retrieval tasks. Therefore, in the MBSO, a multi-layer encoding method is used. It is divided into three layers. Firstly, two layers represent the storage task sequences and the retrieval task sequences. In order to observe the number of tasks in the sequences, the integer encoding method is used. The third layer encoding records the number of the optimal I/O station returned to by the stacker. The left I/O station and the right I/O station are recorded as 0 and 1, respectively.

To facilitate understanding, assume that the S/R machine receives a batch consisting of five storage operations and three retrieval operations. The encoding sequence is {42315/231/011010}, the sequence of storage tasks is Chrom1 = [4 2 3 1 5], the sequence of retrieval tasks is Chrom2 = [2 3 1], and the storage and retrieval I/O stations of the S/R machine are Point = [0 1 1 0 1 0].

After the S/R machine receives the task execution sequence, it starts the decoding process. Starting from its initial starting point, from the Point from left to right, the S/R machine respectively chooses the storage and retrieval tasks from Chrom1 and Chrom2 form the DC operations and the final storage or retrieval tasks are performed as SC operations. Every time the DC and SC operations are completed, the algorithm will record the I/O station docked by the stacker to the Point.

4.2.2. Initialization of Population

As previously stated, the S/R machine scheduling is a particular TSP problem in the DE-AS/RS. In [26], a multi-layer integer encoding technique is employed, symbolized by non-repeating sequences of retrieval and storage task numbers. The preliminary solutions are typically crucial in engineering applications, particularly in scheduling problems, as they have an impact on the ultimate optimization outcomes [27]. An initial population of high quality can increase the algorithm’s speed and likelihood of finding the best or most satisfactory solution.

Based on the model and characteristics of the AS/RS problem, we designed an improved method for generating better initial solutions, combining heuristic sorting and 3-opt local search techniques. The specific steps are as follows:

Step 1: Generate initial solutions with heuristic sorting

First, all tasks are sorted in ascending order based on their completion times to obtain the initial solutions. This heuristic sorting method can quickly generate superior solutions to be used as high-quality individuals in the initial population.

Step 2: Optimize with 3-opt local search

Next, the initial solutions obtained from the heuristic sorting are further optimized using the 3-opt local search algorithm. The 3-opt algorithm significantly improves the quality of the initial solutions and reduces scheduling time by performing broader adjustments and exchanges within the solution space.

Step 3: Select high-quality individuals

The high-quality individuals optimized by the 3-opt local search are added to the initial population. These optimized solutions not only have good initial quality but also expedite the process and enhance the likelihood of achieving optimal or satisfactory results.

Step 4: Enhance population diversity

To enhance the diversity of the population and avoid local optima, the remaining individuals are generated randomly. This mixed generation method ensures that the population contains high-quality initial solutions while maintaining sufficient diversity, thereby improving the overall performance of the algorithm.

4.2.3. Solution Clustering

Making the solution converge to a limited region is the aim of solution clustering. Various clustering algorithms can be applied within the BSO algorithm for this purpose [22,28]. The original BSO algorithm utilizes the basic k-means clustering method. Nonetheless, because of the weak robustness of the k-means algorithm, it is prone to getting stuck in local optimal solutions, with the cluster centers significantly influencing the clustering outcomes.

Therefore, an objective space clustering based on optimal scheduling time in BSO is proposed to cluster the DE-AS/RS. The detailed process for implementing the object space clustering algorithm is outlined below.

Step 1: Select m individuals from the current n individuals as clustering centers.

Step 2: Evaluate an individual and calculate the calculated Euclidean distance between the fitness value of the current individual and the fitness value of each cluster center.

Step 3: Assign the individual to a cluster with the smallest Euclidean distance from its cluster center.

Step 4: Repeat step 2 and step 3 until all individuals are clustered.

4.2.4. Generation of New Individuals

The core of the MBSO is used to generate new individuals by the way of thinking. Like other intelligent algorithms, intergroup and intragroup interactions are considered [23,29]. By maintaining the diversity of the population, the MBSO can efficiently find the optimal solution when solving the scheduling problems in the DE-AS/RS. In this paper, operators for mutation with the greedy strategy and improved order crossover are used to generate individuals.

(1) Mutation with the greedy strategy

For the individual in the cluster, the reverse mutation operation is adopted for the storage task sequence, and two positions in the storage task sequence are randomly selected. Then, the task sequence between two positions is reversed. The mutation process is shown in Figure 3.

Secondly, the greedy strategy is adopted for the retrieval task sequence. According to the storage task sequence, starting the first storage task from left to right, the greedy strategy selects the closest task from the retrieval task sequence to the first storage task according to Equation (1) and forms the first DC cycle; then, they are deleted from the original retrieval sequence until all DC cycles are completed and only SC operations are left.

(2) Improved order crossover operations

In order to enhance the variety of the population, a more effective order crossover method is implemented to refresh the population during the initial phase. This exchange rule is illustrated by an example below.

Step 1: Suppose that two task sequences, P1 and P2, are selected from the two clusters.

Step 2: Two crossover points are randomly selected from the two parents, and the crossover parts are marked as seg1 and seg2. The dotted circles are the components of the seg1 and seg2.

Step 3: Delete the same parts as seg2 and seg1 in P1 and P2, and obtain the remaining parts temp1 and temp2. Related operations are shown in Figure 4.

Step 4: Seg1 is regarded as a complete gene fragment and inserted between any two genes in temp2 in sequence, and multiple offspring can be produced. Then, seg1 is inverted and the same operation is performed. The offspring produced by the two operations are recorded as son1. Similarly, temp1 and seg2 can also produce multiple offspring, which are recorded as son2.

Step 5: The optimal individuals were selected from the set of son1 and son2, and they are inherited by the next operations.

4.2.5. Update Individuals

The new individuals generated above are compared with the individuals in each cluster center, and the individuals with better fitness values are retained. The optimal individual of each generation and the number of the docking station for each DC job and SC job are stored in the corresponding array in turn, and each row represents the optimal operation path of each iteration of the algorithm.

4.2.6. Output Optimal Solution

When the termination condition is met, the global optimal path and the corresponding optimal objective function are output.

The flow of the MBSO with hierarchical clustering is shown in Figure 5.

4.3. The Computational Complexity of the Algorithm

The importance of analyzing an algorithm’s computational complexity lies in its ability to help assess and evaluate the algorithm’s performance. Computational complexity includes both time complexity and space complexity, typically denoted by the symbol “O”. Time complexity describes the relationship between the algorithm’s execution time and the size of the input, while space complexity describes the relationship between the amount of memory required by the algorithm and the size of the input. Below, we will analyze the time and space complexity of the MBSO algorithm to gain insights into its efficiency and resource requirements.

The MBSO is an optimization algorithm used to solve the DE-AS/RS inbound and outbound scheduling problem. Its computational complexity primarily focuses on three aspects: objective space clustering, crossover operations, and mutation operations.

(1) Objective Space Clustering:

Time Complexity: This mainly depends on calculating the distance between each individual and the clustering centers. With two clustering centers defined in the algorithm, the time complexity for calculating distances to both centers is O(2n). Ignoring the constant factor, the simplified time complexity is O(n).

Space Complexity: This is determined by storing the positions of the clustering centers and intermediate results during calculations. With only two clustering centers, the space complexity for storing the centers is O(1). Additionally, the space required for storing each individual’s information leads to an overall space complexity of O(n).

(2) Crossover Operations:

Time Complexity: The crossover operations involve traversal and insertion, with each step having a time complexity of O(n).

Space Complexity: This is mainly determined by the number of parent and offspring individuals stored. Although the number of offspring is twice that of the parents, the overall storage requirement remains in the same dimension, so the space complexity is O(n).

(3) Mutation Operations:

Time Complexity: Mutation involves reversing the storage task sequence, resulting in a time complexity of O(n).

Space Complexity: The space complexity for mutations is determined by storing intermediate results and individual information, which is also O(n).

In summary, the overall time complexity of the MBSO is O(n), and its space complexity is O(n). This analysis helps in understanding the practical feasibility of implementing the MBSO algorithm in various industrial settings and its potential limitations.

5. Experiments and Discussion

In order to prove the practicability of the storage and retrieval scheduling model in the DE-AS/RS and the effectiveness of the MBSO, several specific scheduling problems of AS/RSs are given. The number of columns of each storage rack is 60, the number of layers of each storage rack is 12, and therefore there is a total of 720 cargo spaces. In the DE-AS/RS, the I/O stations are located at the lower left (Called IO_L) and lower right (Called IO_R) corners of the shelves, respectively, and the basic parameters of the DE-AS/RS are shown in

Table 1. Parameters of enterprise warehouse.

Index	Parameters
IOL station coordinates	(0, 1)
IOR station coordinates	(61, 1)
The length/width/height of the AS/RS (m/m/m)	1/1/1
The horizontal velocity of the S/R machine (m/s)	2
The vertical velocity of the S/R machine (m/s)	1

. The serial number of each storage and retrieval task and the specific storage location coordinate information during operation are shown in

Table 2. I/O operation order task sequence and the location coordinates.

Storage Tasks				Retrieval Tasks
Num	Coordinates	Num	Coordinates	Num	Coordinates	Num	Coordinates
1	(40, 6)	9	(43, 11)	1	(35, 11)	9	(5, 11)
2	(24, 8)	10	(39, 10)	2	(19, 8)	10	(45, 4)
3	(32, 12)	11	(20, 5)	3	(50, 12)	11	(58, 7)
4	(15, 11)	12	(25, 2)	4	(23, 9)	12	(18, 10)
5	(34, 4)	13	(42, 6)	5	(9, 7)	13	(32, 8)
6	(42, 1)	14	(6, 3)	6	(9, 5)
7	(28, 3)	15	(45, 8)	7	(27, 6)
8	(25, 12)			8	(52, 9)

.

To validate the superiority of the proposed MBSO, we selected the improved genetic algorithm (NIGA) [30] and the basic BSO algorithm for comparison, both of which are used to solve the AS/RS scheduling problem. The experimental case is based on the specific company scheduling instance shown in Table 2. The parameter settings of the algorithms are crucial to the optimization performance of the intelligent optimization algorithms. Therefore, the parameters of the MBSO, BSO algorithm, and NIGA in this study were set based on the relevant literature [31,32]. After thorough analysis and manual tuning, the parameters for the three algorithms were determined as shown in

Table 3. Algorithm-related parameter settings.

Algorithm	Related Parameters
MBSO	MacIter = 100; Population = 200; p_one = 0.5 p_two = 0.5; cluster_num = 5; p_replace = 0.1 p_one_center = 0.4; p_two_center = 0.6
BSO
NIGA	MacIter = 100; Population = 200; pc = 0.6, pm = 0.2

. The full names of the algorithm parameters are listed in

Table 4. The full names of the relevant parameters.

Full Names	Relevant Parameters
Maximum number of iterations	MacIter
Number of clusters	cluster_num
Replace cluster center probability	p_replace
Choose a clustering probability	p_one
Choose two clustering probabilities	p_two
Choose a cluster center	p_one_center
Select two cluster centers	p_two_center

. To ensure fairness in the experimental comparison, all algorithms were implemented in Matlab 2018b and tested on a computer with an Intel Core i5-12400 processor running at 2.50 GHz and 16 GB of RAM.

The dynamic average convergence curves of the MBSO, BSO algorithm, and NIGA for the case in Table 2 are shown in Figure 6, with each algorithm set to a fixed iteration number of 100. From Figure 6, it is evident that for the storage and retrieval tasks in Table 2, the proposed MBSO converges at the 17th iteration, achieving a task scheduling optimization time of 435. In contrast, the scheduling optimization times of the NIGA and BSO algorithm continue to converge with increasing iterations but eventually settle at local optima. The NIGA converges at the 28th iteration with a scheduling optimization time of 442, while the BSO algorithm converges at the 38th iteration with a scheduling optimization time of 437. These results demonstrate that the MBSO outperforms the NIGA and BSO algorithm in terms of convergence speed and final optimization results. The MBSO not only achieves convergence in fewer iterations but also shows a significant advantage in optimization time. In comparison, while the NIGA and BSO algorithm gradually converge, they ultimately only reach local optima and exhibit slower convergence speeds. This indicates that the MBSO possesses higher efficiency and stronger global search capability in addressing AS/RS scheduling optimization problems.

To further verify the optimization effect and stability of the proposed MBSO, the three algorithms were run randomly 10 times, with the specific data shown in

Table 5. Simulation results of three algorithms running 10 times at random.

Number of Experiments	1	2	3	4	5	6	7	8	9	10
NIGA	441	442	442	435	442	441	445	445	442	442
BSO	438	437	439	435	437	440	435	441	435	437
MBSO	435	435	435	435	435	435	435	435	435	435

. From Table 5, it can be seen that the MBSO consistently achieved a scheduling optimization time of 435 in all 10 runs, demonstrating the best and most stable optimization performance. BSO was second, achieving the optimal scheduling optimization time of 435 in 3 out of 10 runs, with the other 7 runs reaching different local optimization times. The NIGA performed the worst, achieving the optimal completion time only once, with the other nine runs exceeding 440 in optimization time. In summary, the experimental results show that the proposed MBSO algorithm has better and more stable optimization capabilities compared to BSO and the NIGA in solving the DE-AS/RS scheduling problem. This is because the MBSO employs object space clustering to obtain near-global optimal solutions and utilizes a combination of greedy strategy mutation operations and improved crossover operations to generate new individuals during the optimization process. This ultimately allows the MBSO to have better convergence speed and optimization ability when optimizing the DE-AS/RS scheduling problem, enabling it to escape local optima more quickly. After solving the tasks in Table 2 with the MBSO, we recorded the optimized storage and retrieval task sequences and the I/O stations where the S/R machine docked each time in

Table 6. I/O task sequence and its corresponding I/O counter sequence.

Num	Storage Task Serial Number	Retrieval Task Serial Number	I/O Station	Num	Storage Task Serial Number	Retrieval Task Serial Number	I/O Station
0	/	/	0 *	9	10	3	0 *
1	2	4	1 *	10	14	6	0 *
2	7	7	1 *	11	12	5	0 *
3	3	1	0 *	12	4	9	1 *
4	11	2	0 *	13	13	11	1 *
5	1	10	0 *	14	6	/	1 *
6	5	13	0 *	15	15	/	1 *
7	8	12	1 *
8	9	8	0 *

Note: 0 * and 1 * in the table represent IO_L and IO_R, respectively.

and plotted the S/R machine’s travel path in Figure 7. For ease of display, use a solid green line to highlight the running path of the stacker to execute SC operation. Similarly, the green lines in the following figure represent the same meanings.

To further verify the practicability of the DE-AS/RS in this paper and the superiority of the strategy where the R/S machine can choose the closer I/O station after optimizing, we have compared the scheduling optimization results between the DE-AS/RS and the traditional AS/RS whose I/O station is set at the left or the right end of the shelf. The MBSO mentioned in this paper is used as a benchmark to optimize the scheduling of the task orders in Table 2. The dynamic average convergence curves of the scheduling instance between the AS/RS with IO_L and the AS/RS with IO_R are shown in Figure 8a,b. Figure 9a,b show the optimal paths of the S/R machines of two AS/RSs.

From Figure 8a,b, the same conclusion can be obtained: No matter which type I/O of the AS/RS, compared with BSO and the NIGA, the MBSO proposed in this paper has better optimization results and faster convergence speed. Additionally, it can be seen from Figure 9a,b that the tasks in Table 2 are relatively evenly distributed in the shelf in the warehouse. In the AS/RS with IO_L and the AS/RS with IO_R, the MBSO obtains the optimal running time of the stacker to be 505 s and 516.5 s, respectively, and the two results are not much different. But the DE-AS/RS scheduling in this paper takes less time than them. And the optimization efficiency is increased by 13.9% and 15.8%, respectively.

In order to further explore the potential advantages of the DE-AS/RS, this paper continues to use these algorithms to optimize instance 1 (the storage and retrieval tasks and their coordination are shown in

Table 7. The information of storage and retrieval tasks for instance 1.

Storage Tasks				Retrieval Tasks
Num	Coordinates	Num	Coordinates	Num	Coordinates	Num	Coordinates
1	(41, 6)	9	(29, 11)	1	(32, 11)	9	(5, 11)
2	(16, 8)	10	(39, 6)	2	(4, 8)	10	(13, 4)
3	(32, 12)	11	(20, 5)	3	(58, 7)	11	(50, 12)
4	(15, 11)	12	(25, 12)	4	(26, 9)	12	(52, 9)
5	(34, 1)	13	(42, 4)	5	(9, 7)	13	(19, 8)
6	(21, 1)	14	(11, 3)	6	(9, 2)
7	(6, 3)	15	(44, 8)	7	(27, 6)
8	(25, 2)			8	(18, 10)

) and instance 2 (the storage and retrieval tasks and their coordination are shown in

Table 8. The information of storage and retrieval tasks for instance 2.

Storage Tasks				Retrieval Tasks
Num	Coordinates	Num	Coordinates	Num	Coordinates	Num	Coordinates
1	(40, 7)	9	(43, 11)	1	(35, 11)	9	(5, 11)
2	(35, 6)	10	(39, 10)	2	(19, 8)	10	(23, 6)
3	(53, 12)	11	(40, 5)	3	(50, 12)	11	(58, 7)
4	(30, 12)	12	(37, 2)	4	(15, 9)	12	(18, 10)
5	(20, 4)	13	(42, 6)	5	(49, 7)	13	(32, 8)
6	(42, 1)	14	(6, 3)	6	(59, 5)
7	(38, 3)	15	(13, 12)	7	(27, 10)
8	(25, 12)			8	(52, 9)

), where the storage and retrieval tasks of the two cases are closely distributed and unevenly located in the warehouse. All algorithms are tested on the same computer running the same operating system in order to guarantee the fairness of the comparison for the same situations. For the task scheduling of instance 1, the dynamic convergence curves of the three warehouse models optimized by the NIGA, BSO, and the MBSO are shown in Figure 10b,d,f. And their corresponding optimal paths of the R/S machine obtained by the MBSO are shown in Figure 10a,c,e. Similarly, for scheduling of instance 2, the curves of the above warehouse optimized by these three algorithms are shown in Figure 11b,d,f, whose corresponding optimal paths of the R/S machine obtained by the MBSO are shown in Figure 11a,c,e.

Obviously, from the results presented in Figure 10 and Figure 11, it can be intuitively seen that for instance 1 and instance 2, the MBSO proposed in this paper is more accurate and faster to converge to the optimal solution than BSO and the NIGA. Moreover, from Figure 10a,c,e, it can be observed that the proportion of the tasks distributed on the left side of the shelf in the warehouse is obviously larger than that on the right side. Therefore, for a batch of task orders of instance 1, the scheduling time of the AS/RS with IO_L is less than the one with IO_R, but the DE-AS/RS scheduling in this paper takes less time than them. And the optimization efficiency is increased by 10.4% and 30.2%, respectively. On the contrary, in Figure 11a,c,e, the number of tasks on the right shelf was significantly more than that on the left shelf. For instance 2, the scheduling time of the AS/RS with IO_R is less than the one with IO_L, but the scheduling time of the DE-AS/RS is still the least among the three warehouse modes. The optimization efficiency is increased by 16.5% and 2.3%, respectively. The above results are due to the more reasonable and efficient layout structure and configuration of the DE-AS/RS. Compared with the traditional single-sided I/O station AS/RS layout, the stacker crane needs to move throughout the entire warehouse to complete all tasks in the single-sided I/O station layout, which increases the travel distance and time. Especially when tasks are concentrated on the side far away from the I/O station, the scheduling efficiency will be significantly reduced. In the DE-AS/RS layout with dual I/O stations, the movement distance of the stacker crane is significantly reduced, making the scheduling process more efficient. For tasks distributed on both sides of the warehouse, the stacker crane can choose the nearest I/O station to reduce overall running time.

The conclusions obtained from Figure 10 and Figure 11 are similar to the previous conclusions. Compared with the NIGA and BSO, the MBSO once again demonstrates the best overall performance in optimizing AS/RS scheduling problems. The outstanding performance of the MBSO is attributed to targeting space clustering based on optimal scheduling time and improved cross-mutation operations. The advantage of the DE-AS/RS lies in its dual I/O station layout, which effectively reduces the movement distance of the stacker crane.

Overall, the excellent scheduling optimization capability of the MBSO combined with the reasonable layout of the DE-AS/RS not only significantly improves the efficiency of storage and retrieval scheduling problems in the warehouse but also effectively solves the problem of uneven task distribution, enabling the entire system to demonstrate higher performance and better robustness in dealing with complex scheduling problems.

6. Conclusions

Based on the analysis of the DE-AS/RS layout structure and operation characteristics, the SC and DC operation modes suitable for the R/S machine in the DE-AS/RS are proposed. The total time required for the machine to execute the sequence of storage and retrieval tasks is used as the evaluation standard to model the stacker’s scheduling path.

According to the characteristics of the DE-AS/RS, a multi-layer encoding method is adopted. In addition, the selection of the optimal I/O stations returned by the stacker each time also included in the optimization model of the scheduling path, which makes the scheduling optimization more effective.

Addressing the shortcomings of the NIGA and the BSO algorithm, which tend to fall into local minima and exhibit low convergence efficiency when solving task scheduling in AS/RSs, we propose the MBSO. This modified algorithm incorporates a mutation operation based on a greedy strategy and an improved order crossover operation during the new individual generation process. The MBSO is designed to optimize the travel path model of the S/R machine in the DE-AS/RS, yielding the optimal sequence of retrieval/storage tasks and their corresponding I/O stations. Simulation tests compare the MBSO with the NIGA and the BSO algorithm, verifying the correctness of the established travel path model for the stacker. The results demonstrate the applicability and superiority of the MBSO in optimizing the stacker dispatch path.

This work still needs some improvement. It assumes a warehouse layout with two I/O stations and a stacker. However, in actual working conditions, more stackers and I/O stations may be needed in warehouses to improve working efficiency, and their positions may also be more flexible. In addition, future work will explore the potential of combining the MBSO with other optimization algorithms to further improve convergence speed and optimization effectiveness. This includes researching how to introduce other algorithms during algorithm execution to enhance the performance of the MBSO so as to more effectively solve scheduling problems when facing more complex and varied practical working conditions.