Bi-Objective Circular Multi-Rail-Guided Vehicle Scheduling Optimization Considering Multi-Type Entry and Delivery Tasks: A Combined Genetic Algorithm and Symmetry Algorithm

Abstract

Circular rail-guided vehicles (RGVs) are widely used in automated warehouses, and their efficiency directly determines the transportation efficiency of the entire system. The congestion frequency of RGVs greatly increases when facing dense multi-type entry and delivery tasks, affecting overall transportation efficiency. This article focuses on the RGV scheduling problem of multi-type parallel transportation tasks in a real-world automated warehouse, considering maximizing efficiency while reducing energy consumption and thus establishing the RGV scheduling optimization model. At the same time, an improved genetic algorithm (GA) based on symmetry selection function and offspring population structure symmetry is proposed to solve the above RGV scheduling problem, achieving the model solution. The case study demonstrates the superiority of the proposed method in breaking local optima and achieving bi-objective optimization in genetic algorithms.

1. Introduction

The application of automation in warehouses has enhanced the efficiency of transportation. The circular RGV system plays a crucial role in the transportation process. A circular RGV system allows multiple RGVs to operate simultaneously, transporting products among several units. However, considering that different types of entry and delivery transportation tasks intersect, improper RGV scheduling always easily leads to traffic congestion, severely impacting transportation efficiency. Additionally, frequent acceleration and deceleration of RGVs always result in significant energy consumption. To enhance efficiency and reduce energy consumption, a bi-objective optimization approach focusing concurrently on efficiency and energy consumption metrics is imperative during the scheduling of RGVs [1]. The scheduling of RGVs has garnered significant interest among scholars, with some studies intently focusing on crafting individual scenarios for entry and delivery operations, while others delved into establishing more easily solvable models and enhancing the performance of optimization algorithms.

In recent years, many scholars have conducted multidimensional research on the analysis and optimization of RGV scheduling problems. The work in [2] conducted a simulation analysis on the optimization problem of circular RGV quantity. The simulation results indicated that the picking throughput did not exhibit a linear increase with the augmentation of RGV quantity under congestion conditions. In [3,4,5], the authors formulated a dynamic multi-vehicle entry and delivery problem under a hybrid predictive adaptive control scheme and combined genetic algorithm (GA) and fuzzy clustering to solve the problem. Furthermore, it is reported in [6] that an improved GA is proposed for the linear reciprocating RGV system to improve transportation efficiency.

Many articles have conducted research on the RGV scheduling optimization problem in terms of models. The work in [7,8,9,10] provided a dynamic programming model that greatly improved the output efficiency of industrial production. Several works like [11,12,13,14,15] established a mixed integer programming model to solve the optimal scheduling scheme for the workshop. The work in [16,17] proposed a resource-integrated scheduling model for production equipment and bi-directional RGVs. The work aimed to minimize the longest completion time.

Many scholars used GAs to find the optimal solution for RGV scheduling problems. The GA was first proposed in [18] and has undergone several key improvements [19,20]. In addition, scholars have also proposed solutions for optimizing GAs. Several studies like [21,22,23,24] introduced new GAs to find a global optimal solution, avoiding falling into local minima. A self-adaptive dimensionality reduction genetic optimization algorithm (ADRGA) is proposed to address the issue of dimensionality in GA optimization [25]. In [26], researchers proposed a hybrid GA for addressing a mixed flow shop scheduling problem with real-world constraints, integrating the Nawaz–Enscore–Ham (NEH) heuristic, local search algorithms and machine assignment rules focused on minimizing total tardiness. Some enhanced hybrid GAs are developed to improve the capability of solving production scheduling problems in [27,28]. In [29], researchers proposed the chaotic particle swarm optimization algorithm into the RGV scheduling optimization model and designed a multi-step processing mechanism to improve the reliability and accuracy of calculation. Some works like [30,31,32] introduced new structures into GA to enhance the solution search for technician routing and scheduling problems, achieving superior outcomes. The authors in [33] proposed a method called random partial optimization cyclic shift crossover operator to address the issue of local optimization in traditional GAs. Researchers also used partitioning methods to partition the transmission regions of genes and calculate the shortest logistics scheduling time [34]. Additionally, the work in [35] proposed a feasible adaptive change rule for element encoding based on the decreasing element encoding GA, which changes the element number according to the value of fitness function.

Some research mainly focused on different RGV scheduling problems, including linear reciprocating RGV systems and circular RGV systems, static velocity RGVs and dynamic velocity RGVs [3,4,5], and analyzed the difficulty of solving problems and actual transportation effects [2]. Additionally, some scholars considered different constraints like response time of RGVs, priority tasks and failure rate, establishing a mixed integer programming model [11,12,13,14,15] or a dynamic programming model [7,8,9,10]. The GA has excellent performance in solving NP-hard problems and can solve complex optimization problems such as RGV scheduling. So many scholars have made various improvements to GAs, such as combining the simulated annealing algorithm (SAA) and improving population structure [30,31,32]. However, there are three main gaps in the current relevant research. There is no research on RGV scheduling problems for large-scale dense transportation scenarios with multi-type entry and delivery tasks. Additionally, energy consumption is an essential indicator for measuring operation quality in practical situations, yet it has not received attention in this field. Improved GAs have been widely applied to solve circular RGV scheduling problems. However, these improved GAs usually output local optimum in solving the scheduling optimization problem of circular RGVs.

This article addresses circular multi-RGV scheduling optimization problems considering multi-type entry and delivery tasks. This research aims to improve transportation efficiency and reduce equipment energy consumption. The main contribution of this article consists of three aspects:

• The trade-off between energy consumption and efficiency is first evaluated in a RGV scheduling problem.
• A mathematical model is proposed for a general multi-RGV scheduling optimization problem considering multi-type entry and delivery tasks.
• A combined GA and symmetry algorithm is proposed for the above scheduling model to find the optimal solution.

This study enriches the theoretical research of multi-RGV scheduling problems and widens the application scenarios of RGV scheduling theory. In addition, this study provides reference solutions for similar problems encountered in real-world warehouse logistics transportation, thereby optimizing transportation processes, improving transportation efficiency and reducing transportation costs.

2. Problem Description and Analysis

The main focus of this article is to study the scheduling system of circular RGVs in real-world factory A. The layout of warehouse lanes is illustrated in Figure 1.

2.1. Scenario Description

There are six RGVs on the circle track, each with individually controllable speeds for transportation and loading/unloading. The stacker crane passage in the three-dimensional warehouse consists of six lanes, with the entrance in Zone 1. RGVs perform delivery tasks at $O_1,O_2,O_3,O_4,O_5,O_6$ and entry tasks at $I_1,I_2,I_3,I_4,I_5,I_6$. These task locations alternate (see Figure 1), and they are spaced 3 m apart from each other. In Zone 2, two lifts $X_1$ and $X_2$ transport full pallets to the third floor for delivery tasks and return empty pallets. Each elevator cannot simultaneously receive goods from two RGVs. The task of the latter RGV must wait until the entire process of the former RGV’s task at the elevator is completed. The task flow is shown as follows:

• RGV receives pallets with goods from the task port $O_1,O_2,O_3,O_4,O_5,O_6$.
• RGV delivers them to elevator $X_1,X_2$.
• RGV receives empty pallets and transports them to the entry task port $I_1$, $I_2$, $I_3$, $I_4$, $I_5$, $I_6$ after the elevator transfer is completed.

2.2. Constraints and Assumptions

The constraints of the RGV scheduling problem in this article are as follows:

• RGVs can only travel counterclockwise along the track.
• Each RGV can only pick up or deliver goods once per cycle.
• RGVs maintain a safe distance from each other.

Additionally, to simplify the model, the following assumptions are made:

• There are stacking machine buffer zones in these aisles, where RGVs can pick up and deliver goods without waiting.
• RGVs cannot run empty during operation.
• The downtime for RGV failures, communication times between different system parts and deceleration times on curves are negligible.
• Goods in the warehouse are of the same type and are not distinguished between aisles.

2.3. Scheduling Model of a Circular RGV System

We construct a discrete nonlinear mixed integer programming model for the RGV scheduling optimization problem mentioned above. The parameters used in the model are shown in

Table 1. Explanation of parameters used in the model.

Parameter	Description
i	Index of RGVs.
$i^{\prime }$	The RGV following the i-th RGV.
j/$j^{\prime }$	The task index for the i-th/$i^{\prime }$-th RGV.
$j_{last}$/$j_{last}^{\prime }$	The previous task for the i-th/$i^{\prime }$-th RGV.
$m_{ij}^o$	Delivery location that the i-th RGV transports the j-th task, $m_{ij}^o\in \{1,2,3,\dots \dots ,n\}$, n is the number of lanes.
$m_{ij}^i$	Entry location that the i-th RGV transports the j-th task, $m_{ij}^i\in \{1,2,3,\dots \dots ,n\}$, n is the number of lanes.
$m_{ij}^d$	Hoist location that the i-th RGV transports the j-th task, $m_{ij}^d\in \{1,2,3,\dots \dots ,k\}$, k is the number of elevators.
$v_i^{run}$/$v_{i^{\prime }}^{run}$	The i-th/$i^{\prime }$-th RGV’s operating speed, $v_i^{run}\in [0,3]$, $v_{i^{\prime }}^{run}\in [0,3]$.
$v_i^{load}$/$v_{i^{\prime }}^{load}$	Loading and unloading speed of the i-th/$i^{\prime }$-th RGV, $v_i^{load}\in [0,3]$, $v_{i^{\prime }}^{load}\in [0,3]$.
$\mathbb{M}$	The set of waypoints for the j-th task, $\mathbb{M}=\{m_{ij}^o,m_{ij}^i,m_{ij}^d\}$.
$t_{ij}^{os}$/$t_{ij}^{oe}$	The starting/ending time of j-th task for the i-th RGV at the delivery point.
$t_{ij}^{is}$/$t_{ij}^{ie}$	The starting/ending time of j-th task for the i-th RGV at the entry point.
$t_{ij}^{ds}$/$t_{ij}^{de}$	The starting/ending time of j-th task or the i-th RGV at the hoist point.
$t_{i{j}_{last}}^{end}$/$t_{i{j}_{last}^{\prime }}^{end}$	The completion time of previous task step for the i-th/$i^{\prime }$-th vehicle.
$S_{ij}$	The distance traveled by the i-th RGV to reach the j-th task point.
$S_{i^{\prime }{j}^{\prime }}$	The distance traveled by the $i^{\prime }$-th RGV to reach the $j^{\prime }$ task point.
$t_{up}$	Elevator operation time.
$t_i^{run}$	The running time for the i-th vehicle without encountering congestion.
$t_i^{jam}$	The time wasted due to congestion for the i-th vehicle.
$t_i^{go}$	The total running time of the i-th RGV.
$t_i^{load}$/$t_{i^{\prime }}^{load}$	The loading and unloading time of the i-th/$i^{\prime }$-th vehicle.
$t_i^j$	Time that the i-th RGV performs a task at task point j.
$t_{i^{\prime }}^{j^{\prime }}$	Time that the $i^{\prime }$-th RGV performs a task at task point $j^{\prime }$.
$E_i$	The additional kinetic energy consumption during the acceleration and deceleration of the i-th RGV.
$\theta$	Safe distance.
$S_1$	The distance from the delivery operation point to the processing operation point.
$S_2$	The distance from processing task point to entry task point.
$S_3$	The distance from entry operation point to delivery operation point.

. Among these parameters, $i,i^{\prime },j,j^{\prime },j_{last},j_{last}^{\prime }$ are indexes, and $m_{ij}^o$, $m_{ij}^i$, $m_{ij}^d$, $v_i^{run}$, $v_{i^{\prime }}^{run}$, $v_i^{load}$, $v_{i^{\prime }}^{load}$ are decision variables of the model.

This objective is to maximize efficiency and minimize energy consumption. Given the RGV system above, the objective function is described as

$$f_1=min\sum _{i=1}^6(\mathit{num\_jam}·{\left(v_i^{run}\right)}^2+4·\mathit{num\_mission}·{\left(v_i^{load}\right)}^2),$$

(1)

$$f_2=min\left\{max\left\{t_i^{jam}+t_i^{run}+4·\mathit{num\_mission}·t_i^{load}+t_{up}\right\}\right\},(i=1,2,3,4,5,6).$$

(2)

In Equation (1), $\mathit{num\_jam}$ represents the number of traffic jams for the i-th RGV, and $\mathit{num\_mission}$ represents the amount of transportation tasks for the i-th RGV. In Equation (2), the function first finds the maximum transportation time for RGVs and then determines the shortest time among all options.

The time constraints for the transportation task of the i-th RGV are as follows:

$$t_{ij}^{oe}\le t_{ij}^{ds},t_{ij}^{de}\le t_{ij}^{is},$$

(3)

$$t_{ij}^{os}\le t_{i^{\prime }j}^{os},t_{ij}^{ds}\le t_{i^{\prime }j}^{ds},t_{ij}^{is}\le t_{i^{\prime }j}^{is},$$

(4)

$$t_{ij}^{oe}={\displaystyle \frac{S_1}{v_i^{run}}}+t_{ij}^{os}+t_i^{load},$$

(5)

$$t_{ij}^{ie}={\displaystyle \frac{S_2}{v_i^{run}}}+t_{ij}^{is}+t_i^{load},$$

(6)

$$t_{ij}^{de}={\displaystyle \frac{S_3}{v_i^{run}}}+t_{ij}^{ds}+2t_i^{load}+t_{up},$$

(7)

$$t_i^{run}+{\displaystyle \frac{\theta }{v_{i^{\prime }}^{run}}}=t_{i^{\prime }}^{run}.$$

(8)

Equations (3) and (4) constrain the time sequence of three steps in an entry/delivery task; Equations (5)–(7) define the process of three steps separately; Equation (8) represents the impact of safety distance using transportation time between different RGVs.

Traffic congestion only requires consideration of two consecutive RGVs due to the inability of RGVs to overtake on the track. Generally, the judgment of RGV traffic congestion has two scenarios as follows:

Scenario 1: Rear vehicle task index is ahead of the front vehicle task index, i.e.,

$${\displaystyle \frac{S_{ij}}{v_i^{run}}}+t_i^j+t_{i{j}_{last}}^{end}\le {\displaystyle \frac{S_{i^{\prime }{j}^{\prime }}}{v_{i^{\prime }}^{run}}}+t_{i{j}_{last}^{\prime }}^{end},(j\le j^{\prime }).$$

(9)

Scenario 2: Rear vehicle task index is behind the front vehicle task index, i.e.,

$${\displaystyle \frac{S_{ij}}{v_i^{run}}}+t_i^j+t_{i{j}_{last}}^{end}\ge {\displaystyle \frac{S_{i^{\prime }{j}^{\prime }}}{v_{i^{\prime }}^{run}}}+t_{i{j}_{last}^{\prime }}^{end}+t_{i^{\prime }}^{j^{\prime }},(j\ge j^{\prime }).$$

(10)

When Equations (9) or (10) are satisfied, it can be confirmed that a traffic jam has occurred. Once a traffic jam has occurred, increment the congestion count by one and calculate the congestion time.

3. Algorithm Design

This article proposes improvements to the genetic algorithm for specifically solving the problem described in the chapter above. The improvements encompass the encoding scheme, selection method, mutation mode and population updating approach.

3.1. Gene Encoding Method

The chromosome encoding for RGV task points adopts integer encoding, with sequential numerical labels assigned to the task lanes and elevators. Lanes are labeled 0–5, while elevators are labeled 6 and 7. Binary encoding is utilized for RGV speed. The maximum travel speed and load/unload speed of RGV are 3 m/s. It is represented by five binary bits. The decoding of speed can achieve a precision of 0.1 m/s. The encoding process is illustrated in Figure 2. The first digit of the task encoding part in the figure represents the number of the RGV, and subsequent encodings indicate the order in which the RGV receives the task encoding. The number of tasks and the number of RGVs can be adjusted arbitrarily.

3.2. Fitness Function Design

The fitness function is designed based on the two objective functions mentioned in Equations (1) and (2).

According to duality, $f_2=max\left\{t_i^{go}\right\},(i=1,2,3,4,5,6)$ is equivalent to $min\left\{{\displaystyle \frac{1}{t_i^{go}}}\right\},$ $(i=1,2,3,4,5,6)$. It is feasible to use the ratio of two objective functions to represent the overall fitness comprehensively. For the dual objectives of time and energy consumption in this article, we need to maximize efficiency and minimize energy consumption. Efficiency can be represented by $min\left\{{\displaystyle \frac{1}{t_i^{go}}}\right\}, (i=1,2,3,4,5,6)$. When the fitness function adopts the ratio of efficiency to energy consumption, both increasing efficiency and decreasing energy consumption can improve fitness. However, since the efficiency objective is typically more emphasized in general scenarios, it is necessary to increase the influence of efficiency in the fitness calculation. By treating $f_2$ as a constant and taking the derivative of $f_1$, it can be observed that we have reduced the rate at which fitness decreases with increasing energy consumption. This function is expressed as

$$Fitness={\displaystyle \frac{f_2}{\sqrt{f_1}}},$$

(11)

$${\displaystyle \frac{f_2}{\sqrt{f_1}}}={\displaystyle \frac{min\left\{\frac{1}{t_i^{jam}+t_i^{run}+4·\mathit{num\_mission}·t_i^{load}+t_{up}}\right\}}{\sqrt{{\sum }_{i=1}^6[\mathit{num\_jam}·{\left(v_i^{run}\right)}^2+4·\mathit{num\_mission}·{\left(v_i^{load}\right)}^2}]}}.$$

(12)

3.3. Individual Selection Mode

This article proposes a new selection method that involves two stages of selection. The first selection adopts the roulette wheel selection, where selection probabilities are set based on the proportion of individual fitness in total fitness [19]. The second combines the idea of elite retention, screening again on the basis of one selection. The number of individuals selected in the two-stage process is set based on empirical methods. The second selection probability adopts a new selection function as

$$P={\displaystyle \frac{1}{1+e^{-arctanh\left(x\right)}}}.$$

(13)

The function introduces the sigmoid function from deep learning. In order to extend the function domain into $(-\infty ,+\infty )$, the hyperbolic tangent function is used to ensure that the probability distribution of selection covers the entire curve. The function graph is shown in Figure 3. For those individuals possessing low fitness levels, their selection probability weight is decreased, making it more challenging for them to remain in the selection pool; whereas, for individuals exhibiting high fitness levels, their probability weight undergoes an increase.

The second selection is proposed to filter the selected individuals based on the first selection. Firstly, calculate the average fitness of the population selected in the first round. Then, compute the difference between average fitness and each individual to obtain the deviation from the mean. The minimum and maximum deviations are used as the endpoints of domain interval, which is then mapped to (−1, 1) according to $y=-arctanh\left(x\right)$. After this mapping, the sigmoid function can be applied to calculate the probability of each individual being selected.

The implementation of the program is shown in Algorithm 1. The input to the algorithm includes population, number of individuals to select and fitness of each individual. When the number of individuals selected in both the first and second rounds of selection reaches respective preset numbers, the algorithm terminates and outputs the selected individuals from both rounds.

3.4. Gene Crossover Mode

Due to the length of individual genes, single-point crossover has minimal impact on fitness, resulting in slower convergence speeds. However, excessive gene crossover can reduce the convergence quality of the GA. A two-segment gene crossover method is adopted to avoid local optimal solutions and increase the population’s genetic diversity for better convergence performance. This method randomly selects four nodes on the same location of both parents’ genes and performs crossovers. The process is illustrated in Figure 4. The two colors here represent the two chromosomes before crossing.

Algorithm 1 Selection algorithm

Input: population, num_select, scores

Output: selected, save_ind

1: initialize cumulative_probs, selected, selected_indices, pro, save_ind, selected_scores  2: total_fitness = sum(scores)  3: for each fitness ∈ scores do  4:    proportion = fitness / total_fitness  5:    cumulative_probs append proportion  6: end for  7: for each i ∈ num_select do  8:    rand_num = random ∈ (0,1)  9:    for each i ∈ cumulative_probs do 10:      if rand_num < prob then 11:         selected append individual 12:      end if 13:      break 14:    end for 15: end for 16: ave_fitness = sum(selected_scores)/len(selected_scores) 17: for each selectscores ∈ selected_scores do 18:    diff_result = selectscores - ave_fitness 19: end for 20: diffmax = 1/max(diff_result) 21: diffmin = 1/abs(min(diff_result)) 22: for each diffitness ∈ diff_result do 23:    if diffitness ≤ 0 then 24:      pro append $1/(1+exp(-np.arctanh$(diffitness·diffmin))) 25:    else 26:      pro append $1/(1+exp(-np.arctanh$(diffitness·diffmax))) 27:    end if 28: end for 29: select elite as select_ind according to pro 30: return selected, save_ind

3.5. Mutation Mode

A semi-random mutation approach is adopted for a regionally specified mutation to reduce the probability of infeasible solutions during mutation. Similar to the crossover, single-point mutation has a minimal impact on fitness. Randomly generating multiple mutation positions on the chromosome can accelerate the convergence speed of GAs. The operation is illustrated in Figure 5, and this mutation is primarily targeted at the chromosomes with mixed encoding. Orange represents the mutated gene.

The implementation of the program is shown in Algorithm 2. The inputs of the algorithm are mutation probability, individuals to be mutated and the number of segments for the concatenation process during encoding. For each segment of an individual, the algorithm chooses a location randomly and generates a random number between 0 and 1. If this number is less than the mutation probability, mutation is executed on that location, and then the process proceeds to the next segment of the individual in the same manner. The algorithm stops when it has traversed all segments of each individual to be mutated.

Algorithm 2 Mutation algorithm

Input: individual, mutation_rate, num_RGV

Output: individual_mutation

1: n = num_RGV  2: reshape individual into n lines  3: gene = individual  4: for each a ∈ lines do  5:    for each b ∈ num_mutation do  6:      j = $random.randint$(1, $len$(each_lines))  7:      i = $random\in$ (0,1)  8:      if a < mutation_rate then  9:         if i % 2 = 1 and i <= 2·num_mission_av then 10:           $gene\left[i\right]\left[j\right]=random.randint(0,6)$ 11:         end if 12:         if i!=0 and i%2 = 0 and i <= 2·num_mission_av then 13:           $gene\left[i\right]\left[j\right]=random.randint(6,8)$ 14:         end if 15:         if i>2·num_mission_av then 16:           $gene\left[i\right]\left[j\right]=random.randint(0,2)$ 17:         end if 18:      end if 19:    end for 20: end for 21: return ndividual_mutation

3.6. Population Update Mode

The offspring population is divided into elite individuals, extensively crossed individuals and new-generation, symmetrizing the population update process. Elite individuals refer to the individuals retained through secondary selection; extensively crossed individuals refer to individuals obtained after crossover and mutation among first-selection individuals; new-generation refers to the individuals obtained through crossover and mutation among elite individuals.

3.7. Execution Steps and Flowchart

The steps of the procedure are as follows:

• Initialize the population and simulation parameters: population size (popsize), crossover probability (pc) and mutation probability (pm).
• Calculate each individual’s fitness in the population.
• Judge if the iteration count has been reached. If so, output the global optimal solution. If not, proceed to the following steps.
• Screen out the excellent individuals and elite individuals from the population through two rounds of selection. Individuals with poor fitness are screened out according to a specific ratio.
• Perform multi-point crossover on the population of excellent individuals to obtain extensively crossed individuals.
• Execute mutation operations on the population obtained in Step 5.
• Perform crossover operations on the elite individuals to obtain new-generation.
• Combine elite individuals, extensively crossed individuals and new-generation, and generate additional individuals to join the population, ensuring population size remains unchanged.

Figure 6 presents the process in the form of a flowchart.

4. Case Study

This article use a real-world automated circular RGV system in factory A’s warehouse to validate the feasibility and effectiveness of the proposed solution. The basic parameters of the RGV system are presented in

Table 2. Parameters of the circular RGV system.

Length of Track 1	Length of Track 2	Length of Track 3	Safe Distance	Number of RGVs	Number of Lane Entrances	Elevation Time	Max Speed
89.6 m	8 m	82 m	3 m	6	6	10 s	3 m/s

. The program is written in Python 3.10 (Python Software Foundation, Wilmington, DE, USA), and the simulation environment is set up on a Windows 10 system with a Gen Intel(R) Core(TM) i5-13400 (Intel Corporation, Santa Clara, USA) processor running at 2.5 GHz and 32 GB of RAM. The initial population size is set to 100, with a first selection probability of 0.7, a second selection probability of 0.25, a mutation probability of 0.01 and an iteration count of 1000. After the population update, the new individuals are regenerated to maintain a population size of 100. The selection parameters represent the proportion of each selection stage during the population update. The initial value of the selection probability is determined by the fitness of the randomly generated initial population. Parameters may not necessarily be optimal, and empirical methods are generally used to set them in genetic algorithms.

4.1. Results of the Proposed Framework

Table 3. Comparison of algorithm iteration parameters.

			Iterations
	0	200	400	00	800	1000
Fitness/$\times {10}^{-4}$	0.10	2.00	2.12	2.10	2.30	2.33
Congestion/n	54.20	7.31	5.04	4.30	4.05	3.54
Finish time/s	38,661	641	765	634	652	623
Energy/${(\mathrm{m}/\mathrm{s})}^2$	491	144	100	97	111	101

shows the result of the proposed framework. During the RGV transportation, neglecting the loss of converting electrical energy into kinetic energy, the energy consumption of the entire system mainly comes from the kinetic energy consumption during transportation. Therefore, since the weight of RGV is unknown, assuming it is 2, according to equation $E=0.5$ ${\mathrm{mv}}^2$, the energy parameter can be represented by the square of velocity when calculating energy consumption. The initial fitness is $0.101159\times {10}^{-4}$; the initial average frequency of congestion is 54.2; the initial maximum completion time is 38,661 s; and the initial energy consumption is 491 ${(\mathrm{m}/\mathrm{s})}^2$. After iteration, fitness reaches $2.32764\times {10}^{-4}$, the average congestion frequency finally converges to 4, the maximum completion time reduces to 630 s, and the energy consumption decreases to 100 ${(\mathrm{m}/\mathrm{s})}^2$.

4.2. Comparison of Algorithm Results

Table 4. Comparison of algorithm iteration parameters.

				Iterations
		0	200	400	600	800	1000
Improved	Fitness/$\times {10}^{-4}$	0.10	2.00	2.12	2.10	2.30	2.33
	Congestion/n	54.2	7.31	5.04	4.3	4.05	3.54
	Finish time/s	38,661	641	765	634	652	623
	Energy/${(\mathrm{m}/\mathrm{s})}^2$	491	144	100	97	111	101
Unimproved	Fitness/$\times {10}^{-4}$	0.10	1.66	1.70	1.68	1.71	1.69
	Congestion/n	54.20	11.10	11.64	12.15	11.00	11.40
	Finish time/s	38,661	727	702	700	726	716
	Energy/${(\mathrm{m}/\mathrm{s})}^2$	491	238	222	220	232	222

shows the comparative data between the improved and original GA for every 200 iterations in the graph above.

As shown in Figure 7, the red line represents the variation tendency of the improved genetic algorithm, while the black line represents the unimproved algorithm.

The unimproved algorithm converges at 230 iterations with a final fitness value of 1.7. The improved algorithm converges at 250 iterations, initially to a fitness value of 2. However, at 750 iterations, it escapes from a local optimum and converges again to a higher fitness value of 2.3.

As indicated by the figure, although the convergence speed of the improved GA is slightly reduced (by about 10%) compared to the original version, the final fitness convergence value has increased by 30%. Additionally, the improved algorithm demonstrates an enhanced ability to break out of local optima. Therefore, the improved algorithm exhibits superior performance compared to the traditional genetic algorithm.

To further illustrate the algorithms’ performance, this article also compares the number of traffic jams, maximum completion time and energy consumption between the improved and unimproved algorithms. As shown in Figure 8, the unimproved algorithm also converges at about 230 iterations, ultimately converging to 11. The improved algorithm starts converging at 250 iterations and ultimately converges to 3.5. The trend of congestion graph is opposite to fitness, but they exhibit consistent characteristics. Overall, in terms of fitness convergence value, efficiency and energy consumption convergence value and changes in traffic congestion frequency, the improved genetic algorithm outperforms the original algorithm.

4.3. Optimization under Different RGV Speeds

When the RGV number is 6, the variation in fitness and traffic jams over iterations in speed optimization is shown in Figure 9 and Figure 10. The red line represents the iterative process with dynamic speed, while the black line represents fixed speed.

Figure 9 depicts the variation in the population’s average fitness over iterations for both dynamic speed and fixed speed. The dynamic speed scheduling begins to converge at around 100 iterations and ultimately converges to a value of 2.3. After 200 iterations, the fitness value remains essentially unchanged, indicating good convergence properties and a relatively fast convergence speed. In bi-objective optimization, the fitness improvement for fixed-speed calculations is significantly smaller than that for dynamic speed calculations.

Figure 10 illustrates the variation of an average number of traffic jams in the population over iterations for dynamic speed and fixed speed scheduling. The trend of traffic jams is opposite to the trend of fitness over iterations.

Comparing Figure 9 and Figure 10, the scheduling approach with dynamic speed shows better performance.

Table 5. Comparison of algorithm iteration parameters.

				Iterations
		200	400	600	800	1000
Improved	Fitness/$\times {10}^{-4}$	2.03	2.21	2.20	2.21	2.29
	Congestion/n	7.93	5.67	6.04	5.01	4.43
Unimproved	Fitness/$\times {10}^{-4}$	0.82	0.78	0.78	0.80	0.80
	Congestion/n	19.70	22.01	21.45	18.02	19.57

shows the data for every 200 iterations in the graph above.

4.4. Optimization of RGV Numbers

To investigate the impact of RGV numbers on total completion time and transportation energy consumption, this section designed 13 test cases with different RGV scales (N = 2, 4, …, 14) as the study subjects. Studying one RGV is meaningless because it is impossible to perform scheduling optimization on a single RGV. The improved GA is applied to solve these cases, and each case’s total completion time and congestion count are calculated. The results are presented in Figure 11, Figure 12, Figure 13 and Figure 14.

Figure 11 shows that when the number of RGVs is less than 7, the fitness is significantly affected by RGV number. However, when the number of RGVs exceeds 7, increasing RGVs has a relatively small impact on fitness. Overall, when the RGV number is greater than 4, the fitness of improved algorithm is higher than that of original algorithm.

In Figure 12, the number of traffic jams shows an upward trend as the RGV number increases, but the increase in the improved algorithm is relatively slower. The curve indicates that when the number of RGVs is small, there is no significant difference in traffic jams between the two algorithms. However, the improved algorithm demonstrates an advantage with increased RGV numbers. Furthermore, the figure also reveals that an RGV count of 7 serves as a demarcation point for this curve: when RGV is less than 7, the change in the number of traffic jams is relatively tiny, but when the number exceeds 7, the number of traffic jams rapidly increases with the increase in RGV number.

Figure 13 shows the trend in maximum completion time. The maximum completion time of the improved algorithm for different numbers of RGVs is generally lower than the unimproved algorithm. This suggests that the improved algorithm is more efficient in minimizing the maximum completion time. After the number of RGVs reaches 7, the maximum completion time shows a relatively small variation. The stabilization of maximum completion time after 7 RGVs indicates that adding more RGVs beyond this point may not significantly reduce the overall completion time, highlighting the efficiency of the improved algorithm in balancing workload and resource allocation.

Based on the information in Figure 14, when the number of RGVs is less than 8, both the improved and unimproved algorithms’ energy consumption changes similarly. However, when the RGV number exceeds 8, the unimproved algorithm exhibits a significant upward trend, while the improved algorithm’s upward trend is relatively more gradual.

In summary, the improved algorithm demonstrates superior overall performance and adaptability to large-scale, bi-objective scheduling optimization problems. Furthermore, evaluating Figure 11, Figure 12, Figure 13 and Figure 14 comprehensively, optimal solutions are achieved when RGV numbers are 6, 7 and 8.

5. Conclusions and Future Research

This article has established a mathematical optimization model for the circular RGV system scheduling of multi-type entry and delivery tasks. Considering multiple influencing factors in a real-world factory, a bi-objective improved genetic algorithm has been proposed based on symmetry to solve the scheduling problem. According to the simulation of practical cases, the following conclusions are drawn:

• The optimization model for a circular RGV system that considers energy consumption and speed of RGVs can more realistically address the RGV scheduling optimization issues in the real-world warehouse environment.
• Drawing on the symmetry experience, the selection process has been improved by adopting a secondary selection mechanism to screen and retain elites. A new probability selection function has been introduced to improve the convergence speed of GAs. Targeted chromosome crossover and mutation methods have been used to avoid becoming stuck in local optima.
• An analysis of the RGV numbers’ impact on total completion time and energy consumption has been conducted, providing a reference for the parameter design of circular RGV systems.

Using the improved GA, we are able to obtain more accurate convergence results compared to the original algorithm. In addition, taking into account energy consumption, we can demonstrate that dynamic speed transportation has better optimizability compared to fixed speed transportation. Additionally, for a three-dimensional warehouse with a determined number of lanes, the optimal number of RGVs considering both energy consumption and efficiency is similar to the number of lanes.

However, RGV systems are quite complex and require the consideration of factors such as system response time, failure rates and anti-interference capability. The application of the method is limited by these factors. Therefore, the next step is to investigate the robustness and resilience of circular RGV systems under scenarios involving RGV failures and design a solution for equipment malfunction. These studies will effectively improve the anti-interference ability of automated warehouse transportation systems and effectively dispose various emergencies.