Multi-objective AGV scheduling in an automatic sorting system of an unmanned (intelligent) warehouse by using two adaptive genetic algorithms and a multi-adaptive genetic algorithm.

Automated guided vehicle (AGV) is a logistics transport vehicle with high safety performance and excellent availability, which can genuinely achieve unmanned operation. The use of AGV in intelligent warehouses or unmanned warehouses for sorting can improve the efficiency of warehouses and enhance the competitiveness of enterprises. In this paper, a multi-objective mathematical model was developed and integrated with two adaptive genetic algorithms (AGA) and a multi-adaptive genetic algorithm (MAGA) to optimize the task scheduling of AGVs by taking the charging task and the changeable speed of the AGV into consideration to minimize makespan, the number of AGVs used, and the amount of electricity consumption. The numerical experiments showed that MAGA is the best of the three algorithms. The value of objectives before and after optimization changed by about 30%, which proved the rationality and validity of the model and MAGA.

Introduction

The competition in the logistics industry has become increasingly fierce with the development of global e-commerce. As those who can provide logistics services more quickly will seize more market share, the emergence of intelligent warehouses and unmanned warehouses has been of great help. In these warehouses, some links, even the whole process, do not require manual participation. Therefore, logistics companies can achieve higher efficiency and solve the problem of recruitment to a certain extent [1]. As a result, the emergence of these warehouses is regarded as the gospel of logistics companies. Automated guided vehicle (AGV) is a logistics transport vehicle with high safety performance and powerful functions, which can genuinely achieve unmanned operation. The use of AGVs in the warehouse will make the whole logistics process optimized, bringing a significant change to the logistics industry. Using AGVs that can work 24 hours a day in an automatic sorting system—the most time-consuming process in warehousing operations—will not only shortens the overall time used but also improves work efficiency. Therefore, a good AGV scheduling scheme will enable consumers' orders to be sorted and delivered to customers earlier, which will significantly enhance consumer experience and improve consumer satisfaction. Although there have been many studies on AGV scheduling, they are mostly concentrated in the manufacturing field, especially in the FMS [2-15]. In the logistics industry, the application of AGVs is still a new trend. Besides, the AGV scheduling remains an open research area due to the different facts to be considered in the scheduling process, for example, objectives, limitations, and considerations.

In the earlier works, the scheduling objective was mostly based on minimizing the makespan to ensure the operating efficiency of the multi-AGV system [2-5,10,11,15-17]. However, these studies ignored the impact of the number of AGVs used in the AGV system, for AGV is a relatively expensive device. If the number of AGVs used can be reduced, many costs will be saved [18-20]. Therefore, subsequent studies began to consider the number of AGVs to guide the actual scheduling while minimizing the makespan [21-23]. Moreover, another practical problem of AGV scheduling has been often neglected, namely the charging process of AGV. Most studies did not consider the battery charge of AGV, which was equivalent to assuming that AGV will not stop working even if its electric power is insufficient. Only in recent years, some studies have begun to add the quantity of electricity of AGV as a constraint to the scheduling considerations [14,24]. However, this is still not considered comprehensively, for the restriction of electricity is only to make AGVs in the system unable to complete tasks or sort orders when they are power off. In fact, they can be charged first and then go back to work. Consequently, a sound AGV scheduling system not only needs to have a job assignment but also should assign a charging task when the power of AGV is insufficient to meet the power demand of its subsequent work.

What is more important is that there is a common assumption in almost AGV scheduling studies that the AGV runs at a constant speed. However, the speed of the AGV is adjustable and controllable, so it is more practical to take the speed of the AGV into the consideration of the scheduling system. This consideration can give birth to a third objective—minimizing the amount of electricity consumed by all AGVs. The less power the system consumes, the fewer carbon emissions, which not only affects the cost of the operation but also takes responsibility for environmental protection.

The final decision is a trade-off among these three objectives (minimization of the makespan, the number of AGVs used, and the amount of electricity consumed). Minimizing the makespan means that more AGVs need to be arranged and those AGVs must be fast. The higher the AGVs’ speeds are, the more electricity they consume per unit time. If the number of AGVs is minimized, the makespan will increase. The speed of the AGV will have to be as high as possible to complete all tasks, resulting in more electricity consumption. Minimizing the amount of electricity consumed by all AGVs requires that the AGV should not be too fast, which means that more AGVs will be needed, and the makespan will not be short. It is a practical and attractive study to find a balance between these three objectives.

To address these issues, considering the charge of the battery in AGV, the speed of the AGV, job assignment and charging task, a multi-objective mathematical model was developed to minimize the makespan, number of AGVs, and amount of electricity consumed by all AGVs to schedule AGVs in logistics sorting operations in this study.

Heuristic algorithm is the most commonly used method in AGV scheduling research, among which genetic algorithm (GA) is the most common [12, 25–33], and adaptive genetic algorithm (AGA) is an improvement of traditional genetic algorithm. The main idea of it is to adjust adaptively the genetic parameters, which greatly improves the convergence accuracy of the genetic algorithm and increases the convergence speed. There are two kinds of adaptive genetic algorithms that are used frequently and perform quite well. The first is an adaptive genetic algorithm that generates adaptive crossover and mutation rate based on fitness values [34]; the second is based on the information entropy, especially the population entropy which is used to reflect population diversity [35]. We have noticed that some scholars have used the multi-phase hybrid approach to solve the complex model, which makes us deeply inspired [36]. So in this study, a multi-adaptive genetic algorithm (MAGA) that combines the characteristics of the two adaptive genetic algorithms was proposed and was compared with these two adaptive genetic algorithms in optimization.

The rest of the research is organized as follows. Section 2 shows the application scenario and the mathematical model. Section 3 is the design of the multi-adaptive genetic algorithm (MAGA). Section 4 demonstrates the experimentation and analysis. Moreover, a conclusion is drawn in section 5.

Problem descriptions and assumptions

Facility layout

In the intelligent warehouse of logistics or e-commerce company, taking the unmanned automatic warehouse operation process of Jingdong and Cainiao as the example, after the consumer places an online order, the Automated Storage and Retrieval System (AS/RS) in the warehouse will pick out the goods that the consumer needs according to the order, and transport them to the packaging area by the conveyor belt. Then after the packaging, the package will be transported to the sorting area by the conveyor belt, and then the AGV will perform the sorting operation. The layout of the work scene is shown in Fig 1. The responsibility of the AGV is to transport the parcels transported by different conveyors (shipment ports) to the corresponding delivery point according to the information of the express waybill in the sorting area. Below the delivery ports are funnels and conveyors that collect the packages and transport them to the distribution area.

Fig 1

The layout of the AGV sorting area.

Model derivation

This section develops a mathematical model of AGV scheduling based on three objectives. These three objectives are: (1) minimizing the makespan, (2) minimizing the number of AGVs used, and (3) minimizing the amount of electricity consumed by all AGVs.

The assumptions and limitations for the model development are as follows:

The AGVs are parked in the home until scheduling commands are assigned.

The travel speed of the AGV can be scheduled, and each order corresponds to an independent AGV speed.

The research environment is a free-form grid-like road network, and the road width is sufficient, so there are no traffic problems, collision, deadlock, or conflict.

When the AGV sorts the order, it takes the shortest path. The shortest path distance is uniquely determined by the beginning and ending points of the AGV.

The loading and unloading time of the AGV is short and negligible.

The AGV can stay in the loading/unloading position (shipment port/delivery port).

Each AGV only loads one package at a time, that is, each AGV can only perform the sorting work of one order at the same time.

Notations

MS—Makespan

NA—The number of AGVs assigned to the order sorting

E—The amount of electricity consumed by all AGVs

m—The total number of orders

O—Order number i

i,j—Index of orders, i, j = 1, 2, 3, …, m

I—The set of all orders

n—The total number of AGVs in the warehouse

A—AGV number a

a—Index of AGVs, a = 1, 2, 3, …, n

A—The set of all AGVs

S—The shipment port of O

D—The delivery port of O

H—AGV charging area

T—A assigned to the order sorting

assigned to complete the sorting of O

—The speed of A when it is assigned to complete the sorting of O,

assigned to complete the sorting of O after completing the sorting of O

tO—The time when the package in O arrive at S (i.e., the earliest time when O can be sorted)

tST—The time when the order sorting system starts to perform its tasks (the beginning of the whole system)

tST—The time when O gets started to be sorted

tCT—The time required to complete the sorting of O (i.e., the time required for the AGV to take the package from S and transport it to D)

dCT—The distance that the AGV needs to complete the sorting of O (i.e., the distance from S to D)

CA—The current position of A

tCA—The time when A is in the current position

—The time required for A to arrive at the shipment port of O from the current position (i.e., the time required for A to arrive at S from CA)

—The distance from the current position of A to the shipment port of O (i.e., the distance from CA to S)

—The time required for A to arrive at the shipment port of O from the delivery port of O (i.e., the time required for A to arrive at S from D)

—The distance from the shipment port of O to the delivery port of O (i.e., the distance from S to D)

—The initial state of charge (SOC) of A’s battery

EA—The current state of charge (SOC) of A’s battery

ECAa—The charging capacity of A

—The final state of charge (SOC) of A’s battery

—To complete the sorting of O, it is necessary to perform a charging task first when EA is insufficient

—The time required for A to perform the charging task ()

—The charging time for A in the charging task ()

—The round-trip time for A to travel between the work area (CA) and the charging area (H)

—The distance that A travels between the work area (CA) and the charging area (H)

—The amount of electricity required for A to complete the sorting of O from the current position and return to the charging area (H)

μ—The ratio of power consumption to time when AGV’s speed is (i.e., μ is the amount of power consumed per unit time at speed ), since , μ

—The minimum power consumption required for A to move from the delivery port of O (D) to the charging area (H)

Variables—Sorting assignment

Use the 0–1 variable as the index for order assignment:

Use the 0–1 variable T to help calculate the number of AGVs used in sorting systems. As long as AGV-Aa participates in order sorting, T equals 1:

Use the 0–1 variable as the index for order sequencing:

Variables—Speed

The time AGV-Aa needed to pick up packages, the time AGV-Aa needed to to sort orders and its charging time are determined by its adjustable speed:

Objective function

Makespan is the time when the last order is sorted minus the start time of the system:

Eq 9 is used to calculate the number of AGVs involved in sorting:

The power consumption is calculated by the initial values, charge values and residual value of AGVs’ batteries. It can also be calculated by the sorting order time and the power consumption coefficient:

Subject to:

Where Eqs 11, 12 and 13 are time constraints, respectively ensuring that the start time of the order-sorting is later than the arrival time of the order, the arrival time of the AGV, and the completion time of the previous order. Eq 14 is the time of the charging task, which is the power constraint of the system. If the power of the AGV is insufficient to complete the order and return to the charging area, the AGV will need to perform the charging task. Eq 15 calculates the amount of electricity required for the AGV to complete the sorting of O and return to the charging area. Eq 16 determines that each order can and can only be executed once by one AGV. Eq 17 makes sure that the responsible order sequence is deterministic for each AGV.

Multi-objective evaluation

Pareto efficiency implies that resources are allocated in the most efficient manner [37]. The overall fitness function is expressed as follows

Where ω is the weight of the α objective function (Σω = 1), and θ is the coefficient of the α objective function for obtaining a range of approximate values among the objectives.

In this study, the weights of the three targets were 0.6, 0.24, and 0.16, respectively. To have similar ranges of variation for the three objectives, the adjustment coefficients were 1, 30, and 20, respectively. The following formula calculates the overall fitness function.

Algorithm design

In recent years, there have been many improvements in adaptive genetic algorithms. The two most commonly used ones are:

Adaptive adjustment of crossover and mutation rate based on individual fitness values to reduce damage to good genes (AGA1).

Adding population entropy to the algorithm, adaptively adjusting the crossover and mutation rate based on the population entropy to maintain the diversity of the population and improve the global search ability of the algorithm (AGA2).

It is not enough to consider the diversity of the population. The variety of individual genes also has a substantial impact. Therefore, it is necessary to add the operation of improving the genetic diversity to the algorithm. Besides, the crossover and mutation rates can be adaptively adjusted in combination with the population entropy and the individual fitness value to ensure that the population is diverse and those competent individuals have a higher probability of entering the next generation.

In this research, we combined the characteristics of these two adaptive genetic algorithms and improved them. A multi-adaptive genetic algorithm (MAGA) is proposed as follows.

Chromosome representation and encoding

The number of genes in the chromosome is twice the number of orders. The first half of the chromosome indicates the assignment of the order sorting, and the second half indicates the speed at which the AGV sorts the order. For example, if the order number is 30, the number of genes on the chromosome will be 60. The chromosome is shown in Fig 2, which indicates that order No. 1 is sorted by AGV No. 9 at a speed of 1.0m/s, order No. 2 is sorted by AGV No. 8 at a speed of 1.2m/s, and order No. 30 is sorted by AGV No. 3 at a speed of 1.5 m/s.

Fig 2

A random chromosome.

Fitness function

The total fitness value is expressed by F, and it will be calculated based on Eq 20.

Initializing population

The initial population is randomly generated, but it must be ensured that the initial population has the appropriate diversity, that is the population entropy cannot be too small.

The representation and calculation of population entropy

The solution space is divided into M non-overlapping regions (Q1, Q2, …, QM). pi (i = 1, 2, …, M) is used to indicate the probability that an individual in the population belongs to Qi. The population entropy of the tth generation (S(t)) can be expressed as:

In order to estimate the population entropy, the range [(1−α)F, (1+α)F] is used instead of the solution space, where Fmin is the minimum value of the fitness value from the initial iteration to the tth generation, Fmax is the maximum one, and α (0 < α < 0.1) is an expansion coefficient. Then divide the entire interval into N (N is the number of individuals in the population) regions. Use l (i = 1, 2, …, N) to denote the number of individuals whose fitness value belongs to the ith region. The estimated value of p is calculated using

Substituting the estimated value of p () into Eq 21 yields the estimate of the population entropy ():

Selection and elitism

In this study, we use the Roulette Wheel Selection [38] to select. The best individual in each generation is transferred directly to the next generation in the elitism step.

Crossover

This study employs a multi-point crossover. Since the first half of the chromosome represents the sorting assignment, and the second half represents the speeds of AGVs, there is a correlation between the two. In order to ensure that the proper individual is not destroyed by crossover operation, after the first half of its chromosome exchanges fragments (sequence of genes) with the other chromosome, the corresponding part of the latter half has to exchange fragments. For example, if the order number is 30, the number of genes on the chromosome will be 60. When genes between the 2nd and 30th on chromosomes are crossed over, genes between the 32nd and the 60th need to be correspondingly crossed over, as shown in Fig 3.

Fig 3

An example of crossover.

The crossover rate (pc) of this study is multi-adaptively adjusted. Firstly, the basic number of crossover rate of the tth generation (pc1) is determined according to the population entropy by using

Where Smax is the maximum possible value of the population entropy, that is, Smax = lnN. pc2 and pc3 are parameters that can be adjusted. When the population diversity becomes smaller, the basic number of the crossover rate becomes larger to increase the diversity of the population.

Secondly, the crossover rate of an individual (pc) is determined according to the fitness value of the individual by Eq 26.

Where F is the larger fitness value of the two crossing-over individuals. If the individual fitness value is large, the crossover rate will be low, so that the structure of a good individual can be destroyed as little as possible. γ is an adjustment coefficient to ensure that individuals with small current fitness values are also likely to enter the next generation.

Mutation

In order to maintain the diversity of the population and individuals, the 0–1 variable (Xk) was added to help. If the individual i is different from the individual j in the kth gene, then Xk = 1 otherwise Xk = 1. The degree of diversity of the kth gene of all individuals of the tth generation population (Ykt) can be expressed as

In the mutation operation, the position of the mutation is determined according to the value of Ykt—

the smaller the genetic diversity, the greater the probability that the gene position is selected to perform the mutation operation (based on the Roulette Wheel Selection).

The mutation rate (pm) of this study is also multi-adaptively adjusted. Firstly, the basic number of the mutation rate of the tth generation (pm1) is calculated according to the population entropy by Eq 28.

Where pm2 and pm3 are parameters that can be adjusted. If the population diversity becomes small, pm1 will become large, which is conducive to the generation of new individuals and increase the diversity of the population.

Then the mutation rate of an individual (pm) is determined according to the individual fitness value using

Where F is the fitness value of the mutated individual. If the individual fitness value is getting larger, the mutation rate will be lower, so that the gene of the excellent individual can be protected.

The characteristics of the Multi-Adaptive Genetic Algorithm (MAGA) proposed in this study are summarized as follows.

The crossover rate and mutation rate are multiple adaptively adjusted.

The site of the mutation is not random but varies according to the degree of the individual genetic diversity.

The initial population is to meet certain conditions, which is conducive to finding a better solution faster.

Thus MAGA is an algorithm with an excellent ability of global optimization, and it can make good results on the problem of a time-cost optimization solution for its convergence rate.

Computational results and discussion

To validate the model, two-scale numerical experiments have been conducted. In the first experiment, there were 30 orders (O1, O2, …, O30) needed to be sorted, 10 AGVs (A1, A2, …, A10) in the warehouse, and the AGV speed varied from 0.5m/s to 1.5m/s. While in the second experiment, there were 50 orders (O1, O2, …, O50), 20 AGVs (A1, A2, …, A20) and the AGV speed adjustable range was also 0.5m/s-1.5m/s.

Taking the Home as the coordinate origin, a coordinate system is established to determine the coordinates of the shipment port and the delivery port, shown in Fig 4, and then the moving distance of AGVs (each unit on the x-axis or y-axis stands for 2 meters) is calculated. For example, the coordinates of the Home are (0, 0), the coordinates of the No. 3 shipping port are (-10, 7), the coordinates of the No. 6 shipping port are (10, 11), the coordinates of the No. 2 delivery port are (-6, 2), the coordinates of the No. 10 delivery port are (-8, 2), and the coordinates of the No. 53 delivery port are (6, 12) (Fig 3 is only a schematic diagram of half of the warehouse, so there are 12 shipment ports and 108 delivery ports). An example of the data content of the order is shown in Table 1.

Fig 4

The upper half of the warehouse coordinate system.

Table 1

Table of order information.

Order ID	Shipment port No.	Delivery port No.	Order arrival time (tOi)
1	12	29	2019-04-10-13:10:00
2	5	10	2019-04-10-13:10:00
3	3	67	2019-04-10-13:10:01

The superiority of the proposed algorithm was verified by comparing the following algorithms:

AGA1: Adaptive adjustment of crossover and mutation rates based on individual fitness values, p = pF/(γF), pm = pF/(γF).

AGA2: Adaptive adjustment of crossover and mutation rates based on the population entropy, p = p+p(1−β), p = p+p(1−β).

MAGA: The hybrid improvement algorithm proposed in Section 3 of this study, p = pF/(γF), p = p+p(1−β), pm = pF/(γF), pm1 = pm2+pm3(1−β).

In order to enhance the representativeness of the experiments, a control with sufficient AGV power and insufficient AGV power was set in both experiments. Based on the experimental approach, the best settings are as follows:

Experiment 1

AGA1: pc1 = 0.6, p = 0.04, γ = 2.

AGA2: pc1 = 0.6, pc2 = 0.3, pm1 = 0.04, pm2 = 0.06, α = 0.05.

MAGA: pc2 = 0.6, pc3 = 0.3, pm2 = 0.04, pm3 = 0.06, γ = 2, α = 0.05.

The algorithms were run 10 times, with each run of a population size of 60 in 350 iterations and their results are shown in Table 2.

Table 2

Test results of optimization algorithms for Experiment 1.

Algorithms		Experiment 1-1(fully charged)			Experiment 1-2(low battery)
		The best result	Mean result	Mean CPU time	The best result	Mean result	MeanCPU time
AGA1	f(x)	185.9	189.4	409.6 Sec	203.9	207.2	397.3 Sec
	MS	146.2	139.6		171.5	168.2
	NA	8	8.8		8	8.6
	E	12.69	13.21		13.57	13.85
AGA2	f(x)	183.1	186.6	411.8 Sec	195.9	201.9	410.6 Sec
	MS	119.5	130.6		163.2	168.2
	NA	10	9.3		8	8.5
	E	12.32	12.91		12.62	12.44
MAGA	f(x)	171.9	176.3	415.2 Sec	182.6	187.5	412.5 Sec
	MS	118.6	124.4		146.1	158.7
	NA	9	8.9		8	8.2
	E	11.22	11.73		11.67	10.39

Experiment 2

AGA1: pc1 = 0.7, p = 0.07, γ = 2.

AGA2: pc1 = 0.7, pc2 = 0.2, pm1 = 0.07, pm2 = 0.03, α = 0.08.

MAGA: pc2 = 0.7, pc3 = 0.2, pm2 = 0.07, pm3 = 0.03, γ = 2, α = 0.08.

The algorithms were run 10 times, with each run of a population size of 100 in 500 iterations, and their results are shown in Table 3.

Table 3

Test results of optimization algorithms for Experiment 2.

Algorithms		Experiment 2-1(fully charged)			Experiment 2-2(low battery)
		The best result	Mean result	Mean CPU time	The best result	Mean result	MeanCPU time
AGA1	f(x)	274.0	278.2	1609.6 Sec	319.6	325.3	1604.1 Sec
	MS	131.5	147.9		192.6	189.3
	NA	18	17.1		17	17.5
	E	20.48	20.74		25.5	26.8
AGA2	f(x)	271.5	274.5	1611.7 Sec	294.4	301.3	1599.7 Sec
	MS	149.8	168.6		156.2	176.5
	NA	16	14.7		18	16.9
	E	20.74	21.10		22.21	23.03
MAGA	f(x)	256.5	258.3	1620.3 Sec	274.2	279.1	1634.8 Sec
	MS	133.3	136.7		184.9	182.1
	NA	16	15.9		13	13.5
	E	19.15	19.30		21.78	22.70

The performances of the three algorithms in the experiments are shown in Figs 5–8. The changes in population entropy are shown in Fig 9.

Fig 5

Performances of the different algorithms for Experiment 1–1.

Fig 8

Performances of the different algorithms for Experiment 2–2.

Fig 9

Population entropy of the different algorithms for all Experiments.

It can be seen from Figs 5–9 that the AGA1 performs well in the convergence rate because the crossover rate and mutation rate of AGA1 are adaptively adjusted based on the individual fitness value. If the individual fitness value is large, the rates of crossover and mutation will be low, so that the great gene sequence is more natural to enter the next generation without being destructed. Therefore, the application of AGA1 can help find a suitable solution quickly. However, the shortcomings of AGA1 is also apparent, that is, the global search ability is not strong so that it is easy to run into the local optimization solution. The mean value of f(x) and the value of population entropy shown in these figures can be well supported by this point. The population entropy of AGA1 decreases rapidly with the increase of iteration numbers, and the mean value of population and optimal value are very close after 150 iterations, which indicates that individuals in the population have converged and it is challenging to produce better individuals.

The performance of AGA2 is precisely the opposite of AGA1. AGA2 has a stronger global search ability but a slower convergence rate. Since the crossover rate and mutation rate of AGA2 are adaptively adjusted based on the value of the population entropy, if the population entropy becomes small, the crossover rate and the mutation rate will increase, thereby generating more new individuals to increase the diversity of the population. It can be seen from Fig 9 that the population entropy of AGA2 has been maintained at a relatively high level. So, if AGA2 with a more diverse population is applied, there will be a higher probability of finding a better solution. However, the high population entropy of AGA2 means that individuals are scattered, so it is difficult for the good genes of different gene positions to converge on the same individual. Moreover, the crossover rate and the mutation rate are only related to the current population entropy, which results in the difficulty to retain the high fitness value of the individual genetic structure. Therefore, the convergence speed of AGA2 is slow.

Among the three algorithms, the MAGA algorithm performs best. It adaptively adjusts the cardinality of population crossover rate and mutation rate, and selects the gene position with low genetic diversity to perform mutation operation with higher probability, so that the population entropy is maintained at the appropriate level. What is more, the individual crossover rate and mutation rate can be further changed with the change of fitness value to protect the current excellent gene and good gene structure, and thus guarantee the retaining of MAGA’s good global search ability and convergence rate.

In several experiments, the optimal distribution results obtained by MAGA are shown in Figs 11–14. The scheduling results are shown in Gantt charts, and the legend is shown in Fig 10.

Fig 11

Gantt chart of the schedule of the example 1–1 after optimization by MAGA.

Fig 14

Gantt chart of the schedule of the example 2–2 after optimization by MAGA.

Fig 10

The legend of the Gantt chart.

The effectiveness of MAGA is verified by comparing data in the AGV scheduling system before and after the optimization (shown in Table 4). Taking Experiment 2–2 as an example, Fig 15 shows the changes of various data in the system when using MAGA to optimize.

Table 4

Data comparison before and after optimization by MAGA.

Experiment		Mean value of 60 or 100 scheduling schemes before optimization	After optimization	Percentage of optimization
1–1	f(x)	290.3	171.9	40.79%
	MS	297.9	118.6	60.19%
	NA	9.6	9	6.25%
	E	13.31	11.22	15.70%
1–2	f(x)	385.5	182.6	52.64%
	MS	443.3	146.1	67.04%
	NA	9.4	8	14.89%
	E	16.13	11.67	27.65%
2–1	f(x)	412.9	256.5	37.88%
	MS	344.2	133.3	61.27%
	NA	18.3	16	12.57%
	E	23.24	19.15	17.60%
2–2	f(x)	503.8	274.2	45.57%
	MS	460.8	184.9	59.87%
	NA	18.4	13	29.35%
	E	29.69	21.78	26.64%

Fig 15

Data changes in the system for Experiment 2–2 optimization by MAGA.

It can be seen that the model and algorithms proposed in this study reduced the makespan while reducing the number of AGVs, indicating that the operational efficiency of the AGV is significantly improved. Besides, the power consumption was reduced while the makespan was reduced, indicating that the mathematical model arranged reasonable AGVs to sort the orders at reasonable speeds.

In summary, the mathematical model and optimization algorithms, especially the MAGA, have achieved success in reducing makespan, reducing the number of AGVs used, and reducing the total power consumption in AGV scheduling. What is more, the average optimization range is around 30%, which verifies the effectiveness of the model and the algorithms.

Conclusion

This research focused on the multi-objective AGV scheduling in an automatic sorting system using two different adaptive genetic algorithms (AGA) and one multi-adaptive genetic algorithm (MAGA). Taking into account of changes in AGV battery power and AGV speed, combined with order assignment, charging task assignment and speed determination, a multi-objective mathematical model was developed to minimize the makespan, number of AGVs used, and amount of electricity consumed by all AGVs to guide AGV scheduling in logistics sorting operations. Comparative numerical experiments were carried out, and the near-optimum schedules of the multi-objective function were successfully obtained. These schedules make it clear with regard to which order is sorted by which specific AGV at what speed. The comparison of the results of the three algorithms showed that the MAGA is superior to the other two adaptive genetic algorithms. A careful comparison of the data before and after optimization revealed that the sorting of all orders could be completed at suitable speeds by fewer AGVs, which results in a shorter makespan, a smaller number of AGVs used, and less total power consumption. Moreover, the value of each objective optimized was reduced by about 30%, which emphatically proved the effectiveness of the model and MAGA.

The major contributions of this paper are as follows:

The work discussed in this paper can effectively improve the operation efficiency of AGV in enterprise intelligent warehouses.

The multi-adaptive genetic algorithm proposed in this paper has strong global search ability and fast optimization speed, which may help other scholars get the approximate optimal solution faster and better under the same conditions.

The scheduling model developed in this paper may inspire AGV scheduling in different research fields, such as FMS, to take speed as one of the variables and energy consumption as one of the objectives.

Programming codes for MAGA.

(PDF)

Click here for additional data file.

Order data.

(XLSX)

Click here for additional data file.

17 Jun 2019

PONE-D-19-14670

Multi-objective AGV scheduling in an automatic sorting system of an unmanned (intelligent) warehouse by using two adaptive genetic algorithms and a multi-adaptive genetic algorithm

PLOS ONE

Dear Mr. Liu,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

We would appreciate receiving your revised manuscript by Aug 01 2019 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Yong Wang

Academic Editor

PLOS ONE

Journal Requirements:

1.  When submitting your revision, we need you to address these additional requirements.

Please ensure that your manuscript meets PLOS ONE's style requirements, including those for file naming. The PLOS ONE style templates can be found at

http://www.journals.plos.org/plosone/s/file?id=wjVg/PLOSOne_formatting_sample_main_body.pdf and http://www.journals.plos.org/plosone/s/file?id=ba62/PLOSOne_formatting_sample_title_authors_affiliations.pdf

2.  We suggest you thoroughly copyedit your manuscript for language usage, spelling, and grammar. If you do not know anyone who can help you do this, you may wish to consider employing a professional scientific editing service.

Whilst you may use any professional scientific editing service of your choice, PLOS has partnered with both American Journal Experts (AJE) and Editage to provide discounted services to PLOS authors. Both organizations have experience helping authors meet PLOS guidelines and can provide language editing, translation, manuscript formatting, and figure formatting to ensure your manuscript meets our submission guidelines. To take advantage of our partnership with AJE, visit the AJE website (http://learn.aje.com/plos/) for a 15% discount off AJE services. To take advantage of our partnership with Editage, visit the Editage website (www.editage.com) and enter referral code PLOSEDIT for a 15% discount off Editage services.  If the PLOS editorial team finds any language issues in text that either AJE or Editage has edited, the service provider will re-edit the text for free.

Upon resubmission, please provide the following:

The name of the colleague or the details of the professional service that edited your manuscript

A copy of your manuscript showing your changes by either highlighting them or using track changes (uploaded as a *supporting information* file)

A clean copy of the edited manuscript (uploaded as the new *manuscript* file)

3.  Please amend either the title on the online submission form (via Edit Submission) or the title in the manuscript so that they are identical.

4.   Thank you for stating the following in the Competing Interests section: "The authors have declared that no competing interests exist."

We note that one or more of the authors are employed by a commercial company:

"Beijing capital international airport Company Limited".

a) Please provide an amended Funding Statement declaring this commercial affiliation, as well as a statement regarding the Role of Funders in your study. If the funding organization did not play a role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript and only provided financial support in the form of authors' salaries and/or research materials, please review your statements relating to the author contributions, and ensure you have specifically and accurately indicated the role(s) that these authors had in your study. You can update author roles in the Author Contributions section of the online submission form.

Please also include the following statement within your amended Funding Statement.

“The funder provided support in the form of salaries for authors [insert relevant initials], but did not have any additional role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript. The specific roles of these authors are articulated in the ‘author contributions’ section.”

If your commercial affiliation did play a role in your study, please state and explain this role within your updated Funding Statement.

b)  Please also provide an updated Competing Interests Statement declaring this commercial affiliation along with any other relevant declarations relating to employment, consultancy, patents, products in development, or marketed products, etc.

Within your Competing Interests Statement, please confirm that this commercial affiliation does not alter your adherence to all PLOS ONE policies on sharing data and materials by including the following statement: "This does not alter our adherence to  PLOS ONE policies on sharing data and materials.” (as detailed online in our guide for authors http://journals.plos.org/plosone/s/competing-interests) . If this adherence statement is not accurate and  there are restrictions on sharing of data and/or materials, please state these. Please note that we cannot proceed with consideration of your article until this information has been declared.

Please include both an updated Funding Statement and Competing Interests Statement in your cover letter. We will change the online submission form on your behalf.

Please know it is PLOS ONE policy for corresponding authors to declare, on behalf of all authors, all potential competing interests for the purposes of transparency. PLOS defines a competing interest as anything that interferes with, or could reasonably be perceived as interfering with, the full and objective presentation, peer review, editorial decision-making, or publication of research or non-research articles submitted to one of the journals. Competing interests can be financial or non-financial, professional, or personal. Competing interests can arise in relationship to an organization or another person. Please follow this link to our website for more details on competing interests: http://journals.plos.org/plosone/s/competing-interests

Additional Editor Comments (if provided):

Please see the reviewers' comments and revise your manuscript point-by-ponit according to reviewers' comments.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: No

**********

4. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Authors may revise this paper according to suggestions below:

1. The major contributions of this paper are not clear. Author should specifically describe them.

2. Authors may polish writing of this paper.

Reviewer #2: The research is interesting and meaningful. The makespan, the number of AGVs, and electricity consumption were taken into consideration in the scheduling process for unmanned warehouse. A multi-adaptive genetic algorithm was proposed to solve the model. The structure is clear. However, before recommending it for publication, there still need several clarifications that are listed below.

(1) In the part of Model derivation, some expressions of formulas are not standard, particularly the objective function. And equation (1) is quite confusing. I suggest explaining the equations (1)-(10), as well as other unexplained formulas, to allow a better understanding experience.

(2) Is there any criteria for choosing the weights and coefficients of the three objectives?

(3) There are lots of mistakes in the paper (e.g., some data in Table 4 is incorrect; Fig. 3 is in consistent with the description; the title of figures are not standardized), please check the whole carefully.

(4) Please put your contributions clear in the part of conclusion.

(5) The language should be improved.

(6) Please cite the following references:

Wang et al., 2017. Profit Allocation in Collaborative Multiple Centers Vehicle Routing Problem. Journal of Cleaner Production, 144:203-219.

**********

6. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #1: No

Reviewer #2: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org. Please note that Supporting Information files do not need this step.

28 Jul 2019

We wish to thank you for the time and effort you have spent reviewing our paper. We are pleased to note that you have found our research work interesting and also pointed out some problems to help us improve the quality of our work.

Our itemized response to your questions, comments and suggestions:

Reviewer #1: Authors may revise this paper according to suggestions below:

1. The major contributions of this paper are not clear. Author should specifically describe them.

Response:

We have described the contribution of this paper in the conclusion section before, but it may not be prominent enough. We have rewritten the content in separate paragraphs in revised manuscript (line 473 – line 481).

The major contributions of this paper are as follows:

� The work discussed in this paper can effectively improve the operation efficiency of AGV in enterprise intelligent warehouses.

� The multi-adaptive genetic algorithm proposed in this paper has strong global search ability and fast optimization speed, which may help other scholars get the approximate optimal solution faster and better under the same conditions.

� The scheduling model developed in this paper may inspire AGV scheduling in different research fields, such as FMS, to take speed as one of the variables and energy consumption as one of the objectives.

2. Authors may polish writing of this paper.

Response:

We have polished writing of this paper with the help of English teachers and our native English speaking friends.

Reviewer #2: The research is interesting and meaningful. The makespan, the number of AGVs, and electricity consumption were taken into consideration in the scheduling process for unmanned warehouse. A multi-adaptive genetic algorithm was proposed to solve the model. The structure is clear. However, before recommending it for publication, there still need several clarifications that are listed below.

(1) In the part of Model derivation, some expressions of formulas are not standard, particularly the objective function. And equation (1) is quite confusing. I suggest explaining the equations (1)-(10), as well as other unexplained formulas, to allow a better understanding experience.

Response:

We have rewritten this part according to the reviewer’s suggestion

We have modified the expression of makespan, adjusted the order of the equations (1)-(10), and added explanations of them in revised manuscript (line 193 – line 215).

(2) Is there any criteria for choosing the weights and coefficients of the three objectives?

Response:

The weights are determined by the degree of attention paid to the three objectives in the actual intelligent warehouse and the core idea of Pareto optimality.

The coefficients are used to adjust the values of the three objectives to have similar ranges of variation, so they are determined based on the range of variation of the three objectives.

(3) There are lots of mistakes in the paper (e.g., some data in Table 4 is incorrect; Fig. 3 is in consistent with the description; the title of figures are not standardized), please check the whole carefully.

Response:

Thank you for pointing them out. We are very sorry for our incorrect writing, and we have revised them.

(4) Please put your contributions clear in the part of conclusion.

Response:

We have rewritten our contributions in separate paragraphs in the part of conclusion (line 473 – line 481).

The major contributions of this paper are as follows:

� The work discussed in this paper can effectively improve the operation efficiency of AGV in enterprise intelligent warehouses.

� The multi-adaptive genetic algorithm proposed in this paper has strong global search ability and fast optimization speed, which may help other scholars get the approximate optimal solution faster and better under the same conditions.

� The scheduling model developed in this paper may inspire AGV scheduling in different research fields, such as FMS, to take speed as one of the variables and energy consumption as one of the objectives.

(5) The language should be improved.

Response:

We have polished writing of this paper with the help of English teachers and our native English speaking friends.

(6) Please cite the following references:

Wang et al., 2017. Profit Allocation in Collaborative Multiple Centers Vehicle Routing Problem. Journal of Cleaner Production, 144:203-219.

Response:

We read the article carefully and found it very helpful to us - some of our revisions are based on the inspiration of this article. And we have cited it in revised manuscript(line 99 – line 101).

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

19 Aug 2019

PONE-D-19-14670R1

Multi-objective AGV scheduling in an automatic sorting system of an unmanned (intelligent) warehouse by using two adaptive genetic algorithms and a multi-adaptive genetic algorithm

PLOS ONE

Dear Mr. Liu,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

We would appreciate receiving your revised manuscript by Oct 03 2019 11:59PM. When you are ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter.

To enhance the reproducibility of your results, we recommend that if applicable you deposit your laboratory protocols in protocols.io, where a protocol can be assigned its own identifier (DOI) such that it can be cited independently in the future. For instructions see: http://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols

Please include the following items when submitting your revised manuscript:

A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). This letter should be uploaded as separate file and labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. This file should be uploaded as separate file and labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. This file should be uploaded as separate file and labeled 'Manuscript'.

Please note while forming your response, if your article is accepted, you may have the opportunity to make the peer review history publicly available. The record will include editor decision letters (with reviews) and your responses to reviewer comments. If eligible, we will contact you to opt in or out.

We look forward to receiving your revised manuscript.

Kind regards,

Yong Wang

Academic Editor

PLOS ONE

Additional Editor Comments (if provided):

Please see the reviewer's comments and revise your paper, thanks.

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #2: (No Response)

Reviewer #3: (No Response)

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #2: Yes

Reviewer #3: Partly

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #2: Yes

Reviewer #3: N/A

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #2: Yes

Reviewer #3: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #2: Yes

Reviewer #3: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #2: I have no further comments. The authors have answered all my questions. This paper can be accepted.

Reviewer #3: The manuscript addresses an optimal loading model for AGV scheduling in an automatic

sorting system by using improved adaptive genetic algorithms. The topic of this paper is interesting. However, this paper has a number of major shortcomings that need to be addressed:

(1) The contributions of this paper to the literature should be clear.

(2) Authors are kindly encouraged to add some latest literature on the heuristic algorithms, optimization model of scheduling problem and vehicle routing problem.

Zhang, D., Wang, X., Li, S., Ni, N., Zhang, Z., 2018a. Joint optimization of green vehicle scheduling and routing problem with time-varying speeds. PLoS One, 1-13.

Zhang, D.Z., Zou, F.Z., Li, S.Y., Zhou, L.Y., 2017. Green Supply Chain Network Design with Economies of Scale and Environmental Concerns. J. Adv. Transp., 1-14.

Yong Wang, Xiaolei Ma, Maozeng Xu, Yinhai Wang, Yong Liu. 2015. Vehicle Routing Problem based on A Fuzzy Customer Clustering Approach for Logistics Network Optimization, Journal of Intelligent&Fuzzy Systems, 29: 1427-1442.

(3) In the section of “Algorithm design”, authors are kindly encouraged to describe clear on the repairing the infeasible solution.

(4) In the section of “Computational results and discussion “, authors are kindly encouraged to give some comparative analysis on the two proposed algorithms (i.e. AGA and MAGA)with other optimization solver ( e.g. Cplex ) on the small-size instances.

**********

7. PLOS authors have the option to publish the peer review history of their article (what does this mean?). If published, this will include your full peer review and any attached files.

If you choose “no”, your identity will remain anonymous but your review may still be made public.

Do you want your identity to be public for this peer review? For information about this choice, including consent withdrawal, please see our Privacy Policy.

Reviewer #2: No

Reviewer #3: No

[NOTE: If reviewer comments were submitted as an attachment file, they will be attached to this email and accessible via the submission site. Please log into your account, locate the manuscript record, and check for the action link "View Attachments". If this link does not appear, there are no attachment files to be viewed.]

While revising your submission, please upload your figure files to the Preflight Analysis and Conversion Engine (PACE) digital diagnostic tool, https://pacev2.apexcovantage.com/. PACE helps ensure that figures meet PLOS requirements. To use PACE, you must first register as a user. Registration is free. Then, login and navigate to the UPLOAD tab, where you will find detailed instructions on how to use the tool. If you encounter any issues or have any questions when using PACE, please email us at figures@plos.org. Please note that Supporting Information files do not need this step.

17 Oct 2019

Replies to the reviewers’ comments

Reviewer #2: I have no further comments. The authors have answered all my questions. This paper can be accepted.

Response:

We appreciate your approval of our paper. Thank you for your suggestions and guidance on our article before.

Reviewer #3: The manuscript addresses an optimal loading model for AGV scheduling in an automatic sorting system by using improved adaptive genetic algorithms. The topic of this paper is interesting. However, this paper has a number of major shortcomings that need to be addressed:

(1) The contributions of this paper to the literature should be clear.

Response:

We have rewritten our contributions in a separate paragraph in the part of conclusion (line 473 – line 481).

The major contributions of this paper are as follows:

The work discussed in this paper can effectively improve the operation efficiency of AGV in enterprise intelligent warehouses.

The multi-adaptive genetic algorithm proposed in this paper has strong global search ability and fast optimization speed, which may help other scholars get the approximate optimal solution faster and better under the same conditions.

The scheduling model developed in this paper may inspire AGV scheduling in different research fields, such as FMS, to take speed as one of the variables and energy consumption as one of the objectives.

(2) Authors are kindly encouraged to add some latest literature on the heuristic algorithms, optimization model of scheduling problem and vehicle routing problem.

Zhang, D., Wang, X., Li, S., Ni, N., Zhang, Z., 2018a. Joint optimization of green vehicle scheduling and routing problem with time-varying speeds. PLoS One, 1-13.

Zhang, D.Z., Zou, F.Z., Li, S.Y., Zhou, L.Y., 2017. Green Supply Chain Network Design with Economies of Scale and Environmental Concerns. J. Adv. Transp., 1-14.

Yong Wang, Xiaolei Ma, Maozeng Xu, Yinhai Wang, Yong Liu. 2015. Vehicle Routing Problem based on A Fuzzy Customer Clustering Approach for Logistics Network Optimization, Journal of Intelligent&Fuzzy Systems, 29: 1427-1442.

Response:

We read these articles carefully and we have cited them in revised manuscript(line 92 – line 94).

(3) In the section of “Algorithm design”, authors are kindly encouraged to describe clear on the repairing the infeasible solution.

Response:

We would like to explain: each solution in this paper consists of two halves, the first half is the allocation of the order sorting work, and the second half is the speed of the AGV when the order is sorted.

If a random solution is an infeasible solution, then there should be one of two situations: battery limited or time conflict. However, the charging task is considered in the mathematical model of this paper. As long as the power of the AGV is insufficient, the charging task will be triggered: the charging time will be no longer 0, it will be calculated (line220). So, there will be no limited of battery.

In addition, the various times (t〖ST〗_i, t〖CT〗_i,…) in this paper are calculated by the composition of the solution (line 202-207, line 216-221), so the task time of different orders sorted by the same AGV does not conflict.

So, even if it is extreme to assign all orders to the same AGV to complete the order picking, it is still a feasible solution with the longer charging time. Therefore, the solutions generated under the rules set in this paper are all feasible solutions, and there are no infeasible solutions.

(4) In the section of “Computational results and discussion “, authors are kindly encouraged to give some comparative analysis on the two proposed algorithms (i.e. AGA and MAGA)with other optimization solver ( e.g. Cplex ) on the small-size instances.

Response:

Thank you for pointing it out. In fact, at the earliest time, we really thought about doing such a comparison. Our idea at the time was to use LINGO to compare with our proposed MAGA. But after the experiment, we found that the effect of using LINGO is very unsatisfactory. Because the research in this paper not only involves the assignment of order sorting, but also the speed is variable, even if the solution space of the small-scale example reaches 1.75*10^61, it is very difficult to solve with LINGO. So in the end, we chose to use our proposed MAGA to compare with the traditional AGA proposed by others. Therefore, we hope that you understand that there is no good effect in using Cplex. We think that our algorithm (MAGA) is better than AGA, and this is enough to reflect the superior performance of the MAGA.

Submitted filename: Response to Reviewers.docx

Click here for additional data file.

21 Nov 2019

Multi-objective AGV scheduling in an automatic sorting system of an unmanned (intelligent) warehouse by using two adaptive genetic algorithms and a multi-adaptive genetic algorithm

PONE-D-19-14670R2

Dear Dr. Liu,

We are pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it complies with all outstanding technical requirements.

Within one week, you will receive an e-mail containing information on the amendments required prior to publication. When all required modifications have been addressed, you will receive a formal acceptance letter and your manuscript will proceed to our production department and be scheduled for publication.

Shortly after the formal acceptance letter is sent, an invoice for payment will follow. To ensure an efficient production and billing process, please log into Editorial Manager at https://www.editorialmanager.com/pone/, click the "Update My Information" link at the top of the page, and update your user information. If you have any billing related questions, please contact our Author Billing department directly at authorbilling@plos.org.

If your institution or institutions have a press office, please notify them about your upcoming paper to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, you must inform our press team as soon as possible and no later than 48 hours after receiving the formal acceptance. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information, please contact onepress@plos.org.

With kind regards,

Yong Wang

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

27 Nov 2019

PONE-D-19-14670R2

Multi-objective AGV scheduling in an automatic sorting system of an unmanned (intelligent) warehouse by using two adaptive genetic algorithms and a multi-adaptive genetic algorithm

Dear Dr. Liu:

I am pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please notify them about your upcoming paper at this point, to enable them to help maximize its impact. If they will be preparing press materials for this manuscript, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

For any other questions or concerns, please email plosone@plos.org.

Thank you for submitting your work to PLOS ONE.

With kind regards,

PLOS ONE Editorial Office Staff

on behalf of

Dr. Yong Wang

Academic Editor

PLOS ONE