Improved artificial bee colony algorithm for air freight station scheduling

: Aiming at improving the operating efficiency of air freight station, the problem of optimizing the sequence of inbound/outbound tasks meanwhile scheduling the actions of elevating transfer vehicles (ETVs) is discussed in this paper. First of all, the scheduling model in airport container storage area, which considers not only the influence of picking sequence, optimal ETVs routing without collision, but also the assignment of input and output ports, is established. Then artificial bee colony (ABC) is proposed to solve the above scheduling issue. For further balancing the abilities of exploration and exploitation, improved multi-dimensional search (IMABC) algorithm is proposed where more dimensions will be covered

global best position to guide the following updating processes in onlooker bee phase which could enhance the local search ability. Literature [21] proposed two search strategies with the help of the information of the best-so-far solution and the mean of two random solutions. In [22], three search strategies with different characteristics were employed to construct a strategy candidate pool, and the Parzen window method was applied to select the best candidate individuals. Two new search strategies were adopted based on the k-neighborhood of best solution in [23] which didn't need to calculate the selection probability. Different from the above literatures, from the perspective of improving the search mechanism, Zhang et al. [24] proposed a new search mechanism named full-dimensional ABC algorithm (fdABC), it was executed with all dimensions of each solution thus the search area could be expanded and the probability of obtaining the optimal solution could be improved. Although the proposed algorithm possessed better optimization performance especially for high-dimensional optimization problems, the time cost increased substantially because more dimensions needed to be updated compared with traditional ABC. In this paper, improvement on the search mechanisms is introduced and applied to solve the scheduling problem in air freight station.
The main features that distinguish this study from previous studies are presented below. a) For solving the tasks scheduling problem in airport container storage area with several I/O ports, the model considering the assignment of I/O ports, as well as the optimization of picking sequence and ETVs' travelling route is first established. The proposed modeling method could be applied to all scheduling problems of storage systems in fast moving consumer goods industry, e-commerce industry and logistics industry; b) In order to balance the abilities of exploration and exploitation of ABC algorithm, two improvements on search mechanisms are proposed. Different from original ABC that only one dimension is randomly updated, improvement is realized by saving the most valuable dimension of current solution and guiding the direction of subsequent exploration. Related improvement could be adopted to solve all optimization and scheduling problems in engineering fields.
The rest of this paper is organized as follows: Section 2 introduces the scheduling model with dual ETVs in the freight station of Luoyang Beijiao Airport. Then ABC and IMABC algorithms are proposed in Section 3, and their optimization performance is verified on benchmark functions. In Section 4, the improved algorithms are applied to the dual ETVs scheduling problem and their effectiveness is proved. Finally, the above work is summarized.

System description
The air freight station in Luoyang Beijiao Airport consists of three parts, which are container storage area, bulk cargo storage area, and unhandled cargo area respectively as shown in Figure 1.
As the core of the whole system, the container storage area is used for handling the containerized cargoes, which are unloaded from aircraft in the airside or inbounded from the bulk cargo storage area in the landside. It is a three-dimensional warehouse with two rows of shelves and 16 I/O ports, each row has eight layers and 60 columns, the total slots are 60 × 8 × 2 = 960. Two ETVs are employed for handling cargoes between the 14 I/O ports in airside and the two I/O ports in landside, and each ETV is responsible for half of the shelf and I/O ports.
The operational process of the two ETVs is depicted as Figure 2 (some columns and I/O ports are omitted). ETVs start from R1 and R3 respectively, the 1# ETV picks up cargo at entrance R2, and stores it at I1. And then, it retrieves outbound cargo from O2 and delivers it to the exit C1. The same actions are executed with 2# ETV from R3 to C2.  Obviously, in order to improve the efficiency of cargo transportation in the container storage area, the sequence of inbound/outbound tasks as well as the corresponding I/O ports need to be scheduled, meanwhile the actions of ETVs should be optimized to obtain the minimum running time.

Model formulations
For solving the scheduling problem, the objective function should be established firstly based on the notations defined in Table 1.  If there are several inbound and outbound tasks in the storage area, the objective of optimized problem which is named as fitness function is to minimize the total execution time Hmax between two ETVs under the constraints in Eq (2).
The first constraint in Eq (2) ensures that the columns between two ETVs at any time are no less than four, which could avoid collision with each other. As the task set is divided into two equal parts and assigned to two ETVs, the second constraint avoids repeated allocation. The time in Eq (3) is defined as the summation of execution time and of tasks assigned to the specific ETV, it is the total time cost for picking up, releasing as well as transporting all assigned cargoes. Here the execution time and corresponding to different tasks can be obtained from Table 2 (Only the values corresponding to the first five layers and six columns are listed), which are calculated from Eqs (4) and (5). The first row and first column of Table 2 respectively represent the number of difference of rows and columns between the start position and destination in the shelf.    For solving the above nonlinear scheduling problem, an effective optimization algorithm should be introduced.

ABC algorithm
ABC is an optimization algorithm based on the intelligent foraging behavior of honeybee swarm, where the bee colony consists of three groups: employed bees, onlooker bees and scout bees. The position of a food source represents an optimal solution of the specific problem, and its quality could be evaluated through the fitness value of the corresponding solution [9,25].
The algorithm starts iterative optimization from employed bee phase, it executes a crossover and mutation process with one randomly chosen companion, and the new solution is updated based on as shown in Eq (6). Then the fitness value of each solution fitnessm could be calculated, and the onlooker bee randomly chooses to exploit or not around corresponding employed bee with the probability Pm defined as Eq (7). If the current mth solution to be exploited cannot improve for several iterations, it will be abandoned, and a scout bee corresponding to a new randomly produced solution will replace it.
(−1, 1) is a random number between is the number of food sources as well as the number of employed bees.

The improved ABC algorithm
In ABC algorithm, for each solution, only one dimension is randomly selected and updated according to Eq (6) in employed bee phase and onlooker bee phase. In this case, the updated dimension may be different in each iteration and the optimal dimension obtained in the previous iteration is likely to be omitted in the following iterations. Thus, the search toward the possible optimal solutions is unable to be continued, the optimization accuracy and the convergence speed will be affected.
In [23], full dimensional search strategy (fdABC) is introduced to traverse all dimensions of the solution and select the optimal dimension for further exploration, therefore the search could be extended and the possibility of obtaining optimal solution will be improved, but the optimization time increases inevitably. Another improvement called random multi-dimensional artificial bee colony algorithm (RmdABC) is mentioned in [26], the key improvement of the strategy is to randomly select several different dimensions from {1,2, … , } for one solution, and execute the updated process with Eq (6) in the employed bee and onlooker bee phases. Obviously, it randomly traverses any several dimensions of the solution in each iteration, and fewer dimensions are updated compared with fdABC, as the result, its time complexity could be greatly improved.
In order to further balance the abilities of exploration and exploitation, IMABC strategy is proposed where more valuable dimensions of solution will be picked out and saved in the external set which is used to guide the subsequent exploration. The operations of IMABC are shown as Figure 3 and outlined as follows: 1) In the first iteration, all dimensions of the solution are searched, and new solutions are generated with Eq (6) in employed bee and onlooker bee phases.
2) The fitness values of the generated solutions need to be compared with the optimal one, and if the new solution is superior to the old one, the solution in that dimension could be recognized as having the potential to be optimized, thus the optimal solution should be substituted with the generated one, the valuable dimension will be recorded in the external set and the flag is set to be one. 3) In employed bee and onlooker bee phases, after updating all dimensions of the solution, the value of flag will be checked. If the flag sign is equal to one, it means there is at least one dimension has been updated. If flag sign is zero, it demonstrates that the exploration is failed, and the number of iteration trial should increase by 1.
if == 1 then = 0; When the whole cycle is greater than or equal to the value of MaxCycle, if trial < Limit, the above updating operations based on the stored optimal dimensions should be performed iteratively. If trial > Limit, the value of trial resets back to zero and a new random solution will be generated to replace the old one. Obviously, more dimensions will be explored in IMABC, it could cover more solution space and the ability of exploration could be improved compared with ABC algorithm. Meanwhile, with the help of the stored optimal dimension searched so far, the speed towards the global optimal solution could be accelerated and the ability of exploitation will be enhanced compared with fdABC and RmdABC. Therefore, the performance of the exploration and exploitation could be balanced. Evaluate the fitness value fitness(Solm) with Eq (7)

Performance evaluation
In order to evaluate the performance of the proposed IMABC algorithm, nine CEC 2017 benchmark functions as listed in Table 3 are employed, where f1(x), f2(x) and f6(x) are unimodal functions, and the others are multi-modal functions. All simulations are executed on an Intel Core i7-8750H CPU with 8G RAM, the population size is 200, the dimension of solution is set to 60, 80 and 100 respectively, the number of maximum iterations is set as being 1000, and the limit used in scout bee phase is taken 100. Independent experiments are run 20 trials, the indices including the mean and standard deviation (Mean ± std dev) which reflect the quality of solution and the stability of algorithm, the average running time (Aver-R), the shortest running time (Best-R) and fitness value of the optimal solution (Best-F) are selected to evaluate the optimization performance of different algorithms.  For functions f3(x), f5(x), f6(x), f7(x) and f8(x), the values of all four indices in each dimension corresponding to IMABC are smallest, which means IMABC possesses the highest optimization accuracy, the fastest convergence speed as well as the best stability when solves the above functions. Another improved algorithm RmdABC gets smaller fitness values and Mean ± std dev values, but it takes longer time compared with the one of ABC. On the other hand, it improves the optimization efficiency of fdABC at the expense of a worse solution. In other words, RmdABC balances the exploration and exploitation abilities of fdABC and ABC.
In addition, IMABC obtains the lowest values of Mean ± std dev and Best-F, but it takes longer running time to obtain optimal solution than ABC algorithm for f2(x), which means that the stability and global search ability of IMABC are the best, while its convergence performance is better than fdABC and RmdABC because it updates less dimensions. Besides that, as the results of optimizing functions f1(x), f4(x) and f9(x), fdABC obtains the best fitness value as well as best stability among all algorithms as it executes full-dimensional search and IMABC takes much less time to get the optimal solution.
The curves of fitness values for functions f3(x) under different dimensions are depicted as Figures 4-6, and the same conclusions as mentioned above could be obtained.      From the analysis above, it can be seen that the proposed IMABC algorithm is able to produce better solutions with higher stability compared with ABC, and costs shorter computational time than other improved algorithms. RmdABC which is deduced from fdABC could reduce its time cost, but the quality of solution becomes poor because it updates less dimensions. Therefore, the proposed IMABC and RmdABC algorithms can balance exploration and exploitation abilities of ABC and fdABC.

Task set schedule with improved ABC algorithms
The performance of proposed algorithms has been evaluated by solving complex mathematical problems above, and they will be applied to solve the scheduling problem in the container storage area of Luoyang Beijiao Airport in this section.
There For this constrained optimization problem as shown in Eqs (1) and (2), the solution is the sequence of inbound and outbound tasks with dual ETVs, which corresponds to the minimum total time cost as Eq (1). In other words, it is a discrete optimization problem, integer encoding scheme mentioned in [27] is introduced and random numbers between -10 and 10 are assigned to each dimension of the optimized solutions, after sorting them in ascending order based on their values, the corresponding scheduling scheme as well as the optimal solution could be obtained. Furthermore, the constraint conditions as Eq (2) should be checked for each obtained scheduling scheme in the iterative optimization procedure. If the constraints corresponding to the generated solution are satisfied, the solution will be saved and used for further exploration, otherwise a new solution needs to be introduced.
Comparative studies among four algorithms, including ABC, fdABC, RmdABC and IMABC, are executed for the above scheduling problem. The swarm size of all algorithms is set to 200 with 60 dimensions, the maximum local search time is 50, the stopping criterion is set to 1000 generations. Initial populations are generated through uniformly random sampling from the search space. Each algorithm is independently tested 20 times. The experiments are performed with an Intel Core i7-8750H CPU and 8GB of RAM. Table 7. Assigned or current positions of inbound or outbound tasks.   Table 8 presents the optimization results (fitness value which is the scheduling time corresponding to the optimal solution) in the first ten trials. It can be seen that all proposed algorithms are able to produce high-quality solutions for scheduling problem, the performance of IMABC is better than RmdABC and ABC.  Table 9 lists four important indices, they are the average time needed to execute the sequence corresponding to optimal solution in 20 trials (Avg), the fitness value of the best solution (Min), the fitness value of the worst solution (Max) and the corresponding running time (CPU time). It is clear that the last two proposed algorithms possess better performance as the first three indices decrease by 6% at most compared with ABC, but the time needed to obtain the optimal solution is longer. It means RmdABC and IMABC can keep the balance between the optimization accuracy and convergence speed compared with ABC. Moreover, because fdABC covers more dimensions than other mentioned algorithms, the fitness values are better than RmdABC, but the running time is 1.97 and 2.35 times of RmdABC and IMABC. For the two proposed algorithms, the fitness value obtained by IMABC was a 1% reduction with respect to RmdABC and the searching time with IMABC is 83% of RmABC, obviously the optimization ability of IMABC is better than RmABC.
The conclusion also can be obtained from Figure 7 which depicts the average fitness values of the optimization problem, the curve slope of fitness values corresponding to the three improved algorithms is relatively steeper as compared with ABC, which proves their convergence. And the fitness values corresponding to IMABC converges at about 100 iterations, thus its ability of convergence could be proved.    Figure 8 and Table 10 show the corresponding trajectories of two ETVs and the sequence of task set optimized with IMABC algorithm respectively. It is clear that the dual ETVs could execute the scheduling tasks successfully without conflicting with each other.
From the above results, the following conclusions could be obtained: 1) IMABC are valid for solving the complex scheduling problem and they can improve the searching ability of traditional ABC algorithm. 2) fdABC possesses the excellent exploration ability, IMABC could keep the balance between the optimization accuracy and convergence speed compared with ABC and fdABC under this scheduling background.
3) For IMABC, with the help of the current optimal dimensions, the convergence speed could be improved compared with fdABC and RmdABC.

Conclusions
In this paper, in order to improve the efficiency in the container storage area of airport cargo terminal, a study on scheduling of cargoes sequence and the action of ETV with ABC algorithms was performed. The dual ETVs scheduling model was established which considered the assignment of I/O ports as well as the trajectory of two ETVs with the constraint of avoiding collisions, and improved IMABC algorithm was proposed to solve this scheduling problem more effective. The computational experiments were carried out and the results proved the proposed algorithm could effectively avoid conflicts and generate optimal scheduling sequences.
ABC and corresponding improved algorithms have been proved to be effective in solving the scheduling problem, but the time cost for obtaining the optimal solution is high because of the iterative calculations. Improving its efficiency with appropriate methods, such as parallelization, can make the algorithm more useful especially in the scheduling problems. In addition, how to choose appropriate control parameters for different optimization issues is a problem in ABC and improved algorithms as with other metaheuristic algorithms. This problem has not been investigated sufficiently in the literature. Therefore, designing a general principle for tuning the control parameters of ABC can be addressed as a searching subject in future studies.