Test task scheduling problem (TTSP) is a complex optimization problem and has many local optima. In this paper, a hybrid chaotic multiobjective evolutionary algorithm based on decomposition (CMOEA/D) is presented to avoid becoming trapped in local optima and to obtain high quality solutions. First, we propose an improving integrated encoding scheme (IES) to increase the efficiency. Then ten chaotic maps are applied into the multiobjective evolutionary algorithm based on decomposition (MOEA/D) in three phases, that is, initial population and crossover and mutation operators. To identify a good approach for hybrid MOEA/D and chaos and indicate the effectiveness of the improving IES several experiments are performed. The Pareto front and the statistical results demonstrate that different chaotic maps in different phases have different effects for solving the TTSP especially the circle map and ICMIC map. The similarity degree of distribution between chaotic maps and the problem is a very essential factor for the application of chaotic maps. In addition, the experiments of comparisons of CMOEA/D and variable neighborhood MOEA/D (VNM) indicate that our algorithm has the best performance in solving the TTSP.
Test task scheduling problem (TTSP) is an essential part of the automatic test system for improving throughput, reducing time, and optimizing resource allocation. Similar to other scheduling problems, the TTSP is one kind of combination optimization problems. It is illustrated to be an NPhard problem through the analysis of the nature of the problem carried out by many researchers [
Recently, many intelligent methods are used for solving the TTSP and other similar scheduling problems based on the problems’ character. All these kinds of researches focus on improving the searching ability of the algorithm and obtaining optimal or nearoptimal solutions for the scheduling problem. There are two basic strategies. One is to propose an improvement algorithm based on the original algorithm, such as variable neighborhood multiobjective optimization algorithm based on decomposition (VNM) for the multiobjective test task scheduling problem [
Different from the hybrid method using different kinds of algorithms, using chaos in the evolutionary process represents its advantages in improving the searching ability. Lu et al. proposed a chaotic nondominated sorting genetic algorithm for the multiobjective test task scheduling problem and validated the best performance in convergence and diversity through the experiment and analysis [
For the TTSP, notice that the previous studies mostly aim at a single objective and only few papers focus on the multiobjective problem [
In this paper, we propose a chaotic multiobjective evolutionary algorithm based on decomposition (CMOEA/D) for the TTSP. Ten chaotic maps are embedded in three different phases in the evolutionary process. The aim is to give guidance for the choice of chaotic maps and phases based on the framework of MOEA/D for the TTSP.
First, the chromosomeencoding scheme is very important for the problem description and the operation in the evolutionary process. For the TTSP, there are different kinds of encoding strategies, like task sequencing list (TSL) (or the operations list coding (OLC)), matrixencoding, and integrated encoding scheme (IES). TSL and matrixencoding are not acceptable if a task can be tested on more than one set of instruments. IES can overcome this problem, but it cannot realize the selection operation under equal probability. Therefore, in this paper, the improved IES is proposed by changing the schemes selection method of every test task. As a result, the equal probability is realized.
Then, ten chaotic maps are embedded in MOEA/D independently in three phases. The ten chaotic maps are baker’s map, cat map, circle map, cubic map, Gauss map, ICMIC map, logistic map, sinusoidal map, tent map, and Zaslavskii map. Three phases are initial population, crossover operator, and mutation operator. Four benchmarks of the TTSP are used to evaluate the performance of the proposed algorithm. They are
The performance metrics hypervolume (HV) and
From the results of experiments, it can be seen that the chaotic map embedded MOEA/D has good performance to solve the TTSP for both small and large scale problems. Different kinds of chaotic maps have different performances in different phases of MOEA/D, but ICMIC map and circle map in initial population, crossover operator, and mutation operator have the best performance. The experiments for comparisons of CMOEA/D and VNM show that our algorithm performs better than the VNM in solving the TTSP. The evidence, the chaotic map is an effective and efficient method for solving the problem with local optima, is validated. The similarity degree of distribution between chaotic maps and the problem is a very essential factor for the application of chaotic maps.
The rest of the paper is organized as follows. Section
Recently, chaotic sequences have been integrated in the evolutionary process through two types of operations. One is using chaotic maps to replace random sequences. Another is to replace the genetic operations. These two kinds of operations always appear in the same algorithm at once.
In detail, all the operations can be divided into seven cases. They are population initialization, setting crossover probability, setting crossover position, setting crossover operator, setting mutation probability, setting mutation operator, and increasing chaotic disturbance. The performance of different operations is totally different. For example, adopting chaotic maps in the initialization can maintain the population diversity. The aim of using chaotic maps to replace standard mutation operator is to avoid the search being trapped in local optima.
For the scheduling problem, the situation is the same as the above in both the single objective and the multiobjective scheduling problems. For the single objective scheduling problem, Cheng et al. used the hybrid genetic algorithm and chaos to optimize the hydropower reservoir operation [
For the multiobjective scheduling problem, Niknam et al. [
All these researches, despite the single objective or the multiobjective problem in these scheduling fields, have the same features. The chaotic maps are used for improving the searching ability of the evolutionary algorithm. However, most of researches only used one kind of chaotic maps embedded in special phases of the algorithm, and comprehensive analysis is inefficient. In fact, different kinds of the scheduling problems have different characters, and different chaotic maps have different effects on the algorithms and the problems. Our work will focus on the analysis and design of chaotic multiobjective algorithm for the TTSP. We investigate the guidance for solving the TTSP.
The aim of the TTSP is to organize the execution of
The TTSP has two types of constraints: the restriction on resources and the precedence constraint between the tasks. The restriction on resources can be expressed as follows:
The precedence constraint between the tasks can be represented as follows:
In this study, we consider two objective functions. The model is defined as follows:
subject to
The first objective function minimizes the maximal test completion time and the second objective function minimizes the mean workload of the instruments. Here,
Constraint (
Ten chaotic maps including both onedimensional maps and twodimensional maps are introduced in this section. Each one has specific features, and different chaotic maps combined with optimization algorithms have different results (Table
The list of chaotic maps.
Chaotic map  Formula  Dimensions  Range 

Baker’s map 

2 



Arnold’s cat map 

2 



Circle map 

1  (0, 1) 




Cubic map 

1  (0, 1) 




Gauss map 

1  (0, 1) 


ICMIC map 

1 





Logistic map 

1  (0, 1) 




Sinusoidal map 

1  (0, 1) 


Tent map 

1  (0, 1) 


Zaslavskii map 

2 


There are two problems for these chaotic maps. One is that the range of ICMIC and Zaslavskii maps is not (0, 1). As a result, the generated chaotic sequences need the scale transformation. Another is some maps, like tent map, have fixed points. Therefore, jumping out from fixed points is necessary for maintaining the chaos characteristics.
Figure
Distribution of different chaotic maps.
Integrated encoding scheme (IES) proposed by our previous research [
Here, we use an example with four tasks and four instruments for illustration of the role of IES. The detail is in Table
A TTSP with four tasks and four instruments.








5 


3  





4 


1  





2 





4 


3  


7 
The main concept of the IES is to use the relationships between the decision variables to express the sequence of tasks and use the values of the variables to represent the occupancy of the instruments for each task. This concept is illustrated in Table
Example of the integrated encoding scheme.
Decision variables 
0.8147  0.9058  0.1270  0.6324 

Tast sequence 
3  4  1  2 

1  1  2  1 






2  4  3  4 
The entries in the first row are the decision variables, which range between 0 and 1. They are sorted in ascending order. The rank of every variable denotes a test task index in the sequence. Thus, the second row (or the task sequence) is obtained. On the other hand, the instrument assignment can also be obtained from the decision variables. If we want to know which instruments will be occupied by the task
Here,
However, this encoding scheme has one defect that all schemes are selected with unequal probability. For example, for one task
The integrated encoding scheme.
Decision variables 






1  2  3 
As seen from Table
Then, the value of
The improving integrated encoding scheme.
Decision variables 






1  2  3 
As seen from Table
The multiobjective evolutionary algorithm based on decomposition is originated from Tchebycheff decomposition. It decomposes a multiobjective problem into a number of scalar optimization subproblems and optimizes them simultaneously. Each subproblem is bound with a weight vector and is optimized by using the information from its several neighbor subproblems [
In this paper, chaotic variables are used instead of random variables in MOEA/D. Ten chaotic maps are embedded in MOEA/D to replace the random operation. Three key phases in evolutionary algorithms, initialization, crossover, and mutation, are chosen to be embedded with chaos. Different chaotic maps have different formulas and characters. Here, we use sinusoidal map [
For example, we assume
Here, the initial population is generated by chaos maps. For example, if the sinusoidal map is used for initialization,
In this paper, a differential evolution (DE) operator is adopted. In the DE operator, each child individual
Here,
In this paper, a polynomial mutation operator is adopted. For a solution
Here,
We carry out four types of experiments to illustrate the performances of the mentioned approaches. Experiment 1 shows the effectiveness of the improving encoding method based on one large scale TTSP. Experiment 2 aims to solve a small scale TTSP benchmark to measure the performance of the evolutionary algorithm using chaotic maps in three phases. Experiment 3 is similar to experiment 2, except that it aims to solve the large scale TTSP. In both experiments 2 and 3, ten chaotic maps are embedded in three different phases in the original MOEA/D algorithm. Each time, only one parameter is modified. The Pareto set (PF) is used to show the effect firstly. Then, the performance metrics HV and
The parameters for all experiments are shown in Table
The setting of parameters.







250  

100  

6  20  30  40 
CR  0.9  

1/6  1/20  1/30  1/40 
This experiment shows the effectiveness of the improving encoding method in solving the TTSP. The instance is based on a large scale TTSP
Comparison of different encoding methods in solving the TTSP.
We can find from the Pareto front that the improving encoding method obtains better convergence of the solutions of the TTSP. The equal probability also helps the algorithm to obtain good convergence. Therefore, the improving IES is used in the following experiments because of the efficiency.
This experiment is carried out to show the effectiveness of CMOEA/D for the small scale TTSP
Comparison of different chaotic maps for crossover.
Comparison of different chaotic maps for initialization.
Comparison of different chaotic maps for mutation.
For the convenience, the algorithms with different combinations of chaotic maps and phases are named as “CMOEA/D[phase][chaotic map].” The ten chaotic maps (baker, cat, circle, cubic, Gauss, ICMIC, logistic, sinusoidal, tent, and Zaslavskii) are denoted by
According to the name role, Figure
For the small scale TTSP, the performance for convergence is not very obvious from the Pareto set. The solutions obtained from the original algorithm and the chaos embedded algorithm overlap each other. However, the diversity of the solutions obtained from the chaos embedded algorithm is better than the original algorithm.
This experiment is carried out to show the effectiveness of CMOEA/D for three large scale problems, TTSP
Comparison of different chaotic maps for crossover for
Comparison of different chaotic maps for initialization for
Comparison of different chaotic maps for mutation for
Comparison of different chaotic maps for crossover for
Comparison of different chaotic maps for initialization for
Comparison of different chaotic maps for mutation for
Comparison of different chaotic maps for crossover for
Comparison of different chaotic maps for initialization for
Comparison of different chaotic maps for mutation for
For the large scale TTSP, both the convergence and diversity of solutions are improved significantly. Almost every chaotic map has good performance for the improvement, but the performance is not stable and positive for some chaotic maps. For example, the tent, baker, and cat maps even have negative effects for the solutions under some situations.
Based on the above experiments, we use the statistical data of the comprehensive metric HV and convergence metric
Here,
The value
The average values of performance metrics HV and
The average value of HV.















 
Baker  0.3312 

0.3312  0.3240  0.3312 

0.7471 

0.7471 

0.7471 

Cat  0.3312 

0.3312 

0.3312 

0.7471  0.7460  0.7471 

0.7471  0.7370 
Circle  0.3312 

0.3312 

0.3312 

0.7471 

0.7471 

0.7471 

Cubic  0.3312 

0.3312  0.3305  0.3312 

0.7471  0.7411  0.7471  0.7448  0.7471 

Gauss  0.3312 

0.3312 

0.3312 

0.7471 

0.7471  0.7294  0.7471 

ICMIC  0.3312 

0.3312 

0.3312 

0.7471 

0.7471  0.7465  0.7471 

Logistic  0.3312 

0.3312  0.3302  0.3312 

0.7471 

0.7471 

0.7471 

Sinusoidal  0.3312 

0.3312 

0.3312 

0.7471 

0.7471 

0.7471 

Tent  0.3312 

0.3312 

0.3312 

0.7471  0.7442  0.7471 

0.7471  0.7448 
Zaslavskii  0.3312  0.3276  0.3312 

0.3312 

0.7471  0.7466  0.7471 

0.7471  0.7433 
















 


Baker  0.5326  0.4976  0.5326 

0.5326 

0.7588 

0.7588  0.7168  0.7588  0.7395 
Cat  0.5326  0.5283  0.5326 

0.5326 

0.7588 

0.7588  0.7289  0.7588 

Circle  0.5326 

0.5326  0.5160  0.5326 

0.7588 

0.7588  0.7250  0.7588  0.7538 
Cubic  0.5326  0.5164  0.5326  0.5314  0.5326  0.5195  0.7588  0.7576  0.7588  0.7551  0.7588  0.7357 
Gauss  0.5326  0.5167  0.5326  0.5194  0.5326  0.5176  0.7588  0.7513  0.7588 

0.7588  0.7353 
ICMIC  0.5326 

0.5326  0.5105  0.5326 

0.7588 

0.7588  0.7370  0.7588  0.7569 
Logistic  0.5326  0.5261  0.5326 

0.5326  0.5155  0.7588  0.7360  0.7588  0.7064  0.7588 

Sinusoidal  0.5326  0.5152  0.5326  0.5201  0.5326  0.5181  0.7588 

0.7588  0.6875  0.7588  0.7440 
Tent  0.5326  0.5312  0.5326 

0.5326  0.5187  0.7588 

0.7588  0.7144  0.7588 

Zaslavskii  0.5326  0.4911  0.5326  0.5028  0.5326  0.5150  0.7588  0.7506  0.7588 

0.7588 

The average value of

















Baker  0.0400 

0.0400  0.0000  0.0000 

0.1753 

0.3741  0.3233  0.3815  0.3572 
Cat  0.0000  0.0000  0.0000 

0.0000 

0.2800 

0.2873 

0.4660  0.1102 
Circle  0.0500 

0.0367 

0.0250 

0.2496 

0.2713 

0.2672 

Cubic  0.0900  0.0833  0.0250 

0.0750 

0.3770  0.3111  0.2801 

0.2964 

Gauss  0.0250 

0.0250  0.0250  0.0250 

0.2982 

0.4517  0.1346  0.2475 

ICMIC  0.0000 

0.0450 

0.0250 

0.3017 

0.3097 

0.2637 

Logistic  0.0900  0.0583  0.0250 

0.0000 

0.1479 

0.2874 

0.2338 

Sinusoidal  0.0250 

0.0500 

0.0500  0.0250  0.2652 

0.3466 

0.3333  0.3158 
Tent  0.0000 

0.0500  0.0000  0.0250 

0.4352  0.2210  0.2500 

0.3911  0.2708 
Zaslavskii  0.0000  0.0000  0.0000 

0.0250  0.0250  0.2804  0.2911  0.3236  0.2762  0.3490  0.2720 




















Baker  0.5861  0.2929  0.5208  0.2383  0.4730  0.4039  0.2847 

0.6541  0.1514  0.4676  0.3046 
Cat  0.4136  0.3699  0.4242  0.3134  0.3993 

0.5515  0.2536  0.5098  0.1996  0.4573  0.2652 
Circle  0.2573 

0.4339  0.3152  0.3287  0.3220  0.4333 

0.4678  0.3517  0.3883  0.3801 
Cubic  0.4172  0.2880  0.4579  0.3487  0.3989  0.3229  0.4228  0.2989  0.5586  0.2629  0.4272  0.3358 
Gauss  0.4304  0.3084  0.4930  0.2412  0.4560  0.3533  0.3788  0.3788  0.5212  0.4023  0.3629 

ICMIC  0.3204 

0.5540  0.2483  0.4886  0.3886  0.3237 

0.6220  0.2232  0.3899  0.3902 
Logistic  0.3935  0.3342  0.5051  0.2571  0.4668  0.3420  0.4870  0.3450  0.5793  0.2007  0.3341 

Sinusoidal  0.4075 

0.5211  0.2960  0.4670  0.3727  0.4508  0.3314  0.7251  0.1100  0.4354  0.4003 
Tent  0.3002 

0.4465  0.3732  0.4913  0.2751  0.4500  0.2876  0.5798  0.2495  0.3538 

Zaslavskii  0.6872  0.1425  0.5136  0.3129  0.5040  0.3512  0.3968  0.3680  0.2626 

0.4932  0.3487 
As shown in Tables
In most cases, the best performance in Table
The visualized result.

















 
Baker  ++  ++  ++  ++  
Cat  ++  ++  ++  ++  
Circle  ++  ++  ++  ++  ++  ++  ++  ++  
Cubic  ++  ++  
Gauss  ++  ++  ++  ++  
ICMIC  ++  ++  ++  ++  ++  ++  ++  
Logistic  ++  ++  ++  ++  ++  
Sinusoidal  ++  ++  ++  ++  
Tent  ++  ++  ++  ++  
Zaslavskii  ++  ++ 
The results show that circle map and ICMIC map in all phases especially in crossover operator have the best performance. Cubic map and logistic map in mutation operator, Gauss map in crossover operator and mutation operator, sinusoidal map in crossover operator and initial population, baker’s map in crossover operator, and Zaslavskii map in initial population have a better effect. In addition, cat map in initial population and mutation operator also has a little bit of effect.
In order to show the above results in an intuitive way, the boxplots of the performance metric
The boxplots of
The boxplots of
The boxplots of
Overall, chaotic maps for crossover and mutation operators are helpful for preventing the solutions from trapping in the local optima and have significant improvement on the evolutionary algorithms based on the decomposition for solving the TTSP. Circle map and ICMIC map have the best performance in ten maps especially. Cubic map, logistic map, Gauss map, and sinusoidal map have better contribution in solving those TTSPs.
We discuss and explore the reason for these conclusions based on the above results. We focus on the distribution of solutions of the TTSP.
We calculate the feasible solutions of a small scale TTSP
Exhaustive result for TTSP
The chaotic map has the nature to avoid becoming trapped in local optima. The TTSP has many local optima. All the experiments illustrate the fact that using chaotic maps embedded with the evolutionary algorithm can help the TTSP to obtain good solutions. In addition, the process of crossover and mutation is important for jumping out of local optima. The experiments also validate this fact.
Furthermore, chaotic maps have a superior effect on escaping from local optima, but not all of them are effective. We want to find the relationship from the distribution. The distribution of every chaotic map is shown in Figure
Referring to Table
We take TTSP
The value of HV.















 
1  0.4067  0.4928  0.4067  0.4582  0.4067  0.4642  0.4016  0.4718  0.4016  0.4576  0.4016  0.4189 
2  0.4018  0.4844  0.4018  0.4756  0.4018  0.4746  0.4148  0.4511  0.4148  0.4449  0.4148  0.4521 
3  0.4552  0.4776  0.4552  0.4582  0.4552  0.4706  0.4729  0.4729  0.4729  0.4937  0.4729  0.4222 
4  0.4258  0.4907  0.4258  0.4789  0.4258  0.4643  0.3881  0.4853  0.3881  0.4521  0.3881  0.4286 
5  0.4283  0.5055  0.4283  0.4607  0.4283  0.4755  0.4010  0.4598  0.4010  0.4674  0.4010  0.4666 
6  0.4492  0.4690  0.4492  0.4738  0.4492  0.4889  0.4155  0.4389  0.4155  0.4496  0.4155  0.4354 
7  0.4560  0.4540  0.4560  0.4612  0.4560  0.4871  0.3900  0.4521  0.3900  0.4688  0.3900  0.4578 
8  0.4306  0.4765  0.4306  0.4688  0.4306  0.4789  0.3968  0.4426  0.3968  0.4703  0.3968  0.4750 
9  0.4460  0.4872  0.4460  0.4632  0.4460  0.4740  0.4235  0.4585  0.4235  0.4500  0.4235  0.4468 
10  0.4458  0.4897  0.4458  0.4803  0.4458  0.4767  0.4002  0.4406  0.4002  0.4465  0.4002  0.4537 


Average  0.4345 

0.4345 

0.4345 

0.4104 

0.4104 

0.4104 

Times  1 

0 

0 

0 

0 

1 

The value of

















1  0.0000  1.0000  0.0000  0.8333  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  0.9000 
2  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
3  0.0000  0.9000  0.3750  0.5000  0.1667  0.7000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
4  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
5  0.0000  1.0000  0.0000  0.9000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
6  0.0000  0.1667  0.0833  0.1667  0.0000  1.0000  0.0000  0.8571  0.0000  1.0000  0.1000  0.8571 
7  0.4444  0.0000  0.0000  0.2000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
8  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 
9  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000  0.0000  0.7778  0.0000  0.7778 
10  0.0000  1.0000  0.1000  0.5714  0.0833  0.5714  0.0000  1.0000  0.0000  1.0000  0.0000  1.0000 


Average  0.0444 

0.0558 

0.0250 

0.0000 

0.0000 

0.0100 

Times  1 

0 

0 

0 

0 

0 

Comparison of VNM and three variants of CMOEA/D for
Comparison of VNM and three variants of CMOEA/D for
It shows that the solutions obtained by the CMOEA/D dominate most of the solutions obtained by the VNM in the above figures. The values of
A short summary can be obtained according to the above experiments and analyses. The improving encoding method is effective for solving the TTSP. In addition, the effectiveness of the multiobjective evolutionary algorithm based on decomposition using chaotic maps, which have nonuniform distributions, is illustrated for TTSP. Furthermore, the comparisons of CMOEA/D and VNM indicate that our algorithm has the best performance for solving the TTSP. The fact, the chaotic map is an effective and efficient method for solving the problem with local optima, is illustrated again.
The TTSP is a complex combinational optimization problem and has many local optima. This paper focuses on the chaotic multiobjective evolutionary algorithm based on decomposition for solving the TTSP. The improving encoding method is proposed to increase the encoding efficiency. Ten chaotic maps are embedded in three phases of MOEA/D to solve the TTSP, and the results show that the proposed algorithm can prevent solutions from falling into local optima. The performance metrics HV and
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the anonymous reviewers for their helpful comments in improving their paper. This research is supported by the National Natural Science Foundation of China under Grant no. 61101153.