Test task scheduling problem (TTSP) is a typical combinational optimization scheduling problem. This paper proposes a variable neighborhood MOEA/D (VNM) to solve the multiobjective TTSP. Two minimization objectives, the maximal completion time (makespan) and the mean workload, are considered together. In order to make solutions obtained more close to the real Pareto Front, variable neighborhood strategy is adopted. Variable neighborhood approach is proposed to render the crossover span reasonable. Additionally, because the search space of the TTSP is so large that many duplicate solutions and local optima will exist, the Starting Mutation is applied to prevent solutions from becoming trapped in local optima. It is proved that the solutions got by VNM can converge to the global optimum by using Markov Chain and Transition Matrix, respectively. The experiments of comparisons of VNM, MOEA/D, and CNSGA (chaotic nondominated sorting genetic algorithm) indicate that VNM performs better than the MOEA/D and the CNSGA in solving the TTSP. The results demonstrate that proposed algorithm VNM is an efficient approach to solve the multiobjective TTSP.
During recent decades, the manufacturing of electronic devices has become highly integrated and increasingly complex. As a result, the resource and time consumption expended on the test of electronic devices became a crucial problem in engineering application. Therefore, the research for improving the test efficiency is a topic that has attracted extensive attention. To address this situation, the objective of this research is to solve the test task scheduling problem (TTSP) more efficiently.
The goal of the TTSP is to arrange the execution of
TTSP, FJSP, and most scheduling problems belong to combinational optimization problems. For combinational optimization problems, the search space is too large that the best solution cannot be obtained by adopting the method of enumeration for even smallscale problem. Therefore, the intelligent algorithms based on integer programming model are devoted to solving these kinds of problems. Take FJSP as the example; genetic algorithm (GA) [
Different from the research of FJSP, the research of TTSP is relatively few because of the development of automatic test system. However, there are still some achievements in TTSP. Xia et al. [
There are also some intelligent algorithms used to solve power dispatch problems and other scheduling problems. For example, oppositionbased learning is employed in oppositionbased gravitational search algorithm (OGSA) to solve optimal reactive power dispatch [
In summary, most of the researches of scheduling problems focus on the singleobjective problem or adopt weighted sum approach to convert the multiobjective problem into a singleobjective problem. However, the weighting coefficients are difficult to choose, and human factors will greatly impact the performance of the algorithms. In fact, there are another two kinds of methods for solving the multiobjective problem. One method is the nonPareto approach utilizing operators for processing the different objectives in a separated way. Another is the Pareto approaches which are directly based on the Pareto optimality concept. They aim at satisfying two goals: converging towards the Pareto front and also obtaining diversified solutions scattered all over the Pareto front. Those two kinds of methods mainly rely on the performance and strategies of the algorithms used in the multiobjective problems.
In this paper, the method based on Tchebycheff decomposition for multiobjective functions was adopted and the algorithm named MOEA/D is used to solve TTSP. MOEA/D is a typical evolutionary algorithm based on decomposition proposed by Zhang and Li [
The scheduling problems, such as TTSP, FJSP, and TSP, and power dispatching problem are a branch of combinational optimization problems. Because of the properties of the combinational optimization problems, the final best solutions only account for a rather small subset of the search space. How to avoid the solutions obtained being trapped in local optima is the key to improve the ability of algorithms to deal with combinational optimization problems. Considering the fact that the size of the neighborhood is important in MOEA/D [
The organization of this paper is as follows. A brief introduction of TTSP is introduced in Section
The goal of the TTSP is to arrange the execution of
Petri net [
The Petri net model for one task TTSP.
In Figure
Graph theory [
The Graph model for TTSP.
Graph theory model can only be adopted by typical optimization methods. With the increment of the scale of TTSP, the computation expense will greatly increase, but typical optimization methods are not suitable for largescale TTSP problem. Therefore, Graph theory model cannot solve largescale TTSP also.
TTSP is a typical integer programming problem. For the integer programming model for TTSP, the TTSP can be described as follows [
In general, task
The following describes the restriction of resources:
Basic hypothesis includes three factors. At a given time, an instrument can only execute one task; each task must be completed without interruption once it starts. Assume
The objective functions are very important in the study of multiobjective optimization problem. The makespan is very important in scheduling problems such as TTSP and FJSP, because the completion time is an essential factor for scheduling problem in product process. In additional, for TTSP, the test instruments have high integration, and the test instruments have become increasingly expensive. Therefore, the demand for reducing the workload of the instruments and increasing the service life of the test instruments has great significance in TTSP. Therefore, our work focuses on two main objectives. One is to minimize the maximal test completion time, and the other is to minimize the mean workload of the instruments. These objectives are represented by
In this section, we proposed a variable neighborhood MOEA/D algorithm (VNM). To obtain solutions close to the real Pareto Front (PF) of the TTSP, two strategies are adopted. The variable neighborhood strategy helps to make the crossover span more reasonable. Moreover, Gauss mutation is adopted at the beginning of the iteration to maintain the diversity of the population.
The VNM is an evolutionary algorithm based on decomposition. The main strategy of the VNM is to decompose a multiobjective optimization problem into a number of scalar optimization subproblems and optimize these subproblems simultaneously. The decomposition method used is the Tchebycheff approach [
Let
The main procedure of the VNM can be described as shown in Figure
The main procedure of the VNM.
In the part of parameter setting, the iteration number
The crossover operation in VNM is as follows.
For each individual
The main idea of VNM is given above. Two improvements are involved in the VNM algorithm. Variable neighborhood strategy is adopted to make the crossover span more reasonable. Moreover, Starting Mutation is used to enhance the diversity of the population.
In the VNM, the size of the neighborhood
The three curves are shown in Figure
Three controlling curves for the neighborhood size.
The TTSP represents a typical combinational optimization problem. The final best solutions may be limited to only several points in the solution space. Because of the neighborhood updating effect of the VNM, there will be many duplicate solutions so that the crossover operation will have little effect. Therefore, how to maintain the diversity of the population is the key question for enhancing the algorithm effect.
Motivated by the ideology above, a starting Gauss mutation is adopted at the beginning of the iteration. For a solution
The convergence analysis of VNM in this section provides the theory ground for its application. The convergence behavior of VNM is analyzed according to the Markov Chain and the transfer matrix, respectively.
This section proposes the basic theories of convergence and proves the strong and weak convergence of VNM from the perspective of Markov Chain.
There is a global optimal solution set
A detailed demonstration for the convergence of MOEA has been proposed in paper [
Based on Theorems
It is defined as
Based on Bayesian, we have
Similarly, it is defined as
By Bayesian formula, we have
This part focuses on the elitist strategy and proves that the VNM converges to the global optimum from the perspective of transfer matrix.
According to the previous description of VNM, the extended transition matrices for crossover
In VNM, the populations go through Gauss mutation
Computational experiments are carried out to compare the approaches and to evaluate the efficiency of the proposed method. There are two objectives: to minimize the makespan and the mean workload of the instruments. In this section, the performance metric, coverage metric
The instance of 40 tasks with 12 instruments.
Task  Scheme  Resource  Time 




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5 
Parameters setting.
Population  Generation 





100  250  0.5  1  1  0.05 
For multiobjective optimization, the convergence to the Paretooptimal set is the most important target to be considered. There are mainly two metrics to evaluate the convergence. One is convergence metric
Assume that
In this section, three curves are designed and tested to identify the best
Comparison of influence of three curves for
Average  Times  


0.2213  13 







0.2069  14 
Comparison of influence of three curves for
Average  Times  


0.244178  14 







0.242146  12 
The boxplot of three curves for
The boxplot of three curves for
In order to verify the improvement of VNM,
Comparison of VNM and MOEA/D for 30 * 12 instance.
Average  Times  





0.2104  15 
Comparison of VNM and MOEA/D for 40 * 12 instance.
Average  Times  





0.1949  10 
The results of the two independent experiments for comparison of VNM and MOEA/D are shown in Figures
The comparison of VNM and MOEA/D for
The comparison of VNM and MOEA/D for
Figures
The boxplot of comparison of VNM and MOEA/D for
The boxplot of comparison of VNM and MOEA/D for
In this section, the VNM is compared with the CNSGA for TTSP. CNSGA is based on NSGAII. NSGAII has been successfully applied to job shop scheduling problems [
There are two chaotic sequences, logistic map and cat map, and the chaotic sequences can be applied in three positions, population initialization, crossover, and mutation. Therefore, there are six combinations for CNSGA. The nomenclatures for six variants of CNSGA are shown in Table
Nomenclature for six variants of the CNSGA.
The logistic map  The cat map  

Initial population 


Crossover operator 


Mutation operator 


Comparison of VNM and six variations of CNSGA for 30 * 12 instance.
Average  Times  





0.2200  14 




0.2648  16 




0.2248  14 




0.2210  15 




0.2288  16 




0.2525  15 
Comparison of VNM and six variations of CNSGA for 40 * 12 instance.
Average  Times  





0.2264  13 




0.2282  14 




0.2338  12 




0.2138  14 




0.2169  15 




0.2055  13 
Figures
The comparison of VNM and three variations with logistic map for CNSGA for
The comparison of VNM and three variations with cat map for CNSGA for
The results of the comparison of VNM and CNSGA for the
The comparison of VNM and three variations with logistic map for CNSGA for
The comparison of VNM and three variations with cat map for CNSGA for
In addition, box plots are used to display the performances of the algorithms. The box plots of
The boxplot of VNM and three variations with logistic map for CNSGA for
The boxplot of VNM and three variations with cat map for CNSGA for TTSP
The boxplot of VNM and three variations with logistic map for CNSGA for
The boxplot of VNM and three variations with cat map for CNSGA for
From the box plots of
The variable neighborhood provides a strategy to improve the performance of the algorithm. For different problems with different scales, the controlling curves for the neighborhood size will be different. The Starting Mutation can be also applied to solve other optimization problem in the evolution process. The strategies proposed in this paper can be investigated in other scheduling problem similar to TTSP.
How to improve the test efficiency is more and more important in modern industry. TTSP has important application value in modern manufacturing process. TTSP is combinational optimization problem. The final best solutions only account for a rather small subset of the search space. In order to help the solutions avoid being trapped in local optima, this paper proposed a new genetic evolutionary multiobjective optimization algorithm (VNM) to solve the TTSP. The variable neighborhood and Starting Mutation strategies are adopted in VNM to make the crossover span more suitable and improve the diversity of population. Three controlling curves for neighborhood size are studied. The experimental results have shown that the monotonic parabolic has the best performance. From the experiment conducted for comparison of VNM and MOEA/D, we see that the improved algorithm has made great progress in solving the TTSP problem. And the experiment conducted for comparison of VNM and CNSGA also shows that the improved algorithm is superior to CNSGA in solving TTSP. VNM can also be applied to solve other combinational optimization problems such as FJSP and TSP. Future work will focus on two objectives: the precedence constraint will be added to the TTSP, and information regarding bottleneck tasks will be considered.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the National Natural Science Foundation of China under Grant no. 61101153 and the National 863 HiTech R and D Plan under Grant 2011AA110101.