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Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem



Hui Lu, Ruiyao Niu, Jing Liu and Zheng Zhu
 
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ABSTRACT

Task scheduling problem is one of the key technologies for automatic test systems. This study proposes a novel and integrated multicriteria decision mechanism called the chaotic non-dominated sorting genetic algorithm plus analytic hierarchy process (CNSGA-AHP) for the automatic test task scheduling problem (ATSP). This mechanism contains two parts: the multiobjective optimisation algorithm CNSGA and the decision making method AHP. CNSGA hybrids chaotic sequences based on the logistic map and the non-dominated sorting genetic algorithm II (NSGA-II) to avoid becoming trapped in local optima. It is responsible for the search process and obtains a set of compromise solutions. However, getting this set does not completely solve the problem. A best compromise solution still must be chosen out of that set. Thereupon, AHP is used for the final decision making process and chooses a best schedule from the solutions obtained by CNSGA. The applied AHP can handle uncertainty, make the consistency check easily to pass and reduce the workload of decision-makers. A real-world ATSP abstracted from a missile system is applied to verify the effectiveness of CNSGA-AHP. Results show that CNSGA-AHP is very concise and suitable for the ATSP.

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Hui Lu, Ruiyao Niu, Jing Liu and Zheng Zhu, 2012. Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem. Information Technology Journal, 11: 1449-1455.

DOI: 10.3923/itj.2012.1449.1455

URL: https://scialert.net/abstract/?doi=itj.2012.1449.1455
 
Received: February 28, 2012; Accepted: April 24, 2012; Published: July 03, 2012



INTRODUCTION

Automatic test technology has been used in aerospace systems to ensure safety and in some modern manufacturing processes to improve the production efficiency. One important feature of the Next Generation Automatic Test System (NGATS) is parallel testing (Curry et al., 2006). Parallel testing makes it possible to test more than one task at the same time. It can improve the throughput by taking full advantage of the test hardware. On the other hand, it increases the complexity of the system (Anderson, 2003). Therefore, an effective and efficient dispatching method for the Automatic Test Task Scheduling problem (ATSP) is significant. However, a concrete optimised algorithm has not been fully elucidated (Zhou et al., 2009). Hence, this study is motivated to perform.

The mean work of dispatching method for the ATSP is allocating a task onto the available instruments to improve the throughput, reduce the test time and optimise the resource allocation (Xia et al., 2007a). Many meta-heuristics methods and evolutionary algorithms have been proposed to solve the scheduling problems. The Simulated Annealing (SA), the Tabu Search (TS) and the Genetic Algorithm (GA) are generally used to deal with the single objective problems. For example, Liaw (2003) proposed a tabu search approach together with the optimal timing algorithm for solving the Open Shop Scheduling problem (OSSP). For the multiobjective problems, the Strength Pareto Evolutionary algorithm (SPEA), the Pareto Archived Evolutionary Strategy (PAES) and the non-dominated sorting genetic algorithm II (NSGA-II) (Deb et al., 2002) are widely used. For example, Moradi et al. (2011) used NSGA-II and Non-ranking Genetic Algorithm (NRGA) for solving the Flexible Job-shop Scheduling Problem (FJSP). Although, most of these multiobjective optimisation approaches have been successfully used in real-world problems, one aspect that is often disregarded in the current research is the decision making process (Fernandez et al., 2010). The multiobjective optimisation approaches always generate an approximation of the Pareto frontier, but getting this set does not completely solve the problem. A best compromise solution still must be chosen out of that set (Fernandez et al., 2011). That is to say, both the multiobjective optimisation algorithms (search process) and the decision maker’s preferences (decision making process) should be considered to solve a multiobjective optimisation problem thoroughly. The combination of these two processes can be classified into three categories (Coello et al., 2007): the a priori, the progressive and the a posteriori decision.

In the a priori decision, the decision-maker (DM) is consulted before starting the search process. The DM’s preferences guide the search until the algorithm converges to a single point which may result in a half-baked search. The progressive decision is an interactive method. In the progressive decision, the DM participates in the course of an iterative optimisation process, gives information about the preferences to guide the search until the DM is satisfied with the current solution. In this process, the DM should always be present, which is not convenient. In the a posteriori decision, a multiobjective optimisation algorithm is usually used to obtain a set of compromise solutions. Then, the DM is asked to choose the final solution. The search process and the decision making process are separated and the alteration of one process will not influence the other process. The a posteriori decision is used in this paper because of its flexibility.

A multicriteria decision mechanism called the chaotic non-dominated sorting genetic algorithm plus analytic hierarchy process (CNSGA-AHP) is proposed to solve the ATSP. As mentioned above, the multicriteria decision mechanism CNSGA-AHP is made up of two parts: the multiobjective optimisation algorithm CNSGA and the decision making method AHP (Saaty, 1980). CNSGA incorporates chaotic sequences based on the logistic map into NSGA-II. NSGA-II is a typical representative of the multiobjective optimisation algorithms, which is effective and can maintain a better spread of solutions. Xu et al. (2006) used NSGA-II in the design of standalone hybrid wind/photovoltaic power systems. Chaos has some special characteristics such as the ergodic property and the stochastic and is highly sensitive to initial conditions. Because of these properties, optimisation algorithms based on chaos is presented to avoid becoming trapped in local optima (Li and Wang, 2011; Ren and Zhong, 2011). Thus, CNSGA can achieve better results. AHP is extensively adopted in multicriteria decision making (MCDM) and has successfully been applied to the ranking process of decision making problems (Saaty, 1996). Barker and Zabinsky (2011) used AHP for reverse logistics and Okeola and Sule (2012) applied AHP to study the urban water supply scheme. AHP has the following advantages (Duran and Aguilo, 2008), (1) it is the only known MCDM model that can measure the consistency in the decision maker’s judgments, (2) it helps dissect the problem and structure it into a rational decision hierarchy, which makes the decision process easy to handle, (3) DMs can derive weights of criteria and scores of alternatives from comparison matrices by pairwise comparisons instead of quantify weights/scores directly, (4) it can hybridise with other techniques to handle more difficult problems, (5) it is easier to understand and can effectively handle both qualitative and quantitative data. However, in many practical situations, DMs may have some uncertain information and cannot assign exact numerical values to the comparison judgments. Then, some researches applied fuzzy AHP as an extension of conventional AHP to handle uncertainty. For example, Duran (2011) proposed a fuzzy-based AHP for comparative evaluation of a number of Computerised Maintenance Management Systems (CMMS) alternatives, Wang et al. (2011) presented an intuitionistic fuzzy AHP approach and Pan (2012) proposed an evaluation method for Smart Logistics based on benchmarking analysis and fuzzy AHP method.

Other researches considered the improvement of consistency check and the scale measurement. In CNSGA-AHP, the decision information can be quantified and the above issues are actually a matter of indifference. That is to say, the uncertainty can be handled, the consistency check can be passed easily and the scale measurement can be more exact. CNSGA-AHP is thus a suitable multicriteria decision mechanism for the ATSP.

PROBLEM FORMULATION

The ATSP is to assign the execution of n tasks on m instruments. In this problem, there are a set of tasks Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem and a set of instruments Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem (Xia et al., 2007b). The notifications Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem are used to present the test time, the test start time and the test completion time of task tj tested on ri, respectively. A variable Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem is defined to express whether the task tj occupies the instrument ri. Its definition is as follows:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(1)

Generally speaking, a task occupies more than one instrument.

In typical cases, some instruments could have redundancies, which make it possible for a task to choose the instruments. The alternative schemes for tj is expressed as Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem where kj is the number of schemes of tj. The numbers of schemes that correspond to every task can make up a set Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem. Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem is represented as Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem and ujk is the number of instruments for Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem. If Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem, tj and tj* have resource conflicts. Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem is the test time of tj for Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem.

In addition, hypotheses considered in this paper are the following (Xia et al., 2007b):

All tasks and instruments are available at time 0
At a given time, an instrument can execute only one task
Each task must be completed without interruption once it starts
Assume Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem and Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem to simplify the problem

In this study, there are two objectives should be minimised: the maximal test completion time (makespan) f1(x) and the mean workload of the instruments f2(x).

The definition of f1(x) is:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(2)

where, Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem is the test completion time of tj for Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem and meets the equation:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(3)

The definition of f2(x) is:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(4)

where, Q is the parallel steps and its initial value is 1. Assign the instruments for all of the tasks, if Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem, Q = Q+1.

MULTICRITERIA DECISION MECHANISM CNSGA-AHP FOR THE ATSP

In CNSGA-AHP, multiobjective optimisation methods CNSGA can provide valuable trade-off information among the conflict objectives and multicriteria decision making approach AHP helps DMs choose the most desirable and satisfactory alternative. Before explaining the details of CNSGA-AHP, some concepts of Multiobjective Optimisation Problem (MOP) and multicriteria decision making should be introduced briefly.

Multiobjective optimisation problem: A multiobjective optimisation problem can be described, without loss of generality, as follows (Zhang and Li, 2007):

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(5)

where, Ω is the decision space and the function F means a mapping from Ω to RM. RM is the objective space, with M objective functions.

Multicriteria decision making: In MCDM, there are a number of options called alternatives and an ultimate goal. The DMs should choose from the alternatives to reach the goal according to some problem-based evaluation criteria. These criteria make different contribution to the goal and have different importance. DMs try to compare the effects of alternatives and achieve best quality of the final result.

CNSGA

Chaotic sequence: A chaotic sequence is generated by a chaotic map which is an evolution function that exhibits some sort of chaotic behaviour. The logistic map is a well-known one-dimensional chaotic map and its definition is as follows:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(6)

where, λt is the value of the chaotic variable λ at the tth iteration and μ is the so-called bifurcation parameter of the system (μ∈[0, 4]) (Hong et al., 2011). In this study, μ = 4, then λ0.∉{0.25, 0.5, 0.75}.

Implementation of CNSGA: The main procedure of CNSGA is similar with NSGA-II, but some details are modified by chaos:

The main loop of CNSGA begins with the combination of the current and previous populations and the calculation of the non-dominated fronts. Once all the Pareto fronts have been determined, the new population is obtained. Finally, using this population, individuals are selected, crossed and mutated to create a new population. This process is described in Fig. 1.
Problem encoding is a representation in a chromosome, which is relative to the conditions of the problem being considered. Designing a suitable encoding approach for the ATSP is very important. In this study, a new encoding scheme, called the Integrated Encoding Scheme (IES), is proposed.

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
Fig. 1: Main loop of CNSGA

  This encoding scheme can use only one chromosome to contain the information about both the processing sequence of the tasks and the occupancy of the instruments for each task. The main concept of the IES is to use relationships between the decision variables to express the sequence of tasks and the values of the variables to represent the occupancy of the instruments for each task. Firstly, the decision variables are sorted in ascending order and the rank of every variable denotes a test task index in sequence. Then, the instrument assignment can also be obtained from the decision variables according to the equation k = [xijx10]mod kj+1. Where, xij represents the decision variable that corresponds to tj and kj is the number of schemes of tj. The IES never generates a duplication of a certain task, can always get feasible instrument assignments, makes full use of decision variables and improves the encoding efficiency:

The fitness function is used to measure the solutions. In this paper, the makespan and the mean workload of the instruments are the objectives. Their calculation is shown in Fig. 2.
The Simulated Binary Crossover (SBX) operator is used in CNSGA. Pc is the crossover probability and ηc is the distribution index for the crossover operator.
The polynomial mutation is applied with some modification. For a solution xs, the polynomial mutation is described as follows:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(7)

where, Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem and Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem are the upper and lower bound of xs, respectively.

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(8)

where, Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem, pm is the mutation probability and ηm is the distribution index for the mutation operator

AHP: It decomposes a complex issue into several levels, rigorously defines the manager priorities and computes weights associated to the alternatives. The output of AHP is a ranking indicating the overall preference for each decision alternative. The main steps of AHP include:

Step 1: Define the problem and determine its goal. For the ATSP, this goal is to obtain a most satisfactory schedule

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
Fig. 2: Calculation of the objective values

Table 1: Pairwise comparison scale for AHP preferences (Saaty, 1980)
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Step 2: Define the structure of the hierarchy of that problem from the top goal through intermediate levels of criteria, subcriteria and actors to the lowest level of alternatives. In this paper, the criteria (C) are the makespan (C1) and the mean workload of the instruments (C2) and the alternatives (A) are solutions obtained by CNSGA
Step 3: Construct a set of pairwise comparison matrices for each lower level in a hierarchy and make all the pairwise comparisons by using the relative scale measurement shown in Table 1. In our work, fmax and fmin are the maximum and minimum of the fitness function, respectively. The comparisons can be quantified as follows:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
(9)

where, aij is an element of the pairwise comparison matrix, which indicates the importance of ith factor over jth. fi and fj are the fitness values of Ai and Aj.

Table 2: Average random consistency ratio (RI) (Saaty, 1980)
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Step 4: Compute the consistency ratio (CR) for each pairwise comparison matrix to measure the consistency of the subjective ranking. Firstly, using the biggest eigenvalue λmax of a pairwise comparison matrix to calculate the consistency index (CI) by CI = (λmax-n)/(n-1), where n is the matrix size. The CR is calculated by comparing the CI with a random index (RI), CR = CI/RI. RI is the consistency index of a randomly generated pairwise comparison matrix and its value depending on the matrix size is shown in Table 2. If CR≤0.10, the judgment matrix is acceptable. Otherwise it is considered inconsistent and the matrix should be improved.
Step 5: Calculate the relative weights and determine the ranking of the alternatives. The lambda max technique is commonly used, which defines a vector of weights as the normalised eigenvector w corresponding to the largest eigenvalue λmax. If the combining vector of weights of the (k-1)th level W(k-1) and the vector of weights of the kth level wk is obtained, the combining vector of weights of the kth level Wk can be calculated by Wk = wkW(k-1), k>1.

RESULTS AND DISCUSSION

Experiments are designed to verify the effectiveness of CNSGA-AHP. The applied ATSP in Table 3 depends on a real-world instance that is abstracted from a missile system. There are 20 tasks to be tested and 8 instruments can be used. It was solved by CNSGA. The parameter settings of CNSGA are shown in Table 4 and the obtained compromise solutions are shown in Fig. 3. The objective values of these compromise solutions are [(39, 14.357), (34, 19.546), (45, 14.286), (52, 13.667), (38, 15.539), (32, 20.000), (33, 19.636), (35, 18.182), (59, 13.533), (65, 12.813)].

Then, AHP is applied to make the final decision. The structure of AHP is shown in Fig. 4.

Firstly, the pairwise comparison matrix for criteria with respect to goal was constructed in Table 5. Generally speaking, for the ATSP, makespan is more important than the mean workload of the instruments. Thus, the rating 5 was considered.

Table 3: A real-world ATSP with 20 tasks and 8 instruments
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
Fig. 3: Comparison solutions obtained by CNSGA

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem
Fig. 4: Structure of the AHP

For matrix Goal-C, the biggest eigenvalue λmax = 2 and the CI = 0. Therefore, the judgment matrix was acceptable. The vector of weights W2 = w2 = (0.833, 0.167)T.

Table 4: The parameter settings of CNSGA
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Table 5: Pairwise comparison matrix Goal-C
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Table 6: Pairwise comparison matrix C1-A
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Table 7: Pairwise comparison matrix C2-A
Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Secondly, the pairwise comparison matrix for alternatives with respect to the criterion was created according to the Eq. 9. Both the makespan (C1) and the mean workload of the instruments (C2) are considered and the pairwise comparison matrices are shown in Table 6 and 7.

For matrix C1-A, the biggest eigenvalue λmax = 10.338, the CI = 0.038 and CR = 0.025<0.10. Therefore, the judgment matrix was acceptable. The vector of weights corresponding to C1 was:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

For matrix C2-A, the biggest eigenvalue λmax = 10.404, the CI = 0.045 and CR = 0.030<0.10. Therefore, the judgment matrix was acceptable. The vector of weights corresponding to C2 was:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

Thus, the vector of weights of the third level was:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

The combining vector of weights was:

Image for - Multicriteria Decision Mechanism CNSGA-AHP for the Automatic Test Task Scheduling Problem

The ranking of all the alternatives were 6, 3, 7, 8, 5, 1, 2, 4, 10 and 9. Therefore, the most satisfactory solution was (32, 20.000). Because the makespan is more important, the ranking of the alternatives is almost corresponding to the ranking of the makespan. Only the solution (65, 12.813) was reverse due to its lowest mean workload.

The above experiments show that CNSGA-AHP is very effective and suitable for the ATSP. It can generate a set of compromise solutions and then choose a most satisfactory solution from them scientifically.

CONCLUSIONS

This study proposes a multicriteria decision mechanism CNSGA-AHP for the automatic test task scheduling problem. In CNSGA-AHP, the multiobjective optimisation algorithm CNSGA is used to generate a set of compromise solutions. It incorporates chaotic sequences based on the logistic map into NSGA-II to avoid becoming trapped in local optima and thus achieve better solutions. On the other hand, the widely applied decision making method AHP is used to choose a most satisfactory schedule from the solutions obtained by CNSGA. CNSGA-AHP is a novel and integrated method combined the search process and the decision making process flexible. It can handle uncertainty, make the consistency check easily to pass and reduce the workload of DMs. Experiment results show that CNSGA-AHP is very concise and suitable for the ATSP.

ACKNOWLEDGMENTS

This research is supported by the Natural Science Foundation of China under Grant No.61101153 and the National 863 Hi-Tech R and D Plan under Grant 2011AA110101.

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