Research Article
QoS-Aware Multiobjective Optimization Algorithm for
Web Services Selection with Deadline and Budget Constraints
Xianzhong Han, Yingchun Yuan, Chen Chen, and Kejian Wang
College of Information Science & Technology, Agriculture University of Hebei, Baoding 071001, China Correspondence should be addressed to Yingchun Yuan; nd hd [email protected]
Received 20 February 2014; Accepted 24 July 2014; Published 19 August 2014 Academic Editor: Hongxing Wei
Copyright © 2014 Xianzhong Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The problem of QoS-aware multiobjective optimization is an important issue for Web services selection in distributed computing environment. In this paper, a novel algorithm called MOASS (multiobjective optimization algorithm for web service selection) is proposed through analyzing the genetic operators such as constraint handling, the initial population generation, fitness assignment, and diversity preservation. Compared with MOEAWP (Yu et al., 2007), simulation results show that the feasible objective region can be filled uniformly with the optimal solutions obtained by MOASS under different test applications. In the case of higher constraints especially, MOASS can obtain more high-quality and evenly distributed nondominated solutions than MOEAWP.
1. Introduction
Service-oriented architecture (SOA) has been a major soft-ware framework for distributed applications such as e-business and enterprise systems. These distributed applica-tions, that is, composition services, can be dynamically and flexibly composed by integrating existing component services developed independently [1]. Usually, service providers can offer some different service versions with similar functional-ity and different QoS (Qualfunctional-ity of Service) levels. QoS for Web services refer to various nonfunctional characteristics such as response time, execution price, availability, and reliability [2]. It is the way to select the best services to compose an efficient and complex service with QoS assurance that matters.
QoS-oriented service selection is essentially a multi-objective optimization problem, and it is usually an NP-hard problem. In the present study, multiple objectives are usually combined into a scalar objective function via a predetermined aggregating function for optimization. Wen et al. [3] presented a QoS objective function with high performance/cost ratio. Zeng et al. [4] transformed the five QoS attributes into a scalar linear weighted sum objective function; the above approach is also found in the litera-tures [5–7]. Although the method is simple to implement
with low time complexity, it can only obtain a particular optimum point on the tradeoff surface. This drawback of such an approach is that the weights are difficult to determine precisely, especially when there is insufficient information or knowledge concerning the optimization problem.
Multiobjective optimization problems involve simulta-neous optimization of several incommensurable and often competing objectives. Often, there is no single optimal solution, but rather a set of alternative solutions. They are known as Pareto-optimal solutions. Multiobjective evolu-tionary algorithms seem to be particularly suited for solving this kind of optimization problems [8]. Based on multiob-jective evolutionary algorithm, several Web service selection algorithms are developed [9–11]. Yu et al. [12] proposed a workflow execution planning approach using multiobjective evolutionary algorithm (MOEAWP), which developed on basis of NSGAII [13]. It implemented simultaneously to optimize the execution time and service price while meeting time and cost constraints. This class of methods can generate a set of nondominated solutions in single run and it is easy for users to select the most suitable solution for the current issue.
Following the guideline of literature [12], we proposed a new Pareto-based multiobjective optimization algorithm
Volume 2014, Article ID 361298, 7 pages http://dx.doi.org/10.1155/2014/361298
for Web service selection (MOASS), which implements the simultaneous optimization of execution time and cost while meeting deadline and budget constraints. Experimental results show that the population can converge and uniformly distribute in the Pareto front.
The rest of this paper is organized as follows.Section 2 defines the optimization problem for Web service selection, the constraints definition, and the problem assumptions.
Section 3 describes the proposed algorithm MOASS for
Web service selection. Section 4 shows the performance evaluation and comparison of two algorithms. The paper is concluded inSection 5.
2. Problem Description
Following the literature [12], a Web service composition application considered in this paper can be described as a workflow application, which is modeled as DAG (directed acyclic graph). Let 𝐺 = {𝑉, 𝐸} denote the DAG graph, where node set𝑉 = {1, 2, . . . , 𝑛}denotes the set of tasks (or activities), and directed arc (edge) set𝐸represents intertask data or control dependencies. Task 1 and𝑛are dummy single-start and single-terminal nodes. Each arc(𝑖, 𝑗) ∈ 𝐸represents the precedence constraint from𝑖to𝑗; that is, task𝑗cannot start before𝑖has finished.
For any activity𝑖in graph𝐺, there are many candidate services which can fulfill𝑖. All candidate services for𝑖 are called the service pool of activity𝑖, denoted as SP(𝑖). Let𝑙(𝑖) denote the service pool length; that is,𝑙(𝑖) = |SP(𝑖)|. In the same pool, different services have different processing times and costs. Generally, the faster a service executes, the more a user should pay [12]. Let 𝑠𝑖𝑗 be the 𝑗th service name for conducting activity𝑖,𝑡𝑖𝑗 the processing time on service𝑠𝑖𝑗, and𝑐𝑖𝑗 the cost for executing𝑖on service𝑠𝑖𝑗. Thus, 3-tuple (𝑠𝑖𝑗, 𝑡𝑖𝑗, 𝑐𝑖𝑗, )denotes the𝑗th service that can fulfill activity𝑖. The service pool of activity 𝑖can be expressed as SP(𝑖) = {(𝑠𝑖𝑗, 𝑡𝑖𝑗, 𝑐𝑖𝑗, ) | 1 ≤ 𝑗 ≤ 𝑙(𝑖)}. The service pool of dummy activities is empty.
For any application𝐺 = (𝑉, 𝐸), one of its solutions can be obtained if each task allocates a service instance using some selection policy. Let𝑆be the solution space of𝐺. For ∀𝑥 ∈ 𝑆, the required execution time is denoted as 𝑇(𝑥), and the required total execution cost is denoted as𝐶(𝑥). Let 𝑓𝑖(1 ≤ 𝑖 ≤ 𝑛)be the completion time of task𝑖; then𝑇(𝑥) = 𝑓𝑛.Let𝑧𝑖𝑘 be 0-1 variable. It means that its value is 1 when node𝑖chooses service level𝑘to perform it and 0, otherwise; then𝐶(𝑥) = ∑𝑖∈𝑉∑1≤𝑘≤𝑙(𝑖)𝑧𝑖𝑘𝑐𝑖𝑘.
Let 𝐷 be the given time constraint (deadline), which is also the latest completion time of task 𝑛. Let 𝐵 be the given cost constraint (budget), which is the upper limit of the required total cost. The objective of this problem is to select an appropriate service level for each task so that the duration and the total cost can be simultaneously minimized while meeting the given deadline and budget constraints.
Assume that the number of service providers is large enough to accommodate the number of tasks that need to be run simultaneously at any given time, so that the waiting time for an available server is negligible. Thus, each service can
provide the expected execution time of any task at any service level. Users can choose properly service levels for these tasks. The constrained multiobjective optimization problem can be formulated as follows: Minimize 𝑦 = 𝑓 (𝑥) = (𝑇 (𝑥) , 𝐶 (𝑥)) , 𝑥 ∈ 𝑆, (1) s.t. 𝑙(𝑖) ∑ 𝑘=1 𝑧𝑖𝑘= 1, 𝑖 ∈ 𝑉, (2) 𝑓𝑖≤ 𝑓𝑗−∑𝑙(𝑖) 𝑘=1 𝑡𝑖𝑘𝑧𝑖𝑘, ∀ (𝑖, 𝑗) ∈ 𝐸, (3) 𝑇 (𝑥) < 𝐷, (4) 𝐶 (𝑥) < 𝐵, (5) 𝑧𝑖𝑘∈ {0, 1} , 𝑖 ∈ 𝑉, 1 ≤ 𝑘 ≤ 𝑙 (𝑖) . (6) The objective function (1) corresponds to minimizing the duration and the total cost. Equation (2) ensures that exactly only one service should be chosen for each task. Constraint (3) meets precedence constraints. Constraint (4) meets the given deadline constraints. Constraint (5) guarantees that the required total cost is less than or equal to budget𝐵. Constraint (6) guarantees that𝑧𝑖𝑘is 0-1 variable.
3. MOASS Algorithm
A new approach, called MOASS (multiobjective optimization algorithm for web service selection) is proposed in this paper. MOASS uses a mixture of established and new techniques in order to find multiple Pareto-optimal solutions in parallel.
The flow of the MOASS algorithm is as follows.
Step (1). Handle constraints based on penalty functions.
Step (2). Encode the problem and determine the population size.
Step (3). Generate an initial population 𝑃 and create the empty external nondominated set𝑃.
Step (4). Copy nondominated members of𝑃to𝑃. Step (5). Operate crossover and mutation on population𝑃. Step (6). Update population𝑃and preserve its diversity. Step (7). Calculate the fitness of each individual in𝑃as well as in𝑃.
Step (8). Select individuals from populations𝑃and𝑃 (mul-tiset union), until the mating pool is filled.
Step (9). If the maximum number of generations is reached, then stop; otherwise go to Step (5).
In MOASS algorithm, every chromosome is encoded by a permutation of integer; each integer represents a selected service instance for executing its corresponding task. We
apply two-point crossover and one-point mutation operators as usual. In Step (8), binary tournament selection with replacement is used. In the next four subsections, all tech-nologies applied in MOASS are described in detail.
3.1. Handling Constraints. The most common way of han-dling constraints is by using penalty techniques like exterior penalty method [14]. Two penalty functions are developed to handle the deadline and budget constraints. Let𝐶𝐵(𝑥)be the budget penalty function and let𝑇𝐷(𝑥)be the deadline penalty function; they are defined as follows:
𝐶𝐵(𝑥) = {𝐶 (𝑥) − 𝐵 if𝐶 (𝑥) > 𝐵
0 otherwise,
𝑇𝐷(𝑥) = {𝑇 (𝑥) − 𝐷0 ifotherwise𝑇 (𝑥) > 𝐷.
(7)
Since satisfying deadline and budget requirements is the primary goal of the Web service selection, the penalty is added separately to the optimization objectives. The new objective function is defined as follows:
Minimize𝑔 (𝑥) = (𝑦1(𝑥) , 𝑦2(𝑥)) , 𝑥 ∈ 𝑆, (8) where𝑦1(𝑥) = 𝑇(𝑥) + 𝑇𝐷(𝑥)and𝑦2(𝑥) = 𝐶(𝑥) + 𝐶𝐵(𝑥). 3.2. Determining the Population Size. In the MO evolu-tion process, populaevolu-tion size has great influence on the exploration efficiency of the solution space. It is difficult to determine an optimal population size. A fixed value of population size is set in literature [12]. For the optimization problem considered in this paper, let𝑁be the number of nodes in a service composition application and let 𝑀 = max{𝑙(𝑖) | 𝑖 ∈ 𝑉} be the maximum length of all service pools; the optimization problem may have𝑁𝑀solutions at most. It is shown that the solution space for each optimization problem is dependent on the composition scale and the length of service pool. Considering specific example of the optimization problem and the effective exploration of the solution space, the population size, denoted as𝐿(𝑃), can be defined as follows:
𝐿 (𝑃) = 𝑀 + 𝑁 . (9)
External population size, denoted as𝐿(𝑃), can be defined by the user. In this paper, it depends on the test application; that is,𝐿(𝑃)is equal to𝑁.
3.3. Fitness Assignment. A fitness function is used to measure the quality of the solutions according to the given optimiza-tion objectives. Considering the optimum and diversity of solutions, a method called Hybrid Nondominated Sorting and Niche Technique (HNSNT) is proposed to calculate the fitness of individuals. Including the evolutionary population and external population, the procedure is a three-stage process and is described as follows.
Step 1(Pareto-based sorting). First, it selects nondominated solutions in population𝑃∪𝑃as the first level nondominated
Dominated individuals Nondominated individuals Fx(n) f2 d(n)1 d(n) d(n)2 Fz(n) F (n) y f1 d(n)min
Figure 1: The tradeoff curve of two-objective optimization.
front, denoted asℎ𝑖= 1. The nondominated solutions in the rest of population are selected as the next level nondominated front, denoted asℎ𝑖 = 2. The procedure is repeated until all individuals in the population are classified.
Step 2(compute dynamic niche radius). Niche techniques aim at preserving diversity in population, and the key problem is to determine the niche radius. Based on literature [15], an approximate calculation method for dynamic niche radius is given. As pointed out in [15], the tradeoffs for an𝑚-objective optimization problem is in the form of an (𝑚−1)-dimensional hypersurface. For the two-objective optimization problem in this paper especially, its one-dimensional surface is actually a curve.
Let𝑑(𝑛)be the diameter of the hypersphere at generation 𝑛; it is the interpolated distance of the tradeoff curve covered by the nondominated individuals as shown in Figure 1. Obviously, computing accurately the distance is a complex procedure. However, the length of curve can be approximated by the average distance between the shortest distance𝑑(𝑛)min and the longest distance𝑑(𝑛)max= 𝑑(𝑛)1 + 𝑑2(𝑛); that is,
𝑑(𝑛)≈ (𝑑 (𝑛) min+ 𝑑max(𝑛) ) 2 , (10) where𝑑(𝑛)min = √(𝑑1(𝑛)) 2 + (𝑑(𝑛) 2 ) 2 .
Let 𝜎share(𝑛) denote the niche radius; its value at each generation population can be dynamically determined by
𝜎(𝑛)share= 𝑑(𝑛)
(𝑃 ∪ 𝑃). (11)
Step 3(compute the similarity of individuals). Let𝐷𝑖be the similarity; the processes are as follows.
(1) For each𝑗 ∈ 𝑃 ∪ 𝑃,
𝑚 = 0; (12)
for each𝑗 ∈ 𝑃 ∪ 𝑃&&𝑗 ̸= 𝑖,
𝑑𝑖𝑗= √(𝑦2(𝑖) − 𝑦2(𝑗))2+ (𝑦1(𝑖) − 𝑦1(𝑗))2, sh(𝑑𝑖𝑗) = { { { { { 1 − ( 𝑑𝑖𝑗 𝜎share(𝑛) ) if𝑑𝑖𝑗< 𝜎 (𝑛) share 0 otherwise, 𝑚+ =sh(𝑑𝑖𝑗) ; (13) (2)𝐷𝑖= 𝑚/|𝑃 ∪ 𝑃|.
Therefore, the fitness function of each individual can be defined as follows:
fitness𝑖= 1
(ℎ𝑖× 𝐷𝑖). (14)
3.4. Preserving the Diversity of Population. The external population 𝑃 stores the current nondominated solutions, which should be distributed evenly along the Pareto front. Meantime, it also participates in selection. So pruning the external nondominated set while maintaining its character-istics might be necessary or even mandatory.
An approach, called NTSC (niche technique of similarity based on crowding), is given. Let 𝑗 be a nondominated individual in 𝑃, and let 𝑖 be an individual in external population𝑃. All the individuals in𝑃that are dominated by𝑗are denoted as a set𝐾and then the NTSC procedure is described as follows.
Step (1). Get any untreated individual𝑗in𝑃. Step (2). Initialize set𝐾empty.
Step (3). If∃𝑖 ∈ 𝑃dominates𝑗, then go to Step (11).
Step (4). Search all individuals in𝑃dominated by𝑖and add them to set𝐾.
Step (5). Remove those individuals of set𝐾from𝑃and add 𝑗to𝑃.
Step (6). If𝑃< 𝐿(𝑃), then go to Step (11).
Step (7). Compute the similarity𝐷𝑘of individual𝑘 ∈ 𝑃via Step (3) inSection 3.3.
Step (8). If there is an infeasible solution in𝑃, then go to Step (10).
Step (9). Delete the feasible individual with the highest value 𝐷𝑘, 𝑘 ∈ 𝑃.
Step (10). Delete the infeasible individual with the highest value𝐷𝑘, 𝑘 ∈ 𝑃.
Step (11). If all individuals in𝑃have been treated, then stop; otherwise, go to Step (1).
In this approach, nondominated infeasible solutions are stored when the number of feasible solutions is less than the upper limit of nondominated set. When updating, the infeasible solutions are preferentially eliminated from exter-nal population.
4. Experimental Results
The proposed algorithm (MOASS) is compared with MOEAWP algorithm [12], which is developed on NSGAII. The strategy of the population size in MOASS can also be adopted by MOEAWP. All these algorithms are coded in Java and performed on a Pentium IV with a 2.93 GHz processor and 512 MB RAM using the operating system Window XP.
4.1. Parameter Setting. Since no standard test instances are available to test the presented algorithms, a DAG graph random generator is developed to generate application DAG with different scales and different structures [16]. The number of nodes varies between 30 and 120 with an increment of 30; that is,𝑉 = {30, 60, 90, 120}. The maximum out-degree of the DAG ranges from 1 to 3; that is,𝑜𝑢𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 = {1, 2, 3}. For each task in a DAG application, the length of its service pool length is 20. For those services in the same service pool, time, and cost quality parameters are randomly generated, but their execution costs are inversely proportional to their processing times.
The behaviors of algorithms are also observed at three constraint levels, namely, relaxed constraint, medium con-straints, and tight constraint. The relaxed constraint level assumes that users require relatively large deadline and bud-get, while tight constraint level assumes that users require low deadline and budget. The relaxed/tight deadlines and budgets of an application are determined by the maximum and minimum time/cost for the service composition execution. The deadline𝐷and budget𝐵are defined as follows:
𝐷 = 𝑇min+ 𝜃 × (𝑇max− 𝑇min) , (15) 𝐵 = 𝐶min+ 𝜃 × (𝐶max− 𝐶min) , (16) where𝑇minis the minimum completion time of a composite instance and 𝑇max is the maximum completion time of a composite instance when all tasks selected are the slowest service for execution. The 𝐶min and 𝐶max have the same meaning of𝑇minand𝑇max. The value of𝜃varied from 0.3 to 0.7 to generate the tight, medium, and relaxed deadlines and budgets.
On all test applications, 2000 generations were simulated per optimization run. The probabilities of crossover (two-point) and mutation were fixed (0.8 and 0.1, resp.). The sizes
Table 1: The parameters setting and the experimental results of performance measures.
𝑁 𝑀 𝑃 𝑃 𝑈
MOA 𝐶(𝑋MOA, 𝑋MOE) 𝐶(𝑋MOE, 𝑋MOA) 𝑈MOA 𝑈MOE 𝐼MOA 𝐼MOE
30 20 50 30 Tight 0.867 0.027 0.771 0.520 17.3 5.4 Middle 0.937 0.019 0.613 0.673 24.5 29.9 Relaxed 0.695 0.134 0.621 0.623 26.9 29.9 60 20 80 60 Tight 1 0 0.601 0.442 25.65 3.25 Middle 1 0 0.582 0.645 47.6 59.95 Relaxed 0.872 0.041 0.602 0.646 53.75 60 90 20 110 90 Tight 0.974 0.008 0.609 0.866 45.35 14.17 Middle 0.992 0.002 0.601 0.637 71.8 89.97 Relaxed 0.697 0.159 0.636 0.650 80.8 89.97 120 20 140 120 Tight 0.950 0 0.601 0.935 53.6 17.6 Middle 1 0 0.591 0.650 98.9 113.2 Relaxed 0.912 0.051 0.612 0.642 110 120 Aver. 0.908 0.037 0.62 0.661 54.68 52.78
of evolutionary population and external population were determined. These parameters are shown inTable 1.
4.2. Performance Measures. In this section, three different quantitative interpretations of statistical MO optimization performances are applied to validate the proposed MOASS, together with its comparison with MOEAWP [12].
(1) Coverage of Two Sets (C) [15]. The coverage of two sets (𝐶) is a measure to compare the domination of two populations in a pairwise manner. Let𝑋, 𝑋⊆ 𝑋be two sets of decision vectors, the function𝐶maps the ordered pair(𝑋, 𝑋)to the interval[0, 1]. Consider
𝐶 (𝑋, 𝑋) = {𝑎
∈ 𝑋; ∃𝑎∈ 𝑋: 𝑎≺ 𝑎 }
𝑋 . (17)
The value 𝐶(𝑋, 𝑋) = 1 means that all solutions in 𝑋 are dominated by or equal to solutions 𝑋. While 𝐶(𝑋, 𝑋) = 0means none of the points in𝑋are covered by the population 𝑋. Since 𝐶(𝑋, 𝑋) is not necessarily equal to𝐶(𝑋, 𝑋), both𝐶(𝑋, 𝑋)and𝐶(𝑋, 𝑋)have to be considered.
(2) Uniform Distribution (U) [17]. An important metric in MO optimization concerns the distribution of individuals. The measure formula is referenced in literature [17]. The met-ric is able to measure the spread of nondominated individuals effectively. The smaller its value, the better uniformity for the nondominated set.
(3) The Number of Optimal Solutions (I). The metric concerns the numbers of nondominated individuals searched at the last generation. If the metric value is bigger, the algorithm should have better performance.
4.3. Experimental Results. Based on the three performance measures, the proposed MOASS has been compared with MOEAWP [12] for all above test applications. In our proposed method, four members of initial population were produced by MPBL [16], where𝜃 ∈ {0, 0.7, 0.8, 1}in (16). The MOEAWP and MOASS were independently executed 40 times for each test application. The arithmetic means of these results using the metrics discussed above are shown inTable 1, where MOA denotes the new approach and MOE denotes the compared method.
For the measure of𝐶, it can be seen from the results in Table 1that MOASS has the higher mean value as compared to MOEAWP in different situations. The average value is up to 90.8%. Also, MOASS can almost dominate MOEAWP when the constraint is higher (middle, tight). The measure value, especially, is equal to 1 in case of𝑁 = 60and higher (middle, tight) constraints. It clearly indicates that MOASS has the capability of providing useful nondominated solutions.
With respect to the number of optimal solutions (𝐼), it can be seen that MOASS can obtain a large number of nondom-inated solutions when the constraint is tight. The MOEAWP algorithm obtained more solutions than the proposal with medium and relaxed constraints. This indicates that MOASS has better performance than MOEAWP in tighter constraint. Although the number of solutions obtained by MOEAWP is more than those of MOASS, the solutions can almost be dominated by MOASS.
In the context of𝑈-measure of the nondominated pop-ulation, MOASS gives the lower value of𝑈than MOEAWP except for two test examples; that is,𝑁 = 30and𝑁 = 60with tight constraint. It is shown that MOASS has more capability to distribute the nondominated individuals evenly. This is because that MOEAWP does not have any operation to take care of the population distribution, whereas our approach applied the niche technologies to both fitness assignment and preservation of population diversity. So it can discover missing tradeoffs at each generation.
8000 10000 12000 14000 16000 190 200 210 220 230 240 250 EC ET MOASS MOEAWP
(a) Tight constraint (𝜃 = 0.3)
0 5000 10000 15000 20000 25000 30000 150 200 250 300 350 400 EC ET MOASS MOEAWP (b) Medium constraint (𝜃 = 0.5) 0 10000 20000 30000 40000 100 200 300 400 500 600 EC ET MOASS MOEAWP (c) Relaxed constraint (𝜃 = 0.7)
Figure 2: The Pareto-front with𝑁 = 60and different constraints.
The running results of the second group and the fourth group of test problems inTable 1are randomly sampled. The population distribution of Pareto optimal solutions in the objective domain isshown in Figures 2 and 3 (where EC
ET 17500 20000 22500 25000 27500 30000 280 310 340 370 400 EC MOASS MOEAWP
(a) Tight constraint (𝜃 = 0.3)
0 10000 20000 30000 40000 50000 200 300 400 500 600 EC ET MOASS MOEAWP (b) Medium constraint (𝜃 = 0.5) 0 20000 40000 60000 80000 200 300 400 500 600 700 800 EC ET MOASS MOEAWP (c) Relaxed constraint (𝜃 = 0.7)
Figure 3: The Pareto-front with𝑁= 120and different constraints.
denotes execution cost𝐶(𝑥)and ET denotes execution time 𝑇(𝑥)). In general, the purpose of producing these figures is to visually inspect the performance of various algorithms in terms of thepopulation distribution, since quantitative
performance alone may provide insufficient information for full observation and understanding. As shown in Figures2 and3, MOASS is more capable of filling up the gap along the discovered Pareto front in all test application, especially those test examples with higher constraints. Although, MOEAWP has a good performance under the relaxed constraint, it is still slightly inferior to the MOASS.
5. Conclusion
A novel multiobjective optimization algorithm (named MOASS) is proposed for solving constrained multiobjective Web service selection problem. In handing constraints, two time and cost penalty functions are added separately to the two objective functions without liner summation. The rithm generates initial population by existing heuristic algo-rithm to achieve broader explorations in boundary regions. The population size depended on the complexity of test applications. The algorithm also applies the niche technology to both fitness assignment and population diversity, which aims to discover missing tradeoffs at each generation.
Furthermore, extensive quantitative comparisons bet-ween MOASS and MOEWP on given test applications have been performed. Experimental results show that MOASS per-formed better than MOEAWP in searching and maintaining the uniform distribution of the nondominated solutions on all the test applications. Under tight and medium constraints especially, MOASS algorithm can discover more optimal solutions distributed uniformly along the tradeoff surface.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by National Natural Science Foun-dation of China under Grant (no. 60873236) and Natural Science Foundation of Hebei Province in China under Grants (no. F2009000653 and no. F201204089). In addition, our research is also supported by the Youth Science Research Foundation of Hebei Province in China under Grant (no. 2010251).
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