TA-ABC: Two-Archive Artificial Bee Colony for Multi-objective Software Module Clustering Problem

3.1 Multi-objective Formulation

In multi-objective SMCP, more than one and less than or equal to three objectives are optimized. It determines a clustering solution x* for which

where M and f_i represent the number of objective functions and i^th objective function, respectively. Q is the number of equality constraints; P is the number of inequality constraints, and xiU and xiLrepresent the upper and lower bounds of the decision variable x_i.

3.2 Module Clustering Objective Functions

The main goal of software module clustering is to improve the quality of clustering by optimizing various conflicting software attributes. Praditwong et al. [43] have proposed two multi-objective formulations (i.e. ECA and MCA) that capture attributes of a well-clustered software system. Moreover, these formulations also help in guiding the optimization process towards good clustering. The objective functions defined under MCA and ECA formulations are as follows: (1) maximization of cohesion (i.e. sum of intracluster edges), (2) minimization of coupling (i.e. sum of intercluster edges), (3) minimization of number of clusters, (4) maximization of MQ, (5) minimization of the number of isolated clusters, and (6) minimization of the differences between maximum size cluster and minimum size cluster.

The MCA formulation includes the objective function numbered with 1, 2, 3, 4, and 5, and ECA formulation includes the objective function numbered with 1, 2, 3, 4, and 6. The computations of all identified objective functions except the MQ objective are straightforward. The computation of MQ is defined as follows:

where i is the number of intracluster edges and j is that of intercluster edges of cluster k for an un-weighted MDG, while for weighted MDG, i represents the total weight of intracluster edges and j represents total weight of intercluster edges of cluster k.

4.1 The Basic Concept of ABC Algorithm

The ABC, a meta-heuristic algorithm based on the behavior of bees, has gained wide attention and has been demonstrated to be effective and well situated for solving the various types of optimization problems in science and engineering fields [11, 23, 25, 31]. The main steps of the basic ABC algorithm are as follows:

• Population initialization phase: The initial population of the basic ABC algorithm is generated by a random process. Let vi={v_i1, v_i2, …, v_in} represent the i^th food source in the population with n number of decision variables. To initialize the population, each food source is generated as follows:

  (3)vi j=vi jmin+(vi jmax−vi jmin)×r,    j=1, . . . ,n;     i=1, . . . , SN,

  where vi jmaxand vi jminrepresent the upper and lower bounds for the decision variable j, respectively, and r is used a uniform random number in [0, 1].

• Employed bee phase: In the employed bee phase, the i^th food source v_ij of the population is assigned to the i^th employed bee, which generates a new neighboring solution around the assigned food source as follows.

  (4)vnew j=vi j+U(−1,1)×(vi j−vk j)

  where i∈{1,…,SN}, and k∈{1,…,SN}∧k≠i is a randomly chosen food source. After generating new solution v_new it is evaluated and compared to v_i then the solution with the higher fitness value is selected.

• Onlooker bee phase: The onlooker bees make a decision on food sources whether to select or not the food source selected by the employed bees. To perform this, the onlooker bees use the probability values, calculated using Eq. (9), to select the food source for discovering promising regions in the search space.

  (5)pi=fiti∑i=nSNfiti

  where fit_i is the fitness value of the i^th food source.

• Scout bee phase: If a food source cannot be further improved through a limited iteration, then the food source is supposed to be abandoned and a randomly produced food source will be replaced with it.

The above basic ABC algorithm was designed to solve the single-objective optimization problems that have the continuous decision variables; hence, the original form of the algorithm cannot be directly used for solving the combinatorial/discrete multi-objective optimization problems. The M-SMCP is a discrete multi-objective optimization problem; therefore, in this work, some alterations to the basic ABC algorithm have been done for making it suitable for the M-SMCPs.

4.2 Proposed TA-ABC Algorithm

This section presents a TA-ABC approach to solve M-SMCPs. The proposed TA-ABC has the following main features:

• The approach can work efficiently for more than three objective functions.

• The approach provides a good balance between exploration and exploitation.

• The proposed approach can be easily implemented.

To impart the above features, the TA-ABC algorithm exploits the concepts of two-archive technique [44] and indicator based ranking [48]. The combination of both these concepts makes TA-ABC algorithm perform efficiently in case of more than three objective functions. On the other hand, use of ABC algorithm concepts makes the approach free from many parameters and perform good exploitation and exploration of the search space. The flow chart of TA-ABC is given in Figure 1.

Figure 1:

Flow Chart of Proposed TA-ABC Algorithm.

The working of TA-ABC method is divided into six parts: food source representation, Population initialization, Send employed bees, Send onlooker bees, Send scout bees, and Update the archive. The detailed explanations of these parts are provided in the subsequent subsections.

4.2.1 Food Source Representation

To solve M-SMCPs with the TA-ABC algorithm, its solution requires to be modeled in a proper way, so as it can be solved efficiently. In the search-based techniques, the solution is encoded as a string of (typically binary) numbers. In our TA-ABC approach, each module clustering solution is encoded as a string of integer numbers instead of binary numbers. In the integer encoding, a single integer perturbation can separate a module clustering solution into two distinct module clustering solutions, while binary representation requires a large number of perturbations. Hence, in the integer encoding, individual module clustering solutions are a smaller distance from one another, which significantly increases the power of exploration and exploitation [10].

Let {m₁, m₂,…m_N} be the set of n number of modules in the software system. Then the solution is represented as a vector of n_m=n integers (m=[m₁, m₂,…m_N]). In this representation, the value 0<m_i≤n of the i^th module indicates the cluster to which the i^th module is assigned. A clustering solution with the same value for all the modules means that all modules are grouped in the same cluster, while a clustering solution containing all possible values (from 1 to n) denotes that each cluster holds only a single module. To demonstrate it, let us consider a hypothetical software system depicted in Figure 2.

Figure 2:

Representation of a Simple Food Source (i.e. Software Clustering Solution).

In Figure 2 the clustering solution (i.e. food source) of software system contains eight modules (i.e. numbered with 1–8) distributed in three clusters, namely, C₁, C₂, and C₃. Hence, it can be represented as a vector C=[1, 1, 2, 2, 2, 3, 3, 3], where modules 1 and 2 are in cluster C₁, modules 3, 4, and 5 are in cluster C₂, and modules 6, 7, and 8 are in cluster C₃.

5.1 Test Problems

In this paper, varieties of MDGs of software systems with different characteristics are used. There are two types of MDGs (weighted and un-weighted) used to evaluate the proposed approach. Table 1 provides a brief description about the number of modules and links of MDGs of considered software systems. In un-weighted MDGs, each connection (link) represents the existence of a unidirectional variable or a method reference between two modules. In weighted MDGs each connection contains weights which are calculated according to the number of the unidirectional variables and method references between modules. Larger connection weights specify more interconnection strength between modules and increase probability that it should be placed in the same cluster.

5.2 Research Questions

In our study, we evaluate the performance of our proposed TA-ABC approach for M-SMCPs by finding out whether it could generate the good modularization in terms of various structural quality metrics (i.e. MQ, coupling, and cohesion) compared to other existing algorithms. In addition to structural quality metrics, we also used the IGD [50], HV [49], spread performance metric [43], Pareto optimality [43], and execution time to compare the algorithms. The major goal of our study is to address the following research questions.

RQ1. MQ value as assessment criterion: How well does the proposed TA-ABC perform when compared against TAA and NSGA-II algorithms using the MQ as the assessment criterion?

RQ2. Coupling as assessment criterion: How well does the proposed TA-ABC perform when compared against TAA and NSGA-II algorithms in terms of coupling?

RQ3. Cohesion as assessment criterion: How well does the proposed TA-ABC perform when compared against TAA and NSGA-II algorithms in terms of cohesion?

RQ4. Pareto optimality as assessment criterion: How well does the TA-ABC algorithm perform at producing good approximations to the Pareto front compared to TAA and NSGA-II algorithms?

RQ5. IGD, hypervolume, and spread as assessment criterion: How well does the proposed TA-ABC perform when compared against TAA and NSGA-II algorithms in terms of IGD, HV, and spread as the assessment criterion?

Note that the IGD metric corresponds to the average Euclidean distance separating each reference solution (true Pareto front) from its closest non-dominated one (Pareto front obtained by the algorithm). For each studied software project, we use the set of Pareto optimal solutions produced by all algorithms over all runs as a true Pareto front.

5.3 Competitor Algorithms and Parameter Setup

This subsection provides a brief description about competitor algorithms with their parameter settings that have been used in this study. The TAA and NSGA-II are two popular algorithms which have been used to solve the M-SMCPs by the previous researchers [4, 5, 43]. In this paper, the results of the TA-ABC are compared with TAA and NSGA-II. The parameter settings of TAA and NSGA-II algorithms are the same as suggested in [5, 43]. Different search-based optimization approaches usually consume different amounts of fitness computations. To make a fair comparison between such meta-heuristic algorithms, an equal number of fitness function computations is allowed to each algorithm. The number of fitness evaluations (N_FE) for the TA-ABC approach is computed using the following method: N_FE≤(SN+SN+1)*MCN+SN, where SN and MCN is the number of onlooker bees and maximum number of iterations, respectively. The parameter for NSGA-II is the same as TAA. The parameter values of the algorithms are assigned according to the number of modules (N) in the problem instances. The crossover and mutation operators are single-point crossover and single-point mutation, respectively. The mutation probability is set as 0.004*log₂ (N). The crossover probability is set as 0.8 for population size less than 100, otherwise 1.0. The maximum number of generations, population size, and total archive size is 200N, 10N, and 10N, respectively. The limit parameter for the TA-ABC is set as (D*PS)/2, where D and PS is the dimension of the problem and population size, respectively.

5.4 Collecting Results from Experiment

The search-based optimization algorithms are stochastic in nature; i.e. they can produce different values on each run. We collect the results of each algorithm on each MDG by executing 30 times, following the same approach as discussed in [5, 43].

6.1 The MQ Value as Assessment Criterion

This section presents the results of the experiments that answer the RQ1. To answer this research question, we compared the TA-ABC with TAA and NSGA-II algorithms over seven un-weighted and 10 weighted MDGs with MCA and ECA multi-objective formulations in terms of MQ values.

Table 2 presents the MQ values obtained by TA-ABC, TAA, and NSGA-II algorithms with MCA formulation.

The 8^th and 9^th columns in the table denote the p-values (p-value below 0.05 is considered statistically significant). The symbols [−] denote that the result is significantly in favor of TA-ABC compared to corresponding approach, symbol [+] denotes opposite, and symbol [≈] is used when there is not a significant favor to any of the approaches. First, if we compare the MQ results of the TA-ABC approach with the TAA approach on un-weighted MDGs, the results show that the TA-ABC approach outperforms TAA approach in five MDGs out of seven MDGs. There are four cases where TA-ABC performs significantly better than TAA approach. Hence, there is good evidence to suggest that for un-weighted MDGs, the TA-ABC approach outperforms the TAA approach. Similarly, for weighted MDGs, the results provide sufficient evidence that the TA-ABC out performs the TAA approach over most of the cases except one case. That is, TA-ABC approach beats TAA approach in nine weighted MDGs, including seven in which the results are statistically significant. Second, if we compare the results of the TA-ABC approach with NSGA-II approach, the results show that the TA-ABC approach outperforms the NSGA-II approach for both weighted and un-weighted MDGs.

Table 3 presents the results obtained by TA-ABC, TAA, and NSGA-II with ECA formulation on weighted and un-weighted datasets. The results provided in Table 3 clearly indicate that the TA-ABC approach outperforms TAA and NSGA-II in most of the cases. The MQ results obtained by the TA-ABC approach and TAA approach over un-weighted MDG show that the TA-ABC approach outperforms the TAA approach in six MDGs out of seven MDGs. There are four cases where the TA-ABC approach performs significantly higher compared to TAA approach. However, for weighted software applications, the results indicate that the TA-ABC approach outperforms the TAA approach over most of the cases except one case. That is, TA-ABC approach performs significantly better than TAA approach in all seven software applications. The comparison results of the TA-ABC with NSGA-II approach shows that the TA-ABC approach outperforms NSGA-II approach in all cases for both weighted and un-weighted software applications.

6.2 Coupling as an Assessment Criterion

The coupling values obtained by TA-ABC, TAA, and NSGA-II approaches on un-weighted and weighted MDGs with MCA formulation are shown in Table 4 and with ECA formulation in Table 5. From Table 4 for un-weighted MDGs it can be observed that TA-ABC approach obtained higher values of coupling than TAA approach in all seven cases, out of which five cases are significantly in favor of TA-ABC. For weighted MDGs, the TA-ABC outperforms TAA in all 10 cases in which six cases are statistically significant. However, TA-ABC performs significantly better than NSGA-II in all cases for un-weighted as well as weighted MDGs.

Table 5 presents the results comparing the performance of the TA-ABC approach and the TAA approach, and TA-ABC approach and NSGA-II approach with ECA formulation in terms of coupling measure. Coupling results shown in Table 5 for un-weighted MDGs indicate that the TA-ABC outperforms TAA in five cases, out of which three cases are statistically significant in favor of TA-ABC. For weighted MDGs, the TA-ABC outperforms TAA in nine cases out of which seven cases are statistically significant in favor of TA-ABC. However, the coupling results of the TA-ABC and NSGA-II show that the TA-ABC approach performs significantly better than NSGA-II in all cases for un-weighted and weighted MDGs.

6.3 Cohesion as an Assessment Criterion

This section compares TA-ABC algorithm with TAA and NSGA-II approach, i.e. how each of the multi-objective approaches performs in terms of cohesion as an assessment criterion using MCA and ECA formulations. Table 6 presents the cohesion results of the TA-ABC approach, TAA, and NSGA-II with MCA formulation. The cohesion results obtained from TA-ABC and TAA approaches over un-weighted MDGs given in Table 6 show that the TA-ABC outperforms TAA in all cases out of which five cases are significantly better. For weighted MDGs, the TA-ABC outperforms TAA in all cases out of which eight cases are significantly better. The cohesion results for TA-ABC and NSGA-II clearly show that the TA-ABC performs NSGA-II algorithm significantly better in all cases of weighted and un-weighted MDGs.

Table 7 presents the results comparing the performance of {TA-ABC approach, TAA approach} and {TA-ABC approach, NSGA-II approach} in terms of cohesion measure using ECA formulation. First, if we compare the cohesion values of TA-ABC approach and TAA approach, the results shown in Table 7 indicate that the TA-ABC approach outperforms TAA in five cases out of the seven un-weighted MDGs, out of which three cases are significantly better. In weighted MDGs, the TA-ABC approach outperforms TAA in nine out of 10 cases, out of which seven cases are significantly better. Second, if we compare the TA-ABC approach and NSGA-II approach, the comparison results indicate that the TA-ABC approach significantly outperforms NSGA-II in all cases for weighted and un-weighted MDGs.

6.4 Pareto Optimality as Assessment Criterion

This section compares the TA-ABC algorithm with TAA and NSGA-II in terms of how well each performs at producing good approximations to the Pareto front. Table 8 presents the dominance relationship for the results obtained from TA-ABC and TAA with both MCA and ECA formulations. This dominance relationship is used to compare any two solutions in multi-objective space. In this table, A denotes the TA-ABC with MCA, B denotes the TA-ABC with ECA, C denotes the TAA with MCA, and D denotes the TAA with ECA. The heading N_XY indicates the number of solutions generated by approach X that are dominated by solutions produced by Y. In comparison, the approach X is better than approach Y if N_XY is small and N_YX is large.

Table 8 shows that the number of solutions produced by {TA-ABC with ECA} outperform {TA-ABC with MCA} in all of the problems studied. The {TA-ABC with MCA} outperforms {TAA with MCA} for un-weighted problems (four out of seven problems), while, in weighted systems, the {TA-ABC with MCA} outperforms the {TAA with MCA} in only one problem. The {TA-ABC with MCA} outperforms {TAA with ECA} for un-weighted problems (three out of seven problems), while in weighted systems, the {TA-ABC with ECA} outperforms the {TAA with MCA} in eight out of 10 problems. The {TA-ABC with ECA} outperforms {TAA with MCA} in all un-weighted and weighted problems. Similarly, the {TA-ABC with ECA} outperforms {TAA with ECA} in all un-weighted and weighted problems. These findings taken together indicate that the TA-ABC is better than TAA.

Table 9 presents the dominance relationship for the results obtained from TA-ABC and NSGA-II with both MCA and ECA formulations. In this table, P denotes the TA-ABC with MCA, Q denotes the TA-ABC with ECA, R denotes the NSGA-II with MCA, and S denotes the NSGA-II with ECA.

Table 9 shows that the number of solutions produced by {TA-ABC with ECA} outperforms {TA-ABC with MCA} in all of the problems studied. The {TA-ABC with MCA} outperforms {NSGA-II with MCA} for un-weighted problems (five out of seven problems), while in weighted systems, the {TA-ABC with MCA} outperforms {NSGA-II with MCA} in only two problems. The {TA-ABC with MCA} outperforms {NSGA-II with ECA} for un-weighted problems (two out of seven problems), while in weighted systems, the {TA-ABC with ECA} outperforms the {NSGA-II with MCA} in seven of 10 problems. The {TA-ABC with ECA} outperforms {NSGA-II with MCA} in all un-weighted and weighted problems. Similarly, the {TA-ABC with ECA} outperforms {NSGA-II with ECA} in all un-weighted and weighted problems. These findings taken together indicate that the TA-ABC is better than NSGA-II.

6.5 IGD, Hypervolume, and Spread as Assessment Criteria

In the previous sections, we compared our TA-ABC algorithm with other existing algorithms (i.e. TAA and NSGA-II) in terms of structural quality metrics (i.e. MQ, coupling, and cohesion) and Pareto optimality. In this section, we compare the TA-ABC algorithm with existing algorithms in terms of IGD, HV, and spread values for both MCA and ECA formulations. The symbol [−] denotes that the result is significantly in favor of TA-ABC compared to corresponding approach, symbol [+] denotes opposite, and symbol [≈] is used when there is not a significant favor to any of the approaches. Table 10 presents IGD values of the results obtained through TA-ABC, TAA, and NSGA-II over weighted and un-weighted software projects. Tables 11 and 12 report the statistics of the HV and spread of the results obtained through TA-ABC, TAA, and NSGA-II algorithms, respectively. For the MCA and ECA formulations, IGD statistics given in Table 10 indicates that TA-ABC outperforms other algorithms in most of the cases for weighted and un-weighted MDGs. Additionally, TAA seems to be better than NSGA-II in most of the cases. The results based on the HV metric show that TA-ABC performs better than A-TAA and NSGA-II in most of the cases. Results in Table 12 indicate that the spread values achieved with the Pareto front generated by the TA-ABC algorithm is lower than those of TAA and NSGA-II, and in most of the cases it has the significantly lower values.