a bee colony optimization algorithm for code ...

Dr. Arvinder Kaur et al. / International Journal of Engineering Science and Technology (IJEST)

A BEE COLONY OPTIMIZATION ALGORITHM FOR CODE COVERAGE TEST SUITE PRIORITIZATION DR. ARVINDER KAUR Associate Professor University School of Information Technology, Guru Gobind Singh Indraprastha University, Dwarka, Sec 16C Delhi, India

SHIVANGI GOYAL Research Scholar University School of Information Technology, Guru Gobind Singh Indraprastha University, Dwarka, Sec 16C Delhi, India

Abstract: Regression testing is the verification process of the modified software in the maintenance phase. Due to the budget and time constraints regression test suite size and its selection is a critical issue for testers. A Bee Colony Optimization (BCO) algorithm for the regression test suite prioritization has been developed and is presented in this paper. The behavior of two types of bees; scout bees and forager bees have been observed and used for test suite prioritization. The proposed algorithm has been explained using examples based on maximum code coverage. The Average Percentage of Conditions Covered (APCC) metrics have been used to show the effectiveness of proposed algorithm. Keywords: Bee Colony Optimization, Regression Testing. 1. Introduction After development and release, software undergo regress maintenance phase of ten to fifteen years. Modifications in software may be due to change in customer’s requirements or change in technology/platform. This leads to release of numerous versions/editions of the existing software. Also in case of the version/edition’s test only modified and affected parts are to be tested to impart confidence in the modified software which is the process of regression testing. During development phase, time and budget of software permits for its thorough testing but same is not the case for regression testing. So, effective and intelligent prioritization of the actual software’s test suite has to be done to make remarkable savings of time and budget. Several attempts have been made in finding techniques/algorithms for the regression test suite prioritization. The time-constrained regression testing can be reduced to NP-complete problem (0/1 knapsack problem) [Alspaughy et al (2007)] which can be solved using algorithms such as Greedy algorithms [Li et al (2007)], nonevolutionary algorithms [Byson (1995)], [Crawford and Williams (1985)], evolutionary algorithms [Al-Salami (2009)], optimization algorithms [Singh (2010)] etc. The Optimization Algorithms are mainly used where problem is optimized based on some given function or criteria available in sample space. Hence optimization algorithms are quite effective in test suite prioritization of modified software. In this paper the optimization phenomenon used is the Bee Colony Optimization System where nature of bees have been observed, studied and mapped to find prioritization of the modified software’s test suite. Automated software testing has been the area of concern for big software development companies. In automation testing development and execution of test cases can be done unattended. Also, comparison of actual results to expected results and logging status can be determined. It is the automation of the manual testing process. It improves accuracy; increases test coverage, saves time and money and helps developers and testers to

ISSN : 0975-5462

Vol. 3 No. 4 April 2011

2786


develop test cases quickly and efficiently. This algorithm automates the test suite prioritization process as per the bee colony optimization criteria. Organization of the paper is as follow. Related work is presented in section 2, BCO method has been explained in section 3. Section 4 presents the formalized BCO algorithm for regression test suite prioritization, in section 4.2 the explanation of the algorithm is given, 5 presents the prioritization of test cases based on total code coverage with an example demonstrating the working of the BCO algorithm in section 5.1. Section 6 gives the comparison of various prioritization approaches to BCO. And finally in the succeeding sections threats to validity, conclusion and future work are presented. 2. Related Work In this section an overview of the related work of test case prioritization techniques and areas of application of Bee Colony Optimization technique has been presented. The prioritization Problem has been addressed by many researchers and they have proposed various techniques for it. Many techniques used for prioritization are “total fault-detection technique” [Rothermal et al (1999)] in which our aim is to detect all the faults inserted in the given code. For effective searching and selection of useful test cases in the process of prioritization some algorithm applied are: “Greedy algorithms” [Li et al (2007)], where test cases are picked using greedy approach i.e. picking the best one first, “Evolutionary algorithms” [Al-Salami (2009)] where test cases evolves with each other to make a prioritized test suite, “Non-evolutionary algorithms” [Byson (1995)], [Crawford and Williams (1985)] where goal based prioritization is done and “Ant colony optimization algorithms” [Singh et al (2010)] where the concept of natural ants has been used to address the test case prioritization problem. In regression testing version specific criteria is useful in some conditions when organization doesn’t want testing to be overbudget, in such cases “Version specific algorithms” [Singh et al (2008)] are useful for prioritization of only specific kinds/versions of test suits unlike general specific test suite prioritization which are meant for any types of regression test suite prioritization. There is another criterion to solve prioritization problem that is to analyze the relationship between variables being changed and their usage in other/new variables that are done using “Variable analysis algorithms” [Singh et al (2008)]. Bee Colony Optimization is an emerging field for researchers. It has been applied to areas like “Optimization of problems” [Saab et al ] where the optimal solution is searched on the basis of some known function. “Travelling Salesman Problem” [Wong et al (2008)] which is NP-Hard combinatorial problem where an optimal path is searched from source to destination. “Scheduling problems” [Chong et al (2006)] like job shop scheduling problems, task scheduling, project scheduling, process scheduling etc. and for solving other combinatorial problems like “Generation of pair-wise test sets” [McCaffrey (2009)] where automated test case generation is done. The concept of BCO is used for solving “Sudoku Puzzles” [Pacurib et al (2009)], and routing problems where optimized path for better routing is searched using “Routing Algorithm” [Rahmatizadeh et al (2009)]. Moreover, bee colony optimization has also been applied in the area of “Web Search” for effective search engine mechanism [Navrat et al (2009)]. The bee colony optimization has also been used for understanding the concept of software test suite prioritization [Mala and Mohan (2009)]. The bee’s concept has also been used for optimization approaches in the field of engineering [Yang (2005)]. 3. Bee Colony Optimization (natural phenomenon of bee colony) Bee Colony Optimization is the name given to the colony formed by the mutual understanding of the natural bees in the food foraging process. In spite all the creatures on this earth follow one or the other mechanism/process to find food that suite them and these mechanism are found in insects like ants, bees or cockroaches. Honey bee comb build-up and management is a classic example of teamwork, experience, co-ordination and synchronization. The way the honey bees find, build and maintain their comb and hive is remarkable. These are the factors which have given rise to interest of researchers to find solutions to their problems. In a natural honey bee hive there are a variety of bees with specific role(s) to perform. There are three types of bees: 1. Queen Bee (one) She is responsible to lay eggs which are used to build new colonies. 2. Male Drone Bees (many) These are responsible for mating with the queen bee. 3. Worker Bees (thousands) These bees perform all the maintenance and management jobs in the hive. There are two types of worker bees, namely scout bees and forager bees, which perform the scout behavior and forager behavior respectively, which are collectively responsible for the working of the honey bee colonies.

ISSN : 0975-5462


2787


3.1. The Scout Behavior In the scout behavior there are following steps: i. The scout bees start from the hive in search of food source randomly. ii. They keep on this exploration process until they are out of energy/tired and return back to the hive. iii. When they return back to the hive, they share their experience and knowledge with the forager bees by performing the mechanism called “waggle dance”. (Waggling is a form of dance in circular direction in the shape of digit 8. It is repeated again and again by a bee. Its intensity and direction gives the idea of food source quality and food source location respectively to other bees). iv. It is the means by which the bees communicate. It is used to convey the parameters like foods Source Quality, distance of food source from hive, Location of food source w.r.t. sun, to guide the path to the available forager bees. v. These steps of the scout bees constitute the first step of the BCO process called the “Path Construction”. 3.2. The Forager Behavior In the Forager behavior, the forger bees do the following: i. The Forager bees observe and learn the steps done by the scout bees while waggling so as to ease their journey. ii. Then these forager bees go to the food sources as guided by the scout bees to exploit them iii. These forms the second step of the BCO process called the “Path Restructuring”. 4. Proposed BCO Algorithm Artificial bee colony system has been modeled in [Owaied and Saab (2008)]. This paper presents a new optimization algorithm that maps the food foraging behavior of natural bees as a procedure to prioritize regression test suite’s test cases. This algorithm takes the test cases as food sources, the number of conditions detected upon execution time of each test case as the food source quality and number of test cases as the number of scout bees, number of forager bees which makes the initial population. Following is the basic flow of the BCO Algorithm:

Figure 1. Basic flow of BCO Algorithm.

Following is the list of notations that are being used in the proposed algorithm: i. Nbee : Total number of bees (Initial Population) ii. NSB : Number of scout bees iii. NFB : Number of forager bee iv. Ei : Energy at ith iteration v. FSQi : ith Food Source Quality

ISSN : 0975-5462


2788


vi. vii. viii. ix. x. xi.

SBi FBi PSBi PFBi ET CC

: : : : : :

ith Socut Bee ith Forager Bee Path of SBi Path of FBi Execution time of test case Number of Condition covered

In execution of the algorithm following two tables will be formed: 1. PSB Table Path of Scout Bee Table, as seen in Table 2 contains all the paths visited (explored) by the SB. It is used for the path construction phase of SB. This table will act as input for the path restructuring phase which is done by FB. 2. PFB Table Path of Forager Bee Table, as seen in Table 3 contains all the paths selected (exploited) by the FB. An account of the total ET of each path is also kept in this table. This table helps in final path selection. 4.1. BCO Algorithm Following are the proposed algorithm’s main steps: 1) Assumptions i. The food sources have been identified. ii. The distance between hive to food source, food source to food source, food source to hive is unity. iii. Any food source is visited only once. iv. Each SBi will start the random search from ith food source. v. Each FBi will exploit only SBi path. vi. E0 (Initial energy level of of SB and FB) is 100%. vii. Energy reduction rate is set up such that only 50% of the test suite gets executed. viii. FSQi=number of conditions covered/Execution Time=CC/ET (1) ix. Stopping criteria i. Total number of condition covered, ii. Ei = 0; Ei =1- i*energy reduction rate. (2) 2) Initializations i. Nbee = NSB + NFS ii. NSB = NFB = number of test cases in the test suite under test. iii. E0 = 100%. 3) BCO Module START (1)Repeat NSB times(Path construction) { Repeat until stopping criteria is met Each SBi starts exploring from ith food source randomly and forms PSBi (Table 2). } (2) Repeat NFB times (Path Structuring) Each FBi starts exploiting PSBi to form PFBi (Table 3). (3) Final Path Selection Choose path PFBi with minimum ET. STOP 4.2. Explanation of the BCO Algorithm 1. Path Construction This is the first step in the execution of the algorithm. This step is executed by the scout bees, NSB times, in order to explore all the food sources (Test Case) available. The SBi starts its exploration process from the ith food source (Test Case). After this they explore the test cases randomly until any one of the stopping criterion is met. Each path PSBi constructed by the SBi is entered into the PSB Table. 2. Path Restructuring In the next step the forager bees exploit the PSBs. This step uses the PSB table as input. Each FBi starts exploiting the PSBi to form the PFBi. Here each FBi chooses the test case (food source) with best FSQi value. The succeeding test cases are chosen such that which detects conditions those are not detected by the existing test cases. The condition makes sure that only those test cases are executed which are effective in code detection

ISSN : 0975-5462


2789


and total code coverage (stopping criteria). Also the summation of the ET of all the test cases present in a particular PFBi is done to get that path’s total ET. All these entries are maintained in PFB Table. 3. Path Selection This is the final step of the algorithm done by the FBi. From the PFB table select the PFBi with minimum Total ET. 4.3. Analysis of the BCO algorithm The running time of BCO Algorithm is bounded by time required to construct path, plus exploring each path ‘n’ number of times and cost for path structuring. For the population of size ‘n’, the path construction requires O(n) operations . All ‘n’ particles in population will go through ‘n’ iterations for completing the process of path construction in the proposed BCO algorithm. For the path structuring step all the ‘n’ individuals will go for n iterations to structure the path. Hence, taking O(n2) operation complexity. For the final path selection step only minimization process will take place which takes O(n) operations. Therefore, the best running time of proposed algorithm is O(n2) + O(n2)+ O(n), which makes the final running time as O(n2). 5. Prioritization based on Maximum Code Coverage Prioritization based on structural testing allows prioritize/reorder the test cases based on the internal workings of the code. Structural testing is also called path testing since test cases are chosen that cause paths to be taken from different parts (structure) of the program. There are certain levels of coverage in path testing namely statement coverage, branch coverage, condition coverage and path coverage. Each level of coverage indicates the level of testing. As we move towards the higher level, more difficult and complete the testing process becomes. The aim of path testing is to attain the highest level of coverage that is condition coverage. The logical flow of a program can be analyzed using graphical representation known as the flow graph. Flow graph generation is the first step of path testing. Then the decision to decision (DD) path graph is generated from the flow graph. It is used to find independent paths. In DD path graph all the sequential nodes are clubbed into a single node leaving behind the decision nodes. An independent path is any path through the DD path graph that introduces at least one new statement node or condition node. Therefore, we need to execute all the independent paths at least once during the process of path testing. 5.1. Example 1 The example taken for code coverage is the triangle problem which takes the three sides (a positive integer in the range of 0 to 100) of the triangle as input and gives the output as scalene, isosceles, equilateral, not a triangle and invalid inputs according to the input. [Aggarwal and Singh (2001)]. The DD Path graph of the triangle problem is given in Figure 2. The test cases, inputs, expected outputs, independent paths, conditions, nodes and statements covered are shown in the Table 1.

Figure 2. DD Path graph of Triangle problem.

Following are the independent paths of the triangle problem: i. ABFGNPQR ii. ABCDEGHJKMQR iii. ABCDEGHIMQR iv. ABCDEGNOQR

ISSN : 0975-5462


2790


v. vi. vii.

ABCEGNPQR ABCDEGHJLMQR ABFGNOQR

Table 1. Test cases with inputs, outputs, independent path, numbers of conditions covered, statements covered and nodes covered.

TEST CASE

X

Y

Z

EXPECTE D OUTPUT

INDEPENDENT PATH

T1

50

50

0

ABFGNPQR

T2 T3 T4 T5 T6 T7

50 50 50 50 50 50

50 50 50 50 50 50

1 2 50 99 100 101

T8

50

0

50

T9 T10 T11 T12 T13 T14

40 50 50 50 50 50

20 1 2 99 100 101

30 50 50 50 50 50

T15

0

50

50

T16 T17 T18 T19 T20

1 20 80 100 101

50 40 50 50 50

50 30 80 50 50

INVALID INPT ISO ISO EQUI ISO NT A TRI INVALID INPT INVALID INPT SCALENE ISO ISO ISO NT A TRI INVALID INPT INVALID INPT ISO SCALENE EQUI NT A TRI INVALID INPT

CONDI TION( S) COVE RED 3

NODE(S) COVERE D

STATEM ENT(S) COVERE D

8

21

ABCDEGHJKMQR ABCDEGHJKMQR ABCDEGHIMQR ABCDEGHJKMQR ABCDEGNOQR ABCEGNPQR

5 5 4 5 4 4

12 12 11 12 10 9

23 23 22 23 21 20

ABFGNPQR

3

8

21

ABCDEGHJLMQR ABCDEGHJKMQR ABCDEGHJKMQR ABCDEGHJKMQR ABFGNOQR ABCEGNPQR

5 5 5 5 3 4

12 12 12 12 8 9

24 23 23 23 20 21

ABFGNPQR

3

8

21

ABCDEGHJKMQR ABCDEGHJLMQR ABCDEGHIMQR ABCDEGNOQR ABCEGNPQR

5 5 4 4 4

12 12 11 9 9

23 24 22 21 21

Step 1. Scout Bee Performs Path Construction and creates PSB Table, Table 2.

ISSN : 0975-5462


2791


Table 2.Path Constructed by scout bee. (PSB Table).

SBi SB1 SB2 SB3 SB4 SB5 SB6 SB7 SB8 SB9 SB10 SB11 SB12 SB13 SB14 SB15 SB16 SB17 SB18 SB19 SB20

PSBi T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20

T3 T15 T1 T13 T6 T19 T1 T2 T17 T20 T1 T1 T9 T1 T20 T15 T6 T1 T16 T2






T18 T12 T15 T20 T5 T9

T16 T8 13 T10 T2

T12 T17 T10 T18

T16 T18 T7 T13

T19 T2 T13

T10 T17 T20

T7 T18

T13 T9

T19

T13

T17

T16

T1

T3

T16 T12

Step 2. Forger bee performs path restructuring and creates PFB Table, Table 3. Table 3.Path restructured by Forager bee. (PFB Table).

FBi FB1 FB2 FB3 FB4 FB5 FB6 FB7 FB8 FB9 FB10 FB11 FB12 FB13 FB14 FB15 FB16 FB17 FB18 FB19 FB20

PFBi T3 T2 T3 T2 T5 T6 T7 T2 T9 T10 T11 T12 T9 T14 T15 T16 T17 T18 T9 T20


Step 3. Forger bee performs final path selection. Hence, following are the final prioritization 1. T3 T17 T18 2. T2 T9 T4 3. T3 T9 T20 4. T2 T17 T18 5. T2 T9 T4 6. T9 T5 T18 7. T17 T19 T18

ISSN : 0975-5462


T19 T6 T19 T19 T6 T6 T2

T19 T6 T19 T19 T8

T14 T7 T18 T20 T13

T1 T15 T1 T15 T16

T13 T13 T13 T13 T18

T18 T6 T16 T3 T9 T19 T6 T6 T3 T19 T2 T6 T18 T19

T13 T7

T9 T8

T13

T17 T7 T7 T7 T9 T5 T18 T20 T13 T20 T18

T18 T13 T17 T13 T13 T17 T13 T15 T17 T1 T1

T14 T7 T18 T20 T7 T7 T20


T1 T15 T1 T15 T8 T13 T15

T13 T8 T18 T17 T13 T13

T13 T13 T13 T13 T13 T8 T13

2792


8. T9 T16 T19 T18 T20 T1 T13 9. T16 T15 T14 T19 T18 T13 T17 10. T17 T19 T18 T2 T20 T15 T13 Hence, we have achieved maximum code coverage by getting only 7 test cases in the prioritized test suite for the test suite of 20 test cases. (We are getting so many paths because there is no ET in this example. Otherwise the path with minimum ET could have been chosen.) 6. Comparison The example mentioned in section 5.1 is compared with respect to the following ordering: No order, Reverse order, Random order, Optimal order of the test cases. The orderings with respect to these approaches for example in section 5.1 is listed in Table 4. These approaches are compared by calculating Average Percentage Condition Coverage (APCC) [Askarunisa (2009)] in Table 5 for the maximum code covered concept. APCC = [1 – ({TC1 + TC2 + ………..+ TCm}/mn) + (1/2n)]

(3)

Where, T - The test suite under evaluation m - The number of conditions contained in the program under test P. n - The total number of test cases in the test suite and TCi - The position of the first test in T that exposes ith condition. The results obtained for the example explained above have been plotted in Figure. 3. These show that the APCC values for the prioritization achieved using BCO are comparable to that obtained with the optimum ordering. Table 4. Order of test cases for various prioritization approaches for example 1 of maximum code coverage.

No Order

Reverse Order

Random Order

Optimum Order

BCO Order






Table 5. APFD values of various prioritization approaches for example 1 of maximum code coverage.

Technique No Order Random Order Reverse Order Optimal Order BCO Order

ISSN : 0975-5462

APCC % 90 71 89.2 91.7 91.7


2793


Figure 3. APCC chart for example 1 of maximum code coverage.

7. Threats to Validity The artificial BCO algorithm which has been presented here has certain short comings which are as follows: 1. In the natural BCO the number of scout bees is 5-10% of forager bees which are assumed to be equal in the algorithm proposed here. 2. The concept of waggle dance was not required so has not been incorporated in the algorithm which is used as a means of communication in the natural BCO. 3. The energy reduction rate has been chosen such that only 50% of the test suite gets executed which leaves the remaining test suite unexplored. Since it executes upto 50% of the test suite, manual execution of small test suites is possible. As the size of test suite increases the manual execution of the test suite becomes difficult and cumbersome. 8. Conclusion and Future Work An artificial bee colony optimization algorithm for the regression test suite prioritization has been developed and presented by studying the natural food foraging behavior of bees. This algorithm makes effective use of the path construction (exploration) and path structuring (exploitation) phenomenon of scout bees and forager bees for the prioritization of the test suite of the modified code. The proposed BCO algorithm has been explained using example on maximum code coverage. The effectiveness of this algorithm has well been demonstrated using APCC metric. The results of prioritization are comparable to the optimal ordering values. Due to the manual execution of this algorithm, it has been applied to test suites with less number of test cases. So to apply it to larger programs, there is a need to automate this algorithm. Therefore an automation tool for the complete usage of the algorithm is being developed. 9. References [1] [2] [3]

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

Aggarwal, K. K.; Singh, Y. (2001): A book on software engineering. New Age International (P) Ltd.; Publishers, 4835/24, Ansari Road, Daryaganj, New Delhi. AL-Salami, N.M.A. (2009): Evolutionary Algorithm Definition, American J. of Engineering and Applied, Science Publications, pp.789-795. Alspaughy, S.; Walcotty, K. R.; Belanichz, M.; Kapfhammerz, G. M.; Soffa, M. L. (2007): Efficient Time-Aware Prioritization with Knapsack Solvers, Proceedings of the ASE 2007 Workshop on Empirical Assessment of Software Engineering Languages and Technologies. Atlanta, Georgia, pp. 1-6. Askarunisa, A.; Shanmugapriya, L.; Ramaraj, N. (2009): Cost and Coverage Metrics for Measuring the Effectiveness of Test Case Prioritization Techniques, INFOCOMP Journal of Computer Science, pp. 1-10. Byson, N. (1995): A goal programming method for generating priorities vectors, Journal of Operational Research Society, Palgrave Macmillan Ltd.,Houndmills, Basingstoke, Hampshire, RG21 6XS, England, pp. 641-648. Chong, C. S. et. al. (2006): A Bee Colony Optimization Algorithm to Job Shop Scheduling, Journal Winter Simulation Conference, Monterey, CA, pp. 1954-1961. Crawford, G.; Williams, C. (1985): A note on the analysis of subjective judgment matrices, Journal of Mathematical Psychology, The Rand Corporation, USA, pp. 387-405. Li, Z.; Harman, M.; Hierons, R.M. (2007): Search algorithms for regression test case prioritization, IEEE Transactions on Software Engineering, San Francisco, CA, USA, pp. 225-237. Mala, D. J.; Mohan, V. (2009): ABC Tester - Artificial Bee Colony Based Software Test Suite Optimization Approach, International Journal of Software Engineering, Sprinter Global Publication, pp. 15-43. McCaffrey, J. D. (2009): Generation of pair wise test sets using a simulated Bee Colony Algorithm, 10th IEEE International Conference, IEEE Press Piscataway, NJ, USA, pp. 115-119. Navrat, P.; Jelinek, T.; Jastrzembska, L. (2009): Bee hive at work: A problem solving, optimizing mechanism, In Proceedings of Nature & Biologically Inspired Computing, IEEE Conferences, Coimbatore, pp. 122-127. Owaied, H.; Saab, S. (2008): Modeling Artificial Bees Colony System, ICAI, Proceedings of the 2008 International Conference on Artificial Intelligence, Las Vegas, Nevada, USA, pp.443-446. Pacurib, J. A.; Seno, G.M.M.; Yusiong, J.P.T. (2009): Solving Sudoku Puzzles Using Improved Artificial Bee Colony Algorithm, Fourth International Conference on Innovative Computing, Information and Control, Kaohsiung, Taiwan, pp. 885-888. Rahmatizadeh, Sh.; Shah-Hosseini, H.; Torkaman, H. (2009): The ant-bee routing algorithm: A new Agent based Nature-Inspired Routing Algorithm, Journal of Applied Sciences, Addison Wesley Publishers, Harlow. Esses. UK, pp. 983-987. Rothermel, G. Untch, R.H.; Chu, C.; Harrold, M.J. (1999): Test Case Prioritization: An Empirical Study, In Proceedings of the International Conference on Software Maintenance, Oxford, UK, pp. 179-188. Saab, S. M.; El-Omari, N. K. T.; Owaied, H. H. Developing Optimization Algorithm Using Artificial Bee Colony System.

ISSN : 0975-5462


2794


[17] Singh, Y.; Kaur, A.; Suri, B. (2006): A New Technique for Version – Specific Test Case Selection and Prioritization for Regression Testing, Journal of the CSI, pp 23-32. [18] Singh, Y.; Kaur, A.; Suri, B. (2008): Regression Test Selection And Prioritization Using Variables: Analysis And Experimentation, New Age International Publishers, New Delhi, pp. 1-15. [19] Singh, Y.; Kaur, A.; Suri, B. (2010): Test Case Prioritization Using Ant Colony optimization, Association in Computing Machinery ,Newsletter ACM SIGSOFT Software Engineering Notes, New York, NY, USA, pp 1-7. [20] Wong, L.P.; low, M. Y. H.; Chong, C. S. (2008): A Bee Colony Optimization Algorithm for Travelling Salesman Problem, Second Asia International Conference on Modeling & Simulation, IEEE Computer Society, Washington, DC, USA, pp. 818-823. [21] Yang, X.S. (2005): Engineering Optimizations via Nature-Inspired Virtual Bee Algorithms, Artificial Intelligence and Knowledge Engineering Applications: A Bio Inspired Approach, Lecture notes in computer science, Springer Berlin / Heidelberg, pp. 317-323.

ISSN : 0975-5462


2795

a bee colony optimization algorithm for code ...

a bee colony optimization algorithm for code ...

Suggest Documents