A Comparative Study of The

A Comparative Study of the Bees Algorithm as a Tool for Function Optimisation

D.T. Pham, M. Castellani

School of Mechanical Engineering, University of Birmingham,

Birmingham B15 2TT, UK.

[ [email protected]; [email protected]]

TWO-DIMENSIONAL MINIMISATION BENCHMARKS

ONLINE APPENDIX A.1. General features

All functions are defined within the interval:

I ={xR2;, where i=1,2}.

    For all functions, f(x )=0 where x =(x 1,x 2) is the global minimum point.

A.2. Multi-modal functions

A.2.1. Function 1

This function maps two “holes” on opposite sides of the search space.

A.2.2. Function 2

This function is characterised by four large secondary basins, and a narrow and steep hole near the top-left corner of the search space where the global minimum is located. The secondary minima are arranged on a grid. Figure A.1: function 1. The red arrow indicates the global optimum.

Figure A.2: function 2. The red arrow indicates the global optimum. A.2.3. Function 3

This function is characterised by eleven large secondary basins, and a narrow and steep hole near the top-right corner of the search space where the global minimum is located. The secondary minima are arranged on a grid.

A.2.4. Function 4

This function is characterised by eleven large secondary basins, and a narrow and steep hole at the origin of the search space where the global minimum is located. The secondary minima are arranged on a grid. Figure A.3: function 3. The red arrow indicates the global optimum.

Figure A.4: function 4. The red arrow indicates the global optimum. A.2.5. Function 5

This function is characterised by eleven large secondary basins, and a narrow and steep hole near the top-left corner of the search space where the global minimum is located. The secondary minima are not arranged on a grid.

A.2.6. Function 6

This function is characterised by eleven large secondary basins, and a narrow and steep hole at the origin of the search space where the global minimum is located. The secondary minima are not arranged on a grid. Figure A.5: function 5. The red arrow indicates the global optimum. Figure A.6: function 6. The red arrow indicates the global optimum.

A.3. Minimum surrounded by flat surface

A.3.1. Function 7

This function maps a multi-modal search surface. The global minimum lies in a hole far from the origin, and characterised by a large flat step and a steep and narrow ending. Four secondary minima are arranged on a grid. A.3.2. Function 8

This function maps a multi-modal search surface. The global minimum lies in a hole far from the origin, and characterised by a large flat step and a steep and narrow ending. Four secondary minima are not arranged on a grid. Figure A.7: function 7. The red arrow indicates the global optimum.

Figure A.8: function 8. The red arrow indicates the global optimum. A.4. Narrow valleys

A.4.1. Function 9

This function maps two valleys on opposite sides of the search space.

A.4.2. Function 10

This function maps two pairs of valleys of opposite slopes that represent competing basins of attraction. The minimum is located near the borders of the search surface at the end of the narrowest valley. The four valleys join at the origin, where a further basin is located.

Figure A.9: function 9. The red arrow indicates the global optimum.

Figure A.10: function 10. The red arrow indicates the global optimum.

A.4.3. Function 11

This function maps a narrow parabolic valley surrounded by a large flat surface. The valley is located in the half plane of positive x1 values (x1>0). The other half of the fitness landscape is covered by a sliding plane.

Figure A.11: function 11. The red arrow indicates the global optimum. A.5. Wavelike

A.5.1. Function 12

This function combines two cosinusoidal functions. Each function depends on one of the two input variables, and its amplitude increases linearly with the associated variable. The global minimum is in a narrow hole that is added to one of the “pockets” of the search surface. Function 8 has an overall unimodal characteristic corresponding to a plane slanted toward the positive values of the two input variables. The optimum is far from the origin and does not correspond to the minimum of the unimodal characteristic (i.e. the slanted plane).

A.5.2. Function 13

This function combines two sinusoidal functions. The two functions have constant amplitude and variable period. The global minimum is in a narrow hole that is added to one of the “pockets” of the search surface. Figure A.12: function 12. The red arrow indicates the global optimum.

Figure A.13: function 13. The red arrow indicates the global optimum. A.6. “Noisy” unimodal

A.6.1. Function 14

This function has an overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 10% of that of the unimodal curve. The peak lies far from the origin.

A.6.2. Function 15

This function has an overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 25% of that of the unimodal curve. The peak lies far from the origin. Figure A.14: function 14. The red arrow indicates the global optimum. The magnitude of the noise component can be appreciated from the thickness of the borders of the fitness landscape.

Figure A.15: function 15. The red arrow indicates the global optimum. The magnitude of the noise component can be appreciated from the thickness of the borders of the fitness landscape. A.6.3. Function 16

This function has overall unimodal behaviour with a cosinusoidal noise component. The magnitude of the noise component corresponds to 40% of that of the unimodal curve. The peak lies far from the origin.

Figure A.16: function 16. The red arrow indicates the global optimum. The magnitude of the noise component can be appreciated from the thickness of the borders of the fitness landscape.

A.6.4. Function 17

This function is similar to function 15 but the period of the cosinusoidal noise component is multiplied by a factor 4. A.6.5. Function 18

This function is similar to function 15 but the period of the cosinusoidal noise component is divided by a factor 4.

Figure A.17: function 17. The red arrow indicates the global optimum. The frequency of the noise signal can be appreciated from the roughness of the fitness landscape. Figure A.18: function 18. The red arrow indicates the global optimum. The frequency of the noise signal can be appreciated from the roughness of the fitness landscape.