I need the algorithm using ant colonization algorithm

I need the algorithm using ant colonization algorithm
Optimization Functions For this project, 40 test challenging benchmark functions ( Liang, Qu et al. 2013 , Liang, Qu et al. 2013 , Wahab, Nefti-Meziani et al. 2015 ) are collected. You are required to implement nature-inspired algorithm to find the optimal value of any 20 optimization functions of your choice from the list given below for your class project. There are several characteristics of the test functions that feature the complexity of the functions’ landscape like modality, separability, scalability ( Dieterich and Hartke 2012 ) . Modality is defined by the number of ambiguous peaks in the function landscape. A function is called separable if the variables of the solution are independent and can be optimized separately. On the other hand, the inseparable class of functions are highly difficult to solve as the variables are interrelated and affect each other. The functions that are extendable to arbitrary dimensionality are known as scalable function, otherwise are called non-scalable. The scalability often introduces high extent of complexity to the search space such as some test function landscapes become multimodal from unimodal when the number of dimensions is scaled up. There are some additional features that can make a function landscape even more challenging while searching for minima such as flat surface (less informative area), basins and narrow curved valley. However, some algorithms are found to exploit several shortcomings ( Liang, Suganthan et al. 2005 ) of popularly used test functions such as the global optimum with same values for different variables ( Zhong, Liu et al. 2004 ) , global optimum lying in the center of the landscape, at the origin ( Zhong, Liu et al. 2004 ) , along the axes ( Leung and Wang 2001 , Van den Bergh and Engelbrecht 2004 ) , on the bounds ( Coello, Pulido et al. 2004 ) etc. Thereafter, the conventional test functions are rotated and/or shifted before using them to test optimization algorithms ( Liang, Suganthan et al. 2005 , Qin, Huang et al. 2009 , Liang, Qu et al. 2013 , Liang, Qu et al. 2013 , Yu and Li 2015 ) . In addition to using rotated and shifted functions, we hybridized functions according to ( Liang, Qu et al. 2013 ) to generate highly complex real-world optimization problems to test the extended-capacity of the algorithms. We collected the rotation matrix, shifting vector and shuffled-indices vector for rotation, shifting and hybridization, respectively from a shared link of data used in CEC 2014 1 . Among 40 functions, 27 are classical functions whereas 13 functions are rotated and/or shifted and hybridized to construct modified functions. Depending on the three basic properties (modality, separability, and scalability) mentioned above, we classify the 27 functions under consideration into six groups (Group I through VI). The rest of the 13 modified functions are categorized into three groups according to their complexities. The definition of the functions under different groups with their corresponding search ranges ([lb ,ub ] ) of solution variable ( X ), global optimum ( X¿ ) and the function value at global optimum, f¿¿ ) are listed as following: Group I: Unimodal, Separable and Scalable functions 1 http://web.mysites.ntu.edu.sg/epnsugan/PublicSite/Shared Documents/Forms/AllItems.aspx 1. Sphere function, f Sphere( X ) = ∑ i = 1d x i2 [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f Sphere ( X ¿ ) = 0 2. Cigar or Bent Cigar function, fCigar (X)= x12+10 6∑i=2 d xi2 [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f Cigar ( X ¿ ) = 0 3. Discus function, f Discus ( X ) = 10 6 x 12 + ∑ i = 2d x i2 [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f Discus ( X ¿ ) = 0 4. Rotated Hyper-ellipsoid (RHE) function, f RHE ( X ) = ∑ i = 1d ∑ j = 1i x j2 [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f RHE ( X ¿ ) = 0 Group II: Unimodal, Inseparable and Non-scalable 5. Zettl function, f Zettl ( X ) = 1 4 x 1 + ( x 12 − 2 x 1 + x 22 ) 2 [lb ,ub ]= [−5,5]d , X¿=[−0.0299 ,0]T , fZettl (X¿)=−0.003791237 6. Leon function, fLeon (X)=100 (x2− x12)2+(1− x1)2 [lb ,ub ]= [−1.2 ,1.2 ]d , X ¿ = [ 1 , 1 ] T , f Leon ( X ¿ ) = 0 7. Easom function, fEasom (X)=−cos (x1)cos (x2)e[−(x1−π)2−(x2−π)2] [lb ,ub ]= [−100 ,100 ]d , X¿=[π,π]T , f Easom ( X ¿ ) = − 1 Group III: Unimodal, Inseparable and Scalable 8. Zakharov function, fZakharov (X)=∑i=1 d xi2+( 1 2∑i=1 d ixi) 2 +( 1 2∑i=1 d ixi) 4 [lb ,ub ]= [−5,10 ]d , X ¿ = [ 0 , … , 0 ] T , f Zakharov ( X ¿ ) = 0 9. Schwefel 1.2 function, f Schwefel 1.2 ( X ) = ∑ i = 1d ( ∑ j = 1i x j ) 2 [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f Schwefel 1.2 ( X ¿ ) = 0 10. Schwefel 2.2 function, f Schwefel 2.2 ( X ) = ∑ i = 1d | x i | + ∏ i = 1d | x i | [lb ,ub ]= [−100 ,100 ]d, X¿=[0,… ,0]T , fSchwefel 2.2 (X¿)=0 Group IV: Multimodal, Separable and Scalable 11. Rastrigin function fRastrigin (X)=10 d+∑i=1 d [xi2− 10 cos (2πxi)] [lb ,ub ]= [−5.2 ,5.2 ]d , X¿=[0,… ,0]T , f Rastrigin ( X ¿ ) = 0 12. Schwefel 2.26 function fSchwefel 2.6 (X)=418.9829 d−∑i=1 d xisin (√|xi|) [lb ,ub ]= [−500 ,500 ]d , X ¿ = [ 420.9687 , … , 420.9687 ] T , f Schwefel 2.6 ( X ¿ ) = 0 13. Michalewicz function fMichalewicz (X)=−∑i=1 d sin (xi)sin 2s ( ixi2 π ) steepness ,s=10 [lb ,ub ]=[0,π]d , X ¿ = [ 2.20 , 1.57 ] T , d = 2 f Michalewicz ( X ¿ ) = − 1.8013 , − 4.687658 ,−9.66015 , d= 2,5,10 14. Styblinski-Tang (ST) function f ST ( X ) = 1 2 ∑ i = 1d ( x i4 − 16 x i2 + 5 x i ) [lb ,ub ]=[−5,5]d , X ¿ = [ − 2.903534 , … , − 2.903534 ] T , f ST ( X ¿ ) = − 39.16599 d Group V: Multimodal, Inseparable and non-Scalable 15. Schaffer F2 function fScafferF 2(X)=0.5 + sin 2(x12− x22)−0.5 [1+0.001 (x12+x22)] 2 [ lb , ub ] = [ − 100 , 100 ] d , X¿=[0,0]T , f ScafferF 2 ( X ¿ ) = 0 16. Schaffer F6 function fScafferF 6(X)=0.5 +sin 2(√x12+x22)−0.5 [1+0.001 (x12+x22)] 2 [ lb , ub ] = [ − 100 , 100 ] d , X¿=[0,0]T , f ScafferF 6 ( X ¿ ) = 0 17. Bird function fBird (x)=(x1− x2)2+sin (x1)e[1−cos (x2)]2 +cos (x2)e[1−sin(x1)]2 [lb , ub ] = [ − 2 π , 2 π ] d , X¿=[4.701056 ,3.152946 ]T , X ¿ = [ − 1.582142 , − 3.130247 ] T , f Bird ( x ¿ ) = − 106.7645367198034 18. Levy 13 function f Levy 13 ( X ) = sin 2 ( 3 π x 1 ) + ( x 1 − 1 ) 2[ 1 + sin 2 ( 3 π x 2 )] +(x2−1)2[1+sin 2(2πx2)] [lb ,ub ]=[−10 ,10 ]d , X ¿ = [ 1 , 1 ] T , fLevy 13(X¿)=0 19. Carrom Table (CT) function fCarromTable (X)= −1 30 e 2|1−√x12+x22 π |cos 2(x1)cos 2(x2) [lb ,ub ]=[−10 ,10 ]d , X¿=[±9.646157 ,±9.646157 ]T f CarromTable ( X ¿ ) = − 24.1568155 Group VI: Multimodal, Inseparable and Scalable 20. Ackley function fAckely (X)=− ae(−b√1d∑i=1 dxi2)− e(1d∑i=1 dcos (cxi))+a+e1 a=20 ,b= 0.2 ,c= 2π [lb ,ub ]=[−32 ,32 ]d , X¿=[0,… ,0]T , f Ackley ( X ¿ ) = 0 21. Rosenbrock function f Rosenbrock ( X ) = ∑ i = 1d − 1 [ 100 ( x i + 1 − x i2 ) 2 + ( x i − 1 ) 2] [ lb , ub ] = [ − 30 , 30 ] d , X¿=[1,… ,1]T , f Rosenbrock ( X ¿ ) = 0 22. Griewank function f Griewank ( X ) = ∑ i = 1d x i2 4000 − ∏ i = 1d cos ( x i√ i ) + 1 [lb ,ub ]= [− 600 ,600 ]d , X ¿ = [ 0 , … , 0 ] T , f Griewank ( X ¿ ) = 0 23. Sine Envelope Sine Wave (SESW) function f SESW ( X ) = ∑ i = 1d − 1 ( sin 2 (√ x 12 + x 22 ) − 0.5 [ 1 + 0.001 ( x 12 + x 22 )] 2 + 0.5 ) [ lb , ub ] = [ − 100 , 100 ] d , X¿=[0,… ,0]T , fSESW (X¿)= 0 24. Trigonometric function f Trigonometric( X ) = ∑ i = 1d [ m + i ( 1 − cos x i ) − sin x i − ∑ j = 1d cos x j ] 2 [lb ,ub ]=[−1000 ,1000 ]d , X¿=[0,… ,0]T , f Trigonometric ( X ¿ ) = 0 25. Levy function fLevy (X)=sin 2(πy1)+∑i=1 d−1 [(yi−1)2(1+10 (sin 2yi+1))]+¿¿ ( y n − 1 ) 2( 1 + sin 2 ( 2 π y n )) , y i = 1 + 1 4 ( x i + 1 ) [lb ,ub ]=[−50 ,50 ]d , X ¿ = [ − 1 , … , − 1 ] T , f Levy ( X ¿ ) = 0 26. Schaffer F7 function f ScafferF 7 ( X ) = [ 1( d − 1 ) ∑ i = 1d − 1 (√ y i + sin ( 50 y i 0.2 )√ y i )] 2 , y i = √ x i2 + x i + 12 [lb ,ub ]=[−100 ,100 ]d , X ¿ = [ 0 , … , 0 ] T , f ScafferF 7 ( X ¿ ) = 0 27. Lunacek Function fLunacek (x)=[min ({∑i=1 d (xi− μ1)2 },{1.d+s∑i=1 d (xi− μ2)2 })]+¿ 10 ∑ i = 1d 1 − cos 2 π ( x i − μ 1 ) μ1=2.5 ,s= 1− 1 2√d+20 −8.2 ,μ2=− √ μ12−1 s [lb ,ub ]=[−10 ,10 ]d , X ¿ = [ μ 1 , … , μ 1 ] T , f Lunacek ( X ¿ ) = 0 Group VII: Rotated functions Rotation of the landscape of the functions using an orthogonal rotation matrix ( R ) ( Salomon 1996 ) can prevent the optimum from lying along the coordinate axes. Here, we rotate 7 scalable (2 unimodal and 5 multimodal) functions. 28. Rotated Cigar function, fR−Cigar (Z)=fCigar (Z) , Z= R×X [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , fR−Cigar (X¿)=0 29. Rotated Discus function, fR−Discus (Z)=fDiscus (Z) , Z = R × X [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , f R − Discus ( X ¿ ) = 0 30. Rotated Rastrigin function f R − Rastrigin ( Z ) = f Rastrigin ( Z ) , Z= R׿ [lb ,ub ]= [−5.2 ,5.2 ]d, X¿=[0,… ,0]T , fR−Rastrigin (X¿)= 0 31. Rotated Ackley function fR−Ackley (Z)=fAckley (Z) , Z= R׿ [lb ,ub ]= [−32 ,32 ]d , X ¿ = [ 0 , … , 0 ] T , f R − Ackley ( X ¿ ) = 0 32. Rotated Griewank function fR−Griewank (Z)= fGriewank (Z) , Z= R׿ [lb ,ub ]= [− 600 ,600 ]d , X ¿ = [ 0 , … , 0 ] T , fR−Griewank (X¿)=0 33. Rotated Schaffer F7 function fR−SchafferF 7(Z)= fSchafferF 7(Z) , Z = R × X [lb ,ub ]= [−100 ,100 ]d , X¿=[0,… ,0]T , fR−SchafferF 7(X¿)=0 34. Rotated Lunacek function fR−Lunacek (Z)= fLunacek (Z) , Z = R × ¿ [lb ,ub ]= [−10 ,10 ]d , X¿=[μ1,… ,μ1]T , f R − Lunacek ( X ¿ ) = 0 Group VIII: Shifted and Rotated functions Shifting operation moves the global optimum to a new random position ( onew ) from the old optimum ( o old ) using Eq. 1 to avoid the limitation of having same values of the variables at optima or having optimum at the center. f(Z),Z= R×(X− onew +oold) (1 ) 35. Shifted and Rotated Rastrigin function fSR−Rastrigin (Z)=fRastrigin (Z) , Z= R׿ [lb ,ub ]= [−5.2 ,5.2 ]d , X¿=[0,… ,0]T , f SR − Rastrigin ( X ¿ ) = 0 36. Shifted and Rotated Ackley function f SR − Ackley ( Z ) = f Ackley ( Z ) , Z= R׿ [lb ,ub ]= [−32 ,32 ]d , X ¿ = [ 0 , … , 0 ] T , f SR − Ackley ( X ¿ ) = 0 37. Shifted and Rotated Lunacek function fSR−Lunacek (Z)= fLunacek (Z) , Z= R׿ [ lb , ub ] = [ − 10 , 10 ] d , X ¿ = [ μ 1 , … , μ 1 ] T , fSR−Lunacek (X¿)= 0 Group IX: Hybrid functions Hybrid function defined by Eq. 2 represents real-world optimization problems where different subcomponents of the variables can have different characteristics. For these functions, the variables are randomly shuffled and divided into subcomponents and then different basic functions are used for different subcomponents. In our hybrid functions, we have further rotated the basic functions. f( Z ) = f 1 ( R 1 Z 1 ) + f 2 ( R 2 Z 2 ) + … + f n ( R n Z n ) (2) Here, Z=[Z1,Z2,… ,Zn] Z1=[xS1,… ,xSd1] , Z2=[xSd1+1,… ,xSd1+d2] ,… Zn=[xS∑i=1n−1di+1,… ,xSd] n is number of basic functions S= random permutation (1:d) fi is the i th basic function di is the dimension of the i th basic function, ∑ i = 1n d i = d p i is the percentage of the variables for i th basic function d1= ⌈p1×d⌉ , d 2 = ⌈ p 2 × d ⌉ , …, d n = d − ∑ i = 1n − 1 d i 38. Hybrid function – 1 fHybrid −1(Z)= fRastrigin (R1Z1)+fCigar (R2Z1)+fGriewank (RnZ1) [ p 1 , p 2 , p 3 ] = [ 1 3 , 1 3 , 1 3 ] for d = 30 [ p 1 , p 2 , p 3 ] = [ 2 5 , 1 5 , 2 5 ] for d = 50 [lb ,ub ]= [−100 ,100 ]d , f Hybrid − 1 ( X ¿ ) = 0 39. Hybrid function – 2 fHybrid −2(Z)= fSphere (R1Z1)+fAckley (R2Z1)+fSchafferF 7(RnZ1) [ p 1 , p 2 , p 3 ] = [ 1 3 , 1 3 , 1 3 ] for d = 30 [ p 1 , p 2 , p 3 ] = [ 2 5 , 1 5 , 2 5 ] for d = 50 [lb ,ub ]= [−100 ,100 ]d , f Hybrid − 2 ( X ¿ ) = 0 40. Hybrid function – 3 fHybrid −3(Z)= fDiscus (R1Z1)+fRosenbrock (R2Z1)+fGriewank (RnZ1) [ p 1 , p 2 , p 3 ] = [ 1 3 , 1 3 , 1 3 ] for d = 30 [ p 1 , p 2 , p 3 ] = [ 2 5 , 1 5 , 2 5 ] for d = 50 [lb ,ub ]= [−100 ,100 ]d, f Hybrid − 3 ( X ¿ ) = 0 References: Liang, J., et al. (2013). “Problem definitions and evaluation criteria for the CEC 2014 special session and competition on single objective real-parameter numerical optimization.” Computational Intelligence Laboratory, Zhengzhou University, Zhengzhou China and Technical Report, Nanyang Technological University, Singapore . Liang, J., et al. 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