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As mentioned in Section 4, constructing a CMOP includes constructing both objective functions and constraint functions. According to Eq. (7), we suggest nine multiobjective functions, including convex, concave, and discrete PF shapes, to construct CMOPs. A set of difficulty adjustable constraint functions is generated by Eq. (7). Nine difficulty adjustable and scalable CMOPs, called DAS-CMOP1-9, are generated by combining the suggested objective functions and the generated constraint functions. The detailed definitions of DAS-CMOP1-9 are given in Table 2.

Table 2:
DAS-CMOPs Test suite: The objective functions and constraint functions of DAS-CMOP1-9.
ProblemObjectivesConstraints
DAS-CMOP1 $minf1(x)=x1+g(x)minf2(x)=1-x12+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=(e-g(x))(g(x)-d)≥0ck+2(x)=((f1-pk)cosθk-(f2-qk)sinθk)2/ak2+((f1-pk)sinθk+(f2-qk)cosθk)2/bk2≥rpk=[0,1,0,1,2,0,1,2,3],ak2=0.3,bk2=1.2,θk=-0.25πqk=[1.5,0.5,2.5,1.5,0.5,3.5,2.5,1.5,0.5]a=20,d=0.5,η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP2 $minf1(x)=x1+g(x)minf2(x)=1-x1+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP1
DAS-CMOP3 $minf1(x)=x1+g(x)minf2(x)=1-x1+0.5*|sin(5πx1)|+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP1
DAS-CMOP4 $minf1(x)=x1+g(x)minf2(x)=1-x12+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=(e-g(x))(g(x)-d)≥0ck+2(x)=((f1-pk)cosθk-(f2-qk)sinθk)2/ak2+((f1-pk)sinθk+(f2-qk)cosθk)2/bk2≥rpk=[0,1,0,1,2,0,1,2,3],ak2=0.3,bk2=1.2,θk=-0.25πqk=[1.5,0.5,2.5,1.5,0.5,3.5,2.5,1.5,0.5]a=20,d=0.5,η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP5 $minf1(x)=x1+g(x)minf2(x)=1-x1+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP4
DAS-CMOP6 $minf1(x)=x1+g(x)minf2(x)=1-x1+0.5*|sin(5πx1)|+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP4
DAS-CMOP7 $minf1(x)=x1*x2+g(x)minf2(x)=x2*(1-x1)+g(x)minf3(x)=1-x2+g(x)whereg(x)=(n-2)+∑j=3n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=cos(aπx2)-b≥0c3(x)=(e-g(x))(g(x)-d)≥0ck+3(x)=∑j=1,j≠k3fj2+(fk-1)2-r2≥0c7(x)=∑j=13(fj-13)2-r2≥0a=20,d=0.5,k=1,2,3η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP8 $minf1(x)=cos(0.5πx1)*cos(0.5πx2)+g(x)minf2(x)=cos(0.5πx1)*sin(0.5πx2)+g(x)minf3(x)=sin(0.5πx1)+g(x)whereg(x)=(n-2)+∑j=3n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP7
DAS-CMOP9 $minf1(x)=cos(0.5πx1)*cos(0.5πx2)+g(x)minf2(x)=cos(0.5πx1)*sin(0.5πx2)+g(x)minf3(x)=sin(0.5πx1)+g(x)whereg(x)=∑j=3n(xj-cos(0.25jnπ(x1+x2)))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP7
ProblemObjectivesConstraints
DAS-CMOP1 $minf1(x)=x1+g(x)minf2(x)=1-x12+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=(e-g(x))(g(x)-d)≥0ck+2(x)=((f1-pk)cosθk-(f2-qk)sinθk)2/ak2+((f1-pk)sinθk+(f2-qk)cosθk)2/bk2≥rpk=[0,1,0,1,2,0,1,2,3],ak2=0.3,bk2=1.2,θk=-0.25πqk=[1.5,0.5,2.5,1.5,0.5,3.5,2.5,1.5,0.5]a=20,d=0.5,η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP2 $minf1(x)=x1+g(x)minf2(x)=1-x1+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP1
DAS-CMOP3 $minf1(x)=x1+g(x)minf2(x)=1-x1+0.5*|sin(5πx1)|+g(x)whereg(x)=∑j=1n(xj-sin(0.5πx1))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP1
DAS-CMOP4 $minf1(x)=x1+g(x)minf2(x)=1-x12+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=(e-g(x))(g(x)-d)≥0ck+2(x)=((f1-pk)cosθk-(f2-qk)sinθk)2/ak2+((f1-pk)sinθk+(f2-qk)cosθk)2/bk2≥rpk=[0,1,0,1,2,0,1,2,3],ak2=0.3,bk2=1.2,θk=-0.25πqk=[1.5,0.5,2.5,1.5,0.5,3.5,2.5,1.5,0.5]a=20,d=0.5,η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP5 $minf1(x)=x1+g(x)minf2(x)=1-x1+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP4
DAS-CMOP6 $minf1(x)=x1+g(x)minf2(x)=1-x1+0.5*|sin(5πx1)|+g(x)whereg(x)=(n-1)+∑j=2n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP4
DAS-CMOP7 $minf1(x)=x1*x2+g(x)minf2(x)=x2*(1-x1)+g(x)minf3(x)=1-x2+g(x)whereg(x)=(n-2)+∑j=3n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ $c1(x)=sin(aπx1)-b≥0c2(x)=cos(aπx2)-b≥0c3(x)=(e-g(x))(g(x)-d)≥0ck+3(x)=∑j=1,j≠k3fj2+(fk-1)2-r2≥0c7(x)=∑j=13(fj-13)2-r2≥0a=20,d=0.5,k=1,2,3η=(b+1)/2,ζ=exp(d-e),γ=2r$
DAS-CMOP8 $minf1(x)=cos(0.5πx1)*cos(0.5πx2)+g(x)minf2(x)=cos(0.5πx1)*sin(0.5πx2)+g(x)minf3(x)=sin(0.5πx1)+g(x)whereg(x)=(n-2)+∑j=3n(xj-0.5)2-cos(20π(xj-0.5))n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP7
DAS-CMOP9 $minf1(x)=cos(0.5πx1)*cos(0.5πx2)+g(x)minf2(x)=cos(0.5πx1)*sin(0.5πx2)+g(x)minf3(x)=sin(0.5πx1)+g(x)whereg(x)=∑j=3n(xj-cos(0.25jnπ(x1+x2)))2n=30,x∈[0,1]n$ They are the same as those of DAS-CMOP7

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