A novel immune dominance selection multi-objective optimization algorithm for solving multi-objective optimization problems

Abstract

In this paper, we propose a novel immune dominance selection multi-objective optimization algorithm (IDSMOA) to solve multi-objective numerical and engineering optimization problems in the real world. IDSMOA was inspired by the mechanism that controls how B cells and T cells differentiate, recombine, and mutate self-adjustably to produce new lymphocytes matching antigens with high affinity, then how lymphocytes cooperatively eliminate antigens. The main idea of IDSMOA is to promote 2 populations, population B and population T, to coevolve through an immune selection operator, immune clone operator, immune gen operator, and memory selection operator to produce satisfying Pareto front. Therefore, several operators enable IDSMOA to exploit and excavate the search space, and decrease the number of dominance resistant solutions (DRSs). We compared IDSMOA performance with 3 known multi-objective optimization algorithms and IDSMOA without the combination operator in simulation experiments optimizing 12 benchmark functions. Our simulations indicated that IDSMOA is a competitive optimization tool for multi-objective optimization problems.

Appendices

Appendix A: Benchmark function

• (a) DEB

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$
  $$f_{2} \left( {\vec{{x}}} \right)=\left( {1+10x_{2}} \right)\left[ {1-\left( {\frac{x_{1}} {1+10x_{2}} } \right)^{2}-\frac{x_{1}} {1+10x_{2}} \sin \left( {8\pi x_{1}} \right)} \right] $$

  variable number is 2, x _i ∈[0,1](i=1,2)

• (b) KUR

  $$f_{1} \left( {\vec{{x}}} \right)=\sum\limits_{i=1}^{m-1} {\left( {-10\exp \left( {-0.2} \right)\sqrt {{x_{i}^{2}} +x_{i+1}^{2}} } \right)} $$
  $$f_{2} \left( {\vec{{x}}} \right)=\sum\limits_{i=1}^{m} {\left( {\left| {x_{i}} \right|^{0.8}+5\sin {x_{i}^{3}}} \right)} $$

  variable number is 3, x _i ∈[−5,5](i=1,2,3)

• (c) ZDT1

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$

  $f_{2} \left ({\vec {{x}}} \right )=g\left ({\vec {{x}}} \right )\left [ {1-\sqrt {x_{1} /g\left ({\vec {{x}}} \right )}} \right ]$, where $g\left ({\vec {{x}}} \right )=1+9\sum \limits _{i=2}^{m} {x_{i} /\left ({m-1} \right )}$ variable number is m=30, x _i ∈[0,1](i=1,2,⋯,m)

• (d) ZDT2

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$

  $f_{2} \left ({\vec {{x}}} \right )=g\left ({\vec {{x}}} \right )\left [ {1-\left ({x_{1} /g\left ({\vec {{x}}} \right )} \right )^{2}} \right ]$, where $g\left ({\vec {{x}}} \right )=1+9\sum \limits _{i=2}^{m} {x_{i} /\left ({m-1} \right )}$ variable number is m=30, x _i ∈[0,1](i=1,2,⋯,m)

• (e) ZDT3

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$
  $$f_{2} \left( {\vec{{x}}} \right)=g\left( {\vec{{x}}} \right)\left[ {1-\sqrt {x_{1} /g\left( {\vec{{x}}} \right)} -x_{1} /g\left( {\vec{{x}}} \right)\sin \left( {10\pi x_{1}} \right)} \right] $$

  where $g\left ({\vec {{x}}} \right )=1+9\sum \limits _{i=2}^{m} {x_{i} /\left ({m-1} \right )}$ variable number is m=30, x _i ∈[0,1](i=1,2,⋯,m)

• (f) ZDT4

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$

  $f_{2} \left ({\vec {{x}}} \right )=g\left ({\vec {{x}}} \right )\left [ {1-\sqrt {x_{1} /g\left ({\vec {{x}}} \right )}} \right ]$, where $g\left ({\vec {{x}}} \right )=1+10\left ({m-1} \right )+\sum \limits _{i=2}^{m} {\left [ {{x_{i}^{2}} -10\cos \left ({4\pi x_{i}} \right )} \right ]}$ variable number is m=10, x ₁∈[0,1], x _i ∈[−5,5](i=2,3,⋯,m)

• (g) ZDT6

  $$f_{1} \left( {\vec{{x}}} \right)=1-\exp \left( {-4x_{1}} \right)\sin^{6}\left( {4\pi x_{1}} \right) $$

  $f_{2} \left ({\vec {{x}}} \right )=g\left ({\vec {{x}}} \right )\left [ {1-\left ({f_{1} \left ({\vec {{x}}} \right )/g\left ({\vec {{x}}} \right )} \right )^{2}} \right ]$, where $g\left ({\vec {{x}}} \right )=1+9\left [ {\sum \limits _{i=2}^{m} {x_{i} /\left ({m-1} \right )}} \right ]^{0.25}$ variable number is m=10, x _i ∈[0,1](i=1,2,⋯,m)

• (h) DTLZ1

  $$f_{1} \left( {\vec{{x}}} \right)=\frac{1}{2}x_{1} x_{2} {\cdots} x_{M-1} \left( {1+g\left( {X_{M}} \right)} \right) $$
  $$f_{2} \left( {\vec{{x}}} \right)=\frac{1}{2}x_{1} x_{2} {\cdots} \left( {1-x_{M-1}} \right)\left( {1+g\left( {X_{M}} \right)} \right) $$
  $$\vdots $$
  $$f_{M-1} \left( {\vec{{x}}} \right)=\frac{1}{2}x_{1} \left( {1-x_{2}} \right)\left( {1+g\left( {X_{M}} \right)} \right) $$
  $$f_{M} \left( {\vec{{x}}} \right)=\frac{1}{2}\left( {1-x_{1}} \right)\left( {1+g\left( {X_{M}} \right)} \right) $$

  where $g\left ({X_{M}} \right )=100\{ | {X_{M}} |+\sum \limits _{x_{i} \in X_{M}} [ (x_{i} -0.5 )^{2}-\cos (20\pi ({x_{i} -0.5} ) ) ] \}$ variable number is (M+|X _M |−1), x _i ∈[0,1](i=1,2,⋯,(M+|X _M |−1))

• (i) DTLZ2

  $$f_{1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\cos \left( {x_{2} \pi /2} \right) $$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2} \pi /2} \right)\cos \left( {x_{M-1} \pi /2} \right) $$
  $$f_{2} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\cos \left( {x_{2} \pi /2} \right)$$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2} \pi /2} \right)\sin \left( {x_{M-1} \pi /2} \right) $$
  $$\vdots $$
  $$f_{M-1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\sin \left( {x_{2} \pi /2} \right) $$
  $$f_{M} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\sin \left( {x_{1} \pi /2} \right) $$

  where $g\left ({X_{M}} \right )=\sum \limits _{x_{i} \in X_{M}} {\left ({x_{i} -0.5} \right )^{2}}$ variable number is (M+|X _M |−1), x _i ∈[0,1](i=1,2,⋯,(M+|X _M |−1))

• (j) DTLZ3

  $$f_{1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\cos \left( {x_{2} \pi /2} \right)$$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2} \pi /2} \right)\cos \left( {x_{M-1} \pi /2} \right) $$
  $$f_{2} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\cos \left( {x_{2} \pi /2} \right)$$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2} \pi /2} \right)\sin \left( {x_{M-1} \pi /2} \right) $$
  $$\vdots $$
  $$f_{M-1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1} \pi /2} \right)\sin \left( {x_{2} \pi /2} \right) $$
  $$f_{M} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\sin \left( {x_{1} \pi /2} \right) $$

  where

  $$g\left( {X_{M}} \right)\,=\,100\left\{ {\left| {X_{M}} \right|\,+\,\sum\limits_{x_{i} \in X_{M}} {\left[ {\left( {x_{i} \,-\,0.5} \right)^{2}\,-\,\cos \left( {20\pi \left( {x_{i} \,-\,0.5} \right)} \right)} \right]}} \right\} $$

  variable number is (M+|X _M |−1), x _i ∈[0,1](i=1,2,⋯,(M+|X _M |−1))

• (k) DTLZ4

  $$f_{1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1}^{\alpha} \pi /2} \right)\cos \left( {x_{2}^{\alpha} \pi /2} \right)$$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2}^{\alpha} \pi /2} \right)\cos \left( {x_{M-1}^{\alpha} \pi /2} \right) $$
  $$f_{2} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1}^{\alpha} \pi /2} \right)\cos \left( {x_{2}^{\alpha} \pi /2} \right)$$
  $$\qquad\qquad{\cdots} \cos \left( {x_{M-2}^{\alpha} \pi /2} \right)\sin \left( {x_{M-1}^{\alpha} \pi /2} \right) $$
  $$\vdots $$
  $$f_{M-1} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\cos \left( {x_{1}^{\alpha} \pi /2} \right)\sin \left( {x_{2}^{\alpha} \pi /2} \right) $$
  $$f_{M} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)\sin \left( {x_{1}^{\alpha} \pi /2} \right) $$

  where $g\left ({X_{M}} \right )=\sum \limits _{x_{i} \in X_{M}} {\left ({x_{i} -0.5} \right )^{2}}$, α=100 variable number is (M+|X _M |−1), x _i ∈[0,1](i=1,2,⋯,(M+|X _M |−1))

• (l) DTLZ6

  $$f_{1} \left( {\vec{{x}}} \right)=x_{1} $$
  $$f_{2} \left( {\vec{{x}}} \right)=x_{2} $$
  $$\vdots $$
  $$f_{M-1} \left( {\vec{{x}}} \right)=x_{M-1} $$
  $$f_{M} \left( {\vec{{x}}} \right)=\left( {1+g\left( {X_{M}} \right)} \right)h\left( {f_{1} ,f_{2} ,{\cdots} ,f_{M-1} ,g} \right) $$

  where $g\left ({X_{M}} \right )=1+\frac {9}{\left | {X_{M}} \right |}\sum \limits _{x_{i} \in X_{M}} {x_{i}}$

  $$h\left( {f_{1} ,f_{2} ,{\cdots} ,f_{M-1} ,g} \right)=M-\sum\limits_{i=1}^{M} {\left[ {\frac{f_{i}} {1+g}\left( {1+\sin \left( {3\pi f_{i}} \right)} \right)} \right]} $$

  variable number is (M+|X _M |−1), x _i ∈[0,1](i=1,2,⋯,(M+|X _M |−1))

Appendix B: Measurement index

• (1) coverage index

Supposing A and B stand for Pareto front set, coverage index I _C (A,B) is calculated as follows:

$$I_{C} \left( {A,B} \right)\overset{\text{def}}{=} \frac{\left| {\left\{ {\vec{{b}}\in B;\exists \vec{{a}}\in A:\vec{{a}}\succ \vec{{b}}} \right\}} \right|}{\left| B \right|} $$

where I _C (A,B)=1 means decision making vectors in B are dominated by decision making vectors in A, I _C (A,B)=0 means decision making vectors in A are dominated by decision making vectors in B.

• (2) convergence index

Supposing $P^{\ast } =\left \{ {\vec {{p}}_{1} ,\vec {{p}}_{2} ,\ldots ,\vec {{p}}_{\left | {P^{\ast } } \right |}} \right \}$ is ideal Pareto front set with distributing evenness, $A=\left \{ {\vec {{a}}_{1} ,\vec {{a}}_{2} ,{\ldots } ,\vec {{a}}_{A}} \right \}$ is approximate Pareto front set calculated by optimization algorithm.

For every solution $\vec {{a}}_{i}$ in set A, the minimal euclidean distance in P ^∗ after normalization processing is calculated as follows:

$$d_{i} =\min\limits_{j=1}^{| {P^{\ast} } |} \sqrt{\sum\limits_{m=1}^{k} {\left( {\frac{f_{m} \left( {\vec{{a}}_{i}} \right)-f_{m} \left( {\vec{{p}}_{j}} \right)}{f_{m}^{\max} -f_{m}^{\min}} } \right)^{2}}} $$

where $f_{m}^{\max }$ and $f_{m}^{\min }$ stand for maximum and minimum of objective m in set P ^∗. Then convergence index is average of all point after normalization processing in set A, which is calculated as follows:

$$C\left( A \right)\overset{\text{def}}{=} \frac{\sum\limits_{i=1}^{\left| A \right|} {d_{i}} } {\left| A \right|} $$

Convergence index stands for distance between approximate Pareto front and ideal Pareto front. Therefore the less convergence index, the better solution and the closer to ideal Pareto front.

• (3) spacing index

Supposing A is approximate Pareto front, and spacing index S is defined as follows:

$$S\overset{\text{def}}{=} \sqrt {\frac{1}{\left| A \right|-1}\sum\limits_{i=1}^{\left| A \right|} {\left( {\bar{{d}}-d_{i}} \right)^{2}}} $$

where

$$\begin{array}{@{}rcl@{}} d_{i} &=&\min\limits_{j} \left\{ {\sum\limits_{m=1}^{k} {\left| {f_{m} \left( {\vec{{a}}_{i}} \right)-f_{m} \left( {\vec{{a}}_{j}} \right)} \right|}} \right\},\vec{{a}}_{i} ,\vec{{a}}_{j} \in A, i,\\ j&=&1,2,{\ldots} ,\left| A \right| \end{array} $$

$\bar {{d}}$ is the average of all d _i and k is objective number. S=0 means that nondominated solutions distributing is even in objective space.