# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm
## 3 Related Work and the Motivation of This Paper
### Related Work of Five-Elements Cycle Optimization (FECO)
\begin{cases}F_{1}(k)=w_{gp}\cdot ln[\frac{M_{5}(k)}{M_{1}(k)}]-w_{rp}\cdot ln [\frac{M_{4}(k)}{M_{1}(k)}]\\ \phantom{F_{1}(k)=w_{gp}\cdot ln[\frac{M_{1}(k)}{M_{1}(k)}]-w_{rp}\cdot ln[ \frac{M_{1}(k)}{M_{3}(k)}]}\\ F_{2}(k)=w_{gp}\cdot ln[\frac{M_{1}(k)}{M_{2}(k)}]-w_{rp}\cdot ln[\frac{M_{5}(k )}{M_{2}(k)}]\\ \phantom{F_{2}(k)=w_{gp}\cdot ln[\frac{M_{2}(k)}{M_{3}(k)}]-w_{ra}\cdot ln[ \frac{M_{2}(k)}{M_{4}(k)}]}\\ F_{3}(k)=w_{gp}\cdot ln[\frac{M_{2}(k)}{M_{3}(k)}]-w_{rp}\cdot ln[\frac{M_{1 }(k)}{M_{3}(k)}]\\ \phantom{F_{3}(k)=w_{gp}\cdot ln[\frac{M_{3}(k)}{M_{4}(k)}]-w_{ra}\cdot ln[ \frac{M_{3}(k)}{M_{5}(k)}]}\\ F_{4}(k)=w_{gp}\cdot ln[\frac{M_{3}(k)}{M_{4}(k)}]-w_{rp}\cdot ln[\frac{M_{2 }(k)}{M_{4}(k)}]\\ \phantom{F_{4}(k)=w_{gp}\cdot ln[\frac{M_{4}(k)}{M_{5}(k)}]-w_{ra}\cdot ln[ \frac{M_{4}(k)}{M_{1}(k)}]}\\ F_{5}(k)=w_{gp}\cdot ln[\frac{M_{4}(k)}{M_{5}(k)}]-w_{rp}\cdot ln[\frac{M_{3 }(k)}{M_{5}(k)}]\\ \phantom{F_{5}(k)=w_{gp}\cdot ln[\frac{M_{4}(k)}{M_{5}(k)}]-w_{rp}\cdot ln[ \frac{M_{5}(k)}{M_{5}(k)}]}\\ \phantom{F_{5}(k)=w_{gp}\cdot ln[\frac{M_{4}(k)}{M_{5}(k)}]-w_{rp}\cdot ln[ \frac{M_{3}(k)}{M_{5}(k)}]}\\ \phantom{F_{5}(k)=w_{gp}\cdot ln[\frac{M_{4}(k)}{M_{5}(k)}]-w_{ra}\cdot ln[ \frac{M_{5}(k)}{M_{2}(k)}]}\end{cases}  \tag{6}\]  
where each \(F_{i}(k)(i\in\{1,2,3,4,5\})\) of the Equation (6) is composed of the four parts \(F_{i\_1}(k)\), \(F_{i\_2}(k)\), \(F_{i\_3}(k)\), \(F_{i\_4}(k)\) described above. The \(w_{gp}\), \(w_{rp}\), \(w_{ga}\), \(w_{ra}\) represent weight coefficients. Considering that the four segments' forces are not exactly the same, the weighting coefficients \(w_{gp}\), \(w_{rp}\), \(w_{ga}\) and \(w_{ra}\) are defined to be positive numerical values in the range of \([0,1]\)[20]. In this paper, we set the same weighting coefficients as in the literature [20]. The mass of each element changes under the generation and restriction force at every iteration as the following Equation (7):  
\[M_{i}(k+1)=M_{i}(k)\cdot f_{\Gamma}(F_{i}(k)),\;i\in\{1,2,3,4,5\} \tag{7}\]where \(f_{\Gamma}(F_{i}(k))\) is a nonlinear function, which is used to express the variation of elements' mass under the forces. Extending the five elements to \(L\) elements, Equations (6) and (7) can be expressed as Equation (8), which is named FECM, as proposed in Reference [19]:  
\[ \begin{cases}F_{i}(k)=w_{gp}\cdot ln[\dfrac{M_{i-1}(k)}{M_{i}(k)}]-w_{rp}\cdot ln [\dfrac{M_{i-2}(k)}{M_{i}(k)}]-w_{ga}\cdot ln[\dfrac{M_{i}(k)}{M_{i+1}(k)}]-w_{ ra}\cdot ln[\dfrac{M_{i}(k)}{M_{i+2}(k)}]\\ M_{i}(k+1)=M_{i}(k)\cdot\dfrac{2}{1+\exp(-F_{i}(k))}\end{cases}  \tag{8}\]  
where \(i\in\{1,2,\ \ldots,L\}\). The subscripts of \(M_{i-2}(k)\), \(M_{i-1}(k)\), \(M_{i}(k)\), \(M_{i+1}(k)\), and \(M_{i+2}(k)\) represent the \(i\) circulates in the order of \(1,2,\ldots,L\). Based on the FECM described above, Reference [19] proposed FECO for solving single-objective problems and the capability of FECO for solving TSP problems and continuous function optimization problems was verified in Reference [20]. In FECO, the elements are generally divided into \(q\) different cycles and each cycle has \(L\) elements, where the value of \(L\times q\) is equivalent to the population size in the evolution algorithm and each element represents an individual in the population.