# 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)  
The literature [19] first puts forward the FECM, which draws on the dynamic system equilibrium relationship described by the five elements of ancient Chinese philosophy. The five elements are metal, wood, water, fire and earth. As depicted in Figure 2, there are two main relationships, generating interaction and restricting interaction, between the five elements. As shown in Figure 2, the outer circle represents the generating interaction: metal produces water, water produces wood, wood produces fire, fire produces earth and earth in its turn produces metal. The generating interaction is analogous to a mother-child relationship, which means children rely on their mother's nutrition to grow and become strong. The inner circle represents the restricting interaction: wood inhibits earth, earth inhibits water, water inhibits fire, fire inhibits metal and metal in its turn inhibits wood. The restricting interaction as the same as the grandparent-grandchildren relationship, where the grandmother limits the development of grandchildren. Through the generation and restriction, all the elements are mutually restricted and promoted by each other, forming a dynamic equilibrium relationship. As mentioned above, each element is influenced by the other four in a certain way, and this way is the force relationship between the five elements. The FECM proposed in Reference [19] is described as follows. Suppose a dynamic system consists of the five elements shown in Figure 2. Here we use \(X_{i}(k)\) to represent the \(i\)-th element at time \(k\), where \(i\in\{1,2,3,4,5\}\) represents the element within each cycle. Each of the elements has its own mass and the force powered by other elements in the  
Figure 2: The generating and restricting interactions between the five elements. same cycle. The mass of the five elements (metal, water, wood, fire, earth) at time \(k\) is defined as \(M_{i}(k)\) and the force exerted on the \(i\)-th element by the other four elements at time \(k\) is denoted by \(F_{i}(k)\), which is corresponding to \(M_{i}(k)\). Each \(F_{i}(k)\) of the elements is composed of four parts (taking the element \"metal\" for example), the first part is the force of generation from its parent element (earth), denoted by \(F_{3\_1}(k)\), and this segment of force is positive because the generation effect from its parent element makes it stronger. The value of \(F_{3\_1}(k)\) is determined by the mass of both wood \((M_{3}(k))\) and water \((M_{2}(k))\). When \(M_{2}(k)>M_{3}(k)\), the wood element will become stronger under the generation effect from its parent element. Furthermore, the bigger \(M_{2}(k)\) is, or the smaller \(M_{3}(k)\) is, the greater \(F_{3\_1}(k)\) will be. Nevertheless, if \(M_{2}(k)<M_{3}(k)\), which means wood element was stronger than its parent element, water element will be injured so that \(F_{3\_1}(k)\) will reverse. Here, we compute the formula of \(F_{3\_1}(k)\) the same as in the literature [19]. The second part is the restriction force from its grandparent element (fire), denoted by \(F_{3\_2}(k)\). This segment of force should be negative, because the restriction effect from an element's grandparent element makes it weaker. In the same way, the value of \(F_{3\_2}(k)\) is determined by the mass of both wood \((M_{3}(k))\) and metal \((M_{1}(k))\). Similarly, the third part is the generation force to its child element (water), denoted by \(F_{3\_3}(k)\) and the fourth segment is the restriction force to its grandchild element (wood), denoted by \(F_{3\_4}(k)\). Thus \(F_{3}(k)\) is the weighted sum of the above four segments. If we extend the wood instance \(F_{3}(k)\) to an arbitrary force \(F_{i}(k)(i\in\{1,2,3,4,5\})\), Equation (6) can be obtained [19]:  
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