# Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP)

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of the Art

#### Sperm Swarm Optimization Algorithm

_{1}and C

_{2}numerical variables, that take random values in the range of 0 to 4. The adaptiveness of the SSO algorithm is based on two factors (i.e., temperature and pH). These variables take a range of numerical variables randomly based on the following rules:

- The normal pH value of a healthy female reproductive system is around 4.5–5.5 [24]. However, low pH of mucus acidic may destroy and deactivate the motility of sperm. Therefore, during the ovulation, the pH of vaginal acid or acidic is in the range of 7 to 4, which is very appropriate for sperm motility and is considered very alkaline and non-toxic to sperm [25]. The pH value of the female reproductive system is affected by the type of food consumed [26] and emotional or mood status of the female, such as happiness or sadness, etc. [27]. Based on this information, we can estimate that the value of pH will be varied in the range of 7 to 14.
- The temperature of the female reproductive system plays a significant role of determining the sperms movement direction. The scientists found that the sperm head acts like a temperature sensor to search for a warmer area (the egg location) [28]. Furthermore, the sperm head can response and sense to a temperature difference of <0.0006 °C [28]. The temperature inside the vagina can change based on women status. The average temperature inside the vagina is approximately between 35.1 and 37.4 °C [29]. However, this temperature may reach 38.5 °C in some cases due to vaginal blood pressure circulation [30].Based on this information, we can estimate that the value of temperature will be varied in the range of 35.1–38.5.

- The initial velocity of the sperm: is the velocity that acquired by each sperm after the ejaculation in the cervix zone. The sperm swarms take random positions inside the cervix and their velocity is affected by the pH value in that position. This velocity can be represented in the following rule:$$Initial\_Velocity=D\cdot {V}_{i}\cdot {Log}_{10}(pH\_{Rand}_{1})$$
- D: is the velocity damping factor, which is a random number between 0 and 1.
- V
_{i}: is the sperm velocity. - pH_Rand
_{1}: is the pH value, which is a random number in the range of (7, 14).

- Personal sperm current best solution: is the best solution that achieved so far by the sperm. As we mentioned previously, sperm head acts like a temperature sensor, which prefers to swim towards warmer temperatures (egg location) [28]. In addition, sperm moves forward to search on the guidance (higher concentrations of molecules) that produced and released by the egg, which this guidance knew as Chemo-taxis [31]. From this information, we can realize that the sperm will not swim backward to the cervix, but, will go forward towards warmer temperatures (the egg location in fallopian tubes). Based on that, this position can be achieved by comparing the sperm current position on X-axis and Y-axis with a sperm past position that is stored in the memory. The past position can be replaced by the current position if the current position is better than the past position. Personal sperm current best solution can be represented in the following equation:$$\begin{array}{l}Current\_Best\_Solution\text{}=Lo{g}_{{}_{10}}(pH\_Ran{d}_{2})\cdot \\ Lo{g}_{{}_{10}}(Temp\_Ran{d}_{1})\cdot (sb\_solution[]-current[])\text{}\end{array}$$
- sb_ solution []: is the best solution that has been achieved so far, which denoted by Sperm Best (sb_ solution).
- pH_Rand
_{2}: is the pH value, which is a random number in the range of (7, 14). - Temp_Rand
_{1}: is the area temperature, which is a random number in range (35.1, 38.5).

- Global best value is used to determine which sperm’s data is currently closest to the target (at the end, this sperm will represent as the winner). The value of sperm global best value is represented as the following equation:$$\begin{array}{l}Global\_Best\_Solution=Lo{g}_{{}_{10}}(pH\_Ran{d}_{3})\cdot \\ Lo{g}_{{}_{10}}(Temp\_Ran{d}_{2})\cdot (sgb\_solution[]-current[])\end{array}$$
- sgb_solution[]: is the best solution of any sperm has achieved so far which denoted by Sperm Global Best (sgb_solution).
- pH_Rand
_{3}: is the pH value, which is a random number in the range of (7, 14). - Temp_Rand
_{2}: is the area temperature, which is a random number in the range of (35.1, 38.5). - current[]: is the current best solution represented by the following equation.

$$current[]=current[]+v[]\text{}$$$$\begin{array}{l}v[]=Initial\_Velocity+Current\_Best\_Solution\text{}\\ +Global\_Best\_Solution\end{array}$$

Algorithm 1 Sperm swarm optimization (SSO) |

1: Begin |

2: Step 1: initialize positions for all sperms. |

3: Step 2: for i=1:number of sperm do |

4: Step 3: evaluate the fitness for each sperm |

5: If obtained fitness > sperm best solution then |

6: Set the current value as the sperm best solution |

7: End if |

8: End for |

9: Step 4: choose the sperm global best solution based on the winner. |

10: Step 5: for i=1:number of sperm Do |

11: Do the swim using velocity update rule |

12: Update sperm location on the search space |

13: End for |

14: Step 6: for i=1:number of sperm Do |

15: Apply mutation operation on the sperm value |

16: End for |

17: Step 7: while maximum iterations not achieved return to step 2 and repeat until reaching the |

17: maximum number of iterations. |

18: End procedure. |

## 3. Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP)

- Domination: x
_{1}, may dominate a position vector, x_{2}(x_{1}< x_{2}), if and only if ${f}_{k}({x}_{1})\le {f}_{k}({x}_{2}),\forall k=1,\cdot \cdot \cdot \cdot \cdot {n}_{k}$ and ${f}_{k}({x}_{1})<{f}_{k}({x}_{2}),$ for at least one k [33]. - Pareto optimal: A vector ${x}^{*}$ $\in $ F is defined as Pareto-optimal if no vector exists x $\in $ F such that ${f}_{k}(x)\le {f}_{k}({x}^{*}),\text{}k\in N,\text{}and\text{}{f}_{m}(x){f}_{m}({x}^{*})$ for at least one $m\in N$. An objective vector ${z}^{*}=f\left({x}^{*}\right)$ is called Pareto optimal if the corresponding vector ${x}^{*}$ is the Pareto optimal. The set of the Pareto optimal decision vectors ${x}^{*}$ $\in $ F is denoted by $P\subseteq F$ [34].
- ${x}^{*}$ $\in $ F is the Pareto optimal of a position vector if no position vector that dominates it, $x\ne {x}^{*}$ $\in $ F. The Pareto optimal solution is non-dominated solution, which not dominated by other solutions [34].
- Pareto optimal set: is a set containing all the Pareto optimal vectors ${\mathrm{P}}_{\mathrm{s}}=\{{x}_{\circ}|\neg \exists {x}_{{}_{1}\prec}{x}_{\circ}\}$ [35].
- Pareto front (PF): is defined as $\mathrm{PF}=\{({f}_{1}(x),{f}_{{}_{2}}(x),\cdot \cdot \cdot \cdot \cdot {f}_{{}_{N}(x))|}x\in {p}_{s}\}$ [35].

_{1}$\in -$ dominance a decision vector x

_{2}for some $\in >0$ if and only if: ${f}_{i}({x}_{1})/(1+\in )\le {f}_{i}({x}_{2}),{\forall}_{i}=1,.....,m$ and ${f}_{i}({x}_{1})/(1+\in )\le {f}_{i}({x}_{2}),$ for at least $i=1,.....,m$. $\in -$ value is a user defined value, which is used to determine the size of the final external archive. In our algorithm like [11], we use the same value of for all objective functions, which are changed based on the amount of points in the final Pareto front.

Algorithm 2 Mutation |

1: Begin |

2: Step 1: for i = 0 to population size do |

3: If (i % 3 == 0) then |

4: Sperms_ mutated with a non-uniform mutation operator |

5: Else if (i % 3 == 1) then |

7: Sperms_ mutated with a uniform mutation operator |

8: Else |

9: Sperms_ without mutation |

10: End if |

11: End for |

12: End procedure. |

_{1}, pH_Rand

_{2}, and pH_Rand

_{3}are the pH values of the visited regions, which take a value in the range of (7, 14). The process of updating the personal sperm current best solution is based on two cases; first, if the value of personal sperm current best solution is dominated by the new sperm; second, if the personal sperm current best solution with the new sperm value are non-dominated with respect to each other. Later on, if all sperms have been updated, the SoW will be updated too, which the sperms that achieve the new positions that are better than the old positions will have the possibility to join the winner set. After that, the $\in -archive$ will take the place on the procedure to be updated. Finally, the crowding values of SoW are processed to be updated, as many of the winners are eliminated in case of exceeding the determined size of the winners set. This process is repeated many times until it reached the determined number of iterations (i

_{max}). The parameter needed by this algorithm are (1) S_size (swarm size), (2) i (number of iterations), (3) mutRate (mutation rate, which is automatically computed), and (4) $\in $, which is the value of the bounding size of the $\in -archive$.

Algorithm 3 Multi-objective optimization algorithm based on sperm fertilization procedure (MOSFP) |

1: Begin |

2: Step 1: initialize positions for all sperms. |

3: Step 2: initialize Winners. |

4: Step 2: archive the Winners in $\in -archive$ |

5: Step 3: crowd the winners using crowding operation. |

6: Step 4: define counter (i) and define number of maximum iterations (i_{Max}). |

7: Step 5: do //this do is a do - while |

8: For <each sperm> do |

9: Select winner from the sperm swarm |

10: Update sperms positions using the predefined sperm velocity update rule (perform swim) |

11: Perform mutation procedure (Algorithm 1) |

12: Evaluate the fitness for each sperm |

13: Update personal sperm current best solution |

14: End for |

15: Update Set of Winners (SoW) |

16: Archive winner in $\in -archive$ |

17: Crowd the SoW using crowding operation |

18: Update value of counter (i) |

19: Step 6: While i < i_{Max} |

20: Step 7: archive results in $\in -archive$ |

21: End procedure. |

## 4. Comparison Strategy

- Minimize the distance between the global Pareto front (the well-known Pareto front of any problem) and the Pareto front that produced by our approach.
- Maximize the spread of the solutions that produced by our approach, so uniform distribution of vectors can be achieved.
- Maximize the convergence of algorithm to obtain a good quality of the Pareto optimal set found.

**Inverted Generational Distance (IGD):**the generational distance measurement was proposed by Van Veldhuizen et al. [46,47,48]. This measurement is used to estimate how far the Pareto front of an algorithm from the true Pareto front of any problem. This metric can be calculated by:

_{i}is the Euclidean distance that measured in object space between the true Pareto front of any problem and the vectors that generated by any algorithm. It is clear that IGD metric should be near or equal to zero. In other meaning, if IGD = 0, all of the elements generated are in the true Pareto front of the problem [49,50]. We use IGD to test the first issue, which is mentioned previously, by comparing a true Pareto front of a problem (the reference Pareto front) with a Pareto front that produced by any algorithm of the same problem.

**Spread (SP):**is the diversity metric that is used to measure the extent of spread and distribution of solutions in the set S of optimal solution. Figure 2 illustrates the spread metric of five non-dominated solutions of S, which are spread in two cases. In the first case shown in Figure 2a, there is a poor spread, but a good distribution, in which S does not contain the radical points, such as (1, 0), (0, 1) on the 2-dimensional Pareto front. On the other hand, in the second case shown in Figure 2b, there is unfavorable distribution, but a good spread of an optimal solution set [51]. We use SP to test the second issue that mentioned previously. SP can be measured by the following equation [52]:

- ${d}_{i}$: is the Euclidean distance measured based on the distance between consecutive solutions,
- $\overline{d}$: is the average of ${d}_{i}$
- $df$ and ${d}_{l}$: are the minimum Euclidean distance measured based on the distance between solutions in S to the radical (bounding) solutions of the Pareto front (P).

**Epsilon (**$\in $

**):**is a binary indicator operates by considering all of the objectives to give a factor by which an approximation set is worse than another. Formally, let M and N be two approximation sets, then $\in \left(M,N\right)$ equals the minimum factor $\in $, which for any solution in set M there is at least one solution in N that is not considered as worse by a factor of $\in $ considering all the objects [53]. This metric is very important to determine the quality of the obtained solution set by each algorithm [54]. We calculate $\in $ to test the third issue that mentioned previously.

**Non-dominated Sorting Genetic Algorithm (NSGA-II):**is one of the most popular algorithms in MOO area. This algorithm performs a set of operators that deduced from the single objective Genetic Algorithm (GA). These operations are selection, crossover and mutation operators [9]. The pseudo-code of NSGA-II is summarized in Algorithm 4 [8].**Algorithm 4**Non-dominated sorting genetic algorithm II (NSGA-II)1: **Begin**2: **Step 1:**initialize Population3: Generate random population – size M 4: **Step 2:**evaluate objective values5: **Step 3:**assign rank (level) based on Pareto dominance-“sort”6: **Step 4:**generate child population7: **Step 5:**binary tournament selection8: **Step 6:**recombination and mutation9: **Step 7:****for**i=1 to number of generations**do**10: With parent and child population 11: Assign rank (level) based on Pareto – “sort” 12: Generate sets on non-dominated fronts 13: Loop (inside) by adding solutions to next generation 14: Starting from the “first” front until M individuals found 15: Determine crowding distances between points on each front 16: Select points (elitist) on the lower front (with lower rank)and are outside a crowding 16: distance 17: Create next generation 18: Binary tournament Selection 19: Recombination and mutation 20: Increment generation index 21: **End for**22: **End Procedure**.**Strength Pareto Evolutionary Algorithm 2 (SPEA2):**is a multi-objective algorithm, which is created as an improvement version of strength Pareto evolutionary algorithm (SPEA) algorithm [56]. This algorithm proposed by Zitzler et al. [57]. The procedure of this algorithm works based on dominance, which each single individual dominates or is dominated by other solutions. The nearest neighbor technique is used to guide the search. In addition, this algorithm operates based on preserving the boundary solutions by using archive truncation method [58]. Algorithm 5 summarizes the pseudo-code of this algorithm [59].**Algorithm 5**Strength Pareto evolutionary algorithm 2 (SPEA2)1: **Begin**2: **Step 1:**initialize population P3: **Step 2:**evaluate objective functions4: **Step 3:**create external archive A5: **Step 4:****for**i=1 to number of generations**do**6: Compute fitness of individual in P and A 7: Add non-dominated individuals from P and A 8: **If**capacity of A is exceeded**Then**9: Remove individuals from A by truncation operator 10: **End If**11: Perform binary tournament selection to create mating pool 12: Perform crossover 13: Perform mutation 14: **End for**15: **End Procedure.****Optimized Multi-objective Particle Swarm Optimization (OMOPSO):**proposed by Sierra et al. [11]. This algorithm performs a set of operators to solve MOOPs, such as crowding the global best solutions that known as leaders, performs mutation, after that, performs an archive of the leaders. The pseudo-code of OMOPSO is summarized in Algorithm 6.In order to compare our algorithm with the previously mentioned algorithms, we use two techniques, including, quantitative and qualitative tests. For the quantitative test, we adopt the IGD, SP, and $\in $ measures, while for the qualitative test, we compare between the quality of Pareto fronts that is achieved by our algorithm and the Pareto fronts that achieved by the other algorithms. For these purposes, two test suites called Zitzler-Deb-Thiele (ZDT), and Walking-Fish-Group (WFG) have been used in this work.**Zitzler-Deb-Thiele (ZDT) Test Suite:**this suite is proposed by Zitzler et al. [60]. ZDT suite is the most widely utilized suite of benchmark functions in evolutionary algorithms and swarm intelligence algorithms literature. The characteristics of ZDT problems are analyzed in Table 1. From the table, we can notice that the geometry of ZDT1 is convex, and the geometry of ZDT2 is concave. In addition, ZDT3 is disconnected on both the Pareto front and the Pareto optimal set, while the other functions consist of convex or concave components. The ZDT benchmark functions share many characteristics, including, how multimodality can cause a disconnected Pareto problems such as in ZDT3, and many-to-one Pareto problem, such as in ZDT6. These functions utilize only a single parameter, which means that these functions own only one variable. There are many advantages of using ZDT problem as a benchmark function for MOOAs, including, that this suite is well defined and employed in many research papers, which facilitate the comparisons with new MOOA. In addition, the Pareto optimal fronts of this suite are easy to apply and to understand [61]. For these reasons, we choose this suite to test our proposed algorithm.**Algorithm 6**Optimized multi-objective particle swarm optimization (OMOPSO)1: **Begin**2: **Step 1****:**initialize swarm and leaders. Send leaders to $\in $$archive$3: **Step 2:**crowding(leaders), g = 04: **Step 3:****While**g < max number of iterations (gmax)5: **For**<each particle>**do**6: Select leader. Flight. Mutation. Evaluation. Update pbest 7: End for 8: Update leaders, Send leaders to $\in $-archive 9: Crowding(leaders), g++ 10: **End while**11: **Step 4****:**Report results in $\in $-archive12: **End Procedure**.**Walking-Fish-Group (WFG) Test Suite:**WFG suite consists of nine problems introduced by [62]. The characteristics of these problems are outlined in Table 2 [60]. WFG1 and WFG7 are the only both unimodal and separable. WFG6 and WFG9 are the non-separable reduction problems, and they are more difficult than that of WFG3 and WFG2. WFG4 is a multimodality problem with a large “hill sizes” and it is more difficult than that of WFG9. WFG1 employs different parameters by using dissimilar weights in its variable weighted sum reduction. The parameters of WFG7 are dependent on the position and distance-related parameters, whereas WFG9 is more complex, which its distance-related parameters depend on other distance-related parameters. WFG8 is different than WFG9, wherein its distance-related parameters depend on other position and distance-related parameters. WFG5 is highly deceptive function and can be more difficult than that of WFG9.

## 5. Experimental Test and Results

**A. ZDT Test Suites**

**B. WFG Test Suites**

## 6. Discussion

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Part I: Zitzler-Deb-Thiele (ZDT) Test Suite**

No | Problem | Model | Domain |

1 | ZDT1 | ${f}_{1}(x)={x}_{1}$ ${f}_{2}(x)=g(x)\cdot (2-\sqrt{{f}_{1}(x)/g(x)})$ $g(x)=1+\frac{9}{n-1}{\displaystyle \sum _{i=2}^{n}{x}_{i}^{2}}$ | $[0,1]\times {[-1,1]}^{n}$ |

2 | ZDT2 | ${f}_{1}(x)={x}_{1}$ ${f}_{2}(x)=g(x)\cdot (2-{({f}_{1}(x)/g(x))}^{2})$ $g(x)=1+\frac{9}{n-1}{\displaystyle \sum _{i=2}^{n}{x}_{i}^{2}}$ | $[0,1]\times {[-1,1]}^{n}$ |

3 | ZDT3 | ${f}_{1}(x)={x}_{1}$ $\begin{array}{l}{f}_{2}(x)=g(x)\cdot (2-\sqrt{{f}_{1}(x)/g(x)}\\ -({f}_{1}(x)/g(x))\cdot \mathrm{sin}(10\pi {f}_{1}))\end{array}$ $g(x)=1+\frac{9}{n-1}{\displaystyle \sum _{i=2}^{n}{x}_{i}^{2}}$ | $[0,1]\times {[-1,1]}^{n}$ |

4 | ZDT6 | ${f}_{1}(x)=1-{e}^{-4{x}_{1}}$ ${f}_{2}(x)=g(x)\cdot (2-{({f}_{1(x)}/g(x))}^{2})$ $g(x)=1+\frac{9}{n-1}\cdot {\displaystyle \sum _{i=2}^{n}{x}_{i}^{2}}$ | $[0,1]\times {[-1,1]}^{n}$ |

**Part II: Walking-Fish-Group (WFG) Test Suite**

No. | Problem | Type | Model |

5 | WFG1 | Shape | ${h}_{m=1:M-1}=conve{x}_{m}$ ${h}_{m}=mixe{d}_{M}(with\text{}\alpha =1\text{}and\text{}A=5)$ ${t}_{i=1:k}^{1}={y}_{i}$ |

t^{1} | ${t}_{i=k+1:n}^{1}=s\_linear({y}_{i},\text{}0.35)$ ${t}_{i=k+1:n}^{1}=s\_linear({y}_{i},\text{}0.35)$ | ||

t^{2} | ${t}_{i=1:k}^{2}={y}_{i}$ ${t}_{i=k+1:n}^{2}=b\_flat({y}_{i},\text{}0.8,\text{}0.75,\text{}0.85)$ | ||

t^{3} | ${t}_{i=1:n}^{3}=b\_poly({y}_{i},\text{}0.02)$ | ||

t^{4} | $\begin{array}{l}{t}_{i=1:M-1}^{4}=r\_sum(\{{y}_{(i-1)k}/(M-1)+1,\cdot \cdot \cdot \cdot ,{y}_{ik}/(M-1)\},\\ \{2((i-1)k/(M-1)+1),\cdot \cdot \cdot \cdot ,2ik/(M-1)\})\end{array}$ ${t}_{M}^{4}=r\_sum(\{{y}_{k+1,\cdot \cdot \cdot \cdot ,{y}_{n}}\},\{2(k+1),\cdot \cdot \cdot ,2n\})$ | ||

6 | WFG2 | Shape | ${h}_{m=1:M-1}=conve{x}_{m}$ ${h}_{m}=dis{c}_{M}(with\text{}\alpha =\beta =1\text{}and\text{}A=5)$ |

t^{1} | As t^{1} from WFG1. (linear shift) | ||

t^{2} | ${t}_{i=1:k}^{2}={y}_{i}$ ${t}_{i=k+1:k+1/2}^{2}=r\_nonsep(\{{y}_{k}+2(i-k)-1,{y}_{k}+2(i-k)\},\text{}2)$ | ||

t^{3} | $\begin{array}{l}{t}_{i=1:M-1}^{3}=r\_sum(\{{y}_{(i-1)k}/(M-1)+1,\cdot \cdot \cdot \cdot ,{y}_{ik}/(M-1)\},\\ \{1,\cdot \cdot \cdot \cdot ,1\})\end{array}$ ${t}_{M}^{3}=r\_sum(\{{y}_{k+1,\cdot \cdot \cdot \cdot ,{y}_{k+l/2}}\},\{1,\cdot \cdot \cdot ,1\})$ | ||

7 | WFG3 | Shape | ${h}_{m=1:M}=linea{r}_{m}(Degenerate)$ |

t^{1:3} | As t^{1:3} from WFG2. (linear shift, non-separable reduction, and weighted sum reduction.) | ||

8 | WFG4 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | ${t}_{i=1:n}^{1}=s\_multi({y}_{i},\text{}30,\text{}10,\text{}0.35)$ | ||

t^{2} | ${t}_{i=1:M-1}^{2}=s\_sum(\{y(i-1)k/(M-1)+1{,}_{\cdot \cdot \cdot \cdot},{y}_{ik}/(M-1)\},\{1,\cdot \cdot \cdot \cdot ,1\})$ ${t}_{M}^{2}=s\_sum(\{{y}_{k+1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{n}\},\{1,\cdot \cdot \cdot \cdot ,1\})$ | ||

9 | WFG5 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | ${t}_{i=1:n}^{1}=s\_decept({y}_{i},\text{}0.35,\text{}0.001,\text{}0.05)$ | ||

t^{2} | As t^{2} from WFG4 (Weighted sum reduction.) | ||

10 | WFG6 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | As t^{1} from WFG1. (Linear shift.) | ||

t^{2} | ${t}_{i=1:M-1}^{2}=s\_sum(\{{y}_{(i-1)k/(M-1)+1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{ik/(M-1)}\},{\{}_{k/(M-1)}\})$ ${t}_{M}^{2}=s\_sum(\{{y}_{k+1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{n}\},l)$ | ||

11 | WFG7 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | ${t}_{i=1:k}^{1}=b\_param({y}_{i},\text{}s\_sum(\{{y}_{i+1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{n}\},\{1,\cdot \cdot \cdot \cdot ,1\}),\text{}\frac{0,98}{49.98},\text{}0.02,\text{}50)$ ${t}_{i=k+1:n}^{1}={y}_{i}$ | ||

t^{2} | As t^{1} from WFG1. (Linear shift). | ||

t^{3} | As t^{2} from WFG4. (weighted sum reduction.) | ||

12 | WFG8 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | ${t}_{i=1:k}^{1}={y}_{i}$ ${t}_{i=k+1:n}^{1}=b\_param({y}_{i},\text{}r\_sum(\{{y}_{1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{i-1}\},\{1,\cdot \cdot \cdot \cdot ,1\}),\text{}\frac{0,08}{49.98},\text{}0.02,\text{}50)$ | ||

t^{2} | As t^{1} from WFG1. (Linear shift). | ||

t^{3} | As t^{2} from WFG4. (weighted sum reduction.) | ||

13 | WFG9 | Shape | ${h}_{m=1:M}=concav{e}_{m}$ |

t^{1} | ${t}_{i=1:n-1}^{1}=b\_param({y}_{i},\text{}r\_sum(\{{y}_{i+1}{,}_{\cdot \cdot \cdot \cdot ,}{y}_{n}\},\{1,\cdot \cdot \cdot \cdot ,1\}),\text{}\frac{0.98}{49.98},\text{}0.02,\text{}50)$ ${t}_{i=n}^{1}=yn$ | ||

t^{2} | ${t}_{i=1:k}^{2}=s\_decept({y}_{i},\text{}0.35,\text{}0.001,\text{}0.05)$ ${t}_{i=k+1:n}^{2}=s\_multi({y}_{i},\text{}30,95,\text{}0.35)$ | ||

t^{3} | As t^{2} from WFG6. (non-separable reduction.) |

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**Figure 1.**The winner and other sperms reach the egg [21].

**Figure 2.**Diversity metrics illustration of two components (spread and distribution) [37]: (

**a**) good distribution, but poor spread; (

**b**) poor distribution, but good spread.

**Table 1.**Analysis of ZDT benchmark functions [60].

Name | ZDT1 | ZDT2 | ZDT3 | ZDT6 | ||||
---|---|---|---|---|---|---|---|---|

Objective | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} |

R3: # Parameters | 1 | ✓ | 1 | ✓ | 1 | ✓ | 1 | ✓ |

F2: Separability | S | S | S | S | S | S | S | S |

F5: Modality | U | U | U | U | U | M | U | U |

R1: No Extremal | ✗ | ✗ | ✗ | ✗ | ||||

R2: No Medial | ✓ | ✓ | ✓ | ✓ | ||||

R5: Diss. Domains | ✗ | ✗ | ✗ | ✗ | ||||

R6: Diss. Ranges | ✗ | ✗ | ✓ | ✓ | ||||

R7: Optima known | ✓ | ✓ | ✓ | ✓ | ||||

F1: Geometry | convex | concave | disconnected | concave | ||||

F3: Bias | - | - | - | + | ||||

F4a: Pareto Many-to-one | - | - | - | + | ||||

F4b: Flat Regions | - | - | - | - |

**Table 2.**Analysis of WFG benchmark functions [60].

Name | Objective | F2: Separability | Modality | Bias | Geometry |
---|---|---|---|---|---|

WFG1 | f_{1}:_{M} | S | U | Polynomial, flat | Convex, Mixed |

WFG2 | f_{1}:_{M-1}f _{M} | NS | U | − | Convex, disconnected |

NS | M | ||||

WFG3 | f_{1}:_{M} | NS | U | − | Linear, degenerate |

WFG4 | f_{1}:_{M} | S | M | − | Concave |

WFG5 | f_{1}:_{M} | S | D | − | Concave |

WFG6 | f_{1}:_{M} | NS | U | − | Concave |

WFG7 | f_{1}:_{M} | S | U | Parameter dependent | Concave |

WFG8 | f_{1}:_{M} | NS | U | Parameter dependent | Concave |

WFG9 | f_{1}:_{M} | NS | M, D | Parameter dependent | Concave |

Parameters | MOSFP | OMOPSO | NSGA-II | SPEA2 |
---|---|---|---|---|

Population size | 100 | 100 | 100 | 100 |

Archive size | (winner) 100 | 100 | (Elite) 100 | 100 |

Mating pool size | - | - | - | 100 |

Maximum generation | 5000 | 5000 | 5000 | 5000 |

Crossover probability | - | - | 0.9 | 0.9 |

Mutation probability | 1/d where d is the variable code size |

**Table 4.**Comparison of experimental results between multi-objective optimization algorithm based on sperm fertilization procedure (MOSFP), optimized multi-objective particle swarm optimization (OMOPSO), non-dominated sorting genetic algorithm II (NSGA-II), and strength Pareto evolutionary algorithm 2 (SPEA2) in terms of epsilon ($\in $) spread (SP), and inverted generational distance (IGD) for Zitzler-Deb-Thiele (ZDT) test functions.

Method | Metrics | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\in $ | SP | IGD | ||||||||||

Best | Worst | Aver. | Med. | Best | Worst | Aver. | Med. | Best | Worst | Aver. | Med. | |

ZDT1 | ||||||||||||

MOSFP | 5.01 × 10^{−3} | 5.57 × 10^{−3} | 5.27 × 10^{−3} | 5.25 × 10^{−3} | 2.76 × 10^{−2} | 9.85 × 10^{−2} | 6.23 × 10^{−2} | 6.22 × 10^{−2} | 3.44 × 10^{−5} | 3.61 × 10^{−5} | 3.51 × 10^{−5} | 3.51 × 10^{−5} |

OMOPSO | 4.92 × 10^{−3} | 5.40 × 10^{−3} | 5.15 × 10^{−3} | 5.15 × 10^{−3} | 3.29 × 10^{−2} | 8.74 × 10^{−2} | 5.54 × 10^{−2} | 5.52 × 10^{−2} | 3.39 × 10^{−5} | 3.60 × 10^{−5} | 3.48 × 10^{−5} | 3.48 × 10^{−5} |

NSGA-II | 1.11 × 10^{−1} | 2.67 × 10^{−1} | 1.61 × 10^{−1} | 1.51 × 10^{−1} | 5.84 × 10^{−1} | 8.17 × 10^{−1} | 6.80 × 10^{−1} | 6.87 × 10^{−1} | 7.65 × 10^{−4} | 1.92 × 10^{−3} | 1.18 × 10^{−3} | 1.14 × 10^{−3} |

SPEA2 | 1.60 × 10^{−1} | 4.58 × 10^{−1} | 2.82 × 10^{−1} | 2.74 × 10^{−1} | 8.92 × 10^{−1} | 6.11 × 10^{−1} | 7.43 × 10^{−1} | 7.35 × 10^{−1} | 1.04 × 10^{−3} | 2.71 × 10^{−3} | 1.76 × 10^{−3} | 1.72 × 10^{−3} |

ZDT2 | ||||||||||||

MOSFP | 5.018 × 10^{−3} | 5.65 × 10^{−3} | 5.27 × 10^{−3} | 5.28 × 10^{−3} | 2.75 × 10^{−2} | 9.95 × 10^{−2} | 5.62 × 10^{−2} | 5.41 × 10^{−2} | 3.41 × 10^{−5} | 3.60 × 10^{−5} | 3.49 × 10^{−5} | 3.49 × 10^{−5} |

OMOPSO | 4.93 × 10^{−3} | 5.32 × 10^{−3} | 5.13 × 10^{−3} | 5.14 × 10^{−3} | 2.45 × 10^{−2} | 8.30 × 10^{−2} | 5.53 × 10^{−2} | 5.56 × 10^{−2} | 3.43 × 10^{−5} | 3.56 × 10^{−5} | 3.48 × 10^{−5} | 3.47 × 10^{−5} |

NSGA-II | 3.46 × 10^{−1} | 1.457 | 6.32 × 10^{−1} | 5.88 × 10^{−1} | 6.38 × 10^{−1} | 1.011 | 8.33 × 10^{−1} | 8.32 × 10^{−1} | 1.54 × 10^{−3} | 9.44 × 10^{−3} | 3.10 × 10^{−3} | 2.75 × 10^{−3} |

SPEA2 | 5.21 × 10^{−1} | 1.775 | 1.107 | 1.051 | 8.14 × 10^{−1} | 1.0162 | 9.38 × 10^{−1} | 9.51 × 10^{−1} | 2.50 × 10^{−3} | 1.16 × 10^{−2} | 6.34 × 10^{−3} | 5.31 × 10^{−3} |

ZDT3 | ||||||||||||

MOSFP | 3.37 × 10^{−3} | 5.28 × 10^{−3} | 4.13 × 10^{−3} | 4.11 × 10^{−3} | 6.99 × 10^{−1} | 7.02 × 10^{−1} | 7.01 × 10^{−1} | 7.01 × 10^{−1} | 2.78 × 10^{−5} | 3.08 × 10^{−5} | 2.91 × 10^{−5} | 2.91 × 10^{−5} |

OMOPSO | 3.54 × 10^{−3} | 5.25 × 10^{−3} | 4.18 × 10^{−3} | 4.06 × 10^{−3} | 6.99 × 10^{−1} | 7.01 × 10^{−1} | 7.00 × 10^{−1} | 7.00 × 10^{−1} | 2.74 × 10^{−5} | 3.04 × 10^{−5} | 2.86 × 10^{−5} | 2.86 × 10^{−5} |

NSGA-II | 1.51 × 10^{−1} | 3.36 × 10^{−1} | 2.10 × 10^{−1} | 2.01 × 10^{−1} | 7.82 × 10^{−1} | 9.39 × 10^{−1} | 8.63 × 10^{−1} | 8.62 × 10^{−1} | 3.96 × 10^{−4} | 1.40 × 10^{−3} | 8.73 × 10^{−4} | 8.87 × 10^{−4} |

SPEA2 | 1.91 × 10^{−1} | 6.06 × 10^{−1} | 2.92 × 10^{−1} | 2.86 × 10^{−1} | 7.88 × 10^{−1} | 9.37 × 10^{−1} | 8.64 × 10^{−1} | 8.62 × 10^{−1} | 7.43 × 10^{−4} | 1.98 × 10^{−3} | 1.30 × 10^{−3} | 1.31 × 10^{−3} |

ZDT6 | ||||||||||||

MOSFP | 5.32 × 10^{−3} | 6.53 × 10^{−3} | 5.90 × 10^{−3} | 5.91 × 10^{−3} | 1.33 × 10^{−1} | 1.81 × 10^{−1} | 1.55 × 10^{−1} | 1.55 × 10^{−1} | 3.21 × 10^{−5} | 3.35 × 10^{−5} | 3.27 × 10^{−5} | 3.27 × 10^{−5} |

OMOPSO | 4.73 × 10^{−3} | 9.00 × 10^{−3} | 5.59 × 10^{−3} | 5.42 × 10^{−3} | 5.59 × 10^{−2} | 2.53 × 10^{−1} | 1.13 × 10^{−1} | 1.08 × 10^{−1} | 3.09 × 10^{−5} | 3.41 × 10^{−5} | 3.19 × 10^{−5} | 3.17 × 10^{−5} |

NSGA-II | 2.34 × 10^{−2} | 3.53 × 10^{−2} | 2.71 × 10^{−2} | 2.65 × 10^{−2} | 6.04 × 10^{−2} | 1.36 | 7.41 × 10^{−1} | 9.53 × 10^{−1} | 4.67 × 10^{−5} | 1.37 × 10^{−3} | 2.03 × 10^{−4} | 1.60 × 10^{−4} |

SPEA2 | 1.85 × 10^{−2} | 3.84 × 10^{−1} | 6.98 × 10^{−2} | 6.09 × 10^{−2} | 9.03 × 10^{−1} | 1.436 | 1.332 | 1.345 | 1.99 × 10^{−4} | 2.29 × 10^{−4} | 2.04 × 10^{−4} | 2.03 × 10^{−4} |

**Table 5.**Comparison of experimental results between MOSFP, OMOPSO, NSGA-II, and SPEA2 in terms of epsilon ($\in $ ) spread (SP), and inverted generational distance (IGD) for WFG test functions.

Method | Metrics | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\in $ | SP | IGD | ||||||||||

Best | Worst | Aver. | Med. | Best | Worst | Aver. | Med. | Best | Worst | Aver. | Med. | |

WFG1 | ||||||||||||

MOSFP | 8.59× 10^{−1} | 1.199 | 1.092 | 1.094 | 8.01 × 10^{−1} | 1.057 | 9.31 × 10^{−1} | 9.31 × 10^{−1} | 2.72 × 10^{−3} | 4.93 × 10^{−3} | 4.33 × 10^{−3} | 4.37 × 10^{−3} |

OMOPSO | 7.68 × 10^{−2} | 1.022 | 1.61 × 10^{−1} | 1.45 × 10^{−1} | 7.41 × 10^{−1} | 1.238 | 8.42 × 10^{−1} | 8.33 × 10^{−1} | 3.04 × 10^{−5} | 3.49 × 10^{−3} | 1.05 × 10^{−4} | 4.81 × 10^{−5} |

NSGA-II | 1.441 | 2.020 | 1.740 | 1.746 | 8.63 × 10^{−1} | 1.030 | 9.37 × 10^{−1} | 9.33 × 10^{−1} | 3.93 × 10^{−3} | 6.17 × 10^{−3} | 5.06 × 10^{−3} | 5.07 × 10^{−3} |

SPEA2 | 1.562 | 2.227 | 1.923 | 1.922 | 8.08 × 10^{−1} | 1.266 | 1.080 | 1.070 | 4.46 × 10^{−3} | 7.02 × 10^{−3} | 5.77 × 10^{−3} | 5.73 × 10^{−3} |

WFG2 | ||||||||||||

MOSFP | 1.14 × 10^{−2} | 1.92 × 10^{−2} | 1.42 × 10^{−2} | 1.39 × 10^{−2} | 7.57 × 10^{−1} | 7.96 × 10^{−1} | 7.77 × 10^{−1} | 7.74 × 10^{−1} | 5.07 × 10^{−5} | 7.37 × 10^{−5} | 5.91 × 10^{−5} | 5.90 × 10^{−5} |

OMOPSO | 7.48 × 10^{−3} | 1.04 × 10^{−2} | 9.01 × 10^{−3} | 8.94 × 10^{−3} | 7.56 × 10^{−1} | 7.59 × 10^{−1} | 7.57 × 10^{−1} | 7.57 × 10^{−1} | 3.81 × 10^{−5} | 4.22 × 10^{−5} | 4.01 × 10^{−5} | 4.00 × 10^{−5} |

NSGA-II | 1.59 × 10^{−2} | 8.15 × 10^{−1} | 4.83 × 10^{−1} | 7.99 × 10^{−1} | 7.86 × 10^{−1} | 1.006 | 8.59 × 10^{−1} | 8.50 × 10^{−1} | 6.21 × 10^{−5} | 1.77 × 10^{−3} | 1.06 × 10^{−3} | 1.73 × 10^{−3} |

SPEA2 | 1.64 × 10^{−2} | 8.18 × 10^{−1} | 5.03 × 10^{−1} | 8.01 × 10^{−1} | 8.01 × 10^{−1} | 1.076 | 9.27 × 10^{−1} | 9.22 × 10^{−1} | 6.85 × 10^{−5} | 1.77 × 10^{−3} | 1.11 × 10^{−3} | 1.73 × 10^{−3} |

WFG3 | ||||||||||||

MOSFP | 1.55 × 10^{−2} | 1.81 × 10^{−2} | 1.81 × 10^{−2} | 1.66 × 10^{−2} | 2.40 × 10^{−2} | 6.63 × 10^{−2} | 4.22 × 10^{−2} | 4.29 × 10^{−2} | 4.48 × 10^{−5} | 4.81 × 10^{−5} | 4.62 × 10^{−5} | 4.61 × 10^{−5} |

OMOPSO | 1.39 × 10^{−2} | 1.51 × 10^{−2} | 1.43 × 10^{−2} | 1.43 × 10^{−2} | 1.54 × 10^{−2} | 4.85 × 10^{−2} | 2.95 × 10^{−2} | 2.89 × 10^{−2} | 4.37 × 10^{−5} | 4.64 × 10^{−5} | 4.49 × 10^{−5} | 4.5 × 10^{−5} |

NSGA-II | 3.37 × 10^{−2} | 7.05 × 10^{−2} | 4.60 × 10^{−2} | 4.43 × 10^{−2} | 2.62 × 10^{−1} | 4.10 × 10^{−1} | 3.44 × 10^{−1} | 3.44 × 10^{−1} | 7.06 × 10^{−5} | 1.35 × 10^{−4} | 8.55 × 10^{−5} | 8.34 × 10^{−5} |

SPEA2 | 2.74 × 10^{−2} | 1.12 × 10^{−1} | 4.87 × 10^{−2} | 4.44 × 10^{−2} | 1.70 × 10^{−1} | 3.09 × 10^{−1} | 2.16 × 10^{−1} | 2.16 × 10^{−1} | 5.99 × 10^{−5} | 1.32 × 10^{−4} | 8.58 × 10^{−5} | 8.60 × 10^{−5} |

WFG4 | ||||||||||||

MOSFP | 2.87 × 10^{−2} | 1.91 × 10^{−1} | 4.71 × 10^{−2} | 4.16 × 10^{−2} | 3.10 × 10^{−1} | 4.38 × 10^{−1} | 3.72 × 10^{−1} | 3.74 × 10^{−1} | 1.13 × 10^{−4} | 1.55 × 10^{−4} | 1.25 × 10^{−4} | 1.24 × 10^{−4} |

OMOPSO | 1.25 × 10^{−2} | 4.47 × 10^{−2} | 1.90 × 10^{−2} | 1.52 × 10^{−2} | 9.02 × 10^{−2} | 2.43 × 10^{−1} | 1.38 × 10^{−1} | 1.27 × 10^{−1} | 8.13 × 10^{−5} | 1.14 × 10^{−4} | 8.86 × 10^{−5} | 8.51 × 10^{−5} |

NSGA-II | 6.26 × 10^{−2} | 1.55 × 10^{−1} | 1.09 × 10^{−1} | 1.10 × 10^{−1} | 4.55 × 10^{−1} | 6.69 × 10^{−1} | 5.56 × 10^{−1} | 5.53 × 10^{−1} | 2.90 × 10^{−4} | 4.30 × 10^{−4} | 3.60 × 10^{−4} | 3.54 × 10^{−4} |

SPEA2 | 3.01 × 10^{−2} | 4.17 × 10^{−1} | 1.04 × 10^{−1} | 7.02 × 10^{−2} | 2.46 × 10^{−1} | 3.51 × 10^{−1} | 3.00 × 10^{−1} | 3.02 × 10^{−1} | 1.01 × 10^{−4} | 1.74 × 10^{−4} | 1.16 × 10^{−4} | 1.14 × 10^{−4} |

WFG5 | ||||||||||||

MOSFP | 4.99 × 10^{−2} | 6.52 × 10^{−2} | 5.61 × 10^{−2} | 5.59 × 10^{−2} | 9.78 × 10^{−2} | 1.76 × 10^{−1} | 1.40 × 10^{−1} | 1.40 × 10^{−1} | 2.66 × 10^{−4} | 4.24 × 10^{−4} | 3.00 × 10^{−4} | 2.95 × 10^{−4} |

OMOPSO | 4.93 × 10^{−2} | 9.35 × 10^{−2} | 5.73 × 10^{−2} | 5.71 × 10^{−2} | 9.05 × 10^{−2} | 3.47 × 10^{−1} | 1.19 × 10^{−1} | 1.18 × 10^{−1} | 2.57 × 10^{−4} | 6.84 × 10^{−4} | 3.05 × 10^{−4} | 2.95 × 10^{−4} |

NSGA-II | 5.30 × 10^{−2} | 1.33 × 10^{−1} | 1.04 × 10^{−1} | 1.04 × 10^{−1} | 3.15 × 10^{−1} | 4.55 × 10^{−1} | 3.88 × 10^{−1} | 3.85 × 10^{−1} | 3.14 × 10^{−4} | 1.15 × 10^{−3} | 8.11 × 10^{−4} | 8.26 × 10^{−4} |

SPEA2 | 9.73 × 10^{−2} | 1.50 × 10^{−1} | 1.19 × 10^{−1} | 1.17 × 10^{−1} | 2.29 × 10^{−1} | 3.34 × 10^{−1} | 2.82 × 10^{−1} | 2.81 × 10^{−1} | 7.32 × 10^{−4} | 1.30 × 10^{−3} | 9.75 × 10^{−4} | 947 × 10^{−4} |

WFG6 | ||||||||||||

MOSFP | 1.55 × 10^{−2} | 2.08 × 10^{−2} | 1.70 × 10^{−2} | 1.69 × 10^{−2} | 8.58 × 10^{−2} | 1.52 × 10^{−1} | 1.20 × 10^{−1} | 1.22 × 10^{−1} | 4.98 × 10^{−5} | 5.46 × 10^{−5} | 5.21 × 10^{−5} | 5.21 × 10^{−5} |

OMOPSO | 1.29 × 10^{−2} | 1.45 × 10^{−2} | 1.36 × 10^{−2} | 1.36 × 10^{−2} | 7.70 × 10^{−2} | 1.42 × 10^{−1} | 1.05 × 10^{−1} | 1.06 × 10^{−1} | 4.71 × 10^{−5} | 5.16 × 10^{−5} | 4.90 × 10^{−5} | 4.88 × 10^{−5} |

NSGA-II | 3.49 × 10^{−2} | 1.30 × 10^{−1} | 6.10 × 10^{−2} | 5.75 × 10^{−2} | 3.19 × 10^{−1} | 5.17 × 10^{−1} | 3.92 × 10^{−1} | 3.86 × 10^{−1} | 7.59 × 10^{−5} | 5.32 × 10^{−4} | 1.64 × 10^{−4} | 1.45 × 10^{−4} |

SPEA2 | 3.41 × 10^{−2} | 2.13 × 10^{−1} | 8.74 × 10^{−2} | 7.79 × 10^{−2} | 2.52 × 10^{−1} | 6.56 × 10^{−1} | 3.61 × 10^{−1} | 3.44 × 10^{−1} | 7.17 × 10^{−5} | 4.08 × 10^{−4} | 1.76 × 10^{−4} | 1.67 × 10^{−4} |

WFG7 | ||||||||||||

MOSFP | 1.47 × 10^{−2} | 1.89 × 10^{−2} | 1.57 × 10^{−2} | 1.56 × 10^{−2} | 9.38 × 10^{−2} | 1.44 × 10^{−1} | 1.20 × 10^{−1} | 1.20 × 10^{−1} | 4.94 × 10^{−5} | 5.38 × 10^{−5} | 5.14 × 10^{−5} | 5.14 × 10^{−5} |

OMOPSO | 1.29 × 10^{−2} | 1.45 × 10^{−2} | 1.36 × 10^{−2} | 1.36 × 10^{−2} | 8.68 × 10^{−2} | 1.38 × 10^{−1} | 1.10 × 10^{−1} | 1.09 × 10^{−1} | 4.71 × 10^{−5} | 5.10 × 10^{−5} | 4.93 × 10^{−5} | 4.93 × 10^{−5} |

NSGA-II | 2.98 × 10^{−2} | 7.71 × 10^{−2} | 4.23 × 10^{−2} | 3.98 × 10^{−2} | 3.13 × 10^{−1} | 4.48 × 10^{−1} | 3.82 × 10^{−1} | 3.79 × 10^{−1} | 6.75 × 10^{−5} | 8.45 × 10^{−5} | 7.47 × 10^{−5} | 7.42 × 10^{−5} |

SPEA2 | 2.85 × 10^{−2} | 3.34 × 10^{−1} | 8.82 × 10^{−2} | 7.23 × 10^{−2} | 2.41 × 10^{−1} | 4.41 × 10^{−1} | 2.92 × 10^{−1} | 2.89 × 10^{−1} | 6.00 × 10^{−5} | 3.34 × 10^{−4} | 7.35 × 10^{−5} | 6.85 × 10^{−5} |

WFG8 | ||||||||||||

MOSFP | 3.05 × 10^{−1} | 4.82 × 10^{−1} | 3.66 × 10^{−1} | 3.66 × 10^{−1} | 6.30 × 10^{−1} | 1.071 | 7.54 × 10^{−1} | 7.34 × 10^{−1} | 1.73 × 10^{−3} | 2.18 × 10^{−3} | 2.02 × 10^{−3} | 2.02 × 10^{−3} |

OMOPSO | 5.31 × 10^{−2} | 4.89 × 10^{−1} | 3.78 × 10^{−1} | 4.87 × 10^{−1} | 4.05 × 10^{−1} | 7.83 × 10^{−1} | 5.06 × 10^{−1} | 4.89 × 10^{−1} | 2.33 × 10^{−4} | 2.11 × 10^{−3} | 1.79 × 10^{−3} | 2.08 × 10^{−3} |

NSGA-II | 1.94 × 10^{−1} | 7.27 × 10^{−1} | 4.62 × 10^{−1} | 4.97 × 10^{−1} | 6.60 × 10^{−1} | 9.34 × 10^{−1} | 7.95 × 10^{−1} | 7.94 × 10^{−1} | 1.10 × 10^{−3} | 2.61 × 10^{−3} | 2.26 × 10^{−3} | 2.37 × 10^{−3} |

SPEA2 | 3.99 × 10^{−1} | 8.15 × 10^{−1} | 6.14 × 10^{−1} | 6.00 × 10^{−1} | 6.45 × 10^{−1} | 9.09 × 10^{−1} | 7.67 × 10^{−1} | 7.58 × 10^{−1} | 2.04 × 10^{−3} | 2.68 × 10^{−3} | 2.44 × 10^{−3} | 2.46 × 10^{−3} |

WFG9 | ||||||||||||

MOSFP | 3.59 × 10^{−2} | 1.05 × 10^{−1} | 9.26 × 10^{−2} | 9.28 × 10^{−2} | 1.37 × 10^{−1} | 2.27 × 10^{−1} | 1.84 × 10^{−1} | 1.83 × 10^{−1} | 8.15 × 10^{−5} | 9.25 × 10^{−5} | 8.69 × 10^{−5} | 8.70 × 10^{−5} |

OMOPSO | 1.53 × 10^{−2} | 8.35 × 10^{−2} | 7.58 × 10^{−2} | 7.81 × 10^{−2} | 6.31 × 10^{−2} | 1.20 × 10^{−1} | 9.51 × 10^{−2} | 9.56 × 10^{−2} | 5.32 × 10^{−5} | 5.77 × 10^{−5} | 5.50 × 10^{−5} | 5.50 × 10^{−5} |

NSGA-II | 8.76 × 10^{−2} | 1.76 × 10^{−1} | 1.09 × 10^{−1} | 1.04 × 10^{−1} | 3.05 × 10^{−1} | 4.39 × 10^{−1} | 3.63 × 10^{−1} | 3.65 × 10^{−1} | 8.17 × 10^{−5} | 1.19 × 10^{−4} | 9.87 × 10^{−5} | 9.81 × 10^{−5} |

SPEA2 | 8.86 × 10^{−2} | 2.44 × 10^{−1} | 1.30 × 10^{−1} | 1.21 × 10^{−1} | 2.29 × 10^{−1} | 3.25 × 10^{−1} | 2.86 × 10^{−1} | 2.86 × 10^{−1} | 7.76 × 10^{−5} | 1.54 × 10^{−4} | 9.38 × 10^{−5} | 9.32 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Shehadeh, H.A.; Ldris, M.Y.I.; Ahmedy, I.
Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP). *Symmetry* **2017**, *9*, 241.
https://doi.org/10.3390/sym9100241

**AMA Style**

Shehadeh HA, Ldris MYI, Ahmedy I.
Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP). *Symmetry*. 2017; 9(10):241.
https://doi.org/10.3390/sym9100241

**Chicago/Turabian Style**

Shehadeh, Hisham A., Mohd Yamani Idna Ldris, and Ismail Ahmedy.
2017. "Multi-Objective Optimization Algorithm Based on Sperm Fertilization Procedure (MOSFP)" *Symmetry* 9, no. 10: 241.
https://doi.org/10.3390/sym9100241