Solving constrained engineering design problems with multi-objective artificial algae algorithm

Abstract


Introduction
Optimization problems with multiple objectives are called multi-objective optimization problems (MOOPs), and simultaneous optimization of these objectives is called multiobjective optimization (MOO).Real-world problems are generally in the type of NP-hard MOOPs.NP-hard means that it cannot be proven that there is a solution in polynomial time, or that the algorithms that can solve it efficiently are not known [1]; examples of these problems are found in engineering design, product and process design, land-use planning, management science, economics etc.In MOOPs, objective functions are generally inversely proportional.In other words, obtaining a satisfactory solution for an objective function result in a poor solution for the other objective function.Thus, it is not possible to obtain a global best solution as in single-objective problems, but instead, a solution set consisting of the best solutions is obtained.
Engineering design problems are one of the most important real-world problems, many of which are constrained problems [44].Since optimal solutions of engineering design problems are difficult to find, metaheuristic algorithms are used in most studies [40], [45], [46].
The test set consists of 14 problems, two unconstrained and twelve constrained.The MOAAA was compared with NSGA-II, MOCell, MOVS, IBEA and PAES algorithms.The results obtained showed that the MOAAA performed better than the comparison algorithms on the used problem set.
The study is organized as follows.In section 2, MOOPs, the concept of Pareto and performance metrics are presented.In section 3, AAA is summarized.The MOAAA is explained in detail in section 4. Section 5 shows the results and performance analysis of the algorithms used on the problems.Finally, section 6 details conclusion and recommendations for further studies.

The major addition of the research
The MOAAA is a recently proposed technique for the solution of MOOPs.The MOAAA was first tested on unconstrained MOOPs and produced successful results.EPSILON and iv.
IGD.The obtained metric results show that the MOAAA is generally superior to the comparison algorithms.
2 Multi-objective optimization, the pareto theorem and performance metrics

Multi-objective optimization problems
MOOPs are mathematically defined as follows.[28,47]: Where  is the number of functions,  is the number of inequality constraints,  is the number of equality constraints, and  is the number of decision variables.Also,   assigns the  ℎ inequality constraint, ℎ  assigns the  ℎ equality constraint,  is a candidate agents in the search area,   is lower bound and   is upper bound of the  ℎ decision variable.

Pareto theorem
In the MOOPs, since functions are generally inversely proportional to each other, numerous solutions that produce different values for different objective functions are formed.Researchers generally use the Pareto Theorem to determine the best ones by comparing these solutions.Where  is the ideal set of solutions -Those inside the search space that do not violate the problem constraints-and ,  ∈ , the Pareto Theorem consists of the following 4 rules [47]: 1. Pareto-dominance: If solution A is not poorer than solution B for any objective and is better at least in one objective, it dominates solution B and it is denoted as  ≺ .For a minimization problem, the mathematical notation of this rule is given in Equation (2).
If solutions A and B produce better values than each other in any objective, these are named as non-dominated solutions.
2. Pareto-optimal (PO): If no element in Q dominates solution A, it means that A is a PO solution.
3. Pareto-optimal-set (PS): A set of position vectors in the search area of PO solutions in Q.
4. Pareto-optimal-front (PF): A set of position vectors in the objective space of solutions in PS.

Performance metrics
In MOO, Pareto-optimal-fronts generated by the algorithms (Pareto front-estimated,  ) are expected to ideally estimate the true Pareto-optimal-fronts (Pareto-front-true,   ).The mathematical formulas of the performance metrics are given below [47], [48].

Artificial algae algorithm (AAA)
The AAA was inspired by the behaviors of the real algae, such as turning towards the source of light, growth by photosynthesis, reproduction by mitosis after reaching a sufficient size and adaptation to medium for survival.AAA was initially applied to solve single-objective problems and achieved quite successful results.In Figure 1, the main steps of the AAA are given; detailed information about the AAA can be obtained from [50].

Multi-objective artificial algae algorithm (MOAAA)
New multi-objective algorithms are proposed by rearranging metaheuristic algorithms that succeeded in the field of singleobjective optimization using suitable strategies (Pareto-based, decomposition-based etc.).Unfortunately, implementation of these strategies is not sufficient for the success of the new algorithms because, while single-objective algorithms are intended to find a single point that produces the best solution, multi-objective algorithms are intended to find the best set of solutions (Pareto-front).Therefore, the ability of singleobjective algorithms to distribute solutions needs to be improved.The arrangements for using the artificial algae algorithms in solving multi-objective problems are explained below.

Non-domination rank & Crowding-distance strategies
Non-domination rank (NDR): Non-dominated solutions were mentioned while explaining Pareto-dominance in section 2.2.When non-domination rank (NDR) strategy is used with NSGA-II [3], each set consisting of non-dominated solutions is indicated by a different number.The first front (FR1) in the population contains the best solution and is called the Paretofront (PF).Selecting solutions with a small front number as the parent solution, or transferring them to the next generation, contributes to the convergence performance of the algorithm.

Crowding-distance (CRD):
The NDR information is not sufficient to choose a solution from two coexisting solutions in the same front.Therefore, Deb et al. proposed the CRD strategy to determine which choice of solution will have a valuable contribution to the diversity.The CRD is calculated as follows: i.
The values obtained for each objective function by each solution on the same front are sorted in ascending order, ii.
The CRD values of the outermost solutions are assigned as infinite, iii.
The CRD values of the solutions in between are calculated for each objective function by normalizing the difference between the two closest neighboring solutions.The CRD value of a non-extreme solution, with the total of M objective functions, is calculated as in Eq. 9.
In MOAAA, NDR and CRD are used in two cases: i.
When the parent algal colony cells are selected using binary tournament selection, ii.
When the best N of the 2N solutions-the main population (N) and the child solutions (N)-is transferred to the next generation at the end of each iteration.

Calculation of quality ranking (QR)
Real algae grow by photosynthesizing as they approach a light source.The algae that are closer to the light source will grow more because the rate of photosynthesis will increase.This is modeled in AAA as follows: the algal colony is initialized with the same sizes (Greatness) (Algorithm-1 Step 2), the sizes are increased at every iteration according to the values of Greatness and the objective function (Algorithm-1, Step 3).The values of the objective function are normalized in the calculateGreatness(Gi, f(Xi)) function.A scaler value representing the quality of the solutions is needed for the normalization, so using the objective vector in MOO is not applicable.Calculation of quality ranking (QR), which calculates the quality rankings of the solutions according to NDR and CRD values, is proposed to overcome this problem.In this calculation, QR values of the extreme solutions in the first front are assigned as 1, and other solutions are sorted in descending order according to CRD values and their QR values are increased by 1.The same procedure is repeated for all remaining solutions, where the QR value of the extreme solutions in the second front is 1 more than the highest QR value in the first front.Figure 2 gives an example of how the QR values are calculated.
Figure 2. Calculation of QR value.

Polynomial mutation (PM)
In the original AAA, the evolutionary process and adaptation allow the failed algal colony cells to be influenced by the most successful colony cells to go to better locations.While the strategies comparing the failed solutions to the most successful one have a positive contribution to the convergence process, they have a negative effect on diversity performance in multiobjective optimization algorithms.Therefore, polynomial mutation [51], which contributes to the diversity, is added in MOAAA instead of the evolutionary process and adaptation.Main steps of the MOAAA is given in Figure 3.

Experimental environment
This study was carried out in the jMetal 4.5 environment, which is a multi-objective optimization software package coded in Java.While NSGA-II, PAES MOCell and IBEA algorithms used in the study were available in the jMetal package, MOVS and MOAAA were coded by the authors.Engineering design problems and the KITA problem were also coded by the authors and added to the package.Since the size of the problems was small (between 2 to 7 dimesions), the maximum function evaluation numbers (maxFES) of the algorithms were kept relatively low.All operations were repeated 50 times for 4000 maxFES.A parameter analysis study was also carried out in order to determine the optimal values of the K and le parameters used in the MOAAA for the problem set in this study.In the original AAA algorithm, parameter K was used as 2 and parameter le as 0.3.In this study, the K parameter was kept constant at 2, the le parameter was increased by 0.1 between 0.1 and 1, and 10 different MOAAA versions were run on the problem set with 50 repetitions.The results of the obtained solution sets in EPSILON metric were ranked by Friedman test.The obtained results showed that the parameter le obtained the most successful results for the value of 0.3.Secondly, the parameter le was kept constant at 0.3 and the parameter K was tested by increasing it from 1 to 5. Obtained results showed that K parameter gives the most successful results for 2 values.The population was taken as 100 for all algorithms; other parameters are set to default values as given in Table 3.The Pareto-front-true (PFt) solutions for the problems were obtained to calculate these indicators.PFt solutions of the problems in the jMetal package were taken from the software website [53].PFt solutions of the problems coded by the authors were obtained by merging the Pareto-front-estimated (PFe) solutions that were generated by solving each problem 50 times for all algorithms and separating the non-dominated solutions (maximum 500 solutions) within these.The values obtained by the algorithms for the four performance metrics are given in Tables 4 to 7. The values are expected to be high for HV and low for the others.For readability, the two best results are highlighted in dark gray (the best) and light gray (the second best).
The SPREAD metrics in Table 5 show that the MOCell is the most successful algorithm by taking first or second place in 13 (9+4) problems.The MOAAA, NSGA-II and PAES takes first or second place in 10 (4+6), 4 (1+3) and 1 (0+1) problems respectively.
The IGD metrics in Table 7 show that, the MOAAA is the most successful algorithm by taking first or second place in 9 (7 + 2) problems.NSGA-II, MOCell and MOVS takes first or second place in 8 (3+5), 5 (3+2) and 5 (1+4) problems respectively.In Table 8, the Friedman test [54] results, which compare the average rankings of the algorithms, are given for each performance metric.In the Friedman test, it is desirable that the average ranking is high for HV and low for others.According to the results, the MOAAA has the best ranking for all metrics except the SPREAD metric.The MOAAA has the second-best ranking for the SPREAD metric.The NSGA-II has the secondbest ranking of all the metrics, except for SPREAD.The MOCell has the best ranking for SPREAD.

Conclusions and recommendations
In this study, the performance of the MOAAA, a recently proposed multi-objective optimization algorithm, has been tested for constrained benchmarks and engineering design problems.The test set consisted of 14 well-known problems.
The values obtained for HV, SPREAD, EPSILON and IGD from the MOAAA test set are compared with the well-known NSGA-II, PAES, MOCell, IBEA algorithms and the recently proposed MOVS algorithms.When the Friedman test, which compares the average rankings of the algorithms was applied, it was observed that MOAAA had the best ranking of all metrics, except for SPREAD.Furthermore, when the Pareto fronts and the boxplots were analyzed, it is seen that MOAAA was a consistent and stable algorithm that successfully estimated the Pareto fronts.Finally, the Wilcoxon rank sum test showed that MOAAA is a unique algorithm that produces statistically significant results different to the compared algorithms.The results show that MOAAA is an alternative method that generates successful results in solving real-world multiobjective problems.
In further studies, researchers could suggest modifications to enhance the distribution performance-SPREAD-of the MOAAA, or use MOAAA in solving discrete, dynamic or hybrid multiobjective real-world problems.

Author contribution statement
In this study, ÖZKIŞ focused on forming the idea, conducting experimental studies and evaluating the results; BABALIK, on the other hand, contributed to the review of the literature, spelling and checking the article in terms of content.

Figure 1 .
Figure 1.Main steps of the AAA.
, SPREAD, EPSILON and HV quality indicators were used to compare the performances of the algorithms.Table 3. Parameter values of algorithms used in runs.
Distribution of   over   .The quality of the algorithms was determined by comparing the   solutions they produced.When the   solutions produced by two different algorithms are similar, it is not possible to distinguish the better one by observation.Therefore, researchers have developed mathematical performance metrics that calculate convergence and diversity of the   solutions.Some of these metrics calculate either convergence or diversity, while others calculate both.The inverted generational distance (IGD), Hypervolume (HV), EPSILON and SPREAD metrics used in this study are explained below: ]: It evaluates the quality of distribution of the objective function vectors in   .It is calculated by using the distance ( (,   )) of the vectors in   to each other and to extreme solution vectors ( 1 ,  2 , … ,   ) in   .It is used to compute the diversity performance of a   , • EPSILON[48]: It evaluates the minimum distance which is needed for converting every solution in   with a view to it is able to dominate the   of the problem, • IGD [48, 49]: It is used to measure the average from   to   .

5 Results and performance analysis 5.1 The test set
[52]test set consists of 14 different multi-objective optimization problems: 7 benchmarks and 7 engineering designs.While twelve of these problems have various constraints, 2 of them are unconstrained problems.The number of objective, constraint and decision variables of the problem set are given in Table2.The mathematical formulas of the problems are given in [1],[11],[27],[28],[52].

Table 2 .
Benchmarks and engineering desing problems.

Table 4 .
HV metrics of the algorithms.

Table 5 .
SPREAD metrics of the algorithms.

Table 6 .
EPSILON metrics of the algorithms.

Table 7 .
IGD metrics of the algorithms.
B: Best.S: Second best.

Table 8 .
12e average rankings of the algorithms for metrics.The Wilcoxon's rank sum test was applied at a 95% confidence level to show whether the metric values obtained by the MOAAA are statistically different to the other algorithms or not.If the p (probability) value is smaller than 0.05, it is denoted by "+" and shows that MOAAA statistically differs from the other algorithms.The Wilcoxon rank sum test results in Tables 9 to12show that results produced by the MOAAA are statistically different compared to the other algorithms.Estimated Pareto fronts (PFe) obtained by the algorithms are compared with the real Pareto fronts (PFt) in Figure4.When the figures are examined, it is observed that the proposed MOAAA generally estimates the PFt more successfully than the other algorithms.The box plot showing the results of the problems executed 50 times for each metric is given in Figure5.When the box plot is examined, it is seen that MOAAA is a successful algorithm that produces robust results in solving the multi-objective optimization problems in the test set.

Table 9 .
Wilcoxon's rank sum test results for HV metric.

Table 10 .
Wilcoxon's rank sum test results for SPREAD metric.

Table 11 .
Wilcoxon's rank sum test results for EPSILON metric.

Table 12 .
Wilcoxon's rank sum test results for IGD metric.