Pareto based differential evolution with homeostasis based mutation

Abstract

Differential evolution algorithm is a very popular nature inspired algorithm and it is used for solving single objective optimization problems. Due to its high convergence rate and ability to produce diverse solutions, it has been extended to solve multiobjective optimization problems. Many versions of DE were proposed however multiobjective versions, where concept of Pareto optimality is used, are most popular. Pareto Based Differential Evolution (PBDE) is one of them. Although performance of this algorithm is very good, yet its convergence rate can be further improved by minimizing the time complexity of nondominated sorting and by improving the diversity among solutions. This has been implemented by using efficient nondominated algorithm whose time complexity is better than the previous one and a new mutation scheme is implemented in DE which can provide more diversity among solutions. The proposed variant adds one more vector named as Homeostasis mutation vector in the existing mutation vectors to provide more bandwidth for selecting effective mutant solutions. The proposed approach provides more promising solutions to guide the evolution and helps DE escaping the situation of stagnation. Performance of proposed algorithm is evaluated on twelve benchmark test functions (bi-objective and tri-objective) on Pareto-optimal front. Performance of proposed algorithm is compared with other state-of-the-art algorithms on five multiobjective evolutionary algorithms (MOEAs). The result verifies that our proposed Homeostasis mutation strategy performs better than other state-of-the-art variants.

Keywords

Adaptation optimization multi-objective evolutionary algorithm

1 Introduction

DE was proposed by Rainer Storn and Kenneth Price [28] to solve optimization problems. The theory of DE is inspired from the biological phenomenon of natural selection and evolution. It updates randomly generated initial population using three operators mutation, crossover, and selection. Due to its high convergence, many attempts have been made to extend this algorithm for multiobjective optimization problems. First attempt was made by Babu et al. [5] where he used weight based method for solving multiobjective problem by single objective DE algorithm. This algorithm was focused to get one optimum solution for the problem and the concept of Pareto front was completely missing. It was felt that Pareto based approaches may produce more suitable solutions for multiobjective problems. Many new variants suggested that Pareto based DE algorithm (PBDE) [20] is very easy and popular as it uses a Pareto based ranking method to produce new offspring. Although performance of this algorithm is very good yet its convergence rate can be further improved by minimizing the time complexity of nondominated sorting and by producing more diverse solution.

The most time consuming part of PBDE algorithm is nondominated sorting. Any reduction in the time complexity of nondominated sorting can enhance the convergence of PBDE algorithm. Moreover other problem which is associated with PBDE is stagnation that means the solutions produced by PBDE may get stuck in local optimum which occurs due to lack of diversity in newly generated solutions. Since in any evolutionary algorithm, mutation operator is responsible for maintaining diversity among solutions, this shows that diversity provided by classical mutation operator is not sufficient, it should be modified so that diverse solutions may be generated and stagnation problem can be handled. This removal of stagnation problem will also improve the convergence speed and diversity of PBDE algorithm.

Therefore to prevent solutions on converging local optimal solutions and to improve the convergence rate of the algorithm, a new variant named multiobjective differential evolution MODE with Homeostasis based mutation(MODE-HBM) is proposed. In MODE-HBM nondominated sorting algorithm with new mutation strategy is used. This variant is inspired from the process of Homeostasis whose whole sole purpose is to maintain the internal environment of the body. The purpose of this variant is to improve the convergence rate of the algorithm and to maintain the same diversity which was created during initial generations by adding additional vectors in mutation operator. In proposed algorithm, diversity is considered as internal environment. To check the performance of proposed Homeostasis variant, this algorithm is compared with other state-of-the-art algorithms. Performance analysis shows that this variant better perform other variants of DE algorithm.

The rest of this paper is organized as follows, in Section 2, background details are discussed. In Section 3, related works are elaborated, In Section 4 proposed Homeostasis variant are discussed and in Section 5 experimental result and analysis is shown and finally in Section 6 conclusions are drown.

2 Background details and related work

2.1 Multiobjective optimization problem

Multiobjective optimization problems (MOPs) involve simultaneous optimization of multiple conflicting objective functions. The nature of multiobjective optimization problem is very different than the single-objective optimization. In single objective optimization, the target of optimization algorithm is to capture global optimum solution as early as possible. However multiobjective optimization algorithm has to capture a set of solutions which are superior to the rest of solutions, in the search space. These solutions are called as nondominated solutions. The other solutions are known as dominated solutions. All nondominated solutions are equally acceptable to various users. Preference of one solution over the other requires problem knowledge and a number of problem-related factors. Thus, one solution provided by a designer may not be acceptable to another designer or in a changed environment. Therefore, it may be useful to have solutions of thePareto-optimal.

Multi-objective optimization problem (MOP) [2] can be defined as the problem of finding all solutions and $Minimize f (x) = (f_{1} (x_{1}), f_{2} (x_{2}), \dots f_{k} (x_{n}))^{T}$ (1) $x ε ω$ (2) Where ω is the decision space and x = (x₁, x₂, …, x_n) ^T is the vector of decision variable (n is the number of variable). Therefore, f (x), which contains k functions.

2.2 Pareto dominance

Pareto dominance deb et al. [27] is a set of solutions which are performing better than the other solutions and hence said to be Pareto dominated over those outcomes. For example, a solution set x = (x₁, x₂, …, x_n) ^T will be said to be dominating over y = (y₁, y₂, …, y_n) ^T if the condition below: for all i εε (1, 2, …, k) : f_i (x) ≤ f_i (y) there exist i ε (1, 2, …, k) : f_i (x) < f_i (y).

2.3 Nondominated sorting

The non-dominated sorting is very important part of many multiobjective optimization algorithms. Sometimes we are interested in identification of nondominated front. Algorithms used for the identification of nondominated fronts are required to classify the population into two sets-the non-dominated set and the remaining dominated set. However, all nondominated sorting algorithms classify all solutions into various non-domination levels. Such algorithms sort population according to an ascending level of non-domination. The classification of solutions in various levels is called nondominated sorting and it consumes most of the time of any optimization algorithm.

In 2002, Deb et al. proposed a new approach for reducing the time of nondominated sorting and it takes O (MN²) for sorting the entire solutions into different Pareto fronts.

Till 2015, this algorithm was the best algorithm for nondominated sorting but now a new nondominated sorting algorithm is proposed by X Zhang et al. [26] which takes less number of comparisons by removing unnecessary comparisons. This algorithm is named as efficient approach for nondominated sorting abbreviated as [ENS].

2.4 DE algorithm

Differential Evolution is a stochastic population-based algorithm for continuous optimization problems. It performs extremely well on a wide variety of test suites. In DE, mutation and crossover operators are used to generate trail vectors and then selection operator is applied to identify the best solution between parent and trail vectors. These selected solutions will be the part of new generation. If population consists of NP parameter vectors $α_{i, G} = α_{i}^{1},_{G}, α_{i}^{2},_{G}, . . . . . . . α_{i}^{D},_{G}$ (3) where i = 1, 2, 3, … NP, D denotes dimension and G denotes generation. The operators and random parameters used in DE are mutation, crossover and selection which are described as shown below.

Mutation. A mutation strategy can be used to generate mutant vector γ_i,G+1 from target vector α_i,G. Simplest mutation strategy is also known as DE/rand/1 mutation strategy. Following this strategy many variation of mutation operators were suggested. Those mutation strategies are represented with different names. Some important mutation strategies are highlighted below. These strategies [1 , 33] generate donor vectors in DE as follows:

DE/best/1: $γ_{i, G} = α_{b e s t, G} + δ_{1} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G})$ (4)

DE/rand-to-best/l:

$\begin{matrix} γ_{i, G} & = & α_{i, G} + δ_{1} \cdot (α_{b e s t, G} - α_{i, G}) \\ + δ_{2} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G}) \end{matrix}$ (5)

DE/best/2:

$\begin{matrix} γ_{i, G} & = & α_{b e s t, G} + δ_{1} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G}) \\ + δ_{2} \cdot (α_{r_{3}^{i}, G} - α_{r_{4}^{i}, G)} \end{matrix}$ (6)

DE/rand/2:

$\begin{matrix} γ_{i, G} & = & α_{r_{1}^{i}, G} + δ_{1} \cdot (α_{r_{2}^{i}, G} - α_{r_{3}^{i}, G}) \\ + δ_{2} \cdot (α_{r_{4}^{i}, G} - α_{r_{5}^{i}, G)} \end{matrix}$ (7)

Where α_best denotes best vector, α_{r
₁}, α_{r
₂}, α_{r
₃}, α_{r
₄} and α_{r
₅} are random vectors, γ_i,G denotes mutant vector, i denotes number of population (NP), G denotes generation and δ₁ and δ₂ are mutation factors.

Crossover. After mutation, crossover operator is used to create a trial vector β_j,i,G+1 for each pair of target vector α_j,i,G and its corresponding mutant vector γ_j,i,G+1. Commonly used crossover schemes in DE are Binomial and exponential crossovers. The binomial crossover is as follows. $β_{i, G + 1} = (β_{1 i, G + 1}, β_{2 i, G + 1}, \dots, β_{Di, G + 1})$

Where $β_{j, i, G + 1} = {\begin{matrix} γ_{j, i, G + 1}, & if r (j) \leq CR or j = rn (i) \\ α_{j, i, G}, & if r (j) > CR and j \neq rn (i) \end{matrix}$ (8)

Where j = 1,2,.... D, r(j) ε [0,1], jth evaluation of a uniform random generator number, Cr is the crossover constant ε [0,1], rn(i) ε (1,2,...D) is a randomly chosen index which ensures that β_j,i,G+1 gets at least one element from γ_j,i,G+1, otherwise, no new parent vector would be produced and then the generated population would not be altered.

Selection. ‘Survival of the fittest’ is the basic principle of selection. The fitness of trial and that of target vector is compared and solution with better fitness is selected for the next generation. The selection operator for minimization problems is expressed as follows:

$\begin{matrix} α_{i, G + 1} \\ = {\begin{matrix} β_{i, G + 1}, & if f (β_{i, G + 1}) < f (α_{i, G}) for \min prob . \\ α_{i, G}, & otherwise \end{matrix} \end{matrix}$ (9)

For j = 1, 2, …, D. iff the trail vector β_i,G+1 gives better fitness value than α_i,G then β_i,G+1 is set to α_i,G+1; otherwise, the old value α_i,G is continued.

2.5 Crowding distance

Crowding distance method is a popular approach for checking the diversity among the solutions. It was proposed by K. Deb et al. in 2002 [27] in which the calculation of crowding distance was done by dividing the entire search space dⁿ into subspaces, where d is the depth parameter and n is the number of decision variables, and by updating the subspaces dynamically.

The overall crowding distance value is calculated as the sum of individual distance values corresponding to each objective. Each objective function is normalized before calculating the crowding distance.

3 Related works

Differential Evolution (DE) was proposed by Storn and Price in 1995 [33]. It is a well known optimization technique which works on very simple logic and takes reasonable amount of time in capturing nearly global optimum solution.

Early attempts to solve multiobjective optimization problems were made by Babu et al and Li at el. [5, 6]. They used weighted and decomposition based approach to convert a multiobjective problem into a single objective problem. Finally DE was used to handle single objective optimization problem. Then after it was felt that the concept of Pareto based ranking might be more suitable for multiobjective problems. Hence many researchers proposed multiobjective variants of DE, some of them are listed below.

Chang et al. [4] proposed a new multiobjective variant of DE in which DE/rand/1/bin mutation strategy was implemented with external archive to store nondominated solutions. In this approach selection operator was implemented in such a manner that not only selects the nondominated solutions but also maintain the diversity among solutions.

Abbass et al. [1] proposed a Pareto differential evolution [PDE] algorithm in 2001 where initial population was generated by Gaussian distribution. Dominated solutions generated after mutation was used for recombination and selection. These steps were repeated until the optimal Pareto front is generated. After some time, he introduced a new variant of PDE algorithm named as SPDE algorithm [2] where self adaptive strategy was implemented in crossover and mutation. Another important variant of PDE, MPANN [3] was created by Abbass where neural network was used for local convergence.

Generalized Differential Evolution (GDE) was suggested by Kukkonen and Lampinen [7] to solve global and multiobjective optimization problems (either constrained or unconstrained). In its first version, Pareto dominance was used as a selection criterion to choose solutions from old population member and the trial vector. Further to promote better distribution among nondominated solutions GDE2 was proposed [8].

ε-MyDE [40] by Santana-Quintero and Coello Coello implement the concept of epsilon nondominated sorting to create Pareto front. It takes less time as compared to other algorithms for finding out good solutions.

A new multiobjective version of differential evolution was proposed by Portilla Flores et al. [9] to solve multiobjective problems. Alike GDE Pareto Dominance was used as a selection criterion to choose solutions from old population member and the trial vector.

In 2010, QU et al. [10] proposed a new multiobjective differential evolution algorithm where diversity enhancement procedure was used to avoid local convergence. The randomly generated parameter vectors provide large differential vectors which improve the global search ability and avoid the algorithm becoming trapped in an inferior local Pareto front.

In 2010, Wang et al. [11], proposed a multiobjective version of self-adaptive differential evolution algorithm which incorporate Pareto dominance approach to solve multiobjective optimization problems. A self-adaptive strategy is used to set the values of parameters F and CR. This parameter setting strategy improved the robustness of DE greatly. Moreover a new strategy named as crowding entropy tactic was used by Wang for maintaining the diversity among the nondominated solutions, this method was better as compared to the method which was proposed by K. Deb.

Non dominated Sorting Differential Evolution (NSDE) was proposed by R Angira and B V Babu [12] to solve optimization problems. Approach of NSDE is a very similar to NSGA-II algorithm, however in NSDE operators of GA are replaced with the operators of Differential Evolution.

In 2011, Li et al. [13] proposed some adaptive strategies for the identification of operators and to control the parameters CR and F of DE algorithm. These strategies were used with Pareto dominance strength to improve the convergence rate of multiobjective algorithm. Moreover to obtain better spread of solutions, he used a tree neighborhood density estimator. Additionally, a novel replacement mechanism is proposed, based on a three step comparison procedure.

MOEA/D was proposed by Zhang and Li [14] in 2007 as a alternative to NSGA-II algorithm. This algorithm was claimed to be effective and efficient algorithm. Unlike other multiobjective MODE variants, MOEA/D works on decomposition method. To solve any multiobjective problem, first of all, the problem is decomposed in many scalar problems and then optimization of each scalar problem is performed simultaneously. Later a new version is suggested by the same authors in 2009 [15]. To overcome the drawbacks of new MOEA/D, UMODE/D was proposed by Y. Y. Tan et al. [16]. Two modifications were implemented in this algorithm. First for uniform distribution of subpopulation, uniform design method is used. Moreover Local search is implemented with simplified quadratic approximation using three best points.

In 2005, a new version of multiobjective DE known as DEMO was proposed by Robic and Filipic [17]. In DEMO, a parent population is replaced by its child only if child dominates parent otherwise this newly created candidate is added to the existing pool. Finally nondominated sorting with crowding distance is used to reduce the size of increased population.

Adaptive differential evolution algorithm (ADEA) for multiobjective problem was proposed by Qian et al. in 2008 [19]. Alike li et al. [13], the value of F was adjusted by checking the current Pareto fronts and the diversity among the current solutions. A combination of Pareto-based ranking and crowding distance sorting are used for selection. ADEA was able to produce good convergence rate and diversity among solutions.

Pareto based multiobjective differential evolution algorithm was suggested by Xue et al. [20]. Alike other multiobjective variants of DE, concept of Pareto based ranking and crowding distance based approach were used to capture Pareto fronts.

In 2015, Chen et al. [21], proposed a combination of DE and simulated annealing algorithm for solving multiobjective problems. Here simulated annealing was used to guide the algorithm to identify all probable areas where solution may exist.

Wang et al. 2011 [44], proposed a composite DE i.e. CoDE, which used three trail vector generations strategies proposed earlier for various DE variants i.e. rand/1/bin, rand/2/bin and current - to - rand/1 and the three control parameter settings i.e. [F = 1.0, Cr = 0.1], [F = 1.0, Cr = 0.9] and [F = 0.8, Cr = 0.2]. The proposed scheme had been shown to perform better than JADE, jDE, SADE, EPSDE, CLPSO, CMA-ES and GL-25.

Adaptive mutation operator was used by Shi et al. [22] to avoid premature convergence by adaptively tuning the mutation scale factor F. Moreover ε-dominance strategy was used to update the archive that stores the nondominated solutions. Experiments was conducted on five widely used multiple objective benchmark functions and better convergence and diversity was predicted.

A modified differential evolution algorithm was propose by Ali et al. [23] to solve single objective optimization problems. For improving the performance of DE algorithm concept of opposition based learning was introduced in it. Further this author extended MDE and proposed a new Multi-Objective Differential Evolution Algorithm named as (MODEA) [24]. Opposition-based learning with random localization in mutation was used to improve the performance of this algorithm. In addition to it, a new selection mechanism was proposed for generating a well distributed Pareto optimal front. The performance of algorithm was compared with other algorithms and it was proved to perform better in comparison to other ones.

In 2009, Adeyemo and Otieno [18], suggested to solve multiobjective problems via adjusting the selection scheme of traditional DE and named the algorithm as multiobjective differential evolution algorithm.

In 2014, Chen et al. [25] proposed a new variant of multiobjective differential evolution (MODE) and named it MODE-RMO for short. A new theory was proposed in this paper which states that good individuals contain good information often have more chance to be utilized to guide other individuals. The results demonstrate that MODE-RMO can generate Pareto optimal fronts with satisfactory convergence and diversity.

4 Proposed method: MODE-HBM

This section presents the proposed algorithm. Firstly we introduced the Homeostasis mutation operator. Further this operator is used in MODE to produce its new variant named multiobjective differential evolution with Homeostasis based mutation (MODE-HBM). This proposed algorithm is to improve the convergence rate and diversity of MODE.

4.1 Homeostasis mutation operator

DE suffers from the problem, as its solutions start stagnating as number of generation increases. We have developed a new variant of DE. This variant is created by updating the mutation operator of DE algorithm. This modification is done because in evolutionary algorithms mutation operator is used to provide diversity among solutions. As mutation operator of DE is not very effective in handling stagnation problem. Therefore some modifications are required in the mutation operator of DE algorithm so that the problem of stagnation can be solved and convergence rate of algorithm can be improved. The basic mutation strategy of DE simply adds the difference of two random vectors into third parameter vector which are selected from current population. This mutation operator is designed to provide diversity among solutions so that whole search area can be explored and good solutions from all areas can be identified during selection. In initial generation this mutation operator provides required diversity but as number of generation increases this operator becomes ineffective and looses its diversity. The reason of loss in diversity lies in selection of vectors which always select good solutions from the pool of selected solutions. After some generations, these solutions start converging to a particular region and looses their diversity. This shows that some additional vectors are required to provide diversity when generation increases.

In this approach, an additional pair of vectors is generated from the whole search space to maintain the diversity among solutions. This process is very similar to Homeostasis as it keeps the internal environment of DE algorithm constant by adding an additional mutation function ψ for adding diversity as and when required.

Homeostasis process maintain the internal environment of the human body by adding different parameter values depending on the nature of the problem and available computing resources [39]. Similar to this process, we have created two Homeostasis random vectors(HRVs) HRV1 and HRV2 as mutation strategies. This is done to maintain the diversity provided during initial generations till end. In some strategies these additional vectors provide new areas for exploration while for others these vectors may be used for exploitation. These improved mutation strategies are named by their basic mutation strategies. This is the common procedure for creating new Homeostasis variant of DE. To generate Homeostasis mutation operator of DE/best/1, difference of Homeostasis vectors are added in to a existing strategy. First random difference vector is created and then the difference of Homeostasis random vectors is added to basic mutation strategy in mutation operators. We Define the general difference vector and add the Homeostasis ψ (means difference of Homeostasis vectors) $γ_{i, G} = α_{b e s t, G} + δ_{1} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G}) + ψ$ (10)

Where $ψ = δ_{2} \cdot (α_{{HRV}_{1}^{i}, G} - α_{{HRV}_{2}^{i}, G})$ .

Generate the new mutant vector:

$\begin{matrix} γ_{i, G} & = & α_{b e s t, G} + δ_{1} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G}) \\ + δ_{2} \cdot (α_{{HRV}_{1}^{i}, G} - α_{{HRV}_{2}^{i}, G}) \end{matrix}$ (11)

4.2 Multiobjective differential evolution with Homeostasis based mutation

The process of multiobjective optimization is very different from single objective optimization. Therefore for solving multiobjective problems, A multiobjective variant of Homeostasis DE has to be created. This Homeostasis variant is created by updating the operators of PBDE algorithm. In PBDE, nondominated sorting was used to identify solutions of the next generation. So any reduction in time complexity of nondominated sorting algorithm, will also improve the performance of algorithm. We have used efficient nondominated sorting [27] to perform nondominated sorting of the solutions. This sorting technique is the best technique for nondominated sorting and takes very less time as compared to other ones. In addition to it, diversity preservation is also required to catch optimal Pareto front as early as possible. So operators of single objective variant is modified so that they can work with multiobjective problem. The detailed explanation of each operator is shown below.

Homeostasis mutation. Homeostasis mutation was suggested to produce diversity among solutions. From Equation 11, it is clear that it has two components. One which comes from population itself and other which is a Homeostasis component that has to be generated from whole search space. Alike PBDE, first of all some random solutions are generated. Then we did nondominated sorting to check which solutions are dominated and which are nondominated. All nondominated solutions which belongs to first Pareto front are used to search other potential regions in the search space. This is done by implementing Homeostasis mutation for these solutions. Other dominated solutions which belong to other fronts are guided towards the best solution that is at the least distance from the solution.

So all solutions of 1st front will be guided by this equation

$\begin{matrix} γ_{i, G + 1} & = & α_{b e s t, G} + δ_{1} \cdot (α_{r_{1}^{i}, G} - α_{r_{2}^{i}, G}) \\ + δ_{2} \cdot (α_{{HRV}_{1}^{i}, G} - α_{{HRV}_{2}^{i}, G}) \end{matrix}$ (12)

Where α_best, G will be solution itself. α_r₁,G, α_r₂,G, G will be the two vectors those will be selected from the entire population. α_{HRV
₁} and α_{HRV
₂} will be generated from the entire search space and will be added to provide diverse solutions in the search space.

Similarly for other solutions, this equation will be used $γ_{i, G + 1} = α_{b e s t, G} + δ_{1} \cdot (α_{r_{2}^{i}, G} - α_{r_{3}^{i}, G})$ (13)

All dominated solutions will be guided by the best solution, which will be very close to the solution. Hence to check which best solution is close to the dominated solution, Euclidian distance of the solution from first front solutions is calculated. The solution with the least distance will work as α_best,G. A similar to the mutation vector of best individuals, two solutions will be selected from the population.

Crossover. There is no change in crossover operator. The trial vector is merged with mutation vector and target vector is generated. However the value of Cr is kept high for the nondominated solutions whereas for other solutions this value is kept low. This is done so that all areas near good solutions can be exploited reasonably.

Selection. There is no change in selection strategy. We have implemented the same strategy which was implemented in PBDE. First the solution with higher fitness will be moved to the new population. In case of tie in the rank of solutions, the solution with lower crowding distance is selected.

Pseudo code for MODE-HBM as shown in Algorithm 1.

Algorithm 1 MODE-HBM

1: Input:

2: f_m, m = 1, 2, . . . , M. MOP with M objectives

3: S^D = Search space of D decision variables

4: ITR = Maximum Number of Iterations

5: Output:

6: New population

7: Begin Procedure

8: for i = 1 to N

9: Initialize each solution α_i randomly with in respective search space range.

10: Apply nondominated sorting and assign ranks to solutions according to their pareto front number

11: Apply Homeostasis mutation

12: Apply crossover

13: Evaluate each solutions fitness for f_m (α_i), m = 1, 2, . . . , M

14: Apply selection and first keep good rank solutions, in case of tie solution choose solution with higher crowding distance

15: End for

16: End Procedure

5 Test functions and experimental setup

5.1 Test problem

The performance of proposed MODE-HBM algorithm is compared with other multiobjective variants of DE on five benchmark functions of ZDT series ZDT1,ZDT2, ZDT3, ZDT4 and ZDT6 [34 –38] and seven test problem of DTLZ series DTLZ1, DTLZ2, DTLZ3, DTLZ4, DTLZ5, DTLZ6 and DTLZ7 [30]. Benchmark functions of ZDT series are bi-objectives while benchmark problems of DTLZ series are tri-objectives. Results generated by proposed algorithm are compared with other state-of-the-art versions of MOEA Algorithm like MOEA-D [15], MOEA [43], MOPSO [42], MODE [5] and MODE-RMO [25]. The characteristics of these benchmark functions are explained in Table 2.

Table 1
Control parameter

Sr. No. Parameter Description Value

name

1 Pop Population Size 100

2 m No. of objectives Test Problem

3 n No. of variables Test Problem

4 ITR No. of maximum iteration 200

5. δ1 & δ2 Mutation Factor 0.1 to 2.0

6 Cr Crossover Rate 0.1 to 1.0

Sr. No.	Parameter	Description	Value
1	Pop	Population Size	100
2	m	No. of objectives	Test Problem
3	n	No. of variables	Test Problem
4	ITR	No. of maximum iteration	200
5.	δ1 & δ2	Mutation Factor	0.1 to 2.0
6	Cr	Crossover Rate	0.1 to 1.0

Table 2

ZDT and DTLZ Problems

Sr. No.	Objective	Characteristics	Domain
	problem		variable
1	ZDT1	Convex	[0, 1]
2	ZDT2	Nonconvex	[0, 1]
3	ZDT3	Convex disconnected	[0, 1]
4	ZDT4	Non convex	[0, 1]
5	ZDT6	Non convex nonuniformely spaced	[0, 1]
6	DTLZ1	Linear	[0, 1]
7	DTLZ2	Concave	[0, 1]
8	DTLZ3	Concave	[0, 1]
9	DTLZ4	Concave	[0, 1]
10	DTLZ5	-	[0, 1]
11	DTLZ6	Nonuniform	[0, 1]
12	DTLZ7	Disconnected	[0, 1]

ZDT Family. ZDT series proposed by Zitzler [34 –38], contains six benchmark problems. We have tested five benchmark functions. The Pareto optimal fronts of its problems are explained in Table 2.

DTLZ Family. The DTLZ is very popular series for testing the performance of multiobjective algorithm. It was proposed by Deb et al. [29 , 30–32] this series include many benchmark functions and they are scalable to any number of objective functions. This is very important characteristic of this series.

5.2 Performance metric

During comparisons, we have used three performance metrics inverted generational distance (IGD), generational distance (GD) and spacing (Sp) metric of algorithms. These metrics are widely used for checking the performance of standard MODE algorithms by calculating the values of the performance distance metrics. Details of these metrics are shown below.

5.2.1 Inverted Generational Distance(IGD)

IGD metric was proposed by Zitzler et al. [36], this is a distance performance measure for MOEA. IGD measure both convergence speed and diversity for multi object optimization problem. IGD for algorithms can be calculated by this formula:

$\begin{matrix} IGD (Pop, Pop *) \\ = \sqrt{\sum_{x ε_{Pop *}} Dis (x, Pop)^{2} / | Pop * |)} \end{matrix}$ (14)

Where Dis (x, Pop) describe the minimum Euclidean Distance between solution x and the solution of Pop and |Pop * | is measure of the convergence speed and diversity of Pop. An algorithm with smaller value of IGD metric will be good as compared to other algorithms.

5.2.2 Generation Distance (GD)

GD metric was proposed by Veldhuizen et al. [35], to check the closeness of nondominated set from Pareto optimal front. This metric calculates the distance between the nondominated solution set Pop and Pareto optimal front Pop*. The process is explained below:

$\begin{matrix} GD (Pop, Pop *) \\ = \sqrt{\sum_{x ε_{Pop}} Dis (x, Pop *)^{2} / | Pop |)} \end{matrix}$ (15)

Where Dis (x, Pop *) is the Euclidean Distance between each solution and nearest solution of Pareto front for measuring the objective space. GD metric measures the nearest of Pareto front set. Minimum value of GD metric value shows a better adjacency set.

5.2.3 Spacing (Sp)

Spacing (Sp) metric was proposed by J. Schott [41], to check the closeness of nondominated set from Pareto optimal front. The process is explained below: $Sp = \sqrt{1 / (n - 1) \sum_{i = 1}^{n}} (\bar{D} - D_{i})^{2},$ (16) $D_{i} = \min_{i}, j \neq i (\sum_{k = 1}^{k} | f_{i}^{k} - f_{j}^{k} |)$

Where n is the number of vectors in the Pareto front. k is the number of objectives and all D_i mean of $\bar{D}$ . Sp metric value is aimed to be minimum.

5.3 Experimental setup and parameter settings

Proposed algorithm is tested on bi-objective and tri-objective provided in the Table 2, population size was taken as 100 and all algorithms were run 200 generations respectively, other parameters of algorithms are listed in Table 1.

All experiments are performed using MATLAB 2013(a) running over MS Windows7 on machine with core-i5 @2.1 GHz processor and 4 GB RAM.

5.4 Experimental results and discussion

Convergence rate and closeness to optimal Pareto fronts are used to compare MODE-HBM algorithms with other state-of-the-art algorithms. The performance matrices IGD, GD and Sp are calculated for all variants. From the Tables 3, 4 and 5 it is clear that the performance of MODE-HBM is better than other algorithms. It was able to capture good solutions which were very close to optimal Pareto front and was sufficiently diverged. Moreover results on ZDT and DTLZ series shows that this algorithm is robust and scalable for any number of objective functions. From the Figs. 1 and 2, it is clear that it was able to capture solutions in discrete functions also. Each objective problem has run 15 times independently, and the statistics values including “Mean” value and standard deviation (SD) are shown in Tables 3, 4 and 5. Table 3 shows the results of convergence metric IGD obtained by six algorithms. MODE-HBM achieves better IGD mean values in comparison to MOEA-D, MOEA, MOPSO, MODE and MODE-RMO for all problems.

Table 3
Comparing result of MODE-HBM with other state-of-the-art algorithm for performance metric IGD

Objectives/ MOEA-D MOEA MOPSO MODE MODE-RMO MODE-HBM

Algorithm Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD

ZDT1 0.020363 0.005126 0.001091 0.009124 0.011074 0.002519 0.00129 0.000623 0.000992 0.000067 0.000987 0.000062

ZDT2 0.017281 0.006981 0.0054443 0.005027 0.009007 0.000846 0.002012 0.000253 0.001601 0.000647 0.001045 0.000528

ZDT3 0.011242 0.005218 0.0062455 0.005819 0.008996 0.000562 0.002481 0.000814 0.001768 0.000891 0.001219 0.000638

ZDT4 0.013179 0.052382 0.0044752 0.009218 0.008638 0.009475 0.000181 0.000864 0.000167 0.000756 0.000164 0.000872

ZDT6 0.058259 0.008911 0.0063074 0.003645 0.009021 0.000628 0.007173 0.000451 0.005678 0.000816 0.001043 0.000816

DTLZ1 0.04499 0.006213 0.006667 0.000416 0.008249 0.000421 0.000337 0.000088 0.000324 0.000091 0.000314 0.000079

DTLZ2 0.045134 0.008131 0.005502 0.000519 0.001008 0.000846 0.000828 0.000039 0.000836 0.000058 0.000798 0.000045

DTLZ3 0.655015 0.823146 0.38725 0.812454 0.081496 0.092195 0.079676 0.110061 0.083978 0.926845 0.074305 0.088912

DTLZ4 0.0060344 0.000281 0.001714 0.000814 0.009038 0.000846 0.000795 0.000069 0.000833 0.000098 0.000795 0.000055

DTLZ5 0.65601 0.063145 0.874714 0.061274 0.793228 0.056421 0.67397 0.02358 0.691799 0.081456 0.590000 0.04534

DTLZ6 0.70613 0.081467 0.097142 0.003198 0.098937 0.002186 0.064354 0.002549 0.065957 0.002845 0.062878 0.001558

DTLZ7 0.531248 0.062385 0.04443 0.009121 0.010484 0.009125 0.001141 0.000684 0.001051 0.000941 0.001012 0.000918

Objectives/	MOEA-D	MOEA	MOPSO	MODE	MODE-RMO	MODE-HBM
ZDT1	0.020363	0.005126	0.001091	0.009124	0.011074	0.002519	0.00129	0.000623	0.000992	0.000067	0.000987	0.000062
ZDT2	0.017281	0.006981	0.0054443	0.005027	0.009007	0.000846	0.002012	0.000253	0.001601	0.000647	0.001045	0.000528
ZDT3	0.011242	0.005218	0.0062455	0.005819	0.008996	0.000562	0.002481	0.000814	0.001768	0.000891	0.001219	0.000638
ZDT4	0.013179	0.052382	0.0044752	0.009218	0.008638	0.009475	0.000181	0.000864	0.000167	0.000756	0.000164	0.000872
ZDT6	0.058259	0.008911	0.0063074	0.003645	0.009021	0.000628	0.007173	0.000451	0.005678	0.000816	0.001043	0.000816
DTLZ1	0.04499	0.006213	0.006667	0.000416	0.008249	0.000421	0.000337	0.000088	0.000324	0.000091	0.000314	0.000079
DTLZ2	0.045134	0.008131	0.005502	0.000519	0.001008	0.000846	0.000828	0.000039	0.000836	0.000058	0.000798	0.000045
DTLZ3	0.655015	0.823146	0.38725	0.812454	0.081496	0.092195	0.079676	0.110061	0.083978	0.926845	0.074305	0.088912
DTLZ4	0.0060344	0.000281	0.001714	0.000814	0.009038	0.000846	0.000795	0.000069	0.000833	0.000098	0.000795	0.000055
DTLZ5	0.65601	0.063145	0.874714	0.061274	0.793228	0.056421	0.67397	0.02358	0.691799	0.081456	0.590000	0.04534
DTLZ6	0.70613	0.081467	0.097142	0.003198	0.098937	0.002186	0.064354	0.002549	0.065957	0.002845	0.062878	0.001558
DTLZ7	0.531248	0.062385	0.04443	0.009121	0.010484	0.009125	0.001141	0.000684	0.001051	0.000941	0.001012	0.000918

Table 4

Comparing result of MODE-HBM with other state-of-the-art algorithm for performance metric GD

Objectives/	MOEA-D		MOEA		MOPSO		MODE		MODE-RMO		MODE-HBM
Algorithm	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
ZDT1	0.010918	0.007363	0.002633	0.0008524	0.010573	0.002531	0.00241	0.0005614	0.001803	0.000316	0.001871	0.000289
ZDT2	0.054443	0.006882	0.003336	0.054443	0.009186	0.000513	0.00384	0.0008015	0.003067	0.000564	0.001935	0.000484
ZDT3	0.0062455	0.000542	0.0016182	0.000561	0.009007	0.002534	0.003155	0.0003486	0.002199	0.000761	0.001439	0.000219
ZDT4	0.0044752	0.085263	0.000575	0.000691	0.008788	0.009618	0.000162	0.000346	0.000174	0.000638	0.000156	0.000309
ZDT6	0.063074	0.008321	0.21209	0.025316	0.008638	0.000925	0.013405	0.006218	0.010891	0.002671	0.001895	0.000435
DTLZ1	0.054699	0.002135	0.006667	0.000541	0.008799	0.000271	0.000224	0.000048	0.000217	0.000082	0.000205	0.000059
DTLZ2	0.041475	0.00921	0.005211	0.000458	0.008249	0.000723	0.000636	0.000036	0.000608	0.000031	0.000598	0.000018
DTLZ3	10.23111	12.55015	1.35902	2.88725	7.501008	7.81493	5.259173	5.79615	10.264273	10.83978	4.98208	5.074305
DTLZ4	0.072809	0.002357	0.035932	0.005271	0.008496	0.000335	0.000581	0.000082	0.000594	0.000079	0.000576	0.000051
DTLZ5	0.049469	0.065601	0.470014	0.04718	0.090308	0.003256	0.073165	0.003955	0.07571	0.001725	0.073706	0.001239
DTLZ6	0.655106	0.036235	0.577305	0.050012	0.791063	0.056318	0.411013	0.030256	0.422482	0.055952	0.407147	0.024818
DTLZ7	0.718624	0.028965	0.004443	0.000621	0.008937	0.000928	0.007324	0.000247	0.006906	0.000851	0.006393	0.000547

Table 5

Comparing result of MODE-HBM with other state-of-the-art algorithm for performance metric Sp

Objectives/	MOEA-D		MOEA		MOPSO		MODE		MODE-RMO		MODE-HBM
Algorithm	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
ZDT1	0.21129	0.02153	0.002513	0.000847	0.041156	0.012048	0.051053	0.007254	0.002528	0.000795	0.002071	0.000842
ZDT2	0.47551	0.04213	0.004276	0.000915	0.011845	0.008412	0.003691	0.000834	0.003109	0.000655	0.003063	0.000615
ZDT3	0.19744	0.07136	0.010058	0.008721	0.089524	0.024161	0.011076	0.002484	0.005209	0.001064	0.004968	0.000971
ZDT4	0.44976	0.65142	0.005253	0.008091	0.053694	0.064215	0.673392	0.85123	0.157994	0.456284	0.005134	0.007289
ZDT6	0.2767	0.0814	0.0069182	0.000421	0.004896	0.0008423	0.015286	0.005617	0.001712	0.000843	0.001617	0.000315
DTLZ1	3.3925	2.58464	0.000597	0.000148	0.031156	0.007618	0.025774	0.007284	0.00273	0.000941	0.014001	0.002584
DTLZ2	0.34301	0.02451	0.004259	0.00613	0.009182	0.000814	0.004942	0.000871	0.004562	0.001162	0.003887	0.001638
DTLZ3	4.1847	4.5124	0.005017	0.008941	0.089562	0.094156	0.588918	0.728405	0.280573	0.721354	0.131505	0.842164
DTLZ4	0.14419	0.03476	0.007489	0.000175	0.085356	0.006417	0.005333	0.001108	0.006841	0.004856	0.005155	0.001951
DTLZ5	0.067738	0.006142	0.006341	0.002014	0.001896	0.000942	0.005435	0.001024	0.005797	0.002102	0.005234	0.001528
DTLZ6	0.20768	0.06184	0.006341	0.003148	0.030516	0.011583	0.034505	0.008124	0.006053	0.000912	0.029239	0.003514
DTLZ7	0.42153	0.01873	0.005889	0.003012	0.024915	0.006814	0.006628	0.001178	0.006376	0.001854	0.005181	0.001617

Fig.1

Pareto optimal solution MODE-HBM comparing with state-of-the-art algorithm on ZDT series.

Fig.2

Pareto optimal solution MODE-HBM comparing with state-of-the-art algorithm on DTLZ series.

The results of convergence metric GD obtained by six algorithms are as given in the Table 4. MODE-HBM achieves better GD mean values in comparison to MOEA-D, MOEA, MOPSO, MODE and MODE-RMO for all problems except ZDT1 and DTLZ3.

Tables 5 shows the results of space metric Sp obtained by six algorithms. MODE-HBM achieves better Sp mean values in comparison to MOEA-D, MOEA, MOPSO, MODE and MODE-RMO for all problems except DTLZ1, DTLZ3 and DTLZ6.

5.5 Effect of various parameters on the performance of algorithm

Population Size: Population size is very important parameter for this algorithm. For bi-objective like ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 and tri-objective like DTLZ1, DTLZ2, DTLZ3, DTLZ4, DTLZ5, DTLZ6 and DTLZ7 population size 100 is sufficient for capturing targeted solutions in all functions. This is due to the design of adaptation operator. If we take large size we may do exploitation and exploration around many areas.

Number of Generations: Number of generations is not that important for this algorithm. A small number of generations like 50 may work with any number of functions (bi-objective and tri-objective).

Control Parameter: The value of control parameters, mutation factor and crossover probability are used for controlling the exploitation and exploration power of algorithm.

6 Conclusion

Proposed work implements new mutation strategy to overcome the problems associated with existing MODE algorithms. Searching capability of MODE-HBM has been improved by introducing a new Homeostasis mutation operator that is capable to do more exploration and exploitation. From the Tables 3, 4 and 5 it is clear that on all dimensions the proposed MODE-HBM is better than all latest variants of DE optimization algorithms. The performance of this variant was almost very good in all kind of functions and a very little downfall have been seen in some functions. In future we can use distribution based strategy to initialize the population as initial strategy. These strategies might be more useful in providing diversity among different solutions.

References

Abbass

H.A.

, Sarker

and Newton

, PDE: A Pareto-frontier differential evolution approach for multi-objective optimization problems, In: Proceedings of the Congress on Evolutionary Computation 2001 (CEC2001), vol. 2, 2001, pp. 971–978.

Abbass

H.A.

, The self-adaptive Pareto differential evolution, In: Proceedings of the Congress on Evolutionary Computation 2002 (CEC2002), vol. 1, 2002, pp. 831–836.

Abbass

H.A.

, A memetic pareto evolutionary approach to artificial neural networks, AI 2001: Advances in Artificial Intelligence, Springer, Berlin Heidelberg, 2001, pp. 1–12.

Chang

C.S.

, Xu

D.Y.

and Quek

H.B.

, Pareto-optimal set based multiobjective tuning of fuzzy automatic train operation for mass transit system, Electric Power Applications, IEE Proceedings146(5) (1999). IET.

Babu

B.V.

and Mathew

, Leenus Jehan, Differential Evolution for Multi-Objective Optimization, In Proceedings of the 2003 Congress on Evolutionary Computation (CEC’2003), volume 4, Canberra, Australia, 2003, pp. 2696–2703. IEEE Press.

and Zhang

, A multiobjective differential evolution based on decomposition for multiobjective optimization with variable linkages, Parallel problem solving from nature-PPSN IX, SpringerBerlin Heidelberg, 2006, pp. 583–592.

Kukkonen

and Lampinen

, An extension of generalized differential evolution for multi-objective optimization with constraints, Parallel Problem Solving from Nature-PPSN VIII, SpringerBerlin Heidelberg, 2004.

Kukkonen

and Lampinen

, GDE3: The third evolution step of generalized differential evolution, Evolutionary Computation, 2005 The 2005 IEEE Congress on, Vol. 1, 2005, IEEE.

Flores

E.A.P.

, Integracion Simultanea de Aspectos Estructurales y Dinamicos para el Diseno Optimo de un Sistema de Transmission de Variacian Continua, Diss PhD thesis, Departamento de Ingenieria Electrica, Seccion de Mecatronica, CINVESTAV-IPN, Mexico, DF, Mexico, 2006.

10.

Bo-Yang

Q.U.

and Suganthan

P.-N.

, Multiobjective differential evolution with diversity enhancement, Zhejiang Univ-Sci C (Comput & Electron) 201011(7) (2010), 538–543.

11.

Wang

Y.-N.

, Wu

L.-H.

and Yuan

X.-F.

, Multi-objective selfadaptive differential evolution with elitist archive and crowding entropy-based diversity measure, Soft Computing14 (2010), 193–209.

12.

Angira

and Babu

B.V.

, Non-dominated Sorting Differential Evolution (NSDE): An Extension of Differential Evolution for Multi-objective Optimization, IICAI, 2005.

13.

, Fialho

and Kwong

, Multi-objective differential evolution with adaptive control of parameters and operators, Learning and Intelligent Optimization, Springer, Berlin Heidelberg, 2011, pp. 473–487.

14.

Qingfu

and Li

, MOEA/D: A multiobjective evolutionary algorithm based on decomposition, Evolutionary Computation, IEEE Transactions on11(6) (2007), 712–731.

15.

and Zhang

, Multi-objective optimization problems with complicated Pareto sets, MOEA/D and NSGA-II, IEEE Transactions on Evolutionary Computation13(2) (2009), 284–302.

16.

Tan

Y.Y.

, Jiao

Y.C.

, Li

and Wang

X.K.

, A modification to MOEA/D-DE for multiobjective optimization problems with complicated Pareto sets, Information Sciences213 (2012), 14–38.

17.

Robic

and Filipic

, DEMO: Differential evolution for multiobjective optimization, In: Proceedings of the 3rd International Conference on Evolutionary MultiCriterion Optimization (EMO 2005), LNCS 3410, 2005, pp. 520–533.

18.

Adeyemo

J.A.

and Otieno

F.A.O.

, Multi-objective differential evolution for solving engineering problems, Journal of Applied Sciences9(20) (2009), 3652–3661.

19.

Qian

and Li

, Adaptive differential evolution algorithm for multiobjective optimization problems, Applied Mathematics and Computation201 (2008), 431–440.

20.

Xue

, Sanderson

A.C.

and Graves

R.J.

, Paretobased multi-objective differential evolution, Evolutionary Computation, 2003 CEC’03, The 2003 Congress on, Vol. 2, 2003, IEEE.

21.

Chen

, Lin

, Zeng

, Zhang

and Si

Y.-W.

, Modified differential evolution algorithm using a new diversity maintenance strategy for multi-objective optimization problems, Applied Intelligence43(1) (2015), 49–73. Harvard.

22.

Shi

, Chen

, Jiang

, Shen

and Shangce-Gao , Handling Multiobjectives with Adaptive Mutation Based ∈-Dominance Differential Evolution, ICSI-CCI 2015, Part I, LNCS 9140, 2015, pp. 523–532.

23.

Ali

, Pant

and Abraham

, A modified differential evolution algorithm and its application to engineering problems, In: Proceedings of International Conference of Soft Computing and Pattern Recognition (SoCPaR-2009), 2009, pp. 196–201.

24.

Ali

, Siarry

and Pant

, An efficient differential evolution based algorithm for solving multi-objective optimization problems, Eur J Oper Res217(2) (2012), 404–416.

25.

Chen

, Du

and Qian

, Multi-objective differential evolution with ranking-based mutation operator and its application in chemical process optimization, Chemometrics and Intelligent Laboratory Systems136 (2014), 85–96.

26.

Zhang

, Tian

, Cheng

and Jin

, An efficient approach to nondominated sorting for evolutionary multiobjective optimization, Evolutionary Computation, IEEE Transactions on19(2) (2015), 201–213.

27.

Deb

, Pratap

, Agarwal

and Meyarivan

, A fast and elitist multi-objective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation6(2) (2002), 182–197.

28.

Storn

and Price

, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization11(4) (1997), 341–359.

29.

Bader

and Zitzler

, HypE: An algorithm for fast hyper volume-based manyobjective optimization, TIK Report 286, Computer Engineering and Networks Laboratory (TIK), ETH Zurich, 2008.

30.

Deb , Thiele

, Laumanns

and Zitzler

, Scalable multiobjective optimization test problems, Proceedings of the Congress on Evolutionary Computation (CEC-2002), Honolulu, USA, 2002, pp. 825–830.

31.

Deb

, Thiele

, Laumanns

, et al., Scalable test problems for evolutionary multi-objective optimization, Technical report, Computer Engineering and Networks Lab (TIK), ETH Zurich, Switzerland, 2001.

32.

Schaffer

J.D.

, Multiple objective optimization with vector evaluated genetic algorithm, In: Proceedings of the First International Conference on Genetic Algorithms, 1985, pp. 93–100.

33.

Storn

and Price

, Differential Evolution-A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces, International Computer Science Institute, Berkeley. Berkeley, CA, 1995.

34.

Van Veldhuizen

D.A.

and Lamont

G.B.

, Multiobjective evolutionary algorithm research: A history and analysis, Technical report, Department of Electrical and Computer Engineering. Graduate School of Engineering, Air Force Inst. Technol, Wright Patterson 1998.

35.

Van

Veldhuizen D.A.

and Lamont

G.B.

, On measuring multiobjective evolutionary algorithm performance, In: Proceedings of the Congress on Evolutionary Computation 2000 (CEC’2000), vol. 1, 2000, pp. 204–211.

36.

Zitzler

and Thiele

, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transactions on Evolutionary Computation3(4) (1999), 257–271.

37.

Zitzler

, Thiele

, Laumanns

, Fonseca

C.M.

and da Fonseca

V.G.

, Performance assessment of multiobjective optimizers: An analysis and review, IEEE Trans Evol Comput7(2) (2003), 117–13.

38.

Zitzler

, Deb

and Thiele

, Comparison of multiobjective evolutionary algorithms: Empirical results, Evol Comput8(2) (2000), 173–195.

39.

Das

, Mullick

S.S.

and Suganthan

P.N.

, Recent advances in differential evolution-An updated survey, Swarm and Evolutionary Computation (27) (2016), 1–30.

40.

Quintero

L.V.S.

and Coello Coello

C.A.

, Development of techniques to improve computational efficiency in multi-objective evolutionary algorithms, Thesis submit in Centro De Investigation Y De Estudious Avanzados Del Instituto Politecnico Nactional Departamento De Computation, Mexico, 2008.

41.

Schott

J.R.

, Fault tolerant design using single and multicriteria genetic algorithm optimization, Technical report, DTIC Document, 1995.

42.

Leung

M.-F.

, Ng

S.-C.

, Cheung

C.-C.

and Lui

A.K.

, A new algorithm based on PSO for multi-objective optimization, IEEE Congress on Evolutionary Computation (CEC), 2015.

43.

, Zhang

, Kwong

, Li

and Wang

, Stable matching-based selection in evolutionary multiobjective optimization, IEEE Transaction On Evolutionary Computation18(6) (2014), 909–923.

44.

Wang

, Cai

and Zhang

, Differential evolution with composite trial vector generation strategies and control parameters, IEEE Transactions on Evolutionary Computation15(1) (2011), 55–66.

Pareto based differential evolution with homeostasis based mutation

Abstract

Keywords

1 Introduction

2 Background details and related work

2.1 Multiobjective optimization problem

2.3 Nondominated sorting

2.4 DE algorithm

3 Related works

4 Proposed method: MODE-HBM

4.1 Homeostasis mutation operator

5.1 Test problem

Table 1 Control parameter Sr. No. Parameter Description Value name 1 Pop Population Size 100 2 m No. of objectives Test Problem 3 n No. of variables Test Problem 4 ITR No. of maximum iteration 200 5. δ1 & δ2 Mutation Factor 0.1 to 2.0 6 Cr Crossover Rate 0.1 to 1.0

5.2.1 Inverted Generational Distance(IGD)

5.4 Experimental results and discussion

6 Conclusion

References

Table 1
Control parameter

Sr. No. Parameter Description Value

name

1 Pop Population Size 100

2 m No. of objectives Test Problem

3 n No. of variables Test Problem

4 ITR No. of maximum iteration 200

5. δ1 & δ2 Mutation Factor 0.1 to 2.0

6 Cr Crossover Rate 0.1 to 1.0