Using an outward selective pressure for improving the search quality of the MOEA/D algorithm
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Abstract
In optimization problems it is often necessary to perform an optimization based on more than one objective. The goal of the multiobjective optimization is usually to find an approximation of the Pareto front which contains solutions that represent the best possible tradeoffs between the objectives. In a multiobjective scenario it is important to both improve the solutions in terms of the objectives and to find a wide variety of available options. Evolutionary algorithms are often used for multiobjective optimization because they maintain a population of individuals which directly represent a set of solutions of the optimization problem. multiobjective evolutionary algorithm based on decomposition (MOEA/D) is one of the most effective multiobjective algorithms currently used in the literature. This paper investigates several methods which increase the selective pressure to the outside of the Pareto front in the case of the MOEA/D algorithm. Experiments show that by applying greater selective pressure in the outwards direction the quality of results obtained by the algorithm can be increased. The proposed methods were tested on nine test instances with complicated Pareto sets. In the tests the new methods outperformed the original MOEA/D algorithm.
Keywords
Multiobjective optimization Evolutionary algorithms MOEA/D algorithm Selective pressure1 Introduction
Algorithms based on Pareto domination relation use it for numerical evaluation of specimens or for selection. Pareto domination relation is used for evaluation of specimens in algorithms such as SPEA [32] and SPEA2 [33]. The approach in which Pareto domination is used for selection is used for example in NSGA [23] and NSGAII [7].
The multiobjective evolutionary algorithm based on decomposition (MOEA/D) was proposed by Zhang and Li [27]. Contrary to the algorithms that base their selection process on the Pareto domination relation the MOEA/D algorithm is a decompositionbased algorithm. In this algorithm the multiobjective optimization problem is decomposed to a number of singleobjective problems. To date the MOEA/D algorithm has been applied numerous times and was found to outperform other algorithms not only in the case of benchmark problems [17, 27], but also in the case of combinatorial problems [21] and various applications to reallife problems. The applications in which the MOEA/D algorithm was used include multiband antenna design [9], deployment in wireless sensor networks [15], air traffic optimization [3], and route planning [24].
Since the MOEA/D algorithm was introduced many modified versions were proposed. Some approaches aim at achieving better performance by exploiting the advantages of various scalarization functions used in the MOEA/D framework. Ishibuchi et al. [12] proposed adaptive selection of either the weighted sum scalarization or Tchebycheff scalarization depending of the region of the Pareto front. An algorithm in which both scalarizations are used simultaneously was also proposed [13]. Other authors attempt using autoadaptive mechanisms in the context of the MOEA/D framework. For example, in the paper [28] a MOEA/DDRA variant is proposed in which a utility value is calculated for each subproblem and computational resources are allocated to subproblems according to their utility values.
There are also hybrid algorithms in which the MOEA/D is combined with other optimization algorithms. Martinez and Coello Coello proposed hybridization with the nonlinear simplex search algorithm [26]. Li and LandaSilva combined the MOEA/D algorithm with simulated annealing [16] and applied their approach to combinatorial problems. Another approach based on PSO and decomposition was proposed in [1]. In the case of combinatorial optimization a hybridization with the ant colony optimization (ACO) was attempted [14] in order to implement the “learning while optimizing” principle.
The main motivation for this paper is to research the possibility of increasing the diversity of the search, namely extending the Pareto front, by directing the selective pressure towards outer regions of the Pareto front. While it is a different approach than those discussed above it can be easily combined with many of the methods by other authors. The presented algorithm can be hybridized with other optimization algorithms and local search procedures in the same way as the original MOEA/D algorithm. For example, the approach proposed in [26] employs a simplex algorithm as a local search procedure to improve the solutions found by the MOEA/D algorithm. It works at a different level than the approach proposed in this paper which modifies the working of the MOEA/D algorithm itself. Therefore, hybrid approaches, such as described in papers [1, 14, 16, 26] could use the weight generation scheme presented in this paper instead of the original one. Adaptive allocation of the resources discussed in [28] could also be attempted with the algorithm presented in this paper.
2 Increasing the outward selective pressure
The MOEA/D algorithm uses weight vectors to aggregate objectives of a multiobjective optimization problem into a single, scalar objective. Clearly, the coordinates of a weight vector \(\lambda ^{(i)}\) determine the importance of each objective \(f_j\), \(j \in \{1, \ldots , m\}\) in \(i\)th scalar subproblem.
Solutions in the neighborhood \(B(i)\) participate in information exchange with the solution of a subproblem parameterized by weight vector \(\lambda ^{(i)}\). Thus, one can expect that the distribution of weight vectors in the neighborhood \(B(i)\) may influence the selective pressure exerted on specimens that represent solutions of the \(i\)th subproblem.
Such placement of weight vectors in the neighborhood of the \(\lambda ^{(0)}\) vector causes a situation in which specimens assigned to the 0th subproblem can only exchange information with specimens that are evaluated using weight vectors pointing slightly downwards. Correspondingly, specimens that solve the subproblem \(N1\) can only exchange information with specimens that are evaluated using weight vectors pointing slightly to the left. In both situations it can be expected that an inward selective pressure will be exerted on specimens on the edges of the Pareto front preventing them from extending towards higher values of objectives \(f_1\) and \(f_2\). Other specimens located at positions closer than \(T / 2\) from the edge can also be influenced by such pressure, which can be expected to be smaller if a specimen is located farther from the edge.
Obviously, the selective pressure influences the behavior of the population and therefore it can be expected to influence also the shape of the Pareto front attained by the algorithm. The methods presented in this paper were developed based on an assumption that directing the selective pressure outwards will make the Pareto front spread wider. This assumption was positively verified by a preliminary test performed on a simple benchmark problem. The construction of this benchmark function and the results of the preliminary test are presented in Sect. 3.
In this paper three methods of directing the selective pressure outwards are proposed: “fold”, “reduce” and “diverge”. The first two of these methods differ from the standard MOEA/D algorithm in the way in which the neighborhoods are defined. In the “diverge” method the neighborhoods are defined in the same way as in the standard MOEA/D algorithm but weight vectors are calculated in a different way.
2.1 The “fold” method
In the “fold” method weight vectors are generated using the same procedure as proposed by the authors of the MOEA/D algorithm in [27]. This weight vector generation procedure is described in Sect. 1. It generates all the possible \(m\)element combinations of numbers from the set (7) that sum to 1 as described by Eq. (8).
After the neighborhood construction procedure described above, the neighborhood \(B(i)\) contains only the “real” weight vectors and the vectors placed on the edge of the set of “real” weight vectors have multiple copies in \(B(i)\). Parent selection is performed by selecting elements from \(B(i)\) with uniform probability. In neighborhoods \(B(i)\) corresponding to weight vectors placed on the edge of the weight vector set some “virtual” weight vectors are replaced by “real” weight vectors. The selective pressure towards the center of the Pareto front is thus reduced.
2.2 The “reduce” method
In the “reduce” method the weight vectors are generated using a similar procedure as in the “fold” method. From the set of “real” and “virtual” weight vectors a preliminary neighborhood \(B'(i)\) of a weight vector \(\lambda _i\) is generated by considering \(T\) closest neighbors of \(\lambda _i\) (both “real” and “virtual” weight vectors). From these \(T\) vectors only the “real vectors” are retained in \(B(i)\). Contrary to the “fold” method no multiple copies of the weight vectors are placed in \(B(i)\). The neighborhood \(B(i)\) of a vector \(\lambda _i\) placed on the edge of the set of the “real” weight vectors looks as shown in Fig. 5. Therefore, the size of the neighborhood is decreased for such \(\lambda _i\) compared to the original MOEA/D algorithm.
2.3 The “diverge” method

\(\alpha \)  a parameter that controls the divergence of weight vectors.
Similarly as the reference point \(z^*\) the nadir point \(z^\#\) is updated during the runtime of the algorithm. It contains the worst objective values found so far by the algorithm. These are usually the objective values attained during the first generations of the evolution.
In the “diverge” method neighborhoods of weight vectors are defined in the same way as in the standard MOEA/D algorithm. The parent selection procedure is also the same as in the standard MOEA/D algorithm.
3 Preliminary tests
In the previous section three methods of directing the selective pressure outwards are proposed: “fold”, “reduce” and “diverge”. The design of these methods was based on a hypothesis that directing the selective pressure outwards will make the Pareto front spread wider. To verify this assumption a preliminary test was performed on a simple benchmark problem with a symmetric Pareto front designed in such a way that the difference in the Pareto front extent should easily be visible. Also, the results produced by the MOEA/D algorithm for this benchmark problem are rather poor considering the simplicity of the problem. The solutions found by the MOEA/D algorithm tend to concentrate in the middle of the actual Pareto front. The solutions at the edges of the actual Pareto front are not found at all by the unmodified MOEA/D algorithm.
The minimum and the maximum values attained along each of the objectives by each method in preliminary tests on a problem with a symmetric Pareto front
Algorithm  min \(f_1\)  max \(f_1\)  min \(f_2\)  max \(f_2\) 

MOEA/D  0.6015  2.2048  0.5840  2.2063 
MOEA/D—fold  0.5188  2.2659  0.5131  2.3757 
MOEA/D—diverge \((\alpha =0.1)\)  0.0332  2.7339  0.0519  2.7330 
MOEA/D—diverge \((\alpha =0.2)\)  0.0108  2.7394  0.0109  2.7375 
MOEA/D—diverge \((\alpha =0.3)\)  0.0061  2.7394  0.0065  2.7370 
MOEA/D—diverge \((\alpha =0.4)\)  0.0045  2.7377  0.0046  2.7390 
MOEA/D—diverge \((\alpha =0.5)\)  0.0023  2.7395  0.0040  2.7394 
MOEA/D—reduce  0.1893  2.6765  0.1286  2.6611 
The Pareto fronts presented in Fig. 7 and the values presented in Table 1 clearly show that the unmodified MOEA/D algorithm performed worst in the preliminary test. The “fold” method seems to be only slightly better. On the other hand the “reduce” method improved the results significantly and the “diverge” method produced even more widely spread Pareto fronts. Thus, the results of the preliminary tests support the hypothesis that the diversity of the search can be improved by directing the selective pressure outwards to the edges of the Pareto front.
4 Experimental study
The experiments were performed in order to verify if increasing the outward selective pressure improves the results obtained by the algorithm. The experiments were performed on test problems with complicated Pareto sets F1–F9 described in [17]. Some of these test problems were also used during the CEC 2009 MOEA competition [29], namely F2 (as UF1), F5 (as UF2), F6 (as UF8), and F8 (as UF3).
The performance of the standard MOEA/D algorithm was compared to the performance of the three methods of directing the selective pressure outwards: “fold”, “reduce” and “diverge” described in Sect. 2. The “diverge” method requires setting the value of the parameter \(\alpha \) which determines by how much the weight vectors diverge to the outside. To test the influence of this parameter on the algorithm performance the value of this parameter was set to \(\alpha = 0.1\), \(0.2\), \(0.3\), \(0.4\) and \(0.5\).
For each data set and each method of increasing the outward selective pressure 30 iterations of the test were performed. For comparison, tests using the standard version of the MOEA/D algorithm were also performed.
4.1 Performance assessment
In the case of multiobjective optimization problems evaluating the results obtained by an algorithm is more complicated than in the case of singleobjective problems. Solutions produced by each run of an algorithm are characterized by two or more conflicting objectives and thus they may not be directly comparable. A common practice is to evaluate the entire set of solutions using a certain indicator which represents the quality of the whole solution set.
In two and three dimensions the hypervolume corresponds to the area and volume respectively. Better Pareto fronts are those that have higher values of the HV indicator. The hypervolume indicator is commonly used in the literature to evaluate Pareto fronts and it has good theoretical properties. It has been proven [10] that maximizing the hypervolume is equivalent to achieving Pareto optimality. To the best of the knowledge of the author, this is the only currently known measure with this property.
Better Pareto fronts are those that have lower values of the IGD indicator. The IGD indicator measures both the diversity and the convergence of \(P\). Calculating the IGD indicator requires generating a set of points that are uniformly distributed along the true Pareto front of a given multiobjective optimization problem. This can be a significant obstacle in the case of reallife optimization problems for which the true Pareto front is not known. For benchmark problems the true Pareto front is usually known so the points in the reference set \(P^*\) can be set exactly on this front.
4.2 Parameter settings
The algorithm used in the experiments is based on the version of the MOEA/D algorithm described in [17] which is the same paper in which the test problems were proposed. For the “fold”, “reduce” and “diverge” methods neighborhood construction and specimen selection procedures were changed as described in Sect. 2. Other aspects of how the algorithm worked were not changed with respect to the standard MOEA/D algorithm.
For each test problem and for each tested method proposed in this paper the algorithm was run for \(N_{gen} = 500\) generations. Since each of the algorithm versions performs the same number of objective function evaluations the total number of objective function evaluations was the same in each test. In the case of problems F1–F9 the size of the population was set to \(N = 300\) specimens for 2objectives and to 595 for 3objectives. This corresponds to the weight vector step size \(H = 299\) and \(H = 33\) respectively. Both the number of generations and the population size were the same as used in the original paper on the problems with complicated Pareto sets [17].
The neighborhood size was set to \(T = 20\). The version of the MOEA/D algorithm proposed in [17] uses two parameters that are intended to prevent premature convergence of the population. These parameters are \(\delta \) and \(\eta _T\). During parent selection, solutions are selected from the neighborhood \(B(i)\) with probability \(\delta \) and from the entire population with probability \(1  \delta \). The \(\delta \) parameter was set to \(0.9\). The \(\eta _T\) parameter determines the maximum number of solutions that can be replaced by a new child solution. This parameter is used to prevent a situation in which too many solutions in a certain neighborhood are replaced by a single child solution. The \(\eta _T\) parameter was set to 2.
Algorithm parameters (note, that population size increases with the increasing number of objectives)
Parameter name  Value 

Number of generations \((N_{gen})\)  500 
Population size \((N)\)  
(2 objectives)  300 
(3 objectives)  595 
Weight vector step size \((H)\)  
(2 objectives)  299 
(3 objectives)  33 
Neighborhood size \((T)\)  20 
The probability that parent solutions are selected from the neighborhood \((\delta )\)  \(0.9\) 
The maximum number of solutions that can be replaced by a child solution \((\eta _T)\)  2 
Crossover probability for the DE operator \((CR)\)  \(1.0\) 
Differential weight for the DE operator \((F)\)  \(0.5\) 
Mutation probability \((P_{mut})\)  \(1/n^{(*)}\) 
Mutation distribution index \((\eta _{mut})\)  20 
Decomposition method  Tchebycheff 
5 Experiments
In the experiments solutions of the test problems F1–F9 were generated using the standard MOEA/D algorithm and using the three methods of directing the selective pressure outwards: “fold”, “reduce” and “diverge” described in Sect. 2. The “diverge” method is parameterized by the parameter \(\alpha \) which controls by how much the weight vectors diverge to the outside. In the experiments the value of this parameter was set to \(\alpha = 0.1\), \(0.2\), \(0.3\), \(0.4\) and \(0.5\). Each algorithm was run 30 times for each test problem. The number of generations in each run was \(N_{gen} = 500\). Solutions generated by the algorithms were compared using the hypervolume (HV) and Inverted Generational Distance (IGD) indicators described in Sect. 4.1.
From the figures that present the dynamic behavior of the hypervolume indicator and from the boxplots that present the distribution of the hypervolume values it can be observed that the standard MOEA/D algorithm was outperformed by other algorithms  most often the ones using the “diverge” method. To validate this observation statistical testing was performed in which the results produced by each of the algorithms were compared to those produced by the standard MOEA/D algorithm. In most cases the normality of the distribution of hypervolume values cannot be guaranteed, therefore a test that does not require this assumption had to be chosen. In this paper the Wilcoxon signed rank test [20] introduced in [25] was used. This test was recommended in a recent survey [8] as one of the methods suitable for statistical comparison of results produced by metaheuristic optimization algorithms. The Wilcoxon test does not assume the normality of data distributions and has a null hypothesis which states the equality of medians.
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F1 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  3.8601  –  – 
MOEA/D—fold  3.8618  3.4053e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.1)\)  3.8640  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  3.8637  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  3.8627  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  3.8625  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  3.8621  1.7344e\(\)006  Significant 
MOEA/D—reduce  3.8608  0.042767  Significant 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F2 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  19.9983  –  – 
MOEA/D—fold  19.8955  0.22888  Worse 
MOEA/D—diverge \((\alpha =0.1)\)  20.1269  0.00024118  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  20.1537  8.9187e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  20.0784  0.0049916  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  20.1473  0.00013595  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  20.1273  0.0011138  Significant 
MOEA/D—reduce  19.9243  0.33886  Worse 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F3 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  7.2661  –  – 
MOEA/D—fold  7.3140  0.37094  Insignificant 
MOEA/D—diverge \((\alpha =0.1)\)  7.3550  1.1265e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  7.3518  6.9838e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  7.3412  0.0043896  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  7.3439  0.0003065  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  7.3406  0.0018326  Significant 
MOEA/D—reduce  7.2312  0.36004  Worse 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F4 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  10.1963  –  – 
MOEA/D—fold  10.1920  0.58571  Worse 
MOEA/D—diverge \((\alpha =0.1)\)  10.2334  0.0043896  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  10.2353  3.1123e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  10.2295  0.0082167  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  10.2213  0.0011973  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  10.1986  0.00066392  Significant 
MOEA/D—reduce  10.2032  0.57165  Insignificant 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F5 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  6.8026  –  – 
MOEA/D—fold  6.8383  0.50383  Insignificant 
MOEA/D—diverge \((\alpha =0.1)\)  6.8371  0.70356  Insignificant 
MOEA/D—diverge \((\alpha =0.2)\)  6.7938  0.68836  Worse 
MOEA/D—diverge \((\alpha =0.3)\)  6.8208  0.74987  Insignificant 
MOEA/D—diverge \((\alpha =0.4)\)  6.7846  0.74987  Worse 
MOEA/D—diverge \((\alpha =0.5)\)  6.8244  0.41653  Insignificant 
MOEA/D—reduce  6.8481  0.55774  Insignificant 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F6 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  6,194.4912  –  – 
MOEA/D—fold  6,194.0290  1.7344e\(\)006  Worse 
MOEA/D—diverge \((\alpha =0.1)\)  6,194.5119  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  6,194.5090  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  6,194.5032  5.2165e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  6,194.4949  0.0082167  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  6,194.4882  0.90993  Worse 
MOEA/D—reduce  6,194.2985  1.7344e\(\)006  Worse 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F7 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  100.4091  –  – 
MOEA/D—fold  100.4219  0.76552  Insignificant 
MOEA/D—diverge \((\alpha =0.1)\)  100.5664  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  100.5616  2.8434e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  100.5594  1.9209e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  100.5525  1.7344e\(\)006  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  100.5559  2.3534e\(\)006  Significant 
MOEA/D—reduce  100.4240  0.11561  Insignificant 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F8 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  72.4230  –  – 
MOEA/D—fold  72.3700  0.027029  Worse 
MOEA/D—diverge \((\alpha =0.1)\)  72.8925  0.0064242  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  72.9374  0.00061564  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  72.7774  0.010444  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  72.8966  0.0001057  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  72.7943  0.0031618  Significant 
MOEA/D—reduce  72.5166  0.45281  Insignificant 
Median hypervolume values obtained at the last (500th) generation by each of the algorithms for the F9 test problem
Algorithm  Median  Comparison to MOEA/D  

p value  Interpretation  
MOEA/D  17.9766  –  – 
MOEA/D—fold  18.0039  0.76552  Insignificant 
MOEA/D—diverge \((\alpha =0.1)\)  18.1837  1.3601e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.2)\)  18.1596  4.8603e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.3)\)  18.1485  4.0715e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.4)\)  18.1640  1.2381e\(\)005  Significant 
MOEA/D—diverge \((\alpha =0.5)\)  18.1373  6.8923e\(\)005  Significant 
MOEA/D—reduce  18.0549  0.075213  Insignificant 
The summary of the results of statistical tests performed for each of the algorithms working on each of the test problems
Algorithm  F1  F2  F3  F4  F5  F6  F7  F8  F9 

MOEA/D—fold  +  \(\)  #  \(\)  #  \(\)  #  \(\)  # 
MOEA/D—diverge \((\alpha =0.1)\)  +  +  +  +  #  +  +  +  + 
MOEA/D—diverge \((\alpha =0.2)\)  +  +  +  +  \(\)  +  +  +  + 
MOEA/D—diverge \((\alpha =0.3)\)  +  +  +  +  #  +  +  +  + 
MOEA/D—diverge \((\alpha =0.4)\)  +  +  +  +  \(\)  +  +  +  + 
MOEA/D—diverge \((\alpha =0.5)\)  +  +  +  +  #  \(\)  +  +  + 
MOEA/D—reduce  +  \(\)  +  #  #  \(\)  #  #  # 
The algorithm based on the “diverge” method is parameterized by the parameter \(\alpha \) which controls the divergence of the weight vectors. In the experiments the value of this parameter was set to \(\alpha = 0.1\), \(0.2\), \(0.3\), \(0.4\) and \(0.5\). Among these values the value of \(\alpha = 0.2\) yielded the best results in the case of test problems F2, F4 and F8. For all the other problems the “diverge” method produced the best results for \(\alpha = 0.1\).
Median values of the Inverted Generational Distance (IGD) indicator obtained at the last (500th) generation by each of the tested algorithms for the F1–F3 test problems
Algorithm  F1  F2  F3 

MOEA/D  \(0.575 \times 10^{3}\)  \({\varvec{2.108 \times 10^{2}}}\)  \(0.794 \times 10^{2}\) 
MOEA/D—fold  \({\varvec{0.547 \times 10^{3}}}\)  \(2.382 \times 10^{2}\)  \(0.895 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.1)\)  \(0.795 \times 10^{3}\)  \(2.749 \times 10^{2}\)  \({\varvec{0.631 \times 10^{2}}}\) 
MOEA/D—diverge \((\alpha =0.2)\)  \(0.930 \times 10^{3}\)  \(3.267 \times 10^{2}\)  \(0.726 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.3)\)  \(1.143 \times 10^{3}\)  \(3.817 \times 10^{2}\)  \(0.985 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.4)\)  \(1.277 \times 10^{3}\)  \(4.165 \times 10^{2}\)  \(1.074 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.5)\)  \(1.483 \times 10^{3}\)  \(4.575 \times 10^{2}\)  \(1.286 \times 10^{2}\) 
MOEA/D—reduce  \(0.579 \times 10^{3}\)  \(2.385 \times 10^{2}\)  \(0.896 \times 10^{2}\) 
Median values of the Inverted Generational Distance (IGD) indicator obtained at the last (500th) generation by each of the tested algorithms for the F4–F6 test problems
Algorithm  F4  F5  F6 

MOEA/D  \(1.004 \times 10^{2}\)  \(1.269 \times 10^{2}\)  \({\varvec{0.669 \times 10^{2}}}\) 
MOEA/D—fold  \(0.839 \times 10^{2}\)  \({\varvec{1.219 \times 10^{2}}}\)  \(2.316 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.1)\)  \({\varvec{0.483 \times 10^{2}}}\)  \({\varvec{1.219 \times 10^{2}}}\)  \(0.791 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.2)\)  \(0.604 \times 10^{2}\)  \(1.318 \times 10^{2}\)  \(0.940 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.3)\)  \(0.788 \times 10^{2}\)  \(1.478 \times 10^{2}\)  \(1.056 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.4)\)  \(0.864 \times 10^{2}\)  \(1.587 \times 10^{2}\)  \(1.211 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.5)\)  \(1.017 \times 10^{2}\)  \(1.632 \times 10^{2}\)  \(1.326 \times 10^{2}\) 
MOEA/D—reduce  \(0.685 \times 10^{2}\)  \(1.185 \times 10^{2}\)  \(1.423 \times 10^{2}\) 
Median values of the Inverted Generational Distance (IGD) indicator obtained at the last (500th) generation by each of the tested algorithms for the F6–F9 test problems
Algorithm  F7  F8  F9 

MOEA/D  \(2.226 \times 10^{3}\)  \(4.624 \times 10^{2}\)  \(2.871 \times 10^{2}\) 
MOEA/D—fold  \({\varvec{2.138 \times 10^{3}}}\)  \(6.184 \times 10^{2}\)  \(3.067 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.1)\)  \(2.673 \times 10^{3}\)  \(5.439 \times 10^{2}\)  \(3.400 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.2)\)  \(4.518 \times 10^{3}\)  \(6.122 \times 10^{2}\)  \(4.501 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.3)\)  \(4.143 \times 10^{3}\)  \(5.419 \times 10^{2}\)  \(4.529 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.4)\)  \(6.171 \times 10^{3}\)  \(6.568 \times 10^{2}\)  \(4.146 \times 10^{2}\) 
MOEA/D—diverge \((\alpha =0.5)\)  \(5.773 \times 10^{3}\)  \(7.313 \times 10^{2}\)  \(5.213 \times 10^{2}\) 
MOEA/D—reduce  \(2.725 \times 10^{3}\)  \({\varvec{4.552 \times 10^{2}}}\)  \({\varvec{2.751 \times 10^{2}}}\) 
6 Conclusion
In this paper three approaches to increasing the selective pressure to the outside of the Pareto front were proposed: the “fold”, “reduce” and “diverge” methods. The investigated methods change the way in which weight vectors affect the working of the multiobjective evolutionary algorithm based on decomposition (MOEA/D). The motivation for increasing the outwards selective pressure is that in the standard version of the algorithm the specimens which are placed near the edges of the Pareto front can only exchange information with specimens that direct their search to the inside of the Pareto front. Such mechanism can be expected to decrease the capability of the algorithm to extend the area of the search.
In order to validate the proposed methods experiments on nine test problems F1–F9 were performed. These test problems were specifically proposed in [17] as benchmarks for testing the MOEA/D algorithm on problems with complicated Pareto sets. The quality of Pareto fronts found by the tested algorithms was evaluated using the hypervolume indicator. The results of the experiments show that the algorithm based on the “diverge” method in which weight vectors are pointed outwards was able to significantly improve the optimization results for all the test problems except F5. In the case of the 3dimensional problem F6 the improvement was obtained for the values of the parameter \(\alpha < 0.5\). For all the problems the best results for the “diverge” method were obtained when the parameter \(\alpha \) was set either to \(0.1\) or to \(0.2\). A possible explanation of these results is that high values of the \(\alpha \) parameter cause the algorithm to put too much pressure on extending the search instead of improving the objectives. Other algorithms, namely “fold” and “reduce” were able to improve the results for some of the test problems but without statistical significance. The only exception is the F1 problem for which all the proposed methods performed significantly better than the standard MOEA/D algorithm. To the contrary, on the F5 problem no significant improvement has been observed, even though some of the tested methods produced better results than the standard MOEA/D algorithm. Distributions of hypervolume values presented in Fig. 34 show that for the F5 problem the quality of results produced by all the algorithms varies greatly between runs. This makes it hard to determine if the observed differences in median values are caused by a better working of a particular algorithm or by coincidental variation of the performance of the search. It is worth noticing that also the authors of the MOEA/D algorithm observed in [17] that this algorithm performs poorly on the test problem F5.
In general, the proposed approach based on extending the selective pressure outwards seems to improve the quality of the results. On most of the test problems the improvement obtained by the “diverge” method is statistically significant at the confidence level \(0.05\). The “fold” and “reduce” methods performed worse than the “diverge” method with the only exception of the F5 test problem for which the “reduce” method performed best. This suggests that it is not enough to only increase the probability of choosing the outermost weight vectors in the selection process. There seems to be a significant benefit in explicitly directing the search towards the outside of the Pareto front.
Further work may include hybridization of the proposed approach with other optimization algorithms and local search methods. For example, hybrid approaches presented in [1, 14, 16, 26] could be combined with the weight generation scheme proposed in this paper. Local search procedures work on solutions already generated by the main evolutionary algoritm and usually are executed as a separate subroutine. Therefore, it is possible to choose independently which weight generation scheme to use and what kind of solution improvement procedure to employ for a given problem.
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