# On the effect of normalization in MOEA/D for multi-objective and many-objective optimization

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## Abstract

The frequently used basic version of MOEA/D (multi-objective evolutionary algorithm based on decomposition) has no normalization mechanism of the objective space, whereas the normalization was discussed in the original MOEA/D paper. As a result, MOEA/D shows difficulties in finding a set of uniformly distributed solutions over the entire Pareto front when each objective has a totally different range of objective values. Recent variants of MOEA/D have normalization mechanisms for handling such a scaling issue. In this paper, we examine the effect of the normalization of the objective space on the performance of MOEA/D through computational experiments. A simple normalization mechanism is used to examine the performance of MOEA/D with and without normalization. These two types of MOEA/D are also compared with recently proposed many-objective algorithms: NSGA-III, MOEA/DD, and \(\theta \)-DEA. In addition to the frequently used many-objective test problems DTLZ and WFG, we use their minus versions. We also propose two variants of the DTLZ test problems for examining the effect of the normalization in MOEA/D. Test problems in one variant have objective functions with totally different ranges. The other variant has a kind of deceptive nature, where the range of each objective is the same on the Pareto front but totally different over the entire feasible region. Computational experiments on those test problems clearly show the necessity of the normalization. It is also shown that the normalization has both positive and negative effects on the performance of MOEA/D. These observations suggest that the influence of the normalization is strongly problem dependent.

### Keywords

Evolutionary multi-objective optimization (EMO) Many-objective optimization Objective space normalization MOEA/D Decomposition-based algorithms## Introduction

Recently, many-objective optimization has received a lot of attention in the evolutionary multi-objective optimization (EMO) community, where optimization of four or more objectives is called many-objective optimization [14, 15, 23]. Many-objective problems present a number of challenges [10, 19] to the EMO community such as the deterioration in search ability of Pareto dominance-based algorithms [6, 29] and the increase in computation time of hypervolume-based algorithms [2, 3]. For many-objective problems, it has been demonstrated in the literature [10, 12] that MOEA/D [27] works well in comparison with Pareto dominance-based and hypervolume-based algorithms in terms of their search ability and computation time. As a result, a number of EMO algorithms have been proposed for many-objective problems based on the same or similar framework as MOEA/D (e.g., NSGA-III [5], MOEA/DD [16], I-DBEA [1], and \(\theta \)-DEA [26]).

Various approaches to the improvement of MOEA/D have also been proposed in the literature (for detail, see a review [22]). One important research issue is the specification of a scalarizing function [11, 21, 24]. The specification of weight vectors [8] and their adjustments [20] is also important. Other important issues include solution–subproblem matching [17, 18] and resource allocation to subproblems [28]. Whereas the normalization of the objective space is also important, this issue has not been actively studied in the literature [4]. One exception is Bhattacharjee et al. [4], where a six-sigma-based method was proposed for removing the influence of dominance-resistant solutions on nadir point calculation.

In this paper, we examine the effect of the normalization on the performance of MOEA/D. The frequently used basic version of MOEA/D [27] has no normalization mechanism, whereas the normalization was discussed in the original MOEA/D paper [27]. As a result, MOEA/D is often outperformed by other EMO algorithms with normalization mechanisms in their applications to test problems, where each objective has a totally different range of objective values. For example, in the WFG4-9 test problems [9], the range of the Pareto front on the *i*th objective is [0, 2*i*]. This means that the tenth objective \(f_{10}\) has a ten times wider range than the first objective \(f_{1}\). It is difficult for MOEA/D without normalization to find a set of uniformly distributed solutions over the entire Pareto front for such a many-objective test problem. In this paper, we combine a simple normalization mechanism to MOEA/D in order to compare the performance between MOEA/D with and without normalization.

This paper is organized as follows. First, we briefly explain MOEA/D and a simple normalization mechanism. Next, we demonstrate that the combined normalization mechanism has positive and negative effects on the performance of MOEA/D through computational experiments on the DTLZ1-4 [7] and WFG4-9 [9] test problems with three to ten objectives. We also discuss why the normalization deteriorates the performance of MOEA/D on some test problems. Then, we create two variants of the DTLZ1-4 test problems to further examine the effect of the normalization in MOEA/D. Each test problem in one variant has objective functions with totally different ranges. More specifically, the range of the Pareto front on the *i*th objective is [0, \(\alpha ^{i-1}\)], where \(\alpha \) is a non-negative parameter (e.g., 2 and 10). The other variant has a kind of deceptive nature. Whereas the range of each objective is totally different over the entire feasible region, the Pareto front has the same range for each objective. In other words, each objective has totally different values in early generations and similar values after enough generations. Experimental results on the new test problems clearly show the necessity of the normalization as well as its negative effects. Finally, we report experimental results on minus versions [13] of the DTLZ1-4 and WFG4-9 test problems. The best results on almost all minus test problems are obtained by MOEA/D with normalization among MOEA/D, NSGA-III, MOEA/DD, and \(\theta \)-DEA.

## MOEA/D and normalization

*m*-objective minimization problem:

*i*th objective to be minimized (\(i= 1, 2, \ldots , m\)), \({{\varvec{x}}}\) is a decision vector, and \({{\varvec{X}}}\) is a feasible region of \({{\varvec{x}}}\) in the decision space.

*H*is a positive integer for determining the resolutions of the generated weight vectors. MOEA/D generates all weight vectors satisfying these relations. The population size is the same as the total number of the generated weight vectors. This is because each single-objective problem with a different weight vector has a single solution.

*i*th objective \(f_{i}({{\varvec{x}}})\) among non-dominated solutions in the current population (\(i = 1, 2, \ldots , m\)), respectively. In this setting, \({{\varvec{z}}}_{L} = (z_{1L}, \ldots , z_{mL})\) and \({{\varvec{z}}}_{U} = (z_{1U}, \ldots , z_{mU})\) can be viewed as estimated ideal and nadir points from the non-dominated solutions in the current population, respectively. Using \(z_{iL}\) and \(z_{iU}\), the objective value \(z_{i}\) of the

*i*th objective \(f_{i}({{\varvec{x}}})\) is normalized as

## Experimental results on DTLZ and WFG

Let us examine the performance of MOEA/D with and without the simple normalization mechanism through computation experiments on frequently used many-objective test problems in the literature: DTLZ1-4 [7] and WFG4-9 [9] with 3, 5, 8, and 10 objectives. The range of each objective on the Pareto front of each test problem is as follows: [0, 0.5] in DTLZ1, [0, 1] in DTLZ2-4, and [0, 2*i*] for \(i = 1, 2, \,\ldots , m\) in WFG4-9. Thus, the normalization of the objective space is needed in WFG4-9, whereas it is not needed in DTLZ1-4.

- 1.
Normalization: NSGA-III and \(\theta \)-DEA have a sophisticated normalization mechanism while MOEA/DD and MOEA/D in this paper do not have a normalization mechanism.

- 2.
Generation update: NSGA-III and \(\theta \)-DEA use a \((\mu + \mu )\) ES model, while MOEA/D and MOEA/DD use a \((\mu + 1)\) ES model.

- 3.
Assignment of a new solution to weight vectors: a new solution is assigned to the single nearest weight vector in NSGA-III, MOEA/DD and \(\theta \)-DEA, while a new solution is compared to all solutions in its neighborhood in MOEA/D.

- 4.
Replacement of solutions: only a single solution can be replaced with a new solution in NSGA-III, MOEA/DD, and \(\theta \)-DEA, while multiple solutions in the neighborhood can be replaced with a new solution in MOEA/D.

- 5.
Handling of weight vectors outside the Pareto front: no new solutions are assigned to those weight vectors in NSGA-III, MOEA/DD, and \(\theta \)-DEA (i.e., some weight vectors may have no solutions and some others may have multiple solutions), while each weight vector always has a single solution and continues to search for its best solution in MOEA/D.

- 6.
Basic mechanisms of fitness evaluation: the PBI function (MOEA/D), Pareto dominance, the number of assigned solutions to each weight vector and the distance to the closest weight vector (NSGA-III), and Pareto dominance, the number of assigned solutions to each weight vector and PBI (MOEA/DD and \(\theta \)-DEA).

*n*, where

*n*is the number of decision variables. The simulated binary crossover with the distribution index 20 is used with the crossover probability 1.0. The number of weight vectors is specified as 91 (three objectives), 210 (five objectives), 156 (eight objectives), and 275 (ten objectives).

Normalized average hypervolume values

Problem | Objectives | MOEA/D | Normalized MOEA/D (\(\varepsilon = 10^{-6}\)) | Normalized MOEA/D (\(\varepsilon = 1\)) | MOEA/DD | NSGA-III | \(\theta \)-DEA |
---|---|---|---|---|---|---|---|

DTLZ1 | 3 | 0.9976 | 0.9963 | 0.9976 | | 0.9954 | 0.9978 |

5 | 0.9999 | 0.9992 | 0.9997 | | 0.9996 | 0.9998 | |

8 | 0.9993 | 0.3827 | 0.9985 | 0.9998 | 0.9999 | | |

10 | 0.9999 | 0.3829 | 0.9998 | 0.9999 | | | |

DTLZ2 | 3 | 0.9996 | 0.9994 | 0.9995 | | 0.9985 | 0.9993 |

5 | 0.9997 | 0.9996 | 0.9996 | | 0.9965 | 0.9993 | |

8 | 0.9998 | 0.9383 | 0.9271 | | 0.9953 | 0.9995 | |

10 | | 0.9767 | 0.8650 | | 0.9975 | 0.9996 | |

DTLZ3 | 3 | 0.9961 | 0.9902 | 0.9624 | | 0.9955 | 0.9967 |

5 | 0.9984 | 0.9882 | 0.5262 | | 0.9955 | 0.9988 | |

8 | 0.9006 | 0.7676 | 0.4812 | | 0.9907 | 0.9971 | |

10 | 0.9757 | 0.8241 | 0.6730 | | 0.9972 | 0.9994 | |

DTLZ4 | 3 | 0.7425 | 0.4122 | 0.3442 | | 0.9339 | 0.9422 |

5 | 0.9299 | 0.5606 | 0.4284 | | 0.9997 | | |

8 | 0.9585 | 0.3519 | 0.3273 | | 0.9997 | | |

10 | 0.9912 | 0.5542 | 0.3577 | | 0.9999 | | |

WFG4 | 3 | 0.9393 | 0.9469 | 0.9463 | 0.9875 | 0.9987 | |

5 | 0.8946 | 0.9854 | 0.9850 | 0.9790 | 0.9984 | | |

8 | 0.5974 | 0.9277 | 0.9922 | 0.9346 | 0.9992 | | |

10 | 0.5809 | | 0.9985 | 0.8926 | 0.9973 | 0.9977 | |

WFG5 | 3 | 0.9547 | 0.9656 | 0.9631 | 0.9851 | 0.9995 | |

5 | 0.9060 | 0.9957 | 0.9894 | 0.9727 | 0.9994 | | |

8 | 0.6812 | 0.5417 | 0.9966 | 0.9272 | 0.9998 | | |

10 | 0.6442 | 0.5739 | | 0.8892 | 0.9994 | 0.9994 | |

WFG6 | 3 | 0.9524 | 0.9597 | 0.9609 | 0.9881 | 0.9993 | |

5 | 0.8360 | 0.9946 | 0.9943 | 0.9770 | 0.9992 | | |

8 | 0.3859 | 0.2252 | 0.9973 | 0.9104 | 0.9985 | | |

10 | 0.3512 | 0.3274 | | 0.8713 | 0.9939 | | |

WFG7 | 3 | 0.8472 | 0.8782 | 0.8771 | 0.9863 | 0.9976 | |

5 | 0.8248 | 0.9914 | 0.9901 | 0.9721 | 0.9974 | | |

8 | 0.4159 | 0.1617 | 0.9971 | 0.9213 | 0.9989 | | |

10 | 0.3866 | 0.3970 | | 0.9072 | 0.9985 | 0.9990 | |

WFG8 | 3 | 0.9440 | 0.9547 | 0.9597 | 0.9857 | 0.9964 | |

5 | 0.7928 | | 0.9945 | 0.9703 | 0.9938 | 0.9944 | |

8 | 0.1684 | 0.1207 | | 0.9572 | 0.9958 | 0.9985 | |

10 | 0.1362 | 0.1121 | | 0.9620 | 0.9870 | 0.9877 | |

WFG9 | 3 | 0.8613 | 0.8652 | 0.8717 | 0.9890 | 0.9997 | |

5 | 0.8439 | 0.9617 | 0.9539 | 0.9402 | 0.9902 | | |

8 | 0.5225 | 0.2536 | 0.9218 | 0.8735 | 0.9866 | | |

10 | 0.4897 | 0.1216 | 0.8855 | 0.8234 | 0.9889 | |

The performance of each approach is evaluated by calculating the average hypervolume value over 51 runs. Before calculating the hypervolume, the objective space is normalized using the true Pareto front of each test problem, so that the ideal and nadir points are \((0, 0, \ldots , 0)\) and \((1, 1, \ldots , 1)\). This information is used only for the hypervolume calculation after the execution of each algorithm. The reference point for the hypervolume calculation is specified as \((1.1, 1.1, {\ldots }, 1.1)\) in the normalized objective space.

Experimental results are shown in Table 1. For easy reading of the experimental results, the average hypervolume value of each algorithm on each test problem is normalized using the best result in each row, as shown in Table 1. Thus, the best value for each test problem is always 1.0000, as shown in Table 1. The best and worst results are highlighted by the bold font and the underline, respectively.

The performance of MOEA/DD in Table 1 clearly shows the necessity of normalization. Whereas the best results are obtained from MOEA/DD for most test problems in DTLZ1-4, NSGA-III, and \(\theta \)-DEA outperform MOEA/DD in their applications to WFG4-9. This is because MOEA/DD has no normalization mechanism, whereas the search in NSGA-III and \(\theta \)-DEA is performed in the normalized objective space. However, when they were applied to the normalized WFG4-9 in [13], the best results were obtained from MOEA/DD.

MOEA/D with no normalization mechanism also shows clear performance deterioration for WFG4-9, as shown in Table 1. By combining the simple normalization mechanism with \(\varepsilon = 10^{-6}\) into MOEA/D, its performance is improved for the three-objective and five-objective WFG4-9 test problems, as shown in Table 1. However, the simple normalization mechanism with \(\varepsilon = 10^{-6}\) deteriorates the performance of MOEA/D for almost all of the other test problems. Especially, negative effects of the simple normalization mechanism on the performance of MOEA/D are clearly observed for DTLZ1-4, as shown in Table 1.

By increasing the value of \(\varepsilon \) from \(\varepsilon = 10^{-6}\) to \(\varepsilon = 1\) in Table 1, the severe negative effects of the simple normalization mechanism with \(\varepsilon = 10^{-6}\) are remedied for most test problems except for DTLZ3 and DTLZ4. In the next section, we discuss why the performance of MOEA/D on some test problems is severely degraded by the simple normalization mechanism.

## Negative effect of normalization

Let us examine the search behavior of MOEA/D with the simple normalization mechanism \((\varepsilon = 10^{-6})\) on the ten-objective DTLZ1 and WFG9 test problems, where the negative effects of the normalization are severe, as shown in Table 1. In Fig. 5a, we show the obtained solution set by a single run of MOEA/D without normalization on the ten-objective DTLZ1 in the normalized objective space \([0, 1]^{10}\). The single run with the median average hypervolume value is selected from the 51 runs as a representative run. The corresponding result by MOEA/D with normalization is shown in Fig. 5b, where the diversity of solutions is severely degraded by normalization.

In the same manner as in Fig. 5, we show the obtained solution sets on the ten-objective WFG9, as shown in Fig. 6. Figure 6 also shows that the diversity of solutions is severely degraded by normalization. For comparison, we show the results by MOEA/D with normalization in the case of \(\varepsilon = 1\), as shown in Fig. 7. Comparison between Fig. 6a without normalization and Fig. 7b with normalization shows clear positive effects of the normalization when \(\varepsilon \) is specified as \(\varepsilon = 1\).

Since the denominator in the simple normalization mechanism is \((z_{iU}-z_{iL}) + \,\varepsilon \), the objective space is severely rescaled by the normalization when \(\varepsilon \) is very small. As a result, the diversity of solutions is severely deteriorated in Figs. 8b and 9b within the first ten generations. When \(\varepsilon \) is not very small (e.g., \(\varepsilon =1\), as shown in Fig. 7), the problem of very small values of \((z_{iU}-z_{iL})\) is remedied.

Let us further discuss why unnecessary normalization decreases the diversity of solutions. Figure 10 shows the relation between the best solution for the PBI function with the weight vector (0.9, 0.1) for three specifications of \(\theta \) when the Pareto front is linear. When \(\theta \) is large, the optimal solution is obtained on the search line specified by the weight vector, as shown in Fig. 10b, c. When \(\theta \) is small, the best solution is not on the search line specified by the weight vector, as shown in Fig. 10a. Thus, a large value of \(\theta \) such as \(\theta = 5\) is frequently used in the literature. However, a large value of \(\theta \) makes the better region than the current solution very small (i.e., inside the red contour line in Fig. 10, see also Fig. 2). This leads to slow convergence especially in the case of many-objective optimization [10]. Thus, an appropriate specification of \(\theta \) is very important and difficult in MOEA/D with the PBI function.

Another potential difficulty of using a small value of \(\varepsilon \) is that objective values of dominated solutions may become very large when the width (\(z_{iU}-z_{iL}\)) is small. For example, when \(z_{iL} = 0.10\), \(z_{iU} = 0.11\), and \(z_{i}\) = 1.00, these values are normalized to 0.00, 1.00, and 89.99 by the formulation \((z_{i} - 0.10)/(0.01 + \varepsilon )\) with \(\varepsilon =10^{-6}\). This difficulty is remedied by increasing the value of \(\varepsilon \). For example, \(z_{iL} = 0.10, z_{iU} = 0.11\) and \(z_{i} = 1.00\) are normalized to 0.00, 0.01, and 0.89, respectively, when \(\varepsilon =1\).

However, in this case (i.e., when \(\varepsilon \) is not very small), we have a different type of difficulty: the width of the Pareto front on each objective in the normalized objective space is different [i.e., the nadir point in the normalized objective space is not (1, 1, \(\ldots \) , 1)]. For example, let us assume that the Pareto front is a line between (0, 3) and (1, 0) in the original two-dimensional objective space. The Pareto front is in a subspace \([0, 1]\times [0, 3]\). This Pareto front is normalized to a line between (0, 1) and (1, 0) when \(\varepsilon \) is very small. That is, the subspace \([0, 1]\times [0, 3]\) including the Pareto front is normalized to \([0, 1]\times [0, 1]\). However, the same Pareto front is normalized to a line between (0.0, 0.75) and (0.5, 0.0) when \(\varepsilon = 1\). That is, the subspace \([0, 1]\times [0, 3]\) is normalized to \([0.0, 0.5]\times [0.0, 0.75]\). As shown by this simple example, a hyper-rectangle including the Pareto front of an *m*-objective problem is not normalized to a unit hypercube \([0, 1]^{m}\) when \(\varepsilon \) cannot be negligible. This negative effect of using a large value of \(\varepsilon \) is illustrated in Fig. 12 using experimental results on the three-objective WFG4 test problem by the three variants of MOEA/D: no normalization, normalization with \(\varepsilon = 10^{-6}\) and \(\varepsilon = 1\). When no normalization mechanism is used in Fig. 12a, many solutions are obtained around the bottom-left corner of the Pareto front. When \(\varepsilon \) is very small in Fig. 12b, well-distributed solutions are obtained. However, when \(\varepsilon \) is not very small (i.e., \(\varepsilon = 1\)) in Fig. 12c, we can see that more solutions are obtained around the bottom-left corner of the Pareto front than the other two corners (e.g., compare the \(3\times 3\) solutions around each corner).

## New many-objective test problems

We have already explained that the simple normalization mechanism has both positive and negative effects on the performance of MOEA/D. In this section, we propose new many-objective test problems to further examine the effect of the normalization. Our new test problems are generated by slightly modifying the DTLZ1-4 test problems as follows.

*i*th objective has an \(\alpha ^{i-1}\) times larger range than the first objective. The newly created test problems from DTLZ1-4 using this objective function are referred to as the rescaled DTLZ1-4 test problems. In our computational experiments, \(\alpha \) is specified as \(\alpha = 10\).

We perform computational experiments on the rescaled and deceptive rescaled DTLZ1-4 test problems in the same manner as in the above-mentioned computational experiments on DTLZ1-4 and WFG4-9. Experimental results are summarized in Table 2. The average hypervolume value by each algorithm on each test problem is normalized by the best result. The best result for each test problem is highlighted by the bold font in Table 2, while the worst result is underlined.

Normalized average hypervolume values on the new test problems

Problem | Objectives | MOEA/D | Normalized MOEA/D (\(\varepsilon = 10^{-6}\)) | Normalized MOEA/D (\(\varepsilon = 1\)) | MOEA/DD | NSGA-III | \(\theta \)-DEA |
---|---|---|---|---|---|---|---|

Rescaled DTLZ1 | 3 | 0.3276 | 0.9973 | 0.1575 | 0.6492 | 0.9992 | |

5 | 0.3531 | 0.9997 | 0.1613 | 0.4632 | 0.9990 | | |

8 | 0.2291 | 0.7780 | 0.1416 | 0.3502 | 0.9947 | | |

10 | 0.2287 | 0.7287 | 0.2496 | 0.3482 | 0.9975 | | |

Rescaled DTLZ2 | 3 | 0.2832 | | 0.9778 | 0.5318 | 0.9991 | 0.9998 |

5 | 0.1940 | | 0.9965 | 0.3075 | 0.9870 | 0.9998 | |

8 | 0.1148 | 0.8259 | 0.8049 | 0.1621 | 0.9953 | | |

10 | 0.1092 | 0.9475 | 0.7955 | 0.1558 | 0.9976 | | |

Rescaled DTLZ3 | 3 | 0.1834 | 0.9953 | 0.9099 | 0.5071 | 0.9976 | |

5 | 0.1855 | 0.9893 | 0.7738 | 0.3252 | 0.9824 | | |

8 | 0.1148 | 0.8656 | 0.2277 | 0.1467 | 0.7214 | | |

10 | 0.1093 | 0.6570 | 0.1505 | 0.1508 | 0.9294 | | |

Rescaled DTLZ4 | 3 | 0.2418 | 0.3561 | 0.2491 | 0.5257 | | 0.9920 |

5 | 0.1911 | 0.3808 | 0.1613 | 0.3310 | 0.9997 | | |

8 | 0.1141 | 0.3220 | 0.1336 | 0.1511 | 0.9972 | | |

10 | 0.1091 | 0.3415 | 0.1115 | 0.1463 | 0.9999 | | |

Deceptive rescaled DTLZ1 | 3 | 0.8579 | 0.9512 | 0.9136 | | 0.9960 | 0.9974 |

5 | 0.9080 | 0.9996 | 0.9200 | 0.9714 | 0.9617 | | |

8 | | 0.0000 | 0.9959 | 0.7548 | 0.2640 | 0.5702 | |

10 | | 0.0000 | 0.9827 | 0.7875 | 0.1620 | 0.2732 | |

Deceptive rescaled DTLZ2 | 3 | 0.8936 | 0.9367 | 0.8576 | | 0.9889 | 0.9948 |

5 | 0.7898 | 0.9069 | 0.3273 | 0.8518 | 0.9473 | | |

8 | 0.7913 | 0.3393 | 0.6748 | 0.6794 | 0.8234 | | |

10 | 0.9243 | 0.3229 | | 0.6925 | 0.6250 | 0.9975 | |

Deceptive rescaled DTLZ3 | 3 | 0.6596 | 0.7294 | 0.6243 | | 0.9461 | 0.9428 |

5 | 0.5444 | 0.9514 | 0.7551 | 0.9023 | 0.8624 | | |

8 | | 0.0362 | 0.4996 | 0.2603 | 0.0012 | 0.1405 | |

10 | | 0.0297 | 0.3635 | 0.3415 | 0.0040 | 0.0339 | |

Deceptive rescaled DTLZ4 | 3 | 0.6529 | 0.3731 | 0.2346 | | 0.9588 | 0.9745 |

5 | 0.9010 | 0.2834 | 0.2409 | 0.9936 | 0.9707 | | |

8 | 0.9554 | 0.1694 | 0.1369 | 0.7511 | 0.9194 | | |

10 | | 0.1617 | 0.1483 | 0.7817 | 0.9720 | 0.9272 |

Normalized average hypervolume values on the minus test problems

Problem | Objectives | MOEA/D | Normalized MOEA/D (\(\varepsilon = 10^{-6}\)) | Normalized MOEA/D (\(\varepsilon = 1\)) | MOEA/DD | NSGA-III | \(\theta \)-DEA |
---|---|---|---|---|---|---|---|

Minus DTLZ1 | 3 | 0.9594 | 0.9584 | 0.9606 | 0.9129 | | 0.9191 |

5 | | 0.9607 | 0.9616 | 0.5667 | 0.7184 | 0.5097 | |

8 | 0.1402 | 0.8952 | 0.8957 | 0.1704 | | 0.8332 | |

10 | 0.0744 | 0.6903 | 0.6892 | 0.0921 | | 0.3926 | |

Minus DTLZ2 | 3 | 0.9972 | | 0.9979 | 0.9889 | 0.9904 | 0.9940 |

5 | 0.8972 | | 0.9998 | 0.4772 | 0.7963 | 0.7514 | |

8 | 0.9133 | 0.9656 | | 0.3958 | 0.6710 | 0.5354 | |

10 | 0.7771 | 0.9346 | | 0.2424 | 0.9217 | 0.7531 | |

Minus DTLZ3 | 3 | 0.9994 | 0.9995 | | 0.9910 | 0.9948 | 0.9987 |

5 | 0.9256 | | 0.9995 | 0.4637 | 0.7496 | 0.7580 | |

8 | 0.9469 | | 0.9969 | 0.3936 | 0.6517 | 0.6118 | |

10 | 0.8422 | 0.9986 | | 0.2696 | 0.8382 | 0.9285 | |

Minus DTLZ4 | 3 | 0.8627 | 0.9816 | 0.8474 | 0.9910 | 0.9972 | |

5 | 0.8146 | | 0.8342 | 0.4764 | 0.8174 | 0.7732 | |

8 | 0.4505 | | 0.6369 | 0.4501 | 0.9927 | 0.9147 | |

10 | 0.3698 | 0.8860 | 0.9417 | 0.4217 | 0.9801 | | |

Minus WFG4 | 3 | 0.9907 | 0.9993 | | 0.9596 | 0.9593 | 0.9948 |

5 | 0.7978 | | 0.9858 | 0.6427 | 0.7371 | 0.8224 | |

8 | 0.1020 | | 0.8518 | 0.0390 | 0.8236 | 0.8396 | |

10 | 0.0385 | | 0.7760 | 0.0117 | 0.9266 | 0.4467 | |

Minus WFG5 | 3 | 0.9886 | | 0.9998 | 0.9730 | 0.9658 | 0.9943 |

5 | 0.7971 | | 0.9807 | 0.7064 | 0.7391 | 0.6840 | |

8 | 0.1399 | | 0.8606 | 0.0982 | 0.7293 | 0.7465 | |

10 | 0.0472 | | 0.7664 | 0.0395 | 0.8073 | 0.4733 | |

Minus WFG6 | 3 | 0.9863 | | 0.9998 | 0.9722 | 0.9854 | 0.9955 |

5 | 0.7868 | | 0.9830 | 0.7017 | 0.7582 | 0.7317 | |

8 | 0.1328 | | 0.8514 | 0.1209 | 0.7477 | 0.6654 | |

10 | 0.0480 | | 0.7818 | 0.0428 | 0.9258 | 0.3733 | |

Minus WFG7 | 3 | 0.9866 | | 0.9997 | 0.9517 | 0.9490 | 0.9945 |

5 | 0.7925 | | 0.9912 | 0.6557 | 0.6815 | 0.6902 | |

8 | 0.0975 | | 0.7460 | 0.0773 | 0.7888 | 0.7182 | |

10 | 0.0365 | | 0.6784 | 0.0343 | 0.9392 | 0.4894 | |

Minus WFG8 | 3 | 0.9849 | | 0.9983 | 0.9752 | 0.9900 | 0.9946 |

5 | 0.7955 | | 0.9866 | 0.7515 | 0.7823 | 0.7081 | |

8 | 0.1499 | | 0.8669 | 0.2144 | 0.7837 | 0.7046 | |

10 | 0.0514 | | 0.7610 | 0.0868 | 0.9441 | 0.4050 | |

Minus WFG9 | 3 | 0.9692 | 0.9851 | 0.9854 | 0.9406 | 0.9789 | |

5 | 0.7911 | | 0.9827 | 0.6965 | 0.8083 | 0.7161 | |

8 | 0.1344 | | 0.8334 | 0.1414 | 0.8336 | 0.7614 | |

10 | 0.0453 | | 0.7317 | 0.0564 | 0.8693 | 0.4880 |

## Experimental results on minus test problems

The minus version of DTLZ and WFG was proposed as many-objective test problems with inverted triangular Pareto fronts in [13]. The minus version can be easily formulated by changing each objective in DTLZ and WFG from \(f_{i}({{\varvec{x}}})\) to \(-f_{i}({{\varvec{x}}})\). This modification is the same as changing from “minimization of \(f_{i}({{\varvec{x}}})\)” to “ maximization of \(f_{i}({{\varvec{x}}})\)” in DTLZ and WFG. The main feature of the minus DTLZ and WFG test problems is that their Pareto fronts are inverted triangular, whereas the Pareto fronts of DTLZ and WFG are triangular. We perform computational experiments on the minus DTLZ1-4 and WFG4-9 test problems in the same manner as in the previous computational experiments on DTLZ1-4 and WFG4-9.

Comparison between the two normalization strategies on DTLZ1-4 and WFG4-9

Problem | Objectives | Based on non-dominated solutions | Based on all solutions | ||||
---|---|---|---|---|---|---|---|

\(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | \(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | ||

DTLZ1 | 3 | 0.9968 | 0.9963 | | 0.9094 | 0.9212 | |

5 | 0.5014 | 0.9992 | 0.9997 | 0.1392 | 0.2431 | | |

8 | 0.2815 | 0.3827 | 0.9985 | 0.1000 | 0.1329 | | |

10 | 0.2523 | 0.3829 | 0.9998 | 0.1066 | 0.1095 | | |

DTLZ2 | 3 | 0.9994 | 0.9994 | | 0.9993 | 0.9993 | |

5 | 0.9995 | | | 0.9474 | 0.9821 | | |

8 | 0.3857 | 0.9383 | 0.9271 | 0.1191 | 0.1941 | | |

10 | 0.5454 | 0.9767 | 0.8650 | 0.1065 | 0.3532 | | |

DTLZ3 | 3 | 0.9899 | 0.9902 | 0.9624 | 0.9111 | 0.9111 | |

5 | 0.9856 | 0.9882 | 0.5262 | 0.1119 | 0.1119 | | |

8 | 0.3544 | 0.7676 | 0.4812 | 0.0984 | 0.0984 | | |

10 | 0.3288 | 0.8241 | 0.6730 | 0.0937 | 0.0937 | | |

DTLZ4 | 3 | 0.3213 | 0.4122 | 0.3442 | 0.7130 | 0.6517 | |

5 | 0.3010 | 0.5606 | 0.4284 | | 0.9538 | 0.9368 | |

8 | 0.2556 | 0.3519 | 0.3273 | | 0.9636 | 0.9404 | |

10 | 0.3068 | 0.5542 | 0.3577 | | 0.9964 | 0.9960 | |

WFG4 | 3 | 0.9470 | 0.9469 | 0.9463 | 0.9457 | 0.9457 | |

5 | 0.9854 | 0.9854 | 0.9850 | 0.9866 | 0.9865 | | |

8 | 0.3519 | 0.9277 | 0.9922 | 0.0992 | 0.0992 | | |

10 | 0.4854 | | 0.9985 | 0.0939 | 0.0986 | 0.9989 | |

WFG5 | 3 | | | 0.9631 | 0.9638 | 0.9645 | 0.9619 |

5 | 0.9957 | 0.9957 | 0.9894 | | | 0.9903 | |

8 | 0.1979 | 0.5417 | | 0.9343 | 0.9767 | | |

10 | 0.0874 | 0.5739 | | 0.6801 | 0.7632 | | |

WFG6 | 3 | 0.9597 | 0.9597 | 0.9609 | 0.9617 | | 0.9586 |

5 | | | 0.9943 | 0.9935 | 0.9934 | 0.9945 | |

8 | 0.2038 | 0.2252 | 0.9973 | 0.0967 | 0.0967 | | |

10 | 0.2286 | 0.3274 | 1.0000 | 0.0915 | 0.0915 | | |

WFG7 | 3 | 0.8783 | 0.8782 | 0.8771 | 0.8741 | 0.8740 | |

5 | 0.9914 | 0.9914 | 0.9901 | | | 0.9917 | |

8 | 0.1571 | 0.1617 | 0.9971 | 0.0987 | 0.0987 | | |

10 | 0.2408 | 0.3970 | 1.0000 | 0.0939 | 0.0977 | | |

WFG8 | 3 | 0.9547 | 0.9547 | | 0.9560 | 0.9560 | 0.9592 |

5 | 1.0000 | | 0.9945 | 0.9991 | 0.9991 | 0.9950 | |

8 | 0.1204 | 0.1207 | 1.0000 | 0.1118 | 0.1114 | | |

10 | 0.1126 | 0.1121 | 1.0000 | 0.1032 | 0.1030 | | |

WFG9 | 3 | 0.8652 | 0.8652 | | 0.8704 | 0.8704 | 0.8599 |

5 | 0.9616 | 0.9617 | 0.9539 | | | 0.9611 | |

8 | 0.1434 | 0.2536 | 0.9218 | 0.2261 | 0.1900 | | |

10 | 0.0833 | 0.1216 | 0.8855 | 0.1469 | 0.1464 | |

Comparison between the two normalization strategies on the rescaled and deceptive rescaled DTLZ1-4

Problem | Objectives | Based on non-dominated solutions | Based on all solutions | ||||
---|---|---|---|---|---|---|---|

\(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | \(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | ||

Rescaled DTLZ1 | 3 | | 0.9973 | 0.1575 | 0.9109 | 0.9217 | 0.8928 |

5 | 0.8424 | | 0.1613 | 0.1382 | 0.2202 | 0.7555 | |

8 | 0.3062 | | 0.1416 | 0.1000 | 0.1090 | 0.4345 | |

10 | 0.2747 | | 0.2496 | 0.0998 | 0.0909 | 0.4095 | |

Rescaled DTLZ2 | 3 | | | 0.9778 | | | 0.9940 |

5 | 0.9999 | | 0.9965 | 0.9651 | 0.9999 | 0.9965 | |

8 | 0.4432 | | 0.8049 | 0.1043 | 0.1264 | 0.4963 | |

10 | 0.8817 | | 0.7955 | 0.0938 | 0.0982 | 0.4312 | |

Rescaled DTLZ3 | 3 | | 0.9953 | 0.9099 | 0.9161 | 0.9160 | 0.9950 |

5 | 0.9923 | 0.9893 | 0.7738 | 0.1122 | 0.1122 | | |

8 | 0.0367 | | 0.2277 | 0.0986 | 0.0986 | 0.3895 | |

10 | 0.0641 | | 0.1505 | 0.0938 | 0.0938 | 0.3785 | |

Rescaled DTLZ4 | 3 | 0.3573 | 0.3561 | 0.2491 | 0.6600 | 0.6868 | |

5 | 0.3184 | 0.3808 | 0.1613 | 0.9607 | | 0.9422 | |

8 | 0.2207 | 0.3220 | 0.1336 | | 0.9753 | 0.3159 | |

10 | 0.1469 | 0.3415 | 0.1115 | | 0.9976 | 0.3267 | |

Deceptive rescaled DTLZ1 | 3 | 0.9536 | | 0.9136 | 0.7535 | 0.7399 | 0.8987 |

5 | 0.3207 | | 0.9200 | 0.0480 | 0.0259 | 0.9077 | |

8 | 0.0000 | 0.0000 | | 0.0000 | 0.0000 | 0.9908 | |

10 | 0.0000 | 0.0000 | 0.9827 | 0.0000 | 0.0000 | | |

Deceptive rescaled DTLZ2 | 3 | 0.9300 | 0.9367 | 0.8576 | 0.6952 | 0.6814 | |

5 | 0.9594 | 0.9069 | 0.3273 | 0.1157 | 0.1392 | | |

8 | 0.3038 | 0.3393 | 0.6748 | 0.0561 | 0.0494 | | |

10 | 0.3171 | 0.3229 | 1.0000 | 0.0031 | 0.0015 | | |

Deceptive rescaled DTLZ3 | 3 | | 0.7294 | 0.6243 | 0.6289 | 0.6095 | 0.6073 |

5 | | 0.9514 | 0.7551 | 0.0012 | 0.0010 | 0.7758 | |

8 | 0.0000 | 0.0362 | 0.4996 | 0.0000 | 0.0000 | | |

10 | 0.0000 | 0.0297 | 0.3635 | 0.0000 | 0.0000 | | |

Deceptive rescaled DTLZ4 | 3 | 0.3534 | 0.3731 | 0.2346 | 0.6371 | | 0.6341 |

5 | 0.1947 | 0.2834 | 0.2409 | 0.8687 | 0.8230 | | |

8 | 0.1259 | 0.1694 | 0.1369 | 0.9262 | | 0.9620 | |

10 | 0.1306 | 0.1617 | 0.1483 | 0.8767 | 0.9593 | |

From careful examinations of Table 3, we can observe that MOEA/D without normalization outperforms MOEA/DD on most minus DTLZ1-4 test problems, while MOEA/DD was the best on DTLZ1-4 in Table 1. We can also see that MOEA/D without normalization also outperforms NSGA-III and \(\theta \)-DEA on some minus DTLZ1-4 test problems. Similar observations were reported in [13]. The reason for the inferior performance of those state-of-the-art algorithms (i.e., NSGA-III, MOEA/DD and \(\theta \)-DEA) is the inverted triangular shape of the Pareto fronts of minus DTLZ1-4. As pointed out in [13], since the simplex lattice structure of weight vectors is triangular, a large number of weight vectors exist outside the inverted triangular Pareto front of each of the minus DTLZ1-4 test problems. No solutions are assigned to those weight vectors in NSGA-III, MOEA/DD, and \(\theta \)-DEA, whereas each weight vector always has a single solution and continues to search for its best solution in MOEA/D. This difference in the handling of weight vectors outside the Pareto front has a significant influence on the performance comparison results, as shown in Table 3 (for details, see [4, 13]). In [4], the simultaneous use of triangular and inverted triangular lattice structures of weight vectors in MOEA/D was proposed to remedy the high dependency of the performance of MOEA/D on the shape of the Pareto front.

The inferior performance of NSGA-III, MOEA/DD, and \(\theta \)-DEA in Table 3 can be explained in this manner using the shape of the Pareto front of each minus test problem. However, it is still totally unclear why the simple normalization mechanism improves the performance of MOEA/D on the minus DTLZ1-4 test problems, where the normalization of the object space is not needed.

## Further computational experiments

Comparison between the two normalization strategies on the minus DTLZ1-4 and WFG4-9

Problem | Objectives | Based on non-dominated solutions | Based on all solutions | ||||
---|---|---|---|---|---|---|---|

\(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | \(\varepsilon = 10^{-12}\) | \(\varepsilon = 10^{-6}\) | \(\varepsilon = 1\) | ||

Minus DTLZ1 | 3 | 0.9584 | 0.9584 | | 0.9596 | 0.9596 | 0.9604 |

5 | 0.9607 | 0.9607 | 0.9616 | | | 1.0379 | |

8 | 0.8952 | 0.8952 | | 0.8480 | 0.8480 | 0.8460 | |

10 | | | 0.6892 | 0.5502 | 0.5502 | 0.5507 | |

Minus DTLZ2 | 3 | | | 0.9979 | 0.9982 | 0.9982 | 0.9993 |

5 | 0.9996 | | 0.9998 | 0.8943 | 0.8943 | 0.8951 | |

8 | 0.9656 | 0.9656 | | 0.8979 | 0.8979 | 0.9024 | |

10 | 0.9346 | 0.9346 | | 0.7807 | 0.7807 | 0.8039 | |

Minus DTLZ3 | 3 | 0.9995 | 0.9995 | | 0.9998 | 0.9998 | 0.9996 |

5 | | | 0.9995 | 0.9227 | 0.9227 | 0.9221 | |

8 | | | 0.9969 | 0.9371 | 0.9371 | 0.9387 | |

10 | 0.9986 | 0.9986 | | 0.8282 | 0.8282 | 0.8281 | |

Minus DTLZ4 | 3 | 0.9818 | 0.9816 | 0.8474 | 0.9452 | 0.9302 | 0.8298 |

5 | | 1.0000 | 0.8342 | 0.9960 | 0.9958 | 0.9815 | |

8 | | 1.0000 | 0.6369 | 0.6954 | 0.6876 | 0.4797 | |

10 | | 0.8860 | 0.9417 | 0.5223 | 0.5155 | 0.4368 | |

Minus WFG4 | 3 | 0.9993 | 0.9993 | 1.0000 | 0.9994 | 0.9994 | |

5 | | | 0.9858 | 0.9353 | 0.9353 | 0.9288 | |

8 | 1.0008 | | 0.8518 | 0.9365 | 0.9365 | 0.8143 | |

10 | 1.0002 | | 0.7760 | 0.8822 | 0.8824 | 0.7160 | |

Minus WFG5 | 3 | 1.0000 | 1.0000 | 0.9998 | 1.0010 | 1.0010 | |

5 | | | 0.9807 | 0.9301 | 0.9301 | 0.9184 | |

8 | 0.9995 | | 0.8606 | 0.9622 | 0.9622 | 0.8307 | |

10 | | | 0.7664 | 0.8929 | 0.8929 | 0.7207 | |

Minus WFG6 | 3 | | | 0.9998 | 0.9999 | 0.9999 | 0.9997 |

5 | 0.9998 | | 0.9830 | 0.9123 | 0.9123 | 0.9042 | |

8 | | 1.0000 | 0.8514 | 0.9433 | 0.9433 | 0.8185 | |

10 | 0.9997 | | 0.7818 | 0.8888 | 0.8888 | 0.7116 | |

Minus WFG7 | 3 | | 1.0000 | 0.9997 | 0.9957 | 0.9957 | 0.9977 |

5 | | | 0.9912 | 0.9240 | 0.9241 | 0.9176 | |

8 | 0.9995 | | 0.7460 | 0.9806 | 0.9803 | 0.7863 | |

10 | | 1.0000 | 0.6784 | 0.9606 | 0.9607 | 0.6692 | |

Minus WFG8 | 3 | | | 0.9983 | 0.9997 | 0.9998 | 0.9989 |

5 | 0.9998 | | 0.9866 | 0.9209 | 0.9209 | 0.9115 | |

8 | | 1.0000 | 0.8669 | 0.9424 | 0.9424 | 0.8181 | |

10 | 0.9986 | | 0.7610 | 0.8677 | 0.8677 | 0.6984 | |

Minus WFG9 | 3 | 0.9851 | 0.9851 | | 0.9840 | 0.9841 | 0.9836 |

5 | | | 0.9827 | 0.9529 | 0.9529 | 0.9411 | |

8 | | | 0.8334 | 0.9940 | 0.9941 | 0.8526 | |

10 | | 1.0000 | 0.7317 | 0.9769 | 0.9765 | 0.7757 |

## Conclusions

In this paper, we clearly demonstrated the necessity of the objective space normalization in MOEA/D. Good results were not obtained from MOEA/D without normalization for the WFG4-9, rescaled DTLZ1-4, and minus WFG4-9 test problems. We examined the effect of the normalization by combining a simple normalization mechanism into MOEA/D. It was demonstrated through computational experiments that the normalization has both positive and negative effects on the performance of MOEA/D. That is, the performance of MOEA/D was improved by the incorporation of the simple normalization mechanism for some test problems but degraded for other test problems. Our experimental results on the newly created deceptive rescaled DTLZ1-4 test problems suggest the existence of negative effects of the normalization on the performance of not only MOEA/D but also NSGA-III and \(\theta \)-DEA. Further examinations are needed for evaluating positive and negative effects of the normalization on the performance of those recently proposed many-objective algorithms. In our computational experiments on the minus DTLZ1-4 and WFG4-9 test problems, the best results were obtained for most test problems by MOEA/D with normalization. This observation suggests the existence of positive effects of the normalization even for test problems that do not need any normalization. Further examination of positive effects of the normalization for such a test problem is also an interesting future research topic.

- 1.
Development of an effective and robust normalization mechanism for MOEA/D: the observed negative effects of the simple normalization mechanism on the performance of MOEA/D throughout this paper clearly suggest the necessity of such a normalization mechanism in MOEA/D.

- 2.
Improvement of the normalization mechanisms in NSGA-III and \(\theta \)-DEA: our experimental results on the deceptive rescaled DTLZ1-4 test problems suggest that the normalization mechanisms in NSGA-III and \(\theta \)-DEA can be further improved.

- 3.
Examination on positive effects of the simple normalization mechanism on the performance of MOEA/D: Our experimental results on the minus DTLZ1-4 and WFG4-9 test problems pose the following question: Why MOEA/D with normalization can outperform MOEA/D without normalization, MOEA/DD, NSGA-III and \(\theta \)-DEA for both the minus DTLZ1-4 and WFG4-9 test problems whereas the minus DTLZ1-4 test problems need no normalization of the objective space.

## Notes

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