# Many-Objectives Optimization: A Machine Learning Approach for Reducing the Number of Objectives

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- can be applied independently on the type and the size of the data and the shape of the Pareto-optimal front,
- is independent from the choice/definition of the algorithm parameters,
- considers the relations DVs-DVs and objectives-objectives (and not only the relations between the DVs and objectives), and
- can provide explainable results for a DM that is a non-expert in optimization or machine learning.

## 2. Machine Learning Approach

#### 2.1. Concepts

#### 2.2. FS-OPA

_{i}. The elements d

_{ij}of the distance matrix correspond to a measure of dissimilarities between objects xi and x

_{j}, according to some given metric. The matrix is broken down into a tree, where the distance between any two objects (leaves) corresponds to the sum of the lengths of the branches connecting them. Finally, the third step merges objects strongly connected (according to the tree topology) into a community, generating a set of different similarity clusters.

#### 2.3. Comparison of FS-OPA with NL-MVU-PCA for MaOPs Data-Driven Structural Learning

^{3}) since n ⩽ l (as in leave-one-out resampling) [34]. Moreover, l = M in a space analysis only uses objectives. Thus, the time complexities of FS-OPA and NL-MVU-PCA have a ratio (n + M)/M

^{4}(l/q

^{3}) of running time for l = M in the worst case (in the usual case).

#### 2.4. FS-OPA Framework

- choose the objective(s) of the less distant clusters;
- choose one objective of the more distant (single) cluster;
- choose one objective from each of the remaining clusters.

- choose the objective(s) of the less distant clusters;
- choose one objective of the more distant (single) cluster;
- choose objective(s) from each of the remaining clusters taking into account, also, the phylogram and the knowledge of the DM(s) about the process.

## 3. Examples of Application: DTLZ Benchmark Problems

_{1}, f

_{2}, f

_{3}, f

_{4}, f

_{5}, f

_{6}, f

_{7}, f

_{8}, and f

_{9}are linearly correlated in DTLZ5). A random population of size 31 with samples normalized and Euclidian distance was used to obtain a distance matrix. SS procedure in Figure 3 was not applied. The output of Figure 5A shows variables and objectives arranged into a phylogram with leaf nodes (the objects under analysis) composing clusters (similarly to the end of the pipeline in Figure 1)—they are identified by the same color.

_{1}, …, f

_{9}are partitioned into three neighbor clusters ({f

_{1}, f

_{2}}, {f

_{3}, f

_{4}, f

_{5}, f

_{6}}, and {f

_{7}, f

_{8}, f

_{9}}) in the phylogram structure; while f

_{10}is together with the leaf nodes, corresponding to variables. The phylogram structure aggregates f

_{1}, …, and f

_{9}into the same subtree, while f

_{10}is isolated from the other objectives in the complementary subtree. The unique node with the label "100" (another type of result from a tree consensus) splits the phylogram into those two subtrees. Such a label (“100”) means that the leaf nodes f

_{10}and x

_{1}, ..., x

_{10}, and f

_{10}were in the same subtree (with the remaining leaf nodes in the complementary subtree) in 100% of all the constructed phylograms, independently of each subtree topology in a phylogram. Such an interpretation suggests a hypothesis: f

_{10}is weakly correlated to the other objectives, which are significantly associated with themselves. Thus, f

_{10}and one of the other objectives could compose an essential objective set; this result is consistent with the DTLZ5 problem structure.

_{11}, to generate samples outside POF, as samples used to construct a phylogram from Figure 5A. The phylogram from Figure 5B shows that f

_{10}is isolated in a subtree, while f

_{1}, …, f

_{9}are in the complementary subtrees. Such a result suggests that f

_{10}and f

_{1}(for example) would enable proper POF estimates; this result agrees with the DTLZ5(2,10) problem structure.

- DTLZ1: {f
_{1}}, {f_{2}, f_{3}, f_{4}, f_{5}}, {f_{6}, f_{7}, f_{8}}, {f_{9}} and {f_{10}}; - DTLZ2: {f
_{1}}, {f_{2}, f_{3}, f_{4}, f_{5}}, {f_{6}, f_{7}, f_{8}}, {f_{9}} and {f_{10}}; - DTLZ3: {f
_{1}}, {f_{2}, f_{3}, f_{4}}, {f_{5}, f_{6}, f_{7}, f_{8}}, {f_{9}} and {f_{10}}; - DTLZ4: {f
_{1}}, {f_{2}, f_{3}, f_{4}}, {f_{5}, f_{6}} and {f_{7}, f_{8}, f_{9}, f_{10}};

## 4. Polymer Extrusion Problem

#### 4.1. The Problem to Solve

_{m}is the melt density, k

_{m}is the melt thermal conductivity, h is the melting entropy, C

_{m}and C

_{s}are specific heat of melt and solids, respectively, T

_{m}is the melting temperature, η is the melt viscosity, T

_{so}and T

_{c}are the solids and the barrel temperatures, $\dot{\gamma}$ is the shear rate, T is the melt temperature in each node of the mesh, T

_{avg}is the average temperature of the melt, V

_{z}is the melt velocity in the Z direction, V

_{s}is the solid velocity in the y direction, and V

_{bx}is the barrel velocity in the X direction.

#### 4.2. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The tree-steps of the pipeline DAMICORE (reproduced from [18]).

**Figure 3.**SS procedure that obtains the selected samples is shown in Figure 2.

**Figure 5.**Phylogram and the clusters found: (

**A**) for unconstrained DTLZ5 with 10 objectives and (

**B**) for constrained DTLZ5 (2,10).

**Figure 7.**Single screw extrusion: (

**A**) plasticating phases; (

**B**) melting mechanism in CS (left) and MBS (right); (

**C**) specific system geometry used in the calculations.

**Figure 8.**Phylograms for Cases 1 and 4 (Table 3).

**Figure 9.**Pareto-optimal fronts after 100 generations for the pair of objectives identified in Figure 8 for Case 1.

**Figure 10.**Pareto-optimal fronts after 100 generations for the pair of objectives identified in Figure 8 for Case 7.

**Table 1.**NL-MVU-PCA and FS-OPA for multidimensional data-driven structural learning applied to real-world MaOPs.

Category | Types | NL-MVU-PCA | FS-OPA | |
---|---|---|---|---|

Analyses | Objective-objective | X | X | |

Variable-variable | X | |||

Variable-objective | X | |||

Objective space reduction | X | |||

Sensitivity | X | |||

Priors | Kernel function usage | X | Not necessary | |

Parameter optimization | X ^{#} | Not necessary | ||

Variable andobjectiverepresentation | Continuous | X | X | |

Discrete (integers, real intervals) | X | X | ||

Ordinal | X | X | ||

Nominal | X | X | ||

Mixed | X | |||

Explainability | Implicit | X | ||

Explicit (The Why) | X | |||

User-friendliness | Stakeholders can easily run FS-OPA and understand results even for a large number of variables and/or objectives | X | ||

Scalability | Time-complexity | Usual cases | O(M^{3}q^{3}) * | O(l^{3}) ** |

The worst case | O(M^{6}) | O(nl^{2} + l^{3}) | ||

Sample-size support | Empirical | Theoretical and empirical |

^{#}Reference [5] shows that one can avoid parameter optimization for a new problem by choosing q = M − 1 for NL-MVU-PCA.

Objectives | Aim | x_{min} | x_{max} |
---|---|---|---|

Output—Q (kg/hr) | Maximize | 1 | 20 |

Length for melting—L (m) | Minimize | 0.1 | 0.9 |

Melt temperature—T (°C) | Minimize | 150 | 210 |

Power consumption—Power (W) | Minimize | 0 | 9200 |

WATS | Maximize | 0 | 1300 |

Viscous dissipation—Viscous | Minimize | 0.9 | 1.2 |

Case | Operating Conditions | Decision Variables | ||||
---|---|---|---|---|---|---|

N (rpm) | Tb1 (°C) | Tb2 (°C) | Tb3 (°C) | Geometry | ||

1 | Constant | 40 | 140 | 150 | 160 | Table 4 |

2 | Constant | 60 | 140 | 150 | 160 | Table 4 |

3 | Constant | 80 | 140 | 150 | 160 | Table 4 |

4 | Variable | [40, 80] | [140, 160] | [150, 170] | [160, 200] | Table 4 |

Screw Type | Decision Variables | ||||||||
---|---|---|---|---|---|---|---|---|---|

CS | case | L1 | L2 | H1 | H3 | P | e | ||

MBS | L1_ | L2_ | H1_ | H3_ | P_ | e_ | Hf | wf | |

Interval | [0, 1] | [100, 400] | [170, 400] | [18, 22] | [22, 26] | [25, 35] | [3, 4] | [0.1, 0.6] | [3, 4] |

‘Q’ | ‘L’ | ‘T’ | ‘Power’ | ‘WATS’ | ‘TTb’ | Average | |
---|---|---|---|---|---|---|---|

‘Q’ | 0.00 | 0.07 | 0.73 | 0.27 | 0.27 | 0.73 | 0.345 |

‘L’ | 0.07 | 0.00 | 0.73 | 0.27 | 0.27 | 0.73 | 0.345 |

‘T’ | 0.73 | 0.73 | 0.00 | 0.67 | 0.67 | 0.07 | 0.478 |

‘Power’ | 0.27 | 0.27 | 0.67 | 0.00 | 0.07 | 0.67 | 0.325 |

‘WATS’ | 0.27 | 0.27 | 0.67 | 0.07 | 0.00 | 0.67 | 0.325 |

‘TTb’ | 0.73 | 0.73 | 0.07 | 0.67 | 0.67 | 0.00 | 0.478 |

‘Q’ | ‘L’ | ‘T’ | ‘Power’ | ‘WATS’ | ‘TTb’ | Average | |
---|---|---|---|---|---|---|---|

‘Q’ | 0.00 | 0.08 | 1.00 | 0.42 | 0.42 | 1.00 | 0.480 |

‘L’ | 0.08 | 0.00 | 1.00 | 0.42 | 0.42 | 1.00 | 0.480 |

‘T’ | 1.00 | 1.00 | 0.00 | 0.83 | 0.83 | 0.08 | 0.620 |

‘Power’ | 0.42 | 0.42 | 0.83 | 0.00 | 0.08 | 0.83 | 0.430 |

‘WATS’ | 0.42 | 0.42 | 0.83 | 0.08 | 0.00 | 0.83 | 0.430 |

‘TTb’ | 1.00 | 1.00 | 0.08 | 0.83 | 0.83 | 0.00 | 0.620 |

**Table 7.**Performance comparison using Hypervolume and IGD for the total number of objectives and the automatic reduction to four and three objectives (between brackets the standard deviation, and loss percentage relative to six objectives) for the four cases studied.

Case Study | Metric | 6 Objectives | 4 Objectives (Q, Power, WATS, T) | 3 Objectives (Q, WATS, T) |
---|---|---|---|---|

1 | HV | 0.21518 (0.008145) | 0.19148 (0.012324) −11.0% | 0.02555 (0.024707) −88.1% |

IGD | 0.10966 (0.004972) | 0.11159 (0.003607) −1.76% | 0.66727 (0.143866) −508% | |

2 | HV | 0.23233 (0.013760) | 0.20867 (0.010411) −10.2% | 0.04689 (0.028991) −79.8% |

IGD | 0.10966 (0.004972) | 0.11205 (0.003526) −2.17% | 0.69042 (0.105262) −529% | |

3 | HV | 0.24809 (0.006384) | 0.21932 (0.014301) −11.6% | 0.04598 (0.0285391) −81.5% |

IGD | 0.11076 (0.005756) | 0.11326 (0.006315) −2.25% | 0.69042 (0.105262) −523% | |

4 | HV | 0.24809 (0.006384) | 0.22911 (0.009955) −7.7% | 0.01967 (0.011256) −92.1% |

IGD | 0.11076 (0.005756) | 0.11431 (0.007949) −3.21% | 0.72078 (0.046369) −550% |

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**MDPI and ACS Style**

Gaspar-Cunha, A.; Costa, P.; Monaco, F.; Delbem, A.
Many-Objectives Optimization: A Machine Learning Approach for Reducing the Number of Objectives. *Math. Comput. Appl.* **2023**, *28*, 17.
https://doi.org/10.3390/mca28010017

**AMA Style**

Gaspar-Cunha A, Costa P, Monaco F, Delbem A.
Many-Objectives Optimization: A Machine Learning Approach for Reducing the Number of Objectives. *Mathematical and Computational Applications*. 2023; 28(1):17.
https://doi.org/10.3390/mca28010017

**Chicago/Turabian Style**

Gaspar-Cunha, António, Paulo Costa, Francisco Monaco, and Alexandre Delbem.
2023. "Many-Objectives Optimization: A Machine Learning Approach for Reducing the Number of Objectives" *Mathematical and Computational Applications* 28, no. 1: 17.
https://doi.org/10.3390/mca28010017