MOO-Synthetic

Problem Difficulty Classification

Difficulty Mode	Training Set	Testing Set
easy	First 80% of problems sorted by complexity	Last 20% of problems sorted by complexity
difficult	First 20% of problems sorted by complexity	Last 80% of problems sorted by complexity

Note: Problems are sorted by complexity (n_obj × n_var). When difficulty is ‘all’, both sets contain all 187 problems.

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MOO-Synthetic is constructed by mixing 4 well-known multi-objective problem sets: ZDT, UF, DTLZ and WFG. In total, we have constructed 187 problem instances. Their objective numbers range from 2 ~ 10, dimensions range from 6D ~ 38D.

• Paper：

  • ZDT

  • UF

  • DTLZ

  • WFG

• Code Resource： MOO-Synthetic

Overview

Problem Set	Number of Problems	Objectives	Dimension	Source
ZDT	5	2	10 or 30	Comparison of Multiobjective Evolutionary Algorithms: Empirical Results
UF	10	2 or 3	30	CEC’2009 Test Instances
DTLZ	46	2–10	6–29	Scalable MOO Test Problems
WFG	117	2–10	12–38	A Review of Multiobjective Test Problems

Detailed Configurations

ZDT Problems

• ZDT1–ZDT3: 30 dimensions, 2 objectives

• ZDT4, ZDT6: 10 dimensions, 2 objectives

• ZDT5 (binary problem) is excluded.

UF Problems

Problem	Objectives	Dimension
UF1–UF7	2	30
UF8–UF10	3	30

DTLZ Problems

Problem Type	Configurations (Objectives, Dimensions)
DTLZ1	(2,6), (3,7), (5,9), (7,11), (8,12), (10,14)
DTLZ2/4/6	(2,11), (3,11), (3,12), (5,14), (7,16), (8,17), (10,19)
DTLZ3/5	(2,11), (3,12), (5,14), (7,16), (8,17), (10,19)
DTLZ7	(2,21), (3,22), (5,24), (7,16), (7,26), (8,27), (10,29)

WFG Problems

Objectives	Dimensions
2	12, 22
3	12, 14, 24
5	14, 18, 28
7	16
8	24, 34
10	28, 38

Dataset Split Strategy

To assess the generalization capabilities of algorithms, the dataset can be split based on problem difficulty, defined as:

difficulty = number_of_objectives × dimension

Then, sort all 178 problems by this value:

• Easy Mode:

  • Training set: first 80% (142 problems)

  • Test set: remaining 20% (36 problems)

• Difficult Mode:

  • Training set: first 20% (36 problems)

  • Test set: remaining 80% (142 problems)

Notes

If you would like, we can also provide:

• Sample loading scripts (e.g., Python)

• Visualization tools for Pareto fronts

• JSON/CSV versions of the benchmark metadata