# Definition of the DTLZ 5 - 7 Problems

For a multi-objective optimization task I want to use the DTLZ5, DTLZ6 and DTLZ7-problems definied by Deb et al. in their Paper "Scalable Multi-Objective Optimization Test Problems".

There are multiple libraries which implemented these problems like Shark (C++), jMetal (Java), deap (Python) and pagmo (C++ and Python).

They all share their code on GitHub, but unfortunately I may only post two links here, for which I chose jMetal to illustrate the problems that occur in all of these implementations.

## DTLZ5

In the paper,

$$g(\text{x}_M)=\sum_{x_i \in \text{x}_M} x_i^{0.1}$$

while in jMetal it's

$$g(\text{x}_M)=\sum_{x_i \in \text{x}_M} (x_i - 0.5)^{2}$$

There is no equivalent to this in the paper, although for DTLZ2 the $g$-function looks this way.

## DTLZ6

In short: DTLZ6 in jMetal is DTLZ5 in the paper.

## DTLZ7

I could not find any implementation of the DTLZ7 problem like it is stated in the paper. In contrast, I found at least one paper and other scientific work that referred to the $h$-value of DTLZ7 like "Dynamic Multiobjective Optimization Problems: Test Cases, Approximations, and Applications" by Farina et al. where Deb is even a co-author. Additionally, there is a definition of the DTLZ7-problem on the website of the ETH Zürich where the three co-authors worked. These two sources as well as the libraries listed above describe/implement DTLZ6 instead of DTLZ7.

## TL;DR

• problem implemented as DTLZ5 is not part of the paper
• implemented DTLZ6 equals DTLZ5 from the paper
• same for DTLZ7 and DTLZ6
• there is no implementation of DTLZ7 out there

## The Question

Is there any paper I could not find that makes my observations obsolete or why do all these sources differ from the paper?

• What exactly is your question? – Paul Nov 3 '15 at 15:51
• My question was if I was getting anything wrong or all the libraries I found. Apparently, the answer is that both ways are valid because there are two different papers with different definitions. – kreluros Nov 3 '15 at 16:11