# 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

## 1 Answer

The answer is, as expected, another paper. It was published one year before and has the same title. In this paper, there are in total eight DTLZ problems, and DTLZ5 from this paper is not part of the one from 2002. Renumbering the problems led to my confusion.