Generation of instance files

Generation of subproblems/subsets

We use the R package gMOIP to generate subproblems. A subproblem is generated such that all nondominated points is integer and in the hypercube \([0, 10000]^p\). A configuration is defined using:

• Number of objectives (\(p\)).
• Number of nondominated points (card).
• Generation method which is either
  • Points generated on the upper (u) part of a sphere resulting in many unsupported points (see Fig. 1).
  • Points generated between to hyperplanes in the middle (m) of the hypercube, resulting in both supported and unsupported points near to the hull (see Fig. 2).
  • Points generated on the lower (l) part of a sphere resulting in many supported points (see Fig. 1).

That is a subproblem can be identified using filename Lyngesen24-sp-<p>-<card>-<gen-method>_<id>.json where id denote the instance id for the same configuration.

For each subproblem we calculate the statistics:

• Classification of points into supported extreme (se), supported non extreme (sne), unsupported (us).
• Number of
  • points (cardinality)
  • supported, extreme and unsupported
  • min and max value for each objective \(i = 1,\ldots p\).
  • width \(w_i\) = \(max_i-min_i\) for each objective \(i = 1,\ldots p\).

A few plots

In total there are 600 subproblems.

Generation of MS problems

The problems generated consists of \(2 \leq S\leq 5\) subproblems and is generated so provide a good test bed for the research questions. The naming convention is Lyngesen24-msp-<p>-<cards>-<gen-methods>_<S>_<id>.json

The following instance/problem groups are generated given:

• \(p=2,\ldots, 5\). [4 options]
• \(S=2,\ldots 5\) where \(S\) is the number of subproblems. [4 options]
• All subproblems have the same method config or half have method u and l. If \(S\) is odd then pick random which method should be used most. [4 options]
• Five instances for each config. [5 options]