This report contains different plots and tables that may be relevant for analyzing the results. Given a problem consisting of \(m\) subproblems with \(Y_N^s\) given for each subproblem \(s\), we use a filtering algorithm to find \(Y_N\).

The following instance/problem groups are generated given:

  • \(p=2,\ldots, 5\).
  • \(S=2,\ldots 5\) where \(S\) is the number of subproblems.
  • All subproblems have the same method config or half have method u and l.
  • Five instances for each config.

1280 problems have been solved.

1 Size of \(Y_N\)

What is \(|Y_N|\) given the different methods of generating the set of nondominated points for the subproblems?

1.1 Summary tables

Below summary tables with summary results for different aggregation levels are given (card = |Y_N|).

1.2 Regression fit

Let us try to fit the results using function \(y=c_1 s^{(c_2p)} S^{c_3p}\) (different functions was tried and this gave the highest \(R^2\)) for each method:

## # A tibble: 4 × 15
##   method fit    tidied   r.squared adj.r.squared sigma statistic   p.value    df
##   <chr>  <list> <list>       <dbl>         <dbl> <dbl>     <dbl>     <dbl> <dbl>
## 1 L      <lm>   <tibble>     0.867         0.866 1.06      1031. 1.90e-139     2
## 2 M      <lm>   <tibble>     0.802         0.800 1.26       641. 4.46e-112     2
## 3 UL     <lm>   <tibble>     0.923         0.923 0.744     1902. 2.77e-177     2
## 4 U      <lm>   <tibble>     0.954         0.954 0.530     3312. 3.54e-213     2
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## #   df.residual <int>, nobs <int>
## # A tibble: 4 × 4
##   method    c1     c2    c3
##   <chr>  <dbl>  <dbl> <dbl>
## 1 L       74.0 0.0870 1.25 
## 2 M       71.1 0.0903 1.16 
## 3 UL      29.2 0.125  1.09 
## 4 U       21.5 0.137  0.974

The parameters estimates are:

2 Relative size of \(Y_N\)

3 Nondominated points classification

We classify the nondominated points into, extreme, supported non-extreme and unsupported.

3.1 Summary tables

Below summary tables with summary results for different aggregation levels are given. Only instances which have been classified are included.

Relative number of extreme (min and max included):

## # A tibble: 1 × 3
##   minPctEx avePctExt maxPctEx
##      <dbl>     <dbl>    <dbl>
## 1 0.000461    0.0449    0.330
## # A tibble: 4 × 4
##   method minPctEx avePctExt maxPctEx
##   <chr>     <dbl>     <dbl>    <dbl>
## 1 L      0.00443     0.0761    0.302
## 2 UL     0.00635     0.0719    0.330
## 3 M      0.000461    0.0205    0.147
## 4 U      0.00196     0.0132    0.104

4 Problems solved for the analysis

5 Generator sets - Summary tables

Below summary tables with summary results for different aggregation levels are given.

6 Reduced generator - Summary tables

Below summary tables with summary results for different aggregation levels are given.

A summary table showing average relative number of redundant vectors (\(q^m\)), for different percentages of non-extreme nondominated vectors known (lambda), and different percentages of non-extreme nondominated vectors known of other subproblems (gamma).

A summary table for showing average relative number of redundant vectors (\(q^m\)), for each size \(S \in \{2,4\}\) for different percentage of non-extreme nondominated vectors known (lambda).

A summary table showing average relative number of redundant vectors (\(q^m\)), for each size \(S \in \{2,4\}\) for different percentage of non-extreme nondominated vectors known (lambda), and for different lower bound sets \({L}^s = \text{conv}({Y}^s_\mathtt{SE})\) (CONV) and \({L}^s = {Y}^s_\mathtt{N}\) (ALL).

A summary table showing the average relative number of redundant vectors (\(q^m\)) avg_q, average minimum generator set sizes avg_MGS_size and average reduced generator set sizes ´avg_RGS_size´ for each \(S \in \{2,4\}\), for different subproblem sizes sp_card \(|{Y}^s_\mathtt{N}| \in \{50,100,200,300\}\). The cases with no redundant ND vectors are filtered out.