This report contains different preliminary 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.

1235 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?

## # A tibble: 4 × 3
##   method mean_card     n
##   <chr>      <dbl> <int>
## 1 l       1506031.   315
## 2 m        572395.   310
## 3 u        109824.   310
## 4 ul       206763.   300

We use plots to check for effects of \(p\), \(m\) and subset size:

1.1 Regression fit

Let us try to fit the results using function \(y=c_1 s^{(c_2p)} m^{c_3p}\) (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>     0.856         0.855 1.05       929. 3.90e-132     2
## 2 m      <lm>   <tibble>     0.768         0.766 1.25       507. 5.07e- 98     2
## 3 ul     <lm>   <tibble>     0.903         0.903 0.747     1389. 1.89e-151     2
## 4 u      <lm>   <tibble>     0.947         0.947 0.527     2759. 6.52e-197     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       83.0 0.0836 1.24 
## 2 m       89.2 0.0847 1.11 
## 3 ul      30.1 0.117  1.12 
## 4 u       23.5 0.135  0.955

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.

Relative number of extreme:

## # 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