Author: Lars Relund Nielsen
Date: 2026-03-01

# Plotting tri-objective models

#### Lars Relund Nielsen

#### 2026-03-01

With `gMOIP` you can make plots of the criterion space for tri-objective models (linear programming (LP), integer linear programming (ILP), or mixed integer linear programming (MILP)). This vignette gives examples on how to make plots of the criterion space.

First we load the package:

    library(gMOIP)

The criterion space can be plotted for tri-objective models. An example with many unsupported:

    view <- matrix( c(0.333316594362259, 0.938472270965576, -0.0903875231742859, 0, 0.83994072675705, -0.339126199483871, -0.423665106296539, 0, -0.428250730037689, 0.0652943551540375, -0.901297807693481, 0, 0, 0, 0, 1), nc = 4)
    loadView(v = view)
    set.seed(1234)
    pts <- genNDSet(3, 20, argsSphere = list(below = FALSE), dubND = FALSE)
    pts <- classifyNDSet(pts[,1:3])
    head(pts)
    #>   z1 z2 z3    se   sne    us cls
    #> 1 48 87 17  TRUE FALSE FALSE  se
    #> 2 14 59 83  TRUE FALSE FALSE  se
    #> 3 70 75 12  TRUE FALSE FALSE  se
    #> 4 98 36 55 FALSE FALSE  TRUE  us
    #> 5 91 40 24  TRUE FALSE FALSE  se
    #> 6 34 93 31  TRUE FALSE FALSE  se
    ini3D(argsPlot3d = list(xlim = c(min(pts[,1])-2,max(pts[,1])+2),
       ylim = c(min(pts[,2])-2,max(pts[,2])+2),
       zlim = c(min(pts[,3])-2,max(pts[,3])+2)))
    plotPoints3D(pts[pts$se,1:3], argsPlot3d = list(col = "red"))
    plotPoints3D(pts[!pts$sne,1:3], argsPlot3d = list(col = "black"))
    plotPoints3D(pts[!pts$us,1:3], argsPlot3d = list(col = "blue"))
    plotCones3D(pts[,1:3], rectangle = TRUE, argsPolygon3d = list(alpha = 1, color = "cornflowerblue"))
    plotHull3D(pts[,1:3], addRays = TRUE, argsPolygon3d = list(alpha = 0.25, color = "red"), useRGLBBox = TRUE)
    finalize3D(argsAxes3d = list(edges = "bbox"))

Example with many supported:

    loadView(v = view)
    pts <- genNDSet(3, 10, argsSphere = list(below = TRUE), dubND = FALSE)
    pts <- classifyNDSet(pts[,1:3])
    ini3D(argsPlot3d = list(xlim = c(min(pts[,1])-2,max(pts[,1])+2),
       ylim = c(min(pts[,2])-2,max(pts[,2])+2),
       zlim = c(min(pts[,3])-2,max(pts[,3])+2)))
    plotPoints3D(pts[pts$se,1:3], argsPlot3d = list(col = "red"))
    plotPoints3D(pts[!pts$sne,1:3], argsPlot3d = list(col = "black"))
    plotPoints3D(pts[!pts$us,1:3], argsPlot3d = list(col = "blue"))
    plotCones3D(pts[,1:3], rectangle = TRUE, argsPolygon3d = list(alpha = 1, color = "cornflowerblue"))
    plotHull3D(pts[,1:3], addRays = TRUE, argsPolygon3d = list(alpha = 0.25, color = "red"), useRGLBBox = TRUE)
    finalize3D(argsAxes3d = list(edges = "bbox"))

## Classifying

Non-dominated points can be classified using `classifyNDSet`:

    pts <- matrix(c(0,0,1, 0,1,0, 1,0,0, 0.5,0.2,0.5, 0.25,0.5,0.25), ncol = 3, byrow = TRUE)
    open3d()
    #> null
    #>   32
    ini3D(argsPlot3d = list(xlim = c(min(pts[,1])-2,max(pts[,1])+2),
      ylim = c(min(pts[,2])-2,max(pts[,2])+2),
      zlim = c(min(pts[,3])-2,max(pts[,3])+2)))
    plotHull3D(pts, addRays = TRUE, argsPolygon3d = list(alpha = 0.5), useRGLBBox = TRUE)
    pts <- classifyNDSet(pts[,1:3])
    pts
    #>     z1  z2   z3    se   sne    us cls
    #> 1 0.00 0.0 1.00  TRUE FALSE FALSE  se
    #> 2 0.00 1.0 0.00  TRUE FALSE FALSE  se
    #> 3 1.00 0.0 0.00  TRUE FALSE FALSE  se
    #> 4 0.50 0.2 0.50 FALSE FALSE  TRUE  us
    #> 5 0.25 0.5 0.25 FALSE  TRUE FALSE sne
    plotPoints3D(pts[pts$se,1:3], argsPlot3d = list(col = "red"))
    plotPoints3D(pts[!pts$sne,1:3], argsPlot3d = list(col = "black"))
    plotPoints3D(pts[!pts$us,1:3], argsPlot3d = list(col = "blue"))
    plotCones3D(pts[,1:3], rectangle = TRUE, argsPolygon3d = list(alpha = 1))
    finalize3D()
    rglwidget(reuse = F)

    pts <- genNDSet(3,20, dubND = FALSE)[,1:3]
    open3d()
    #> null
    #>   35
    ini3D(argsPlot3d = list(xlim = c(0,max(pts$z1)+2),
      ylim = c(0,max(pts$z2)+2),
      zlim = c(0,max(pts$z3)+2)))
    plotHull3D(pts, addRays = TRUE, argsPolygon3d = list(alpha = 0.5))
    pts <- classifyNDSet(pts[,1:3])
    pts
    #>    z1 z2 z3    se   sne    us cls
    #> 1   4 33 53  TRUE FALSE FALSE  se
    #> 2  21 38 13  TRUE FALSE FALSE  se
    #> 3  19 13 51  TRUE FALSE FALSE  se
    #> 4  16 47 15  TRUE FALSE FALSE  se
    #> 5  58  5 33  TRUE FALSE FALSE  se
    #> 6   1 49 56  TRUE FALSE FALSE  se
    #> 7  43 12 20  TRUE FALSE FALSE  se
    #> 8  29 48  6  TRUE FALSE FALSE  se
    #> 9  23 10 42  TRUE FALSE FALSE  se
    #> 10  6 63 32  TRUE FALSE FALSE  se
    #> 11 31 49  5  TRUE FALSE FALSE  se
    #> 12  9 45 23  TRUE FALSE FALSE  se
    #> 13  2 41 46  TRUE FALSE FALSE  se
    #> 14 35 11 25  TRUE FALSE FALSE  se
    #> 15 49  2 42  TRUE FALSE FALSE  se
    #> 16 39  2 47  TRUE FALSE FALSE  se
    #> 17 18 14 48  TRUE FALSE FALSE  se
    #> 18 61  4 37 FALSE FALSE  TRUE  us
    #> 19 28 32 11  TRUE FALSE FALSE  se
    #> 20 34  8 32  TRUE FALSE FALSE  se
    plotPoints3D(pts[pts$se,1:3], argsPlot3d = list(col = "red"))
    plotPoints3D(pts[!pts$sne,1:3], argsPlot3d = list(col = "black"))
    plotPoints3D(pts[!pts$us,1:3], argsPlot3d = list(col = "blue"))
    finalize3D()
    rglwidget(reuse = F)

The classification is done using the `qhull` algorithm that find the convex hull of the points including the rays. If a vertex then if must be supported extreme. Next we use the `inHull` algorithm to find out if the remaining are on the border or not (supported non-extreme and unsupported).