Plotting bi-objective models

With gMOIP you can also make plots of the criterion space for bi-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 both the solution and criterion space.

Bi-objective models with three variables

We define functions for plotting the solution and criterion space:

plotSol <- function(A, b, type = rep("c", ncol(A)),
                        faces = rep("c", ncol(A)),
                        plotFaces = TRUE, labels = "numb")
{
   loadView(v = view, close = F, zoom = 0.75)
   plotPolytope(A, b, type = type, faces = faces, labels = labels, plotFaces = plotFaces, 
                argsTitle3d = list(main = "Solution space"))
}

plotCrit <- function(A, b, obj, crit = "min", type = rep("c", ncol(A)), addTriangles = TRUE, 
                     labels = "numb") 
{
    plotCriterion2D(A, b, obj, type = type, crit = crit, addTriangles = addTriangles, 
                   labels = labels) + 
      ggplot2::ggtitle("Criterion space")
}

Four examples are given. A few plots of the solution space are made interactive to illustrate the functionality.

Example 1

We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with three variables:

Ab <- matrix( c(
   1, 1, 2, 5,
   2, -1, 0, 3,
   -1, 2, 1, 3,
   0, -3, 5, 2
), nc = 4, byrow = TRUE)
A <- Ab[,1:3]
b <- Ab[,4]
obj <- matrix(c(1, -6, 3, -4, 1, 6), nrow = 2)

We load the preferred view angle for the RGL window:

view <- matrix( c(-0.452365815639496, -0.446501553058624, 0.77201122045517, 0, 0.886364221572876,
                  -0.320795893669128, 0.333835482597351, 0, 0.0986008867621422, 0.835299551486969,
                  0.540881276130676, 0, 0, 0, 0, 1), nc = 4)
loadView(v = view)

LP model (solution space):

plotSol(A, b)

LP model (criterion space):

plotCrit(A, b, obj, addTriangles = FALSE)

ILP model (solution space):

plotSol(A, b, type = c("i","i","i"))

ILP model (criterion space):

plotCrit(A, b, obj, type = c("i","i","i"))

MILP model with variable 2 and 3 integer (solution space):

plotSol(A, b, type = c("c","i","i"))

MILP model with variable 2 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","i"))

MILP model with variable 1 and 3 integer (solution space):

plotSol(A, b, type = c("i","c","i"))

MILP model with variable 1 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","i"))

MILP model with variable 1 and 2 integer (solution space):

plotSol(A, b, type = c("i","i","c"))

MILP model with variable 1 and 2 integer (criterion space):

plotCrit(A, b, obj, type = c("i","i","c"))

MILP model with variable 1 integer (solution space):

plotSol(A, b, type = c("i","c","c"), plotFaces = F)

MILP model with variable 1 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","c"))

MILP model with variable 2 integer (solution space):

plotSol(A, b, type = c("c","i","c"), plotFaces = F)

MILP model with variable 2 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","c"))

MILP model with variable 3 integer (solution space - Interactive plot):

plotSol(A, b, type = c("c","c","i"), plotFaces = F)

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MILP model with variable 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","c","i"))

Example 2

We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with three variables:

A <- matrix( c(
   -1, 1, 0,
   1, 4, 0,
   2, 1, 0,
   3, -4, 0,
   0, 0, 4
), nc = 3, byrow = TRUE)
b <- c(5, 45, 27, 24, 10)
obj <- matrix(c(1, -6, 3, -4, 1, 6), nrow = 2)

We load the preferred view angle for the RGL window:

view <- matrix( c(0.976349174976349, -0.202332556247711, 0.0761845782399178, 0, 0.0903248339891434,
                  0.701892614364624, 0.706531345844269, 0, -0.196427255868912, -0.682940244674683,
                  0.703568696975708, 0, 0, 0, 0, 1), nc = 4)
loadView(v = view)

LP model (solution space):

plotSol(A, b)

LP model (criterion space):

plotCrit(A, b, obj, addTriangles = FALSE)

ILP model (solution space):

plotSol(A, b, type = c("i","i","i"))

ILP model (criterion space):

plotCrit(A, b, obj, type = c("i","i","i"))

MILP model with variable 2 and 3 integer (solution space):

plotSol(A, b, type = c("c","i","i"))

MILP model with variable 2 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","i"))

MILP model with variable 1 and 3 integer (solution space):

plotSol(A, b, type = c("i","c","i"), plotFaces = FALSE)

MILP model with variable 1 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","i"))

MILP model with variable 1 and 2 integer (solution space):

plotSol(A, b, type = c("i","i","c"))

MILP model with variable 1 and 2 integer (criterion space):

plotCrit(A, b, obj, type = c("i","i","c"))

MILP model with variable 1 integer (solution space):

plotSol(A, b, type = c("i","c","c"), plotFaces = F)

MILP model with variable 1 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","c"))

MILP model with variable 2 integer (solution space - interactive plot):

plotSol(A, b, type = c("c","i","c"), plotFaces = F)

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MILP model with variable 2 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","c"))

MILP model with variable 3 integer (solution space - Interactive plot):

plotSol(A, b, type = c("c","c","i"))

MILP model with variable 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","c","i"))

Example 3

We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with three variables:

A <- matrix( c(
   1, 1, 1,
   3, 0, 1
), nc = 3, byrow = TRUE)
b <- c(10, 24)
obj <- matrix(c(1, -6, 3, -4, 1, 6), nrow = 2)

We load the preferred view angle for the RGL window:

view <- matrix( c(-0.812462985515594, -0.029454167932272, 0.582268416881561, 0, 0.579295456409454,
                  -0.153386667370796, 0.800555109977722, 0, 0.0657325685024261, 0.987727105617523,
                  0.14168381690979, 0, 0, 0, 0, 1), nc = 4)
loadView(v = view)

LP model (solution space):

plotSol(A, b)

LP model (criterion space):

plotCrit(A, b, obj, addTriangles = FALSE)

ILP model (solution space):

plotSol(A, b, type = c("i","i","i"))

ILP model (criterion space):

plotCrit(A, b, obj, type = c("i","i","i"))

MILP model with variable 2 and 3 integer (solution space):

plotSol(A, b, type = c("c","i","i"))

MILP model with variable 2 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","i"))

MILP model with variable 1 and 3 integer (solution space):

plotSol(A, b, type = c("i","c","i"), plotFaces = FALSE)

MILP model with variable 1 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","i"))

MILP model with variable 1 and 2 integer (solution space):

plotSol(A, b, type = c("i","i","c"))

MILP model with variable 1 and 2 integer (criterion space):

plotCrit(A, b, obj, type = c("i","i","c"))

MILP model with variable 1 integer (solution space):

plotSol(A, b, type = c("i","c","c"), plotFaces = F)

MILP model with variable 1 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","c"))

MILP model with variable 2 integer (solution space):

plotSol(A, b, type = c("c","i","c"), plotFaces = F)

MILP model with variable 2 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","c"))

MILP model with variable 3 integer (solution space - Interactive plot):

plotSol(A, b, type = c("c","c","i"), plotFaces = F)

MILP model with variable 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","c","i"))

Example 4

We define the model \(\max \{cx | Ax \leq b\}\) (could also be minimized) with three variables:

A <- matrix( c(
3, 2, 5,
2, 1, 1,
1, 1, 3,
5, 2, 4
), nc = 3, byrow = TRUE)
b <- c(55, 26, 30, 57)
obj <- matrix(c(1, -6, 3, -4, 1, -1), nrow = 2)

We load the preferred view angle for the RGL window:

view <- matrix( c(-0.412063330411911, -0.228006735444069, 0.882166087627411, 0, 0.910147845745087,
                  -0.0574885793030262, 0.410274744033813, 0, -0.042830865830183, 0.97196090221405,
                  0.231208890676498, 0, 0, 0, 0, 1), nc = 4)
loadView(v = view)

LP model (solution space):

plotSol(A, b)

LP model (criterion space):

plotCrit(A, b, obj, addTriangles = FALSE)

ILP model (solution space):

plotSol(A, b, type = c("i","i","i"), labels="n")

ILP model (criterion space):

plotCrit(A, b, obj, type = c("i","i","i"), labels="n")

MILP model with variable 2 and 3 integer (solution space):

plotSol(A, b, type = c("c","i","i"), labels="n")

MILP model with variable 2 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","i"), labels="n")

MILP model with variable 1 and 3 integer (solution space):

plotSol(A, b, type = c("i","c","i"), plotFaces = FALSE, labels="n")

MILP model with variable 1 and 3 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","i"), labels="n")

MILP model with variable 1 and 2 integer (solution space):

plotSol(A, b, type = c("i","i","c"), labels="n")

MILP model with variable 1 and 2 integer (criterion space):

plotCrit(A, b, obj, type = c("i","i","c"), labels="n")

MILP model with variable 1 integer (solution space):

plotSol(A, b, type = c("i","c","c"), labels="n")

MILP model with variable 1 integer (criterion space):

plotCrit(A, b, obj, type = c("i","c","c"), labels="n")

MILP model with variable 2 integer (solution space):

plotSol(A, b, type = c("c","i","c"), plotFaces = F, labels="n")

MILP model with variable 2 integer (criterion space):

plotCrit(A, b, obj, type = c("c","i","c"), labels="n")

MILP model with variable 3 integer (solution space - Interactive plot):

plotSol(A, b, type = c("c","c","i"), labels="n")

MILP model with variable 3 integer (criterion space):

plotCrit(A, b, obj, type = c("c","c","i"), labels="n")

Plotting bi-objective models

Bi-objective models with two variables

Bi-objective models with three variables

Example 1

Example 2

Example 3

Example 4