Create a plot of the criterion space of a bi-objective problem

Usage

plotCriterion2D(
  A,
  b,
  obj,
  type = rep("c", ncol(A)),
  nonneg = rep(TRUE, ncol(A)),
  crit = "max",
  addTriangles = FALSE,
  addHull = TRUE,
  plotFeasible = TRUE,
  latex = FALSE,
  labels = NULL
)

Arguments

A: The constraint matrix.
b: Right hand side.
obj: A p x n matrix(one row for each criterion).
type: A character vector of same length as number of variables. If entry k is 'i' variable \(k\) must be integer and if 'c' continuous.
nonneg: A boolean vector of same length as number of variables. If entry k is TRUE then variable k must be non-negative.
crit: Either max or min (only used if add the iso-profit line).
addTriangles: Add search triangles defined by the non-dominated extreme points.
addHull: Add the convex hull and the rays.
plotFeasible: If True then plot the criterion points/slices.
latex: If true make latex math labels for TikZ.
labels: If NULL don't add any labels. If 'n' no labels but show the points. If equal coord add coordinates to the points. Otherwise number all points from one.

Value

The ggplot object.

Note

Currently only points are checked for dominance. That is, for MILP models some nondominated points may in fact be dominated by a segment.

Author

Lars Relund lars@relund.dk

Examples

### Set up 2D plot
# Function for plotting the solution and criterion space in one plot (two variables)
plotBiObj2D <- function(A, b, obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = "numb",
   addTriangles = TRUE,
   addHull = TRUE)
{
   p1 <- plotPolytope(A, b, type = type, crit = crit, faces = faces, plotFaces = plotFaces,
                      plotFeasible = plotFeasible, plotOptimum = plotOptimum, labels = labels)
   p2 <- plotCriterion2D(A, b, obj, type = type, crit = crit, addTriangles = addTriangles,
                         addHull = addHull, plotFeasible = plotFeasible, labels = labels)
   gridExtra::grid.arrange(p1, p2, nrow = 1)
}


### Bi-objective problem with two variables
A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)

## LP model
obj <- matrix(
   c(7, -10, # first criterion
     -10, -10), # second criterion
   nrow = 2)
plotBiObj2D(A, b, obj, addTriangles = FALSE)


# \donttest{
## ILP models with different criteria (maximize)
obj <- matrix(c(7, -10, -10, -10), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)))

obj <- matrix(c(3, -1, -2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)))

obj <- matrix(c(-7, -1, -5, 5), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)))

obj <- matrix(c(-1, -1, 2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)))
#> Warning: ggrepel: 70 unlabeled data points (too many overlaps). Consider increasing max.overlaps


## ILP models with different criteria (minimize)
obj <- matrix(c(7, -10, -10, -10), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)), crit = "min")

obj <- matrix(c(3, -1, -2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)), crit = "min")

obj <- matrix(c(-7, -1, -5, 5), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)), crit = "min")

obj <- matrix(c(-1, -1, 2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = rep("i", ncol(A)), crit = "min")
#> Warning: ggrepel: 70 unlabeled data points (too many overlaps). Consider increasing max.overlaps



# More examples
## MILP model (x1 integer) with different criteria (maximize)
obj <- matrix(c(7, -10, -10, -10), nrow = 2)
plotBiObj2D(A, b, obj, type = c("i", "c"))

obj <- matrix(c(3, -1, -2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = c("i", "c"))

obj <- matrix(c(-7, -1, -5, 5), nrow = 2)
plotBiObj2D(A, b, obj, type = c("i", "c"))

obj <- matrix(c(-1, -1, 2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = c("i", "c"))


## MILP model (x2 integer) with different criteria (minimize)
obj <- matrix(c(7, -10, -10, -10), nrow = 2)
plotBiObj2D(A, b, obj, type = c("c", "i"), crit = "min")

obj <- matrix(c(3, -1, -2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = c("c", "i"), crit = "min")

obj <- matrix(c(-7, -1, -5, 5), nrow = 2)
plotBiObj2D(A, b, obj, type = c("c", "i"), crit = "min")

obj <- matrix(c(-1, -1, 2, 2), nrow = 2)
plotBiObj2D(A, b, obj, type = c("c", "i"), crit = "min")

# }

### Set up 3D plot
# \donttest{
# Function for plotting the solution and criterion space in one plot (three variables)
plotBiObj3D <- function(A, b, obj,
                        type = rep("c", ncol(A)),
                        crit = "max",
                        faces = rep("c", ncol(A)),
                        plotFaces = TRUE,
                        plotFeasible = TRUE,
                        plotOptimum = FALSE,
                        labels = "numb",
                        addTriangles = TRUE,
                        addHull = TRUE)
{
   plotPolytope(A, b, type = type, crit = crit, faces = faces, plotFaces = plotFaces,
                plotFeasible = plotFeasible, plotOptimum = plotOptimum, labels = labels)
   plotCriterion2D(A, b, obj, type = type, crit = crit, addTriangles = addTriangles,
                   addHull = addHull, plotFeasible = plotFeasible, labels = labels)
}

### Bi-objective problem with three variables
loadView <- function(fname = "view.RData", v = NULL) {
   if (!is.null(v)) {
      rgl::view3d(userMatrix = v)
   } else {
      if (file.exists(fname)) {
         load(fname)
         rgl::view3d(userMatrix = view)
      } else {
         warning(paste0("Can'TRUE load view in file ", fname, "!"))
      }
   }
}

## Ex
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)
3D plot

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)

# LP model
plotBiObj3D(A, b, obj, crit = "min", addTriangles = FALSE)


# ILP model
plotBiObj3D(A, b, obj, type = c("i","i","i"), crit = "min")


# MILP model
plotBiObj3D(A, b, obj, type = c("c","i","i"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","c","i"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","i","c"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","c","c"), crit = "min")

plotBiObj3D(A, b, obj, type = c("c","i","c"), crit = "min")

plotBiObj3D(A, b, obj, type = c("c","c","i"), crit = "min")



## Ex
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)
3D plot

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)

# LP model
plotBiObj3D(A, b, obj, crit = "min", addTriangles = FALSE, labels = "coord")


# ILP model
plotBiObj3D(A, b, obj, type = c("i","i","i"))


# MILP model
plotBiObj3D(A, b, obj, type = c("c","i","i"))

plotBiObj3D(A, b, obj, type = c("i","c","i"), plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("i","i","c"))

plotBiObj3D(A, b, obj, type = c("i","c","c"), plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("c","i","c"), plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("c","c","i"))



## Ex
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)
3D plot

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)

# LP model
plotBiObj3D(A, b, obj, crit = "min", addTriangles = FALSE, labels = "coord")


# ILP model
plotBiObj3D(A, b, obj, type = c("i","i","i"), crit = "min", labels = "n")


# MILP model
plotBiObj3D(A, b, obj, type = c("c","i","i"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","c","i"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","i","c"), crit = "min")

plotBiObj3D(A, b, obj, type = c("i","c","c"), crit = "min")

plotBiObj3D(A, b, obj, type = c("c","i","c"), crit = "min", plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("c","c","i"), crit = "min", plotFaces = FALSE)



## Ex
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)
3D plot

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)

# LP model
plotBiObj3D(A, b, obj, crit = "min", addTriangles = FALSE, labels = "coord")


# ILP model
plotBiObj3D(A, b, obj, type = c("i","i","i"), crit = "min", labels = "n")


# MILP model
plotBiObj3D(A, b, obj, type = c("c","i","i"), crit = "min", labels = "n")

plotBiObj3D(A, b, obj, type = c("i","c","i"), crit = "min", labels = "n", plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("i","i","c"), crit = "min", labels = "n")

plotBiObj3D(A, b, obj, type = c("i","c","c"), crit = "min", labels = "n")

plotBiObj3D(A, b, obj, type = c("c","i","c"), crit = "min", labels = "n", plotFaces = FALSE)

plotBiObj3D(A, b, obj, type = c("c","c","i"), crit = "min", labels = "n")

# }