inst/examples/ex_polytope.R

In gMOIP: Tools for 2D and 3D Plots of Single and Multi-Objective Linear/Integer Programming Models

#### 2D examples ####
# Define the model max/min coeff*x st. Ax<=b, x>=0
A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)

## LP model
# The polytope with the corner points
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL,
   argsFaces = list(argsGeom_polygon = list(fill = "red"))
)
# With optimum and labels:
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord",
   argsOptimum = list(lty="solid")
)
# Minimize:
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "min",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "n"
)
# Note return a ggplot so can e.g. add other labels on e.g. the axes:
p <- plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
)
p + ggplot2::xlab("x") + ggplot2::ylab("y")

# More examples
\donttest{
## LP-model with no non-negativity constraints
A <- matrix(c(-3, 2, 2, 4, 9, 10, 1, -2), ncol = 2, byrow = TRUE)
b <- c(3, 27, 90, 2)
obj <- c(7.75, 10)
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   nonneg = rep(FALSE, ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)



## The package don't plot feasible regions that are unbounded e.g if we drop the 2 and 3 constraint
A <- matrix(c(-3,2), ncol = 2, byrow = TRUE)
b <- c(3)
obj <- c(7.75, 10)
# Wrong plot
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)
# One solution is to add a bounding box and check if the bounding box is binding
A <- rbind(A, c(1,0), c(0,1))
b <- c(b, 10, 10)
plotPolytope(
   A,
   b,
   obj,
   type = rep("c", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = FALSE,
   labels = NULL
)


## ILP model
A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)
# ILP model with LP faces:
plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("c", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord",
   argsLabels = list(size = 4, color = "blue"),
   argsFeasible = list(color = "red", size = 3)
)
#ILP model with IP faces:
plotPolytope(
   A,
   b,
   obj,
   type = rep("i", ncol(A)),
   crit = "max",
   faces = rep("i", ncol(A)),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
)


## MILP model
A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE)
b <- c(3,27,90)
obj <- c(7.75, 10)
# Second coordinate integer
plotPolytope(
   A,
   b,
   obj,
   type = c("c", "i"),
   crit = "max",
   faces = c("c", "i"),
   plotFaces = FALSE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord",
   argsFeasible = list(color = "red")
)
# First coordinate integer and with LP faces:
plotPolytope(
   A,
   b,
   obj,
   type = c("i", "c"),
   crit = "max",
   faces = c("c", "c"),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
)
# First coordinate integer and with LP faces:
plotPolytope(
   A,
   b,
   obj,
   type = c("i", "c"),
   crit = "max",
   faces = c("i", "c"),
   plotFaces = TRUE,
   plotFeasible = TRUE,
   plotOptimum = TRUE,
   labels = "coord"
)
}



#### 3D examples ####
\donttest{
# Ex 1
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)
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 <- c(20, 10, 15)
# LP model
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord")
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord",
             argsFaces = list(drawLines = FALSE, argsPolygon3d = list(alpha = 0.95)),
             argsLabels = list(points3d = list(color = "blue")))
# ILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj)
# MILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE)
plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)

# Ex 2
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)
A <- matrix( c(
   1, 1, 1,
   3, 0, 1
), nc = 3, byrow = TRUE)
b <- c(10, 24)
obj <- c(20, 10, 15)
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord")
# ILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj)
# MILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE)
plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)

# Ex 3
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)
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 <- c(5, 45, 15)
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord")
# ILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj)
# MILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE)
plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","c","i"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)

# Ex 4
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)
Ab <- matrix( c(
   1, 1, 2, 5,
   2, -1, 0, 3,
   -1, 2, 1, 3,
   0, -3, 5, 2
   #   0, 1, 0, 4,
   #   1, 0, 0, 4
), nc = 4, byrow = TRUE)
A <- Ab[,1:3]
b <- Ab[,4]
obj = c(1,1,3)
plotPolytope(A, b, plotOptimum = TRUE, obj = obj, labels = "coord")
# ILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","i"), plotOptimum = TRUE, obj = obj)
# MILP model
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","c","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","i","i"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotOptimum = TRUE, obj = obj)
plotPolytope(A, b, faces = c("c","c","c"), type = c("i","i","c"), plotFaces = FALSE)
plotPolytope(A, b, type = c("i","c","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, type = c("c","i","c"), plotOptimum = TRUE, obj = obj, plotFaces = FALSE)
plotPolytope(A, b, faces = c("c","c","c"), type = c("c","c","i"), plotOptimum = TRUE, obj = obj)
}