knitr::opts_chunk$set( collapse = TRUE, comment = "#>", warning=FALSE, message=FALSE, include = TRUE, out.width = "99%", fig.width = 8, fig.align = "center", fig.asp = 0.62 ) if (!requireNamespace("rmarkdown", quietly = TRUE) || !rmarkdown::pandoc_available("1.14")) { warning(call. = FALSE, "These vignettes assume rmarkdown and pandoc version 1.14 (or higher). These were not found. Older versions will not work.") knitr::knit_exit() }

With `gMOIP`

you can make 2D plots of the the polytope/feasible region/solution space of a linear programming (LP), integer linear programming (ILP) model, or mixed integer linear programming (MILP) model. This vignette gives examples on how to make plots given a model with two variables.

First we load the package:

```
library(gMOIP)
```

We define the model $\max {cx | Ax \leq b}$ (could also be minimized) with 2 variables:

A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE) b <- c(3,27,90) obj <- c(7.75, 10) # coefficients c

Plots are created using function `plotPolytope`

which outputs a `ggplot2`

object.

Let us consider different plots of the polytope of the LP model with non-negative variables ($x \in \mathbb{R}_0, x \geq 0$):

# The polytope with the corner points p1 <- plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = NULL ) + ggplot2::ggtitle("Feasible region only") p2 <- plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) + ggplot2::ggtitle("Solution LP max") p3 <- plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "min", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) + ggplot2::ggtitle("Solution LP min") p4 <- plotPolytope( A, b, obj, type = rep("c", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) + ggplot2::xlab("x") + ggplot2::ylab("y") + ggplot2::ggtitle("Solution (max) with other axis labels") gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

You may also consider a 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 )

Note The package don't plot feasible regions that are unbounded e.g if we drop the second and third constraint we get the wrong plot:

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 )

You may also use e.g `lpsolve`

to check if the solution is unbounded.

If we add integer constraints to the model ($x\in\mathbb{Z}$) you may view the feasible region different ways:

A <- matrix(c(-3,2,2,4,9,10), ncol = 2, byrow = TRUE) b <- c(3,27,90) obj <- c(7.75, 10) p1 <- plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = "n" ) + ggplot2::ggtitle("Relaxed region with LP faces") p2 <- plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("i", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = FALSE, labels = "n" ) + ggplot2::ggtitle("Relaxed region with IP faces") p3 <- plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) + ggplot2::ggtitle("Optimal solution (max)") p4 <- plotPolytope( A, b, obj = c(-3, 3), type = rep("i", ncol(A)), crit = "max", faces = rep("i", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) + ggplot2::ggtitle("Other objective (min)") gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

Finally, let us have a look at a 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) p1 <- plotPolytope( A, b, obj, type = c("c", "i"), crit = "max", faces = c("c", "c"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) + ggplot2::ggtitle("Second coordinate integer (LP faces)") p2 <- plotPolytope( A, b, obj, type = c("c", "i"), crit = "max", faces = c("c", "i"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) + ggplot2::ggtitle("Second coordinate integer (MILP faces)") p3 <- plotPolytope( A, b, obj, type = c("i", "c"), crit = "max", faces = c("c", "c"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n" ) + ggplot2::ggtitle("First coordinate integer (LP faces)") p4 <- plotPolytope( A, b, obj, type = c("i", "c"), crit = "max", faces = c("i", "c"), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "coord" ) + ggplot2::ggtitle("First coordinate integer (MILP faces)") gridExtra::grid.arrange(p1, p2, p3, p4, nrow = 2)

If you write a paper using LaTeX, you may create a TikZ file of the plot for LaTeX using

library(tikzDevice) tikz(file = "plot_polytope.tex", standAlone=F, width = 7, height = 6) plotPolytope( A, b, obj, type = rep("i", ncol(A)), crit = "max", faces = rep("c", ncol(A)), plotFaces = TRUE, plotFeasible = TRUE, plotOptimum = TRUE, labels = "n", latex = TRUE ) dev.off()

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