Abstract: Most real-world optimization problems are of a multi-objective nature, involving objectives which are conflicting and incomparable. Solving a multi-objective optimization problem requires a method which can generate the set of rational compromises between the objectives. In this paper, we propose two distinct bound set based branch-and-cut algorithms for bi-objective combinatorial optimization problems, based on implicitly and explicitly stated lower bound sets, respectively. The algorithm based on explicitly given lower bound sets computes for each branching node a lower bound set and compares it to an upper bound set. The implicit bound set based algorithm, on the other hand, fathoms branching nodes by generating a single point on the lower bound set for each local nadir point. We outline several approaches for fathoming branching nodes and we propose an updating scheme for the lower bound sets that prevents us from solving the bi-objective LP-relaxation of each branching node. To strengthen the lower bound sets, we propose a bi-objective cutting plane algorithm that dynamically adjusts the weights of the objective functions such that different parts of the feasible set are strengthened by cutting planes. In addition, we suggest an extension of the branching strategy “Pareto branching”. Extensive computational results obtained for the bi-objective single source capacitated facility location problem prove the effectiveness of the algorithms.

More over the research paper “An approximate dynamic programming approach for sequential pig marketing decisions at herd level” have been published in European Journal of Operational Research

Abstract: One of the most important operations in the production of growing/finishing pigs is the marketing of pigs for slaughter. While pork production can be managed at different levels (animal, pen, section, or herd), it is beneficial to consider the herd level when determining the optimal marketing policy due to inter-dependencies, such as those created by fixed transportation costs and cross-level constraints. In this paper, we consider sequential marketing decisions at herd level. A high-dimensional infinite-horizon Markov decision process (MDP) is formulated which, due to the curse of dimensionality, cannot be solved using standard MDP optimization techniques. Instead, approximate dynamic programming (ADP) is applied to solve the model and find the best marketing policy at herd level. Under the total expected discounted reward criterion, the proposed ADP approach is first compared with a standard solution algorithm for solving an MDP at pen level to show the accuracy of the solution procedure. Next, numerical experiments at herd level are given to confirm how the marketing policy adapts itself to varying costs (e.g., transportation cost) and cross-level constraints. Finally, a sensitivity analysis for some parameters in the model is conducted and the marketing policy found by ADP is compared with other well-known marketing polices, often applied at herd level.

]]>`gMOIP`

has been updated to version 1.3.0 and now can plot 3D models too.
The package can make 2D and 3D plots of the polytope of a linear programming (LP), integer linear programming (ILP) model, or mixed integer linear programming (MILP) model with 2 or 3 variables, including integer points, ranges and iso profit curve. Moreover you can also make a plot of the bi-objective criterion space and the non-dominated (Pareto) set for bi-objective LP/ILP/MILP programming models. Figures can be prepared for LaTeX and can automatically be transformed to TikZ using package `tikzDevice`

.

You can install the latest stable release from CRAN:

`install.packages("gMOIP")`

Alternatively (recommended), install the latest development version from GitHub:

```
install.packages("devtools")
devtools::install_github("relund/gMOIP",build_vignettes = TRUE)
```

First we load the package:

```
#' Function for loading missing packages. Install a package from CRAN if not already installed.
#'
#' @param packages String vector with package names
#'
#' @return NULL (invisible)
#' @export
#'
#' @examples loadPackages(c("MASS", "ggplot2", "tikzDevice"))
loadPackages <- function(packages) {
newP <- packages[!(packages %in% installed.packages()[,"Package"])]
if(length(newP)) install.packages(newP, repos = "http://cran.rstudio.com/")
lapply(packages, library, character.only = TRUE)
invisible(NULL)
}
loadPackages("gMOIP")
```

Next, let us have a look at some examples.

We define a function for grouping plots of the solution and criterion space (you may just use functions `plotPolytope`

and `plotCriterion2D`

for single plots):

```
loadPackages("gridExtra") # to combine two plots into one
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) +
ggplot2::ggtitle("Solution space")
p2 <- plotCriterion2D(A, b, obj, type = type, crit = crit, addTriangles = addTriangles,
addHull = addHull, plotFeasible = plotFeasible, labels = labels) +
ggplot2::ggtitle("Criterion space")
gridExtra::grid.arrange(p1, p2, nrow = 1)
}
```

Let us define the constraints:

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

First let us have a look at a LP model (maximize):

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

Note the non-dominated (Pareto) set consists of all supported extreme non-dominated points (illustrated with triangles) and the line segments between them (supported non-dominated points).

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)))
```

Note the non-dominated set consists of all points in black (with shape supported extreme = triangle, supported non-extreme = round, unsupported = round (not on the border)). The triangles drawn using the extreme non-dominated points illustrate areas where unsupported non-dominated points may be found. A point in the solution space is identified in the criterion space using the same number.

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")
```

MILP model (\(x_1\) 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"))
```

Note the solution space now consists to segments and hence the non-dominated set may consist of points and segments (open and closed). Note these segments are not highlighted in the current version of `gMOIP`

.

MILP model (\(x_2\) 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")
```

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")
}
```

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 3D 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 – interactive plot):

`plotSol(A, b)`

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Note this plot is interactive (try using your mouse over the plot). We will do this for a few 3D plots to illustrate functionality.

LP model (criterion space):

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

ILP model (solution space):

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