labsimplex: Simplex Optimization Algorithms for Laboratory and Manufacturing Processes

Description

The labsimplex package implements the simplex optimization algorithms firstly proposed by Spendley et al. (1962) <doi:10.1080/00401706.1962.10490033> and later modified by Nelder and Mead (1965) <doi:10.1093/comjnl/7.4.308> for laboratory and manufacturing processes. The package also provides tools for graphical representation of the simplexes and some example response surfaces that are useful for illustrating the optimization process.

Details

A simplex is a geometric element defined as the simpler polytope possible in an n-dimensional space. If the space has n dimensions, the simplexes there will have n+1 corners called vertexes. The simplexes in two and three-dimensional spaces are the well-known triangle and tetrahedron, respectively.
In the simplex optimization algorithms, the experimental variables are represented by the dimensions in the abstract space. Each vertex in the simplex represents an experiment, then the coordinates of the vertex represent the values for the variables in that experimental setting. The experiments must be performed and a response must be assigned to each vertex. In the optimization process, one of the vertexes is discarded in favor of a new one that must be evaluated. In the first simplex, the vertex with the worst response is discarded. The second worst vertex in this simplex is discarded in the following simplex and the procedure is repeated until the optimum is reached or a response good enough is obtained. The process of discarding a vertex and generating a new one is known as a movement of the simplex.
In this document, the words vertex and experiment are used interchangeably. The same applies to dimensions and experimental variables.

labsimplex functions

This package uses list objects of class 'smplx' to store the simplex information, including all the coordinates of the vertexes and their responses.
The labsimplex functions can generate a new 'smplx' class object, assing responses to the vertices to generate the next one and to visualize different spatial representations of the n-dimensional simplex in 2D or 3D projections. Detailed information can be found by typing vignette('labsimplex').

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

References

Nelder, J. A., and R. Mead. 1965. “A Simplex Method for Function Minimization.” The Computer Journal 7 (4): 308–13.

Spendley, W., G. R. Hext, and F. R0. Himsworth. 1962. “Sequential Application of Simplex Designs in Optimization and Evolutionary Operation.” Technometrics 4 (4): 441–61.

Multivariate functions that define hypothetical response surfaces.

Description

The functions in this section simulate the yield of hypothetical chemical reactions as a function of temperature, pH, and concentration (the latter only for exampleSurfaceR3()). Those functions are useful to illustrate most concepts of the simplex optimization algorithms implemented in the labsimplex package, as shown in the vignentte of the package. This vignette can be visualized by running vignette('labsimplex').

Usage

exampleSurfaceR2(x1, x2, noise = 0)

exampleSurfaceR2.2pks(x1, x2, noise = 0)

exampleSurfaceR3(x1, x2, x3, noise = 0)

Arguments

Details

Parameters x1, x2, and x3 may be supplied as vectors in which case all must have the same length.
Boundary values are proposed consistently with real-life limitations in aqueous media. If such boundaries are violated in the variables input, a negative result without physical meaning is returned. This negative value represents an infinitely bad response that will force the simplex to move in another direction.

Value

exampleSurfaceR2(x1, x2, noise = 0) defines a response surface with one maximum at pH 10 and 300 K.

exampleSurfaceR2.2pks(x1, x2, noise = 0) defines a response surface with global and local maxima at pH 10 and 300 K and pH 4.5 and 340 K, respectively.

exampleSurfaceR3(x1, x2, x3, noise = 0) defines a response surface with one maximum at pH 10, 300 K and a concentration of 0.5.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

cntr, prspctv and exampleOptimization

Examples

  exampleSurfaceR2(x1 = 320, x2 = 4.5)
  exampleSurfaceR2(x1 = c(310, 320), x2 = c(4.5, 5.8))
  exampleSurfaceR2(x1 = c(310, 320), x2 = c(4.5, 5.8), noise = 5)
  exampleSurfaceR2.2pks(x1 = 320, x2 = 4.5)
  exampleSurfaceR2.2pks(x1 = c(310, 320), x2 = c(4.5, 5.8))
  exampleSurfaceR2.2pks(x1 = c(310, 320), x2 = c(4.5, 5.8), noise = 5)
  exampleSurfaceR3(x1 = 320, x2 = 4.5, x3 = 0.3)
  exampleSurfaceR3(x1 = c(310, 320), x2 = c(4.5, 5.8), x3 = c(0.3, 0.5))
  exampleSurfaceR3(x1 = c(310, 320), x2 = c(4.5, 5.8), x3 = c(0.3, 0.5),
                   noise = 5)

Adds the simplex movements to a response surface contour

Description

The function complements the contour plot produced by using cntr function. Given a contour plot and a simplex (an object of class smplx) the function adds the simplex movements to the contour plot to illustrate the optimization process and the path that was followed.

Usage

addSimplex2Surface(p, simplex)

Arguments

Value

a ggplot object with the optimization path over the contour plot provided.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

cntr exampleOptimization

Examples

simplex <- exampleOptimization(surface = exampleSurfaceR2,
                               centroid = c(7, 340),
                               stepsize = c(1.2, 15))
p <- cntr(surface = exampleSurfaceR2)
p <- addSimplex2Surface(p = p, simplex = simplex)
print(p)

Modify the coordinates of given vertexes of a simplex

Description

Changes the coordinates of previously generated vertexes when slight differences were impossible to avoid at the moment of setting the experiment variables (e.g. small differences in mass components when preparing a mixture). This function allows the correction of the vertexes of a simplex in order to produce movements of the simplex based on the actual coordinates.

Usage

adjustVertex(simplex, newcoords, overwrite = FALSE)

Arguments

Value

An object of class smplx with the modified simplex information.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

Examples

simplex <- labsimplex(n = 3, start = c(7, 25, 0.15),
                      stepsize = c(0.2, 5, 0.02))
adjustVertex(simplex = simplex, newcoords = list(Vertex.1 = c(7.1, NA, NA),
                                                 Vertex.3 = c(6.9, NA, 0.155)),
             overwrite = TRUE)

Contour plot of example response surfaces

Description

Plots a ggplot with the contour of the bivariate example response surfaces included in the package.

Usage

cntr(surface, length = 150, noise = 0, x1lim = c(278, 365), x2lim = c(0, 14))

Arguments

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

References

H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016.

Examples

  p <- cntr(surface = exampleSurfaceR2, length = 200)
  print(p)

Performs a complete simplex optimization over a response surface

Description

The function uses the information in a simplex or creates a new one by using the defined centroid and step-size to perform a simplex optimization using the responses produced in the example response surfaces included in the package.

Usage

exampleOptimization(surface, simplex = NULL, centroid = c(7, 340),
  stepsize = c(0.6, 10), algor = "fixed", experiments = 17, noise = 0)

Arguments

Value

An object with class smplx with the simplex optimization data.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

Generates the new vertex of a simplex optimization

Description

Gives the coordinates for the new vertex that must be performed based on the responses for the vertexes on the current simplex and considering the optimization criteria.

Usage

generateVertex(simplex, qflv = NULL, crit = "max", algor = "fixed",
  overwrite = FALSE)

Arguments

Details

When minimization is the criteria, the algorithm will tend to approach zero. If negative responses are possible and the most negative value is desired, a very large negative number must be provided in crit parameter.

Value

An object of class smplx with the new simplex information including the conditions for the new experiment to be permormed.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

Examples

  simplex <- labsimplex(n = 3, centroid = c(320, 7, 0.4),
                        stepsize = c(35, 2, 0.3))
  ## The experiments must be performed and the responses passed to qflv.
  ## Here we get the responses by using an example response surface
  ## included in the package:
  ##
  ## Initially, the response must be provided for all the vertexes
  response <- exampleSurfaceR3(x1 = simplex$coords[, 1],
                               x2 = simplex$coords[, 2],
                               x3 = simplex$coords[, 3])
  simplex <- generateVertex(simplex = simplex, qflv = response)

  ## After this, the last vertex is the only one that must be evaluated
  response <- exampleSurfaceR3(x1 = simplex$coords[nrow(simplex$coords), 1],
                               x2 = simplex$coords[nrow(simplex$coords), 2],
                               x3 = simplex$coords[nrow(simplex$coords), 3])
  simplex <- generateVertex(simplex = simplex, qflv = response)

  ## Alternatively the simplex object can overwrite the older one:
  generateVertex(simplex = simplex, qflv = response, overwrite = TRUE)

Generates a simplex object

Description

The simplex (a list with class smplx) contains the coordinates of the n+1 vertices that define a simplex in an n-dimensional space. By default, the function produces a regular simplex centered at the origin. The coordinates of the regular simplex are transformed into the real variables space by using the information of the start or centroid and step-size. The only non-optional parameter is n that relates the simplex dimensionality. Once the simplex is generated, the experiments under the conditions indicated for each variable at each vertex must be carried and the response obtained. Those responses are assigned to the smplx object at the moment of generating the new vertex (see generateVertex).

Usage

labsimplex(n, start = NULL, centroid = NULL, stepsize = NULL,
  usrdef = NULL, var.name = NULL)

Arguments

Details

The regular simplex coordinates are generated following the general algorithm for the cartesian coordinates of a regular n-dimensional simplex. This algorithm considers that all vertices must be equally distanced from simplex centroid and all angles subtended between any two vertexes and the centroid of a simplex are equal to arccos(-1/n).
If the vertexes coordinates are manually given (in usr.def parameter), the function checks if the faces produced belong to different hyperplanes. This avoids the generation of a degenerated simplex.

Value

An object of class smplx with the information of the new simplex.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

References

Nelder, J. A., and R. Mead. 1965. “A Simplex Method for Function Minimization.” The Computer Journal 7 (4): 308–13.

Spendley, W., G. R. Hext, and F. R0. Himsworth. 1962. “Sequential Application of Simplex Designs in Optimization and Evolutionary Operation.” Technometrics 4 (4): 441–61.

Examples

  simplex <- labsimplex(n = 3)
  simplex <- labsimplex(n = 3, centroid = c(350, 7, 0.4),
                        stepsize = c(35, 2, 0.3),
                        var.name = c('temperature', 'pH', 'concentration'))
  simplex <- labsimplex(n = 3, usrdef = rbind(c(390, 8, 0.2), c(330, 8, 0.2),
                                   c(330, 6, 0.6), c(330, 6, 0.1)))
  ## Not run: 
    ## User defined coordinates may define a degenerated simplex:
    simplex <- labsimplex(n = 3,
                          usrdef = rbind(c(390, 8, 0.3), c(340, 8, 0.3),
                                         c(355, 8, 0.3), c(340, 5, 0.1)))
  
## End(Not run)

Draws a two dimentional plot of the vertexes in a simplex

Description

The function generates a 2D plot of the vertexes in a simplex optimization when simplex dimensionality is at least 2. When dimensionality is higher than 2, the plot produced is a projection of the selected variables.

Usage

## S3 method for class 'smplx'
plot(x, sel.dim = NULL, all.ver = TRUE, all.lin = TRUE,
  expand = TRUE, exp.fac = 1.5, ...)

Arguments

Details

For 3D representations of simplexes with dimensionality higher than 2 you can use plotSimplex3D.

Value

2D plot of the simplex coordinates.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

plotSimplex3D

Examples

  plot(x = labsimplex(n = 2, centroid = c(7, 340), stepsize = c(1.2, 15)))

  ## Several options are posible when visualizing higher order simplexes
  plot(x = labsimplex(n = 3))
  plot(x = labsimplex(n = 3), sel.dim = c(2, 3))

  ## Simplex movements can be visualized after some experiments has been
  ## performed
  simplex <- exampleOptimization(surface = exampleSurfaceR2,
                                 centroid = c(7, 340),
                                 stepsize = c(1.2, 15), experiments = 16)
  plot(x = simplex)

Draws a three dimentional plot of the vertexes in a simplex

Description

The function generates a 3D plot of the vertexes in a simplex optimization when simplex dimensionality is at least 3. When dimensionality is higher than 3, the plot produced is a projection of the selected variables.

Usage

plotSimplex3D(simplex, sel.dim = NULL, all.ver = TRUE, all.lin = TRUE,
  main = NULL, angle = 30, ...)

Arguments

Value

3D plot of the simplex coordinates.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

plot.smplx

Examples

  plotSimplex3D(simplex = labsimplex(n = 3, centroid = c(350, 11, 0.7),
                                     stepsize = c(10, 0.5, 0.1),
                                     var.name = c('temperature', 'pH',
                                                  'concentration')))

  ## Several options are posible when visualizing higher order simplexes
  plotSimplex3D(simplex = labsimplex(n = 8))
  plotSimplex3D(simplex = labsimplex(n = 8), sel.dim = c(4, 6, 8))

  ## Simplex movements can be visualized after some experiments has been
  ## performed
  simplex <- exampleOptimization(surface = exampleSurfaceR3,
                                 centroid = c(350, 11, 0.7),
                                 stepsize = c(10, 0.5, 0.1),
                                 experiments = 18)
  plotSimplex3D(simplex = simplex, angle = 80)

Plots the response versus the vertex number of a simplex optimization.

Description

The function generates a ggplot object from an object with class smplx. The response is plotted against the vertex number.

Usage

plotSimplexResponse(x, ...)

Arguments

Details

If the simplex object being ploted was obtained using a variable size algorithm, some experimental points could be disregarded and will be shown with a red mark indicating that the vertex was not used in the obtention of new vertexes.

Value

Plot of response against vertex number.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

Examples

  simplex <- exampleOptimization(surface = exampleSurfaceR3,
                                 centroid = c(350, 11, 0.7),
                                 stepsize = c(10, 0.5, 0.1),
                                 experiments = 18, algor = 'variable')
  plotSimplexResponse(simplex)

S3 method print for simplex objects

Description

Prints simplex information.

Usage

## S3 method for class 'smplx'
print(x, extended = FALSE, conventions = TRUE, ...)

Arguments

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

3D perspective plot of example response surfaces

Description

Plots a persp plot of the bivariate example response surfaces included in the package.

Usage

prspctv(surface, length = 45, noise = 0, x1lim = c(278, 365),
  x2lim = c(0, 14), par = NULL, theta = 22, phi = 15, shade = 0.2,
  ticktype = "detailed", ...)

Arguments

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

Examples

  prspctv(surface = exampleSurfaceR2.2pks)
  prspctv(surface = exampleSurfaceR2.2pks, theta = 35, phi = 25,
          expand = 0.75, xlab = 'Temperature (K)', ylab = 'pH',
          zlab = 'Yield (%)')

Exports the information contained in an object of class smplx.

Description

Creates a text file with extension .smplx that contains the complete information contained in a simplex (an object with class smplx, see labsimplex). This file allows the continuation of an optimization proccess when the experiments take too long and multiple R sessions are required. The file produced is also useful to share the information of the optimization process. The exported simplex can be later imported with simplexImport.

Usage

simplexExport(simplex, filename = NULL, direc = NULL)

Arguments

Value

Generates a .smplx file containing all the information required to continue with the optimization process after the experiments have been carried.

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

simplexImport

Examples

  
  simplex <- labsimplex(n = 5)
  simplexExport(simplex = simplex)
  

Imports the information contained in a .smplx file.

Description

The function reads and (optionally) loads into the environment the simplex (object of class smplx) contained in a .smplx file that was previously created using simplexExport.

Usage

simplexImport(filename, aut.load = TRUE, name = NULL)

Arguments

Value

A smplx class object with the complete information of the simplex

Author(s)

Cristhian Paredes, craparedesca@unal.edu.co

Jesús Ágreda, jagreda@unal.edu.co

See Also

simplexImport

Examples

  
  simplexR2 <- exampleOptimization(surface = exampleSurfaceR2)
  simplexExport(simplex = simplexR2)
  rm(simplexR2)
  simplexImport(filename = "simplexR2")