limSolve (version 1.4)

linp: Linear Programming.

Description

Solves a linear programming problem, $$\min(\sum {Cost_i.x_i})$$ subject to $$Ex=f$$ $$Gx>=h$$ $$x_i>=0$$ (optional)

This function provides a wrapper around lp (see note) from package lpSolve, written to be consistent with the functions lsei, and ldei. It allows for the x's to be negative (not standard in lp).

Usage

linp(E=NULL, F=NULL, G=NULL, H=NULL, Cost,
  ispos = TRUE, int.vec=NULL, verbose=TRUE, ...)

Arguments

E

numeric matrix containing the coefficients of the equality constraints $Ex=F$; if the columns of E have a names attribute, they will be used to label the output.

F

numeric vector containing the right-hand side of the equality constraints.

G

numeric matrix containing the coefficients of the inequality constraints $Gx>=H$; if the columns of G have a names attribute, and the columns of E do not, they will be used to label the output.

H

numeric vector containing the right-hand side of the inequality constraints.

Cost

numeric vector containing the coefficients of the cost function; if Cost has a names attribute, and neither the columns of E nor G have a name, they will be used to label the output.

ispos

logical, when TRUE then the unknowns (x) must be positive (this is consistent with the original definition of a linear programming problem).

int.vec

when not NULL, a numeric vector giving the indices of variables that are required to be an integer. The length of this vector will therefore be the number of integer variables.

verbose

logical to print error messages.

...

extra arguments passed to R-function lp.

Value

• a list containing:
• X
  vector containing the solution of the linear programming problem.
• residualNorm
  scalar, the sum of absolute values of residuals of equalities and violated inequalities. Should be very small or zero for a feasible linear programming problem.
• solutionNorm
  scalar, the value of the minimised Cost function, i.e. the value of $\sum {Cost_i.x_i}$.
• IsError
  logical, TRUE if an error occurred.
• type
  the string "linp", such that how the solution was obtained can be traced.

References

Michel Berkelaar and others (2007). lpSolve: Interface to Lpsolve v. 5.5 to solve linear or integer programs. R package version 5.5.8.

See Also

ldei, lsei, lp the original function from package lpSolve Blending, a linear programming problem.

Examples

#--------------------------------------------
# Linear programming problem 1, not feasible
#--------------------------------------------

# maximise x1 + 3*x2 
# subject to
#-x1 -x2    < -3
#-x1 + x2   <-1
# x1 + 2*x2 < 2
# xi>0
G    <- matrix(nrow=3,data=c(-1,-1,1, -1,1,2))
H    <- c(3,-1,2)
Cost <- c(-1,-3)
(L<-linp(E=NULL,F=NULL,Cost=Cost,G=G,H=H))
L$residualNorm

#--------------------------------------------
# Linear programming problem 2, feasible
#--------------------------------------------

# minimise x12 + 8*x13 + 9*x14 + 2*x23 + 7*x24 + 3*x34  
# subject to:
#-x12             + x23 + x24       = 0
#     - x13       - x23       + x34 = 0
# x12 + x13 + x14                   > 1
#             x14       + x24 + x34 < 1
# xi>0
A    <- matrix(nrow=2,byrow=TRUE,data=c(-1,0,0,1,1,0,
                                        0,-1,0,-1,0,1))
B    <- c(0,0)
G    <- matrix(nrow=2,byrow=TRUE,data=c(1,1,1,0,0,0,
                                        0,0,-1,0,-1,-1))
H    <- c(1,-1)
Cost <- c(1,8,9,2,7,3)
(L<-linp(E=A,F=B,Cost=Cost,G=G,H=H))
L$residualNorm

#---------------------------------------------
# Linear programming problem 3, no positivity
#---------------------------------------------
# minimise x1 + 2x2 -x3 +4 x4
# subject to:
# 3x1 + 2x2 + x3 + x4 = 2
#  x1 +  x2 + x3 + x4 = 2

E <- matrix(ncol=4, byrow=TRUE,
            data=c(3,2,1,4,1,1,1,1))
F <- c(2,2)

G <-matrix(ncol=4,byrow=TRUE,
           data=c(2,1,1,1,-1,3,2,1,-1,0,1,0))
H <- c(-1, 2, 1)
Cost <- c(1,2,-1,4)
linp(E=E,F=F,G=G,H=H,Cost,ispos=FALSE)