solveLP: Solve Linear Programming / Optimization Problems

Description

Minimizes (or maximizes) \(c'x\), subject to \(A x <= b\) and \(x >= 0\).

Note that the inequality signs <= of the individual linear constraints in \(A x <= b\) can be changed with argument const.dir.

Usage

solveLP( cvec, bvec, Amat, maximum = FALSE,
   const.dir = rep( "

Arguments

cvec

vector \(c\) (containing \(n\) elements).

bvec

vector \(b\) (containing \(m\) elements).

Amat

matrix A (of dimension \(m \times n\)).

maximum

logical. Should we maximize or minimize (the default)?

const.dir

vector of character strings giving the directions of the constraints: each value should be one of "<," "<=," "=," "==," ">," or ">=". (In each pair the two values are identical.)

maxiter

maximum number of iterations.

zero

numbers smaller than this value (in absolute terms) are set to zero.

tol

if the constraints are violated by more than this number, the returned component status is set to 3.

dualtol

if the constraints in the dual problem are violated by more than this number, the returned status is non-zero.

lpSolve

logical. Should the package 'lpSolve' be used to solve the LP problem?

solve.dual

logical value indicating if the dual problem should also be solved.

verbose

an optional integer variable to indicate how many intermediate results should be printed (0 = no output; 4 = maximum output).

Value

solveLP returns a list of the class solveLP containing following objects:

opt

optimal value (minimum or maximum) of the objective function.

solution

vector of optimal values of the variables.

iter1

iterations of Simplex algorithm in phase 1.

iter2

iterations of Simplex algorithm in phase 2.

basvar

vector of basic (=non-zero) variables (at optimum).

con

matrix of results regarding the constraints: 1st column = maximum values (=vector \(b\)); 2nd column = actual values; 3rd column = differences between maximum and actual values; 4th column = dual prices (shadow prices); 5th column = valid region for dual prices.

allvar

matrix of results regarding all variables (including slack variables): 1st column = optimal values; 2nd column = values of vector \(c\); 3rd column = minimum of vector \(c\) that does not change the solution; 4th column = maximum of vector \(c\) that does not change the solution; 5th column = derivatives to the objective function; 6th column = valid region for these derivatives.

status

numeric. Indicates if the optimization did succeed: 0 = success; 1 = lpSolve did not succeed; 2 = solving the dual problem did not succeed; 3 = constraints are violated at the solution (internal error or large rounding errors); 4 = simplex algorithm phase 1 did not find a solution within the number of iterations specified by argument maxiter; 5 = simplex algorithm phase 2 did not find the optimal solution within the number of iterations specified by argument maxiter.

lpStatus

numeric. Return code of lp (only if argument lpSolve is TRUE).

dualStatus

numeric. Return code from solving the dual problem (only if argument solve.dual is TRUE).

maximum

logical. Indicates whether the objective function was maximized or minimized.

Tab

final 'Tableau' of the Simplex algorith.

lpSolve

logical. Has the package 'lpSolve' been used to solve the LP problem.

solve.dual

logical. Argument solve.dual.

maxiter

numeric. Argument maxiter.

Details

This function uses the Simplex algorithm of George B. Dantzig (1947) and provides detailed results (e.g. dual prices, sensitivity analysis and stability analysis). If the solution \(x=0\) is not feasible, a 2-phase procedure is applied. Values of the simplex tableau that are actually zero might get small (positive or negative) numbers due to rounding errors, which might lead to artificial restrictions. Therefore, all values that are smaller (in absolute terms) than the value of zero (default is 1e-10) are set to 0. Solving the Linear Programming problem by the package lpSolve (of course) requires the installation of this package, which is available on CRAN (http://cran.r-project.org/src/contrib/PACKAGES.html#lpSolve). Since the lpSolve package uses C-code and this (linprog) package is not optimized for speed, the former is much faster. However, this package provides more detailed results (e.g. dual values, stability and sensitivity analysis). This function has not been tested extensively and might not solve all feasible problems (or might even lead to wrong results). However, you can export your LP to a standard MPS file via writeMps and check it with other software (e.g. lp_solve, see ftp://ftp.es.ele.tue.nl/pub/lp_solve). Equality constraints are not implemented yet.

References

Dantzig, George B. (1951), Maximization of a linear function of variables subject to linear inequalities, in Koopmans, T.C. (ed.), Activity analysis of production and allocation, John Wiley \& Sons, New York, p. 339-347.

Steinhauser, Hugo; Cay Langbehn and Uwe Peters (1992), Einfuehrung in die landwirtschaftliche Betriebslehre. Allgemeiner Teil, 5th ed., Ulmer, Stuttgart.

Witte, Thomas; Joerg-Frieder Deppe and Axel Born (1975), Lineare Programmierung. Einfuehrung fuer Wirtschaftswissenschaftler, Gabler-Verlag, Wiesbaden.

Examples

Run this code

# NOT RUN {
## example of Steinhauser, Langbehn and Peters (1992)
## Production activities
cvec <- c(1800, 600, 600)  # gross margins
names(cvec) <- c("Cows","Bulls","Pigs")

## Constraints (quasi-fix factors)
bvec <- c(40, 90, 2500)  # endowment
names(bvec) <- c("Land","Stable","Labor")

## Needs of Production activities
Amat <- rbind( c(  0.7,   0.35,   0 ),
               c(  1.5,   1,      3 ),
               c( 50,    12.5,   20 ) )

## Maximize the gross margin
solveLP( cvec, bvec, Amat, TRUE )


## example 1.1.3 of Witte, Deppe and Born (1975)
## Two types of Feed
cvec <- c(2.5, 2 )  # prices of feed
names(cvec) <- c("Feed1","Feed2")

## Constraints (minimum (<0) and maximum (>0) contents)
bvec <- c(-10, -1.5, 12)
names(bvec) <- c("Protein","Fat","Fibre")

## Matrix A
Amat <- rbind( c( -1.6,  -2.4 ),
               c( -0.5,  -0.2 ),
               c(  2.0,   2.0 ) )

## Minimize the cost
solveLP( cvec, bvec, Amat )

# the same optimisation using argument const.dir
solveLP( cvec, abs( bvec ), abs( Amat ), const.dir = c( ">=", ">=", "<=" ) )


## There are also several other ways to put the data into the arrays, e.g.:
bvec <- c( Protein = -10.0,
           Fat     =  -1.5,
           Fibre   =  12.0 )
cvec <- c( Feed1 = 2.5,
           Feed2 = 2.0 )
Amat <- matrix( 0, length(bvec), length(cvec) )
rownames(Amat) <- names(bvec)
colnames(Amat) <- names(cvec)
Amat[ "Protein", "Feed1" ] <-  -1.6
Amat[ "Fat",     "Feed1" ] <-  -0.5
Amat[ "Fibre",   "Feed1" ] <-   2.0
Amat[ "Protein", "Feed2" ] <-  -2.4
Amat[ "Fat",     "Feed2" ] <-  -0.2
Amat[ "Fibre",   "Feed2" ] <-   2.0
solveLP( cvec, bvec, Amat )
# }

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