FCLP.fixedBeta: Solves a Fuzzy Linear Programming problem with fuzzy constraints.

Description

The goal is to solve a linear programming problem having fuzzy constraints. $$Max\, f(x)\ or\ Min\ f(x)$$ $$s.t.:\quad Ax<=b+(1-\beta)*t$$ Where $t$ means we allow not to satisfy the constraint, exceeding the bound $b$ at most in $t$.

FCLP.fixedBeta uses the classic solver (simplex) to solve the problem with a fixed value of $\beta$.

FCLP.sampledBeta solves the problem in the same way than FCLP.fixedBeta but using several $\beta's$ taking values in a sample of the $[0,1]$ inteval.

Usage

FCLP.fixedBeta(
  objective,
  A,
  dir,
  b,
  t,
  beta = 0.5,
  maximum = TRUE,
  verbose = TRUE
)
FCLP.sampledBeta(
  objective,
  A,
  dir,
  b,
  t,
  min = 0,
  max = 1,
  step = 0.25,
  maximum = TRUE,
  verbose = TRUE
)

Value

FCLP.fixedBeta returns the solution for the given beta if the solver has found it or NULL if not.

FCLP.sampledBeta returns the solutions for the sampled $\beta's$ if the solver has found them. If the solver hasn't found solutions for any of the $\beta's$ sampled, return NULL.

Arguments

objective: A vector $(c_1, c_2, \ldots, c_n)$ with the objective function coefficients $f(x)=c_1 x_1+\ldots+c_n x_n$.
A: Technological matrix of Real Numbers.
dir: Vector of strings with the direction of the inequalities, of the same length as b and t. Each element of the vector must be one of "=", ">=", "<=", "<" or ">".
b: Vector with the right hand side of the constraints.
t: Vector with the tolerance of each constraint.
beta: The value of $\beta$ to be used.
maximum: TRUE to maximize the objective function, FALSE to minimize the objective function.
verbose: TRUE to show aditional screen info, FALSE to hide aditional screen info.
min: The lower bound of the interval to take the sample.
max: The upper bound of the interval to take the sample.
step: The sampling step.

References

Verdegay, J.L. Fuzzy mathematical programming. In: Fuzzy Information and Decision Processes, pages 231-237, 1982. M.M. Gupta and E.Sanchez (eds).

Delgado, M. and Herrera, F. and Verdegay, J.L. and Vila, M.A. Post-optimality analisys on the membership function of a fuzzy linear programming problem. Fuzzy Sets and Systems, 53:289-297, 1993.

Examples

Run this code

## maximize:   3*x1 + x2
## s.t.:       1.875*x1   - 1.5*x2 <= 4 + (1-beta)*5
##              4.75*x1 + 2.125*x2 <= 14.5 + (1-beta)*6
##               x1, x2 are non-negative real numbers

obj <- c(3, 1)
A <- matrix(c(1.875, 4.75, -1.5, 2.125), nrow = 2)
dir <- c("<=", "<=")
b <- c(4, 14.5)
t <- c(5, 6)
valbeta <- 0.5
max <- TRUE

FCLP.fixedBeta(obj, A, dir, b, t, beta=valbeta, maximum = max, verbose = TRUE)

FCLP.sampledBeta(obj, A, dir, b, t, min=0, max=1, step=0.25, maximum = max, verbose = TRUE)