# FCLP.Beta: Solves a Fuzzy Linear Programming problem with fuzzy... In FuzzyLP: Fuzzy Linear Programming

## Description

The goal is to solve a linear programming problem having fuzzy constraints.

Max f(x) or Min f(x)

s.t.: Ax<=b+(1-β)*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 β.

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

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```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 ) ```

## Arguments

 `objective` A vector (c1, c2, …, cn) with the objective function coefficients f(x)=c1*x1+…+cn*xn. `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 β 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.

## 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 β's if the solver has found them. If the solver hasn't found solutions for any of the β's sampled, return NULL.

## 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.

`FCLP.classicObjective`, `FCLP.fuzzyObjective`

`FCLP.fuzzyUndefinedObjective`, `FCLP.fuzzyUndefinedNormObjective`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```## 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) ```

### Example output

```[1] "Solution is optimal."
beta       x1        x2 objective
[1,]  0.5 3.606188 0.1744023  10.99297
beta       x1        x2 objective
[1,] 0.00 4.315789 0.0000000 12.947368
[2,] 0.25 4.000000 0.0000000 12.000000
[3,] 0.50 3.606188 0.1744023 10.992968
[4,] 0.75 3.164557 0.4556962  9.949367
[5,] 1.00 2.722925 0.7369902  8.905767
```

FuzzyLP documentation built on April 11, 2021, 5:06 p.m.