Description Usage Arguments Value References See Also Examples
The goal is to solve a linear programming problem having Trapezoidal Fuzzy Numbers as coefficients in the objective function (f(x)=c1*x1+…+cn*xn).
Max f(x) or Min f(x)
s.t.: Ax<=b
FOLP.multiObj
uses a multiobjective approach. This approach is based on each βcut
of a Trapezoidal Fuzzy Number is an interval (different for each β). So the problem may be
considered as a Parametric Linear Problem. For a value of β fixed, the problem becomes a
Multiobjective Linear Problem, this problem may be solved from different approachs, FOLP.multiObj
solves it using weights, the same weight for each objective.
FOLP.interv
uses an intervalar approach. This approach is based on each βcut
of a Trapezoidal Fuzzy Number is an interval (different for each β). Fixing an β,
using interval arithmetic and defining an order relation for intervals is posible to compare intervals,
this transforms the problem in a biobjective problem (involving the minimum and the center of intervals).
Finally FOLP.interv
use weights to solve the biobjective problem.
FOLP.strat
uses a stratified approach. This approach is based on that βcuts are a
sequence of nested intervals. Fixing an β two auxiliary problems are solved, the first
replacing the fuzzy coefficients by the lower limits of the βcuts, the second doing the
same with the upper limits. The results of the two auxiliary problems allows to formulate a new
auxiliary problem, this problem tries to maximize a parameter λ.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 
objective 
A vector (c1, c2, ..., cn) of Trapezoidal Fuzzy Numbers with the objective function coefficients f(x)=c1*x1+…+cn*xn. Note that any of the coefficients may also be Real Numbers. 
A 
Technological matrix of Real Numbers. 
dir 
Vector of strings with the direction of the inequalities, of the same length as 
b 
Vector with the right hand side of the constraints. 
maximum 

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. 
w1 
Weight to be used, 
FOLP.multiObj
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.
FOLP.interv
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.
FOLP.strat
returns the solutions and the value of λ 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. A greater value of λ may be interpreted as the
obtained solution is better.
Verdegay, J.L. Fuzzy mathematical programming. In: Fuzzy Information and Decision Processes, pages 231237, 1982. M.M. Gupta and E.Sanchez (eds).
Delgado, M. and Verdegay, J.L. and Vila, M.A. Imprecise costs in mathematical programming problems. Control and Cybernetics, 16 (2):113121, 1987.
Bitran, G.. Linear multiple objective problems with interval coefficients. Management Science, 26(7):694706, 1985.
Alefeld, G. and Herzberger, J. Introduction to interval computation. 1984.
Moore, R. Method and applications of interval analysis, volume 2. SIAM, 1979.
Rommelfanger, H. and Hanuscheck, R. and Wolf, J. Linear programming with fuzzy objectives. Fuzzy Sets and Systems, 29:3148, 1989.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  ## maximize: [0,2,3]*x1 + [1,3,4,5]*x2
## s.t.: x1 + 3*x2 <= 6
## x1 + x2 <= 4
## x1, x2 are nonnegative real numbers
obj < c(FuzzyNumbers::TrapezoidalFuzzyNumber(0,2,2,3),
FuzzyNumbers::TrapezoidalFuzzyNumber(1,3,4,5))
A<matrix(c(1, 1, 3, 1), nrow = 2)
dir < c("<=", "<=")
b < c(6, 4)
max < TRUE
# Using a Multiobjective approach.
FOLP.multiObj(obj, A, dir, b, maximum = max, min=0, max=1, step=0.2)
# Using a Intervalar approach.
FOLP.interv(obj, A, dir, b, maximum = max, w1=0.3, min=0, max=1, step=0.2)
# Using a Stratified approach.
FOLP.strat(obj, A, dir, b, maximum = max, min=0, max=1, step=0.2)

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