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Interval linear systems as a necessary step in fuzzy linear systems
International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies
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Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation
For mathematical programming (MP) to have greater impact as a
decision tool, MP software systems must offer suitable support in
terms of model communication and modelling techniques. In this
paper modelling techniques that allow logical restrictions to be
modelled in integer programming terms are described and their
implications discussed. In addition it is demonstrated that many
classes of non-linearities which are not variable separable may be
after suitable algebraic manipulation put in a variable separable
form. The methods of reformulating the fuzzy linear programming
problem as a Max-Min problem is also introduced. It is shown that
analysis of bounds plays a key role in the following four important
contexts: model reduction, reformulation of logical restrictions
as 0-1 mixed integer programs, reformulation of nonlinear programs
as variable separable programs and reformulation of fuzzy linear
programs. It is observed that as well as incorporating an
interface between the modeller and the optimiser there is a need to
make available to the modeller software facilities which support the
model reformulation techniques described here
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