A study on multi-criteria structural optimum design using qualitative and fuzzy reasoning (Q1329523)

The authors study the formulation and solution of multi-criteria decision-making problems in the setting of qualitative modelling and employing some mechanisms of approximate reasoning. The standard nonlinear optimization problem discussed here is of the form, \(\min_{x_ i\in X_ i} {\mathbf F}({\mathbf x})\) subject to \(g_ j({\mathbf x})\leq 0\), \(j=1,2,\dots, M\) where the vector \[ {\mathbf F}({\mathbf x})= [f_ i({\mathbf x}) f_ 2({\mathbf x})\dots f_ L({\mathbf x})]^ T \] summarizes the individual objective functions. The design variables \({\mathbf x}\) are assumed to take on some discrete values. The qualitative optimization introduces the basic notions of qualitative modelling such as qualitative sensitivity, qualitative optimality, and qualitative tradeoff ratio. The crux of the computations relies on suitable algebraic operations (such as multiplication and addition) carried out on symbolic quantities and the use of linguistic terms (fuzzy sets) in the quantification of the variables. For instance, the sensitivity of \(h({\mathbf x})\) with respect to \(x_ k\) is defined accordingly, \[ \Biggl[ {{\partial h} \over {\partial x_ k}}\Biggr]= \text{sign} \Biggl( {{\partial h} \over {\partial x_ k}}\Biggr) \left\{ \begin{smallmatrix} \text{small} (x_ k)\\ \text{same} (x_ k)\\ \text{large} (x_ k) \end{smallmatrix} \right\} \] with small, same, and large being some fuzzy sets coming from the fuzzy language of the linguistic terms. Similarly, the algorithm of fuzzy reasoning is based upon the language of fuzzy variables. The paper includes a complete design example of a parabolic antenna.