Lexmaxmin in fuzzy multiobjective decision-making

Werners has taken lexmaximin (lexicographically extended maxmin) optimization as a starting point for fuzzy multiobjective decision problems in order to cover both the maxmin aggregation operator from fuzziness as well as the sine-quanon prerequisite of Pareto optimality. There are two other criteria, one sharper, the other less sharp, than lexmaxmin, which could serve the same purpose: ‘No-reason-for-regret’ (NR), which was previously introduced by the author of the present paper, and Heindl’s criterion. The paper discusses interrelations between Pareto, maxmin, NR, lexmaxmin, and Heindl’s criterion. It is shown that the multiobjective functions involved in any of these models are strictly quasiconcave if their component functions are quasiconcave and strictly quasiconcave. The component functions are given on a convex subset of a real linear space and take on values from a total order. It is briefly discussed that any strictly quasiconcave function with values from a connected quasiorder can be handled by maximum search methods of the Fibonacci type. Maxmin and lexmaxmin belong to this class. There is a ‘lexicographic’ representation given for Heindl’s relation. For NR, the quasiconcavity and strict quasiconcavity conditions, together with convexity of the feasible set, can be relaxed to ‘semilocal quasiconcavity on a locally star shaped feasibile set’. It is shown that NR and Heindl’s criterion (and thus lexmaxmin, too) are equivalent under some of these generalized conditions. For biobjective problems, a similar remark holds for a combination of Pareto and maxmin as compared with NR. There is a close connection between Heindl’s relation and the tradeoff-compromise set of Ecker and Shoemaker.