When xt'=zi' and, for any xt t left in set Xt, there exists irt-,, for which x~1\~=x, ' Vi<l-|. If ~-I is left in X~-, , we again consider
zl i, zt Β§ = , and so on, comparing them, not with xlt ,'-, ~ but with z-, -If the values are the same, we proceed as described above.
If x~\, is unacceptable, we put r~-~=| and increase the superscript of the ([--2)-th component to the minimum extent needed for it not to be the same as the values of the previous components, and so on.
If even the first component goes outside the acceptable range, it remains either to widen the acceptable range or to return to the vector obtained at step 3.
Step 5. step 4 can be repeated several times with different acceptable ranges, e.g., discarding all the time the values of components realizing min;.~[~(xl), as long as it is possible to construct in the relevant X~ a vector x satisfying the constraints.
Obviously, if, for any i~[ , all the values of /t(x,) in Xf are distinct, then algorithm 7 at step 3 gives the Pareto-optimal vector with respect to the system of criteria (~i(z,)li~I), while otherwise (~(x,")]i~]) is weakly Pareto (in the set of constraints of problem 8)). Thus) if we interpret function Fas a convolution of criteria ~ then, even at the first iteration, algorithm 7 gives not too bad a result, which is certainly important in actual problems.
The above methods of solving problem (1)--( 3), ( 8) enable us to deal with optimization or at least improvement of a whole range of complex problems with different practical applications.
The isolation in discrete optimization, and in particular, multicriterion, problems with monotonic constraints, of the rain (max) functions, when the criteria are written in accordance with relation ( 6), immediately reduces the inspection of possible versions in accordance with (7), thus speeding the search for the solution.
REFERENCES
i. GERMEIER YU.B., Introduction to the theory of operations research (Vvedenie v teoriyu issledovaniya operatsii), Nauka, Moscow, 1969. 2. KREINES M.G. and ZHUKOVSKII V.D., Foundations of the optimal choice of diagnostic methods, in: Mathematical methods of optimization and applications to large economic and technical systems (Matem. metody optimizatsii i ikh prilozheniya v bol'shikh ekonomich, i technich, sistemakh), TsEMI Akad. NAUK SSSR, Moscow, 1980. 3. NOVIKOVA N.M., The control of a system of radio stations, in: Mathematical methods in operations research (Matem. metody v issl. operatsii), Izd-vo MGU, Moscow, 1981.