New conditions for testing necessarily/possibly efficiency of non-degenerate basic solutions based on the tolerance approach

In this paper, a specific type of multiobjective linear programming problem with interval objective func- tion coefficients is studied. Usually, in such problems, it is not possible to obtain an optimal solution which optimizes simultaneously all objective functions in the interval multiobjective linear programming (IMOLP) problem, requiring the selection of a compromise solution. In conventional multiobjective pro- gramming problems these compromise solutions are called efficient solutions. However, the efficiency cannot be defined in a unique way in IMOLP problems. Necessary efficiency and possible efficiency have been considered as two natural extensions of efficiency to IMOLP problems. In this case, necessarily ef- ficient solutions may not exist and the set of possibly efficient solutions usually has an infinite number of elements. Furthermore, it has been concluded that the problem of checking necessary efficiency is co- NP-complete even for the case of only one objective function. In this paper, we explore new conditions for testing necessarily/possibly efficiency of basic non-degenerate solutions in IMOLP problems. We show properties of the necessarily efficient solutions in connection with possibly and necessarily optimal solu- tions to the related single objective problems. Moreover, we utilize the tolerance approach and sensitivity analysis for testing the necessary efficiency. Finally, based on the new conditions, a procedure to obtain some necessarily efficient and strictly possibly efficient solutions to multiobjective problems with interval objective functions is suggested.This research was partly supported by the Spanish Ministry of Economy and Competitiveness (project ECO2017-88883-R ) and by the Fundação para a Ciência e a Tecnologia (FCT) under project grant UID/Multi/00308/2019 . This work has been also partly sup- ported by the Consejería de Innovación, Ciencia y Empresa de la Junta de Andalucía (PAI group SEJ-532 ). Carla Oliveira Henriques also acknowledges the training received from the University of Malaga PhD Programme in Economy and Business [Programa de Doctorado en Economía y Empresa de la Universidad de Malaga]. José Rui Figueira acknowledges the support from the FCT grant SFRH/BSAB/139892/2018 under POCH Program and to the DOME (Discrete Optimization Methods for Energy management) FCT Re- search Project (Ref: PTDC/CCI-COM/31198/2017)