Abstraet Since Briec developed a family of the least dista[tice based ineficiency measures satisfying weak monotonicity over weakly eMcient frontier, the existence of a least distance based eMciency measure satisfying strong monotonicity on the strongly eficient frontier is still an open problem. This paper gives a negative answer to the open problem and its relaxed open problem. Modifying Briec's ineficiency measures gives an alternative solution to the relaxed open problem, that can be used for theDretical and practical applications.