  Existence of an increasing quasi-concave value function consistent with given
preference information is an important issue in various fields including
Economics, Multiple Criteria Decision Making, and Applied Mathematics. In this
paper, we establish necessary and sufficient conditions for existence of a
value function satisfying aforementioned properties. This leads to an
operational, tractable and easy to use test for checking the existence of a
desirable value function. In addition to developing the existence test, we
construct consistent linear and non-linear desirable value functions.
