In this paper we dene a special interval uncertainty by extending the well known concept of fuzzy intervals (or, fuzzy numbers), dening a new concept of extended fuzzy interval and its subspace: extended linear fuzzy interval. We present some examples and derive basic properties. Then we demonstrate an application of the new concept by formulating the well known Optimal Allocation Problem (OAP) and its solution under uncertainty. We formulate the corresponding optimization problem with the extended fuzzy interval coecients and also extended fuzzy interval variables and derive its optimal solution in the form of extended fuzzy intervals. Two numerical examples are presented in order to illustrate particular problems and solution concepts. The main innovation highlights of this paper may be formulated as follows: We generalize the well known concept of the fuzzy interval, i.e. fuzzy set of the real numbers R with triangular or trapezoidal membership function, by dening the new concept of extended fuzzy interval - EFI. Then we dene a special subspace of extended fuzzy intervals with piece-wise linear membership functions called extended linear fuzzy interval - ELFI. We derive basic properties of the newly dened concepts and present some examples illustrating particular new properties. We demonstrate application possibilities and advantages of the new concept by an optimal allocation problem with the extended fuzzy interval coecients as well as variables. We present two numerical examples illustrating a particular problem of allocation type and corresponding solution concepts.