<p>This article demonstrates that a type-2 fuzzy set is a useful and insightful way to model optimization problems under uncertainty. Toward this objective, a new representation of interval-valued fuzzy sets based on constraint functions, called generalized fuzzy intervals, is developed for the analysis of possibilistic optimization problems in which the parameters are generalizations of interval-valued fuzzy numbers. The aim of this study is to represent interval-valued fuzzy sets in a way that possibilistic optimization models solved with generalized fuzzy intervals result in increased information about the risk associated with proposed actions that might be taken based on a solution strategy. This methodology uses the penalty method to reduce a possibilistic Linear Programming Problem (LPP) into a nonlinear optimization problem, whose results are compared with the resolution of the LPP by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-levels whose solutions are defuzzified via centroid and, possibility and necessity, taking as reference the solution of the deterministic LPP. An application of a supplemental diet is presented to an individual who has been diagnosed with nutritional deficiencies, with varying amounts of nutrients in relation to one of their preferred foods, which is mathematically described by interval-valued fuzzy number. The results obtained in solving the possibilistic optimization problem with the penalty method allow the individual to determine the best diet to meet their daily nutrient needs at the lowest cost in relation to the other methods presented.</p>