Possibility engineering is a variant of decision analysis that focuses on highlighting a robust set of acceptable courses of action that satisfy a decision maker’s value constraints rather than identifying and prescribing a single, optimal course of action. A shift from a conventional decision-analytic approach to a possibility engineering approach can facilitate problem resolution by sparing decision makers from misconceptions surrounding the term “decision analysis” and freeing them from its demands for structural precision, preferential fidelity, and prescription adherence. In this work, I will develop possibility engineering as a framework for resolving problems of choice under uncertainty. This development will include motivation for adopting a possibility engineering framework, establishment of synthetic probabilities and synthetic decision trees as core framework elements, formalization of concatenation and clone operations for matrices, and ultimately establishment of the possibility matrix as the mechanism for partitioning the set of all available courses of action in a manner consistent with the decision maker's value constraints.