(Joint work in progress with David Blanc and Jim Turner) Blanc-Markl and then Blanc-Chacholski, gave a definition of higher homotopy operations in terms of the W-construction of Boardman-Vogt. However, the combinatorial complexity of building maps inductively through the skeleta of the W-construction has made it difficult to work with. However, by focusing on a simplicial presentation, these skeleta decompose in terms of certain cones on convex polytopes, so maps out of their boundary (given by the induction hypothesis) may be viewed as maps from spheres, and filling them can be viewed as choosing a null homotopy of the restriction to the boundary sphere. In special cases, here called minimal, the induction step can be further reduced to a question of producing certain simple commutative diagrams involving cones, suspensions and the Moore chains construction in simplicial spaces. This allows the collaborators to relate vanishing of higher homotopy operations with vanishing of their Andre-Quillen cohomology obstructions to realizing diagrams of $\Pi$-algebras.