Quantification of assembly difficulty (referred to as an index of difficulty - ID) is an important step in finding an optimum assembly plan. Assembly tasks can be analyzed into at least ten elemental factors that are quantified in terms of their IDs. Given a set of parts to be assembled, assembly ID for each task is computed based on a number of geometrical and operational properties. The objective of the optimization problem here is to minimize the assembly ID and categorize parts/subassemblies based on their preferred direction of assembly in consideration of possible re-orientation of the base part. The problem formulation assumes that the preferred direction of assembly is vertically down consistent with manual and most automatic assembly protocols. In this paper, we establish heuristics to optimize the number of degrees of freedom required in a re-orienting fixture and mathematically derive the requirements for such a fixture. This study considers the assembly of a small engine because of the variety of ideally rigid parts involved.