When designing surveys, survey organizations must consider numerous design features that may have a substantial and differential impact on both data quality and survey costs. They must recognize that surveys are inherently multipurpose and that a potentially long list of constraints (e.g., minimum sample sizes for domains) must be satisfied. A typical approach is to optimize an objective function subject to constraints on costs and quality. However, as the list of constraints lengthens and the cost and quality structures become more complex, finding a solution to this optimization problem (i.e., choosing the appropriate set of design features) while satisfying all of the constraints becomes increasingly challenging. This paper reviews the methods by which survey designers have attempted to satisfy multiple constraints while optimizing some function of data quality and survey costs.