The 1971 Calvert Cliffs decision declared that the National Environmental Policy Act "mandates a rather finely tuned and 'systematic' balancing analysis." This spurred the development of a wide range of analytic methods for systematically "balancing apples and oranges" in environmental impact analysis. Many recent power plant impact statements explicitly weigh environmental, technical, economic, and safety factors in choosing a site. Some of the multi-objective methods used are theoretically rigorous; others are less so. Brookhaven National Laboratory has completed a two-year project in which results and theoretical validity of a large number of methods for choosing weights and combining such considerations were compared. Panels of siting experts applied the methods to siting problems in Long Island, Maryland, and the western United States. Three common decision rules--weighting summation, the power law, and exclusionary screening--often chose strikingly different locations. Simple, commonly applied methods for weight selection chose weights consistently differing from the results of techniques that, when applied correctly, yield valid weights. "Valid weights" accurately represent trade-offs decision makers are willing to make. Choice of decision rule and weight-selection technique can make an important difference in siting decisions, as well as in other decisions involving multiple objectives. Users of such methods in environmental impact assessment should take considerable care in choosing and applying them.