In many situations (particularly ones where consultants are called in to!), there are many, often conflicting, objectives. For instance, in an advertising campaign, there might be a number of different market segments to reach. Or, in our fire station problem, there might be two types of objectives: minimizing response time and minimizing service cost. The models we have seen, however, only allow one objective. How can they be adapted to handle multiple objectives?
There are a number of fundamental problems when there are multiple objectives. For instance, consider the case where there are a number of decision makers, each with a preference ordering over a number of alternatives. Our goal is to choose the ``fair'' alternative that aggregates the preferences of the decision makers. This is an example of multiple criteria decision making (each decision makers represents one criteria), and we need to balance those objectives in a fair way. A very common example of this is voting for a leader, say the president of the United States. In the most recent (1996) election, there were three candidates for president. Is the US electoral system a ``fair'' system? What do we mean by fairness? The field of study that addresses these problems is called social choice, and is filled with pessimistic results. In general, under any minimal set of axioms of fairness, no fair voting rule exists. For instance, in the US system, it is possible that, in a three way race, a candidate who would be either of the other two if they ran alone, would still lose the election. This seems to violate some fundamental ideas of what it means to be the fair winner.
These issues of aggregating views about alternatives is difficult even with a single decision maker (or a group trying to reach consensus). Imagine trying to locate an ``obnoxious facility'', like a waste disposal plant. There are many factors that go into such a decision. These might include distance from highly populated areas, transportation costs, land costs, geological stability, and so on. Is there any organized way that one might think about determining the relative importance of these factors and then go about comparing alternative sites? One technique that is used is the Analytic Hierarchy Process (AHP), which uses very simple calculations to try to put numerical values on factors and alternatives.
Finally, how can we use our models in more complicated situations, where the choice is not just a number of alternatives, but rather an entire decision set, say represented by a linear program. Can we use linear programming to explore alternative decisions that trade off various objectives?