Multiobjective Optimization

return on his or her investment with a small risk of incurring a loss; an oncologist prescribes radiotherapy to a cancer patient so as to destroy the tumor without causing damage to healthy organs; an airline manager constructs schedules that incur small salary costs and that ensure smooth operation even in the case of disruptions. All three decision makers (DMs) are in a similar situation—they need to make a decision trying to achieve several conflicting goals at the same time: The highest return investments are in general the riskiest ones, tumors can always be destroyed at the expense of irreversible damage to healthy organs, and the cheapest schedules to operate are ones that leave as little as possible time between flights, wreaking havoc to operations in the case of unexpected delays. Moreover, the investor, the oncologist, and the airline manager are all in a situation where the number of available options or alternatives is very large or even infinite. There are infinitely many ways to invest money and infinitely many possible radiotherapy treatments, but the number of feasible crew schedules is finite, albeit astronomical in practice. The alternatives are therefore described by constraints, rather than explicitly known: the sums invested in every stock must equal the total invested; the radiotherapy treatment must meet physical and clinical constraints; crew schedules must ensure that each flight has exactly one crew assigned to operate it. Mathematically, the alternatives are described by vectors in variable or decision space; the set of all vectors satisfying the constraints is called the feasible set in decision space. The consequences or attributes of the alternatives are described as vectors in objective or outcome space, where outcome (objective) vectors are a function of the decision (variable) vectors. The set of outcomes corresponding to feasible alternatives is called Articles