Multi-objective decision problems

Previously, we saw what a Multi-objective decision problem: Problems with 2 or more objectives that can conflict with each other. Some building blocks of this type of problem are:

• Decisions: Alternative strategies that decision makers want to explore.
• Objectives: Measures to rank the desirability of a certain decision.
• Constraints: Condition that needs to be satisfied, often non-negotiable.

General form of a multi-objective optimization problem:

where:

Basically, we are mapping the decision variable space into the objective function space:

It is very easy to optimize a problem where we only have one objective. However, for multiple objectives, we want to find the set of solutions that define the best possible tradeoffs between objectives.

But, when is a solution optimal? Pareto's optimality:

• Premise: “A solution is Pareto optimal when improvements in one objective would degrade the performance in at least another objective”
• Purpose: To rank alternatives in multi-objective problems.

For example, suppose we want to select a vehicle based on cost, comfort and environmental impact (3 objectives):

We use heuristic methods (e.g., evolutionary algorithms) for multi-objective optimization to find the best tradeoffs:

• Insensitive to discontinuities or shape of the Pareto front.