Overview of Multi-objective Optimization Approach
For single-objective optimization problems, the notion of optimality is very well defined as the minimum or maximum value of some given objective function is sought. In multi-objective (or vector) optimization problems, the notion of optimality is not obvious because of the presence of multiple, incommensurable and conflicting objectives. In general, there is no single optimal (non – dominated or superior) solution that simultaneously yields a minimum (or maximum) for all objective functions. The Pareto optimality concept has been introduced as the solution to multi-objective optimization problems (Koski 1984; Duckstein 1984). A maintenance strategy x* is said to be a Pareto optimum if and only if there exists no maintenance strategy in the feasible set of maintenance alternatives that may yield an improvement of some criterion without worsening at least one other criterion. The multi-objective maintenance optimization problem can be mathematically stated as
where: x* = vector of optimum solutions; f = vector of optimization objectives (e. g. condition rating, maintenance cost, user cost, etc.); Cmt (xj) =maintenance cost of project xj at time t ; Cmax= available
budget; fl= subset of the bridge network that at time t contains deficient bridge decks having a condition rating (CR) above a specified threshold value (requiring maintenance); N= entire set of bridge deck projects within the network.
The concept of Pareto optimality mentioned above, may be stated mathematically as follows (Koski 1984; Lounis and Cohn 1993):
x* = Pareto optimum (8a)
if fi(x) < fi(x*) for i=1,2,…,m (8b)
and fk(x) < fk(x*) for at least one k (8c)
In general, for a multi-objective optimization problem, there are several Pareto optima, and the problem is to select the solution that achieves the best compromise between all competing objectives. Such a solution is referred to as “satisficing” solution in the multi-objective optimization literature (Koski 1984; Lounis and Cohn 1995). The determination of this satisficing solution is discussed in the next section.