One potential approach to solving multiple criteria optimization
problems is to use an "a posteriori" method that approximates the
efficient frontier. A common difficulty in this type of approach,
however, is evaluating the quality of approximate solutions, since
sets of multiple solutions are generated by "a posteriori" methods.
This necessitates a robust measure to evaluate sets of non-dominated
solutions. Furthermore, a robust measure could play an important role
in metaheuristic optimization (i.e., evolutionary algorithms,
simulated annealing, etc.) for "tuning" various parameters. This
measure could also be used as a "stopping criteria" of such
heuristics. Integrated Preference Functional (IPF) is presented for these
purposes. IPF is a set functional that, given a weight density function
provided by a decision maker and a discrete set of solutions for a
particular problem, assigns a numerical value to that solution set.
IPF is compared to that of other measures appearing in the literature
through numerical experiments---specifically, we use two "a posteriori"
solution techniques based on genetic algorithms for a bi-criteria
parallel machine scheduling problem and evaluate their performance (in
terms of solution quality) using different measures. Experimental results
show that the IPF measure evaluates the solution quality of approximations
robustly (i.e., similar to visual comparison results) while other
alternative measures can misjudge the solution quality. The computational
effort to obtain IPF is negligible for bi-criteria cases. For three or more
(k) objective cases, however, the exact calculation of IPF is computationally
demanding since this requires high (k) dimensional integration. We suggest a
theoretical framework to obtain IPF for k (>= 3) objectives and build a
durable exact method to obtain IPF mainly using the methods of convex-hull,
extreme points enumeration, and triangulation of convex polytope in
computational geometry. We also resort to a sampling method since the exact
method is computationally expensive. A Monte Carlo approximation method of
IPF is experimented and its performance is compared to the exact value of IPF.
We find that the number of weight vectors in R^k needed to estimate IPF
within a specific level of error ratio is independent to the number of
objectives or number of non-dominated points in a set. The required number
of weight vectors increases exponentially as the error ratio we would require
decreases. Finally, we recommend a desirable number of weight vectors
required to estimate IPF within 5% and 1% error ratio with 5% significance
level.