Functions r1, r2 and r3 calculate the R1, R2 and R3
indicator respectively for two sets of points x and y.
r1(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...) r2(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...) r3(x, y, ip = NULL, np = NULL, lambda = NULL, utility = "tschebycheff", ...)
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Single numeric indicator value.
Consider two Pareto-front (approximation) sets \(X = \{x_1, \ldots, x_{|X|}\}\) and \(Y = \{y_1, \ldots, y_{|Y|}\}\) for some optimization problem with \(m\) objectives. Furthermore, let \(U\) be a set of utility functions each of the form \(u : R^m \to R\) which maps each point in the objective space into a so-called measure of utility. Let \(p\) be a probability distribution over \(U\). We denote by \(u^{*}(A) = \max_{x \in A} u(x)\) the maximal value reached by some utility function \(u\) on a set \(A\).
The R1-indicator measures the probability that approximation set \(X\) is better than approximation set \(Y\) by integrating over the utility function \(u \in U\). More precisely, it is defined as $$ R1(X, Y, U, p) = \int_{u \in U} C(X, Y, u)p(x)du $$ where \(C(X, Y, u) = 1\) if \(u^{*}(X) > u^{*}(Y)\), \(C(X, Y, u) = 1/2\) if \(u^{*}(X)=u^{*}(Y)\) and \(C(X, Y, u) = 0\) otherwise is termed the outcome function.
Interpretation: if \(R1(X, Y, U, p) > 0.5\), \(X\) is better than \(Y\). Owing to the definition it holds that $$ R1(X, Y, U, p) = 1 - R1(X, Y, U, p) $$ and therefore \(X\) and \(Y\) cannot outperform each other simultaneously.
In practice this defintion is not of much use. Therefore, a finite discrete set of utility functions is used with the uniform probability distribution. Then, \(R1\) can be written as $$ R1(X, Y) = \frac{1}{|U|}\sum_{u \in U} C(X, Y, u). $$ The same approach is used for \(R2\) and \(R3\) which will be discussed in the following.
Classical utility functions involve weighted sum, Tschebycheff and augmented
Tschbycheff. All three are availabe via the argument utility in the
implementation. As an example, the set of (weighted) Tschebycheff utility
functions is \(U_{\infty} = (u_{\lambda})\) where
$$
u_{\lambda}(x) = -\max_{j=1, \ldots, m} \left(\lambda_j \cdot |x_j - r_j|\right)
$$
where \(r \in R^m\) is the ideal point or an approximation thereof and
\(\lambda \in R^m\) is a weight vector with \(\lambda_j \geq 0\) for \(j=1,\ldots,m\)
and \(\sum_{j=1}^{m} \lambda_j = 1\). Further details are beyond the scope
of this documentation (see [1] for in-depth information).
The R2-indicator instead considers the expected utility values. It is defined as $$ R2(X, Y, U, p) = E(u^{*}(X)) - E(u^{*}(Y)) = \int_{u \in U} (u^{*}(A) - u^{*}(B))p(u)du. $$ Interpretation: here, since the measure is defined as the difference of expectations approximation set \(X\) is considered better than set \(Y\) if \(R(X, Y) > 0\).
Eventually, indicator R3 considers the ratios of best utility function values: $$ R3(X, Y, U, p) = E\left(\frac{u^{*}(Y) - u^{*}(X)}{u^{*}(Y)}\right) = \int_{u \in U} \frac{u^{*}(Y) - u^{*}(X)}{u^{*}(Y)}p(u)du. $$ For further details we refer the reader to Hansen and Jaszkiewicz [1].
[1] M. P. Hansen and A. Jaszkiewicz. 1998. Evaluating the quality of approximations to the nondominated set. Imm-rep-1998-7. Institute of Mathematical Modeling, Technical University of Denmark.