Combinations of OMPs

By definition, OMPs are combinations of BPQs. The implicit value of an OMP $p$ equals the combination $b_1\oplus\ldots\oplus b_n$ of its constituent BPQs $b_1,\ldots,b_n$. This property allows OMPs to be defined as functions such that an OMP $P=b_1\oplus\ldots\oplus b_n$ is a function $f_P: \mathds{B}\mapsto\mathds{N}: b\rightarrow f_P(b)$ where $\mathds{B}$ is the set of BPQs, $\mathds{N}$ is the set of natural numbers and $f_P(b)$ equals the number of occurrences of $b$ in $b_1,\ldots,b_n$.

For example, let $P_{\text{model}}$ denote the OMP associated with the scenario model that contains three logistic population growth models ($b_{21}$), two Holling predation model ($b_{13}$) and one competition model ($b_{31}$). Therefore,

$$P_{\text{model}}=b_{21}\oplus b_{21}\oplus b_{21}\oplus b_{13}\oplus b_{13}\oplus b_{31}$$

and hence:

$$f_{P_{\text{model}}}(b)= \begin{cases}3 & \text{if }b=b_{21}\\ 2 & \text{if }b=b_{13}\\ 1 & \text{if }b=b_{31}\\ 0 & \text{otherwise} \end{cases}$$

By describing OMPs as functions, the concept of combinations of OMPs becomes clear. For two OMPs $P_1$ and $P_2$, the combined preference $P_1\oplus P_2$ is defined as:

$$f_{P_1\oplus P_2}:\mathds{B}\mapsto\mathds{N}:b\rightarrow f_{P_1\oplus P_2}(b)=f_{P_1}(b)+f_{P_2}(b)$$

Note that the combination operator $\oplus$ is assumed to be commutative, associative and strictly monotonic ( $P\prec P\oplus P$). The latter assumption is made to better reflect the ideas underpinning conventional utility calculi [1].