Multicriteria Decision Analysis (MCDA)

Stochastic multicriteria acceptability analysis (SMAA) is a collection of multicriteria decision analysis (MCDA) methods that considers user preference as a weight space to rank the proposed projects from most preferred to least preferred. The SMAA algorithms produce two indices, rank acceptability (RA) and central weight vector (CW), that help the user to understand the problem in conjunction with the preference. The RA index describes the combination of user preferences resulting in a certain rank for a project. The CW describes the typical preferences that results in a project being favored in ranking.

The proposed framework prioritizes n different areas of interest (AoIs) based on five conservation goals and 22 data measures (Figure 17), as previously described (see Stakeholder Engagement and Design Decisions). With the user input of n AoI footprints and user preferences/weights for the conservation goals, the CPT will extract 22 ecological and socioeconomic measures (Table 4) from the database and these measures will be analyzed using the SMAA algorithm to produce the decision support suggestions.

Figure 17. Conservation Prioritization Tool architecture.

The SMAA algorithm assumes a discrete decision-making problem where a set of n potential footprints or AoI P= {p1,...,pi,...pn} are assessed based on a set of m criteria C= {c1,…,ci,...cm}. The evaluation of an AoI pi on criterion cj is denoted by cj(pi). The SMAA then ranks n projects based on user preferences represented as m criteria by considering a weight vector w and a real-valued utility function u(pi,w) for i=1,..,n. The decision model is based on the utility function. Each project is represented by a set of measurements corresponding to a priority attribute and its corresponding preference by the user, then the utility function (equation 1) evaluates each project by a real number from 0 to 1. The shape of the utility function decides the evaluation of measurements and corresponding preference. In this work, three types of utility functions were considered: 1) Linear - Higher the better, 2) Linear - Lower the better, and 3) Piecewise Linear - Higher the better.

Equation 1:

$$u(p_i,w)=\sum_{m}^{j=1} (c_j (p_i))w_j$$

User preferences represented as weights is an m dimensional vector from the space defined by W (equation 2). The weights are normalized and considered to be non-negative.

Equation 2:

$$W=w\epsilon R^{m}:w_j\geq 0\ and\ \sum_{m}^{j=1}w_j=1$$

Imprecise and uncertain criteria values are represented as stochastic variables ξij. The users' unknown preferences are represented by a weight distribution with a join density fW(w) in the feasible weight space W. With that SMAA derives RA and CW. For this purpose, a ranking function (equation 3) and rank weights (equation 4) are defined.

Equation 3:

$rank(i,\xi,w)=1+\sum_{k=1}^{n}\rho (u(\xi_k,w))> (u(\xi_i,w))$

where ρ(true) = 1 and ρ(false) = 0.

Equation 4:

$W_{i}^{r}=\left \{ \right.w\epsilon W:rank(i,\xi,w)=r\left. \right \}$

Rank Acceptability

The RA index describes the share of parameter values positioning an AoI xi, with a rank of r (equation 5). The best AoIs are the ones with high RA for the best ranks.

Equation 5:

$b_{i}^{r}=\int_{\xi \epsilon \chi }^{}f_\chi (\xi)\int_{w\epsilon W_{i}^{1}(\xi )}^{}fw(w)\ dw\ d\xi$

Central Weights

Equation 6:

$w_{i}^{k}=\int_{\xi \varepsilon \chi }^{}f_\chi (\xi )\int_{w\epsilon w_{i}^{1}(\xi )}f_W(w)w\ dw\ d\xi /b_{i}^{1}$

The CW is defined as the expected center of gravity of the favorable weight space (equation 6). The CW describes the weights of a user supporting an AoI with the assumed preference model.