Ecological benefit quantification based on numerous performance measures is a nontrivial issue. There is a plethora of performance measures (PMs) and habitat suitability indices that have evolved for Everglades restoration. These lead to a very high dimensional optimization problem that is conceptually and mathematically intractable. The simplicity of just trying to attain the predicted Natural System Model (NSM) stage given current conditions and projected rainfall is consequently, very attractive as a yardstick for developing the seasonal or real time optimization plans. However, particularly for real time operation, this strategy may not guarantee improvement of real outcomes in terms of the various PMs and Habitat Suitability Indices (HSIs). On the other hand, trade-offs between and interpretation of the very large number of state variables/targets/ PMs are a concern.
Analytic Hierarchy Process
Analytic Hierarchy Process (AHP) is a quantitative technique of subjective judgments (Saaty, 1980; Saaty and Alexander, 1981). AHP have won popularity in the cases where multi-objectives (i.e., multiple criteria or multi-attribute decision making) have to be satisfied thus AHP provides means for decision making when there are several alternatives.
Let be the number of items (i.e., performance measures) being considered with the goal of providing and quantifying judgments on the relative weight, importance, and priority of each item with respect to the rest. One has to set the problem in hand as a hierarchy where the overall objective of the restorations plans is the topmost node. Subsequent nodes represent criteria and sub-objectives used in arriving at this decision.
The bottom level of the hierarchy is composed of the alternatives, from which the benefits have to be quantified and the choice has to be made which are the items we need to elicit. Then, pair-wise comparisons are carried out between each two items with respect to their contribution towards the factor from the level immediately upper level. For instance given elements and stakeholders or decision makers need to decide which is more important with respect to the given factor and the level of importance. The preference strength is expressed on a ratio scale of 1-9 (Saaty, 1980). The resulting matrix of the pairwise comparison is necessarily reciprocal with and . If judgments are completely consistent then just by providing the first row of; one could deduce the rest of the elements because of the transitivity of the relative importance of the items. Practically consistency is not assumed and the process of comparison for each column of the matrix is carried out. Suppose that at the end of the comparisons, we have filled the matrix A with the exact relative weights; if we multiply the matrix with the vector of preferences we obtain:
or (1).
To recover the overall score from the matrix of ratios the homogeneous linear equations from (1) must be solved. Since is an eigenvalue of thus we would have a nontrivial solution, unique to within a multiplicative constant, with all positive entries. Since we don’t start with precise judgments of the rations in thebut estimates of them which is different than the actual weights’ ration.
The small perturbation in the coefficient implies perturbations in the resulting eigenvalues. Again, from the nature of the matrix the maximum eigenvalue has to equal and its corresponding eigenvector gives a unique estimate of the underlying ratio scale between the elements of the studied case. The consistency index could be easily measured to evaluate the degree of uniqueness in the solution and it measure the error due to inconsistency. In literature there are some threshold consistency ratios that could be estimated and they provide means for recommendation regarding the pair-wise comparison matrix. This consistency ratio CR simply reflects the consistency of the pair-wise judgments and shows the degree to which various sets of importance relativities can be reconciled into a single set of weights and to verify the goodness of the judgments (Satty, 1981; Ahti and Raimo, 2001). This process is described for only one level in the hierarchy. Hierarchical composition should be used to combine all the levels. The sum is taken over all weighted eigenvector entries corresponding to those in the lower level, and so on, resulting in a global priority vector for the lowest level of the hierarchy.
The Limitations of AHP includes the following:
1. It is not robust to errors in the pair-wise comparison.
2. It can not be tested for statistical significance.
3. Variation due to uncertainty is not contemplated, analyzed or quantified.
4. The preference scale in the comparisons is being made within a limited range to allow perception to be sensitive to make a distinction and it doesn’t account to managerial aspects such as importance of alternatives and the selection of the subject-matter experts.
In order to overcome the limitations of AHP we propose to utilize a modified version of AHP. In addition, in an effort to answer the concern of how could ecological performance measures be incorporated into decision support tools that have reasonably simple output formats, here we propose a framework to judge project components and policies based in probabilistic preference elicitation (See Figure 1). This is done in the following four stages:
1. Planning
a. Problem recognition
b. Perform PCA to evaluate the interaction and relationships between PMs. In reviewing the Interim Operational Plan (IOP) and HSI reports
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Figure 1: Integrated framework for ecological benefit quantification.
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c. from the ENP and the SFWMD we have noted that at a given location, and for a given time period, many of the proposed PMs, HSIs and water levels are highly correlated. Indeed, many of the PMs and HSI's are monotonic functions of the same target variable (e.g., hydroperiod duration). To include spatial performance measure in PCA one may resort to aggregate the spatial patterns into a unified measure of the spatial heterogeneity –one has to think about indicative ways of doing it.
d. Select the important PMs for further analysis in light of the results of PCA.
e. Select a group of subject matter experts. This multi-disciplinary team is required to assign and asses the preferences in the selected PMs. Here efforts should be made to assess different alternatives of discernments. Many interested and qualified parties should be incorporated in this process to later provide probabilistic analysis.
f. Refine the preference assignment to attain conformity with PCA. For instance enhancement of a PC that carries positive and negative weights implies that there is deterioration of another PM. By such analyses it will be easy to spot the orthogonal PMs.
Then Pareto dominance concept of multiobjective optimization could be used as a weighing scheme (Figure 2). This weighing will guarantee that there is no counter-interaction between the performance measures.
Figure 2: Multiobjective optimization based on Pareto dominance concept.
As illustrated in the plot, high preference of solution #1 favors PMi and deteriorates PMj and vice versa with high preference of solution #2. In the case of conflicting PMs the best preference assignment technique is the one that favors both with no counter-interaction which is #3. The same could be carried out in any high dimensions. So that will be one benefit of principal components analysis.
g. Define scope and boundaries for AHP: define restoration alternatives and objectives…etc.
h. Decompose the problem into hierarchy and strategy analysis: once PCA is provided to structure the relations between PMs. Brainstorming between decision makers to prioritize the objectives and the alternatives and their associated measure of performance. In other words, because projects address a wide range of objectives (i.e., multipurpose) an initial selection of pertinent ecological objectives must be made at the beginning of the process. Only those that provide ecological benefits are included in this analysis. Affinity diagrams to organize the resulting grouping. In the strategy analysis will enable accounting for multiobjective aspects and the decision makers’ acceptance of the desired state. In other words, incorporation of new information will eventually provide improved information to be used by the project managers to better predict the ecological benefits or adverse affects of management scenarios. This might be critical to ensure consistent decision making by the many stakeholders working to restore ecological sustainability.
2. Simulation
a. Perform pairwise comparison: this could be done at different levels using the preferential judgment of the mater-experts according to their belief and personal experience. The pairwise comparison indicates how much more important ith PM than jth performance measure. This will lead to the construction of the composite priority vector of restoration plans’ importance.
b. Define probability distribution and probability matrix: people perception changes from one to another and the ambiguity with judgments result in a significant uncertainty. This is carried out to address the fact that asking people to give preference weights to so many PMs is a difficult task and it involves randomness. The Probability matrix is reciprocal and square matrix of distributions of the pairwise preference ratio determined in the previous step.
c. Replicate the eigenvectors: AHP estimated the eigenvector of one realization of the decision hierarchy to produce the priority vector. Monte Carol type of simulation for this step will result in an estimate of the probabilities associated with the composite priorities. Consequently statistical tools could be used to select the most appropriate measure.
d. Consistency index: to test the reliability of the analysis. There is a theoretical upper bound threshold between consistency of a reciprocal matrix and its corresponding average random consistency.
e. Sensitivity analysis: to understand the source of variation by estimating the overall contribution of the variance of the probabilistic judgment to the variance of the consistency index. With this sensitivity analysis is possible to judge the influence of the preferential judgments on the ranking of the PMs. One can rank the benefits of each PM according to the impact of their associated preference on the consistency index. The product of such analysis is a chart depicting which judgment is the most important in the quantification of the benefits. This is used to further investigate the quality of the assigned judgments and to discard or modify it if necessary. If one to obtain equity and fairness the decision makers judgments have to have equal importance. Providing that backpropagation of uncertainty kind of analysis might be used here.
3. Analysis
a. confirmation Confidence interval analysis
b. Analysis of variance
c. Visualize results of “what if” scenarios and the sensitivity analysis.
4. Confirmation and quantification
a. Confirmation of the results
b. Final decision on the quantification, importance, and interaction of PMs.
References:
Saaty, T.L.,1980. The Analytic Hierarchy Process. McGraw Hill, New York.
Saaty, T.L., and J. M. Alexander, 1981. Thinking with Models. Pergamon Press.
Banuelasy, R. and J. Antony, 2004. Modified analytic hierarchy process to incorporate uncertainty and managerial aspects. International Journal of Production Research, 42 (18), pp: 3851–3872.
Ahti A. S. and P. H. Raimo, 2001. Preference Ratios in Multiattribute Evaluation (PRIME)—Elicitation and Decision Procedures Under Incomplete Information. IEEE Transactions on Systems, Man, and Cybernetics, 31 (6).
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