CVEN 5393 – April 25, 2011

1. CVEN 5393 – April 25, 2011 Multi-objective optimization Systems Approach to Decision-making Multi-criteria Analysis Techniques Graphical Policy Analysis Tool Exam, presentations, etc.

2. Multi-objective Optimizationrefer to Revelle et al., Chapter 5 Several objectives which could be conflicting: Maximize reservoir.storage (for water supply) and Minimize reservoir.storage (for flood control) Or, a non-commensurate example: Maximize economic efficiency and Maximize environmental benefits The multi-objective analysis is to understand the tradeoffs among objectives and to find the set of solutions for which we can demonstrate that no better solutions exist. Best available set of solutions is call the set of noninferior solutions.

3. noninferior solutions A solution to a problem having multiple and conflicting objectives is noninferior if there exists no other feasible solution with better performance with respect to any one objective, without having worse performance in at least one other objective. Also called nondominatedorpareto optimal set

4. Method to find the noninferior set: Graphical Method

5. Northeast Corner Rule: A feasible solution to a 2-objective (maximization) opt problem is noninferior if there does not exist a feasible solution in the northeast corner of a quadrant centered at that point.

6. Method to find the noninferior set: Weighting Method

10. Constraint MethodSee Revelle et al. Chp. 5 133-138 Another method for generating the noninferior set in objective space: The constraint method. After solving p individual models to identify the solution that optimizes each, one objective function is selected to be optimized, with the others included in the constraint set with RHS set so as to restrain the value of the objective function that was selected for optimization. By iteratively solving this modified formulation and because, as with the weighting method, each solution to the modified problem is a noninferior solution to the original problem, an approximation of the noninferior set in objective space can be generated.

11. Complex water resources problems What if our problem is not algebraic? Instead the relationships among the decision variables are defined by a complex physical process model with many successive logical step, and possibly also includes logical operating rules and perhaps the objectives are computed by post-processing algorithms like a habitat simulation model? How can we handle that?

12. Systems Approach to Decisions Modeling Activities Multi-Criteria DecisionAnalysis Tools • Define objectives • Generate alternatives • Identify performance criteria and define measurement of these[quantitative and/or qualitative] • Obtain values of performance criteria for each alternative • Evaluation the alternatives • Transform into commenurate units • Weight the criteria • Rank or score the options • Sensitivity analysis to understand robustness of results • Make a decision

13. Bruen 2002

14. Multi-criteria Analysis (MCA) • Water resource management decisions are typically guided by multiple objectives measured in different units. MCA represents a body of techniques potentially capable of improving the transparency , auditability and analytic rigour of these decisions. • The MCA framework ranks or scores the performance of alternative decision options against multiple criteria which are typically measured in different units. • MCA emerged as a decision analysis technique in the 1960s and 1970s, partly resulting from the rapid growth of operations research in WWII. Some early applications of MCA were in military planning. Today MCA is an established methodology with dozens of books, thousands of applications, dedicated scientific journal, software packages and university courses. • It has received particular attention in water resource management, partly because water policy is seldom guided by a single objective.

15. Multi-criteria Analysis (MCA) MCA is a decision model that contains: • A set of decision options which need to be ranked or scored by the decision maker • A set of criteria, typically measured in different units; and • A set of performance measures, which are the raw scores for each decision option against each criterion. Example: To address anticipated population increases, a city is considering increasing future water supplies by: A: raise height of dam on existing reservoir to withdraw more from river B: install a number of local water re-use treatment plants and distribution C: Install wells and pump to increase the supply

16. MCA Example: 3 alternatives (raise dam, water reuse, gw pumping) • Criteria: Economic benefits Water Supply security Environmental protection Public Health Social Impacts • Performance Measures (Indicators) – non-commensurate Capital costs Maintenance costs Reliability of supply Risk of system failures Quality of water delivered; risk to drinking water quality Effects on fish Effects on birds Effects on riparian vegetation Effects on recreation (fishing, boating) Public perception and acceptance ……

17. MCA Model Structure The MCA model is represented by an evaluation matrix Xof n decision options and m criteria xij is the raw performance score for decision option i with respect to criterion j W is a vector containing m weights; wj denotes the weight assigned to the jth criterion The MCA algorithms will define, by some means, one or both of these functions: ri = f1(X, W) ui = f2(X,W) Where ri is the rank of decision option i and ui is the overall performance score of option i. The nature of f1 and f2 varies with the different MCA techniques.

18. Types of MCA Techniques • Multi-criteria value functions (Multi-attribute Utility Theory) Evaluate each alternative as the sum of the weighted performance scores. ui is the overall performance score of the ith alternative. The weights wj sum to 1 and vij is a transformed performance score for xij on a scale of 0 to 1 where 1 represents best performance. (or weighted product) • Pairwise comparisons, esp Analytic Hierarchy Processs (AHP) Decompose problem into a hierarch of smaller problems, comparing criteria and alternatives in every unique pair. The comparisons can be made to attain criteria weights and decision option performance scores.

19. Types of MCA Techniques • Distance to ideal point methods Identify ideal and anti-ideal values for the criteria. They then identify the decision options that are closest to the ideal and furthest from the anti-ideal (or can substitute the min and max criterion values). Often used methods in water resources are Compromise Programming and TOPSIS (Technique for Order-Preference by Similarity to Ideal Solution). • Outranking Approaches Involve identifying every pair of decision options i and i′ and apply some type of utility function, which contains criteria weights, to determine the amount option i outperforms i′. Widely used methods/software using this approach are PROMETHEE and ELECTRE. • Fuzzy Set Analysis Fuzzy set theory is based on a gradual transition from one class to another. Items can have partial membership in multiple sets. This can be particularly powerful in handling uncertainty inherent in MCA problems. Fuzzy approaches may apply concept from the other MCA methods.

20. Hajkowicz and Collins, 2006

21. Example: AHP http://en.wikipedia.org/wiki/Talk:Analytic_Hierarchy_Process/Example_Leader

23. 1/9  A is absolutely less important than B 9  A is absolutely more important than B

24. Eigenvalues

27. Multi-attribute evaluation of ecosystem management for the Missouri River system (Prato, T., 2003 Ecological Economics 45, p.297-309) CWCP – Current Water Control Policy establishes guidelines for water releases from the six mainstem reservoirs to balance flood control, navigation, irrigation, hydropower, water supply, WQ, recreation and F&W. Alternatives: MCP – Modified Conservation Plan: adaptive management; increased drought conservation measures, Ft. Peck dam release changes, unbalancing upper 3 reservoirs GPA, GPB, GPC and GPD All 4 of these alternatives incorporate all MCP options plus a range of special flow patterns from Gavin’s Point Dam designed to improve habitat for threatened fish and birds. Each of these 4 alternatives has a different pattern of spring rise for sturgeon spawning and lower summer flows for tern and plover nesting and shallow water habitat for young pallid sturgeon.

29. Attribute Weights Attribute weights cannot be agreed on by very large groups of stakeholders with diverse interests, but is possible to reach consensus on weights for relatively small groups. Several weighting schemes can be maintained and the preferences can be voted on or commented on later during the public comment period of the EIS.

31. Determine Values of Attributes • Use Interval standardization to transform the raw values, xij, to to unit-less, standardized valued sijfor the jth attribute for the ith alternative. • Utility scores, vi, for each of the n alternatives are calculated using a linear additive utility function: Where m is number of attributes, Wj is the weight for the jth attribute, 0 ≤ wj ≤ 1 and sum of weights for all the attributes = 1.

32. Estimation of Attributes The attributes for each alternative were estimated using: • River System modeling • Stream habitat models • Water quality models • Indices of biological integrity • Crop and soil models Values of attributes were expressed as a percentage deviation from the corresponding value of the attribute for the CWCP. For example relative attributes for the 5 management alternatives range from -32% for navigation to 74% for tern and plover habitat. A negative value indicates the attribute is lower and a positive value indicates the attribute is higher with the alternative than with CWCP.

34. Relative utility scores are calculated by substituting the relative attributes for an alternative and hypothetical attribute weights into the utility function. Results are shown above. A positive score implies the alternative is preferred to CWCP and a negative score implies CWCP is preferred to the alternative.

35. Reasons to use MCA in Water Management and Planning • Transparency and accountability to procedures • Conflict resolution (objective decision tool) • Stakeholder engagement (inclusion of stakeholder views and values is informative) • MCA uses formal axioms of decision theory to inform choice – helps ensure the analysis is logical and robust. Research shows that methods are equally good and give similar results.

36. Decision-Making Under Uncertainty

37. 1-hour Exam next Tuesday at 2pm – more information will be emailed to you • Projects – questions? • Presentations Sat April 30 1:00-4:30pm • ½ hour per presentation including questions • You will be graded on listening and asking questions as well as your own presentation • Turn in presentation and all supporting data, models, etc. • You will fill out a short questionaire grading your team members • Online course evaluations • Other questions?