CVEN 5393 – April 1, 2013

Topics Revisit multi-objective optimization

Heuristics and Metaheuristics – a conceptual overview

Multi-criteria decision analysis

Multiobjective Optimization Topics

• Multiple objectives multiple solutions • Domination and Pareto Optimality

(noninferior set) • Methods of solving

– Scaling (reduce to multiple single opt problems) – Goal Programming – Heuristics and Meta-heuristics

Pareto optimality

Noninferior == Non-dominated == Pareto Optimal solution: 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. Pareto Front – set of all possible pareto-optimal solutions Multiobjective Optimization aims to deduce the pareto front or a “near” pareto front. (Decision Makers use decision analysis techniques to identify the most desireable solution)

Example problem with many solutions • 2 objectives for a cantilever to carry a given force:

minimize weight and minimize deflection • Decision variables: length; diameter (continuous; many values) • Constraints: stress <= max; cantilever equations

Goals of the MOO Solution:

• Converge on the pareto optimal front • Maintain as diverse a distribution as possible

Solution Methods: Classical

“Scaling” approach: reduce to many single objective problems Weighting method and Constraint method Problems: • Don’t have good control over the objective space

searched • Approach will not work on non-convex parts of

the tradeoff curve • These work mainly for well-behaved problems

with continuous, differentiable constraints, objectives

Simulation and optimisation modelling approach for operation of the Hoa Binh reservoir, Vietnam by Long Le Ngo, Henrik Madsen, Dan Rosbjerg, J. of Hydrology (2007)

Non-smooth pareto front indicates variables that are not continuous as may be seen in complex operational models (this is result of “simulation-based optimization”

The balanced optimum typically has both objectives changing at close to the same rate

“Difficult” optimization problems are characterized by:

• Many decision variables • discontinuous, combinatorial • Objective difficult to calculate • Severely constrained • Feasibility difficult to determine • Multiple objectives • Stochastic variables In such cases, the usual mathematical programming approaches are not likely to be successful. Instead use…

Heuristics – a conceptual overview Heuristic optimization algorithms are artificial intelligence search methods that can be used to find the optimal decisions for designing or managing a wide range of complex systems. A heuristic technique is a method which seeks good (i.e.. near optimal) solutions at a reasonable computational cost without being able to guarantee optimality. • For many problems there is no feasible way to guarantee

finding the optimal solution. • Heuristics are a very useful tool for finding good (near

optimal) solutions.

Heuristic Terminology Search is the term used for constructing/improving solutions to obtain the optimum or near-optimum. Solution Encoding (representing the solution) (e.g. a set of values of decision variables)

Neighborhood Nearby solutions Move Transforming current solution to another

(usually neighboring) solution

Evaluation The solutions’ feasibility and objective function value

Heuristic Terminology • Heuristics can be constructive (build a solution piece by piece) or

improvement (take a solution and alter it to find a better solution).

• Many constructive heuristics are greedy or myopic, that is, they take the best thing next without regard for the rest of the solution.

• Search techniques may be deterministic - always arrive at the same final solution through the same sequence of solutions, although they may depend on the initial solution (e.g., FIFO, LIFO, Tabu search, greedy heuristics) or stochastic where the solutions considered and their order are different depending on random variables (e.g., simulated annealing, ant colony optimization and genetic algorithms.

• Search techniques may be local - they find the nearest optimum which may not be the real optimum (e.g., greedy heuristic (local optimizers), or global - they find the true optimum even if it involves moving to worst solutions during search (non-greedy).

Meta-Heuristics

An iterative generation process which guides a subordinate heuristic by combining intelligently different concepts derived from classical heuristics, artificial intelligence, biological evolution, natural and physical sciences for exploring and exploiting the search spaces using learning strategies to structure information in order to find efficiently near-optimal solutions.

- Osman and Kelly, 1996

Meta-heuristics inspired by nature

• Simulated Annealing (SA) – molecular/crystal arrangement during cool down

• Genetic Algorithms (GA) – biological evolution • Tabu Search (TS) – long and short term

memory • Particle Swarm Optimization (PSO) – individual

and group behavior using communication between agents

Advantages of Meta-heuristics • Very flexible • Often global optimizers • Often robust to problem size, problem instance and

random variables • May be only practical alternative

Disadvantages of Meta-heuristics • Often need problem specific information / techniques • Optimality (convergence) may not be guaranteed • Lack of theoretic basis • Different searches may yield different solutions to the

same problem (stochastic) • Stopping criteria • Multiple search parameters

Classifications of metaheuristics http://en.wikipedia.org/wiki/Metaheuristic

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 conditional logical steps, logical operating rules and perhaps the objectives are computed by post-processing algorithms like a habitat simulation model? How can we find the “best” solution or set of solutions?

Simulation-based optimization

Water Resources Model Physical processes, operating rules and policies, Power generation projected changes in supply and demand; hydrology (historic, climate change, stochastic) Costs and benefits – algorithms to compute the values of performance criteria

Decision Variables Reservoir and structure sizes Hydroplant capacities Operating policy Demand schedule (crops, etc.) Development schedule Constraints on variables

Performance Criteria (values of objectives) Reliability of Supply Environmental compliance or benefits Energy produced; reliability of energy Flood damage reductions Benefit/Cost ratio

Search algorithm: look for values of decision variables for best outcome of objectives

Pareto Optimal Set

Multi-objective decision-making

From notes from Shoemaker CS5722 MIT 2011

Multi-criteria decision making

Modeling Activities

The decision-making process

Multi-Criteria Decision Analysis Tools

• Define objectives • Generate alternatives via MOO OR Stakeholder meetings • 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

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.

Multi-criteria Analysis (MCA) MCA is a decision model that contains: 1. A set of decision options which need to be ranked or scored by the

decision maker 2. A set of criteria, typically measured in different units; and 3. 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

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 ……

MCA Model Structure The MCA model is represented by an evaluation matrix X

of 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.

Types of MCA Techniques 1. 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)

2. 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.

Types of MCA Techniques 3. 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).

4. 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.

5. 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.

Hajkowicz and Collins, 2006

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

1/9 A is absolutely less important than B 9 A is absolutely more important than B

Eigenvalues

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.

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.

Determine Values of Attributes 1. Use Interval standardization to transform the raw values, xij, to to unit-

less, standardized valued sij for the jth attribute for the ith alternative.

2. 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.

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.

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.

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.