MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering.7 Water Resource Systems Lecture 5 Multiobjective Optimization and Utility Oct., 006 Multiobjective problems Benefits and costs are often incommensurate (measured in different units) are they may accrue to different parties (equity issues): Eamples: Benefits Costs Hydropower output (MWhrs, $) Loss of species habitats or recreational opportunities (Units???) Additional crop revenues Reduced crop revenues for for upstream farmers benefiting downstream farmers with from a water diversion ($) less water ($) Information provided by Sampling cost ($) a field monitoring program (Units??) Multiobjective analysis recognizes this by revealing tradeoffs among different objectives. Etension of the crop allocation eample Etend previous eample by considering objectives maimization of crop revenue and minimization of pesticide concentration in groundwater: Decision variables: = mass of Crop grown (tonnes = 0 kg) = mass of Crop grown (tonnes = 0 kg)

2 Maimize, Minimize, such that : Crop revenue ($) Pesticide concentration in groundwater (ppm) Water constraint (0 Land constraint (ha) Minimum production constraint (tonnes) m All constraints and the feasible region are the same as before. /season) It is convenient to transform the problem so that both objectives are maimized. Call the negative of pesticide concentration environmental quality : Maimize, Maimize, that : F (, ) = 6 + Crop revenue ($) F (, ) = - 5 Water constraint (0 Land constraint (ha) Minimum production constraint (tonnes) Environmental quality (-ppm) m /season) There is a tradeoff between the revenue and environmental quality objectives : As and/or increases crop revenue increases environmental quality decreases (and vice versa) F (, ) (5, 0) Inferior (0, 0) (0,5) (40, 5) (, ) (0, 8) (40, 6) Infeasible Pareto frontier F

3 The nature of the tradeoff is revealed in plot of F vs F : Each feasible solution corresponds to a single point in the F - F plane. If a solution is inferior it is possible to increase one of the objectives without decreasing the other. Non-inferior (Pareto optimal) solutions lie on the Pareto frontier which forms a boundary separating inferior and infeasible solutions. Different Pareto optimal solutions represent different tradeoffs between the two objectives if one objective is increased by moving to another Pareto solution the other objective cannot increase (and usually decreases). How can we identify the Pareto frontier in general? Best alternative is usually to carry out a parametric analysis: Treat all but one objective (F i, i =,, N) in an N-objective problem as constraints with specified right-hand values for F,, F N. Maimize the remaining objective F. As the right-hand side values F,, F N are changed the solutions trace out the Pareto frontier. In the eample, treat crop production objective as a constraint and maimize environmental quality F as a function of F : Maimize, such that : F (, - - F ) = - 5 Crop production must be at least F Water constraint (0 Land constraint (ha) Minimum production constraint (tonnes) m /season) The Pareto frontier can be obtained in GAMS by solving the above problem in a loop which varies F from 75 (the minimum feasible Pareto value) to 440 (the maimum feasible Pareto value). Same result is obtained if we treat environmental quality as a constraint and maimize crop production F as a function of F. Above concepts apply equally well to nonlinear and discrete multi-objective optimization problems.

4 Different types of tradeoffs: F Here small improvements in environmental quality have a large adverse impact on F Here small improvements in revenue have a large adverse impact on environmental Knee looks like best compromise Tradeoff is the same No obvious compromise! Utility F F Tradeoff curves do not tell us which Pareto optimal solution to adopt. One approach for finding a single optimum solution is to identify a utility (or preference) function. The utility function defines combinations of F, F,, F N values that a particular party (individual, group, etc.) finds equally acceptable. Contours of constant utility are called indifference curves. F Indifference curves (contours of equal utility) Increasing utility Pareto frontier Maimum utility Pareto solution Pareto curve can be viewed as an equality constraint in a new optimization problem where we seek to maimize utility. Then maimum utility solution lies at the point where the gradients to the utility function and Pareto frontier constraint point in the same direction. Utility functions are difficult to measure, although economists have developed indirect ways to estimate them from surveys. F A typical eample of a two-objective utility function U ( F, F ) the Cobbs-Douglas function: that may be fit to survey data is 4

5 α β U ( F, F ) = F F where α and β are specified (or fit) non-negative coefficients The dependence of the utility function on any given objective value is typically nonlinear. Utility and Risk For the crop allocation eample, consider the dependence of utility on revenue F for fied environmental quality F. To eamine effects of uncertain F epand U(F ) in a Taylor series around mean revenue F : U U U ( F ) = U ( F ) + ( F F ) + ( F F ) F F Mean of this epression is: + Κ U U ( F ) = U ( F ) + σ +Κ where σ = variance of F F F F F F When there is no uncertainty: σ = 0 U ( F ) = U ( F ). When there is uncertainty: sign of U / F. Three possibilities: σ > 0 relationship between U ( F ) and U ( F ) depends on Risk averse: U(F ) is concave, U / F < 0 mean utility is lower when F is uncertain (risk lowers utility) Risk neutral: U(F ) is linear, U / F = 0 mean utility is the same when F is uncertain (risk has no effect on utility) Risk seeking: U(F ) is conve, U / F > 0 mean utility is higher when F is uncertain (risk raises utility). Utility is often a concave function of revenue (decision-maker is risk averse) for sufficiently large revenue. In the crop allocation eample this could reflect the fact that the marginal utility gained by having more revenue gradually decreases as environmental quality declines. 5

6 Eample: Consider a risk adverse farmer faced with uncertain revenue because of uncertainty in the farm water supply. F has possible values F ± δf, each with probability =. U Average utility when revenue is certain Average utility when revenue is uncertain Concave utility function (risk averse) Uncertainty lowers average utility F δf F F + δf F Probabilities: certain values uncertain values Suppose the (concave) utility function for this risk adverse farmer is U ( F ) = ln( F ). The farmer can sell a crop option for a price P before the growing season starts. The option guarantees the farmer revenue P. The actual value of the crop is either F + δf or F δf, depending on uncertain water availability. What price is the farmer willing to accept for the option? Suppose F = $ 000, δ F = $ 00 If farmer sells the option for price P the mean (certain) utility is U ( F ) = ln( P) If farmer does not sell the option and accepts risk the mean utility is: U F ) = ln( F + δ F) + ln( F δf) = ( = Equate these two mean utilities and solve for P: P = ep( 6.89) = $ So the farmer is willing to sell the crop option for P = $98.40 rather to obtain epected revenue of $000. The risk premium is $7.60. If the farmer is risk neutral he would require that P = $000 and the risk premium would be zero. 6