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MIT OpenCourseWare http://ocw.mit.edu 1.020 Ecology II: Engineering for Sustainability Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.1.020 Ecology II: Engineering for Sustainability Lectures 08_20 Multiple Objectives, Pareto Optimality Motivation/Objective Develop a way to compare values of different resource uses. Consider tradeoff between using limited water for farm revenue vs. using water for preservation of the riparian ecosystem. Approach 1. Introduce a riparian ecological abundance measure as a second objective in the resource allocation problem of Lectures 8_16 and 08_17. This simplified measure assumes abundance is a linear function of the flow downstream of the irrigation diversion. We would like to maximize both revenue and abundance but these objectives conflict. 2. Include abundance objective as a constraint in the resource allocation problem and use MATLAB to evaluate the tradeoff between revenue and diversity. 3. Display tradeoff as a Pareto frontier (set of Pareto optimal solutions). 4. Consider how tradeoff depends on technological inputs (e.g. water requirement, yield). Concepts and Definitions Needed: Multiobjective optimization problem: )( =xFMaximizerevxRevenue ($) Objective function 1 )( =xFMaximizeabdxEcological abundance (unitless 0-1) Objective function 2 x= vector of decision variables Decision variables Such that following constraints hold for each resource: Resource used (x) ≤ Resource available Inequality constraints Upper and lower bounds on x Inequality constraints Physical constraints (e.g. mass, energy balance) Equality constraints We seek Pareto optimal solutions (solutions where one objective can be improved only at the expense of the other). Other solutions are either infeasible or inferior. For 2 crop example identify Pareto frontier by converting one objective (Objective 2) to a constraint: abdminabdFxF ≥)(.Pareto frontier is vs curve. Points along frontier correspond to particular solutions (x). revFabdminFModify optimization problem of Lecture 08_16 & 08_17 to include upstream water limit and abundance constraint: Maximize iiirevxYpxF∑==21i )( ][21Dxxx =, Area crop i (ha), D = diversion to farm (MCM season=ix-1) iiiixdYY −=0 = nominal yield (tonne ha0iY-1 season-1) =id yield reduction coef (tonne ha-2 season-1) Constraints: (MCM = 106 m3) , =iw Water rqmt crop i (MCM ha-1 season-1 ) 16Supply = upstream flow (MCM seasonUD ≤-1 ) Water: = water diverted to farm (MCM seasonDxwiii≤∑=21-1 ) Land: = land available (ha) availiiLx ≤∑=21 Abundance: abdminFDU ≥− ][β →abdminFUD−≤ββ Nonnegativity: xi ≥ 0 i = 1, 2, 3 Input Arrays for MATLAB (quadprog): Quadprog format: xfHxxxFMinimizeTTrevx+=21)( Find decision variables x that minimize )(xFrev:such that Inequality constraints bAx ≤ Equality constraints eqeqbxA = Lower and upper bound constraints ublbxxx ≤≤For multiobjective problem (converted to minimization problem): ]0[202101YpYpf −= ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=000000022211dpdpH⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=β00011110021wwA ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=abdminavailFULUbβ0=lbx [0 0 0] []===ubeqeqxbA (unused) Make sure that H is a symmetric matrix. Plot of vs gives Pareto frontier. revFabdminF Tradeoff Results Note dependence of tradeoff curve problem inputs (farm inputs, β, etc.)


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MIT 1 020 - Multiple Objectives, Pareto Optimality

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