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.MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering 1.020 Ecology II: Engineering for Sustainability Problem Set 6 – Resource Allocation Due: 5PM Weds. April 30, 2008 A. Introduction The objective of this problem is to allocate agricultural water to maximize profits, subject to environmental constraints. B. Problem Description There are 3 farms, all growing rice. Water is allocated by an irrigation district to each farm to maximize district income (over all 3 farms) The maximum land available for cultivation is specified for each farm. Yield is enhanced by fertilizer application. Fertilizer cost is negligible Total nitrogen runoff must not exceed a specified threshold. Variable definitions: 3 Objective: Maximize F =∑ pYi Li = Net district revenue ($ season-1) i=1 p = Rice price ($ tonnes-1) Li = Crop area for Farm i (ha) Yi = Y0i +γ iFi = Net yield Farm i (tonnes ha-1 season-1) Y0i = Nominal yield Farm i (tonnes ha-1 season-1) γ i = Fertilizer enhanced yield coefficient for Farm i (tonnes crop (kg fertilizer)-1) Fi = Amount of fertilizer applied to Farm i (kg ha-1 season-1) Ni =ηi Fi Lmax i = Nitrogen runoff from Farm i (kg season-1) Lmax i = Maximum land area Farm i [Note: As a simplification, above expression assumes fertilizer is applied to entire farm] ηi = Fraction of applied nitrogen that runs off Farm i (unitless) 3 R =∑ Ni = Total nitrogen runoff (kg season-1) i= 1 3 W =∑Wi Li = Total water used (MCM season-1) i=1 Wi= Unit water requirement Farm i (MCM ha-1 season-1) 1Resource and environmental constraints: Water: WavailW ≤ Land: iL iLmax≤ for each Farm i Nitrogen runoff: maxRR ≤ C. Inputs Price p = 200 $ tonne-1 Nominal rice yield Y0i tonnes ha-1 Farm1 Farm2 Farm3 100 70 90 Fertilizer coefficient γ i : tonnes kg-1 Farm1 Farm2 Farm3 0.9 1.2 1.1 Nitrate runoff fraction ηi : unitless Farm1 Farm2 Farm3 0.4 0.35 0.45 Water requirement Wi : m season-1 Farm1 Farm2 Farm3 1.0 1.2 0.9 Resource limits Resource Symbol Value Units Water Wavail 2.4 MCM season-1 Land Farm1 max,1L 200 ha Land Farm2 max,2L 100 ha Land Farm2 max,3L 150 ha N Runoff Limit maxR 0-5000 kg season-1 2D. Problem Set Tasks *** All of the information requested below should be compiled in a single MS Word (or equivalent) file with all your team member names clearly identified in the file name and submitted via the 1.020 Stellar site *** 1. Use MATLAB’s quadprog function to find the set of crop land areas and amount of fertilizer applied that maximizes revenue when Rmax= 5000 kg season-1. [HINT: Make sure that you include all land, water, and nitrogen runoff constraints. Also, make sure that the Hessian matrix you construct is symmetric]. 2. Determine the increase in revenue obtained if Rmax is reduced. Do this by plotting 1) the revenue and 2) the shadow price of the nitrogen runoff constraint vs. Rmax over the range Rmax= 0 to Rmax= 5000. Use the MATLAB function subplot to put the two plots in the same figure window. 3. Explain why the shadow price of the runoff constraint increases as the maximum permitted nitrogen increases [Hint: Examine the fertilizer yield coefficients as well as changes in cropland allocations to see how the benefits of fertilizer increase as greater amounts are permitted].