CIVL4440, Fall 2015 - HOMEWORK 2 DUE DATE: 2 OCT 2015, FRIDAY, DURING LECTURE. Part a (modification of 2-5 from Haith, 1982) Old Nasty is a company operating a plant producing 100 m 3 /day of wastewater that is discharged into Very Clean River. The wastewater contains 1 kg/m 3 of YUK, a toxic substance. The local environmental protection agency has imposed a limit on the plant prohibiting discharge of more than 20 kg/day of YUK. Old Nasty has investigated two methods for reducing its discharge of YUK. Method 1 is land disposal, which costs X 1 2 / 20 in $/day, where X 1 is the volumetric flow of wastewater (m 3 /day) disposed of on the land. With this method, 20% of the YUK applied to the land will eventually drain into the stream, that is, 80% of the YUK is removed from the wastewater and trapped among soil particles. Existing regulations prohibit the addition of more than 60 kg/day of YUK onto the land that will remain on the land indefinitely. Method 2 is a chemical treatment system that costs $1.50 /m 3 of wastewater treated. The chemical treatment system has an efficiency of e = 1.0 – 0.005 X 2 , where X 2 is the amount of wastewater treated, in m 3 /day, using this method. For example, if X 2 = 50 m 3 /day, then e = 1.0 – 0.005(50) = 0.75, that is 75% of the YUK in X 2 is removed. Construct an optimization model to determine the cheapest approach for Old Nasty to treat its discharge of YUK to meet regulatory requirements. Part b (modification of 2-3 from Haith, 1982) The runoff from 100 ha of cropland transports phosphorus into a lake, leading to eutrophication. Three crops are grown on the cropland. Let p i be the amount of phosphorus entering the lake in kg/ha/yr from crop i. For example, if there are 30 ha of each crop, the total phosphorus entering the lake is 30 p 1 + 30 p 2 + 30 p 3 . Due to environmental concerns, the total amount of phosphorus entering the lake must not exceed 1000 kg/yr. The farmers using the 100 ha of cropland require minimum quantities of each crop, L i . The net profit from crop i is R i (X i ) in $/yr where X i is the amount of land in ha dedicated to crop i. Crop i p i (kg/ha/yr) L i (ha) R i (X i ) ($/yr) 1 10 30 1000 X 1 1/2 2 18 10 3000 X 2 1/3 3 9 10 1200 X 3 1/2 Develop an optimization model to find the optimal combination of X 1 , X 2 and X 3 that gives maximum overall profit while meeting all the constraints described above, and any physical constraints not specifically mentioned but relevant. Part c For the problem in part b, show the feasible region on a graph (in other words, the decision space of the problem) with all the constraints included. Show the feasible region for X 3 = 20 ha with the horizontal axis representing X 1 and the vertical axis representing X 2 . From the graph of the feasible region, determine, roughly, the optimal combination of X 1 and X 2 given X 3 = 20 ha. Part d (*** NOT FOR GRADING ***) Consider again the optimization problem from part a. (i) Use the method of Lagrange multipliers to determine a set of simultaneous equations that solves to give stationary points, of which the one resulting in the least objective function value is the optimum of the problem. (ii) If given that the optimum is X 1 = 60 and X 2 = 40, which constraints are binding, and which non-binding? What are the values of the Lagrange multiplier and slack/surplus variable associated with each constraint?

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CIVL4440, Fall 2015 - HOMEWORK 2 DUE DATE: 2 OCT 2015, FRIDAY, DURING LECTURE. Part a (modification of 2-5 from Haith, 1982)

Old Nasty is a company operating a plant producing 100 m3/day of wastewater that is discharged into Very Clean River. The wastewater contains 1 kg/m3 of YUK, a toxic substance. The local environmental protection agency has imposed a limit on the plant prohibiting discharge of more than 20 kg/day of YUK.

Old Nasty has investigated two methods for reducing its discharge of YUK. Method 1 is land disposal, which costs X1

2 / 20 in $/day, where X1 is the volumetric flow of wastewater (m3/day) disposed of on the land. With this method, 20% of the YUK applied to the land will eventually drain into the stream, that is, 80% of the YUK is removed from the wastewater and trapped among soil particles. Existing regulations prohibit the addition of more than 60 kg/day of YUK onto the land that will remain on the land indefinitely.

Method 2 is a chemical treatment system that costs $1.50 /m3 of wastewater treated. The chemical treatment system has an efficiency of e = 1.0 – 0.005 X2, where X2 is the amount of wastewater treated, in m3/day, using this method. For example, if X2 = 50 m3/day, then e = 1.0 – 0.005(50) = 0.75, that is 75% of the YUK in X2 is removed.

Construct an optimization model to determine the cheapest approach for Old Nasty to treat its discharge of YUK to meet regulatory requirements. Part b (modification of 2-3 from Haith, 1982)

The runoff from 100 ha of cropland transports phosphorus into a lake, leading to eutrophication. Three crops are grown on the cropland. Let pi be the amount of phosphorus entering the lake in kg/ha/yr from crop i. For example, if there are 30 ha of each crop, the total phosphorus entering the lake is 30 p1 + 30 p2 + 30 p3. Due to environmental concerns, the total amount of phosphorus entering the lake must not exceed 1000 kg/yr. The farmers using the 100 ha of cropland require minimum quantities of each crop, Li. The net profit from crop i is Ri (X i) in $/yr where Xi is the amount of land in ha dedicated to crop i.

Crop i pi (kg/ha/yr) Li (ha) Ri (Xi) ($/yr)

1 10 30 1000 X1 1/2

2 18 10 3000 X2 1/3

3 9 10 1200 X3 1/2

Develop an optimization model to find the optimal combination of X1, X2 and X3 that gives maximum overall profit while meeting all the constraints described above, and any physical constraints not specifically mentioned but relevant. Part c

For the problem in part b, show the feasible region on a graph (in other words, the decision space of the problem) with all the constraints included. Show the feasible region for X3 = 20 ha with the horizontal axis representing X1 and the vertical axis representing X2. From the graph of the feasible region, determine, roughly, the optimal combination of X1 and X2 given X3 = 20 ha. Part d (*** NOT FOR GRADING ***)

Consider again the optimization problem from part a. (i) Use the method of Lagrange multipliers to determine a set of simultaneous equations that solves to give stationary points, of which the one resulting in the least objective function value is the optimum of the problem. (ii) If given that the optimum is X1 = 60 and X2 = 40, which constraints are binding, and which non-binding? What are the values of the Lagrange multiplier and slack/surplus variable associated with each constraint?