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Samuel Sandoval Solis, PhDAssistant Professor

University of California, DavisDepartment of Land, Air and Water Resources

Optimization

ESM-121 Water Science and Management

Presentation 5 of 6

Allocate reservoir release Rt to 3 users and provide instream flow Qt

Operating Policy Allocation Policy

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Operating Policy

Allocation Policy

• Address “What if …” questions

• What will likely happen• Include larger hyd, econ,

and env. data• i.e. “evaluate change given

a design or policy”

• Best answer: - “Maximize the Net Benefits …” or - “Minimize the shortages”

• Look for the best (ideal) operation

• Perfect foresight

x1t x2t x3t Rt t 1,2,

Maximize Bi (xit )i1

3

t1

T

St1 St It Rt t 1,2,

St K t 1,2,

Optimization model

Constraints

ObjectiveBenefits:

Decision Variables:

Bi (xit )

xit

Optimization model

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• Simulation models: Predict response to given design• Optimization models: Identify optimal design or

operation

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0

subject to

minimize

x

cbAx

X

)f(

x

x

xObjective function

Decision variables

Constraint set

x

f(x)

x*

minimum

Find the decision variables, x, that optimize (maximize or minimize) an objective function

x*

x

f(x)

Xa b

X={x: a<x< b}Feasible region

1. Identify the Objective Function: Maximize or Minimize?

2. Define Objective Function (Write the Obj. Funct. Eq.)

3. Define the Constraints (Write the Constraints Eqs.)

4. Define the Feasible Region

5. Obtain the vertices of the feasible region

6. Substitute vertices in the Objective Function

7. Select the data where the value is Maximized/Minimized

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Objective: Maximize the Profits

Irrigation project

1800 acre-feet of water per year

Decision variables

xA = acres of Crop A to plant?

xB = acres of Crop B to plant?

1,800 acre feet = 2,220,267 m3

400 acre = 1,618,742 m2

Crop A Crop B

Water requirement (Acre feet/acre) 3 2

Profit ($/acre) 300 500

Max area (acres) 400 600

180023

0

600

0

400

toSubject

500300Maximize

BA

B

B

A

A

BA

xx

x

x

x

x

xxZ

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180023

0

600

0

400

toSubject

500300Maximize

BA

B

B

A

A

BA

xx

x

x

x

x

xxZ

180023

0

600

0

400

toSubject

500300Maximize

BA

B

B

A

A

BA

xx

x

x

x

x

xxZ

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180023

0

600

0

400

toSubject

500300Maximize

BA

B

B

A

A

BA

xx

x

x

x

x

xxZ

Point XA XB Z=300XA+500XB

a 0 0

b 0 600

c 200 600

d 400 300

e 400 0

Z=$100,000=300xA +500xB Z=$200,000=300xA +500xB

Z=$300,000=300xA +500xB Z=$360,000=300xA +500xB

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Parameter Units Value

Q1 m3/s 10

Q2 m3/s 12

Q3 m3/s 13

W1 kg/day 250,000

W2 kg/day 80,000

P1 mg/l 32

P2max mg/l 20

P3max mg/l 20

Objective: Minimize the Cost

Cost of 100% Waste Removal (WR)

C1=$200K

C2=$100K

Decision variables

x1 = % of WR at site 1; 0.8≤x1≤1.0

x2 = % of WR at site 2; 0≤x2≤1.0

X1+1.3x2≥1.8

8.13.1

1

0

1

8.0

toSubject

100;200:

)()(

Minimize

21

2

2

1

1

21

2211

xx

x

x

x

x

CCWhere

xCxCZ

Z

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8.13.1

1

0

1

8.0

toSubject

100;200:

)()(Minimize

21

2

2

1

1

21

2211

xx

x

x

x

x

CCWhere

xCxC

8.13.1

1

0

1

8.0

toSubject

100;200:

)()(Minimize

21

2

2

1

1

21

2211

xx

x

x

x

x

CCWhere

xCxC

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Point X1 X2 Z=200X1+100X2

a 0.8 0.78

b 0.8 1.0

c 1.0 1.0

d 1.0 0.62

8.13.1

1

0

1

8.0

toSubject

100;200:

)()(Minimize

21

2

2

1

1

21

2211

xx

x

x

x

x

CCWhere

xCxC

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1 ft = 0.3048 m

1 m3 = 28.3168x10 -3 ft3

1 m3 = 35.3147 ft3

1 ha = 10,000 m2

1 acre = 43,560 ft2

= 0.4047 ha= 4047 m2

1 gal = 3.785x10 -3 m3

= 3.785 L

1 m3 = 8.11x10-4 af109 m3 = 8.11x105 af1 km3 = 0.811 maf

1 m3 = 264 gal109 m3 = 264x109 gal1 km3 = 264 bg1 km3/yr = 0.7234 bgd