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(v) LP Applications: Water

Resources Problems

D Nagesh Kumar, IIScWater Resources Planning and Management: M3L5

Linear Programming and

Applications

Objectives


To formulate LP problems

To discuss the applications of LP in

Deciding the optimal pattern of irrigation

Water quality management

2

Introduction


LP has been applied to formulate and solve several types of

problems in engineering field

LP finds many applications in the field of water resources

which include

Planning of urban water distribution

Reservoir operation

Crop water allocation

3

Example - Irrigation Allocation Consider two crops 1 and 2. One unit of crop 1 brings four units of profit

and one unit of crop 2 brings five units of profit. The demand of

production of crop 1 is A units and that of crop 2 is B units. Let x be the

amount of water required for A units of crop 1 and y be the same for B

units of crop 2.

The linear relations between the amounts of crop produced (i.e.,

demands A and B) and the available water (i.e., x and y) for two crops

are shown below

A = 0.5(x - 2) + 2

B = 0.6(y - 3) + 3


Example - Irrigation Allocation …

Solution:

Objective: Maximize the profit from crop 1 and 2

Maximize f = 4A + 5B;

Expressing as a function of the amount of water,

Maximize f = 4[0.5(x - 2) + 2] + 5[0.6(y - 3) + 3]

f = 2x + 3y + 10



subject to

; Maximum availability of water

; Minimum amount of water required for crop 1

; Minimum amount of water required for crop 2

The above problem is same as maximizing

f ’ = 2x + 3y

subject to same constraints.

10 yx

2x

3y



Changing the problem into standard form by introducing slack variables

S1, S2, S3

Maximize f ’ = 2x + 3y

subject to

x + y + S1 =10

-x + S2 = -2

-y + S3 = -3

This model is solved using simplex method



The final tableau

is as shown

The solution is x = 2; y = 8; f ’ = 28

Therefore, f = 28+10 = 38

Water allocated to crop A is 2 units and to crop B is 8 units and total

profit yielded is 38 units.


Example – Water Quality Management

Waste load allocation for water quality management in a river system

can be defined as

Determination of optimal treatment level of waste, which is discharged

to a river

By maintaining the water quality standards set by Pollution Control

Agency (PCA), through out the river

Conventional waste load allocation involves minimization of treatment

cost subject to the constraint that the water quality standards are not

violated


Example - Water Quality Management …

Consider a simple problem of M dischargers, who discharge waste into

the river, and I checkpoints, where the water quality is measured by

PCA

Let xj be the treatment level and aj be the unit treatment cost for jth

discharger (j=1,2,…,M)

ci be the dissolved oxygen (DO) concentration at checkpoint i

(i=1,2,…,I), which is to be controlled

Decision variables for the waste load allocation model are xj

(j=1,2,…,M).



Objective function can be expressed as

Relationship between the water quality indicator, ci (DO) at

a checkpoint and the treatment level upstream to that

checkpoint is linear (based on Streeter-Phelps Equation)

Let g(x) denotes the linear relationship between ci and xj.

Then,

1

M

j j

j

Minimize f a x

( ) ,i jc g x i j



Let cP be the permissible DO level set by PCA, which is to be

maintained through out the river

Therefore,

Model can be solved using simplex algorithm which will give the

optimal fractional removal levels required to maintain the water quality

of the river

i Pc c i


LINEAR PROGRAMMING SOFTWARES

MMO Software (Dennis and Dennis, 1993)

An MS-Dos based software to solve various types of problems


Opening Screen of MMO

MMO Software

Press any key to see Main menu screen of MMO

Use arrow keys from keyboard to select different models.

Select “Linear Programming” and press enter. Two options will appear as follows:

SOLUTION METHOD: GRAPHIC/ SIMPLEXD Nagesh Kumar, IIScWater Resources Planning and Management: M3L514

Main Menu Screen of MMO

MMO Software

SIMPLEX Method using MMO

Select SIMPLEX in Linear Programming option of MMO software.

Screen for “data entry method” will appear


Screen for “Data Entry Method”


Data entry may be done by either of two different ways.

Free Form Entry: Write the equation when prompted for input.

Tabular Entry: Data can be input in spreadsheet style. Only the coefficients

are to be entered, not the variables.

Consider the problem


0,

6

,52

,5

32

21

21

21

1

21

xx

xx

xx

xtoSubject

xxZMaximize

Screen after Entering the Problem


Once the problem is run, it will show the list of slack, surplus and artificial

variables

There are three additional slack variables in the above problem.

Press any key to continue


List of slack, surplus and artificial variables


It will show three different options

1. No Tableau: Shows direct solutions

2. All Tableau: Shows all simplex tableau one by one

3. Final Tableau: Shows only the final simplex tableau

directly

Final solution is ;

;


Different Options for Simplex Solution667.15Z

333.21 x 667.32 x

Final Simplex Tableau

MATLAB Toolbox for Linear Programming

Very popular and efficient

Includes different types of optimization techniques

To use the simplex method

set the option as

options = optimset ('LargeScale', 'off', 'Simplex', 'on')

then a function called ‘linprog’ is to be used


MATLAB Toolbox for Linear Programming…


MATLAB Documentation for Linear Programming

MATLAB Toolbox for Linear Programming - Example


Let us consider the same problem as before

Note: The maximization problem should be converted to

minimization problem in MATLAB

0,

6

,52

,5

32

21

21

21

1

21

xx

xx

xx

xtoSubject

xxZMaximize

23

Example…

Thus,

tscoefficienCost%32 f

sconstraintoftsCoefficien%

1

2

0

1

1

1

A

sconstraint ofsidehandRight%655b

variablesdecisionofsLowerbound%00lb


24

Example…


MATLAB code

clear all

f=[-2 -3]; %Converted to minimization problem

A=[1 0;-1 2;1 1];

b=[5 5 6];

lb=[0 0];

options = optimset ('LargeScale', 'off', 'Simplex', 'on');

[x , fval]=linprog (f , A , b , [ ] , [ ] , lb );

Z = -fval %Multiplied by -1

x

Solution

Z = 15.667 with x1 = 2.333 and x2 = 3.667

LINGO


• Tool to solve linear, nonlinear, quadratic, stochastic and integer optimization

models

• Can be downloaded from http://www.lindo.com

• Key benefits of LINGO are:

• Easy model expression

• Convenient data options

• Powerful solvers

• Extensive documentation and help.

http://www.lindo.com/

LINGO…


Consider the same problem

LINGO formulation is:0,

6

,52

,5

32

21

21

21

1

21

xx

xx

xx

xtoSubject

xxZMaximize

Max = 2*x+ 3*y;

x<=5;

x-2*y>=-5;

x+y<=6;

x>=0;

y>=0;

LINGO…


Solution report from LINGO

Solution:

Z = 15.667

x1 = 2.333 and

x2 = 3.667


Thank You

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