1. Linear Programming and Applications
(v) LP Applications: Water Resources Problems
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

2. Objectives To formulate LP problems
To discuss the applications of LP in
Deciding the optimal pattern of irrigation
Water quality management
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

3. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

4. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

5. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

6. Example - Irrigation Allocation …
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.
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

7. Example - Irrigation Allocation …
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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

8. Example - Irrigation Allocation …
The final tableau
is as shown
The solution is x = 2; y = 8; f’ = 28
Therefore, f = = 38
Water allocated to crop A is 2 units and to crop B is 8 units and total profit yielded is 38 units.
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

9. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

10. 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).
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

11. Example - Water Quality Management …
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,
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

12. Example - Water Quality Management …
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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

13. LINEAR PROGRAMMING SOFTWARES
MMO Software (Dennis and Dennis, 1993)
An MS-Dos based software to solve various types of problems
Opening Screen of MMO
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

14. 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/ SIMPLEX
Main Menu Screen of MMO
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

15. Screen for “Data Entry Method”
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”
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

16. SIMPLEX Method using MMO
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
Screen after Entering the Problem
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

17. SIMPLEX Method using MMO
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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

18. SIMPLEX Method using MMO
It will show three different options
No Tableau: Shows direct solutions
All Tableau: Shows all simplex tableau one by one
Final Tableau: Shows only the final simplex tableau directly
Final solution is ; ;
Different Options for Simplex Solution
Final Simplex Tableau
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

19. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

20. MATLAB Toolbox for Linear Programming…
MATLAB Documentation for Linear Programming
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

21. MATLAB Toolbox for Linear Programming…
MATLAB Documentation for Linear Programming
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

22. 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
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

23. Example… Thus, Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

24. Example… MATLAB code Solution
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 = with x1 = and x2 = 3.667
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

25. LINGO
Tool to solve linear, nonlinear, quadratic, stochastic and integer optimization models
Can be downloaded from
Key benefits of LINGO are:
Easy model expression
Convenient data options
Powerful solvers
Extensive documentation and help.
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

26. Consider the same problem
LINGO…
Consider the same problem
LINGO formulation is:
Max = 2*x+ 3*y;
x<=5;
x-2*y>=-5;
x+y<=6;
x>=0;
y>=0;
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

27. LINGO… Solution: Solution report from LINGO Z = 15.667 x1 = 2.333 and
Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc

28. Thank You Water Resources Planning and Management: M3L5
D Nagesh Kumar, IISc