1. Water Resources Development and Management Optimization (Linear Programming)
CVEN 5393
Feb 14, 2011

2. History of optimization
The history of optimization techniques is that of operations research.
The beginning of Operations Research (OR) has generally been attributed to the military services early in World War II.
When the war ended, the success of OR in the war effort spurred interest in applying OR outside the military as well.
Two other factors that played a key role in the rapid growth of OR during this period.
One was the substantial progress that was made early in improving the techniques of OR.
The other was the onslaught of the computer revolution.
(4) In the field of civil and environmental engineering, the most active applications OR have taken place in water resources and water environment.

3. Optimization modeling
Concept of optimazation model
The general form of an optimization model can be expressed by the following model.
(1.1)
Subject to
(1.2)
Where F(X) is objective function and gi(X)=bi are constraint conditions.

4. Types of optimazation models
Uniobjective model
The model expressed by Equations (1.1) and (1.2) has only a single objective and is called Uniobjective model
Multiobjective model
Subject to
(1.3)
(1.4)

5. Types of optimazation models
Deterministic model
If a model does not take in account the uncertainty in mode variables or parameters, it is called a deterministic model.
Stochastic model
A model which explicitly considers the uncertainty of a system is called stochastic models.

8. Linear Programming (LP)
The development of LP has been ranked among the most important scientific advances of the mid -20th century.
The adjective linear means that all the mathematical functions in an LP model are required to be linear functions. The word programming does not refer here to computer programming; rather, it’s essentially a synonym for planning.
The most common type of LP application involves the general problem of allocating limited resources among competing activities in a best possible way.
LP has numerous other important applications as well. In fact, any problem whose mathematical model fits the very general format for the linear programming model is a linear programming problem.

15. Blending Water Supply

17. Example 2: Cleaning Up the River

18. These two functions are as follows.

20. Transportation/Shipping Example
The Fagersta Steelworks currently is working two mines to obtain its iron ore. This iron ore is shipped to either of two storage facilities. When needed, it then is shipped on to the company’s steel plant. The diagram below depicts this distribution network, where M1 and M2 are the two mines, S1 and S2 are the two storage facilities, and P is the steel plant. The diagram also shows the monthly amounts produced at the mines and needed at the plant, as well as the shipping cost and the maximum amount that can be shipped per month through each shipping lane.
Management now wants to determine the most economic plan for shipping the iron ore from the mines through the distribution network to the steel plant.

21. (a) Formulate a linear programming model for this problem.
The decision variables are defined as follows:
xm1-s1 : number of units (tons) shipped from Mine 1 to Storage Facility 1,
xm1-s2 : number of units (tons) shipped from Mine 1 to Storage Facility 2,
xm2-s1 : number of units (tons) shipped from Mine 2 to Storage Facility 1,
xm2-s2 : number of units (tons) shipped from Mine 1 to Storage Facility 2,
xs1 -p : number of units (tons) shipped from Storage Facility 1 to the Plant,
xs2 -p : number of units (tons) shipped from Storage Facility 2 to the Plant.
The total shipping cost is:
Z = xm1-s xm1-s xm2-s xm2-s xs1-p xs2-p

22. The constraints we need to consider are:
Supply constraint on M1 and M2:
xm1-s1 + xm1-s2 = 40
xm2-s1 + xm2-s2 = 60
Conservation-of-flow constraint on S1 and S2:
xm1-s1 + xm2-s1 - xs1-p = 0
xm1-s2 + xm2-s2 - xs2-p = 0
Demand constraint on P:
xs1-p + xs2-p = 100
Capacity constraints:
xm1-s1  30, xm1-s2  30
xm2-s1  50, xm2-s2  50
xs1-p  70, xs2-p  70
Non-negativity constraints:
xm1-s1  0, xm1-s2  0
xm2-s1  0, xm2-s2  0
xs1-p  0, xs2-p  0

23. The resulting linear programming model for this problem is:
Minimize Z = 2000 xm1-s xm1-s xm2-s xm2-s2
+ 400 xs1-p xs2-p,
subject to
xm1-s1 + xm1-s2 = 40
xm2-s1 + xm2-s2 = 60
xm1-s1 + xm2-s1 - xs1-p = 0
xm1-s2 + xm2-s2 - xs2-p = 0
xs1-p + xs2-p = 100
xm1-s1  30, xm1-s2  30
xm2-s1  50, xm2-s2  50
xs1-p  70, xs2-p  70
and
xm1-s1  0, xm1-s2  0
xm2-s1  0, xm2-s2  0
xs1-p  0, xs2-p  0

24. Additional LP Problems
Water Resources Problems
Reservoir Systems
Water Quality
Industrial Applications / Graphical Solutions

34. Common Terminology for LP Model
Objective Function:
The function being maximized or minimized is called the objective function.
Constraint:
The restrictions of LP Model are referred to as constraints.
The first m constraints in the previous model are sometimes called functional constraints.
The restrictions xj ≤ 0 are called nonnegativity constraints.
Feasible Solution:
A feasible solution is a solution for which all the constraints are satisfied.
A feasible solution is located in the feasible region. An infeasible solution is outside the feasible region.
Infeasible Solution:
An infeasible solution is a solution for which at least one constraint is violated.
Feasible Region:
The feasible region is the collection of all feasible solutions.

35. Common Terminology for LP Model
No Feasible Solutions:
It is possible for a problem to have no feasible solutions.
An Example
Fig The Wyndor Glass Co. problem would have no feasible solutions if the constraint 3x1 + 5x2 ≤ 50 were added to the problem.
In this case, there is no feasible region

36. Common Terminology for LP Model
Optimal Solution:
An optimal solution is a feasible solution that has the maximum or minimum of the objective function.
Multiple Optimal Solutions:
It is possible to have more than one optimal solution.
An Example
Fig The Wyndor Glass Co. problem would have multiple optimal solutions if the objective function were changed to Z = 3x1 + 2x2

37. Common Terminology for LP Model
Unbounded Objective:
If the constraints do not prevent improving the value of the objective function indefinitely in the favorable direction, the LP model is called having an unbounded objective.
An Example
Fig The Wyndor Glass Co. problem would have no optimal solutions if the only functional constrait were x1 ≤ 4, because x2 then could be increased indefinitely in the feasible region without ever reaching the maximum value of Z = 3x1 + 2x2

38. Common Terminology for LP Model
Corner-Point Feasible (CPF) Solution:
A corner-point feasible (CPF) is a solution that lies at a corner of the feasible region.
Fig The five dots are the five CPF solutions for the Wyndor Glass Co. problem

39. Common Terminology for LP Model
Relationship between optimal solutions and CPF solutions :
Consider any linear programming problem with feasible solutions and a bounded feasible region. The problem must posses CPF solutions and at least one optimal solution. Furthermore, the best CPF solution must be an optimal solution. Therefore, if a problem has exactly one optimal solution, it must be a CPF solution. If the problem has multiple optimal solutions, at least two must be CPF solutions.
(2,6)
(4,3)
The modified problem has multiple optimal solution, two of these optimal solutions , (2,6) and (4,3), are CPF solutions.
The prototype model has exactly one optimal solution, (x1, x2)=(2,6), which is a CPF solution

40. Tabular Standard Form of an LP Model

41. Matrix Standard Form of an LP Model
To help you distinguish between matrices, vectors, and scalars, we use BOLDFACE CAPITAL letters to represent matrices, bold lowercase letters to represent vectors, and italicized letters in ordinary print to represent scalars.