Water Resources Development and Management - PowerPoint Presentation

Slide1

Water Resources Development and ManagementOptimization(Linear Programming)

CVEN 5393

Feb

25, 2013Slide2

Acknowledgements Dr. Yicheng Wang (Visiting Researcher, CADSWES during Fall 2009 – early Spring 2010) for slides from his Optimization course during Fall 2009Introduction to Operations Research by Hillier and Lieberman, McGraw HillSlide3

Today’s LectureSimplex MethodRecap of algebraic formSimplex Method in Tabular formSimplex Method for other formsEquality ConstraintsMinimization Problems(Big M and

Twophase

methods)

Sensitivity / Shadow Prices

R-resources / demonstrationSlide4

(2)

Setting Up the Simplex Method

Original Form of the Model

Augmented Form of the ModelSlide5

Simplex Method (Tabular Form)Slide6

The

Simplex Method in Tabular Form

The tabular form is more convenient form for performing the required calculations. The logic for the tabular form is identical to that for the algebraic form.Slide7

Summary of the Simplex Method in Tabular Form

TABLE 4.3b The Initial Simplex TableauSlide8

TABLE 4.3b The Initial Simplex TableauSlide9

Iteration1Slide10
Slide11
Slide12
Slide13

Iteration2

Step1: Choose the entering basic variable to be

x

1

Step2: Choose the leaving basic variable to be

x

5Slide14

Step3: Solve for the new BF solution.

Optimality test: The solution (2,6,2,0,0) is optimal.

The new BF solution is (2,6,2,0,0) with Z =36Slide15

Tie

Breaking in the Simplex Method

Tie for the Entering Basic Variable

The answer is that the selection between these contenders may be made arbitrarily.

The optimal solution will be reached eventually, regardless of the tied variable chosen.

Tie for the Entering Basic VariableSlide16

Tie for the Leaving Basci Variable-Degeneracy

If two or more basic variables tie for being the leaving basic variable, choose any one of the tied basic variables to be the leaving basic variable. One or more tied basic variables not chosen to be the leaving basic variable will have a value of zero.

If a basic variable has a value of zero, it is called degenerate.

For a degenerate problem, a perpetual loop in computation is theoretically possible, but it has rarely been known to occur in practical problems. If a loop were to occur, one could always get out of it by changing the choice of the leaving basic variable.Slide17

No Leaving Basic Variable – Unbounded Z

If every coefficient in the pivot column of the simplex tableau is either negative or zero, there is no leaving basic variable. This case has an unbound objective function Z

If a problem has an unbounded objective function, the model probably has been misformulated, or a computational mistake may have occurred.Slide18

Multiple Optimal Solution

In this example, Points C and D are two CPF Solutions, both of which are optimal. So every point on the line segment CD is optimal.

Fig. 3.5 The Wyndor Glass Co. problem would have multiple optimal solutions if the objective function were changed to Z = 3

x

1

+ 2

x

2

C

(2,6)

E

(4,3)

Therefore, all optimal solutions are a weighted average of these two optimal CPF solutions.Slide19

Multiple Optimal Solution

Any linear programming problem with multiple optimal solutions has at least two CPF solutions that are optimal.

All optimal solutions are a weighted average of these two optimal CPF solutions.

Consequently, in augmented form,

any linear programming problem with multiple optimal solutions has at least two BF solutions that are optimal. All optimal solutions are a weighted average of these two optimal BF solutions.Slide20

Multiple Optimal SolutionSlide21
Slide22

These two are the only BF solutions that are optimal, and all other optimal solutions are a convex combination of these .Slide23

Sensitivity(Shadow Prices & Significance)Slide24

Shadow

Prices

(0)

(1)

(2)

(3)

Resource

b

i

= production time available in Plant

i

for the new products.

How will the objective function value change if any

b

i

is increased by 1 ?Slide25

b

2

: from 12 to 13

Z

: from 36 to 37.5

△

Z=3/2

b

1

: from 4 to 5

Z

: from 36 to 36

△

Z=0

b

3

: from 18 to 19

Z

: from 36 to 37

△

Z=1Slide26
Slide27

indicates that adding 1 more hour of production time in Plant 2 for the two

new products would increase the total profit by $1,500.

The constraint on resource 1 is

not

binding

on the optimal solution, so there is a surplus of this resource. Such resources are called

free goods

The constraints on resources 2 and 3 are

binding constraints

. Such resources are called

scarce goods

.

H

(0,9)Slide28

Sensitivity

Analysis

Maximize

b, c, and a are parameters whose values will not be known exactly until the alternative given by linear programming is implemented in the future.

The main purpose of sensitivity analysis is to identify the sensitive parameters.

A parameter is called a sensitive parameter if the optimal solution changes with the parameter.Slide29

How are the sensitive parameters identified?

In the case of

b

i

, the shadow price is used to determine if a parameter is a sensitive one.

For example, if > 0 , the optimal solution changes with the

b

i

.

However, if = 0 , the optimal solution is not sensitive to at least small changes in

b

i

.

For

c

2

=5, we have

c

1

=3 can be changed to any other value from 0 to 7.5 without affecting the optimal solution (2,6)Slide30

Parametric

Linear Programming

Sensitivity analysis involves changing one parameter at a time in the original model to check its effect on the optimal solution.

By contrast,

parametric linear programming

involves the systematic study of how the optimal solution changes as many of the parameters change simultaneously over some range.Slide31

Adapting Simplex to other forms(Minimization, equality and greater than equal constraints, )Slide32

Adapting

to Other Model Forms

Original Form of the Model

Augmented Form of the ModelSlide33

Artificial-Variable Technique

The purpose of artificial-variable technique

is to obtain an initial BF solution.

The procedure is to construct an artificial problem that has the same optimal solution as the real problem by making two modifications of the real problem.Slide34

Augmented Form of the Artificial Problem

Initial Form of the Artificial Problem

The Real ProblemSlide35

The feasible region of the Real Problem

The feasible region of the Artificial ProblemSlide36

Converting Equation (0) to Proper Form

The system of equations after the artificial problem is augmented is

To algebraically eliminate from Eq. (0), we need to subtract from Eq. (0) the product, M times Eq. (3)Slide37

Application of the Simplex Method

The new Eq. (0) gives Z in terms of just the nonbasic variables (

x

1

,

x

2

)

The coefficient can be expressed as a linear function aM+b, where a is called multiplicative factor and b is called additive term.

When multiplicative factors a’s are not equal, use just multiplicative factors to conduct the optimality test and choose the entering basic variable.

When multiplicative factors are equal, use the additive term to

conduct the optimality test and choose the entering basic variable.

M only appears in Eq. (0), so there’s no need to take into account M when conducting the minimum ratio test for the leaving basic variable.Slide38

Solution to the Artificial ProblemSlide39

Functional Constraints in

≥

FormSlide40

The Big M method is applied to solve the following artificial problem (in augmented form)

The minimization problem is converted to the maximization problem bySlide41

Solving the Example

The simplex method is applied to solve the following example.

The following operation shows how Row 0 in the simplex tableau is obtained.Slide42
Slide43

The Real Problem

The Artificial ProblemSlide44

The Two-Phase Method

Since the first two coefficients are negligible compared to M, the two-phase method is able to drop M by using the following two objectives.

The optimal solution of Phase 1 is a BF solution for the real problem, which is used as the initial BF solution.Slide45

Summary of the Two-Phase MethodSlide46

Phase 1 Problem (The above example)

Example:

Phase2 Problem

Example:Slide47

Solving Phase 1 ProblemSlide48

Preparing to Begin Phase 2Slide49

Solving Phase 2 ProblemSlide50

How to identify the problem with no feasible solutons

The artificial-variable technique and two-phase method are used to find the initial BF solution for the real problem.

If a problem has no feasible solutions, there is no way to find an initial BF solution.

The artificial-variable technique or two-phrase method can provide the information to identify the problems with no feasible solutions.Slide51

To illustrate, let us change the first constraint in the last example as follows.

The solution to the revised example is shown as follows.