simplex theory method of solving

Chapter 5 The Theory of Simplex Method

1. FOUNDATIONS OF THE SIMPLEX

METHOD

2. THE REVISED SIMPLEX METHOD

3. A FUNDAMENTAL INSIGHT

The Theory Simplex Method

5.1 FOUNDATIONS OF THE SIMPLEX METHOD

Reading assignment: Read and understand the material covered in Section 5.1 of the text book

(Foundations of the Simplex Method).


5.2 THE REVISED SIMPLEX METHOD

The simplex method presented in previous lectures is a straight forward algebraic procedure.

However, this way of executing is not the most efficient computational procedure for computers

– because it computes and stores many numbers that are not needed at the current iteration and that may not even become relevant for decision making at subsequent iterations.

The only pieces of information relevant at each iteration are

– the coefficients of the nonbasic variables in Eq. (0),

– the coefficients of the entering basic variable in the other equations, and

– the right-hand sides of the equations.

It would be very useful to have a procedure that could obtain this information efficiently without computing and storing all the other coefficients.


The revised simplex method

explicitly uses matrix

manipulations, so it is necessary

to describe the problem in matrix

notation as shown →

c is a row vector

x, b, and 0 are column vectors

A is the matrix

Where:


To obtain the augmented form, we need column vector of slack variables

So that the constraints become

where I is the m ×m identity matrix, and the null vector 0 now

has n + m elements.


Solving for a Basic Feasible Solution

One of the key features of the revised simplex method involves the

way in which it solves for each new BF solution after identifying its

basic and nonbasic variables.

Given these variables, the resulting basic solution is the solution of

the m equations:

in which the n nonbasic variables from the n + m elements of

are set to zero.


Eliminating these n variables by equating them to zero leaves a set of m equations

in m unknowns (the basic variables). This set of equations can be denoted by

where the vector of basic variables

is obtained by eliminating n nonbasic variables from

and the basis matrix

is obtained by eliminating the columns

corresponding to the coefficients of nonbasic variables from [ A, I ]



To solve BxB = b both sides are multiplied by B1

B1

BxB = B1b

xB = B1b

The value of the objective function for this basic solution is then

Z = cBxB = cBB1b

Where cB is the vector whose elements are the objective function coefficients for

the corresponding elements of xB.




18233

1222

41

53

521

42

31

21

xxx

xx

xx

toSubject

xxZMaximize

)(

)(

)(

:


Iteration 0 (Solving for the initial BF solution)


Iteration 1 (Solving for BF solution after x2 and x4 are identified as

entering and leaving basic variables, respectively)

so


Iteration 2 (Solving for BF solution after x1 and x5 are identified as

entering and leaving basic variables, respectively)

so


Matrix Form of the Current Set of Equations

For the original set of equations, the matrix form is

The algebraic operations performed by the simplex method (multiply an equation

by a constant and add a multiple of one equation to another equation)

are expressed in matrix form by multiplying both sides of the original set of

equations by the appropriate matrix.

After any iteration, xB = B1b and Z = cBxB , so the right-hand sides of the new

set of equations have become


Because we perform the same series of algebraic operations on both sides of the

original set of operations, we use this same matrix that multiplies the original right-

hand side to multiply the original left-hand side. Consequently, since

the desired matrix form of the set of equations after any iteration is



Iteration

0

any

Optimal



Matrix Form of the Current Set of Equations – Illustration using the results obtained at Iteration 2 on Slide 11

xB = B1b

Z = cBxB


Matrix Form of the Current Set of Equations – Illustration using the results obtained at Iteration 2 on Slide 11


Same as shown in the final simplex tableau in the previous lecture slides.


The Overall Procedure

Two Key Implementations

1. The first is that only B1 needs to be derived to be able to calculate all the numbers in the simplex tableau from the original parameters (A, b, cB) of the problem.

2. The second is that any one of these numbers can be obtained individually, usually by performing only a vector multiplication (one row times one column) instead of a complete matrix multiplication.

Therefore, the required numbers to perform an iteration of the simplex method can be obtained as needed without expending the computational effort to obtain all the numbers.


Summary of the Revised Simplex Method

1. Initialization: Same as for the original simplex method.

2. Iteration:

– Step 1: Determine the entering basic variable: Same as for the original simplex method.

– Step 2: Determine the leaving basic variable: Same as for the original simplex method,

except calculate only the numbers required to do this [the coefficients of the entering basic variable in every equation but Eq. (0), and then, for each strictly positive coefficient, the right-hand side of that equation].

– Step 3: Determine the new BF solution: Derive B1 and set xB = B1b.

3. Optimality test: Same as for the original simplex method, except calculate only the numbers required to do this test, i.e., the coefficients of the nonbasic variables in Eq. (0).


Example: Wyndor Glass Co. problem

The initial basic variables are the slack variables

5

4

3

xxx

xB

Initialization: Because the initial B1 = I, no calculations are needed to obtain the numbers required to identify the entering and leaving basic variable.

Remember:


Iteration 1: x2 is the entering basic variable (most negative)

Coefficients of x2 in Eq. (1, 2, and 3) are: Minimum Ratio test

Thus, the new set of basic variable is

4

~

12 / 2 min ( )

18 / 2

x is leaving

5

2

3

xxx

xB

Optimality test:

Coefficients of non-basic variables x1 and x2 are -3 and -5 respectively:

Hence, the initial BF solution is not optimal.


Iteration 1: (Contd.)

Updating the matrices:

Optimality test:

Coefficients of non-basic variables x1 and x4 are calculated as follows:

Hence, the initial BF solution is not optimal.


Iteration 2:

Using these coefficients of the nonbasic variables in Eq. (0), since only x1 has a negative coefficient, we begin the next iteration by identifying x1 as the entering basic variable. To determine the leaving basic variable, we must calculate the coefficients of x1 in Eq. 1, 2, and 3 These coefficients can be obtained from B1A by performing only the relevant parts of the matrix multiplications as follows By using the right side column for the current BF solution (the value of xB) just given for iteration 1, minimum ratio test can be done

Thus, x5 is the leaving basic variable

~

~

~

~

~

~

AB

3

0

1

3

0

1

110

02

10

001

1

min/

~

/

36

14


Iteration 2: (Contd.) The new set of basic variables is therefore The updated matrices are:

1

2

3

x

x

x

xB

1 0 1

0 2 0

0 2 3

B

1

1 1/ 3 1/ 3

10 0

2

0 1/ 3 1/ 3

B

2

6

2

18

12

4

31310

02

10

31311

1

//

//

bBxB

36

2

6

2

3501

],,[xcbBcZ BBB


Optimality test: To apply the optimality test, we have to find that the coefficients of

the nonbasic variables (x4 and x5) in Eq. (0) from cBB1 by performing

only the relevant parts of the matrix multiplications as follows Because both coefficients (3/2, and 1) are nonnegative, the current solution (x1 = 2, x2 = 6, x3 = 2, x4 = 0, x5 = 0) is optimal and the procedure terminates.

123

3131

02

1

3131

3501 ,/~,

//~

~

//~

],,[BcB


Final Observations

The matrix form of the simplex method has the advantage of carrying less amount of information from one iteration to the next.

It is only necessary to know the current B-1 and cBB-1 which appears in the slack variable portion of the current simplex tableau.

However, a drawback is that we need to calculate B-1 in every iteration, which may turn out to be a tedious task if the inverse is to be calculated from scratch.

Fortunately, a more efficient approach to obtain the inverse matrix is the revised simplex method .

If the time allows, we will return to discuss the mechanism of the revised simplex method at a later stage.

simplex theory method of solving

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