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Title: Chapter%203%20Linear%20Programming%20Methods

1
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Chapter 3 Linear Programming Methods
2

Introduction
• Whereas the simplex method searches for the
optimal solution at the boundary of the feasible
region, the latest developments are based on the
idea of following a path through the interior of
the feasible region until the optimum is reached.
These so-called interior point methods are of
great theoretical importance because they provide
a bound on the computational effort required to
solve a problem that is a polynomial function of
its size.
• No polynomial bound is available for the simplex
algorithm. Nevertheless, simplex codes have
proven to be highly efficient in practice and
remain at the center of virtually all commercial
optimization packages.

3

Standard Form of LP
• The standard form of the model, sometimes
referred to as the canonical form, is written as

(1) (2) (3)
Max Z s.t.
Where all bi are nonnegative.
4
• TheoryIf a finite optimal solution exists, there
is an optimal solution at one of the extreme
points or vertices of the feasible region.
• Definition 1 Let P be a polyhedron in
n-dimensional space, written as . A
vector x is an extreme point of P if we cannot
find two vectors y, , both different from
x, and a scalar such that x ?y (1
-?)z.

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• In other words, x is an extreme point if it
doesn't lie on the line between any two points in
P.
• Definition 2 Let a be a nonzero row vector in
n-dimensional space, written as , and
let b be a scalar.
• a. The set x ax b is called a
hyperplane.
• b. The set x ax?b is called a half
space.

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• Note that a hyperplane is the boundary of a
corresponding halfspace. For a polyhedron in
n-dimensional space, an extreme point is the
intersection of n non-coplanar hyperplanes. (If
more than n hyperplanes are intersected, the
extreme point is degenerated.)
• For the linear programming problem defined by
Equations (1) to (3), some of these hyperplanes
will be of the form 0.

8
PREPARING THE MODEL
• The objective must be to maximize.
• The objective function must be linear in the
variables and must not contain any constant
terms.
• All variables must be restricted to be
nonnegative.
• Each constraint must be written as a linear
equation with the variables on the left of the
equal sign and a positive constant on the right.

9
• Minimize ltgt
Maximize -
• ltgt
• ltgt
• For unrestricted variable ,
• where 0 and

10
• When more than one variable is unrestricted,
• where and

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Example (Transformation)
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Letand
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Maximize Subject to
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GEOMETRIC PROPERTIES OF LPs
• When solving two-dimensional problems
graphically, we saw that solutions were always
found at extreme points of the feasible region.
These solutions are called basic solutions.
• In a problem with n variables and m
constraints, a basic solution is determined by
identifying m variables as basic, setting the
remaining n - m (nonbasic) variables equal to
zero, and solving the resultant set of
simultaneous equations.

16
• In order for these equations to have a unique
solution and hence correspond to an extreme
point, care must be taken in choosing which
variables to make basic.

17
Linear Independence
• Let us consider a system of m linear equations
in m unknowns, written as Axb.

18
• For this system to have a unique solution, the
matrix A must be invertible or nonsingular that
is, there must exist another m m matrix B such
that AB BA I, where I is the m x m identity
matrix. Such a matrix B is called the inverse of
A and is unique. It is denoted by .

19
Definition 3 Let be a
collection of k column vectors, each of dimension
m. We say that these vectors are linearly
independent (LI)if it is not possible to find k
real numbers not all zero such
that , where 0 is
the m-dimensional null vector otherwise, they
are called linearly dependent (LD).
20
• Theorem 1 Let A be a square matrix. The
following
• statements are equivalent.
• The determinant of A is nonzero.
• The matrix A is invertible, as is its transpose
.
• The rows and columns of A are linearly
independent.
• For every vector b, the linear system Ax b has
a unique solution.

21
Basic Solution
• Consider a system of equations Ax b, where x
is an n-dimensional vector, b is an m-dimensional
vector, and A is an m n matrix. Suppose that
from the n columns of A we select a set of m
linearly independent columns (such a set exists
if the rank of A is m).

22
• Definition 4 Given a set of m simultaneous
linear equations in n unknowns, let B be any
nonsingular m m matrix made up of columns of A.
If all the n - m components of x not associated
with columns of B are set equal to zero, the
solution to the resulting set of equations is
said to be a basic solution with respect to the
basis B.
• The components of x associated with columns of
B are called basic variables.

23
• Given the assumption that A has full row rank,
the system Ax b always has a solution and, in
fact, will always have at least one basic
solution however, the basic variables in a
solution are not necessarily all nonzero.
• Definition 5 A degenerate basic solution is
said to occur if one or more of the basic
variables in a basic solution have values of
zero.

24
• Definition 6 A vector x S x
Ax b, x?0 is said to be feasible to the linear
programming problem in standard form a feasible
solution that is also basic is said to be a basic
feasible solution (BFS).If this solution is
degenerate, it is called a degenerate basic
feasible solution.

25
Foundations
• Theorem 2 Given a linear program in standard
form, where A is an m n matrix of rank m,
• a. if there is a feasible solution, there is
a BFS
• b. if there is an optimal feasible solution,
there is an optimal BFS.
• In general, if there are m constraints and n
variables, the number of basic solutions will be
bounded by the number of ways to choose m basic
variables (or, equivalently, the n -m nonbasic
variables) from the n variablesi.e.,

26
Example
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The upper bound on the number of basic solutions
is computed to be
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• Theorem 3 For any linear program, there is a
unique extreme point in the feasible region S
x Ax b, x?0 corresponding to a BFS.
In addition, there is at least one BFS
corresponding to each extreme point in S.

30
• When more than one BFS maps into a specific
extreme point, at least one of the binding
constraints at that extreme point is redundant.
It could be either one of the structural
constraints or one of the nonnegativity
restrictions.
• In either case, the BFSs would all be
degenerate.

31
• Definition 7 For a linear program in standard
form with m constraints, two basic feasible
solutions are said to be adjacent if they have m
- 1 basic variables in common.
• For example, referring to Figure 3.3 we see
that points 2 and 8 are adjacent. Given that m
3, these points should have 3-12 basic
variables in common to be adjacent according to
Definition 7. (which ones?)

32
gt If x is better than all its adjacent BFSs,
then x is an optimal BFS. (why?)
gt Convexity of Linear programming.
33
Simplex Algorithm
• Step 1 Find an initial BFS to the linear
program. As the algorithm progresses, the most
recent BFS will be called the incumbent.
• Step 2 Determine if the incumbent is optimal. If
not, move to an adjacent BFS that has a larger z
value.
• Step 3 Return to Step 2, using the new BFS as
the incumbent.

34
• Although the number of BFSs could be huge, the
simplex algorithm is seen in practice to be
highly efficient. Vanderbei 1996 has
empirically demonstrated that in the absence of
degeneracy, approximately 0.5(m n) BFSs have to
be examined before optimality is confirmed (when
degeneracy is present, this number doubles)

35
Simplex Tableau
36
Move all variables to the left hand side of
, and all constants to the right hand side of
.
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The marginal effects of increasing can be
described by the following partial derivatives,

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• In general terms, the equations are written as
• where Q is the set of nonbasic variables.

42
What is the Basic Solution?
• From the general expression of the simplex form
we read the objective function value and the
values of the basic variables as

43
Is the basic solution feasible?
• In general, a basic solution is feasible if

44
Is the basic feasible solution optimal?
• Optimality condition
• Why?
• Increasing the value of any nonbasic variable
(from its current value 0) can not improve the
current objective function value.

45
How Can a Nonoptimal Feasible Solution Be
Improved?
• The current solution can be improved by
increasing any nonbasic variable with a negative
coefficient in Equation E0.
• In the general case, all nonbasic variables with
lt 0 are candidates for conversion to basic
variables.

46
How Much Should the Nonbasic Variable Be Changed?
• When gt 0, a positive value of will
drive the corresponding basic variable to zero.
When more than one row has this characteristic,
the nonbasic variable can increase only to the
value of the minimum positive ratio if
feasibility is to be guaranteed. Thus the
nonbasic variable may increase by

(min. ratio test)
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How Is a New Simplex Form Obtained?
• The process of obtaining the new tableau is
called pivoting or pricing out the entering
column. It is simply a series of linear
operations on the equations that results in the
simplex form of the new set of basic variables.

49
Pivoting
Suppose the rth row is selected by the min. ratio
test. Then,
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Before Pivoting
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After Pivoting
0.5
0.5
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A Complete Example
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3
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Another Example
• Max Z3x15x2
• s.t. x1 ?4
• 2x2 ?12
• 3x1 2x2 ?18
• x1 ,x2 ?0

59
Convert to Canonical Form
• Max Z3x15x2 0x3 0x40x5
• s.t. x1 x3 4
• 2x2 x4 12
• 3x1 2x2 x5 18
• x1 ,x2 , x3 ,x4, x5 ?0

60
Convert to System of Equations
• Z-3x1-5x2 0x3 0x40x5 0
• x1 x3 4
• 2x2 x4 12
• 3x1 2x2 x5 18
• Perform elementary row operations (e.r.o.) to
• keep it in a standard for and try to maximize Z.
• Maintain implicitly x1 ,x2 , x3 ,x4, x5 ?0

61
Tableau form
row basic x1 x2 x3 x4 x5 RHS
0 Z -3 -5 0 0 0 0
1 x3 1 0 1 0 0 4
2 x4 0 2 0 1 0 12
3 x5 3 2 0 0 1 18
row basic x1 x2 x3 x4 x5 RHS
0 Z -3 0 0 5/2 0 30
1 x3 1 0 1 0 0 4
2 x2 0 1 0 1/2 0 6
3 x5 3 0 0 -1 1 6
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Tableau form
row basic x1 x2 x3 x4 x5 RHS
0 Z 0 0 0 3/2 1 36
1 x3 0 0 1 1/3 -1/3 2
2 x2 0 1 0 1/2 0 6
3 x1 1 0 0 -1/3 1/3 2
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SPECIAL SITUATIONS
• Tie for the Entering Variable
• When the steepest ascent rule is used for
selecting the variable that will enter the basis,
if there is a tie for the most negative value,
any one of the tied variables may be chosen
arbitrarily.

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• Tie for the Leaving Variable
• The basic variable for any one of the tied
rows may be chosen arbitrarily to leave the
basis.
• If more than one row has the same positive
ratio, more than one basic variable will go to
zero at the next iteration. (degenerate BFS)

65
• Definition 8 A basic solution is
said to be degenerate if the number of structural
constraints and nonnegativity conditions active
at x is greater than n.
• In two dimensions, a degenerate basic solution
occurs at the intersection of three or more
lines in three dimensions, a degenerate solution
occurs at the intersection of four or more
planes.
• The effect of degeneracy is to interrupt the
steady progress of the simplex algorithm toward
optimality.

66
Example
67
• When degeneracy occurs, the possibility arises
previously encountered basis. (Cycling)
• Nevertheless, cycling can be eliminated by
modifying the rules used to select the entering
and leaving variables.

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• The following rules are based on the indices of
the problem variables (j 1,..., n) and are
attributable to Bland 1977.
• Rule for selecting the entering variable From
all the variables having negative coefficients in
row 0, select as the entering variable the one
with the smallest index,
• Rule for selecting the leaving variable Use the
standard ratio test to select the leaving
variable, but if there is a tie in the ratio
test, select from the tied rows the variable
having the smallest index.

69
Alternative optima
• Although the simplex method automatically stops
when the first optimal BFS is reached, others can
be found by forcing a nonbasic variable with zero
reduced cost to enter the basis.
• When more than one optimal BFS exists, any
weighted average of them is also optimal, where
the weights are nonnegative and sum to 1. Such a
weighted linear average is called a convex
combination.

70
Unbounded Solution
• When there is a nonbasic variable with a negative
coefficient in row 0 and all the other
coefficients in that column are nonpositive, the
value of that variable can be increased
indefinitely without driving any basic variable
to zero. The objective function can then be made
arbitrarily large. This condition signals an
unbounded solution, and there is no optimal
solution for the model. The algorithm must stop
if such an event occurs.

71
Example