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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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The following are the topics to be discussed under
Iterative Improvement:
Definition of iterative improvement algorithm
design technique
Advantages of iterative improvement algorithm
design technique.
Application of simplex method.
Application of Geometric Interpretation of linear
programming
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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a type of algorithm design technique for solving
optimization problems
starts with initial state, and changes it iteratively
for improvement to find the best or the least value
Advantages:
Use very little memory (usually constant amount),because it only stores the current state
Often finds reasonable solutions in large or infinite
(continuous) state
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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identified as important in the 1950’s, and a
general solution called the simplex method was
developed
actually name of a branch of applied mathematics
that deals with solving optimization problems of a
particular form
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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The generic interpretation of Linear Programming
It must be a maximization problem.
All constraints (except the non-negativityconstraints) must be in the form of liner
equations.
All the variables must be required to be non-
negative.
Thus, the general linear programming problem in
standard form with m constraints and n unknowns
(n ≥ m) is:
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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a classic method used for solving linear
programming problems
considered as the one of the most importantalgorithms ever developed
developed by a U.S. mathematician named George
B. Danzig in 1947
generates a sequence of adjacent points of the
problem’s feasible region with improving values of
the objective function until no further
improvement is possible
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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Every linear programming problem can be
represented in the following:
The possible solution to be given problem is any
point x, y) that satisfies all the constraints of the
problem’s possible region is the set of all its
feasible points.
Maximize: 3x + 5y
Subject to: x + y ≤ 4
x + 3y ≤ 6
x≥0, y≥0
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Design and Analysis of Algorithms
* Property of STI
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It is instructive to visualize on:
A line divides the plane into two half-planes,
whereas for all the points have ax + by < c and on
the other, ax + by > c.
Therefore, the set of points defined by inequality x
+ y ≤ 4 encompasses the points on and below the
line x + 3y = 6.
ax + by = c
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Design and Analysis of Algorithms
* Property of STI
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The points of the feasible region must satisfy all
the constraints of the problem.
The feasible region attained by the intersection of
these two half-planes and the first quadrant of the
Cartesian plane defined by non-negative
constraints x ≥ 0, y ≥ 0.
The feasible region for the problem is the convex
polygon with the vertices (0,0), (4,0), (0,2) and
(3,1).
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Design and Analysis of Algorithms
* Property of STI
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The last point, which is the point of intersection of
the lines x + y = 4 and x + 3y = 6 is achieved by
solving the system of the two linear equations.
To find the optimal solution, a point in the feasible
region with the largest value of the objective
function: z = 3x = 5y.
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Design and Analysis of Algorithms
* Property of STI
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Are there feasible solutions for which the value of
the objective function equals, say, 20?
The points (x,y) for which the objective function, x= 3x + 5y, is equal to 20 form the line, 3x + 5y =
20.
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Design and Analysis of Algorithms
* Property of STI
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There are infinitely many feasible points for which
the objective function is equal to, say, 10.
These are the intersection points of the line 3x +
5y = 20 and 3x + 5y = 10 with the feasible region.
To apply the simplex method to a linear
programming problem, it has to be represented in
a special form called the standard form.
The standard form has the following requirements:
It must have a
maximum
problem.
All the constraints
except the non-
negativity
constraints) must
be in the form of
linear equations
.
All the variables
must be required
to be non-
negative
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Design and Analysis of Algorithms
* Property of STI
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The general linear programming problem in
standard form with m, constraints and n
unknowns (n≥m) is:
Any linear programming problem can be
transformed into an equivalent problem in
standard form.
If an objective function needs to be minimized, it
can be replaced by the equivalent problem of
maximizing the same objective function with all its
coefficients cj replaced by –cj, j = 12….n.
Maximize: c1x1 + ….+cnxn
Subject to:
ai1x1 + …. + ainxn = bi for i=1,2,….m
x1≥0, ……xn≥0.
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Design and Analysis of Algorithms
* Property of STI
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Example: Two inequalities of the given problem
can be transformed, respectively, into the
following equations:
If this is not the case in an initial statement of a
problem, an unconstrained variables x j can be
replaced by the difference between two new
nonnegative variables.
Thus, the given problem in standard form is the
following linear programming problem in fourvariables:
x + y + u = 4 where u ≥ 0and
x + 3y + v = 6 where v ≥ 0
x j = x’ j – x’’j, x’ j ≥0,x’’ j≥0.
Maximize: 3x + 5y + 0u + 0v
Subject to: x + y + u = 4
x + 3y + v = 6
x, y, u, v ≥ 0
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Design and Analysis of Algorithms
* Property of STI
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It is easy to see that if we find an optimal
solution (x*, y*, u*, v*) to a given problem, it
can be obtain by simply ignoring its last two
coordinates.
Specifically, we can rewrite the system of
constraint equations of the problem:
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Design and Analysis of Algorithms
* Property of STI
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If all the coordinates of a basic solution are
nonnegative, the basic solution is called a basic
feasible solution.
Example: If we set variables x and y to zero and
solve the resulting system u and v, we obtain thebasic feasible solution (0, 0, 4, 6); otherwise, we
obtain the basic solution (0, 4, 0, -6), which is not
feasible.
Each such point can be represented by a simplex
tableau, a table storing the information about thebasic feasible solution corresponding to the
extreme point.
Example: The simplex tableau for (0,0,4,6) is:
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Design and Analysis of Algorithms
* Property of STI
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A simplex tableau for a linear programming
problem in standard form with n unknown and m
linear equality constraints ( n ≥ m) has m + 1 rows
and n + 1 columns.
Each of the first m rows of the table contains the
coefficients of a corresponding constraint
equation with the last column’s entry containing
the equations right-hand side.
The columns, except the last one, are labeled bythe name variables.
The rows in the last column are labeled by the
basic variables of this solution.
The columns labeled by the basic variables form n-by-m identity matrix.
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Design and Analysis of Algorithms
* Property of STI
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Th e Sim p lex M eth o d
c The last row of a simplex tableau is calledobjective row , which is initialized by thecoefficients of the objective function with theirsigns reversed and the value of the objectivefunction at the initial point.
c The figure below is the outline of the Simplexmethod:
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Design and Analysis of Algorithms
* Property of STI
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the problem of finding a feasible flow in an s-t
network with the largest possible flow value for a
given weight function [www.answers.com]
a structured on a network but here the arc
capacities, or upper bounds, are the only relevant
parameters; the problem is to find the maximum
flow possible from some given source node to a
given sink node [www.me.utexas.edu]
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Design and Analysis of Algorithms
* Property of STI
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Example 1:
The illustration shows that all arc costs are zero,
but the cost on the arc leaving the sink is set to -1.
Since the goal of the optimization is to minimizecost, the maximum flow possible is delivered to
the sink node.
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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Solution:
The figure shows that the maximum flow from
node 1 to node 8 is 30.
The heavy arcs on the figure are called minimal
cut.
The sum of the capacities of the arcs on the
minimal cut equals the maximum flow is a famous
theorem of network theory called the max flow min
cut theorem.
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Design and Analysis of Algorithms
* Property of STI
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Example 2:
We will assume that the transportation network
can be represented by a connected weighted
digraph with a scenario where an oil company has
the following pipeline network.
Each pipeline is labeled with its maximum flow
rate (in thousands of gallons per hour).
Question: What is the maximum possible flow rate
from A to G?
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Iterative Improvement Algorithm Design Technique
Design and Analysis of Algorithms
* Property of STI
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Assumptions of Maximum Flow Problem:
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It ti I t Al ith D i T h i
Design and Analysis of Algorithms
* P t f STI
Some Applications of Maximum Flow Problem:
1. Maximize the flow through a distribution
network.2. Maximize the flow through a company’s
supply network from its vendors to its
processing facilities.
3. Maximize the flow of oil through a system of
pipelines.
4. Maximize the flow of water through a systemof aqueducts.
5. Maximize the flow of vehicles through a
transportation network.