#### Transcript Document

```Linear Programming Models:
Graphical Method
from the companion CD - Chapter 2 of the book:
Balakrishnan, Render, and Stair,
2nd ed., Prentice-Hall, 2007
http://www.stmartin.edu/
Rev. 2.3 by M. Miccio on January 20, 2015
Fundamental Theorem
of Linear Programming
(stated here in two variables)
A linear expression ax + by, defined over a closed
bounded convex set S whose sides are line segments,
takes on its maximum value at a vertex of S and its
minimum value at a vertex of S. If S is unbounded,
there may or may not be an optimum value, but if there
is, it occurs at a vertex.
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LP
The graphical method
Using a graphical presentation
we can represent all the constraints,
the objective function, and the three
types of feasible points.
from the companion CD of the book:
Lawrence and Pasternack, “Applied Management Science: Modeling, Spreadsheet Analysis,
and Communication for Decision Making”, 2nd Edition, © 2002 John Wiley & Sons Inc.
3
Example LP Model Formulation:
The Product Mix Problem
Decision: How much to make of > 2 products?
Objective: Maximize profit
Constraints: Limited resources
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Example: Flair Furniture Co.
Two products: Chairs and Tables
Decision: How many of each to make this
month?
Objective: Maximize profit
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Flair Furniture Co. Data
Tables
Chairs
(per table)
(per chair)
Profit
Contribution
\$7
\$5
Hours
Available
Carpentry
3 hrs
4 hrs
2400
Painting
2 hrs
1 hr
1000
Other Limitations:
• Make no more than 450 chairs
• Make at least 100 tables
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Decision Variables:
T = Num. of tables to make
C = Num. of chairs to make
Objective Function: Maximize Profit
Maximize \$7 T + \$5 C
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Constraints:
• Have 2400 hours of carpentry time
available
3 T + 4 C < 2400 (hours)
• Have 1000 hours of painting time available
2 T + 1 C < 1000 (hours)
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More Constraints:
• Make no more than 450 chairs
C < 450
(num. chairs)
• Make at least 100 tables
T > 100
(num. tables)
Nonnegativity:
Cannot make a negative number of chairs or tables
T>0
C>0
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Model Summary
z = 7T + 5C max!
(profit)
Subject to the constraints:
3T + 4C < 2400
(carpentry hrs)
2T + 1C < 1000
(painting hrs)
T
C < 450
(max # chairs)
> 100
(min # tables)
T, C > 0
(nonnegativity)
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Graphical Method
Graphing an LP model helps provide insight into LP models
and their solutions:
A straight line is plotted in place of each disequation
A convex and bounded set (hopefully) is generated
An ideal line, that is a family of parallel lines, is drawn to
represent the objective function
The optimum is found at the interception of the ideal line
with a vertex
 While this can only be done in two dimensions, the same properties apply to all LP
models and solutions.
LP-2D.avi
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Carpentry
Constraint Line
C
3T + 4C = 2400
Infeasible
> 2400 hrs
600
Intercepts
(T = 0, C = 600)
(T = 800, C = 0)
Feasible
< 2400 hrs
0
0
800 T
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C
1000
Painting
Constraint Line
2T + 1C = 1000
600
Intercepts
(T = 0, C = 1000)
(T = 500, C = 0)
0
0
500
800 T
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Max Chair Line
C
1000
C = 450
Min Table Line
600
450
T = 100
Feasible
0
Region
0 100
500
800 T
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C
Objective Function
Line
z = 7T + 5C Profit
500
Optimal Point
(T = 320, C = 360)
400
300
200
100
0
15
0
100
200
300
400
500 T
New optimal
point
T = 300, C = 375
C
Need at least 75
more chairs than
tables
500
400
T = 320
C = 360
No longer
feasible
C > T + 75
or
300
C – T > 75
200
100
0
16
0
100
200
300
400
500 T
LP-2D conclusion
Extreme Point theorem :
Any LP problem with a nonempty bounded
feasible region has an optimal solution;
moreover, an optimal solution can always be
found at an (or at least one) Corner Point
(extreme point) of the problem's feasible region.
• Optimal Solution: The corner point with
the best objective function value is optimal
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LP-2D with MatLab®
LINEAR PROGRAMMING WITH MATLAB
course by Edward Neuman
Department of Mathematics
Southern Illinois University at Carbondale
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Function drawfr
function drawfr(c, A, rel, b)
% Graphs of the feasible region and the line level
% of the LP problem with two legitimate variables
%
%
min (max)z = c*x
%
Subject to Ax <= b (or Ax >= b),
%
x >= 0
% Enter a sequence of instructions like these into the COMMAND
WINDOW:
%
c=[1 2];
%
A=[-1 3; 1 1; 1 -1; 1 3; 2 1];
%
rel='<<<>>';
%
b=[10; 6; 2; 6; 4];
% NB:
% b must be a COLUMN vector
% components of b vector can
% indifferently be <0 or >0
It is possible to read the vertex
coordinates in the Figure window by
Tools >>> Data Cursor
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Function drawfr
Cautions in its use
function drawfr(c, A, rel, b)
%
%
%
min (max)z = c*x
Subject to Ax <= b (or Ax >= b),
x >= 0
 Negative value of a resource
drawfr accepts one or more negative resource in the vector b
 Unbounded Feasible Region
drawfr is currently unable of drawing an unbounded region
 Equality constraints
A constraint of the type
ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn = bi
must be transformed in 2 constraints of the type
ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≤ bi’
ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≥ bi’’
with bi’ ≈< bi ≈< bi’’
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LP-2D aids from Internet
http://www.authorstream.com/Presentation/bsndev-242949-linearprogramming-entertainment-ppt-powerpoint/
http://www.authorstream.com/Presentation/bsndev-242950-linearprogramming-example-2-entertainment-ppt-powerpoint/
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LP-2D aids from a movie
LP-2D.avi
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