Linear Programming Models: Graphical Method © 2007 Pearson Education from the companion CD - Chapter 2 of the book: Balakrishnan, Render, and Stair, “Managerial.

Linear Programming Models:Graphical Method

© 2007 Pearson Education

from the companion CD - Chapter 2 of the book:Balakrishnan, Render, and Stair,

“Managerial Decision Modeling with Spreadsheets”, 2nd ed., Prentice-Hall, 2007

http://www.stmartin.edu/

Rev. 2.3 by M. Miccio on January 20, 2015

Fundamental Theorem of Linear Programming

2

(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.

LPThe graphical method

3

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.

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. DataTables

(per table)

Chairs(per chair)

Hours Available

Profit Contribution

$7 $5

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 > 0C > 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)

C < 450 (max # chairs)

T > 100 (min # tables)

T, C > 0 (nonnegativity)

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Graphical MethodGraphing an LP model helps provide insight into LP models and their solutions:A straight line is plotted in place of each disequationA convex and bounded set (hopefully) is generatedAn ideal line, that is a family of parallel lines, is drawn to represent the objective functionThe 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.

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LP-2D.avi

Carpentry

Constraint Line

3T + 4C = 2400

Intercepts

(T = 0, C = 600)

(T = 800, C = 0)

0 800 T

C

600

0

Feasible

< 2400 hrs

Infeasible

> 2400 hrs

3T + 4C = 2400

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Painting

Constraint Line

2T + 1C = 1000

Intercepts

(T = 0, C = 1000)

(T = 500, C = 0)

0 500 800 T

C1000

600

0

2T + 1C = 1000

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0 100 500 800 T

C1000

600

450

0

Max Chair Line

C = 450

Min Table Line

T = 100

Feasible

Region

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0 100 200 300 400 500 T

C

500

400

300

200

100

0

Objective Function Line

z = 7T + 5C Profit

7T + 5C = $2,100

7T + 5C = $4,040

Optimal Point(T = 320, C = 360)7T + 5C

= $2,800

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0 100 200 300 400 500 T

C

500

400

300

200

100

0

Additional Constraint

Need at least 75 more chairs than tables

C > T + 75

or

C – T > 75

T = 320C = 360

No longer feasible

New optimal point

T = 300, C = 375

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LP-2D conclusion

• Optimal Solution: The corner point with the best objective function value is optimal

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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.

LP-2D with MatLab®

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LINEAR PROGRAMMING WITH MATLABcourse by Edward Neuman Department of Mathematics

Southern Illinois University at Carbondale

Function drawfr

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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 activating in the menu bar:Tools >>> Data Cursor

Function drawfrCautions in its use

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function drawfr(c, A, rel, b)

% min (max)z = c*x % Subject to Ax <= b (or Ax >= b), % x >= 0

Negative value of a resourcedrawfr accepts one or more negative resource in the vector b

Unbounded Feasible Regiondrawfr is currently unable of drawing an unbounded region

Equality constraintsA constraint of the typeai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn = bi

must be transformed in 2 constraints of the typeai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≤ bi’

ai1 x1 + ai2 x2 +.....+ aij .xj+.....+ ain xn ≥ bi’’

with bi’ ≈< bi ≈< bi’’

LP-2D aids from Internet

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http://www.authorstream.com/Presentation/bsndev-242949-linear-programming-entertainment-ppt-powerpoint/

http://www.authorstream.com/Presentation/bsndev-242950-linear-programming-example-2-entertainment-ppt-powerpoint/

LP-2D aids from a movie

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LP-2D.avi