Linear Programming _Graph (1)

7/28/2019 Linear Programming _Graph (1)

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To accompany Quantitative Analysis

for Management, 8e

by Render/Stair/Hanna7-1

2003 by Prentice Hall, Inc.

Upper Saddle River, NJ 07458

Linear Programming

Models: Graphical

and Computer

Methods


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Examples of SuccessfulLP Applications

1. Development of a production schedule

that will satisfy future demands for a

firms production and at the same timeminimize total production and inventory

costs

2. Selection of the product mix in a

factory to make best use of machine-

hours and labor-hours available while

maximizingthe firms products


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Examples of SuccessfulLP Applications

3. Determination of grades of petroleum

products to yield the maximum profit

4. Selection of different blends of raw

materials to feed mills to produce

finished feed combinations at minimum

cost

5. Determination of a distribution system

that will minimize total shipping cost

from several warehouses to various

market locations


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Requirements of a LinearProgramming Problem

All problems seek to maximize or

minimize some quantity (the

objective function).

The presence of restrictions orconstraints, limits the degree to

which we can pursue our objective.

There must be alternative courses ofaction to choose from.

The objective and constraints in

linear programming problems must

be expressed in terms of linear

equations or inequalities.


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Basic Assumptions ofLinear Programming

Certainty

Proportionality

Additivity

Divisibility

Nonnegativity


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Flair Furniture CompanyData - Table 7.1

Hours Required to Produce One Unit

Department

T

Tables

C

Chairs

Available

Hours ThisWeek

Carpentry

Painting

&Varnishing

4

2

3

1

240

100

Profit Amount $7 $5

Constraints: 4T + 3C 240 (Carpentry)

2T + 1C 100 (Paint & Varnishing)

Objective: Max: 7T + 5C


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Flair Furniture CompanyConstraints

Number of Tables

120

100

80

60

40

20

0

NumberofChairs

20 40 60 80 100

Painting/Varnishing

Carpentry


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Flair Furniture CompanyFeasible Region

120

100

80

60

40

20

0

NumberofChairs

20 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

Feasible

Region


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Flair Furniture CompanyIsoprofit Lines

Number of Tables

Num

berofChairs

120

100

80

60

40

20

020 40 60 80 100

Painting/Varnishing

Carpentry

7T+ 5C= 210

7T+ 5C= 420


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Flair Furniture CompanyOptimal Solution

NumberofChairs

120

100

80

60

40

20

020 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

Solution

(T= 30, C= 40)

Isoprofit Lines


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Flair Furniture CompanyOptimal Solution

NumberofChairs

120

100

80

60

40

20

020 40 60 80 100

Number of Tables

Painting/Varnishing

Carpentry

Solution

(T= 30, C= 40)

Corner Points

1

2

3

4


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Flair Furniture - Excel


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Holiday Meal TurkeyRanch

(C)

(B)

toSubject

:Minimize

48900

3

1

2

2

2

11/2

+

X

XX

A)(XX:

XX


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Holiday Meal Turkey

Problem

Corner Points


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Holiday Meal Turkey

Problem

Isoprofit Lines


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Special Cases in LP

Infeasibility

Unbounded Solutions

Redundancy/Degeneracy

More Than One Optimal Solution


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A Problem with NoFeasible Solution

X2

X1

8

6

4

2

0

2 4 6 8

Region Satisfying

3rd Constraint

Region Satisfying First 2 Constraints


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A Problem with aRedundant ConstraintX2

X1

30

25

20

15

10

5

0

5 10 15 20 25 30

Feasible

Region

2X1

+X2

< 30

X1 < 25

X1 +X2 < 20

Redundant

Constraint


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An Example of AlternateOptimal Solutions

8

7

6

5

4

3

2

1

01 2 3 4 5 6 7 8

Optimal Solution Consists ofAll Combinations ofX1 and

X2 Along theAB Segment

Isoprofit Line

for $12

Overlays Line

Segment

Isoprofit Line for $8A

B

AB


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Sensitivity Analysis

Changes in the Objective

Function Coefficient

Changes in Resources (RHS)

Changes in Technological

Coefficients


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Changes in the TechnologicalCoefficients for High Note

Sound Co.

X1

StereoReceivers

60

40

20

0

CD Players

20 40

X2

(a) Original Problem

3X1 + 1X2 < 60

Optimal Solution

a2X1 + 4X2 < 80

b

c

20 40

X2

X1

(c) Change in Circle

Coefficient

3X1 + 1X2 < 60

Optimal Solution

f

2X1 + 5X2 < 80

g

c

CD Players


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T Q tit ti A l i 2003 b P ti H ll I

Part A, November 2005

Documents

Linear Programming _Graph (1)