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(1)

Linear Programming:

Computer Solution and

Sensitivity Analysis

(2)

Chapter Topics

Computer Solution

(3)

Early linear programming used lengthy manual

mathematical solution procedure called the Simplex

Method (See CD-ROM Module A).

Steps of the Simplex Method have been programmed in

software packages designed for linear programming

problems.

Many such packages available currently.

Used extensively in business and government.

Text focuses on Excel Spreadsheets and QM for

Windows.

(4)

Beaver Creek Pottery Example

Excel Spreadsheet

–

Data Screen (1 of 6)

(5)

Beaver Creek Pottery Example

“Solver” Parameter Screen

(2 of 6)

(6)

Exhibit 3.3

Beaver Creek Pottery Example

(7)

Beaver Creek Pottery Example

“Solver” Settings (4 of 6)

(8)

Exhibit 3.5

(9)

Beaver Creek Pottery Example

Answer Report (6 of 6)

(10)

Linear Programming Problem: Standard Form

Standard form

requires all variables in the constraint equations to

appear on the left of the inequality (or equality) and all numeric

values to be on the right-hand side.

Examples:

x

3

x

1

+ x

2

must be converted to x

3

- x

1

- x

2

0

x

1

/(x

2

+ x

3

)

2 becomes x

1

2 (x

2

+ x

3

)

(11)

Beaver Creek Pottery Example

QM for Windows (1 of 5)

(12)

Beaver Creek Pottery Example

QM for Windows

–

Data Set Creation (2 of 5)

(13)

Beaver Creek Pottery Example

QM for Windows: Data Table (3 of 5)

(14)

Beaver Creek Pottery Example

QM for Windows: Model Solution (4 of 5)

(15)

Beaver Creek Pottery Example

QM for Windows: Graphical Display (5 of 5)

(16)

Sensitivity analysis

determines the effect on the optimal solution

of

changes in parameter values

of the objective function and

constraint equations.

Changes may be reactions to anticipated uncertainties in the

parameters or to new or changed information concerning the

model.

Beaver Creek Pottery Example

(17)

Maximize Z = $40x

1

+ $50x

2

(18)

Maximize Z =

$100x

1

+ $50x

2

subject to: x

1

+ 2x

2

 40

4x

1

+ 3x

2

120

x

1

, x

2

 0

Figure 3.2 Changing the x

1

Objective Function Coefficient

Beaver Creek Pottery Example

(19)

Maximize Z = $40x

1

+

$100x

2

subject to: x

1

+ 2x

2

 40

4x

1

+ 3x

2

120

x

1

, x

2

 0

Figure 3.3 Changing the x

2

Objective Function Coefficient

Beaver Creek Pottery Example

(20)

The

sensitivity range

for an objective function coefficient is the

range of values

over which the current optimal solution point will

remain optimal

.

The sensitivity range for the x

i

coefficient is designated as c

i.

(21)

objective function Z = $40x

1

+ $50x

2

sensitivity range for:

x

1

: 25

c

1

66.67

x

2

: 30

c

2

80

Figure 3.4 Determining the Sensitivity Range for c

Objective Function Coefficient

(22)

Minimize Z = $6x

1

+ $3x

2

Fertilizer Cost Minimization Example (3 of 3)

(23)

Exhibit 3.12

Objective Function Coefficient Ranges

(24)

Exhibit 3.13

Objective Function Coefficient Ranges

(25)

Exhibit 3.14

Objective Function Coefficient Ranges

QM for Windows Sensitivity Range Screen (3 of 3)

Sensitivity ranges

for objective

(26)

Changes in Constraint Quantity Values

Sensitivity Range (1 of 4)

The

sensitivity range for a right-hand-side

value is the

range of values over which the quantity’s value can change

(27)

Changes in Constraint Quantity Values

Increasing the Labor Constraint (2 of 4)

Maximize Z = $40x

1

+ $50x

2

subject to: x

1

+ 2x

2

+ s

1

= 40

4x

1

+ 3x

2

+ s

2

= 120

x

1

, x

2

 0

(28)

Changes in Constraint Quantity Values

Sensitivity Range for Labor Constraint (3 of 4)

(29)

Changes in Constraint Quantity Values

Sensitivity Range for Clay Constraint (4 of 4)

(30)

Exhibit 3.15

(31)

Exhibit 3.16

(32)

Changing individual constraint parameters

Adding new constraints

Adding new variables

(33)

Other Forms of Sensitivity Analysis

Changing a Constraint Parameter (2 of 4)

Maximize Z = $40x

1

+ $50x

2

subject to: x

1

+ 2x

2

 40

4x

1

+ 3x

2

120

x

1

, x

2

 0

(34)

Adding a new constraint to Beaver Creek Model:

0.20x

1

+ 0.10x

2

 5 hours for packaging

Original solution: 24 bowls, 8 mugs, $1,360 profit

Exhibit 3.17

Other Forms of Sensitivity Analysis

(35)

Adding a new variable to the Beaver Creek model, x

3

, for a third

Solving model shows that change has no effect on the original solution

(i.e., the model is not sensitive to this change).

(36)

Defined as the

marginal value

of one additional unit

of resource.

The

sensitivity range

for a constraint quantity value is

(37)

Maximize Z = $40x

1

+ $50x

2

subject to:

x

1

+ 2x

2

40 hr of labor

4x

1

+ 3x

2

120 lb of clay

x

1

, x

2

0

Exhibit 3.18

(38)

Excel Sensitivity Report for Beaver Creek Pottery

Solution Screen (2 of 2)

(39)

Two airplane parts: no.1 and no. 2.

(40)

Maximize Z = $650x

1

+ $910x

2

(41)

Example Problem

(42)

Gambar

Optimal Solution PointFigure 3.1
Figure 3.2 Changing the x1 Objective Function Coefficient
Figure 3.3 Changing the x2 Objective Function Coefficient
Figure 3.4 Determining the Sensitivity Range for c1 Copyright © 2010 Pearson Education, Inc
+6

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