第五章线性规划的what-if分析

7/27/2019 What-If

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Table of Contents

Chapter 5 (What-If Analysis for Linear Programming)

Continuing the Wyndor Case Study (Section 5.2) 5.2

Changes in One Objective Function Coefficient (Section 5.3) 5.3 – 5.9Simultaneous Changes in Objective Function Coefficients (Section 5.4) 5.10 – 5.17

Single Changes in a Constraint (Section 5.5) 5.18 – 5.23

Simultaneous Changes in the Constraints (Section 5.6) 5.24 – 5.26

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Wyndor (Before What-If Analysis)

3

4

5

6

7

89

10

11

12

B C D E F G

Doors Window s

Unit Profit $300 $500

Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12Plant 3 3 2 18 <= 18

Doors Window s Total Profit

Units Produced 2 6 $3,600

Hours Used Per Unit Produced

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Using the Spreadsheet to do Sensitivity Analysis

3

4

5

6

7

8

9

10

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12

Plant 3 3 2 18 <= 18




The profit per door has been revised from $300 to $200.

No change occurs in the optimal solution.

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3

4

5

6

7

89

10

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12Plant 3 3 2 18 <= 18




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The profit per door has been revised from $300 to $1,000.

The optimal solution changes.

3

4

5

6

78

9

10

11

12

B C D E F G

Doors Window s

Unit Profit $1,000 $500

Hours Hours

Used Available

Plant 1 1 0 4 <= 4Plant 2 0 2 6 <= 12

Plant 3 3 2 18 <= 18




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Using Solver Table to do Sensitivity Analysis

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4

5

6

7

8

9

10

11

12

1314

15

16

17

18

19

20

21

22

2324

25

26

27

28

B C D E F G

Doors Windows


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12

Plant 3 3 2 18 <= 18

Doors Windows Total Profit


Unit Profit Total

for Doors Doors Windows Profit

2 6 $3,600

$100

$200

$300

$400

$500$600

$700

$800

$900

$1,000


Optimal Units ProducedSelect

hese cells

(B18:E28),

before

choosing

he Solver

Table.

16

17

18

C D E

Total

Doors Window s Profit

=DoorsProduced =WindowsProduced =TotalProfit

Optimal Units Produced

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16

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18

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20

21

22

23

24

25

26

2728

B C D E

Unit Profit Total

for Doors Doors Window s Profit

2 6 $3,600

$100 2 6 $3,200

$200 2 6 $3,400

$300 2 6 $3,600

$400 2 6 $3,800

$500 2 6 $4,000

$600 2 6 $4,200

$700 2 6 $4,400

$800 4 3 $4,700

$900 4 3 $5,100$1,000 4 3 $5,500


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Using the Sensitivity Report to Find the Allowable Range

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$C$12 Units Produced Doors 2 0 300 450 300

$D$12 Units Produced Window s 6 0 500 1E+30 300

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Graphical Insight into the Allowable Range

The two dashed lines that pass through the solid constraint boundary lines are

the objective function lines when P D (the unit profit for doors) is at an endpoint

of its allowable range, 0 ≤ P D ≤ 750.

W

D

(2, 6) is optimal for 0< PD < 750

PD = 0 (Profit = 0 D + 500 W)

PD = 300 (Profit = 300 D + 500 W)

PD = 750 (Profit = 750 D + 500 W)

Line A

Line C

Line B

0 2 4 6

2

4

6

8

Production rate for doors

Production ratefor windows

Feasibleregion

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The profit per window has been revised from $500 to $400.


3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12

Plant 3 3 2 18 <= 18




7/27/2019 What-If



The profit per door has been revised from $300 to $600.The profit per window has been revised from $500 to $300.

The optimal solution changes.

3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 4 <= 4

Plant 2 0 2 6 <= 12

Plant 3 3 2 18 <= 18




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3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

19

20

21

B C D E F G H I

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12

Plant 3 3 2 18 <= 18

Doors Windows Total ProfitUnits Produced 2 6 $3,600

Total Profit Unit Profit for Windows

$3,600 $100 $200 $300 $400 $500

$300

Unit Profit $400

for Doors $500

$600

Hours Used Per Unit

Select

hese cells

(C17:H21),

before

choosing

he Solver

Table.

17

C

=TotalProfit

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16

17

18

19

20

21

B C D E F G H

Total Profit Unit Profit for Windows

$3,600 $100 $200 $300 $400 $500

$300 $1,500 $1,800 $2,400 $3,000 $3,600

Unit Profit $400 $1,900 $2,200 $2,600 $3,200 $3,800

for Doors $500 $2,300 $2,600 $2,900 $3,400 $4,000

$600 $2,700 $3,000 $3,300 $3,600 $4,200

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24

25

26

27

28

29

B C D E F G H

Units Produced (Doors, Windows) Unit Profit for Windows

(2, 6) $100 $200 $300 $400 $500

$300 (4, 3) (4, 3) (2, 6) (2, 6) (2, 6)

Unit Profit $400 (4, 3) (4, 3) (2, 6) (2, 6) (2, 6)

for Doors $500 (4, 3) (4, 3) (4, 3) (2, 6) (2, 6)

$600 (4, 3) (4, 3) (4, 3) (4, 3) (2, 6)

25

C

="(" & DoorsProduced & ", " & WindowsProduced & ")

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The 100 Percent Rule

The 100 Percent Rule for Simultaneous Changes in Objective Function

Coefficients: If simultaneous changes are made in the coefficients of the objectivefunction, calculate for each change the percentage of the allowable change

(increase or decrease) for that coefficient to remain within its allowable range. If

the sum of the percentage changes does not exceed 100 percent, the original

optimal solution definitely will still be optimal. (If the sum does exceed 100

percent, then we cannot be sure.)

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Graphical Insight into 100 Percent Rule

W

D0 2 4 6

2

4

6

8


Production rate

for windows

Feasible

region

10

Objective func tion line now is

Profit = $3150 = 525 D + 350 W

since PD = $525, PW = $350.

Entire line segment is optimal

(4, 3)

(2, 6)

8

The estimates of the

unit profits for doors

and windows change

to P D = $525 and

P W = $350, which lies

at the edge of what is

allowed by the 100

percent rule.

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Graphical Insight into 100 Percent Rule

When the estimates of the unit profits for doors and windows change to P D =

$150 and P W = $250 (half their original values), the graphical method shows

that the optimal solution still is ( D, W ) = (2, 6) even though the 100 percent

rule says that the optimal solution might change.

0 2 4 6

2

4

6

8

(2, 6)

Feasibleregion

Optimal solution


Production ratefor windows

Profit = $1800 = 150D+ 250 W

8

W

D

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The hours available in plant 2 have been increased from 12 to 13.The total profit increases by $150 per week.

3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 1.667 <= 4

Plant 2 0 2 13 <= 13

Plant 3 3 2 18 <= 18


Units Produced 1.667 6.5 $3,750


7/27/2019 What-If



The hours available in plant 2 have been further increased from 13 to 18.The total profit increases by $750 per week ($150 per hour added in plant 2).

3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 0 <= 4

Plant 2 0 2 18 <= 18

Plant 3 3 2 18 <= 18




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The hours available in plant 2 have been further increased from 18 to 20.The total profit does not increase any further.

3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 0 <= 4

Plant 2 0 2 18 <= 20

Plant 3 3 2 18 <= 18




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3

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5

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31

32

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34

35

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4

Plant 2 0 2 12 <= 12

Plant 3 3 2 18 <= 18



Time Available Total Incremental

in Plant 2 (hours) Doors Windows Prof it Prof it

2 6 $3,600

4 4 2 $2,200

5 4 2.5 $2,450 $250

6 4 3 $2,700 $250

7 3.667 3.5 $2,850 $150

8 3.333 4 $3,000 $150

9 3 4.5 $3,150 $150

10 2.667 5 $3,300 $150

11 2.333 5.5 $3,450 $15012 2 6 $3,600 $150

13 1.667 6.5 $3,750 $150

14 1.333 7 $3,900 $150

15 1 7.5 $4,050 $150

16 0.667 8 $4,200 $150

17 0.333 8.5 $4,350 $150

18 0 9 $4,500 $150

19 0 9 $4,500 $0

20 0 9 $4,500 $0



Select

these cell s

(B18:E35),

before

choosing

the Solver

Table.

7/27/2019 What-If


Using the Sensitivity Report

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$C$12 Units Produced Doors 2 0 300 450 300

$D$12 Units Produced Window s 6 0 500 1E+30 300

Constraints

Final Shadow Constraint Allowable AllowableCell Name Value Price R.H. Side Increase Decrease

$E$7 Plant 1 Used 2 0 4 1E+30 2

$E$8 Plant 2 Used 12 150 12 6 6

$E$9 Plant 3 Used 18 100 18 6 6

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Graphical Interpretation of the Allowable Range

0 2 4 6

2

4

6

8

2 W = 6 Profit = 300 (4) + 500 (3) = $2,7

2 W = 18 Profit = 300 (0) + 500 (9) = $4,500

2 W = 12 Profit = 300 (2) + 500 (6) = $3,600

(4, 3)

(2, 6)

Feasible

region for

original problem

Line B

Line A (D = 4)

Line C (3 D + 2 W = 18)

10

(0, 9)

D

W


Production r atefor windows

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One available hour in plant 3 has been shifted to plant 2.The total profit increases by $50 per week.

3

4

5

6

7

8

910

11

12

B C D E F G

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 1.333 <= 4

Plant 2 0 2 13 <= 13

Plant 3 3 2 17 <= 17


Units Produced 1.333 6.5 $3,650


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11

12

13

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18

19

20

2122

23

24

25

26

B C D E F G H

Doors Window s


Hours Hours

Used Available

Plant 1 1 0 2 <= 4 Total (Plants 2 & 3)

Plant 2 0 2 12 <= 12 30

Plant 3 3 2 18 <= 18


Units Produced 2.000 6 $3,600

Time Available Time Available Total Incremental

in Plant 2 (hours) in Plant 3 (hours) Doors Window s Profit Profit

2 6 $3,600

12 18 2 6 $3,600

13 17 1.333 6.5 $3,650 $5014 16 0.667 7 $3,700 $50

15 15 0 7.5 $3,750 $50

16 14 0 7 $3,500 -$250

17 13 0 6.5 $3,250 -$250

18 12 0 6 $3,000 -$250



Select

these cell s

(C19:F26),before

choosing

the Solver

Table.

7/27/2019 What-If


The 100 Percent Rule

The 100 Percent Rule for Simultaneous Changes in Right-Hand Sides: The

shadow prices remain valid for predicting the effect of simultaneously changingthe right-hand sides of some of the functional constraints as long as the changes

are not too large. To check whether the changes are small enough, calculate for

each change the percentage of the allowable change (decrease or increase) for that

right-hand side to remain within its allowable range. If the sum of the percentage

changes does not exceed 100 percent, the shadow prices definitely will still be

valid. (If the sum does exceed 100 percent, then we cannot be sure.)

第五章 线性规划的what-if分析

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第五章线性规划的what-if分析