Table of Contents Chapter 5 (What-If Analysis for Linear Programming) Continuing the Wyndor Case...

Table of ContentsChapter 5 (What-If Analysis for Linear Programming)

Continuing the Wyndor Case Study (Section 5.2)

5.2Changes in One Objective Function Coefficient (Section 5.3)

5.3–5.9Simultaneous Changes in Objective Function Coefficients (Section 5.4)

5.10–5.17Single Changes in a Constraint (Section 5.5)

5.18–5.23Simultaneous Changes in the Constraints (Section 5.6)

5.24–5.26

Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

Wyndor (Before What-If Analysis)

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B C D E F GDoors Windows

Unit Profit $300 $500Hours HoursUsed Available

Plant 1 1 0 2 <= 4Plant 2 0 2 12 <= 12Plant 3 3 2 18 <= 18

Doors Windows Total ProfitUnits Produced 2 6 $3,600

Hours Used Per Unit Produced

5-2

Using the Spreadsheet to do Sensitivity Analysis

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The profit per door has been revised from $300 to $200.No change occurs in the optimal solution.

5-3


The profit per door has been revised from $300 to $500.No change occurs in the optimal solution.

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5-4


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

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Unit Profit $1,000 $500Hours HoursUsed Available




5-5

Using Solver Table to do Sensitivity Analysis

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Unit Profit Totalfor Doors Doors Windows Profit

2 6 $3,600$100$200$300$400$500$600$700$800$900

$1,000


Optimal Units ProducedSelect these cells (B18:E28), before choosing the Solver Table.

161718

C D ETotal

Doors Windows Profit=DoorsProduced =WindowsProduced =TotalProfit

Optimal Units Produced

5-6


16171819202122232425262728

B C D EUnit Profit Totalfor Doors Doors Windows 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


5-7

Using the Sensitivity Report to Find the Allowable Range

Variable CellsFinal 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 Windows 6 0 500 1E+30 300

5-8

Graphical Insight into the Allowable Range

The two dashed lines that pass through the solid constraint boundary lines are the objective function lines when PD (the unit profit for doors) is at an endpoint of its allowable range, 0 ≤ PD ≤ 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

5-9


The profit per door has been revised from $300 to $450.The profit per window has been revised from $500 to $400.No change occurs in the optimal solution.

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5-10


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.

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5-11


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B C D E F G H IDoors Windows




Total Profit Unit Profit for Windows$3,600 $100 $200 $300 $400 $500$300

Unit Profit $400for Doors $500

$600

Hours Used Per Unit

Select these cells (C17:H21), before choosing the Solver Table.

17C

=TotalProfit

5-12


161718192021

B C D E F G HTotal 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 $3,800for Doors $500 $2,300 $2,600 $2,900 $3,400 $4,000

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

5-13


242526272829

B C D E F G HUnits 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)

25C

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

5-14

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 objective function, 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.)

5-15

Graphical Insight into 100 Percent Rule

W

D0 2 4 6

2

4

6

8


Production rate

for windows

Feasible

region

10

Objective function 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 PD = $525 and PW = $350, which lies at the edge of what is allowed by the 100 percent rule.

5-16

Graphical Insight into 100 Percent Rule

When the estimates of the unit profits for doors and windows change to PD = $150 and PW = $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)

Feasible region

Optimal solution


Production rate for windows

Profit = $1800 = 150D + 250 W

8

W

D

5-17


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

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Plant 1 1 0 1.667 <= 4Plant 2 0 2 13 <= 13Plant 3 3 2 18 <= 18

Doors Windows Total ProfitUnits Produced 1.667 6.5 $3,750


5-18


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).

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5-19


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

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5-20


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Time Available Total Incrementalin Plant 2 (hours) Doors Windows Profit Profit

2 6 $3,6004 4 2 $2,2005 4 2.5 $2,450 $2506 4 3 $2,700 $2507 3.667 3.5 $2,850 $1508 3.333 4 $3,000 $1509 3 4.5 $3,150 $15010 2.667 5 $3,300 $15011 2.333 5.5 $3,450 $15012 2 6 $3,600 $15013 1.667 6.5 $3,750 $15014 1.333 7 $3,900 $15015 1 7.5 $4,050 $15016 0.667 8 $4,200 $15017 0.333 8.5 $4,350 $15018 0 9 $4,500 $15019 0 9 $4,500 $020 0 9 $4,500 $0



Select these cells (B18:E35), before choosing the Solver Table.

5-21

Using the Sensitivity Report

Variable CellsFinal 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 Windows 6 0 500 1E+30 300

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell 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

5-22

Graphical Interpretation of the Allowable Range

0 2 4 6

2

4

6

8

2 W = 6 Profit = 300 (4) + 500 (3) = $2,700

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 rate for windows

5-23


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

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Plant 1 1 0 1.333 <= 4Plant 2 0 2 13 <= 13Plant 3 3 2 17 <= 17

Doors Windows Total ProfitUnits Produced 1.333 6.5 $3,650


5-24


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B C D E F G HDoors Windows


Plant 1 1 0 2 <= 4 Total (Plants 2 & 3)Plant 2 0 2 12 <= 12 30Plant 3 3 2 18 <= 18

Doors Windows Total ProfitUnits Produced 2.000 6 $3,600

Time Available Time Available Total Incrementalin Plant 2 (hours) in Plant 3 (hours) Doors Windows Profit Profit

2 6 $3,60012 18 2 6 $3,60013 17 1.333 6.5 $3,650 $5014 16 0.667 7 $3,700 $5015 15 0 7.5 $3,750 $5016 14 0 7 $3,500 -$25017 13 0 6.5 $3,250 -$25018 12 0 6 $3,000 -$250



Select these cells (C19:F26), before choosing the Solver Table.

5-25

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 changing the 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.)

5-26

Table of Contents Chapter 5 (What-If Analysis for Linear Programming) Continuing the Wyndor Case...

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