mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2008 5.1 table of contents chapter 5 (what-if...

© The McGraw-Hill Companies, Inc., 2008

5.1McGraw-Hill/Irwin

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

Sensitivity Analysis (UW Lecture) 5.27–5.43These slides are based upon a lecture to first-year MBA students at the University of Washington that discusses sensitivity analysis for linear programming models (as taught by one of the authors).



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



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.




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

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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






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




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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




Using the Sensitivity Report to Find the Allowable Range

Adjustable 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



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




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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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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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




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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




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 & ")"



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.)



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.



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




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





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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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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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.



Using the Sensitivity Report


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



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




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





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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.



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.)



A Production Problem

Weekly supply of raw materials:

6 Large Bricks

Products:

TableProfit = $20 / Table

ChairProfit = $15 / Chair

8 Small Bricks



Sensitivity Analysis Questions

• With the given weekly supply of raw materials and profit data, how many tables and chairs should be produced? What is the total weekly profit?

• What if one more large brick were available. How much would you be willing to pay for it?

• What if an additional two large bricks were available (to make a total of 9). How much would you be willing to pay for these two additional bricks?

• What if the profit per table were now $25. (Assume now there are only 6 large bricks again.) How many tables and chairs should now be produced?

• What if the profit per table were now $35. How many tables and chairs should now be produced?



Graphical Solution (Original Problem)

Maximize Profit = ($20)T + ($15)Csubject to

2T + C ≤ 6 large bricks2T + 2C ≤ 8 small bricks

andT ≥ 0, C ≥ 0.

1 2 3 4 5 6

1

2

3

4

T

C

2T + 2C < 8 small bricks

2T + C < 6 large bricks

Z = ($20)T + ($15)C = $70

Optimal Solution (2, 2). Profit = $70



7 Large Bricks



andT ≥ 0, C ≥ 0.

1 2 3 4 5 6

1

2

3

4

T

C



Z = ($20)T + ($15)C = $75

Old Optimal Solution (2, 2). Profit = $70

New Optimal Solution (3, 1). Profit = $75




9 Large Bricks



andT ≥ 0, C ≥ 0.

1 2 3 4 5 6

1

2

3

4

T

C



Z = ($20)T + ($15)C = $80






$25 Profit per Table



andT ≥ 0, C ≥ 0.

1 2 3 4 5 6

1

2

3

4

T

C



Z = ($25)T + ($15)C = $80

Optimal Solution (2, 2). Profit = $80






andT ≥ 0, C ≥ 0.

1 2 3 4 5 6

1

2

3

4

T

C


Z = ($35)T + ($15)C = $105





Generating the Sensitivity Report

After solving with Solver, choose “Sensitivity” under reports:

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B C D E F GTables Chairs

Profit $20.00 $15.00

Total Used AvailableLarge Bricks 2 1 6 <= 6Small Bricks 2 2 8 <= 8

Tables Chairs Total ProfitProduction Quantity: 2 2 $70.00

Bill of Materials



The Sensitivity Report


Cell Name Value Cost Coefficient Increase Decrease$C$11 Production Quantity: Tables 2 0 20 10 5$D$11 Production Quantity: Chairs 2 0 15 5 5


Cell Name Value Price R.H. Side Increase Decrease$E$7 Large Bricks Total Used 6 5 6 2 2$E$8 Small Bricks Total Used 8 5 8 4 2

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Profit $20.00 $15.00



Bill of Materials



The Sensitivity Report





The solutionAllowable range

(Solution stays the same)

Usage of the resource(Left-hand-side of constraint)

Allowable range(Shadow price is valid)

Increase in objective function value per unit increase in right-hand-side (RHS)∆Z = (shadow price)(∆RHS)




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Profit $35.00 $15.00



Bill of Materials


Cell Name Value Cost Coefficient Increase Decrease$C$11 Production Quantity: Tables 3 0 35 1E+30 5$D$11 Production Quantity: Chairs 0 -2.5 15 2.5 1E+30


Cell Name Value Price R.H. Side Increase Decrease$E$7 Large Bricks Total Used 6 17.5 6 2 6$E$8 Small Bricks Total Used 6 0 8 1E+30 2



7 Large Bricks





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Profit $20.00 $15.00



Bill of Materials



9 Large Bricks

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Profit $20.00 $15.00



Bill of Materials


Cell Name Value Cost Coefficient Increase Decrease$C$11 Production Quantity: Tables 4 0 20 1E+30 5$D$11 Production Quantity: Chairs 0 -5 15 5 1E+30


Cell Name Value Price R.H. Side Increase Decrease$E$7 Large Bricks Total Used 8 0 9 1E+30 1$E$8 Small Bricks Total Used 8 10 8 1 8



100% Rule for Simultaneous Changesin the Objective Coefficients

For simultaneous changes in the objective coefficients, if the sum of the percentage changes does not exceed 100%, the original solution will still be optimal. (If it does exceed 100%, we cannot be sure—it may or may not change.)





Examples: (Does solution stay the same?)Profit per Table = $24 & Profit per Chair = $13Profit per Table = $25 & Profit per Chair = $12Profit per Table = $28 & Profit per Chair = $18



100% Rule for Simultaneous Changesin the Right-Hand-Sides

For simultaneous changes in the right-hand-sides, if the sum of the percentage changes does not exceed 100%, the shadow prices will still be valid. (If it does exceed 100%, we cannot be sure—they may or may not be valid.)





Examples: (Are the shadow prices valid? If so, what’s the new total profit?)(+1 Large Brick) & (+2 Small Bricks)(+1 Large Brick) & (–1 Small Brick)



Summary of Sensitivity Report for Changes in the Objective Function Coefficients

• Final Value– The value of the decision variables (changing cells) in the optimal solution.

• Reduced Cost– Increase in the objective function value per unit increase in the value of a zero-

valued variable (for small increases)—may be interpreted as the shadow price for the nonnegativity constraint.

• Objective Coefficient– The current value of the objective coefficient.

• Allowable Increase/Decrease– Defines the range of the coefficients in the objective function for which the current

solution (value of the decision variables or changing cells in the optimal solution) will not change.



Summary of Sensitivity Report for Changes in the Right-Hand-Sides

• Final Value– The usage of the resource (or level of benefit achieved) in the optimal solution—the

left-hand side of the constraint.

• Shadow Price– The change in the value of the objective function per unit increase in the right-hand-

side of the constraint (RHS):∆Z = (Shadow Price)(∆RHS)

(Note: only valid if change is within the allowable range—see below.)

• Constraint R.H. Side– The current value of the right-hand-side of the constraint.

• Allowable Increase/Decrease– Defines the range of values for the RHS for which the shadow price is valid and

hence for which the new objective function value can be calculated. (NOT the range for which the current solution will not change.)

mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2008 5.1 table of contents chapter 5 (what-if...

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