Session 2a. Decision Models -- Prof. Juran2 Overview Sensitivity Analysis –Goal Seek and Data...

Session 2a

Decision Models -- Prof. Juran

2

Overview

• Sensitivity Analysis– Goal Seek and Data Table– Marketing and Finance examples

• Call Center LP• More Sensitivity Analysis

– SolverTable


3

Sensitivity Analysis

• How do key outputs change in response to changes in inputs?

• Which inputs are the most important?

• How robust is our decision?


4

Finance Example

• A European call option on a stock earns the owner an amount equal to the price at expiration minus the exercise price, if the price of the stock on which the call is written exceeds the exercise price. Otherwise, the call pays nothing.

• A European put option earns the owner an amount equal to the exercise price minus the price at expiration, if the price at expiration is less than the exercise price. Otherwise the put pays nothing.


5

Finance Example

• The Black-Scholes formula calculates the price of a European options based on the following inputs: – today's stock price– the duration of the option (in years)– the option's exercise price– the risk-free rate of interest (per year)– the annual volatility (standard deviation) in

stock price


6

Managerial Problem Definition

How do the parameters in Black-Scholes affect the option price?


7

FormulationT h e B lack -Sch o les m o d el :

21 dNEedSNC rt

w h ere: S = cu r ren t stock p r ice E = exercise p r ice r = r isk -free rate o f retu rn σ2 = v ar ian ce o f th e stock ’s retu rn t = tim e to exp i ration

d1 =

t

trES

2

2

2ln

d2 = td 21

N (d) = p robabi l i ty th at z < d


8

1

234567

89

10111213

141516

A B C D E F G HInputs

Type of option (1 for call, 2 for put) 1Stock price 35Exercise price 40Duration (years) 0.5Riskfree interest rate 0.05Volatility 0.4

Quantities for Black-Scholes formulad1 -0.242 N(d1) 0.404d2 -0.525 N(d2) 0.300

Option price 2.456

=(LN(B3/ B4)+(B6+B7^2/ 2)*B5)/ (B7*SQRT(B5))

=B10-SQRT(B7^2*B5)

=IF(B2=1,NORMSDIST(B10),NORMSDIST(-B10))

=IF(B2=1,NORMSDIST(B11),NORMSDIST(-B11))

=IF(B2=1,B3*E10-B4*EXP(-B5*B6)*E11,-(B3*E10-B4*EXP(-B5*B6)*E11))

Solution Methodology

Notice the use of “if” statements in cells E10:E11 and B13, so that the same model can be used for both puts and calls.


9

Data Table

• Similar to copying a formula over many cells, but better for complicated functions (e.g. Black-Scholes)

• Specify Row and/or Column Input Cells

• Tricky to learn, but worth it


10


1

234567

89

10111213

141516

A B C D EInputs

Type of option (1 for call, 2 for put) 1Stock price 35Exercise price 40Duration (years) 0.5Riskfree interest rate 0.05Volatility 0.4

Quantities for Black-Scholes formulad1 -0.242 N(d1) 0.404d2 -0.525 N(d2) 0.300

Option price 2.456

Volatility Price2.456

=B13


11

Solution Methodology1516171819202122232425262728293031323334353637

A BVolatility Price

2.4560.010.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00


12



13


A BVolatility Price

2.4560.01 0.0000.05 0.0000.10 0.0710.15 0.3120.20 0.6640.25 1.0750.30 1.5180.35 1.9810.40 2.4560.45 2.9390.50 3.4260.55 3.9170.60 4.4100.65 4.9030.70 5.3970.75 5.8900.80 6.3820.85 6.8730.90 7.3620.95 7.8501.00 8.335


14

Conclusions

Price vs. Volatility

$-

$2.00

$4.00

$6.00

$8.00

$10.00

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Volatility

Pri

ce


15

Conclusions

Option Price vs. Current Stock Price

$-

$10.00

$20.00

$30.00

$40.00

$50.00

$60.00

$70.00

$- $10.00 $20.00 $30.00 $40.00 $50.00 $60.00 $70.00 $80.00 $90.00 $100.00

Current Stock Price

Op

tio

n P

ric

e


16

Conclusions

Option Price vs. Duration

$-

$5.00

$10.00

$15.00

$20.00

$25.00

0 1 2 3 4 5 6 7 8 9 10

Duration

Op

tio

n P

ric

e


17

Marketing Example

• Microsoft is trying to determine whether to give a $10 rebate, a $6 price cut, or have no price change on a software product.

• Currently 40,000 units of the product are sold each week for $45.

• The variable cost of the product is $5. • The most likely case appears to be that a $10

rebate will increase sales 30% and half of all people will claim the rebate.

• For the price cut, the most likely case is that sales will increase 20%.


18


Under what circumstances should Microsoft offer the rebate, and under what circumstances should they offer the price cut? (Or should they do neither?)


19

Formulation

Decision variables: 3 possible marketing policies.

Objective: Maximize Profit.

Constraints:Various assumptions have been made (current sales level, current cost structure, consumer behavior in response to marketing policies).


20

Formulation

U n d e r t h e c u r r e n t p o l i c y ,

P r o fi t = V a r i a b l e R e v e n u e – V a r i a b l e C o s t = V o l u m e * ( P r i c e – V a r i a b l e C o s t ) 5$45$000,40 000,600,1$


21

Formulation

U n d e r t h e r e b a t e p o l i c y :

P r o fi t = V a r i a b l e R e v e n u e – V a r i a b l e C o s t – R e b a t e C o s t = V o l u m e * ( P r i c e – V a r i a b l e C o s t ) – ( C l a i m V o l u m e * R e b a t e )

10$*5.0*3.1*000,405$45$*3.1*000,40 000,820,1$


22

Formulation

W i t h t h e p r i c e c u t :

P r o fi t = V a r i a b l e R e v e n u e – V a r i a b l e C o s t = V o l u m e * ( P r i c e – V a r i a b l e C o s t ) 5$39$*000,40*2.1 000,632,1$


23


9

10111213141516

17

181920

A B C D E F G HInputsCurrent sales 40000Current price $45Unit variable cost $5

Data on rebatesAmount of rebate $10Pct taking advantage 50%

Increase in sales 30.00%

Data on price cutAmount of cut $6Increase in sales 20%

ProfitsCurrent $1,600,000

With rebate $1,820,000

With price cut $1,632,000

=B2*(B3-B4)

=((B2*(1+B9))*(B3-B4))-((B2*(1+B9)*B8)*B7)

=B2*(1+B13)*(B3-B12-B4)


24

• Under current assumptions, the rebate policy appears to be optimal.

• How sensitive is this result to possible errors in our assumptions?

• Specifically, how wrong could we be as to the 30% assumption and still be correct in using the rebate?

• What is the point of indifference between the rebate and the price cut?


25

Goal Seek

• Similar to Solver, but simpler• Specify a Target Cell and a

Changing Cell• “Value” must be a number (not a

cell reference)


26

Goal Seek


27


12345678

9

10111213141516

17

18

A B C D E F GInputsCurrent sales 40000Current price $45Unit variable cost $5

Data on rebatesAmount of rebate $10Pct taking advantage 50%

Increase in sales 16.57%


ProfitsCurrent $1,600,000

With rebate $1,632,000

With price cut $1,632,000

Use Goal Seek to make the value in cell B17 equal to 1632000 (the value in B18), using cell B9 as the changing cell.


28

Conclusions and Recommendations

• Go with the rebate as long as the increase in sales is expected to be at least 16.57%.

• Under current assumptions, Microsoft would earn $1,820,000 profit (an improvement of $220,000).


29

What If?

• Important parameters are not known; they are only estimates.

• How robust is the rebate strategy?


30

Two-Way Data Table123456789

101112131415161718

A B C D E F G H I JInputs Best policy RebateCurrent sales 40000Current price $45Unit variable cost $5

Data on rebatesAmount of rebate $10Pct taking advantage 50%Increase in sales 30%


ProfitsCurrent $1,600,000With rebate $1,820,000With price cut $1,632,000

=IF(B16=MAX(B16:B18),"Current",IF(B17=MAX(B16:B18),"Rebate","Price cut"))


31

Two-Way Data Table123456789101112131415161718


Data on rebates Two-way data table for best policyAmount of rebate $10 Increase from rebate (along side) and from price cut (along top)Pct taking advantage 50% Rebate 10% 15% 20% 25% 30%Increase in sales 30% 15%

20%Data on price cut 25%Amount of cut $6 30%Increase in sales 20% 35%

40%ProfitsCurrent $1,600,000With rebate $1,820,000With price cut $1,632,000


=E1


32

Two-Way Data Table


33

Two-Way Data Table

123456789101112131415161718


Data on rebates Two-way data table for best policyAmount of rebate $10 Increase from rebate (along side) and from price cut (along top)Pct taking advantage 50% Rebate 10% 15% 20% 25% 30%Increase in sales 30% 15% Rebate Rebate Price cut Price cut Price cut

20% Rebate Rebate Rebate Price cut Price cutData on price cut 25% Rebate Rebate Rebate Rebate Price cutAmount of cut $6 30% Rebate Rebate Rebate Rebate RebateIncrease in sales 20% 35% Rebate Rebate Rebate Rebate Rebate

40% Rebate Rebate Rebate Rebate RebateProfitsCurrent $1,600,000With rebate $1,820,000With price cut $1,632,000

Unless Microsoft thinks the sales increase from a price cut will be high and the sales increase from a rebate will be low,

it looks like the rebate is the way to go.


=E1


34

Conclusions and Recommendations

• Unless Microsoft thinks the sales increase from a price cut will be high and the sales increase from a rebate will be low, it looks like the rebate is the way to go.


35

Call Center Example

• For a telephone survey, a marketing research group needs to contact at least 150 wives, 120 husbands, 100 single adult males, and 110 single adult females.

• It costs $2 to make a daytime call and (because of higher labor costs) $5 to make an evening call.

• Because of a limited staff, at most half of all phone calls can be evening calls.


36

Call Center Example

Person Responding Percentage of Daytime Calls Percentage of Evening Calls Wife 30 30 Husband 10 30 Single male 10 15 Single female 10 20 None 40 5


37


We want to minimize the total cost of completing the survey, subject to the various probabilities of reaching certain types of people at certain times of the day, costs of making calls, and minimum requirements for numbers of calls to certain demographic groups.


38

Formulation

Decision VariablesWe need to decide how many evening calls and how many daytime calls to make.

ObjectiveMinimize the total cost.

ConstraintsWe need to contact 150 wives, 120 husbands, 100 single adult males, and 110 single adult females. At most half of all phone calls can be evening calls.


39

FormulationDecision VariablesX1 = Daytime Calls, X2 = Evening Calls

ObjectiveMinimize Z = 2X1 + 5X2

Constraints0.30X1 + 0.30X2 ≥ 1500.10X1 + 0.30X2 ≥ 1200.10X1 + 0.15X2 ≥ 1000.10X1 + 0.20X2 ≥ 1101X1 ≥ 1X2

1X1, 1X2 ≥ 0


40


1011

12

131415161718192021

22

232425

A B C D E F G HPercentages Daytime Evening

Wife 30% 30%Husband 10% 30%

Single male 10% 15%Single female 10% 20%

None 40% 5%Sum 100% 100%

Cost/call 2.00$ 5.00$

Daytime Evening Total

Calls made 1 1 2

<=Max evening calls 1

Contacts Made RequiredWife 0.6 >= 150

Husband 0.4 >= 120Single male 0.25 >= 100

Single female 0.3 >= 1100 0

Total cost 7.00$

=SUM(B12:C12)

=0.5*D12

=SUMPRODUCT($B$12:$C$12,B5:C5)

=SUMPRODUCT($B$12:$C$12,B9:C9)


41



42



43


131415161718192021

22

A B C DPercentages Daytime Evening

Wife 30% 30%Husband 10% 30%

Single male 10% 15%Single female 10% 20%

None 40% 5%Sum 100% 100%

Cost/call 2.00$ 5.00$

Daytime Evening TotalCalls made 900 100 1000

<=Max evening calls 500

Contacts Made RequiredWife 300 >= 150

Husband 120 >= 120Single male 105 >= 100

Single female 110 >= 1100 0

Total cost 2,300.00$


44

Optimal Solution

Make 900 Daytime calls and 100 Evening calls.

Total cost = $2,300.


45

SolverTable

• Similar to Data Table; works with Solver

• Solves optimization problems repeatedly and automatically

• One or two inputs can be varied


46

Example: Sensitivity to Calling Costs

• Starting with the optimal solution to the initial problem, use the SolverTable add-in to investigate changes in the unit cost of either type of call.

• Specifically, investigate changes in the cost of a daytime call, with the cost of an evening call fixed, to see when (if ever) only daytime calls or only evening calls will be made.


47



48



49

SolverTable Output

123456789

1011

12

131415161718192021

22

F G H IDaytime Evening Total Cost

0 1200 0 -$ 1 1200 0 1,200.00$ 2 900 100 2,300.00$ 3 700 200 3,100.00$ 4 400 400 3,600.00$ 5 400 400 4,000.00$ 6 400 400 4,400.00$ 7 400 400 4,800.00$ 8 400 400 5,200.00$ 9 400 400 5,600.00$

10 400 400 6,000.00$

11 400 400 6,400.00$ 12 400 400 6,800.00$ 13 400 400 7,200.00$ 14 400 400 7,600.00$ 15 400 400 8,000.00$ 16 400 400 8,400.00$ 17 400 400 8,800.00$ 18 400 400 9,200.00$ 19 400 400 9,600.00$

20 400 400 10,000.00$


50

ConclusionsSensitivity Analysis

0

200

400

600

800

1000

1200

1400

$- $1.00 $2.00 $3.00 $4.00 $5.00 $6.00 $7.00 $8.00 $9.00 $10.00

Cost per Daytime Call

Ca

lls M

ad

e

$-

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

Tota

l Co

st

Daytime

Evening

Total Cost


51

Conclusions

If daytime calls are very inexpensive, we can dispense with evening calls altogether. However, we will always have to make at least 400 daytime calls, no matter how expensive they are.


52

ConclusionsSensitivity Analysis

0

200

400

600

800

1000

1200

1400

$- $1.00 $2.00 $3.00 $4.00 $5.00 $6.00 $7.00 $8.00 $9.00 $10.00

Cost per Evening Call

Ca

lls M

ad

e

$-

$500

$1,000

$1,500

$2,000

$2,500

$3,000

Tota

l Co

st

Daytime

Evening

Total cost


53

Summary

• Sensitivity Analysis– Goal Seek and Data Table– Marketing and Finance examples

• Call Center LP• More Sensitivity Analysis

– SolverTable

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Session 2a. Decision Models -- Prof. Juran2 Overview Sensitivity Analysis –Goal Seek and Data...

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