Session 2a

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

Decision Models -- Prof. Juran

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?

Decision Models -- Prof. Juran

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.

Decision Models -- Prof. Juran

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

Decision Models -- Prof. Juran

6

Managerial Problem Definition

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

Decision Models -- Prof. Juran

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 iration

d1 = t

trES

2

2

2ln

d2 = td 21

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

Decision Models -- Prof. Juran

8

123456789

10111213141516

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.

Decision Models -- Prof. Juran

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

Decision Models -- Prof. Juran

10

Solution Methodology

12345678910111213141516

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

Decision Models -- Prof. Juran

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

Decision Models -- Prof. Juran

12

Solution Methodology

Decision Models -- Prof. Juran

13

Solution Methodology1516171819202122232425262728293031323334353637

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

Decision Models -- Prof. Juran

14

ConclusionsPrice vs. Volatility

$-

$2.00

$4.00

$6.00

$8.00

$10.00

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

Volatility

Pric

e

Decision Models -- Prof. Juran

15

ConclusionsOption 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

Opt

ion

Pric

e

Decision Models -- Prof. Juran

16

ConclusionsOption 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

Opt

ion

Pric

e

Decision Models -- Prof. Juran

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%.

Decision Models -- Prof. Juran

18

Managerial Problem Definition

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

Decision Models -- Prof. Juran

19

FormulationDecision 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).

Decision Models -- Prof. Juran

20

FormulationU 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$000,40 000,600,1$

Decision Models -- Prof. Juran

21

FormulationU 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$

Decision Models -- Prof. Juran

22

FormulationW 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$

Decision Models -- Prof. Juran

23

Solution Methodology123456789

1011121314151617181920

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

=B2*(B3-B4)

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

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

Decision Models -- Prof. Juran

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?

Decision Models -- Prof. Juran

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)

Decision Models -- Prof. Juran

26

Goal Seek

Decision Models -- Prof. Juran

27

Solution Methodology123456789101112131415161718

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%

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

ProfitsCurrent $1,600,000With rebate $1,632,000With 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.

Decision Models -- Prof. Juran

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).

Decision Models -- Prof. Juran

29

What If?• Important parameters are not

known; they are only estimates.• How robust is the rebate strategy?

Decision Models -- Prof. Juran

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%

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

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"))

Decision Models -- Prof. Juran

31

Two-Way Data Table123456789101112131415161718

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

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

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

=E1

Decision Models -- Prof. Juran

32

Two-Way Data Table

Decision Models -- Prof. Juran

33

Two-Way Data Table123456789101112131415161718

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

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.

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

=E1

Decision Models -- Prof. Juran

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.

Decision Models -- Prof. Juran

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.

Decision Models -- Prof. Juran

36

Call Center ExamplePerson 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

Decision Models -- Prof. Juran

37

Managerial Problem Definition

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.

Decision Models -- Prof. Juran

38

FormulationDecision 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.

Decision Models -- Prof. Juran

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 ≥ 1X21X1, 1X2 ≥ 0

Decision Models -- Prof. Juran

40

Solution Methodology123456789

10111213141516171819202122232425

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 TotalCalls 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)

Decision Models -- Prof. Juran

41

Solution Methodology

Decision Models -- Prof. Juran

42

Solution Methodology123456789

10111213141516171819202122

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$

Decision Models -- Prof. Juran

43

Optimal Solution

Make 900 Daytime calls and 100 Evening calls.

Total cost = $2,300.

Decision Models -- Prof. Juran

44

SolverTable• Similar to Data Table; works with

Solver• Solves optimization problems

repeatedly and automatically• One or two inputs can be varied

Decision Models -- Prof. Juran

45

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.

Decision Models -- Prof. Juran

46

Solution Methodology

Decision Models -- Prof. Juran

47

Solution Methodology

Decision Models -- Prof. Juran

48

SolverTable Output123456789

10111213141516171819202122

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$

Decision Models -- Prof. Juran

49

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

Cal

ls M

ade

$-

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

$7,000

Tota

l Cos

t

DaytimeEveningTotal Cost

Decision Models -- Prof. Juran

50

ConclusionsIf 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.

Decision Models -- Prof. Juran

51

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

Cal

ls M

ade

$-

$500

$1,000

$1,500

$2,000

$2,500

$3,000

Tota

l Cos

t

DaytimeEveningTotal cost

Decision Models -- Prof. Juran

52

Summary• Sensitivity Analysis

– Goal Seek and Data Table– Marketing and Finance examples

• Call Center LP• More Sensitivity Analysis

– SolverTable