Session 2a
Overview
• Sensitivity Analysis
– Goal Seek and Data Table
– Marketing and Finance examples
• Call Center LP
• More Sensitivity Analysis
– SolverTable
Decision Models -- Prof. Juran
2
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
3
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
4
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
5
Managerial Problem Definition
How do the parameters in Black-Scholes
affect the option price?
Decision Models -- Prof. Juran
6
Formulation
The Black-Scholes model:
C  SN d 1   Ee  rt N d 2 
where:
S
E
r
σ2
t
d1
d2
N(d)
Decision Models -- Prof. Juran
= current stock price
= exercise price
= risk-free rate of return
= variance of the stock’s return
= time to expiration
2
 S    
ln    r 
t

2 
E 
=
 2t
2
d


t
1
=
= probability that z < d
7
Solution Methodology
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
Inputs
1
35
40
0.5
0.05
0.4
Type of option (1 for call, 2 for put)
Stock price
Exercise price
Duration (years)
Riskfree interest rate
Volatility
C
D
E
F
G
H
=IF(B2=1,NORMSDIST(B10),NORMSDIST(-B10))
=(LN(B3/B4)+(B6+B7^2/2)*B5)/(B7*SQRT(B5))
Quantities for Black-Scholes formula
d1
d2
Option price
-0.242
-0.525
=B10-SQRT(B7^2*B5)
2.456
N(d1)
N(d2)
0.404
0.300
=IF(B2=1,NORMSDIST(B11),NORMSDIST(-B11))
=IF(B2=1,B3*E10-B4*EXP(-B5*B6)*E11,-(B3*E10-B4*EXP(-B5*B6)*E11))
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
8
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
9
Solution Methodology
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Type of option (1 for call, 2 for put)
Stock price
Exercise price
Duration (years)
Riskfree interest rate
Volatility
B
Inputs
1
35
40
0.5
0.05
0.4
Quantities for Black-Scholes formula
d1
d2
-0.242
-0.525
Option price
2.456
Volatility
Decision Models -- Prof. Juran
Price
2.456
C
D
E
N(d1)
N(d2)
0.404
0.300
=B13
10
Solution Methodology
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Decision Models -- Prof. Juran
A
Volatility
B
Price
2.456
0.01
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
11
Solution Methodology
Decision Models -- Prof. Juran
12
Solution Methodology
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
Decision Models -- Prof. Juran
A
Volatility
0.01
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
B
Price
2.456
0.000
0.000
0.071
0.312
0.664
1.075
1.518
1.981
2.456
2.939
3.426
3.917
4.410
4.903
5.397
5.890
6.382
6.873
7.362
7.850
8.335
13
Conclusions
Price vs. Volatility
$10.00
$8.00
Price
$6.00
$4.00
$2.00
$0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Volatility
Decision Models -- Prof. Juran
14
Conclusions
Option Price vs. Current Stock Price
$70.00
$60.00
Option Price
$50.00
$40.00
$30.00
$20.00
$10.00
$$-
$10.00
$20.00
$30.00
$40.00
$50.00
$60.00
$70.00
$80.00
$90.00
$100.00
Current Stock Price
Decision Models -- Prof. Juran
15
Conclusions
Option Price vs. Duration
$25.00
Option Price
$20.00
$15.00
$10.00
$5.00
$0
1
2
3
4
5
6
7
8
9
10
Duration
Decision Models -- Prof. Juran
16
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
17
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
18
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).
Decision Models -- Prof. Juran
19
Formulation
Under the current policy,
Profit = Variable Revenue – Variable Cost
= Volume*(Price – Variable Cost)
 40 ,000$45  $5 
 $1,600,000
Decision Models -- Prof. Juran
20
Formulation
Under the rebate policy:
Profit = Variable Revenue – Variable Cost – Rebate Cost
= Volume*(Price – Variable Cost) – (Claim Volume*Rebate)
  40 ,000 * 1.3  * $45  $5    40 ,000 * 1.3 * 0.5  * $10 
 $1,820,000
Decision Models -- Prof. Juran
21
Formulation
With the price cut:
Profit = Variable Revenue – Variable Cost
= Volume*(Price – Variable Cost)
 1.2 * 40 ,000  * $39  $5 
 $1,632,000
Decision Models -- Prof. Juran
22
Solution Methodology
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
Inputs
Current sales
Current price
Unit variable cost
Data on rebates
Amount of rebate
Pct taking advantage
Increase in sales
B
D
E
F
G
H
40000
$45
$5
$10
50%
30.00%
Data on price cut
Amount of cut
Increase in sales
Profits
Current
With rebate
With price cut
C
$6
20%
=B2*(B3-B4)
$1,600,000
$1,820,000
$1,632,000
Decision Models -- Prof. Juran
=((B2*(1+B9))*(B3-B4))-((B2*(1+B9)*B8)*B7)
=B2*(1+B13)*(B3-B12-B4)
23
• 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
24
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
25
Goal Seek
Decision Models -- Prof. Juran
26
Solution Methodology
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Inputs
Current sales
Current price
Unit variable cost
Data on rebates
Amount of rebate
Pct taking advantage
Increase in sales
Data on price cut
Amount of cut
Increase in sales
Profits
Current
With rebate
With price cut
Decision Models -- Prof. Juran
B
C
D
E
F
G
40000
$45
$5
$10
50%
16.57%
Use Goal Seek to make the value in cell
B17 equal to 1632000 (the value in B18),
using cell B9 as the changing cell.
$6
20%
$1,600,000
$1,632,000
$1,632,000
27
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
28
What If?
• Important parameters are not known;
they are only estimates.
• How robust is the rebate strategy?
Decision Models -- Prof. Juran
29
Two-Way Data Table
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Inputs
Current sales
Current price
Unit variable cost
B
C
40000
$45
$5
Data on rebates
Amount of rebate
Pct taking advantage
Increase in sales
$10
50%
30%
Data on price cut
Amount of cut
Increase in sales
$6
20%
Profits
Current
With rebate
With price cut
D
Best policy
E
Rebate
F
G
H
I
J
=IF(B16=MAX(B16:B18),"Current",IF(B17=MAX(B16:B18),"Rebate","Price cut"))
$1,600,000
$1,820,000
$1,632,000
Decision Models -- Prof. Juran
30
Two-Way Data Table
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
A
Inputs
Current sales
Current price
Unit variable cost
B
C
40000
$45
$5
Data on rebates
Amount of rebate
Pct taking advantage
Increase in sales
$10
50%
30%
Data on price cut
Amount of cut
Increase in sales
$6
20%
Profits
Current
With rebate
With price cut
D
Best policy
E
Rebate
F
G
H
I
J
=IF(B16=MAX(B16:B18),"Current",IF(B17=MAX(B16:B18),"Rebate","Price cut"))
Two-way data table for best policy
Increase from rebate (along side) and from price cut (along top)
Rebate
10%
15%
20%
25%
30%
15%
20% =E1
25%
30%
35%
40%
$1,600,000
$1,820,000
$1,632,000
Decision Models -- Prof. Juran
31
Two-Way Data Table
Decision Models -- Prof. Juran
32
Two-Way Data Table
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
A
Inputs
Current sales
Current price
Unit variable cost
B
C
40000
$45
$5
Data on rebates
Amount of rebate
Pct taking advantage
Increase in sales
$10
50%
30%
Data on price cut
Amount of cut
Increase in sales
$6
20%
Profits
Current
With rebate
With price cut
D
Best policy
E
Rebate
F
G
H
I
J
=IF(B16=MAX(B16:B18),"Current",IF(B17=MAX(B16:B18),"Rebate","Price cut"))
$1,600,000
$1,820,000
$1,632,000
Decision Models -- Prof. Juran
Two-way data table for best policy
Increase from rebate (along side) and from price cut (along top)
Rebate
10%
15%
20%
25%
30%
15%
Rebate
Rebate Price cut Price cut Price cut
20% =E1
Rebate
Rebate
Rebate Price cut Price cut
25%
Rebate
Rebate
Rebate
Rebate Price cut
30%
Rebate
Rebate
Rebate
Rebate
Rebate
35%
Rebate
Rebate
Rebate
Rebate
Rebate
40%
Rebate
Rebate
Rebate
Rebate
Rebate
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.
33
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
34
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
35
Call Center Example
Person Responding
Wife
Husband
Single male
Single female
None
Percentage of Daytime Calls
30
10
10
10
40
Decision Models -- Prof. Juran
Percentage of Evening Calls
30
30
15
20
5
36
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
37
Formulation
Decision Variables
We need to decide how many evening calls and how
many daytime calls to make.
Objective
Minimize the total cost.
Constraints
We 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
38
Formulation
Decision Variables
X1 = Daytime Calls, X2 = Evening Calls
Objective
Minimize Z = 2X1 + 5X2
Constraints
0.30X1 + 0.30X2 ≥ 150
0.10X1 + 0.30X2 ≥ 120
0.10X1 + 0.15X2 ≥ 100
0.10X1 + 0.20X2 ≥ 110
1X1 ≥ 1X2
1X1, 1X2 ≥ 0
Decision Models -- Prof. Juran
39
Solution Methodology
A
Percentages
Wife
Husband
Single male
Single female
None
Sum
B
Daytime
30%
10%
10%
10%
40%
100%
1
2
3
4
5
6
7
8
9
Cost/call
$
2.00
10
11
Daytime
12
Calls made
1
13
14 Max evening calls
15
16
Contacts
Made
17
Wife
0.6
18
Husband
0.4
19
Single male
0.25
20
Single female
0.3
21
0
22
Total cost
$
7.00
23
24
25
Decision Models -- Prof. Juran
C
Evening
30%
30%
15%
20%
5%
100%
$
D
E
F
G
H
5.00
Evening
1
<=
1
>=
>=
>=
>=
Total
2
=SUM(B12:C12)
=0.5*D12
Required
150
120
100
110
0
=SUMPRODUCT($B$12:$C$12,B5:C5)
=SUMPRODUCT($B$12:$C$12,B9:C9)
40
Solution Methodology
Decision Models -- Prof. Juran
41
Solution Methodology
A
Percentages
Wife
Husband
Single male
Single female
None
Sum
B
Daytime
30%
10%
10%
10%
40%
100%
1
2
3
4
5
6
7
8
9
Cost/call
$
2.00
10
11
Daytime
12
Calls made
900
13
14 Max evening calls
15
16
Contacts
Made
17
Wife
300
18
Husband
120
19
Single male
105
20
Single female
110
21
0
22
Total cost
$ 2,300.00
Decision Models -- Prof. Juran
C
Evening
30%
30%
15%
20%
5%
100%
$
D
5.00
Evening
100
<=
500
>=
>=
>=
>=
Total
1000
Required
150
120
100
110
0
42
Optimal Solution
Make 900 Daytime calls and 100
Evening calls.
Total cost = $2,300.
Decision Models -- Prof. Juran
43
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
44
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
45
Solution Methodology
Decision Models -- Prof. Juran
46
Solution Methodology
Decision Models -- Prof. Juran
47
SolverTable Output
F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Decision Models -- Prof. Juran
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
G
Daytime
1200
1200
900
700
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
H
Evening
0
0
100
200
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
I
Total Cost
$
$ 1,200.00
$ 2,300.00
$ 3,100.00
$ 3,600.00
$ 4,000.00
$ 4,400.00
$ 4,800.00
$ 5,200.00
$ 5,600.00
$ 6,000.00
$ 6,400.00
$ 6,800.00
$ 7,200.00
$ 7,600.00
$ 8,000.00
$ 8,400.00
$ 8,800.00
$ 9,200.00
$ 9,600.00
$ 10,000.00
48
Conclusions
Sensitivity Analysis
1400
$7,000
1200
$6,000
Daytime
Evening
$5,000
Total Cost
800
$4,000
600
$3,000
400
$2,000
200
$1,000
0
Total Cost
Calls Made
1000
$$-
$1.00
$2.00
$3.00
$4.00
$5.00
$6.00
$7.00
$8.00
$9.00
$10.00
Cost per Daytime Call
Decision Models -- Prof. Juran
49
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.
Decision Models -- Prof. Juran
50
Conclusions
Sensitivity Analysis
1400
$3,000
1200
$2,500
1000
800
Daytime
$1,500
Evening
600
Total cost
Total Cost
Calls Made
$2,000
$1,000
400
$500
200
0
$$-
$1.00
$2.00
$3.00
$4.00
$5.00
$6.00
$7.00
$8.00
$9.00
$10.00
Cost per Evening Call
Decision Models -- Prof. Juran
51
Summary
• Sensitivity Analysis
– Goal Seek and Data Table
– Marketing and Finance examples
• Call Center LP
• More Sensitivity Analysis
– SolverTable
Decision Models -- Prof. Juran
52