Chapter 27 - Introduction to Uncertainty Analysis

EMSE 388 – Quantitative Methods in Cost Engineering Financial Models Using Simulation and Optimization

Lecture Notes by Instructor: Dr. J. Rene van Dorp Chapter 27 - Page 276 Source: Financial Models Using Simulation and Optimization by Wayne Winston

CHAPTER 27 : Introduction to UNCERTAINTY ANALYSIS

UNCERTAINTY ANALYSIS VERSUS SENSITIVITY ANALYSIS

MODEL =F(X,Y,Z)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.20 0.40 0.60 0.80 1.00

INPUT UNCERTAINTY

OUTPUT

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.20 0.40 0.60 0.80 1.00

L UM

UM

UML

L

Uncertainty Analysis = Quantificationof Output Uncertainty given Modeland Input Uncertainty

Sensitivity Analysis = Sensitivity ofOutput Parameter to change in oneparameter keeping others constant.

MS1

X

Y

Z



MONTE CARLO SIMULATION/INTEGRATION

MODEL =F(X,Y,Z)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.20 0.40 0.60 0.80 1.00

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

0.00 0.20 0.40 0.60 0.80 1.00

INPUT UNCERTAINTY

OUTPUT

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.20 0.40 0.60 0.80 1.00

X

Y

Z

Sample X1,Y1,Z1 O1

O

Calculate

Sample X2,Y2,Z2 O2Calculate

Sample X3,Y3,Z3 O3Calculate

STATIS

T ICS

ETC ...



Description Case Study:

You need to determine how many Year 2006 Calendars you need to order in August 2005. It costs $2.00 to order each calendar and you can sell a calendar for $4.50. After January 1, 2006 left over calendars are returned for $0.75. Suppose you decide to order X calendars in August and the actual Demand equals D. What would be your profit?

Total Cost = X·$2.00 Full Price Revenue = Min(X,D)·$4.50

Salvage Revenue = 1[D,∞](X) ·(X-D)·$0.75

Total Revenue = Full Price Revenue – Salvage Revenue Total Profit = Total Revenue – Total Cost.



EXAMPLE CALCULATION:

Quantity Ordered 100 Total Cost $200.00Quantity demanded 200 Full Price Revenue $450.00Sales price $4.50 Salvage Revenue $0.00Salvage value $0.75 Total Revenue $450.00Purchase price $2.00 Total Profit $250.00

Using DataTable we can graph profit as function of

Order Quantity X with a given Demand D.

$0.00

$100.00

$200.00

$300.00

$400.00

$500.00

$600.00

0 100 200 300 400 500

Order Quantity

Prof

it



CONCLUSION: PROFIT is maximized when Order Quantity exactly equals the demand!

Thus, if you know the demand you would order the same amount (NO SURPRISE)

BUT!

The demand is uncertain and can only take the values 100,150, 200, 250, 300

AS A RESULT:

For any given Order Quantity X, the Profit is Uncertain as well.

• Input Parameter: Demand • Output Parameter: Profit • Model: Profit Calculation with given demand D and Order Quantity X



The demand for the calendar up to start of the new year calendar is uncertain and follows a discrete distribution.

Suppose you order 200 Calendars. What is the uncertainty distribution of the Profit?

Demand

100

150

200

250

300

(0.30)

(0.20)

(0.30)

(0.15)

(0.05)0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

100 150 200 250 300

Demand

Prob

abili

ty



0

0.1

0.2

0.3

0.4

0.5

0.6

$125.00 $312.50 $500.00

Profit

Prob

abili

ty

Demand

100

150

200

250

300

(0.30)

(0.20)

(0.30)

(0.15)

(0.05)

Order Quantity = 200

$125.00

$312.50

$500.00

$500.00

$500.00

EMV=$350.00



The decision problem at hand is to select that order quantity that maximizes the expected profit. It only makes sense to only consider ordering 100, 150, 200, 250 or 300 Calendars.

WHY?

CONCLUSION: Our input uncertainty model greatly simplified our problem

Demand 100 150 200 250 300 Expected Profit

100 $250.00 $250.00 $250.00 $250.00 $250.00 $250.00150 $187.50 $375.00 $375.00 $375.00 $375.00 $318.75200 $125.00 $312.50 $500.00 $500.00 $500.00 $350.00250 $62.50 $250.00 $437.50 $625.00 $625.00 $325.00300 $0.00 $187.50 $375.00 $562.50 $750.00 $271.88

Order Quantity

CONCLUSION: ORDER 200 Calendars!



CONTINUOUS DEMAND The assumption of discrete demand is somewhat unrealistic. Suppose for example demand is Normal Distributed with a mean of 200 and a standard deviation of 30.

INTERMEZZO: THE NORMAL DISTRIBUTION • D∼ Ν(µ, σ):

2

2

2)(

21),|( σ

πσσµ

ud

Y edf−

−⋅

⋅=

• E[Y] = µ

• Var(Y) = 2σ • Some handy rules of thumb:

68.0)Pr( ≈+<<− σµσµ D 95.0)22Pr( ≈+<<− σµσµ D 99.0)33Pr( ≈+<<− σµσµ D



Probability Density Function - N(2,0.5)

00.10.20.30.40.50.60.70.80.9

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

≈ 68%≈ 95%

≈ 99%



• 68 % Credibility Interval for Demand: [170,230] • 95 % Credibility Interval for Demand: [140,260] • 99 % Credibility Interval for Demand: [110,290]

• Due to out INPUT UNCERTAINTY MODEL, we need to consider all possible

values for ORDER QUANTITY and not just the five values we had before.

CONTINUOUS RANDOM NUMBER GENERATION

Suppose X is a CONTINUOUS random

variable with cumulative distribution function F

)()Pr( xFxX =≤



A random Variable Z would also have cumulative distribution function F if: Pr( ) ( )Z z F z≤ =

0.000.100.200.300.400.500.600.700.800.901.00

100 150 200 250 300

x

y=F(x)



CONCLUSION: CDF is Continuous Strictly Increasing Function. Therefore

F(x) has a well defined inverse function: F-1(y)=x

THEOREM :

Let X be a continuous random variable with cdf F(x). Let U be a uniform random variable on [0,1].

Let Z be the random variable, such that: Z= F-1(U)

Z is a continuous random variable with cdf F(z).



PROOF: Pr{Z ≤ z}= Pr{ F-1(U) ≤ z }

Because F is a strictly increasing function we now have

Pr{Z ≤ z}= Pr{ F[F-1(U)] ≤ F[z] }

But F[F-1(U)]=U, hence

Pr{Z ≤ z}= Pr{ U ≤ F[z] } = F(z),

Because for a uniform U on [0,1] we know that Pr{U≤ u}=u.



0.000.100.200.300.400.500.600.700.800.901.00

100 150 200 250 300

x=F-1(u)

STEP 1: SampleRealization u fromUniform RandomVariable U

1

0

STEP 2: Calculaterealization x=F-1(u) from RandomVariable

SAMPLING ALGORITHM



HOMEWORK: PROVE THE FOLLOWING THEOREM

THEOREM :

Let X be a continuous random variable with cdf F(x).

Let Z be the random variable, such that: Z= F(X)

Prove that Z is a uniform random variable on [0,1].



MACRO CODE IN EXCEL

Sub CreateSample() ' ' CreateSample Macro ' Macro recorded 4/4/2005 by . '

' For i = 1 To 500 Sheets("Profit Sample").Cells(1, 5).Value = i Sheets("Profit Sample").Cells(i, 2).Value =

Sheets("Profit Model Normal").Cells(12, 6).Value Calculate Next i

End Sub



ESTIMATION OF EMPIRICAL CONTINUOUS CDF

Y = PROFIT 1. Given data: iy , i = 1,…,n. 2. Order data such that

(1) (2) ( 1) ( )n ny y y y−< < < < 3. Set:

nyYyF 1)Pr()( )1()1( =≤=

nyYyF 2)Pr()( )2()2( =≤= ; …; n

nyYyF nn1)Pr()( )1()1(

−=≤= −−

( ) ( )( ) Pr( ) 1n nnF y Y yn

= ≤ = =

4. Plot the points (1) (1) ( ) ( )( , ( )), , ( , ( ))n ny F y y F y in a graph. 5. Connect these points by a straight line.



ESTIMATION OF M-POINT EMPIRICAL PROBABILITY MASS FUNCTION (or HISTOGRAM)

APPROACH: Develop a DISCRETE APPROXIMATION of CONTINUOUS PDF by assigning probability mass on the interval [a,b] to the midpoint of this interval i.e. (a+b)/2. NOTE: Pr(Y ∈ [a,b]) = F(b)-F(b)

M-point approximation method of PDF 1. Given data: iy , i = 1,…,n. 2. Order data such that

(1) (2) ( 1) ( )n ny y y y−< < < < 3. Calculate

( ) (1)(1)

nj

y yz y j

m−

= + ⋅ , j=1,…,m



4. For every jz determine )(iy such that

( ) ( 1)i j iy z y +< <

5. Set for j=0,1, …, m

( )jiF zn

=

6. Set for j=1, …, m

11Pr( ) ( ) ( ), 1, ,

2j j

j j

z zY F z F z j m−

−

+= = − =

STEPS 4, 5 and 6 can be executed in EXCEL using the FREQUENCY FUNCTION.



PROFIT DISTRIBUTION: You are 90% sure that your Profit will be larger than $356 (see Spreadsheet)

Distribution appears to have a SPIKE WHY?

Profit Distribution (after simulation of 500 replications)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

$212

$241

$271

$301

$330

$360

$389

$419

$448

$478

Profit

Prob

abili

ty



MODEL

Total Cost = X·$2.00 Full Price Revenue = Min(X,D)·$4.50 Salvage Revenue = 1[D, ∞](X) ·(X-D)·$0.75 or Salvage Revenue = 1[0,X](D) ·(X-D)·$0.75 Total Revenue = Full Price Revenue – Salvage Revenue Total Profit = Total Revenue – Total Cost.

Setting ORDER QUANTITY X=200 it follows that

SALVAGE REVENUE = 0 when D>200



X Pr(D<=X) Pr(D>X)200 50% 50%

PROFIT when D>200:

Quantity Ordered 200 Total Cost $400.00Quantity demanded 250 Full Price Revenue $900.00Sales price $4.50 Salvage Revenue $0.00Salvage value $0.75 Total Revenue $900.00Purchase price $2.00 Total Profit $500.00

HENCE :

Pr(Profit=$500.00)=50.00%



Calculating Mean Profit and Uncertainty for

a number of different order quantities

$0.00

$100.00

$200.00

$300.00

$400.00

$500.00

$600.00

$700.00

150 170 190 210 230 250

Order Quantity

Pro

fit

5 % Bound Mean Profit 95% Bound



Order Quantity 5% Bound Expected Profit 95% Bound

150 $330.82 $372.45 $375.00

160 $339.70 $396.34 $400.00

170 $312.83 $415.10 $425.00

180 $303.25 $431.59 $450.00

190 $259.06 $439.54 $475.00

200 $273.81 $452.57 $500.00

210 $254.89 $459.41 $525.00

220 $252.73 $462.75 $550.00

230 $240.43 $452.70 $575.00

240 $231.94 $443.16 $600.00

250 $187.48 $432.00 $625.00

Conclusion: Set order quantity at 220.

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Chapter 27 - Introduction to Uncertainty Analysis