# CHAPTER 27 Introduction to UNCERTAINTY ANALYSIS UNCERTAINTY ANALYSIS by giz44836

VIEWS: 6 PAGES: 25

• pg 1
```									EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

CHAPTER 27 : Introduction to UNCERTAINTY ANALYSIS

UNCERTAINTY ANALYSIS VERSUS SENSITIVITY ANALYSIS
INPUT UNCERTAINTY
Uncertainty Analysis = Quantification
of Output Uncertainty given Model
and Input Uncertainty
3 .5 0

3 .0 0

2 .5 0

2 .0 0

1.5 0

1.0 0

0 .5 0

0 .0 0
0 .0 0   LM 0 .2 0    0.4 0     0 .60   U  0 .8 0      1.00
X
Sensitivity Analysis = Sensitivity of
Output Parameter to change in one
parameter keeping others constant.
3.50

3.00

2.50

2.00

1.50

1.00

0.50

0.00
0.00       L  0.20      0.40     M U
0.60       0.80        1.00

Y
OUTPUT
MODEL =
2 .5 0

1.4 0
2 .0 0

F(X,Y,Z)
1.2 0
1.5 0
1.0 0

1.0 0                                                                                                 0 .8 0

0 .5 0
0 .6 0

L                           MU
0 .4 0
0 .0 0
0 .0 0    0 .2 0   0 .4 0   0 .60    0 .8 0      1.00                                           0 .2 0

Z
0 .0 0
0 .0 0   S1 M
0 .2 0   0.4 0   0 .60   0 .8 0    1.0 0

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                                                                                              Chapter 27 - Page 276
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

MONTE CARLO SIMULATION/INTEGRATION

INPUT UNCERTAINTY                                                                               Calculate
Sample X1,Y1,Z1                                O1

STATISTICS
3.50

3.00

2.50
Calculate
2.00

1.50                                                                   Sample X2,Y2,Z2                                O2
1.00

0.50

X
0.00

Calculate
0.00        0.20      0.40      0 .60      0 .80       1.0 0

Sample X3,Y3,Z3                                O3
3 .5 0

ETC ...
3 .0 0

2 .5 0

2 .0 0

1.5 0

1.0 0

0 .5 0

0 .0 0
0 .0 0    0 .2 0    0 .4 0   0 .6 0     0 .8 0      1.0 0

Y
OUTPUT
MODEL =
2 .50

1.4 0
2 .00
1.2 0

1.50
F(X,Y,Z)                  1.0 0

1.00                                                                                               0 .8 0

0 .6 0
0 .50

0 .4 0
0 .00
0.0 0    0 .2 0   0 .4 0   0 .60    0 .8 0     1.0 0                                           0 .2 0

Z
0 .0 0

O
0 .0 0   0 .2 0   0 .40   0 .6 0   0 .80    1.0 0

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                                                                                            Chapter 27 - Page 277
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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.

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 278
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

EXAMPLE CALCULATION:

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

Using DataTable we can graph profit as function of
Order Quantity X with a given Demand D.
\$600.00
\$500.00
\$400.00
Profit

\$300.00
\$200.00
\$100.00
\$0.00
0       100           200         300   400    500
Order Quantity

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                Chapter 27 - Page 279
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 280
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

(0.30)         100                         0.35
0.3

(0.20)         150
0.25

Probability
0.2
Demand                      (0.30)         200                         0.15
0.1
(0.15)         250                         0.05
0
(0.05)         300                                100   150    200      250           300
Demand

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

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                 Chapter 27 - Page 281
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

(0.30)   100
\$125.00
(0.20)   150
EMV=\$350.00                          \$312.50
(0.30)   200
Order Quantity = 200                                     \$500.00
(0.15)   250
Demand                        \$500.00
(0.05)   300   \$500.00

0.6

0.5
Probability

0.4
0.3

0.2
0.1

0
\$125.00          \$312.50      \$500.00
Profit

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                               Chapter 27 - Page 282
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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        \$250.00
150             \$187.50         \$375.00       \$375.00         \$375.00   \$375.00        \$318.75
200             \$125.00         \$312.50       \$500.00         \$500.00   \$500.00        \$350.00
250              \$62.50         \$250.00       \$437.50         \$625.00   \$625.00        \$325.00
300               \$0.00         \$187.50       \$375.00         \$562.50   \$750.00        \$271.88
Order Quantity

CONCLUSION:
ORDER 200 Calendars!

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                    Chapter 27 - Page 283
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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):
( d −u ) 2
1         −
fY (d | µ ,σ ) =             ⋅e         2σ 2
σ 2 ⋅π
• E[Y] = µ

• Var(Y) = σ 2

• Some handy rules of thumb:

Pr( µ − σ < D < µ + σ ) ≈ 0.68
Pr( µ − 2σ < D < µ + 2σ ) ≈ 0.95
Pr( µ − 3σ < D < µ + 3σ ) ≈ 0.99

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                               Chapter 27 - Page 284
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

Probability Density Function - N(2,0.5)

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.00      0.50 1.00            1.50      2.00 2.50   3.00 3.50   4.00

≈ 68%
≈ 95%
≈ 99%

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                             Chapter 27 - Page 285
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

• 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( X ≤ x ) = F ( x )

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 286
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

1.00
0.90
0.80
0.70
y=F(x)                   0.60
0.50
0.40
0.30
0.20
0.10
0.00
100                 150                200       250           300

x

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                             Chapter 27 - Page 287
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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).

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 288
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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.

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 289
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

SAMPLING ALGORITHM
1                  1.00
0.90
0.80
STEP 1: Sample                       0.70
Realization u from                   0.60
Uniform Random                       0.50
Variable U                           0.40
0.30
0.20
0.10
0                  0.00
100               150          200         250            300

STEP 2: Calculate
realization x=F-1(u)           x=F-1(u)
from Random
Variable

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                               Chapter 27 - Page 290
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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].

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 291
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 292
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

ESTIMATION OF EMPIRICAL CONTINUOUS CDF

Y = PROFIT

1. Given data: yi , i = 1,…,n.
2. Order data such that
y(1) < y(2) <                 < y( n −1) < y( n )
3. Set:
1
F ( y(1) ) = Pr(Y ≤ y(1) ) =
n
2                                           n −1
F ( y( 2 ) ) = Pr(Y ≤ y( 2 ) ) =               F ( y( n−1) ) = Pr(Y ≤ y( n −1) ) =
n ; …;                                        n
n
F ( y( n ) ) = Pr(Y ≤ y( n ) ) = = 1
n
4. Plot the points ( y(1) , F ( y(1) )),                 , ( y( n ) , F ( y( n ) )) in a graph.

5. Connect these points by a straight line.

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                    Chapter 27 - Page 293
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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: yi , i = 1,…,n.
2. Order data such that
y(1) < y(2) <                      < y( n −1) < y( n )
3. Calculate
y( n ) − y(1)
z j = y(1) + j ⋅                         , j=1,…,m
m

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                           Chapter 27 - Page 294
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

4. For every         zj    determine
y (i ) such that
y(i ) < z j < y(i +1)
5. Set for j=0,1, …, m
i
F (z j ) =
n
6. Set for j=1, …, m

z j −1 + z j
Pr(Y =                         ) = F ( z j ) − F ( z j −1 ), j = 1,   ,m
2

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

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                Chapter 27 - Page 295
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

PROFIT DISTRIBUTION: You are 90% sure that your Profit
will be larger than \$356 (see Spreadsheet)
Profit Distribution (after simulation of 500 replications)

0.60

0.50

0.40
Probability

0.30

0.20

0.10

0.00
\$212

\$241

\$271

\$301

\$330

\$360

\$389

\$419

\$448

\$478
Profit

Distribution appears to have a SPIKE
WHY?

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                                    Chapter 27 - Page 296
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

Lecture Notes by Instructor: Dr. J. Rene van Dorp                             Chapter 27 - Page 297
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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

PROFIT when D>200:

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

HENCE :

Pr(Profit=\$500.00)=50.00%

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                 Chapter 27 - Page 298
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

Calculating Mean Profit and Uncertainty for

a number of different order quantities

\$700.00

\$600.00

\$500.00

\$400.00
Profit

\$300.00

\$200.00

\$100.00

\$0.00
150           170          190     210   230    250
Order Quantity
5 % Bound                  Mean Profit         95% Bound
Lecture Notes by Instructor: Dr. J. Rene van Dorp                                                 Chapter 27 - Page 299
Source: Financial Models Using Simulation and Optimization by Wayne Winston
EMSE 388 – Quantitative Methods in Cost Engineering
Financial Models Using Simulation and Optimization

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.

Lecture Notes by Instructor: Dr. J. Rene van Dorp                                              Chapter 27 - Page 300
Source: Financial Models Using Simulation and Optimization by Wayne Winston

```
To top