CHAPTER 27 Introduction to UNCERTAINTY ANALYSIS UNCERTAINTY ANALYSIS by giz44836

VIEWS: 6 PAGES: 25

									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

								
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