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Problem 1

Problem 1. Given a high-resolution computer image of a map of an irregularly shaped lake with several islands, determine the water surface area. Assume that the x-y coordinates of every point on the map can be measured at an acceptable level of resolution. Suggest alternative solution approaches!.

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Problem 1

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  1. Problem 1 • Given a high-resolution computer image of a map of an irregularly shaped lake with several islands, determine the water surface area. Assume that the x-y coordinates of every point on the map can be measured at an acceptable level of resolution.Suggest alternative solution approaches!

  2. Monte Carlo Simulation Y Step 1:Enclose the area of interest in the smallest rectangle of known dimensions X and Y. Set j = 1, S = 0, and choose a large value for N where:j = trial numberS = number of trials resulting in a hit on the water surface areaN = total number of trials X 0

  3. RN y RN x Monte Carlo Simulation Y Step 2:Generate a uniformly distributed random number, RNxover the length of X. Step 3:Generate another uniformly distributed random number, RNyover the length of Y. X 0

  4. Y RN y X 0 RN x Monte Carlo Simulation Step 4:If the point of intersection, (RNx, RNy), falls on the water surface area, add 1 to S. Step 5:Add 1 to j. If j > N, go to Step 6; otherwise, go to Step 2.

  5. Y RN y X 0 RN x A S = N X Y Monte Carlo Simulation Step 6:The estimate of the water surface area, A, is given by: NOTE: As N , A  true value of the area

  6. Problem 2 • Use the fundamental theory and logic of the Monte Carlo Simulation technique to estimate the area under the Sine curve over 0 and . y 1 Sin (x) 0 

  7. Monte Carlo Simulation y Step 1:Enclose the area of interest in the smallest rectangle of known dimensions  and 1. Set j = 1, S = 0, and choose a large value for N where:j = trial numberS = number of trials with a hit on the area under the Sin (x) curveN = total number of trials 1 Sin (x) 0 

  8. Monte Carlo Simulation y Step 2:Generate a uniformly distributed random number, RNxover the length of . Step 3:Generate another uniformly distributed random number, RNy over the length of 1. 1 RNy Sin (x) 0 RNx 

  9. Monte Carlo Simulation y Step 4:If the point of intersection, (RNx, RNy), falls on or below the Sin (x) curve (i.e., if RNy ≤ Sin (RNx)), add 1 to S. Step 5:Add 1 to j. If j > N, go to Step 6; otherwise, go to Step 2. 1 RNy Sin (x) 0 RNx 

  10. A S = N X Y Monte Carlo Simulation y Step 6:The estimate of the area, A, is given by: NOTE: As N , A  2 (true value of the area) 1 RNy Sin (x) 0 RNx 

  11. Problem 3 • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

  12. Monte Carlo Simulation • Step 1:Set j = 1 and choose a large value for N where:j = trial numberN = total number of trials • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

  13. Monte Carlo Simulation • Step 2:Generate a proper random number, RN1 , over 0 and 1. • Step 3:Generate another proper random number, RN2 , over 0 and 2. • Step 4:Generate another proper random number, RN3 , over 2 and 3. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

  14. Monte Carlo Simulation • Step 5:Substitute RN1 for X1 , RN2 for X2 , and RN3 for X3 in the objective function. Store its value in Z(j) and record the corresponding values for X1 , X2 , and X3. • Step 6:Add 1 to j. If j > N, go to Step 7; otherwise, go to Step 2. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

  15. Monte Carlo Simulation • Step 7:The approximate solution of the problem is determined by the values of X1 ( = RN1 ), X2 ( = RN2 ), and X3 ( = RN3 ), which correspond to the maximum value of { Z(j), j = 1, 2, 3, ..., N }. • NOTE: As N , X1 1, X2 2, and X3 3. • Use the fundamental theory and logic of the Monte Carlo Simulation technique to solve the following optimization problem:Maximize Z = ( eX1+ X2 ) 2 + 3 ( 1 – X3 ) 2Subject to: 0 ≤ X1 ≤ 1 0 ≤ X2 ≤ 2 2 ≤ X3 ≤ 3

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