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Unformatted text preview: DMOR OEM 2007 Homework 2 Solution Key Solution problem 1: (questions, comments, suggestions please to Zeki Caner Taskin, firstname.lastname@example.org ) a) The idea is to define a pair of decision variables for each absolute value term. Define decision variables e + i and e- i i = 1 , . . . , 8 that represent the amount of positive and negative discrepancy, respectively, between the amount of fuel consumption predicted by the model for data point i , and the realized fuel consumption. Using these decision variables, we can linearize the objective function using these decision variables as follows. min e + 1 + e- 1 + e + 2 + e- 2 + + e + 8 + e- 8 ( A ) s.t. e- 1- e + 1 = 1100- 17 p 1- 22 p 2 e- 2- e + 2 = 1420- 25 p 1- 22 p 2 . . . e- 8- e + 8 = 2850- 60 p 1- 41 p 2 e- i , e + i The structure of the LP model guarantees that at most one of e + i and e- i can be positive in an optimal solution, and hence e + 1 + e- 1 = | 1100- 17 p 1- 22 p 2 | , and so on. The optimal solution is p 1 = 23 . 98 , p 2 = 34 . 41, with an objective function value of 412 . 26. Another way of thinking about this is the following. Ideally, we want to find p 1 , p 2 so that our model can perfectly explain the relationship between altitude, distance and fuel usage. We can define the following LP model for finding p 1 , p 2 . min ( B ) s.t. 0 = 1100- 17 p 1- 22 p 2 0 = 1420- 25 p 1- 22 p 2 . . . 0 = 2850- 60 p 1- 41 p 2 Since we just want to find a feasible pair of multipliers p 1 , p 2 to be used in our linear regression, we dont need an objective function. Any feasible solution of this model is a good set of multipliers that can be used in the linear regression model to explain the relationship between altitude, distance and fuel usage. However, it is highly unlikely that a perfectly linear relationship exists between those factors. In other words, the LP presented above is highly likely to be infeasible. Therefore we convert the constraints to soft constraint presented in class, and associate e + i and e- i i = 1 , . . . , 8 variables with each constraint to measure the amount of violation. We set the objective function to minimize the total amount of violation. The resulting model is exactly the same as Model A. b) Even though this part looks nonlinear, it is not. The trick is that all nonlinear terms are just functions of the data, not the decision variables. Therefore, we 1 can just calculate the values of D/A and D 2 for each data point, and then use the same modeling idea as in part a). min e + 1 + e- 1 + e + 2 + e- 2 + + e + 8 + e- 8 s.t. e- 1- e + 1 = 1100- (17 / 22) p 1- 17 2 p 2 e- 2- e + 2 = 1420- (25 / 22) p 1- 25 2 p 2 . . . e- 8- e + 8 = 2850- (60 / 41) p 1- 60 2 p 2 e- i , e + i The optimal solution is p 1 = 1329 . 59 , p 2 = 0 . 25, with an objective function value of 1357 . 78. Since the sum of errors in the first model is less, it is a better model in explaining the relationship between the factors given....
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- Fall '09