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Unformatted text preview: ENGRI 1101 Engineering Applications of OR Fall 2008 Homework 8 Homework 8 Solutions 1. The following the the illustration of the graphical representation of the problem: (a) After graphing the constraints, the feasible region is shown in yellow. We can see from the graph that the constraint y 4 is redundant. If we removed the constraint (whose boundary is represented by a cyan-colored horizontal line in the picture) the feasible region would not change. Another way we can determine the constraint y 4 is redundant by adding x + y 5 and y- x 2 to get 2 y 7, or y 3 . 5. Therefor having y 4 is redundant because we have y 3 . 5 from the combination of two other constraints. (b) To solve the linear program, we can draw the line x + 2 y = c , where c = the objective function value. We want to maximize the objective function value, so we push the line parallel to x +2 y = c as we increase c . Once we are only touching the feasible region at one point, we cant push the objective up any higher, and thus we have reached the maximum value for c , the optimal objective function value. In this example, as the picture demonstrates, the optimal solution occurs where the lines x + y = 5 and y- x = 2 meet. Solving these two equations for x and y (or by simply looking at the graph), we find that x = 1 . 5 and y = 3 . 5 is the optimal solution. Thus the optimal solution is x + 2 y = c = 8 . 5. 2. After introduction of the slack variables, the LP can be written as: 1 (a) max 5 x 1 +4 x 2 s.t.- 5 x 1 +3 x 2 + x 3 = 3 5 x 1 +3 x 2 + x 4 = 18 x 1 + x 5 = 3 x 1 , x 2 , x 3 , x 4 , x 5 = 0 To start simplex method, we need to write the problem in dictionary form. We will use the slack variables as the LHS variables, and proceed from there: Initial dictionary and associated solution: max z = 5 x 1 +4 x 2 s.t. x 3 = 3 +5 x 1- 3 x 2 x 4 = 18- 5 x 1- 3 x 2 x 5 = 3- x 1 (Here and in the following I am omitting the constraints requiring that all variables are greater than or equal to zero these are implicit when the problem is written in dictionary form.)when the problem is written in dictionary form....
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- Spring '05