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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
- Optimization, 8%, 40%, objective function value, $1,000,000