ex1-sol10

# ex1-sol10 - AMS 341(Spring 2010 Exam 1 Solution notes Estie...

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Unformatted text preview: AMS 341 (Spring, 2010) Exam 1 - Solution notes Estie Arkin Mean 72.16, median 71, high 100, low 36. 1. (9 points) Consider the following LP: min z =- 2 x 1 + x 3 s . t .- x 1 + 5 x 2 + x 3 1 x 1- 2 x 2 + x 3 = 10 x 1 ,x 3 x 2 unrestricted Rewrite the LP in standard form. min z =- 2 x 1 + x 3 s . t .- x 1 + 5 x 2- 5 x 2 + x 3- e 1 = 1 x 1- 2 x 2 + 2 x 2 + x 3 = 10 x 1 ,x 3 ,x 2 ,x 2 2. (15 points) Consider the feasible region given by the following constraints: (It may be helpful to sketch it and/or put it into standard form.) x 1 + x 2 6 (1) x 1 2 (2) x 2 4 (3) x 1 (4) x 2 (5) In standard form, there are 3 constraints (and the non negativity constraints) and 5 variables, so there should be 3 basic variables and 2 non basic variables. x 1 + x 2 + s 1 = 6 (6) x 1 + s 2 = 2 (7) x 2 + s 3 = 4 (8) x 1 ,x 2 ,s 1 ,s 2 ,s 3 (9) (a). Is the point x 1 = 0, x 2 = 4 a feasible point? Is it a basic solution? Yes, it is feasible (constraints are satisfied) and basic, x 2 ,s 1 ,s 2 are basic, x 1 ,s 3 non basic. (b). Is the point x 1 = 2, x 2 = 2 a feasible point? Is it a basic solution? Yes, it is feasible (constraints are satisfied) but not basic, since x 1 ,x 2 ,s 1 ,s 3 are all positive, so we would need to have 4 basic variables, but there are only 3 constraints. 1 (c). The point x 1 = 2, x 2 = 4 is a basic feasible solution. Is it a degenerate basic feasible solution? Yes it is feasible and is a degenerate BFS. x 1 ,x 2 are basic, and all slack variables are equal to zero, one of them must be basic, and therefore it is degenerate. (Basic variable equals to zero.) 3. (15 points) You are given the tableau for a max problem. Give conditions on the unkowns a 1 ,a 2 ,a 3 ,b,c that make the following true. Your conditions should be as general as possible (dont just give an example, such as a 1 = 3.) z x 1 x 2 x 3 x 4 x 5 RHS 1 c 2 10-1 a 1 1 4 a 2-4 1 1 a 3 3 1 b (a). The current BFS is optimal. To be feasible, b 0, for optimality,...
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ex1-sol10 - AMS 341(Spring 2010 Exam 1 Solution notes Estie...

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