p2 - would it be desirable to buy some? Why? (h) (10) A new...

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Summer 2006 Prelim #2 – June 9, 2006 Closed book. Justify all work. 1. Consider the following LP and its optimal tableau. Here x 4 and x 5 are slack variables for the Frst and second inequalities, respectively. max 2 x 1 + x 2 - x 3 s.t. x 1 + 2 x 2 + x 3 8 - x 1 + x 2 - 2 x 3 4 x 1 , x 2 , x 3 0 Basis x 1 x 2 x 3 x 4 x 5 (–z) 0 -3 -3 -2 0 -16 x 1 1 2 1 1 0 8 x 5 0 3 -1 1 1 12 All changes mentioned below refer to the original problem data. (a) (5) Determine (by inspection) B - 1 and c B for the optimal basis. (b) (5) By how much must c 3 be increased in order that the problem will have an alternative optimal basis? (c) (10) ±or what range of values of c 1 does the basis ( x 1 , x 5 ) remain optimal? (d) (5) Does the above basis remain optimal if the (original) coe²cient of x 3 in the Frst constraint is changed from 1 to -1? (e) (10) ±or what range of values of b 2 (the original r.h.s. coe²cient for the 2nd constraint) does basis ( x 1 , x 5 ) remain optimal? (f) (15) Show that for this LP the basis ( x 1 , x 5 ) remains optimal for any nonnegative r.h.s. vector, i.e., whenever b 1 0 , b 2 0 . (g) (5) If an external agent o³ers to supply additional units of resource 2 at a price of 3 per unit,
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Unformatted text preview: would it be desirable to buy some? Why? (h) (10) A new product with resource coecients ( 2 ) is proposed, where is an arbitrary constant. What must the objective coecient for this activity be in order that it be proFtable to produce some of the new product? Why is your answer independent of ? 2. (a) (10) Give brief (but precise) mathematical deFnitions for: a convex set; an extreme point; a basic feasible solution. (b) (25) Suppose S and T are convex sets with S T and x is a point which is in both S and T . or the following statements either prove valid or demonstrate false by example (a diagram will suce to show false): (i) If x is an extreme point of S , then x is also an extreme point of T . (ii) If x is an extreme point of T , then x is also an extreme point of S ....
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