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 coe²cients ( 2 α ) is proposed, where α is an arbitrary constant. What must the objective coe²cient for this activity be in order that it be proFtable to produce some of the new product? Why is your answer independent of α ? 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 su²ce 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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This note was uploaded on 10/04/2010 for the course ORIE 3300 taught by Professor Todd during the Spring '08 term at Cornell.

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