Answer the following questions along with a concise explanation with respect to the linear program to maximize cx subject to x∈X={x:Ax=b,x≥0}, where A is m×n of rank m<n. ) a3 J' j( |' _! t; h In a simplex tableau, if z_j-c_j=-7 for a nonbasic variable x_j, what is the change in objective value when x_j enters the basis given that the minimum ratio is 3 in the pivot? + Z! O0 t7 \# B; m- x z( k* C If an extreme point is optimal, then is it possible that not all z_j-c_j≥0 for an associated basis?6 F) g- ^# Z6 |
If there exists a d such that Ad=0,d≥0, and cd≥0, then is the optimal objective value unbounded? + B* L" g6 `, @& i- j, \ Let x ̅ be a feasible solution with exactly m positive components. Is x ̅ necessarily an extreme point of X?7 M3 c: q: q" \/ ^4 K9 |. @& `) T$ \
If a nonbasic variable x_k has z_k-c_k=0 at optimality, then can one claim that alternative optimal solutions exist? ; s! c; r, k5 Q/ K3 l3 w! D If x_1 and x_(2 )are adjacent points and if B_1 and B_2 are respective associated bases, then these bases are also adjacent. True or false? Explain. 6 M+ O/ O# g3 ~9 B1 n Is it possible for an optimal solution to have more than m positive variables? 4 q7 o5 ?; Q7 F7 o# m Suppose that n=m+1. What is the least upper bound on the number of extreme points and feasible bases? 1 J; h9 B0 f X2 |1 V) s( ^7 g' r1 L A p-dimensional polyhedron can have at most p extreme directions. True or false? Explain. R9 E) L8 ^# ~4 M Let x ̅ be an extreme point having (m-1) positive components. Then there are (p+1) bases associated with this extreme point, where p=n-m. True or false? (Assume that Ax=b does not imply any variable to be a constant) Explain. % {3 F* m% q9 H' l, l% S5 N) i3 B/ }) i2 M4 i( `6 Y
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发表于 2014-11-5 20:50
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