Let B2 to be Birkhoff polytope of order 2, that is the collection of all 2 x 2 matrices with nonnegative entries whose rows and columns cach sum to one. Suppose cij are real numbers such that c12 = c21 < c11 < ¢22. Consider the following LPP: maximize: C1I11+¢12#12+¢21T21 + €22F22 Il1 T12 subject to: e B2 121 122 #|1, F12; F21, T22 > 0 (a) Write this problem in canonical form {r = (r11,12, 21, #22) E Rª | Ar = b, x > 0}. Hint: A should be a 3 x 4 matrix (after you climinate one row which is lincarly dependent on the others). (b) List all BFS. Remark: compare your answer to BFS of B2 in HW2. (c) Find the mnaximizer r* and the maximum cost for this LPP.
Let B2 to be Birkhoff polytope of order 2, that is the collection of all 2 x 2 matrices with nonnegative entries whose rows and columns cach sum to one. Suppose cij are real numbers such that c12 = c21 < c11 < ¢22. Consider the following LPP: maximize: C1I11+¢12#12+¢21T21 + €22F22 Il1 T12 subject to: e B2 121 122 #|1, F12; F21, T22 > 0 (a) Write this problem in canonical form {r = (r11,12, 21, #22) E Rª | Ar = b, x > 0}. Hint: A should be a 3 x 4 matrix (after you climinate one row which is lincarly dependent on the others). (b) List all BFS. Remark: compare your answer to BFS of B2 in HW2. (c) Find the mnaximizer r* and the maximum cost for this LPP.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(7) Let B2 to be Birkhoff polytope of order 2, that is the collection of all 2 x 2 matrices with
nonnegative entries whose rows and columns cach sum to one. Suppose c;; are real numbers
such that c12 = c21 < ¢11 < ¢22. Consider the following LPP:
maximize: c1711+¢12F12+ C21T21 + C22T22
I12
subject to:
E B2
I21
122
T11, F12, E21, 122 > 0
(a) Write this problem in canonical form {r = (x11, F12, 121, 122) E Rª | Ar = b, x > 0}.
Hint: A should be a 3 x 4 matrix (after you eliminate one row which is linearly
dependent on the others).
(b) List all BFS. Remark: compare your answer to BFS of B2 in HW2.
(c) Find the maximizer r* and the maximum cost for this LPP.
(d) For the maximizer r*, show that for all basic directions D; at x* the reduced cost
"D; < 0.
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