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# Is the di erence in length between the total track

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Unformatted text preview: is the di erence in length between the total track times on CD1 and CD2, i.e. min z . The assignment of songs to the CD's must satisfy the following conditions: (a) Each CD must have exactly two ballads. (b) CD 1 must have at least three hit songs. (c) Either song 5 or song 6 must be on CD 1. (d) If songs 2 and 4 are on CD 1, then song 5 must be on CD 2. Song Type Length (in minutes) 1 Ballad 4 2 Hit 5 3 Ballad 3 4 Hit 2 5 Ballad 4 6 Hit 3 7 Unspeci ed 5 8 Ballad and hit 4 De ne suitable variables and write down constraints so that the problem can be solved using integer programming. DO NOT SOLVE THE RESULTING PROBLEM. (10 marks) AMA405 4 Continued AMA405 4 (i) The table below represents the optimal tableau for a certain linear pro- gramming problem. Basic z x 1 x 2 x 3 x 4 x 5 Soln z row 1 1 = 2 3 x 2 1 1 1 1 2 x 1 1 1 1 = 2 1 1 x 5 5 = 2 2 1 1 Describe any interesting features of this solution giving your interpretation of the consequences of any such peculiarity. (3 marks) (ii) Consider the following linear programming problem in which x 2 is the start- ing basis vector max z = c T 1 x 1 + c T 2 x 2 subject to Ax 1 + Ix 2 = b Prove that at the optimal solution BB 1 A c T 1 T and c T B 1 c T 2 T (7 marks) (iii) Consider the primal problem max z = f ( x ) subject to g ( x ) x 2 R n Write down the theorem specifying the Wolfe dual problem. (4 marks) (iv) With reference to the Wolfe dual, derive carefully the dual of the primal Linear Programming problem max x z = c T x such that Ax b x . Prove that, if a primal Linear Programming problem has an unbounded feasible optimal solution, then the dual problem has no feasible solution. (11 marks) AMA405 5 Turn Over AMA405...
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