Question

Question 6: You have been assigned to arrange the songs on the cassette version of Tarkan's latest album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette must total between 14 and 16 minutes in length. The length and type of each song are given in Table below. The assignment of songs to the tape must satisfy the following conditions: • Each side must have exactly two ballads. . Side 1 must have at least three hit songs. Either song 5 or song 6 must be on side 1 If songs 2 and 4 are on side 1, then song 5 must be on side Explain how you could use an integer programming formulation to determine whether there is an arrangement of songs satisfying these restrictions Song Type Length Ballad Ballad (in minutes) 4 5 3 2 4 3 2 3 4 5 6 7 8 Hit Ballad Hit Ballad and hit

i want only objective function and constraints. not a Lp solver.

Answer 1

Decision Variables:

Let X1,X2, X3,X4, ....... X8 be the Binary variable to represent the placement on songs 1 to 8 on Side 1 of Casette;

such that:

Xi = 1 ; if song i is placed on casette side 1

   = 0 ; if song i is placed on casette side 2 ;

Objective: To Minimize the sum product of song assignment and the Length of the song.

Min Z = 4X1 + 5X2 + 3X3 + 2X4 + 4X5 + 3X6 + 5X7 + 4X8

Such that:

4X1 + 5X2 + 3X3 + 2X4 + 4X5 + 3X6 + 5X7 + 4X8 >= 14

4X1 + 5X2 + 3X3 + 2X4 + 4X5 + 3X6 + 5X7 + 4X8 <= 16

Subject to constraints:

Each side to have exactly 2 balads: Songs 1,3,5 & 8 are balads

X1 + X3 + X5 + X8 = 2

Side 1 to have at least 3 hit songs: Songs 2,4,6 & 8 are hit songs

X2 + X4 + X6 + X8 >= 3

Either Song 5 or Song 6 to be on Side 1:

X5 + X6 = 1

If Songs 2 and 4 are on side 1, Song 5 must be on side 2:

X2 + X4 + X5 <= 2