Question

The Record-a Song Company has contracted with a rising star to record eight songs. The duration of the different songs are 8, 3,5,5,9,6,7 and 12 minutes, respectively. Record-a-Song uses two-sided cassette tape for the recording. Each side has a capacity of 30 minutes. The company would like to distribute the songs between the two sides such that the length of the songs on each side is about the same. Formulate the problem as an integer programming model and find the optimum solution.

It would also be very helpful for a step by step explanation of the process in solver mode in Excel

Answer 1

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The total duration of the eight songs for the given song is 55 minutes.

Consider the variables as

S₁ , S₂, S₃, S₄, S₅, S₆, S₇, and S₈.

Based on that write the objective of the function:

8S₁ +  3S₂ + 5S₃ + 5S₄ + 9S₅ + 6S₆ + 7S₇ + 12S₈ = 55

Consider the following constraints based on the allocation of duration.

Constraint 1:

8S₁ + 5S₃ + 9S₅ + 6S₆ <= 30

Constraint 2:

3S₂ + 5S₄ + 7S₇  + 12S₈ <= 30

We can see that the calculated value is in decimals. But, we cannot split the song into two CD’s.

Hence, round the values to nearest of 1.

Thus, the value of constraint 1 is and the value of constraint 2 is 27 and 28 min.

The optimal solution of the linear programming is 55 minutes.

We can distribute the songs into two CD’s with the duration of 27 and 28 min.

So, the songs with 8 min, 5 min, 9 min, and 6 min will be distributed in first CD and the songs with 3 min, 5 min, 7 min, and 12 min will be distributed in the second CD.