# The advertising manager of a magazine faces the following problem: For week t, t = 1, 2 . . . , 13, she has been allocated

The advertising manager of a magazine faces the following problem: For week t, t = 1, 2 . . . , 13, she has been allocated a maximum of nt pages to use for advertising. She has received requests r1,r2, . . . ,rB for advertising, bid rk indicating:

i) the initial week ik to run the ad,

ii) the duration dk of the ad (in weeks),

iii) the page allocation ak of the ad (half-, quarter-, or full-page),

iv) a price offer pk .

The manager must determine which bids to accept to maximize revenue, subject to the following restrictions:

i) Any ad that is accepted must be run in consecutive weeks throughout its duration.

ii) The manager cannot accept conflicting ads. Formally, subsets Tj and T¯ j for j = 1, 2, . . . , n of the bids are given, and she may not select an ad from both Tj and T¯ (j = 1, 2, . . . , n). For example, if T1 = {r1,r2}, T¯ 1 = {r3,r4,r5}, and bid r1 orr2 is accepted, then bids r3,r4, or r5 must all be rejected; if bid r3,r4, or r5 is accepted, then bids r1 and r2 must both be rejected.

iii) The manager must meet the Federal Communication Commission’s balanced advertising requirements. Formally, subsets Sj and S¯ j for j = 1, 2, . . . , m of the bids are given; if she selects a bid from Sj , she must also select a bid from S¯ j (j = 1, 2, . . . , m). For example, if S1 = {r1,r3,r8} and S= {r4,r6}, then either request r4 or rmust be accepted if any of the bids r1,r3, or r8 are accepted.

Formulate as an integer program.