Uhura Enterprises provides consulting services for customers. Consultants are brought in for two-hour shifts. A consultant will spend one full hour with a customer. The customers remain for that hour and then leaves. Each consultant is paid $25 per hour. Let Xi represent the number of consultants starting their shift at hour i, (i = 1,2,3,4). The firm wishes to minimize salary costs while satisfying demand. Customer demand is as follows.
1:00-2:00 4
2:00-3:00 5
3:00-4:00 10
4:00-5:00 4
5:00-6:00 8
Suppose that if consultants start shifts at both 1:00 and 2:00, then the firm will incur an additional fixed cost of $100 (for secretarial support). Let Y be a binary (0-1) variable that equals 1 if this event occurs, and 0 otherwise. Which of the following would "activate" Y (i.e., which would force Y to equal 1 when it is supposed to)? (In the options below, Y and qi are binary (0-1) variables.)
a) X1 + X2 ≤ 5000Y
b) X1 ≤ 5000Y and X2 ≤ 5000Y
c) X1 + X2 ≥ 5000Y
d) X1 + X2 ≥ Y
e) X1 ≤ 5000q1 and X2 ≤ 5000q2 and q1 + q2 ≤ 1 + Y
Answer: e