QBUS6820: Prescriptive Analytics Tutorial 10 Semester 1, 2024
Problem 1:
The Royal Seas Company runs a three-night cruise to the Carribean from Port Canaveral. The
company wants to run TV ads promoting its cruises to high-income men, high-income women, and
retirees. The company has decided to consider airing ads during prime time, afternoon soap operas
and during the evening news. The number of exposures (in millions) expected to be generated by
each type of ad in each of the company’s target audiences is summarized in the following table.
Prime Time Soap Operas Evening News
High Income Men 6 3 6
High Income Women 3 4 4
Retirees 4 7 3
Ads during prime time, the afternoon soaps, and the news hour cost $120,000, $85,000 and $100,000
each respectively. Royal Seas wants to achieve the following goals:
• Goal 1: To spend approximately $900,000 of TV advertising
• Goal 2: To generate aproximately 45 million exposures among high-income men
• Goal 3: To generate approximately 60 million exposures among high-income women
• Goal 4: To generate approximately 50 million exposures among retirees
(a) Formulate a GP model for this problem. Assume overachievement of the first goal is equally as
undesirable as underachievement of the remaining goals on a percentage deviation basis.
(b) Implement your model in a spreadsheet and Python and solve it.
(c) What is the optimal solution?
(d) What solution allows the company to spend as close to $900,000 as possible without exceeding
this amount?
Solution:
(a) Let xi be the number of ads to air during period i for i ∈ {1, 2, 3}, where 1, 2, 3 stand for
prime time, soap operas and evening news respectively.
min
o1
900
+
u2
45
+
u3
60
+
u4
50
(1)
s.t. 120x1 + 85x2 + 100x3 + u1 − o1 = 900 (2)
6x1 + 3x2 + 6x3 + u2 − o2 = 45 (3)
3x1 + 4x2 + 4x3 + u3 − o3 = 60 (4)
4x1 + 7x2 + 3x3 + u4 − o4 = 50 (5)
xi ≥ 0, xi is an integer, ∀i (6)
oj , uj ≥ 0, ∀j (7)
The objective (1) minimises the total undesired deviations. Constraints (2) - (5) are the
goals. Constraint (6) and (7) are the integer and non-negative constraints.
(b) See Canvas.
(c) The optimal solution is x1 = 0, x2 = 5, x3 = 5.
(d)
min u1 (8)
s.t. 120x1 + 85x2 + 100x3 + u1 = 900 (9)
xi ≥ 0, xi is an integer, ∀i (10)
u1 ≥ 0 (11)
There are multiple optimal solutions. An optimal solution should have u1 = 0, i.e. all
$900,000 budget has been spent.