Question 1.
DynaTune is trying to determine how to allocate its
$145,000 advertising budget for a new product. Radio and television are the two
media outlets selected for advertising. The table below shows the costs of
advertising in these two media and the estimated number of new customers
reached by increasing amounts of advertising.
|
Media Type & Number of Ads |
Number New Customers Reached |
Cost per Ad |
|
Radio 1-10 |
900 |
$1,000 |
|
Radio 11-20 |
700 |
$900 |
|
Radio 21-30 |
400 |
$800 |
|
Television 1-5 |
10,000 |
$12,000 |
|
Television 6-10 |
7,500 |
$10,000 |
|
Television 11-15 |
5,000 |
$8,000 |
For example, each of the first 10 ads the company places on the radio will cost
$1000 and is expected to reach 900 new customers. Each of the next 10 will cost
$900 and is expected to reach 700 new customers. Assume the company will
purchase no more than 30 radio ads and no more than 15 television ads.
An LP to maximize the number of new customers reached is
formulated below:
Let R1 = # radio ads in 0-10 range
R2 = # radio ads in 11-20 range
R3 = # radio ads in 21-30 range
T1 = # TV ads in 0-5 range
T2 = # TV ads in 6-10 range
T3 = # TV ads in 11-15 range
Maximize 900R1 + 700R2 + 400R3 + 10000T1 + 7500T2 + 5000T3
Subject to:
R1 <= 10 (upper limit for type 1 radio ads)
R2 <= 10 (upper limit for type 2 radio ads)
R3 <= 10 (upper limit for type 3 radio ads)
T1 <= 5 (upper limit for type 1 TV ads)
T2 <= 5 (upper limit for type 2 TV ads)
T3 <= 5 (upper limit for type 3 TV ads)
R1 + R2 + R3 <= 30 (radio ads purchase limit)
T1 + T2 + T3 <= 15 (TV ads purchase limit)
1000R1 + 900R2 + 800R3 + 12000T1 + 10000T2 + 8000T3 <= 145000 (budget constraint)
R1, R2, R3, T1, T2, T3 <= 0
(a)
Set up and solve the problem on a
spreadsheet.
(b)
What is the optimal
solution? Explain the rationale for the
solution.
(c)
Suppose the number of new
customers reached by 11-15 television ads is 7500 (instead of 5000), will the
optimal solution in (b) still be applicable?
(d)
Suppose the number of new
customers reached by 6-10 television ads is 6500 (instead of 7500), will the
optimal solution in (b) still be applicable?
(e)
What will the number of new
customers reached be if the advertising budget is increased by $25,000?
(f)
What will the number of new
customers reached be if the advertising budget is reduced to $135,000?
Question
2.
Montana Pulp (MP) produces recycled paperboard by
combining 4 grades of recycled paper stock. Each grade of stock has a different
strength, color, and texture. The strength, color, and texture of the finished
paperboard are a weighted average of those characteristics of the paper stock
inputs. The table provides the characteristics of the paper inputs and their
costs per ton. An order for 500 tons of paperboard with a
strength of at least 7, a color of at least 5, and a texture of at least
6 has been received. MP needs to find the minimal-cost mix of inputs required
to satisfy this order.
|
Paper Stock |
Strength |
Color |
Texture |
Cost/Ton |
|
1 |
8 |
9 |
8 |
$150 |
|
2 |
6 |
7 |
5 |
$110 |
|
3 |
5 |
5 |
6 |
$90 |
|
4 |
3 |
4 |
5 |
$50 |
An LP to minimize the cost of the orders is formulated as:
Let x1 = tons of paper stock 1 to use
x2 = tons of paper stock 2 to use
x3 = tons of paper stock 3 to use
x4 = tons of paper stock 4 to use
Minimize 150x1 + 110x2 + 90x3 + 50x4
Subject to:
x1 + x2 + x3 + x4 = 500 (order size)
(8x1 + 6x2 + 5x3 + 3x4)/500 >= 7 (strength requirement)
(9x1 + 7x2 + 5x3 + 4x4)/500 >= 5 (color requirement)
(8x1 + 5x2 + 6x3 + 5x4)/500 >=6 (texture requirement)
x1, x2, x3, x4 >= 0 (non-negativity)
(a)
Set up and solve the problem on a
spreadsheet.
(b)
What is the optimal
solution? Explain the rationale for the
solution.