Question- Your run a petroleum company that produces motor oil in four plants and ships it to four distributors D1, D2, ..., D4). The details of shipping and producing are on the table below:
|
Plant
|
Production
Capacity (gallons)
|
Production Cost
per gallon
|
Impurities
Level
|
Shipping cost [attributors per gallon
|
|
D1
|
D2
|
D3
|
D4
|
|
Northeast
|
1000
|
$12.40
|
12
|
$ 1.20
|
$ 1.75
|
$ 2.35
|
$ 2.85
|
|
Southeast
|
1250
|
$1135
|
15
|
$ 1.95
|
$ 1.35
|
$ 1.75
|
$ 2.15
|
|
Northwest
|
950
|
$10.85
|
18
|
$ 2.45
|
$ 1.50
|
$ 2.10
|
$ 1.95
|
|
Southwest
|
1200
|
$12.05
|
12
|
$2.75
|
$ 2.25
|
$ 2.00
|
$ 1.45
|
The next table contains the dist 'baton' demand for the motor oil and how their demand can Increase due to advertising. For example, distributor D1 will accept up to 700 gallons of motor oil, plus 0.05 gallons for every dollar you spend on national advertising. Advertising is not separated by distributor: a single expenditure affects all distributors simultaneously. Thus, If you spend $100 on advertising, D1 will accept up to 700 + (0.05)(100) = 705 gallons, D2 will accept up to 600 + (0.1)(100) = 610 gallons, D3 will accept up to 555 gallons, and D4 will accept up to 687.5 gallons.
|
D1
|
D2
|
D3
|
D4
|
|
Base Demand (gallons)
|
700
|
600
|
550
|
675
|
|
Advertising Sensitivity
|
0.05
|
0.1
|
0.05
|
0.125
|
|
Max Impurities level
|
15.0
|
15.0
|
14.0
|
15.5
|
"Max impurities" Indicates the maximum average Impurity level allowed for shipments to each distributor. For Instance, the shipments from the four plants to D1, when mixed together, should have an average impurity level of at most 15.0.
You have at most $59,000 to spend on production, shipping and advertising, and all the distributors pay you $28.50 per gallon. How can you maximize your profits?
a. Draw the network that illustrates the dynamics of this problem.
b. Develop a linear program formulation for this problem (write decision variables, objective function and constraints).
Hint: If needed, you con hove constraints regarding how much you spend. So, the amount you spend shows up both In the objective function (as costs) and in the constraints.
c. Solve the problem using Excel (you can use the Excel file provided that contains all the data).
d. Now, let us study the influence of advertising. How much investing in advertising did increase the total revenue? That is, if the company had not invested in advertising, how much would have been its revenue?