For each problem below, hand in an algebra formulation, and also solve using
Excel and Solver, handing in the standard printouts.
Q1. (25 points) Problem 11 on page 50 of the course pack
(Eli Daisy).
Q2. (35 points) Problem 60 on page 77 of
the course pack (Candy Kane cosmetics).
Q3. (40 points) Your firm makes fluorescent paint pigments in four plants and ship them to four distributors (abbreviated "D1" through "D4"), as follows:
Unit |
Unit Shipping Cost To |
||||||
Plant |
Capacity |
Cost |
Impurities |
D1 | D2 | D3 | D4 |
Northeast | 1000 | $ 12.40 | 12 | $ 1.20 | $ 1.75 | $ 2.35 | $ 2.85 |
Southeast | 1250 | $ 11.55 | 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 distributors' demand for the pigments is as follows:
D1 | D2 | D3 | D4 | |
Max Impurities | 15.0 | 15.0 | 14.0 | 15.5 |
Base Demand | 700 | 600 | 550 | 675 |
Advertising Sensitivity | 0.05 | 0.1 | 0.05 | 0.125 |
For example, distributor D1 will accept up to 700 units of pigment, plus 0.05 units 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 units, D2 will accept up to 600 + (0.1)(100) = 610 units, D3 will accept up to 555 units, and D4 will accept up to 687.5 units. "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 unit. How can you maximize your profits?
Note: this problem combines blending, transportation, and elements of the "pickles" problem. Do not be worried if your solution contains a lot of fractions. To save typing, you may download the data for this problem here.