What is the minimal shipping cost and the associated optimal shipping schedule?A new type of tanker trucks can completely eliminate the yield loss. If the manufacturer can replace its fleet with the new trucks, how much can it save on shipping cost? How should it change the optimal shipping schedule?

1. Shipping Schedule
A manufacturer has three plants and four warehouses. Each plant has a potential capacity, expressed in cartons of product, and each warehouse has a demand requirement for the week that must be met. There are 12 possible shipment routes, and for every plant–warehouse combination, the unit shipping cost is known. The following table provides the given information.
(From) plant Atlanta Boston Chicago Denver Capacity
Minneapolis $0.60 $0.56 $0.22 $0.40 16,000
Pittsburgh 0.36 0.30 0.28 0.58 16,000
Tucson 0.65 0.68 0.55 0.42 16,000
Requirement 8000 10,000 12,000 9000
The product is subject to evaporation in the tanker trucks used for distribution purposes. The average amount of evaporation depends the distance traveled and the average temperature. The following table shows the corresponding yield loss as it has been observed to occur on each of its shipping routes.
(From) plant Atlanta Boston Chicago Denver
Minneapolis 0.24 0.19 0.07 0.11
Pittsburgh 0.10 0.05 0.04 0.15
Tucson 0.26 0.41 0.32 0.27
(a) What is the minimal shipping cost and the associated optimal shipping schedule?
(b) A new type of tanker trucks can completely eliminate the yield loss. If the manufacturer can replace its fleet with the new trucks, how much can it save on shipping cost? How should it change the optimal shipping schedule?
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2. Sites Selection
An owner of a motel chain seeks to expand its business. The owner has identified nine potential locations where a motel might be located. However, some of these locations are within 30 miles of each other, and the owner prefers to separate motels in the chain by at least 30 miles to avoid cannibalizing demand. The information about location conflicts has been obtained from a map:
Potential site Potential sites within 30 miles
1 2
2 1, 5, 7
3 5, 6, 8, 9
4 6, 7
5 2, 3, 6, 8
6 3, 4, 5, 7
7 2, 4, 6
8 3, 5
9 3
(a) [5pts] The owner wants to build motels on as many sites as possible, while adhering to the 30-mile requirement. Which sites should be selected?
Hint: we can build the table for the conflicts: the numbered rows and columns correspond to the various motel sites. An entry of 1 appears in row i and column j if sites i and j are within 30 miles
of each other.
1 2 3 4 5 6 7 8 9
1 1 1
2 1 1 1 1
3 1 1 1 1 1
4 1 1 1
5 1 1 1 1 1
6 1 1 1 1 1
7 1 1 1 1
8 1 1 1
9 1 1
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