"A) A city is reviewing the location of its fire stations. The city is made up of a number of neighborhoods, as illustrated in the figure below.
A fire station can be placed in any neighborhood. It is able to handle the fires for both its neighborhood and any adjacent neighborhood (any neighborhood with a non-zero border with its home neighborhood). The objective is to minimize the number of fire stations used.
Solve this problem. Which neighborhoods will be hosting the firestations?
B) Ships are available at three ports of origin and need to be sent to four ports of destination. The number of ships available at each origin, the number required at each destination, and the sailing times are given in the table below.
Origin Destination Number of ships available
1 2 3 4
1 5 4 3 2 5
2 10 8 4 7 5
3 9 9 8 4 5
Number of ships required 1 4 4 6 Develop a shipping plan that will minimize the total number of sailing days.
C) The following diagram represents a flow network. Each edge is labeled with its capacity, the maximum amount of stuff that it can carry.
a. Formulate an algebraic model for this problem as a maximum flow problem.
b. Develop a spreadsheet model and solve this problem. What is the optimal flow plan for this network? What is the optimal flow through the network?"