Question 1. A distribution company delivers supplies to six different customers (A-F). The company is looking to locate a new warehouse near these customers. Find the optimal location for that warehouse that minimizes the sum of the distance travelled to all customers each week. In each case below, show the optimal solution (x*, y*), and the objective function value f(x*, y*). Distance units are in miles.

a) Assume Euclidean distance and complete one iteration of the manual procedure assuming a starting location of (4,4). Use Excel to find the optimal location

b) Assume that the company instead wants to minimize the maximum rectilinear distance to their customers. Where should they put the warehouse? What is the maximum distance travelled?

Question2. Suppose the company decided to build multiple warehouses but the criterion is that each warehouse must be able to reach a customer within 5 minutes. The matrix below shows six potential sites, the cost to build on each site, and a matrix of 0-1 indicators as to whether or not that site is within 5 minutes of each customer. Find the optimal number of warehouses to build (and where they should be) such that costs are minimized while ensuring each customer is served. Indicate which locations are chosen and total cost. Use the greedy algorithm covered in class. Is this optimal?

	Potential Warehouse Locations
Customer	1	2	3	4	5	6
A	1	0	1	1	1	0
B	1	0	0	0	1	1
C	0	1	0	1	0	0
D	0	1	1	0	1	0
E	0	1	1	0	0	1
F	1	0	0	0	0	1
Cost (millions)	$ 3.70	$ 2.80	$ 3.10	$ 1.90	$ 2.70	$ 3.20

\begin{tabular}{|c|c|c|c|} \hline Customer & Deliveries/Week & ai & bi \\ \hline A & 12 & 6 & 6 \\ \hline B & 9 & 2 & 1 \\ \hline C & 4 & 1 & 7 \\ \hline D & 8 & 8 & 4 \\ \hline E & 7 & 5 & 2 \\ \hline F & 6 & 2 & 7 \\ \hline Total & 46 & & \\ \hline \end{tabular} Existing Customers \begin{tabular}{|c|c|c|c|} \hline Customer & Deliveries/Week & ai & bi \\ \hline A & 12 & 6 & 6 \\ \hline B & 9 & 2 & 1 \\ \hline C & 4 & 1 & 7 \\ \hline D & 8 & 8 & 4 \\ \hline E & 7 & 5 & 2 \\ \hline F & 6 & 2 & 7 \\ \hline Total & 46 & & \\ \hline \end{tabular} Existing Customers