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Oliveira Office Supply distributes specialty papers to big-box stores in twelve major U.S. metropolitan areas and plans to consolidate its warehouses into two national distribution centers. To identify suitable locations for the centers, Oliveiras distribution manager maps the twelve stores on a two-dimensional grid so that coordinates (x_k, y_k) can be associated with each store k. Besides, the logistics manager points out that the trucks make different number of trips to the various stores each year. Thus, a better proxy for total distribution cost would be obtained by weighting the Euclidean distance from each store k to its assigned distribution center by the annual number of trips to store k (represented as n_k). The coordinates of the twelve stores and their annual number of trips are shown in the following table.

Store k	Coordinates		Trips
	xk	yk	nk
1	2	96	21
2	5	41	12
3	20	10	20
4	44	48	15
5	60	58	27
6	100	4	8
7	122	94	21
8	138	80	16
9	150	40	10
10	170	18	18
11	182	2	25
12	190	56	14

Please develop an optimization model to determine the optimal locations of two distribution centers in integer x-y coordinates that minimize the total annual distribution cost from the two centers to the twelve stores.

1. Define the decision variables and write down the detailed mathematical formulation for Oliveiras optimization problem.
2. Solve the optimization problem by Excel Solver. Write down Oliveiras detailed optimal decisions and optimal total annual distribution cost.
3. Now suppose Oliveira restricts the locations of the two distribution centers to the locations of the twelve stores. In other words, the locations of the two distribution centers must be selected from the locations of the twelve stores. Besides, the two distribution centers must locate at different places. With the above restrictions, re-do parts a) and b).

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