$1)$ Manufacturing company plans to build a warehouse somewhere near to its three existing plants$($facilities$).$ The coordinates of the existing plants and the number of trips to the warehouse a day are given below.

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a$)$ Determine the location for the warehouse which minimizes the total cost.$($minisum$)(5)$

b$)$ Calculate the total cost if the warehouse is built there. $(5)$

c$)$ The management has decided that in order to reduce the total costs it may build more than $1$ warehouse. Determine the optimal number of warehouses and which plants should be served by which warehouse. It costs $40$ units to build each warehouse. Please consider and calculate the total costs of following cases and find which case total cost is minimum. $(20)$