Using the Dinko$$s Facility Location Model

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$1.$ Suppose Evelyn chooses not to open both facilities immediately.$$ Instead she will open a single facility and wait to see whether it is successful before opening the second.

a$)$ If you followed the directions above, you found that the best location for a single facility is the corner of $22$nd Avenue and $25$th Street.$$ If this location is a success, where should Evelyn place the second facility?$($Use your work area for two facilities.$$ Enter the location $(22,25)$ into cells C$40$:D$40,$ and evaluate potential locations for the second Dinko$$s by changing cells C$41$:D$41.)$

b$)$ The instructions above also directed you to find the best locations for two facilities when they are positioned simultaneously.$$ These locations were the corner of $20$th Avenue and $35$th Street, and the corner of $15$th Avenue and $17$th Street.$$ Which of these facilities should be opened first?$$ Explain.

c$)$ How and why does the pair of locations identified in part $($a$)$ differ from the pair of locations in part $($b$)?$ If Evelyn thinks it is very likely that she will open both facilities, which pair of locations should she choose?

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$2.$ a$)$ We used the metropolitan distance metric rather than the Euclidean distance.$$ Is the Euclidean distance between two buildings longer or shorter than the metropolitan distance?

b$)$ When placing a single facility, our objective function minimized the weighted average distance between the facility and the ten office buildings.$$ What is the optimal location if the objective function minimizes the maximum distance between the facility and the ten office buildings?$$ What is the value of the objective function $($i$.$e$.$ the maximum distance$)$ at this optimal location?

c$)$ Under what circumstances does it make sense to choose an objective function that minimizes the maximum distance between a facility and its customers?$$ Give an example.

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$3.$ Suppose Evelyn is not restricted to the locations listed in Table $2$; instead she can place her facilities anywhere in the city.$$ Try using Solver to identify the best location for a single facility.$$ Use your work area for one facility, and set up the Solver Parameters window as shown below.

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This is not a linear program, so choose $$GRG Nonlinear$$ as the Solving Method in the Solver Options window.

a$)$ Enter the initial location $(1,1)$ into cells C$19$:D$19,$ and run Solver.$$ Then enter the initial location $(40,40),$ and re$-$run Solver.$$ Solver will most likely return different solutions.$$ Why does this happen?

b$)$ Since we are not limited to the five locations listed in Table $2,$ you might argue that we should include the constraints: $$to keep the solution in the downtown district.$$ Why are these constraints unnecessary?