Consider the single facility rectilinear location problem. There are six $(6)$ exist$$ing facilities, numbered $1$ through $6,$ which will interact with the new facility. The $($x$,$y$)$ coordi$$nates of the existing facilities are given as follows: $1.(2,2),2.(4,5),3.(4,8),4.(8,5),5.(12,8),$ and $6.(12,2).$ The new facility will have "equal interaction" with each existing facility.

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In answering the following parts, when applicable you must clearly show the feasible region, and in those cases where more than one solution exists, you must show all points that are optimum. In presenting your answers, please use graph paper $($or engineering paper$).$ Make sure you still show all your calculations on a separate page.

$($a$)$ Determine the location and the objective function$/$value of the minisum optimum.

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$($b$)$ Determine the location and the objective function value of the minimax optimum.

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$($c$)$ Is there a point which is optimum for both the minisum and the minimax objec$$tive$?$ Why or why not?

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$($d$)$ Suppose the optimum minisum location$($s$)$ you found in part $($a$)$ is not usable. Given the minisum objective, determine the best location for the new facility among the following three points:

X$.(3,5),$ Y$.(4,7),$ and Z$.(10,5).$

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$($e$)$ Suppose the maximum distance to any existing facility must be less than or equal to $9$ rectilinear distance units. For the minisum objective, determine the optimum location and the objective function value at the optimum location.

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$($f$)$ Suppose the distance to each existing facility must be greater than or equal to $3$ rectilinear distance units. For the minisum objective, determine the optimum location and the objective function value at the optimum location.

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$($g$)$ Suppose the distance to each existing facility must be less than or equal to $3$ rectilinear distance units. For the minisum objective, determine the optimum location and the objective function value at the optimum location.