Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are given in Figure 31. Town 1 is at (10, 20); town 2 is at (60, 20); town 3 is at (40, 30); town 4 is at (80, 60). Town 1 averages 20 fires per year; town 2, 30 fires; town 3, 40 fires; and town 4, 25 fires. The county wants to build the fire station in a location that minimizes the average distance that a fire engine must travel to respond to a fire. Since most roads run in either an east–west or a north–south direction, we assume that the fire engine can only do the same. Thus, if the fire station were located at (30, 40) and a fire occurred at town 4, the fire engine would have to travel (80 30) (60 40) 70 miles to the fire. Use linear programming to determine where the fire station should be located. (Hint: If the fire station is to be located at the point (x, y) and there is a town at the point (a, b), define variables e, w, n, s (east, west, north, south) that satisfy the equations x a w e and y b n s. It should now be easy to obtain the correct LP formulation.)
Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are given in Figure 31. Town 1 is at (10, 20); town 2 is at (60, 20); town 3 is at (40, 30); town 4 is at (80, 60). Town 1 averages 20 fires per year; town 2, 30 fires; town 3, 40 fires; and town 4, 25 fires. The county wants to build the fire station in a location that minimizes the average distance that a fire engine must travel to respond to a fire. Since most roads run in either an east–west or a north–south direction, we assume that the fire engine can only do the same. Thus, if the fire station were located at (30, 40) and a fire occurred at town 4, the fire engine would have to travel (80 30) (60 40) 70 miles to the fire. Use linear programming to determine where the fire station should be located. (Hint: If the fire station is to be located at the point (x, y) and there is a town at the point (a, b), define variables e, w, n, s (east, west, north, south) that satisfy the equations x a w e and y b n s. It should now be easy to obtain the correct LP formulation.)
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