Using the 2010 County Sorted 250.grt data set which represents the populations of the 250 most populous counties in the contiguous United States in 2010,
(a) Solve the P-median problem for P = 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 using SITATION. Solve the problem optimally using the Lagrangian solution option. If you use the default parameters, the solution times should be very small. For each value of P, record the demand-weighted total distance. Also, for each value of P, use SITATION to plot a map of the results and save the 14 maps.
(b) The total demand for this data set should be 187,866,509. Use this value to convert the demand-weighted total distances in part (a) into demand-weighted average distances. Note that by computing the demand-weighted average distances in this way, you will avoid some of the rounding that may be present when you simply record the demand-weighted average distance reported by SITATION.
(c) Use the values found in part (b) to regress the demand-weighted average distance against the number of facilities. You should estimate a regression line of the form Dem Wtd Avg Dist = að Þ number of facilities b . What are the values of a and b? What is the R2 value for the line that you have computed?
(d) How does the value of b compare to the theoretical value of b ¼ 0:5 that we obtained in Chapter 1 for the analytic model in which we used Manhattan distances in a diamond with uniformly distributed demands? Why do you think the value found in part (c) is different from the theoretical value?
(e) Using the regression found in part (c) estimate the demand-weighted average distance that you would get if you used eight facilities.
(f) Solve the 8-median problem for this data set. How well does your prediction from part (d) compare to the exact value?
(g) Use the maps that you saved in part (a) to generate an animated GIF of the solutions as you progress from 1 to 50 sites. Use only the 14 maps that you saved in part (a)