Question

you have five errands to run around town in no particular order. you must visit the post office (p), bank (b), dry cleaners (d), hospital (h), and market (m). a weighted graph is shown with the distances (in miles) to the locations shown on the edges. a,b,h,p,m,d,a use the nearest neighbor method to find an approximate solution. let the letter (a) represent home. type the correct path. (use a comma to separate vertices as needed.) using the brute force method, one optimal solution is home, bank, hospital, post office, market, dry cleaners, and home. how does the length of the route found above compare to this solution? select the correct answer below and fill in the answer box(es) to complete your choice. a. the distance found using the nearest neighbor method, the distance is miles and using the brute force method, the distance is miles. b. the distance found using the nearest neighbor method is larger than the distance using the brute force method. using the nearest neighbor method, the distance is miles and using the brute force method, the distance is miles. c. the distances are both miles.

Explanation:

Step1: Find the path using Nearest Neighbor Method

Start at \(A\) (home). The nearest neighbor from \(A\) is \(B\) (distance \(2\)). From \(B\), the nearest is \(H\) (distance \(8\)). From \(H\), the nearest is \(P\) (distance \(9\)). From \(P\), the nearest is \(M\) (distance \(5\)). From \(M\), the nearest is \(D\) (distance \(8\)). Then back to \(A\). The total distance is \(2 + 8+9 + 5+8=32\) miles.

Step2: Analyze the Brute - Force Method (optimal solution)

The optimal path (by checking all possible Hamiltonian circuits) is \(A - B - H - P - M - D - A\). The distance is \(2+8 + 9+5+7+7 = 38\) miles.

Answer:

A. The distance found using the Nearest Neighbor Method is \(32\) miles and using the Brute Force Method, the distance is \(38\) miles.