Service Woes at Speedy Delivery: Teacher Resources 2. Extension 1: A more complicated network
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- Scarlett Sims
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1 Extensions our extensions are provided in the packet. The first extension is a more difficult version of the sample problem, because the network contains more nodes and arcs. The second requires the construction of a table of distances (similar to tables found on maps or in atlases) from the graph in the Extension 1. The third extension asks students to construct a weighted network from a table of distances. inally, the fourth further complicates the Extension 1 by adding one-way streets to that network. Extension contains one-way streets. This aspect of the problem is captured through the concept of adjacent nodes. or example, if you are at node J in this network, node K is not an adjacent node due to the one-way arc from node K to J. In extension, students can find the required shortest route without finding the shortest route to each of the other nodes. With a little extra effort, the algorithm can be continued to generate the shortest path to every other node in the network.
2 Service Woes at Speedy Delivery: Teacher Resources 2 Extension 1: A more complicated network Because of the success of its quick delivery time, the Speedy Delivery Company has expanded operations. Headquarters is still located at node A, but some additional locations have been added to the delivery area. It was :0 P.M. and Mr. I.M. Lait has just made a delivery to a new customer at point H. Harry Upp contacts I.M. Lait and informs him that the Pane N. Butt Co. at point N, also a new customer, needs something picked up before P.M. Refer to network below, and find the shortest time to go from point H to point N. ill in the graph and the table provided according to the algorithm. Can Mr. Lait reach the company in time? 9 C B A H D J E G L K 9 N M Adjacent Circled Nodes Path Total Node (uncircled) (from H) Time
3 Service Woes at Speedy Delivery: Teacher Resources Extension 2: The Other Way Around Perhaps you have seen a table showing the travel times between major cites on a map. Using the weighted network from figure 2 in the Speedy Delivery problem, make such a table. Be sure to show the travel times between every pair of nodes in the network. C B A D E G A B C D E G A B --- C --- D --- E G
4 Service Woes at Speedy Delivery: Teacher Resources Extension : Cities in Michigan Helen Wheels drives a truck "over the road" between the cities listed in the mileage table below. She would like to develop a schedule where no city-to-city mileage exceeds 100 miles. a. Based upon the mileages given in the table, draw a graph with vertices representing cities and edges representing trips less than 100 miles. If your original graph has edges intersecting, reposition the vertices and redraw the graph to eliminate the intersections. b. Using your graph find the shortest route from Kalamazoo to lint. c. Using your graph find the shortest route from Detroit to each city listed in the table. Ann Arbor Detroit lint Grand Rapids Kalamazoo Lansing Ann Arbor * Detroit 1 * lint 2 * Grand Rapids * 0 91 Kalamazoo * 8 Lansing *
5 Service Woes at Speedy Delivery: Teacher Resources Extension : One-way Streets In the network below, some of the routes are one-way streets. They are marked with arrows. Headquarters is still located at point A. It was :0 p.m. and Mr. I.M. Lait has just made a delivery to a new customer at point H. Harry Upp contacts I.M. Lait and informs him that the Pane N. Butt Co. at point N, also a new customer, needs something picked up before p.m. Using the graph and chart below, find the shortest time to go from point H to point N. ill in the graph and the table provided according to the algorithm. Can Mr. Lait reach the company in time? 9 C B A H D J E G L K 9 N M Adjacent Circled Nodes Path Total Node (uncircled) (from H) Time
6 Service Woes at Speedy Delivery: Teacher Resources Homework Exercises The exercises which follow are standard shortest path problems which are set in contexts taken from, or similar to, those in the case studies. You may want to provide students with some of the information in the case studies for background. Homework problem number 1 includes the concept of one-way streets and raises the possibility that there is no route between two nodes. Homework problem number 2 introduces a real-world context for a large network. It also asks the students to discuss how contextual issues such as time of day might affect the choice of the best travel route between a plant and a warehouse.
7 Service Woes at Speedy Delivery: Teacher Resources 1. In the diagram below the arrows indicate one-way streets. "Hot Meals & Wheels" must make deliveries at locations 1 and. ind the shortest route from node 1 to node. What can be observed about the best path from node to node 1?
8 Service Woes at Speedy Delivery: Teacher Resources 8 2. The Acme Bottling Company recently had a grand opening for their new warehouse in Soda City. Nate Carbo, the bottling manager, feels that the present route being used by the trucks supplying the new warehouse can be improved. Trucks currently take the route OAS where O is the plant at O.K.Cola Junction and S is the new warehouse. Although the roads are relatively straight and well maintained, Nate objects to the distance traveled. Recent studies indicate that the cost of transportation is greatly affected by the distances the trucks travel. Transportation costs primarily consist of maintenance, replacement, and driver wages. Because the fleet makes many trips to Soda City annually, considerable cost reductions can be expected for each mile the route is shortened. The diagram below is a representation of the road network between O.K. Cola Junction and Soda City. Although the arcs of the network are all straight lines, the actual roads may have many curves or hills. It is safe to assume that the rate of speed along any road is the same. The diagram shows the distance, in miles, of each road segment. O 10 B A C 8 D E G H 11 J S a. Use the shortest-route method to find the best route from O.K. Cola Junction to Soda City. b. What other considerations should Mr. Carbo make before authorizing a new route? How might time of day affect the choice of preferred route?
9 Service Woes at Speedy Delivery: Teacher Resources 9 Solution to Extensions Extension #1 9 C B A H D J E G L K 9 N M Adjacent Circled Nodes Path Total Node (uncircled) (fromh) Time H A HA 9 J HJ J HJ 1 K HJK 12 A B HAB 1 HA 1 K L HJKL 18 B HJB 1 C HJC 20 D HJD 1 G HJG 20 L HJL 1 B C HABC 19
10 Service Woes at Speedy Delivery: Teacher Resources 10 D C HJDC 21 E HJDE 21 G HJDG 20 L G HJLG 22 M HJLM 22 C None G E HJGE 2 M HJGM 2 N HJGN 29 E N HJDEN 28 M N HJLMN 2 N endpoint of optimal route The shortest time from H to N is 2 and the path is HJLMN. Extension #2 A B C D E G A B C D E G Extension # a.) GR 91 0 L AA 2 K D
11 Service Woes at Speedy Delivery: Teacher Resources 11 adjacent circled nodes path total node (uncircled) (from A) distance b.) K G KG 0 L KL 8 G L KGL 11 L A KLA 1 KL 1 D KLD 18 The shortest route from Kalamazoo to lint is 1 miles.
12 Service Woes at Speedy Delivery: Teacher Resources 12 adjacent circled nodes path total node (uncircled) (from A) distance c.) D D 2 A DA 1 L DL 90 A DA 10 L DAL 12 L DAL 1 L G DLG 181 K DLK 18 K G DLKG 218 Detroit to Ann Arbor: Detroit to lint: Detroit to Lansing: Detroit to Grand Rapids: Detroit to Kalamazoo: 1 miles 2 miles 90 miles 181 miles 18 miles
13 Service Woes at Speedy Delivery: Teacher Resources 1 Extension # Adjacent Circled Nodes Path Total Node (uncircled) (from H) Time H A HA 9 J HJ J HJ 1 K wrong way on one-way street A B HAB 1 wrong way on one-way street B HJB 1 C HJC 20 D wrong way on one-way street G HJG 20 L wrong way on one-way street B C HABC 19 C D HABCD 2 G D HJGD 2 E HJGE 2 L HJGL 2 M HJGM 2 N HJGN 29 D E wrong way on one-way street E N HJGEN 1 M L HJGML 29 N HJGMN 28 Answer: Path HJGMN. Time: 28 minutes. Mr. Lait will reach the company at :8 pm.
14 Service Woes at Speedy Delivery: Teacher Resources 1 Homework Solutions 1. Circled Node Adjacent Node Path Total Time The shortest route from location 1 to location is found by taking the path 1 2, which leads to a total time of 1 minutes. 2. Circled Node Adjacent Node Path Total Time O A OA 10 B OB B A OBA 11 C OBC 9 D OBD 8 E OBE 12 D E OBDE 1 H OBDH 19 C A OBCA 12 E OBCE 1 G OBCG 2 A OA 19 E G OBEG 18 H OBEH 1 H J OBEHJ 20 G S OBEGS 2 S OAS 0 J S OBEHJS 2 a. There are two possible paths providing the shortest route: paths OBEGS and OBEHJS. Both paths are 2 miles long. The old path ( OAS ) is 0 miles long. b. It is not possible in this network of one-way streets to travel from node to node 1. Lead a discussion of experiences in which students had difficulty in getting to their destinations because the streets seemed always to be one-way in the wrong direction.
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