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Deck 7: Network Flow Models
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Question
PAWV Power and Light has contracted with a waste disposal firm to have nuclear waste from its nuclear power plants in Pennsylvania disposed of at a government-operated nuclear waste disposal site in Nevada. The waste must be shipped in reinforced container trucks across the country, and all travel must be confined to the interstate highway system. The government insists that the waste transport must be completed within 42 hours and that the trucks travel through the least populated areas possible. The following network shows the various interstate segments the trucks might use from Pittsburgh to the Nevada waste site and the travel time (in hours) estimated for each road segment.
The approximate population (in millions) for the metropolitan areas the trucks might travel through are as follows:
Determine the optimal route the trucks should take from Pittsburgh to the Nevada site to complete the trip within 42 hours and expose the trucks to the least number of people possible.
The approximate population (in millions) for the metropolitan areas the trucks might travel through are as follows:
Determine the optimal route the trucks should take from Pittsburgh to the Nevada site to complete the trip within 42 hours and expose the trucks to the least number of people possible.
Question
In 1862, during the second year of the Civil War, General Thomas J. "Stonewall" Jackson fought a brilliant military campaign in the Shenandoah Valley in Virginia. One of his victories was at the Battle of McDowell. Using the following figure and your imagination, determine the shortest path and how long it will take (in days) for General Jackson to move his army from Winchester to McDowell to fight the battle:
Question
In 1861, Western Union wanted to develop a transcontinental telegraph network to connect all the forts and trading posts from St. Louis to Oregon that are shown on the network, in problem, which had grown into communities. Multiplying the days shown on the network by 10 as a rough estimate of the miles between each node, develop a minimal spanning tree for this network that connects the different communities with the minimum amount of telegraph line.
Problem
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
Problem
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
Question
Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to each of the other six nodes (2, 3, 4, 5, 6, and 7):
Question
The Voyager spacecraft has been transported by an alien being to the Delta Quadrant of space, millions of light-years from Earth. Captain Janeway and her crew are attempting to plot the shortest course home. Following is a network of the possible routes through the Delta Quadrant, where the nodes are different worlds, planetary systems, and space anomalies, and the values on the branches are the actual times, in years, between nodes. Determine the shortest route from the Voyager 's present location (at node 1) to Earth.
Question
Given the following network, with the indicated flow capacities of each branch, determine the maximum flow from source node 1 to destination node 6 and the flow along each branch:
Question
Suntrek, based in China, is a global supplier of denim jeans for apparel companies around the world. It purchases raw cotton from producers in Arkansas, Mississippi, and Texas, where it is picked, ginned, and baled and then transported by flatbed trucks to ports in Houston, New Orleans, Savannah, and Charleston, where it is loaded into 80-foot containers and shipped to factories overseas. It ships the cotton it has purchased from the U.S. ports to overseas ports in Shanghai, Karachi, and Saigon, where its denim fabric factories are also located. After Suntrek manufactures denim fabric at these factories it loads it back into containers and ships it to its denim jeans manufacturing facilities in Taiwan, India, Japan, Turkey, and Italy. It supplies containers with finished denim jeans to its customers' distribution centers in the United States in New York and New Orleans, and in Europe in Liverpool and Marseilles. A critical component in its supply chain logistics is having enough available containers when and where they are needed in order to ultimately provide quick delivery to its market-conscious jeans company customers. As such, at any given time the company will often select which of its fabric and jeans manufacturing locations it will use for an order based on available containers, especially if the customer demands quick delivery. Suntrek has contracted with its U.S. cotton broker for bales of cotton that will fill 200 containers. Sixty containers of cotton bales will be loaded at Houston, 40 containers at Savannah, 70 containers at New Orleans, and 30 containers will be loaded at Charleston, which will ultimately fill orders for 200 containers of finished jeans to be delivered to its customers' distribution centers in the United States and Europe-50 containers to New York, 50 to Liverpool, 60 to Marseilles, and 40 to New Orleans. Following are the available containers at each port or manufacturing facility.
Determine the container shipments through the shipping network at each port or manufacturing facility and indicate if there are enough containers in the network to meet the demand of Suntrek's customers.
Determine the container shipments through the shipping network at each port or manufacturing facility and indicate if there are enough containers in the network to meet the demand of Suntrek's customers. Question
John Clooney, a bush pilot in Alaska, makes regular charter flights in his floatplane to various towns and cities in western Alaska. His passengers include hunters, fishermen, backpackers and campers, and tradespeople hired for jobs in the different localities. He also carries some cargo for delivery. The following network shows the possible air routes between various towns and cities John might take (with the times, in hours). For safety reasons, he flies point-to-point, flying over at least one town along a route, even though he might not land there. In the upcoming week John has scheduled charter flights for Kotzebue, Nome, and Stebbins. Determine the shortest route between John's home base in Anchorage and each of these destinations.
Question
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 7 and the flow along each branch:
Question
On December 16, 1944, in the last year of World War II, two German panzer armies, supported by a third army of infantry, together totaling more than 250,000 men, staged a massive counteroffensive in northern France, overwhelming the American First Army in the Ardennes. The offensive emanated from the German defensive line along the Our River, north of the city of Luxembourg, and was directed almost due west toward Namur and Liége in Belgium. The result, after several days of fighting, was a huge "bulge" in the Allied line and, therefore, this last major ground battle of World War II became known as the Battle of the Bulge.
On December 20, General Dwight D. Eisenhower, Supreme Allied Commander, called on General George Patton to attack the German offensive with his Third Army, which was then situated near Verdun, approximately 100 miles due south of the German left flank. Patton's immediate objective was to relieve the 101st Airborne and elements of Patton's own 9th and 10th Armored Divisions surrounded at Bastogne. Within 48 hours, on December 22, Patton was able to begin his counteroffensive, with three divisions totaling approximately 62,000 men.
The winter weather was cold with snow and fog, and the roads were icy, making the movement of troops, tanks, supplies, and equipment a logistical nightmare. Nevertheless, on December 26 Bastogne was relieved, and on January 12, 1945, the Battle of the Bulge effectively ended in one of the great Allied victories of the war.
General Patton's staff did not have knowledge of the maximal flow technique nor access to computers to help plan the Third Army's troop movements during the Battle of the Bulge. However, the figure on the following page shows the road network between Verdun and Bastogne, with (imagined) troop capacities (in thousands) along each road branch between towns. Using the maximal flow technique (and your imagination), determine the number of troops that should be sent along each road in order to get the maximum number of troops to Bastogne. Also indicate the total number of troops that would be able to get to Bastogne.
On December 20, General Dwight D. Eisenhower, Supreme Allied Commander, called on General George Patton to attack the German offensive with his Third Army, which was then situated near Verdun, approximately 100 miles due south of the German left flank. Patton's immediate objective was to relieve the 101st Airborne and elements of Patton's own 9th and 10th Armored Divisions surrounded at Bastogne. Within 48 hours, on December 22, Patton was able to begin his counteroffensive, with three divisions totaling approximately 62,000 men.
The winter weather was cold with snow and fog, and the roads were icy, making the movement of troops, tanks, supplies, and equipment a logistical nightmare. Nevertheless, on December 26 Bastogne was relieved, and on January 12, 1945, the Battle of the Bulge effectively ended in one of the great Allied victories of the war.
General Patton's staff did not have knowledge of the maximal flow technique nor access to computers to help plan the Third Army's troop movements during the Battle of the Bulge. However, the figure on the following page shows the road network between Verdun and Bastogne, with (imagined) troop capacities (in thousands) along each road branch between towns. Using the maximal flow technique (and your imagination), determine the number of troops that should be sent along each road in order to get the maximum number of troops to Bastogne. Also indicate the total number of troops that would be able to get to Bastogne.
Question
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
Question
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 6 and the flow along each branch:
Question
The Pearlsburg (West Virginia) Rescue Squad serves a mountainous, rural area in southern West Virginia. The only access to the homes, farms, and small crossroad communities and villages is a network of dirt, gravel, and poorly paved roads. The rescue squad had just returned from an emergency at Blake's Crossing, and two of the squad members, Melanie Hart and Ben Cross, were cleaning up and getting the truck back in proper order for the next call.
"You know, Ben," said Melanie, "that was close. We could have lost little Randy if we had been a few minutes later in getting to their farm."
"Yep," said Ben.
After a moment Melanie continued, "I was sort of wondering about the route Dave took to get there. It seems to me if we had gone around by Cedar Creek, we could have gotten there a few minutes sooner. And as close as things got, a few minutes could have made a big difference, don't you think"
"Yep," said Ben.
"Well, I was wondering, Ben, why don't we study all these different ways to get to the little communities and farms around here so we will always know what the quickest way to all the different places is"
Ben thought a moment before answering. "It would take us a while to time all the different ways you could get from here to everywhere we go."
"That's true," Melanie answered, "but I've been studying a way to work out this sort of problem in one of my college courses. All we have to do is get the times it takes to travel each piece of road between all these little communities, and I think I can do the rest. Will you help me"
"Sure," Ben said.
"Okay, then, here's what we'll do. I'll write down all the routes you should time, and I'll time the rest all the way over to Holbrook."
Here is the combined list of times (in minutes) Melanie and Ben compiled for all the route segments between Pearlsburg and Holbrook:
Determine the shortest routes from Pearlsburg to all the different communities and farms visited by the rescue squad.
"You know, Ben," said Melanie, "that was close. We could have lost little Randy if we had been a few minutes later in getting to their farm."
"Yep," said Ben.
After a moment Melanie continued, "I was sort of wondering about the route Dave took to get there. It seems to me if we had gone around by Cedar Creek, we could have gotten there a few minutes sooner. And as close as things got, a few minutes could have made a big difference, don't you think"
"Yep," said Ben.
"Well, I was wondering, Ben, why don't we study all these different ways to get to the little communities and farms around here so we will always know what the quickest way to all the different places is"
Ben thought a moment before answering. "It would take us a while to time all the different ways you could get from here to everywhere we go."
"That's true," Melanie answered, "but I've been studying a way to work out this sort of problem in one of my college courses. All we have to do is get the times it takes to travel each piece of road between all these little communities, and I think I can do the rest. Will you help me"
"Sure," Ben said.
"Okay, then, here's what we'll do. I'll write down all the routes you should time, and I'll time the rest all the way over to Holbrook."
Here is the combined list of times (in minutes) Melanie and Ben compiled for all the route segments between Pearlsburg and Holbrook:
Determine the shortest routes from Pearlsburg to all the different communities and farms visited by the rescue squad. Question
A new police car costs the Bay City Police Department $26,000. The annual maintenance cost for a car depends on the age of the car at the beginning of the year. (All cars accumulate approximately the same mileage each year.) The maintenance costs increase as a car ages, as shown in the following table:
In order to avoid the increasingly high maintenance costs, the police department can sell a car and purchase a new car at the end of any year. The selling price for a car at the end of each year of use is also shown in the table. It is assumed that the price of a new car will increase $500 each year. The department's objective is to develop a car replacement schedule that will minimize total cost (i.e., the purchase cost of a new car plus maintenance costs minus the money received for selling a used car) during the next 6 years. Develop a car replacement schedule, using the shortest route method.
In order to avoid the increasingly high maintenance costs, the police department can sell a car and purchase a new car at the end of any year. The selling price for a car at the end of each year of use is also shown in the table. It is assumed that the price of a new car will increase $500 each year. The department's objective is to develop a car replacement schedule that will minimize total cost (i.e., the purchase cost of a new car plus maintenance costs minus the money received for selling a used car) during the next 6 years. Develop a car replacement schedule, using the shortest route method. Question
A new stadium complex is being planned for Denver, and the Denver traffic engineer is attempting to determine whether the city streets between the stadium complex and the interstate highway can accommodate the expected flow of 21,000 cars after each game. The various traffic arteries between the stadium (node 1) and the interstate (node 8) are shown in the following network:
The flow capacities on each street are determined by the number of available lanes, the use of traffic police and lights, and whether any lanes can be opened or closed in either direction. The flow capacities are given in thousands of cars. Determine the maximum traffic flow the streets can accommodate and the amount of traffic along each street. Will the streets be able to handle the expected flow after a game
The flow capacities on each street are determined by the number of available lanes, the use of traffic police and lights, and whether any lanes can be opened or closed in either direction. The flow capacities are given in thousands of cars. Determine the maximum traffic flow the streets can accommodate and the amount of traffic along each street. Will the streets be able to handle the expected flow after a game
Question
Frieda Millstone and her family live in Roanoke, Virginia, and they are planning an auto vacation across Virginia, their ultimate destination being Washington, DC. The family has developed the following network of possible routes and cities to visit on their trip:
The time, in hours, between cities (which is affected by the type of road and the number of intermediate towns) is shown along each branch. Determine the shortest route that the Millstone family can travel from Roanoke to Washington, DC.
The time, in hours, between cities (which is affected by the type of road and the number of intermediate towns) is shown along each branch. Determine the shortest route that the Millstone family can travel from Roanoke to Washington, DC.
Question
Solve Problem from Chapter to determine the shortest route for the "Weekend Rider" bus service from Tech to DC that will also result in at least 30 passengers.
Problem
The bus company in Draper operates a "Weekend Rider" bus service from the Tech campus to Washington, DC. The bus seats 55 passengers and selects the shortest route to DC that will also result in at least 30 passengers. The bus can travel through 11 different cities and towns depending on the number of passengers it expects to attract. The following network shows the different connecting pickup points and the distance between them in miles:
The bus service expects to pick up the following passengers at each possible stop: 20 at stop 1 (Tech), 4 at stop 2, 3 at stop 3, 5 at stop 4, 5 at stop 5, 7 at stop 6, 6 at stop 7, 4 at stop 8, 8 at stop 9, 6 at stop 10, 6 at stop 11, and 5 at stop 12. Determine the shortest route between Tech (at 1) to DC (at 13) that will result in at least 30 passengers. Compare this route to the shortest route without the required number of passengers, and indicate how many passengers the bus would have with this route.
Problem
The bus company in Draper operates a "Weekend Rider" bus service from the Tech campus to Washington, DC. The bus seats 55 passengers and selects the shortest route to DC that will also result in at least 30 passengers. The bus can travel through 11 different cities and towns depending on the number of passengers it expects to attract. The following network shows the different connecting pickup points and the distance between them in miles:
The bus service expects to pick up the following passengers at each possible stop: 20 at stop 1 (Tech), 4 at stop 2, 3 at stop 3, 5 at stop 4, 5 at stop 5, 7 at stop 6, 6 at stop 7, 4 at stop 8, 8 at stop 9, 6 at stop 10, 6 at stop 11, and 5 at stop 12. Determine the shortest route between Tech (at 1) to DC (at 13) that will result in at least 30 passengers. Compare this route to the shortest route without the required number of passengers, and indicate how many passengers the bus would have with this route.
Question
The FAA has granted a license to a new airline, Omniair, and awarded it several routes between Los Angeles and Chicago. The flights per day for each route are shown in the following network:
Determine the maximum number of flights the airline can schedule per day from Chicago to Los Angeles and indicate the number of flights along each route.
Determine the maximum number of flights the airline can schedule per day from Chicago to Los Angeles and indicate the number of flights along each route.
Question
The Roanoke, Virginia, distributor of Rainwater Beer delivers beer by truck to stores in six other Virginia cities, as shown in the following network:
The mileage between cities is shown along each branch. Determine the shortest truck route from Roanoke to each of the other six cities in the network.
The mileage between cities is shown along each branch. Determine the shortest truck route from Roanoke to each of the other six cities in the network.
Question
A developer is planning a development that includes subdivisions of houses, cluster houses, townhouses, apartment complexes, shopping areas, a daycare center and playground, a community center, and a school, among other facilities. The developer wants to connect the facilities and areas in the development with the minimum number of streets possible. The following network shows the possible street routes and distances (in thousands of feet) between 10 areas in the development that must be connected:
Determine a minimal spanning tree network of streets to connect the 10 areas, and indicate the total streets (in thousands of feet) needed.
Determine a minimal spanning tree network of streets to connect the 10 areas, and indicate the total streets (in thousands of feet) needed.
Question
The National Express Parcel Service has established various truck and air routes around the country over which it ships parcels. The holiday season is approaching, which means a dramatic increase in the number of packages that will be sent. The service wants to know the maximum flow of packages it can accommodate (in tons) from station 1 to station 7. The network of routes and the flow capacities (in tons of packages per day) along each route are shown in the following network:
Determine the maximum tonnage of packages that can be transported per day from station 1 to station 7 and indicate the flow along each branch.
Determine the maximum tonnage of packages that can be transported per day from station 1 to station 7 and indicate the flow along each branch.
Question
The plant engineer for the Bitco manufacturing plant is designing an overhead conveyor system that will connect the distribution/inventory center to all areas of the plant. The network of possible conveyor routes through the plant, with the length (in feet) along each branch, follows:
Determine the shortest conveyor route from the distribution/inventory center at node 1 to each of the other six areas of the plant.
Determine the shortest conveyor route from the distribution/inventory center at node 1 to each of the other six areas of the plant.
Question
One of the opposing forces in a simulated army battle wishes to set up a communications system that will connect the eight camps in its command. The following network indicates the distances (in hundreds of yards) between the camps and the different paths over which a communications line can be constructed:
Using the minimal spanning tree approach, determine the minimum distance communication system that will connect all eight camps
Using the minimal spanning tree approach, determine the minimum distance communication system that will connect all eight camps
Question
The traffic management office in Richmond is attempting to analyze the potential traffic flow from a new office complex under construction to an interstate highway interchange during the evening rush period. Cars leave the office complex via one of three exits, and then they travel through the city streets until they arrive at the interstate interchange. The following network shows the various street routes (branches) from the office complex (node 1) to the interstate interchange (node 9):
All intermediate nodes represent street intersections, and the values accompanying the branches emanating from the nodes represent the traffic capacities of each street, expressed in thousands of cars per hour. Determine the maximum flow of cars that the street system can absorb during the evening rush hour.
All intermediate nodes represent street intersections, and the values accompanying the branches emanating from the nodes represent the traffic capacities of each street, expressed in thousands of cars per hour. Determine the maximum flow of cars that the street system can absorb during the evening rush hour.
Question
The Burger Doodle restaurant franchises in Los Angeles are supplied from a central warehouse in Inglewood. The location of the warehouse and its proximity, in minutes of travel time, to the franchises are shown in the following network:
Trucks supply each franchise on a daily basis. Determine the shortest route from the warehouse at Inglewood to each of the nine franchises.
Trucks supply each franchise on a daily basis. Determine the shortest route from the warehouse at Inglewood to each of the nine franchises.
Question
The management of the Dynaco manufacturing plant wants to connect the eight major manufacturing areas of its plant with a forklift route. Because the construction of such a route will take a considerable amount of plant space and disrupt normal activities, management wants to minimize the total length of the route. The following network shows the distance, in yards, between the manufacturing areas (denoted by nodes 1 through 8):
Determine the minimal spanning tree forklift route for the plant and indicate the total yards the route will require.
Determine the minimal spanning tree forklift route for the plant and indicate the total yards the route will require.
Question
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
Question
The Petroco gasoline distributor in Jackson, Mississippi, supplies service stations in nine other southeastern cities, as shown in the following network:
The distance, in miles, is shown on each branch. Determine the shortest route from Jackson to the nine other cities in the network.
The distance, in miles, is shown on each branch. Determine the shortest route from Jackson to the nine other cities in the network.
Question
Several oil companies are jointly planning to build an oil pipeline to connect several southwestern, southeastern, and midwestern cities, as shown in the following network:
The miles between cities are shown on each branch. Determine a pipeline system that will connect all 10 cities, using the minimum number of miles of pipe, and indicate how many miles of pipe will be used.
The miles between cities are shown on each branch. Determine a pipeline system that will connect all 10 cities, using the minimum number of miles of pipe, and indicate how many miles of pipe will be used.
Question
Suppose in Problem that the processing cost per unit at each plant is different because of different machinery, workers' abilities, overhead, and so on, according to the following table:
There are no differentiated costs at stage 5, the distribution center. Given a budget of $700,000, determine the maximum number of units that can be processed through the five-stage manufacturing process.
Problem
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
There are no differentiated costs at stage 5, the distribution center. Given a budget of $700,000, determine the maximum number of units that can be processed through the five-stage manufacturing process.Problem
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
Question
The Hylton Hotel has a limousine van that transports guests to various business and tourist locations around the city. The following network indicates the different routes the limousine could follow from the hotel at node 1 to the nine locations (nodes 2 through 10):
The values on each branch in the network are the distances, in miles, between the locations. Determine the shortest route from the hotel to each of the nine destinations and indicate the distance for each route.
The values on each branch in the network are the distances, in miles, between the locations. Determine the shortest route from the hotel to each of the nine destinations and indicate the distance for each route.
Question
A major hotel chain is constructing a new resort hotel complex in Greenbranch Springs, West Virginia. The resort is in a heavily wooded area, and the developers want to preserve as much of the natural beauty as possible. To do so, the developers want to connect all the various facilities in the complex with a combination walking-riding path that will minimize the amount of pathway that will have to be cut through the woods. The following network shows possible connecting paths and corresponding distances (in yards) between the facilities:
Determine the path that will connect all the facilities with the minimum amount of construction and indicate the total length of the pathway.
Determine the path that will connect all the facilities with the minimum amount of construction and indicate the total length of the pathway.
Question
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 10 and the flow along each path:
Question
A steel mill in Gary supplies steel to manufacturers in eight other midwestern cities by truck, as shown in the following network:
The travel time between cities, in hours, is shown along each branch. Determine the shortest route from Gary to each of the other eight cities in the network.
The travel time between cities, in hours, is shown along each branch. Determine the shortest route from Gary to each of the other eight cities in the network.
Question
The Barrett Textile Mill is remodeling its plant and installing a new ventilation system. The possible ducts connecting the different rooms and buildings at the plant, with the length (in feet) along each branch, are shown in the following network:
Determine the ventilation system that will connect the various rooms and buildings of the plant, using the minimum number of feet of ductwork, and indicate how many feet of ductwork will be used.
Determine the ventilation system that will connect the various rooms and buildings of the plant, using the minimum number of feet of ductwork, and indicate how many feet of ductwork will be used.
Question
A manufacturing company produces different variations of a product at different work centers in its plant on a daily basis. Following is a network showing the various work centers in the plant, the daily capacities at each work center, and the flow of the partially completed products between work centers:
Node 1 represents the point where raw materials enter the process, and node 15 is the packaging and distribution center. Determine the maximum number of units that can be completed each day and the number of units processed at each work center.
Node 1 represents the point where raw materials enter the process, and node 15 is the packaging and distribution center. Determine the maximum number of units that can be completed each day and the number of units processed at each work center.
Question
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
Question
The town council of Whitesville has decided to construct a bicycle path that will connect the various suburbs of the town with the shopping center, the downtown area, and the local college. The council hopes the local citizenry will use the bike path, thus conserving energy and decreasing traffic congestion. The various paths that can be constructed and their lengths (in miles) are shown in the following network:
Determine the bicycle pathway that will require the minimum amount of construction to connect all the areas of the town. Indicate the total length of the path.
Determine the bicycle pathway that will require the minimum amount of construction to connect all the areas of the town. Indicate the total length of the path.
Question
The following network shows the major roads that would be part of the hurricane evacuation routes for Charleston, South Carolina:
The destination is the interchange of Interstate 26 and Interstate 526, north of Charleston. The nodes represent road intersections, and the values at the nodes are the traffic capacities, in thousands of cars. Determine the maximum flow of cars that can leave Charleston during a hurricane evacuation and the optimal flow along each road segment (branch 1).
The destination is the interchange of Interstate 26 and Interstate 526, north of Charleston. The nodes represent road intersections, and the values at the nodes are the traffic capacities, in thousands of cars. Determine the maximum flow of cars that can leave Charleston during a hurricane evacuation and the optimal flow along each road segment (branch 1).
Question
The members of the Vistas Club are planning a month long hiking and camping trip through the Blue Ridge Mountains, from north Georgia to Virginia. There are a number of trails through the mountains that connect at various campgrounds, crossings, and cabins. The following network shows the different connecting trails and the distance of each connecting branch, in miles:
Determine the shortest path from node 1 in north Georgia to node 13 in Virginia and indicate the total mileage of the trail.
Determine the shortest path from node 1 in north Georgia to node 13 in Virginia and indicate the total mileage of the trail.
Question
Determine the minimal spanning tree for the network in Problem.
Problem
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
Problem
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
Question
In Problem in Chapter, KanTech Corporation ships electronic components in containers from seaports in Europe to U.S. ports, from which the containers are transported by truck or rail to inland ports and then transferred to an alternative mode of transportation before being shipped to KanTech's U.S. distributors. Using the shipping capacity at each port, develop a maximal flow network to determine the number of containers to ship through each port to meet demand at the distribution centers.
Problem
KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/ visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs ($/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:
Formulate and solve a linear programming model to determine the optimal shipments from each point of embarkation to each destination along this supply chain that will result in the minimum total shipping cost.
Problem
KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/ visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs ($/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:
Formulate and solve a linear programming model to determine the optimal shipments from each point of embarkation to each destination along this supply chain that will result in the minimum total shipping cost. Question
Kathleen Taylor, a student at Tech, is planning to visit her sister, Lindsey, who is living in Toulouse, France, over the summer break. She is going to fly from Dulles Airport in Washington to Charles de Gaulle Airport in Paris, and because of the time changes, this travel will take a full day. In Paris, Kathleen is going to spend 2 nights and 1 full day before taking the train to Toulouse. Kathleen has never been to Paris, so she wants to spend her 1 day there seeing as many of the famous attractions as she can, including the Eiffel Tower, the Louvre, Notre Dame Cathedral, the Arch de Triumph, the Pantheon, and the Palace at Versailles. She plans to stay at a youth hostel very near Sacre Coeur in Montmartre and from there use the Paris Metro to visit as many of the sites as she can in a day. She has downloaded a detailed Metro map from the French rail Web site at http://www.ratp.fr and has discovered that the Metro system is huge, with almost 250 stations and 14 lines throughout Paris.
There's no question that Kathleen can get to all the sites by the Metro, but she is concerned about her limited time frame and her ability to get from location to location quickly. She has determined the following information regarding the average times (in minutes) between stations for each line:
She's also guessing that the subway stops about 1 minute at each station. If she has to change lines, she assumes it will take her at least 5 minutes. She plans to leave early in the morning, when the sites open, and she has no specific time she must be back to the hostel.
Kathleen, a business student, would like to use some of kind of logical, systematic approach to help her plan her movement using the Metro around the city to the different sites. Help Kathleen develop a route around the city to each of the sites she wants to see, starting from her youth hostel in Montmartre, for the day she'll be in Paris. Do you think she'll be able to see all the sites she wants to see
There's no question that Kathleen can get to all the sites by the Metro, but she is concerned about her limited time frame and her ability to get from location to location quickly. She has determined the following information regarding the average times (in minutes) between stations for each line:
She's also guessing that the subway stops about 1 minute at each station. If she has to change lines, she assumes it will take her at least 5 minutes. She plans to leave early in the morning, when the sites open, and she has no specific time she must be back to the hostel.Kathleen, a business student, would like to use some of kind of logical, systematic approach to help her plan her movement using the Metro around the city to the different sites. Help Kathleen develop a route around the city to each of the sites she wants to see, starting from her youth hostel in Montmartre, for the day she'll be in Paris. Do you think she'll be able to see all the sites she wants to see
Question
Hard Rock Concrete Supply makes concrete at its plant in Centerville, Virginia, and delivers it to construction sites throughout the metropolitan Washington, DC, area. The following network shows the possible routes and distances (in miles) from the concrete plant to seven construction sites:
Determine the shortest route a concrete truck would take from the plant at node 1 to node 8 and the total distance for this route, using the shortest route method.
Determine the shortest route a concrete truck would take from the plant at node 1 to node 8 and the total distance for this route, using the shortest route method.
Question
State University has decided to reconstruct the sidewalks throughout the east side of its campus to provide wheelchair access. However, upgrading sidewalks is a very expensive undertaking, so for the first phase of this project, university administrators want to make sure they connect all buildings with wheelchair access with the minimum number of refurbished sidewalks possible.
Following is a network of the existing sidewalks on the east side of campus, with the feet between each building shown on the branches:
Determine a minimal spanning tree network that will connect all the buildings on campus with wheelchair access sidewalks and indicate the number of feet of sidewalk.
Following is a network of the existing sidewalks on the east side of campus, with the feet between each building shown on the branches:
Determine a minimal spanning tree network that will connect all the buildings on campus with wheelchair access sidewalks and indicate the number of feet of sidewalk.
Question
In the novel Around the World in 80 Days by Jules Verne, Phileas T. Fogg wagered four of his fellow members of the Reform Club in London £5,000 apiece that he could travel around the world in 80 days. This would have been an astounding feat in 1872, the year in which the novel is set. Then, the fastest modes of travel were rail and ship; however, much of the world still traveled by coach, wagon, horse, elephant, or donkey, or on foot. Phileas Fogg was not a frivolous man. He would not have undertaken such a large wager if he had not carefully researched the feasibility of such a trip and been confident of his chances. Although he undoubtedly analyzed various routes to circumnavigate the globe, he did not have knowledge of techniques such as the shortest route method, nor did he have a computer to help him select an optimal route. If he had, he might have chosen a route that would have completed his trip in less than 80 days or possibly would not have been so quick to make his wager. On the next page is a network of the various routes of the day for traveling around the world from London. Fogg traveled eastward. The travel time, in days, is shown on each branch. Travel times are based as much on available modes of transportation at the time as on distance. Using the shortest route method (and the computer), select the best route for Mr. Fogg. Would the route you determined have won him his wager
Question
George is camped deep in the jungle, and he wants to make his way back to the coast and civilization. Each of the paths he can take through the jungle has obstacles that can delay him, including hostile natives, wild animals, dense forests and vegetation, swamps, rivers and streams, snakes, insects, and mountains. Following is the network of paths that George can take, with the time (in days) to travel each branch:
Determine the shortest (time) path for George to take and indicate the total time.
Determine the shortest (time) path for George to take and indicate the total time.
Question
Tech wants to develop an area network that will connect its server at its computer and satellite center with the main campus buildings to improve Internet service. The cable will be laid primarily through existing electrical tunnels, although some cable will have to be buried underground. The following network shows the possible cable connections between the computer center at node 1 and the various buildings, with the distances, in feet, along the branches:
Determine a minimal spanning tree network that will connect all the buildings and indicate the total amount of cable that will be needed to do so.
Determine a minimal spanning tree network that will connect all the buildings and indicate the total amount of cable that will be needed to do so.
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Deck 7: Network Flow Models
1
PAWV Power and Light has contracted with a waste disposal firm to have nuclear waste from its nuclear power plants in Pennsylvania disposed of at a government-operated nuclear waste disposal site in Nevada. The waste must be shipped in reinforced container trucks across the country, and all travel must be confined to the interstate highway system. The government insists that the waste transport must be completed within 42 hours and that the trucks travel through the least populated areas possible. The following network shows the various interstate segments the trucks might use from Pittsburgh to the Nevada waste site and the travel time (in hours) estimated for each road segment.
The approximate population (in millions) for the metropolitan areas the trucks might travel through are as follows:
Determine the optimal route the trucks should take from Pittsburgh to the Nevada site to complete the trip within 42 hours and expose the trucks to the least number of people possible.
The approximate population (in millions) for the metropolitan areas the trucks might travel through are as follows:
Determine the optimal route the trucks should take from Pittsburgh to the Nevada site to complete the trip within 42 hours and expose the trucks to the least number of people possible.Solve the network flow model problem using excel solver. Follow the given steps to find out the optimal route for the trucks.
Step 1: Formulate an excel sheet using given values and formulas:
Step 2: Formulated spreadsheet is shown below:
Access "Excel solver" from menu bar and enter the following values in the pop-up window:
Click on "Solve" option. Then, the following pop up would appear:
Now, click on "OK" button. It would give the following results:
The optimal route for the truck is 1-2-11-15-16-19-23-28-31-33. The least number of people exposed to the trucks are 7.03 million.
Step 1: Formulate an excel sheet using given values and formulas:
Step 2: Formulated spreadsheet is shown below: Access "Excel solver" from menu bar and enter the following values in the pop-up window: Click on "Solve" option. Then, the following pop up would appear: Now, click on "OK" button. It would give the following results: The optimal route for the truck is 1-2-11-15-16-19-23-28-31-33. The least number of people exposed to the trucks are 7.03 million. 2
In 1862, during the second year of the Civil War, General Thomas J. "Stonewall" Jackson fought a brilliant military campaign in the Shenandoah Valley in Virginia. One of his victories was at the Battle of McDowell. Using the following figure and your imagination, determine the shortest path and how long it will take (in days) for General Jackson to move his army from Winchester to McDowell to fight the battle:
First, count the number of networks among the given nodes. The networks are the connections represented by distances among the nodes.
The following represents the networks or branches and the distances between them:
Now, to solve this transportation problem, use the QM for Window software (provided with the book) following these instructions:
First, open the software; under module, select Networks. Next, open a blank document and select "shortest route". A menu will open; enter the number of branches which is 34 for this problem. Select OK and a window will show in order to enter the distances. Enter the values; then, select solve on the top right corner and a series of windows with solutions will open. The window titled: Assignment Minimum distance matrix.
The shortest route from node 1 to node 14 (because McDowell is at 14) would be the following; notice also that the "Networks Result" window will show the proper order:
Network 1 (Romney) to 3 (Moorefield) to 11 (Franklin) to 14 (McDowell) is 13
The following represents the networks or branches and the distances between them:
Now, to solve this transportation problem, use the QM for Window software (provided with the book) following these instructions:First, open the software; under module, select Networks. Next, open a blank document and select "shortest route". A menu will open; enter the number of branches which is 34 for this problem. Select OK and a window will show in order to enter the distances. Enter the values; then, select solve on the top right corner and a series of windows with solutions will open. The window titled: Assignment Minimum distance matrix.
The shortest route from node 1 to node 14 (because McDowell is at 14) would be the following; notice also that the "Networks Result" window will show the proper order:
Network 1 (Romney) to 3 (Moorefield) to 11 (Franklin) to 14 (McDowell) is 13
3
In 1861, Western Union wanted to develop a transcontinental telegraph network to connect all the forts and trading posts from St. Louis to Oregon that are shown on the network, in problem, which had grown into communities. Multiplying the days shown on the network by 10 as a rough estimate of the miles between each node, develop a minimal spanning tree for this network that connects the different communities with the minimum amount of telegraph line.
Problem
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
Problem
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
Step 1: Draw the network diagram for the given problem as shown below:
Step 2: Arbitrarily select the node1 to start. Select the closest node or the shortest node to 1 to join the spanning tree. The unconnected node closest to node 1 is 4. Connect node 1 to node 4 as shown below:
Step 3: The spanning tree consists of two nodes: 1 and 4. Select the closest not to 1 and 4.The unconnected node closest to either node 1 or 4 is 6. Connect node 4 to node 6 as shown below:
Step 4: The spanning tree consists of three nodes: 1, 4 and 6. Select the closest not to 1, 4 or 6.The unconnected node closest to node 1, 4 6 is 5. Connect node 6 to node 5 as shown below:
Step 5: The spanning tree consists of four nodes: 1, 4, and 6 and5. Select the closest not to 1, 4, 6 or 5.The unconnected node closest to node 1, 4 6, 5 is 3. Connect node 6 to node 5 as shown below:
Step 6: The spanning tree consists of four nodes: 1, 4, 6, 5, 2 and 3. Select the closest not to 1, 4, 6, 5 or 3.The unconnected node closest to node 1, 4 6, 5 and 3 is 7. Connect node5 to node 7 as shown below:
Step 7: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2 and 7. Select the closest not to 1, 4, 6, 5, 3 and 7.The unconnected node closest to node 1, 4, 6, 5, 3 and 7is 8. Connect node 7 to node 8 as shown below:
Step 8: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7 and 8. Select the closest not to 1, 4, 6, 5, 3, 2, 7 and 8.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7 and 8 is 9. Connect node 8 to node 9 as shown below:
Step 9: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8 and 9. Select the closest not to 1, 4, 6, 5, 3, 2, 7, 8 and 9.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8 and 9 is 11. Connect node 9 to node 11 as shown below:
Step 10: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11 is 13. Connect node 11to node 13 as shown below:
Step 11: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13 is 13. Connect node 13to node 15 as shown below:
Step 12: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, and 15.
Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13 and 15. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13 and 15 is 12. Connect node 13to node 12 as shown below:
Step 13: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12 is 15. Connect node 12 to node 15 as shown below:
Step14: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12 is 14. Connect node 15 to node 14 as shown below:
Step 15: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, and 14. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, and 14. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 is 16. Connect node 14 to node 16 as shown below:
Step 16: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16 is 17. Connect node 15 to node 17 as shown below:
A minimum spanning tree has (V-1) edges where V is the number of vertices.
Therefore,
Number of lines = (17-1)
= 16 lines.
Therefore, the minimum number of telegraph line that will connect all the communities in the above graph is 16 lines.
Step 2: Arbitrarily select the node1 to start. Select the closest node or the shortest node to 1 to join the spanning tree. The unconnected node closest to node 1 is 4. Connect node 1 to node 4 as shown below: Step 3: The spanning tree consists of two nodes: 1 and 4. Select the closest not to 1 and 4.The unconnected node closest to either node 1 or 4 is 6. Connect node 4 to node 6 as shown below: Step 4: The spanning tree consists of three nodes: 1, 4 and 6. Select the closest not to 1, 4 or 6.The unconnected node closest to node 1, 4 6 is 5. Connect node 6 to node 5 as shown below: Step 5: The spanning tree consists of four nodes: 1, 4, and 6 and5. Select the closest not to 1, 4, 6 or 5.The unconnected node closest to node 1, 4 6, 5 is 3. Connect node 6 to node 5 as shown below: Step 6: The spanning tree consists of four nodes: 1, 4, 6, 5, 2 and 3. Select the closest not to 1, 4, 6, 5 or 3.The unconnected node closest to node 1, 4 6, 5 and 3 is 7. Connect node5 to node 7 as shown below: Step 7: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2 and 7. Select the closest not to 1, 4, 6, 5, 3 and 7.The unconnected node closest to node 1, 4, 6, 5, 3 and 7is 8. Connect node 7 to node 8 as shown below: Step 8: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7 and 8. Select the closest not to 1, 4, 6, 5, 3, 2, 7 and 8.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7 and 8 is 9. Connect node 8 to node 9 as shown below: Step 9: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8 and 9. Select the closest not to 1, 4, 6, 5, 3, 2, 7, 8 and 9.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8 and 9 is 11. Connect node 9 to node 11 as shown below: Step 10: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8, 9 and 11 is 13. Connect node 11to node 13 as shown below: Step 11: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13.The unconnected node closest to node 1, 4, 6, 5, 3, 2, 7, 8, 9, 11 and 13 is 13. Connect node 13to node 15 as shown below: Step 12: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, and 15. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13 and 15. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13 and 15 is 12. Connect node 13to node 12 as shown below:
Step 13: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15 and 12 is 15. Connect node 12 to node 15 as shown below: Step14: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, and 12 is 14. Connect node 15 to node 14 as shown below: Step 15: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, and 14. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, and 14. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 is 16. Connect node 14 to node 16 as shown below: Step 16: The spanning tree consists of four nodes: 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16. Select the closest not 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16. The unconnected node closest to nodes 1, 4, 6, 5, 3, 2, 7, 8, 9, 11, 13, 15, 12, 14 and 16 is 17. Connect node 15 to node 17 as shown below: A minimum spanning tree has (V-1) edges where V is the number of vertices.Therefore,
Number of lines = (17-1)
= 16 lines.
Therefore, the minimum number of telegraph line that will connect all the communities in the above graph is 16 lines.
4
Given the following network with the indicated distances between nodes (in miles), determine the shortest route from node 1 to each of the other six nodes (2, 3, 4, 5, 6, and 7):
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5
The Voyager spacecraft has been transported by an alien being to the Delta Quadrant of space, millions of light-years from Earth. Captain Janeway and her crew are attempting to plot the shortest course home. Following is a network of the possible routes through the Delta Quadrant, where the nodes are different worlds, planetary systems, and space anomalies, and the values on the branches are the actual times, in years, between nodes. Determine the shortest route from the Voyager 's present location (at node 1) to Earth.
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6
Given the following network, with the indicated flow capacities of each branch, determine the maximum flow from source node 1 to destination node 6 and the flow along each branch:
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Suntrek, based in China, is a global supplier of denim jeans for apparel companies around the world. It purchases raw cotton from producers in Arkansas, Mississippi, and Texas, where it is picked, ginned, and baled and then transported by flatbed trucks to ports in Houston, New Orleans, Savannah, and Charleston, where it is loaded into 80-foot containers and shipped to factories overseas. It ships the cotton it has purchased from the U.S. ports to overseas ports in Shanghai, Karachi, and Saigon, where its denim fabric factories are also located. After Suntrek manufactures denim fabric at these factories it loads it back into containers and ships it to its denim jeans manufacturing facilities in Taiwan, India, Japan, Turkey, and Italy. It supplies containers with finished denim jeans to its customers' distribution centers in the United States in New York and New Orleans, and in Europe in Liverpool and Marseilles. A critical component in its supply chain logistics is having enough available containers when and where they are needed in order to ultimately provide quick delivery to its market-conscious jeans company customers. As such, at any given time the company will often select which of its fabric and jeans manufacturing locations it will use for an order based on available containers, especially if the customer demands quick delivery. Suntrek has contracted with its U.S. cotton broker for bales of cotton that will fill 200 containers. Sixty containers of cotton bales will be loaded at Houston, 40 containers at Savannah, 70 containers at New Orleans, and 30 containers will be loaded at Charleston, which will ultimately fill orders for 200 containers of finished jeans to be delivered to its customers' distribution centers in the United States and Europe-50 containers to New York, 50 to Liverpool, 60 to Marseilles, and 40 to New Orleans. Following are the available containers at each port or manufacturing facility.
Determine the container shipments through the shipping network at each port or manufacturing facility and indicate if there are enough containers in the network to meet the demand of Suntrek's customers.
Determine the container shipments through the shipping network at each port or manufacturing facility and indicate if there are enough containers in the network to meet the demand of Suntrek's customers. Unlock Deck
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8
John Clooney, a bush pilot in Alaska, makes regular charter flights in his floatplane to various towns and cities in western Alaska. His passengers include hunters, fishermen, backpackers and campers, and tradespeople hired for jobs in the different localities. He also carries some cargo for delivery. The following network shows the possible air routes between various towns and cities John might take (with the times, in hours). For safety reasons, he flies point-to-point, flying over at least one town along a route, even though he might not land there. In the upcoming week John has scheduled charter flights for Kotzebue, Nome, and Stebbins. Determine the shortest route between John's home base in Anchorage and each of these destinations.
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9
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 7 and the flow along each branch:
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10
On December 16, 1944, in the last year of World War II, two German panzer armies, supported by a third army of infantry, together totaling more than 250,000 men, staged a massive counteroffensive in northern France, overwhelming the American First Army in the Ardennes. The offensive emanated from the German defensive line along the Our River, north of the city of Luxembourg, and was directed almost due west toward Namur and Liége in Belgium. The result, after several days of fighting, was a huge "bulge" in the Allied line and, therefore, this last major ground battle of World War II became known as the Battle of the Bulge.
On December 20, General Dwight D. Eisenhower, Supreme Allied Commander, called on General George Patton to attack the German offensive with his Third Army, which was then situated near Verdun, approximately 100 miles due south of the German left flank. Patton's immediate objective was to relieve the 101st Airborne and elements of Patton's own 9th and 10th Armored Divisions surrounded at Bastogne. Within 48 hours, on December 22, Patton was able to begin his counteroffensive, with three divisions totaling approximately 62,000 men.
The winter weather was cold with snow and fog, and the roads were icy, making the movement of troops, tanks, supplies, and equipment a logistical nightmare. Nevertheless, on December 26 Bastogne was relieved, and on January 12, 1945, the Battle of the Bulge effectively ended in one of the great Allied victories of the war.
General Patton's staff did not have knowledge of the maximal flow technique nor access to computers to help plan the Third Army's troop movements during the Battle of the Bulge. However, the figure on the following page shows the road network between Verdun and Bastogne, with (imagined) troop capacities (in thousands) along each road branch between towns. Using the maximal flow technique (and your imagination), determine the number of troops that should be sent along each road in order to get the maximum number of troops to Bastogne. Also indicate the total number of troops that would be able to get to Bastogne.
On December 20, General Dwight D. Eisenhower, Supreme Allied Commander, called on General George Patton to attack the German offensive with his Third Army, which was then situated near Verdun, approximately 100 miles due south of the German left flank. Patton's immediate objective was to relieve the 101st Airborne and elements of Patton's own 9th and 10th Armored Divisions surrounded at Bastogne. Within 48 hours, on December 22, Patton was able to begin his counteroffensive, with three divisions totaling approximately 62,000 men.
The winter weather was cold with snow and fog, and the roads were icy, making the movement of troops, tanks, supplies, and equipment a logistical nightmare. Nevertheless, on December 26 Bastogne was relieved, and on January 12, 1945, the Battle of the Bulge effectively ended in one of the great Allied victories of the war.
General Patton's staff did not have knowledge of the maximal flow technique nor access to computers to help plan the Third Army's troop movements during the Battle of the Bulge. However, the figure on the following page shows the road network between Verdun and Bastogne, with (imagined) troop capacities (in thousands) along each road branch between towns. Using the maximal flow technique (and your imagination), determine the number of troops that should be sent along each road in order to get the maximum number of troops to Bastogne. Also indicate the total number of troops that would be able to get to Bastogne.
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11
From 1840 to 1850, more than 12,000 pioneers migrated west in wagon trains. It was typically about a 2,000-mile journey, and pioneers averaged about 10 miles per day. One pioneer family, the Smiths, is planning to join a wagon train traveling west to Oregon. The destination is Fort Vancouver, which is near present-day Portland. The family plans to join a wagon train in St. Louis. However, the trains follow various trails west that are mostly determined by the location of forts and trading posts along the way. The Smith family wants to choose a wagon train that will get them to Oregon in the shortest amount of time. They have checked around with the different wagon train leaders plus immigrants, soldiers, fur traders, and scouts who have previously made the trip west, and from the information they have gathered, they have developed the following network, with estimated times (in days) along each branch:
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
a. Determine the shortest route for the Smiths from St. Louis to Ft. Vancouver.
b. Another family, the Steins, is leaving from Ft. Smith, Arkansas. Determine the shortest route for the Steins to Ft. Vancouver, Oregon.
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12
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 6 and the flow along each branch:
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13
The Pearlsburg (West Virginia) Rescue Squad serves a mountainous, rural area in southern West Virginia. The only access to the homes, farms, and small crossroad communities and villages is a network of dirt, gravel, and poorly paved roads. The rescue squad had just returned from an emergency at Blake's Crossing, and two of the squad members, Melanie Hart and Ben Cross, were cleaning up and getting the truck back in proper order for the next call.
"You know, Ben," said Melanie, "that was close. We could have lost little Randy if we had been a few minutes later in getting to their farm."
"Yep," said Ben.
After a moment Melanie continued, "I was sort of wondering about the route Dave took to get there. It seems to me if we had gone around by Cedar Creek, we could have gotten there a few minutes sooner. And as close as things got, a few minutes could have made a big difference, don't you think"
"Yep," said Ben.
"Well, I was wondering, Ben, why don't we study all these different ways to get to the little communities and farms around here so we will always know what the quickest way to all the different places is"
Ben thought a moment before answering. "It would take us a while to time all the different ways you could get from here to everywhere we go."
"That's true," Melanie answered, "but I've been studying a way to work out this sort of problem in one of my college courses. All we have to do is get the times it takes to travel each piece of road between all these little communities, and I think I can do the rest. Will you help me"
"Sure," Ben said.
"Okay, then, here's what we'll do. I'll write down all the routes you should time, and I'll time the rest all the way over to Holbrook."
Here is the combined list of times (in minutes) Melanie and Ben compiled for all the route segments between Pearlsburg and Holbrook:
Determine the shortest routes from Pearlsburg to all the different communities and farms visited by the rescue squad.
"You know, Ben," said Melanie, "that was close. We could have lost little Randy if we had been a few minutes later in getting to their farm."
"Yep," said Ben.
After a moment Melanie continued, "I was sort of wondering about the route Dave took to get there. It seems to me if we had gone around by Cedar Creek, we could have gotten there a few minutes sooner. And as close as things got, a few minutes could have made a big difference, don't you think"
"Yep," said Ben.
"Well, I was wondering, Ben, why don't we study all these different ways to get to the little communities and farms around here so we will always know what the quickest way to all the different places is"
Ben thought a moment before answering. "It would take us a while to time all the different ways you could get from here to everywhere we go."
"That's true," Melanie answered, "but I've been studying a way to work out this sort of problem in one of my college courses. All we have to do is get the times it takes to travel each piece of road between all these little communities, and I think I can do the rest. Will you help me"
"Sure," Ben said.
"Okay, then, here's what we'll do. I'll write down all the routes you should time, and I'll time the rest all the way over to Holbrook."
Here is the combined list of times (in minutes) Melanie and Ben compiled for all the route segments between Pearlsburg and Holbrook:
Determine the shortest routes from Pearlsburg to all the different communities and farms visited by the rescue squad. Unlock Deck
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14
A new police car costs the Bay City Police Department $26,000. The annual maintenance cost for a car depends on the age of the car at the beginning of the year. (All cars accumulate approximately the same mileage each year.) The maintenance costs increase as a car ages, as shown in the following table:
In order to avoid the increasingly high maintenance costs, the police department can sell a car and purchase a new car at the end of any year. The selling price for a car at the end of each year of use is also shown in the table. It is assumed that the price of a new car will increase $500 each year. The department's objective is to develop a car replacement schedule that will minimize total cost (i.e., the purchase cost of a new car plus maintenance costs minus the money received for selling a used car) during the next 6 years. Develop a car replacement schedule, using the shortest route method.
In order to avoid the increasingly high maintenance costs, the police department can sell a car and purchase a new car at the end of any year. The selling price for a car at the end of each year of use is also shown in the table. It is assumed that the price of a new car will increase $500 each year. The department's objective is to develop a car replacement schedule that will minimize total cost (i.e., the purchase cost of a new car plus maintenance costs minus the money received for selling a used car) during the next 6 years. Develop a car replacement schedule, using the shortest route method. Unlock Deck
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15
A new stadium complex is being planned for Denver, and the Denver traffic engineer is attempting to determine whether the city streets between the stadium complex and the interstate highway can accommodate the expected flow of 21,000 cars after each game. The various traffic arteries between the stadium (node 1) and the interstate (node 8) are shown in the following network:
The flow capacities on each street are determined by the number of available lanes, the use of traffic police and lights, and whether any lanes can be opened or closed in either direction. The flow capacities are given in thousands of cars. Determine the maximum traffic flow the streets can accommodate and the amount of traffic along each street. Will the streets be able to handle the expected flow after a game
The flow capacities on each street are determined by the number of available lanes, the use of traffic police and lights, and whether any lanes can be opened or closed in either direction. The flow capacities are given in thousands of cars. Determine the maximum traffic flow the streets can accommodate and the amount of traffic along each street. Will the streets be able to handle the expected flow after a game
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16
Frieda Millstone and her family live in Roanoke, Virginia, and they are planning an auto vacation across Virginia, their ultimate destination being Washington, DC. The family has developed the following network of possible routes and cities to visit on their trip:
The time, in hours, between cities (which is affected by the type of road and the number of intermediate towns) is shown along each branch. Determine the shortest route that the Millstone family can travel from Roanoke to Washington, DC.
The time, in hours, between cities (which is affected by the type of road and the number of intermediate towns) is shown along each branch. Determine the shortest route that the Millstone family can travel from Roanoke to Washington, DC.
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17
Solve Problem from Chapter to determine the shortest route for the "Weekend Rider" bus service from Tech to DC that will also result in at least 30 passengers.
Problem
The bus company in Draper operates a "Weekend Rider" bus service from the Tech campus to Washington, DC. The bus seats 55 passengers and selects the shortest route to DC that will also result in at least 30 passengers. The bus can travel through 11 different cities and towns depending on the number of passengers it expects to attract. The following network shows the different connecting pickup points and the distance between them in miles:
The bus service expects to pick up the following passengers at each possible stop: 20 at stop 1 (Tech), 4 at stop 2, 3 at stop 3, 5 at stop 4, 5 at stop 5, 7 at stop 6, 6 at stop 7, 4 at stop 8, 8 at stop 9, 6 at stop 10, 6 at stop 11, and 5 at stop 12. Determine the shortest route between Tech (at 1) to DC (at 13) that will result in at least 30 passengers. Compare this route to the shortest route without the required number of passengers, and indicate how many passengers the bus would have with this route.
Problem
The bus company in Draper operates a "Weekend Rider" bus service from the Tech campus to Washington, DC. The bus seats 55 passengers and selects the shortest route to DC that will also result in at least 30 passengers. The bus can travel through 11 different cities and towns depending on the number of passengers it expects to attract. The following network shows the different connecting pickup points and the distance between them in miles:
The bus service expects to pick up the following passengers at each possible stop: 20 at stop 1 (Tech), 4 at stop 2, 3 at stop 3, 5 at stop 4, 5 at stop 5, 7 at stop 6, 6 at stop 7, 4 at stop 8, 8 at stop 9, 6 at stop 10, 6 at stop 11, and 5 at stop 12. Determine the shortest route between Tech (at 1) to DC (at 13) that will result in at least 30 passengers. Compare this route to the shortest route without the required number of passengers, and indicate how many passengers the bus would have with this route.
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18
The FAA has granted a license to a new airline, Omniair, and awarded it several routes between Los Angeles and Chicago. The flights per day for each route are shown in the following network:
Determine the maximum number of flights the airline can schedule per day from Chicago to Los Angeles and indicate the number of flights along each route.
Determine the maximum number of flights the airline can schedule per day from Chicago to Los Angeles and indicate the number of flights along each route.
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19
The Roanoke, Virginia, distributor of Rainwater Beer delivers beer by truck to stores in six other Virginia cities, as shown in the following network:
The mileage between cities is shown along each branch. Determine the shortest truck route from Roanoke to each of the other six cities in the network.
The mileage between cities is shown along each branch. Determine the shortest truck route from Roanoke to each of the other six cities in the network.
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20
A developer is planning a development that includes subdivisions of houses, cluster houses, townhouses, apartment complexes, shopping areas, a daycare center and playground, a community center, and a school, among other facilities. The developer wants to connect the facilities and areas in the development with the minimum number of streets possible. The following network shows the possible street routes and distances (in thousands of feet) between 10 areas in the development that must be connected:
Determine a minimal spanning tree network of streets to connect the 10 areas, and indicate the total streets (in thousands of feet) needed.
Determine a minimal spanning tree network of streets to connect the 10 areas, and indicate the total streets (in thousands of feet) needed.
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21
The National Express Parcel Service has established various truck and air routes around the country over which it ships parcels. The holiday season is approaching, which means a dramatic increase in the number of packages that will be sent. The service wants to know the maximum flow of packages it can accommodate (in tons) from station 1 to station 7. The network of routes and the flow capacities (in tons of packages per day) along each route are shown in the following network:
Determine the maximum tonnage of packages that can be transported per day from station 1 to station 7 and indicate the flow along each branch.
Determine the maximum tonnage of packages that can be transported per day from station 1 to station 7 and indicate the flow along each branch.
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22
The plant engineer for the Bitco manufacturing plant is designing an overhead conveyor system that will connect the distribution/inventory center to all areas of the plant. The network of possible conveyor routes through the plant, with the length (in feet) along each branch, follows:
Determine the shortest conveyor route from the distribution/inventory center at node 1 to each of the other six areas of the plant.
Determine the shortest conveyor route from the distribution/inventory center at node 1 to each of the other six areas of the plant.
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23
One of the opposing forces in a simulated army battle wishes to set up a communications system that will connect the eight camps in its command. The following network indicates the distances (in hundreds of yards) between the camps and the different paths over which a communications line can be constructed:
Using the minimal spanning tree approach, determine the minimum distance communication system that will connect all eight camps
Using the minimal spanning tree approach, determine the minimum distance communication system that will connect all eight camps
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24
The traffic management office in Richmond is attempting to analyze the potential traffic flow from a new office complex under construction to an interstate highway interchange during the evening rush period. Cars leave the office complex via one of three exits, and then they travel through the city streets until they arrive at the interstate interchange. The following network shows the various street routes (branches) from the office complex (node 1) to the interstate interchange (node 9):
All intermediate nodes represent street intersections, and the values accompanying the branches emanating from the nodes represent the traffic capacities of each street, expressed in thousands of cars per hour. Determine the maximum flow of cars that the street system can absorb during the evening rush hour.
All intermediate nodes represent street intersections, and the values accompanying the branches emanating from the nodes represent the traffic capacities of each street, expressed in thousands of cars per hour. Determine the maximum flow of cars that the street system can absorb during the evening rush hour.
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25
The Burger Doodle restaurant franchises in Los Angeles are supplied from a central warehouse in Inglewood. The location of the warehouse and its proximity, in minutes of travel time, to the franchises are shown in the following network:
Trucks supply each franchise on a daily basis. Determine the shortest route from the warehouse at Inglewood to each of the nine franchises.
Trucks supply each franchise on a daily basis. Determine the shortest route from the warehouse at Inglewood to each of the nine franchises.
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26
The management of the Dynaco manufacturing plant wants to connect the eight major manufacturing areas of its plant with a forklift route. Because the construction of such a route will take a considerable amount of plant space and disrupt normal activities, management wants to minimize the total length of the route. The following network shows the distance, in yards, between the manufacturing areas (denoted by nodes 1 through 8):
Determine the minimal spanning tree forklift route for the plant and indicate the total yards the route will require.
Determine the minimal spanning tree forklift route for the plant and indicate the total yards the route will require.
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27
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
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28
The Petroco gasoline distributor in Jackson, Mississippi, supplies service stations in nine other southeastern cities, as shown in the following network:
The distance, in miles, is shown on each branch. Determine the shortest route from Jackson to the nine other cities in the network.
The distance, in miles, is shown on each branch. Determine the shortest route from Jackson to the nine other cities in the network.
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29
Several oil companies are jointly planning to build an oil pipeline to connect several southwestern, southeastern, and midwestern cities, as shown in the following network:
The miles between cities are shown on each branch. Determine a pipeline system that will connect all 10 cities, using the minimum number of miles of pipe, and indicate how many miles of pipe will be used.
The miles between cities are shown on each branch. Determine a pipeline system that will connect all 10 cities, using the minimum number of miles of pipe, and indicate how many miles of pipe will be used.
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30
Suppose in Problem that the processing cost per unit at each plant is different because of different machinery, workers' abilities, overhead, and so on, according to the following table:
There are no differentiated costs at stage 5, the distribution center. Given a budget of $700,000, determine the maximum number of units that can be processed through the five-stage manufacturing process.
Problem
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
There are no differentiated costs at stage 5, the distribution center. Given a budget of $700,000, determine the maximum number of units that can be processed through the five-stage manufacturing process.Problem
The Dynaco Company manufactures a product in five stages. Each stage of the manufacturing process is conducted at a different plant. The following network shows the five different stages and the routes over which the partially completed products are shipped to the various plants at the different stages:
Stage 5 (at node 10) is the distribution center in which final products are stored. Although each node represents a different plant, plants at the same stage perform the same operation. (For example, at stage 2 of the manufacturing process, plants 2, 3, and 4 all perform the same manufacturing operation.) The values accompanying the branches emanating from each node indicate the maximum number of units (in thousands) that a particular plant can produce and ship to another plant at the next stage. (For example, plant 3 has the capacity to process and ship 7,000 units of the product to plant 5.) Determine the maximum number of units that can be processed through the five-stage manufacturing process and the number of units processed at each plant.
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31
The Hylton Hotel has a limousine van that transports guests to various business and tourist locations around the city. The following network indicates the different routes the limousine could follow from the hotel at node 1 to the nine locations (nodes 2 through 10):
The values on each branch in the network are the distances, in miles, between the locations. Determine the shortest route from the hotel to each of the nine destinations and indicate the distance for each route.
The values on each branch in the network are the distances, in miles, between the locations. Determine the shortest route from the hotel to each of the nine destinations and indicate the distance for each route.
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32
A major hotel chain is constructing a new resort hotel complex in Greenbranch Springs, West Virginia. The resort is in a heavily wooded area, and the developers want to preserve as much of the natural beauty as possible. To do so, the developers want to connect all the various facilities in the complex with a combination walking-riding path that will minimize the amount of pathway that will have to be cut through the woods. The following network shows possible connecting paths and corresponding distances (in yards) between the facilities:
Determine the path that will connect all the facilities with the minimum amount of construction and indicate the total length of the pathway.
Determine the path that will connect all the facilities with the minimum amount of construction and indicate the total length of the pathway.
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33
Given the following network, with the indicated flow capacities along each branch, determine the maximum flow from source node 1 to destination node 10 and the flow along each path:
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34
A steel mill in Gary supplies steel to manufacturers in eight other midwestern cities by truck, as shown in the following network:
The travel time between cities, in hours, is shown along each branch. Determine the shortest route from Gary to each of the other eight cities in the network.
The travel time between cities, in hours, is shown along each branch. Determine the shortest route from Gary to each of the other eight cities in the network.
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35
The Barrett Textile Mill is remodeling its plant and installing a new ventilation system. The possible ducts connecting the different rooms and buildings at the plant, with the length (in feet) along each branch, are shown in the following network:
Determine the ventilation system that will connect the various rooms and buildings of the plant, using the minimum number of feet of ductwork, and indicate how many feet of ductwork will be used.
Determine the ventilation system that will connect the various rooms and buildings of the plant, using the minimum number of feet of ductwork, and indicate how many feet of ductwork will be used.
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36
A manufacturing company produces different variations of a product at different work centers in its plant on a daily basis. Following is a network showing the various work centers in the plant, the daily capacities at each work center, and the flow of the partially completed products between work centers:
Node 1 represents the point where raw materials enter the process, and node 15 is the packaging and distribution center. Determine the maximum number of units that can be completed each day and the number of units processed at each work center.
Node 1 represents the point where raw materials enter the process, and node 15 is the packaging and distribution center. Determine the maximum number of units that can be completed each day and the number of units processed at each work center.
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37
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
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38
The town council of Whitesville has decided to construct a bicycle path that will connect the various suburbs of the town with the shopping center, the downtown area, and the local college. The council hopes the local citizenry will use the bike path, thus conserving energy and decreasing traffic congestion. The various paths that can be constructed and their lengths (in miles) are shown in the following network:
Determine the bicycle pathway that will require the minimum amount of construction to connect all the areas of the town. Indicate the total length of the path.
Determine the bicycle pathway that will require the minimum amount of construction to connect all the areas of the town. Indicate the total length of the path.
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39
The following network shows the major roads that would be part of the hurricane evacuation routes for Charleston, South Carolina:
The destination is the interchange of Interstate 26 and Interstate 526, north of Charleston. The nodes represent road intersections, and the values at the nodes are the traffic capacities, in thousands of cars. Determine the maximum flow of cars that can leave Charleston during a hurricane evacuation and the optimal flow along each road segment (branch 1).
The destination is the interchange of Interstate 26 and Interstate 526, north of Charleston. The nodes represent road intersections, and the values at the nodes are the traffic capacities, in thousands of cars. Determine the maximum flow of cars that can leave Charleston during a hurricane evacuation and the optimal flow along each road segment (branch 1).
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40
The members of the Vistas Club are planning a month long hiking and camping trip through the Blue Ridge Mountains, from north Georgia to Virginia. There are a number of trails through the mountains that connect at various campgrounds, crossings, and cabins. The following network shows the different connecting trails and the distance of each connecting branch, in miles:
Determine the shortest path from node 1 in north Georgia to node 13 in Virginia and indicate the total mileage of the trail.
Determine the shortest path from node 1 in north Georgia to node 13 in Virginia and indicate the total mileage of the trail.
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41
Determine the minimal spanning tree for the network in Problem.
Problem
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
Problem
Determine the shortest route from node 1 (origin) to node 12 (destination) for the following network. Distances are given along the network branches:
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42
In Problem in Chapter, KanTech Corporation ships electronic components in containers from seaports in Europe to U.S. ports, from which the containers are transported by truck or rail to inland ports and then transferred to an alternative mode of transportation before being shipped to KanTech's U.S. distributors. Using the shipping capacity at each port, develop a maximal flow network to determine the number of containers to ship through each port to meet demand at the distribution centers.
Problem
KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/ visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs ($/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:
Formulate and solve a linear programming model to determine the optimal shipments from each point of embarkation to each destination along this supply chain that will result in the minimum total shipping cost.
Problem
KanTech Corporation is a global distributor of electrical parts and components. Its customers are electronics companies in the United States, including computer manufacturers and audio/ visual product manufacturers. The company contracts to purchase components and parts from manufacturers in Russia, Eastern and Western Europe, and the Mediterranean, and it has them delivered to warehouses in three European ports, Gdansk, Hamburg, and Lisbon. The various components and parts are loaded into containers based on demand from U.S. customers. Each port has a limited fixed number of containers available each month. The containers are then shipped overseas by container ships to the ports of Norfolk, Jacksonville, New Orleans, and Galveston. From these seaports, the containers are typically coupled with trucks and hauled to inland ports in Front Royal (Virginia), Kansas City, and Dallas. There are a fixed number of freight haulers available at each port each month. These inland ports are sometimes called "freight villages," or intermodal junctions, where the containers are collected and transferred from one transport mode to another (i.e., from truck to rail or vice versa). From the inland ports, the containers are transported to KanTech's distribution centers in Tucson, Pittsburgh, Denver, Nashville, and Cleveland. Following are the handling and shipping costs ($/container) between each of the embarkation and destination points along this overseas supply chain and the available containers at each port:
Formulate and solve a linear programming model to determine the optimal shipments from each point of embarkation to each destination along this supply chain that will result in the minimum total shipping cost. Unlock Deck
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43
Kathleen Taylor, a student at Tech, is planning to visit her sister, Lindsey, who is living in Toulouse, France, over the summer break. She is going to fly from Dulles Airport in Washington to Charles de Gaulle Airport in Paris, and because of the time changes, this travel will take a full day. In Paris, Kathleen is going to spend 2 nights and 1 full day before taking the train to Toulouse. Kathleen has never been to Paris, so she wants to spend her 1 day there seeing as many of the famous attractions as she can, including the Eiffel Tower, the Louvre, Notre Dame Cathedral, the Arch de Triumph, the Pantheon, and the Palace at Versailles. She plans to stay at a youth hostel very near Sacre Coeur in Montmartre and from there use the Paris Metro to visit as many of the sites as she can in a day. She has downloaded a detailed Metro map from the French rail Web site at http://www.ratp.fr and has discovered that the Metro system is huge, with almost 250 stations and 14 lines throughout Paris.
There's no question that Kathleen can get to all the sites by the Metro, but she is concerned about her limited time frame and her ability to get from location to location quickly. She has determined the following information regarding the average times (in minutes) between stations for each line:
She's also guessing that the subway stops about 1 minute at each station. If she has to change lines, she assumes it will take her at least 5 minutes. She plans to leave early in the morning, when the sites open, and she has no specific time she must be back to the hostel.
Kathleen, a business student, would like to use some of kind of logical, systematic approach to help her plan her movement using the Metro around the city to the different sites. Help Kathleen develop a route around the city to each of the sites she wants to see, starting from her youth hostel in Montmartre, for the day she'll be in Paris. Do you think she'll be able to see all the sites she wants to see
There's no question that Kathleen can get to all the sites by the Metro, but she is concerned about her limited time frame and her ability to get from location to location quickly. She has determined the following information regarding the average times (in minutes) between stations for each line:
She's also guessing that the subway stops about 1 minute at each station. If she has to change lines, she assumes it will take her at least 5 minutes. She plans to leave early in the morning, when the sites open, and she has no specific time she must be back to the hostel.Kathleen, a business student, would like to use some of kind of logical, systematic approach to help her plan her movement using the Metro around the city to the different sites. Help Kathleen develop a route around the city to each of the sites she wants to see, starting from her youth hostel in Montmartre, for the day she'll be in Paris. Do you think she'll be able to see all the sites she wants to see
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44
Hard Rock Concrete Supply makes concrete at its plant in Centerville, Virginia, and delivers it to construction sites throughout the metropolitan Washington, DC, area. The following network shows the possible routes and distances (in miles) from the concrete plant to seven construction sites:
Determine the shortest route a concrete truck would take from the plant at node 1 to node 8 and the total distance for this route, using the shortest route method.
Determine the shortest route a concrete truck would take from the plant at node 1 to node 8 and the total distance for this route, using the shortest route method.
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45
State University has decided to reconstruct the sidewalks throughout the east side of its campus to provide wheelchair access. However, upgrading sidewalks is a very expensive undertaking, so for the first phase of this project, university administrators want to make sure they connect all buildings with wheelchair access with the minimum number of refurbished sidewalks possible.
Following is a network of the existing sidewalks on the east side of campus, with the feet between each building shown on the branches:
Determine a minimal spanning tree network that will connect all the buildings on campus with wheelchair access sidewalks and indicate the number of feet of sidewalk.
Following is a network of the existing sidewalks on the east side of campus, with the feet between each building shown on the branches:
Determine a minimal spanning tree network that will connect all the buildings on campus with wheelchair access sidewalks and indicate the number of feet of sidewalk.
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46
In the novel Around the World in 80 Days by Jules Verne, Phileas T. Fogg wagered four of his fellow members of the Reform Club in London £5,000 apiece that he could travel around the world in 80 days. This would have been an astounding feat in 1872, the year in which the novel is set. Then, the fastest modes of travel were rail and ship; however, much of the world still traveled by coach, wagon, horse, elephant, or donkey, or on foot. Phileas Fogg was not a frivolous man. He would not have undertaken such a large wager if he had not carefully researched the feasibility of such a trip and been confident of his chances. Although he undoubtedly analyzed various routes to circumnavigate the globe, he did not have knowledge of techniques such as the shortest route method, nor did he have a computer to help him select an optimal route. If he had, he might have chosen a route that would have completed his trip in less than 80 days or possibly would not have been so quick to make his wager. On the next page is a network of the various routes of the day for traveling around the world from London. Fogg traveled eastward. The travel time, in days, is shown on each branch. Travel times are based as much on available modes of transportation at the time as on distance. Using the shortest route method (and the computer), select the best route for Mr. Fogg. Would the route you determined have won him his wager
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47
George is camped deep in the jungle, and he wants to make his way back to the coast and civilization. Each of the paths he can take through the jungle has obstacles that can delay him, including hostile natives, wild animals, dense forests and vegetation, swamps, rivers and streams, snakes, insects, and mountains. Following is the network of paths that George can take, with the time (in days) to travel each branch:
Determine the shortest (time) path for George to take and indicate the total time.
Determine the shortest (time) path for George to take and indicate the total time.
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48
Tech wants to develop an area network that will connect its server at its computer and satellite center with the main campus buildings to improve Internet service. The cable will be laid primarily through existing electrical tunnels, although some cable will have to be buried underground. The following network shows the possible cable connections between the computer center at node 1 and the various buildings, with the distances, in feet, along the branches:
Determine a minimal spanning tree network that will connect all the buildings and indicate the total amount of cable that will be needed to do so.
Determine a minimal spanning tree network that will connect all the buildings and indicate the total amount of cable that will be needed to do so.
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