Design and development process

Requirements: . OVERVIEWAnalyze motorized traffic flows for a transportation department using a weighted graph. Then streamline the design and development process and improve scheduling for several systems engineering projects using PERT analysis and critical path management. For this assessment you will demonstrate your ability to use graphs and trees in application modeling analysis and optimization. Context The Assessment 5 Context document contains information about graphs including graph applications and representations ASSESSMENT INSTRUCTIONS

PART 1: USING GRAPHS AND MATRICES TO REPRESENT SYSTEMSAssume you have been recently hired by the Department of Transportation (DoT) to analyze motorized vehicle traffic flows. Your initial goal is to analyze the traffic and traffic delays in a large metropolitan area. You choose to use a weighted graph to represent this particular scenario. Remember that a graph is a collection of nodes (or vertices) and edges. Each edge will have a corresponding weight.Describe how you would construct such a weighted graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Justify your design. Feel free to include an example in your description.Describe an adjacency matrix you might create to represent this graph. What do the rows represent? What do the columns represent? What do the entries in the matrix represent? Feel free to create an example or create an illustration to better explain yourself.Suppose you wish to identify the shortest travel time between nodes on the graph. What type of path would represent such a route? Describe this path in detail.Suppose that the DoT is considering repairing some of the roads in the metropolitan area. But first the DoT wishes to assess each and every road to determine which of the roads require repair. You have reported to the manager of the repair project and noted that you can use your graphical representation and graph-based methods to suggest the most efficient route the inspection crew can take to inspect all the roads. Describe in detail and name the type of the particular path on the graph that would represent this most efficient route. Consider all aspects of travel time undertaken by the inspection crew. Justify your answer.Suppose that the DoT is planning to widen roads that create a bottleneck in the flow of traffic. As such you pose this problem as a max flow/min cut problem. You have determined that roads are in need of widening if the traffic flow on the roads is at maximum capacity; these roads are essentially the bottlenecks of the road system. These bottleneck roads can be identified using the Max Flow/Min Cut theorem.Using the graph representation of the road system explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut.

PART 2: GRAPHING CRITICAL PATHS Assume you have been recently hired as a systems engineering manager. One of the main goals in your new position is to streamline the design and development process and improve scheduling. As such you wish to perform PERT analysis and critical path management for each project under your supervision. In critical path management a schedule of events or tasks is listed and sequential dependencies between events are identified. In this context the critical path is the path of sequentially dependent events that has the longest total time (aggregation of time for all events in the sequence). You decide to represent the events including their estimated times and sequential dependencies using a directed graph.Describe how you would construct such a graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Is this graph directed or undirected? Justify this design.Suppose you wish to identify the critical path. What type of path is this? Describe and name the type of path within the context of the graph. Explain how you will use graphs and graph-based methods to find the critical path. Justify your answer.Create an example graph that has at least five events and at least seven dependencies. Identify the critical path Describe how you would construct such a weighted graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Justify your design. Feel free to include an example in your description. Describe an adjacency matrix you might create to represent this graph. What do the rows represent? What do the columns represent? What do the entries in the matrix represent? Feel free to create an example or create an illustration to better explain yourself. Suppose you wish to identify the shortest travel time between nodes on the graph. What type of path would represent such a route? Describe this path in detail. Suppose that the DoT is considering repairing some of the roads in the metropolitan area. But first the DoT wishes to assess each and every road to determine which of the roads require repair. You have reported to the manager of the repair project and noted that you can use your graphical representation and graph-based methods to suggest the most efficient route the inspection crew can take to inspect all the roads. Describe in detail and name the type of the particular path on the graph that would represent this most efficient route. Consider all aspects of travel time undertaken by the inspection crew. Justify your answer. Suppose that the DoT is planning to widen roads that create a bottleneck in the flow of traffic. As such you pose this problem as a max flow/min cut problem. You have determined that roads are in need of widening if the traffic flow on the roads is at maximum capacity; these roads are essentially the bottlenecks of the road system. These bottleneck roads can be identified using the Max Flow/Min Cut theorem.Using the graph representation of the road system explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut. Using the graph representation of the road system explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads. Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut. Describe how you would construct such a graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Is this graph directed or undirected? Justify this design. Suppose you wish to identify the critical path. What type of path is this? Describe and name the type of path within the context of the graph. Explain how you will use graphs and graph-based methods to find the critical path. Justify your answer. Create an example graph that has at least five events and at least seven dependencies. Identify the critical path Discrete Structures Scoring GuideCRITERIADISTINGUISHEDDiscriminate between graphs and trees.Discriminates between graphs and trees and identifies or rejects appropriate uses for each.Apply graph-based methods to real-world problems.Applies graph-based methods to real-world problems and explains the application in the context of the problem.Describe adjacency matrices as representations of graphs.Describes adjacency matrices as representations of graphs and constructs an example or relates matrix representations to other graph representations.Synthesize exemplar graphs to model real-world problems.Synthesizes exemplar graphs to model real-world problems and clearly relates this graph to the real-world problem.Explain optimization problem using graphs: linear/integer programming.. Explains optimization problem using graphs: linear/integer programming and synthesizes an example with solution. Discrete Structures Scoring GuideCRITERIADISTINGUISHEDDiscriminate between graphs and trees.Discriminates between graphs and trees and identifies or rejects appropriate uses for each.Apply graph-based methods to real-world problems.Applies graph-based methods to real-world problems and explains the application in the context of the problem.Describe adjacency matrices as representations of graphs.Describes adjacency matrices as representations of graphs and constructs an example or relates matrix representations to other graph representations.Synthesize exemplar graphs to model real-world problems.Synthesizes exemplar graphs to model real-world problems and clearly relates this graph to the real-world problem.Explain optimization problem using graphs: linear/integer programming.. Explains optimization problem using graphs: linear/integer programming and synthesizes an example with solution. Describe how you would construct such a weighted graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Justify your design. Feel free to include an example in your description.Describe an adjacency matrix you might create to represent this graph. What do the rows represent? What do the columns represent? What do the entries in the matrix represent? Feel free to create an example or create an illustration to better explain yourself.Suppose you wish to identify the shortest travel time between nodes on the graph. What type of path would represent such a route? Describe this path in detail.Suppose that the DoT is considering repairing some of the roads in the metropolitan area. But first the DoT wishes to assess each and every road to determine which of the roads require repair. You have reported to the manager of the repair project and noted that you can use your graphical representation and graph-based methods to suggest the most efficient route the inspection crew can take to inspect all the roads. Describe in detail and name the type of the particular path on the graph that would represent this most efficient route. Consider all aspects of travel time undertaken by the inspection crew. Justify your answer.Suppose that the DoT is planning to widen roads that create a bottleneck in the flow of traffic. As such you pose this problem as a max flow/min cut problem. You have determined that roads are in need of widening if the traffic flow on the roads is at maximum capacity; these roads are essentially the bottlenecks of the road system. These bottleneck roads can be identified using the Max Flow/Min Cut theorem.Using the graph representation of the road system explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut. Using the graph representation of the road system explain how the Max Flow/Min Cut theorem can be used to identify the bottleneck roads.Create an example graph one might use to analyze this network as a Max Flow problem. Include at least 8 nodes and 16 edges. Include capacities and flows and identify the max flow and/or min cut. Describe how you would construct such a graph. What do the nodes represent? What do the edges represent? What values would the weights represent? Is this graph a tree? Is this graph directed or undirected? Justify this design.Suppose you wish to identify the critical path. What type of path is this? Describe and name the type of path within the context of the graph. Explain how you will use graphs and graph-based methods to find the critical path. Justify your answer.Create an example graph that has at least five events and at least seven dependencies. Identify the critical path Discrete Structures Scoring GuideCRITERIADISTINGUISHEDDiscriminate between graphs and trees.Discriminates between graphs and trees and identifies or rejects appropriate uses for each.Apply graph-based methods to real-world problems.Applies graph-based methods to real-world problems and explains the application in the context of the problem.Describe adjacency matrices as representations of graphs.Describes adjacency matrices as representations of graphs and constructs an example or relates matrix representations to other graph representations.Synthesize exemplar graphs to model real-world problems.Synthesizes exemplar graphs to model real-world problems and clearly relates this graph to the real-world problem.Explain optimization problem using graphs: linear/integer programming.. Explains optimization problem using graphs: linear/integer programming and synthesizes an example with solution.