Lesson 4.2. Critical Paths

Transcription

1 Lesson. ritical Paths It is relatively easy to find the shortest time needed to complete a project if the project consists of only a few activities. ut as the tasks increase in number, the problem becomes more difficult to solve by inspection alone. In the 9s the U.S. government was faced with the need to complete very complex systems such as the U.S. Navy Polaris Submarine project. In order to do this efficiently, a method was developed called PRT (Program valuation and Review Technique). This technique targeted tasks that were critical to the earliest completion of the project. The path of targeted tasks from the start to the finish of the project became known as the critical path. Recall the graph in Lesson. that represented the entral High yearbook project. How might you go about finding a systematic way to identify the critical path for this project? To do this, an earliest-start time (ST) for each task must be found. The ST is the earliest that an activity can begin if all the activities preceding it begin as early as possible. To calculate the ST for each task, begin at the start. Then label each vertex with the smallest possible time that is needed before the task can begin. The label for in igure. is found by adding the ST of to the day that it takes to complete task ( + = ). Task cannot be completed until both predecessors, and, have been completed. Hence, cannot begin until days have passed. In the case of the yearbook staff, the earliest time in which the project can be completed is days. s paradoxical as it may seem, the least amount of time that it takes to complete all of the tasks in the project corresponds to the time it takes to complete the longest path through the graph from start to finish.

2 hapter raphs and Their pplications path with this longest time is the desired critical path. In igure., the critical path is -H-. () () () H () () () () () () igure.. Yearbook diagram showing the earliest-start time for each task. xample. opy the graph and label the vertices with the ST for each task. Then determine the earliest completion time for the project. ll times are in minutes.. Identify the critical path. Solution:. () () () () () () (9)

3 Lesson. ritical Paths The earliest time in which the project can be completed is minutes.. Since the critical path is the longest path from the start to the finish, the critical path is --. If it is desirable to cut the completion time of a project, it can be done by shortening the length of the critical path once it is found. In the preceding example, one way to shorten the time it takes to complete the project is to cut the time it takes to complete task. If task s time is cut from minutes to minutes, the completion time for the project is cut to minutes. The efficient management of large projects like the construction of a building requires the use of critical path analysis. xercises. omplete the following. Vertex arliest- Time Minimum project time = ritical path(s) =

4 hapter raphs and Their pplications In xercises and, list the vertices of the graphs and give their earliest-start time, as in xercise. etermine the minimum project time and all of the critical paths.. 9 H. J H I K L. Using the information from the following table, construct a graph and label each of the vertices with its earliest-start time. etermine the minimum project time and critical path. Task Time Prerequisites None None,,,,,

5 Lesson. ritical Paths 9. a. opy the graph and label each vertex with its earliest-start time. b. How quickly can the project be completed? c. etermine the critical path. d. What happens to the minimum project time if task s time is reduced to 9 days? To days? e. Will the project time continue to be affected by reducing the time of task? xplain why or why not.. onstruct a graph with three critical paths.. etermine the minimum project time and the critical path for the following graph. 9

6 hapter raphs and Their pplications. Task Time Prerequisites None None,, H a. raw a graph using the information in the table. b. Label each vertex with its earliest-start time. c. etermine the minimum project time. d. etermine the critical path(s). 9. In the following graph, each vertex has been label with its ST, and the critical path is marked. () () () () (9) () () a. Task can begin as early as day 9. If it begins on day 9, when will it be completed? If it begins on day? On day? What will happen if it begins on day? b. What is the latest day on which task can begin if task is to begin on day? If an activity is not on the critical path, it is possible for it to start later than its earliest-start time and not delay the project. The latest a task can begin without delaying the project s minimum completion time is known as the latest-start time (LST) for the task. or example, the LST for is day.

7 Lesson. ritical Paths c. In order to find the LST for vertex, the times of the two vertices and need to be considered. Since vertex is on the critical path, the latest it can start is day. or to begin on time, what is the latest day on which can begin? In part b, you found that the latest can start is day. In that case, what is the latest can begin? rom this information, what is the latest (LST) that can begin without delaying either task or?. To find the LST for each task, it is necessary to begin with the and work through the graph in reverse order to the. ach of the vertices in the following graph is labeled with its ST. The LSTs for several of the tasks have been calculated and are shown below the STs on the vertices. ind the LSTs for the remaining tasks. () () () 9 9 () ( ) ( ) ( ). Write an algorithm to find the LSTs for the tasks in a graph. Test your algorithm on the graph in xercise. ( ) H () Project. Interview the yearbook sponsors in your school to find out how they organize the publication of your school s yearbook. reate a task table that shows the approximate times and prerequisite tasks that must be completed before your yearbook can go to the publisher. esign a graph with the ST for each task, and identify the critical path. Modeling Project. Use the Internet or other sources to research and report on businesses or people who use PRT or similar evaluation techniques such as antt harts to model project planning.