Topic6 TheCriticalPathMethod

# Topic6 TheCriticalPathMethod - Operations Management 33:386...

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Operations Management 33:386 Fall 2004 Farid Alizadeh Project Scheduling And The Critical Path Method (CPM) Last updated on 10/22/2008 In project scheduling we often have a series of tasks or activities each requiring a certain amount of time to complete. Furthermore, some activities need the completion of one or more other activities before they can be started. (Study the Widgetco problem on page 97 of the coursepack for a concrete example.) We wish to study the earliest time the project can be completed. Also, if the time requirement of each task can be reduced with bounds and at a cost, we wish to see if we can complete the project within a given deadline as economically as possible. The project scheduling problem can be represented by an acyclic network. In this model activities are represented by nodes and if one activity can only start after another we draw an arrow from the first to the second. For example, the Widgetco activities can be represented by the following diagram. Figure 1: precedence representation of Wigetco example Here the node labeled ‘s’ represents a dummy activity which indicates the start of the project; and the node labeled “f” is a dummy activity that indicates end of the project. The durations of both start and end activities equal zero. s A B C D F E f G

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The arcs represent dependency. So the fact that activity C must follow activities A and B are represented by arcs from A and B to C. Also if an activity does not depend on any other activity we make it dependent on the start node. Similarly if an activity is an end- node , that is no other activity depends on it, we make the finish node dependent on it. For instance in Widgetco problem activities A and B depend on no other activities, so we draw arrows from s to A and to B. And no activity depends on F, so we draw an arrow from F to finish node f. The network that comes out of these dependencies is called acyclic because it does not contain any cycles, that is starting from any node we cannot follow a set of arcs and eventually end up where it started. Here is an example of a network that is not acyclic: Starting from C we can walk to B then to A then to D and back to C again. This network cannot possibly model any sort of dependency in a consistent way. If it were indeed a dependency network, then B would depend on C and A would depend on B, and D would depend on A and C would depend on D, which would ultimately imply C need C to complete before it could start! With this set up it is now possible to determine what the minimum time required to complete the project. We model the problem with two different approaches, both of which will give the same answer but with slightly different byproducts. Minimum Completion Time with finish times Here we can assign to each activity (node) i a decision variable: y i = the finish time of activity i .
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