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J E Beasley OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research OR. They are now available for use by any students and teachers interested in OR subject to the following conditions.
A full list of the topics available in OR-Notes can be found here. Network analysis - activity on node In the network diagram shown below, for the problem we considered beforeeach node circle represents an activity and is labelled with the activity number and the associated completion time shown in brackets after the activity number.
This network is an activity on node AON network. In constructing the network we: Note here that all arcs have arrows attached to them indicating the direction the project is flowing in.
One tip that I find useful in drawing such diagrams is to structure the positioning of the nodes activities so that the activities at the start of the project are at the left, the activities at the end of the project at the right, and the project "flows" from left to right in a natural fashion.
Once having drawn the network it is a relatively easy matter to analyse it using a dynamic programming algorithm to find the critical path. However we will not consider this algorithm in any detail here but will instead use the computer package to solve the problem. Note here one key point, the above network diagram assumes that activities not linked by precedence relationships can take place simultaneously e.
Essentially the above diagram is not needed for a computer - a computer can cope very well indeed better with just the lists of activities and their precedence relationships we had before.
The above diagram is intended for people. Consider what might happen in a large project - perhaps many thousands or tens of thousands of activities and their associated precedence relationships.
Do you think it would be possible to list those out without making any errors? Obviously not - so how can we spot errors? Looking at long lists in an attempt to spot errors is just hopeless. With a little practice it becomes easy to look at diagrams such as that shown above and interpret them and spot errors in the specification of the activities and and their associated precedence relationships.
Package solution The problem as represented by the network diagram was solved using the packagethe input being shown below. Note that we have chosen to enter the problem as a deterministic CPM problem. The various input options shown below will become more familiar to you as we progress.
The numbers entered following the above screen are as below. The output from the package is shown below. In the output we have a "Project Completion Time" of 24 weeks. This means that if all the activities take exactly as long as expected the minimum time in which we can complete the project complete all activities whilst obeying the precedence relationships is 24 weeks.
Note here that we have implicitly assumed in calculating this figure of 24 weeks that we have sufficient resources to enable activities to be carried out simultaneously if required e. Problems where this assumption does not hold are considered here.
In the column headed "Slack" we have, for each activity in turn, the amount of time that the activity can be delayed without altering increasing the overall project completion time. If delays occur in two or more activities then we must either analyse the effect on the project by hand, or rerun the problem with new data.
Many textbooks also refer to slack by the term "float". Activities with a slack of zero are called critical activities since they must all be completed on time to avoid increasing the overall project completion time.
Hence, for this network, activities 1, 3, 5, 7, 8, 9 and 11 are the critical activities. Note here that 1 3 5 7 8 9 11 constitutes a path in the network diagram from the initial node node 1 to the final node node This is no accident because for any network there will always be a path of critical activities from the initial node to the final node.
Such a path is called the critical path. More strictly the definition of a critical path is a path of activities, each pair of activities in the path directly connected via a precedence relationship arcfrom the start initial node to the end final node of the project, where the completion times of the activities on the path sum to the overall minimum project completion time.
All activities in this path must be critical by definition. The output also lists, for each activity: All immediate predecessors must be finished before an activity can start. If the earliest start and latest start times are the same then the activity is critical. As with start times, the activity is critical if the earliest finish and latest finish times are the same.
For example, suppose in the above network activity 10 took 3 weeks to complete i.A profit center is a branch or division of a company that is accounted for on a standalone basis for profit calculation. A profit center is responsible for generating its own results and earnings. In a formal (business) letters you have to put the address of the receiver on the top of the letter as well.
I suggest the top, left corner as shown in the example. There is a simple reason why. Many people use envelopes with a transparent window so that they don't have to put an address label on the outide.
This place is common in Germany. Ralf Dillerup with expertise in: Business Administration, Business Ethics and Industrial Organization.
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Please choose your country Europe. Deutschland. Capital employed, also known as funds employed, is the total amount of capital used for the acquisition of profits. It is the value of all the assets employed in a business, and can be calculated.