On an aon graph, a is a sequence of nodes and arrows within the activity network.

7.12 Arrow diagrams

The basic arrow diagram

The process for making a basic arrow diagram is as follows:

•

Using the output from a tree diagram, (including the constraints), note each of the main activities/tasks on the top of a separate Post-it note.

•

Note all of the tasks necessary for the main activities on additional Post-it notes.

•

Arrange the notes in time order (left to right) based on when the task should take place to allow subsequent tasks to be completed. Tasks that can take place in parallel, i.e. at the same time, should be arranged vertically.

•

Remove any duplicate tasks, i.e. those necessary for two or more subsequent tasks.

•

Connect the tasks with arrows and nodes. The convention for drawing arrow diagrams is:

▪

Events are noted as circles. An event is either the finish or start of a task (although you can use ‘dummy events’ to get the diagram to work).

▪

Tasks are drawn on the arrows.

•

Move the notes around to reduce the overall time for completion.

Note 1: ‘Dummy’ events can be useful for separating simultaneous tasks or controlling the sequence of events.

Note 2: It may be possible to split some tasks up to reduce the total time needed, i.e. it may be possible to start a subsequent task before the preceding task is fully complete.

•

Connect the events to complete the diagram.

An example of a simple arrow diagram for a trial implementation of SPC is shown at upper right. These are the same data as used for the Gantt chart on the left.

Putting times in the diagram

After the standard arrow diagram is completed it is possible to extend this by inserting the time required for each task and using this time to find the critical path for the project. This is shown on the lower right for the trial implementation of SPC with the ‘critical path’ marked in bold, this is the shortest time for completion of the project and activities making up the critical path must be protected to ensure that the project is finished on time.

The convention for abbreviations used in the critical path calculations are:

•

DR - the duration of the task (in days, weeks or months but the units must remain consistent).

•

ES - the earliest start time of the task, i.e. sum of the times for the tasks needed before this task can start.

•

LS - the latest start time of the task to maintain schedule adherence.

•

EF - the earliest finish time of the task, i.e. ES + DR.

•

LF - the latest finish time of the task, i.e. the latest finish time of the task to maintain schedule adherence.

•

SL - the ‘slack’ time of the task, i.e. LS - ES or LF - EF.

“A project is only ever late because nobody knew it was going to be late until too late.”

Henrique Neto

The arrow diagram can be simple or complex but either way it provides an invaluable planning tool for delivering complex or simple projects on-time - on budget is another topic altogether!

4.5.1 Precedence diagram method (PDM)

In the past the arrow diagram method (ADM) was commonly used, but with the increase in computer-based scheduling tools, its use has declined. This method employs a dashed line, called a dummy activity, with time zero, to represent a dependency between tasks without any actual activity. For complicated activities, these dummy activities are not practical. Therefore, the use of a precedence diagram is preferred over the ADM.

The current most common method is to represent each activity by a rectangle, as shown in Fig. 4.5, and to define the relationship between activities by connecting them with arrows. The rectangle in detail is illustrated in Fig. 4.6. From this figure, the relations between activities can be seen, as activity (C) starts after finishing the activities (D) and (B).

In addition, Fig. 4.6 shows data written inside the rectangle of the activity, including the activity ID, its time duration, and then the early start time (ES), early finish (EF) time, and also the latest start (LS) and latest finish (LF) times.

Limitations

The limitations of the graphical method are basically the size of the bar chart paper and therefore, the number of activities. Most programmes are drawn on either A1 or A0 size paper and the number of different activities must be compressed into the 840 mm width of this sheet. (It may, of course, be possible to divide the network into two, but then the interlinking activities must be carefully transferred.) Normally, the division between bars is about 6 mm, which means that a maximum of 120 activities can be analysed. However, bearing in mind that in a normal network, 30% of the activities are dummies, a network of 180–200 activities could be analysed graphically on one sheet.

Briefly, the mode of operation is as follows:

Draw the network in arrow diagram or precedence format and write in the activity titles (Fig. 24.1 or 24.2). Although a forward pass has been carried out on both these diagrams, this is not necessary when using the graphical method of analysis.

Insert the durations.

List the activities on the left-hand vertical edge of a sheet of graph paper (Fig. 21.11) showing:

activity title

duration (in days, weeks, etc.)

node no (only required when using these for bar chart generation)

Draw a time scale along the bottom horizontal edge of the graph paper.

Draw a horizontal line from day 0 of the first activity, which is proportional to the duration (using the time scale selected), for example 6 days would mean a line six divisions long (Fig. 24.3). To ease identification an activity letter or no. can be written above the bar.

Repeat this operation with the next activity on the table starting on day 0.

When using activity on arrow networks, mark dummy activities by writing the end time of the dummy next to the start time of the dummy, for example 4 → 7 would be shown as 4,7 (Fig. 21.13).

All subsequent activities must be drawn with their start time (start day no.) directly below the end time (end day no.) of the previous activity having the same time value (day no.).

If more than one activity has the same end time (day no.), draw the new activity line from the activity end time (day no.) furthest to the right.

Proceed in this manner until the end of the network.

The critical path can now be traced back by following the line (or lines), which runs back to the start without a horizontal break.

The break between consecutive activities on the bar chart is the free float of the preceding activity.

The summation of the free floats in one string, before that string meets the critical path, is the total float of the activity from which the summation starts, for example in Fig. 21.11 the total float of activity K is 1 + 1 + 2 = 4 days, the total float of activity M is 1 + 2 + 3 days, and the total float of activity N is 2 days.

The advantage of using the start and end times (day nos.) of the activities to generate the bar chart is that there is no need to carry out a forward pass. The correct relationship is given automatically by the disposition of the bars. This method is therefore equally suitable for arrow and precedence diagrams.

An alternative method can, however, be used by substituting the day numbers by the node numbers. Clearly, this method, which is sometimes quicker to draw, can only be used with arrow diagrams, because precedence diagrams do not have node numbers. When using this method, the node numbers are listed next to the activity titles (Fig. 24.5) and the bars are drawn from the starting node of the first activity with a length equal to the duration. The next bar starts vertically below the end node with the same node number as the starting node of the activity being drawn.

As with the day no. method, if more than one activity has the same end node number, the one furthest to the right must be used as a starting time. Fig. 24.4 shows the same network with the node numbers inserted, and Fig. 24.5 shows the bar chart generated using the node numbers.

Fig. 24.6 shows a typical arrow diagram, and Fig. 24.7 shows a bar chart generated using the starting and finishing node numbers. Note that these node numbers have been listed on the left-hand edge together with the durations to ease plotting.

Basic advantages

The advantages of a Lester diagram are as follows:

Faster to draw than precedence diagram – about the same speed as an arrow diagram;

As in a precedence diagram:

Total float is vertical difference;

Free float is horizontal difference;

Room under arrow for duration and total float value;

Logic lines can cross the activity arrows;

Requires less space on paper when drafting the network;

Good for examinations due to speedy drafting and elimination of node boxes;

Can be updated for progress by ‘redding’ up activity arrows as arrow diagram;

Uses same procedures for computer inputting as precedence networks;

Output from computer similar to precedence network;

Can be used on a grid;

Less chance of error when calculating backward pass due to all lines emanating from one node point instead of one of the four sides of a rectangular node;

Shows activity as flow lines rather than points in time;

Looks like an arrow diagram, but is in fact more like a precedence diagram;

No risk of individual link lines being merged into a thick black line when printed out and

No possibility of creating the type of logic error often associated with ladders.

The risk of creating an unwanted restraint is greatly reduced.

19.4 Activity on Arrow Networks

In activity on arrow diagrams, activities are represented in arrows. Nodes are considered to be events. The very first event is the “Start” event. Very last event is the “End” event. An activity cannot stat until the event prior to that activity is accomplished. For an event to be completed, all activities coming to that event (node) should be completed.

Fig. 19.3 shows an activity on arrow network. It has three activities. (activity A, B, and C). It also has three events (START event, Event F, Event G, and END event).

Activity D cannot start until event G is accomplished. Event G is accomplished when both activities A and A have been completed.

Practice Problem 19.7

Activity on arrow diagram is shown below. Find the duration of the project.

19.4.3 Resource Leveling

We may develop a fast schedule that completes the project on time. But what about the resources? Does the contractor have resources (equipment and manpower) to do two or three activities at the same time? If the contractor does not have enough equipment and manpower to do two or three simultaneous activities, then resources should be increased. This can be done by renting new machines and hiring new personnel. Renting more machines will be an expensive thing to do. Hiring new people can also be a problem. It is not easy to find people who have suitable expertise.

Hence, the next option is to manipulate the activities.

Construction resources are labor, material, and equipment. In construction scheduling, conflicts can arise when activities compete for common resources that are available in limited quantities. After development of the critical path schedule, a resource utilization chart is developed.

Resources need to be allocated evenly during the lifetime of the project. In Fig. 19.4, when activities B and D are conducted, demand for resources increases. It is important to level the resources during the project duration. In many cases, it is not an easy task. In Fig. 19.4, it is possible to stretch activities B and D and shorten the duration of activities A and F.

Fig. 19.4. Resource utilization chart.

Increase the duration of activities B and D. This can be done by reducing the daily quantity of resources.

Decrease the duration of activities A and F. This can be done by increasing the daily quantity of resources.

It is possible to level resources by manipulating duration of activities. In some cases it is possible to move activities around for the purpose of resource leveling.

Various computer algorithms are developed to level resources without affecting the schedule.

Competition for resources among activities:

Resource leveling procedure: Following procedure is normally adopted.

Construct the critical path schedule.

Develop the resource schedule.

Move around the activities to level the resources (Fig. 19.5).

Fig. 19.5. Resource utilization chart (after resource leveling).

Practice Problem 19.10

Contractor has developed durations and resources required for construction of two slabs.

Resources required for each activity is given below:

Activity 1: Formwork slab in building A: 1 foreman, 8 carpenters, 6 laborers.

Activity 2: Rebar installation of slab in building A: 1 foreman, 5 iron workers, 3 laborers.

Activity 3: Concreting of slab in building A: 1 foreman, 8 concrete masons, 6 laborers.

Activity 4: Formwork of slab in building B: 1 foreman, 8 carpenters, 6 laborers.

Activity 5: Rebar installation of slab in building B: 1 foreman, 5 iron workers, 3 laborers.

Activity 6: Concreting of slab in building B: 1 foreman, 8 concrete masons, 6 laborers.

Activity 7: Opening ceremony.

Activity 1: Duration 7 days

Activity 2: Duration 10 days

Activity 3: Duration 7 days

Activity 4: Duration 12 days

Activity 5: Duration 11 days

Activity 6: Duration 7 days

Activity 7: Duration 1 day

Logic of activities:

Activities 1 and 4 can start at any time.

Predecessor of activity 2 is activity 1.

Predecessor of activity 3 is activity 2.

Predecessor of activity 5 is activity 4.

Predecessor of activity 6 is activity 5.

Predecessors of activity 7 are 3 and 6.

Maximum resources available at a given time:

2 foremen

8 carpenters

10 iron workers

16 concrete masons

12 laborers

What is the project duration?

Solution

There are only eight carpenters available. Hence, activities 1 and 4 cannot go parallel.

There are 10 ironworkers available. Hence, there are enough ironworkers to conduct activities 2 and 5 in parallel.

There are 16 concrete masons available. Hence, activities 3 and 6 can be done parallel.

Hence, new network can be drawn as shown.

There are two paths.

Path 1, 2, 3, 7

Path 1, 4, 5, 6, 7

Duration ofpath2,2,3,7= 7+10+7+1=25daysDurationofpath2,4,5,6,7=7 +12+11+7+1=38days

Projectduration= 38days

14.3 Series Payments (A)

Business loans are paid using series payments. Assume a contractor obtained a loan and agreed to pay the loan on a yearly basis. This could be represented in an arrow diagram. Money coming into the contractor is represented with an upward arrow and money paid is represented with a downward arrow.

P = present worth (shown with an upward arrow to indicate money coming in)

A = series payments (shown with a downward arrow to indicate money going out)

P=A1+in−1/i1+in

Practice Problem 14.3

A contractor obtained a 3 million dollar loan from a bank at an interest rate of 4% and agreed to make yearly payments for 5 years. What is his yearly payment?

Solution

P = 3,000,000, i = 4%, n = 5

P=A1+in−1/i1+in

3,000,000=A1+0.045−1/0.041+0.0453,000,000=A1.045−1/0.041.0453,000,000=A×4.45A= 3,000,000/4.45=674,157

Documents

Documents issued during Phase V include operating and maintenance procedures, the results of a formal acceptance test and warranties from the contractor and subcontractors.

Precedence or activity on node diagrams

Some planners prefer to show the interrelationship of activities by using the node as the activity box and interlinking them by lines. The durations are written in the activity box or node, and are therefore called activity on node (AoN) diagrams. This has the advantage that separate dummy activities are eliminated. In a sense, each connecting line is, of course, a dummy because it is timeless. The network produced in this manner is also called variously a precedence diagram or a circle and link diagram. Precedence diagrams have a number of advantages over arrow diagrams in that

No dummies are necessary.

They may be easier to understand for people familiar with flowsheets.

Activities are identified by one number instead of two so that a new activity can be inserted between two existing activities without changing the identifying node numbers of the existing activities.

Overlapping activities can be shown very easily without the need for the extradummies shown in Fig. 19.25.

Analysis and float calculation (see Chapter 21) is identical to the methods employed for arrow diagrams and if the box is large enough, the earliest and latest start and finishing times can also be written.

A typical precedence network is shown in Fig. 19.27 where the letters in the box represent the description or activity numbers. Durations are shown above-centre, and the earliest and latest starting and finish times are given in the corners of the box, as explained in the key diagram. The top line of the activity box gives the earliest start (ES), duration (D) and earliest finish (EF).

The bottom line gives the latest start and the latest finish. Therefore,

LS=LF−D.

The centre box is used to show the total float.

ES is of course the highest EF of the previous activities leading into it, that is the ES of activity E is 8, taken from the EF of activity B.

LF is the lowest LS of the previous activity working backwards, that is the LF of A is 3, taken from the LS of activity B.

The ES of activity F is 5 because it can start after activity D is 50% complete, that is

ES of activity D is 3.

Duration of activity D is 4.

Therefore, 50% of duration is 2.

Therefore, ES of activity F is 3 + 2 = 5.

Sometimes, it is advantageous to add a percentage line on the bottom of the activity box to show the stage of completion before the next activity can start (Fig. 19.28). Each vertical line represents 10% completion. In addition to showing when the next activity starts, the percentage line can also be used to indicate the percentage completion of the activity as a statement of progress once work has started as shown in Fig. 19.29.

There are four other advantages of the precedence diagram over the arrow diagram.

The risk of making logic errors is virtually eliminated. This is because each activity is separated by a link so that the unintended dependency on another activity is just not possible.

This is made clear by referring to Fig. 19.30, which is the precedence representation of Fig. 19.25.

As can be seen, there is no way for an activity like ‘Level bottom’ in Stage I to affect activity ‘Hand trim’ in Stage III, as is the case in Fig. 19.30.

In a precedence diagram, all the important information of an activity is shown in a neat box.

A close inspection of the precedence diagram (Fig. 19.31) shows that in order to calculate the total float, it is necessary to carry out the forward and backward pass. Once this has been done, the total float of any activity is simply the difference between the latest finishing time (LF) obtained from the backward pass and the earliest finishing time (EF) obtained from the forward pass.

On the other hand, the free float can be calculated from the forward pass alone, because it is simply the difference of the earliest start (ES) of a subsequent activity and the earliest finishing time (EF) of the activity in question.

This is clearly shown in Fig. 19.31.

Despite the earlier-mentioned advantages, which are especially appreciated by people familiar with flow diagrams as used in manufacturing industries, many prefer the arrow diagram because it resembles more closely to a bar chart. Although the arrows are not drawn to scale, they do represent a forward-moving operation and, by thickening up the actual line in approximately the same proportion as the reported progress, a ‘feel’ for the state of the job is immediately apparent.

One major practical disadvantage of precedence diagrams is the size of the box. The box has to be large enough to show the activity title, duration and earliest and latest times so that the space taken up on a sheet of paper reduces the network size. By contrast, an arrow diagram is very economical, as the arrow is a natural line over which a title can be written, and the node need be no larger than a few millimetres in diameter – if the coordinate method is used.

The difference (or similarity) between an arrow diagram and a precedence network can be seen most easily by comparing the two methods in the following example. Fig. 19.32 shows a project programme in AoA format and Fig. 19.33 the same programme as a precedence diagram, or AoN format. The difference in the area of paper required by the two methods is obvious (see also Chapter 33).

Fig. 19.33 shows the precedence version of Fig. 19.32.

In practice, the only information necessary when drafting the original network is the activity title, duration and, of course, the interrelationships of the activities. A precedence diagram can therefore be modified by drawing ellipses just big enough to contain the activity title and duration, leaving the computer (if used) to supply the other information at a later stage. The important thing is to establish an acceptable logic before the end date and the activity floats are computed. For explaining the principles of network diagrams in textbooks (and in examinations), letters are often used as activity titles, but in practice when building up a network, the real descriptions have to be used.

An example of such a diagram is shown in Fig. 19.34. Care must be taken not to cross the nodes with the links and to insert the arrowheads to ensure the correct relationship.

One problem of a precedence diagram is that when large networks are being developed by a project team, the drafting of the boxes takes up a lot of time and paper space, and the insertion of links (or dummy activities) becomes a nightmare, because it is confusing to cross the boxes, which are in fact nodes. Therefore, it is necessary to restrict the links to run horizontally or vertically between the boxes, which can lead to congestion of the lines, making the tracing of links very difficult.

When a large precedence network is drawn by a computer, the problem becomes even greater because the link lines can sometimes be so close together that they will appear as one thick black line. This makes it impossible to determine the beginning or end of a link, thus nullifying the whole purpose of a network, that is to show the interrelationship and dependencies of the activities (see Fig. 19.35).

For small networks with fewer dependencies, precedence diagrams are no problem, but for networks with 200–400 activities per page, it is a different matter. The planner must not feel restricted by the drafting limitations to develop an acceptable logic, and the tendency by some irresponsible software companies to advocate eliminating the manual drafting of a network altogether must be condemned. This manual process is after all the key operation for developing the project network and the distillation of the various ideas and inputs of the team. In other words, it is the thinking part of network analysis. The number crunching can then be left to the computer.

Once the network has been numbered and the times or durations added, it must be analysed. This means that the earliest start and completion dates must be ascertained and the floats or ‘spare times’ calculated. There are three main types of analysis:

Arithmetical

Graphical

Computer

Since these three different methods (although obviously giving the same answers) require very different approaches, a separate chapter is devoted to each technique (see Chapters 21, 22 and 24).

Constraints

By far the most common logical constraint of a network is as given in the examples on the previous pages, that is ‘finish to start’ (F–S) where activity B can only start when activity A is complete. However, it is possible to configure other restraints. These are start to start (S–S), finish to finish (F–F) and start to finish (S–F). Fig. 19.36 shows these less usual constraints, which are sometimes used when a lag occurs between the activities. Analysing a network manually with such restraints can be very confusing, and should there be a lag or delay between any two activities, it is better to show this delay as just another activity. In fact, all these three less usual constraints can be redrawn in the more conventional finish to start (F–S) mode as shown in Fig. 19.37.

Figure 19.36. Dependencies.

Figure 19.37. Alternative configurations.

When an activity can start before the previous one has been completed, that is, when there is an overlap, it is known as lead. When an activity cannot start until part of the previous activity has been completed, it is called a lag.