7.02 Project networks

When completing tasks and projects there are often activities (sometimes called events/items) that need completing before others can happen. For example, the task of making a pot of tea. We need to fill the jug, boil the water, put tea in the teapot, add the water to the teapot and let the tea infuse. Then we pour the tea into the teacup, and add milk and sugar before drinking.

There are a number of items here that can only be completed after some of the others have been done. A simple example is that we cannot boil the water until after we have filled the kettle. The tasks that must be completed earlier are called predecessors, and the immediate predecessors are also called dependencies.

We can create a directed network graph for the "making tea" situation. We call this a project network.

We can also represent the same information in a table called an activity table.

This network shows us that you can add the milk and sugar in either order, but that you shouldn't stir and drink your tea until you have added both. It also shows us that both filling the kettle and putting the tea in the teapot have no predecessors.

We can also add timings to each of the activities, making the graph weighted.

These weightings can be included in the activity table.

Examples

Example 1

Given the following network, fill in the activity chart.

List the duration and direct dependencies for each activity in the table below.

Activity	Duration	Dependencies
A
B
C
D
E
F
G
H

If an activity has no dependencies, write X. If there are multiple dependencies, write them in the same box separated by commas.

Worked Solution

Create a strategy

Use the weights of the edges for the durations, and the previous connected edge for the dependencies.

Apply the idea

We will move from left to right along the network.

Activities A, B, and C have no edges to the left of them that they are connected to. So these activities do not have dependencies. Their durations are their weights next to the letters.

Activity	Duration	Dependencies
A	5	X
B	8	X
C	6	X
D	5	A
E	6	B
F	3	D, E
G	6	C
H	1	F, G

Activity D is connected to activity A on its left, so it's dependency is A.

Activity E is connected to activity B on its left, so it's dependency is B.

Activity F is connected to activities D and E on its left, so it's dependencies are D and E.

Activity G is connected to activity C on its left, so it's dependency is C.

Activity H is connected to activities F and G on its left, so it's dependencies are F and G.

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Example 2

Draw a network representation for the activity table below.

Activity	Duration	Dependencies
A	1	-
B	5	A
C	3	A
D	2	A
E	4	C
F	6	C
G	8	B, E
H	4	B, E
I	9	F, D
J	5	H, I

Worked Solution

Create a strategy

Start the network by drawing the vertex with no dependencies, then use the dependencies column to see how the activities are connected.

Apply the idea

The only activity without any dependencies is A so our first vertex on the left end of the network should have one arrow coming out of it for activity A with duration 1.

Activities B,C, and D all depend on A, so the vertex at the end of activity A should have 3 arrows coming out of it with activities B-5,C-3, and D-2 on them.

Activities E and F depend on C, so the vertex at the end of activity C should have 2 arrows coming out of it with activities E-4 and F-6 on them.

Activities G and H depend on both B and E, so the arrows with those activities need to meet at a vertex which should then have 2 arrows coming out of it with activities G-8 and H-4 on them.

Activity I depends on both F and D, so the arrows with those activities need to meet at a vertex which should then have 1 arrow coming out of it with activity I-9 on it.

Activity J depends on both H and I, so the arrows with those activities need to meet at a vertex which should then have 1 arrow coming out of it with activity J-5 on it.

The graph should end at one vertex, which is connected to any activities that are not dependencies of any other activity. By looking at the table we see that activities G and J do not appear in the dependencies column. So these activities should join at the final vertex. The final network is below:

Consider the following activity table:

Tasks C and D are both dependent on the activities A and B. To represent this in a network diagram the tasks A and B terminate at the same vertex and then tasks C and D start from this vertex.

The activity table below is very similar but this time only task C requires both A and B, whereas task D has only A as its dependent activity.

To represent this situation correctly we need to introduce a dummy activity. This is an activity with zero weight often represented by a dashed line and used for the purpose of correctly displaying dependencies.

On the network diagram below a dummy activity has been added from the completion of task A to the completion of task B. Note the direction of the arrow, this now correctly indicates task D only requires activity A, whereas task C requires the completion of both activities A and B.

In a further example below, notice that tasks D and E share the dependent activity B but do not share other dependent activities.

To represent this situation we add two dummy activities emanating from the completion of task B. Dummy activities can be labelled with the weight zero and task(s) they represent the completion of as shown below.

In general, dummy activities are required to represent a situation where two or more tasks share some but not all of a set of dependent activities.

Examples

Example 3

Task	Time	Dependencies
A	5	-
B	5	-
C	4	B
D	8	A
E	3	D,G
F	1	E
G	2	C
H	3	D
J	4	H
K	3	F,J
L	4	E

This precedence table has been provided to control the movement of stock within a store during a refit. The junior manager has been given the task of planning the operation so that the total time taken by the project is kept to a minimum and that staff are used most efficiently. All time is given in hours.

The network shown below requires a dummy task to be added to correctly represent the network.

Select which two nodes should be joined with a dummy task. The order indicates the direction of the arrow.

A

7 to 4

B

4 to 5

C

4 to 7

D

5 to 4

Worked Solution

Create a strategy

Consider which activities have a common dependency in the table.

Apply the idea

Activities E and H are both dependent on activity D. In the network, activity H is shown to be dependent on D but activity E is not connected to D.

So we need to dummy activity that connects activity D to activity E by connecting node 4 to 7.

The answer is option C.