D1 4.1, D1 4.2 - Activity networks

Precedence and immediate predecessors

A precedence constraint says that an activity must finish before another can start. An immediate predecessor is a directly required preceding activity, excluding earlier activities whose completion is already forced through another listed predecessor. A dash means no predecessor: the activity may start when the project starts.

Worked example 1: direct and inherited requirements

Activities $A,B$ may start immediately. $C$ requires $A$. $D$ requires $A,B$. $E$ requires $C,D$. Write the immediate-predecessor table.

Plan. Enter direct requirements and leave inherited earlier activities out.

Answer. The table above. In particular $E$'s row is $C,D$, not $A,B,C,D$.

Check. Completing $C$ guarantees $A$ has finished; completing $D$ guarantees both $A,B$ have finished. Nothing from the full history has been lost.

Worked example 2: removing redundant entries

Use the revised requirements, keeping $E$ dependent on completion of $C,D$.

Plan. Follow the new $C$-before-$D$ link before deciding which table entries are immediate.

$C$ already requires $A$, so $D$'s immediate predecessors are $B,C$, with $A$ inherited through $C$. Once $D$ finishes, $C$ has finished too, so $E$'s only immediate predecessor is now $D$.

Answer. Rows $A:\text{none}, B:\text{none}, C:A, D:B,C, E:D$.

Check. Both versions still force $A,B,C,D$ before $E$. The revised version additionally forces $C$ before $D$, exactly the one new requirement.

Activity on arc

An activity is a labelled directed arc. An event is a vertex: it occurs only when all incoming activities have finished. Every outgoing activity must wait for that event. The event itself takes no time.

Worked example 3: joining two branches

Represent $A:\text{none}, B:\text{none}, C:A, D:B, E:C,D$ as an activity-on-arc network.

Plan. Build the independent branches and merge only at $E$'s start.

Use arcs

These are the arcs in the diagram. $A$ finishes at event $2$, making $C$ available; $B$ finishes at event $3$, making $D$ available. Event $4$ waits for both $C,D$.

Answer. The five listed arcs, with start $1$ and finish $5$.

Check. $C$ does not wait for $B$, and $D$ does not wait for $A$. Only $E$ has both branches as prerequisites.

Worked example 4: an independent terminal activity

Construct the revised network.

Plan. Keep $E$'s start after $C$, and move $D$'s finish to the project finish.

Use

Answer. Event $4$ now depends only on $C$; finish event $5$ waits for both $D,E$.

Check. $E$ can start while $D$ is still in progress, but the project cannot finish until $D$ has finished. Leaving $D:3\to4$ would incorrectly retain $D$-before-$E$.