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CRITICAL PATH METHODS
With this lecture we move into a different area of Operational Research that we have already touched in previous lectures: graphs and networks. We will start with the oldest algorithm: CPM, the Critical Path Method. This is used to schedule projects, particularly construction projects. In the lectures we will use some simple examples in order to learn the rules, as they are implemented in computer programs. The material will closely follow Taha, chapter 6, which you can find in the library of the Sciences Faculty. In the book you will find many examples on which you can practice.
In CPM we have a set of activities consuming time and resources and we need to schedule the activities. We follow several steps:
- Define activities, precedence relationships, and time requirements.
- Translate the project into a project showing precedence relationships.
- Solve the network to create a time schedule for the complete project.
Example: publishing a book
Code Activity Predecessor Duration
A Manuscript proof reading by editor - 3
B Sample pages prepared by typesetter - 2
C Book cover design - 4
D Preparation of artwork for book figures - 3
E Author’s approval of edited manuscript and sample pages
A,B 4
F Book typesetting E 2
G Authors checks typeset pages F 2
H Author checks artwork D 1
I Production of printing plates G,H 2
J Book production and binding C,I 4
The network is represented by nodes and arcs. We can represent activities by means of arcs, or by means of nodes. Since we are following Taha we will do what he does, and represent activities by means of arcs (arrows) starting in a node and finishing in another node.
The network can also be represented by means of a matrix, where the rows indicate the origin of the arcs and the columns the end points of the arcs. This allows us to use symbolic mathematical methods in the study of the network. The matrix associated with the above network is:
A B C D E F G H I J
A 0 0 0 0 1 0 0 0 0 0
B 0 0 0 0 1 0 0 0 0 0
C 0 0 0 0 0 0 0 0 0 1
D 0 0 0 0 0 0 0 1 0 0
E 0 0 0 0 0 1 0 0 0 0
F 0 0 0 0 0 0 1 0 0 0
G 0 0 0 0 0 0 0 0 1 0
H 0 0 0 0 0 0 0 0 1 0
I 0 0 0 0 0 0 0 0 1 0
J 0 0 0 0 0 0 0 0 0 0
We impose ourselves some rules in order to draw the network:
Rule 1. Each activity is represented by means of an oriented arc (arrow) in the network. Rule 2. Each activity must have two distinct end nodes. Rule 3. Link activities taking into account precedence relationships.
To apply the rules we may have to create dummy activities (zero duration) and additional nodes
This representation satisfies rule 1, because each activity is represented by an arrow, but breaks rule 2 because the two activities end up in the same node. Two activities can start in the same node, or can finish in the same node, but they cannot start and end in the same node. To solve this situation we create a new node and a dummy activity of zero duration. Dummy activities are represented by means of dotted lines.
If we need to have a situation such as the one below
We can also solve it by means of a dummy activity.
To identify the critical path we first calculate the E (^) i for each event. This is done in the
“forward pass”.
Having calculated the values of Ei , we calculate the values of the Fj in the “backward pass”.
We will first think in terms of the forward pass. Consider the situation below where several activities need to be completed before the activity that starts in even j can take place.
The earliest moment when we arrive at node p is Ep, the earliest time when we arrive at node q is Eq, and the earliest time when we arrive at node v is E (^) v. From p to j it takes Dpj units of time, so the earliest time we can arrive at node j from this route is E (^) p+D (^) pj. From q to j it takes D (^) qj units of time, so the earliest time we can arrive at node j from node q is Eq+Dqj. The earliest time we can arrive at node j from v is Ev, and it takes D (^) vj
to reach j, so the earliest time we can reach node j from this route is Ev+D (^) vj. To start the activity commencing at j activities (p,j), (q,j), and (v,j) must have been completed, so we need to wait until the slowest activity has been completed. Therefore,
Ej = max (Dpj+E (^) p, Dqj+E (^) q,…., Dvj+Ev)
These are simple calculations. We will perform them in the network associated with the music concert. In the figure I have replaced the names of the activities with their duration, and the name of the node with its earliest starting time.
Having calculated the earliest times at which an activity can start, we now concentrate on latest times it can start. We now start at the end. In the above network, we start at time 130.
The latest time at which p can start is L (^) p, if it starts later, the project will be delayed.
The same is true of q, with a latest starting time of L (^) q, and of V with a latest starting time of Lv. L (^) j must, therefore, be the smallest between L (^) p-D (^) jp, Lq-D (^) jq, and L (^) v-D (^) jv. In order not to delay any later activity. Hence we write:
Lj = min (L (^) p-D (^) jp, Lq-D (^) jq,...... Lv-D (^) jv)
Applying this rule to the above network (event codes have been replaced by latest starting times) The The critical path is formed by the sequence of events whose earliest starting time and latest finishing time coincide. This is a peculiar network in which there are quite a lot of critical activities.
We have two final definitions,
Total float = Latest finishing time j –Earliest starting time i – Duration (i,j)
Total float indicates the delay acceptable in non-critical activities without delaying the whole project.
Free float = Earliest Starting Time i – Earliest Starting time j – Duration (i,j)
The free float is the delay that a non-critical activity can accept without delaying other non-critical activities.
We can also produce a time schedule for the project, and a Gant diagram, but this is too complicated for me to produce using the word processor. I will explain in the class.
Cecilio Mar Molinero
