Project scheduling, also known as project time management,

deals with the issue of when project activities will be performed by project resources.

The project management Institute in the project management body of knowledge defines

project time management as

the processes required to manage the time to completion of the project,

including the timely completion of

project scope and the timely completion of product scope.

In other words, the project is completed when

all its activities are completed and all its deliverables are delivered and accepted.

Information needed for basic project scheduling

includes at least a list of project activities,

the estimated duration of each activity,

and the precedence relations among the activities.

Advance scheduling takes into account additional information such as uncertainty,

availability of resources, cash flow considerations and availability of funds.

In basic scheduling, we assume that all precedence relations are finished to start.

In other words, an activity can start as soon as all its predecessors are finished.

In reality, other precedence relations exist as well.

And activities may overlap and be performed in parallel or partially in parallel.

In basic scheduling, the assumption is that

all the resources required to perform the activities are

available instantaneously as well as the funds needed to pay for these resources.

Network models are commonly used for scheduling.

A network is composed of nodes in arcs.

In the Activity On Nodes or AON presentation,

each node corresponds to an activity and each arc represents precedence relations.

Another network model, is the Activities On Arc or AOA presentation,

where each node corresponds to an arc in each node

represents an event such as the completion of an activity.

A network based model known as

the Critical Path Method CPM was developed at the late 50s.

The first step in applying the Critical Path Method is a forward pass.

In which the early start and the early finish time for each activity are calculated.

Each activity is scheduled to start as early as possible.

Thus activities without predecessors start as the project starts.

Say at time zero.

While every other activities starts when all its predecessors finish.

So its start time is the latest among the early finish time of its predecessors.

The only finish of the activity is its early start plus its duration.

The second step is a backward pass,

where each activity starts as late as possible without

delaying the project finished time thus the late finish

of activities they do not have successors is

the early finish time of the project or the project due date if a due date is defined.

The late finish time of all other activities is the

earliest among the late start time of its successors.

The late start time of each activity is its late finish time minus its duration.

The slack of each activity is calculated as

the difference between its late start time and its early start time,

or the difference between its late finish time and its early finish time.

The slack of each activity tells us by

how much time the activity can be delayed without delaying the project.

Activities that have a slack equal to zero cannot be delayed.

These activities form the longest sequence of

activities connecting the start of the project to its end.

This longest sequence of activities connecting the start of

the project to its end is known as the critical path,

and the activities in the critical parts are known as critical activities.

Critical Path Method or CPM,

is designed to find a critical part of

a project and the slack of non-critical activities.

A popular scheduling model is a Gantt chart developed by Henry Gantt about 100 years ago.

The Gantt Chart is a simple graphical scheduling model.

The analysis is identical to the forward pass of the critical path method.

The horizontal axis or the x axis,

represent time and the activities are represented by horizontal bars.

The start time and end time of each activity are

the start time and end time of the corresponding bar.

And the length of each bar corresponds to the duration of the activity.

The simplicity of the Gantt chart makes its very popular.

But like the Critical Path Method,

it has several drawbacks.

It does not take resource availability into account.

It does not take cash considerations into account.

It ignores uncertainty inactivity duration and in resource availability.

It assumes that each activities start as early as possible.

The duration of each activity is an important input to both CPM and the Gantt analyses.

This duration is estimated based on

past experience in the form of real data or some expert opinion.

Like any estimate, this estimate is subject to estimation airhole due to uncertainty.

As a result of these uncertainties,

the actual start time of activities as well as the endtime and

the project completion date may differ from the results of CPM and Gantt analysis,

creating the risk of project delays.

To manage these risks,

special techniques are used.

A popular tool to risk analysis is the Monte Carlo simulation.

Stochastic activity duration is represented by a distribution.

The activity duration is not a symmetrical distribution,

as an activity may take

a very long time but it cannot take very short time close to zero.

And no activity duration can take negative values.

Thus the distribution of activity duration is skewed.

A common way to present

the activity duration distribution is known as the three point estimate.

The three points are the optimistic,

most likely and pessimistic duration.

The optimistic duration, is

the shortest duration in which it is possible to perform the activity.

The most likely duration,

represents the duration most commonly seen if the activitiy is performed multiple times.

The pessimistic duration, is

the longest duration that it might take to perform the activity.

And therefore to estimate the risk of delays,

led to the development of statistical tools for scheduling.

A popular tool is based on Monte Carlo simulation.

The Monte Carlo simulation is a statistical tool that simulates

project instances or realisations by randomly generating

the duration of activities based on

the three point estimate and

the assumption that activity duration follow a specific distribution,

say a triangular distribution.

Each project realisation is presented as a set of activities,

the precedence relations among the activities and the duration of

each activity that was randomly

generated from the assumed distribution of the activity duration.

Each project realisation is analyzed by

the critical path method to find the critical path and the critical activities.

The result of the Monte Carlo simulation is the distribution of

project duration and the frequencies that each activity was on the critical path.

This is known as the criticality index of activities.

The criticality index of activities,

is a good indication of the risk associated with each activity.

If the criticality index is high,

it means that the activity was frequently on

the critical path in the Monte Carlo simulation.

If a thousand simulation runs are performed,

it means that 1000 different realisations of the project were

generated and the duration of each activity was randomly generated in each realization.

If an activity was on the critical path in 270 realisations out of 1000,

then its criticality index is 27 percent.

It is convenient to present the results of Monte Carlo simulation as a distribution.

Each bar represents a probability of the project finishes in that period of time.

The cumulative distribution showed the probabilities of

the project duration will be equal to or less than the specified duration.

The scheduling tools discussed in this this lecture, the Gantt Chart,

the Critical Path Method and the Monte Carlo simulation,

simplify the real project scheduling problem by ignoring resource and funds.

In reality, resource constraints as well as

funding or cash flow constraints are present in most projects.

In the following lecturers,

we will introduce resources and budgeting considerations to

the project scheduling problem.