# Project Scheduling

The project scheduling models are used to find the (expected) project completion time for either a PERT (activity on arc, AOA) network or a CPM (activity on node, AON) network. For both networks either one or three time estimate problems can be represented.

There are four models which are common. These models can be changed without starting anew by using the method box.

## Single time estimate and triple time estimate PERT

Consider a project given by the precedence diagram below and the table of times beneath the graph.
 Task Start Node End Node Optimistic Most Likely Pessimistic A B C D E 1 2 3 3 4 2 3 4 6 6 2 4 10 3 4 12 5 23 5 7 25 6 28 7 9

Data

The screen below contains a triple time estimate PERT data screen for our example. In both PERT representations the network is defined by giving the starting node and ending node for each task. The network type is given by the box on the left above the data. In this first example we are using the start node/end node representation for activities.

The data consists of

Activity name. Activities can be named. In PERT the name is not used while in CPM (Precedence list) the names are critical. In PERT, the 'real' name of an activity is given by its starting and ending node labels.

Starting node The number of the node at which the activity starts is to be given here. We remind you that the numbers in a PERT diagram serve as labels rather than numerical values. The labeling is arbitrary. That is, the first node can be 1 or 2 or 90.

Ending node. The number of the node at which the activity ends is to be given here. The number can not be the same as the starting node number. Also, two activities can not have the same pair of starting and ending node numbers. Also, transitivity must be obeyed. That is, an error will occur if laws of transitivity are violated such as by the pairs (1,2; 2,3; 3,1).

Time estimate. In the single time problem it is necessary to give only one time estimate for each task.

Optimistic time. This appears only in the 3 time estimate option from the submenu.

Most likely time. This time must be entered in either the one or three time estimate version.

Pessimistic time. This appears only in the three time estimate version.

### The Solution

The exact solution screen depends on whether it is a one estimate or three estimate problem. In the screen below we present a sample three estimate problem since this contains all of the output information. The one time estimate problem has less information.

Time. If the three time estimate version is used then a single time estimate is computed and printed for each activity. The formula used is the traditional formula of (a + 4b + c)/6 where a is the optimistic time, b is the most likely time and c is the pessimistic time. For example, in the screen the time used for activity 1-2 is

(2 + 4*12 + 25)/6 = 75/6 = 12.5.

Early start(ES). For each activity its early start is computed. For example, the early start for activity 3-6 is 17.5. The column named time is used for this computation.

Early Finish(EF). For each activity its early finish is computed. In the example, the early finish for activity 3-6 is 22.5. The early finish is, of course, the early start plus the activity time. For example, the early finish of 3-6 is its early start of 17.5 plus the 5 from the time column.

Late start(LS). For each activity its late start is computed. In the example the late start for activity 3-6 is 36.

Late finish(LF). For each activity its late finish is computed.

Slack. For each activity its slack (late start- early start or late finish - early finish) is computed. In the example the slack for activity 3-6 is

41-22.5 = 18.5 or 36-17.5 = 18.5.

Standard deviation. For the three time estimate model the standard deviation of each activity is listed. The standard deviation is given by pessimistic-optimistic divided by 6. In the example, the standard deviation of 3-6 is

(7-3)/6=.67.

Project completion time. The (expected) time at which the project should be completed is given. In the example this time is 41.

Project standard deviation. If the three time estimate module is chosen the project standard deviation is printed. It is computed as the square root of the project variance which is computed as the sum of the variances of all critical activities.

Note that in general there are problems in defining the variance of project completion time. In addition, some books vary in the manner of computation. This program will overestimate the standard deviation if there is more than one critical path. Your text likely does not explain what to do when more than one critical path exists.

There is available a table that displays the computations of the tasks' times, standard deviations and variances as illustrated below.

It is possible to display Gantt Charts for the project as shown below.

## CPM

The critical path module has the data input in a fashion nearly identical to the assembly line balancing module. The initial data screen appears as given below.
 Task Time Precedences design program document test advertise 25 30 22 10 30 design design program design

Task names. Tasks can be named up to 8 characters long. The usual naming conventions are true. That is, upper and lower case do not matter but spaces within a name do.

Predecessors. The predecessors are listed here. Enter one predecessor per spreadsheet cell with up to seven predecessors per activity. It is sufficient to enter only the immediate predecessors.

The data is displayed below.

Rather than displaying the solution to this problem we show the precedence graph that the software can display.

### Crashing

Given below is an example of project management with crashing. The four columns of data are the standard columns for this type of problem - the normal time and normal cost for each activity as well as the crash time and crash cost for each activity. The crash time must be less than or equal to the normal time and the crash cost must be greater than or equal to the crash cost.

The results are as follows. The software finds the normal time of 16 days and the minimum time of 12 days. For each activity the computer finds the cost of crashing per period (Crash cost-Normal Cost)/(Normal time-Crash time), which activities should be crashed and by how much and the prorated cost of crashing.

## Budgeting

The software has a model for determining the amount of money that will be spent over a project's lifetime. The data is the activity cost as shown below. An early start budget and a late start budget can be computed.

The regular solution screen is given but two others are also available. Below we show a part of the early start budget. In addition the graph contains the early start and late start budget for the entire project.

## Normal Distribution

Project management is an area where the normal distribution calculator is useful.

The mean and standard deviation from the project are automatically filled in. We have several options in terms of what we may want to compute. For example, we can compute the probability of finishing within 50 days, or a 95% confidence interval for finishing the project or alternatively we can compute how many days to allow to be 90% sure of finishing within that time. We have chosen a 95% confidence interval and the solution is given below.

We are 95% confident that the project will be completed in 31 to 51 days.