#Operations Research_OR

#Network Analysis

#PERT and CPM

#Project Management

#PERT Chart

#Network Diagram

#Scheduling

Project Evaluation and Review Technique (PERT)

Draw Network Diagram and trace the Critical Path. Also Also find Total, Interfering, Free and Independent floats/slacks :

Activity

Name: A B C D E F G

Activity

Node: 1-2 1-3 1-4 2-5 3-5 4-6 5-6

estimated

Duration

(weeks)

Optimistic 1 1 2 1 2 2 3

Most Likely 1 4 2 1 5 5 6

Pessimistic 7 7 8 1 14 8 15

(a) Draw a network diagram of the activities in the project.

(b) Find the expected duration and variance for each activity. What is the expected project length?

(c) Calculate the variance and standard deviation of the project length. What is the probability that the project will be completed –

(i) At least 4 weeks earlier than the expected time?

(ii) No more than 4 weks later than the expected time?

(d) The project due date is 19 weeks, what is the probability of not meeting the date?

The main objective in the analysis through PERT is to find out the completion for a particular event within specified date. The PERT approach takes into account the uncertainties. PERT was developed in the context where many activities associated with the project had never been attempted previously (Research and Development projects). PERT was developed to handle projects where the time duration for each activity is no longer just a single time estimate, but is a random variable that is characterized by some probability distribution – usually Beta Distribution. To estimate the parameters (mean and variance) of the Beta Distribution, the PERT requires 3 time estimates for each activity. From these time estimates a single time is estimated – Expected time. The three time values are associated with each activity

1. Optimistic time – It is the shortest possible time in which the activity can be finished. It assumes that everything goes very well. This is denoted by t0.

2. Most likely time – It is the estimate of the normal time the activity would take. This assumes normal delays. If a graph is plotted in the time of completion and the frequency of completion in that time period, then most likely time will represent the highest frequency of occurrence. This is denoted by tm.

3. Pessimistic time – It represents the longest time the activity could take if everything goes wrong. As in optimistic estimate, this value may be such that only one in hundred or one in twenty will take time longer than this value. This is denoted by tp. In PERT calculation, all values are used to obtain the percent expected value.

Expected time – It is the average time an activity will take if it were to be repeated on large number of times and is based on the assumption that the activity time follows Beta distribution, this is given by

Expected Time = te = {(t0 + tp)/2 + 2tm}/3 = ( t0 + 4 tm + tp ) / 6

The calculations are based on the analogy of the normal distribution. In a normal distribution, 99% of the area under the normal curve is within + 3σ from the mean. That means fall within the Range Appx 6 SD.

Thus, 6σ = tp – t0 (i.e the Range of the possible time duration)

So, SD = σ = (tp – t0)/6

Hence, the variance of the activity is given by σ^2 = [(tp – to) / 6]2

If we assume that the duration of the activities are independent random variables, then

(i) The expected completion time of the project which can be obtained by adding the expected time of each activity lying on the Critical Path; and

(ii) The variance of the total duration of the Critical Path is the sum total of the variances on the Critical Path. σc^2 = Σσ^2 and σc = √Σσ^2

Since we believe the uncertainty in the completion of a project and we expect variation in the activity duration, we can calculate the probability of ‘completing the project in desired/scheduled time’ and we can also calculate other possible or desired situations. This we can do approximating the normal distribution and, thus, due to the central limit theorem.

The Probability of completing the project in desired/scheduled time would be – Probability {Z = (Ts – Te)/ σc}

OR, Operations Management, Math, Network Analysis, PERT, CPM, Project Management, PERT Chart, Network Diagram, Critical Path, Optimistic, Pessimistic, Most likely, Mean, Variance, Probability, Beta Distribution, Normal Distribution, Activity, Event, Merge Event, Burst Event, Predecessor, Successor, Statistics, OM, Operations Management, MBA, MCA, CA, CS, CWA, BBA BCA, BCom, MCom, GRE, GMAT, Grade 11, Grade 12, Class 11, Class 12, IAS, CAIIB, FIII, IBPS, BANK PO, UPSC, CPA, CMA

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