CRITICAL PATH METHOD & PERT

1. Meaning of Critical Path

A network consists of chains of activities called paths of network. Addition of durations of activities on any path gives the duration of that path. the comparison of durations of the paths identifies a path whose duration is the longest. It is this path, the path with the longest duration which sets the overall duration of the project is called critical path. And the activities on the critical path are known as critical activities.

2. Characteristic Of Critical Path

Identifying critical path is of great importance as it determines duration of the project. if any activity on this path is delayed, then the entire project will be delayed

Some important characteristics of critical path are:.

1. Every network has a critical path 2. It is possible to have more than one critical paths3. A critical path is one of the connecting links between the first & the last event.4. A critical path may have lesser number of activities compared to non-critical paths. 5. A critical path may run through a dummy activity.

3. Benefits Of Critical Path

Major benefits of identifying the critical path are

1. Critical path helps to identify a set of activities and events which are critical and as such must be carefully monitored and controlled.

2. Mere allocation of additional resources does not help to reduce duration of the project. To shorten the time of a project, some of activities on the critical path must be shortened.

3. Certain resources (men, machines and money) are generally common to different activities. Critical path identifies the activities to be given preference in allocation of resources.

4. Each and every activity of the project need not be controlled. If critical activities are started and completed on time, the project automatically gets completed on schedule. Since critical activities are few in number identification of critical path helps to exercise control by exception.

4. How To Identify Critical Path ?

Critical path in a small network can be identified by performing the following four steps:

Enumerate all the paths in the network List down the activities on each of the above paths Sum up the times of the activities along each path. Compare the duration of the paths to identify a path (s) whose duration is the longest. It is this

path which is called critical path. Showing critical path in the network

Critical path is shown in the network by either red line or by double line or by thick line.

5. Non Critical Paths And Float Times

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All activity chains which do not lie on the critical path are called non-critical paths and the activities along these paths are called non-critical activities. Logically it would not affect the project completion time if such activities take little longer time than planned.

Although non-critical activities can safely be allowed to take longer than planned, yet it is important that the extended time should not result in total chain time of activities to exceed lime along the parallel critical path. It is, therefore, necessary to know the spare time available along a non-critical path. The spare time which is called float or slack can be obtained by subtracting the “non-critical path time from the “parallel critical path / time”.

6. Critical Path In A Big Network

In a small network, it is a simple process to identify the critical path by tracing and comparing all the paths in the network. As the number of activities increases, it becomes very difficult and lime consuming to find the critical path by complete enumeration or inspection. Therefore, for larger networks, a more systematic procedure is needed to determine the critical path.

The most commonly used method employs two sets of calculations: forward pass computation and backward pass computation.

The forward pass computation begins from start event moves towards the end event of the project network. It determines the earliest expected time for each event, called earliest starting (event) time (TE).

The backward pass computation begins from the end event and move backward to the start event of the pro network. It determines the latest all time for each event, called latest Completion (event ) time (TL). Earliest event time (TE) represents earliest possible occurrence time of the activity emanating from event. Latest event time represents latest allowable occurrence time of activities terminating into the event.

7. Analysis Of Activity Durations Based On Computations

Mere computation of event time is not sufficient. Equally important is the task of establishing the date at which each activity should start and end to maintain

1. Earliest Start time (ES). The network logic indicates that an activity can not commence until its preceding event is completed. This implies that the earliest start time of an activity equals earliest event time (TE)of the tail event.

2. Earliest finish time (EF) equals the earliest start time of the activity plus duration of the activity emanating from the tail event.

3. Latest finish time (LF) is the latest time of the head event. 4. Latest start time (LS) is the latest finish time of the activity minus duration of the activity converging on the head event.

Identifying Critical Path Based On Computations

Once activity durations have been worked out, the critical path can be identified by comparing the “earliest finish times” and “latest finish times’ of the activities. Clearly activities whose earliest finish times and latest finish times are equal will constitute the critical path.

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8. Activity Cost Slope

Activity Cost Slope is the rate of increase in the cost activity per unit with a decrease in time. A necessary measure for the analysis is the calculation of the cost slope for each activity. The cost slope indicates the additional cost incurred per unit of time saved in reducing duration of art activity. It can be understood more clearly by considering following diagram:

D

Cost Slope

C

O B A

Let OA represent the normal time duration for completing a job and OC the normal cost involved to complete the job. Assume that the management wishes to reduce time of completing the job to OB from normal time OA. Therefore, under such a situation the cost of the project increases and it goes up to say OD (Crash Cost). This only amounts to saying that by reducing the time period by AB the cost has increased by the an CD. The rate of increase in the cost of activity per unit with a decrease in time is known as activity cost slope and is described as below:

Since the project duration is determined by the length of the critical path, it can be shortened by reducing the time of critical activities. As the objective of the management is to reduce the project duration at the lowest possible cost, the critical activity with the lowest cost slope is selected for crashing.

The amount of time by which an individual activity can be reduced is limited by its crash time. However, during such a crashing other factors must be taken into account. For example, the next longest route may also become critical. There will be then two critical paths and any further reduction in project time must occur on both the paths for overall project time to be reduced. The process is continued in this manner and the sum of direct and indirect costs i.e. the total for each completion time are tabulated. The optimum schedule corresponds to minimum total cost.

9. Activity Float Analysis

Float of an activity represents the excess of available time over its duration. Float is an important concept in project planning. It allows planners to

1. Decide priorities in allocation of resources. 2. Transfer resources from less pressing areas to more pressing areas. 3. Minimize requirements of a resource. 4. Prevent peaks and valleys in requirements of a resource.

10. Types of floats

Float is mainly of Three types: Total float, Free float & Independent float. All activities lying on the non-critical paths have total float and some of them may also free floats. Total float and free float have following significance.

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1. Total Float

It signifies the maximum delay that can be permitted in the completion of the activity without affecting the project completion Total float can be interpreted in two ways: (I) The maximum time available to delay the commencement of an activity. (ii) The maximum expansion of the duration of the activity. (iii)Some combination of decisions (i) and (ii)

If total float is used up an activity, that activity and subsequent activities in the chain become critical.

Total float may be viewed as the maximum leeway available to an activity when all preceding activities occur at the earliest start time and all succeeding activities occur at the latest start time. Total float of an activity thus can be obtained as under:

Total float = Latest occurrence time of the succeeding event minus Earliest occurrence time of the preceding event minus Duration of the activity.

2. Free Float

Free float is the amount of time an activity can be delayed without affecting the commencement of a succeeding activity at its earliest start time but may affect the float of previous activity. Free float results when all preceding activities occur at the earliest event times and all succeeding activities also occur at the earliest event times.

Free float = Earliest occurrence time of the succeeding event minus Earliest occurrence time of the preceding event minus Duration of the activity

3. Independent Float

Sometimes, it may be desirable to know what spare time is present in an activity if it is started as late as possible and finished as early as possible. This characteristic is known as independent float.

Independent float therefore, is the amount of time an activity can be delayed when all preceding activities are completed as late as possible and all succeeding activities are completed as early as possible. Independent float thus neither affects the float of preceding activities nor that of succeeding activities. Independent float = Earliest occurrence time of the succeeding event minus Latest occurrence time of the preceding event minus Duration of the activity

11. Node Labeling

The Fulkerson’s Rule

The Fulkerson’s rule consists of following four steps: Once the network is drawn, it is a good practice to label the events systematically. As a general rule, the numbering system must ensure that a) the event numbers as far as possible reflect logical relationships of the activities. b) for a given activity, the number of its tail event is higher than that of its head event.

A very simple and logical approach suggested by D.R. Fulkerson called Fulkerson’s rule is the commonly accepted technique for numbering.

1. Identify an initial event and number it as “1’’. (An initial event is one which has arrows originating from it and none entering it).

2. Delete the arrows emerging from event “1’’ so as to create one or more initial events.

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3. Number these new initial events as 2, 3 4. Repeat steps (ii) and (iii) until last event is obtained which has no arrows emerging out of it,

12. PERT Concept Of Multiple Times

One major difference between PERT and CPM is the formers adaptability for the projects where high degree of uncertainty prevail and activity times during performance are expected to vary considerably for certain activities. The concept of multiple times (three time estimates) was evolved in PERT to reduce the incidence of uncertainty in project planning. The three time estimates are

a) Optimistic time (a)

This indicates the minimum time that an activity can take if everything goes smoothly The chances that such a time could i be shorter would be one in hundred or less. The optimistic time is represented by ‘a’.

b) Pessimistic time (b)

It indicates the maximum time an activity can take under adverse conditions. The chances that time could even be longer would be one in hundred or less. The pessimistic time is denoted by ‘b’.

c) Most likely time (m)

This indicates the time an activity can take most often if it is repeated again & again under the same conditions Most likely is denoted by “m’.

Three time estimates are not directly entered into the network. They are transformed into an expected time using the statistical relation given as Te = a + 4m + b / 6

13. Differences Between CPM & PERT

1. PERT is event oriented while CPM is activity oriented2. PERT provides for an allowance for uncertainty while CPM does not(PERT makes three time

estimates for each activity, while CPM makes only one time estimate)3. Activity times in CPM technique are related to costs while it is not so in PERT since it is event

oriented4. PERT is applied generally to the projects of non-repetitive nature whereas the CPM is applied

where the projects are of repetitive time where time estimates can be made on the basis of experience.

14. Crashing

Crashing is employed when we want to shorten the project completion time by spending extra resources i.e., ultimately money. In real life, it is always possible to employ more resources or book a lightening call. Therefore we have to estimate the crashing limit for each activity as also extra money for crashing each activity. Once these estimates are available, any analysis can be made to determine the time cost trade off curve i.e., what (cost) it takes to crash the project to a given duration.

In many situations, there may be compelling reasons to complete the project earlier than the originally estimated duration of the critical path on the basis of normal activity times or in many cases, the execution of project gets delayed due to certain reasons. We have to reduce the duration of future activities so that the project is completed earlier/or as per schedule.

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Under these circumstances, additional resources can be used to expedite certain activities resulting in earlier completion of the project. This shortening of activity times, which usually can be achieved by adding resources such as manpower of overtime is referred to as crashing the activity time However, Since the additional resource associated with crashing activity times usually result in added costs, the management would want to identify the least cost activities to crash and the amount of duration by which activities may be crashed to meet the desired project completion date.

In order to determine just where and how much to crash activity times, management would need following information:

15. Objectives Of Project Crashing

1. To complete the project in the least possible time2. To effect cost of economy (to reduce the project cost below its normal cost).3. To expedite one or more unfinished activities when earlier critical activities have taken more than

estimated time and thereby prevent lateness penalties and cost overruns.4. To reduce idle time of the facilities in the non-critical paths and achieve uniformity in requirement

of resources5. To release the facilities more quickly for transfer to other profitable projects6. To enhance reputation of the firm and thereby improve its competitive position

A major advantage of CPM is its ability to evolve a relationship between time and cost of the activity and it is this concept which is exploited in project crashing.

16. UP-dating of the Project:

A network designed during planning may not adhere to the schedule when put to work because situations sometimes keep on changing & during execution they are different than those assumed at the stage of planning.

In spite of best efforts, some activities need more time than originally planned or some new activities crop-up. To ensure that schedule time is maintained, it is necessary to review the progress of the project & redrafting the network according to latest requirements. If it is not possible to delay the project, then activities on new critical path are accelerated by allocating extra resources in order to adhere to committed date. Hence the process of up to date the network diagram of the project by incorporating in it the changes which has occurred due to re-planning & rescheduling is called updating. There is no rule about the time to go for updating. The frequency of updating may be more when the project duration is small because few slippages in detecting the progress will affect the project as a whole, as the time for absorbing such slippages is less. However to add dynamism to the nature & progress of work, updating may be carried out as frequently as possible & viable.

CONCEPTS

1. Activity. It is a time consuming job or task that is a key sub-part of the total project. It is a clear definable portion of a project that requires for its completion, the consumption of resources, and time in particular.

2. Activity Cost Slope (AC / AT) is the additional cost to be incurred to reduce the duration of an activity by one unit of time. ( Activity cost slope of an activity = Crash cost - Normal cost / Normal time - Crash time )

3. Activity Time. Physical time required to complete an activity.

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4. Activity Duration. In CPM, the best estimate of the time to complete an activity. In PERT, the expected time or average time to complete an activity.

5. Activity List. A list of the jobs in a project with their immediate predecessors, expected times and resources required.

6. Backward Pass. A procedure that moves from the end of the network to the beginning of the network. It is used in determining the latest finish and start times.

7. Crashing. It is the process of reducing the total time that it takes to complete a project by expending additional funds. A term in the CPM method describing the process of reducing the time required to complete an activity.

8. Crash Time : In CPM, the minimum possible time for completion of an activity, corresponding to maximal resource concentration.

9. Crash cost of an activity represents the lowest possible direct cost required to complete an activity within its crashed time.

10. Crash duration of the project refers to the irreducible minimum duration of the project and it occurs when all activities on the critical path have been crashed. Crash time of the project, therefore, is sum of crashed times of the activities on the critical path

11. Critical Activity. An activity becomes critical if delay in its estimated time duration delays the whole project to that extent.

12. Critical Path. The series of activities that have a zero slack. It is the longest time path through the network. A delay for an activity that is on the critical path will delay the completion of the entire project.

13. Critical Path Analysis. An analysis that determines! the total project completion time, the critical path for project, and the slack, ES, EF, LS, and LF for every activity.

14. CPM. An acronym for Critical Path Method, a method for scheduling and controlling projects

15. Dummy activity. In most projects many activities can be performed concurrently or simultaneously. It is possible that two activities could be drawn by the same beginning and end events. In situations where two or more activities can be performed concurrently, the concept of dummy activity is introduced to resolve this problem. Therefore there will be only one activity between two events.

As a result of using the dummy activity, other activities can be identified by unique end events. Dummy activities consume no time or resources. By convention, dummy activities are represented by a dashed arrows on the project network and are Inserted in the network to clarify activity pattern in the following situations

To make activities with common starting and finishing events distinguishable, and to identify and maintain the proper precedence relationship between activities that are not connected by events.

When two (or more) activities run exactly in parallel such that they would both start at the same node (event) and finish at the same node, a dummy would be inserted between the end of one of the activities and the common finishing node. This is to ensure that each activity has a unique description when referred to by its start and finish node numbeRs.

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In some situations, a dummy activity is necessary to fully and accurately represent the logic of a series of activities. For example, if activity C follows activity A and activity D follows both activities A and B, this cannot be represented without the aid of a dummy.

Dummies are often used to improve the layout of a network when they may not be strictly necessary to represent the logic involved. This often happens at the start or finish of a network where a number of activities either start from a certain point or coverage to a particular point. In this way, the need for curved’ activities is avoided.

16. Earliest Activity Finish Time (EF) : The earliest time that an activity can be finished without violation of precedence requirements.

17. Earliest Activity Start Time (ES). The earliest time that an activity can be started without violation of precedence requirements.

18. Event. An event represents the start or completion of some activity and as such it consumes no time. It is also known as Node. An activity begins and ends with an event. A point in time that marks the beginning or ending of an activity.

19. Expected Activity Time. The average time that it should take to complete an activity. Te = (a + 4m + b).

20. Expected Duration Of The Project refers to the sum of the normal time of the activities on the critical path. It is in other words the duration the critical path

21. Forward Pass : A procedure that moves from the beginning of a network to the end of the network. it is used in determining earliest activity start times and earliest finish times.

22. Immediate Predecessors. Those activities that must be completed immediately prior to the start of the activity in question

23. Latest Activity Finish Time (LF) : The latest that an activity can finish from the beginning of the project without causing a delay in the completion of the entire project.

24. Latest Activity Start Time (LS) : The latest that an activity can start from the beginning f the project without causing a delay in the completion of the entire project.

25. Latest Allowable Time. The latest time that the event can be delayed without delaying the completion of the entire project.

26. Most Likely Time (m). The time required to complete an activity under normal circumstances.

27. Network Diagram. : A graphical display of a project that contains both activities and events.

28. Normal cost of an activity represents the lowest possible direct cost (cost of materials, labour, tools, equipment, outside operation cost, etc.) required to complete an activity within its normal time.

29. Normal time of an activity represents the expected duration of the activity. Such a time may be based on single time estimate (CPM approach) or three time estimate (PERT approach).

30. Optimistic Time is the shortest amount of time that could be required to complete the activity.

31. PERT An acronym for Programme Evaluation & Review Technique a method for scheduling & controlling projects.

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32. Pessimistic Time: It is the greatest amount of time that could be required to complete the activity.

33. Predecessor Activity : It is an activity that must occur before another activity.

34. Project Direct Cost implies the sum total of normal costs of all activities comprising the project.

35. Project Indirect Cost refers to the cost related to overall duration of project. Project indirect costs include costs of indirect labour, salaries and customary overheads, interest payable on the capital, penalty charges for delays in completing the project on time, loss of business(sales) and others.

The project indirect costs are generally a function of the time the project takes to complete. Thus, shorter the period,, the lesser the overhead charges. However it may be recognized that the direct cost of performing an activity would tend to increase if we desire to perform it in a time shorter than what it requires. .

36. Resource Optimisation : It is the process of manipulation of the network to try to ensure that the resources required and available are in balance.

37. Slack. The amount of time by which the start of an activity may be delayed without affecting the overall duration of the project. Slack is equal to the latest start time minus the earliest start time, or the latest finish time minus the earliest finish time. Slack is computed in relation to events.

38. Successor Activity. An activity that must occur after another activity.

39. Time Cost Relationship In PERT: The Project cost like time management plays an important role in any project. The project costs can be classified into two groups’ i.e. project direct cost and project indirect costs.

Project direct costs represent the sum of the direct activity costs (Direct activity costs are the costs which can be directly identifiable with the activities) whereas Project indirect costs are the costs related to the duration of the project. Project indirect costs increase /decrease directly with increase/decrease in the project completion time. The indirect costs include costs of indirect labour, salaries and customary overheads, interest payable on the capital, penalty due to late completion of the project, loss of business and otheRs.

Project total cost (sum of project direct costs and project indirect cost) therefore, are different for different durations of the project. The duration of the project in which sum of these two cost is minimum is called Optimal Project time.

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QUESTIONS ON CPM & PERT ( University Papers )

Oct 2009Activity Predecessorrs Time Estimates In Weeks

    Optimistic Most Likely PessimisticA ----- 3 6 9B ----- 2 5 8C A 2 4 6D B 2 3 10E B 1 3 11F C / D 4 6 8G E 1 5 15

1. Draw the project network.2. Find the critical path & expected project completion time3. What is the probability that the project will be completed in 18 weeks4. What will be the project duration so that the project manger is confident with 95 % that the project

will be completed on schedule?

April 2009A company has taken up a special project consisting of 8 activities whose 3 point time estimates are listed as below. The activities are marked with their node numbers.

Time estimates in weekActivity node

numbersOptimistic Time Most Likely Time Pessimistic Time

1 – 2 1 3 51 – 3 2 4 62 - 5 3 5 72 – 4 5 6 75 – 6 5 7 94 – 6 6 8 103 – 6 7 9 116 - 7 2 3 4

1. Draw the PERT diagram for the project and identify Critical Path.2. Prepare a chart to show estimated time for each activity and standard deviation and variance for

critical activities from the time estimates given above3. If 21 weeks deadline is imposed, what is the probability that the project will be finished within tat

time?4. If the project manager wants to be 99 % certain that the project should be completed on schedule

what will the project duration?

April 2009

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Activities involved in a small project are given below along with relevant information

Activity 1 - 2 1 - 3 2 - 3 2 - 4 3 - 4 4 - 5Duration in days 20 25 10 12 6 10

Construct the network and find Critical PathFind and tabulate Earliest starting time, Earliest finish time, Latest start time, Latest finish time and total Floats. Verify the critical path found earlier.,April 2008

ACTIVITY TIME ESTIMATES IN DAYS

  OPTIMISTIC MOST LIKELY PESSIMISTIC

1 – 2 2 5 14

1 – 6 2 5 8

2 – 3 5 11 29

2 – 4 1 4 7

3 – 5 5 11 17

4 – 5 2 5 14

6 – 7 3 9 27

5 – 8 2 2 8

7 – 8 7 13 31

1. Draw the project network.2. Find the critical path & expected project completion time3. What is the probability that the project will be completed in 38 days4. What will be the project duration so that the project manger is confident with 94.5 % that the

project will be completed on schedule?

Oct.2007

ACTIVITY TIME ESTIMATES IN WEEKS

  PESSIMISTICMOST

LIKELY OPTIMISTIC

1 – 2 21 7.5 3

1 – 3 27 8 3

2 – 4 8 8 8

2 – 5 3.5 2 0.5

3 – 5 10 10 10

4 – 5 1.7 1 0.3

4 – 6 9 7.5 3

5 – 6 5 3 1

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1. Draw the project network and identify all paths through it.2. What is the project completion time?3. Find the approximate probability of completing the project not more than 4 weeks later than

expected4. What is the probability of completing the project in 20 weeks?5. If the project manager has the option of speeding up activity 4 – 5 or 4 – 6 what should be his

choice?April 2007

ACTIVITY PREDECESSORRS TIME ESTIMATES IN DAYS

   MOST

LIKELY OPTIMISTIC PESSIMISTIC

A --- 3 1 7

B A 6 2 14

C A 3 3 3

D B / C 10 4 22

E B 7 3 15

F D / E 5 2 14

G D 4 4 4

1. Draw the project network.2. Find the critical path & expected project completion time And next most critical path3. What is the probability that the project will be completed in 30 days4. What will be the project duration so that the project manger is confident with 95 % that the project

will be completed on schedule?5. IF the fixed cost of the project is rs.10,00,000 and variable cost is Rs 8000 per day then find the

amount the firm should bid under this policy of 95 % confidence of completion?

Oct 2006Three time estimates of two adjacent activities of a project are ( 4, 8 ,12) and ( 8,12,16). What is the expected duration of the project? what is the chance that the project will be completed on or before 18 weeks? Project has only two activities.

April 2006

ACTIVITY PREDECESSORRS TIME ESTIMATES IN DAYS

    OPTIMISTICMOST

LIKELY PESSIMISTIC

A --- 3 6 15

B --- 5 11 17

C --- 4 19 28

D A 16 20 30

E A 7 10 13

F B / D 6 10 20

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G C / A 10 20 36

1. Draw the network diagram and calculate the expected duration of all the activities2. Find the expected duration of the project with 50 % and 75% chances of its completion.3. If a penalty of Rs.10000 per day is to be imposed what is the probability that more than Rs.20000

penalty will have to be paid?

Oct. 2005Estimated duration in weeks

Activity Optimistic time Most likely time Pessimistic time1 – 2 1 1 71 – 3 1 4 71 – 4 2 2 82 – 5 1 1 13 – 5 2 5 144 – 6 2 5 85 - 6 3 6 15

1. Draw the project network. Find the critical path & expected project completion time2. Find the expected duration and variance for each activity. Calculate the variance and standard

deviation of the project length.3. What is the probability that the project will be completed a) at least 4 weeks earlier than the

expected time and b) not more than 4 weeks later.4. If the project due date is 19 weeks what is the probability of not meeting the due date?

April 2005ACTIVITY PREDECESSORRS TIME ESTIMATES IN WEEKS

    OPTIMISTIC MOST LIKELY PESSIMISTICA -- 4 7 16B -- 1 5 15C A 6 12 30D A 2 5 8E C 5 11 17F D 3 6 15G B 3 9 27H E / F 1 4 7I G 4 19 28

1. Draw the project network. Find the critical path & expected project completion time2. Prepare the activity schedule for the project3. Determine the mean project completion time4. Find the probability that the project is completed in 36 weeks.

Oct. 2004Activity Time in weeks

1 - 2 41 – 3 12 - 4 13 – 4 13 – 5 64 – 9 5

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5 – 6 45 – 7 86 – 8 17 – 8 28 – 9 1

8 – 10 89 - 10 71. Draw a network diagram. Find the critical path & expected project completion time2. Calculate the earliest starting time and latest finish time for each activity3. Calculate the total and independent floats for each activity

April 2004ACTIVITY PREDECESSORRS TIME ESTIMATES IN WEEKS

    OPTIMISTICMOST

LIKELY PESSIMISTICA -- 2 3 4B -- 8 8 8C A 7 9 11D B 6 6 6E C 9 10 11F C 10 14 18G C / D 11 11 11H F / G 6 10 14I E 4 5 6J I 3 4 5K H 1 1 1

1. Draw the PERT network diagram Compute the slack for each activity and determine critical path.2. As per the contract a penalty of Rs 5000 is to be charged for any delay beyond 37 weeks. what is

the probability that the company will have to pay a maximum penalty of Rs.15000 Oct. 2003

Activity Time(hours)1 – 2 21 – 3 21 - 4 12 – 5 43 – 6 83 – 7 54 – 6 35 – 8 16 – 9 57 – 8 48 - 9 3

1. Draw a network diagram. Find the critical path , sub critical path & expected project completion time.

2. Calculate the earliest starting time, earliest finish time, latest starting time and latest finish time for each activity Calculate and tabulate total float, free float , interfering float and independent floats

April 2003

ACTIVITYPredecessor

activity TIME ESTIMATES IN WEEKS    OPTIMISTIC MOST LIKELY PESSIMISTIC

A - 4 6 8B A 5 7 15

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C A 4 8 12D B 15 20 25E B 10 18 26F C 8 9 16G E 4 8 12H D / F 1 2 3I G / H 6 7 8

A ) The Company’s management has established a required 52 week completion time for the project. Can this be achieved? Include probability information in your discussion.

1. What recommendation do you have if the 52 week completion time is required? B) If the management request that the activity times be shortened to provide an 80 % chance of meeting 50 week completion time.

2. if the variance in the project completion time is the same as found as above, how much should the expected project completion time be shortened to achieve the goal of an 80 % chance of completion within 50 weeks.

 Activity

Crashed Activity (weeks)

Normal Cost ( Rs. )

Crashed Cost ( Rs. )

A 1 – 2 4 1000 1900

B 2 – 3 6 1000 1800

C 2 – 4 4 1500 2700

D 3 – 6 15 2000 3200

E 3 – 5 15 5000 8000

F 4 – 6 8 3000 4100

G 5 – 7 6 8000 10250

H 6 – 7 1 5000 6400

I 7 – 8 5 10000 12400

April 2002

ACTIVITY PREDECESSORRS TIME ESTIMATES IN WEEKS

    OPTIMISTICMOST

LIKELY PESSIMISTIC

A --- 2 2 8B --- 2 5 8C --- 3 3 9D A 2 2 2E B 3 6 15F C 3 6 9G D / E 4 7 16H F / G 2 3 4

1. Draw the project network. Find the critical path & expected project completion time

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2. What duration will have 95 % confidence for project completion? (given area under normal curve from Z = 0 to Z = 1.65 is 0.45)

PROBLEMS ON CRASHING

Oct 2009

ActivityImmediate Preceding

Activity Normal Crash  Time(Days) Cost(Rs.) Time(Days) Cost(Rs.)

A --- 3 140 2 210B --- 6 215 5 275C --- 2 160 1 240D A 4 130 3 180E A 2 170 1 250F A 7 165 4 285G B / D 4 210 3 290H C / E 3 110 2 160

1. Draw a PERT diagram2. Find the critical path & expected project completion time3. What is the minimum project completion time after crashing the activities involved under the

project and the associated cost of completing the project?

Oct.2008

Activity Normal Crash

  Time(Days) Cost(Rs.) Time(Days) Cost(Rs.)

1 – 2 5 170 4 240

1 – 3 9 310 6 550

1 – 4 6 80 4 200

2 – 4 10 130 8 230

3 - 4 6 110 4 290

The indirect cost is estimated at Rs.120 per day of the project duration

1. What is the optimum project duration ie. duration at total minimum costs?

April 2008

 Activity Normal Time(Days) Crash Time(Days) Cost ( Rs per day.)1 – 2 9 6 201 – 3 8 5 251 – 4 15 10 30

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2 – 4 5 3 103 – 4 10 6 154 - 5 2 1 40

1. Draw a network diagram. Find the critical path & expected project completion time2. If overhead costs are Rs 60 per day and assuming normal project cost as Rs.1000,what is the

optimum project length and minimum total cost?Oct.2007

ActivityImmediate

Preceding Activity Normal Crash    Time(Days) Cost(Rs.) Time(Days) Cost(Rs.)

A --- 6 10000 4 14000B --- 4 5000 3 8000C A 3 4000 2 5000D B 8 1000 3 6000E B 14 9000 6 13000F C / D 8 7000 4 8000

Overhead costs amount to Rs.1000 per week.

1. Crash the project to optimal extent. what will be the optimal project duration and total minimum cost?

2. What will be the critical activities after such crashing?3. What will be the minimum project duration and corresponding total project costs?

April 2007

Activity Normal Time(Days) Crash Time(Days) Cost ( Rs per day.)1 – 2 9 6 201 – 3 8 5 251 – 4 15 10 302 – 4 5 3 103 – 4 10 6 154 - 5 2 1 40

1. Draw the network diagram and find the normal project length.2. What is normal project duration and total cost?3. What is optimal project duration and total cost?4. What is minimum project duration and total cost?5. What additional cost for minimum project duration?

Oct 2006

ActivityImmediate

Preceding ActivityNormal

Time(Days)Crash

Time(Days)Cost(Rs.) per unit time

(day)reductionA --- 8 6 50B --- 4 2 100C A 2 1 40D B 10 5 60E A 5 1 25F C / D 3 1 10

Direct project cost is Rs.580 and indirect cost is Rs. 70 per day.

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1. Draw a network diagram. Find the critical path & expected project completion time.2. Identify the lowest cost and associated time3. Determine the optimum project length and optimum cost after crashing

Oct 2006 ( Resource Planning)

From the following PERT diagram using the concept of forward and backward computation, Find the minimum number of machines required for the projects for its activities : 2 – 5,3 – 7, and 8 – 9 so as not to delay the project. Can these activities be completed by one machine. 4

2 1 2 5 4

1 8 3 3 5

April 2006 ( Resource Planning)

Activity Time1 – 2 21 – 4 21 - 7 12 – 3 43 – 6 14 – 5 54 – 8 85 – 6 46 – 9 37 – 9 38 - 9 5

1. Draw a network diagram. Find the critical path & expected project completion time2. Calculate the floats and determine subcritical path3. Activities 2 – 3, 4 – 5,and 6 – 9 each require one unit of key machine to complete it. The cost of

the machine does not permit to acquire another unit. you are asked to opine that availability of one unit of the machine is enough to complete the activities in question. Justify your opinion.

Oct 2005Activity Time1 – 2 21 – 3 71 - 4 82 – 5 33 – 5 6

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1

2 5

3 7 8

4 6 9

3 – 6 103 – 7 44 – 6 65 – 7 26 – 8 57 - 8 6

1. Determine earliest starting time / earliest finish time / latest starting time / latest finish time / free floats / total floats and independent floats for each activity.

April 2005

ActivityImmediate Preceding

Activity Normal Crash    Time(weeks) Cost(Rs.) Time(weeks) Cost(Rs.)

A ---- 3 80 2 190B --- 8 6 6 10C B 6 100 4 120D B 5 40 2 100E A 13 30 10 90F A 4 150 4 150G F 2 12 1 14H C / E / G 6 35 4 45I F 2 70 1 80

1. Draw a network diagram. Find the critical path & expected project completion time2. If a deadline of 17 weeks is imposed for completion of the project which activity will be crashed,

what should be the additional cost and what would be the critical activities of the crashed network after crashing?

Oct. 2004

ActivityImmediate Preceding

Activity Normal Crash    Time(Days) Cost(Rs.) Time(Days) Cost(Rs.)

A --- 4 60 3 90B --- 6 150 4 250C --- 2 38 1 60D A 5 150 3 250E C 2 100 2 100F A 7 115 5 175G D / E / B 4 100 2 240

Indirect cost is as follows

Days Cost in Rs.15 60014 50013 40012 25011 17510 1009 758 50

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7 356 25

1. Draw an arrow diagram for the project and determine the project duration which will return in minimum total project cost.

April 2004

ActivityPredecessor

activity TimeA  -- 2B A 3C A 4D B / C 6E  -- 2F E 8

1. Draw a network diagram. Find the critical path & expected project completion time2. Calculate the earliest starting time and earliest finish time for each activity3. Calculate the slack for each activity

April 2003

Activity Normal Crash

  Time(Days) Cost(Rs.) Time(Days) Cost(Rs.)

1 – 2 8 100 6 200

1 – 3 4 150 2 350

2 – 4 2 50 1 90

2 – 5 10 100 5 400

3 – 4 5 100 1 200

4 - 5 3 80 1 100

Indirect cost is Rs.70 per day.1. Draw a network diagram2. Crash systematically the activities and determine the optimal project duration and cost

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