Concepts & Steps of Operations Research April 2014 Exam Network Analysis

Objectives of network analysis:

1. Minimization of total project completion time
2. Minimization of total project completion cost
3. Minimization of cost for a given total time
4. Minimization of time for a given total cost
5. Minimization of idle resources
6. Minimization of conflicting production schedule

 

Application areas:-

1. Planning
2. Scheduling
3. Executing
4. Monitoring
5. Controlling

 

Fundamental concepts:

1. Activity:         A task or an operation which requires expenditure of time & resources for its completion.

In network analysis an activity is represented by an arrow.  The arrow shows the direction of activity.  The arrow consists of a head & a tail.  An activity cannot begin until the preceding activity has been completed.

 

a) Preceding activity:  The activity which immediately precedes activity under examination & which needs to be completed to start activity under examination.

{Also called Predecessor.}

 

b) Succeeding activity: The activity which immediately succeeds present activity & which cannot be started until present activity is completed.

{Also called Successor.}

 

c) Concurrent activity: The activity which is not affected by the start or completion of present activity.  Concurrent activities can be accomplished simultaneously.

 

2. Event: An event represents a specific accomplishment in the project.  It takes place at a particular instance of time and doesn’t consume time or resource.  It is a time oriented reference point.  It signifies the end of one activity & start of another activity.

In network analysis an event is represented by a circle, which is called a node.

 

3. Dummy activity: In a project there are many activities which can be performed concurrently.  It is also possible that they have the same start event & end event.  Even then these activities need to be represented separately in the network analysis.  This is done with the help of dummy activity.  Dummy activity doesn’t consume any time or resource.  It is represented by b broker arrow.

Purpose:

To distinguish two or more activities having a common start event & end event

To identify & maintain proper precedence relationship between activities which are not connected by events.

 

 

Guidelines for constructing Network Diagrams:

1. Each activity is represented by only one arrow.
2. An arrow can represent only one activity
3. If two or more activities are identified with the same start & end events.  Dummy activity / activities  need to be introduced in the network
4. Before an activity can be started all preceding activities must be completed
5. The flow of diagram is from left to right.
6. The arrows are not to the scale
7. Arrows should be straight
8. Unless absolutely unavoidable, arrows should not be crossed.

 

TIME ESTIMATES IN NETWORK ANALYSIS 

 

Activity time:

Time required by an activity to get completed under normal conditions.

Objectives of time analysis:

1. To determine total completion time for the project.
2. To determine Earliest time when each activity can start.
3. To determine Latest time when each activity can start
4. To determine Float for each activity, i.e. amount of time by which the completion of an activity.
5. Identification of critical activities.
6. Identification of critical path.

 

Project :

A project can be defined as a combination of inter – related activities that must be executed in an order before the entire task can be completed within (i)  a specified time (ii)  at a specified cost and (iii)  meeting the performance standards.

Examples of a project are :

(i)      Introduction of a new product in the market.

(ii)     Construction of a new bridge over a river or construction of a multi-storied building.

 

Path :

A path is defined as an unbroken sequence of activities directed from the origin node to the terminal node.

 

Activity :

An activity in a project is usually viewed as a job requiring time and resources for its completion. An arrow is commonly used to represent an activity with its head indicating the direction of progress in the project. The head of the arrow indicates where the task ends the tail where the task begins.

 

Event :

An event represents a point in time that signifies the completion of some activities and the beginning of the new ones. E.g. Wall built, Debtors verified, Foundations dug etc.

An event is represented in a network by a circle or node e.g. The beginning and end point of an activity are thus described by two events usually known as the tail and head events.

 

Dummy Activity :

It is an activity which does not consume time or resources. It is used merely to show clear, logical dependencies between activities so as to violate the rules for drawing networks. It is represented on a network by a dotted arrows.

 

Network :

This is the combination of activities, dummy activities and events in logical sequence according to the rule for drawing networks. A project network consists of numbered circles that are interconnected by arrows. The circles are called nodes and the arrows connecting the nodes are called branches or arcs. In project network the arcs corresponding to activities and the nodes corresponding to events.

 

Critical Path :

The critical path of a network gives the shortest time in which the whole project can be completed. It is the chain of activities with the longest duration times. There may be more than one critical path in a network and it is possible for the critical path to run through a dummy activity.

Activities on the critical path are known as critical activities.

An activity is said to be critical if a delay in its start will cause a delay in the completion date of the entire project. A none-critical activity is such that the time between its earliest start and its latest completion dates (as allowed by the project) is longer than its actual duration. In this case the non-critical activity is said to have a slack or float time.

 

RULES FOR DRAWING NETWORKS :

The following rules are all logically based and should be thoroughly learned before attempting to draw networks.

1)      A complete network should have only one point of entry – a START event and only one point of exist – a FINISH event.

2)      Each activity is represented by one and only one arrow in the network. Every activity        must have one preceding or ‘tail’ event and one succeeding or ‘head’ event. Note that          many activities may use the same tail event and many may use the same head event.

However no two activities can be identified by the same head and tail events. When two or  more parallel activities in a project have the same head and tail events, DUMMY ACTIVITIES are needed in constructing the network.

Dummy activities are also useful in establishing logic relationships in the arrow diagram that cannot otherwise be represented correctly. Such dummy activities are known as LOGIC DUMMIES. E.g. Suppose that in a certain project, jobs A and B must proceed C while job E is preceded by job B only.

(3)     No activity can start until its tail event is reached.

(4)     An event is not complete until all activities leading into it are complete.

This is an important rule an invariably has to be applied in examination questions.

(5)     ‘Loops’ i.e. a series of activities which lead back to the same event are not allowed because the essence of networks is a progression of activities always moving onwards in time.

“LOOPS” NOT TO BE USED :

6)      All activities must be tied into the network i.e. they must contribute to the progression        or be discarded as irrelevant. Activities which do not link into the overall project are       termed ‘danglers’.

 

DANGLER NOT TO BE USED :

To ensure the correct precedence relationship in the arrow diagram, the following questions must be answered as every activity is added to the network.

(a)     What activities must be completed immediately before this activity can start?

(b)     What activities must follow this activity?

(c)     What activities must occur concurrent with this activity?

 

CONVENTIONS FOR DRAWING NETWORKS :

(a)     Networks proceeds from left to right.

(b)     Networks are not drawn to scale i.e. the length of the arrow does not represent time elapsed.

(c)     Arrows need not be drawn in the horizontal plane but unless it is totally unavoidable they should proceed from left to right.

(d)     If they are not already numbered, events or nodes should be progressively numbered from left to right. Simple networks may have events numbered in simple numeric        progression i.e. 0, 1, 2, 3 etc. but larger, more realistic networks may be numbered in fives i.e. 0, 5, 10, 15 etc. or ‘tens’ i.e. 0, 10, 20, 30 etc.

This enables additional activities to be inserted subsequently without affecting the numbering sequence of the whole project.

 

ACTIVITY IDENTIFICATION :

Activities may be identified in several ways and students should familiarize themselves with the various methods so that unfamiliar presentation does not cause confusion. Typical of the methods to be found include :

(a)     Shortened description of the job e.g. plaster wall, order timber etc.

(b)     Alphabetic or numeric code.

e.g. A, B, C etc or 100, 101, 108 etc.

(c)     Identification by the tail and head event numbers

e.g. 1-2, 2-3, 2-5 etc.

 

FLOAT :

Float or spare time can only be associated with activities which are non-critical. By definition, activities on the critical path cannot have float. There are four types of floats. Total Float, Free Float, Interfering Float and Independent Float.

(i)      Total Float :

The total float of an activity represents the amount of time by which an activity can be delayed without delaying the project completion date. In order words, it refers to the amount of free time associated with an activity which can be used before, during or after the performance of this activity. Total float is the positive difference between the earliest finish time and the latest finish time or the positive difference between the earliest start time and the latest start time of an activity depending upon which way it is defined.

 

(ii)     Free Float :

Free float is that portion of the total float within an activity which can be manipulated without affecting the float of subsequent activities. It is computed for an activity by subtracting the head event slack from its total float. The head event slack is the difference between the latest and earliest event timings of head event of an activity that is its (L-E)

 

(iii)    Interfering Float :

Utilisation of the float of an activity may affect the float times of the other activities in the network. Interfering float is that part of the total which causes a reduction in the float of the successor activities. It is the difference between the latest finish time of the activity in question and the earliest start time of the following activity or zero, whichever is larger. It indicates that portion of the float of an activity which cannot be consumed without affecting adversely the float of the subsequent activities.

 

(iv)    Independent Float :

This is the amount of time an activity can be delayed when all preceding activities are completed as late as possible and all succeeding activities started as early as possible. Independent float therefore does not affect float of either preceding or subsequent activities. It is computed by subtracting the tail event slack from the free float of the activity. If the result is negative it is taken as zero.

For examination purposes the most important type of float is total float because it is involved with the overall project duration. On occasions the term ‘Float’ is used without qualification. In such cases assume that Total Float is required.

 

COSTS AND NETWORKS :

A further important features of network analysis is concerned with the cost of activities and of the project as a whole. This is sometimes known as PERT / COST.

The primary object of network cost analysis is to be able to calculate the cost of various project durations. The normal duration of a project incurs a given cost and by more labour, working overtime, more equipment etc., the duration could be reduced but at the expense of higher costs. Some ways of reducing the project duration will be cheaper than others and network cost analysis seeks to select the cheapest way of reducing the overall duration.

A common feature of many projects is a penalty clause for delayed completion and / or a bonus for earlier completion. In examination questions, network costs analysis often combined with a penalty and / or bonus situation with the general aim of calculating whether it is worthwhile paying extra to reduce the project time so as to save penalty.

(a)     Normal Cost :

The costs associated with a normal time estimate for an activity. Often the “normal” time estimate is set at the point where resources (men, machines etc.) are used in the most efficient manner.

(b)     Crash Cost :

The cost associated with the minimum possible time for an activity. Crash costs, because of extra wages, overtime premiums, extra facility are always higher than normal costs.

(c)     Crash time :

The minimum possible time that an activity is planned to take. The minimum time is invariably brought about by the application of extra resources, e.g. more labour or machinery.

(d)     Cost slope :

This is the average cost of shortening an activity by one time unit (day, week, month as appropriate). The cost slope is generally assumed to be linear.

(e)     Activity direct costs :

The cost of materials, equipment and direct labour (payroll, overtime, hiring and firing costs) required in performing an activity is called its direct cost. If the activity in question is subcontracted and is being performed in its entirety by a contractor then the activity direct cost is equal to the price of the subcontract. The sum total of the direct costs of all the activities of a project is the project direct cost.

(f)      Project indirect costs :

The project indirect costs imply the overhead charges related to the project which include the supervision and other charges, late completion penalties and awards for early completion, and so on.

The project indirect costs are generally a function of the time the project takes to complete. Thus the shorter the period the lesser the overhead charges. However it may also be recognised 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. Thus, if a project is set to be performed in a shorter time, then the direct costs would be larger and we can economise on the indirect costs. Similarly, a slower pace of work might, to some extent, mean lower direct costs, coupled with higher indirect costs.

 

CRASHING NETWORK IN EXAMINATION :

a)      Only critical activities affect the project duration so take care not to crash non-critical activities.

b)      The minimum possible project duration is not necessarily the most profitable option. It may be cost effective to pay some penalties to avoid higher crash costs.

c)      If there are several independent critical paths then several activities will need to be crashed simultaneously. If there are several critical paths which are not separate i.e. they share an activity or activities then it may be cost effective to crash the shared activities even though they may not have lowest cost slopes.

 

Summary for Crashing :

(a)     Cost analysis of network seeks the cheapest ways of reducing project times.

(b)     The crash cost is the cost associated with the minimum possible time for an activities, which is known as the crash time.

(c)     The average cost of shortening an activity by one time period (day, week etc.) is known as the cost slope.

(d)     Least cost scheduling finds the cheapest of reducing the overall project time by reducing the time of the activity on the critical path with the lowest cost slope.

(e)     The total project cost includes ALL activity costs not just those on the critical path.

(f)      The usual assumption is that the cost slope is linear. This need not be so and care should be taken not to make the linearity assumption when circumstances point to some other conclusion.

(g)     Dummy activities have zero slopes and cannot be crashed.

The resources (men of varying skills, machines of all types, the required materials, finance, and space) used in a project are subject to varying demands and loadings as the project proceeds. Management need to know what activities and what resources are critical to the project duration and if resource limitations (e.g. shortage of materials, limited number of skilled craftsmen) might delay the project. The also wish to ensure, as far as possible constant work rates to eliminate paying overtime at one stage of a project and having short time working at another stage.

 

BASIC TIME ANALYSIS (PERT ANALYSIS)

In many situations, activity times can be estimated accurately. For example, in projects such as construction or maintenance, manager may have sufficient experience or historical data to provide activity time estimates that are fairly accurate. In addition, the nature of these activities may have low variability and thus time would be relatively constant. In other cases, activity times are uncertain and perhaps best described by a range of possible values. In these instances the uncertain activity times are treated as random variables with associated probability distributions.

The most common method of dealing with uncertain activity times is to obtain three time estimates for each activity. The three estimates are :

(1)     Optimistic Time (a or to) :

This is the shortest time an activity can take to complete. For this estimate, no provision for daily as or setbacks are made. It represents an ideal estimate.

(2)     Most Likely Time (m or tm) :

This refers to the time that would be expected to occur most often if the activity were frequently repeated under exactly the same conditions. It assumes that things go in a normal way with few setbacks. It is the modal time.

(3)     Pessimistic Time (b or tp) :

This is the longest time the activity could take to finish. If everything went wrong and abnormal situations prevailed, this would be the time estimate.

Using the values of a, b and m, the expected time of various activities and their standard deviations are calculated as follows. The three time estimates are reduced into a single expected time (Tei) with the weighted average formula :

The standard deviation, s i, of the completion time of an activity is calculated as follows :

From this, the variance  =    calculated

Once the expected time of the activities are obtained, the critical path of the project network is determined using these time estimates. Having found the critical path, the PERT methodology assumes that the sum of the mean times and the summation of the variances of critical jobs would yield the expected project duration and its variance.

 

 

 

TIPS TO DRAW NETWORK DIAGRAM:

1)         Activities A and B start together

2)         A and B start together, C starts after A is over. D and E start after B is over.

3)         J starts after G and H are completed.

4)         A and B starts together. C starts after A is over. D and E starts after B is over. F starts after C and E are over.

5)         A and B start together, C starts after both A and B are over.

6)         A and B starts together, C starts after A and B are over. D starts after A is over.

7)         A and B starts together. C starts after A and B are over. D starts after A is over. E starts after B is over.

 

Activity:

Ans.    An activity which does not consume any resource but only shows the technological dependence is called as ‘Dummy Activity”.

            In Network diagram, No to activies can have same start event and same and event. Such a situation if exist can be avoided using a dummy activity. It is the activity with time duration ‘0’ (zero).

 

Forward Pass :

Ans.    The initial event of the network is assigned time zero and then the earlies time at which the next event in project can occur is computed by using expected duration. It is denoted by ‘T_E’

            This gives the earliest time at which 2^nd event can occure using the same procedure the earliest time for all events can be computed. If there are more than one activities coming to a particular event then earliest time for that event is taken as the maximum of the T_E values among the different activities. Thus by taking the T_E value for the start event as zero, we compute the T_E values for all the events in the network proceding in the forward direction. Hence the calculation of earliest time for the different events in the network is called as ‘FORWARD PASS’.

BACKWARD PASS : The project completion time is called as the ‘CONTRACUTAL OBLIGATION TIME’.  is the latest time (T_L) by which the last event in the project should occure.

            If the contractual obligation Time is not specified, the earliest time (T_E) for the last event should be the value assume by the latest time (T_L) of the last event.

Once the T_L value for the last event is known, we proceed in the backward direction through the network calculating the values of T_L for the earlier events till the start event is reached. If there are more than one activities immerging from an event. The T_L values for that event is taken as the minimum of the T_L values among different activities. This is called ‘BACKWARD PASS’.

 

In PERT 3 different time estimates are defined for each activities.

Ans.    (i)      OPTIMISTIC TIME ESTIMATE :

          This is the shortest time within which an activity can be completed.

Conditions better than normal are expected during the execution of activity and no provisions are made for delay. This time estimate is usually denoted by ‘t_o’

(ii)     PESSIMISTIC TIME ESTIMATE :

          This is the maximum time required for an accomplish on of an activity i.e. It is the time required if the activity is performed under totally abnormal or adverse condition.

It is denoted by ‘t_p’  

(iii)    MOST LIKELY TIME ESTIMATE :

          This is the estimate of the type which an activity will require under normal circumstances. It is usually denoted by ‘t_m’.

The frequency curve of the activity time estimate resemble the b distribution curve with t_o and t_p as an extreme point and tm as the nodal value. Hence the mean time require for the completion of an activity is given by.

 

Critical path:

1. Critical Events:  The events with zero slack time are known as critical events.

Slack of an event is the difference between the latest and earliest occurrence times of        the event.

Slack (i) = Li – Ei

1. Critical Activities: The activities with zero total float are known as critical activities.  The difference between the latest finish time and the earliest start time is total float.  This difference indicates the amount of time by which the activity can be delayed without affecting the total project duration.
2. Critical Path:  The sequence of critical activities in a network is called critical path.  The critical path is the longest path in the network from the starting event to the ending event and defines the minimum time required to complete the project.

 

Characteristics of critical path:

1. If total completion time of the project is to be reduced, some of the activities on the critical path need to be shortened.
2. Application of additional resources on other activities will not give the desired result, if critical path is not shortened before that.

 

Limitations or Drawbacks of CPM:

1.  It is assumed that time required for completion of each activity is known with certainly.  This may not be true in real life situations.
2. Statistical analysis is not applied in determining time required for activities.  Time estimates are not based on statistical analysis.
3. If there are any changes in the network entire evaluation of the project has to be repeated & critical path has to be determined again.
4. CPM is applicable only for those situations which have a definite start & end point.

 

 

PROGRAM EVALUATION & REVIEW TECHNIQUE (PERT)

PERT concept of multiple time estimates:-

{to}1. Optimistic time (a):- This indicates the minimum time that an activity can take.  This          is possible if everything happens as per planning.

{tm}2. Most likely time (m):- This indicates the time an activity will take most often when it         is repeated a no. of times under the same conditions.

{tp}3.  Pessimistic time (b):- This indicates the maximum time an activity can take under   adverse conditions.

Expected time =    te =  a+4m+b/6

Standard deviation & variance in PERT:-

Standard Deviation of each activity   =   b – a / 6

PERT COST

 

1. Direct costs: Cost of raw materials, labour, power, direct OH etc.
2. Indirect Costs: Cost of supervision, project administration, office expenditure, penalty for delay in project completion etc. Indirect costs are directly proportional to time.
3. Total cost: Total of direct & indirect costs.
4. Total Direct Cost of Project: Total of direct costs of each individual activity.

 

Time – Cost Trade-off:

It is assumed that for each activity, there in an activity duration for which the direct cost is minimum. If the activity takes more time, indirect cost increases due to inefficient use of resources. If the activity has to be completed in less time, extra resources are required which results in increase in direct cost.

 

Normal time: –

Activity time which corresponds with minimum direct cost is called normal time.

 

Normal cost: –

The direct cost associated with normal time (minimum direct cost) is called normal cost. There is a limit to which activity time can be reduced by application of extra resources.

 

Crash time: –

The minimum time (reduced time) required for completion of the activity by applying extra resources.

 

Crash cost:-

The cost associated with crash time is called crash cost.

The cost slope indicates the extra cost required per unit time (days / weeks etc) to decrease to completion period of an activity.

Hence, cost slope is Extra cost per day / per week / per month.

If the total project duration is to be reduced, completion time of some or all the activities in the project should be reduced. The activities should be selected in such a way that the increase in project cost is minimum.  This selection is done on the basis of cost slope.

Those activities are selected for which cost slope is minimum,

This process of reducing total project duration by reducing activity timing is called crashing of project network.

 

UNIVERSITY THEORY QUESTIONS

 

1. What is Dummy activity? Explain its use in network Analysis.(Apr 2002)

A Dummy activity is an imaginary activity. It does not exist in the Project activities. It is used in the network diagram to show dependency relationship or connectivity between two or more activities. It is represented by a dotted arrow.

Say in a network following is the dependency relationship-

Activity           Preceding Activity

D                                 B

E                                  B, C

Activity D follows activity B. Activity E follows both the activities B & C. Hence, for drawing E, we need to show a connection between B & C. Hence, from B we will draw a Dummy activity on C.

 

2. Forward and backward pass in CPM / PERT. (Apr 2003) (Oct 2006) 

We use forward and backward pas in CPM / PERT to find Earliest and Latest occurrence times of events.

Forward pass calculations are used to find Earliest occurrence times of events (Ei). Forward pass calculations are from Left to Right. Start value is equal to zero. (Earliest occurrence time of 1^st event = 0).

When two or more activities merge in an event, the maximum value is taken as the Earliest occurrence time for that event.

Forward pass time= Earliest time of Tail event + Activity time

Backward pass calculations are from Right to Left. Backward pass calculations are used to find Latest occurrence times of events (Li). For the last event, Latest time = Earliest time. If there is more than one activity coming back in an event, in backward pass we take minimum value.

Backward pass time= Latest time of Head event – Activity time

 

 

3. Three time estimates in PERT and their relationship with expected time and its variance in the project. (Apr 2003) (Oct 2005) (Apr 2006)

 

In PERT, there are three time estimates –

1. Optimistic time (shortest possible time ) Denoted by “a” or “to”
2. Most likely time (middle value) Denoted by “m” or “tm”
3. Pessimistic time (longest possible time) Denoted by “b” or “tp”

 

From these three time estimates, we can calculate Expected time of an activity, Standard deviation of activity and Variance of activity.

1. Expected time of the activity = (a + 4m + b ) / 6
2. Standard deviation of the activity = (b – a) / 6
3. Variance of an activity = Square of standard deviation
4. Variance of critical path = Total of variances of all critical activities

 

 

4. Compare PERT and CPM.  (Oct 2003) 

CPM	PERT
1. In CPM there is only one time given for each activity.	1. In PERT there are three time estimates. Optimistic time, Most likely time and Pessimistic time. From these, we find Expected time for completion of an activity (te).
2. After drawing the network we find critical path.	2. We draw the network using Expected time (te) and then we find critical path.
3. There is no probability information in CPM.	3. Probability is associated with Project completion time (Te). The probability of completing the project in that duration is 50% (0.5).
4. There is no standard deviation and variance for activities. We assume that all activities will be completed in specified time.	4. We find standard deviation and variance of critical activities and from that we find variance and standard deviation of critical path.
5. As there is no probability, there is no relation with normal distribution table.	5. We can find “z” value and from “z” value the probability of completing the project in specified time duration.

 

 

5. Sub critical path in PERT / CPM analysis.  (Oct 2003)

Sub critical path means the second longest path in the network. It is shorter than the critical path but longer than all other paths in the network. All activities on the sub critical path are not critical. Some of the activities may be critical.

 

6. Explain Time-Cost trade off in CPM.  (Oct 2003)

Time – Cost trade off means we are ready to bear a higher cost for completing the project in a shorter period of time. This can be done by employing extra resources to complete activities in a shorter period of time. This is called crashing of network. It may not be possible to crash all activities in the network.

As the duration of project completion reduces, direct cost of project (activities) increases. Indirect cost reduces because it is on a per day basis. Hence, as duration reduces, indirect cost will reduce. The effect on total cost may be increase or decrease. We have to find that duration for which Total cost of project is minimum. This is called optimal cost. Corresponding duration is optimal time of project completion.

 

7. Dangling event and Dummy activity in network diagram.  (Apr 2004)

Dangling event – It means an event which is not connected to another event by an activity. An activity is merging into the event, but no activity is starting or emerging from that event (Except the last event of the network.) Hence, the event becomes detached from the network.

Dummy activity – A Dummy activity is an imaginary activity. It does not exist in the Project activities. It is used in the network diagram to show dependency relationship or connectivity between two or more activities. It is represented by a dotted arrow.

Say in a network following is the dependency relationship-

Activity           Preceding Activity

D                                 B

E                                  B, C

 

Activity D follows activity B. Activity E follows both the activities B & C. Hence, for drawing E, we need to show a connection between B & C. Hence, from B we will draw a Dummy activity on C.

 

8. Events and activities in PERT & CPM.  (Oct 2004)

Activities-

1. Activities represent actions which are represented by Arrows.
2. Activities consume resources and time.
3. Every activity has a Tail event (start) and Head event (end).
4. Activities can be critical or non critical. Critical activities represent critical path.
5. Critical activities have zero floats.

Events –

1. Events represent point of time. They are represented by Nodes or circles.
2. Events do not consume any time or resource.
3. An event represents start or end of an activity.
4. An event may be critical or non critical. Critical events have zero slacks. Critical events connect critical activities.

 

9. Distinguish between free float and independent float. (Apr 2005)

Free float: Free float is present if an activity is started at Earliest starting time but it can be finished before Earliest finishing time. (FF = EF – ES – time required for activity.)

Independent float: Independent float is present if an activity is started at Latest starting time but it can be finished before Earliest finishing time. (IF = EF – LS – time required for activity)

 

10. Necessary and sufficient conditions for critical path in PERT / CPM. (Apr 2006)

Necessary conditions: If Earliest occurrence time equals Latest occurrence time for Tail and Head event of the activity, then it possibly can be a critical activity.

Sufficient condition:   If the necessary condition is satisfied and ES and LS of activity are same, also EF and LF of activity is same; then the activity is definitely a critical activity.

 

11. Uses of slacks and floats in PERT and CPM. (Apr 2006) 

Slacks:            Slacks are for events. Slacks of all critical events are zero. Slack is the difference in Earliest occurrence time and Latest occurrence time for the event. It represents the margin by which the event occurrence can be delayed.

Floats: Floats are for activities. Total, Free, Independent and Interfering floats are types of floats. Floats represent the margin by which activity start times can be delayed without affecting project completion time.

Floats for critical activities are zero. They can not be delayed by any margin.

 

12. Critical path in PERT / CPM analysis. (Oct 2006)

Critical path indicates the path containing all critical activities. The critical path is the longest path in the network. All activities on the critical path have zero floats and zero tail and head slacks. For critical activities Earliest and Latest starting and finishing times are equal.