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Critical Path Method (CPM): Diagram, Chart, schedule, calculator & solution in Network analysis of Project Management

Significance of  Network Analysis

The technique of Critical path method (CPM) is devised to plan and manage all types of construction projects. It is known by various terms such as network analysis, critical path analysis, critical path scheduling, least-cost estimating and scheduling, etc. The term scheduling has been defined as a complex activity for setting the starting date and completion date of any project. Whenever a project is initiated, it is desirable to fix up a target date of completion, and to meet the target, it becomes necessary to plan all the activities that comprise the project in a systematic manner.

The adoption of CPM for various construction projects has started only from 1960 or so. But its success for small as well as large projects has opened vast field of its application. The CPM technique is fast developing and many improvements are being introduced to increase the utility of the method. The CPM has proved to be an useful and a powerful tool in the hands of construction engineers, planners, cost estimators, managers and accountants.

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Data For Network Analysis

To analysis and calculate CPM for a particular project the following data are to be collected.: 

  • Calculating costs of all possible items of the project by the process of quantity surveying, and
  • Estimating the normal time required to complete each operation. 

Thus, the normal cost and time data for each operation are worked out. The   basic information thus collected is then reviewed and revised various possible steps which might be taken to finish up the project in the best economic way. The variations which might be considered include use of alternative equipment, changing gangs of workers, adopting better construction methods, increasing working hours, etc.

The relevant variations are then checked on critical path network diagrams and ultimately, the optimum solution to the time-cost problems of the project is obtained. This whole procedure is termed network analysis and it can be carried out with the knowledge of simple only.

If the project involves a vast number of alternatives and operations, the process of network analysis can be carried out with the help of computers.

 Features of Network Planning

Following are the features of network planning: 

  • (i)It expresses the project in a graphical form.
  •  (ii)It forms a basic document for the preparation of work schedules of  different tasks and activities connected with the project.
  • (iii) It gives an overall picture at a glance of the whole project and indicates the inter-relationships between various activities, jobs, and events of the project. 

(iv) It helps in ascertaining activities over a critical path and at the same  time, the tolerable slackness or delay for other activities can also be known from the study of the network diagram of the project.

(v) It is a flexible self-adjusting technique and it can be easily modified for various reasons such as mistakes in original calculations, strikes of laborers, new rules and regulations, availability of resources, etc. 

(vi) It serves as a check on the time of completion with respect to the cost and hence, it rants optimum utilization of resources.

 (vii) It serves as a medium of communication for various groups associated in the execution of the project. 

(viii) The available resources can be diverted and utilized advantageously over the activities along the critical path for the project. 

Rules for Network Diagram

Following are the general rules to be followed in the case of a network diagram:

(i)A dummy activity should be introduced to bring out parallel features of the activities or multiple dependencies of events.

(ii) An Activity following the dummy activity cannot be started until the dummy activity is completed. A dummy activity does not require any time and it is represented by dotted lines with duration zero in a network diagram.

Every activity is preceded as well as followed by an event. 

(iii) Every event is preceded as well as followed by one or more than one activity, excepting of course the starting and ending events.

(iv) If there are variations in quantities and qualities of resources, there will be changes in the time of performance and cost of activity.

(v) It is possible to reduce the time of the activity by increasing the cost of that activity. 

(vi) It must be possible to break down the project into various suitable parts or stages which can be easily defined. 

(vii) It must be possible to clearly define the beginning and end of the project. 

(viii) It must be possible to estimate with reasonable accuracy the performance time of each part or stage of the project.

 (ix) No activity can start unless and until its preceding events have occurred. 

(x) No event can occur unless and until all the activities leading to it have been completed. 

(xi) Non-critical paths permit utilization of the available resources for the critical path activities.

(xi) The closed loops cannot be incorporated in a critical path network. cil The length of activity line has no bearing on time estimate of the activity unless the network plan is in the form of a square diagram or time-grid  diagram. 

(xiv) The performance of activity along the critical path must be carried out with utmost care and caution. Otherwise the project as a whole will be delayed.

(xv) There will be a change in the critical path of the project, if non-critical activities become critical during the progress of project.

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Procedure for Critical Path Method (CPM) 

The procedure for CPM consists of the following six steps: 

(1) Project breakdown 

(2) Network diagram

(3) Uses of data with respect to time and cost

(4) Determination of critical path

(5) Activity times and floats

 (6) Scheduling. 

I want to show briefly each of the above step here:  

(1) Project breakdown: The project is broken down into suitable processes or operations which are essential for its completion. Following factors are considered while deciding the degree of breakdown of the project: 

(1) broad general sequence of the job; 

(ii) costing data required by management;

  • item-wise cost of all the items;
  •  location of the work on the site;
  • nature of work involved; 
  • type of labour to be handled; etc..

The term activity is used to indicate each such process or operation and the term event is used to denote the completion of an activity. Thus, the time is required to complete an activity, whereas an event requires no time in itself. Further, the events are separated from one another by the activities. 

After preparing the list of activities, the next step is to decide the relationship between all these activities and then to arrange them in proper sequence or order. For this purpose, each activity is subjected to the following three tests: 

(i) What activities must precede? 

(ii) What activities must follow?

(iii) What activities may be con-current? 

Thus, each activity is properly scrutinised and examined and then placed at its order in the network diagram. If there are any constraints or hurdles or overlapping of work, the activities are further sub-divided and ultimately, a smooth flow diagram of activities of the project is obtained.

(2) Network diagram: In order to exhibit the diagrammatic relationship between various activities and events of a particular project in a logical sequence, the network diagram is drawn. Following are the two types of network diagrams:

(1) Arrow diagram

(2) Circle diagram. 

(i) Arrow diagram: In this type of network diagram, an arrow represents one activity and each circle represents an event. This network diagram is also known as the activity-oriented network and its salient features are as follows: 

(a) A separate arrow is drawn to represent each individual activity.

(b) Each event and each activity are numbered for the purpose of identification on the network.

(c) It is understood that the start of all activities leaving a circle presumes the completion of all activities entering that circle. 

(d) The length, orientation, and shape of the arrow have no significance. It merely indicates the passage of time in the direction of the arrow ahead.

(e) The order of numbering is such that the number at the tail of the arrow is lower than the number at the head of the arrow. Thus, the project starts: from the first event and it proceeds, event by event, to the completion of the work. The beginning of an activity is a tail event while the completion of an activity is a head event. Thus, the head event always has a number higher than that of the tail event.

(f) It is assumed that time flows from left to right.

It becomes necessary to use a connecting arrow, known as dummy, to show the logical sequence of events between A and D. It should be noted that the dummy activities are indicated by dotted lines and they have zero cost and zero time.

Fig1. Arrow diagram
Arrow Diagram
Fig.2,3,4,5 arrow digram

Fig1. shows that A must precede B and B must precede C.

Fig2 shows A must precede both B and C.
Fig3. shows that A and B must Pricede C.
Fig 4. shows that A must precede C and B must precede D.
Fig 5. shows that A must precede C and D; and B must precede D.

(ii) Circle diagrams: 

 In this type of network diagram, a circle represents one activity and the arrows represent logical time relationships between activities. This network diagram is also known as the event-oriented network or precedence diagram. 

Fig 6 a, 7 and 8. Circle diagram.
Fig. 9 and 10. Circle diagram.

The elements of a circle diagram are shown in fig. 6 to fig. 10 for similar identical conditions of an arrow diagram as shown in fig. 1 to fig. 5. It may be noted that the circle diagram avoids the necessity of dummy activities. In this diagram, each activity is identified with a single reference number only. It is not possible to draw circle diagrams on a time scale and in the case of a complex project, it becomes difficult to follow a circle diagram.

 Hence, the resort is usually not made to the circle diagram in most of the CPM construction planning techniques. 

Thus, the arrow diagram is preferred to the circle diagram in practice. Further, the time required to complete each activity is also shown on the network diagram. 

Now, for each activity, the cost of that activity is related to a certain time during which it should be completed. If the time of completion of the activity changes, there is variation in cost of that activity also. If this information, known as the utility data, is also included in the network diagram, it is termed as a network model. 

(3) Uses of data with respect to time and cost:

The direct cost of every activity is related to the time of completion of that activity by every possible method of carrying out that activity. The information regarding detailed time and cost of each activity is known as utility data and by the application of CPM, the utility data are suitably analyzed to arrive at the optimum cost of each activity in relation to its optimum time of completion. Such a solution lies somewhere between two extremes, namely, the least-cost solution and the least-time solution. 

The least-cost solution is referred to as the all-normal solution and it gives the duration of activity with the lowest possible direct cost. If the duration of entire project is to be reduced, some of the activities will have to be speeded up or crashed. If all the activities of the project are crashed, it is known as all-crash solution and it will indicate the lowest possible time of completion of the project. It is quite evident that the all-crash solution will be uneconomical because of overtime, use of more equipment, etc. 

The utility data for each project is prepared and a time is assigned for each activity on the network by using the all-normal utility data.

(4) Determination of critical path:

The next important step is the determination of critical path. Fig. 11 shows the first draft of network diagram of simple hypothetical project involving 7 activities. Fig.12 shows the length of time which might be expressed in hours, days or weeks, required for the completion of work involved in a particular activity. Then, by proceeding through the events in numerical order, the Earliest Finish Time, briefly written as EFT, required for all the activities entering each event is worked out by simple additions and it is recorded in the left side of the time box, as shown in fig. 12. 

Fig.12. Timing The Network

Thus, finally EFT of the last event of the network is worked out and it indicates the earliest possible time in which the project can be completed. In this particular case, it works out to, say 36 weeks. 

If the project is to be completed by EFT of the last event, the reverse process is started and subtracting the durations of each activity, the Least Finish Time, briefly written as LFT, permissible for each event is worked out. The figure LFT for each event is entered in the right side of the time box, as shown in fig. 13. 

As seen in fig. 13, there are twee figures in each time box, giving EFT and LFT for every event. The difference between these two figures gives the time available for delays and it is known as float. For some events, the time for the earliest finish as well as time for the latest finish are the same which means that these events have no float at all. Such events are known  as the critical events and if the project is to be completed as per schedule, these critical events must be finished in time without any delay.

Fig13. Critical path for all normal duration

 The path joining these critical events is known as the critical path is Longest path(in time) from start to finish  and for the network under consideration, it is shown by thick lines in fig. 13. Thus, the path joining the events 1, 3, 4 and 7 is the critical path and the activities lying along the critical path are known as the critical activities. 

(5) Activity times and floats: The activity times and floats of every event are then worked out. The floats may be divided into following three types: 

(i) Total float: The term total float, briefly written as TF, is used to mean the full amount of time by which a particular activity can be delayed without causing any effect on the completion period of the project.

(ii) Free float: The term free float, briefly written as FF, is used to mean the amount of time by which the start of the activity may be delayed without interfering with the start of any succeeding activity. Thus, it is clear that the free float is smaller than the total float.

  • Interfering float: The term interfering float, briefly written as IF, is used to mean the difference between the total float and the free float of an activity. If full use is made of IF, the subsequent activities in the chain will become critical and if it is exceeded, the duration of the project will increase. 

Thus, from the study of the network, the earliest start time and the latest start time, briefly written as EST and LST respectively, of every activity are worked out. Following equations show the relationship between various activity times and floats:

EST= EFT of a tail event

EFT= EST+ duration of activity 

LFT=  LFT of head event

LST=  LFT- duration of activity 

TF= LFT- EFT or =  LST -EST 

FF= EST of following activity -EFT of activity 


Significance of floats: 

The path which has zero floats is called the critical path. Total float affects both the previous and the following activities. For problem shown in fig. 13 and table 1, total float of activity C is 4. Its significance is as follows:

When TF is zero, it indicates critical activity and cannot be delayed. If it is delayed, then it will delay the whole project. When TF is negative, it indicates that the project is bound to be delayed by the same period of delay. Negative float is considered as zero.

Free float (FF) is always smaller than TF. It can be used without affecting subsequent activities. If it is necessary to increase the duration of time, independent float (IF) can be used without necessitating any re-planning.

Thus, floats represent following two points: 

  • Under-utilized resources
  • Flexibility of an activity. 

(6) Scheduling:

A schedule is then prepared showing all the time and float calculations. Table 1 shows the scheduling of activities of the network under consideration. 

Table 1: scheduling of activities

The critical activities i.e. activities without any float can easily be identified and if the project is to be completed in time, the critical activities should not be delayed. under any circumstances. Similarly, for other activities, the permissible delay can be ascertained and float time for such activities serves as a safety margin. Thus, the tolerable delays of activities along non-critical paths can be worked out. The information and data obtained in scheduling of activities help a lot to the construction manager for planning the working of the project for its successful completion in time.

Advantages Of Critical path method, CPM 

The adoption of CPM technique in the construction industry is rapidly increasing because of its following advantages. 

  • If something goes wrong with the planning of project, it can be easily identified and then, concentration may be made to correct the same.
  • It assists in preparation of the most economical time table for all the operations of the project.
  • It assists in the selection of the best combination of equipment and labour so as to finish up the project in time.
  • It determines the activities and operations which are to be controlled with respect to their time of completion for the successful finish of the project as a whole.
  •  It helps in working out the effect of variations such as extra work, change of order of work, etc. upto the time of completion and upon the cost of project.
  •  It is an open ended process giving sufficient flexibility to the management  to suit their requirements and obligations.
  •  It makes the most economical use of the available resources i.e. labour, equipment, finance, etc. 
  • It permits systematic reviewing of the project at various stages and accordingly, the allowances may be made to accommodate uncertainties which were not  thought of in original planning.
  •   It rationalizes construction costing and financing. 
  • It shows the activities and operations which can be reasonably delayed without showing any impact on the final completion period of the project.
  •  The study of information and data available from this method suggests alternative schemes also.
  • The use of this method makes it possible to work out the optimum project duration. It merely represents a logical mathematical model of the project.

Uses of CPM

The CPM is becoming useful in a variety of industries, especially building industry. The study and knowledge of this new technique is, therefore, of vital importance to engineers, surveyors, contractors and clerks of works.

 In addition, this technique is becoming popular among the professionals who are intimately related with building industry such as financiers, cost accountants, superintendents, office managers, etc. 

As time passes, the CPM technique may also become a useful tool in the hands of lawyers because it will certainly provide a sound mathematical approach to work out compensation for delays, extra items, wage changes, etc. In general, it may be stated that the CPM will be useful in all activities which require planning, scheduling, comparison of alternatives, finance, cost recording and efficient management.

Application of CPM in project management 

The application of CPM in the project management is becoming essential as it provides the inter-linking and determination of inter-dependency of various activities involved in a project. It further provides a diagrammatical form of the project as a whole and from the study of CPM chart, it becomes easy to identify critical works or items.

Further, the possible bottle-necks likely to hamper the progress of the project can be visualized well in advance and remedial measures may be taken to set the matter right in time. 

Following are the four important stages of a project where the CPM can be applied:

(i) Planning of the project as a whole. 

(ii) Planning of individual tenders for various stages involved in the project.

 (iii) Scheduling for the invitation of various tenders. 

(iv)  Planning of various stages in the construction work and determining extension of time, if necessary. 

CPM  for determining extension of time

 In all contracts, it is generally agreed that the contractor shall be given an  extension of time for reasons which are beyond his control. Such causes can be of the following categories:

 (i) acts of nature; 

(ii) delay in supply of tools, if any, by the owner; 

  • late handling of site to the contractor; 
  •  ordering of extra items not covered under the contract;
  • supply of materials by sub-contractors appointed by the owner;
  • unforeseen circumstances; etc. 

In the conventional system, each item contributing to the delay of work is considered separately by the deciding authority and such an important issue of extension of time, as such, is decided in an ad-hoc manner without any scientific approach to the problem. On the other hand, if the contract is based on the CPM technique, the CPM network will provide a guide line for calculating delays and analyse their overall effect on the completion of the project.

 Hence, the contractor will be prevented from claiming an extension of time on the grounds that cannot be on the CPM network.

 At the same time, the deciding authority will get ample data, information, and facts to recommend suitable extension of time. Thus, justified the deciding authority will be in a position to arrive at a decision that will be fair, judicious, sagacious, and sound in nature. 

Difficulties in implementation of the CPM 

Following are the three important difficulties which are experienced in implementation of the CPM in this country: 

(1) Lack of knowledge of the subject 

(2) Literature on the subject

(3) Unwillingness of contractors. 

(1) Lack of knowledge of the subject:

The persons who are supposed to know the subject are not given opportunities, either at study level or in practical Seld and hence, the engineers, surveyors, etc. are not trained for the successful implementation of the CPM technique. 

(2) Literature on the subject:

This new technique is widely and exhaustively explained in various books on the subject by foreign authors.

(3) Unwillingness of contractors:

The structure of the construction activity in poor countries is such that it does not inspire the contractors to adopt this technique. The important factors leading to the unwillingness of contractors for this technique are as follows: 

  • Capital of contractor:

Most of the contractors start the work with limited capital of their own and they assume that there will be free circulation of this capital among various contracts under them as anticipated, which, in most of the cases, does not come true.. 

(ii) Expected return on investment: It is the natural tendency of a contractor to employ his key labour force and equipment for items giving him more return, rather than to those items where the expected return for such key labour force and equipment will be less. 

(iii) Financial soundness of the contractor: In most cases, it is found that the contractors are granted projects which are much beyond their financial ability of the contractor. Hence, such contractors will not be able to invest the money as planned in the CPM network. 

Let us hope that the difficulties encountered above are solved in near future and the advantage of this technique is taken by engineers, surveyors, contractors, and all others concerned with the building industry all over the world. The sooner it occurs, the better will it be for the building industry and engineering profession. 

IF you want any Numerical and Practical solutions to CPM please contact me. Thank you!

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