FORWARD & BACKWARD PLANNING5.1

PLANNING

  A network diagram can be built in one of two ways. One approach would be to start from an initial or starting event and build up the events and activities until the end event is reached. In this process the planner asks himself, the questions "what event comes next" and "what events can take place concurrently?". However it is also possible to think backward. This means having the goal in view the planner works backward asking himself "if we want to achieve this what should have taken place?" In this manner we can come to the initial event. In practice however we may not strictly adhere to either the forward planning or backward planning but a combination of both the systems may be adopted the network being traversed back and forth several times until it is found satisfactory.

 

5.2

Earliest event time

  An event is the beginning or an end of an activity. In a network where each activity is given a duration, we can speak of the time when an event can be said to occur. Consider the network below:

  In this network event 1 stands for the beginning of activities 1-2 and 1-3 and we can say that event 1 occurs at time 0. Event 2 represents end of activity 1-2 and the beginning of activity 2-4. Let us assume that the figures shown near the activity is the duration in weeks that the particular activity requires for its execution from beginning to end. Then event 2 can occur at a time equal to 6 weeks. So the event time for event 2 is 6. These event times are shown on top of the nodes. When an event is connected by more than one activity the calculation of event time needs extra care. Consider the event 7 which is connected by two activity paths 1-2, 2-4, 4-7 and 1-3, 3-5 and 5-7. By choosing the first path 1-2 and 4-7, event 7 can occur at TE = 32 weeks. By definition no event can be a considered reached until all activities leading to it are completed. Therefore event 7 cannot be considered reached until activities 1-3, 3-5 and 5-7 which require longer time for completion are completed. The earliest time for occurrence of event 7 is therefore 32 weeks and not 29 weeks. In other words the earliest time of occurrence of an event is that which is obtained by considering that path which requires the longest time for completion. Consider event 9. There are again two alternative paths for reaching the event 9 : One along 7-9 and the other along 3-6, 6-8 and 8-9. The first alternative path requires 44 weeks and the second requires 46 weeks. The longer of these two viz. 46 weeks is therefore the event time of event 9. A simple rule can be formulated for convenience from the above example.

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  Firstly to evaluate TE Tor any event.TE( successor event) = TE(predessor event)+ TE(activity). The TE for successor event ±e obtained by adding to the event time of predecessor event the time required by the activity connecting the two events. When there are more than one predecessor event for a successor event the rule is  TE(successor event) is the maximum of TE(predeces8or)+TE(activity). The TE

is  shown as indicating that this is obtained by "Forward Pass".5.3

Latest Allowable Occurrence Time

  In 5.2, we have considered the idea of "Earliest Event Time". Now we consider another idea regarding the estimate of time. The latest time by which an event must occur to keep the project on schedule is called the "Latest allowable occurrence time". This is denoted by TL. To understand this idea, let us assume that it has been agreed to complete a given project within a certain allotted time called the "contractual obligation time". This time refers to the occurrence of the end event. For calculating the latest allowable occurrence time, we have to start from the end event.

 

  Let us assume that 4 is the end event. Now, consider the event 3. The activity 1-4 which itself consumes 19 weeks has to be completed before event 4 can occur. Therefore if the event 4 has to occur at the required time the latest allowable time for event 3 to occur is (45 - 19) =26 weeks.

5.4

In a network where an event has more than one predecessor events one has to be more careful. Consider the above network. Let us assume that in this case the earliest event time of the end event and the given project time are the same. (This may not always be the ease. The management may require that the completion time for the project be reduced. However such eases will be discussed later The earliest event times for all the events have been worked out. The latest allowable occurrence time for the project i.e. for the end event to occur is 42 weeks. Now going backwards, consider event 5. The activity 5-6 requires 6 weeks for completion. Therefore the latest occurrence time for event 5 is (42 - 6=) 36 weeks. Note that for event 5, the earliest event time and the latest allowable occurrence time are the same. Now let us go to event 4. The event 4 can be approached in two ways :

  (i) via 6-4 and (ii) via 6-5 and 5-4. Alternative (i) will give the latest allowable occurrence time for event 4 as (42-8a=) 34 weeks while as alternative (ii) will give the latest allowable occurrence time as (36-4=) 32 weeks being the latest allowable occurrence time of event 5 minus the duration of activity 5-4. Thus we now have two latest allowable occurrence times for the event 4 viz. 34 weeks and 32 weeks. For completion of the project in time obviously the earlier of the two times will be necessary and therefore the latest allowable occurrence time for event 4 is 32 weeks.

5.5

Rule for evaluating latest allowable occurrence time for events.

  Latest allowable occurrence time for a predecessor event is the lowest of the values obtained by subtracting the duration of an activity from the occurrence time of the successor event.

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5.6 Let us now prepare a network diagram for a small project and study it in detail. A

company has decided to construct a guest house of its own on a plot of land it already has in its possession. We want to prepare a network diagram for this project. Let us first list out the various activities involved.1. Get requirements finalized.2. Survey the plot.3. Get an outline plan from Architect.4. Get the outline plan approved.5. Get rough cost estimate.6. Approval to estimate.7. Get detailed plans and estimate from Architect.8. Approval to detailed plans and estimate.9. Approval to plans from Municipality.10. Prepare tenders for construction of building.11. Invite tenders for construction of building.12. Fix agency for construction of building.13. Construct building.14. Get detailed drawings for furniture.15. Invite quotations for furniture.16. Fix agency for furniture.17. Order furniture.18. Prepare furniture.19. Select light fixtures.20. Order light fixtures.21. Get light fixtures.22. Get drawings for garden.23. Engage gardener.24. Prepare garden.25. Install furniture.26. Install light fixtures.27. Opening ceremony for the guest house or site clearanceNote that in the overall project, construction of the building is only one of the activities requiring 24 weeks duration. This single activity itself can be considered as a project in itself and a separate network diagram can be prepared for it.  For the project under consideration, the "Earliest Time" (i.e. the earliest time that an event can occur) and the "Latest Time" (i.e. the latest allowable occurrence time at which an event may be allowed to occur) for each and every event have been worked out and shown on the Network Diagram itself. As already discussed, the Earliest Time is obtained by the forward pass and the Latest Time is obtained by the backyard pass. instead of writing TE and TL for earliest time and latest time respectively it is convenient to show them in a rectangle with an arrow pointing forward and a rectangle with an arrow pointing backward respectively as shown below :

 Means Earliest Time of 23.

 means Latest time of 23.

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5.7

The first step in the whole process of network planning is of course to draw the network diagram. In order to make use of this diagram for planning this network diagram must now be analyzed. The analysis of the network diagram will provide data which will enable the planner of the project as well as the project manager to set up a precise schedule for the project and monitor and control its progress. The analysis of the network is relatively simple in principle but involves the performance of many arithmetical calculations. The number of calculations will depend on the size of the network and the amount and type of information required from the analysis. The calculations may be carried out manually or by computer according to circumstances.

5.8

The first step in the analysis of the network diagram is to calculate the earliest possible and the latest allowable occurrence times at which each event can occur. We have already seen in paras 5.2 and 5.3 how these times can be calculated. However the procedure is once again summarized below as it can bear repetition. The earliest time at which an event can occur is determined by the longest time path from the start event upto the event in question because there must be time to complete the longest chain of activities before that event can be reached. The latest time an event can occur if the end of the project is not to be delayed is determined by the longest time path from the terminal event back to the event in question, because time must be allowed to complete the longest chain of activities from that event to the terminal event. The earliest time of the start event is taken as zero. The earliest possible time for each succeeding event path can be calculated by successively adding intervening activity durations. Where there is more than one possibility for the earliest time of an event representing more than one path leading to that event the largest total time (representing the longest path) is the earliest event time. i.e. Earliest event time = maximum of earliest time of preceding event + intervening activity duration. If the end of the project is not to be delayed beyond the earliest completion time, the latest time for the terminal event must be the same as its earliest time. The latest allowable time tor each preceding event along each path. backward from the terminal event can be calculated by successively deducting intervening activity durations. Where there is more than one possibility for the latest time of an event, representing more than one path back to that event, the smallest possible time (representing the longest time path to that event) is the latest allowable occurrence time for the event, i.e. Latest allowable occurrence time of an event = minimum of latest occurrence time of succeeding event - intervening activity duration.

5.9

For each event of the Network Diagram in Fig.5.6. The earliest event time and latest allowable occurrence time can now be worked out. The table below shows both these times for each event :

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EVENT SLACK & FLOAT

5.10

EVENT SLACK

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  It may be seen from the above table that there are certain events for which the earliest possible time of occurrence and the latest allowable time of occurrence are the same. These events and the activities connecting these events are said to be critical. There are other events for which the earliest possible occurrence time and the latest allowable occurrence time are not the same. The activities between such events are non critical. Such events have a slack which is simply the difference between the latest allowable occurrence time and the earliest possible occurrence time of an event. The idea of slack is quite easy to understand. In the above example, event 25 can occur at the earliest possible time of 27 weeks after the beginning of the project. However this event has the latest allowable occurrence time of 41 weeks meaning thereby that this event nay occur at 4l weeks from the start of the project without affecting or delaying the completion time of the project.

 

  This means that even if the event occurs at any time between 27 weeks and 41 weeks from the start of the project the completion time of the project of 42 weeks will not be delayed. Event 25 may not occur at the earliest possible time of 25 weeks from the start but may be delayed upto 14 weeks without affecting the total project time.

5.11

The slack for each event can now be verified. Slacks for all the events worked out in the example may now be verified. There are quite a few events which have a slack.

5.12

An activity is critical if there is no flexibility at which it may be carried out. That is to say if(a) it is immediately preceded and followed by a critical event and (b) the difference between the times of the events immediately preceding and immediately succeeding the activity is equal to the activity duration. The chain of critical activities from the start event to the terminal event is the longest time path through the network. This chain or sequence of critical activities governs the length of the project as a whole and is therefore called the "Critical Path". The project can be shortened only by shortening the critical path and effort applied to shortening the non-critical activities will have no effect on the length of the project. Event times are of great value because instead of having only one or a few target dates to work to in the project, the event times provide target dates for each and every activity in the network. This makes control much easier and helps to ensure that the forecast project completion dates are achieved.

5.13

Now from events let us divert our attention to activities. The project manager will always want to know at what times activities can be started and finished. In case of activities we can work out the following different times,(a) Earliest starting time :An activity cannot start until the event from which its arrow emanates has been reached. The Earliest Starting Time for an activity is therefore the earliest  time of the preceding event.

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(b) Earliest Finishing Time :The earliest finishing time for an activity is easily its earliest starting time plus its own duration.(c) Latest Finishing Time :The activity must finish before its terminating event is reached. The Latest Finishing Time is therefore the latest allowable occurrence time of the terminating event.(d) Latest Starting Time :The Latest Starting T k of an activity is its Latest Finishing Time minus its own duration. For each activity therefore there are four times to be calculated.(i) Earliest Starting Time = earliest possible time of occurrence of the preceding event.(ii) Earliest Finishing Time = earliest starting Time + Activity duration.(iii) Latest Finishing Time = latest allowable occurrence time of the following or succeeding event.(iv) Latest Starting Time = latest allowable occurrence time of the succeeding event minus activity duration.

5.14

Activity Float

  (a) Total FloatThe total float available for an activity is the total available flexibility within which the activity must be carried out if it is not to affect the overall duration of the project. In order to avoid affecting the time for the project, each activity must be carried out between the EST and LFT. The total flexibility for each activity is therefore the interval between the EST and LFT minus the activity duration i.e.TF = LFT - EST - DTF = LFT - (EST + D)TF = LFT - EFTThe total float for an activity is the difference between its earliest and latest starting times. The critical activities are those with zero Total Float.(b) Free Float :The free float for an activity is the flexibility in the time it may be carried out if all immediately succeeding activities are carried out at their EST i.e., the time the activity can be delayed without delaying any succeeding activity. If the succeeding activities start at their EST , then the activity under consideration has flexibility only to the extent that must be carried out in the time interval between its own EST and the EST of the succeeding activities. The free float for an activity is therefore this time interval minus its own activity duration i.e.Free Float = EST succeeding - EST - D= EST succeeding - (EST + D)= EST succeeding - EFTThat is to say that the Free Float for an activity is the difference between its EFT and the EST of the succeeding activities. The free float for an activity can never be greater than its Total Float. Total Float is always associated with sequences rather than individual activities and when advantage is taken of  the Total Float associated with one activity this will reduce or wipe out the Total Float associated with subsequent activities in the sequence.(c) Independent FloatThe Independent Float for an activity is the float which does not affect the float of a succeeding activity nor is affected by the utilization of the total available float of the preceding activity. It is the difference between the latest

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finishing time for the preceding activity and the earliest starting time of the succeeding activity minus its own activity duration. Independent Float = TE succeeding - TL preceding - DThus the independent float identifies the activities which even if delayed or even if started late will not affect the total float of either the preceding or the succeeding activities.

5.15

Now in order to have a better grasp of these various times associated with activities and also the various kinds of floats for activities a hypothetical Network Diagram prepared for a Project is taken and all these times and floats are calculated in a Tabular Form below :

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5.16  

  It may be seen that activity 20-60 can start after a period of 2 weeks from the start of the project. The duration of the activity itself is only 4 weeks. The latest allowable occurrence time of the event 60 or in other words the latest finishing time of the activity 20-60 is 11 weeks. This means that this activity 20-60 can start at any time after 2 weeks from the start of the project and must be completed at 11 weeks from the start. This in turn means that there is a total time of 9 weeks for carrying out this activity. But the activity itself requires only 4 weeks to be carried out. This means that there is a spare time of 5 weeks available for the activity which is called the float for that activity. The activity may either start late or start early & finish late or may be carried out in such as way that it may even consume 9 weeks instead of 4 weeks.

 

  In order to understand what is meant by free float consider the above diagram. The free float is that part of the total float which can be consumed by it without affecting the EST of the succeeding activity. The time available from the EST of activity 20-60 to EST of the succeeding activity 60-70 is 2 to 10 i.e. 8 weeks. Out of this period the activity itself requires only 4 weeks for its execution; therefore the remaining period of (8 - 4 =) 4 weeks is the Free Float. Note that the total float is 5 weeks out of which the free float is 4 weeks.

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  Consider activity 6-8 in the above diagram. The preceding activity 4-6 has the latest finishing time of 15 weeks and the succeeding activity 8-9 has the earliest starting time of 26. The difference between these two is 11 weeks. The duration of the activity itself is only 7 weeks. Therefore the independent float is (11 - 7 =) 4 weeks. This independent float of the activity is such that it does not affect the latest finishing time of the preceding activity nor the earliest starting time of the succeeding activity.

5.17

We shall now take preparation of a network diagram for the construction of a guest house costing roughly about Rs 600,000/- consisting of ground and one upper storey with open foundation. Before we draw the network let us first list out activities involved.1 . Moving in.2. Line out.3. Excavation for foundation pits.4. Lean concrete.5. Fixing formwork.6. Fixing reinforcement for footings and plinth columns.7. Concreting of footings and plinth columns.8. Backfilling of foundations.9. Formwork for plinth beams.10. Reinforcement for plinth beams.11. Concreting plinth beams.12. Block work upto plinth.13. Murrum filling in plinth.14. Cement Concrete bedding.15. Reinforcement for ground floor columns.16. Formwork for ground floor columns.17. Concreting ground floor columns.18. Centering and formwork for first slab and beams.19. Fix reinforcement for first slab and beams.20. Concreting first slab and beams.21. Reinforcement for first floor columns.22. Formwork for first floor columns.23. Concreting first floor columns.24. Release centering of first slab and beams.25. Centering and formwork for second slab and beams.26. Block work for ground floor.27. Fixing reinforcement for second slab.28. Concreting second slab.

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29. Fixing of electrical conduits, drainage and water pipes on ground floor.30. Internal plaster on ground floor.31. Flooring on ground floor.32. Releasing centering of 2nd slab.33. Block work on first floor.34. Fixing electrical conduits, drainage and water pipes on first floor.35. R.C.C parapets on terrace.36. Curing of second slab.37. Internal plaster on first floor.38. Flooring on first floor.39. external plaster.40. Waterproofing on terrace.4l . Fixing door shutters.42. fixing aluminum windows.43. Internal painting.44. External painting.45. External drainage and water supply.46. Constructing septic tank and soak pit.47. Constructing compound wall and gates.48. Constructing garage.49. Constructing asphaltic drive way.50. Laying concrete tiles in open area.51. Site clearance.52. Spread sand.53. Move out.

5.18

In case of all these activities it is desirable, before starting preparation of the Network Diagram, to have a table giving quantity of work involved for each activity, the time required in man-hours, the number of men proposed to be employed and the number of days (duration) for which these men will be employed. A sample of such a table is given below. For preparation of such a table the task work for each trade like excavator, mason, carpenter, concreter etc. will have to be decided from previous experience and also the machinery available such as bulldozers, cranes, hoists, concrete mixing plants, dumpers etc.

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CALENDAR DATES

5.19 COMPUTING CALENDAR DATES

  In CPM planning all durations are in working days. However the plan has to be implemented by the calendar year, The working days can be converted into calendar dates by use of a specially set up calendar as shown below :

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6.1  The network diagram and the table give all the information about the activity times which is required for preparing a schedule for the project. The floats show to what extent the timing of the activities can be altered either accidentally or by design without delaying the completion of the project. The project planners therefore know from the results of the network analysis the extent to which they may vary the timing of any activity while preparing the schedule for the project and this is particularly helpful when considering smoothing the demand for resources which is discussed later.

6.2 A critical activity is one which is contained between a pair of critical events and whose duration is equal to the difference between the times of the two events. The critical path is the chain of events passing through all these critical activities. Therefore if the project is to be completed according to the network plan there is no choice in the timing of the critical activities as they have no float. 

6.3 The critical path forms the longest chain of activities from the start to the end event and therefore governs the overt time of the project. Along part of its length the critical path may be split into two or more parallel paths and even dummy activities may form part of the critical path. 

6.4 The identification of the critical path is clearly a very important first step in the planning and management of a project. Its great advantage is it will

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immediately direct planning and management attention to those activities which govern the completion date of the project.

6.5 The activities in the project can be ranked in ascending order of total float that is in ascending order of importance to the early completion of the project and these floats will range from zero upwards. Attention of the project manager will be required to be concentrated not only on the zero float activities but also on the small float "near critical" activities. These may be normally relatively few  say not more than about 5 per cent of all the activities in the network. 

6.6 The first attempt at analyzing a network may bring out a series of anomalies which will have to be rectified before a proper analysis can be carried out. These anomalies may include duplication in numbering of events or of activities or a "dangle" i. e. an arrow whose head has been left in the air instead of being connected to an event or loops going round and round. When these anomalies have been rectified by referring back to the network itself it will be possible to complete the analysis and identify the critical activities. This will further afford concentration of attention on those areas of the project where refinement of logic and durations is most important to the preparation of a successful plan for the execution of the project .

6.7 Negative float

  It may often be the case that external factors may require fixed target dates to be imposed on intermediate events in the network as well as on the start and end events. Such imposed target dates as an effect of certain unavoidable external factors may now be substituted for the latest times for those particular events. If the target date for any event is earlier than the calculated latest time for that event then the network as it stands may not meet the target date. Some activities will have a negative float and it is then these activities which must be shortened if the negative float is to be eliminated and target dates are to be met. Examples of such early target dates for particular activities are :(i) In a multistoried building it is many times required that the ground floor (many times to be used for shops) is complete in all respects by a particular date irrespective the completion of the whole building.(ii) Suppose a concrete plant being used on a particular work is required to be shifted to another far away place by a particular date and therefore all concreting work is required to be completed by a particular date apart from the programme of completion for the whole building.(iii) For an industrial shed with attached offices, the sheds have to be ready by a particular date as the machinery has to be housed in the shed. The office portion may be completed at a later date.(iv) Particular foundations have to be completed by particular dates otherwise the pits get flooded on account of high tide. Supposing in the above network diagram event 50 is required to occur at 6 weeks from the start. It may be noted that the event 50 had a slack of 1 week (TL = 8,  TE = 7). Now TL  being = 6, TL - TE = 6 - 7 = -1 i. e. the event 50 will now have a negative slack of 1 week. The activities 30-50 and 10-50 will therefore be affected and each of these will now have a negative float of 1 week. Since TE = 0 cannot bechanged, this will only mean that either activity 10-30 or 30-50 or both together will have to reduce their duration by 1 week. In practice a number of such restraints may operate creating negative floats. Such negative floats must be eliminated before the network is taken up for analysis.

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6.8 One of the steps towards refining the network will be to confirm that the estimates of duration for the critical and near critical activities are accurate. Usually there is a tendency to provide an allowance for contingencies in preparing these time estimates. In other cases it may be necessary to take up some of the long duration activities and "explode" them into a more detailed network to ensure that the original estimate is achievable in practice. For example for the construction of the building proper in the project "Guest House" vide para 5.6, the activity "Construct building" is assumed to require a duration of 24 weeks. It may be necessary to prepare a separate network diagram for this activity to see that the time estimate of 24 weeks is reasonably accurate.

6.9  Another important point is re-examination of the methods of working implied by the logical sequences of critical and near critical activities in the network. The best methods of operation including all the possible alternatives and change in the resources and machinery utilized will have to be carefully examined. The cheapest method for each activity may not always be chosen if a costing exercise can demonstrate that the higher cost of an alternative method will be justified by savings resulting from earlier completion of the project. One such example is pre-casting instead of in situ work. Any change in the methods or procedures may require a change in the logical relationships between arrows in the network which would therefore have to be revised. Another refinement of the network diagram can be made by replacing single arrows or groups of arrows with overlapping arrows and possibly considerable saving in time can be achieved through this process of overlapping. The duration of most activities can usually be reduced at some additional coat. If the cost savings, direct and indirect, from advancing the project completion date are greater than the extra coat of accelerating a particular activity then the extra coat may be worthwhile. By a process of alternately refining and re-analyzing the network, during which the apparent critical path may shift from one sequence to another, a network may ultimately be evolved which may meet the objectives. For a large project (say 1000 activities) the network may have be analyzed five or six times. The network at this stage will still not take into account any limitations in the availability of resources which may exist in actual practice.

PROJECT COST ANALYSIS7.1 It is worthwhile examining how a decrease or increase in the total duration for

the completion of a project may cause increase or decrease in the cost of the project.

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7.2 Consider the example of digging the foundation for a water tank. Let us assume that the activity for this particular operation in a network diagram assumes a duration of 10 day. It has been assumed that the excavation work is to be carried out with one machine and one operator. Let us assume that that hiring charges for one machine are Rs 200/- per day and the operator cost is Rs 40/- per day. The total cost for the activity is therefore Rs 2,400/-. Let us assume that by using one additional machine and one additional operator at the same daily rates, the job can be finished in 6 days. The total cost will then work out to Rs 2280/-.  So there is no advantage in employing an additional machine with an operator, However assuming  that the operator is prepared to work over time for half a day at an additional payment of Rs 40/-  but without any extra charges for the machine  and the work can be completed in 7 days, then the total cost will work out to 7 x 80 = Rs 560.07 x 200 = Rs 1400Total = Rs 1960which will give a saving of 3 days in time and also a saving of Rs. 440/_ in cost.  Note that in these discussions we  have taken into consideration only the direst costs involved in carrying out work of that activity.

7.3 However usually the cost of a project goes up if the project time is reduced. The graph below is therefore normal.

 

  Associated with a given activity is a normal time for its completion point 'A' and a crash time for its completion point 'B'. The crash time limit Imposes the condition that the duration cannot be further reduced. This is represented in the figure by making the coat infinite at the crash time limit. Similarly by extending the duration beyond the normal time the cost will not be reduced. This is shown by making the curve asymptotic at point 'A'. In fact in the normal course of activities the cost may go up with further decrease in time beyond 'A'. An approximation may be made to make things easier and that is to assume that the variation of cost between points 'A' and 'B' is not curvilinear but follows a straight line.Then the crash slope =  (crash cost - normal cost) ÷ (normal time - crash time)The cost represented by the curve or straight line in the above diagram is the direct cost of executing the job such as those of labour, machinery, material and other resources.

7.4 On a big project however there are other indirect costs involved such as general administration, overheads, depreciation, insurance etc. It is reasonable to assume that these indirect costs are linearly proportional to time as shown in the graph below :

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  The total project cost would be the sum of the direct and indirect costs. If the direct and indirect costs and the total cost are shown on the same graph, the resulting diagram will be a curve.

 

  Such a curve will have a point where the tangent would be horizontal. At this point the total of direct and indirect costs will b® minimum. The duration corresponding to this point will be the optimum duration time and the cost the optimum cost for the project. In the process of decreasing the project time, the aim is to achieve the optimum duration.

7.5 Let us take an actual example.

 

  A particular project has a normal duration of 18 days and the normal cost is Rs 15000/- The crash duration is 12 days and the crash cost is Rs 30000/- Let us assume that the direct cost variation between these two extremes is curvilinear tending to become asymptotic at both the points. The direct cost which is

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directly proportional to time has a slope of Rs 5,OO0/- ÷ 6 per day. The total cost becomes a curve which changes slope from +ve to -ve at a point which has a duration of 16 days and the optimum cost is Rs 2,300/- Such cost considerations are of great importance in big projects involving direct and indirect costs.

7.6 Contracting a network :

  It is now necessary to consider by using the network diagram how the project duration can be reduced and at what cost. It is necessary to have a systematic approach for reducing the project duration at the minimum extra cost.

 

  The table below gives the various activities, their normal durations in days, their corresponding normal costs in Rs their crash durations and the crash costs. A straight line variation for the cost-time relation is assumed so that for each activity the cost slope is constant. The cost slope for each activity is also given in the table.

 

  The indirect cost is Rs l60/- per day. The normal duration For the project as worked from the Critical Path is 25 days. The critical path is marked in red and the activities forming the critical path are marked red in the above table. The network diagram is drawn to scale below :

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  For the normal duration of 25 days the total direct cost is Rs 7,560/- Supposing we want to find out the minimum time during which the project can be completed. Taking the crash times of ail the critical activities, the minimum duration works out to 15 days. (Note that even after crashing all the critical activities the critical path remains the same). The direct cost for this duration works out to Rs 9,330/-. It must be noted that it is not necessary to crash the activities which are not on the critical path unless it becomes necessary to crash them as they now come on the critical path. The management may decide to reduce the project time for two reasons (i) If the completion date is required to be brought down. For example if a certain building has to be occupied on a particular auspicious day, then to adhere to that target date crashing of certain activities may become necessary. In such a case it may be necessary to crash the activities by incurring extra expenditure and the management would like to know how much will be the extra cost involved to expedite the project and complete it by the given date. (ii) The second reason could be to reduce the overall cost itself. This may become particularly relevant when the indirect costs in proportion to the direct costs are quite high. In the above example, cost slopes of activities 1-2, 2-3, 2-4, 3-4 and 4-5 are lower than the indirect cost of Rs 160/- per day. In such a situation the management will be interested in cutting down the project duration thereby reducing the total indirect cost and therefore the total project cost. This is done systematically in the following manner.The first step is to identify the activities which have cost slopes less than the indirect cost. These activities must be on the critical path so that by reducing their durations we can reduce the project duration. In the above case these activities are 1-2. 2-3. 3-4 and 4-5. Now amongst these activities we must consider first that activity which has the lowest cost slopes. Therefore arrange these activities first in the order of increasing cost slopes. Activity 1-2 has, the lowest cost slope of Rs 40/- per day.This activity has a normal duration of 3 days and crash duration of 2 days so we cut down 1 day by crashing this activity and the extra cost is Rs 40/-. The next activity in the order of increasing cost slopes is activity 4-5 which has a cost slope of Rs 85/- per day. The maximum saving in time possible by crashing this activity is 2 days and the extra cost involved is (85 x 2=) Rs 170/-. It is necessary to examine whether by reducing the durations of these two activities a new critical path has come up or a sub critical path is established. Now we take activity 2-3 which has a cost slope of Rs 90/- per day. This activity can be contracted by 2 days at an extra cost of Dh.180/-. Thus in crashing 3 activities 1-2, 4-5 and 2-3, the project duration has been cut down by 5 days at a total extra cost of (40 +170 +180 =) Rs 390/-. The new network diagram is as under :

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  The next activity in order of increasing cost slopes is 3-4 with a cost slope of Rs 100/- per day. This activity duration can be cut down by 4 days from 8 days to 4 days. However if we have a look at the network diagram we shall see that activity 2-4 has a total float of 3 days only. Hence If we cut down the duration of activity 3-4 by more than 3 days, it will affect activity 2-4. By reducing the activity duration of activity 3-4 by 3 days, the project duration can be reduced by 3 days at an extra cost of Rs 300/- but at the same time another critical path or a new sub critical path gets formed as shown below :

 

  Now if we want to contract activity 3-4 by one more day we shall have to simultaneously crash Activity 2-4 which is also now on the critical path. Therefore the cost slope of crashing activity 3-4 beyond 3 days will have to include the cost of crashing activity 2-4 also and the two cost slopes together will be (55 + 100 =) Rs 155/- per day. Therefore the duration of the project is now reduced to 16 days by crashing activities 2-3 and 2-4 simultaneously by one day. Thus the project duration has bean reduced by 9 days at a total extra cost of (390 + 300 +155 -) Rs 845/-. Total saving in indirect cost will be (160 x 9 =) Rs 1440/- i.e. a net saving of (1440 - 845 =) Rs 595/-. The total cost for the project is direct cost Rs 7560/- (normal cost) + + 845 + indirect cost of (160 x l6 =) Rs 2560 /- i.e. Rs 10965/-. The normal cost would have been direct cost Rs 7560/- + indirect cost (25x 160 =) 4000/- i. e. total of Rs 11560/-. Thus a saving of a duration of 9 days and a saving in the overall cost of (11560 - 10965) = Rs 595/- has been obtained by the above method. The value of the network diagram to management is very apparent from the above discussions.

7.7 In analyzing the cost of a project as far as each activity is concerned three factors have to be borne in mind :(i) its normal duration(ii) its crash duration and(iii) the cost slope of the activity i.e. the extra cost per unit time for reducing the normal activity duration to the crash duration.It is also to be borne in mind that though for simplification of the analysis even though we have assumed that the slops of crash curve is constant (i.e. a

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straight line graph) in actual practice it may not be so. In other words the cost of crashing may not be the same for first day as the last day. So far as the project as a whole is concerned the following important points should be noted:(a) The normal project duration is obtained by summing up the normal duration times for all activities along the critical path.(b) The crash project duration is obtained by summing up the crash duration times for all activities along the new critical path. The new critical path obtained on the basis of crash durations for the activities may be quite different from the one obtained by considering the normal durations. There may be even 2 or more critical paths obtained.(c) The total project cost will depend upon the time for the completion of the project. In general if the indirect costs are higher than at least the cost slopes of some of the activities, the total cost of the project including the direct and indirect costs can be reduced by crashing. (d) In crashing activities, we consider them in the order of increasing cost slopes, the activity having the lowest cost slope being the first to be taken up for crashing. It in also necessary to remember that crashing of certain activities may make critical certain activities which were not critical and in working out the extra cost in crashing this may require consideration of the extra, cost of crashing more than one activity simultaneously.

7.8 From the above discussions it will be clear that two types of problems can be solved by resorting to the above described methods : (i) Given a project for which the normal completion time has been worked out. The problem is if this total time is to be reduced by a certain number of days how should it be done so that the extra cost involved is the minimum and also to find out the extra cost involved. In such a kind of problem we consider only the direct costs totally ignoring the indirect costs.(ii) Given a project with the normal completion time. Now knowing the normal cost, crash duration, crash cost, cost slope and indirect cost per day, the problem is to find out the optimum duration so that the total cost including the direct costs and indirect costs, is the minimum and also the optimum cost.

7.9 It must be noted that in all the above discussions it is assumed that the management has available the primary data regarding the costs. If not, it will be very necessary for the management to collect such data in advance from projects already executed. After completion of any project it will be desirable to analyze each and every activity & work out with good accuracy how much it has actually cost to execute a certain item of work and split it further into cost of labour and cost of material. On the basis of these job costs average costs for the usual items of work will have to be worked out & these will form the basis for any network analysis. These job costs will have always to be kept up to date by taking into account the prevailing rates for materials & labour. Also the indirect costs based on actual will have also to be updated so that the optimum duration worked out is not fictitious but real.

UPDATING NETWORK8.1 A network diagram is normally prepared before a project begins. However in a

large number of projects this original network diagram needs to be altered or modified as the work progresses. There may be a number of reasons necessitating such an alteration or revision e.g. (i) non-availability of certain materials such as cement, steel. (ii) non-availability of certain machinery at the proper time.

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(iii) Strike by labour.(iv) Unprecedented rains and floods.(v) Unexpected availability of certain additional or new machinery.(vi) Availability of additional labour force.The network if it is to remain dynamic must reflect these changed conditions. The network diagram must therefore be suitably altered or modified to take into account the new circumstances. This process of incorporating changes and replanting is Called "updating".

8.2 Consider the network diagram of a particular project drawn below :

 

  This is the network diagram as originally planned. The critical path and the earliest times for all events are as shown. The Critical Path is 10-20-30-50-60-70 and the completion time is 36 weeks. When progress was reviewed at the end of 12 weeks from the start it was observed that (i) Activities 10-20, 20-30 and 10-30 were completed.(ii) Activity 30-40 was in progress for 3 weeks and it was expected that it will take another 8 more weeks for completion. (iii) Activity 30-50 was in progress for 3 weeks and it was expected that this activity could be completed in 6 more weeks. (iv) Activity 60-70 could be completed in 7 weeks instead of 8 weeks assumed earlier.The above situation could be considered as leading to a fresh network diagram which would include the whole revised programme including activities already completed.

 

  According to this new network the critical path now lies along 10-20-30-40-70 and the project duration now is 35 weeks instead of 36 weeks as planned earlier.

8.3 The situation could be viewed from another point of view. It can be considered that a new project is being started at the end of 12 weeks with the given uncompleted activities as obtained at this time and the new activity durations as given now. The network diagram on the basis of this new concept has been drawn below :

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8.4 Interestingly the position as reviewed at the end of 12 weeks can also be shown on a bar chart.

 

  The bar chart shows the programme as originally planned and also the actual progress achieved at the end of 12 weeks. The reassessed duration for the completion of the unfinished activities have been shown by dotted lines. The actual progress achieved has been shown by shading below the programme. This diagram shows both the good points as well as the shortcomings of the bar charts in comparison with the Network diagram. The actual progress achieved against the programme can be vividly brought out but the interdependencies of the various activities cannot be well followed from the bar charts.

8.5 How often should a network be updated is an important question. The answer to this question will depend on the complexity of the project and the management. In general it can be said that in the case of projects whose overall duration is small, updating is done more frequently. The reason is that a few slippages occurring in such a small project may considerably affect the project as a whole. In large projects updating may not be necessary at the initial stages since a few slippages can be absorbed in the project. However as the project approaches completion, updating may have to be done more frequently. Of course in any project whenever reassessment is made as the project progresses and the time scheduling is altered as a result of the reassessment, updating will have definitely to be done.

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RESOURCE SCHEDULING9.1 One of the important aspects of an efficient management of a project is the best

use of the resources. A resource is a physical variable such as labour, finance, equipment, space etc. which will impose a limitation on the duration of the

project. When resources are limited (as they always are) and conflicting demands are made for the same type of resource a systematic method for the allocation of resources becomes essential. Resource allocation is nothing but a compromise between conflicting demands and this is done by judgement by the

project manager.

9.2 A project is to be executed for which a network diagram has been drawn. The earliest possible occurrence times and the latest allowable occurrence times

and the floats for the various activities have been worked out. Let us consider only one of the resources required for the project say the skilled and unskilled

labour. The project requires carpenters and coolies for the execution or the project and their requirement for each activity has also been worked out as

under :

 

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9.3  From the two graphs drawn vide Figs. 9.3.C and 9.3.D the variation in demand of the resources will be immediately apparent. On the 7th and 8th days the demand for carpenters is as high as 14 whereas on the 11th, 12th and 13th days it comes down to 2. If the carpenters and coolies are to be hired for the entire project duration of 22 days then for many day most of them will be idle. This will be very uneconomical and wasteful. It is necessary to utilize the resources in a fairly uniform manner. This can be conveniently done with the help of the

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network diagram and the table of analysis giving earliest and latest starting tomes, earliest and latest finishing times and the float for each activity in the project.

9.4 There are two basic approaches looking at the problem. In one approach the project duration is maintained as it is and the resources are allocated to make the demands as even as possible by using the float of the various activities. In the other approach the uniformity of demand on the resource is given more importance & the project duration is allowed to increase if necessary. In the first approach there is a constrain on the operation of resource leveling viz; the project duration time. In the second approach a constrain can be applied by specifying the limit of the available resource & allowing the project duration to exceed what was originally planned as project duration.

9.5 Let us first consider the first approach in which we try to smoothen out the demand on resources within the constraint of not altering the project duration. It is seen from the graphs that the maximum demand on labour is on the 6th, 7th, 8th, 9th and 10th days. The only way to reduce this demand is to utilize the floats of activities. It will therefore be necessary to examine whether during this period any activities have floats. We find that activity 3-0 has a float of 5 days and it is therefore possible to start this activity after the peak demand period. This activity can therefore start on the l6th day instead of on the 7th day (by utilizing the float of 9 days available for this activity). This will reduce the demand for carpenters on 7th and 8th days by 4 so that on these days now 10 carpenters would be enough instead of 14. We further find that activities 1-4, 4-6 and 6-1 together have a float of 5 days. The activities 4-6 and 6-9 may therefore start 5 days later so that the demand on carpenters is more evened out. The time scale network and resource schedule will then appear as under.

 

  The maximum demand on carpenters has now been reduced to 6 and that of coolies to 4. If the management desires to hire labour, it may employ 6 carpenters from 3rd to 16th day and 2 carpenters from 17th to 19th days. Similarly only 4 coolies may be employed for the whole project duration.

9.6  Table 9.6 below shows another technique used to level resources which is basically similar to the procedures set in computerized resource leveling applications. In this example the activities are listed in i minor - j major fashion and are considered for scheduling in this order which avoids invalidating the network logic. Activities are scheduled at their early Start time until maximum load conditions are attained when they may be deferred until the earliest future

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time that adequate resources are again available. In this manner the demand on resources is evened out.

 

9.7 Now let us consider the second approach where the more emphasis is on leveling of resources by extending project duration to the extent necessary. Assuming that only 4 carpenters are to be engaged then perhaps the duration of activity 2-8 which requires 6 carpenters can be increased from 4 to 6 so that the job can be carried out with 4 carpenters. Since the demand on carpenters

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from the 3rd day right upto the 10th day is 6 carpenters and again from 14th day to 17th day again 6 carpenters even after making full use of the floats available, the project duration may have to be increased by 6 days in all if not more than 4 carpenters are available for the project. In such cases the criterions are(i) delay jobs having large floats(ii) if two or more jobs are competing for the same resource their duration is then used as the second criterion. The job with the shortest duration is chosen since by doing so. the delay in the project duration will be minimum.

PLANNING & SCHEDULING10.1 By planning is meant reaching a decision as to "what" is to be done.

Scheduling establishes "when" something is to be done. The planning of a network is carried out in several stages. These can be outlined as follows :

(i) Networks describe an operation by presenting the interdependence of the component activities and the necessary requirements for their execution

and also indicate possible consequences. (ii) Networks can be further developed into bar charts as a basis for

planning of conventional construction programs.(ili) The network can be subjected to regular checking, analyzing and

updating in the light of subsequent experience or new developments. The planner is compelled to think logically because of the inherently sequential

nature of the network. (iv) The network clearly indicates the requirements needed for the execution

Of separate activities & helps develop resource allocation.(v) The network affords precise ordering and delivery of materials.

(vi) A network leads to better utilization of the available equipment.(vii) Networks make it possible to specify the performance of critical

activities on which construction time primarily depends & help to direct attention to measures which will enable the completion time of these activities to be attained and often reduce expenditure on unnecessary

expediting (such as overtime spent on non-critical activities). Expediting of critical activities is the best way to guarantee adherence to construction

time or possibly even to reduce it which in turn contributes to the saving in cost. Delay in the completion of a project is usually accompanied by financial

loss. (viii) Network planning makes it possible at any time during the execution of a project to detect quickly those activities which have not been completed

according to plan and to deal with effects of such alterations on the completion tine and possibly also analyze the reasons contributing to the

delay. (ix) Network planning makes it possible to select quickly those activities in

which an alteration of resources would result in changes of construction time and would help contractors to take steps necessary to make up for the

loss of time.(x) Network programme form the basis for optimization of construction time

and resource planning. (xi) Network programme offer construction Management a means of judging

the effect of decisions and also the possibility of recognizing in time the approaching bottlenecks in the execution of a project.

10.2 Activities forming a Network Diagram will be of different kinds. These can be grouped as under:

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(a) Construction ProcessesThese are what are usually easily understood as activities consuming time, manpower and materials.(b) Obtaining tools and site equipmentWhen executing projects quite often building equipment is involved which may have to be ordered, purchased, transported and assembled. All these require time and must be thought out well in advance and it is therefore desirable to treat them as activities. (c) Site organizationCertain equipment necessary for a job may have to be erected at site and may therefore be provided for as a separate activity.(d) Delivery of materialsObtaining of materials is an important operation and if not thought out in advance may cause bottlenecks and holdups and may therefore be separately provided for as activities.(e) Drawing design and calculationsThough of course these will mainly come from the Architects and Consultants sometimes detailed drawings if not available in time may cause holdups. It is therefore desirable to provide for them as separate activities.(f) Essential waiting periods During construction certain essential waiting periods like setting of concrete, drying of paint, become necessary to be provided for as separate activities.(g) Activities of unknown durationA project may involve certain activities for which it is difficult to estimate the durations at the planning stage. For example a foundation where ground water is required to be controlled. For the activity "control f ground water" a question mark must therefore appear at the planning stage. Perhaps at a later stage with more detailed information being available a better estimate of time for the "control of ground water" could perhaps be made.

10.3 However much time and effort are expended on setting up the plan and howsoever carefully a Network Diagram is prepared, all predictions may not prove correct nor will all contingencies be foreseen. As the project proceeds, it may transpire that some estimates of duration were in error, unpredictable delays may affect other durations, some logical relationships will not nave been quite accurately represented suppliers may default on delivery dates, absenteeism or strikes may reduce the availability of resources and so on. The Network plan provides a means of quickly evaluating the effect of such contingencies, monitoring the progress of the project and simplifying the management control. If an unforeseen contingency arises which causes the completion of an activity to be delayed beyond its planned date then so long as the delay to that activity is less than the float remaining available to it in the plan, there will be no direct effect on the completion date of the project although some re-planning might be required if the delay upsets a finely balanced resource schedule. Due to delays in completion of one or more activities a new critical path may develop. In such circumstances the Project Manager will be quickly made aware of the appearance of new critical path and he may think of the possibility of allocating more resources to certain activities so as to reduce their duration times and restore the project time to its original figure. Adjusting the plan following an unforeseen contingency is in effect re-planning. Although unforeseen contingencies do result in different sequences in the network becoming critical, there are many projects in practice in which once the real critical path has been identified. The same activities remain critical throughout the project duration. In many projects

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most of the activities have large floats so that it requires a substantial delay to a non-critical activity to cause a shift in the critical path. Delays when they do occur do not necessarily require a complete updating & re-analysis of the network . Only when an activity is delayed by an amount greater than its total float & it therefore becomes critical will it be necessary to to examine subsequent activities on the new critical path for the possibility of reducing their durations & thereby restoring their original planned project completion dates. A re-analysis of the network along with re-programming the resource might then be undertaken if necessary to evaluate the effects of accumulated changes and on the results of the re-analysis, a decision made on any remedial action required. Before the progress of the project can be monitored & feedback of progress information is required from the field operation. What is required is something simple namely that at intervals comparing activity by activity the information fed to the field managers from the network plan with the actual situation at the reporting date. This stresses the need to have the whole team to have the basic knowledge of the Network Method right from the top executive down to the site engineer. Once the personnel are trained to have a full understanding of the technique, It will be a gainful exchange by communicating to them all the information from the network which could have a a relevance to their work and decisions and having in return the actual progress vis-à-vis the plan for each activity. It must however be made clear to the field engineers that the float is not to be utilized by them but they must stick to the programmed dates for all activities. The field managers will therefore be reporting progress of each activity against the programmed date and the consumption of resources against estimated targets. The planning office puts all these reports together & monitor the current position of the project to the management.

10.4 Monitoring by computer.

    The scope for manual method of monitoring is limited to relatively small projects. Networks of a few hundred activities can be successfully analyzed by manual method but when resource analysis is required in addition the maximum size of network which can be successfully monitored is small. Computer programme of immense power and versatility are available to carry out the huge volumes of data processing and calculations associated with thorough time and resource analysis of large and complex projects. These programme provide the data processing and reporting facility for all network analysis operations that include : (i) time analysis (ii) resource scheduling according to specified sets of priority rules. Thus a computer can handle very complex situations involving (a) say 100 different resources with each activity employing different levels of these throughout the duration of the project.(b) specifying different levels of availability of each of these resources from time to time. (c) stopping work on any activity at any time and restarting when required.(d) modifying the priority rules to meet special requirements. (iii) Cost analysis linked with time analysis. (iv) analyze any part of the network. (v) provide n wide variety of reports such as (a) tables of activity times, floats, arranged in different groups. (b) bar charts showing each activity in its planned position against a time scale (c) summary reports for key activities and events 

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(d) resource utilization reports. 10.5  Management Control :

No management technique however useful will absolve management from the need to exercise control through making decisions. The technique by providing the relevant Information can enable management to make better Informed decisions and thereby exercise a finer degree of control than would otherwise be possible. The network plan which has evolved acts as a base against which to monitor progress. When things go wrong, the Project Manager can using a computer promptly assess the significance of the changing circumstances and quickly decide how best to restore the situation : say e.g. whether to use a machine instead of manual labour, whether to prefabricate instead of constructing In situ, which activities to accelerate to compensate for the failure of a supplier. These are the kinds of decisions the project Manager will be called upon to make to maintain control of the project. The monitoring of progress through the use of critical path planning technique will enable him to become aware at the earliest possible moment of the problem & contingencies which inevitably arise and to deal with them promptly and efficiently. In this way he will be able to direct the course of the project more assuredly towards the planned completion date and the achievement of the Management's objectives.

10.6 Even though use of Network Planning Techniques is becoming more and more common because of the greater complexities and ever-increasing sizes of projects, there is basically nothing new in this technique. The new technique only provides a visual and numerical means of doing what has been done in the past in embarking on a project, viz. fitting the constituent parts of the project together in their logical sequences and assessing the implications of these sequences on scheduling. These techniques do however enable much larger and more complex projects to be planned in much greater detail and with much greater precision than has been possible with the use of traditional planning systems. The technique provides not only a means of setting- up an initial plan for setting up the project in hand but also provides a system for coordinating the many activities in the project, monitoring their progress against realistic target dates for each and assessing well in advance the effects of unforeseen contingencies on the project plan as a whole so that the necessary action can be taken and the necessary degree of control can be maintained. The technique will not however of itself manage the project nor in any way relieve the management of the responsibility for making decisions. But with the use of Network Planning Techniques management has much more reliable information on which to base its decisions and is in a position better to assess the value to the project as a whole of alternative courses of action. Although the principles of the technique are simple enough, their application in practice is not without problems. Preparing the arrow diagram involves in a sense living through the project in advance. Decisions have to be made at this stage as to the methods and equipment to be employed. These are decisions which however have to be made eventually in any case and the Fact that they are taken early does not make them irrevocable. The arrow diagram incorporating these decisions yields data which are valid only so long as these decisions remain unaltered. The identification of the true sequences and the best way to represent them in the arrow diagram are ever-present problems despite the apparent simplicity of asking questions "what must precede?" and "what must follow?". Endeavoring to show all the interdependencies with an uncompromising exactness tends to lead to more

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detail than is really required. The boundaries of the project itself are not always clearly defined. Everything that is added makes the project larger and more unwieldy, anything that Is left out means a risk of overlooking some vital interdependency which might alter the whole situation. The only solution is to consider the situation on the broadest possible scale at the outset to make sure that everything which might conceivably affect the project has been taken into account. It may then be possible to exclude from the initial arrow diagram those parts of the work which have no vital influence on the project completion date. The first arrow diagram should be prepared quickly. Undue preoccupation with detail in the early stages may turn out to be effort wasted on non-critical part of the project. Refinement of an arrow diagram can come later. The sooner it can be prepared and analyzed oven in an imperfect form, the sooner the arrow diagram will fulfill its primary function of directing attention to the critical areas of the project. It is this feature of the technique and the ability to update quickly and bring to light shifts in the critical areas of the project which enables management to concentrate and control very complex projects with greater ease and efficiency. This alone makes the Network Planning Techniques of unique value to the management.

10.7 The number of activities involved in a project will naturally vary according to the complexity of the project. Normally a project may involve a few hundred activities but complex projects may run into several thousand activities. In such cases attempts are made to break the project into several levels of management. The upper level management may keep control through a summarized overall diagram and the lower level management may control specific areas of the project through several sub-networks.

10.8 The time required for drawing arrow diagrams naturally varies according to the complexity of the project and instances of time quoted vary over a wide range from 100 man-hours for a 400 activity diagram to 600 man-hours for a diagram of the same complexity. The drawing of a network diagram is usually of an iterative nature and the newness of each problem is due to the varying constraints on the time for completion and the resources available. The availability of resources in general is a very heavy constraint which determines the nature of the network diagram in its parallel and serial activities. If the resources are limited, particularly in regard to men and material, more serial activities tend to appear in the network. If time is the constraint then more parallel activities will appear. In practice a compromise will have to be achieved.

10.9 In the United States for complex projects the Government itself is known to have spent as much as 0.1 to 0.5 per cent of the total project cost on preparation of networks with the higher figure for research and development programme. But of course expected returns from such expenditures are far greater in magnitude. Savings from network techniques in better coordination, reduced delays, more accurate forecasts, fewer crash programme etc. are less easily identifiable than the actual amount in Rupees spent on the advantages are often apparent only from the manager's  project wise viewpoint.

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