Review Technique · Activity Cost: It is defined as the cost of performing and completing a particular activity or task. Crash Cost ... The cost slope indicates the additional cost
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Programme Evaluation and Review Technique
Basic Concepts
Estimate of Probability
Due to variability in the activity duration, the total project may not be completed exactly in time. Thus, it is necessary to calculate the probability of actual meeting the schedule time of the project as well as activities. Probability of completing the project by schedule time (Ts) is given by
Z = s e
e
T – T
σ
Te represents the duration on the critical path. Te can be calculated by adding the expected time of each activity lying on the critical path. e represents standard deviation of the critical path. Variance of the critical path can be get by adding variances of critical activities. e is the square root of variance of the critical path.
Expected Time
The expected time (te) is the average time taken for the completion of the job. By using beta-distribution, the expected time can be obtained by following formula.
et = o m pt +4t +t
6
Project Crashing
Project Crashing deal with those situations which will speak of the effect of increase or decrease in the total duration for the completion of a project and are closely associated with cost considerations. In such cases when the time duration is reduced, the project cost increases, but in some exceptional cases project cost is reduced as well. The reduction in cost occurs in the case of those projects which make use of a certain type of resources, for example, a machine and whose time is more valuable than the operator’s time.
Activity Cost : It is defined as the cost of performing and completing a particular activity or task.
Crash Cost (Cc) : This is the direct cost that is anticipated in completing an activity within the crash time.
Crash Time (Ct) : This is the minimum time required to complete an activity.
Normal Cost (Nc) : This is the lowest possible direct cost required to complete an activity.
Normal Time (Nt) : This is the minimum time required to complete an activity at Cost normal cost.
Activity Cost Slope : The cost slope indicates the additional cost incurred per unit of time saved in reducing the duration of an activity. It can be understood more clearly by considering the below figure.
Let OA represent the normal time duration for completing a job and OC the normal cost involved to complete the job. Assume that the management wish to reduce the time of completing the job to OB from normal time OA. Therefore under such a situation the cost of the project increases and it goes up to say OD (Crash Cost). This only amounts to saving that by reducing the time period by BA the cost has increased by the amount CD. The rate of increase in the cost of activity per unit with a decrease in time is known as cost slope and is described as below.
Optimum Duration: The total project cost is the sum of the direct and the indirect costs. In case the direct cost varies with the project duration time, the total project cost would have the shape indicated in the following figure:
At the point A, the cost will be minimum. The time corresponding, to point A is called the optimum duration and the cost as optimum cost for the project.
Resource Smoothing
It is a network technique used for smoothening peak resource requirement during different periods of the project network. Under this technique the total project duration is maintained at the minimum level. For example, if the duration of a project is 15 days, then the project duration is maintained, but the resources required for completing different activities of a project are smoothened by utilising floats available on non-critical activities. These non-critical activities having floats are rescheduled or shifted so that a uniform demand on resources is achieved. In other words, the constraint in the case of resource smoothing operation would be on the project duration time. Resource smoothing is a useful technique or business managers
to estimate the total resource requirements for various project activities. In resources smoothing, the time-scaled diagram of various activities and their floats (if any), along with resource requirements are used. The periods of maximum demand for resources are identified and non-critical activities during these periods are staggered by rescheduling them according to their floats for balancing the resource requirements.
Resource Leveling
It is also a network technique which is used for reducing the requirement of a particular resource due to its paucity. The process of resource levelling utilize the large floats available on non-critical activities of the project and thus cuts down the demand on the resource. In resource levelling, the maximum demand of a resource should not exceed the available limit at any point of time. In order to achieve this, non-critical activities are rescheduled by utilising their floats. Sometimes, the use of resource levelling may lead to enlonging the completion time of the project. In other words, in resource levelling, constraint is on the limit of the resource availability.
Time Scaled Diagrams
Below figure shows the network diagram drawn to a horizonal time scale. The critical path has been arranged as a straight line with non-critical events above or below it. Solid lines represent activities, dotted horizontal lines represent float.
Updating the Network
The progress of various activities in a project network is measured periodically. Normally, either most of the activities are ahead or behind the schedule. It is therefore, necessary to update or redraw the network periodically to know the exact position of completion of each activity of the project. The task of updating the network may be carried out once in a month. Sometimes the updating of the network may provide useful information to such an extent that it may demand the revision of even those very activities which have not started. Even the logic may also change i.e. some of the existing activities may have to be dropped and new activities may be added up. In brief the network should be amended accordingly in the light of new developments. It is also not unlikely that the total physical quantum of work
accomplished at a point of time may exceed what was planned but the progress against the critical path alone may be slower than the scheduled pace.
Variance
Beta distribution is assumed for these “guess estimates” and PERT analysts have found that beta-distribution curve happens to give fairly satisfactory results for most of the activities. For a distribution of this type, the standard deviation is approximately one sixth of the range, i.e.
tS = p ot – t
6
The variance, therefore; is
2tS =
2p ot – t
6
Why PERT?
PERT (Program Evaluation and Review Technique) is more relevant for handing such projects which have a great deal of uncertainty associated with the activity durations. To take these uncertainty into account, three kinds of times estimates are generally obtained. These are: (i) The Optimistic Times Estimate: This is the estimate of
the shortest possible time in which an activity can be completed under ideal conditions. For this estimate, no provision for delays or setbacks are made. We shall denote this estimate by to.
(ii) The Pessimistic Time Estimate: This is the maximum possible time which an activity could take to accomplish the job. If everything went wrong and abnormal situations prevailed, this would be the time estimate. It is denoted by tp.
(iii) The Most Likely Time Estimate: This is a time estimate of an activity which lies between the optimistic and the pessimistic time estimates. It assumes that things go in a normal way with few setbacks. It is represents by tm. Statistically, it is the model value if duration of the activity.
These activity durations follow a probability distribution called Beta Distribution.
Answer The progress of various activities in a project network is measured periodically. Normally, either most of the activities are ahead or behind the schedule. It is therefore, necessary to update or redraw the network periodically to know the exact position of completion of each activity of the project. The task of updating the network may be carried out once in a month. Sometimes the updating of the network may provide useful information to such an extent that it may demand the revision of even those very activities which have not started. Even the logic may also change i.e. some of the existing activities may have to be dropped and new activities may be added up. In brief the network should be amended accordingly in the light of new developments.
It is also not unlikely that the total physical quantum of work accomplished at a point of time may exceed what was planned but the progress against the critical path alone may be slower than the scheduled pace.
Question-2
Write a short note on ‘Activity Cost Slope’
Answer The cost slope indicates the additional cost incurred per unit of time saved in reducing the duration of an activity. It can be understood more clearly by considering the figure.
Let OA represent the normal time duration for completing a job and OC the normal cost involved to complete the job. Assume that the management wish to reduce the time of completing the job to OB from normal time OA. Therefore under such a situation the cost of the project increases and it goes up to say OD (Crash Cost). This only amounts to saving that by reducing the time period by BA the cost has increased by the amount CD. The rate of increase in the cost of activity per unit with a decrease in time is known as cost slope and is described as below.
= Crash Cost – Normal CostNormal Time – Crash Time
Question-3
Explain the terms ‘Resource Smoothing’ and ‘Resource Levelling’.
Answer Resource Smoothing
It is a network technique used for smoothening peak resource requirement during different periods of the project network. Under this technique the total project duration is maintained at the minimum level. For example, if the duration of a project is 15 days, then the project duration is maintained, but the resources required for completing different activities of a project are smoothened by utilising floats available on non critical activities. These non critical activities having floats are rescheduled or shifted so that a uniform demand on resources is achieved. In other words, the constraint in the case of resource smoothing operation would be
on the project duration time. Resource smoothing is a useful technique or business managers to estimate the total resource requirements for various project activities.
In resources smoothing, the time-scaled diagram of various activities and their floats (if any), along with resource requirements are used. The periods of maximum demand for resources are identified and non critical activities during these periods are staggered by rescheduling them according to their floats for balancing the resource requirements.
Resource Leveling
It is also a network technique which is used for reducing the requirement of a particular resource due to its paucity. The process of resource levelling utilize the large floats available on non-critical activities of the project and thus cuts down the demand on the resource. In resource levelling, the maximum demand of a resource should not exceed the available limit at any point of time. In order to achieve this, non critical activities are rescheduled by utilising their floats. Some times, the use of resource levelling may lead to enlonging the completion time of the project. In other words, in resource levelling, constraint is on the limit of the resource availability.
PERT/ CPM Question-4
Under what circumstance PERT is more relevant? How?
Answer PERT (Program Evaluation and Review Technique) is more relevant for handling such projects which have a great deal of uncertainty associated with the activity durations.
To take these uncertainty into account, three kinds of times estimates are generally obtained. These are:
The Optimistic Times Estimate: This is the estimate of the shortest possible time in which an activity can be completed under ideal conditions. For this estimate, no provision for delays or setbacks are made. We shall denote this estimate by to.
The Pessimistic Time Estimate: This is the maximum possible time which an activity could take to accomplish the job. If everything went wrong and abnormal situations prevailed, this would be the time estimate. It is denoted by tp.
The Most Likely Time Estimate: This is a time estimate of an activity which lies between the optimistic and the pessimistic time estimates. It assumes that things go in a normal way with few setbacks. It is represents by tm.
Write short notes on Distinction between PERT and CPM.
Answer The PERT and CPM models are similar in terms of their basic structure, rationale and mode of analysis. However, there are certain distinctions between PERT and CPM networks which are enumerated below:
(i) CPM is activity oriented i.e. CPM network is built on the basis of activities. Also results of various calculations are considered in terms of activities of the project. On the other hand, PERT is event oriented.
(ii) CPM is a deterministic model i.e. it does not take into account the uncertainties involved in the estimation of time for execution of a job or an activity. It completely ignores the probabilistic element of the problem. PERT, however, is a probabilistic model. It uses three estimates of the activity time; optimistic, pessimistic and most likely, with a view to take into account time uncertainty. Thus, the expected duration for each activity is probabilistic and expected duration indicates that there is fifty per probability of getting the job done within that time.
(iii) CPM places dual emphasis on time and cost and evaluates the trade-off between project cost and project item. By deploying additional resources, it allows the critical path project manager to manipulate project duration within certain limits so that project duration can be shortened at an optimal cost. On the other hand, PERT is primarily concerned with time. It helps the manger to schedule and coordinate various activities so that the project can be completed on scheduled time.
(iv) CPM is commonly used for those projects which are repetitive in nature and where one has prior experience of handling similar projects. PERT is generally used for those projects where time required to complete various activities are not known as prior. Thus, PERT is widely used for planning and scheduling research and development project.
Question-6
State any 5 limitations of the assumptions of PERT and CPM.
Answer (i) Beta distribution may not always be applicable
(ii) The formulae for expected duration and standard deviation are simplification. In certain cases, errors due to these have been found up to 33 %
(iii) The above errors may get compounded or may cancel each other
(iv) Activities are assumed to be independent. But the limitations on the resources may not justify the assumption.
(v) It may not always be possible to sort out completely identifiable activities and to state where they begin and where they end
(vi) If there exist alternatives in outcome, they need to be incorporated by way of a decision tree analysis.
(vii) Time estimates have a subjective element and to this extent, techniques could be weak. Contractors can manipulate and underestimate time in cost plus contract bids. In incentive contracts, overestimation is likely.
(viii) Cost-time tradeoffs / cost curve slopes are subjective and even experts may be widely off the mark even after honest deliberations
The Chennai Construction Company is bidding on a contract to install a line of microwave towers. It has identified, the expected duration of the critical path is 18 weeks and the sum of the variances of the activities on the critical path is 9 weeks.
Required Calculate the probability that the project may be completed not earlier than 15 weeks and not later than 21 weeks.
Solution
Probability of Completing the Project by Schedule Time Ts is given by
Z = s e
e
T – T
σ
Probability if the Project is required to be completed in 15 weeks: Probability if the Project is required to be completed in 15 weeks is given by
Z = 15 – 18
3
Z = – 1
Probability (Z = –1) = 0.1587
Probability if the Project is required to be completed in 21 weeks: Probability if the Project is required to be completed in 21 weeks is given by
Z = 21 – 18
3
Z = + 1
Probability (Z = +1) = 0.8413
Probability that the Project may be completed not earlier than 15 weeks and not later than 21 weeks = 0.8413–0.1587
A small project is composed of seven activities, whose time estimates are listed below. Activities are identifies by their beginning (i) and ending (j) note numbers:
Required Assuming that the cost and time required for one activity is independent of the time and cost of any other activity and variations are expected to follow normal distribution, draw a network based on the above data and calculate:
(ii) Expected Cost of Construction of the plant is `80 millions (`5 + `3 + `4 + `9 + `2 +
`12 + `20 + `7 + `14 + `4, millions).
(iii) Expected Time Required to build the plant is 20 months (4 + 6 + 9 + 1, months)
(iv) The variance of the expected time is 9 months. Determined by summing the variance of critical activities (1 + 2 + 5 + 1, months). Standard Deviation of the expected time 3 months (root of variance).
Problem-4
A German Construction Company is preparing a network for laying the foundation of a new science museum. Given the following set of activities, their predecessor requirements and three time estimates of completion time:
Activity Predecessors Time Estimates (Weeks) Optimistic Pessimistic Most Likely
A None 2 4 3 B None 8 8 8 C A 7 11 9 D B 6 6 6 E C 9 11 10 F C 10 18 14 G C,D 11 11 11 H F,G 6 14 10 I E 4 6 5 J I 3 5 4 K H 1 1 1
Required (i) Draw the network and determine the critical path. (ii) If the project due date is 41 weeks, what is the probability of not meeting the due date? (iii) Compute the float for each activity.
Solution
(i) The network for the given problem:
(ii) The Expected Time and Variance for each of the activities (in weeks):
Standard Deviation of the Critical Path ( e ) = 379
= 2.0276
(iii) Probability of not meeting the due date of 41 weeks:
Probability of Completing the Project by Schedule Time Ts is given by Z = s e
e
T – T
σ
Accordingly probability of meeting the due date of 41 weeks is given by Z = 41 – 372.0276
= 1.97
Probability (Z = 1.97) = 0.9756
Probability of not meeting the due date of 41 weeks = 1 – 0.9756 = 0.0244 Or = 2.44% (iv) Calculation of Total Float, Free Float and Independent Float:
Required (i) Draw a network diagram. (ii) Find the critical path after estimating the earliest and latest event times for all
nodes. (iii) Find the probability of completing the project before 31 weeks? (iv) What is the chance of project duration exceeding 46 weeks? (v) What will be the effect on the current critical path if the most likely time of activity
3–5 gets revised to 14 instead of 11 weeks given above?
Solution
(i) The network for the given problem:
(ii) The Expected Time and Variance for each of the activities (in weeks):
Chances that the project will be completed in a period 46 weeks is given by Z
= 46 – 36
5
= 2.00
Probability (Z = 2.00) = 0.9772
Chances of the project duration exceeding 46 weeks = 1 – 0.9772
= 0.0228
Or = 2.28% (v) Effect on the current critical path if the most likely time of activity 3–5 gets
revised to 14: If the most likely time of activity 3–5 gets revised to 14 instead of 11 weeks as given, the expected duration of the activity 3–5 will be
te = o m pt + 4t + t
6
te = 5 4x14 176
= 13 Weeks
Accordingly, expected duration of the activity 3–5 will be 13 weeks instead of 11 weeks calculated earlier. As activity 3–5 lie on the critical path, the project duration will increase by 2 weeks (13-11) and the total project duration will become 38 weeks (36+2).
Problem-6
Consider the following project:
Activity Predecessors Time Estimates (Weeks) Optimistic Most Likely Pessimistic
A None 3 6 9 B None 2 5 8 C A 2 4 6 D B 2 3 10 E B 1 3 11 F C,D 4 6 8 G E 1 5 15
Required Find the critical path and its standard deviation. What is the probability that the project will be completed by 18 weeks?
Required Draw the PERT network. Indicate the expected total slack for each activity and hence indicate the average critical path. What time would you expect if the project to be completed with 99% chance?
Solution
(i) The network for the given problem:
(ii) The Expected Time and Variance for each of the activities:
Required (i) Draw the network diagram. (ii) Calculate the expected time and variance of each activity. (iii) Find out the expected length of critical path and its standard deviation. (iv) Find the probability that the launching will be completed in 27 days. (v) Find the duration, which has 95% probability of completion.
Solution
(i) The network for the given problem:
(ii) The Expected Time and Variance for each of the activities (in days):
Required (i) Draw the network and find out the expected time and variance for each activity. What is
the expected duration for completion of the project? (ii) IT the target time is 22 days, what is the probability of not meeting the target? (iii) Within how many days can the project be expected to be completed with 99 percent
Required (i) Calculate for the each activity, its early start time, early finish time, late start time, late
finish time, total float, free float and independent float. (ii) Identify the critical path. (iii) If the project manager finds that either of the activities 2–6 or 4–5 can each be speeded
up by two days at the same cost, which of the two activities should be speeded up? Explain.
Solution
(i) The earliest start, earliest finish, latest start, latest finish, total float and free float for activities of above network are given in the table below:
6–7 8 27 35 27 35 0 0 0 0 0 (ii) The Critical Path is 1–2–4–5–6–7 with project duration of 35 days.
(iii) Activity 2–6 lies on the path 1–2–6–7 (having duration 30 days) which is not a critical path. If activity 2–6 is speeded up by 2 days, it will not reduce the total project duration.
Activity 4–5 lies on the critical path. If activity 4–5 is speeded up by 2 days, the project duration will come down to 33 days.
Required (i) Draw the network and find the critical paths.
(ii) After 15 days of working, the following progress is noted:
(a) Activities 12, 13 and 14 completed as per original schedule.
(b) Activity 24 is in progress and will be completed in 4 more days.
(c) Activity 36 is in progress and will need 17 more days to complete.
(d) The staff at activity 36 are specialised. They are directed to complete 36 and undertake an activity 67, which will require 7days. This rearrangement arose due to a modification in a specialisation.
(e) Activity 68 will be completed in 4 days instead of the originally planned 7 days.
(f) There is no change in the other activities.
Update the network diagram after 15 days of start of work based on the assumption given above. Indicate the revised critical paths along with their duration.
As the Project Manager of KL Construction Company, you are involved in drawing a network for laying the foundation of a new art museum. The relevant information for all the activities of this project is given in the following table.
Activity Immediate Predecessors
Time Estimates (Weeks) Normal Cost (`)
Crash Cost (`)
Optimistic Most Likely Pessimistic
A None 2 3 4 6,000 8,000 B A 4 5 6 12,000 13,500 C A 3 5 7 16,000 22,000 D A 2 4 6 8,000 10,000 E C, D 1 2 3 6,000 7,500 F B, E 1 3 5 14,000 20,000
(i) Construct the network for the project and determine the critical path and the expected duration of the project.
(ii) The Director of your company is not impressed by your analysis. He draws your attention that the project must be completed by seven weeks and refers to the penalty
clause in the agreement which provides for payment of penalty at the rate of `2,500 for every weeks or part thereof exceeding seven weeks. Your Director also strongly believes that the time duration of various activities of the project can be crashed to their optimistic time estimates with the crashing costs mentioned in the above table. Determine the optimal duration of the project if your objective is to minimise the sum of the project execution cost and the penalty cost.
Solution
The Expected Duration of each activity:-
Activity Time Estimates (Weeks) Expected Time
Optimistic (to) Most Likely (tm) Pessimistic (tp) o m pe
t + 4t + tt =
6
A 2 3 4 3
B 4 5 6 5
C 3 5 7 5
D 2 4 6 4
E 1 2 3 2
F 1 3 5 3
The network for the given problem:
The Critical Path is 1–2–4–5–6 or A–C–E–F with duration of 13 weeks.
As activity E of critical path A–C–E–F has least cost slope, crash activity E by 1 week at a crash cost of `1,500 Revised Project Duration (Critical Path A–C–E–F) = 12 Weeks Total Cost of the Project for the 12 Weeks = Normal Cost + Penalty Cost + Crash Cost = `62,000 + `2,500 5 Weeks + `1,500 = `76,000
Crashing Second Step:
Activity T C/T Remark
A (1–2)
1 2,000 Least Cost Slope
C (2–4)
2 3,000
E (4–5)
1 1,500 Already Crashed
F (5–6)
2 3,000
We shall now crash activity A by one week at a crash cost of `2,000
Now remaining activities C and F (on the critical path A–C–E–F) have cost slope equal to ` 3,000. Crashing of any one of these shall increase the total cost of the project by ` 500 (` 3,000 − ` 2,500) per week.
As our objective is to minimize the sum of the project execution cost and the penalty cost, therefore the Optimal Project Duration is 11 weeks and the Total Minimum Cost is ` 75,500.
Problem-15
The table below provides cost and time estimates of seven activities of a project;
Required (i) Draw the project network corresponding to normal time. (ii) Determine the critical path and the normal duration and normal costs of the project. (iii) Crash the activities so that the project completion time reduces to 9 weeks, with
1–3–5–6 with project duration = 16 Weeks 1–3–4–6 with project duration = 14 Weeks 1–2–4–6 with project duration = 11 Weeks
The critical path is 1–3–5–6. The normal length of the project is 16 days.
Crashing steps so that the project completion time reduces to 9 weeks with minimum additional cost:
Crashing Step 1:
We will first crash the activities on the critical path.
Activity 1–3 of critical path 1–3–5–6 has minimum costs slope. We can crash activity 1–3 by 3 weeks for additional cost of `6,000 (3Weeks × `2,000). Now the project duration is reduced to 13 weeks.
The various paths in the network with revised duration are:
1–3–5–6 with project duration = 13 Weeks 1–3–4–6 with project duration = 11 Weeks 1–2–4–6 with project duration = 11 Weeks
Crashing Step 2:
Crash activity 5–6 by 2 weeks for additional cost of `12,000 (2Weeks × `6,000). Now the project duration is reduced to 11 weeks.
The various paths in the network with revised duration are:
1–3–5–6 with project duration = 11 Weeks 1–3–4–6 with project duration = 11 Weeks 1–2–4–6 with project duration = 11 Weeks
Crashing Step 3:
Now there are three critical paths:
1–3–5–6 with project duration = 11 Weeks 1–3–4–6 with project duration = 11 Weeks 1–2–4–6 with project duration = 11 Weeks
To reduce the project duration further, we crash activity 4–6 by 2 weeks at an additional costs of ` 6,000 (2Weeks × `3,000) and activity 5–6 by two weeks at an additional cost of ` 12,000 (2Weeks × `6,000).
(i) Name the paths and given their total duration. (ii) Give three different ways of reducing the project above duration by four days.
Solution
(i) Assuming that the duration of activity 3–5 is 4 weeks.
The various critical paths are:
1–2–5–8–9 15 Weeks
1–3–4–7–8–9 15 Weeks
1–3–4–6–7–8–9 15 Weeks
1–3–5–8–9 15 Weeks
(ii) As the duration for activity 3–5 is not specified it is open to assume the duration. Three possibilities emerge on the basis of the duration assumed.
a) If the duration assumed is more than 4 weeks then that path (1–3, 3–5, 5–8, 8–9) alone will be critical. In that case any of the activity in the critical path can be selected.
b) If the duration assumed is exactly 4 weeks then it will be one of the 4 critical paths. Since all the paths are critical, reduction is possible by combining activities. The activities can be independent, common to few paths and common to all the paths. The various categories are given below.
NO INDEPENDENT ACTIVITIES------
<Consider all Critical Paths {refer part (i)} simultaneously>
Activity Common to all the paths and no independent activity
Required (i) Draw the project network. (ii) Determine the critical path and its duration. (iii) Find the optimum duration and the resultant cost of the project.
Solution
The network for the given problem:
The Critical Path is 1–2–3–5 with normal duration of 13 weeks. Normal duration cost of the project is `16,625. (`)
Normal Cost 10,125 Indirect Cost [11 Days × `500] 5,500 Crashing Cost [2 Days × `250] 500 Total Cost 16,125
After crashing the activity 2–3 by 2 day, revised position of various paths are as under:
1–2–3–5 with duration 11 days 1–2–5 with duration 11 days 1–4–5 with duration 10 days 1–3–5 with duration 4 days
Crashing Step 2:
1–2 is a common activity in the first two paths with cost slope of ` 500 per day. There is no profit or loss in crashing this activity. Hence we can crash it by one by one day.
(`) Normal Cost 10,125 Indirect Cost [10 Days × `500] 5,000 Crashing Cost [2 Days × `250 + 1 Day × `500] 1,000 Total Cost 16,125
After crashing the activity 1–2 by one day, revised position of various paths are as under:
1–2–3–5 with duration 10 days 1–2–5 with duration 10 days 1–4–5 with duration 10 days 1–3–5 with duration 4 days
Crashing Step 3
To reduce the duration of project further, we are required to consider the activities on all the three paths. These activities may be 3–5, 2–5, and 1–4. If all of these activities are crash by even 1 day each, then the total increase in cost would be `1,250 (`375 + `500 + `375) for saving `500. Accordingly further crashing is not possible. Hence Optimal Project Duration is 10 days with Optimal Cost of `16,125.
Problem-18
The following network and table are presented to you:
T 8 2,250 6 2,750 U 16 1,875 11 2,750 V 14 2,250 9 3,000 W 12 3,000 9 3,750 X 15 1,000 14 2,500 Y 10 2,500 8 2,860
Required Perform step by step crashing and reduce the project duration by 11 days while minimizing the crashing cost. What would be the cost of the crashing exercise?
The critical path is T–U–V–Y with normal duration of 48 weeks.
Particulars T U V Y
Crash Days Possible (∆T) 2 5 5 2
Crash Cost Less Normal Cost (∆C) `500 `875 `750 `360
Crashing Cost per Day [(∆C) / (∆T)] `250 `175 `150 `180
Step I Crash V by 5 Days
---
---
`750
---
Step II Crash U by 5 Days
---
`875
---
---
Step III Crash Y by 1 Day
---
---
---
`180
Minimum Cost of Crashing Exercise is `1,805 (`750 + `875 + `180) for Project Duration of 11 Days.
Problem-19
The Noida Nirman Authority intends to install a road traffic regulating signal in a heavy traffic prone area. The total installation work has been broken down into six activities. The normal duration, crash duration and crashing cost of the activities are expected as given in the following table:
Required (i) Draw the network and find the normal and minimum duration of the work. (ii) Compute the additional cost involved if the authority wants to complete the work in the
Required (i) List the critical paths. (ii) Given that each activity can be crashed by a maximum of one day, choose to crash any
four activities so that the project duration is reduced by 2 days.
Solution
Critical Paths:
All are critical paths:
1–2–5–6 2 Days + 8 Days + 5 Days 15 Days 1–3–5–6 3 Days + 7 Days + 5 Days 15 Days 1–4–5–6 4 Days + 6 Days + 5 Days 15 Days 1–3–4–5–6 3 Days + 1 Days + 6 Days + 5 Days 15 Days
To reduce Project Duration by 2 Days:
Crash Activity 5–6 by 1 Day
Crash Activities 1–2, 1–3, 1–4 by 1 Day
Note: Other Crashing Alternatives are also possible.
Problem-21
The normal time, crash time and crashing cost per day are given for the following network:
Crash Cost at minimum duration of 30 Days is ` 360.
Since the total cost (crashing cost + indirect cost) starts increasing from 30 days, the Optimum Project Duration is 31 days with Crashing Cost of ` 280.
Problem-22
A project with normal duration and cost along with crash duration and cost for each activity is given below:
Required (i) Draw network diagram and identify the critical path. (ii) Find out the total float associated with each activity. (iii) Crash the relevant activities systematically and determine the optimum project
Remark Common Activity Independent Activities Common Activity
As cost per hour for every alternative is greater than `50 i.e. indirect cost per hour. Therefore, any reduction in the duration of project will increase the cost of project completion. Hence, optimum project completion time is 28 hours with cost of `3,375.
The following table shows for each activity needed to complete the road construction project, the normal time, the shortest time in which the activity can be completed and cost per day for reducing the time of each activity. The contract includes a penalty clause of ` 80 per day over 19 days. The overhead cost is `150 per day. The cost of completing the eight activities in normal time is ` 6,000.
Remark Independent Activities Independent Activity + Common Activity*
As crashing cost per day for every alternative is greater than ` 150 i.e. Overhead Cost per day. Therefore, any reduction in the duration of project will increase the cost of project completion.
Hence, the Lowest Cost of Completion is ` 9,150 with the Completion Time of 19 Days.
Problem-24
A project comprised of 10 activities whose normal time and cost are given as follows:
Peak requirement is 11 men and same is required on 7th Day (Refer above Time Scale Diagram).
As only 10 men are available on any day, we have to shift Activity F to 10th Day. Now the project can be completed in 10 days (Refer below Time Scale Diagram).
The global construction company is bidding on a contract to install a line of microwave towers. It has identified the following activities, along with their expected times, predecessor restriction and worker requirements:
The contract specifies that the project must be completed in 28 weeks. This company will assign a fixed number of workers to the project for its entire duration, and so it would like to ensure that the minimum number of workers is assigned and that the project will be completed in the 28 weeks.
Required Find a schedule which will do this.
Solution
Critical Path of the network is 1–3–4–5–6 (i.e., B–E–G–H). The duration of the project is 28 weeks.
It can be seen from below given resource accumulation table and the time-scaled version of the project that demand on the resources is not even on the 15th, 16th, 17th and 18th week the demand of workers is as high as 18 on the 19th, 20th, 21st and 22nd week, it is only 6. If workers are to be hired for the entire project duration for 28 weeks, then during most of the days, they will be idle.
As mentioned above the maximum demand of the resources occurs during 9th weeks to 14 weeks (i.e., 16 workers) and during 15th week to 18th weeks (i.e.18 workers). The activities on these days will have to be shifted depending upon their floats such that the demand comes down. As can be seen from the above time –scaled version, activity C has a float of 14 weeks and activity F has a float of 4 weeks. We will try to shift activity C by Fourteen weeks so that it starts on 23rd week instead of 9th week. This reduces demand of workers from 16 to 12 workers during 9th to 14th week.
Similarly, we will try to shift activity F by four weeks so that it starts on 19th week instead of 15th week. The requirement during 15th to 18th week now reduces to 12 workers instead of 18 workers required earlier.
The modified resource accumulation table and the time-scaled version of the project are given above. As can be seen from the above figure the requirement for workers reduces to 12 as against 18 workers originally estimated. Hence, by judiciously utilizing the float, we can smooth the demand on the resources.
Problem-28
Rearrange the activities suitably for leveling the audit executives with the help of time scale diagram if during first 52 days only 8 to 10 audit executives and during remaining days 16 to 22 audit executives can be made available.
This is a problem of duration constraint of 100 days as also resource constraint (audit executives).
We have to re-arrange the activities so that they can be performed with the given resource availabilities in the stipulated time of 100 days.
Refer Time Scale Diagram-2:
The critical activities 1–2, 2–4, 4–6, 6–7, and 7–8 would be scheduled first. Activity 1–3 is not critical. However scheduling 1–3 even at the scheduled time zero would involve increase of audit executives to 14 on days 21 & 22, which is in excess of availability. We therefore have to resort to do this by doing overtime.
Now activities 2–6 can be delayed by 32 days i.e. instead of starting it on 21st day we can delay it to start on 53rd day.
Similarly we delay activities 4–5 and 5–7 by twelve days each to start on 59th day and 73rd day.
This would ensure that the resources are demanded as per availability and project completion too take place at 100 days.
The following information is given for a certain project:
Activity Normal Duration (Days)
Crash Duration (Days)
Difference (Days)
Normal Cost (`)
Crash Cost (`)
Difference
(`)
Activity slope
(`/day)
I II III = I - II IV V VI = V - IV VII = VI/ III
1–2 9 6 3 640 700 60 20
1–3 8 5 3 500 575 75 25
1–4 15 10 5 400 550 150 30
2–4 5 3 2 100 120 20 10
3–4 10 6 4 200 260 60 15
4–5 2 1 1 100 140 40 40
Required
(i) What is the normal project duration? (ii) Perform step-by step crashing to reduce the project duration by 5 days. What is the
cost incurred for the optional crashing exercised? (iii) Independent of (ii) above, if the Project Manager is able to save as per rates in Column
VII of the above table for every day relaxed for the activities, compute the number of days and associated savings for 5 days of relaxation, in the order of optimality, without extending the project duration as per (i). The Project Manager is interested in this exercise to schedule resources.
Now the various paths in the network with revised duration are:
1–3–4–5 with project duration = 15 Days
1–4–5 with project duration = 15 Days
1–2–4–5 with project duration = 15 Days
Total Crashing Cost = ` 45 + ` 40 + ` 45
= ` 130
Statement Showing “Crashing Cost (5 Days)”
Normal Project Length (Days) Job Crashed Crashing Cost (`)
20 - -
19 3–4 ` 15
18 3–4 ` 30 (` 15 + ` 15)
17 3–4 ` 45 (` 30 + ` 15)
16 4–5 ` 85 (` 45 + ` 40)
15 3–4,1–4 ` 130 (` 85 + ` 15+ ` 30)
(iii) The Project Manager can save some costs by relaxing non-critical activities. Activity 4–5 is a common as well as last activity of the critical path. This particular activity starts on 18th Day of the Project, hence the non-critical activities can be relaxed up to this day i.e. 18th Day.
Among various non-critical activities 1–2, 2–4 and 1–4, activities 1–2 and 2–4 are laying on same path.
It is clear from the table that activity 1–4 has the highest saving per day which is ` 30 and it can be relaxed up to 3 Days {i.e. 4–5 Start Day (18) less 1–4 Duration (15)}.
Between activity 1–2 and 2–4, activity 1–2 has higher saving i.e. ` 20 per day and it can be relaxed up to 4 days {i.e. 4–5 Start Day (18) less 1–2 Duration (9) less 2–4 Duration (5)}. Since there is a specific requirement of ‘maximum 5 Days of Relaxation’, it can be relaxed by 2 Days only.
Statement Showing “Saving Due to Relaxation”
Activity Savings per day (`) No. of Days Total (`)