Lecture 6_15.Uncertainty, Budgets, And Cashflow

Project and Portfolio Management (PPM)

Sudhir M Chadha

(Course of 20 lectures)

Uncertainty

PERT Three Estimate Approach

The duration of each activity is a random variable having some probability distribution.

The three estimates to be obtained for each activity are: Most likely estimate (m) = estimate of the most

likely value of the duration;

Optimistic estimate (o) = Estimate of the duration under the most favourable conditions;

Pessimistic estimate (p) = Estimate of the duration under the most unfavourable conditions

06/10/2015 3 sudhir chadha

= Mean of probability distribution 2 = Variance of the probability distribution

is the average of the various activity durations and is the standard deviation that measures the variability of the durations about the mean.

Model of Probability Distribution

Beta distribution

o p m Elapsed time


Approximate Formulas for and 2

For most probability distributions, such as the -distribution, essentially all the durations would lie in the interval -3 and +3, i.e., the spread of durations is about 6. (For example, for a normal distribution 99.73% of the distribution lies inside this interval).

2 = [(p-o)/6 ]2 = (o+4m+p)/6

Note that the mean and the most likely estimate are not the same, because the possibility of much higher durations pushes the mean up.


Probability Distribution of Project Duration

The next step is to estimate the probability of meeting the deadline of 47 weeks, which requires making three simplifying approximations. For the project duration we need: The mean p, The standard deviation p, The form of the distribution.

The mean critical path is the path through the project network that would be the critical path if the duration of each activity were equal to its mean.


Three Simplifying Approximations

Approximation 1: Assume that the mean critical path will turn out to be the longest path through the project network. As we have just seen this may not be true. This implies that p = Sum of the means of the durations for the activities on the mean

critical path.

Approximation 2: Assume that the duration of the activities on the mean critical path are statistically independent. This would not be true if the same cause produces deviations in the durations of two or more activities. p

2 = Sum of the variances of the durations for the activities on the mean critical path.

Approximation 3: Assume that the form of the distribution of the project duration is the normal distribution (one form of the central-limit theorem). This is justified if the number of activities for the mean critical path is, say, 5.


Probability of Meeting the Deadline

The three simplifying approximations lead to the probability duration of the project being approximated by a normal distribution.

D = Deadline for the project;

Number of standard deviations by which d exceeds p

= (D - p)/p = 1.

P(T d) = Probability that the project duration (T) does not exceed the deadline (D)

D (Deadline) p (Mean)

Project Duration (in weeks)


Managerial Evaluation of PERT/CPM

The P(T d) is only a rough approximation of the true probability of meeting the project deadline. Since the true critical path of the project is usually longer than the mean critical path (i.e., more than 44 weeks), the real probability of meeting the deadline is less than that obtained from using the simplifying assumptions.

The approximations made in estimating the mean and variance of activity durations, as well as the simplifying approximations to get to the project duration probability distribution are questionable. Nevertheless, the method allows an understanding of uncertainty in activity durations.

Another deficiency is that PERT/CPM does not allow an activity to begin until all its immediate predecessors are completely finished. However, an extension called the precedence diagramming method does allow dealing with overlapping activities. For example, although activity H (do the exterior painting) follows activity G (put up the exterior siding) in Reliables project network, it would be better to have a start-to-start relationship with some lag.


Budget and Cash Flow

Project Costs

The following terminology is used for project costs: Baseline costs. The original planned task, resource, or assignment

costs saved as part of a baseline plan; Current (or scheduled) costs. The calculated costs of tasks, resources,

and assignments in a project plan. As adjustments are made to the baseline plan (such as assigning or removing resources) , the recalculated costs are the current costs. The current cost equals the actual cost plus the remaining cost per task, resource, or assignment.

Actual costs. The costs that have been incurred for tasks, resources or assignments. After the project incurs actual costs (typically by tracking actual work), the current cost equals the actual cost plus the remaining cost per task, resource, or assignment.

Remaining costs. The difference between the current or scheduled costs and the actual costs for tasks, resources, or assignments.


Scheduling Project Costs

PERT/Cost is a systematic procedure to help the project manager plan, schedule, and control project costs.

A common assumption when using PERT/Cost is that the costs of performing an activity are incurred at a constant rate throughout its duration, i.e., prorated. This assumption can be changed.

PERT/Cost provides a weekly schedule of expenses so that the project manager can monitor whether the project is staying within budget.

Postponing activities to their latest start times also postpones the costs of these activities, which is helpful when cash is short, but this also increases the risk of missing the scheduled project completion date.


Cumulative Project Costs

The top curve is the schedule of cumulative project costs when all activities begin at their earliest start times.

The bottom curve is the schedule of project costs when all activities begin at their latest start times.

The areas between the two curves shows the only feasible week by week budget that will not delay project completion. -

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

4,500,000

5,000,000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43

ET project cost schedule

LT project cost schedule

Feasible region for project costs


Time-Cost Trade-off

Time-Cost Trade Offs

Crashing an activity refers to special costly measures to reduce the duration of an activity below its normal value.

These special measures might include using overtime, hiring additional temporary help, using special time-saving materials, obtaining special equipment, or anything else to expedite an activity.

Crashing the project refers to crashing a number of activities to reduce the duration of a project to below its normal value.

The CPM method of time-cost tradeoffs is concerned with determining how much (if any) to crash each of the activities to reduce the anticipated duration of the project down to a desired value.


Crashing The normal point on the time-cost graph shows the time (duration) of an activity when it is performed in the normal way. The crash point shows the time and cost when the activity is fully crashed; i.e., it is fully expedited with no cost spared to reduce its duration as much as possible. The options for each activity are to be at its crash point, its normal point, or somewhere on the line segment between these two points.

Normal time Crash time

Normal

Activity duration

Crash cost

Normal cost

Crash

A typical time-cost graph for an activity


Project Network for Reliable

Start Finish A B C

I

D

E

G H

M

F

J

L

N

K

Activity Code

A. Excavate

B. Foundation

C. Rough wall

D. Roof

Activity Code

I. Electrics

J. Wallboard

K. Flooring

E. E plumbing

F. I plumbing

G. E siding

H. E painting

L. I painting

M. E fixtures

N. I fixtures

0 2 4 10

7 6

4

7

6

4

8

0 5

2

9

5


Time-Cost Trade-Off Data Activity Normal

weeks Crash weeks

Normal Cost

Crash Cost Max Reduction in Time

Crash Cost per Week Saved

A 2 1 $180,000 $280,000 1 $100,000

B 4 2 320,000 420,000 2 50,000

C 10 7 620,000 860,000 3 80,000

D 6 4 260,000 340,000 2 40,000

E 4 3 410,000 570,000 1 160,000

F 5 3 180,000 260,000 2 40,000

G 7 4 900,000 1,020,000 3 40,000

H 9 6 200,000 380,000 3 60,000

I 7 5 210,000 270,000 2 30,000

J 8 6 430,000 490,000 2 30,000

K 4 3 160,000 200,000 1 40,000

L 5 3 250,000 350,000 2 50,000

M 2 1 100,000 200,000 1 100,000

N 6 3 330,000 510,000 3 60,000 06/10/2015 18 sudhir chadha

Marginal Cost Analysis

Summing the normal cost and the crash cost in the time-cost trade-off table gives: Sum of normal costs = $4.55 million; Sum of crash costs = $6.15 million; Anticipated duration of the project if all the activities are

performed in the normal way = 44 weeks; If all the activities are fully crashed, then the project duration

(assuming no delays) = 28 weeks.

Marginal cost analysis (MCA) finds the least expensive way to reduce project duration one week at a time.

MCA becomes unwieldy for large networks. Linear programming (LP) provides a more efficient

alternative to marginal cost analysis, for large projects.


Performing MCA on Reliable Project

Activity to

Crash

Crash Cost

AB

CD

GH

M

AB

CEH

M

AB

CEFJK

N

AB

CEFJLN

AB

CIJK

N

AB

CIJLN

40 Weeks 31 43 44 41 42

J $30,000 40 31 42 43 40 41

J 30,000 40 31 41 42 39 40

F 40,000 40 31 40 41 39 40

F 40,000 40 31 39 40 39 40

Cost of the partially crashed project = $4.69m


Linear Programming

Restatement of the problem: Consider the total cost of the project, including the extra cost of crashing

activities. The problem then is to minimize the total cost, subject to the constraint that the project duration is less than or equal to the time desired by the project manager.

The following decisions will appear in the changing cells: The start time of each activity; The reduction in the duration of each activity due to crashing; The finish time of the project (must not exceed 40 weeks for Reliable).

The start-time constraints specify that an activity cannot start until each of its immediate predecessors have finished.

Although the start-time constraints allow a delay in starting an activity, an optimal solution would not allow this to happen for any activity on the critical path.


Conclusions

The CPM method of time-cost trade-offs ignores the considerable uncertainty in activity times, so the predicted project duration under any crashing plan may miss the actual duration by a considerable amount.

Conclusion 1. The plan for crashing the project only provides a 50% chance of actually finishing the project within 40 weeks, so the extra cost of the plan ($140000) is not justified.

It is sometimes useful to postpone a decision on crashing an activity until near its start time. Information on how well the project schedule is progressing can then influence this decision.

Conclusion 2. The extra cost of the crashing plan can be justified if it almost certainly would earn the bonus of $150000 for finishing the project within 40 weeks. Hold the plan in reserve to be implemented if the project is running well ahead of schedule before reaching activity F.

Conclusion 3. The extra cost of part or all of the crashing plan can be easily justified if it likely would make the difference in avoiding the penalty of $300000 for not finishing the project within 47 weeks.. Hold the crashing plan in reserve to be partially or wholly implemented if the project is running far behind schedule before reaching activity F or activity J.


Lecture 6_15.Uncertainty, Budgets, And Cashflow

Documents

probability duration

mean critical path

project deadline

project duration t

normal distribution

true probability

project network

duration optimistic