Chapter 8 - Project Management1 Chapter 8 - Project Management Chapter Topics The Elements of Project Management The Project Network Probabilistic Activity

Chapter 8 - Project Management 1

Chapter 8 - Project Management

Chapter Topics

• The Elements of Project Management

• The Project Network

• Probabilistic Activity Times

• Project Crashing and Time-Cost Trade-Off

• Formulating the CPM/PERT Network as a Linear Programming Model


Project Management

Overview

• Uses networks for project analysis.

• Networks show how projects are organized and are used to determine time duration for completion.

• Network techniques used are

- CPM (Critical Path Method)

- PERT (Project Evaluation and Review Technique)

• Developed during late 1950s.


The Elements of Project Management

• Management is generally perceived as concerned with planning, organizing, and control of an ongoing process or activity.

• Project Management is concerned with control of an activity for a relatively short period of time after which management effort ends

• Primary elements of Project Management to be discussed:

- Project team

- Project planning

- Project control.


The Elements of Project ManagementThe Project team

• Project team typically consists of a group of individuals from various areas in an organization and often includes outside consultants.

• Members of engineering staff often assigned to project work.

• Most important member of project team is the project manager.

• Project manager is often under great pressure because of uncertainty inherent in project activities and possibility of failure.

• Project manager must be able to coordinate various skills of team members into a single focused effort.


Project Planning: PERT/CPM

• PERT– Program Evaluation and Review Technique– Developed by U.S. Navy for Polaris missile project– Developed to handle uncertain activity times

• CPM– Critical Path Method– Developed by Du Pont & Remington Rand– Developed for industrial projects for which activity

times generally were known• Today’s project management software packages have

combined the best features of both approaches.


• PERT and CPM have been used to plan, schedule, and control a wide variety of projects:

– R&D of new products and processes

– Construction of buildings and highways

– Maintenance of large and complex equipment

– Design and installation of new systems

PERT/CPM


• Project managers rely on PERT/CPM to help them answer questions such as:

– What is the total time to complete the project?

– What are the scheduled start and finish dates for each specific activity?

– Which activities are critical and must be completed exactly as scheduled to keep the project on schedule?

– How long can noncritical activities be delayed before they cause an increase in the project completion time?

PERT/CPM


PERT Network: Activity-on-Node Approach

• A PERT network can be constructed to model the precedence of the activities.

• The arcs of the network represent the precedence relationships of the activities.

• The nodes (rectangles) of the network represent activities.

– You will need to add a “Start” and a “Finish” nodes.


• Activity time estimates usually can not be made with certainty.

• In the three-time estimate approach, the time to complete an activity is assumed to follow a Beta distribution.

• An activity i’s mean completion time is:

ti = (ai + 4mi + bi)/6• An activity’s completion time variance is:

2i = ((bi-ai)/6)2

– ai = activitity i’s optimistic completion time estimate– bi = activitity i’s pessimistic completion time estimate– mi = activitity i’s most likely completion time estimate



• In the three-time estimate approach, the critical path is determined as if the mean times for the activities were fixed times.

• The expected project time is the sum of the expected times of the critical path activities.

• The project variance is the sum of the variances of the critical path activities.

• The expected project time is assumed to be normally distributed (based on central limit theorem).



• Step 1: Make a forward pass through the network as follows: For each of these activities, i, compute:– Earliest Start (ES) Time = the maximum of all earliest

finish times for all its immediate predecessors. (For node “START”, this is 0.)

• ESi= Maximum (EFj) for all immediate proceeding activities j.

– Earliest Finish (EF) Time = (Earliest Start Time) + (Time to complete activity i).

• EFi= ESi+ ti

The project completion time is the of the Earliest Finish Times at the “FINISH” node. – This will also be used as Latest Finish Time at

“FINISH” node in the next step.

PERT Analysis Algorithm


• Step 2: Make a backwards pass through the network as follows: Move sequentially backwards from the last node, “FINISH” to its immediate predecessors, etc. At a given node, j, consider all activities immediately following it and compute:– Latest Finish (LF) Time = the minimum of the latest

start times for all activities that immediately follow j. (For node “FINISH”, this is the project completion time.)

• LFj= Minimum (LSi) for all immediate following activities i.

– Latest Start (LS) Time = (Latest Finish Time) - (Time to complete activity j).

• LSj= LFj - tj



• Step 3: Calculate the slack time for each activity by:

Slack = (Latest Start) - (Earliest Start) or

= (Latest Finish) - (Earliest Finish).

A critical path is a path of activities, from node “START” to “FINISH”, with 0 slack times.

– Shared slack is slack available for a sequence of activities.



Example: Riverwalk Associates

Riverwalk Associates is in the business of building elaborate parade floats. Its crew has a new float to build and want to use PERT/CPM to help them manage the project .

The table on the next slide shows the activities that comprise the project. Each activity’s estimated completion time (in weeks) and immediate predecessors are listed as well.

The project manager wants to know the total time to complete the project, which activities are critical, and the earliest and latest start and finish dates for each activity.


• Project activity initial information: Immed. Optimistic Most Likely Pessimistic Activity (i) Predec. Time (weeks) Time (wk.) Time (wk.) A — 4 6 8 B — 1 4.5 5 C A 3 3 3 D A 4 5 6 E A 0.5 1 1.5 F B,C 3 4 5 G B,C 1 1.5 5 H E,F 5 6 7 I E,F 2 5 8 J D,H 2.5 2.75 4.5 K G,I 3 5 7



• Activity Expected Time and Variances

ti = (ai + 4mi + bi)/6 2i = ((bi-ai)/6)2

Activity (i) Expected Time Variance (week2) A 6 4/9

B 4 4/9 C 3 0 D 5 1/9 E 1 1/36 F 4 1/9 G 2 4/9 H 6 1/9 I 5 1 J 3 1/9 K 5 4/9



• PERT Activity Node Representation

A6

ES EFLS LF

Earliest Start Earliest Finish

ExpectedDuration ofthe activity

Latest Start Latest Finish



• PERT Network Representation

A6

C3

B4

E1

D5

F4

G2

J3

H6

I5

K5

FINISH START

0 0



• Earliest/Latest TimesActivity ES EF LS LF Slack A 0 6 0 6 0 *critical

B 0 4 5 9 5 C 6 9 6 9 0 * D 6 11 15 20 9 E 6 7 12 13 6 F 9 13 9 13 0 * G 9 11 16 18 7 H 13 19 14 20 1 I 13 18 13 18 0 * J 19 22 20 23 1 K 18 23 18 23 0 *• The estimated project completion time is t0 = 23 (weeks) at

FINISH.



• Define variables for each activity in the following manner:

ES_i = Earliest Start time for activity i

EF_i = Earliest Finish time for activity i

LS_i = Latest Start time for activity i

LF_i = Latest Finish time for activity i

where i=A, B…K;

and FINISH = the earliest and also latest completion time of the project.

Riverwalk Associates – Linear Programming Form


Minimize ES_A + EF_A + ES_B + EF_B + … + EF_K + FINISH

S.t.EF_A - ES_A >= 6 (“=“ OK)EF_B - ES_B >= 4EF_C - ES_C >= 3EF_D - ES_D >= 5EF_E - ES_E >= 1EF_F - ES_F >= 4EF_G - ES_G >= 2EF_H - ES_H >= 6EF_I - ES_I >= 5EF_J - ES_J >= 3EF_K - ES_K >= 5 (“=“ OK)

ES_C - EF_A >= 0 (Not “=“ )ES_D - EF_A >= 0 ES_E - EF_A >= 0

Riverwalk Associates – LP Model 1 for Earliest Times

ES_F - EF_B >= 0

ES_G - EF_B >= 0

ES_F - EF_C >= 0

ES_G - EF_C >= 0

ES_J - EF_D >= 0

ES_J - EF_H >= 0

ES_H - EF_E >= 0

ES_H - EF_F >= 0

ES_I - EF_E >= 0

ES_I - EF_F >= 0

ES_K - EF_G >= 0

ES_K - EF_I >= 0

FINISH - EF_J >= 0

FINISH - EF_K >= 0


Maximize LS_A + LF_A + LS_B + LF_B + … + LF_K

S.t.

LF_A - LS_A >= 6 (“=“ OK) LF_B - LS_B >= 4

LF_C - LS_C >= 3

LF_D - LS_D >= 5

LF_E - LS_E >= 1

LF_F - LS_F >= 4

LF_G - LS_G >= 2

LF_H - LS_H >= 6

LF_I - LS_I >= 5

LF_J - LS_J >= 3

LF_K - LS_K >= 5 (“=“ OK)

LS_C - LF_A >= 0 (Not just “=“)

LS_D - LF_A >= 0

LS_E - LF_A >= 0

Riverwalk Associates – LP Model 2 for Latest Times

LS_F - LF_B >= 0

LS_G - LF_B >= 0

LS_F - LF_C >= 0

LS_G - LF_C >= 0

LS_J - LF_D >= 0

LS_J - LF_H >= 0

LS_H - LF_E >= 0

LS_H - LF_F >= 0

LS_I - LF_E >= 0

LS_I - LF_F >= 0

LS_K - LF_G >= 0

LS_K - LF_I >= 0

FINISH - LF_J >= 0

FINISH - LF_K >= 0

FINISH = 23 (from the optimal results of Model 1) – this is the main diff.)


• Probability the project will be completed within t1=24 weeks

project time variance 2 = 2A + 2

C + 2F + 2

I + 2K

= 4/9 + 0 + 1/9 + 1 + 4/9 = 2 (weeks-squared)

project time standard deviation = 1.414 (weeks). z1 = (24 - 23)/ = (24-23)weeks/1.414weeks = .71

From the Standard Normal Distribution table: P(z < z1=.71) = .5 + .2611 = .7611

More precisely,P(t < t1) = P(t-t0 < t1-t0) = P[(t-t0)/ < (t1-t0)/]

= P(z < z1=.71) = .7611if we define z= (t-t0)/ and z1 = (t1-t0)/.



Probability Analysis of the Project: Example 2Question: If the mean project completion time is X0 = 25, what is the probability that the project will be completed within X1=30 weeks?

2 = 6.9, = 2.63. Z1 = (X1 - X0)/ = (30 -25)/2.63 = 1.90

•Z1 value of 1.90 corresponds to probability of .4713 in Table A.1, appendix A. Probability of completing project in 30 weeks or less : (.5000 + .4713) = .9713.

•More precisely, P(x < X1) = P(x- X0 < X1 - X0) = P[(x- X0)/ < (X1 - X0)/ ]

= P(z < Z1=1.90) = .9713

if we define

z= (x - X0)/ (new variable)

and

Z1 = (X1 - X0)/ (constant).

Figure 8.14Probability the network will be completed in 30 weeks or less


Question: If the mean project completion time is X0 = 25, what is the probability that the project will be completed within X1=22 weeks?

Z1 = (22 - 25)/2.63 = -1.14

Where Z1 value of 1.14 (ignore negative) corresponds to probability of 0.3729 in Table A.1, appendix A. Probability that customer will be retained is .1271

Figure 8.15Probability the network will be completed in 22 weeks or less

Probability Analysis of the Project: Example 3


Probability Analysis of the Project NetworkCPM/PERT Analysis with QM for Windows

Exhibit 8.1


Project Crashing and Time-Cost Trade-Off: Definition

• Project duration can be reduced by assigning more resources to project activities.

• Doing this however increases project cost.

• Decision is based on analysis of trade-off between time and cost.

• Project crashing is a method for shortening project duration by reducing one or more critical activities to a time less than normal activity time.

• Crashing achieved by devoting more resources to crashed activities.


Crashing Activity Times

• In the Critical Path Method (CPM) approach to project scheduling, it is assumed that the normal time to complete an activity, tj , which can be met at a normal cost, cj , can be crashed to a reduced time, tj’, under maximum crashing for an increased cost, cj’.

• It is assumed that its cost per unit reduction, Kj , is linear and can be calculated by:

Kj = (cj' - cj)/(tj - tj').

E.g.: in the example on the right,

Kj = total crash cost/total crash time

= $2000/5 = $400/wk


Normal CrashNormal Crash ActivityActivity TimeTime CostCost TimeTime CostCostA) Study Feasibility A) Study Feasibility 6 $ 80,000 5 6 $ 80,000 5 $100,000$100,000B) Purchase Building B) Purchase Building 4 100,000 4 4 100,000 4 100,000100,000C) Hire Project Leader C) Hire Project Leader 3 50,000 2 3 50,000 2 100,000100,000D) Select Advertising Staff D) Select Advertising Staff 5 150,000 2 5 150,000 2 300,000300,000E) Purchase Materials E) Purchase Materials 1 180,000 1 1 180,000 1 180,000180,000F) Hire Manufacturing Staff F) Hire Manufacturing Staff 4 300,000 1 4 300,000 1 480,000480,000G) Manufacture Prototype G) Manufacture Prototype 2 100,000 2 2 100,000 2 100,000 100,000H) Produce First 50 Units H) Produce First 50 Units 6 450,000 5 6 450,000 5 800,000800,000 I) Advertising Product 5 350,000 1 I) Advertising Product 5 350,000 1 650,000650,000 J) Assessing User Feedback 3 300,000 3 J) Assessing User Feedback 3 300,000 3 300,000300,000 K) Distributing Product K) Distributing Product 5 550,000 5 5 550,000 5 550,000550,000

Normal Costs and Crash Costs

Crashing Example for Riverwalk Associates


Crashing: The completion time for this project using normal times is 23 weeks. Which activities should be crashed, and by how many weeks, in order for the project to be completed in a Target of 20 weeks?

Let: Yi = the amount of time activity i is crashed.

Then borrow from the LP formulation for the Earliest Times, we have the following…

Crashing Example for Riverwalk Associates


Min 20YA + 50YC + 50YD +

60YF + 350YH + 75YI S.t.

EF_A - ES_A >= 6 - YA

EF_B - ES_B >= 4

EF_C - ES_C >= 3 - YC

EF_D - ES_D >= 5 - YD

EF_E - ES_E >= 1

EF_F - ES_F >= 4 - YF

EF_G - ES_G >= 2

EF_H - ES_H >= 6 - YH

EF_I - ES_I >= 5 - YI

EF_J - ES_J >= 3EF_K - ES_K >= 5

ES_C - EF_A >= 0 (Not “=“ )ES_D - EF_A >= 0 ES_E - EF_A >= 0

Riverwalk Associates – LP Model for Crashing

ES_F - EF_B >= 0

ES_G - EF_B >= 0

ES_F - EF_C >= 0

ES_G - EF_C >= 0

ES_J - EF_D >= 0

ES_J - EF_H >= 0

ES_H - EF_E >= 0

ES_H - EF_F >= 0

ES_I - EF_E >= 0

ES_I - EF_F >= 0

ES_K - EF_G >= 0

ES_K - EF_I >= 0

FINISH - EF_J >= 0

FINISH - EF_K >= 0

FINISH <= 20 (Target)

YA <=1

YC <=1

YD <=3

YF <=3

YH <=1

YI <=4


Project Crashing and Time-Cost Trade-OffProject Crashing with QM for Windows

Exhibit 8.2


Project Crashing and Time-Cost Trade-OffGeneral Relationship of Time and Cost

• Project crashing costs and indirect costs have an inverse relationship.

• Crashing costs are highest when the project is shortened.

• Indirect costs increase as the project duration increases.

• Optimal project time is at minimum point on the total cost curve.

Figure 8.20The time–cost trade-off

Documents

Chapter 8 - Project Management1 Chapter 8 - Project Management Chapter Topics The Elements of Project Management The Project Network Probabilistic Activity