OPERATIONS MANAGEMENT

CHAPTER :PERT & CPMBy :- Prof. B.R.Narielwala

PROJECT DESIGN AND NETWORK ANALYSIS

• Execution of a project follows a definite path of planning, scheduling and controlling.

• Project design defines individual activities which go into corpus of project and their interrelationship.

• Network techniques help in performing various functions efficiently and effectively.

• Framework formulated with detailed sequences and work plan - duration time - to implement project.

• Based on techno-economic analysis, details are worked out to compile sequential activities.

ORIGIN OF PERT AND CPM

PERT, CPM AND MOST ARE THREE DIFFERENT NAMES GIVEN TO NETWORK ANALYSIS

PERT - PROGRAMME EVALUATION AND REVIEW TECHNIQUES• It coordinates and synchronize various parts of overall job, expedite

completion of project to accomplish a pre determined job.• Communication facility to report developments and keep managers

posted and informed.

PERT HAS 2 ASPECTS:

1. EVENTS : Specific completion that occurs at a recognizable point of time.

2. ACTIVITIES : Specific work required to complete specific event.

STEPS IN PERT

1. Development of project work2. Time estimation3. Determination of critical path, event slacks And activity floats4. Development of project schedule5. Calculation of variability duration and Probability of completion in a given time

•In PERT, time is basic measure. It is expressed in calendar Weeks of completion of project

within stipulated time.

•Three time estimates employed are:

1.OPTIMISTIC TIME : Shortest possible time if Everything goes perfectly well.2.PESSIMISTIC TIME : Longest time conceived due to Unusual delays.3.MOST LIKELY TIME : Best estimates of what normally would occur.

CPM : Critical Path Method

CPM has two time cost estimate for each activity : • One for normal situation & • Other for crash situation

Basic assumption is that there is a precise known time for each activity of project

It identifies most critical elements and thus more Attention can be paid to these activities

CPM is difficult to use to control activity as fresh Evaluation is required each time when changes are Introduced into network

CPM is static planning model

DIFFERENCE BETWEEN PERT AND CPM

Though fundamental network of PERT and CPM are identical, there are certain differences in details as per the chart given below:

PERT

1). Event-oriented (time-based)

2). Allowance for uncertainty3). Three time estimates4). Probabilistic model with uncertainty in activity duration

CPM

1). Activity oriented (cost based)2). No such allowance3). Only one time estimate. Emphasis on cost4). Deterministic model with well-known activity time based.

PERT (Continue)…

5). No demarcation for critical/non-critical activities6). Suitable for high precision time estimates7). Time is averaged8). Concept of ‘crashing’ is not applied9). Based on reduction of execution time without too much cost implications

CPM (Continue)…

5). Marking of critical activity6). Suitable for reasonable precision time estimates7). No averaging of time involved8). ‘Crashing concept’ is applied9). Greatest reduction in completion time with least increase in project cost

DIFFERENCE BETWEEN PERT AND CPM

APPLICATION OF NETWORK ANALYSIS

Two types of graphs are used in CPM/PERT:1. Activity on the arrow (aoa) system2. Activity on node (aon) system

AOA SYSTEM : Activity is graphically represented by an Arrow represents the start and the head shows the end of activity.

Where “a” & “b” are activities, this can be represented by theNumbers of nodes like a = (1,2) b = (2,3)

AON SYSTEM : Activities are represented by circles or nodes and arrows are used to show only dependency relationship between the activity nodes. b a c

a b 1 2 3

GRAPHS OF AOA & AON

1 2 3 10 days 8 days

AOA For showing needed time

a,10 b,8

AON Showing time needed for activity

DUMMY ACTIVITIES

53

1 2

4 6

7

a d

h

i e

c

bf

g

“b” is common immediate predecessor of both “d” and “e”, “a” is an immediate predecessor of “d” alone, and “c” is one of “e”. Hence “f” and “g” are dummy jobs introduced to indicate precedence relationship

CRITICAL PATH

1

2

6

54

CRITICAL PATH

3 a, 10 daysc, 20 days

e, 10 days

f, 30 days d, 60 days

b, 10 days

SLACK PERIOD = ( 100 - 40 ) = 60

Longest path in net work 1-4-5-6 (100) & 1-2-3-6 ( 40)

CRASHING CPM NETWORKS

Time estimates are developed under conditions of normal Working and known as normal time estimates. It does not take into consideration overtime working or use of Abnormal resources for completion of an activity.

If it is desired to reduce total duration of the project to be below normal duration, it is necessary to analyze the critical path for a possible reduction of the original time estimates for some of the activities on critical path. ThisReduced time estimates is called ‘crash time estimate’.

STEPS IN NETWORK CRASHING

STEP NO.1 :Determine the time - cost ratio for each activity in the network

TIME-COST RATIO = Increase in cost

Decrease in time = Crash cost - normal cost

Normal time - crash time

STEP NO.2 :

Identify activities on the critical path and select that activity on the critical path which has the smallest time-cost ratio and crash that activity to the extent possible.

STEP NO.3:

Observe whether there is any change in the critical path i.e. any other path becomes critical. If so, calculate time-cost ratio for the activities in the new critical path.

STEP NO.4:

Repeat step no.2 and 3, till activities are crashed to reduce the project duration to the desired time period.

COMMONLY USED NETWORK SYMBOLS:

• Activity• Dummy activity• Event

• Activity “a” must be completed before activity “b” can begin

• Activities “a” and “b” can occur concurrently, but both must be completed before activity “c” can begin

1 2 3

2

13 4

A B

A C

B

SYMBOL MEANING

• Activites “a” and “b” must be completed before activities “c” and “d” can begin, but “c” can begin independently of “d” and vice versa.5

4

3

2

1 A C

B D

• Activities “a” and “b” can occur concurrently, but both must be completed before activity “d” can begin. Activity “e” can occur only after activity “b” is completed. Activity “c” (dotted line with arrow head) is a dummy activity which shows a precedence relationship but has a zero time duration.

A D

B E

C

1). ACTIVITY:- An effort that is required to complete a part of a project

2). EVENT:- A beginning, completion point or milestone accomplishment within the project. An

activity begins and ends with an event- e.g:-

1& 2 – events; A – Activity

3). Predecessor Activity:- An activity that must occur before another activity

Activity ‘a’ is a predecessor activity for activity ‘b’

4). Successor Activity:- An activity that must occur after another activity.

Activity ‘d’ is a successor activity to activity ‘c’.

3

1 2 1 A B

TERMS USED IN NETWORK BASED SCHEDULING TECHNIQUES

( )

1 2 A

5 Activity duration

In CPM, this means the best estimate of the time to complete an activity. In PERT, the expected time or average time to complete an activity.

6 Optimistic time (to)

The time for completing an activity if all goes well, used in PERT.

7 Pessimistic time (tp)

The time for completing an activity, if everything goes wrong used in pert.

8 Most likely time (tm)

The time for completing an activity, that is the consensus best estimates, used in pert.

9 Expected time (te)

The average time for completing an activity.

10 DUMMY ACTIVITY

An activity that consumes no time (zero time duration) but shows precedence between events. DUMMY ACTIVITY

The above network diagram shows that both activities m and n begin with event number 2 and end with event number 3 as a rule, no two activities can be represented by the same set of event numbers. This problem is solved by inserting a dummy activity as shown below:

1 2 3

4 5N

ML

P

11 Earliest start (es)

The earliest that an activity can start, from the beginning of the project.

12 Earliest finish (ef)

The earliest that an activity can finish, from the beginning of the project.

13 Latest start (ls)

The latest that an activity can start, from the beginning of the project, without causing a delay in the completion of the project.

14 Latest finish (lf)

The latest that an activity can finish, from the beginning of the project, without causing a delay in the completion of the project.

15 Slack The amount of time that an activity or group of activities can slip without causing a delay in the completion of the project. It is also know as ‘float’.

16 CRITICAL ACTIVITY

An activity that has no room for schedule slippage, if it slips the entire project completion will slip. An activity with zero slack.

17 CRITICAL PATH

The chain of critical activities for the project and is the longest path through the network. *********************

CPM – TIME ANALYSIS TIME SYMBOLS USED ARE: D DURATION OF ACTIVITY (HOURS OR DAYS OR WEEKS OR

MONDAYS)

ESA EARLIEST (POSSIBLE) STARTING TIME OF THE ACTIVITY.

LSA LATEST (PERMITTED) STARTING TIME OF THE ACTIVITY

EEA EARLIEST (POSSIBLE) ENDING TIME OF THE ACTIVITY

LEA LATEST (PERMITTED) ENDING TIME OF THE ACTIVITY

EE

EARLIEST (POSSIBLE) TIME OF THE EVENT

LE LATEST (PERMITTED) TIME OF THE EVENT

TF TOTAL FLOT OF THE ACTIVITY

FF FREE FLOR OF THE ACTIVITY

IF INDEPENDENT FLOAT OF THE ACTIVITY

MODE OF REPRESENTATION EXPECTED TIME (te) = to + 4 tm + tp 6

ACTIVITY STANDARD DEVIATION :

EVENT STANDARD DEVIATION:

Ei Lt Ej Lj

I j

ASSESSMENT OF FLOATS OR SLACKS

• If difference between earliest starting time (esa) and Latest ending time (lea) of an activity exceeds its duration then the respective activity is not critical i.e. Lea-esa>d

• In such cases certain time reserves are ensuing which are termed as floats or slacks

• Slack analysis is done to indentify sub-critical paths i.e. Which activity is likely to become critical, if slack period moves up beyond available slack period

• Slack analysis is done from two stand points i.e. Event or Activity:

(A) event slack = latest event time - earliest event timeAll events having zero slack represent critical path

(B) Activity slack analysis

- Provides margin of allowance available for commencement and completion of various activities.

- It follows that activities with zero slack value represent activities on critical path

- Three types of activity slacks or floats are identified:(I) Total float or slack(II) Free float or slack(III) Independent float or slack

(I) Total float : It is the amount of time an activity can be delayed beyond its earliest starting time without delaying the project’s completion, if other activities take their estimated duration

Total float for activity = (a-b)* LE(b) - EE (a) - D

(II) Free float : It is the amount of time on the basis of which an activity can be delayed without delaying the early start of a successor activity

* Free float for activity (a-b) = EE (b)- EE(a) -D

(III) Independent float : It indicates the time span by which the activity (a-b) can be expanded/shifted if, for the event (a) the latest and for the event (b) the earliest times of occurrence shall be maintained

* Independent float : EE(b)-LE(a)-D

Independent float can be negative also. Since a negative float has not meaning, independent float is taken as zero, if negative.

RELATIONSHIP BETWEEN 3 TYPES OF FLOATS

EE= Earliest time for eventLE = Latest event time for event

D

D

Total float

Free float

Independent float

Time

EE (i) LE (i) EE (j ) LE (j)

SOLUTION -

(PART – I)

The dummy activity 3-4 has been introduced here because “b” and “c” both precede activity d and activity “c” precedes activity “e”.

Dummyactivity

A B

D

F

H

G E

C

1 2

4 5

7 8

63

(PART – II)

Network diagram - Based on Predecessor Activity

Critical path = Events having no slacks to be connected

And project duration time is 12+4+18+4+6 = 44 Months

= A C F H J

A C

E

H

J

B

IG

(12)

(8)

(12)

(4)

(6)

(9) (5)D(3)

(4)

(18) F

1

0 0

2

12 12

3

20 22

4

16 16

6

21 29

5

34 34

7

38 38

8

44 44

(PART – III)

Step No. : 1 Calculate expected time (te) For each activity

FORMULA : te = to +4tm+tp6

to = Optimistic Time tm = Most likely Time tp = Pessimistic Time

Activity Expected time for the activity (te)

1-2 2+4X5+8 = 5 6

1-4 4+4X19+28 = 18 6

1-5 5 + 4X11+17 = 11 6

2-3 3+3X9+27 = 11 6

2-6 3+4X6+15 = 7 6

3-6 2+4x5+14 = 6 6

4-6 3+4x4+7 = 7 6

5-7 1+4x4+7 = 4 6

5-8 2+4x5+14 = 6 6

6-8 6+4x12+30 =14 6

7-8 2+4x5+8 = 5 6

Step No.2 DRAW PERT NETWORK DIAGRAM

5 7

8

641

2 3

0 0

11 30 15 34

39 39

18 18 25 25

5 8 16 19

1-5

(11)

5-7

(4) 7-8

(5)

6-8

(14)

3-6 (6)

2-3

(11)

1-2 (5)

1-4

(18)

4-6

(7)

2-6

(7)

5-8

(6)

CRITICAL PATH : 1-4 4-6 6-818 + 7 + 14 = 39 weeks

(B) Mean time for each activity expected time = teStandard deviation for each activity = tp-to

ACTIVITY STANDARD DEVIATION (WEEKS) Z 61-2 8-2 = 1 7-8 8-2 = 1

6 6*1-4 28-4 = 4 *4-6 15-3 =2

6 21-5 17-5 = 2

62-3 27-3 = 6

62-6 15-3 = 2

63-6 14-2 = 2

65-7 7-1 = 1

65-8 14-2 = 2

6*5-8 30-6 = 4

6

(C) To determine std. deviation for critical path :

Activity Std. Deviation1-4 1= 44-6 2= 26-8 3= 4

Std. Deviation for critical path=

cp = 12 + 2

2+ 32

= (4)2 + (2)2 + (4)2

= 6 weeks

(D) Calculation of slacks for events :

Event no. Slack (weeks)1 NIL2 8-5 = 33 19-16 = 34 18-18 = NIL5 30-11 = 196 25-25 = NIL7 34-15 = 198 39-39 = 0

(E) To calculate total float, free float and independent float for each activity

Total float for activity (I-j) = (Lej)- (EEi) - DFree float for activity (I-j) = (Eej-Eei) - DIndependent float (I-j) = (Eej - Lei) - D

If D > (EEJ-LEI), Then independent float is negative hence no meaning hence zero.

Activity Total Float Free Float Independent Float1-2 (8-0)-5 = 3 (5-0)-5 = 0 (5-0)-5 = 0*1-4 (18-0)-18 = 0 (18-0)-18 = 0 (18-8)-18 = 01-5 (30-0)-11 = 19 (11-0)-11 = 0 (11-0)-11 = 02-3 (19-5)-11 = 3 (16-5)-11 = 0 (16-8)-11 = (-3)2-6 (25-5)-7 = 13 (25-5)-7 = 3 (25-8)-7 = 103-6 (25-16)-6 = 3 (25-16)-6 = 3 (25-19)-6 = 0*4-6 (25-18)-7 = 0 (25-18)-7 = 0 (25-18)-6 = 15-7 (34-11)-4 = 19 (15-11)-4 = 0 (15-30)-4 = (-)5-8 (39-11)-6 = 22 (39-11)-6 = 22 (39-30)-6 = 3*6-8 (39-25)-14 = 0 (39-25)-14 = 0 (39-25)-5 = 07-8 (39-15)-5 = 19 (39-15)-15 = 19 (39-34)-5 = 0*Floats are nil for critical path

Calculation of float :

Crashing of CPM Network

[Answer 1]

STEP : 1

Time Cost Ratio Crash Cost – Normal Cost Per Week Normal Time – Crash Time

Activity T/C Ratio Per Week (Rs.)

A 2000B 3000C 1000D 1000E 500F 250

STEP 2 : DRAW NETWORK DIAGRAM

NORMAL:CRITICAL PATH B D FDURATION 4 + 8 + 8 = 20 weeksCOST Rs. 36,000/-

CRASH:CRITICAL PATH A C F

4 + 2 + 4 = 10 weeksCOST Rs. 54,000/-

B D F3 + 3 + 4 = 10 weeks

( ) Normal Time Crash Time

34

6

A C

B DE

F

(4)

(14)

(8)(6) (3)

(8)3

24

Step : 3 Normal Cost Rs.36,000Current Project

Completion Time

Critical Path (s)

Activities which can be Crashed

Cost of Crashing Activities Selected Cost Total Cost Time

20 B-D-F B,D,F 3000, 1000250

F 250 36,250 19

19 B-D-F B,D,F ------ “ ------ F 250 36,500 18

18 B-D-FB-E

B,D,FB,E

3000,1000,2503000,500

FE

250500 37250 17

17 B-D-FB-E

B,D,FB,E

3000,1000,2503000,500

FE

250500 38,000 16

16 B-D-FB-E

B,DB,E

3000,10003000,500

DE

1000500 39,500 15

15 B-D-FB-E

B,DB,E

3000,10003000,500

DE

1000500 41,000 14

14 B-D-FB-E

B,DB,E

3000,10003000,500

DE

1000500 42,500 13

13 B-D-FB-E

B,DB,E

3000,10003000,500

DE

1000500 44,000 12

12 B-D-FB-E

B,DB,E

3000,10003000,500

DE

1000500 45,500 11

11B-D-F

B-EB

B,E3000

3000,500-E

500 46,000 10Cannot be crashed further

SUMMARY:

Project Time Direct Cost

Overhead Cost @

Rs.1,000/- per week

Total Cost (Rs.)

10 46,000 10,000 56,00011 45,500 11,000 56,500

Therefore, Optimal Time of Crashing duration 10 weeks Cost Rs. 56,000/-

CRITICAL ACTIVITIES B,E after crashing.

[Answer 2]

STEP : 1

Activity T/C Ratio Per Week (Rs.)

1-2 (A) 201-3 (B) 251-4 (C) 302-4 (D) 103-4 (E) 154-5 (F) 40

STEP : 2 NETWORK DIAGRAM

NORMAL:CRITICAL PATH B E FDURATION 8 + 10 + 2 = 20 days

CRASH:CRITICAL PATH B E F

5 + 6 + 1 = 12 days (Minimum)

( ) Normal Time Crash Time

1

2

4 5

3

5

31

6

10

6

AC

B

D

E

F

(8) (10)

(15)

(5)

(2)

STEP : 3 Project CrashingProject

Completion Time (Days)

Critical Path(s)

Activities which can be

crashed

Cost of crashing

Activity selected Cost Time

L 20 B-E-F B,E,F 25,15,40 E 15 19L-1 19 B-E-F B,E,F 25,15,40 E 15 18L-2 18 B-E-F B,E,F 25,15,40 E 15 17L-3 17 B-E-F

C-FB,E,FC,F

25,15,4030,40

EC

1530 16

L-4 16 B-E-FC-F

A-D-F

B,FC,F

A,D,F

25,4030,40

20,10,40

BCD

253010

15

L-5 15 A-D-FC-F

B-E-F

A,D,FC,FB,F

20,10,4030,4025,40

DCB

103025

14

L-6 14 A-D-FC-F

B-E-F

A,FC,FB,F

20,4030,4025,40

ACB

203025

13

L-7 13 A-D-FC-F

B-E-F

A,FC,FF

20,4030,40

40

ACF

203040

12

L-8 12 A-D-FC-F

B-E-F

A--

20--

A--

20--

11

(No further Crashing)

SUMMARYProject Time Crashing Cost

(Rs.)OverheadCost @ Rs.60/- day

Total Cost

12 385 720 110511 405 660 1065

Optimal length schedule for each jobA 6 daysB 5 daysC 10 daysD 3 daysE 6 daysF 1 day

Minimum Crashing Cost Rs.385Adding overhead Cost Rs.1105Minimum Number of Weeks 12