1 AM ITY G LO BAL BU SIN ESS SC H OOL Bangalore Project Management - CPM/PERT
Jan 14, 2016
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Project Management - CPM/PERT
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AMITY GLOBALBUSINESS SCHOOL BangaloreNetwork Analysis
For planning, scheduling, monitoring and coordinating large and complex projects comprising a number of activities
- Defining the entire job to be done
- Formulating the logical sequence
- Controlling the progress
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AMITY GLOBALBUSINESS SCHOOL BangaloreObjectives
• Minimization of total time
• Minimization of total cost
• Mimimize production delays
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AMITY GLOBALBUSINESS SCHOOL BangaloreApplications
• Construction of bridges
• Assembly line scheduling
• Installation of complex equipment
• Inventory planning and Control
• Construction of residential complex
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AMITY GLOBALBUSINESS SCHOOL BangaloreComponents
• Activity : represented by an arrow, definite starting time, and a point where it ends
- Predecessor Activity : An activity which must be completed before one or more other activities start
- Successor Activity : An activity which started immediately after one or more activities are completed
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- Dummy Activity : any activity which does not consume time or resource ( -----)
• Event : represents the start of an activity and the end of an activity
activity i j
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AMITY GLOBALBUSINESS SCHOOL BangaloreCritical Path Method
• Ei = earliest start time for activity (i,j)
• Li = latest start time for activity (i,j)
• Ej = earliest finish time for activity (i,j)
• Lj = latest finish time for activity (i,j)
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AMITY GLOBALBUSINESS SCHOOL Bangalore Critical Path
• The longest continuous chain of activities through the network starting from the first to last event. ( )
• All activities lying in this critical path are called critical activities as any delay in their execution will lead to a delay in the completion of the entire project.
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AMITY GLOBALBUSINESS SCHOOL BangaloreFloat
• Total float :
The amount of time by which an activity can be delayed without delay in the project completion date
• Free Float :
The portion of the total float within which an activity can be manipulated without affecting floats of the subsequent activities
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Independent Float :
That portion of the total float within which an activity can be delayed for start without affecting floats of the preceding activities.
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AMITY GLOBALBUSINESS SCHOOL BangalorePERT
programme evaluation and review technique
• Completion time is considered to be unknown
• Probability of activity completion time is estimated
– optimistic time (to or a)
- pessimistic time (tp or b )
- normal time / most likely time (tm or m)
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• Expected time te / μ = 1/6 ( to + 4 tm + tp)• Std deviation σ = 1/6 (tp – to)• Variance σ² = ( 1/6 (tp – to))²• Prob of completing the project by
scheduled time (Ts) is given by : Prob ( Z ≤ Ts – μ ) σ
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
For the following PERT diagram : D A 3 4 H 2 B E I K 2 4 5 3 F 7 2 C G 4 J 6
1
2 3
4 7 8 9
5 6
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Compute earliest event time and latest event time
• Critical path and total project duration
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AMITY GLOBALBUSINESS SCHOOL BangaloreForward pass method
• E1 =0
• E2= E1 + t1,2 = 0 + 3
• E3 = E2 + t2,3 = 3 + 4 = 7
• E4 = E1 + t1,4 = 0 + 2 = 2
• E5 = E1 + t1,5 = 0 + 2 = 2
• E6 = max ( E4 + t4,6 ; E5 + t5,6 )
= max ( 9, 6 ) = 9
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• E7 = E4 + t4,7 = 2 + 4 = 6
• E8 = max ( E3 + t3,8 ; E7 + t7,8 )
= max ( 9, 11) = 11
• E9 = max ( E8 + t8,9 ; E6 + t6,9 )
= max ( 14, 15) = 15
Lower left portion
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AMITY GLOBALBUSINESS SCHOOL Bangalore Backward Pass
Method• L9= E9 = 15
• L8 = L9 – t8,9 = 15 -3 =12
• L7 = L8 – t7,8 = 12-5 =7
• L6 = L9 – t 6,9 = 15 – 6 = 9
• L 5 = L6 – t5,5 = 9-4 = 5
• L4 = min( L6- t4,6; L7-t4,7)
min( 9-7 ; 7 - 4) = 2
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• L3 = L8 – t 3,8 = 12 -2 = 10
• L2= L3- t2,3 = 10 -4 = 6
• L1= min ( L2-t1,2 ; L4- t1,4; L5 – t1,5)
= min ( 6-3; 2-2; 5-2 ) = 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
For the following PERT diagram : D A 3 4 H 2 B E I K 2 4 5 3 F 7 2 C G 4 J 6
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0 0
23 6
37 10
42 2
76 7
811 12
915 15
52 5
69 9
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
An established company has decided to add a new product to its line. It will buy the product from a mfg concern, package it, and sell it to a number of distributors selected on a geographical basis. Market Research has indicated the volume expected and the size of sales force reqd. the steps in the following table are to be planned :
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Activity Description Time(wk)A1-2 B 2-3C 2-4D 2-5E 3-6F 4-7G 5-8H 6-7I 6-8J 6-9
Organize sales officeHire salesmenTrain salesmenSelect advertising agencyPlan advertising campaignConduct advertising campaignDesign packageSet up packaging facilitiesPackage initial stocksOrder stock from manufacturer
24221463334
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AMITY GLOBALBUSINESS SCHOOL Bangalore
K7-10L 8-11M 9-12N 10-12
O 11-12P 12-13
Select distributorsSell to distributorsShip stocksCheck inventoryGet customer feedbackRe assess strategy
353233
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AMITY GLOBALBUSINESS SCHOOL Bangalore
What is the critical path
Calculate total and free float
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AMITY GLOBALBUSINESS SCHOOL Bangalore
6 3
2 3 2 2 4 14 4 3 3
2 3 3
3 5
10 0
2
2 2
36 6
620 20
924 28
1231 31
1334 34
44 20
7
23 26
10
26 29
54 20
823 23
11
28 28
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Total float :
The amount of time by which an activity can be delayed without delay in the project completion date
= (Lj – tij) - Ei• Free Float :
The portion of the total float within which an activity can be manipulated without affecting floats of the subsequent activities = (Ej – Ei) - tij
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AMITY GLOBALBUSINESS SCHOOL Bangalore Time EstimatesActi Dur
Earliest
Start Fin
Latest
Start Fin
Float
Total Free
A 2 0 2
0 2 0 0
B 4 2 6 2 6 0 0
C 2 2 4 18 20 16 0
D 2 2 4 18 20 16 0
E 14 6 20 6 20 0 0
F 6 4 10 20 26 16 13=23-4-6
G 3 4 7 20 23 16 16 =23-4-3
H 3 20 23 23 26 3 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activity Duration
Earliest
Start Finish
Latest
Start Finish
Float
I 3
J 4
K 3
L 5
M 3
N 2
O 3
P 3
20 23
20 24
23 26
23 28
24 27
26 28
28 31
31 34
20 23
24 28
26 29
23 28
28 31
29 31
28 31
31 34
0 0
4 0
3 0
0 0
4 4
3 3
0 0
0 0
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ : A project schedule has the
following characteristics :Activity Time(days) Activity Time(days)
1-2 4 5-6 4
1-3 1 5-7 8
2-4 1 6-8 1
3-4 1 7-8 2
3-5 6 8-10 5
4-9 5 9-10 7
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Construct a network diagram
• Compute the total float
• Find the critical path and total project duration
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AMITY GLOBALBUSINESS SCHOOL Bangalore
4 1 5 7
1 1 8 2
1
0 0
2
4 9
31 1
4
5 10
57 7
715 15
910 15
611 16
8
17 17
1022 22
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Critical path is 1-3-5-7-8-10 and the project duration is 22 days
• Total Float = ( Lj-tij ) - Ei
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activit Duratio Earliest start
Earliest finish
Latest start
Latest finish
Total float
1-2 4 0 4 5 9 5
1-3 1 0 1 0 1 0
2-4 1 4 5 9 10 5
3-4 1 1 2 11 10 7
3-5 6 1 7 1 7 0
5-7 8 7 15 7 15 0
5-6 4 7 11 12 16 5
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AMITY GLOBALBUSINESS SCHOOL Bangalore
7-8 2 15 17 15 17 0
6-8 1 11 12 16 17 5
4-9 5 5 10 12 17 7
9 -10 7 10 17 15 22 5
8-10 5 17 22 17 22 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
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AMITY GLOBALBUSINESS SCHOOL Bangalore
steps in developing PERT network• Draw a network diagram
• Time estimates are obtd
- m , a , b
• Calculate te
• Determine the critical path
• Calculate σ²
• Calculate Z
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
Various activities of a small project and other relevant information have been shown in the adjoining table. Using the given info the resulting network is shown in the following figure. Map out the critical path and find the prob of completing the project within 48 days.
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AMITY GLOBALBUSINESS SCHOOL Bangalore
activity Most optimistic time ( in days )
Most likely time m
Most pessimistic
time b
1-2 4 8 12
2-3 1 4 7
2-4 8 12 16
3-5 3 5 7
4-5 0 0 0
4-6 3 6 9
5-7 3 6 9
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AMITY GLOBALBUSINESS SCHOOL Bangalore
5-8 4 6 8
7-9 4 8 12
8-9 2 5 8
9-10 4 10 16
6-10 4 6 8
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Calculate expected time te = a + 4m + b
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Variance σ² = b-a ²
6
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activity a m b te σ²
1-2 4 8 12 8 1.78
2-3 1 4 7 4 1
2-4 8 12 16 12 1.78
3-5 3 5 7 5 0.44
4-5 0 0 0 0 0
4-6 3 6 9 6 1
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AMITY GLOBALBUSINESS SCHOOL Bangalore
5-7 3 6 9 6 1
5-8 4 6 8 6 0.44
7-9 4 8 12 8 1.78
8-9 2 5 8 5 1
9-10 4 10 16 10 4
6-10 4 6 8 6 0.44
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AMITY GLOBALBUSINESS SCHOOL BangaloreEarliest
Start Time• E(μ1)= 0
• E(μ2) = 0+8 = 8
• E (μ3) = 8 + 4 = 12
• E(μ4)= 12 + 8 = 20
• E(μ5) = Max(12+5,20+0) = 20
• E(μ6) = 20+6 = 26
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• E (μ7) = 20 + 6 = 26
• E(μ8)= 20 + 6 = 26
• E(μ9) = Max(26+8, 26+5) = 34
• E(μ10) =Max(34 + 10, 26 + 6 ) = 44
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AMITY GLOBALBUSINESS SCHOOL BangaloreLatest Start
Time• E(L10) = 44
• E(L9) = 44-10 = 34
• E(L8) = 34 – 5 = 29
• E(L7) = 34 – 8 = 26
• E(L6) = 44-10 = 34
• E(L5) = Min (26-6, 29 -6) = 20
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• E(L4) = Min(20 – 0, 36 – 6 ) = 20
• E(L3) = 20 – 5 = 15
• E(L2) = Min(15-4, 20 -12) = 8
• E(L1) = 8 – 8 = 0
Draw network diagram and find critical path
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AMITY GLOBALBUSINESS SCHOOL Bangalore
10 0
28 8
312 15
420 20
520 20
726 26
934 34
828 29
626 38
1044 44
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Critical path is
1---2---4---5---7----9---10
• Expected time in completing the project
8 + 12 + 0 + 6 + 8 + 10 = 44 days
• Project variance
1.78 + 1.78 + 0 +1 + 1.78 + 4 = 10.34
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• To find out the prob of completing the project, within , 48 days
Z = X - X¯ = 48 – 44 = 1.24
σ 3.216
value from normal tables = 0.8925
Hence prob of completing project within 48 days = 89 %
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ (Dummy Variable )
A small project consists of seven activities for which the relevant data are given :
• Draw the network and find the project completion time
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activity Preceding activity
Activity duration ( days )
A - 4
B - 7
C - 6
D A,B 5
E A,B 7
F C,D,E 6
G C,D,E 5
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AMITY GLOBALBUSINESS SCHOOL Bangalore
A D1 G D3
B E F
C D D2
10 0
24 7
37 7
46 14
514 14
614 20
720 20
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AMITY GLOBALBUSINESS SCHOOL Bangalore
CRITICAL PATH IS 1-3-5-7
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
• Q has been commissioned by the food manufacturer Z to carry out market research in a new product development project, prior to a test market launch. The table below lists required activities and their duration in weeks
• Draw the new product development network. State and explain the critical path and its duration
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Prepare a table of the earliest start and finish times, the latest start and finish times and the total and free float
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activity Immediate predecessor
duration
A - 6
B - 4
C A 20
D B 12
E B 11
F C 5
G C,D,E 4
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AMITY GLOBALBUSINESS SCHOOL Bangalore
H E 30
I G 8
J G 10
K F,I 7
L H,J,K 6
M H,J,K 4
N L 10
P M,N 3
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Use only 2 dummy variables
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• F P C A R I K M N
B D G J L S E H
10 0
26 6
34 4
426 26
838 38
626 26
515 15
730 30 9
45 4510
51 51
1161 61
12
64 64
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Activity
duration
ES EF LS LF Tot F Free
A 6 0 6 0 6 0 0
B 4 0 4 0 4 0 0
C 20 6 26 6 26 0 0
D 12 4 16 14 26 10 10
E 11 4 15 4 15 0 0
F 5 26 31 33 38 7 7
G 4 26 30 26 30 0 0
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H 30 15 45 15 45 0 0
I 8 30 38 30 38 0 0
J 10 30 40 35 45 5 5
K 7 38 45 38 45 0 0
L 6 45 51 45 51 0 0
M 4 45 49 57 61 12 12
N 10 51 61 51 61 0 0
P 3 61 64 61 64 0 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Floats are useful to management as they indicate how much an activity can be delayed without delaying the entire project. Resources can be redeployed from an activity with slack to be used elsewhere
• Slack --- events
• Head slack = Lj - Ej
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Tail Slack = Li – Ei
• Total float = Lj – Ei - tij
• Free float = total float – head slack
• Independent float = free float – tail slack
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
Activity Preceding activity
to tm tp
A Nil 4 7 16
B Nil 1 5 15
C A 6 12 30
D A 2 5 8
E C 5 11 17
F D 3 6 15
G B 3 9 27
H E , F 1 4 7
I G,H 4 19 28
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Draw the PERT network diagram• Identify the critical path• Find the expected duration and variance for each
activity. What is expected project length ?• What is the prob that the project is completed
between 35 and 40 weeks ?
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activity to tm tp te σ²
Time estimates(wks)
A 1-2 4 7 16 8 4
B 1-3 1 5 15 6 49 / 4
C 2-4 6 12 30 14 16
D 2-5 2 5 8 5 1
E 4-6 5 11 17 11 4
F 5-6 3 6 15 7 4
G 3-7 3 9 27 11 16
H 6-8 1 4 7 4 1
I 7-8 4 19 28 18 16
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AMITY GLOBALBUSINESS SCHOOL Bangalore
10 0
28 8
36 8
422 22
633 33
513 26
717 19
8
37 3755 55
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Critical path
1---2 ---4 –6---7----8---9
• Expected project length
8 + 14 + 11 + 4 = 55 weeks
• Variance
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Prob that the project will be completed between 35 and 40 weeks
prob { 35 - 55 ≤ Z = Ts – Te ≤ 40 – 55 }
σ σ e σ
= p ( - ≤ Z ≤ )
= P ( - ≤ Z ≤ 0 ) + P ( 0 ≤ Z ≤ 0.6 )
=
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AMITY GLOBALBUSINESS SCHOOL BangaloreCrashing
The process of reducing an activity time by putting extra effort is called crashing the activity. It is the process of shortening a project and is usually achieved by adding extra resources to an activity.
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• Direct costs are the costs directly associated with each activity such as machine costs, labor costs and so on.
• Indirect costs are the costs due to management services, rentals, insurance etc.
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• Duration of any activity cannot be reduced indefinitely
• Crash time represents minimum activity duration time that is possible.
• Activity cost corresponding to the crash time is called the crash cost.
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AMITY GLOBALBUSINESS SCHOOL BangaloreActivity crashing
Act
ivit
y co
st
Activity time
Crashing activity
Crash time
Crash cost
Normal Activity
Normal time
Normal cost
Slope = crash cost per unit time
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• Crashing costs increase as project duration decreases
• Indirect costs increase as project duration increases
• Reduce project length as long as crashing costs are less than indirect costs
Time-Cost Relationship
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Time-Cost Tradeoff
cost
time
Direct cost
Indirect cost
Total project costMin total cost = optimal project time
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Cost slope =
Crash cost – normal cost
Normal time – crash time
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AMITY GLOBALBUSINESS SCHOOL BangaloreQ
The following table gives the activities in a construction project and other relevant information
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Activity Predecessor
Time Direct Costs
Normal Crash Normal Crash
A - 4 3 60 90
B - 6 4 150 250
C - 2 1 38 60
D A 5 3 150 250
E C 2 2 100 100
F A 7 5 115 175
G D,B,E 4 2 100 240
Total 713
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Indirect costs vary as follows :
Days 15 14 13 12 11 10 9 8 7 6
Cost-Rs 600 500 400 250 175 100 75 50 35 25
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Draw an arrow diagram of the network
• Determine the project duration which will result in minimum total cost.
• Activity F & G end at the same time.
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3
24 4
513 13
• 2 4
C 2 E 2
B 4 G
6 9 9 4
A 4 D 5 F 7
10 0
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The critical path is 1-2-4-5.
Critical activities are 1-2, 2-4 and 4-5
Project duration is 13 days
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Activity A B C D E F G
1-2 1-4 1-3 2-4 3-4 2-5 4-5
Cost slope
30 50 22 50 - 30 70
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Among the critical activities, choose activity with least cost slope
i.e A (1-2)
Crash activity A by 4-3 = 1 day.
Project duration reduces by 1 day
= 13-1 = 12days
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Total project cost = total direct cost + increased direct cost due to crashing +
indirect cost for 12 days
= 713 + 1 * 30(cost slope) + 250
= 993
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AMITY GLOBALBUSINESS SCHOOL BangaloreNew network
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AMITY GLOBALBUSINESS SCHOOL Bangalore
23 3
512 12
C 2 E 2
1 6 4 4
0 0 B 8 G
A 3 D 5 F 7
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AMITY GLOBALBUSINESS SCHOOL Bangalore
The critical path is 1-2-4-5 Total time = 12 days;
The critical activities are 1-2, 2-4, 4-5
The first one A 1-2 has already been crashed
From the other two choose the one with the smallest cost slope and crash it.
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Activity D(2-4) is crashed by 5-3=2 days.
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3
23 3
510 10
2 2
C 2 E 2
B 4 G
6 6 6 4
A 3 D 3 F 7
10 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
There are 3 critical paths
1-2-5
1-4-5
1-2-4-5
Each with 10 days
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Total cost = 743 +2*50 +100 = Rs. 943
since there are more than one parallel critical path, crashing must be performed.
As A & D are crashed, crash activities F & G by 2 days each to reduce the project duration by 2 days.
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AMITY GLOBALBUSINESS SCHOOL Bangalore
3
23 3
58 8
2 2
C 2 E 2
B 4 G
6 6 6 2
A 3 D 3 F 5
10 0
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AMITY GLOBALBUSINESS SCHOOL Bangalore
There are 3 critical paths1-4-51-2-51-2-4-5Each with 8 days
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AMITY GLOBALBUSINESS SCHOOL Bangalore
• Total cost = 843 +(2*30 +2*70) +50
= Rs. 1093
Since all the activities in the critical path 1-2-5 are crashed , further crashing is not possible.
Decision : minimum total cost is Rs. 943 and the optimum duration is 10 days.
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Q
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AMITY GLOBALBUSINESS SCHOOL Bangalore
The following table gives data on normal time, and cost and crash time and cost for a project.
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AMITY GLOBALBUSINESS SCHOOL BangaloreActivity Normal Crash
Time(weeks) Cost (INR) Time(weeks) Cost (INR)
1-2 3 300 2 400
2-3 3 30 3 30
2-4 7 420 5 580
2-5 9 720 7 810
3-5 5 250 4 300
4-5 0 0 0 0
5-6 6 320 4 410
6-7 4 400 3 470
6-8 13 780 10 900
7-8 10 1000 9 1200
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Indirect cost is Rs. 50 per week.
1. Draw the network for the project and identify the critical path
2. What are the normal project duration and associated cost
3. Crash the relevant activities systematically and determine the optimal project completion time and cost.
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10 0
2
3 3
4
10 12
36 7
512 12
618 18
722 22
832 32
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AMITY GLOBALBUSINESS SCHOOL Bangalore
The critical path is
1-2-5-6-7-8
Project duration is 32 weeks and associated cost is as follows :
= direct normal cost +indirect cost for 32 weeks
= 4220 + 50*32 = Rs. 5820
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AMITY GLOBALBUSINESS SCHOOL BangaloreCrash cost slope for critical
activities are 1-2 400-300 = 100
3-2
2-5 810 – 720 = 45
9-7
5-6 45
6-7 70
7-8 200
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The minimum crash cost per week is for activity 2-5 & 5-6.
Hence, crash activity 2-5 by 2 weeks.
But, the activity should b reduced only by 1 week – so that another critical path be raised
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• 81
0 0
2
3 3
4
10 11
36 6
511 11
617 17
721 21
831 31
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AMITY GLOBALBUSINESS SCHOOL Bangalore
Network is as shown in previous slide.
New project time is 31 weeks
The critical paths are
1-2-5-6-7-8
1-2-3-5-6-7-8
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AMITY GLOBALBUSINESS SCHOOL Bangalore
The new total cost
= total direct normal cost + increased direct cost due to crashing + indirect cost for 31 weeks.
= (4220 + 1*45) + 31 * 50 = Rs. 5815
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AMITY GLOBALBUSINESS SCHOOL BangaloreWrt new network
Crashed Activity Slope
1-2 100
2-5 Crashed
2-3 No need(not in first path)
3-5 50
5-6 45
6-7 70
7-8 200
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Crash 5-6 by 2 weeks.
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• 8 41
0 0
2
3 3
4
10 11
36 6
511 11
615 15
719 19
829 29
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AMITY GLOBALBUSINESS SCHOOL Bangalore
The new total cost
= total direct normal cost + increased direct cost due to crashing of 5-6 + indirect cost for 29 weeks.
= (4220 + 1*45 + 2 *45) + 29 * 50 = Rs. 5805
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If we go in for further crashing the project cost wil;l go up (Rs.5825), hence stop here.