Analysing the AoA network Project Management

Total Project Time The minimum time in which the project can be completed. Calculation: forward pass Forward pass: calculating the earliest event times (EETs) and the earliest start times (ESTs) of all activities. Earliest Finishing Time = EST + Duration

Critical path Path: continuous series of project activities connected by logical relationships as designated in the project schedule network diagram. Critical path: sequence of activities that has no float time, and that determines the duration of the project. It is the longest path. Activities on the critical path (or paths) are the critical activities. The critical path can be identified by a backward pass, calculating the Latest Event Times (LETs) and the Latest Finishing Times (LFTs). Latest Starting Time = Latest Finishing Time - Duration

Activity times & event times EET = EST of all emerging activities LET = LFT of all entering activities Deadline Activity identifier Duration 1 2 EET LET

TPT EST 0 EFT 14 14 14 1 2 a 14 LST 0 LFT 14 TPT = 14

Float 1 2 Float on activity ‘a’: Float: 6 a 20 EST 0 EFT 14 14 14 6 20 14 14 1 2 a 6 20 LST 6 LFT 20 Float: 6

4 6 5 Calculate the… EET of event 6 LETs of events 4 and 5 22 ? 5 24 6 34 d e 8 10 Calculate the… EET of event 6 LETs of events 4 and 5 ESTs and EFTs of activities ‘d’ and ‘e’ LSTs and LFTs of activities ‘d’ and ‘e’

4 6 5 EST and EFT of ‘d’: 22 and 30 EST and EFT of ‘e’: 24 and 34 8 4 30 d 26 26 34 34 6 34 34 24 24 5 10 34 e 24 24 EST and EFT of ‘d’: 22 and 30 EST and EFT of ‘e’: 24 and 34 LST and LFT of ‘d’: 26 and 34 LST and LFT of ‘e’: 24 and 34

15 ? 30 10 a 10 ? 20 ? 35 12 b 25 ? 8 c

15 20 30 10 a 10 17 20 22 35 12 b 25 18 8 c

Calculate all event and activity times and the float-times, find the critical path

Calculate all event and activity times and the float-times, find the critical path 16 16 16 31 8 24 24 24 39 20 30 29 39 20 20 20 35 35 35 38 38 38 50 51 20 20 20 35 35 36 39 39 39 51 51 35 51 35 51 30 21 51

Activity times and float for the previous diagram EST LST EFT LFT Float A B C D E G H J K

Activity times and float for the previous diagram EST LST EFT LFT Float A 8 16 24 B 20 C 21 30 51 D 35 E 29 39 9 G 36 38 1 H J 31 K 50

Four characteristics of the critical path It starts at the first node It is continuous It ends at the last node It has no float

Floats in the AoA network

Activity and event times Activity times: EST LST EFT LFT Event times: EET LET

Float Time available for an activity or path in addition to its duration. It can be positive or negative It is a property of activities (and available only at given activities) In case of more succeeding activities, the minimum have to be taken. Types of float: Total float Free float

Total float 2 1 4 3 The total float possessed by an activity. Calculation for activity j: Total floatj = LFTj – EFTj Identify the activity/activities possessing float below, and calculate the total float(s). 2 ? Total float = = 35 – 22 = 13 Total float = = 25 – 12 = 13 12 12 35 12 10 12 25 25 1 ? 25 4 ? 22 c a 13 23 35 23 12 15 3 ? 8 35 35 15 15 b 15 d 27 27 Total float = = 27 – 15 = 12 27 Total float = = 35 – 23 = 12 Critical path = ? TFT = ?

Total float 2 1 4 3 The total float possessed by an activity. Calculation for activity j: Total floatj = LFTj – EFTj = LFTj – (ESTj + Dj) Identify the activities possessing float below, and calculate the total float for each. Total float = = 11 – (0+5) = 6 2 5 Total float = 23 – (5+12) = 6 12 5 11 c a 1 4 23 15 8 23 b 3 15 d 15

Calculating free float The FREE FLOAT is the float possessed by an activity which, if used, will not change the float in later activities. Free floatj = ESTj+1 – EFTj = EEThead – EETtail - D Free float = = 5 – 0 – 5 = 0 2 5 Free float = 23 – 5 – 12 = 6 12 5 11 c a 1 4 23 15 8 23 b 3 15 d 15

Positive float on the critical path If the target time (deadline) for the project (or for a part of the project) is grater than TPT, than a float will appear on the critical path(s), too (and also on other paths). The float will be the minimal on the critical path(s).

Negative float Negative float = the time by which activities on the path or paths concerned must be reduced if the TPT is to be met. Negative float is a type of the Total Float. If it appears on both critical and non-critial paths, than the critical path has the greatest negative float (in absolute value). Negative float apperas when the TPT is longer than the project’s target time. 1 ? 2 a 10 3 c 15

Negative float Negative float = the time by which activities on the path or paths concerned must be reduced if the TPT is to be met. Negative float is a type of the Total Float. If it appears on both critical and non-critial paths, than the critical path has the greatest negative float (in absolute value). Negative float apperas when the TPT is longer than the project’s target time. 1 -5 2 10 5 a 3 20 15 c Total float: -5

Total float = 11 – 0 – 5 = 6 Free float = 5 – 0 – 5 = 0 Slack It is the ’float’ measured at events, and not at activities. Slack = LET - EET Slack: 6 Total float = 23 – 5 – 12 = 6 Free float = 23 – 5 – 12 = 6 Total float = 11 – 0 – 5 = 6 Free float = 5 – 0 – 5 = 0 2 5 12 5 11 c a 1 4 23 TF = 0; FF = 0 TF = 0; FF = 0 15 8 23 b 3 15 d Slack: 0 Slack: 0 15 Slack: 0

Readings Lockyer – Gordon (2005) Chapter 13

Analysing the AoN network Project Management

Data on the activity node Activity label & description Total Float EST EFT LST Duration LFT

Total Project Time The shortest time in which the project can be completed. Determined by the critical path. Calculation: forward pass Forward pass: The earliest start times (EST) of all activities are calculated. Trom these the earliest finishing times (EFT) are also calculated

Critical path sequence of activities that has no float time (or has the maximum negative float time in absolute value), and that determines the duration of the project. It is the longest path. Activities on the critical path are the critical activities. The critical path can be identified by a backward pass, calculating the Latest Finishing Times (LFT), and from these the Latest Starting Times (LST).

Floats in AoN Total float: the time by which an activity can be delayed or extended without affecting the TPT. It can be used to delay the start of an activity or to increase its duration. TF = LST - EST Free float: the time by which an activity can be delayed or extended without affecting the start of any succeding activity. FF = ESTj+1 - EFTj

Example: organising a conference Objectives: to organise a 3 days long open scientific conference with 100-200 participants, 30-50 lectures, buffet reception, a conference book of the best studies and TV and radio interviews with some of the most known lecturers. Create the WBS chart and create the task list with estimated durations and precedence relations (in a table form) Plot both the AoA and the AoN diagram Calculate the TPT, identify the critical path, the total, and the free float times.

Organising participants Invitation and marketing Organising interviews Example: WBS Project Book Event management Marketing Editing Publishing Infra-structure Organising participants Arranging event Invitation and marketing Organising interviews Collecting articles Peer reviewing Facilities Staff Materials

Task list with precedence relations Activity label Task description Duration (weeks) Immediate predecessors a Invitation 2 – b Organising participants 4 c Facilities 3 d Staffing e Materials f Collecting articles 6 g Peer reviewing h Organising interviews 1 c, d, e i Publishing 5 j Arranging event h, i Activity label Task description Duration (weeks) Immediate predecessors a Invitation 2 b Organising participants 4 c Facilities 3 d Staffing e Materials h Collecting articles 6 j Peer reviewing k Publishing 5 l Organising interviews 1 m Arranging event Activity label Task description Duration Immediate predecessors a Invitation b Organising participants c Facilities d Staffing e Materials h Collecting articles j Peer reviewing k Publishing l Organising interviews m Arranging event

AoA 4 9 19 3 c 5 10 19 4 d 1 2a 2 3 6 6 10 19 4b 9 20 3e 1h 10 21 1j 6 5 f TPT = 21 i 7 12 8 15 3g CP: a-b-f-g-i-j

AoN TPT: 21 CP: a-b-f-g-i-j c a b d h j e i f g 1 6 9 10 16 19 3 9 2 2 9 a 2 b 2 6 4 d 6 10 15 19 4 h 10 11 9 19 20 1 j 20 21 1 1 e 6 9 10 16 19 3 i 15 20 5 f 6 12 g 12 15 3 TPT: 21 CP: a-b-f-g-i-j

Activity times for the previous diagram (finalize individually) Duration EST LST EFT LFT Total float Free float a 2 b 4 6 c 3 9 16 19 10 1 d e f g h i j

Example 2 (for individual practice) a) Draw the AoA and AoN diagram with the data below: Activity label Duration (weeks) Immediate predecessors a 1 – b 2 c 5 d 3 e f g 4 f, c, d b) Determine the TPT and the critical path and activity floats. c) Compute the EETs, LETs and slack for every node in the AoA & ESTs, LSTs, EFTs, LFTs in the AoN diagram.

Solution: AoA 3 5 1 2 6 7 4 2 b 2 e 2 f 1 7 1 a 5 c 4 g 7 3 d TPT: 11 Float: 0 3 5 Float: 0 2 b 2 e 2 f Float: 0 Slack: 0 Slack: 0 1 2 1 6 7 7 11 1 a 5 c 4 g T. float: 1 F. float: 1 Float: 0 Float: 0 Slack: 0 Slack: 0 Slack: 0 Slack: 0 4 7 3 d T. float: 3 F. float: 3 Slack: 3 TPT: 11 CP: a-b-e-f-g

Solution: AoN TPT: 11 CP: a-b-e-f-g b e f g a c d 1 3 2 3 5 2 5 7 2 1 b 1 3 2 e 3 5 2 f 5 7 2 1 g 7 11 4 a 1 c 1 6 2 7 5 3 d 1 4 3 7 TPT: 11 CP: a-b-e-f-g

Example 3 3 2 b 5 3 e 5 1 2 2 a 2 5 c 5 7 8 1 f 14 2 8 14 4 2 4 g 6 3 d 12 2 h 5 12 Calculate the EETs and LETs. Create a precedence table (with task, duration, immediate predecessor, total and free floats).

Solution Activity Duration Total float Free float a 2 b c 5 1 d 3 e f b c 5 1 d 3 e f g 4 h

‘Crashing’ – reducing task durations by increased costs

Definition of crashing Obtaining reduction in time at an increased cost (increasing the employed resources). Cost-slope: the cost of reducing duration time by a unit of time. Let’s see the following example: a 4 b 2 c e d 5 f 3 c 6 8 1 7 9 2 e 8 10 1 9 11 2 a 4 b 4 6 2 f 11 14 3 d 6 11 5

Procedure for crashing Crash one time unit at a time Only the crashing of critical activities has any effect on TPT Crash that activity first that is the cheapest to reduce in time Be aware of multiple critical paths Stop crashing when: the crash-time is reached at every ‘crashable’ activity, benefits of possible crashing are lower than crashing costs.

Crashing table If the costs to reduce times are known, then a table can be set up showing the relative costs for the reduction in time of each activity by a constant amount. Crash-time is the minimum duration of an activity. It is given by technical factors. Activity (label) Duration (day) Float (day) Crash time Cost-slope (€/day) a 4 2 100 b 150 c 1 110 d 5 3 200 e 160 f 500 Crash time is the minimum duration of an activity (one cannot crash the duration of an activity below the crash time). Benefit of reducing TPT by one day: 400 €/day

Solution method step: identify the critical activities step: find the critical activity with cheapest crash cost, and if its cost slope is lower than the daily benefit from crashing, reduce its duration with one day. If there is no activity to crash, or it is too costly, stop crashing and go to step 4. step: reidentify the critical path, and go back to step two. step: identify the final critical path(s), TPT and the total net benefit of crashing.

Path / activity crashed Path / activity crached Solution Path durations Path / activity crashed normal step 1 step 2 step 3 step 4 step 5 – a d d, c none a-b-c-e-f 13 12 11 10 a-b-d-f 14 Cost: 100 200 310 Cumulated net benefit: 300 600 800 890 Path durations Path / activity crached normal step 1 step 2 step 3 step 4 step 5 Cost: Cumulated net benefit: After crashing: there are two critical paths TPT is 10 days total benefit of crashing is €890

Example 2 (for individual work) b 2 d 2 e 5 a 3 g 3 7 c 3 f 3 Identify the critical path and the TPT.

Example 2 (for individual work) b 3 5 2 d 5 7 2 e 7 12 5 a 3 g 12 15 3 7 c 3 6 1 4 7 f 6 9 3 12 Critcal: a-b-d-e-g TPT: 15 Using tbe table on the next slide, calculate the optimal TPT with crashing.

What is the total profit on crashing? 10 days €3000 Activity (label) Normal duration (day) Float (day) Crash time Cost-slope (€/day) a 3 1 500 b 2 550 c 150 d 5 900 e 4 400 f 100 g 200 Activity (label) Normal duration (day) Float (day) Crash time Cost-slope (€/day) a 3 1 500 b 2 550 c 150 d 5 900 e 4 400 f 100 g 200 Benefit of reducing TPT by one day: 1200 €/day What is the new TPT? What is the total profit on crashing? 10 days €3000

Reading Lockyer – Gordon (2005) Chapter 8 pp. 61-63. & Chapter 14

Thanks for the attention!