LINE DIAGRAMS Learning objectives After this lecture you

LINE DIAGRAMS Learning objectives After this lecture you will be able to: • Prepare and use line diagrams • Demonstrate the effect that making activities ‘continuous’ has on a project • Describe why this method is particularly suitable for repetitive work

Quantity Activity A Activity B 80% 20% Time Day X

Example Project = To build six partly prefabricated houses where each house consists of the following four activities. A B C A = Excavation/Services takes 2 weeks/house B = Footings takes 3 weeks/house C = Superstructure takes 4 weeks/house D = 2 nd Fix takes 2 weeks/house D

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 House 1. A B C D House 2. A B C D House 3. A B C D House 4. A B C D House 5. A B C D House 6. A B C D 31 weeks

House 1 House 2 House 3 House 4 House 5 House 6 0 2 A/2 0 2 2 4 A/2 4 6 A/2 8 10 6 8 A/2 12 14 8 10 A/2 16 18 10 12 A/2 20 22 2 5 B/3 2 5 5 8 B/3 6 9 8 11 B/3 10 13 11 14 B/3 14 17 B/3 18 21 17 20 B/3 5 9 C/4 5 9 9 13 C/4 9 13 13 17 C/4 13 17 17 21 C/4 17 21 21 25 C/4 21 25 25 29 C/4 22 25 25 29 9 11 D/2 11 13 13 15 D/2 15 17 17 19 D/2 19 21 21 23 D/2 23 25 25 27 D/2 27 29 29 31 = Project completion time D/2 29 31

House No. 12 5 10 15 20 20 25 29 31 (Project compl. ) 30 Weeks 6 5 4 A B C 3 2 1 2 5 9 19 D alt. D

Example These are two alternative schedules for building six identical houses. In Schedule 1 it takes 15 weeks to build one house whilst in Schedule 2 it takes 20 weeks to build one house. Which alternative would result in the quickest project completion time if all activities must be continuous? Schedule 1 3 6 houses 2 4 Cycle = 15 weeks Schedule 2 4 6 houses 4 4 Cycle = 20 weeks

Schedule 1 House No. 5 wks 6 10 15 18 20 20 25 30 34 36 35 40 5 4 (3) (2) (4) (2) 3 2 1 8 10 24 26 (4) 45 50 50

Schedule 2 House No. 5 wks 6 10 20 24 25 28 32 36 40 30 35 40 (4) (4) 15 5 4 (4) 3 2 1 4 8 12 16 45 50

Demonstration of affect of continuous activities Say we have a project consisting of six identical houses with each house having the three activities A, B and C. Starting with all activities having a duration of 5 we will check the result of making each activity both longer and shorter. A/5 6 Houses B/5 Cycle time = 15 weeks

All equal: 5 A/5 10 6 5 4 3 2 1 5 10 15 B/5 20 C/5 25 30 Cycle = 15 weeks 35 40 45

Say, we reduce Activity A to only 4 weeks on each of the six houses. What affect will that have on the project completion time?

‘A’ reduced to 4: 5 10 A/4 15 20 B/5 24 30 C/5 34 Cycle = 14 weeks 39 45 6 5 4 3 A saving of only ONE week 2 1 4 9

Say, we increase Activity A to 6 weeks on each of the six houses. What affect will that have on the project completion time?

‘A’ increased to 6: 5 10 A/6 15 20 B/5 25 30 C/5 Cycle = 16 weeks 36 41 46 6 5 4 3 2 A delay of SIX weeks 1 11 16

Say, we reduce Activity B to 4 weeks on each of the six houses. What affect will that have on the project completion time?

‘B’ reduced to 4: 5 10 A/5 15 20 B/4 25 30 C/5 34 Cycle = 14 weeks 40 44 6 5 4 3 2 A DELAY of FOUR weeks! 1 10 14

Say, we increase Activity B to 6 weeks on each of the six houses. What affect will that have on the project completion time?

‘B’ increased to 6: 5 10 A/5 15 20 B/6 25 30 C/5 35 Cycle = 16 weeks 41 46 6 5 4 3 2 A delay of SIX weeks! 1 5 16

Say, we reduce Activity C to 4 weeks on each of the six houses. What affect will that have on the project completion time?

‘C’ reduced to 4: 5 10 A/5 15 20 B/5 25 30 C/4 Cycle = 14 weeks 35 39 6 5 4 3 2 A saving of ONE week! 1 5 15

Finally, say we increase Activity C to 6 weeks on each of the six houses. What affect will that have on the project completion time?

‘C’ increased to 6: 5 10 15 A/5 20 B/5 25 30 C/6 35 Cycle = 16 weeks 46 6 5 4 3 2 A delay of SIX weeks! 1 5 10

SUMMARY Reduce ‘A’ by one week/house: Saving of ONE week for project Extend ‘A’ by one week/house: Delay of SIX weeks for project Reduce ‘B’ by one week/house: Delay of FOUR weeks for project Extend ‘B’ by one week/house: Delay of SIX weeks for project Reduce ‘C’ by one week/house: Saving of ONE week for project Extend ‘C’ by one week/house: Delay of SIX weeks for project CONCLUSION It is difficult to predict the result of changing activity durations when activities must be continuous and it is efficient to have parallel activities.

Project Evaluation and Review Technique (PERT) Similar to the Critical Path Method, but an attempt to estimate the probability of meeting certain deadlines or milestones.

EXAMPLE A B D C E F For each activity we select three possible activity durations O = Optimistic duration M = Most likely duration P = Pessimistic duration (within reason e. g. not incl WW 3 or similar)

Normal distribution

Activity A B C D E F O 1 2 1 1 2 2 M 2 4 2 2 4 3 P 3 (weeks) 7 5 3 6 7

Expected activity duration te = O + 4 x. M + P 6 In our example we get these te-values A. te = (1 + 4*2 + 3)/6 = 2. 00 B. te = (2 + 4*4 + 7)/6 = 4. 17 C. te = ( 1 + 4*2 + 5)/6 = 2. 33 D. te = (1 + 4*2 + 3)/6 = 2. 00 E. te = (2 + 4*4 + 6)/6 = 4. 00 F. te = (2 + 4*3 + 7)/6 = 3. 50

Use the te-values as activity durations and analyse the network A/2. 00 B/4. 17 C/2. 33 D/2. 00 E/4. 00 F/3. 50

Use the te-values as activity durations and analyse the network 0 2. 00 6. 17 8. 50 A/2. 00 B/4. 17 C/2. 33 0 2. 00 6. 17 8. 50 2. 00 4. 00 8. 50 12. 00 D/2. 00 E/4. 00 F/3. 50 2. 50 4. 50 8. 50 12. 00 CP = A – B – C – F Expected project completion time is 12. 00 weeks

STATISTICS If a series of Variables t 1, t 2, t 3, …, tn have the Mean Times te 1, te 2, te 3, …, ten and the Standard Deviations S 1, S 2, S 3, …, Sn, then the expected total time T = t 1 + t 2 + t 3 + …+tn follows a Normal Distribution with a Mean Value Te = te 1 + te 2 + te 3 +…+ ten and the Variance V = S^2 = S 1^2 + S 2^2 + S 3^2 +…Sn^2 Say: te 1 +/- S 1 te 2 +/- S 2 Then: Te = te 1 + te 2 + te 3 +/- S 3 S 1^2 + S 2^2 + S 3^2

So we’ll use the following formula to determine the expected project completion time Te = te +/S 2 To calculate the Standard Deviation for each activity we use This formula S = P – O 6 In our example we get A. S = (3 – 1)/6 = 0. 33 B. S = ( 7 – 2)/6 = 0. 83 C. S = (5 - 1)/6 = 0. 67 D. S = (3 – 1)/6 = 0. 33 E. S = (6 – 2)/6 = 0. 67 F. S = (7 – 2)/6 = 0. 83

For our project we use the critical path activities to calculate the expected project completion time with its standard deviation. Te = 2 + 4. 17 + 2. 33 + 3. 50 +/- 0. 332 + 0. 832 + 0. 672 + 0. 832 = 12. 00 +/- 1. 39 weeks 1 SD = 1. 39 weeks 12. 00 weeks Next we need to look at a Normal Distribution table in order to determine the probability of completing the project with a certain time.

NORMAL DISTRIBUTION TABLE SD . 00 . 01 . 02 . 03 . 04 . 05 . 06 . 07 . 08 . 09 . 0. 5000 . 5040 . 5080 . 5120 . 5160 . 5199 . 5239 . 5279 . 5319 . 5359 . 1. 5398 . 5435 . 5478 . 5517 . 5557 . 5596 . 5636 . 5675 . 5714 . 5753 . 2. 5793 . 5832 . 5871 . 5910 . 5948 . 5987 . 6026 . 6064 . 6103 . 6141 . 3. 6179 . 6217 . 6255 . 6293 . 6331 . 6368 . 6406 . 6443 . 6480 . 6517 . 4. 6554 . 6591 . 6628 . 6664 . 6700 . 6736 . 6772 . 6808 . 6844 . 6879 . 5. 6915 . 6950 . 6985 . 7019 . 7054 . 7088 . 7123 . 7157 . 7190 . 7224 . 6. 7257 . 7291 . 7324 . 7357 . 7389 . 7422 . 7454 . 7486 . 7517 . 7549 . 7. 7580 . 7611 . 7642 . 7673 . 7703 . 7734 . 7764 . 7794 . 7823 . 7852 Read the whole number and the first decimal of the number of Standard Deviations down the left hand column and read the second decimal across to the right. The number at that intersection represents the probability. For example 0. 56 SD has a probability of 0. 7123 or ~71%

SD . 00 . 01 . 02 . 03 . 04 . 05 . 06 . 07 . 08 . 09 . 0 . 5000 . 5040 . 5080 . 5120 . 5160 . 5199 . 5239 . 5279 . 5319 . 5359 . 1 . 5398 . 5435 . 5478 . 5517 . 5557 . 5596 . 5636 . 5675 . 5714 . 5753 . 2 . 5793 . 5832 . 5871 . 5910 . 5948 . 5987 . 6026 . 6064 . 6103 . 6141 . 3 . 6179 . 6217 . 6255 . 6293 . 6331 . 6368 . 6406 . 6443 . 6480 . 6517 . 4 . 6554 . 6591 . 6628 . 6664 . 6700 . 6736 . 6772 . 6808 . 6844 . 6879 . 5 . 6915 . 6950 . 6985 . 7019 . 7054 . 7088 . 7123 . 7157 . 7190 . 7224 . 6 . 7257 . 7291 . 7324 . 7357 . 7389 . 7422 . 7454 . 7486 . 7517 . 7549 . 7580 . 7611 . 7642 . 7673 . 7703 . 7734 . 7764 . 7794 . 7823 . 7852 . 8 . 7881 . 7910 . 7939 . 7967 . 7995 . 8023 . 8051 . 8078 . 8106 . 8133 . 9 . 8159 . 8186 . 8212 . 8238 . 8264 . 8289 . 8315 . 8340 . 8365 . 8389 1. 0 . 8413 . 8438 . 8461 . 8485 . 8508 . 8531 . 8554 . 8577 . 8599 . 8621 1. 1 . 8643 . 8665 . 8686 . 8708 . 8729 . 8749 . 8770 . 8790 . 8810 . 8830 1. 2 . 8849 . 8869 . 8888 . 8907 . 8925 . 8944 . 8962 . 8980 . 8997 . 90147 1. 3 . 90320 . 90490 . 90658 . 90824 . 90988 . 91149 . 91309 . 91466 . 91621 . 91774 1. 4 . 91924 . 92073 . 92220 . 92364 . 92507 . 92647 . 92785 . 92922 . 93056 . 93189 1. 5 . 93319 . 93448 . 93574 . 93699 . 93822 . 93943 . 94062 . 94179 . 94295 . 94408 1. 6 . 94520 . 94630 . 94738 . 94845 . 94950 . 95053 . 95154 . 95254 . 95352 . 95449 1. 7 . 95543 . 95637 . 95728 . 95818 . 95907 . 95994 . 96080 . 96164 . 96246 . 96327 1. 8 . 96407 . 96485 . 96562 . 96638 . 96712 . 96784 . 96856 . 96926 . 96995 . 97062 1. 9 . 97128 . 97193 . 97257 . 97320 . 97381 . 97441 . 97500 . 97558 . 97615 . 97670 2. 0 . 97725 . 97778 . 97831 . 97882 . 97932 . 97982 . 98030 . 98077 . 98124 . 98169 2. 1 . 98214 . 98257 . 98300 . 98341 . 98382 . 98422 . 98461 . 98500 . 98537 . 98574 2. 2 . 98610 . 98645 . 98679 . 98713 . 98745 . 98778 . 98809 . 98840 . 98870 . 98899 2. 3 . 98928 . 98956 . 98983 . 99010 . 99036 . 99061 . 99086 . 99110 . 99134 . 99158 2. 4 . 99180 . 99202 . 99224 . 99245 . 99266 . 99286 . 99305 . 99324 . 99343 . 99361 2. 5 . 99379 . 99396 . 99413 . 99430 . 99446 . 99461 . 99477 . 99420 . 99506 . 99520 2. 6 . 99534 . 99547 . 99560 . 99573 . 99586 . 99598 . 99609 . 99621 . 99632 . 99643 2. 7 . 99653 . 99664 . 99674 . 99683 . 99693 . 99702 . 99711 . 99720 . 99728 . 99736 2. 8 . 99744 . 99752 . 99760 . 99767 . 99774 . 99781 . 99788 . 99795 . 99801 . 99804 2. 9 . 99813 . 99819 . 99825 . 99830 . 99836 . 99841 . 99846 . 99851 . 99856 . 99860 3. 0 . 99865 . 99869 . 99874 . 99878 . 99882 . 99886 . 99889 . 99893 . 99896 . 99900 3. 1 . 99903 . 99906 . 99910 . 99913 . 99915 . 99918 . 99921 . 99924 . 99926 . 99929 3. 2 . 99931 . 99934 . 99936 . 99938 . 99940 . 99942 . 99944 . 99946 . 99948 . 99950 3. 3 . 99952 . 99953 . 99955 . 99957 . 99958 . 99960 . 99961 . 99962 . 99964 . 99965 3. 4 . 99966 . 99968 . 99969 . 99970 . 99971 . 99972 . 99973 . 99974 . 99975 . 99976 3. 5 . 99977 . 99978 . 99979 . 99980 . 99981 . 99982 . 99983

What is the probability of completing the project in 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks?

What is the probability of completing the project in 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks? 1. 00 SD = 0. 8413 = 84% Note that completion times longer than the mean value will have higher than 50% probability whilst shorter completion times will have probabilities of less than 50%. The table is only made up for 50% and greater probabilities.

What is the probability of completing the project in 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks? 1. 00 SD = 0. 8413 = 84% and within 12 weeks plus two SDs, i. e. 14. 78 weeks?

What is the probability of completing the project I 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks? 1. 00 SD = 0. 8413 = 84% and within 12 weeks plus two SDs, i. e. 14. 78 weeks? 2. 00 SD = 0. 97725 = 98%

What is the probability of completing the project I 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks? 1. 00 SD = 0. 8413 = 84% and within 12 weeks plus two SDs, i. e. 14. 78 weeks? 2. 00 SD = 0. 97725 = 98% What about 12 weeks minus one SD, i. e. 10. 61 weeks?

What is the probability of completing the project I 12 weeks plus one Standard Deviation, i. e. 13. 39 weeks? 1. 00 SD = 0. 8413 = 84% and within 12 weeks plus two SDs, i. e. 14. 78 weeks? 2. 00 SD = 0. 97725 = 98% What about 12 weeks minus one SD, i. e. 10. 61 weeks? The table is only set up for durations longer than the 12 weeks i. e. with probabilities higher than 50% so we need to take 1. 0 minus the probability we read in the table, so; 1. 00 SD = 0. 8413 but 1. 0 – 0. 8413 = 0. 1587 = 16%

To find the probability of completion within a certain number of weeks we need to determine how many Standard Deviations that time represents. To finish the project within 12 weeks represents zero Standard Deviations from the Mean value of 12. 00. 0. 00 SD = 0. 500 = 50%

To find the probability of completion within a certain number of weeks we need to determine how many Standard Deviations that time represents. We had T = 12 +/- 1. 39 weeks. To finish the project within 12 weeks represents zero Standard Deviations from the Mean value of 12. 00. 0. 00 SD = 0. 500 = 50% To finish within 13 weeks represents (13 -12)/1. 39 = 0. 72 SD = 0. 7642 = 76%

To find the probability of completion within a certain number of weeks we need to determine how many Standard Deviations that time represents. We had T = 12 +/- 1. 39 weeks. To finish the project within 12 weeks represents zero Standard Deviations from the Mean value of 12. 00. 0. 00 SD = 0. 500 = 50% To finish within 13 weeks represents (13 -12)/1. 39 = 0. 72 SD = 0. 7642 = 76% To finish within 11 weeks represents (12 -11)/1. 39 = 0. 72 SD which has a probability of 0. 7642 (as above) but since 11 is less than 12 the probability will be less than 50% so we get 1 – 0. 7642 = 0. 2358 = 23%

Tutorial example: Line Diagrams One of your colleagues handed you a schedule for a housing project consisting of 48 identical, partly prefabricated, houses. The schedule for each house is linear and contains the following activities with their respective duration (in days). A/3 B/2 C/1 D/4 E/2 F/6 • Determine the shortest possible completion time for the whole project, as it has been scheduled. • It is important that you manage to keep the workforce for the duration of the whole project. It is therefore essential that you can offer the workers continuous employment. How long would it take to complete the project if that condition was applied? • You now realise that activities C and D utilise the same type of trade people, which means that you can distribute the workers between those two activities as you see fit. Assume that after the first 24 houses, you move some workers from activity C to activity D. This has the effect of changing the activity duration for the following 24 houses so that C takes 2 days and D takes 3 days. Determine the effect this has on the completion time for the whole project. • Assume that all the workers, in all activities, were multi-skilled so that you could decide how to distribute them between all activities and thereby control the duration of each activity. The total number of working days per house must remain the same as before. How would you suggest that each activity’s duration should be modified in order to complete the project in the shortest possible time, whilst maintaining continuity? What would that completion time be?

Tutorial: PERT The optimistic (O), most likely (M) and pessimistic (P) activity durations have been given for the project described by the network below. Use the scheduling technique PERT to; • Calculate the probability of the whole project being completed within 15 weeks. • Calculate which project completion time would give you at least a 90% probability of completion (in a whole number of weeks). • Calculate the probability of activity 7 -9 being able to start in week 11. Activity 1 -2 1 -3 2 -5 2 -6 3 -4 4 -7 4 -8 5 -7 6 -7 7 -9 8 -9 O 3 4 2 2 2 1 3 2 2 3 2 M 4 5 3 4 3 1 4 2 3 4 2 P 5 7 5 8 4 1 6 3 4 5 2 3 1 2 4 8 5 7 6 9