上海交通大学：《生产计划与控制 Production Planning and Control》课程教学资源（课件讲稿）chap10_Class3

Production and Operation Managements Project Scheduling Dr.Na GeNG Prof.Zhibin JIANG Department of Industrial Engineering Management Shanghai Jiao Tong University

Production and Operation Managements Dr. Na GENG Prof. Zhibin JIANG Department of Industrial Engineering & Management Shanghai Jiao Tong University Project Scheduling

Content The nature of project management Techniques for project scheduling Resource constraints Activity crashing Incorporating uncertainty in activity times 国 Problems with implementing critical path analysis Monitoring projects 2 上浒充通大粤

Content The nature of project management Techniques for project scheduling Resource constraints Activity crashing Incorporating uncertainty in activity times Problems with implementing critical path analysis Monitoring projects 2

Incorporating uncertainty in activity times Estimating activity duration distribution In PERT model,the duration for each activity is assume to follow Beta Distribution with pdf Tp+qx-to)P-1(4,-x)9-1 f(x)=r(p)r(q) (tp-to)p+q-2 where .totp-the lower and upper control point ·r(m)=6xm-1e-xdx Expected value: p,q-the shape parameters -o+e,-o49 Variance: 5=（6p-to'p+90+q+西 pq 3

Incorporating uncertainty in activity times Estimating activity duration distribution • In PERT model, the duration for each activity is assume to follow Beta Distribution with pdf ൌ ݔ ݂ ݍ൅݌ Г Г ݌ Г ݍ ݐെݔ ௢ ௣ିଵ ݐ ݔ െ௣ ௤ିଵ ݐ ݐ െ௣ ௢ ௣ା௤ିଶ where • t o, tp – the lower and upper control point • Г η ൌ ׬ ଵ஗ିݔ ݔ݀௫݁ି ஶ ଴ • p, q - the shape parameters 3 Expected value: ݐൌߤ ௢ ൅ ݐ ݐ െ௣ ௢ ݌ ݍ൅݌ Variance: ܵ ൌ ݐ ݐ െ௣ ௢ ݍ݌ ଶ ݍ൅݌ 1൅ݍ൅݌ ଶ

Incorporating uncertainty in activity times Estimating activity duration distribution A typical Beta distribution is commonly used to describe the duration of uncertain activities. Optimistic time(A):the duration of an activity if no complications or problems occur. Most likely time (M):the duration is most likely to occur. Pessimistic time (B):the duration of an activity if extraordinary problems arise. 01,Gm,=A+4M+B 350-98 Expected value: 5容9由 A+4M+B sideT ni be g=B-A行5Da5d orr bait 6 1a0800d2v t三 6 P (duration <A)=0.01 P (duration B)=0.01 Standard Variance: 6.01 oldeT B-A Days A M B 0三 Optimistic Most Mean Pessimistic iwL-1-8-O-A 6 Time Likely Time Time Time ong5由1 o noistitn t.出H的a

Incorporating uncertainty in activity times Estimating activity duration distribution • A typical Beta distribution is commonly used to describe the duration of uncertain activities. • Optimistic time (A): the duration of an activity if no complications or problems occur. • Most likely time (M): the duration is most likely to occur. • Pessimistic time (B): the duration of an activity if extraordinary problems arise. 4 Expected value: ൌ ݐ ܤ ൅ ܯ4 ൅ ܣ 6 Standard Variance: ൌ ߪ ܣെܤ 6

Incorporating uncertainty in activity times Project completion time distribution 1)For every activity,obtain estimates of A,M,B. 2)Use t=(A+4M+B)/6 to calculate the expected activity durations, and perform critical path analysis using the expected activity durations t. 3)The expected project completion time T is assumed to be the sum of the expected durations of activities on critical path 4)The variance of project completion time o2r is assumed to be the sum of the variances of activities on the critical path. 5)Probabilities regarding project completion time can be determined from standard normal tables. 5 上浒充通大粤

Project completion time distribution 1) For every activity, obtain estimates of A, M, B. 2) Use t=(A+4M+B)/6 to calculate the expected activity durations, and perform critical path analysis using the expected activity durations t. 3) The expected project completion time T is assumed to be the sum of the expected durations of activities on critical path. 4) The variance of project completion time ߪ ଶ ் is assumed to be the sum of the variances of activities on the critical path. 5) Probabilities regarding project completion time can be determined from standard normal tables. 5 Incorporating uncertainty in activity times

Incorporating uncertainty in activity times able 10.6 Variances and expe cted activity duratio Ex.10.5 Tennis tournament project Time estimates Variance Expected Activity completion time distribution M B duration A 2 3 0.11 2 The expected project completion time T is B 8 11 1.00 8 determined by the critical path A-C-E-I-J,i.e.,20 C 2 3 4 0.11 3 D 1 2 3 0.11 2 days. E 6 9 18 4.00 10 The variance of the project completion time is F 2 4 6 0.44 4 o2x=5.2 G 1 3 11 2.78 4 H 1 1 0.00 1 The Z value for the standard normal deviate is I 2 2 8 1.00 3 calculated 2 2 2 0.00 2 X-u_24-20 2= =1.75 G V5.2 From the standard normal table with Z=1.75,we =5.2 find the probability of completing the project within 24 days to be approximately 0.96. .04 Days T=20 24 FIGURE 16.15 Project completion time dis- tribution. 6 上游充鱼大学

Ex. 10.5 Tennis tournament project completion time distribution • The expected project completion time T is determined by the critical path A-C-E-I-J, i.e., 20 days. • The variance of the project completion time is ߪ ଶ ் ൌ 5.2 • The Z value for the standard normal deviate is calculated ܼ ൌ ܺെߤ ߪ ൌ 24 െ 20 5.2 ൌ 1.75 • From the standard normal table with Z=1.75, we find the probability of completing the project within 24 days to be approximately 0.96. 6 Incorporating uncertainty in activity times AMB A 1 2 3 0.11 2 B 5 8 11 1.00 8 C 2 3 4 0.11 3 D 1 2 3 0.11 2 E 6 9 18 4.00 10 F 2 4 6 0.44 4 G 1 3 11 2.78 4 H 1 1 1 0.00 1 I 2 2 8 1.00 3 J 2 2 2 0.00 2 Activity Variance Expected duration Time estimates able 10.6 Variances and expected activit y duratio

Incorporating uncertainty in activity times A critique of the project completion time analysis The key assumption underlying our analysis leading to a project completion time distribution is that the critical path as calculated from expected activity durations will actually be the true critical path. In reality,the critical path itself is a random variable that is not known for certain until the project is completed. If the expected duration of the critical path is much longer than that of any other path,then the estimates likely will be good. If the project network contains noncritical paths with very little total slack time,these paths may well affect project completion time.This situation is called merge node bias.That is,the project completion node has several paths coming into it,any one of which could be the critical path that determines the project completion time. 7 上浒充通大粤

A critique of the project completion time analysis • The key assumption underlying our analysis leading to a project completion time distribution is that the critical path as calculated from expected activity durations will actually be the true critical path. • In reality, the critical path itself is a random variable that is not known for certain until the project is completed. • If the expected duration of the critical path is much longer than that of any other path, then the estimates likely will be good. • If the project network contains noncritical paths with very little total slack time, these paths may well affect project completion time. This situation is called merge node bias. That is, the project completion node has several paths coming into it, any one of which could be the critical path that determines the project completion time. 7 Incorporating uncertainty in activity times

Incorporating uncertainty in activity times able 10.6 Variances and expected activity duratio Ex.10.6 Tennis tournament project Time estimates Variance Expected Activity completion time distribution M B duration A 2 3 0.11 2 A few days into the project,we discover that activity B 8 11 1.00 8 G may take longer than expected,with revised C 2 3 4 0.11 3 estimates of A =2,M=3,B=28.What effect does D 1 2 3 0.11 2 E 6 9 18 4.00 10 this have on the probability of completing the F 2 4 6 0.44 4 project in 24 days? G 3 11 2.78 4 H 1 1 0.00 1 =7,2=576=18.78 for activity G I 2 2 8 1.00 3 36 2 2 2 0.00 2 Initially,activity G had a 4-day expected duration and total slack of 4 days. 2) C3) D2) 4 With a revised duration of 7 days,activity G is still noncritical,with a TS=1. 日10 However,the large variance of activity G,will have 40 an impact on the likelihood of this path A-C-D-G-I-J becoming critical. Fig.10.4 PERT chart (AON network)for tennis tournament 8 上浒充通大粤

Ex. 10.6 Tennis tournament project completion time distribution • A few days into the project, we discover that activity G may take longer than expected, with revised estimates of A =2, M=3, B=28. What effect does this have on the probability of completing the project in 24 days? • t=7, ߪ ଶ ൌ ଺଻଺ ଷ଺ ൌ 18.78 for activity G • Initially, activity G had a 4-day expected duration and total slack of 4 days. • With a revised duration of 7 days, activity G is still noncritical, with a TS=1. • However, the large variance of activity G, will have an impact on the likelihood of this path A-C-D-G-I-J becoming critical. 8 Incorporating uncertainty in activity times AMB A 1 2 3 0.11 2 B 5 8 11 1.00 8 C 2 3 4 0.11 3 D 1 2 3 0.11 2 E 6 9 18 4.00 10 F 2 4 6 0.44 4 G 1 3 11 2.78 4 H 1 1 1 0.00 1 I 2 2 8 1.00 3 J 2 2 2 0.00 2 Activity Variance Expected duration Time estimates able 10.6 Variances and expected activit y duratio

Incorporating uncertainty in activity times able 10.6 Variances and expected activity duratio Ex.10.6 Tennis tournament project Time estimates Activity Variance Expected completion time distribution M B duration A 2 3 0.11 2 A few days into the project,we discover that activity B 8 11 1.00 8 G may take longer than expected,with revised C 2 3 4 0.11 3 estimates of A=2,M=3,B=28.What effect does D 1 2 3 0.11 2 E 6 9 18 4.00 10 this have on the probability of completing the F 2 4 6 0.44 4 project in 24 days? G 3 11 2.78 4 H 1 1 1 0.00 1 Duration distribution of the path A-C-D-G-I-J: I 2 2 8 1.00 3 T=19,σ2x=20 2 2 2 0.00 2 。2=X-=24-19=1.12 V20 V2) C3) D2) G4) With z=1.12,we found that the probability of completing the project within 24 days is Start I3) 2) approximately 0.87. 38 40 Fig.10.4 PERT chart(AON network)for tennis tournament 9 上浒充通大粤

Ex. 10.6 Tennis tournament project completion time distribution • A few days into the project, we discover that activity G may take longer than expected, with revised estimates of A =2, M=3, B=28. What effect does this have on the probability of completing the project in 24 days? • Duration distribution of the path A-C-D-G-I-J: • T=19, ߪ ଶ ் ൌ 20 • ݖ ൌ ௑ିఓ ఙ ൌ ଶସିଵଽ ଶ଴ ൌ 1.12 • With z=1.12, we found that the probability of completing the project within 24 days is approximately 0.87. 9 Incorporating uncertainty in activity times AMB A 1 2 3 0.11 2 B 5 8 11 1.00 8 C 2 3 4 0.11 3 D 1 2 3 0.11 2 E 6 9 18 4.00 10 F 2 4 6 0.44 4 G 1 3 11 2.78 4 H 1 1 1 0.00 1 I 2 2 8 1.00 3 J 2 2 2 0.00 2 Activity Variance Expected duration Time estimates able 10.6 Variances and expected activit y duratio

Outline The nature of project management Techniques for project management Resource constraints Activity crashing Incorporating uncertainty in activity times © Problems with implementing critical path analysis Monitoring projects 10 上浒充通大粤

Outline The nature of project management Techniques for project management Resource constraints Activity crashing Incorporating uncertainty in activity times Problems with implementing critical path analysis Monitoring projects 10