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Chapter 10 Project Scheduling: PERT/CPM. Project Scheduling with Known Activity Times Project Scheduling with Uncertain Activity Times Considering Time-Cost Trade-Offs. PERT/CPM. PERT Program Evaluation and Review Technique Developed by U.S. Navy for Polaris missile project - PowerPoint PPT Presentation
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1 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Chapter 10Chapter 10Project Scheduling: PERT/CPMProject Scheduling: PERT/CPM
Project Scheduling with Known Activity TimesProject Scheduling with Known Activity Times Project Scheduling with Uncertain Activity Project Scheduling with Uncertain Activity
TimesTimes Considering Time-Cost Trade-OffsConsidering Time-Cost Trade-Offs
2 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM
PERTPERT• Program Evaluation and Review TechniqueProgram Evaluation and Review Technique• Developed by U.S. Navy for Polaris missile Developed by U.S. Navy for Polaris missile
projectproject• Developed to handle uncertain activity timesDeveloped to handle uncertain activity times
CPMCPM• Critical Path MethodCritical Path Method• Developed by Du Pont & Remington RandDeveloped by Du Pont & Remington Rand• Developed for industrial projects for which Developed for industrial projects for which
activity times generally were knownactivity times generally were known Today’s project management software packages Today’s project management software packages
have combined the best features of both have combined the best features of both approaches.approaches.
3 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM
PERT and CPM have been used to plan, PERT and CPM have been used to plan, schedule, and control a wide variety of projects:schedule, and control a wide variety of projects:• R&D of new products and processesR&D of new products and processes• Construction of buildings and highwaysConstruction of buildings and highways• Maintenance of large and complex Maintenance of large and complex
equipmentequipment• Design and installation of new systemsDesign and installation of new systems
4 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM
PERT/CPM is used to plan the scheduling of PERT/CPM is used to plan the scheduling of individual individual activitiesactivities that make up a project. that make up a project.
Projects may have as many as several Projects may have as many as several thousand activities.thousand activities.
A complicating factor in carrying out the A complicating factor in carrying out the activities is that some activities depend on the activities is that some activities depend on the completion of other activities before they can completion of other activities before they can be started.be started.
5 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT/CPMPERT/CPM Project managers rely on PERT/CPM to help them Project managers rely on PERT/CPM to help them
answer questions such as:answer questions such as:• What is the What is the total timetotal time to complete the project? to complete the project?• What are the What are the scheduled start and finish datesscheduled start and finish dates
for each specific activity?for each specific activity?• Which activities are Which activities are criticalcritical and must be and must be
completed exactly as scheduled to keep the completed exactly as scheduled to keep the project on schedule?project on schedule?
• How long can How long can noncritical activitiesnoncritical activities be delayed be delayed before they cause an increase in the project before they cause an increase in the project completion time?completion time?
6 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Project NetworkProject Network
A A project networkproject network can be constructed to model can be constructed to model the precedence of the activities. the precedence of the activities.
The The nodesnodes of the network represent the of the network represent the activities. activities.
The The arcsarcs of the network reflect the precedence of the network reflect the precedence relationships of the activities. relationships of the activities.
A A critical pathcritical path for the network is a path for the network is a path consisting of activities with zero slack.consisting of activities with zero slack.
7 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Frank’s Fine Floats is in the business of Frank’s Fine Floats is in the business of building elaborate parade floats. Frank and his building elaborate parade floats. Frank and his crew have a new float to build and want to use crew have a new float to build and want to use PERT/CPM to help them manage the projectPERT/CPM to help them manage the project . .
The table on the next slide shows the The table on the next slide shows the activities that comprise the project. Each activities that comprise the project. Each activity’s estimated completion time (in days) activity’s estimated completion time (in days) and immediate predecessors are listed as well.and immediate predecessors are listed as well.
Frank wants to know the total time to Frank wants to know the total time to complete the project, which activities are complete the project, which activities are critical, and the earliest and latest start and critical, and the earliest and latest start and finish dates for each activity.finish dates for each activity.
8 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Immediate CompletionImmediate Completion ActivityActivity DescriptionDescription PredecessorsPredecessors Time Time
(days)(days) A Initial Paperwork A Initial Paperwork --- --- 3 3 B Build Body B Build Body A A 3 3 C Build Frame C Build Frame A A 2 2 D Finish Body D Finish Body B B 33 E Finish Frame E Finish Frame C C 7 7 F Final Paperwork F Final Paperwork B,C B,C 3 3 G Mount Body to Frame D,EG Mount Body to Frame D,E 6 6 H Install Skirt on Frame CH Install Skirt on Frame C 2 2
9 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Project NetworkProject Network
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66 FF
33
HH22
EE77
10 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Earliest Start and Finish TimesEarliest Start and Finish Times
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66 FF
33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
11 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Latest Start and Finish TimesLatest Start and Finish Times
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66 FF
33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
6 96 9 9 9 1212
0 30 3
3 53 5
12 12 1818
15 15 1818
16 16 1818
5 125 12
12 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path
Step 3:Step 3: Calculate the slack time for each Calculate the slack time for each activity by: activity by:
SlackSlack = (Latest Start) - (Earliest = (Latest Start) - (Earliest Start), or Start), or
= (Latest Finish) - (Earliest = (Latest Finish) - (Earliest Finish).Finish).
13 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats Activity Slack TimeActivity Slack Time
ActivityActivity ESES EFEF LSLS LFLF SlackSlack A 0 3 0 3 0 (critical)A 0 3 0 3 0 (critical) B 3 6 6 9 3B 3 6 6 9 3 C 3 5 3 5 0 (critical)C 3 5 3 5 0 (critical) D 6 9 9 12 3D 6 9 9 12 3 E 5 12 5 12 0 (critical)E 5 12 5 12 0 (critical) F 6 9 15 18 9F 6 9 15 18 9 G 12 18 12 18 0 (critical)G 12 18 12 18 0 (critical) H 5 7 16 18 11H 5 7 16 18 11
14 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path• A A critical pathcritical path is a path of activities, from the is a path of activities, from the
Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.
• Critical Path: A – C – E – GCritical Path: A – C – E – G
• The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.
• Project Completion Time: 18 daysProject Completion Time: 18 days
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
15 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: Frank’s Fine FloatsExample: Frank’s Fine Floats
Critical PathCritical Path
StartStart FinishFinish
BB33
DD33
AA33
CC22
GG66 FF
33
HH22
EE77
0 30 3
3 63 6 6 96 9
3 53 5
12 12 1818
6 96 9
5 75 7
5 125 12
6 96 9 9 9 1212
0 30 3
3 53 5
12 12 1818
15 15 1818
16 16 1818
5 125 12
16 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
In the In the three-time estimate approachthree-time estimate approach, the time to , the time to complete an activity is assumed to follow a Beta complete an activity is assumed to follow a Beta distribution. distribution.
An activity’s An activity’s mean completion timemean completion time is: is:
tt = ( = (aa + 4 + 4mm + + bb)/6)/6
• aa = the = the optimisticoptimistic completion time estimate completion time estimate• bb = the = the pessimisticpessimistic completion time completion time
estimateestimate• mm = the = the most likelymost likely completion time completion time
estimateestimate
Uncertain Activity TimesUncertain Activity Times
Variance: The measure of uncertainty. Its value largely Variance: The measure of uncertainty. Its value largely affected by the difference between b and a (large affected by the difference between b and a (large differences reflect a high degree of uncertainty in differences reflect a high degree of uncertainty in
actv. timeactv. time
17 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
An activity’s An activity’s completion time variancecompletion time variance is: is: 22 = (( = ((bb--aa)/6))/6)22
• aa = the = the optimisticoptimistic completion time estimate completion time estimate• bb = the = the pessimisticpessimistic completion time completion time
estimateestimate• mm = the = the most likelymost likely completion time completion time
estimateestimate
Uncertain Activity TimesUncertain Activity Times
18 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Uncertain Activity TimesUncertain Activity Times
In the three-time estimate approach, the critical In the three-time estimate approach, the critical path is determined as if the mean times for the path is determined as if the mean times for the activities were fixed times. activities were fixed times.
The The overall project completion timeoverall project completion time is assumed is assumed to have a normal distribution with mean equal to have a normal distribution with mean equal to the sum of the means along the critical path to the sum of the means along the critical path and variance equal to the sum of the variances and variance equal to the sum of the variances along the critical path.along the critical path.
19 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC Associates Example: ABC Associates Consider the following project:Consider the following project: Immed. Optimistic Most Likely PessimisticImmed. Optimistic Most Likely Pessimistic ActivityActivity Predec.Predec. Time (Hr.Time (Hr.) ) Time (Hr.)Time (Hr.) Time (Hr.)Time (Hr.) A A -- 4 -- 4 6 6 8 8 B B -- 1 -- 1 4.5 4.5
5 5 C C A A 3 3 3 3
3 3 D D A 4 5 A 4 5 6 6 E E A 0.5 1 A 0.5 1
1.51.5 F F B,C 3 4 5 B,C 3 4 5 G G B,C B,C 1 1.5 5 1 1.5 5 H H E,F E,F 5 6 7 5 6 7 I I E,F 2 5 8 E,F 2 5 8 J J D,H D,H 2.5 2.75 2.5 2.75
4.5 4.5 K K G,I 3 5 G,I 3 5
77
20 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Project NetworkProject Network
E
Start
A
H
D
F
J
I
K
Finish
B
C
G
66
44
33
55
55
22
44
1166
33
55
21 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates Activity Expected Times and VariancesActivity Expected Times and Variances tt = ( = (aa + 4 + 4mm + + bb)/6 )/6 22 = (( = ((bb--
aa)/6))/6)22
ActivityActivity Expected TimeExpected Time VarianceVariance A A 6 6 4/9 4/9
B B 4 4 4/9 4/9 C C 3 3 0 0 D D 5 5 1/9 1/9 E E 1 1 1/36 1/36 F F 4 4 1/9 1/9 G G 2 2 4/9 4/9 H H 6 6 1/9 1/9 I I 5 5 1 1 J J 3 3 1/9 1/9 K K 5 5 4/9 4/9
22 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates Earliest/Latest Times and SlackEarliest/Latest Times and Slack
ActivityActivity ESES EF EF LSLS LFLF SlackSlack A A 0 6 0 6 0 * 0 6 0 6 0 *
B B 0 4 5 9 5 0 4 5 9 5 C 6 9 6 9 0 *C 6 9 6 9 0 * D D 6 11 15 20 9 6 11 15 20 9 E E 6 7 12 13 6 6 7 12 13 6 F F 9 13 9 13 0 * 9 13 9 13 0 * G 9 11 16 18 7G 9 11 16 18 7 H H 13 19 14 20 1 13 19 14 20 1 I I 13 18 13 18 0 * 13 18 13 18 0 * J J 19 22 20 23 1 19 22 20 23 1 K K 18 23 18 23 0 * 18 23 18 23 0 *
23 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Determining the Critical PathDetermining the Critical Path• A A critical pathcritical path is a path of activities, from the is a path of activities, from the
Start node to the Finish node, with 0 slack Start node to the Finish node, with 0 slack times.times.
• Critical Path: A – C – F – I – KCritical Path: A – C – F – I – K
• The The project completion timeproject completion time equals the equals the maximum of the activities’ earliest finish maximum of the activities’ earliest finish times.times.
• Project Completion Time: 23 hoursProject Completion Time: 23 hours
Example: ABC AssociatesExample: ABC Associates
24 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: ABC AssociatesExample: ABC Associates
Critical Path (A-C-F-I-K)Critical Path (A-C-F-I-K)
E
Start
A
H
D
F
J
I
K
Finish
B
C
G
66
44
33
55
55
22
44
1166
33
55
0 60 60 60 6
9 139 139 139 13
13 1813 1813 1813 18
9 119 1116 1816 18
13 1913 1914 2014 20
19 2219 2220 2320 23
18 2318 2318 2318 23
6 76 712 1312 13
6 96 96 96 9
0 40 45 95 9
6 116 1115 2015 20
25 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Probability the project will be completed within 24 Probability the project will be completed within 24 hrshrs22 = = 22
AA + + 22CC + + 22
FF + + 22HH + + 22
KK
= 4/9 + 0 + 1/9 + 1 + 4/9 = 4/9 + 0 + 1/9 + 1 + 4/9 = 2= 2 = 1.414= 1.414
zz = (24 - 23)/ = (24 - 23)/(24-23)/1.414 = .71(24-23)/1.414 = .71 From the Standard Normal Distribution table: From the Standard Normal Distribution table:
P(z P(z << .71) = .5 + .2612 = .7612 .71) = .5 + .2612 = .7612
Example: ABC AssociatesExample: ABC Associates
Probability completed within 24 hrs = 76.12%Probability completed within 24 hrs = 76.12%
26 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
0 Zzz .00.00 .01.01 .02.02 .03.03 .04.04 .05.05 .06.06 .07.07 .08.08 .09.09
.0.0 .5000.5000 .5040.5040 .5080.5080 .5120.5120 .5160.5160 .5199.5199 .5239.5239 .5279.5279 .5319.5319 .5359.5359
.1.1 .5398.5398 .5438.5438 .5478.5478 .5517.5517 .5557.5557 .5596.5596 .5636.5636 .5675.5675 .5714.5714 .5753.5753
.2.2 .5793.5793 .5832.5832 .5871.5871 .5910.5910 .5948.5948 .5987.5987 .6026.6026 .6064.6064 .6103.6103 .6141.6141
.3.3 .6179.6179 .6217.6217 .6255.6255 .6293.6293 .6331.6331 .6368.6368 .6406.6406 .6443.6443 .6480.6480 .6517.6517
.4.4 .6554.6554 .6591.6591 .6628.6628 .6664.6664 .6700.6700 .6736.6736 .6772.6772 .6808.6808 .6844.6844 .6879.6879
.5.5 .6915.6915 .6950.6950 .6985.6985 .7019.7019 .7054.7054 .7088.7088 .7123.7123 .7157.7157 .7190.7190 .7224.7224
.6.6 .7257.7257 .7291.7291 .7324.7324 .7357.7357 .7389.7389 .7422.7422 .7454.7454 .7486.7486 .7517.7517 .7549.7549
.7.7 .7580.7580 .7611.7611 .7642.7642 .7673.7673 .7703.7703 .7734.7734 .7764.7764 .7794.7794 .7823.7823 .7852.7852
.8.8 .7881.7881 .7910.7910 .7939.7939 .7967.7967 .7995.7995 .8023.8023 .8051.8051 .8078.8078 .8106.8106 .8133.8133
.9.9 .8159.8159 .8186.8186 .8212.8212 .8238.8238 .8264.8264 .8289.8289 .8315.8315 .8340.8340 .8365.8365 .8389.8389
1.01.0 .8413.8413 .8438.8438 .8461.8461 .8485.8485 .8508.8508 .8531.8531 .8554.8554 .8577.8577 .8599.8599 .8621.8621
1.11.1 .8643.8643 .8665.8665 .8686.8686 .8708.8708 .8729.8729 .8749.8749 .8770.8770 .8790.8790 .8810.8810 .8830.8830
1.21.2 .8849.8849 .8869.8869 .8888.8888 .8907.8907 .8925.8925 .8944.8944 .8962.8962 .8980.8980 .8997.8997 .9015.9015
1.31.3 .9032.9032 .9049.9049 .9066.9066 .9082.9082 .9099.9099 .9115.9115 .9131.9131 .9147.9147 .9162.9162 .9177.9177
1.41.4 .9192.9192 .9207.9207 .9222.9222 .9236.9236 .9251.9251 .9265.9265 .9279.9279 .9292.9292 .9306.9306 .9319.9319
1.51.5 .9332.9332 .9345.9345 .9357.9357 .9370.9370 .9382.9382 .9394.9394 .9406.9406 .9418.9418 .9429.9429 .9441.9441
Areas of the cumulative standard normal distribution
Areas of the cumulative standard normal distribution
27 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
0 Z
Areas of the cumulative standard normal distribution(Continue)zz .00.00 .01.01 .02.02 .03.03 .04.04 .05.05 .06.06 .07.07 .08.08 .09.09
1.61.6 .9452.9452 .9463.9463 .9474.9474 .9484.9484 .9495.9495 .950.95055
.9515.9515 .9525.9525 .9535.9535 .9545.9545
1.71.7 .9554.9554 .9564.9564 .9573.9573 .9582.9582 .9591.9591 .959.95999
.9608.9608 .9616.9616 .9625.9625 .9633.9633
1.81.8 .9641.9641 .9649.9649 .9656.9656 .9664.9664 .9671.9671 .967.96788
.9686.9686 .9693.9693 .9699.9699 .9706.9706
1.91.9 .9713.9713 .9719.9719 .9726.9726 .9732.9732 .9738.9738 .974.97444
.9750.9750 .9756.9756 .9761.9761 .9767.9767
2.02.0 .9772.9772 .9778.9778 .9783.9783 .9788.9788 .9793.9793 .979.97988
.9803.9803 .9808.9808 .9812.9812 .9817.9817
2.12.1 .9821.9821 .9826.9826 .9830.9830 .9834.9834 .9838.9838 .984.98422
.9846.9846 .9850.9850 .9854.9854 .9857.9857
2.22.2 .9861.9861 .9864.9864 .9868.9868 .9871.9871 .9875.9875 .987.98788
.9881.9881 .9884.9884 .9887.9887 .9890.9890
2.32.3 .9893.9893 .9896.9896 .9898.9898 .9901.9901 .9904.9904 .990.99066
.9909.9909 .9911.9911 .9913.9913 .9916.9916
2.42.4 .9918.9918 .9920.9920 .9922.9922 .9925.9925 .9927.9927 .992.99299
.9931.9931 .9932.9932 .9934.9934 .9936.9936
2.52.5 .9938.9938 .9940.9940 .9941.9941 .9943.9943 .9945.9945 .994.99466
.9948.9948 .9949.9949 .9951.9951 .9952.9952
2.62.6 .9953.9953 .9955.9955 .9956.9956 .9957.9957 .9959.9959 .996.99600
.9961.9961 .9962.9962 .9963.9963 .9964.9964
2.72.7 .9965.9965 .9966.9966 .9967.9967 .9968.9968 .9969.9969 .997.99700
.9971.9971 .9972.9972 .9973.9973 .9974.9974
2.82.8 .9974.9974 .9975.9975 .9976.9976 .9977.9977 .9977.9977 .997.99788
.9979.9979 .9979.9979 .9980.9980 .9981.9981
2.92.9 .9981.9981 .9982.9982 .9982.9982 .9983.9983 .9984.9984 .998.99844
.9985.9985 .9985.9985 .9986.9986 .9986.9986
3.03.0 .9987.9987 .9987.9987 .9987.9987 .9988.9988 .9988.9988 .998.99899
.9989.9989 .9989.9989 .9990.9990 .9990.9990
3.13.1 .9990.9990 .9991.9991 .9991.9991 .9991.9991 .9991.9991 .999.99922
.9992.9992 .9992.9992 .9993.9993 .9993.9993
3.23.2 .9993.9993 .9993.9993 .9994.9994 .9994.9994 .9994.9994 .999.99944
.9994.9994 .9995.9995 .9995.9995 .9995.9995
3.33.3 .9995.9995 .9995.9995 .9995.9995 .9996.9996 .9996.9996 .999.99966
.9996.9996 .9996.9996 .9996.9996 .9997.9997
3.43.4 .9997.9997 .9997.9997 .9997.9997 .9997.9997 .9997.9997 .999.99977
.9997.9997 .9997.9997 .9997.9997 .9998.9998
28 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
EarthMover is a manufacturer of road EarthMover is a manufacturer of road constructionconstruction
equipment including pavers, rollers, and graders. equipment including pavers, rollers, and graders. TheThe
company is faced with a newcompany is faced with a newproject, introducing a newproject, introducing a newline of loaders. Managementline of loaders. Managementis concerned that the project mightis concerned that the project mighttake longer than 26 weeks totake longer than 26 weeks tocomplete without crashing somecomplete without crashing someactivities.activities.
Example: EarthMover, Inc.Example: EarthMover, Inc.
29 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Immediate Immediate CompletionCompletion
ActivityActivity DescriptionDescription PredecessorsPredecessors Time (wks)Time (wks) A Study Feasibility A Study Feasibility --- ---
6 6 B Purchase Building B Purchase Building A A 4 4 C Hire Project Leader C Hire Project Leader A A 3 3 D Select Advertising StaffD Select Advertising Staff B B 6 6 E Purchase Materials E Purchase Materials B B 3 3 F Hire Manufacturing Staff B,CF Hire Manufacturing Staff B,C 10 10 G Manufacture Prototype E,FG Manufacture Prototype E,F 2 2 H Produce First 50 Units GH Produce First 50 Units G 6 6 II Advertise Product D,G Advertise Product D,G 8 8
Example: EarthMover, Inc.Example: EarthMover, Inc.
30 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
PERT NetworkPERT Network
Example: EarthMover, Inc.Example: EarthMover, Inc.
C
Start
D
E
I
A
Finish
H
G
B
F
6644
331010
33
66
22 66
88
31 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Earliest/Latest TimesEarliest/Latest Times
ActivityActivity ESES EFEF LSLS LFLF SlackSlack A A 0 6 0 6 0 * 0 6 0 6 0 * B B 6 10 6 10 0 * 6 10 6 10 0 * C C 6 9 7 10 1 6 9 7 10 1 D 10 16 16 22 6D 10 16 16 22 6 E E 10 13 17 20 7 10 13 17 20 7 F F 10 20 10 20 0 * 10 20 10 20 0 * G G 20 22 20 22 0 * 20 22 20 22 0 * H H 22 28 24 30 2 22 28 24 30 2 I I 22 30 22 30 0 * 22 30 22 30 0 *
Example: EarthMover, Inc.Example: EarthMover, Inc.
32 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: EarthMover, Inc.Example: EarthMover, Inc.
Critical ActivitiesCritical Activities
C
Start
D
E
I
A
Finish
H
G
B
F
6644
331010
33
66
22 66
880 60 60 60 6
10 2010 20 10 2010 20
20 2220 2220 2220 22
10 1610 1616 2216 22 22 3022 30
22 3022 30
22 2822 2824 3024 30
6 96 9 7 107 10
10 1310 1317 2017 20
6 106 10 6 106 10
33 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Example: EarthMover, Inc.Example: EarthMover, Inc.
CrashingCrashingThe completion time for this project using The completion time for this project using
normalnormaltimes is 30 weeks. Which activities should be times is 30 weeks. Which activities should be crashed,crashed,and by how many weeks, in order for the and by how many weeks, in order for the project to beproject to becompleted in 26 weeks?completed in 26 weeks?
34 Slide
© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Crashing Activity TimesCrashing Activity Times
In the In the Critical Path Method (CPM)Critical Path Method (CPM) approach to approach to project scheduling, it is assumed that the project scheduling, it is assumed that the normal time to complete an activity, normal time to complete an activity, ttj j , which , which can be met at a normal cost, can be met at a normal cost, ccj j , can be crashed , can be crashed to a reduced time, to a reduced time, ttjj’, under maximum crashing ’, under maximum crashing for an increased cost, for an increased cost, ccjj’.’.
Using CPM, activity Using CPM, activity jj's maximum time 's maximum time reduction, reduction, MMj j , may be calculated by: , may be calculated by: MMj j = = ttjj - - ttjj'. '. It is assumed that its cost per unit reduction, It is assumed that its cost per unit reduction, KKj j , is linear and can be calculated by: , is linear and can be calculated by: KKjj = ( = (ccjj' - ' - ccjj)/)/MMjj..
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Example: EarthMover, Inc.Example: EarthMover, Inc.
Normal CrashNormal Crash ActivityActivity TimeTime CostCost TimeTime CostCostA) Study Feasibility A) Study Feasibility 6 $ 80,000 5 6 $ 80,000 5 $100,000$100,000B) Purchase Building B) Purchase Building 4 100,000 4 4 100,000 4 100,000100,000C) Hire Project Leader C) Hire Project Leader 3 50,000 2 3 50,000 2 100,000100,000D) Select Advertising Staff D) Select Advertising Staff 6 150,000 3 6 150,000 3 300,000300,000E) Purchase Materials E) Purchase Materials 3 180,000 2 3 180,000 2 250,000250,000F) Hire Manufacturing Staff F) Hire Manufacturing Staff 10 300,000 7 10 300,000 7 480,000480,000G) Manufacture Prototype G) Manufacture Prototype 2 100,000 2 2 100,000 2 100,000 100,000H) Produce First 50 Units H) Produce First 50 Units 6 450,000 5 6 450,000 5 800,000800,000 I) Advertising Product 8 350,000 4 I) Advertising Product 8 350,000 4 650,000650,000
Normal Costs and Crash CostsNormal Costs and Crash Costs
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Min 20Min 20YYAA + 50 + 50YYCC + 50 + 50YYDD + 70 + 70YYEE + 60 + 60YYFF + 350 + 350YYHH + + 7575YYII s.t. s.t. YYAA << 1 1 XXAA >> 0 + (6 - 0 + (6 - YYII) ) XXGG >> XXFF + + (2 - (2 - YYGG) ) YYCC << 1 1 XXBB >> XXAA + (4 - + (4 - YYBB) ) XXHH >> XXGG + (6 - + (6 - YYHH) ) YYDD << 3 3 XXCC >> XXAA + (3 - + (3 - YYCC) ) XXII >> XXDD + (8 - + (8 - YYII) ) YYEE << 1 1 XXDD >> XXBB + (6 - + (6 - YYDD) ) XXII >> XXGG + (8 - + (8 - YYII)) YYFF << 3 3 XXEE >> XXBB + (3 - + (3 - YYEE)) XXHH << 26 26
YYHH << 1 1 XXFF >> XXBB + (10 - + (10 - YYFF) ) XXII << 26 26 YYII << 4 4 XXFF >> XXCC + (10 - + (10 - YYFF) ) XXGG >> XXEE + (2 - + (2 - YYGG) ) XXii, , YYjj >> 0 0 for all i for all i
Example: EarthMover, Inc.Example: EarthMover, Inc.
Linear Program for Minimum-Cost CrashingLinear Program for Minimum-Cost CrashingLet: Let: XXii = earliest finish time for activity = earliest finish time for activity ii
YYii = the amount of time activity = the amount of time activity ii is crashed is crashed
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End of Chapter 10End of Chapter 10