Chapter 10 Project Scheduling: PERT/CPM

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© 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

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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.

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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

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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.

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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?

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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.

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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.

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Immediate CompletionImmediate Completion

ActivityActivity DescriptionDescription PredecessorsPredecessors Time (days)Time (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 3 3

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

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StartStart FinishFinish

BB33

DD33

AA33

CC22

GG66

FF33

HH22

EE77

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Earliest Start and Finish TimesEarliest Start and Finish Times


BB33

DD33

AA33

CC22

GG66

FF33

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

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Latest Start and Finish TimesLatest Start and Finish Times


BB33

DD33

AA33

CC22

GG66

FF33

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

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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).

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Activity Slack TimeActivity Slack Time

ActivityActivity ESES EFEF LSLS LFLF SlackSlack A 0 3 0 3 0 A 0 3 0 3 0

(critical)(critical) B 3 6 6 9 3B 3 6 6 9 3 C 3 5 3 5 0 C 3 5 3 5 0

(critical)(critical) D 6 9 9 12 3D 6 9 9 12 3 E 5 12 5 12 0 E 5 12 5 12 0

(critical)(critical) F 6 9 15 18 9F 6 9 15 18 9 G 12 18 12 18 0 G 12 18 12 18 0

(critical)(critical) H 5 7 16 18 11H 5 7 16 18 11

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• 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


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Critical PathCritical Path


BB33

DD33

AA33

CC22

GG66

FF33

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

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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

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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


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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.

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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

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Example: ABC AssociatesExample: ABC Associates


E

Start

A

H

D

F

J

I

K

Finish

B

C

G

E

Start

A

H

D

F

J

I

K

Finish

B

C

G

6666

4444

3333

5555

5555

2222

4444

11116666

3333

5555

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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

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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 *

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• 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


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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

E

Start

A

H

D

F

J

I

K

Finish

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C

G

6666

4444

3333

5555

5555

2222

4444

11116666

3333

5555

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

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Probability the project will be completed within 24 Probability the project will be completed within 24 hrshrs

22 = = 22AA + + 22

CC + + 22FF + + 22

HH + + 22KK

= 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 (24-23)/1.414 = .71= .71

From the Standard Normal Distribution From the Standard Normal Distribution table: table:

P(z P(z << .71) = .5 + .2612 = .7612 .71) = .5 + .2612 = .7612


Probability completed within 24 hrs = 76.12%Probability completed within 24 hrs = 76.12%

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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

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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

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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 new

project, introducing a newproject, introducing a new

line of loaders. Managementline of loaders. Management

is concerned that the project mightis concerned that the project might

take longer than 26 weeks totake longer than 26 weeks to

complete without crashing somecomplete without crashing some

activities.activities.

Example: EarthMover, Inc.Example: EarthMover, Inc.

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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


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PERT NetworkPERT Network


C

Start

D

E

I

A

Finish

H

G

B

F

C

Start

D

E

I

A

Finish

H

G

B

F

66664444

333310101010

3333

6666

2222 6666

8888

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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 *


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Critical ActivitiesCritical Activities

C

Start

D

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B

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C

Start

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66664444

333310101010

3333

6666

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88880 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

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CrashingCrashing

The completion time for this project using The completion time for this project using normalnormal

times 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 be

completed in 26 weeks?completed in 26 weeks?

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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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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


Linear Program for Minimum-Cost CrashingLinear Program for Minimum-Cost Crashing

Let: 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

37 37 Slide

Slide


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Chapter 10 Project Scheduling: PERT/CPM

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