Management Science

MS

472.12

10

Fall 2011

Problems

ch 8

 Probabilistic Activity Times

• pr 27

 Network

 ES, EF, LS, LF table

 Probable time

 Standard Deviation

 bonus:

» likelihood of meeting deadline (June 1)

 Crashing

• pr 33

 either AON or AOA

» network

» Excel

» crash network to 26 weeks

» what is the shortest time it can be crashed to

» what would it cost at the shortest time?

Probabilistic Activity Times

estimates not exact

turning guesses into math

 most likely time m

 optimistic time a

 pessimistic time b





a  4m  b

mean (expected time) t 

6

ba

2



variance v   

 6 

Probabilistic Activity Times



Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1

2

3

4

5

6

7

8

9

10

11

Probabilistic Activity Times



Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 8.00

2 6.00

3 3.00

4 4.00

5 3.00

6 4.00

7 2.00

8 7.00

9 4.00

10 4.00

11 10.00

Probabilistic Activity Times



Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 6.00 8.00

2 3.00 6.00

3 1.00 3.00

4 2.00 4.00

5 2.00 3.00

6 3.00 4.00

7 2.00 2.00

8 3.00 7.00

9 2.00 4.00

10 1.00 4.00

11 1.00 10.00

Probabilistic Activity Times

a  4m  b

t

6

Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 6.00 8.00 10.00

2 3.00 6.00 9.00

3 1.00 3.00 5.00

4 2.00 4.00 12.00

5 2.00 3.00 4.00

6 3.00 4.00 5.00

7 2.00 2.00 2.00

8 3.00 7.00 11.00

9 2.00 4.00 6.00

10 1.00 4.00 7.00

11 1.00 10.00 13.00

Probabilistic Activity Times

a  4m  b

t

6

Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 6.00 8.00 10.00 8.00

2 3.00 6.00 9.00 6.00

3 1.00 3.00 5.00 3.00



ba

2

4 2.00 4.00 12.00 5.00

5 2.00 3.00 4.00 3.00 v 

6 3.00 4.00 5.00 4.00  6 

7 2.00 2.00 2.00 2.00

8 3.00 7.00 11.00 7.00

9 2.00 4.00 6.00 4.00

10 1.00 4.00 7.00 4.00

11 1.00 10.00 13.00 9.00

Probabilistic Activity Times



Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 6.00 8.00 10.00 8.00 0.44

2 3.00 6.00 9.00 6.00 1.00

3 1.00 3.00 5.00 3.00 0.44



ba

2

4 2.00 4.00 12.00 5.00 2.78

5 2.00 3.00 4.00 3.00 0.11 v 

6 3.00 4.00 5.00 4.00 0.11  6 

7 2.00 2.00 2.00 2.00 0.00

8 3.00 7.00 11.00 7.00 1.78

9 2.00 4.00 6.00 4.00 0.44

10 1.00 4.00 7.00 4.00 1.00

11 1.00 10.00 13.00 9.00 4.00

Probabilistic Activity Times

   project    activities on criticalpath



Mean

Time Estimates (weeks) Time Variance

Activity a m b t v

1 6.00 8.00 10.00 8.00 0.44

2 3.00 6.00 9.00 6.00 1.00

3 1.00 3.00 5.00 3.00 0.44

4 2.00 4.00 12.00 5.00 2.78

5 2.00 3.00 4.00 3.00 0.11

6 3.00 4.00 5.00 4.00 0.11

7 2.00 2.00 2.00 2.00 0.00

8 3.00 7.00 11.00 7.00 1.78

9 2.00 4.00 6.00 4.00 0.44

10 1.00 4.00 7.00 4.00 1.00

11 1.00 10.00 13.00 9.00 4.00

Project Crashing

turning money into time

 spend money on selected activities

• to reduce total project time



• which ones?

• how much?

 money to spend

 time saved

Project Crashing

all projects have both direct & indirect costs



Total costs



Indirect costs

$







crashing

Direct costs



optimal t

project time

2

8

4

Crashing 1

12

7

12 4

6

house 5 4

4

3

4









Normal

Activity time

1: Design house & obtain financing 12

2: Lay foundation 8

3: Order materials 4

4: Build house 12

5: Select paint 4

6: Select carpet 4

7: Finish work 4

2

8

4

Crashing 1

12

7

12 4

6

5 4

4

3

4









Normal

Activity time cost

1: Design house & obtain financing 12 $3,000

2: Lay foundation 8 $2,000

3: Order materials 4 $4,000

4: Build house 12 $50,000

5: Select paint 4 $500

6: Select carpet 4 $500

7: Finish work 4 $15,000

2

8

4

Crashing 1

12

7

12 4

6

5 4

4

3

4









Normal Crash

Activity time cost time cost

1: Design house & obtain financing 12 $3,000 7

2: Lay foundation 8 $2,000 5

3: Order materials 4 $4,000 3

4: Build house 12 $50,000 9

5: Select paint 4 $500 1

6: Select carpet 4 $500 1

7: Finish work 4 $15,000 3

2

8

12 4

Crashing 1 20 12

7

12 4

6

5 4

4

3

4

12

16





Normal Crash

Activity time cost time cost

1: Design house & obtain financing 12 $3,000 7 $5,000

2: Lay foundation 8 $2,000 5 $3,500

3: Order materials 4 $4,000 3 $7,000

4: Build house 12 $50,000 9 $71,000

5: Select paint 4 $500 1 $1,100

6: Select carpet 4 $500 1 $1,100

7: Finish work 4 $15,000 3 $22,000

2

8

4

Crashing 1

12

7

12 4

6

5 4

4

3

4









Resulting

Normal Crash time cost

Activity time cost time cost saved incurred

1: Design house & obtain financing 12 $3,000 7 $5,000

2: Lay foundation 8 $2,000 5 $3,500

3: Order materials 4 $4,000 3 $7,000

4: Build house 12 $50,000 9 $71,000

5: Select paint 4 $500 1 $1,100

6: Select carpet 4 $500 1 $1,100

7: Finish work 4 $15,000 3 $22,000

per week

2

8

4

Crashing 1

12

7

12 4

6

5 4

4

3

4







assume linear relationship

Resulting

Normal Crash time cost

Activity time cost time cost saved incurred

1: Design house & obtain financing 12 $3,000 7 $5,000 5

2: Lay foundation 8 $2,000 5 $3,500 3

3: Order materials 4 $4,000 3 $7,000 1

4: Build house 12 $50,000 9 $71,000 3

5: Select paint 4 $500 1 $1,100 3

6: Select carpet 4 $500 1 $1,100 3

7: Finish work 4 $15,000 3 $22,000 1

per week

2

8

4

Crashing 1

12

7

12 4

6

5 4

4

3

4

critical path and nodes may change





Resulting

Normal Crash time cost

Activity time cost time cost saved incurred

1: Design house & obtain financing 12 $3,000 7 $5,000 5 $400

2: Lay foundation 8 $2,000 5 $3,500 3 $500

3: Order materials 4 $4,000 3 $7,000 1 $3,000

4: Build house 12 $50,000 9 $71,000 3 $7,000

5: Select paint 4 $500 1 $1,100 3 $200

6: Select carpet 4 $500 1 $1,100 3 $200

7: Finish work 4 $15,000 3 $22,000 1 $7,000

per week

Problems

ch 8

 Probabilistic Activity Times

• pr 27

 Network

 ES, EF, LS, LF table

 Probable time

 Standard Deviation

 bonus:

» likelihood of meeting deadline (June 1)

 Crashing

• pr 33

 either AON or AOA

» network

» Excel

» crash network to 26 weeks

» what is the shortest time it can be crashed to

» what would it cost at the shortest time?