Question

Use a PERT network to establish the expected duration for the project described in the table below:    (SHOW the PERT NETWORK)

Activity	Predecessors	Duration a        m       b			Exp. Duration	Var.	Crash Week 1	Crash Week 2
A	-	3	6	9			700	100
B	A	6	7	14			600	200
C	A	4	6	14			400		600
D	B, C	4	6	8			200		600
E	B, C	6	9	12			400	300
F	C	4	6	14			200	900
G	D, E	3	5	7			400	400
H	E	3	6	9			600	100
I	G	8	8	8			600
J	H	4	10	16			700	300

a.   List the activities on the CRITICAL PATH:………………………  

b. What is the project expected duration?...........

c.   In your table list ALL expected durations and ALL variances.

d.   List is the slack for:   J ……..            I ………            F……….           C ………

e.   List the latest starting time for:    G ……..               F ………                E ……….

f.   The company indicated that they need to have a 90% probability of completing in 39 weeks.      

    What will become of the expected duration to make sure that we meet the 90% probability to

     complete within 39 weeks?......................

   

g.   Because of the new expected duration, (see f), how many weeks are to be crashed?.............

h.   List specifically which activities you would crash, and for how many weeks:           

Activity	# Weeks	Crashing Cost

         

Answer 1

8 6 B 5 G - 8 10 Start А. С E Н. J End 6 7 9 6 F 7 a) Critical path is the path with longest completion time. We will assess all paths and identify critical path with longest duration. Possible paths- A-B-D-G-I: 33 A-B-E-G-I: 36 A-B-E-H-J: 39 A-C-D-G-I: 32 A-C-E-G-I: 35 A-C-E-H-J: 38 A-C-F: 20 Path A-B-E-H-J has longest duration of 39 week Hence, critical path is A-B-E-H-J Project's expected completion time is 39 weeks.

We will use below approach for Slack and Late finish as well as validating critical path and project duration.

Earliest Start Time (ES) = the earliest time that an activity can begin

Earliest Finish Time (EF) = the earliest time that an activity can be completed

EF = ES + duration of activity        

Latest Start Time (LS) = the latest time that an activity can begin without lengthening the minimum project duration

Latest Finish Time (LF) = the earliest time than an activity can be completed without lengthening the minimum project duration

LF = LS + duration of activity   

Duration of the project = the difference between the maximum value of the "latest finish time" of projects and the minimum value of the "earliest start time" of projects

Project duration = max(LF) - min(ES)     

Slack = the amount of time that a project can be delayed without increasing the duration of the project

Slack = LF- LS or EF - ES

Critical path is the path with Total Slack = 0

Calculations are below:

			Expected time = (Optimistic + 4*Most Likely + Pessimistic)/ 6
			Variance = ((Pessimistic - Optimistic)/ 6)2
Activity	Predecessors	Duration	Exp. Duration	Variance	Early Start (ES)	Early Finish (EF)	Late Start (LS)	Late Finish (LF)	Slack	Critical Path
		a	m	b
A	-	3	6	9	6.0	1.00	0	6.0	0.0	6.0	0.0	Y
B	A	6	7	14	8.0	1.78	6.0	14.0	6.0	14.0	0.0	Y
C	A	4	6	14	7.0	2.78	6.0	13.0	7.0	14.0	1.0
D	B,C	4	6	8	6.0	0.44	14.0	20.0	18.0	24.0	4.0
E	B,C	6	9	12	9.0	1.00	14.0	23.0	14.0	23.0	0.0	Y
F	C	4	6	14	7.0	2.78	13.0	20.0	29.0	39	19.0
G	D,E	3	5	7	5.0	0.44	23.0	28.0	24.0	29.0	1.0
H	E	3	6	9	6.0	1.00	23.0	29.0	23.0	29.0	0.0	Y
I	G	8	8	8	8.0	0.00	28.0	36.0	29.0	39.0	3.0
J	H	4	10	16	10.0	4.00	29.0	39.0	29.0	39.0	0.0	Y
		a. Activities on Critical path is A-B-E-H-J
		b. Project expected duration is 39 weeks
		c. Table with expected duration and all variances ->> Updated above
		d. Slack: J=0, I=3, F=19, C=1
		e. Latest starting time for G=24, F=29, E=14

Variance of critical path = 1 + 1.78 +1 + 1+ 4 = 8.78

Standard deviation of critical path = √8.78 = 2.96

f)

For probability we will use z statistics,

Z= (X- Expected time)/ Standard deviation

For 90% probability, looking at z table as given below we will get z value of 1.29

Z= (X- Expected time)/ Standard deviation

1.29 = (39-Expected time)/ 2.96

Expected time = 35.18

Target expected duration to male sure that we meet 90% probability to complete within 39 weeks is 35.18 weeks. (~ 35 week rounded)

g.

To meet new expected duration, 4 weeks duration (approaximately) need to be crashed

(Earlier expected duration was 39 and now required is ~35 weeks)

h.

Now to crash project duration by 4 weeks, we will crash activities on critical path A-B-E-H-J starting with activities with lowest crashing cost.

- Activity E has lowest crashing cost for Week 1 (400), so we crash E by 1 week. New project duration is 39-1=38 weeks. Crashing cost is $400. Critical path is same A-B-E-H-J.

- Next we will crash activity E by second week (300).New project duration is 38-1=37 weeks. Crashing cost is $400+300=700. Critical path is same A-B-E-H-J.

- Next we will crash activity H by 1 Week (600). New project duration is 37-1=36 weeks. Crashing cost is $700+600=$1300. Critical path is same A-B-E-H-J.

- Next we will crash activity H by second Week (100). New project duration is 36-1=35 weeks. Crashing cost is $1300+100=$1400. Critical path is same A-B-E-H-J.

Activity	# Weeks	Crashing Cost
E	1	400
E	1	300
H	1	600
H	1	100
		1400

So total $1400 of crashing cost is needed to reduce project duration to 35 weeks.

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Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive Z-scores v -3.4 -3.3 -3.2 -3.1 -3.0 -29 2.8 -2.7 -2.6 25 -24 0,00 0.0003 0.0005 0.0007 0.0010 0.0013 0.0019 0.0026 0.0035 0.0047 0.0062 0.0082 0.0107 0.0139 0.0179 0.0228 0.0287 0.0359 0.0446 0.0548 0.0668 0.0008 0.0968 0.1151 -2.1 -20 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -10 -0.9 -0.8 07 06 05 04 03 0.01 0.0003 D.DOOS 0.0007 D.0009 0.0013 0.0018 0.0025 0.0034 0.0015 0.0000 0.0080 D.D104 0.0136 0.0174 0.0222 0.0281 0,0351 0,0436 0.0537 0,17655 0.0793 0.0951 0.1131 0.1335 0.1562 D.1814 0.2090 0.2389 0.2709 0.3050 0.3.000 0.3783 0.4168 0.4562 0.4960 0.02 0.0003 0.0005 0.0006 0.0009 0.0013 0.0018 0.0024 0.0033 0.0014 0.0059 0.0078 0.0102 0.0132 0.0170 0.0217 0.0274 0.0344 0.0427 0.0526 0,0643 0.0778 0.0934 0.1112 0.1314 0.1539 0.17 0.2061 0.2358 0.2676 0.3015 0.3372 0.3745 0.4129 0.4522 0.4920 0.03 0.0003 0.0004 0.0006 0.0009 0.0012 0,0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0000 0.0129 0.0166 0.0212 0.0268 0.0336 0,0418 0.0S16 0,0630 0.0764 0.0918 0.1093 0.1292 0.1515 01.1762 0.2033 0.2227 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880 0.04 0.0003 0.004 0.0006 0.0008 0.0012 0.0016 0.0023 0.0031 0.004 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207 0.0262 0,0320 0,0400 0.OSOS 0,0618 0.0749 0.0901 0.1075 0.1271 0.1492 0.1736 0.2005 0.2206 0.2611 0.2916 0.3200 0.3669 0.4052 0.4441 0.4840 0.05 0.0003 0.0014 0.0006 0.00CE 0.0011 0.0016 0.0022 0.0030 0.0010 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202 0.0256 0.0322 0,0401 0.0495 0.0606 0.0738 0.0883 0.1056 0.1251 0.1469 0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801 0.06 0.0003 OC004 0.0006 OLCOCIS OV0011 0.0015 0.0021 000029 0.0009 0.0052 0.0069 0.0091 00119 0.0154 0.0197 0.0250 0.0314 0.0192 0.0485 0.0594 0,0721 000669 0.1038 0.1230 0.1446 0.1695 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761 0,07 0.0003 0.004 0.0005 0.0008 0.0011 0.0015 0.0021 0.0028 0.0038 0.005 0.0068 0.0089 0.0116 0.0LSO 0.0192 0.0244 0.0307 0.0394 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423 0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721 0,08 0.0003 0.004 0.0005 0.0007 0.0010 0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188 0.0239 0.01 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401 0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681 0.00 0.0002 0.0003 0.0005 D.CCO7 D.0010 0.0014 D.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 D.DIR3 0.0233 0,0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0085 0.1170 0.1379 0.1611 D.1 867 0.2148 D.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641 7 0,0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.0 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 0.00 0.5000 0.539 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.9990 0.9993 0.9995 0.9997 0.01 05040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.98% 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.9991 0.9993 0.9995 0.9997 0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0,6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.986R 0.9898 0.9922 0.9941 0.9936 0.9967 0.9976 0.9982 0.9987 0.9991 0.9994 0.9995 0.9997 0.03 0,5120 0.5517 0,5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.823 0.8485 0.870R 0.8907 0.9062 0.9236 0.9370 0.9484 0.9582 0.9664 0,0732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.0006 0.9977 0.04 0,05 05160 05199 O.SSST 0.5596 0.59:18 0.5987 0.6331 0.6.168 0.6700 0.6736 0.705 0.7088 0.7380 0.7422 0.7704 0.7734 0.7995 0.8023 0.8264 0.8299 0.8508 0.8531 0.8720 0.8749 0.8925 0.8944 0.9099 0.9115 0.9251 0.9265 0.9382 0.9394 0.9495 0.9505 0.9591 0.9599 0.9671 0.9678 0.9738 0.9744 0.9793 0.9798 0.9838 0.9842 0.987 0.9878 0.9904 0.9636 09927 0.9929 0.9945 0.9946 0.9959 0.9960 0.999 0.9970 0.9977 0.9978 0.9984 0.9984 0.9988 0.9999 09992 0.9992 0.9994 0.9901 0.9996 0.9996 0.9997 0.9997 0,06 05239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 D.8770 0.8962 0.9131 0.9279 0.9406 0.9515 D.9608 0.9686 1.9750 0.990 0.9846 0.9881 0.9909 1993 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.9992 0.9991 0.9996 0.9997 0,07 OOR 0.5279 0.5319 0.5675 0.5714 0.6064 0,6103 D.6443 0.6430 0.6806 0.6844 0.7157 0.7190 D.7486 0.7517 0.7794 0.7823 0.8078 0.8106 0.8340 0.8365 0.8577 0.8599 D.8700 0.8810 0.8980 0.8997 0.9147 0.9162 1.9202 0.9306 0.9418 0.9429 0.9525 0.9535 0.9616 0.9625 0.9693 0.9699 0.9756 0.9761 1.9806 0.9812 0.9850 0.9854 DORR4 0.9887 0.9911 0.9913 0.9932 0.9934 0.9949 0.9951 0.9962 0.9963 0.9972 0.9973 0.9979 0.9980 0.9985 0.9986 0.9989 09990 0.9992 0.9993 0.999 0.9995 0.0006 0.9976 0.9997 0.9997 0.00 0.5250 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.839 0.8621 0.88 0.915 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.998 0.9986 0.9990 0.9993 0.9998 0.9997 0.9998 0.1357 0.1587 0.1841 0.2119 0.2420 0.2743 0.3085 0.446 0.3821 04207 0.4602 0.5000 0.9983 CU 10 00 0.9988 0.9291 0.9994 0.9996 0.9997 Note that the probabilities given in this table represent the area to the LEFT of the 7-score. The area to the RIGHT of a z-score = 1- the area to the LEFT of the z-score