Question

Sager Products has been in the business of manufacturing and marketing toys for toddlers for the past two decades. Jim Sager, president of the firm, is considering the development of a new manufacturing line to allow it to produce high-quality plastic toys at reasonable prices. The development process is long and complex. Jim estimates that there are five phases involved and multiple activities for each phase.

Phase 1 of the development process involves the completion of four activities. These activities have no immediate predecessors. Activity A has an optimistic completion time of 2 weeks, a probable completion time of 3 weeks, and a pessimistic completion time of 4 weeks. Activity B has estimated completion times of 5, 6, and 8 weeks; these represent optimistic, probable, and pessimistic time estimates. Similarly, activity C has estimated completion times of 1 week, 1 week, and 2 weeks; and activity D has expected completion times of 8 weeks, 9 weeks, and 11 weeks.

Phase 2 involves six separate activities. Activity E has activity A as an immediate predecessor. Time estimates are 1 week, 1 week, and 4 weeks. Activity F and activity G both have activity B as their immediate predecessor. For activity F, the time estimates are 3 weeks, 3 weeks, and 4 weeks. For activity G, the time estimates are 1 week, 2 weeks, and 2 weeks. The only immediate predecessor for activity H is activity C. Time estimates for activity H are 5 weeks, 5 weeks, and 6 weeks. Activity D must be performed before activity I and activity J can be started. Activity I has estimated completion times of 9 weeks, 10 weeks, and 11 weeks. Activity J has estimated completion times of 1 week, 2 weeks, and 2 weeks.

Phase 3 is the most difficult and complex of the entire development project. It also consists of six separate activities. Activity K has three time estimates of 2 weeks, 2 weeks, and 3 weeks. The immediate predecessor for this activity is activity E. The immediate predecessor for activity L is activity F. The time estimates for activity L are 3 weeks, 4 weeks, and 6 weeks. Activity M has 2 weeks, 2 weeks, and 4 weeks for the estimates of the optimistic, probable and pessimistic time estimates. The immediate predecessor for activity M is activity G. Activities N and O both have activity I as their immediate predecessor. Activity N has 8 weeks, 9 weeks, and 11 weeks for its three time estimates. Activity O has 1 week, 1 week, and 3 weeks as its time estimates. Finally, activity P has time estimates of 4 weeks, 4 weeks, and 8 weeks. Activity J is its only immediate predecessor.

Phase 4 involves five activities. Activity Q requires activity K to be completed before it can be started. The three time estimates for activity Q are 6 weeks, 6 weeks, and 7 weeks. Activity R requires that both activity L and activity M be completed first. The three time estimates for activity R are 1, 2, and 4 weeks. Activity S requires activity N to be completed first. Its time estimates are 6 weeks, 6 weeks, and 7 weeks. Activity T requires that activity O be completed. The time estimates for activity T are 3 weeks, 3 weeks, and 4 weeks. The final activity for phase 4 is activity U. The time estimates for this activity are 1 week, 2 weeks, and 3 weeks. Activity P must be completed before activity U can be started.

Phase 5 is the final phase of the development project. It consists of only two activities. Activity V requires that activity Q and activity R be completed before it can be started. Time estimates for this activity are 9 weeks, 10 weeks, and 11 weeks. Activity W is the final activity of the process. It requires three activities to be completed before it can be started. These are activities S, T, and U. The estimated completion times for activity W are 2 weeks, 4 weeks, and 5 weeks.

1. Develop a table showing the activities of the project, the duration of each activity, the ES, EF, LS, LF and slack times.
2. Determine the expected completion time for the entire process. Show the variance for each activity and compute for the total project variance.
3. Determine the critical path and identify the activities on the critical path.

Answer 1

The calculations are:

Activity	Immediate predecessor	Optimistic	Most probable	Pessimistic	Expected time	Variance
A	-	2	3	4	3	0.6667
B	-	5	6	8	6.166666667	1.5
C	-	1	1	2	1.166666667	0.1667
D	-	8	9	11	9.166666667	1.5
E	A	1	1	4	1.5	1.5
F	B	3	3	4	3.166666667	0.1667
G	B	1	2	2	1.833333333	0.1667
H	C	5	5	6	5.166666667	0.1667
I	D	9	10	11	10	0.6667
J	D	1	2	2	1.833333333	0.1667
K	E	2	2	3	2.166666667	0.1667
L	F	3	4	6	4.166666667	1.5
M	G	2	2	4	2.333333333	0.6667
N	I	8	9	11	9.166666667	1.5
O	I	1	1	3	1.333333333	0.6667
P	J	4	4	8	4.666666667	2.6667
Q	K	6	6	7	6.166666667	0.1667
R	L,M	1	2	4	2.166666667	1.5
S	N	6	6	7	6.166666667	0.1667
T	O	3	3	4	3.166666667	0.1667
U	P	1	2	3	2	0.6667
V	Q,R	9	10	11	10	0.6667
W	S,T,U	2	4	5	3.833333333	1.5

Activity	Immediate predecessor	Expected time	ES	EF	LS	LF	Slack
A	-	3	0	3	15.5	18.5	15.5
B	-	6.166666667	0	6.166667	12.66667	18.83333	12.66667
C	-	1.166666667	0	1.166667	32	33.16667	32
D	-	9.166666667	0	9.166667	0	9.166667	0
E	A	1.5	3	4.5	18.5	20	15.5
F	B	3.166666667	6.166667	9.333333	18.83333	22	12.66667
G	B	1.833333333	6.166667	8	22	23.83333	15.83333
H	C	5.166666667	1.166667	6.333333	33.16667	38.33333	32
I	D	10	9.166667	19.16667	9.166667	19.16667	0
J	D	1.833333333	9.166667	11	26	27.83333	16.83333
K	E	2.166666667	4.5	6.666667	20	22.16667	15.5
L	F	4.166666667	9.333333	13.5	22	26.16667	12.66667
M	G	2.333333333	8	10.33333	23.83333	26.16667	15.83333
N	I	9.166666667	19.16667	28.33333	19.16667	28.33333	0
O	I	1.333333333	19.16667	20.5	30	31.33333	10.83333
P	J	4.666666667	11	15.66667	27.83333	32.5	16.83333
Q	K	6.166666667	6.666667	12.83333	22.16667	28.33333	15.5
R	L,M	2.166666667	13.5	15.66667	26.16667	28.33333	12.66667
S	N	6.166666667	28.33333	34.5	28.33333	34.5	0
T	O	3.166666667	20.5	23.66667	31.33333	34.5	10.83333
U	P	2	15.66667	17.66667	32.5	34.5	16.83333
V	Q,R	10	15.66667	25.66667	28.33333	38.33333	12.66667
W	S,T,U	3.833333333	34.5	38.33333	34.5	38.33333	0

Expected completion time = 38.33333

the total project variance = 5.3333

The Critical Path is = D - I - N - S - W