MOI UNIVERSITY

2024/2025 Academic Year
Third Year Second Semester Examination
For the Degree of Bachelor of Science in Project Planning and Management and Bachelor of Business Management
Course Code: BBM 351 / BPM 316
Course Title: OPERATIONS RESEARCH
Date: 7th July, 2025
Time: 9:00 A.M. - 12:00 P.M.

QUESTION FOUR

(a) Consider a project consisting of nine jobs with the following precedence relations and time estimates:

Activity	Predecessor	Optimistic Time (a)	Most Probable Time (m)	Pessimistic Time (b)
A	-	2	5	8
B	A	6	9	12
C	A	6	7	8
D	B, C	1	4	7
E	A	8	8	8
F	D, E	5	14	17
G	C	3	12	21
H	F, G	3	6	9
I	H	5	8	11

(i) Construct the project network (4 marks)

We construct the project network using the **Activity-on-Node (AON)** format.

Start | v A / | \ B C E \ / | D | \ | \ | F \ \ \ H | I | Finish

Note: A starts the project; B and C follow A; D follows B and C; E follows A; F follows D and E; G follows C; H follows F and G; I follows H.

(ii) Find the expected duration and the variance of each activity (4 marks)

We use the **PERT formula**:

Expected time (t): (a + 4m + b) ÷ 6
Variance (σ²): ((b - a) ÷ 6)²

Activity	a	m	b	Expected Time	Variance
A	2	5	8	5	1.00
B	6	9	12	9	1.00
C	6	7	8	7	0.11
D	1	4	7	4	1.00
E	8	8	8	8	0.00
F	5	14	17	13	4.00
G	3	12	21	12	9.00
H	3	6	9	6	1.00
I	5	8	11	8	1.00

(iii) What is the expected length of the project and its variance? (4 marks)

Step 1: Forward Pass (Calculate ES and EF)

Activity	ES	EF = ES + t
A	0	5
B	5	14
C	5	12
D	14	18
E	5	13
F	18	31
G	12	24
H	31	37
I	37	45

Step 2: Backward Pass (Calculate LS and LF)

Activity	LF	LS = LF - t
I	45	37
H	37	31
F	31	18
G	31	19
D	18	14
E	18	10
C	14	7
B	14	5
A	5	0

Step 3: Slack and Critical Path

Critical Path: A → B → D → F → H → I

Total Duration: 5 + 9 + 4 + 13 + 6 + 8 = 45 days

Variance: 1 + 1 + 1 + 4 + 1 + 1 = 9.00

(iv) What is the probability of completing the project in 50 days? (3 marks)

We use the **Z-score** to estimate the probability:

Mean (μ) = 45
Standard deviation (σ) = √9 = 3
Target time (T) = 50

Z-score:

Z = (Target time - Mean) ÷ Standard deviation

Z = (50 - 45) ÷ 3 = 5 ÷ 3 = 1.67

From the standard normal distribution table:

Cumulative probability for Z = 1.67 is approximately 0.9525

✅ Answer: 95.25%

(b) Consider the following activities and their durations:

Activity	Preceding Activity	Duration
A	-	3
B	-	4
C	A	9
D	A	3
E	A	8
F	C	2
G	C	3
H	C	2
J	B, D	6
K	F, J	1
L	E, H, G, K	4

(i) Construct CPM network (3 marks)

We construct the CPM network using **Activity-on-Node (AON)** format:

Start | v A / | \ C D E |\ | | F G H L | | | J K | Finish

(ii) Determine the critical path (2 marks)

Critical Path: A → C → F → J → K → L

Total Duration: 3 + 9 + 2 + 6 + 1 + 4 = 25 days

(iii) Compute total floats and free floats for non-critical activities (4 marks)

Activity	Duration	ES	EF	LS	LF	Total Float	Free Float
B	4	0	4	6	10	6	6
D	3	3	6	11	14	8	8
G	3	12	15	13	16	1	1
H	2	12	14	19	21	7	7
E	8	3	11	13	21	10	10

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