The document outlines project network operations management concepts, focusing on creating project network diagrams and calculating essential metrics such as earliest start and finish times (ES and EF), latest start and finish times (LS and LF), slack time, and critical paths. It includes detailed solutions for multiple math problems, allowing users to understand how to identify critical paths, expected project completion times, and slack times for various project activities. The critical paths highlighted demonstrate that they have zero slack time, indicating the most time-sensitive sequences of tasks within the project.

1. Project Networking
Operations Management
MBA, Fareast International University, Dhaka, Bangladesh
Basics of Project Network Math Solution
Considerations in Drawing Project Network Diagram
During Forward Progress
 ES time of the first activity will be 0.
 EF time of an activity will be considered as ES time of next activity (EF ES)
 In case of 2 or more EF, the highest EF will be considered as ES
During Backward Progress
 Ending EF, ES will be considered as ending LS, LF as it is.
 LS time of an activity will be considered as LF time of next activity (LS LF)
 In case of 2 or more LS, the lowest LS will be considered as LF of burst activity.
Critical Path Identification
The critical path is the path(s) of project network in which all the activities have zero (0) slack time
Float/Slack Time Calculation
In project management, Float or slack is the amount of time that a task in a project network can be
delayed without delaying subsequent tasks.
Slack/float time, SL= LS-ES or (LS-EF)
Note that,
At starting point, ES, EF and LS, LF will be same values

2. Math 01
Activity Description Immediate Predecessor Time (Weeks)
A Meet With board - 7
B Hire Coaches A 3
C Reserve Pool A 2
D Announce Program A 1
E Meet with coaches B, D 3
F Order team Members C 4
G Visiting the Project E, F 5
Queries:
a. Draw a project network.
b. Find out ES, EF and LS, LF.
c. Find out the critical path and expected project completion time.
d. Find out the slack time.

3. Solution to the Math Problem 01
Here Given,
Activities Duration
A-----------------7
B-----------------3
C-----------------2
D----------------1
E--------------- -3
F--------------- -4
G----------------5
Here in Network Diagram,
ES=Earliest Start Time.
EF=Earliest Finish Time.
LS=Latest Start Time.
LF=Latest Finish Time.
ES, EF
(7,7+3=10)
LS, LF ES, EF
(10-3=7,10) (10,10+3=13)
ES, EF
ES, EF ES, EF (13,13+5=18)
(0,0+7=7) (7,7+1=8) LS, LF
(13-3=10,13)
ES, EF
LS, LF LS, LF (9,9+4=13) LS, LF
(7-7=0,7) (10-1=9,10) (13, 18)
ES, EF
(7,7+2=9) LS, LF
(13-4=9,13)
LS, LF
(9-2=7,9)
Critical Path & Expected Project Completion Time Calculation
Different Path Completion time(week)
A-B-E-G= 7+3+3+5=18 weeks
A-D-E-G= 7+1+3+5=16 weeks
A-C-F-G= 7+2+4+5=18 weeks
So, the critical paths are A-B-E-G and A-C-F-G as all of activities of these paths has zero slack time and, in both
paths, the expected project completion time is 18 weeks.
Float/Slack Time Calculation
Activities Slack time (LS – ES)
A --------------------- 0-0= 0
B --------------------- 7-7= 0
C --------------------- 7-7= 0
D --------------------- 9-7= 2
E --------------------- 10-10= 0
F --------------------- 9-9= 0
G -------------------- 13-13= 0
A
B
D
C
E
F
G

4. Math 02
Activity Description Immediate Predecessor Time (Weeks)
A Meet With board - 1
B Hire Coaches A 2
C Reserve Pool A 5
D Announce Program B, C 3
E Meet with coaches B 4
F Order team Members C 1
G Visiting the Project D 3
H Plan first Practice E, G, F 1
Queries:
a. Draw a project network.
b. Find out ES, EF and LS, LF.
c. Find out the critical path and expected project completion time.
d. Find out the slack time.

5. Solution to the Math Problem 02
Here Given,
Activities Duration
A ------------------ 1
B ------------------ 2
C ------------------ 5
D ------------------ 3
E ------------------ 4
F ------------------ 1
G ----------------- 3
H ----------------- 1
Here in Network Diagram,
ES=Earliest Start Time.
EF=Earliest Finish Time.
LS=Latest Start Time.
LF=Latest Finish Time.
ES, EF ES, EF
(1, 1+2=3) (3, 3+4=7)
ES, EF LS, LF LS, LF
(0, 0+1=1) (6-2=4, 6) ES, EF (12-4=8, 12) ES, EF
(6, 6+3=9) (9, 9+3=12) ES, EF
(12, 12+1=13)
LS, LF ES, EF LS, LF LS, LF
(1-1=0, 1) (1, 1+5=6) (9-3=6, 9) ES, EF (12-3= 9, 12) LS, LF
(6, 6+1=7) (12,13)
LS, LF LS, LF
(6-5=1, 6) (12-1=11, 12)
Critical Path & Expected Project Completion Time Calculation
Different Path Completion time(week)
A-B-E-H=1+2+4+1=8 weeks
A-B-D-G-H=1+2+3+3+1=10 weeks
A-C-D-G-H=1+5+3+3+1=13 weeks
A-C-F-H=1+5+1+1=8 weeks
So, the critical path is A-C-D-G-H as all of activities of this path has zero slack time and, in this path, the expected
project completion time is 13 weeks.
Float/Slack Time Calculation
Activities Slack (LS – ES)
A ----------------------0-0= 0
B ----------------------4-1= 3
C ----------------------1-1= 0
D ----------------------6-6= 0
E -----------------------8-3= 5
F ----------------------11-6= 5
G ----------------------9-9= 0
H -----------------------12-12= 0
A
B
D
C
E
G
F
H

6. Math 03
Activity Description Immediate Predecessor Time (Weeks)
A Meet With board - 3
B Hire Coaches A 2
C Reserve Pool A 1
D Announce Program A 5
E Meet with coaches B, D 1
F Order team Members C 2
G Visiting the Project E, F 7
Queries:
a. Draw a project network.
b. Find out ES, EF and LS, LF.
c. Find out the critical path and expected project completion time.
d. Find out the slack time.

7. Solution to the Math Problem 03
Here Given,
Activities Duration
A ------------------ 3
B ------------------ 2
C ------------------ 1
D ------------------ 5
E ------------------ 1
F ------------------ 2
G ----------------- 7
Here in Network Diagram,
ES=Earliest Start Time.
EF=Earliest Finish Time.
LS=Latest Start Time.
LF=Latest Finish Time.
ES, EF ES, EF
(3, 3+2= 5) (8, 8+1= 9)
LS, LF LS, LF
(8-2= 6, 8) (9-1= 8, 9) ES, EF
ES, EF (9, 9+7= 16)
(0, 0+3= 3) ES, EF
(3, 3+5= 8)
LS, LF
LS, LF (9, 16)
(3-3= 0, 3) LS, LF ES, EF
(8-5= 3, 8) (4, 4+2= 6)
ES, EF
(3, 3+1= 4)
LS, LF
(9-2= 7, 9)
LS, LF
(7-1= 6, 7)
Critical Path & Expected Project Completion Time Calculation
Different Path Completion time(week)
A-B-E-G=3+2+1+7= 13 weeks
A-D-E-G=3+5+1+7= 16 weeks
A-C-F-G=3+1+2+7= 13 weeks
So, the critical path is A-D-E-G as all of activities of this path has zero slack time and, in this path, the expected
project completion time is 16 weeks.
Float/Slack Time Calculation
Activities Slack (LS – ES)
A ---------------------- 0-0= 0
B ---------------------- 6-3= 3
C ---------------------- 6-3= 3
D ---------------------- 3-3= 0
E ----------------------- 8-8= 0
F ----------------------- 7-4= 3
G ----------------------- 9-9= 0
G
B
C
D
E
F
A

8. Math 04
Activity Description Immediate Predecessor Time (Weeks)
A Meet With board - 3
B Hire Coaches A 2
C Reserve Pool A 7
D Announce Program A 1
E Meet with coaches D, C 11
F Order team Members B 9
G Visiting the Project D 1
H Plan first Practice E, F, G 2
Queries:
a. Draw a project network.
b. Find out ES, EF and LS, LF.
c. Find out the critical path and expected project completion time.
d. Find out the slack time.

9. Solution to the Math Problem 04
Here Given,
Activities Duration
A ------------------ 3
B ------------------ 2
C ------------------ 7
D ------------------ 1
E ------------------ 11
F ------------------ 9
G ----------------- 1
H ------------------2 Here in Network Diagram,
ES=Earliest Start Time.
EF=Earliest Finish Time.
LS=Latest Start Time.
LF=Latest Finish Time.
ES, EF ES, EF
(3, 3+2= 5) (5, 5+9= 14)
LS, LF
LS, LF (21-9= 12, 21)
(12-2= 10, 12)
ES, EF ES, EF ES, EF ES, EF
(0, 0+3= 3) (3, 3+1= 4) (4, 4+1= 5) (21, 21+2= 23)
LS, LF LS, LF LS, LF LS, LF
(3-3= 0, 3) (10-1= 9, 10) (21-1= 20, 21) (21, 23)
ES, EF ES, EF
(3, 3+7= 10) (10, 10+11= 21)
LS, LF LS, LF
(10-7= 3, 10) (21-11= 10, 21)
Critical Path & Expected Project Completion Time Calculation
Different Path Completion time(week)
A-B-F-H=3+2+9+2= 16 weeks
A-D-G-H=3+1+1+2=7 weeks
A-D-E-H=3+1+11+2=17 weeks
A-C-E-H=3+7+11+2= 23 weeks
So, the critical path is A-C-E-H as all of activities of this path has zero slack time and, in this path, the expected
project completion time is 23 weeks.
Float/Slack Time Calculation
Activities Slack (LS – ES)
A 0-0= 0
B 10-3= 7
C 3-3= 0
D 9-3= 6
E 10-10 = 0
F 12-5= 7
G 20-4= 16
H 21-21= 0
A
B
F
H
C E
GD