Predecessors and Successors Node Terminology and CPM Computations
B activity name
A
C
D earliest start time
= maximum EF of all predecessors
(0 if no preds)
ES i EF earliest finish time
= ES + t e
A is a predecessor of B and C.
B and C are predecessors of D.
B and C are successors of A.
A has no predecessors.
D has no successors.
Critical Path Method
From left to right, compute all ES and EF times. Then, from right to left, compute all LF and LS times.
Project completion time is maximum EF value of activities with no successors.
LS t e
LF latest start time
= LF - t e activity time latest finish time
= minimum LS of all successors
(project completion time if no succs.)
slack time = (LF-EF) or (LS-ES)
it is the amount that an activity can be delayed without delaying the project completion time.
critical activities are those with 0 slack.
critical path is the sequence of activities that determine the project length. (made up of activities with no slack.)
Probabilistic Times a = most optimistic time (estimate) m = most likely time (estimate) b = most pessimistic time (estimate) t
 e
2
= expected time (computed)
= variance of activity time (computed)
t e
= a

4 m
 b
6
 2 =
 
6
2
Act. a
Use of Probabilistic Times m b t e
A
B
5
10
7
15
9
26 
7
16
 2
0.444
7.111
.
.
.
Use expected times on network to find critical path.
T
E
= expected project completion time = critical path time from CPM method or sum of expected times for activities on the critical path.

CP
= standard deviation of critical path time
= add up the variances of the activities on the critical path and take the square root of the total.
Probability of Completion by a Given Date
T = given date, such as a due date z = (T - T
E
) / 
CP
Probability is found using normal table or Excel function: =NORMSDIST(z)
Promise Date to Meet Given Probability z value is found using normal table or the Excel function:
=NORMSINV(probability expressed as a decimal)
Promise date = T
E
+ z* 
CP
Management 120
1
Formula Sheets

NT = normal time = time to complete activity under normal conditions
NC = normal cost = cost to complete under normal conditions
CT = crash time = shortest possible time to complete activity
CC = crash cost = cost to complete activity in the shortest possible time
Maximum time reduction = NT - CT
Cost to crash per period = (CC-NC) / (NT-CT)
Activity
A
Normal
Time
(NC)
12
Normal
Cost
(NC)
Crash
Time
(CT)
Crash
Cost
(CC)
12,000 11 13,000
Maximum
Time
Reduction
1
Cost to
Crash per
Period
1,000
B
C
.
9
10
.
50,000
4,000
.
7 64,000 
5
.
.
7,000
.
.
2
5
.
.
7,000
600
.
. . . .
NC and CC are direct costs.
Indirect costs are cost per period of project duration. Penalty costs are per period late.
Finding the minimum cost schedule:
1. Compute max time reduction and cost to crash per period for each activity.
2. Draw network and find all paths and the normal time of each path.
3. Create a crashing table listing each path in a row with it’s normal time.
Paths Normal Time Crash 1 Crash 2 etc….
Crashing cost:
4. Make crashes one at a time: a. find the activity or activities which when crashed would shorten the critical path(s) and whose cost to crash per period is the lowest. If there is more than one critical path, it may require that a combination of activities to be crashed simultaneously. b. determine the number of periods to crash, which is the smallest of three values: i). the maximum reduction remaining for the activity or activities being crashed ii). the reduction possible before making an additional path become critical iii). the reduction possible where crashing costs would be less than indirect and penalty costs. c. write the crash at the top of a new column and reduce any paths containing the crashed activity or activities. d. write the cost of the crash at the bottom of the column. (cost to crash per period * periods crashed)
5. stop when reduction possible is zero or when given due date is met.
6. Find total cost = sum of normal cost column + total crashing cost + indirect costs + penalty costs.
Management 120 2 Formula sheets

Finding the minimum time schedule at least cost:
1. Set all times to crash time and find the critical path and project completion time.
2. Find the non-critical activity with the most expensive crash cost per period.
3. Relax it as much as possible up to the normal time or until no slack is left.
4. Repeat steps 2 and 3 until no more non-critical activities can be relaxed.
Management 120 3 Formula sheets