Critical path method and float

The critical path method (CPM) turns a network of activities and durations into a schedule. It finds the earliest and latest dates each activity can start and finish, the amount of flexibility (float) each activity has, and the critical path: the longest path through the network, which determines the shortest possible project duration. CPM calculations are a staple of the PMP exam, and more importantly they tell a project manager where to focus. This lesson works through the forward and backward passes on the Project Compass fit-out network, explains total and free float, near-critical paths and negative float, and covers how imposed dates and calendars affect the result.

The idea of the critical path

In a network, several paths run from start to finish. Each path's length is the sum of its activity durations. The longest path determines the project duration, because every path must be complete before the project ends. Activities on that path are critical: if any of them is delayed, the project finish is delayed, unless something else is changed. Activities on shorter paths have float, meaning they can slip by some amount without delaying the project.

For the Project Compass fit-out, the paths from A to K are:

Listing paths works for small networks, and it is a useful exam technique when a question gives a small diagram. For anything larger, the forward and backward passes are faster and also give each activity's float.

The forward pass

The forward pass calculates early start (ES) and early finish (EF) for each activity, moving from the start of the network to the end. This course uses the convention that the project starts at time 0, so an activity's early finish is ES plus duration, and a successor's early start equals its predecessor's early finish. Another convention starts at day 1 and adds or subtracts 1 at each step; both give the same critical path and float, and exam questions usually make the convention clear or offer options consistent with one of them.

Rules for the forward pass with finish-to-start relationships:

• The first activity has ES = 0.
• EF = ES + duration.
• An activity's ES equals the largest EF among its predecessors. At a merge point, take the latest predecessor finish, because the activity must wait for all of them.

For Project Compass: A runs 0 to 15. B and C both start at 15; B finishes at 35 and C at 45. D starts at 35 and finishes at 45. E and F start at 45; E finishes at 60, F at 65. G must wait for both E and F, so it starts at 65 (the larger of 60 and 65) and finishes at 80. H waits for C (45) and G (80), so it starts at 80 and finishes at 90. I follows C and runs 45 to 55. J runs 90 to 98. K waits for I (55) and J (98), so it starts at 98 and finishes at 100. The project duration is 100 working days.

The backward pass

The backward pass calculates late finish (LF) and late start (LS), moving from the end of the network to the start. The late dates are the latest an activity can start and finish without delaying the project finish.

• The last activity has LF equal to the project finish (here 100), unless a later target date is imposed.
• LS = LF − duration.
• An activity's LF equals the smallest LS among its successors. At a burst point, take the earliest successor start, because the activity must finish in time for all of them.

Working back: K has LF 100 and LS 98. J and I both have LF 98; J's LS is 90 and I's LS is 88. H has LF 90 and LS 80. G has LF 80 and LS 65. E and F both have LF 65; E's LS is 50, F's LS is 45. D's successors start at the latest at 50 (E) and 45 (F), so D's LF is the smaller, 45, and its LS is 35. C's successors are H (LS 80) and I (LS 88), so C's LF is 80 and LS 50. B has LF 35 and LS 15. A's successors are B (LS 15) and C (LS 50), so A's LF is 15 and LS 0.

Float

Total float (also called slack) is the amount of time an activity can be delayed from its early start without delaying the project finish or violating a schedule constraint. It is calculated as LS − ES or, equivalently, LF − EF. Free float is the amount of time an activity can be delayed without delaying the early start of any successor. It is calculated as the smallest ES of the successors minus the activity's EF.

# Critical path method: forward and backward pass for the Project Compass fit-out
acts = {
    "A": (15, []), "B": (20, ["A"]), "C": (30, ["A"]), "D": (10, ["B"]),
    "E": (15, ["D"]), "F": (20, ["D"]), "G": (15, ["E", "F"]),
    "H": (10, ["C", "G"]), "I": (10, ["C"]), "J": (8, ["H"]), "K": (2, ["I", "J"]),
}
order = ["A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K"]   # topological order
succ = {a: [b for b in acts if a in acts[b][1]] for a in acts}

ES, EF = {}, {}
for a in order:                      # forward pass (day 0 = start of project)
    ES[a] = max((EF[p] for p in acts[a][1]), default=0)
    EF[a] = ES[a] + acts[a][0]
end = max(EF.values())
LS, LF = {}, {}
for a in reversed(order):            # backward pass
    LF[a] = min((LS[s] for s in succ[a]), default=end)
    LS[a] = LF[a] - acts[a][0]

print("Act Dur  ES  EF  LS  LF  TF  FF")
for a in order:
    tf = LS[a] - ES[a]
    ff = min((ES[s] for s in succ[a]), default=end) - EF[a]
    print(f" {a} {acts[a][0]:4d} {ES[a]:3d} {EF[a]:3d} {LS[a]:3d} {LF[a]:3d} {tf:3d} {ff:3d}")
crit = [a for a in order if LS[a] == ES[a]]
print("Project duration:", end, "working days")
print("Critical path:", "-".join(crit))

Act Dur  ES  EF  LS  LF  TF  FF
 A   15   0  15   0  15   0   0
 B   20  15  35  15  35   0   0
 C   30  15  45  50  80  35   0
 D   10  35  45  35  45   0   0
 E   15  45  60  50  65   5   5
 F   20  45  65  45  65   0   0
 G   15  65  80  65  80   0   0
 H   10  80  90  80  90   0   0
 I   10  45  55  88  98  43  43
 J    8  90  98  90  98   0   0
 K    2  98 100  98 100   0   0
Project duration: 100 working days
Critical path: A-B-D-F-G-H-J-K

The results confirm the critical path A-B-D-F-G-H-J-K with zero total float. E has 5 days of total float and 5 days of free float: it can slip 5 days without delaying G. C has 35 days of total float but zero free float, because any delay to C immediately delays I's early start. I has 43 days of both. The difference between total and free float matters when assigning resources: using up C's float costs nothing at the project level, but it removes float from I, which shares the path.

Near-critical paths and negative float

The critical path is not the only path to watch. A near-critical path has little float and can become critical with a small delay. In Project Compass, the path through E has only 5 days of float: if the electrical rough-in slips by more than a week, it becomes critical. Many organizations monitor all paths with float below a threshold, such as 10 days or 10 % of the remaining duration.

Negative float occurs when the calculated early dates are later than a required date. If the library's board imposed a soft opening at day 95, the backward pass would start from 95 instead of 100. Every activity on the critical path would then have −5 days of float, meaning the plan is 5 days too long to meet the target. Negative float is a signal for action: compress the schedule, change logic, add resources or negotiate the date. It is never a signal to adjust the numbers until they look acceptable.

Constraints, calendars and multiple critical paths

Real schedules include imposed constraints such as "start no earlier than" (for example, the lease handover date) and "finish no later than" (a contractual milestone). These override logic in the calculation and can create negative float or hide true float. Use them sparingly and document why each exists. Calendars also matter: an activity done by a contractor who works six days a week, or by staff who do not work on certain holidays, has a different calendar from the project default. Scheduling software handles this, but it means float should be read in the activity's own calendar.

A network can have more than one critical path, when two or more paths have the same longest length. Multiple critical paths increase schedule risk, because a delay on any of them delays the project. It is also possible for the critical path to change during the project, as actual durations differ from estimates. In Project Compass, if the HVAC upgrade (F) finishes 6 days early, the path through E becomes critical. This is why critical path analysis is repeated at every status update rather than done once.

The critical path is not the same as the most important or most difficult activities. Training staff (I) is important to the opening, but with 43 days of float it is not critical. Conversely, a routine activity like demolition (D) is critical because of where it sits in the network. Exam questions sometimes try to blur this distinction.

Using CPM to manage

CPM results guide day-to-day decisions. Focus monitoring on critical and near-critical activities. Use float to schedule non-critical work when resources are available, which is the basis of resource smoothing in lesson 8. When a delay occurs, compare it with the activity's float before deciding whether to act. When asked to accelerate, look at critical activities first, because shortening a non-critical activity does not shorten the project.

Float also belongs to someone, and that question can be contentious. In many construction contracts, float belongs to the project, meaning whoever uses it first benefits. Some contracts specify that float is shared or owned by the owner. Projects should agree how float will be used and reported so that one party does not consume it silently, leaving others to bear the delay.

Critical chain method is a variation that accounts for resource limits and moves safety margins out of individual activities into buffers. Activity estimates are aggressive, and a project buffer protects the finish date, with feeding buffers where non-critical chains join the critical chain. Buffer consumption is monitored rather than individual activity dates. The exam may ask you to recognize the method and its buffers; you are unlikely to need detailed calculations.