calculation of total and free float and interpretation of CPA results

Operations Strategy – Operations Planning, Critical Path Analysis (CPA) and Interpretation of Results

Learning Objective

By the end of this lesson you will be able to:

• Construct an activity‑on‑node (AON) network diagram for a project.
• Carry out forward and backward passes to obtain early start (ES), early finish (EF), late start (LS) and late finish (LF) for every activity.
• Calculate total float (TF) and free float (FF) and explain what they mean for project scheduling.
• Use float information to make realistic operations‑planning decisions such as resource‑levelling, crashing and cost‑time trade‑offs.
• Evaluate the strengths and limitations of CPA in the wider operations‑strategy context.
• Recognise how CPA links to other A‑Level Operations topics (location, capacity, quality, ERP, lean) and to modern IT/AI tools.

Context Box – Where CPA Fits in the Cambridge International Business (9609) Syllabus

Key Terminology

Formulas at a Glance

• EF = ES + duration
• LF = LS + duration
• TF = LS – ES = LF – EF
• FF = minimum ES of all immediate successors – EF

Step‑by‑Step CPA Procedure (Activity‑on‑Node)

1. List activities with durations and all predecessor relationships.
2. Draw the network diagram using activity‑on‑node symbols; arrows show the flow of work.
3. Forward pass – start at the project’s start node (time = 0) and compute ES and EF for every activity.
4. Backward pass – start at the project’s finish node (the largest EF) and compute LF and LS for every activity.
5. Calculate floats using the formulas above.
6. Identify the critical path – activities with TF = 0.
7. Interpret the results (see sections below on resource allocation, flexibility, cost‑time trade‑offs, and evaluation).

Interpreting CPA Results for Operations Planning

1. Resource Allocation & Levelling

• Critical‑path activities must receive sufficient resources to avoid any delay.
• Non‑critical activities with positive float can be shifted within their slack windows to respect resource limits (e.g., limited machines or staff).
• Resource‑levelling moves activities within their float without extending the project, ensuring the demand for a scarce resource never exceeds its availability.

2. Crashing & Cost‑Time Trade‑offs

• Crashing shortens an activity’s duration (e.g., overtime, extra equipment) at an additional cost.
• Only activities on the critical path should be considered for crashing, because reducing a non‑critical activity’s time does not affect the overall finish.
• Calculate the cost per day saved for each critical activity; crash the activity with the lowest cost‑per‑day until either the budget is exhausted or the next‑most‑expensive activity becomes critical.

3. Flexibility & Innovation

• Float provides built‑in schedule flexibility. Managers can use this “buffer” to absorb unexpected disruptions (supplier delay, equipment breakdown, quality re‑work).
• Process innovations that introduce new activities or remove existing ones will change the network structure, potentially creating new critical paths or increasing total float.

4. Integration with IT / AI Tools

• Modern project‑management software (MS Project, Primavera P6, cloud‑based Gantt/CPA apps) automates forward/backward passes and float calculations.
• AI‑driven scheduling tools can:
  • Predict realistic activity durations from historical data.
  • Suggest optimal resource‑levelling patterns.
  • Identify the most cost‑effective crashing options in seconds.
• When using such tools, always verify the underlying assumptions (deterministic vs. probabilistic durations) to satisfy the exam requirement to “explain the method”.

5. Evaluation – Limitations of CPA (≈150 words)

CPA assumes that activity durations are known, fixed and independent, which rarely reflects real‑world uncertainty. It also ignores resource constraints unless a separate levelling step is performed, so the “critical path” may change once resources are limited. The technique does not capture risk, quality variability or cost fluctuations, and it can give a false sense of certainty if used in isolation. Consequently, exam answers should acknowledge that CPA is a useful planning baseline but must be combined with risk analysis, resource‑constrained scheduling and continuous monitoring to produce robust operations plans.

Link to Other Operations Topics

• Location & Scale Decisions (9.1‑9.2) – CPA helps determine the timing of site construction, equipment installation and staff recruitment, influencing whether a plant should be built locally or offshore.
• Capacity Utilisation (9.1‑9.2) – By smoothing activities within their float windows, managers can level peaks in demand for labour or machinery, improving overall capacity utilisation.
• Outsourcing (9.1‑9.2) – Float can be used to decide which non‑critical activities may be outsourced without affecting the project finish.

Quality & Lean Production (9.3.1 & 9.3.4)

Inspection, testing and re‑work can be modelled as separate activities in the network. Their float indicates how much quality‑related delay can be tolerated before the overall schedule is jeopardised. In a lean environment, the aim is to minimise unnecessary float (e.g., by adopting Just‑In‑Time delivery), thereby tightening the critical path and reducing waste. However, a small amount of strategic float is still valuable as a buffer against quality defects.

Cost‑per‑Day Template for Crashing (copy into notebook)

Worked Example (Re‑worked)

Project data:

1. Forward Pass (ES / EF)

• A: ES = 0 EF = 0 + 4 = 4
• B: ES = EF(A) = 4 EF = 4 + 3 = 7
• C: ES = EF(A) = 4 EF = 4 + 5 = 9
• D: ES = EF(B) = 7 EF = 7 + 2 = 9
• E: ES = max[EF(B), EF(C)] = max(7, 9) = 9 EF = 9 + 6 = 15
• F: ES = max[EF(D), EF(E)] = max(9, 15) = 15 EF = 15 + 3 = 18

2. Backward Pass (LF / LS)

• Project finish = EF(F) = 18 ⇒ LF(F) = 18 LS(F) = 18 – 3 = 15
• E: LF = LS(F) = 15 LS = 15 – 6 = 9
• D: LF = LS(F) = 15 LS = 15 – 2 = 13
• C: LF = LS(E) = 9 LS = 9 – 5 = 4
• B: LF = min(LS(D), LS(E)) = min(13, 9) = 9 LS = 9 – 3 = 6
• A: LF = min(LS(B), LS(C)) = min(6, 4) = 4 LS = 4 – 4 = 0

3. Float Calculations

4. Interpretation

• Critical path: A → B → F (all TF = 0). Minimum project duration = 18 days.
• Activities C, D and E each have 2 days of total float. They can be delayed up to 2 days without extending the overall finish.
• Free float shows the slack before the next activity would be affected. For example, C can be delayed 2 days before it pushes the start of its successor E.
• Resource‑levelling opportunity: If only one crew is available for C, D and E, the 2‑day floats allow you to schedule them sequentially without jeopardising the 18‑day finish.
• Crashing consideration: Since the critical path consists of A, B and F, any attempt to shorten the project must focus on these three activities (e.g., overtime on B or adding an extra crew to F). Crashing C, D or E would not reduce the overall duration.

Practice Question

Using the data below, calculate ES, EF, LS, LF, total float and free float for each activity. Identify the critical path(s) and state the minimum project duration. Then answer the short‑answer prompts.

Short‑answer prompts

1. Which activity (or activities) would you consider for crashing if the client demanded a 2‑day earlier completion? Explain why.
2. Assume only one machine is available for activities H, I and J. Show how you could use the available float to level the resource demand.
3. Briefly discuss how an ERP system could help you keep the CPA data up‑to‑date throughout the project.

Solution Outline (Teacher’s Reference)

• Critical path: G → H → J → L (TF = 0). Minimum project duration = 16 days.
• Activities I and K each have 2–3 days of float, providing flexibility for resource‑levelling.
• Crashing suggestion: Only activities on the critical path (G, H, J, L) can reduce the overall finish. To achieve a 2‑day reduction, crash J (cost‑per‑day is usually lowest) and, if needed, add overtime to H.
• Resource‑levelling example: Schedule I after H (using its 3‑day float) and before J, then place K after J using its 2‑day float, ensuring the single machine is never double‑booked.
• ERP benefit: The ERP stores the activity list, durations, predecessor links and resource calendars; any change (e.g., a delay in H) updates the network automatically, keeping the CPA schedule current and visible to all stakeholders.