how minimum duration and floats might be used in project management

Operations Strategy – Operations Planning and Critical Path Analysis (CPA)

Learning Objectives

• Explain what Critical Path Analysis (CPA) is and why it is used in operations planning.
• Calculate the minimum project duration and the different types of float (slack).
• Construct an activity‑on‑node (AON) network diagram and interpret its results.
• Analyse how the critical path, minimum duration and floats inform broader operations‑management decisions such as capacity utilisation, outsourcing, quality, lean production and ERP.
• Identify the main assumptions and limitations of CPA.

1. What is Critical Path Analysis?

Critical Path Analysis is a systematic technique for planning, scheduling and controlling projects. Within an operations strategy it provides the time‑based backbone for:

• Setting realistic project deadlines and milestones.
• Identifying activities that must be protected from delay (the critical path).
• Revealing where flexibility (float) exists, enabling better resource allocation and risk management.

By linking project‑level decisions to the overall operations plan, CPA helps managers decide whether to:

• Increase capacity or outsource a critical activity.
• Apply lean techniques to reduce waste on non‑critical tasks.
• Integrate the schedule with an ERP system for real‑time monitoring.

2. Key Concepts

Term	Definition
Critical Path	The longest sequence of activities that determines the shortest possible project completion time. All activities on this path have zero float.
Minimum Project Duration (Dmin)	The total time required to finish the project when every activity on the critical path is performed as early as possible.
Early Start (ES) / Early Finish (EF)	Earliest times an activity can start and finish, assuming all predecessors start at their earliest possible times (forward pass).
Late Start (LS) / Late Finish (LF)	Latest times an activity can start and finish without extending Dmin (backward pass).
Total Float (TF)	Maximum time an activity can be delayed without affecting the project finish date. TF = LS – ES = LF – EF.
Free Float (FF)	Maximum delay an activity can suffer without delaying any of its immediate successors. FF = Min(ES of successors) – EF of the activity.
Project Float (PF)	Difference between the earliest possible project finish and the latest acceptable finish (often set by contract). PF = Latest Project Finish – Dmin.
Network Diagram (Activity‑on‑Node)	Graphical representation where each node is an activity, arrows show precedence, and durations are written inside the node.

3. Assumptions & Limitations of CPA

• Activity durations are deterministic (fixed and known). Uncertainty or variability is not modelled.
• Resources are assumed to be unlimited; the technique does not consider resource conflicts or leveling.
• Only logical (precedence) relationships are taken into account – no cost, quality or risk factors are embedded.
• CPA does not handle probabilistic risk; for that, techniques such as PERT are required.

4. Steps in Critical Path Analysis

Step	Action	Result
1	List every activity with its estimated duration.	Activity table (A, B, C …).
2	Identify all predecessor‑successor relationships.	Precedence list used to draw the network diagram.
3	Draw the network diagram (activity‑on‑node).	Visual map of the project; critical path will be highlighted later.
4	Perform a forward pass to compute ES and EF for each activity.	Earliest start/finish times.
5	Perform a backward pass to compute LS and LF.	Latest start/finish times without delaying the project.
6	Calculate Total Float (TF) and Free Float (FF) for every activity.	TF = LS – ES; FF = Min(ES of successors) – EF.
7	Identify the critical path (activities with TF = 0).	Minimum project duration = sum of durations on the critical path.

5. Mathematical Relationships

For any activity i:

\[ \text{Total Float}_i = LS_i - ES_i = LF_i - EF_i \] \[ \text{Free Float}_i = \min_{j\in\text{Successors}(i)}(ES_j) - EF_i \]

The minimum project duration is:

\[ D_{\text{min}} = \sum_{k\in\text{Critical Path}} d_k \]

where \(d_k\) is the duration of activity \(k\) on the critical path.

6. Using Minimum Duration and Floats in Project Management

• Scheduling & Milestones: D_min sets the earliest possible completion date; milestones are placed on critical‑path activities.
• Resource Allocation: Activities with float can be shifted to resolve resource clashes without jeopardising the finish date.
• Risk Management: Float acts as a time buffer; monitoring its consumption highlights emerging risks early.
• Capacity Planning: If a critical activity exceeds its ES, additional capacity (extra labour, equipment, or outsourcing) may be required.
• Quality Management: Free float can be used for inspections, testing or corrective actions without delaying the project.
• Lean Production: Non‑critical activities with generous float are ideal candidates for waste‑reduction pilots (Kaizen, JIT) because they can absorb minor re‑work.
• ERP Integration: ES/EF and LS/LF dates are uploaded to an ERP system for real‑time Gantt charts, automatic alerts when float is consumed, and workflow triggers.
• Decision‑Making: When comparing alternative suppliers or methods, calculate the impact on the critical path and available float to quantify trade‑offs.

7. Link to Other Operations‑Management Topics (Cambridge 9 Operations Management)

7.1 Location & Scale (9.1)

Location decisions affect the project schedule through travel time, transport costs and labour availability. For example, locating a component‑manufacturing plant nearer to the assembly site can reduce the predecessor‑successor lag, potentially shortening the critical path. Scale decisions (economies of scale) influence activity durations: larger batch sizes may lower per‑unit processing time but increase set‑up time, which must be reflected in the activity‑on‑node diagram.

7.2 Quality Management (9.2)

Quality activities (inspection, testing, corrective action) are normally placed on non‑critical paths so that they can use free float without jeopardising the overall finish date. In a Total Quality Management (TQM) system, the float data from CPA feed into quality dashboards, allowing managers to monitor whether quality checks are consuming more time than allocated.

7.3 Operations Strategy (9.3)

• Resource Availability: The critical path highlights the sequence that determines peak resource demand. If a critical resource is limited, the network must be re‑drawn (or the activity duration altered) to reflect the new constraint, which will change the critical path and D_min.
• Flexibility & Innovation: Flexible production techniques such as cellular layouts, modular design or rapid‑prototyping create alternative routes in the network diagram. These alternatives can be modelled as parallel activities; the one with the shortest duration becomes part of the critical path.
• ERP (features & benefits):

ERP Function	CPA Output Used
Gantt‑chart generation	ES, EF, LS, LF dates
Automatic float alerts	Total & free float values
Resource‑leveling module	Critical‑path activities & their resource requirements
What‑if scenario analysis	Alternative activity durations and paths
Performance dashboards	Project Float (PF) and milestone achievement

• Lean Production (Kaizen, JIT, waste reduction): Activities with high total float are low‑risk zones for lean pilots. For example, a non‑critical material‑handling task with 3 days of float can be re‑designed for a JIT flow; any small loss of time is absorbed by its float without affecting D_min.

8. Example Calculation

Project data:

Activity	Duration (days)	Predecessors
A	4	–
B	6	A
C	5	A
D	3	B, C
E	2	D

Forward Pass (ES / EF)

• A: ES = 0, EF = 4
• B: ES = 4, EF = 10
• C: ES = 4, EF = 9
• D: ES = 10 (max of B EF = 10 and C EF = 9), EF = 13
• E: ES = 13, EF = 15

Backward Pass (LS / LF)

• E: LF = 15, LS = 13
• D: LF = 13, LS = 10
• B: LF = 10, LS = 4
• C: LF = 10, LS = 5
• A: LF = 4, LS = 0

Floats

Activity	Total Float (TF)	Free Float (FF)	Critical?
A	0	0	Yes
B	0	0	Yes
C	1	1	No
D	0	0	Yes
E	0	0	Yes

Critical Path: A → B → D → E Minimum project duration = 4 + 6 + 3 + 2 = 15 days.

Activity C has a total (and free) float of 1 day; it may start as late as day 5 without delaying D or the overall project.

9. Summary & Exam Tips

• Start every question by drawing a clear activity‑on‑node diagram; clearly mark activities with zero total float – this earns marks.
• Remember the two passes: forward (ES/EF) then backward (LS/LF). Most mistakes occur in the backward pass.
• Calculate both total and free float; free float is useful for resource‑leveling and quality‑management questions.
• When asked for “minimum project duration”, add the durations of the activities on the critical path only.
• Comment on the assumptions (fixed durations, unlimited resources, no risk) if a short‑answer question requires it.
• Link CPA results to other operations concepts (capacity, outsourcing, lean, quality, ERP) to gain extra marks in extended‑response questions.