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4.3 Cumulative Resource S-Curves & Productivity Tracking
AACE Planning and Scheduling Professional PSP
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1 Chapter 0: Introduction & AACE PSP Exam Structure
0.1 Current AACE PSP Exam, Eligibility, and Certification Facts 0.2 Planning and Scheduling Professional Roles and Competencies 0.3 Cost Engineering and Scheduling Terminology per RP 10S-90
2 Chapter 1: Scope Definition & Planning Fundamentals
1.1 Work Breakdown Structure (WBS) Architecture & RP 33R-15 1.2 Organization & Resource Breakdown Structures (OBS, RBS, RAM) 1.3 Project Planning and Activity-Logic Development 1.4 Contract Requirements and Delivery Constraints 1.5 Stakeholder Identification, Needs, and Plan Review 1.6 Constructability Methods and Value Engineering 1.7 Resource Identification and Project Variables 1.8 Scope, Goals, Project Plan, and Phase Definition 1.9 Cost Breakdown Structure and Cost Estimate Development 1.10 Phase Relationships, Baseline Plan, Forecasts, and Risk/Recovery Planning
3 Chapter 2: Critical Path Method Mechanics & Network Logic
2.1 Activity Definition and Logic Relationships 2.2 Activity Durations, Productivity, and Work Calendars 2.3 Forward and Backward Pass Mathematical Mechanics 2.4 Float Calculations and Critical-Path Identification
4 Chapter 3: Advanced Network Techniques & Schedule Constraints
3.1 Schedule Constraints and Their Impact on Dates and Float 3.2 Advanced Logic Modeling: Ladders and Fragnets 3.3 Schedule Levels of Detail and Schedule Basis Documentation 3.4 Schedule Scope and Schedule Specification 3.5 Breakdown-Structure Relationships and the Cost Estimate Model 3.6 Stakeholder Feedback and Types of Schedules 3.7 Activities, Milestones, Schedule Quality, and Basis Documentation
5 Chapter 4: Resource Management & Leveling
4.1 Resource Planning, Allocation, and Histograms 4.2 Resource Leveling vs. Resource Smoothing 4.3 Cumulative Resource S-Curves & Productivity Tracking
6 Chapter 5: Baseline Management & Schedule Updating
5.1 Baseline Establishment, Forecasts, and Controlled Revision 5.2 Progress Measurement and Data Date Mechanics 5.3 Out-of-Sequence Logic: Progress Override vs. Retained Logic 5.4 Schedule Update Review and Quality Verification per RP 53R-06 5.5 Cost/Resource Management, Schedule Change, and Feedback 5.6 Acceleration and Recovery Scheduling per RP 54R-07 5.7 Control Schedules, Variance Analysis, Forecasts, and Management Summaries
7 Chapter 6: Schedule Risk Analysis & Monte Carlo Simulation
6.1 Schedule Risk Principles, Risk Drivers, and Three-Point Estimates 6.2 Monte Carlo Simulation, Criticality, Correlation, and Merge Effects 6.3 Risk Outputs, Confidence Dates, Sensitivity, and Contingency
8 Chapter 7: Earned Value & Earned Schedule Management
7.1 Earned Value Foundations: PV, EV, AC, and Variances 7.2 Traditional EV Performance Indices & Late-Project Quirks 7.3 Earned Schedule (ES) Theory, SV(t), and SPI(t) 7.4 Independent Schedule Forecasting & IEAC(t)
9 Chapter 8: Forensic Schedule Analysis & Delay Claims
8.1 Delay Classification, Criticality, and Contract Questions 8.2 RP 29R-03 Taxonomy, Data Validation, and Method Selection 8.3 TIA, Windowed Analysis, and Subtractive Models 8.4 Concurrency, Pacing, and Float: Technical Analysis Before Entitlement
10 Chapter 9: Executive Communication & Practical Scenario Applications
9.1 Official PSP Memo Writing Framework 9.2 Structured Technical Problem-Solving for PSP Scenarios 9.3 Current Exam Strategy and Core Calculation Review
4.3 Cumulative Resource S-Curves & Productivity Tracking
Key Takeaways Explain This Topic AI
Cumulative resource curves form an S-shape (sigmoidal distribution) because capital projects naturally experience an initial mobilization ramp-up, a linear steady-state peak, and a gradual commissioning ramp-down.
The area bounded by the Early Dates S-curve and Late Dates S-curve forms the 'Banana Curve' envelope; the horizontal distance represents available float, while actual progress positioning diagnoses project health.
Cost engineering evaluates labor performance through the Unit Rate (Labor Hours / Output Quantity) and the Productivity Index (PI = Planned Unit Rate / Actual Unit Rate = Earned Hours / Actual Hours), where values greater than 1.0 indicate superior labor efficiency.
Learning curve theory (Wright's cumulative average model Y = a * X^b) demonstrates that direct labor-hours decrease by a constant percentage every time cumulative production volume doubles.
Short-interval lookahead schedules (3-week to 6-week lookaheads) operationalize the CPM network at the field workface by systematically screening constraints and tracking the Percent Plan Complete (PPC) metric.
Last updated: October 2026
4.3 Cumulative Resource S-Curves & Productivity Tracking
AACE Professional Context: Developing an approved, resource-leveled baseline schedule is only the first step in project controls. Once construction commences, project controls engineers must continuously track physical execution against the plan. In AACE Total Cost Management (TCM), Cumulative Resource S-Curves and Productivity Tracking Metrics serve as the primary diagnostic instruments for detecting emerging variances before they manifest as catastrophic project delays. On the AACE PSP examination, candidates must thoroughly understand how cumulative curves are generated, how to interpret early versus late S-curve envelopes (the "Banana Curve"), how to calculate unit rates and the Productivity Index ( P I PI P I ), how to apply learning curve mathematics, and how to manage field execution using short-interval lookaheads and the Percent Plan Complete ( P P C PPC PPC ) metric.
Cumulative Resource S-Curves: Generation and Geometry
A Cumulative Resource S-Curve is the mathematical integral (running cumulative summation) of period-by-period resource demand or cost expenditures plotted over time:
C ( t ) = ∑ i = 1 t R i C(t) = \sum_{i=1}^{t} R_i C ( t ) = ∑ i = 1 t R i
Where R i R_i R i represents the resource units (labor-hours, machine-hours, or dollars) consumed during period i i i , and C ( t ) C(t) C ( t ) represents the cumulative total expended through period t t t . Cumulative Units / Hours
- ▲
- 100% │ .--- [PLATEAU: Turnover]
- │ . - '
- 80% │ . - '
- │ . - ' STEADY-STATE PRODUCTION
- 60% │ . - ' (Maximum Linear Slope = Peak Burn Rate)
- │ . - '
- 40% │ . - ' <--- POINT OF INFLECTION (d²Y/dt² = 0)
- │ . - '
- 20% │ .-' RAMP-UP (Accelerating Slope: d²Y/dt² > 0)
- └────────────────────────────────────────────────────────► Time
- 0% 30% 70% 100%
- The Three Geometric Zones of the S-Curve
The characteristic sigmoidal (S-shaped) profile emerges naturally from the life cycle of industrial and construction projects:
Mobilization / Ramp-Up Zone (Convex: d 2 Y d t 2 > 0 \frac{d^2Y}{dt^2} > 0 d t 2 d 2 Y > 0 ): During the initial 15% to 30% of project duration, cumulative expenditure grows slowly but with positive acceleration. Site establishment, engineering finalization, material staging, and initial earthworks involve relatively small, specialized crews.
Steady-State Production Zone (Point of Inflection: d Y d t = Maximum \frac{dY}{dt} = \text{Maximum} d t d Y = Maximum ): During the middle 30% to 70% of the project timeline, major multi-discipline construction (structural steel, process piping, electrical equipment, building envelope) proceeds in parallel. The slope of the S-curve reaches its maximum steepness, representing the peak daily burn rate of labor-hours or capital.
Closeout / Ramp-Down Zone (Concave: d 2 Y d t 2 < 0 \frac{d^2Y}{dt^2} < 0 d t 2 d 2 Y < 0 ): As physical installations reach completion, major trades demobilize. Work decelerates into pre-commissioning, hydrotesting, electrical loop checks, punch-list remediation, and regulatory turnover, causing the curve to flatten toward its final 100% cumulative plateau.
The "Banana Curve" Envelope: Early Dates vs. Late Dates
When cumulative resource curves are generated for both the Early Start dates and Late Start dates of a baseline CPM schedule, they form an enclosed envelope known throughout the project controls profession as the Banana Curve . Cumulative Progress %
- ▲
- 100% │ ┌───┐
- │ . - ' │ │
- 80% │ . - ' │ B │ [EARLY DATES S-CURVE]
- │ . - ' . - ' │ A │ (Upper Boundary)
- 60% │ . - ' . - ' │ N │
- │ . - ' . - ' │ A │ [LATE DATES S-CURVE]
- 40% │ .-' . - ' │ N │ (Lower Boundary)
- │ . - ' │ A │
- 20% │ . - ' └───┘
- └────────────────────────────────────────────► Time
- Project Start Project Finish
- ◄──────────────── Total Float ───────────────►
- (Horizontal distance between curves)
- Geometric and Physical Properties of the Envelope
Upper Boundary (Early Dates Curve): Represents cumulative planned progress if every single activity commences at its earliest possible start date ( E S ES ES ). It represents the fastest theoretical pace of work and highest early capital expenditure while preserving 100% of project float.
Lower Boundary (Late Dates Curve): Represents cumulative planned progress if every single activity is delayed to its latest permissible start date ( L S LS L S ) without extending the final project completion date. It reflects the absolute slowest allowable pace of execution.
Horizontal Distance Between Curves: At any given progress percentage (e.g., at 50% completion), the horizontal distance between the Early Curve and Late Curve represents the total float available in the project network at that stage.
Diagnostic Interpretation of Actual Performance
Project controls professionals plot actual progress ( E V hours EV_{\text{hours}} E V hours or physical percent complete) against the baseline Banana Curve envelope to evaluate project health:
Actual Progress Position Diagnostic Evaluation Management Action Required
Tracking Above Early Curve Progress is exceeding early baseline expectations. However, planners must verify that work is proceeding in proper sequence rather than cherry-picking easy tasks, and ensure cash flow financing can support the accelerated burn rate. Verify quality compliance and inspect material supply chains to support accelerated installation.
Comfortably in Upper Envelope (Top 25%–50%) Optimal Project Health. Work is progressing steadily; the project team is executing near early dates while preserving a substantial float buffer against future field disruptions. Maintain existing crew allocations and standard lookahead planning routines.
Drifting Toward Late Curve Emerging Schedule Distress. Activities are consistently starting late or suffering productivity losses. Float is being rapidly consumed across multiple non-critical paths. Perform root-cause analysis; review trade productivity; initiate selective overtime or crew reassignments.
Hugging / Touching Late Curve CRITICAL ALERT. All total float across the project network has been completely exhausted. Every activity has effectively become critical ( T F = 0 TF = 0 TF = 0 ). Any single future delay will directly delay project completion. Issue formal executive notice; freeze baseline change requests; implement schedule recovery measures.
Breaching Below Late Curve SCHEDULE BREACH. The project is mathematically incapable of meeting its contractual completion date under current production rates. Project completion has slipped. Immediate schedule crashing, logic fast-tracking, or formal negotiation of contract extension.
Productivity Measurement & Unit Rate Mechanics
In AACE Total Cost Management, labor productivity is the single greatest variable influencing project profitability and schedule fidelity. In industrial and civil construction, productivity measures the efficiency with which labor-hours are converted into installed physical scope.
Core Cost Engineering Productivity Formulas
Labor Production Rate (Output-to-Input): Production Rate = Quantity of Physical Work Completed Labor-Hours Expended ( e.g., 1.25 Linear Feet of Pipe 1.0 Labor-Hour ) \text{Production Rate} = \frac{\text{Quantity of Physical Work Completed}}{\text{Labor-Hours Expended}} \quad \left(\text{e.g., } \frac{1.25\text{ Linear Feet of Pipe}}{1.0\text{ Labor-Hour}}\right) Production Rate = Labor-Hours Expended Quantity of Physical Work Completed ( e.g., 1.0 Labor-Hour 1.25 Linear Feet of Pipe )
Labor Unit Rate (Labor Norm / Input-to-Output): Unit Rate = Labor-Hours Expended Quantity of Physical Work Completed ( e.g., 0.80 Labor-Hours 1.0 Linear Foot of Pipe ) \text{Unit Rate} = \frac{\text{Labor-Hours Expended}}{\text{Quantity of Physical Work Completed}} \quad \left(\text{e.g., } \frac{0.80\text{ Labor-Hours}}{1.0\text{ Linear Foot of Pipe}}\right) Unit Rate = Quantity of Physical Work Completed Labor-Hours Expended ( e.g., 1.0 Linear Foot of Pipe 0.80 Labor-Hours )
In estimating and cost engineering, unit rates represent the standard "labor norm" (e.g., hours per cubic yard of concrete, hours per ton of structural steel).
Productivity Index (PI) / Labor Performance Factor (PF): Productivity Index (PI) = Planned Unit Rate Actual Unit Rate = Earned Labor-Hours Actual Labor-Hours \text{Productivity Index (PI)} = \frac{\text{Planned Unit Rate}}{\text{Actual Unit Rate}} = \frac{\text{Earned Labor-Hours}}{\text{Actual Labor-Hours}} Productivity Index (PI) = Actual Unit Rate Planned Unit Rate = Actual Labor-Hours Earned Labor-Hours
Where: Earned Labor-Hours = Actual Quantity Completed × Planned Unit Rate \text{Earned Labor-Hours} = \text{Actual Quantity Completed} \times \text{Planned Unit Rate} Earned Labor-Hours = Actual Quantity Completed × Planned Unit Rate Actual Labor-Hours = Actual Craft Hours Expended (from Payroll / Timesheets) \text{Actual Labor-Hours} = \text{Actual Craft Hours Expended (from Payroll / Timesheets)} Actual Labor-Hours = Actual Craft Hours Expended (from Payroll / Timesheets)
Interpreting the Productivity Index (PI) per AACE Standards
{ PI > 1.0 Superior Productivity: Crews are installing scope using fewer hours than budgeted (Favorable). PI = 1.0 On-Target Performance: Actual productivity matches baseline estimate exactly. PI < 1.0 Substandard Productivity: Crews are burning more hours than budgeted (Unfavorable / Cost Overrun). \begin{cases} \text{PI} > 1.0 & \text{Superior Productivity: Crews are installing scope using fewer hours than budgeted (Favorable).} \\ \text{PI} = 1.0 & \text{On-Target Performance: Actual productivity matches baseline estimate exactly.} \\ \text{PI} < 1.0 & \text{Substandard Productivity: Crews are burning more hours than budgeted (Unfavorable / Cost Overrun).} \end{cases} ⎩ ⎨ ⎧ PI > 1.0 PI = 1.0 PI < 1.0 Superior Productivity: Crews are installing scope using fewer hours than budgeted (Favorable). On-Target Performance: Actual productivity matches baseline estimate exactly. Substandard Productivity: Crews are burning more hours than budgeted (Unfavorable / Cost Overrun).
Caution
Exam Trap Alert — Inverse Software Definitions: On the AACE PSP exam, always verify the mathematical formulation! In AACE Total Cost Management, Productivity Index is an efficiency metric where higher is better ( Earned / Actual \text{Earned} / \text{Actual} Earned / Actual or Planned Rate / Actual Rate > 1.0 \text{Planned Rate} / \text{Actual Rate} > 1.0 Planned Rate / Actual Rate > 1.0 ). However, some commercial contractor software defines a "Labor Cost Factor" as Actual / Planned \text{Actual} / \text{Planned} Actual / Planned (where < 1.0 < 1.0 < 1.0 is good). Always check whether the formula places Planned or Actual in the numerator!
Learning Curve Theory: Wright's Cumulative Average Model
When direct craft labor performs highly repetitive operations, productivity systematically improves over time due to worker familiarity, improved tooling organization, supervisory refinement, and reduced hesitation. This phenomenon is modeled in cost engineering using Learning Curve Theory .
Originally formulated by T.P. Wright in 1936, the Wright Cumulative Average Learning Curve Model establishes that:
The Doubling Principle: Every time the cumulative volume of repetitive units produced doubles, the cumulative average labor-hours required per unit decreases by a constant percentage (the Learning Rate).
- The Mathematical Learning Curve Equation
- Y x = a ⋅ X b Y_x = a \cdot X^b Y x = a ⋅ X b
Where:
- Y x Y_x Y x = Cumulative average labor-hours required per unit for the first X X X units
- a a a = Direct labor-hours required to produce the very first unit ( X = 1 X = 1 X = 1 )
- X X X = Cumulative unit number
b b b = Learning curve exponent (slope parameter), calculated as:
b = ln ( L ) ln ( 2 ) b = \frac{\ln(L)}{\ln(2)} b = l n ( 2 ) l n ( L )
Where L L L is the learning rate expressed as a decimal (e.g., 0.80 0.80 0.80 for an 80% learning curve).
Doubling Progression for an 80% Learning Curve ( L = 0.80 L = 0.80 L = 0.80 , Unit 1 = 100 Hours)
Unit 1 ( X = 1 X = 1 X = 1 ): Cumulative Average = 100.0 hours 100.0\text{ hours} 100.0 hours . Total Hours = 100.0 hours 100.0\text{ hours} 100.0 hours .
Unit 2 ( X = 2 X = 2 X = 2 , 1st doubling): Cumulative Average = 100.0 × 0.80 = 80.0 hours/unit 100.0 \times 0.80 = 80.0\text{ hours/unit} 100.0 × 0.80 = 80.0 hours/unit . Total Hours for 2 units = 160.0 hours 160.0\text{ hours} 160.0 hours . (Hours for Unit 2 alone = 160 − 100 = 60.0 hours 160 - 100 = 60.0\text{ hours} 160 − 100 = 60.0 hours ).
Unit 4 ( X = 4 X = 4 X = 4 , 2nd doubling): Cumulative Average = 80.0 × 0.80 = 64.0 hours/unit 80.0 \times 0.80 = 64.0\text{ hours/unit} 80.0 × 0.80 = 64.0 hours/unit . Total Hours for 4 units = 256.0 hours 256.0\text{ hours} 256.0 hours .
Unit 8 ( X = 8 X = 8 X = 8 , 3rd doubling): Cumulative Average = 64.0 × 0.80 = 51.2 hours/unit 64.0 \times 0.80 = 51.2\text{ hours/unit} 64.0 × 0.80 = 51.2 hours/unit . Total Hours for 8 units = 409.6 hours 409.6\text{ hours} 409.6 hours .
Practical Applications and Field Limitations in Construction
Where Applicable: Repetitive modular process skid fabrication, precast bridge beam casting, structural steel pipe rack module assembly, multi-floor repetitive hotel or hospital drywall and rough-in installations.
Where Learning Breaks Down: High craft turnover (loss of trained personnel), design revisions (change orders introducing new configurations), intermittent activity splitting, and extreme adverse weather. Once mechanized operations govern (e.g., machine cycle time of an automated tunneling machine), human learning plateaus.
Short-Interval Lookahead Schedules & Operational Field Planning
A Level 3 or Level 4 CPM schedule updated monthly provides strategic governance, but it is too high-level to control daily craft assignments at the workface. To bridge the gap between baseline CPM logic and daily field execution, project teams utilize Short-Interval Lookahead Schedules (SILAs) and Weekly Work Plans (WWPs) . [STRATEGIC LEVEL] Level 3 CPM Baseline Schedule (Updated Monthly)
- │ (Filtered for Next 3 to 6 Weeks)
- ▼
- [TACTICAL LEVEL] 6-Week Lookahead Schedule (Constraint Screening)
- │ (Screened for Materials, Permits, RFIs, Access)
- ▼
- [OPERATIONAL LEVEL] Weekly Work Plan (WWP) (Last Planner Commitments)
- │ (Daily Execution by Field Foremen)
- ▼
- [CONTROL FEEDBACK] PPC Metric & Root Cause Variance Analysis
- The 3-Week and 6-Week Lookahead Horizon
3-Week Lookahead: Focuses on immediate tactical deployment: current week execution plus the following two weeks. Used in daily superintendent huddles for equipment allocation and trade handoffs.
6-Week Lookahead: The standard window for Constraint Screening . Tasks scheduled to begin 4 to 6 weeks in the future are audited against six critical operational prerequisites:
The Six-Constraint Screening Protocol
Before an activity on the lookahead schedule is released for field execution, the planning team must verify that all six operational constraints are cleared:
Engineering & Information: Approved Issued-For-Construction (IFC) drawings, approved shop drawings, and resolved Requests for Information (RFIs).
Materials & Tagged Equipment: Materials physically delivered, inspected, and staged at the laydown yard with quality release tags.
Labor & Specialized Craft: Qualified, badged craft labor available with required safety and craft certifications.
Tools & Heavy Plant: Cranes, rigging gear, scaffolding, and specialty tools mobilized, tested, and certified.
Work Area Access & Predecessor Handoff: Preceding trade handoffs formally signed off; area cleaned, barricaded, and physically accessible.
Permits & Safety Clearances: Hot work permits, confined space entry authorizations, excavation permits, and environmental clearances secured.
The Last Planner System (LPS) & Percent Plan Complete (PPC)
Developed by the Lean Construction Institute (LCI) and widely integrated into AACE project controls frameworks, the Last Planner System shifts planning focus from what should be done (CPM baseline) to what can be done (constraint-screened) to what will be done (committed field tasks).
At the end of each work week, the project controls team calculates the primary reliability metric, Percent Plan Complete (PPC) :
PPC = Number of Weekly Work Plan Tasks Completed 100% Total Number of Work Plan Tasks Committed × 100 % \text{PPC} = \frac{\text{Number of Weekly Work Plan Tasks Completed 100\%}}{\text{Total Number of Work Plan Tasks Committed}} \times 100\% PPC = Total Number of Work Plan Tasks Committed Number of Weekly Work Plan Tasks Completed 100% × 100%
Rules for PPC Measurement
The 100% Binary Rule: An activity is either 100% complete or it is counted as incomplete (0%). If a task is 95% complete on Friday afternoon, it receives a score of 0 in the PPC numerator. There is no partial credit.
Root Cause Variance Tracking: For every failed commitment, the project team must log the root cause (e.g., material delayed, drawing error, equipment breakdown, inclement weather, trade interference). Weekly tracking of variance root causes enables continuous improvement and eliminates systemic execution bottlenecks before they impact CPM milestone dates.
- Worked Examples: Productivity and Learning Curve Calculations
- Example 1: Productivity Index (PI) Calculation
A mechanical piping contractor is installing carbon steel process piping for a chemical refinery expansion.
Baseline Budget: 2,400 linear feet (LF) of pipe with an approved estimating norm of 0.75 labor-hours per LF . Total Budgeted Hours = 2 , 400 LF × 0.75 MH/LF = 1 , 800 labor-hours \text{Total Budgeted Hours} = 2,400\text{ LF} \times 0.75\text{ MH/LF} = 1,800\text{ labor-hours} Total Budgeted Hours = 2 , 400 LF × 0.75 MH/LF = 1 , 800 labor-hours
Progress Status at Data Date:
Physical installed quantity verified by QA/QC: 1,600 LF .
Actual direct craft labor-hours charged to the control account: 1,500 labor-hours .
Calculation Steps:
Calculate Planned Unit Rate: Planned Unit Rate = 0.75 MH/LF \text{Planned Unit Rate} = 0.75\text{ MH/LF} Planned Unit Rate = 0.75 MH/LF
Calculate Actual Unit Rate: Actual Unit Rate = 1 , 500 actual labor-hours 1 , 600 LF installed = 0.9375 MH/LF \text{Actual Unit Rate} = \frac{1,500\text{ actual labor-hours}}{1,600\text{ LF installed}} = 0.9375\text{ MH/LF} Actual Unit Rate = 1 , 600 LF installed 1 , 500 actual labor-hours = 0.9375 MH/LF
Calculate Earned Hours ( E V hours EV_{\text{hours}} E V hours ): Earned Hours = 1 , 600 LF × 0.75 MH/LF = 1 , 200 earned labor-hours \text{Earned Hours} = 1,600\text{ LF} \times 0.75\text{ MH/LF} = 1,200\text{ earned labor-hours} Earned Hours = 1 , 600 LF × 0.75 MH/LF = 1 , 200 earned labor-hours
Calculate Productivity Index (PI): Productivity Index (PI) = Planned Unit Rate Actual Unit Rate = 0.75 0.9375 = 0.80 \text{Productivity Index (PI)} = \frac{\text{Planned Unit Rate}}{\text{Actual Unit Rate}} = \frac{0.75}{0.9375} = \mathbf{0.80} Productivity Index (PI) = Actual Unit Rate Planned Unit Rate = 0.9375 0.75 = 0.80 Alternatively: PI = Earned Hours Actual Hours = 1 , 200 1 , 500 = 0.80 \text{Alternatively: } \text{PI} = \frac{\text{Earned Hours}}{\text{Actual Hours}} = \frac{1,200}{1,500} = \mathbf{0.80} Alternatively: PI = Actual Hours Earned Hours = 1 , 500 1 , 200 = 0.80
Diagnostic Conclusion: The Productivity Index is 0.80 (significantly below 1.0). The contractor is burning 25% more labor-hours per foot than estimated ( 0.9375 / 0.75 = 1.25 0.9375 / 0.75 = 1.25 0.9375/0.75 = 1.25 ). If uncorrected, the final piping scope will overrun by 450 labor-hours ( 1 , 800 / 0.80 = 2 , 250 forecast hours 1,800 / 0.80 = 2,250\text{ forecast hours} 1 , 800/0.80 = 2 , 250 forecast hours ). Immediate field supervision and craft tooling reviews are required.
Example 2: 85% Learning Curve for Modular Precast Elements
A contractor is fabricating 4 identical precast concrete bridge pier caps for an elevated highway viaduct. Historical estimating records establish an 85% learning curve ( L = 0.85 L = 0.85 L = 0.85 ).
Direct labor required to form, reinforce, and pour the first pier cap ( X = 1 X = 1 X = 1 ): 600 labor-hours .
Calculation Steps:
Calculate Learning Curve Exponent ( b b b ): b = ln ( 0.85 ) ln ( 2 ) = − 0.1625 0.6931 = − 0.2345 b = \frac{\ln(0.85)}{\ln(2)} = \frac{-0.1625}{0.6931} = -0.2345 b = l n ( 2 ) l n ( 0.85 ) = 0.6931 − 0.1625 = − 0.2345
Calculate Cumulative Average Hours for 2 Units (1st Doubling): Y 2 = 600 × 0.85 = 510 hours/unit Y_2 = 600 \times 0.85 = 510\text{ hours/unit} Y 2 = 600 × 0.85 = 510 hours/unit Total Hours for 2 Units = 2 × 510 = 1 , 020 total hours \text{Total Hours for 2 Units} = 2 \times 510 = 1,020\text{ total hours} Total Hours for 2 Units = 2 × 510 = 1 , 020 total hours Hours for Pier Cap 2 Alone = 1 , 020 − 600 = 420 hours \text{Hours for Pier Cap 2 Alone} = 1,020 - 600 = 420\text{ hours} Hours for Pier Cap 2 Alone = 1 , 020 − 600 = 420 hours
Calculate Cumulative Average Hours for 4 Units (2nd Doubling): Y 4 = 510 × 0.85 = 433.5 hours/unit Y_4 = 510 \times 0.85 = 433.5\text{ hours/unit} Y 4 = 510 × 0.85 = 433.5 hours/unit Total Hours for All 4 Pier Caps = 4 × 433.5 = 1 , 734 total labor-hours \text{Total Hours for All 4 Pier Caps} = 4 \times 433.5 = \mathbf{1,734\text{ total labor-hours}} Total Hours for All 4 Pier Caps = 4 × 433.5 = 1 , 734 total labor-hours
Conclusion: Instead of budgeting 2,400 hours ( 4 × 600 4 \times 600 4 × 600 without learning), the scheduler should load the 4-unit activity with 1,734 labor-hours , reflecting a realistic 27.75% efficiency gain over the repetitive production run.