7.3 Earned Schedule (ES) Theory, SV(t), and SPI(t)

7.3 Earned Schedule (ES) Theory, SV(t), and SPI(t)

AACE Professional Context: Traditional Earned Value schedule metrics ($SV$ and $SPI$) are value-denominated and converge toward their planned completion values, limiting their usefulness for interpreting time performance late in a project. In 2003, American quality and project controls engineer Walt Lipke published the seminal paper Schedule is Different, creating Earned Schedule (ES) theory. Earned Schedule is an established time-based extension of EVM, and AACE RP 80R-13 discusses its use in schedule forecasting. It supplies aggregate indicators that complement rather than replace CPM milestone and path analysis. Candidates benefit from understanding the interpolation, $SV(t)$, and $SPI(t)$ calculations and their assumptions.

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The Concept of Earned Schedule (ES)

The fundamental premise of Earned Schedule is remarkably elegant:

Core Principle: Instead of measuring schedule accomplishment along the vertical cost axis (in dollars), Earned Schedule measures accomplishment along the horizontal time axis (in calendar units).

Earned Schedule asks the question: "At what point in time was the currently achieved Earned Value originally planned to be accomplished?"

                         THE EARNED SCHEDULE (ES) DERIVATION

   Cumulative Cost (\$)
        ▲
        │                                            Planned Value (PV) S-Curve
        │                                                    ┌─────────*
        │                                                ┌───┘
        │                                            ┌───┘
   EV ──┼───────────────────────────────────────►*───┘
        │                                       ││
        │                                       ││
        │                                       ││
        │                                       ││
        └───────────────────────────────────────┴┴─────────────────────► Time
                                                ▲▲
                                                ││
                                               ES│
                                                 │
                                                 AT (Actual Time / Data Date)

         ◄───────────────── ES ─────────────────►
         ◄─────────────────────── AT ────────────────────────►
                                                 ◄── SV(t) ──►
                                                  (Delay in Time)

Core Definitions

1. Actual Time ($AT$): The elapsed calendar time from the project start date to the current Data Date (Status Date). $AT$ is expressed in time units such as days, weeks, or months (e.g., $AT = 8.0 \text{ months}$).
2. Earned Schedule ($ES$): The date or elapsed time increment on the baseline schedule when the cumulative Planned Value ($PV$) equals the current Earned Value ($EV$). $ES$ is expressed in the identical time units as $AT$ (e.g., $ES = 6.4 \text{ months}$).

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Mathematical Derivation and Linear Interpolation

Because project controls data is typically recorded in discrete periodic increments (e.g., weekly or monthly status periods), Earned Value ($EV$) rarely falls precisely on an exact integer time period of the Planned Value table. Therefore, linear interpolation is used to determine the exact fractional portion of $ES$.

The Earned Schedule Formula

$\mathbf{ES = C + I}$

Where:

• $C$ (Integer Time Index): The time increment where cumulative $PV$ is less than or equal to current $EV$, but where the next period's $PV$ exceeds $EV$: $PV_C \le EV < PV_{C+1}$
• $I$ (Fractional Interpolation Increment): The proportional distance traversed between period $C$ and period $C+1$: $\mathbf{I = \frac{EV - PV_C}{PV_{C+1} - PV_C}}$

Full Linear Interpolation Expression:

$\mathbf{ES = C + \left[ \frac{EV - PV_C}{PV_{C+1} - PV_C} \right]}$

Note: If $EV \ge BAC$ (project scope is complete), $ES$ equals the baseline Planned Duration ($PD$).

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Time-Based Schedule Metrics: $SV(t)$ and $SPI(t)$

With Earned Schedule ($ES$) established in pure units of time, Lipke defined time-based analogs to the traditional EVM schedule metrics:

1. Time-Based Schedule Variance: $SV(t)$

$\mathbf{SV(t) = ES - AT}$

• Denominated in units of time (calendar days, working days, weeks, or months).
• $SV(t) > 0$ (Positive): Favorable aggregate earned-schedule variance.
• $SV(t) = 0$ (Zero): Earned Schedule equals Actual Time.
• $SV(t) < 0$ (Negative): Unfavorable aggregate earned-schedule variance.
• Example: If $ES = 5.2 \text{ months}$ and $AT = 7.0 \text{ months}$, then: $SV(t) = 5.2 - 7.0 = \mathbf{-1.8 \text{ months}}$ The project controls manager can immediately inform project leadership: "Aggregate Earned Schedule variance is -1.8 months; CPM determines milestone impact." Compare this clarity against traditional EVM reporting: "Schedule Variance is -$142,000," which leaves stakeholders wondering how much time was lost.

2. Time-Based Schedule Performance Index: $SPI(t)$

$\mathbf{SPI(t) = \frac{ES}{AT}}$

• Represents the time efficiency factor of the project.
• $SPI(t) > 1.00$: Favorable. Project is converting calendar time into earned schedule at an accelerated rate.
• $SPI(t) = 1.00$: On schedule.
• $SPI(t) < 1.00$: Unfavorable. Project is experiencing time slippage (e.g., $SPI(t) = 0.75$ indicates the project is earning only 0.75 months of schedule for every 1.0 calendar month expended).

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Why $SPI(t)$ avoids traditional SPI convergence at late completion

The breakthrough achievement of Earned Schedule is solving the late-project mathematical distortion that plagues traditional $SPI$.

Recall from Section 7.2 that when a delayed project reaches final completion ($AT > PD$):

• Traditional $EV$ reaches $BAC$, and $PV = BAC$, forcing traditional $SPI = EV / PV = 1.00$.

Now, analyze the behavior of Earned Schedule under the identical late-project scenario:

• At project finish, all scope is complete, so $EV = BAC$.
• Because $EV = BAC$, the earned time on the baseline PV curve is the full Planned Duration: $ES = PD$.
• However, because the project was late, the elapsed Actual Time exceeds Planned Duration: $AT > PD$.
• Substituting these values into the time-based formulas:

$\mathbf{SV(t)_{\text{final}} = ES - AT = PD - AT_{\text{final}} < 0}$

$\mathbf{SPI(t)_{\text{final}} = \frac{ES}{AT_{\text{final}}} = \frac{PD}{AT_{\text{final}}} < 1.00}$

                  BEHAVIOR AT COMPLETION OF A DELAYED PROJECT
   (Planned Duration = 10 Months | Actual Completion Date = 14 Months | BAC = \$1.0M)

   Metric                     Traditional EVM                   Earned Schedule (ES)
  ──────────────────────────────────────────────────────────────────────────────────
   Schedule Variance          SV = EV - PV                     SV(t) = ES - AT
                              SV = \$1.0M - \$1.0M = \$0         SV(t) = 10 - 14 = -4.0 Months
                              (Falsely indicates on plan!)      (Correctly shows 4 months late!)

   Schedule Index             SPI = EV / PV                     SPI(t) = ES / AT
                              SPI = \$1.0M / \$1.0M = 1.00      SPI(t) = 10 / 14 = 0.714
                              (Falsely indicates 100% on time!) (Correctly preserves 71.4% efficiency!)

Key takeaway: At completion, a late project has ES equal to planned duration and AT equal to actual duration, so SPI(t) remains below 1.00 and SV(t) expresses the duration variance in the chosen time units. CPM remains necessary for milestone and path analysis.

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Step-by-Step Worked Calculation of $ES$, $SV(t)$, and $SPI(t)$

A municipal light rail infrastructure project has a contract baseline budget $BAC$ = $12,000,000 and a Planned Duration of 10 months ($PD = 10$).

Cumulative Baseline Planned Value Table (Monthly Periods):

Status at Data Date = End of Month 7 ($AT = 7.0$ months):

• The field survey confirms physical progress corresponding to an Earned Value ($EV$) of $6,300,000.
• The accounting system books an Actual Cost ($AC$) of $7,000,000.

Step 1: Identify Integer Time Increment $C$

Search the Cumulative $PV$ row to locate the interval that bounds $EV$ = $6,300,000:

• $PV_5$ = $5,200,000
• $PV_6$ = $7,400,000
• Since $PV_5$ ($5.2M) $\le EV$ ($6.3M) $< PV_6$ ($7.4M), the integer period is $C = 5$.

Step 2: Compute Fractional Interpolation Increment $I$

$I = \frac{EV - PV_C}{PV_{C+1} - PV_C} = \frac{\text{\textdollar}6{,}300{,}000 - \text{\textdollar}5{,}200{,}000}{\text{\textdollar}7{,}400{,}000 - \text{\textdollar}5{,}200{,}000} = \frac{\text{\textdollar}1{,}100{,}000}{\text{\textdollar}2{,}200{,}000} = \mathbf{0.50}$

Step 3: Compute Earned Schedule ($ES$)

$ES = C + I = 5 + 0.50 = \mathbf{5.50 \text{ months}}$ Physical Meaning: The work accomplished by the end of Month 7 was originally planned to be reached at Month 5.50.

Step 4: Compute Time-Based Schedule Variance ($SV(t)$)

$SV(t) = ES - AT = 5.50 - 7.00 = \mathbf{-1.50 \text{ months}}$ Physical Meaning: The project is 1.50 months behind schedule in elapsed calendar time.

Step 5: Compute Time-Based Schedule Performance Index ($SPI(t)$)

$SPI(t) = \frac{ES}{AT} = \frac{5.50}{7.00} = \mathbf{0.786}$ Physical Meaning: The project is progressing at 78.6% of the planned temporal schedule rate (earning 0.786 months of schedule per actual month worked).

Step 6: Comparison with Traditional EV Metrics

• Traditional Planned Value at Month 7: $PV_7$ = $9,400,000.
• Traditional Schedule Variance: $SV = EV - PV$ = $6.3M - $9.4M = -$3,100,000.
• Traditional Schedule Index: $SPI = EV / PV = 6.3 / 9.4 = 0.670$.
• Cost Performance Index: $CPI = EV / AC = 6.3 / 7.0 = 0.900$.

Analytical Insight: While traditional EVM reports an alarming -$3,100,000 variance, Earned Schedule reports an aggregate $SV(t)$ of -1.50 months and $SPI(t)$ of 0.786; the CPM network determines the forecast effect on milestones.