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Concept: When work is done out of sequence, knowledge gaps are inevitable. To bridge the gaps, performers must make assumptions about missing inputs. At some point, the missing information becomes available, and when it does, it often differs from the assumptions. That means what has already been done must be reworked.

Rework is doubly wasteful. First, and obviously, re-doing what has already been done is lost time and money—time that could be spent completing new deliverables. Second, and less obviously, the time spent making the assumptions is also lost time and money—it too could have been spent on completing new deliverables. In both cases, performers feel frustration with a work process that is the exact opposite of “one and done”. They want to stop wasting time.

Knowing rework will be required and even knowing which tasks are likely to cause rework are important but only part of what you need to know to stop losing time.  What’s missing is a fix on the impact of rework.

Figure 1

Practice: One key impact of rework is on cost. The Schedule Adherence Index (SAI) offers a way to forecast the cost of rework. The first step is to approximate the amount of rework, specifically the rework created within a single period. A mathematical model is used to make the estimate. [1]

Let SAI₁ and SAI₂ indicate the fraction of work that is likely to require revision at the end of two successive periods. In addition, let C₁ and C₂ represent the fraction of work that is complete at the end of the same two periods. How much rework is incurred during the intervening period?

The amount can be determined using geometry. First, express SAI1, SAI2, C1, and C2 as successive points on a chart (see Figure 2).

Figure 2

The amount of rework equals the area under the curve (the shaded portion of Figure 2). Why think that geometry gives us that information? Here’s an example that offers a rationale.

Say that a car is travelling at a steady speed of 80 kilometers per hour. In a quarter of an hour, how far has it gone? The calculation is straightforward: velocity * time. That is, 80 kph * .25 hrs is 20 km. The answer is the product of the two terms.

Now, let’s express the case graphically (see Figure 3). Say that the x-axis represents elapsed time, and the y-axis represents velocity. In the graph, the velocity is a constant 80 kph, and the time is a quarter of an hour. The area under the curve is a rectangle. So, its area is Length times Width, i.e., 80 kph * .25 hrs for 20 km. Thus, the area under the curve is the product of the same two terms with the same result.

Figure 3

The example suggests that if a result is the product of two variables, they can be represented as the dimensions of a shape with a known formula for area.

Now, let’s apply this logic to the periodic amount of rework. It is the product of two terms related to the period: (1) the fraction of work likely to require revision, and (2) the total amount of work completed in the period.

Given that SAI rarely stays the same from one period to the next, the area under the curve is generally a more complex shape than in Figure 3. It can still be broken into simple geometric shapes, namely, a triangle and a rectangle (see dotted lines in Figure 4), and the same principle can be applied.

Figure 4

For the amount of rework in a period, the area of the rectangle (Width times Length) is given by the following formula:

Equation 18

…where Width is defined as the fraction complete and Length as the amount of SAI₁.

For the amount of rework in a period, the area of the triangle (half of the Base times the Height) is given by the following equation:

Equation 19

…where the Base is defined as the fraction complete and Height as the difference between SAI₂ and SAI₁. [2]

The sum of the two areas gives the area under the curve.

That sum, in turn, reduces to the following equation:

Equation 20  [3]

…and that is the formula for the area of a trapezoid, ((1/2 * (a+b)) * h. Hence, this can be considered as a special case of the trapezoidal approximation
technique.

Given that the amount of rework equals the area under the curve, the cost of rework for the period can now be calculated. That is the topic of the next post.

Appendix:

Definitions:

A_rect= Area of the rectangle

A_tri= Area of the triangle

A_sum= Area of the sum of the A_{rect and} A_tri

Derivation:

Use Equation 18 for the area of the rectangle:

A_rect= ((C2-C1)* SAI₁

Use Equation 19 for the area of the triangle:
Area_tri = ½ * ((C₂-C₁) * (SAI₂-SAI₁))

Sum the two terms:

Area_sum = ((C₂-C₁) * SAI₁) + (1/2 * ((C₂-C₁)*(SAI₂-SAI₁)))

Factor out (C₂-C₁):

Area_sum = (C₂-C₁) * ((1 * SAI₁) + (½ * 1 * (SAI₂-SAI₁)))

Simplify the result:

Area_sum = (C₂-C₁) * (SAI₁ + (½ * (SAI₂ - SAI₁)))

Multiply both sides of the equation by 2:

2 * Area_sum = 2 * ((C₂-C₁) * (SAI₁ + (½ * (SAI₂ - SAI₁))))

Expand the multiplication on the right side:

2 * Area_sum = (2 * (C₂-C₁) * SAI₁) + (2 * (C₂-C₁) * ½ * SAI₂) - (2 * (C₂-C₁) * ½ * SAI₁)

Cancel like terms and simplify:

2 * Area_sum = (2 * (C₂-C₁) * SAI₁) + ((C₂-C₁) * SAI₂) - ((C₂-C₁) * SAI₁)

Factor out (C₂-C₁):

2 * Area_sum = (C₂-C₁) * ((2 * SAI₁) + SAI₂ - SAI₁)

Subtract the redundant term:

2 * Area_sum = (C₂-C₁) * (SAI₁ + SAI₂)

Expand the multiplication on the right side:

2 * Area_sum = ((C₂-C₁) * SAI₁) + ((C₂-C₁) * SAI₂)

Multiply both sides of the equation by ½ and cancel like terms:

Area_sum = (½ * (C₂-C₁) * SAI₁) + (½ * (C₂-C₁) * SAI₂)

Factor out (C₂-C₁):

Area_sum = (C₂-C₁) * ((½ * SAI₁) +   (½ *  SAI₂))

Factor out ½:

Area_sum = (C₂-C₁) * (½ * (SAI₁ + SAI₂))

Rearrange the terms:

Area_sum =  (½ * (SAI₁ + SAI₂)) * (C₂-C₁)

QED

Notes:

[1] This is the second mathematical model introduced in association with rework. Previously, a model was proposed to estimate the total amount of rework. That model, in turn, was used to define the Schedule Adherence Index. For a description of the previous model, click here. For the derivation of SAI, click here.