Two Approaches
The above loglog relationship can be expressed mathematically as follows.
The cumulative average time (or cost) for each of 'n' units up
to the 'n'th unit, when plotted against the number of units on loglog paper,
produces a straight line.
This may be referred to as the "Log Linear  Cumulative Average Approach"
(The LLCA Model). This relationship is useful in forecasting or comparing similar
operations but with significantly different numbers of units involved. It is
also useful in analyzing large amounts of data as, for example, the records of
a large number of units produced from a precasting yard. This is because the
cumulative average curve has considerable power to smooth out the unit data.
It can also be deceptive because this power increases as the quantity increases
(Thomas 1986). It is, therefore, less useful for examining the expectations for
individual units or the latest unit such as would be needed in tracking actual
progress on a construction site.
This has led to a variation of the first relationship which states as follows.
The time (or cost) of the 'n'th unit, when plotted against the
number of units on loglog paper, produces a straight line.
This may similarly be referred to as the "Log Linear  Unit Approach"
(The LLU Model) (Drewin 1982; DSMC 1989). The mathematics of both models are
developed and compared in Appendix 2. Table
2 shows calculations of the time to the nth unit and the time
of the nth unit over a range from one to fifty units for ratios
ranging from 70% to 95% as determined by each approach. The Cumulative Average
figures are shown on white background, while the corresponding Cumulative Unit
figures are shaded. As might be expected, the results of the two approaches are
similar but not identical. The differences in results obtained from the two approaches
vary from about 7% for a repetition of only five units at a 95% productivity
ratio to over 100% for 50 units at a ratio of 70%.
Lp = r

0.950

0.900

0.850

s=logr/log2

0.074

0.152

0.234


CumAv

CumUnit

CumAv

CumUnit

CumAv

CumUnit

n

Tn/U1

Un/U1

Tn/U1

U'n/U1

Tn/U1

Un/U1

Tn/U1

U'n/U1

Tn/U1

Un/U1

Tn/U1

U'n/U1


1

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

5

4.4

0.829

4.7

0.888

3.9

0.675

4.4

0.783

3.4

0.538

4.2

0.686

10

8.4

0.784

9.0

0.843

7.0

0.602

8.1

0.705

5.8

0.452

7.3

0.583

15

12.3

0.760

13.2

0.818

9.9

0.565

11.5

0.663

7.9

0.409

10.1

0.530

20

16.0

0.743

17.2

0.801

12.7

0.540

14.8

0.634

9.9

0.382

12.6

0.495

25

19.7

0.731

21.2

0.788

15.3

0.521

17.9

0.613

11.8

0.362

15.0

0.470

30

23.3

0.721

25.1

0.777

17.9

0.507

20.9

0.596

13.5

0.346

17.3

0.450

35

26.9

0.713

29.0

0.769

20.4

0.495

23.9

0.583

15.2

0.334

19.6

0.434

40

30.4

0.705

32.8

0.761

22.8

0.485

26.7

0.571

16.8

0.323

21.7

0.421

45

34.0

0.699

36.6

0.755

25.2

0.476

29.6

0.561

18.4

0.314

23.8

0.410

50

37.4

0.694

40.3

0.749

27.6

0.469

32.4

0.552

20.0

0.307

25.8

0.400

Lp = r

0.800

0.750

0.700

s=logr/log2

0.322

0.415

0.515


CumAv

CumUnit

CumAv

CumUnit

CumAv

CumUnit

n

Tn/U1

Un/U1

Tn/U1

U'n/U1

Tn/U1

Un/U1

Tn/U1

U'n/U1

Tn/U1

Un/U1

Tn/U1

U'n/U1


1

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

1.0

1.000

5

3.0

0.418

3.9

0.596

2.6

0.314

3.7

0.513

2.2

0.224

3.4

0.437

10

4.8

0.329

6.6

0.477

3.8

0.230

5.9

0.385

3.1

0.152

5.2

0.306

15

6.3

0.287

8.8

0.418

4.9

0.193

7.6

0.325

3.7

0.123

6.6

0.248

20

7.6

0.261

10.8

0.381

5.8

0.171

9.2

0.288

4.3

0.105

7.8

0.214

25

8.9

0.242

12.6

0.355

6.6

0.155

10.5

0.263

4.8

0.094

8.8

0.191

30

10.0

0.228

14.3

0.335

7.3

0.144

11.8

0.244

5.2

0.085

9.7

0.174

35

11.1

0.217

16.0

0.318

8.0

0.135

13.0

0.229

5.6

0.078

10.5

0.160

40

12.2

0.208

17.5

0.305

8.7

0.127

14.1

0.216

6.0

0.073

11.3

0.150

45

13.2

0.200

19.0

0.294

9.3

0.121

15.1

0.206

6.3

0.069

12.0

0.141

50

14.2

0.193

20.5

0.284

9.9

0.116

16.1

0.197

6.7

0.065

12.7

0.134


















= CumAv Approach



= CumUnit Approach

Table 2: Comparison of Cum. Av. and Cum. Unit Productivity from 70% to 90%
In practice, one would select one approach or the other depending on the objective,
and use the corresponding set of ratios. It does mean, however, that
When comparing the learning ratios on different jobs or of different
crews on similar work, the method of calculation must be the same and it must
be specified.
