This paper was first published in the Canadian Journal of Civil Engineering, Vol. 21, 1994 pp 939-953, under the title "A Pragmatic Approach to Using Resource Loading, Production and Learning Curves on Construction Projects". It has been modified only to the extent necessary to make it presentable in web page format. Published here October, 2001.

## Two Approaches

The above log-log 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 log-log paper, produces a straight line.

This may be referred to as the "Log Linear - Cumulative Average Approach" (The LL-CA 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 log-log paper, produces a straight line.

This may similarly be referred to as the "Log Linear - Unit Approach" (The LL-U 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 Cum-Av Cum-Unit Cum-Av Cum-Unit Cum-Av Cum-Unit 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 Cum-Av Cum-Unit Cum-Av Cum-Unit Cum-Av Cum-Unit 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 = Cum-Av Approach = Cum-Unit 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.