ᎏᎏ X CA/U ⫻ X CO/EXCA/E/XCA/U
6.2 Cost Estimation Techniques
Cost estimation is the process of forecasting the present and future cash-flow consequences of engineer- ing designs and investments (Canada et al., 1996). The process is useful if it reduces the uncertainty sur- rounding a revenue or cost element. In doing this, a decision should result that creates increased value relative to the cost of making the estimate. Three groups of estimating have proven to be very useful in preparing estimates for economic analysis: time-series, subjective, and cost engineering techniques.
6.2.1 Time-Series Techniques
Time-series data are revenue and cost elements that are functions of time, e.g., unit sales per month and annual operating cost.
6.2.1.1 Correlation and Regression Analysis
Regression is a statistical analysis of fitting a line through data to minimize squared error. With linear regression, approximated model coefficients can be used to obtain an estimate of a revenue/cost element.
The relationship between x and y used to fit n data points (1ⱕiⱕn) is
y⫽a⫹bx (6.1)
where
x⫽independent variable y⫽dependent variable
x¯⫽average of independent variable y¯⫽average of dependent variable
The mathematical expressions used to estimate a and b in Equation (6.1) are b⫽
a⫽y¯⫺bx¯
(6.2)
The correlation coefficient is a measure of the strength of the relationship between two variables only if the variables are linearly related.
Let
r⫽ (⫺1ⱕrⱕ1) (6.3)
where
Sxy⫽冱i⫽1n xiyi⫺
Sxx⫽冱i⫽1n x2i⫺ (6.4)
Syy⫽冱n
i⫽1
y2i⫺ᎏᎏ冢冱ni⫽1yi冣2
n
冢冱ni⫽1xi冣2
ᎏᎏn
冢冱ni⫽1xi冣冢冱ni⫽1yi冣
ᎏᎏᎏn Sxy
ᎏSxxSyy
冱ni⫽1xiyi⫺x¯冱ni⫽1yi ᎏᎏᎏ冱ni⫽1xi2⫺x¯冱ni⫽1xi
r⫽the correlation coefficient and measures the degree of strength
r2⫽the coefficient of determination that measures the proportion of the total variation that is explained by the regression line
A positive value of r indicates that the independent and the dependent variables increase at the same rate. When r is negative, one variable decreases as the other increases. If there is no relationship between these variables, r will be zero.
6.2.1.2 Exponential Smoothing
An advantage of the exponential smoothing method compared with the simple linear regression for time- series estimates is that it permits the estimator to place relatively more weight on current data rather than treating all prior data points with equal importance. In addition, exponential smoothing is more sensitive to changes than linear regression. However, the basic assumption that trends and patterns of the past will continue into the future is a disadvantage. Hence, expert judgment should be used in interpreting results.
Let
St⫽α⬘xt⫹(1⫺α⬘)St⫺1 (0ⱕα⬘ ⱕ1) (6.5)
where
St⫽forecast for period t⫹1, made in period t α⫽smoothing constant
xt⫽the actual data point in period t
St⫺1⫽forecast for period t, made in period t⫺1
Intermediate choices for α⬘between 0 and 1 provide forecasts that have more or less emphasis on long- run average outcomes vs current outcomes.
6.2.2 Subjective Techniques
These techniques are subjective in nature; however, they are widely used in cases where the current invest- ment events are not well enough understood to apply cost engineering or time-series techniques.
6.2.2.1 The Delphi Method
This method is a progressive practice to develop consensus from different experts. The experts are usu- ally selected on the basis of their experience and visibility in the organization. The Delphi method is usu- ally both complex and poorly understood. In it, experts are asked to make forecasts anonymously and through an intermediary. The process involved can be summarized as follows (Canada et al., 1996):
● Each invited participant is given an unambiguous description of the forecasting problem and the necessary background information.
● The participants are asked to provide their estimates based on the presented problem scenarios.
● An interquartile range of the opinions is computed and presented to the participants at the begin- ning of the second round.
● The participants are asked in the second round to review their responses in the first round in rela- tion to the interquartile range from that round.
● The participants can, at this stage, request additional information. They may maintain or change their responses.
● If there is a significant deviation in opinion in the second round, a third-round questionnaire may be given to the participants. During this round, participants receive a summary of the second- round responses and a request to reconsider and explain their decisions in view of the second- round responses.
6.2.2.2 Technological Forecasting
This method can be used to estimate the growth and direction of a technology. It uses historical informa- tion of a selected technological parameter to extrapolate future trends. This method assumes that factors
that affect historical data will remain constant into the future. Some of the commonly predicted parame- ters are speed, horsepower, and weight. This method cannot predict accurately when there are unforeseen changes in technology interactions.
6.2.3 Cost Engineering Techniques
Cost engineering techniques are usually used for estimating investment and working capital parameters.
They can be easily applied because they make use of various cost/revenue indexes.
6.2.3.1 Unit Technique
This is the most popular of the cost engineering techniques. It uses an assumed or estimated “per unit”
factor such as, for example, maintenance cost per month. This factor is multiplied by the appropriate number of units to provide the total estimate. It is usually used for preliminary estimating.
6.2.3.2 Ratio Technique
This technique is used for updating costs through the use of a cost index over a period of time.
Let
Ct⫽C0冢 冣 (6.6)
where
Ct⫽estimated cost at present time t C0⫽cost at previous time t0 It ⫽index value at time t I0 ⫽index value at time t0 6.2.3.3 Factor Technique
This is an extension of the unit technique in which one sums the product of one or more quantities involving unit factors and adds these to any components estimated directly.
Let
C⫽冱Cd⫹冱fiUi (6.7)
where
C ⫽value estimated
Cd⫽cost of selected components estimated directly fi ⫽cost per unit of component i
Ui⫽number of units of component i 6.2.3.4 Exponential Costing
This is also called “cost-capacity equation.” It is good for estimating costs from design variables for equip- ment, materials, and construction. It recognizes that cost varies as some power of the change in capacity of size.
Let
CA⫽CB冢 冣X⫽CB冢 冣X冢 冣 (6.8)
where
CA⫽cost of plant A CB⫽cost of plant B SA⫽size of plant A SB⫽size of plant B
It ᎏI0 SA ᎏSB SA
ᎏSB
It ᎏI0
It⫽index value at time t I0⫽index value at time t0
X⫽cost-exponent factor, usually between 0.5 and 0.8
The accuracy of the exponential costing method depends largely on the similarity between the two projects and the accuracy of the cost-exponent factor. Generally, error ranges from ±10 to ±30% of the actual cost.
6.2.3.5 Learning Curves
In repetitive operations involving direct labor, the average time to produce an item or provide a service is typically found to decrease over time as workers learn their tasks better. As a result, cumulative average and unit times required to complete a task will drop considerably as output increases.
Let
Yi⫽Y1ib
b⫽ (6.9)
where
Y1⫽direct labor hours (or cost) for the first unit
Yi⫽direct labor hours (or cost) for the ith production unit i ⫽cumulative count of units of output
b ⫽learning curve exponent p ⫽learning rate percentage 6.2.3.6 A Range of Estimates
To reduce the uncertainties surrounding estimating future values, a range of possible values rather than a single value is usually more realistic. A range could include an optimistic estimate (O), the most likely estimate (M), and a pessimistic estimate (P). Hence, the estimated mean cost or revenue value can be estimated as (Badiru, 1996; Canada et al., 1996)
mean value⫽ (6.10)