The last two lessons introduced
breakdown schedule (WBS) and cost breakdown schedule (CBS)
for organizing the estimation of costs for projects and systems,
respectively. A question that remains to be answered is how
does one find estimates for the tasks or components at
the lowest level of these two schedules. For example, the
assembly line project has the WBS below. A top-down approach
will fill in the work package (WP) costs and use them to estimate the
cost of the project, but where do these costs come from?
Later on this page we use a wheel assembly example from a car to illustrate the computations. When the goal is to estimate
the part cost for the assembly alone, we would construct
the CBS shown below. For a bottom-up analysis, how do we
find the costs of the component parts?
Clearly, the estimation of the task costs or
component costs at the lowest level of the breakdown structures
is an important and difficult problem. The accuracy of these
estimates affects the accuracy of the analyses and ultimately
the quality of the decisions that are based on them. We use the
term cost here, but revenues and savings are also critical aspects
of many projects. Estimates of these factors may be even more
The response to questions of estimation accuracy
depends on the situation. For example, a college
graduate with several job offers in different parts of the country,
might want to include in her decision process the cost of housing
in the different locations. She could look on the web and find
a single estimate to represent each location, even though she
knows that the estimate is not fully accurate. Because there are a
variety of other considerations, that single value may be sufficient.
A more complex analysis would take time and likely confuse
Alternatively, consider a homeowner in Houston
deciding whether to flee from an impending hurricane churning
in the Gulf of Mexico. The weather service provides a chart that
gives an estimate of the location of landfall 50 miles
to the east of his home. Should he rest assured that his family
will be safe or should he leave town just in case the path
deviates a few degrees? If the homeowner could trust the
estimate, his decision would be easy. The uncertaintly, though, complicates the situation and he may make the wrong decision..
The response to risk depends on many factors such as our economic circumstances, our personal attitude towards risk, and the range of consequences that may result. It is often helpful to answer the following equestions when formulating a course of action:: How important
is the estimate to the decision that must be made? Can the decision
be put off until more information is available? Can we devise
solutions that will adjust to uncertain events after they occur?
Do we fear making risky decisions? Do we enjoy making risky decisions?
In most cases, the problem of obtaining and using
estimates does not have an obvious or single answer. It is central,
however, to obtaining valid planning models that are used
throughout business. Planning for the future without estimates
of what might happen is impossible. This section introduces some
methods and models that have been used for cost estimation. The
methods that use mathematical models are called cost
estimating relationships or CERs.
Be aware of resources available
to make point estimates of the costs associated with
the components of a project.
Where appropriate, use the CERs described in
the lesson to make estimates.
Use the Estimate add-in to
estimate the net system cost or revenue as a function
Section 4.4 describes Cost Estimating Relationships including
the Power-Sizing Relationship. Section 9.3 describes
the Learning Curve Relationship.
You are a young engineer in charge
of a WP for a project. The manager in charge of the larger task
that includes yours is seeking a cost estimate. He say's "Give
me a number. Just give me a number." How should you respond?
That manager is asking for a point
estimate, a single number to fill in a blank space on the
analysis worksheet. Sometimes we call this a deterministic
managers and most individuals are more comfortable with a point
estimate. It is easy to make a decision with one number, but
what does your manager do if you come up with three numbers,
or perhaps a whole range of numbers?
Some sources of data that you might look into are listed below.
Often these give only point estimates, but some may provide risk
estimates as well.
- Within the firm
- Accounting records may describe similar tasks.
- Formal bid procedures. Companies that sell services
through bids, often have formal procedures for arriving
- Persons in management, engineering,
sales, production, quality, purchasing, and personnel
may provide subjective estimates. The quality of the
estimates depends on experience on similar projects.
- New kinds of projects or components might require basic
- Sources outside the firm
- Cost estimation is important throughout the economy.
In almost every engineering area there is published
as technical directories, buyers indexes, U.S. government
publications, reference books, trade journals and catalogs.
A search on the web will provide links to a variety
- Personal contacts, such as vendors, salespeople, customers,
consultants and even competitors may be rich sources
- Specific items may be estimated by bids from suppliers.
That puts the estimation problem in their hands.
- Cost Estimating Relationships
- These are models based on statistics, empirical observations
or logic that relate costs to other variables in the
system. They might be constructed using internal or external
data and they might be developed internally or obtained
from external sources. We discuss some simple CERs below.
Of course, another technique is to subdivide your work
package into another level and put others in charge of making
estimates that you can use to calculate your own. Then you
can tell each of your subordinates, "Give me a number. Just
give me a number."
The cost and effort involved with obtaining an
estimate might determine how much trouble you want to put into
it. The sensitivity of the solution to the quality of
the estimate may also play a role. There's no point in spending
much time on something that doesn't really matter. How the company
rewards or punishes errors will play a role for subjective methods.
If you are ultimately charged with carrying out the WP and are
judged harshly for cost overruns, you will likely come up with
a conservatively large estimate for cost. On the other hand,
if your manager wants to win some business with a low estimate
for the project, he may put on pressure to have a low estimate
for the cost of your WP.
The time frame of the estimate also makes a difference.
All planning estimates involve future events, but some events
are further away in time than others. As an event becomes more
remote in time, the quality of an estimate is usually reduced,
but remoteness sometimes reduces the penalties for mistakes.
We will see later that the time value of money automatically
reduces the effects of expenditures and receipts far in the future.
We might also ask whether we plan to be around when the future
finally comes to pass. Who will bother to check if our estimate
There are many issues regarding how to make point
estimates, but they must be made when decisions involve factors
that are not known with certainty. Sometimes it is necessary
to make an immediate decision. Judging a competitive ice skater
in the Olympics requires a single score. Several judges make
point estimates, but the score is found by averaging or a more
complicated, but well defined, procedure. Most immediate decisions
require a number, even if that number carries a good deal of
Many of the models proposed for decision
making accept only point estimates. When risk is included, the
models become more complicated, and decisions based on them
are not easy. This course presents both point estimate, or deterministic,
models and models that explicitly consider estimates
One way to create at least the impression
of objectivity is to use models such as
CERs. The book provides some examples in section 4.4. Click
the icon to see a presentation on the subject. The presentation
refers to problems in the document with the link just below.
Open the document and stop the movie when a question is asked.
Try to do the problem and compare your answer with the solution
given in the presentation.
Click the icon to see the questions and answers
used in the presentation.
The following paragraphs summarize formulas for
the relationships in the presentation.
An index is a dimensionless number that
indicates how a cost or a price has changed with time
with respect to a base year.
A famous index that describes the affect
of inflation on consumer prices in the United States is
the Consumer Price Index (CPI). It is meaningless
to compare prices for an item at two different times without
correcting for the effect of inflation. The base year for
the current index is 1983.
The US Bureau of Labor Statistics maintains the CPI. A
text document with monthly values starting from 1913 is
at the the WWW link below. The "-U" in the title indicates
urban. There are several varieties of the CPI,
but the CPI-U is most often quoted.
To estimate a cost of a product consisting
of several components, one might use a factor relationship.
Each component may contribute a term that is independent
of the number of units of the component in the product
and a term that is proportional
to the number of units. The number of
units may measure a characteristic of the system such as
the number of rooms in a house. Our examples are linear
functions, but the factor relationship can include nonlinear
relationships between cost and the number of units.
When the relative costs of two facilities
are related to their relative sizes,
the costs may be described by the power-sizing technique.
When the exponent, x, is less than 1, the model
demonstrates what is called economies of scale. Facilities such as highways,
supermarkets, power plants and many others seem to exhibit
such a characteristi.
Learning Curve Relationship
The learning curve explains the phenomenon
of increased worker efficiency and improved performance
through repetitive production. In the case where resources
equate to time, as assumed in the definitions below, the time required to produce
a unit is reduced with each doubling of the number produced
by the factor s. See textbook for more detail.
The Estimate add-in uses a linear factor
relationship to model the cost of a product that
consists of multiple components. The product is described by
the CBS, as illustrated
in this section by an automobile example.
CBS can be used to estimate product costs for complex systems
consisting of thousands of parts. Certainly this is the case
for automobiles, which have approximately 14,000 parts. We illustrate
the counting feature with a wheel assembly pictured below. It
consists of a tire, hubcap, wheel, rotor, hub and lug nuts.
An automobile has a number of major systems, but we break
out the wheel assembly to illustrate the use of the numbers labeled units
specified as N1 through N4 in the CBS. Fictional costs are
assigned for illustration. Click the icon to open
We use N1 for the number of units produced for
each finished product. Although one would think this number should
be 1, when the process is imperfect, some units produced may
be scrapped. The yield measures this effect and may
have a value less than 1. N1 has the value 1/yield so, if the
yield is less than 1, N1 would be greater than 1.
N2 is the number of level 2 units required for
each unit at level 1. We see in the
example that there are four wheel assemblies for each car. N3
is the number of level 3 units for each unit at level 2.
We see that for each wheel assembly there is one hub assembly.
Finally, N4 is the number of lug nuts required for each hub assembly.
For this wheel, five lug nuts are required.
The yellow colored column labeled units is
the product of these four numbers. Thus we see that there are
20 lug nuts per car. When we assign the unit cost of
$2 for the lug nuts, the total item cost for that component
is $40 per car. The total cost column computes the cost
for 100 cars. The volume is specified at the top of the column.
The provision of unit numbers in the CBS simplifies the computation of product costs for complex
systems. The unit numbers multiply the variable cost but not
the fixed cost of an item.
Fixed and Variable
For an example with fixed costs, let's say that
a machine shop is thinking of building and selling wheel assemblies.
There is a cost of $2000 to set up for production and ordering
costs for tires, wheels, rotors and hub assemblies. These are
fixed costs, independent of the number the shop produces. The
variable costs for these items are the marginal costs per unit
produced. The shop hopes to sell the wheel assemblies for $400
General fixed and variable costs are illustrated
below. Costs that change with the production level are called
variable. Costs that that are independent of the level are fixed.
The data are shown in the CBS below. The shop is
manufacturing only the wheel assembly, so the CBS has only three
levels. The cell at the top right holds the production
volume that may be varied to illustrate the effect of volume
on total cost. Here we show the production and sale of 50 assemblies.
The summary for this solution shows a profit of
$5750 on sales of 50 units with a total revenue of $20,000.
The profit depends a great deal on the quantity
of the assemblies produced and sold. The chart below shows that
with 0 sales there is a loss of $3000. This is the total of the
fixed costs. As sales (and corresponding production) increase
the profit increases linearly. The breakeven point is about
17 sales. This is the point of zero profit.
Of course, one does not expect the profit to grow
at a linear rate for all values of sales because the
market is limited. More realistically, we get an interesting variation when the unit
revenue decreases with sales giving rise to a profit
function that is nonlinear and concave. In such a case, the profit would eventually reach a peak value and then decline.
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