Engineering Finance


The last two lessons introduced the work 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 difficult.

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 the issue.

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.

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  • 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 of demand.

Section 4.4 describes Cost Estimating Relationships including the Power-Sizing Relationship. Section 9.3 describes the Learning Curve Relationship.

4.4 Developing the LCC Model
9.3 Effect of Learning


Point Estimates

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 estimate. Most 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 at costs.
    • 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 research studies.
  • Sources outside the firm
    • Cost estimation is important throughout the economy. In almost every engineering area there is published information, such 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 of sources.
    • Personal contacts, such as vendors, salespeople, customers, consultants and even competitors may be rich sources of information.
    • 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 is correct?

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 uncertainty.

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 of risk.

Cost Models

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.

Cost Estimating Relationships

Click the icon to see the questions and answers used in the presentation.

CER Problems


The following paragraphs summarize formulas for the relationships in the presentation.

Index Relationship

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.

U.S. Consumer Price Index Data

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.

CPI-U: Bureau of Labor Statistics


Factor Relationship

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.

Power-Sizing Relationship
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.
Automobile Example

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.

The 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 the CBS.

Automobile CBS

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 Costs

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 each.

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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Estimates Summary


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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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