5.5 Unit Cost Method of Estimation

For design estimates, the unit cost method is commonly used when the project is decomposed into elements at various levels of a hierarchy as follows:

1. Preliminary Estimates. The project is decomposed into major structural systems or production equipment items, e.g. the entire floor of a building or a cooling system for a processing plant.
2. Detailed Estimates. The project is decomposed into components of various major systems, i.e., a single floor panel for a building or a heat exchanger for a cooling system.
3. Engineer's Estimates. The project is decomposed into detailed items of various components as warranted by the available cost data. Examples of detailed items are slabs and beams in a floor panel, or the piping and connections for a heat exchanger.

1. Subcontractor Quotations. The decomposition of a project into subcontractor items for quotation involves a minimum amount of work for the general contractor. However, the accuracy of the resulting estimate depends on the reliability of the subcontractors since the general contractor selects one among several contractor quotations submitted for each item of subcontracted work.
2. Quantity Takeoffs. The decomposition of a project into items of quantities that are measured (or taken off) from the engineer's plan will result in a procedure similar to that adopted for a detailed estimate or an engineer's estimate by the design professional. The levels of detail may vary according to the desire of the general contractor and the availability of cost data.
3. Construction Procedures. If the construction procedure of a proposed project is used as the basis of a cost estimate, the project may be decomposed into items such as labor, material and equipment needed to perform various tasks in the projects.

Simple Unit Cost Formula

Suppose that a project is decomposed into n elements for cost estimation. Let Qi be the quantity of the ith element and ui be the corresponding unit cost. Then, the total cost of the project is given by:

(5.7)

where n is the number of units. Based on characteristics of the construction site, the technology employed, or the management of the construction process, the estimated unit cost, ui for each element may be adjusted.

Factored Estimate Formula

A special application of the unit cost method is the "factored estimate" commonly used in process industries. Usually, an industrial process requires several major equipment components such as furnaces, towers drums and pump in a chemical processing plant, plus ancillary items such as piping, valves and electrical elements. The total cost of a project is dominated by the costs of purchasing and installing the major equipment components and their ancillary items. Let Ci be the purchase cost of a major equipment component i and fi be a factor accounting for the cost of ancillary items needed for the installation of this equipment component i. Then, the total cost of a project is estimated by:

(5.8)

where n is the number of major equipment components included in the project. The factored method is essentially based on the principle of computing the cost of ancillary items such as piping and valves as a fraction or a multiple of the costs of the major equipment items. The value of Ci may be obtained by applying the exponential rule so the use of Equation (5.8) may involve a combination of cost estimation methods.

Formula Based on Labor, Material and Equipment

Consider the simple case for which costs of labor, material and equipment are assigned to all tasks. Suppose that a project is decomposed into n tasks. Let Qi be the quantity of work for task i, Mi be the unit material cost of task i, Ei be the unit equipment rate for task i, Li be the units of labor required per unit of Qi, and Wi be the wage rate associated with Li. In this case, the total cost y is:

(5.9)

Note that WiLi yields the labor cost per unit of Qi, or the labor unit cost of task i. Consequently, the units for all terms in Equation (5.9) are consistent.

Example 5-7: Decomposition of a building foundation into design and construction elements.

The concept of decomposition is illustrated by the example of estimating the costs of a building foundation excluding excavation as shown in Table 5-5 in which the decomposed design elements are shown on horizontal lines and the decomposed contract elements are shown in vertical columns. For a design estimate, the decomposition of the project into footings, foundation walls and elevator pit is preferred since the designer can easily keep track of these design elements; however, for a bid estimate, the decomposition of the project into formwork, reinforcing bars and concrete may be preferred since the contractor can get quotations of such contract items more conveniently from specialty subcontractors.

TABLE 5-5  Illustrative Decomposition of Building Foundation Costs

Example 5-8: Cost estimate using labor, material and equipment rates.

For the given quantities of work Qi for the concrete foundation of a building and the labor, material and equipment rates in Table 5-6, the cost estimate is computed on the basis of Equation (5.9). The result is tabulated in the last column of the same table.

TABLE 5-6  Illustrative Cost Estimate Using Labor, Material and Equipment Rates

5.6 Methods for Allocation of Joint Costs

One common application is found in the allocation of field supervision cost among the basic costs of various elements based on labor, material and equipment costs, and the allocation of the general overhead cost to various elements according to the basic and field supervision cost. Suppose that a project is decomposed into n tasks. Let y be the total basic cost for the project and yi be the total basic cost for task i. If F is the total field supervision cost and Fi is the proration of that cost to task i, then a typical proportional allocation is:

(5.10)

Similarly, let z be the total direct field cost which includes the total basic cost and the field supervision cost of the project, and zi be the direct field cost for task i. If G is the general office overhead for proration to all tasks, and Gi is the share for task i, then

(5.11)

Finally, let w be the grand total cost of the project which includes the direct field cost and the general office overhead cost charged to the project and wi be that attributable task i. Then,

(5.12)

and

(5.13)

Example 5-9: Prorated costs for field supervision and office overhead

If the field supervision cost is $13,245 for the project in Table 5-6 (Example 5-8) with a total direct cost of $88,300, find the prorated field supervision costs for various elements of the project. Furthermore, if the general office overhead charged to the project is 4% of the direct field cost which is the sum of basic costs and field supervision cost, find the prorated general office overhead costs for various elements of the project.

For the project, y = $88,300 and F = $13,245. Hence:

z = 13,245 + 88,300 = $101,545
G = (0.04)(101,545) = $4,062
w = 101,545 + 4,062 = $105,607

The results of the proration of costs to various elements are shown in Table 5-7.

TABLE 5-7  Proration of Field Supervision and Office Overhead Costs

Example 5-10: A standard cost report for allocating overhead

The reliance on labor expenses as a means of allocating overhead burdens in typical management accounting systems can be illustrated by the example of a particular product's standard cost sheet. Table 5-8 is an actual product's standard cost sheet of a company following the procedure of using overhead burden rates assessed per direct labor hour. The material and labor costs for manufacturing a type of valve were estimated from engineering studies and from current material and labor prices. These amounts are summarized in Columns 2 and 3 of Table 5-8. The overhead costs shown in Column 4 of Table 5-8 were obtained by allocating the expenses of several departments to the various products manufactured in these departments in proportion to the labor cost. As shown in the last line of the table, the material cost represents 29% of the total cost, while labor costs are 11% of the total cost. The allocated overhead cost constitutes 60% of the total cost. Even though material costs exceed labor costs, only the labor costs are used in allocating overhead. Although this type of allocation method is common in industry, the arbitrary allocation of joint costs introduces unintended cross subsidies among products and may produce adverse consequences on sales and profits. For example, a particular type of part may incur few overhead expenses in practice, but this phenomenon would not be reflected in the standard cost report.

TABLE 5-8  Standard Cost Report for a Type of Valve

Source: H. T. Johnson and R. S. Kaplan, Relevance lost: The Rise and Fall of Management Accounting, Harvard Business School Press, Boston. Reprinted with permission.

5.7 Historical Cost Data

Construction cost data are published in various forms by a number of organizations. These publications are useful as references for comparison. Basically, the following types of information are available:

Since the future prices of constructed facilities are influenced by many uncertain factors, it is important to recognize that this risk must be borne to some degree by all parties involved, i.e., the owner, the design professionals, the construction contractors, and the financing institution. It is to the best interest of all parties that the risk sharing scheme implicit in the design/construct process adopted by the owner is fully understood by all. When inflation adjustment provisions have very different risk implications to various parties, the price level changes will also be treated differently for various situations.

5.8 Cost Indices

A price index is a weighted aggregate measure of constant quantities of goods and services selected for the package. The price index at a subsequent year represents a proportionate change in the same weighted aggregate measure because of changes in prices. Let lt be the price index in year t, and lt+1 be the price index in the following year t+1. Then, the percent change in price index for year t+1 is:

(5.14)

or

(5.15)

If the price index at the base year t=0 is set at a value of 100, then the price indices l1, l2...ln for the subsequent years t=1,2...n can be computed successively from changes in the total price charged for the package of goods measured in the index.

The best-known indicators of general price changes are the Gross Domestic Product (GDP) deflators compiled periodically by the U.S. Department of Commerce, and the consumer price index (CPI) compiled periodically by the U.S. Department of Labor. They are widely used as broad gauges of the changes in production costs and in consumer prices for essential goods and services. Special price indices related to construction are also collected by industry sources since some input factors for construction and the outputs from construction may disproportionately outpace or fall behind the general price indices. Examples of special price indices for construction input factors are the wholesale Building Material Price and Building Trades Union Wages, both compiled by the U.S. Department of Labor. In addition, the construction cost index and the building cost index are reported periodically in the Engineering News-Record (ENR). Both ENR cost indices measure the effects of wage rate and material price trends, but they are not adjusted for productivity, efficiency, competitive conditions, or technology changes. Consequently, all these indices measure only the price changes of respective construction input factors as represented by constant quantities of material and/or labor. On the other hand, the price indices of various types of completed facilities reflect the price changes of construction output including all pertinent factors in the construction process. The building construction output indices compiled by Turner Construction Company and Handy-Whitman Utilities are compiled in the U.S. Statistical Abstracts published each year.

Figure 5-7 and Table 5-9 show a variety of United States indices, including the Gross National Product (GNP) price deflator, the ENR building index, the Handy Whitman Utilities Buildings, and the Turner Construction Company Building Cost Index from 1970 to 1998, using 1992 as the base year with an index of 100.

TABLE 5-9  Summary of Input and Output Price Indices, 1970-1998

Note: Index = 100 in base year of 1992.

Figure 5-7  Trends for US price indices.

Figure 5-8  Price and cost indices for construction.

Since construction costs vary in different regions of the United States and in all parts of the world, locational indices showing the construction cost at a specific location relative to the national trend are useful for cost estimation. ENR publishes periodically the indices of local construction costs at the major cities in different regions of the United States as percentages of local to national costs.

When the inflation rate is relatively small, i.e., less than 10%, it is convenient to select a single price index to measure the inflationary conditions in construction and thus to deal only with a single set of price change rates in forecasting. Let jt be the price change rate in year t+1 over the price in year t. If the base year is denoted as year 0 (t=0), then the price change rates at years 1,2,...t are j1,j2,...jt, respectively. Let At be the cost in year t expressed in base-year dollars and At' be the cost in year t expressed in then-current dollars. Then:

(5.16)

Conversely

(5.17)

If the prices of certain key items affecting the estimates of future benefits and costs are expected to escalate faster than the general price levels, it may become necessary to consider the differential price changes over and above the general inflation rate. For example, during the period between 1973 through 1979, it was customary to assume that fuel costs would escalate faster than the general price levels. With hindsight in 1983, the assumption for estimating costs over many years would have been different. Because of the uncertainty in the future, the use of differential inflation rates for special items should be judicious.

Future forecasts of costs will be uncertain: the actual expenses may be much lower or much higher than those forecasted. This uncertainty arises from technological changes, changes in relative prices, inaccurate forecasts of underlying socioeconomic conditions, analytical errors, and other factors. For the purpose of forecasting, it is often sufficient to project the trend of future prices by using a constant rate j for price changes in each year over a period of t years, then

(5.18)

and

(5.19)

Estimation of the future rate increase j is not at all straightforward. A simple expedient is to assume that future inflation will continue at the rate of the previous period:

(5.20)

(5.21)

More sophisticated forecasting models to predict future cost increases include corrections for items such as economic cycles and technology changes.

Example 5-12: Changes in highway and building costs

Table 5-10 shows the change of standard highway costs from 1940 to 1990, and Table 5-11 shows the change of residential building costs from 1970 to 1990. In each case, the rate of cost increase was substantially above the rate of inflation in the decade of the 1970s.. Indeed, the real cost increase between 1970 and 1980 was in excess of three percent per year in both cases. However, these data also show some cause for optimism. For the case of the standard highway, real cost decreases took place in the period from l970 to l990. Unfortunately, comparable indices of outputs are not being compiled on a nationwide basis for other types of construction.

TABLE 5-10  Comparison of Standard Highway Costs, 1940-1990

Source: Statistical Abstract of the United States. GDP deflator is used for the price deflator index.

TABLE 5-11  Comparison of Residential Building Costs, 1970-1990

Source: Statistical Abstract of the United States. GNP deflator is used for the price deflator index.

5.9 Applications of Cost Indices to Estimating

The general conditions for the application of the single parameter cost function for screening estimates are:

1. Exclude special local conditions in historical data
2. Determine new facility cost on basis of specified size or capacity (using the methods described in Sections 5.3 to 5.6)
3. Adjust for inflation index
4. Adjust for local index of construction costs
5. Adjust for different regulatory constraints
6. Adjust for local factors for the new facility

Example 5-13: Screening estimate for a refinery

The total construction cost of a refinery with a production capacity of 200,000 bbl/day in Gary, Indiana, completed in 2001 was $100 million. It is proposed that a similar refinery with a production capacity of 300,000 bbl/day be built in Los Angeles, California, for completion in 2003. For the additional information given below, make an order of magnitude estimate of the cost of the proposed plant.

1. In the total construction cost for the Gary, Indiana, plant, there was an item of $5 million for site preparation which is not typical for other plants.
2. The variation of sizes of the refineries can be approximated by the exponential rule, Equation (5.4), with m = 0.6.
3. The inflation rate is expected to be 8% per year from 1999 to 2003.
4. The location index was 0.92 for Gary, Indiana and 1.14 for Los Angeles in 1999. These indices are deemed to be appropriate for adjusting the costs between these two cities.
5. New air pollution equipment for the LA plant costs $7 million in 2003 dollars (not required in the Gary plant).
6. The contingency cost due to inclement weather delay will be reduced by the amount of 1% of total construction cost because of the favorable climate in LA (compared to Gary).

On the basis of the above conditions, the estimate for the new project may be obtained as follows:

1. Typical cost excluding special item at Gary, IN is

   $100 million - $5 million = $ 95 million

2. Adjustment for capacity based on the exponential law yields

   ($95)(300,000/200,000)0.6 = (95)(1.5)0.6 = $121.2 million

3. Adjustment for inflation leads to the cost in 2003 dollars as

   ($121.2)(1.08)4 = $164.6 million

4. Adjustment for location index gives

   ($164.6)(1.14/0.92) = $204.6 million

5. Adjustment for new pollution equipment at the LA plant gives

   $204.6 + $7 = $211.6 million

6. Reduction in contingency cost yields

   ($211.6)(1-0.01) = $209.5 million

Since there is no adjustment for the cost of construction financing, the order of magnitude estimate for the new project is $209.5 million.

In making a preliminary estimate of a chemical processing plant, several major types of equipment are the most significant parameters in affecting the installation cost. The cost of piping and other ancillary items for each type of equipment can often be expressed as a percentage of that type of equipment for a given capacity. The standard costs for the major equipment types for two plants with different daily production capacities are as shown in Table 5-12. It has been established that the installation cost of all equipment for a plant with daily production capacity between 100,000 bbl and 400,000 bbl can best be estimated by using linear interpolation of the standard data.

TABLE 5-12  Cost Data for Equipment and Ancillary Items

A new chemical processing plant with a daily production capacity of 200,000 bbl is to be constructed in Memphis, TN in four years. Determine the total preliminary cost estimate of the plant including the building and the equipment on the following basis:

1. The installation cost for equipment was based on linear interpolation from Table 5-12, and adjusted for inflation for the intervening four years. We expect inflation in the four years to be similar to the period 1990-1994 and we will use the GNP Deflator index.
2. The location index for equipment installation is 0.95 for Memphis, TN, in comparison with the standard cost.
3. An additional cost of $500,000 was required for the local conditions in Memphis, TN.

1. The costs of the equipment and ancillary items for a plant with a capacity of 200,000 bbl can be estimated by linear interpolation of the data in Table 5-12, and the results are shown in Table 5-13.
   

TABLE 5-13  Results of Linear Interpolation for an Estimation Example