Parametric Cost Estimation

Parametric cost estimates are usually based on mathematical equations or models. Simple mathematical relationships such as linear and nonlinear regression are mainly utilized. Often they are based on historical data from like projects. Tools such as Excel can be utilized easily to fit these relationships. The biggest challenge is determining the relationship between the dependent and independent variables and their range of usefulness.

Drawing Cash Flow Diagrams with a Spreadsheet.

One simple way to draw cash flow diagrams with "arrows" proportional to the size of the cash flows is to use a spreadsheet to draw a stacked bar chart. The data for the cash flows is entered, as shown in the table part of Figure 2-8. Tomake a quick graph, select cells B1 to D8, which are the three columns of the cash flow.Then select the graph menu and choose column chart and select the stack option. Except for labeling axes (using the cells for year 0 to year 6), choosing the scale for the y axis, and adding titles, the cash flowdiagram is done. Refer to the appendix for a review of basic spreadsheet use. (Note: a bar chart labels
periods rather than using an x axis with arrows at times 0, 1, 2 ...)

FIGURE 2-8 Example of cash flow diagram in spreadsheets.

Drawing a Cash Flow Diagram.

The cash flowdiagram shows when all cash flows occur. Look at Figure 2-7 and the $100 positive cash flowat the end of period 2. Fromthe time line one can see that this cash flow can also be described as occurring at the beginning of period 3. Thus, in a CFD the end of period t is the saI}letime as the beginning of period t + 1. Beginning-of-period cash flows (such as rent, lease, and insurance payments) are thus easy to handle: just draw your CFD and put them in where they occur.Thus O&M, salvages, revenues, and overhauls are assumed to be end-of-period cash flows.

The choice of time 0 is arbitrary. For example, it can be when a project is analyzed, when funding is approved, or when construction begins. When construction periods are assumed to be short; first costs are assumed to occur at time 0, and the first annual revenues and costs start at the end of the first period.When construction periods are long, time 0 is usually the date of commissioning-when the facility comes on stream.

 

FIGURE 2-7 An example cash flow diagram(CFD).

Perspective is also important when one is drawing a CFD. Consider the simple transaction of paying $5000 for some equipment. To the firm buying the equipment, the cash flow is a cost and hence negative in sign. To the firm selling the equipment, the cash flow is a revenue and positive in sign. This simple example shows that a consistent perspective is required when one is using a CFD to model the cash flows of a problem. One person's cash outflowis another person's inflow.

Often two or more cash flows occur in the same year, such as an overhaul and an O&M expense or the salvage value and the last year's O&M expense. Combining these into one total cash flowper year would simplify the cash flowdiagram. However, it is better to show each individually, to ensure a clear connection from the problem statement to each cash flowin the diagram.

Categories of Cash Flows.

The expenses and receipts due to engineering projects usually fall into one of the following categories.

First cost = expense to build or to buy and install

Operations and maintenance (O&M) = annual expense, such as electricity, labor, and minor repairs

Salvage value = receipt at project termination for sale or transfer of the equipment (can
be a salvage cost)

Revenues = annual receipts due to sale of products or services

Overhaul = major capital expenditure that occurs during the asset's life

Individual projects will often have specific costs, revenues, or user benefits. For example, annual operations and maintenance (O&M) expenses on an assembly line might be divided into direct labor, power, and other. Similarly, a public-sector dam project might have its annual benefits divided into flood control, agricultural irrigation, and recreation.

ECONOMIC ANALYSIS: CASH FLOW DIAGRAMS.

The costs and benefits of engineering projects occur over time and are summarized on a cash flow diagram (CFD). Specifically,a CFD illustrates the size, sign, and timing of individual cash flows. In this way the CFD is the basis for engineering economic analysis.

A cash flow diagram is created by first drawing a segmented time-based horizontal line, divided into appropriate time units. The time units on the CFD can be years, months, quarters, or any other consistent time unit. Then at each time at which a cash flow will occur, a vertical arrow is added-pointing down for costs and up for revenues or benefits.

These cash flows are drawn to relative scale.

The cash flows are assumed to occur at time 0 or at the end of each period. Consider

Figure 2-7, the CFD for a specific investment opportunity whose cash flows are described as follows:

FIGURE 2-7 An example cash flow diagram(CFD).

⇒ Drawing Cash Flow Diagrams with a Spreadsheet.

⇒ Drawing a Cash Flow Diagram.

⇒ Categories of Cash Flows.

ECONOMIC ANALYSIS: ESTIMATING BENEFITS.

However, engineering economists must often also estimate benefits. Example benefits include sales of products, revenues from bridge tolls--and-electricpower sales, cost reductions from reduced material or labor costs, reduced time spent in trafficjams, and reduced risk of flooding.Many engineering projects are undertaken precisely to secure these benefits.

The cost concepts and cost estimatingmodels can also be applied to economic benefits.

Fixed and variable benefits, recurring and nonrecurring benefits, incremental benefits, and life-cycle benefits all have meaning. Also, issues regarding the type of estimate (rough, semidetailed, and detailed) as well as difficulties in estimation (one of a kind, time and effort, and estimator expertise) all apply directly to estimating benefits. Last, per unit, segmented, and indexed models are used to estimate benefits.The concept of triangulation is particularly important for estimating benefits.

The uncertainty in benefit estimates is also typically asymmetric, with a broader limit for negative outcomes. Benefits are more likely to be overestimated than underestimated, so an example set of limits might be (-50%, +20%). One difference between cost and benefit estimation is that many costs of engineering projects occur in the near future (for design and construction),but the benefits are further in the future.Because benefits are often further in the future, they are more difficult to estimate accurately, and more uncertainty is typical.

The estimation of economic benefits for inclusion in our analysis is an important step that should not be overlooked.Many of the models, concepts, and issues that apply in the estimation of costs also apply in the estimation of economic benefits.

ESTIMATING COST MODELS: Improvement and the Learning Curve.

One commonphenomenon observed, regardless of the task being performed, is that as the number of repetitions increases, performance becomes faster andmore accurate. This is the


concept of  learning and improvement in the  activities that people perfonn. Fromour own experience weall know that our fiftieth repetitionis completed in much less time than we needed to accomplish the task the first time.

The learning curve captures the relationship between task perfonnance and task repetition. In general. as output doubles the-unit production time will be reduced to some fixed percentage. the learning curve percentage or learning curve rate. For example. it may take 300 minutes to produce the third unit in a production run involving a task with a 95% learning time curve. In this case the sixth (2 x 3) unit will take 300(0.95) = 285 minutes to produce. Sometimes the learning curve is also known as the progress curve. improvement curve. experience curve. or manufacturing progress function.

Equation2-4 givesan expressionthat can be used for time estimatingin repetitive tasks.

As just given.a learning curve is oftenreferred to by its percentage learning slope. Thus a curve with b = -0.074 is a 95% learning curve because 2^-0.074 = 0.95. This equation uses 2 because the learning curve percentage applies for doubling cumulative production.

The learning curve exponent is calculated using Equation 2-5.

EXAMPLE 2-9.

Calculate the time required to produce the hundredth unit of a production run if the first unit took 32.0 minutes to produce and the learning curve rate for production is 80%.

SOLUTION

It is particularly important to account for the learning-curve effect if the production run involves a small number of units instead of a large number.When thousands or even millions of units are being produced. early inefficiencies tend to be "averaged out" because of the larger batch sizes. However. in the short run. inefficiencies of the same magnitude can lead to rather poor estimates of production time requirements. and thus production cost estimatesmay be understated.ConsiderExample2-10 and the results that might be observed if the learning-curve effectis ignored.Noticein this example that a "steady state" time is given. Steady state is the time at which the physical constraints of performing the task prevent the achievement of any more learning or improvement.

EXAMPLE 2-10.

Estimate the overall labor cost portion due to a task that has a learning-curve rate of 85% and reaches a steady state value after 16units of 5.0minutes per unit. Labor and benefits are $22 per hour, and the task requires two skilledworkers. The overall production run is 20 units.

SOLUTION  

Because we know the time required for the 16thunit, we can use Equation 2-4 to calculate the time required to produce the first unit.

Now we use Equation 2-4 to calculate the time requirements for each unit in the production run as well as the total production time required.

The Toral cumulative time of the proctuction run is 119.8 minutes (2.0 hours). Thus the total

labor cost wstimate would be:

2.0 hours x $22/hour per worker x 2 workers = $88

If we ignore the learning-curve effect and calculate the labor cost portion based only, on tbe steady
state labor rate, the estimate would be:

0.083 hours/unit x 20 units x $22/hour per worker x 2 workers = $73.04

This estiIIl'£lteis understated by about 20% from what the true cost would be.

ESTIMATING COST MODELS: Triangulation.

Triangulation is used in engineering surveying.A geographical area is divided into triangles from which the surveyor is able to map points within that region by using three fixed points and horizontal angular distances to locate fixed points of interest (e.g., property line referencepoints). Since any point can be located with two lines, the third line represents an extra perspective and check. We will not use trigonometry to arrive at our cost estimates, but we can utilize the concept of triangulation.We should approach our economic estimate from differen tperspectives because such varied perspectives add richness, confidence, and quality to the estimate. Triangulation in cost estimating might involve using different sources of data or using different quantitativemodels to arrive at the value being estimated.

As decisionmakers we should always seek out varied perspectives.

ESTIMATING COST MODELS: Power-Sizing Model.

The power-sizing model is usedto estimatethe costsof industrialplants and equipment.

The model "scales up" or "scales down" known costs, thereby accounting for economies of scale that are common in industrial plant and equipment costs. Consider the cost to build a refinery.Wouldit cost twice as much to build the samefacility with double the capacity? It is unlikely. The power-sizing model uses the exponent (x), called the power-sizing exponent, to reflect economies of scale in the size or capacity:

where x is the power-sizing exponent, costs of A and B are at the same point in time (same dollar basis), and size or capacity is in the same physical units for both A and B.'

The power-sizing exponent (x) can be 1.0(indicating a linear cost-versus-size/capacity relationship) or greater than 1.0 (indicating diseconomies of scale), but it is usually less than 1.0 (indicating economies of scale). Generally the ratio should be less than 2, and it should never exceed 5.This model works best in a "middle" range-not very small or very large size.

Exponent values for plants and equipment of many types may be found in several sources, including industry reference books, research reports, and technical journals. Such exponent values may be found in Perry's Chemical Engineers' Handbook, Plant Design and Economics for Chemical Engineers, and Preliminary Plant Design in Chemical Engineering. Table 2-1 gives power sizing exponent values for several types of industrial
facilities and equipment. The exponent given applies only to equipment within the size range specified.

In Equation 2-3 equipment costs for both A and B occur at the same point in time. This equation is useful for scaling equipment costs but not for updating those costs. When the time of the desired cost estimate is different from the time in which the scaling occurs (per Equation 2-3) cost indexes accomplish the time updating. Thus, in cases like Example 2-8 involving both scaling and updating, we use the power sizing model together with.cost indexes.

 TABLE2-1 Example Power-Sizing Exponent Values

EXAMPLE 2-8.

Based on her work in Example 2-7,Miriam has been asked to estimate the cost today of a 2500 ft^2
heat exchange system for the new plant being-analyzed.'She has the following data.

· Her company paid $50,000.0for a 1000 ft^2 heat exchanger 5 years ago.
· Heat exchangers within this range of capacity have a power sizing exponent (x) of .0.55.
· Five years ago the Heat Exchanger Cost Index (HECI) was 1306; it is 1487 today.

SOLUTION.
Miriam will first use Equation 2-3 to scale up the cost of the 1000 ft^2 exchanger to one that is
2500 ft^2 using the 0.55 power-sizing exponent.

Miriam knows that the $82,8000 reflects only the scaling up of the cost of the 1000 ft^2 model
to a 2500 ft^2 model. Now she will use Equation 2-2 and the HECI data to estimate the cost of a
2500 ft^2 exchanger today. Miriam's cost estimate would be:

ESTIMATING COST MODELS: Cost Indexes.

Cost indexes are numerical values that reflect historical change in engineering (and other) costs. The cost index numbers are dimensionless, and reflect relativeprice change in either. individualcost items (labor,material, utilities) or groupsof costs (consumerprices,producer prices). Indexes can be used to update historical costs with the basic ratio relationship given in Equation 2-2.

Equation 2-2 states that the ratio of the cost index numbers at two points in time (A and B) is equivalent to the dollar cost ratio of the item at the same times (see Example 2-7).

EXAMPLE 2 - 7.

Miriam isinterested in estimatingthe annuallabQrandmaterialcosts for a newproduction facility.

She was able to obtain the following labor and material cost data:

Labor costs
· Lapor cost index value was at 124 ten years ago and is 188 today.
·Annual labor costs for a similarfacility were $575,500 ten years ago.

Material Costs
·Material cost index valuewas at 544 three years ago and is 715 today.
·Annual material costs for a similar facility were $2,455,OPQ three years ago.

SOLUTION.

Miriam will use Equation 2,.2to develop her cost estimat~s for;annual.labor and material costs.

Labor.

Material.

Cost index data are collected and published by several private and public sources in the United States (and world). The U.S. government publishes data through the Bureau of Labor Statistics of the Department of Commerce. The Statistical Abstract of the United-States publishes cost indexes for labor, construction, and materials. Another useful source for engineering cost index data is the Engineering News Record.

ESTIMATING COST MODELS: Segmenting Model.

The segmenting model can be described as "divide and conquer."An estimate is decomposed into its individual components, estimates aremade at those lower levels, and then the estimates are aggregated (added) back together. It is much easier to estimate at the lower levels because they are more readily understood. This approach is common in engineering estimating in many applications and for any level of accuracy needed. In planning the camp trip of Example 2-5, the overall estimate was segmented into the costs for travel, living, and entertainment. The example illustrated the segmenting model (division of the overall estimate into the various categories) togetherwith the unit factormodel to make the subestimates for each category.Example 2-6 provides another example of the segmenting approach.

EXAMPLE 2-6

Clean Lawn Corp. a manufacturer of yard equipment is planning to introduce a new high-end industrial-use lawn mower called the Grass Grabber. The Grass Grabber is designed as a walk- behind self-propelled mower. Clean Lawn engineers have been asked by the accounting depart'ment to estimate the material costs that will make up the new mower.The material cost estimate will be used, along with estimates for labor and overhead to evaluate the potential of this new model.

SOLUTION.

The engineers decide to decompose the design specifications for the Grass Grabber into its subcomponents, estimate the material costs for each of the subcomponents, and then sum these costs up to obtain their overall estimate. The engineers are using a segmenting approach to build up their estimate. Ater careful consideration; the engineers have divided the mower into the following major systems: chassis, drive train, controls, and cutting/collecction system. Each of these isfqrther divided as appropriate, and unit material costs were estimated at this lowest of levels as follows:

The total material cost estimate of $173.45 was calculated by summing up theestimates for each of the four major subsystemlevels (chassis, drive train, controls, and cuttinglcollection system).

It should be noted that this cost represents only thematerial portion of the overall cost to produce the mowers. Other costs would include labor and overhead items.


In Example 2-6 the engineers at Clean Lawn Corp. decomposed the cost estimation problem into logical elements. The scheme they used of decomposing cost items and numbering the material components (A.I, A.I, A.2, etc.) is known as a work break down structure. This technique is commonlyused in engineering cost estimating and project management of large products, processes, or projects. A work break down structure decomposes a large "work package" into its constituent parts which can then be estimated or managed individually. In Example 2-6 the work breakdown structure of the Grass Grabber has three
levels.At the toplevel is the product itself, at the second level are the fourmajor subsystems, and at the third level are the individual cost items. Imagine what the product work break-down structure for aBoeing 777 looks like.Then imagine trying tomanage the 777 design, engineering, construction, and costing without a tool like the work breakdown structure.

ESTIMATING COST MODELS: Per-Unit Model.

The per-unit model uses a "per unit" factor, such as cost per square foot, to develop the estimate desired.

This is a very simplistic yet useful technique, especially for developing estimates of the rough or order-of-magnitude type. The per unit model is commonly used in the construction industry. As an example, you may be interested in a new home that is constructed with a certaintype ofmaterial andhas a specific construction style.Based on this information a contractormay quote a cost of $65 per square foot for your home. If you are interested in a 2000 square foot floorplan, your cost would thus be: 2000 x 65 =$130,000.

Other examples where per unit factors are utilized include

·Service cost per customer
·Safety cost per employee
·Gasoline cost per mile
· Cost of defects per batch
·Maintenance cost per window
· Mileage cost per vehicle
·Utility cost per square foot of floor space
·Housing cost per student

It is important to note that the per-unitmodel does not make allowances for economies of scale (the fact that higher quantities usually cost less on a per-unit basis). In most cases, however, themodel can be effective at getting the decisionmaker "in the ballpark" of likely costs, and it can be very accurate if accurate data are used.

EXAMPLE 2-5.

Use the per-unit model to estimate the cost per student that you will incur for hosting 24 foreign exchange students at a local island campground for 10 days. During camp you are planning the following activities:

·2 days of canoeing
·3 campsite-sponsored day hikes
·3 days at the lake beach (swimming, volleyball, etc.)
·Nightly entertainment

After calling the campground and collecting other information, you have accumulated the follow-
ing data:

· Vanrental from your city to the camp (one way) is $50 per 15 person van plus gas.
· Camp is 50 miles away, the van gets 10 miles per gallon, and gas is $1 per gallon.
· Each cabin at the camp holds 4 campers, and rent is $10 per day per cabin.
· Meals are $10 per day per camper; no outside food is allowed.
· Boat transportation to the island is $2 per camper (one way).
· Insurance/grounds fee/overhead is $fpetday.per camper.
· Canoe rentals are $5 per day per canoe, canoes hold 3 campers.
· Day hikes are $2.50 per camper (plus the cost for meals).
· Beach rental is $25 per group per half-day.
· Nightly entertainment is free.

SOLUTION

Transportation Costs

Living Costs

Entertfiinment Costs

Total cost

Thus, the cost per student would be $3966/24 = $165.25.

Thus, it would cost you $165.25 per student to host the students at theisland campground for the lO-day period. In this case the per-urtitmodel gives you a very detailed cost estimate (although its accuracy depends on the accuracy of your data and assumptions you've made).

ESTIMATING COST MODELS.

This section develops several estimatingmodels that can be used at the rough, semidetailed, or detailed design levels.For rough estimates the models are used with rough data, likewise for detaileddesign estimates they-areused-withdetailed data. The level of detailwill depend upon the accuracy of the model's data.

⇒ Per-Unit Model.

⇒ Segmenting Model.

⇒ Cost Indexes.

⇒ Triangulation.

⇒ Improvement and the Learning Curve.

Estimator Expertise.

Consider two common phrases: The past is our greatest teacher and knowledge is power.

These simple axioms hold true for much of what we encounter during life, and they are true in engineering estimating aswell.Themore experiencedandknowledgeablethe engineering estimator is, the less difficult the estimating process will be, the more accurate the estimate will be, the less likely it is that a major error will occur, and the more likely it is that the estimate will be of high quality.

How is experience acquired in industry? One approach is to assign inexperienced engineers relatively smallerjobs, to create expertise and build familiaritywith products and processes. Another strategy used is to pair inexperienced engineerswith mentorswho have vast technical experience. Technical boards and review meetings conducted to ''justify the numbers" also are used to build knowledge and experience. Finally, many firms maintain databases of their past estimates and the costs that were actually incurred.

Time and Effort Available.

Our ability to develop engineering estimates is constrained by time and person-power availability. In an ideal world, it would cost nothing to use unlimited resources over an extended period of time. However, reality requires the use of limited resources in fixed intervals of time. Thus for a rough estimate only limited effort is used.

Constraints on time and person-power can make the overall estimating task more difficult. If the estimate does not require as much detail (such as when a rough estimate is the goal), then time and personnel constraintsmay not be a factor.When detail is necessary and critical (such as in legal contracts), however, requirements must be anticipated and resource use planned.

One-of-a-Kind Estimates.

Estimated parameters can be for one-of-a-kind or first-runprojects.The first time something is done, it is difficult to estimate costs required to design, produce, and maintain a product over its life cycle. Conside~the. prqjec.tedcost estimates that were developed for the first NASA missions. The U.S. space program initially had no experience with human flight in outer space; thus the development of the cost estimates for design, production, launch, and recovery of the astronauts, flight hardware, and payloads was a "first-time experience."The same is true for any endeavor lacking local or global historical cost data. New products or processes that are unique and fundamentally differentmake estimating costs difficult.

The good news is that there are very few one-of-a-kind estimates to be made in engineering design and analysis. Nearly all new technologies, products, and processes have "close cousins" that have led to their development.The concept of estimation by analogy allows one to use knowledge about well-understood activities to anticipate costs for new activities. Inthe 1950s, at the start of themilitarymissile program, aircraft companies drew on their in-depth knowledge of designing and producing aircraft when they bid on missile contracts. As another example, consider the problem of estimating the production labor requirements for a brand new product, X. A company may use its labor knowledge about Product Y, a similar type product, to build up the estimate for X. Thus, although "first-run" estimates are difficult to make, estimation by analogy can be an effective tool.

Difficulties in Cost Estimation.

Estimating is difficult because the future is unknown. With few exceptions (such as with legal contracts) it is difficult to anticipate future economic consequences exactly. In this section we discuss several aspects of estimating that make it a difficult task.

FIGURE 2-6 Accuracy versus cost trade-off in estimating.

⇒ Estimator Expertise.

⇒ Time and Effort Available.

⇒ One-of-a-Kind Estimates.

Types of Cost Estimate.

The American poet and novelist Gertrude Stein wrote in The Making of Americans in 1925 that "a rose is a rose is a rose is a rose." However,what holds for roses does not necessarily hold for estimates because "an estimate is not an estimate."Ms. Stein was not suggesting that all roses are the same, but it is true that all estimates are not the same. Rather,we can define three general types of estimate whose purposes, accuracies, and underlyingmethods are quite different.

Rough estimates: Order-of-magnitude estimates used for high-level planning, for determining macrofeasibility, and in a project's initial planning and evaluation phases. Rough estimates tend to involve back-of-the-envelope numbers with little detail or accuracy. The intent is to quantify and consider the order of magnitude of the numbers involved. These estimates require minimumresources to develop, and their accuracy is generally -30% to +60%.

Notice the nonsymmetry in the estimating error. This is because decision makers tend to underestimate the magnitude of costs (negative economic effects). Also as Murphy's law predicts, there seem to be more ways for results to be worse than expected than there are for the results to be better than expected.

Semidetailed estimates: Used for budgeting purposes at a project's conceptual or preliminary design stages. These estimates are more detailed, and they require additional time and resources to develop. Greater sophistication is used in developing semidetailed estimates than the rough-order type, and their accuracy is generally -15 to +20%.

Detailed estimates: Usedduringap!oject' sdetailed designandcontractbiddingphases.

These estimates are made from detailed quantitative models, blueprints, product specification sheets, and vendor quotes. Detailed estimates involve the most time and resources to develop and thus are much more accurate than rough or semidetailed estimates. The accuracy of these estimates is generally -3 to +5%.

The upper limits of +60%for rough order, +20%for semi-detailed, and +5%for detailed estimates are based on construction data for plants and infrastructure.

Final costs for software, research and development, and new military weapons often correspond to much higher percentages.

In considering the three types of estimate it is important to recognize that each has its unique purpose, place, and function in a project's life. Rough estimates are used for general feasibility activities, semidetailed estimates support budgeting and preliminary design decisions, and detailed estimates are used for establishing design details and contracts. As one moves from rough to detailed design, one moves from less to much more accurate estimates.

However, this increased accuracy requires added time and resources. Figure 2-6 illustrates the trade-off between accuracy and cost. In engineering economic analysis, the resources spent must be justified by the need for detail in the estimate. As an illustration, during the project feasibility stages we would not want to use our resources (people, time, and money) to develop detailed estimates for unfeasible alternatives that will be quickly eliminated from further consideration.However, regardless of how accurate an estimate is assumed to be, it only an estimate of what the future will be. There will be some error even if ample resources and sophisticatedmethods are used.