Parametric Equations For Estimating Aircraft Airframe Costs
AJO R-1693-1-PA&EFebruary 1976./-*'y/1 ( a tti**- -f./Parametric Equations for EstimatingAircraft Airframe CostsJoseph P. Large, Harry G. Campbell and David CatesA Report prepared for/ w»ASSISTANT SECRETARY OF DEFENSEllÜiJb'cjbJ U'id A(PROGRAM ANALYSIS AND EVALUATION)-RandSANTA MONICA V 4WH,,v.;;ed
· ·THIS DOCUMENT IS BESTQUALITY AVAILABLE. THE COPYFURNISHED TO DTIC CONTAINEDA SIGNIFICANT NUMBER OFPAGESWHICHDONOTREPRODUCE LEGIBLYo
I The research described in this Report was sponsoredby the Office of the Assistant Secretary of Defense(Program Analysis and Evai.iation) under ContractDAHC 15 C 0220. Reports of The Rand Corporationdo not necessarily reflect the opinions or policiesof the sponsors of Rand research.
„m .„I, - ,., -,AJIR-1693-1-PA&EFebruary 1976iParametric Equations for EstimatingAircraft Airframe CostsJoseph P. Large, Harry G. Campbell and David CatesA Report prepared forASSISTANT SECRETARY OF DEFENSE(PROGRAM ANALYSIS AND EVALUATION)RandSANIA MONICA. CA. 4040bAPPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITEDriii i n im TinINrnn --iiiiittifinaaii mat«! r«t —i
BMae ,— -a-wfc-iii*****PREFACE14?'4"%cAfcj. o' *&*This study was sponsored by the Office of the Assistant Secretaryof Defense (Program Analysis and Evaluation) as part of a research program focused on improved methods of estimating the development, procurement, and operating costs of new weapon systems.The purpose of thestudy was to derive equations for estimating the acquisition cost ofaircraft airframes.Such equations are intended primarily for use inlong-range planning, not for contract negotiation or financial management.This report was first distributed in May 1975.In the presentprinting, dated February 1976, the author has supplied supplementarymaterial (Appendix C) illustrating how equations that appear to be comparable on the basis of statistical measurements can give widely different estimates of cost.The repoit should be useful to persons concernedwith the selection, procurement, and production of military aircraft.ina kaMHmmmmm-'-—-*"- —"-— *— "-- '
, ,«,ft«*&*.-V- SUMMARY*4%.Studies pertaining to the selection and acquisition of militaryaircraft generally begin with little more than a statement of theanticipated weight and the desired performance of a proposed aircraft.Even at that point, however, cost is an important consideration, andcost-estimating techniques must be devised that require only the available information, i.e., estimates of a limited number of physical andperformance characteristics.This report presents generalized equa-tions for estimating development and production costs of aircraft airframes on the basis of such characteristics as aircraft weight andspeed.It provides separate equations for the following cost elements:engineering, tooling, nonrecurring manufacturing labor, recurring manufacturing labor, nonrecurring manufacturing material, recurring manufacturing material, flight-test operations, and quality control.Italso provides equations for estimating total program cost and prototypedevelopment cost.The estimating relationships are expressed in the form of exponential equations derived by multiple-regression techniques.Costs orman-hours are related to aircraft characteristics and quantity.earlier Rand work. 'Init was found that the characteristics that bestexplain variations in cost among airframes are airframe unit weightand maximum speed.A determined effort was made in the present studyto find additional characteristics that would make an estimating modelmore flexible and hence better able to deal with characteristics peculiar to individual aircraft.That effort was not productive.Thevariations in cost that are not explained by weight and speed are notexplained by any other objective parameters tested.The equations presented here were derived from cost data on 25military aircraft with first-flight dates from 1953 to 1970.(TheLevenson, G. S., and S. M. Barro, Cost-Ectimatinq Relationshipsfor Aircraft Airframes, The Rand Corporation, RM-4845-PR, February1966.2Levenson, G. S., et al., Cost-Estimating Relationships for Aircraft Airframes, The Rand Corporation, R-761-PR, December 1971.-.,.—,„. .—.—. ,.„.:„.—. -.:,.mHMMI
M 3LS MÜfite'-VI-earlier work included aircraft developed as far back as 1946.)Theaircraft in the sample have airframe unit weights ranging from 5.000to 279,000 lb and maximum speeds ranging from 300 to better than1300 kn.Cost data were obtained directly from the airframe contrac-tors whose aircraft appear in the sample and from standard Departmentof Defense references such as the Cost Information Report (now theContractor Cost Data Reporting System).The report explains the derivation of each of the estimating equations and describes the treatment of the data, the fitting of regression equationsv and the selection of preferred equations.Otherequations with additional explanatory variables re included where thestatistical basis for choosing one equation over another is not strong.A detailed numerical example is included which applies the preferredequations and compares the results to those obtained using severalsets of alternative equations.until rin'n-iin nr-frtof,mMmimimtimmmim&u mI il1r.liln1rr.Mr».Im- IXM—«M «M kattHü
"l&HJF**'- States*w-*i-vii-ACKNOWLEDGMENTSThe authors wish to express their appreciation for the cooperation of the following major airframe companies:Boeing AerospaceCompany, Fairchild Republic Company, General Dynamics Corporation,Grumman Aerospace Corporation, LTV Aerospace Corporation, LockheedCalifornia Company, Lockheed-Georgia Company, McDonnell-Douglas Corporation, Northrop Corporation, and Rockwell-International Corporation.Their willingness to provide data made the analyses described herepossible.Within Rand technical assistance on engineering matters wasprovided by G. K. Smith and T. F. Kirkwood.Smith also acted as tech-nical reviewer along with B. D. Bradley and D. Dreyfuss, and theircomments were particularly helpful.H. E. Boren, Jr., K. Hoffmayer,and F. Kontrovich assisted with data preparation. .,., . , II——.»— .--. . .
aLs . gi»- ;-*r-,IIMuyi—1—SU M,Ji ': ;-ixP- CiCONTENTS»** p47* s c iVby' . ESEARCH PROCEDUREData Acquisition and ReviewAnalytical Techniques5516ENGINEERING18TOOLING23MANUFACTURING LABOR27MANUFACTURING MATERIALS31FLIGHT TEST36QUALITY CONTROL38TOTAL COST40OTHER AVENUES EXPLOREDJudgmentGroupings444446PROTOTYPE AIRCRAFT49CONCLUSIONS53AppendixA. AN ILLUSTRATIVE PKOBLEMB. REGRESSION EQUATIONS AND STATISTICSC. USING ESTIMATING EQUATIONS --*M.- .M- - irmnnirr7riirn5563113mmmmmm*tmm&mmt*mMMmiifm:-"tm
-k.fic«kijI l'.ftiv.*" if;isv3WSR»-' .,,?.;. 1-I.INTRODUCTIONFor many years estimates of aircraft airframe cost were basedmainly on weight.The ARCO factor, for example (which took its namefrom the World War II Aircraft Resources Control Office), stipulatedthat manufacturing hours per pound of airframe could be estimatedsolely on the basis of airframe weight and production quantity.Fora given quantity all aircraft were estimated from the same curve.In the years since then estimators have been searching for otheraircraft characteristics that (1) will, in combination with weight,provide consistently accurate estimates, (2) are logically related tocost, and (3) can easily be determined prior to actual design and development.The third requirement has led to use of characteristicssuch as speed, wetted area, and aspect ratio, rather than those thatrequire more detailed knowledge such as number of engineering drawingsor number of parts.A model published by Rand in 1966 showed thatweight and speed were the only two explanatory variables that met thethree criteria cited.The model produced estimates that were founduseful both by government and by industry, but the feeling lingeredthat it should be possible to achieve greater accuracy by includingnew and different variables.Other companies, looking at ostensibly the same data, had developedmodels using additional variables.Planning Research Corporation, forexample, found time to be an important variable for material costs.Consequently, it was felt that a revised model could perhaps be moreflexible, more responsive to program variations such as type of development program and development schedule.Also, after several yearsthe addition of new aircraft to the U.S. inventory meant that the sample size could be increased and the enlarged sample would be morerepresentative of aircraft likely to be developed in the future.Levenson, G. S., and S. MBarro, Cost-Estimatinj Relationshipsfor Air-craft Airframes, The Rano Corporation, PM-4845-PR, February 1966.2Method:! cf Estimating Fixed-Wing Airframe Costs, Vol. 1 (Revised),PRC-547A, April 1967.-i-an
-2-A revised model was published by Rand in 1971.Several aircraftwere added to the data base, a method for distinguishing between prototype and full development programs was added, and the procedures followedin developing the model were less judgmental.Users pointed out almostimmediately, however, that the model suffered from what were believed tobe shortcomings in the earlier version:(1) the only two major explan-atory variables were weight and speed; (2) all aircraft were lumpedtogether rather than treated as classes (e.g., fighters, bombers, etc.);and (3) no provision was made for taking into account changes in airframe structural materials and manufacturing methods.Consequently,when information on several new aircraft became available and it seemeddesirable to update the data base, OSD (PA&E)M agreed to sponsor a research effort to produce a new estimating model and in the process dealexplicitly with the questions raised earlier.The study plan called for:1.Review of airframe data in the Rand files to ensure accuracyand consistency of definition and acquisition data on newaircraft.2.Consideration of additional explanatory variables that wouldmake the model better able to deal with characteristicspeculiar to individual aircraft, e.g., variable-geometrywing, oversize fuselage.3.Examination of the cost impact of major changes in manufacturing technology over time and of the use of differentstructural materials.Reviewing and expanding the data base turned out to be more of ajob than was anticipated.Data had been collected over a period ofyears from a variety of sources, and to ensure internal consistency itwas necessary to obtain additional cost details from airfrsme contractors."Levenson, G. S., et al., Coat-Estimathuj Itelationshipa for Aircraj't Airframcs, The Rand Corporation, R-761-PR, December 1971.The Office of the Assistant Secretary of Defense ("rogram Analysisand Evaluation).
Our goal was to obtain total contract cost for every contract of interest (i.e., out through the first few huudied aircraft) for over30 military aircraft.In the time available it was not possible toresolve all the questions concerning data, but we believe our datasample is far more complete, comparable, and accurate than those usedin previous analyses.Our search for other explanatory variables that would improve theaccuracy of estimates was less fruitful than we had hoped.The varia-tions in cost that are not explained by weight and speed are not explained by any other objective indexes that we could find.For severalof the cost elements, use of a dummy variable to distinguish cargoaircraft from other types proved beneficial.None of the other 20 orso variables considered, however, satisfied the criteria for inclusion.It was necessary to examine the cost implications of major changesin airframe manufacturing technology and structural materials becausethe data sample consists largely of aluminum aircraft.The shift toother materials such as steel, titanium, and composites raises a question about the value of equations derived from that sample for estimating the cost of future aircraft.Titanium is much more expensive thanaluminuu. and is more difficult to fabricate.The fasteners used tojoin titanium structural parts are many times more expensive on a perunit basis than those customarily used in airframe assembly.On theother hand, adoption of a "unitized design concept" by some producershas reduced the number of parts and fasteners required.Thus the in-crease in fabrication man-hours may be offset by a decrease in assemblyman-hours, and the shift would presumably result in a flattening of thecost-quantity curve.Statistical analysis does suggest a trend towardhigher material costs and reduced manufacturing man-hours, and we discuss some of the qualitative considerations involved later in thisreport.The estimating model developed is similar to previous Rand modelsin that it allows estimates to be made of individual cost elements(except in the case of prototype programs).An additional feature isthat it a.lows estimates to be made of total program cost with noseparation into cost elements.iitMmin MI.IMSeveral contractors suggested that morei
-4-ac;urate estimates could be achieved by estimating at the total program level, and a report on aircraft estimating prepared for the U.S.Navy makes a similar recommendation.1Our conclusion is that theresults obtained from the two methods are comparable; we cannot saythat either method gives consistently better estimates.Noah, J. W., et al., Estimating Aircraft Acquisition Costs byParametric Methods, J. Watson Noah Associates, Inc., FR-103-liSN(abridged), September 1973.
i iffi-' ''' ii--I"I ." — "- 'f-i - -- '-": '— *--'--WifriiiVi iiirTirf*'" * "'" *'""-5-II.RESEARCH PROCEDUREDATA ACQUISITION AND REVIEWThe cost data used in this study were obtained from both government and industry, and within the time available every effort was madeto ensure that the data were complete and comparable.Our goal was toobtain total contract cost for every contract of interest (i.e., outthrough the first few hundred aircraft) and to break that cost downinto all the elements shown on the worksheet in Fig. 1.That amountof detail was not available for some of ehe older aircraft in thesample, but with the generous cooperation of major airframe contractors enough data were obtained to make sure that most costs are included and properly classified.The sample consists of the fore the final analyses were made, all aircraft with first flightdates prior to 1952 (i.e., the B-47, F3D, F-84A, F-H6A, F-86D, r.nd F-89)were deleted from the sample, partially because of problems with thedata and partially because development and production experience on aircraft that old does not appear to be a reliable guide to the future.A basic question when dealing with data recorded by so many different contractors over so many years is whether to use actual costsor normalized costs.Actual costs may reflect problems that are irrel-evant to the task of developing and producing an aircraft.A severelabor shortage, for example, may caus;e a contractor to hire unqualified:,Li. ! !!! I. !
"tiTJT»«!* .Imm.- ***.-,,,."T*-* " p?i* S3s-F" * *ä& .-'' npiprww ,Ji Villi»-6-r-ict)u0Hco-inuXuso300L.UCou o o Hi—1CO3C / oocr3OCcooa.-HMT—1Ureure'0uu 3Cre0V ax:* o-6o Xo7DOouoOC2oT300r- / O)X300cuegVc 2c / -**nuc«30a c3 -'J U0 . c c30 r-U—o» *.1C0 «* —* u c4Jrecut/13,.ijiß01*- oa.en01reTotal recurring J"öcnCu3 U C 3L.U3U01uc0OCexcG a ANonrecurringRecurring3fJue u uuw.uu u v.c0ccl OCCOCOtherNonrecurringRecurringc *uuat - curringiBasic airframeNonrecurringRecurringDCCTotal nonrecurrin; t.uXSystem engineeringNonrecurringRecur ringi u—— H-i l.
.;,- .: ,- Mi, i.i,i ».i—,».iiMii. ;- 1I.I. ITM-,. -*".),- ,n-,iM - " ' i'*-' - -7-workers, with the result that manufacturing man-hours are higher thanthey would be under normal conditions.In anticipating a high produc-tion rate a contractor may expend many more tooling hours than can bejustified by the production rates achieved.Engineering changes andmodifications are a normal part of every program, but extensive changesdue to customer decisions are not.effect of schedule changes.Also, contractors speak of theOn one program the planned delivery sched-ule of one lot .was accelerated by 6 months; on a later lot it wasstretched out by 14 months.Such changes have a disruptive effect thatcan cause a temporary increase in manufacturing man-hours.One point of view is that such problems are a normal part of thebusiness of developing and producing military aircraft.\x allowancefor the cost implications of such problems must be made or an estimatingmodel will consistently underestimate cost.The opposite view is thata contractor estimates the cost of building a certain aircraft in acertain way and at a certain rate, and thus the government shouldobserve the same ground rules in reviewing his estimate—even thoughboth parties know that design changes and schedule changes will occurand cause cost increases.We have chosen to follow a middle course:In general, contractordata are used unchanged, but where a model change (e.g., the changefrom an A model to a B model) has demonstrably caused an increase inman-hours or costs, we have adjusted the data to eliminate that effect.Also, on the specific advice of contractors we have adjusted hours ina very few cases to what are believed to be more reasonable numbers.Our goal was to begin the analysis with a data base that is representative of the costs to be expected in a program with its fair share ofproblems but with no major design changes.Achieving a perfectly consistent data base when the data have beencompiled by so many different contractors is probably impossible because accounting practices differ so greatly among companies.Thegreatest source of potential error is in the treatment of off-sitecosts, e.g., purchased labor, vendor tooling, subcontracts, and outside production.Such costs sometimes turn up in contractor reportsas manufacturing material or other direct charges.They can be identi-fied only by an examination of contractor records, and then, along withd
r9y«m*vve!w-r .** '' 7 ' WS 5p 3P5jBp»i«» «? j;'.--8-all other off-site costs, they must be converted into equivalent onsite labor hours and material costs.Unless we had information tothe contrary we assumed that the subcontractor supplied any necessarymaterial, and total subcontract value was reduced by an amount consistent with the in-plant manufacturing and material dollar distribution.The remaining amount was divided by a composite dollar ratecalculated from the in-plant wage, overhead, and general and administrative rate plus an assumed profit for the contractor.RequirementsConstructing an estimating model would be greatly simplified ifthe only requirement were to estimate total program cost or totaldevelopment and total production costs.For long-range planningstudies, estimates at such aggregated levels may suffice, but they areof little use in understanding why a new program is estimated to costa certain amount.An analyst often wants to be able to compare majorcost elements with their counterparts in previous programs to determinewhether they seem reasonabTe and to make adjustments wherever indicatedby special characteristics of the proposed aircraft.For some purposes, then, it is essential to estimate at the majorcost-element level.In addition, it is desirable to distinguish be-tween nonrecurring and recurring costs.is simple:Conceptually, the distinctionRecurring costs are a function of the number of aircraftproduced; nonrecurring costs are one-time expenditures.In practice,however, the distinction is more difficult because contractors may notkeep track of costs in that way.Some accounts, such as mockups, windtunnel, and static test, are clearly nonrecurring; and others, such asmanufacturing material for production aircraft, are clearly recurring.Engineering and tooling hours are not so easily classified, and contractors appear to have somewhat different views on how to make theseparation.For the older aircraft in the sample the separation isarbitrary because records were not kept that way.After-the-factdeterminations are always open to question, and attempting to dealwith nonrecurring and recurring costs separately introduces a certainamount of error into the data.- i—-- -M« " """ -' lariflni irm-n-Wi iConsequently, we did not attempt to- --"—- —' - -— "MIIITIJ—— - - ——
5 5p"l Jv-' "". ' ., * '-* #f'sJjWSJHStüf'*''''3L"- airTB aifpifme«aaledistinguish between nonrecurring and recurring costs where the distinction seemed unwarranted.However, this does not mean that development and production costscannot be separated.Assuming an aircraft program consisting of 20test aircraft and 250 production aircraft, and using the cost elementsfor which estimating equations are derived, development and productioncosts would consist of the following:Development CostsProduction CootsEngineering for aircraft 1-20Tooling for aircraft 1-20Nonrecurring manufacturing laborRecurring manufacturing labor foraircraft 1-20Nonrecurring manufacturingmaterialsRecurring manufacturing materialsfor aircraft 1-20Flight testEngineering for aircraft 21-250Tooling for aircraft 21-250Recurring manufacturing labor foraircraft 21-250Recurring manufacturing materialsfor aircraft 21-250Appendix A presents an illustrative example that shows in some detailhow this may be done.The example also shows the relative Importanceof each cost element for a hypothetical military aircraft.Dealing with cost elements separately may result in errors becausepossible comDlementaries between some of the elements are not takeninto account.(Heavy investment in tooling should reduce manufacturinglabor hours; extra care in initial engineering should reduce the numberof changes later on; etc.)In addition, personnel at several airframecompanies stated that in their experience a highly aggregated estimating model has been more accurate than detailed models.On the basisof that advice we derived equations for total program cost in additionto the equations for individual cost elements.A second requirement was that the inputs, i.e., the informationto be supplied by the estimator, be readily available.Aircraft char-acteristics such as weight, speed, aspect ratio, and ceiling can bespecified long before engineering development begins, whereas moredetailed information cannot be.«AuHfeaMiiiiBaife MMÜOHMIIAdmittedly, characteristics do change.-'turn-'----
JfSW?" '* *'" "f '-r-«r-- rz&ms»g«?lf -';';:---"»sB»K5«äm«t,'w--10-Weight generally increases and speed sometimes decreases, so informedjudgment concerning the validity of early estimates is important.Estimates involving time are seldom reliable.If date of firstflight or first production aircraft is a required input, early estimates can be off by several years.Length of development program andlength of production program are even less likely to be estimatedaccurately; hence we have avoided these items.Subjective factorssuch as technological advance are also questionable because a priorijudgmencs are often different from ex post facto judgments.We didconsider them, however.A third requirement was that the model distinguish between prototype development programs and full development programs.In the former,a small number of aircraft, usually less than 4, are built with nocommitment to further production—no production planning, limited tooling, and limited systems development.The cost of a prototype programfor the first few aircraft is substantially lower because many costsare deferred until a decision to produce for inventory is made.Totalprogram cost is assumed to be the same for both approaches, but thetime-phasing of cost is different.For planning and budgeting, thatdifference can be important.Aircraft GroupingsPrevious Rand models have not distinguished among types of aircraft; bombers, fighters, cargo aircraft, etc., have all been estimatedby the same equations.Despite the intuitive appeal of stratifyingthe sample in that way, we have not done so for the following reasons.F5rst, when the data were plotted as in Fig. 2, no natural boundariesappeared.Trainers are mixed with fighters, fighters with bombers,and bombers with cargo aircraft.That is not surprising in view ofthe fact that the B-58 and the F-lll are very similar in both weightand speed, and the T-38 is as large as some fighters and faster thanmany.Second, the sample size for individual aircraft types was toosmall to be representative except in the case of fighters.However,because of the general belief that stratification of the sample into* * ii-i-—-mtmmmm
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2*zzzibäs». JLm j r ' ;- '« -» ,.:v ;.- -:: .': ' vif- vT;wrw - ; - . --12-more homogeneous groupings would result in improved estimating equations, we explored a number of possible groupings.In the course ofthe study aircraft were stratified by type (fighter, bomber, cargo),age, speed regime, weight, weight and speed, and structure design loadfactor.Our conclusion is that the total sample is still too small andprobably always will be, because at some point it becomes clear thatexperience with old aircraft is no longer relevant.Using a smallsample of homogeneous aircraft is a good idea if the next aircraft isgoing to be very much like those in the- sample.If, as is usuallythe case, the new aircraft will be substantially different, it isbetter to have a larger group of more diverse aircraft as a datasample.Explanatory VariablesEstimators are continuously searching for a combination of aircraft characteristics that will provide consistently reliable estimatesand be logically related to cost.Weight is a logical variable becauseit is an index of size, and all other things being equal a large aircraft should cost more than a small one.no other variable is universally accepted.Apart from weight, however,Previous Rand studies havefound speed to be a useful variable, but other organizations have foundit to be of no significance.In tins study all the characteristicsbelow were considered: WeightLift-to-drag ratio SpeedLoad factor CeilingWetted area Climb rate Range factorRatio of gross takeoff weight toairframe unit weight Thrust-toweight r tr Wirg loadi Aspect Ta Static thrustWing areaEmpty weight minus structure weightRatio of wetted area to stressdesign weightRatio of wetted area to wing areafhe values of these characteristics are shown in Table 1.— . . —,,. .—MB
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«» Wf llliifpgB j jS- --14-We have a mixed bag of reasons for choosing the parameters listed.Some have been shown in previous work at Ra
in airframe manufacturing technology and structural materials because the data sample consists largely of aluminum aircraft. The shift to other materials such as steel, titanium, and composites raises a ques- tion about the value of equations derived from that sample for estimat- ing the cost
Parametric equations are useful for modeling the path of an object. For instance, in Exercise 59 on page 777, you will use a set of parametric equations to model the path of a baseball. Parametric Equations Jed Jacobsohn/Getty Images 10.6 Rectangular equation: yx x 2 72 (36, 18) (0, 0) (72, 0) t t t 0 Parametric equations: x 24 2t .
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Objective: (10.4) Find d2y/dx2 Given Parametric Equations 14) 0 Objective: (10.4) Find d2y/dx2 Given Parametric Equations 15) 10 (t 8)3 Objective: (10.4) Find d2y/dx2 Given Parametric Equations 16) y 6x Objective: (10.4) Find Equation of Tangent Given Parametric Equations 17) 6 5 Objective: (10.4) Find Length of Parametric Curve I 18) 12"2
Surface is partitioned into parametric patches: Watt Figure 6.25 Same ideas as parametric splines! Parametric Patches Each patch is defined by blending control points Same ideas as parametric curves! FvDFH Figure 11.44 Parametric Patches Point Q(u,v) on the patch is the tensor product of parametric curves defined by the control points
parametric models of the system in terms of their input- output transformational properties. Furthermore, the non-parametric model may suggest specific modifications in the structure of the respective parametric model. This combined utility of parametric and non-parametric modeling methods is presented in the companion paper (part II).
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