AN INVESTIGATION OF CHANGES IN DIRECT LABOR REQUIREMENTS RESULTING FROM CHANGES I N AIRFRAME PRODUCTION RATE by LARRY LACROSS SMITH A DISSERTATION Presented to the Department of Marketing, Transportation and Business Environment and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1976 ii APPROVED: __&/ _ fY~;71~~~RR~ J~.~ a~rn~p~sion~-----~ iii VITA NAME OF AUTHOR: Larry Lacross Smith PLACE OF BIRTH: North Bend, Oregon DATE OF BIRTH: November 23, 1936 UNDERGRADUATE AND GRADUATE SCHOOLS ATTENDED: Oregon State University University of Washington University of Oregon DEGREES AWARDED: Bachelor of Science, 1958, Oregon State University Master of Science, 1963, University of Washington Master of Bus i ness Administration, 1974, University of Oregon AREAS OF SPECIAL INTEREST: Logistics Management Procurement Management Contract Pricing PROFESSIONAL EXPERIENCE: Course Director, School of Systems and Logistics, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1969-1973 AWARDS AND HONORS: Professional Designation in Contract Management, 1971 Outstanding Educator of America, 1972 Professional Designation in Logistics Management, 1973 Beta Gamma Sigma National Honorary Society, 1974 iv PUBLICATIONS: Smith, Larry L .• An Overview of Integrated Logistics Support in the Air Force. Dayton, Ohio: Air Force Institute of Technology, 1971. Noe, Charles G.: Smith, Larry L.; and Jacobs, Grady L. Defense Cost and Price Analysis. 2 vols. Gunter Air Force Station, Alabama: Extension Course Institute, 1973-74. V PREFACE This research topic is selected from a number of suggestions for business research by United States Air Force staff and operational activities. The topics are promulgated by the Air Force Business Research Management Center located at Wright-Patterson Air Force Base in Ohio. The Center serves as a central coordinating activity in assisting the investigator and the office needing the results. Travel funding, expert advice on the problem and assistance in gaining access to data are provided by the Comptroller Directorate of the Aeronautical Systems Division also located at Wright-Patterson AFB. Access to historical data is provided by the McDonnell Aircraft Company, the General Dynamics Corporation and the Boeing Company. The author is indebted to all those individuals who aided in completing this dissertation. Special thanks are due Professor Roy J. Sampson for his service as program advisor to the author and research committee chairman, Professor Harold K. Strom for his thoughtful assistance and Professors Kenneth D. Ramsing and Robert E. Smith for their ideas and consideration. vi CONTENTS VITA . . . . . . . . . . . . . . . . . . . . . iii PREFACE . . . . . . . . . . . . . . . . . . . . V Chapter I. INTRODUCTION AND OVERVIEW . . . . . . . 1 II. AIRFRAME COSTS . . . . . . . . . . . . . 5 Direct Manufacturing Labor Hours . • • 5 Learning Curve Theory . • . . . • . . 8 The Production Rate . . . • . • . . . 11 III. PREVIOUS FINDINGS . . . . . . . . . . . 15 P. Guibert • • . . • • . . • • • • • 16 Joel Dean . • • • • • . • • • • . . 17 Harold Asher • • . • . • . • • • . . 18 Alchian and Allen • . . • • • • • . . 19 Gordon Johnson • . . . . • • • . . • 21 Joseph Orsini .. , . . • . . . • . 22 Fazio and Russell . . . . . . . . • • 25 The Fiscal Year 74 F-4 Aircraft Case • 26 Large, Hoffmayer and Kontrovich • . • 29 Joseph Noah • • • • . • • . • • • • 32 IV. RESEARCH OUTLINE . . . . . . . . . . . . 36 Purpose and Approach • . . • . . • . 36 The Variables . • . . . . • . . • • . 37 A Cumulative Production and Production Rate Cost Model . • • • • . • • . 42 Hypotheses . • . . . • • . . . . . • 47 Data Selection. • . . • • . • . . . • 57 Summary . • • . . . . . . . . . . 58 v. ANALYSIS AND EVALUATION . . . . . . . . 59 The F-4 Program • . • • . • . • • . • 59 The F-102 Program • • . . . • • • . . 103 The KC-1J5A Program • . . . . . . • . 116 IV. SUMMARY AND CONCLUSIONS 138 t t t t t t t t t t t t I I t t I t t t t t t • t • t • I vii APPENDIX A . . . . . . . . . . . . . . . . . . 147 SELECTED REFERENCES . . . . . . . . . . . . . . 154 LIST OF TABLES 1. C-141A Learning and Production Rate Data in Quarterly Structure . . . . . . . 23 2. Variables for Analysis of F-4A, B, C, D, E and F Total Direct Production Hours per Pound . . • . . . . • . • • . • . . • • 6 3 3. Regression Results - Model 1 . . . . . . . 66 4. Predictive Ability - Full Model 1 • • • • • 71 5. Predictive Ability - Reduced Model 1 73 6. Regression Results - Model 2 . . . . . . . 75 7, Predictive Ability - Full Model 2 . . . . . 77 8. Predictive Ability - Reduced Models 2 and 3 78 9, Comparison of Manufacturing and Delivery Rates in Models 2 and 3 •..• 80 10. Regression Results - Model 4 . . . . . . . 82 11. Predictive Ability - Full Model 4 83 12. Predictive Ability - Reduced Models 4 and 5 85 13. Regression Results - Model 5 . . . . . . . 86 14. Predictive Ability - Full Model 5 . . . 87 15. Variables for Analysis of F-4 Fabrication and Assembly Hours per Pound • . . • • • 89 16. Regression Results - Model 6 • • • • • 93 17. Regression Results - Model 7 . . . . . . . 93 18. Predictive Ability - Full Model 6 . . . . . 95 viii LIST OF TABLES-Continued 19. Regression Results - Model 8 . . . . . . . 95 20. Regression Results - Model 9 . . . . . . . 98 21. Predictive Ability - Full Model 8 . . . . . 100 22. Predictive Ability - Full Model 9 . . . . . 100 2J. Predictive Ability - Reduced Models 8 and 9 101 24. Variables for Analysis of F-102A Total Hours per Pound • • • • • • • • • . • . • . • 104 25. Regression Results - Model 10 . . . . . . . 109 26. Predictive Ability - Full Model 10 . . . . 111 27. Predictive Ability - Reduced Model 10 . . . 113 28. Regression Results - Model 11 I I t I I • I 114 29. Predictive Ability - Full Model 11 . . . . 115 JO. Variables for Analysis of KC-1J5A Total Hours 117 Jl. Regression Results - Model 12 . . . . . . . 121 32. Predictive Ability - Full Model 12 . . 125 33. Predictive Ability - Reduced Model 12 • • • 125 J4. Variables for Analysis of KC-135A Fabrication and Assembly Hours . . . . . . . . . 127 35. Regression Results - Model 13 . . . . . . . 129 36. Regression Results - Model 14 . . . . . . . 132 37. Regression Results - Model 15 . . . . . . . 133 J8. Regression Results - Model 16 . . . . . . . 135 39, Regression Model Summary . . . . . . . 143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I INTRODUCTION AND OVERVIEW Each year great sums are expended to produce military aircraft. For example, the United States budget reflects that an estimated ~2.J billion will be needed to pay for new U.S. Air Force combat aircraft purchased in fiscal year (FY) 1976. 1 The FY 1976 budget request for 108 F-15 air superiority aircraft is $1.4 billion. 2 Similarly, some $361 million is requested for production of 53 A-10 close air support aircraft in FY 1976. 3 Programs of such magnitude require careful management. One facet of that task is cost estimating. Program managers must be able to predict costs in order to success- fully advocate, budget, contract for and control the programs. This research is focused on developing a cost 1Executive Office of the President of the United States, Office of Management and Budget, Appendix, The Bud et of the United States Government Fiscal Year 1 6, Washington D.C.: U.S. Government Printing Office, 1975 , p. 300. 2 Prepared statement by Thomas C. Reed, Secretary of the Air Force, before the Committee on Armed Services, U.S. House of Representatives, Washington, D.C., January 29, 1976, quoted in Air Force Policy Letter for Commanders 3-1976, supplement (March 1976): 15. Jibid. 2 estimating technique to use when program requirements change after production of aircraft is begun. At the outset of an aircraft production program, a tentative monthly production schedule for the life of the program is negotiated between the contracting parties. This schedule permits planning for such items as work force buildup, facility and tooling needs and the ordering of long lead time items. Although the planning delivery schedule covers the life of the program, formal contractual agreements between the Department of Defense and manu- facturers usually cover only annual delivery requirements. Delivery requirements for subsequent years are funded through the exercise of options or separate contracts as funds are appropriated by the Congress. These multiple year programs may result in a need to change the production rate. For example, when funding for a particular year is insufficient to cover the production scheduled under an existing production plan, it may be necessary to stretch out the production over a longer time span. A national emergency or changed mission requirement may dictate an accelerated rate of production. When such changes in delivery schedules are required, changes in cost estimates are also required to support contract negotiations and additional funding requests. It is suggested that the rate of production is 3 an important independent variable that can be used to help project the change in costs due to either program accelerations or decelerations. Industrial and governmental cost estimators have traditionally used learning curve techniques to predict the direct labor hours required to produce airframes. 4 These techniques assume that given certain production conditions, the direct labor hours needed to manufacture each airframe decrease in a regular pattern as the cumu- lative number of airframes produced increases. There are a few approaches to pricing changes in the rate of production, most of which are adaptations of learning curve techniques. But there is no generally accepted estimating technique that considers the rate of production on a systematic basis of prescribed data collection, analysis and prediction. The purpose of this research is to develop a procedure to consider the effect of a production rate change on direct production labor requirements for additional air- frame production. The procedure encompasses the needed 4The airframe can be viewed as an accounting entity that encompasses the manufacturer's production responsi- bility. For example, airframe costs would not include the direct labor hours required to produce engines and avionics but would include the hours required to install those components. In contrast, aircraft costs would include all the costs associated with producing the aircraft. 4 elements of data collection, variable formation, data analysis and forecasting. In the following chapter, airframe cost elements are discussed and the topic is narrowed to estimating airframe direct production labor hours. A third chapter summa- rizes previous studies in the area. The fourth chapter outlines the approach for conducting the research and identifies sources of data. An analysis of the data is presented in the fifth chapter. The paper closes with a summary, some conclusions and suggestions for further research. 5 CHAPTER II AIRFRAME COSTS This chapter focuses on three main issues. First the discussion is narrowed to estimating airframe direct manufacturing labor hours. Then a brief overview of learning curve theory is presented. The chapter closes with a discussion of airframe production rate as a cost variable. DIRECT MANUFACTURING LABOR HOURS There are many ways to categorize airframe production costs into manageable elements for the purpose of analysis. One generally accepted procedure is to segregate costs into the elements specified on a Department of Defense Contract Pricing Proposal (DD Form 633).5 These elements include; direct material, material overhead, direct engineering labor, engineering overhead, direct manufacturing labor, manufacturing overhead, general and administrative expenses and profit. One of these cost elements, direct manu- facturing labor, is of particular significance since 5u.s. Department of Defense, Armed Services Procure- ment Regulation, 1973 edition (Washington D.C.; Govern- ment Printing Office, 1973), p. Fl31, 6 indirect costs such as manufacturing overhead and general and administrative expenses are often allocated as a pro- portion of direct cost. Accordingly, an error in estimat- ing the direct manufacturing labor level would lead to a much larger error in the total cost estimate. 6 Direct manufacturing labor cost can be further described as the product of the average labor rate and the labor hours. At a particular facility the average labor rate will vary with the skill mix, the change in labor force size and the passage of time. 7 In the recent past, the labor rate has steadily increased while the value of the dollar decreased and the level of output per worker increased. The problem of forecasting changes in the labor rate is a significant one but it is outside the scope of this research. This study focuses on predicting direct manufacturing labor hours required to produce an airframe. The use of direct manufacturing labor hours as a cost element is widely practiced in the airframe industry. In general, this element includes those hours required to machine, process, fabricate and assemble all integral parts 6u.s. Department of Defense, Armed Services Procure- ment Regulation Manual for Contract Pricing (Washington D.C.: Government Printing Office, 14 February 1969), p. 6-1. 7charles G. Noe, Larry L. Smith and Grady L. Jacobs, Defense Cost and Price Analysis, Vol. 2, (Gunter Air Force Station, Alabama: Extension Course Institute, 1974), pp. 6-11. 7 of the airframe structure. It also includes the instal- lation (but not the production) of all aeronautical accessories and equipment. The hours required to produce raw materials and standard shelf items such as fasteners and gaskets are usually excluded. Direct manufacturing labor hours are those hours of effort which can be readily identified to a given end item or lot of end items through a work order or comparable document. In the airframe industry, direct labor is sometimes further categorized as either fabrication or assembly effort. One reason for this breakout is because the labor required to fabricate parts is usually more machine paced than is the effort required to assemble those parts into major assemblies or an airframe. Thus, the rate of improvement or learning on the building of successive airframes is often viewed as slower for fabrication than for assembly. Fabrication effort can also be viewed as the sum of set-up and run time. Set-up time per unit is inversely related to the size of a production lot release while run time per unit might be expected to follow a shallow learning curve. Assessing the impact of these subelements is difficult since actual set-up and run times are not often individually recorded. It appears that different factors affect the behavior 8 of fabrication hours than affect the behavior of assembly hours. Accordingly, it may be advantageous to separate these elements when analyzing historical data. LEARNING CURVE THEORY Direct manufacturing labor hours in the airframe industry are often predicted using learning curve tech- niques. These techniques are quantitative adaptations of the idea that individuals performing repetitive tasks exhibit a rate of improvement due to increased manual dexterity. Observations of complex airframe assembly operations reveal that management innovations such as work simplification, environment improvement and engineer- ing changes also contribute to the rate of improvement. It is subsequently found that this rate of improvement occurs in a regular pattern and can be predicted by a simple model. One model, the unit learning curve model, can be expressed as; Yx = c 0 • xcl where: Yx represents the direct manufacturing labor hours required to make the xth unit, xis the cumulative unit number, c0 is a coefficient that represents the theoretical number of direct manufacturing labor hours required to make the first unit and C is a coefficient that reflects the rate of ikprovement that exists in a particular manufactur- ing environment. 9 The model is more intuitively appealing when one observes that as the total quantity of units produced doubles, the cost per unit decreases by some constant percentage. 8 The model is also frequently used in a form where they value is expressed as the cumulative average hours rather than the unit hours. Certain elements that characterize the airframe manufacturing environment appear to be important contribu- tors to the cost behavior exhibited by the labor require- ment for airframes, 9 The first is the building of a sizeable, complex end item which requires large numbers of direct labor hours. Another is production that is manually paced rather than machine paced. A third factor is continuity of production. A production break would permit loss of learning through worker dispersal or forgetfulness. Frequent engineering changes seem to be dynamically accommodated by the learning model as long as the changes are not major. It should also be noted that while the learning curve is essentially a trend concept, it is not a time series trend form. Rather, the independent variable is taken 8Ibid., Vol. 1, p. 50, 9Herbert R. Kroeker and Robert Peterson, A Handbook of Learning Curve Techniques (Columbus, Ohio: The Ohio State University Research Foundation, 1961), p, 3, 10 to be the number of opportunities to learn while the d epend en t var.i ab l e i. s cos t i. npu t per un.i t o f pro d uc t i. on. l O Accordingly, 500 airframes produced at the rate of 50 per month would be predicted by the model to require the same number of direct manufacturing labor hours as 500 airframes produced at the rate of ten per month when the same rate of learning (c1 coefficient) is assumed. This lack of sensitivity of the learning curve model to the production rate is a problem if the rate is explicitly changed in midprogram. Logically, direct labor requirements per airframe should change as a result of a forced change in the production rate. The worker who senses the pressure of an increased production rate should be motivated to work faster than the worker who senses a production line slowdown. Higher rates of production should permit greater worker specialization than lower rates where one worker would be expected to accomplish multiple tasks. Tool and tooling set-up costs can be spread over a greater number of units at higher production rates. These factors suggest that the production rate should be considered as a variable in the models used to predict the direct labor hours required to manufacture an airframe. lOibid., p. 1. 11 THE PRODUCTION RATE An airframe production rate can be defined in different ways for different purposes. Four definitions are discussed here. They are a peak rate, an equivalent rate, a lot average manufacturing rate and a delivery rate. A peak production rate is the maximum output rate attained during a program. It is usually expressed as units per month. This variable relates to such investments as size of the facilities and the requirements for tools and tooling. Peak production rate is often used as a variable in a parametric cost model. For exampl e, using multiple regression analysis, Brents relates weight and peak rate to direct labor hours at the JOOth unit for 13 jet fighter airframe programs. 11 He uses the resulting model, hours= Bo . (DCPR weight)Bl • (peak rate)B2 , to estimate direct costs for a new jet fi ghter production program. 12 11Interview with Thomas E. Brents, Jr., Estimating Division, General-Dynamics Corporation, Fort Worth Division, 12 September 1975. 12It is the practice of the Government to furnish many components to a contractor building a new aircraft. Engines, electronic systems, wheels and brakes, tires, batteries, certain instruments and auxiliary power units are illustrative of these kinds of components. To provide a common basis for comparing weight based cost estimating relationships , Government and Industry planners have agreed to a Defense Contractor Planning Report (DCPR) 12 An equivalent rate of production can be used to allow for the fact that work on a specific airframe usually takes place in more than one calendar month. Thus fractional parts of airframes are theoretically produced in different months. Assigning the fraction of each airframe or lot of airframes produced to the proper month and then summing the fractions develops the equivalent airframes produced per month. This method provides a formidable editing problem when manufacturers do not collect direct labor hour data for each airframe or lot of airframes broken out by month. Another problem is that it is difficult to forecast the equivalent rate beyond the historical data. A lot average manufacturing rate is constructed by dividing the number of airframes in a lot by the time required to produce the lot. The lot release date for the first airframe in a lot and the airframe acceptance date for the last airframe in a lot define the extremes of the lot time span. This form of the production rate is easy to construct and appears to match well with lot average labor hour expenditures. However, the averaging process obscures any learning or rate effect within a lot. Thus when the lots are released infrequently, such as once a weight that excludes the Government furnished equipment, fuels and lubricants. The DCPR weight was formerly called the Airframe Manufacturer Planning Report (AMPR) weight. 13 year, the number of observations is limited and the data are somewhat distorted. A delivery rate can be developed by averaging monthly acceptances for each lot. If direct labor requirements are available for each airframe, the actual monthly airframe acceptance rate can also be used to develop cases for analysis. Historical acceptance data are readily available and delivery rates are easily forecasted from contract delivery schedules. Thus the delivery rate is also a possible candidate as a production rate proxy. In this research both the lot average manufacturing rate and delivery rate are examined as proxies for the production rate. Additional discussions of their con- struction and characteristics are included in chapters four and five. The rate of production can be viewed as either a dependent or independent variable. It is a dependent variable when holding the work force constant and allowing the improvement in unit process time due to learning to be realized in terms of an increased output rate. The production rate is more frequently viewed as an independent variable. The contract delivery schedule states the needs of the buyer in terms of a monthly delivery rate. Then the manufacturer balances the 14 scheduling of facilities, the work force and the production rate to best meet that contract delivery schedule. Typically, such production rates are low at the beginning of the program when the hours required per airframe are high. Then the program rate is accelerated to some peak production rate. Near the end of the program the rate is diminished. For the purposes of this study, the rate of production is viewed as an independent variable to be indirectly specified by the purchaser through the contract delivery schedule. 15 CHAPTER III PREVIOUS FINDINGS A number of people have investigated the effects of production rate on unit production costs. By no means have their findings been unanimous. Some have concluded that the effect of production rate on cost is insignificant or unpredictable. Others have concluded that production rate is an important independent variable that improves the accuracy of production cost models. This chapter summarizes contributions to the litera- ture on the topic. The summaries are presented in chronological sequence beginning with an older work by the French writer Guibert. Ideas from the writings of Dean, Asher, Alchian and Allen, Johnson, Orsini and Fazio and Russell are extracted. Some observations from an Air Force pricing and negotiation memorandum are included. A review of two recent works closes the chapter. One is by the team of Large, Hoffmayer and Kontrovich of the Rand Corporation and the other is by Joseph Noah writing under contract with the Navy . 16 P. GUIBERT Guibert sets forth his ideas on planning for the production of airframes in a book published in 1945, In that study he introduces the production rate as a variable affecting the unit labor cost. Guibert considers the production rate to be an important cost determinant because of its proportional relationship to the number of tools required. 13 Guibert observes that the number of airframes pro- duced prior to achieving the peak production rate is approximately equal to the number of airframes in work when the peak rate is attained. He then uses this generalization to derive the cost of a hypothetical air- frame which is a function of the production rate and the flow time at peak production. Guibert's models relate unit production cos t to cumulative quantity at four different rates of production; 10, 20, 50, and 100 units per month . The models are of the form yi = M - (V / (xi - P)) where: yi represents the man hours required to produce unit i and is expressed as the ratio of the actual 13P. Guibert, Mathematical Studies of Aircraft Production, translated into English by the U.S. Air Force, from Le Plan de Fabrication Aeronautigue (Paris: Dunod) 1945, p. 64. 17 cost in hours to the cost of the hypothetical unit, xi represents the cumulative unit number and M, V and Pare constants derived through simultaneous solution of the model with observed values of xi and yi. At any unit of production xi' the models predict higher unit costs at higher rates of production when using Guibert's reported constants. JOEL DEAN Dean explores the relationship between cost and rate of output in his book on managerial economics. Based largely on previous evaluations of empirical data from a hosiery mill, a leather belt shop and a furniture factory that were generated in the 1930's, he concludes that the most important independent variable in determining short run cost is the rate of output. 14 He further observes that unit costs are generally constant over the range of different production rates included in the samples. 15 When selecting the form of cost observations, Dean advises other investigators to analyze costs for an 14Joel Dean, Managerial Economics, (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1951), p. 292, l5Ibid., pp. 272-273, 18 accounting period rather than costs per unit of product. 16 This approach should facilitate editing the input data. However in the case of airframe production, where hours are usually aggregated and reported by production lot, some method of matching the input hours per time period with the output airframes per time period needs to be used. This is a problem because production of a particular airframe often spans more than one month or quarter. HAROLD ASHER Asher writes on the relationship between cost and quantity in the airframe industry. In the course of examining empirical data for many different airframe production programs, he subjectively evaluates the effect of production rate on direct labor hours. His conclusion is that production rate is not a very important predictive variable. Asher says that production rate 11 ••• is felt to be of minor importance, within a certain range of rates of production, and definitely subordinate to the effect of cumulative production. 1117 Asher does not attempt to separate statistically 16Ibid. , p. 286. 17Harold Asher, Cost-Quantity Relationships in the Airframe Industry, R-291 (Santa Monica, California: The Rand Corporation, July, 1956), p. 86. 19 the effect of production rate from the effect of cumulative production on unit cost. He notes however that there are at least two ways in which the rate of production can influence unit labor cost. First, it can affect the number of hours of machine set-up time charged to each unit. Second, it can affect the number of subassemblies employed in the manufacturing process. This in turn affecti the number of hours for subassembly work charged to each unit. He concludes that except for these two effects, there is little reason to expect the unit hours for a 200 unit per month case to be significantly fewer than for a 30 unit per month case. 18 ALCHIAN AND ALLEN Alchian and Allen advance the idea that production cost is dependent on three production variables. The variables are the total volume of the item to be produced, the production rate and the time span from the decision to produce until the first item is output. 19 They suggest that larger total volumes lead to smaller unit costs. This cost behavior is explained as 18Ibid., p . 87 . 19Armen A. Alchian and William R. Allen, University Economics (Bilmont, California: Wadsworth Publishing Company, Inc., 1964), p. 308 . 20 the result of increasing product standardization with increasing total volume. 20 These writers further suggest that unit costs should increase with increasing production rates. This behavior is attributed to the fact that higher production rates require use of more overtime and reliance on less efficient workers. 21 Alchian and Allen view the cost variable as increasing with decreasing initial production span. This behavior is explained as follows. When startup time is compressed, less efficient procedures and equipment are used than if time were allowed to prepare properly for production. 22 These inefficiencies must then be corrected as production progresses resulting in relatively higher unit costs. The writers do not support these ideas with evaluations of data. Nevertheless, the notions that total volume, production rate and startup span explain pro- duction cost behavior may have some application to the airframe industry. 20Ibid., p. 313. 21Ibid., p. 315. 22Ibid., p. 322. 21 GORDON JOHNSON Johnson describes an approach to estimating direct labor requirements when production rate changes occur on rocket motor production lines. 23 He develops a model that incorporates both the rate effect and the learning effect in estimating direct labor hours. Johnson's approach is to regress direct labor hours per month as a linear function of production rate in equivalent units per month and as a power function of cumulative units produced as of the end of each month. In equation form, the model is y = A + Bx + Cx z 1 2 where: y = direct labor hours per month, x6 = production rate in equivalent units per m nth, x? = cumulative units produced as of the end of el:Ich month and A, B, C and Z are model parameters. Johnson tests the model on four sets of rocket motor production data. He reports good results on two of the sets and fair results on one. Johnson explains that the fourth data set may have been distorted due to an in- adequate accounting system. He subsequently uses the 23Gordon J. Johnson, "The Analysis of Direct Labor Costs for Production Program Stretchouts," National Contract Management Journal (Spring 1969): 25-41. 22 model to price the direct labor element in a rocket motor production program stretchout. JOSEPH ORSINI Orsini tests Johnson's rocket motor model on the C-141A airframe production data listed in Table 1 to determine if the model is adaptable to the airframe production industry. 24 In the C-141A program, there is no indication that the production rate is explicitly changed in midprogram although the data reflect that the production rate gradually varies throughout the program. Orsini reports a method for editing lot airframe direct labor hour data into an equivalent quarterly airframe production rate. He first develops an average production rate for each lot by dividing the number of units in a lot by the lot total hours. Then, from a financial management report prepared by the manufacturer, he extracts the number of labor hours expended per quarter per lot. The product of the average lot production rate in airframes per hour and the hours per lot expended in each quarter produces equivalent units per lot per 24Joseph A. Orsini, "An Analysis of Theoretical and Empirical Advances in Learning Curve Concepts Since 1966," GSA/AM/72-12, M.S. Thesis, Air Force Institute of Tech- nology, Wright-Patterson Air Force Base, Ohio, 1970, pp. 5J-80. 23 TABLE 1 C-141A LEARNING AND PRODUCTION RATE DATA IN QUARTERLY STRUCTURE Quarter Direct Hours Equivalent Cumulative per Quarter Production Units Rate 1 72,088.1 .17033 .17033 2 201,565.4 . 47625 .64658 3 406 , 43lL 9 1.0192 1.6658 4 501,005.0 1.2479 2. 9137 5 64-4, 070. 2 1.7871 4.7008 6 944,570.4 3.0159 7,7167 7 1,216,825 . 8 4 . 4853 12.202 8 1,254,365.4 5,9089 18.111 9 1,258,344.3 6.6978 24.809 10 1,517,091.4 9.0960 33.906 11 1,784,800.2 12.217 46.123 12 2,265 ,594.6 18.442 64,565 13 2,397 ,482 .5 21.608 86.173 14 2,453,744.4 25.556 111.73 15 2,364,145.4 27,487 139.22 16 2,137,634.0 26.292 165.51 17 2,280 ,516.7 28.734 194.24 18 2,241,113.5 28.713 222,95 19 1,768,963.7 23.105 246.05 20 1,444 ,044.0 18 .274 264.32 21 1,095,218.6 13 .210 277.53 22 412,769.7 4.9630 282.49 23 71,639.9 .92512 283.42 24 24,199.1 . 31937 283.74 SOURCE: Joseph A. Orsini, "An Analysis of Theoretical and Empirical Advances in Learning Curve Concepts Since 1966, 11 GSA/SM/70-12, M.S. Thesis, Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio, 1970, p . 64. 24 quarter. Summing the equivalent units for all lots in each quarter produces equivalent airframes per quarter. This editing is necessary to fit the data to Johnson's model. Orsini report s that the three variable model developed by Johnson fits the edited C-141A data better than the Johnson model with the production rate term omitted. He concludes from this compari son that rate is a significant factor in determining manufacturing labor hours. For further comparison, Orsini also tests the C-141A data in a linear form of the multiplicative model Y e A= x Bx C 1 2 where: y represents direct labor hours per month, x1 represents the production rate in equivalent units per quarter, x? represents cumulative units pro- duced as of the end of each month and A, Band Care model parameters. This multiplicative model fits the data better than does the Johnson model. Orsini suggests that" . the multiplicative models may be more suitable for analysis than Johnson's model because they eliminate the require- ment for estimating the additional parameter Z. 112 5 25 FAZIO AND RUSSELL Fazio and Russell overview the entire airfra~e production cost estimating problem in terms of the sensitivity of each decision variable to the production rate. 26 Their objective is to find the optimum rate of production given the decision variables of production scheduling, Their approach is analytical as contrasted to the statistical investigations of historical data reported by Johnson and Orsini. Fazio and Russell observe that the number of parallel production lines or stations is an important determinant of direct labor hours per unit. They conclude that the installation of" . duplicate load centers, which may be required for higher rates of production, will in fact reduce the overall rate of learning and thereby increase t otal manufacturing hours for a fixed buy. 1127 They further note that the efficiency of the labor force varies by shift, Accordingly, the direct labor hours required to produce each unit could also be expected to vary with the number of shifts employed. 26Peter F, Fazio and Stephen H. Russell, "An Analyti - cal Approach To Optimizing Airframe Production Costs as a Function of Production Rate,'' SLSR J0-74A, M.S. Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1974, p. 1. 27Ibid., p. 66, THE FISCAL YEAR 74 F-4 AIRFRAME CASE The rate of production of F-4 airframes at the manufacturer's plant was increased at the request of the Air Force. The Government pricing and negotiation memorandum reflects that the Contracting Officer had some problems in pricing the assembly hour increase associated with the changed production rate. "This element of the proposal was one of the more controversial elements of cost in the FY 74 procurement." 28 In this instance , t he manufacturer was requested to increase the production rate from 15 a irframe s per month in 1973 to 18 per mon~h in 1974, The rate had already b~en increased from six airframes per month in 1972 to 15 per month in 1973, The manufacturer indicated that a substantial number of recalls and new hires would be required to meet the i ncrease in delivery rate. Since these personnel would not have worked on the F-4 airframe for some time or not at all. the ir efficiency would not be as high as the personnel currently building the airframe. 29 Thus, the learning curve model alone would not suffice to 28An extract from the Pricing and Negotiat ion . Memorandum for the Fiscal Year 1974 F- 4 airframe procure- ment, unpublished document, Aeronautical Systems Division, Wright-Patterson Air Force Base, Ohio, p. 17, 29Ibid. 27 project the cost of the additional airframes. The manufacturer's approach in supporting his proposal that a higher rate of production requires a higher rate of labor expenditure per aircraft is summarized here. In 1967, there was a production rate i ncrease on the F-4 program from an average of 4J airframes per month to a peak of 67 per month. The manufacturer estimated the direct labor hours that would have been required to produce the additional airframes with the standard learning curve model. This model makes no allowance for a changed rate. These estimated hours were related to the hours actually experienced as the production rate in- creased. The relationship was expressed in terms of a per cent increase in unit labor requirements due to the r ate increase for each lot of airframes. This per cent increase in delivery rate was then adjusted downward to an "equivalent aircraft delivery rate" to compensate " . . . for the le a rning that t akes place each month during the period of increased de- liveries,1130 To achieve this , an equivalent airframe flow time reduction curve based upon a 75 per cent learn- ing factor was developed for the anticipated delivery schedule. This adjustment reflects that additional JOI bid . , p. 18 . 28 learning takes place each month that a given rate of delivery remains above the base rate. The equivalent per cent increase in rate was then plotted against per cent increase in assembly hours. From this graph , the manufacturer estimated the per cent increase in assembly hours from the per cent increase in equivalent delivery rate . Th i s approach was used to support the assembly labor hour portion of the cost proposal . In a note to the Air Fo rce analysis of the proposal, the analyst wrote that the procedure summarized above predicted a higher impact due to changed production rate than was actually experienced in the 1973 procurement at a rate of 15 per month . With this new information, the predicted increase in labor requirements due to the change in production rate was adjusted downward by some 75 per cent. In searching for reasons why the cost est i mating relati onshi p developed by the manufac turer predicted incorrectly , it is noted that it was based on only one set of previous observat ions. In addition, the nominal values of production rate used in constructing t he relat ionship were in the range of 40 to 70 airframes per month while the application of the relationship was t o the range of six to 18 airframes per month . Even though 29 percentage change does not consider order of magnitude, it is possible the prediction was produced outside the relevant range of the relationship. LARGE, HOFFMAYER AND KONTROVICH Three investigators from the Rand Corporation report on a search for parametric cost models that would enable program planners to project the magnitude of cos ts that could be expected at different production rates. 31 They use statistical techniques to examine the effects of airframe production rate and other selected design parameters on ma jor cost elements. The investigation is an aggregative approach where a few descriptors from each of many programs are evaluated simultaneously through multiple regression analysis. The purpose is to develop a general cost model suitable for predicting costs of other programs. Observations from 29 different programs are examined. The model selected for direct manufacturing labor hours is y . A•w B = •s C •rD where : l 31Joseph P. Large, Karl Hoffmayer ar.d Frank Kontrovich, Production Rate and Production Cost, R-1609- PA&E, (Santa Monica, California: The Rand Corporation, December 1974), p. 1. JO y. repres ents the cumulat ive direct manufacturing l~bor hours through unit number i, w represents the program average DCPR weight expressed in pounds, s represents the max imum design airspeed in knots, r represents the production rate expressed as the acceptance span in months for the first i air- frames. For the investigation, i is arbitrarily chosen at 100 or 200 airframes. A, B, C and Dare model parameters. The data are evaluated in the model at y100 • y 200 and y200 - y100 . Then to determine the importance of the · acceptance span as a proxy for the production rate, the data are revaluated in the model without the acceptance span term . This comparison shows that the acceptance span proxy for the production rate is of little value in explaining the variation in cumulative labor hours among the different programs. The investigators conclude that the influence of production rate cannot be predicted with confidence on the basis of the analysis performect. 32 The investigators observe that the programs included in the sample are typical of modern production runs in that the number of airframes produced is low (less than JOO). The production rates are also low at less than lJ airframes per month. Thus their findings are limited to the effects of production rate for programs where total 321arge, Hoffmayer and Kontrovich, pp. 50-51, Jl output amounts to a few hundred airframes or less. The use of an acceptance span in mont hs as a proxy for the production rate is explained to be necessary because within each program the production rate changes a number of times. Thus in order to have a single observation to represent each program production r ate, an averaging approach is needed. This use of an average for the production rate will almost s urely mask the effects of a rate change after a particular program is in progress. Large, Hoffmayer and Kontrovich also evaluate the cost mode l proposed by Johnson f or rocket motor direct cos t estimating and later tested by Ors ini with airframe production data. Using data from seven different airframe production programs, they found that "In none of the programs did the inclusion of production rate improve the coefficient of determination , R2 , by as much as one per cent over what was obta ined using cumulative quanti t y alone. 1133 In one of the four cases whe re the contribution of the rate was found t o be stat is t ically significant, costs were found to i ncrease as rate increase d. Thus, the Johnson model does not appear to be appropriate f or the task at hand . JJibid., p. 49, 32 JOSEPH NOAH Noah reports on a statistical analysis of cost data to discover the effect of production rate on airframe cost. 34 His research includes the majo r cost elements for two fighter airframe production programs, the A-7 and the F- 4 . The programs are characteri zed by the large numbers of airframes produced. The A-7 data cover 1252 airframes which were produced over a period of nine years wh ile the F-4 da t.a span 46L~5 airframes and eight years. Noah analyzes all the major elements of airframe cost but his findings on direc~ labor hours are of par ticular interest here. For the two programs studied, he r eports that the following relationship models the data well: y = e A• x B ·x C · x D 3 whe re: 1 2 y represents the average direct labor hours expended per pound of airframe produce d for each airframe lot, e is the base of t~e natural system of logar i thms, x represents the cumulat i ve volume expres sed as p~unds of airf rame produced t hrough the midpoint of each successive airframe lo t . 34J. ~'/ , Noah, "Resourc e Input vs . Output Rate and Vo l ume in the Airframe Industry ," Draft Technical Report TR- 204-USN , Contract N00014-71 -0119 , (Alexandria, Virg ini8: J . Watson Noah Assoc i a tes, Inc., December 1974 ) , p. 1 . 33 x? represents the production rate expressed as the average pounds of airframe delivered per month for the period spanr.ing the first and last delivery of the lot, x 1 represents the annual volume of aircraft ordered, expressed in airframe pounds and A, B, C and Dare model parameters. The data adjustment from airframes to pounds is accomplished with the observed data element, DCPR weight per airframe, averaged over each production lot. Using this model, Noah reports multiple coefficients of determination for the F-4 and A-7 data of 0,99 and 0.80 respectively. Noah attempts to generalize the cost model by averaging the estimated re gre ssion coefficients derived from the two sets of cost data. He suggests t hat this generalized model can be used to predict the effects on cost of changing the production rate of the F-14 airframe production program. Noah conclude s that "Our findings suggest that delivery rate as a proxy for production rate has a significant effect, and an important one, on the production cost of airframes. 1135 The conversion of airframe numbers to DCPR pounds is a common practice when estimating airframe costs. This practice adjusts the data for variations resulting from 35Ibid., p. 41. J4 minor design changes in the airframe as production pro- ceeds. A change that adds weigh t to an airframe i s assumed to add labor hours t o t he cost of manufacturing that airframe. One problem wi th this conversion is that the direct labor hours required to install Government furnished equipment on the airframe are usually included in the recorded hours required to assemble the airframe. As previously noted , the weight of thos e Government furnished components is excluded from the DCPR weight. This presents a possible data mismatch. The first independent variable, cumulat ive volume, i s analogous t o t he cumulative airframe variable used in learning curve analysis. The cho ice of the lot midpoint of cumulative volume in pounds relates l ogically to the choi ce of average hours per pound per lo t for the dependent variable. The lot ave rage airframe delivery rate is a practical choice to represent the production r ate . The data are easily obtained and manipulated to f orm the variable . However, it appears t hat the average delivery rate variable will lag t he average expenditure of the hours required to produce the delivered airframes . There is no clear indication of the adequacy of the Noah approach. The two programs examined are very l a rge. Noah ' s approach of averaging coefficients derived f rom J5 the two programs to construct a generalized cost model for a third program defies logic. Nevertheless, the best test of any predictive technique is how well it predicts. Until this model is tested on additional programs no firm conclusions can be drawn. SUMMA RY All of the literature reviewed reflects a common interest in the relationship between production rate and direct labor requirements. But the authors do not agree on the form or the importance of that relationship. Some write of increasing unit cost with increasing rate while others express the opposite viewpoint. Some write of no significant effect of rate on unit cost while others suggest that rate is an important independent variable. This research will test th e idea that the production rate changes can explain changes in direct labor requirements. Furthermore, it is postulated that an increase in the rate will cause a decrease in the unit labor requirement within the relevant capacity of a given plant. The next chapter sets forth some theory and an approach for testing these ideas. 36 CHAPTER IV RESEARCH OUTLINE This chapter outlines the procedures used to conduct the research. For ease of presentation, the chapter is divided i nto five sections. First the general purpose and approach are described. Then the variables and some theory on their relationships are discussed. A cost model is suggested to explain the variation in the direct labor hours as production proceeds and the rate changes. A fourth section sets forth hypotheses for testing the adequacy of the model . Included under each hypothes is are tests and conditions for accept ance. The data used in the research are discussed in the final section. PURPOSE AND APPROACH The effect of a change in production rate on the direct labor hours required to manufacture airframes is the focus of this research . Should the findings indicate that production rate provides an explainat ion of variation in unit labor r equirements, the purpose is to develop procedures for collecting, editing and analyzing historical airframe production data. These procedures are designed to 37 assist analys t s in predicting the effects of production rate changes on the direct labor requirement for airframe manufacture. The approach is to evaluate the effects of a pro- duction rate on direct labor requirements within an individual program. This contrasts with developing a general cost model to apply to all airframe production programs. The individual program approach necessitates gathering and evaluating many observations for each program investigated. The desired product from each set of data is a tailored cost model suitable for predicting the effects of a production rate change on the direct labor hours required to manuf acture additional airframes in that program. Each data set should be considered as the basis for a separate investigation. The results of the investigations are not to be aggregated into a generalized cost model . THE VARIABLES The dependent variable to be investigated is the average number of direct labor hours required to manu- facture each pound of airframe. When expressed as a total, this variable includes all the hours re quired by the manufacturer and major subcontractors to f abricate 38 parts, assemble those parts into components and the air- frame and finish the aircraft. The hours required to produce raw materials, bench stock such as rivets and standard fasteners and aeronautical accessories and equipment such as avionics and engines are not included. It has been suggested that direct labor hours for fabrication and asse mbly operations behave differently with respect to cumulative production. 36 Accordingly, when the data permit, the dependent variable is also examined in parts as well as in total . For example, in two programs, the dependent variable is analyzed as fabrication, assembly and total hours required to manu- facture an airframe. Results may also be improved if the data are strati- fied according to different airframe mode l s that are produced simultaneously . When the data permit, this proposition is also tested . The procedures used by manufacturers to collect the hours required to produce an airframe vary in accuracy for different parts of the process. For example , the direct labor hours require d to fabric ate bits and pie ces in the plant are usually prorated to each airf~ame ac co rding to 361nterview with Charles Meranda , Cost Analyst, Comptroller Directorate, Aeronautical Systems Division, Wright-Patterson AFB, Ohio, 9 September 1975. 39 the size of the batch released to the machine shop . On the other hand, t he hours required to assemble those parts into an airframe can often be accurately traced to a specific airframe . The direct labor hours required to manufacture purchased parts such as standard fasteners are often ignored. The hours required for subcontractors to manu- facture major parts and assemblies to the manufacturer's specifications are included bu t frequently they must be estimated from the results of subcontract negotiations. Accordingly, when working with the different levels of data aggregation, one must remember that di ffe rent levels of accuracy accompany the data from the different parts of the process. Many airframe manufacturers collect data by lot or block of airframes. This generalization is particularly valid for the fabrication of parts in a batch operation. When the data are in lot f orm i t is convenient to express the dependent variable as the lot average direct labor hours required to fabricate, assemble or output each airframe. When the data are avai lable by individual a irf rame , it is conve nient to aggre gate the dat~ by sublets where each suble t includes the hours required to produc e the airframes output in a specific time frame such as a month . This action r educes the number of observations to a 40 manageable number and permits a logical matching of the hours expended to a delivery schedule. The design and weight of an airframe often change over the life of a program. In general, the direct labor hours required to manufacture an airframe will increase with an increase in weight. Thus, variation in direct labor hours required to produce an airframe can theo- retically be reduced by expressing those hours on the basis of pounds produced. The average DCPR weight of each lot of airframes is usually available from airframe manu- facturers. In this r eport, the dependent variable is expressed as the unit average direct labor hours per DCPR pound of airframe. The independent variable of primary interest is the production rate. One can define different forms of a production rate that might logically be used to explain variation in the labor hours required to produce an air- frame . For this research it is important to restrict these definitions to the confines of the available data. Since models developed from the research procedure are in- tended to be used for prediction, one must also be able to forec ast the production ra te variable in terms of future output. Finally, the prod~ction rate variable must logically relate to the dependent variable. With these restrictions in mind, two definitions of production rate 41 are selected for testing. They a re the lot average manufacturing rate and the lot average delivery rate. The lot average manufacturing rate is the number of airframes in a production lot divided by the production time span. This span bridge s the lot release date of the first airframe in a lot and the completion date of the last airframe in a lo t . The lo t release date is defined as the date work orders are issued to fabricate the first batch of parts in a lot . The completion date is defined as the date the customer signs for acceptance of the completed aircraft. Construction of the lot average manu- facturing rate variable re quires the lo t size, t he lot release date and the acceptance date as raw data . The lot average delivery rate is the number of air- frames in a lot divided by the time span over which thos e airframes are delivered . The time span is bounded by the dates the first and last a ircraft in the lot are accepted by the customer . Construction of the de livery rate variable requires only the lot size and the accept ance dates as raw data. When direct labor hour data are aggregate d for each airframe accepted in a month, the actual airframe acceptance rate may be used for this variabl e . It is clear that other inde pendent variables may need to be examined simul taneously with the production rate in 42 order to statistically isolate the effect of rate on cost. The cumulative number of airframes produced is one such variable. One method used to explain much variation in the direct labor hours required to produce an airframe is through application of a unit cumulative learning model. One model is yi = BO · x1iB1 • 1oei where: Yi represents the unit average direct labor hours required to output each pound of airframe in lot i, xii represents the cumulative learning achieved on all airframes of the same type through lot i, ei is an error term that accounts for the variation in each lot i that is not explained by the in- dependent variable and BO and B1 are model parameters. When the dependent variable is expressed as lot average direct labor hours per pound of airframe, cumulative production can logically be expressed as one- half the corresponding lot size plus the cumulative total number of airframes produced in prior lots. Th is lot midpoint is selected to represent the entire lot since it matches conceptually with the use of an average to represent the direct labor hours. THE CUMULATIVE PRODUCTIO. AND PRODUCTION RATE COST MODEL This study is based on the assumption that the production rate affects the quantity of di.rect labor hours required t o manufacture an a irframe . Since cumulative pro- duction has be en shown to be a strong explainer of changes in the direct hours required to produce an airframe, addition of a production rate variable to the learning curve model should explain additional variation in the direct labor requirement. Accordingly, a three variable model is suggested as more suitable for explaining and predicting variation in direct labor requirements than is the cumulative learning model. The model suggested is Yi= Bo . xliB1 . x2iB2 . 1oei where: y. represent s the unit average direct labor hours r~quired to output each pound of airframe in lot i, xJ. re presents the cumulative learning achieved on ar± airframes of the same type through lo t i, x 2 . represents the lot i production rate for all aITframes of the same type, e. represents the variation in each dependent viriable that is not explained by the two in- dependent variables and B0 , B1 and B2 are parameters in the model . The production rate i s chosen for i nclusion in the model in the multiplicative form for a number of reasons. Other writers have suggested that it might be a good pred ictor in this application . Mul tiple re gr ession analysis is facilitated by this choice. Final ly, investi- gation of some test data indicates that it works well . An increase in the production rate within the 44 planned plant capacity should cause a dec rease in the hours required to manufac ture each airframe . A rate decrease should produce th e opposite reaction . Th i s theory is supporte d by the ideas of specializing labor , prorating set up time and motivat i ng workers to produce . The management concept of spe ciali zing labor to increase efficiency supports the i dea that the rate . and the direct production labor hours sh ould be inversely related . At higher r ates of production , more workers are added . Th i s pe r mits the foreman to assign each worker fewer tasks which are performed more frequently. Thus , i n addition to the i ncreased profi ciency which accrues due to the learning process , some of the t i me that would be required to change from one task to another i s saved . At decreased rates of production, the oppos ite cos t behavior is expe cted as workers are r emoved from the production line and the remaining worke r s are r equired to perform add itional tasks . At h i ghe r rat es of produc tion , the hours required to set up mach ines f or fabricating airframe parts can be spread over mo r e airframes because of ~he acc ompanying larger batch s i zes . Th is should also contribute to an i nverse r elationsh ip between production rate and labor r equirements . Fi nally, it a ppear s lo~ical that the worker who senses the pressure of a high production rate will be motivated to work faster than the worker who is not pressed to finish a task because the level of effort is diminish- ing . This assumption is supported by the "toe-up" phenomenon familiar to students of the airframe productior. process. The " toe-up" phenomenon is an increas e in the direct hours required to manufacture airframes that is frequently experienced as a production program is concluded and the production rate is decreased toward zero. The C-141A data in Table 1 display this phenomenon. As previously implied, least squares multiple re gres sion analysis is used to examine historical airframe production data in the cumulative production and production rate cost model. To facilitate the regression analysis, the model is transformed by extracting the common logari th~ of each term. The transformed model is log yi = log B0 + B1 l og xli + B2 log x 2i + e .• The model is now linear l. in each term. Some assumptions about the error terms in the model are required to develop tests for the regression results, For this analysis, the error terms are assumed to be normally distributed, not related t o the independent variables or each other, with a me an of zero and a constan t variance equal to that of the dependent variable . 46 Collinearity of the independent variables, cumulative production and production rate, is likely to exist. It is obvious that as the production rate increases, the cumu- lative number of airframes produced also increases. But as production rate decreases, cumulative production con- tinues to increase as long as some airframes are being completed. When the independent variables in a multiple regression are highly correlated with each other, the standard error of individual regression coefficients may become unreliably large . 37 However, it may not alter the predictive power of the total regression model. Since the purpose of developing the model is for prediction, some collinearity is not considered a problem as long as estimates of individual coe fficients do not fail sta- tistical t ests yet to be described. However, collinearity may produce some peculiar effects in regression analysis besides indicat ing unrelia- bility of individual regression coefficients. 38 For ex- ample, the two variables cumulat ive production and pro- duction rate, when expressed as logarithms, may be posi- tively correlated with each other and negatively correlated 3?William A. Spurr and Charles P . Bonini, Statistical Analysis for Business Decisions (Homewood, Illinois: Richard D. Irwin, Inc., 1967), p. 610. JSibid. , p. 611. with the logarithm of hours per pound of airframe. But the net effect of the production r ate , when taking cumulative production into a ccount, may be positive . Based on the earlier dis cus sion of learning curve and production r ate theory, the expected signs of the estimated regress ion coefficients associated with the cumulative production and production rate variables are negative. Yet the above discussion points ou t that if sufficient collinearity exists, the sign of the coefficient associated wi th the weaker independent variable may change to pos i t ive while the model still supports the theory. HYPOTHESES There are thre e hypotheses t o be tested in this re search effort . The f irst is that the product i on rate , when expres s ed in a suitable form in a prope r model, can explain an i mportant part of the variation i~ the dire c t labor re quirements to bu ild an a i rframe , The second i s that the cumulative production and product ion rate cost model can be employed t o explain the variation in the hours required t o fabrica te and assemble the airframe . The third i s that the cos t model is suitable to pr edict the hours r equired to produce additional a i rframes . 48 HYPOTHES IS Ol\1E The firs t hypothesis is that the production rate is an important expla i ner of variation in t ota l direct labor requirements when included in an appropriate model. It can be tested by examining the results of regression analysis of historical data i n the selected model . The model to be used for the tests i s the cumulative pro- duction and production rate cost model . For th is first hypothesis, the dependent variable is the logarithm of total hours per pound rather than fabri- cation or assembly hours . The hours are stratified by homogeneous models of airframes when practical . The independent variables are, in turn, the logarithm of the cumulat ive production lot midpoint and the logarithm of the production rate. Both definitions of the production rate 2re examined 1hen data permit . Th r ee subhypo t heses arc used to examine the mode l in parts. The first i s that cumulative production and pro - duction r ate are related to hours per pound as indicated in the model. This hypothes i s is stated more f ormally in the null and alternate form as (lA) HO : Bl and B = 0 2 Ha : not both B1 and B2 = 0 The null hypothesis is rejected and the alternate 49 hypothesis accepted if it is shown that the statistic F * is larger than the theoretical value of Fat the 0.05 level of significance. In this instance, F * = (SSE(0 ) - SSE(x1 ,x2 ))/2 / SSE(x1 ,x 2)/(n-J) . SSE repres ents the sum of the square of each residual term. The residual term is the difference between the observed value of the hours per pound and the fitted re gression line at the corresponding values of cumulative production and production rate . Then SSE(0) is SSE about the mean value of the de pendent variable . SSE(x1 ,x 2 ) is SSE with both the independent var i ables in the model. The number of observations is r epresented by n. 39 Throughout this discussion, the theoretical percentiles of the F distribut i on are obtained from a table in the Nete r and Wasserman text. 40 It is of primary interest to test the hypothes is that th e produc t ion rate ca n explain additional variation in the hours pe r pound of airframe in the _presence of the cumulative produc tion variable. This second subhypothes is is the equivalent of stating that the B2 coefficient in t he cumulative production and production rate cost model 39Johr1 Neter and William Wasse r man , Applied Linear Statistical Models (Homewood, Illinois: Richard D. Irwin, I nc . , 1974), p . 242 . 40Ibid ., pp . 807- 813, 50 has a non zero value at some predetermined confidence level. In the null and alternate form, the hypothesis is: (lB) H0 : B2 = 0 P.a: B2 I 0 As before, the alternate hypothesis is accepted if the test statistic F * is larger than the theoretical value of Fat the 0.05 level of significance. In this case the F * statistic is calculated by measuring the reduction in the sum of the square of each residual that is attributable to adding the produc tion rate variable t o a reduce d model containing only the cumulative production independent variable. This statistic is calculated from the relationship: F * = (SSE(x1 ) - SSE(x1 ,x2 )) / SSE(x1 ,x2 )/(n-J). SSE(x1 ) represents the SSE for the reduced model, without th e pro d uc t i. on ra t e vari. a bl e. 41 The aptness of the model is evaluated through an analysis of the residuals or the observed errors . If the model i s appropriate for the data, the residuals should reflect the properties assumed for the theoretical error values . Accordingly , the residuals are examined for departures from the assumptions of constant va iance, independence and normality. A third subhypothesis 41 Ibid., pp. 262-264. 51 is advanced to test for aptness of the model: (lC) H0: The model is not appropriate. H: The model cannot be rejected. a The assumption of a constant error variance is evaluated by plotting the residual values against the fitted values of the dependent variable. Lack of a discernable pattern and a scattered distribution of . points is accepted as evidence that the assumption of constant variance is not violated. The assumption of error term independence is checked by plotting the residual values against each of the independent variables. If the residuals fluctuate in a more or less random pattern around a base line of zero, independence is assumed to exist. The residual values may also be plotted in the sequence in which they were gathered against a time variable. This check f or autocorrelation may be some- what distorted in regression analysis of unadjusted labor requirements for airframe manufacture . Peaks in the unit data caused by major engineering or model changes show up as runs of positive and negative residuals. Since these are symptoms similar to that of autocorre- lation , one may be mislead into rejecting a model when it does not exist. Therefore, in testing for autocorre- lation, care is taken to account for alternative 52 sources of runs in t he residuals . The assumption that the e r ror terms are normally distributed is studied by examining the cumulative f r equency di stribution of the residuals when plotted on normal probability paper . A characteristic . of this type of paper is that a normal distribution plots in a straight line . Therefore , the assumvtion of normality is rejected if a plot of the cumulative frequency distribution of the residuals on normal pro bability paper produces substant i al departures from a straight line . The null hypothesis i s rejected and the alternate hypothesis accepted if the subjective tests described above do not indicate substantial departures from the assumptions of constant variance, independence and normality for the calcula ted residuals. Another statistic of interest is the mean squared error (MSE ). Calculated by dividing SSE by the associated degrees of freedom, MSE i s convenient to use when comparing the reduced and full models . Specifically one would expect to obse rve a reduction in lfSS when a production rate variable is added to ~he reduced model . The total sum of square s (SS~O) for the regress · on mode l is calculated by finding the difference between the logarithm of each obs erved value and ~he mean of those values for the dependent variable . The multiple coef- 53 ficient of determination (R2 ) is calculated by subtracting from one the sum of the square of each residual (SSE) from SSTO and dividing that result by the total sum of squares. The resulting ratio represents the variation in the dependent variable that is explained by the regression model . Bu~ since each variable is transformed to logarithms to linearize th e terms, R2 actually represents the variation in the logarithm of the hours per pound explained by the model This transformation to logarithms somewhat obscures the interpretat ion of R2 with respect to the true variable of interest, hours per pound. Therefore, another statistic is introduced that reduces the effect of the transforma - tion. Termed R2 actual, a calculation analogous to that for R2 is performed on the variables expressed in their original form. Specifically, a SSTO actual is calculated by summing the square of the difference between each obs erved value of hours per pound and -their mean . Each observation i s predicted by the model in logarithms and then transformed to the original form . Actual residuals are then calculated, squared and summed producing an SSE actual. The R2 a c tual statistic is calculated by dividing SSE actual by SSTO actual and subt racting the quotient from one . R2 actual represents the variation in the actual hours per pound explained by the cost mode l . 54 If all three subhypotheses are not rejected, the MSE is relatively small, both R2 and R2 actual statistics are high and estimates of the model parameters follow the theory, the model is accepted as suitable for fitting the data. From this conclusion, it follows that the production rate is an important variable to be used in explaining variation in labor requirements to produce airframes. HYPOTHESIS TWO The direct labor hours required to fabricate and assemble an airframe are collected at a number of different process levels. For example, the total direct labor hours required to produce an airframe can be separately identified as in-plant or major subcontractor hours . Within a particular plant, a major portion of the direct labor hours can be identified as fabrication or assembly hours. It is of interest to know if the cumulative production and production rate cos t model can be used to explain the behavior of direct labor hours at sublevels of aggregation. Data at lower levels of aggregation are analyzed using the same cost mode l and regression techniqu~s that are described under the first hypothesis. The model may not be as suitable for this investigation if there is variation in subcontracting as a percentage of 55 the total workload as the program progresses. In order to exclude this source of variation it may be necessary to limit observations to those where major subcontracting is a relatively constant percentage. The only difference between the tests and subhypo- theses described under hypothesis one and those described under hypothesis two is the definition of the dependent variable. Under hypothesis two, the dependent variables are hours per pound required to f abricate or assemble the airframe. The definitions of the independent variables remain the same. For each set of data examined in the ~odel, the subhypotheses are: (2A) HO: Bl and B 2 = 0 H a : not both Bl and B2 = 0 (2B) Ho: B2 = 0 H .. B2 I 0 a (2C) HO: The model is not appropriate. Ha : The model cannot be rejected. The same tests and rejection criteria are us ed to evaluate these subhypotheses as are used to evaluate the first three subhypotheses. 56 HYPOTHESIS THREE One purpose of this research is to develop a model form and define variables so that model parameters can be tail ored to a continuing airframe production program. · These tailored models would then be used to predict the direct labor component of the cost of additional airframes . After successfully developing the models suggested under the fi r st two hypotheses, they should be tested to see how well they p_edict . In a real application of the model, the prediction would be beyond the range of the historical data. The only way to test the accuracy of the prediction would be to wait and see how many hours it takes to build ~he next airframe lot. To simulate this situation , the regress ion coefficients in the model are estimated with the last few observed data points omitte d . Then using this new model, omitted values (which are known but not used in est i matine the model coefficients) are predicted . Comparisons are then drawn between the a ctual and predicted hours as a subj ective measure of predic tive ability. Hypothesis three is that the predictive capability of each model is good for one year i nto the futu r e . Subject ive evaluation of the accuracy of the forecasts is the bas i s for accepting or rejecting the hypothesis, 57 DATA SELECTION The selection of programs to be examined is guided by convenience and accessability of data. There is no random sampling from all possible airframe production programs . Data from the F-4, F- 102, and KC-1J5A production programs are examined. The F-4 data are the most compl e te and comprehensive of the three programs . These data are used to test all three hypotheses. The F-102 data appear to be very accurate but are limited to total hours . Accord i ngly, hypo t hesis two is not eval~ated with F-102 data . The KC-1J5A data are available in both t otal hours and lower process l evel hours. The research procedure is to examine data sets from individual programs through application of the statistics and logic outlined he re . Coefficients are e stimate d from a data set that tailor the proposed cost model to that set. Then the predictive abil ity of that model is tested. Ther e is no intent to develop a generalized cost model, only a generalized approach to building tailored cost models . In this sense, each data set from each program represents a unique population. Therefore, the departure from randomness in selecting the programs should not in itself bias the conclusions . 58 SUf/IJ';iARY The approach i s to use multiple r egres s ion analysis of historical airframe production data to estimate coefficients in the proposed cumulative production and production rate cost model for a number of different data sets. Both statistica l and subject ive tests are used to verify the mode l and the assumption that the production rate is an important explainer of the variation in labor requirements to produce airframes. Finally , the ab ility of the model to predict the cost of new lots of a irframes is tested. The next chapte r d i s cusses the results of ·the res earch . 59 CHAPTER V ANALYSIS AND EVALUATION This chapter is arranged in four sections. The first three are findings about individual programs. Each in- cludes a discussion of the raw data, construction of the variables, results of the hypotheses testing and comments on the findings, The first section reports on the analysis of nine sets of F-4 data. Two sets of F-102A data are examined in the second section while five sets of KC-135A data are analyzed in the third. The chapter closes with a summary of the findings. THE F-4 PROGRAM The F-4 data are the most comprehensive of the three programs evaluated in this research. They include direct labor hours collected at many process levels as well as in total. The data cover 4665 airframes produced in 12 models. These airframes are produced in 57 lots from 1958 to 1975. The DCPR weight of the airframe increases from 15,JOO pounds for early models to over 21,JOO pounds for a more recently produced F-4E. After an initial peak of 71 airframes per month in 1967, the total delivery rate declines to a low of five per month in 1972 and then 60 increases to 20 per month in 1975 . The data are complex with many possible sources of variation for the direct labor hours. But the change in delivery rate over time makes the data unusually interesting for an investigation of the effect of a changing production rate on labor requirements. Because the F-4 data are so comprehensive. evaluations of many different data sets are possible . In this study. evaluations are classified along two major lines. total direct production labor requirements and direct production labor requirements at lower process levels. There are five data sets used to evaluate the research procedure for total hours. Then the procedure is tested using four data sets from lower process levels . COST MODEL 1 - TOTAL HOURS: F-4A-F The first evaluation of hypotheses one and three is accomplished on input data that include the total direct labor hours required to produce F-4A. B, C, D, E and F model airframes in all 57 lots. These airframe models represent a direct evolution of the F-4 and provide an opportunity to examine the entire program for eff ects of the production rate on labor requirements. Labor for rec onnaissance or other fighter versions are excluded from the lot average calculations in an effort to generate a 61 more homogeneous data set. Cost Model 1 has the form y = Bo • x B1 • x B2 · lOe 1 2 as do all the models in this investigation. But in this case the dependent variable is the weighted average direct labor hours required t o produce a pound of airframe in each lot. This variable is calculated by first developing the weighted average direct labor hours per airframe for each lot. Here the average labor requirement for each of the six airframe models is weighted by the corresponding number of airframes in the lot. Similarly, a weighted average DCPR weight is calculated . The average hours per airframe are then divided by the average pounds per air- frame producing the dependent variable, hours per pound. The effects of design changes are assumed to be eliminated by including the DCPR weight in the dependent variable. Cumulative production plot point is the first inde- pendent variable. It is obtained by dividing the total number of airframes in each lot by two and adding the quotient to the cumulative number of airframes already produced. In accordance with established learning curve technique, the plot point for the first lot is adjusted to allow for the steep drop in hours per pound for the first few airframes. This first lot plot point is ex- tracted from learning curve tables using an arbitrarily 62 selected 70 per cent learning factor. 42 The second independent variable is the program lot average delivery rate. It is developed from the actual acceptance schedule as follows. First the months are identified during which the airframes from a particular lot are accepted by the customer. Then the total number accepted during those months is divided by the number of months. This produces a lot ave rage delivery rate that closely approximates the rate that the airframes in the lot move through the plant. The three variables described above are listed in columns 2, 4 and 5 of Table 2. The data are evaluate d through multiple regression analysis in the following regression model: logy = log B0 + B1 log x1 + B2 log x 2 + e. The variables are transformed to logarithms to produce linearity in the model. Some important products of the regression are listed in Table 3. 43 The three part test of hypothesis one indicates that the model fits the data set well. The fact that the F * 42H. E. Boren, Jr. and H. G. Campbell, Learning Curve Tables: Volume II, 70-85 Per Cent Slo~es (Santa Monica, California: The Rand Corporation, April 1970), p. 2. 43Throughout this study the primary regression results are obtained through application of a FORTRAN (WATFIV) program written by the author and tailored to the needs of the research. The program is listed in Appendix A. 63 TABLE 2 VARIABLES FOR ANALYSIS OF F-4A, B, C, D, E and F TOTAL DIRECT PRODUCTION HOURS PER POUND Lot Cumulative Lot Average Lot Average Total Number Production Manufactur- Monthly Production Plot Point ing Rate Delivery Hours per Rate Pound (1) (2) (3) (4) (5) 1 3.0 0.50 maskeda 2 12.5 0.50 3 20.5 0.19 0,86 4 28.0 0.38 1.70 5 40.0 0.58 7.67 6 59.0 1.09 10.67 7 83.0 1.26 10 .00 8 107.0 1.26 6,75 9 131.0 1.41 11.00 10 155.0 1.60 11 . 33 11 179.0 1.71 13.50 12 206.0 2.14 10.75 13 243.0 2.93 12.20 14 287.0 2.75 10.75 15 333.0 2,82 11.50 16 382.0 2,78 13,50 17 432.0 2.63 24.00 18 502.0 5,00 33.33 19 602.0 5.79 35.00 20 717,0 6.32 33.40 21 837.0 6.67 41.25 22 . 957.0 7.06 41.50 23 1082.0 7.65 43 .00 24 1214.5 7,94 37.75 25 1349.5 7.50 44.25 26 1484.5 7.50 51.33 27 1619.5 7.50 55.00 28 1763.5 9.00 62.33 29 1930.0 10.59 63,75 JO 2125.0 12.35 67,75 31 2340.0 12.22 66.75 32 2555.0 11.05 52.33 33 2762.5 9.76 46.20 34 2947.5 7.86 55.67 64 TABLE 2-Continued Lot Cumulative Lot Average Lot Average Total Number Production Manufactur- Monthly Production Plot Point ing Rate Delivery Hours per Rate Pound (1) (2) (3) (4) (5) 35 3110.0 7.62 60.00 rnas keda 36 3252.5 6.25 44.67 37 3375.0 6.oo 39.00 38 3475.0 4.21 40.33 39 3555.0 4.44 39.67 40 3635.0 4.44 31.00 41 3715.0 4.44 25.50 42 3800.0 4.74 21.20 43 3894.0 4.90 25.25 44 3939.0 4.60 28.00 45 4062.5 2.89 23.67 46 4118.0 2.95 15.75 47 4175.0 2.90 13.33 48 4220.5 1.43 6.29 49 4252.5 1.29 5.67 50 4283.0 1.30 6,80 51 4313. 5 1.48 10.67 52 4348.5 1.95 12.25 53 4387.5 2.05 14.50 54 4436.5 3.11 14.80 55 4495.0 3.05 16.40 56 4558.5 3.45 17.20 57 4701.0 3 .13 12.57 SOURCES: Data for calculating the plot point are obtained from the "F-4 Procurement Summary", a single page periodic report developed by Department 926 of the McDonnell Douglas Corporation and dated 3 February 1974. The lot average manufacturing rate is developed from four sources. The lot release date for lots 16 trrough 57 is read from a series of graphs entitled ''F4 Manufacturing Schedule 0 , developed by Department 149A of the McDonnell Douglas Corporation and dated from 24 January 1963 to 11 August 1975. The release dates for lots 2 t hrough 15 are read fro m Section 3.3.2 of MCAIR Report 7290 entitled "F 4 Cost Data" and updated in August of 1975. The lot sizes are read from the "F-4 Procurement Summary" listed above. The lot completion dates are extracted from 65 TABLE 2-Continued McDonnell Douglas Report Numbe r 3.0, file number 3231, entitled "F-4 Scheduled Deliveries vs Actuals ", prepared by Department 013 and revised 7 August 1975 , The program delivery rate is also developed from McDonnell Douglas Report Number J.O described above. The total hours per pound variable is developed from three sources. The first 50 lots of data are extracted from a summary report entitled "McDonnell-Douglas F-4 Airplane Production Man- hours per Pound, F-4B, C, D/E Airplanes" and dated 10 January 1974. The labor hour data for lots 51 through 57 are extracted from Section 2.2.4 of MCAIR report 7290 listed above. DCPR weights for lots 51 through 57 are extracted from a McDonne ll Douglas summary report entitled "DCPR Weight-Pounds per Airplane" and dated 17 November 1975. For airframes 34 through 57, the DCPR weight includes useful load items at 1039 pounds. The hours required for major subcontracting are assumed to be constant for lots 51 through 57 at thB same level as that experienced for lot 50. aThe Total Production Hours per Pound are considered proprietary by the manufacturer. Accordingly, these data are masked in the published version of this dissertation. Access to these data can be provided by the author upon approval of the McDonnell Aircraft Co. representatives in St. Louis. The masked data have been reviewed in the draft dissertation by the Doctoral Dissertation Committee. 66 TABLE 3 REGRESSION RESULTS - MODEL 1 - (F-4A-F Total Hours Versus Plot Point and Delivery Rate) Estimate BO = maske da R2 actual = 0.982 Estimate Bl = -0.261 t ratio B1 = 26.10 Estimate B2 = -0.169 t ratio B2 = 11.27 F * (full) = 1206.57 F(2,54),0.05 = 3.17 F * (incremental) = 122.85 F(l,54),0 . 05 = 4,02 MSE (full) = 0.0016 MSE (reduced) = 0.0053 R2 (full) = 0,978 R2 (reduced) = 0.928 Case Observed Predicted Residual Per Cent Value Value Deviation 1 maskeda masked a maskeda maskeda 2 3 4 g 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 TABLE 3-Continued Case Observed Predicted Residual Per Cent Value Value Deviation 27 maskeda maskeda 28 29 30 31 32 33 34 35 36 37 J8 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 aTotal Production Hours per Pound are considere d proprietary by the manufacturer. Accordingly , thes e data are masked in the published vers ion of this dissertation. Access to these data can be provided by the author upon approval of McDonnell Aircraft Co, representatives in St. Louis. The masked data have been reviewed in the draft dissertation by the Doctoral Dissertation Committee. 68 statistic of 1206.56 is larger than the theoretical F value of J.17 indicates that the set of variables. logarithm plot point and logarithm rate. are related to the dependent variable, logarithm hours per pound. The F * incremental statistic of 122.85 is larger than the theoretical F value of 4.02, This indicates that the logarithm of the delivery rate contributes importantly to the model. All F t~sts are conducted at the 0,05 significance level. An examination of the residuals indicates no serious departures from the assumptions of the model. A plot of the residuals versus the corresponding fitted values of the dependent variables reveals little pattern. A plot of the residuals against each of the independent variables gives no evidence of a relationship. Plotting the cumulative frequency of the residuals on normal probability paper indicates no significant departure from a straight line. A plot of the residuals versus the time sequence in which they were observed presents a suggestion of corre- lation. But a check of the data reveals that much of the pattern is due to introduction of major design changes. For example, a C model airframe is introduced at lot 15 resulting in an increase in average direct labor hour expenditures. This increase causes the immediately suc- ceeding residuals to become positive until the change is 69 absorbed into the process. This cause of patterns in the residuals does not in itself invalidate the assumpt ion of independence. It does indicate that dividing the hours per pound by the DCPR weight does not completely eliminate the effects of major design changes on labor requirements. The theory advanced in the previous chapter for the behavior of the variables is supported by the estimated model coefficients. The estimate of the B1 coefficient is negative indicating that an increase in cumulative production leads to a decrease in unit labor requirements. The B2 coe ffi cient estimate is also negative. This supports the idea that a delivery rate increase (up to plant capacity) can be expected to decrease unit labor requirements. Comparisons of statistics produced from a reduced model (with the delivery rate variable omitted) and a full model support the conclusion that the delivery rate is an important explanatory variable. The MSE calculated from data in the reduced model is 0.0053 as compared to a MSE for the full model of 0.0016. This indicates a 70 per cent reduction in the MSE due to addition of the rate variable. Similarly, the mu ltiple coefficient of determination, R2 , increases 4.98 per cent as the result of adding the rate term to a reduced model. The R2 actual statistic, which is calculated from 70 predicted and actual values in their original form, is 0,982, This ratio indicates that 98,2 per cent of the variation in hours per pound is explained by the cost model, The t ratios for the B1 and B2 estimates are listed with the regression results for readers that prefer this statistic. In this analysis, the incremental test based on the F statistic is equivalent to the familiar t test. Therefore no further comment is offered on the t statistic. These tests sustain the first hypothesis that the production rate, when expressed in the cumulative pro- duction and production rate cost model, explains an important part of the variation in the total direct production labor required to produce F-4 airframes, But this conclusion is of little value if the model does not predict well. One way to test the predictive ability of the full model is to successively remove a few of the most recent cases from the data set, One may then perform the regression analysis with the remaining data to develop new regression coefficients. Using the model developed from the truncated data set, one can forecast the points that are removed. A comparison between the forecast and the observed values gives a measure of the predictive ability of the model tailored from the truncated data set. 71 As successively larger increments of data points are re- moved from the original data set, a family of estimated regression coefficient sets is developed for the cost model. Each of these tailored models can be used to predict some particular future data point. These pre- dictions, compared to an observed value, are the basis for subjective evaluation of the ability of the model to forecast. Table 4 lists the results of estimating the full cumulative production and production rate cost model from successively fewer observations. For example, in the row for 50 cases are estimated regression coefficients and statistics produced from regression analysis of the first 50 of the 57 observations in the data set. Also included TABLE 4 PREDICTIVE ABILITY - FULL MODEL 1 (Total Hours: F-4A-F) Cases Estimate Estimate Estimate Forecast Per Cent Bo Bl B2 Lot 57a Deviation 55 32.654 -0.261 -0.169 2.34 -2.63 50 33.027 -0.266 -0.162 2.31 -1.32 45 32.556 -0.259 -0.172 2.36 -3.50 a The observed value of the dependent variable for lot 57 is 2.28 hours per pound. 72 in the table .are forecasts for lot 57 using the different sets of estimated coefficients. During this program, a new lot is released approximately every fourth month. Therefore removing seven observations and developing the model on 50 cases allows ~bout a 28 month forecast to lot 57. The column captioned "Forecast Lot 57" lists the forecast direct hours per pound after transformation from common logarithms to their original form. The per cent deviation column reflects the difference between the observed and the predicted value of the hours per pound divided by the observed value. These ratios are expressed as a percentage. The choice of lot 57 for the comparison is arbitrary. It is clear that the observed value of hours per pound is a random variable subject to many sources of error. It could well be that choice of another data point for the comparison would give better (or worse) relative results. Nevertheless , the approach provides a stationary mark for the purpose of comparison. As one might anticipate, the model accuracy (as measured at lot 57) decreases as fewer data points are used to estimate the coefficients and the prediction span becomes longer. With 50 cases in the model, where per cent deviation happens to be the smallest, the forecast value for lot 57 is 2.34 as compared to the observed value of 73 of 2.28 hours per pound. When the coefficients are . estimated from 45 cases, the forecast for lot 57 increases to 2.37 hours per pound which is still within 3.5 per cent of the target. A reduced model (without the delivery rate variable) is employed to estimate regression coefficients using the same observation sets used to develop the full model. This reduced model is identical to the unit learning curve model. The results of these evaluations are listed in Table 5. TABLE 5 PREDICTIVE ABILITY - REDUCED MODEL 1 Cases Estimate Estimate Forecast Per Cent BO Bl Lot 57 Deviation 55 33.339 -0.336 1.95 14.5 50 35.138 -0.347 1.86 18 .4 45 37.906 -0.363 1.76 22.8 Once more using the actual direct hours per pound for lot 57 as the prediction target, per cent deviation from the observed value is calculated. The reduced model predicts low in every case. The magnitude of the differences is much larger than those produced from the full model. For example, f or a reduced model developed 74 from 50 observations, the deviation for a prediction of lot 57 is 18.4 per cent. This result is much less accurate than the prediction deviation of l.J2 per cent from the full model. One can conclude from these observations that for this data set, the full cumulative production and production rate cost model is superior in predicting power to the reduced model. Furthermore, the accuracy of prediction of the full model appears to warrant its use as a forecasting tool. Accordingly, hypothesis three is sustained for Model 1. COST MO DELS 2 AND J - TOTAL HOURS: F-4B-F The production rate variable can be expressed in a number of different ways. In Cost Model 1, it is a lot average delivery rate. In Cost Model 2, the production rate is represented by the lot average manufacturing rate. It is constructed by dividing the total number of airframes in a lot by the lot manufacturing time span. This span bridges the date the first work orders are released to fabricate parts for a lot until the date the last airframe in the lot is accepted by the customer. Because this span is so much larger than the span of months during wh ich airframes from a particular lot are delivered, the lot average manufacturing rate is a smaller number than the 75 lot average delivery rate. Cost Model 2 is identical to the first cost model in all other respects. The data used to test the model are listed in columns 2, 3 and 5 of table 2. Observations for lots one and two are not constructed because the lot re- lease dates are not included in the available data. Accordingly, the analysis is accomplished with 55 lots of F-4B, C, D, E and F data. Regression analysis results are listed in Table 6. TABLE 6 REGRESSION RESULTS - MODEL 2 Estimate BO = masked R2 actual = 0.971 Estimate Bl = -0.246 t ratio B1 = 26.74 E:_timate B2 = -0.183 t ratio B2 = 11.37 F (full) = 920.19 F(2,52) ,0.05 = 3.18 F * (incremental) = 129.30 F(l,52) ,0,05 = 4 .02 MSE (full) = 0.0013 MSE (reduced) = 0.0045 R2 (full) = 0,973 R2 (reduced) = 0.904 The statistics indicate that the lot average manu- facturing rate logarithm is also an important variable for explaining variation in the logarithm of hours per pound. The full regression model F * statistic of 920,19 is larger than the theoretical F value of J.18. This indicates that both of the regression coefficients are not zero with a 95 per cent level of confidence. Similarly, the F * statistic of 129.JO produced by incrementally adding the manufacturing rate to the reduced model is larger than 4.02. Again, at the 95 per cent level of conf idence, one can conclude that the B2 coefficient is not zero and that the manufacturing rate is an important variable. The increase in the R2 statistic from 0. 904 to 0.973, which results f rom adding manufacturing rate to the reduced model, provides further evidence that the manufacturing rate variable contributes importantly to the model. Other tests als o validate the model. As in Model 1, the MSE statistic for full regression Model 2 is less than one- third the value produced from the reduced regression model. The high value of R2 actual indicates that the full model explains some 97 per cent of the total variation in the hours per pound for the 55 observations . An analysis of the residuals by the methods outlined previ- ously reveals no serious departures from the assumptions of the model. Accordingly, Model 2 is also considered to be valid and hypothesis one is accepted . The predictive ability of Model 2 is examined by again successively removing observations from the end of the data set, estimating model coefficients from these truncated sets and predicting the hours per pound value 77 for lot 57, The results of this examination for the full model are listed in Table 7, The model appears to be quite TABLE 7 PREDICTIVE ABILITY - FULL MODEL 2 (Total Hours: F-4B-F) Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Lot 57a Deviation 50 22.442 -0,249 -0,180 2,23 2.19 45 22.652 -0.252 -0.176 2,20 3,51 40 20.042 -0,224 -0,216 2.36 -3,51 aThe observed dependent variable for lot 57 (case 55) is 2,28 hours per pound. stable for this test. It predicts within plus or minus 3 . 5 per cent for some 45 months at the arbitrarily selected test points. This stability is further evidenced by the relatively small change in the regression coefficients as up to 15 observations are removed from the original data set, When compared to the predictive ability of a reduced Model 2, the full model is clearly superior. The reduced model test results are listed in Table 8. The family of reduced model coefficients is estimated from data points that are common to those used to develop the coefricients listed in Table 7, Thus the results are d irec tly compara- 78 ble. For example, for coefficients estimated from the TABLE 8 PREDICTIVE ABILITY - REDUCED MODELS 2 AND 3 (F-4B-F Total Hours Versus Plot) Cases Estimate Estimate Forecast Per Cent Bo Bl Lot 57a Deviation 50 28.676 -0.317 1,97 13.60 45 31.944 -0.337 1.86 18.42 40 33,534 -0.J47 1,78 21.93 aThe observed dependent variable for lot 57 (case 55) is 2.28 hours per pound. first 50 cases, the full and reduced models forecast the hours per pound for case 55 (lot 57) at 2.23 and 1,97 respectively. When compared with the observed value of 2.28 hours per pound for lot 57 1 the improvement in pre- dictive ability resulting from adding a manufacturing rate variable to the reduced model is apparent. This same im- provement in predictive ability exists at all of the test points. It is concluded that the predictive ability of full Model 2 is superior to that of the reduced Model 2·. Furthermore, the ability of the full model to predict beyond the range of the data appears to be good. This is evidenced by the sm 11 per cent deviation for predictions 79 of up to 45 months. Accordingly, hypothesis three is also accepted for Model 2. It is d e sira b e to compare the production rate when expressed as a deli ery rate (Model 1) or as a manufactur- ing rate (Model 2). To facilitate the comparison, coef - ficients for a · regres•sion model are estimated. This model is based e same observations as is Model . 1 except that the for lots 1 and 2 are omitted. The data are listed in columns 2, 4 and 5 of Table 2. With this data set, the comparison between the effectiveness of the manufacturing rate and the delivery rate in the cumu- lative production and production rate cost model is simply a comparison between the tests of Models 2 and J. Before this comparison is made, the tests to validate the model and the delivery rate in the model are performed. Because the data used in es t i mating the coefficients for Model 3 are so nearly the s ame as for Model 1, one would expect similar results fr om the tests. This is so, and hypothesis one is sustained for Model J. The primary purpose for constructing Model 3 is to compare the effectiveness be t ween the production rate expressed as a lot average manufacturing rate or delivery rate. Some statistics and characteristics of the models are listed in Table 9 to assist in the comparison. 80 TABLE 9 COMPARISON OF MANUFACTURING AND DELIVERY RATES IN MODELS 2 AND 3 Oost Model 2 3 Production Rate Manufacturing Rate Delivery Rate Estimate of BO masked masked Estimate of Bl -0,246 -0,257 Estimate of B2 -0.183 -0.161 F * (full) . 920.19 742.13 F * (incremental) 129,30 95,20 MSE (full) 0.0013 0.0016 MSE (reduced) 0,0045 0,0045 R2 (full) 0.973 0.966 R2 (reduced) 0 .904 0.904 R2 actual 0,971 0.966 For these data sets of 55 observations, the statistics suggest that the manufacturing rate is the better repre- sentative of the production rate. The MSE (expressed in logarithms) for Model 2 is 0.0003 smaller than that for Model J, The R2 actual statistic for Model 2 is 0 . 005 larger than for Model J. The other measures similarly indicate that manufacturing rate is better. However, the differences are so small that they could easily be due to chance variation in the data. Thus, little confidence is given to the statement that one is better than the other. On a time axis, the manufacturing rate aligns con- 81 ceptually with the average hours required to produce a pound of airframe in a particular lot. Conversely, air- frame delivery as related by the delivery rate must lag the expenditure of the average hours per pound by some months. This observation may provide some clue as to why lot average production rate appears to be the stronger representative of production rate. COST MODELS 4 AND 5 The manufacturer's data collection system for F-4 direct labor hours changed between lots 15 and 16. Although grand total production hours are available, data for the fabrication and assembly portions of the manufacturing process do not match before and after that change point. To provide a basis for comparison of models developed from fabrication hours, assembly hours and total hours, coefficients for a model are developed from total F-4B, C, D, E and F hours per pound for lots 15 through 57, Cost Model 4 is the cumulative production and pro- duction rate cost model with the manufacturing rate form of the x2 variable. The data for lots 16 through 57, listed in columns 2, 3 and 5 of Table 2, are used to estimate the model coefficients. Table 10 presents the regression analysis results. 82 TABLE 10 REGRESSION RESULTS - MODEL 4 (F-4B-F Total Hours) Estimate BO = masked R2 actual = 0.887 Estimate Bl = -0. 230 t ratio B1 = 14.62 Estimate B2 = -0.157 t ratio B2 = 8.41 F * (full) = 112.96 F(2,J9),0.05 = J.24 F * (incremental) = 70.85 F(l,J9).0.05 = 4.09 MSE (full) = 0.0009 MSE (reduced) = 0.0025 R2 (full) = 0,853 R2 (reduced) = 0.585 The statistical t ests of the regression results indicate that the model is appropriate for the set of 42 observations. The F tests indicate that the coefficient estimates are not O with 95 per cent confidence. Exami- nation of the residuals reveals no i mportant departures from assumptions of the model. While the R2 actual statistic of 0,887 leaves some 11 per cent of the variation in hours unexplained, it still indicates that this simple model captures 89 per cent of the variation in a most complex process. Neverthe- less, it is also apparent that the fit is not as good as that of Model 2 which is estimat ed with more observations . Perhaps it is i mportant to include the early observations in models of F-4 data. When comparing the full model with the reduced model, 83 two statistics indicate that delivery rate is an important explainer of direct labor requirements. First, the R2 statistic is improved more than 23 per cent by adding the rate variable to the reduced model. Second, the MSE for the full model is less than one-half the MSE for the reduced model. These and other tests of the three sub- hypotheses provide no basis for rejecting the model. Therefore, hypothesis one is accepted for Model 4. Table 11 lists the results of tests for the pre- TABLE 11 PREDICTIVE ABILITY - FULL MODEL 4 (Total Hours: F-4B-F) Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Lot 57a Deviation 40 19.012 -0.232 -0.156 2.24 1,75 38 19.166 -0.233 -0.156 2.23 2.19 36 19.256 -0.235 -0 .151 2.22 2.63 J4 19.234 -0.236 -0.149 2.21 3.07 32 19.023 -0.240 -0.126 2.16 5.27 JO 18.865 -0.236 -0.136 2.20 3.51 aThe observed value at lot 57 (case 42) is 2.28 hours per pound. dictive ability of- Model 4. The tests show that the model form is well suited to prediction beyond the range of the 84 data. For example, with coefficients estimated from the first 40 of the 42 cases, the model predicts 1.75 per cent high for lot 57, a span of about eight months. With coefficients estimated from 36 of the 42 cases, the prediction accuracy for the 57th lot decreases to 2.63 per cent. The time span for the six lots is some 23 months. The other calculations reflect similar results. For this data set, the procedure appears to be quite stable. As additional cases are removed from each regression analysis, one would anticipate changes in the estimated coefficients and the resulting predictions. Yet as Table 11 reflects, the coefficients do not change very much as observations are removed. The predictions for lot 57 resulting from these different sets of coefficients are also relatively constant. The range for the seven different estimates is 0.08 hours per pound over a time span of about 45 months. The results of the predictive ability test for a reduced Model 4 are listed in Table 12. The new coef- ficients are estimated with the same cases used in the tests reported in Table 11. This permits a comparison of a model that includes the manufacturing rate with one that lacks it. The stabilizing effect of the manufacturing rate on the full model is even more apparent in this comparison. 85 TABLE 12 PREDICTIVE ABILITY - REDUCED MODELS 4 AND 5 (F-4B-F Total Hours versus Plot) Cases Estimate Estimate Forecast Per Cent BO Bl Lot 57a Deviation 40 10.514 -0.187 2.16 5.26 38 10.815 -0 .191 2.15 5.70 36 11.682 -0.202 2.12 7,02 34 12.950 -0.217 2.07 9.21 32 15.118 -0.239 2.00 12.28 30 15.789 -0.246 1.97 13.60 aThe observed value of the dependent variable for lot 57 (case 42) is 2.28 hours per pound. In the reduced model, the per cent deviation of the forecast value from the observed value increases as fewer observations are included in estimating the coef- ficients of the reduced model. As previously noted, per cent deviation for predictions from the full model remain relatively constant. With all 42 observations in the data set, the prediction accuracy of the full model is much better than that of the reduced model. This difference is maintained as different numbers of cases are used to evaluate the model. In addition, the predictions from the full model form are quite accurate when compared to the target 86 observed value. These indications of accuracy permit one to conclude that .the full Model 4 is a good predictor of total hours per pound and that hypothesis three may be accepted. The coefficients of Model 5 are estimated from the same 42 observations used in Model 4 except that the x 2 variable is the delivery rate. The variables are listed in columns 2, 4 and 5 of Table 2 for lots 16 through 57. Results of the regression analysis are listed in Table 13 . TABLE 13 REGRESSION RESULTS - MODEL 5 (F-4B-F Total Hours) Estimate BO = masked R2 actual = o. 854 Estimate Bl = -0.229 t ratio B1 = 13.10 Estimate B2 = -0.136 t ratio B2 = 7.14 F * (full) = 88.96 F(2,39),0.05 = 3.24 F * (incremental) = 50.96 F(1 , 39),o.05 = 4.09 MSE (full) = 0.0011 MSE (reduced) = 0 .0025 R2 (full) = 0.820 R2 (reduced) = 0.585 Tests of the statistics and the residuals reflect that the model is appropriate for the data set. The production rate proxy, delivery rate, is again an im- portant explanatory variable in the model. As with the comparison between Cost J,1 odels 2 and 3, Model 4 with the manufacturing rate variable produces 87 slightly better statistics than Model 5 with the delivery rate variable. In both comparisons, the R2 statistics are 3.3 per cent higher in the models with the manufacturing rate . Similarly, the MSE statistic is smaller for the models with the manufacturing rate. This comparison also favors the manufacturing rate variable as the better production rate representative although the evidence is not conclusive. The results of tests for the predictive ability of full Model 5 are listed in Table 14. The model form appears to forecast well·. This is indicated by pre- TABLE 14 PREDICTIVE ABILI TY - FULL MODEL 5 (Total Hours: F-4B-F) Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Lot 57a Deviation 40 23.050 -0.229 -0.137 2.35 3.07 38 23.064 -0.229 -0.136 2.36 3.51 36 23.176 -0.233 -0.131 2.32 1.75 34 22 . 869 -0 . 235 -0.123 2.30 0.09 32 21.529 -0 . 239 -0.099 2.22 -2.70 30 22.052 -0.234 -0.115 2.28 0,00 aThe observed value for lot 57 (case 42) is 2.28 hours per pound. 88 dictions of within five per cent of the observed value of 2.28 hours per pound for spans of up to 48 months. When compared with the predictive ability of reduced Model 5 (listed in Table 13), the full model is more stable and predicts better. It is concluded that Model 5 is also a good predictive model and hypothesis three is accepted. COST MODELS 6 AND 7 F-4 FABRICATION HOURS PER POUND The second hypothesis concerns the direct labor hours required to produce the airframe at lower process levels. Two additional levels of data aggregation are examined. They are the direct labor hours required to fabricate the airframe parts and the hours required to assemble those parts into an airframe. These data are analyzed in the cumulative production and production rate cost model with the manufacturing rate and the delivery rate serving as alternate representatives of the production rate. Four models, numbered 6 through 9 are tested to determine if hypothesis two is viable. Variables used to test the procedure with the four models are listed in Table 15. Models 6 and 7 are the models with the dependent variable expressed as fabrication hours per pound. The cumulative production variable for the xwo models is the program total cumulative production plot point. The pro- duction rate representative for Model 6 is the manufactur- TABLE 15 VARIABLES FOR ANALYSIS OF F- 4 FABRICATI ON AND ASSEMBLY HOURS PER POUND Lot Cumulative Lot Average Lot Average Fabricat i on Assembly Number Production Monthly Monthly Hours Pe r Hours Per Plot Point Manufactur- Delivery Pound Pound ing Rate Rate (1) (2) (3) (4) (5) (6) 16 382.0 2 . 78 13.50 1.574 1.740 17 432.0 2.63 24. 00 . 1.459 1.783 18 502.0 5.00 33.33 1 .Jll 1 .514 1 9 602.0 5,79 35.00 1.174 1.200 20 717.0 6.32 33.40 1.151 1.054 21 837.0 6 . 67 41.25 1 .103 1.002 22 957.0 7.06 41 .50 0.954 0. 900 23 1082.0 7,65 43.00 0.927 0, 836 24 1214.5 7.94 37.75 0.911 0. 808 25 1349.5 7.50 44.25 0.892 0. 837 26 1484.5 7.50 51. 33 o. 919 0.917 27 1619.5 7.50 55.00 0.891 0. 895 28 1763.5 9 . 00 62.33 0. 849 0. 873 29 1930.0 10.59 63.75 0.795 0. 862 30 2125 . 0 12.35 67.75 0. 808 0. 861 31 2340.0 12 .22 66 , 75 0. 837 0. 883 32 2555 . 0 11.05 52.33 0.873 0 .961 33 2762.5 9.76 46.20 0 ,896 0.966 34 2947,5 7 , 86 55,67 0 . 860 0 . 928 35 3110 . 0 7.62 60.00 0. 859 0, 854 36 3252.5 6.25 44.67 0.803 0. 816 C'O '° TABLE 15-Continued Lot Cumulative Lot Average Lot Average Fabrication Assembly Number Production Monthly Monthly Hours Per Hours Per Plot Point Manufactur- Delivery Pound Pound ing Rate Rate ( 1) (2) (3) (4) (5) (6) 37 3375.0 6.00 39.00 0.816 0,773 38 3475.0 4.21 40.33 0.842 0.765 39 3555,0 4.44 39 .67 0.830 0,723 40 3635,0 4.44 31.00 0,830 0,773 41 3715.0 4.44 25.50 0.813 0,733 42 3800.0 4,74 21.20 0,767 0.721 43 3894.0 4.90 25.25 0.782 o.688 44 3989.0 4.60 28.00 0.815 0.715 45 4062.5 2,89 23.67 0,942 0.689 46 4118.0 2.95 15.75 0.891 0,676 47 4175.0 2.90 13.33 0.928 0.662 48 4220.5 l.4J 6.29 0.969 0.959 49 4252.5 1.29 5.67 1. 010 0.900 50 4283.0 1.30 6.80 0.970 0.924 51 4313.5 1.48 10.67 0.927 0,786 52 4348.5 1.95 12.25 0.907 0 , 861 53 4387,5 2.05 14.50 0.882 0, 847 54 4436.5 3.11 14.80 0.796 0,786 55 4495.0 3.05 16.40 0.763 0,738 56 4558.5 3.45 17.20 0.776 0 , 764 57 4701.0 J. lJ 12.57 0.765 0.745 \.() 0 TABLE 15-Continued SOURCES: Fabrication and assembly hours per lot are extracted from Section 2.2.4 of MCAIR report 7290 entitled "F 4 Cost Data", produced by the McDonnell Douglas Corporation and updated in August of 1975. DCPR weights are read from a McDonnell Douglas summary report entitled "DCPR Weight-Pounds per Airplane" and dated 17 November 1975, Plot points, manufacturing rates and delivery rates are identical to those listed in Table 2. 'I°-' 92 ingrate and for Mode l 7 it is the delivery rate. Obser- vations are taken at lots 16 through 57, Thus Models 6 and 7 are identical except for the different production rate variables, The results of the regression analysis of these models are presented in Tables 16 and 17 for evalua tion and comparison, The statistical tests indicate rejection of the corresponding null subhypotheses for both of the models. The theoretical F value of J .24 is smaller than the F * statistic f or each model, The theoretical F incremental value of 4 .09 is also smaller than the F * incremental statistic in each case. At the outset of this investigation , the labor required to fabricate parts is assumed to be production rate sensitive, Prorating the hours required to set up machine tools and tooling over larger ~uantities at higher production rates can explain this behavior. The assumption holds for the F-4B-F fabrication hours for lots 16 through 51, The sign of the estimated B2 coefficient in Model 6 is negative as anticipated. Furthermore, the explanatory ability of the full model becomes 27 per cent greater than the reduced model. Finally, addition of the manufacturing rate variable to the reduced model causes the MSE to drop some 70 per cent in magnitude. The same type of improvement over reduced model statistics is noted 93 TABLE 16 REGRESSIO N RESULTS - MODEL 6 (F-4 Fabrication Hours) Estimate BO = 6.328 R2 actual = o. 919 Estimate Bl = -0.221 t ratio B1 = 17 .22 Estimate B2 = -0.148 t ratio B2 = 9.75 F * (full) = 155.93 F(2,39),0.05 = J.24 F * (incremental) = 95.10 F(l,39),0.05 = 4.09 MSE (full) = 0.0006 MSE (reduced) = 0,0020 R2 (full) = 0.889 R2 (reduced) = 0.618 TABLE 17 REGRESSION RESULTS - MODEL 7 (F-4 Fabrication Hours) Estimate BO = 7.601 R2 actual = 0.889 Estimate Bl = -0.219 t ratio B1 = 14.71 E!timate B2 = -0.127 t ratio B2 = 7,82 F (full) = 111.60 F(2,39),0,05 = J.24 F * (incremental) = 61.21 F(l,39),0,05 = 4.09 MSE (full) = 0.0008 MSE (reduced) = 0.0020 R2 (full) = 0.851 R2 (reduced) = 0.618 94 for the data in Model 7. The explanatory ability of Models 6 and 7 is also good . As indicat ed by the R2 actual statistic, Model 6 explains 92 per cent of the variance in the hours required to fabr icate parts. The comparable statistic for Mode l 7 is 89 per cent. These other tests of the analysis products also show the production rate as an important variable in explaining variation in fabrication .hours per pound. Therefore, hypothesis two is accepted for Models 6 and 7. As with the other model pairs examined so far, the statistics associated with the model that includes the manufacturing rate are slightly better than those from the model that includes the delivery rate . For example, the R2 actual statis ti c for Model 6 is three per cent higher than the value for Model 7 , The predictive ability of the models is evaluated by estimating the model coefficients with suc cessively smaller data sets. Omitted observations are predicted from each new model as if they were unknown, The observed value for lo t 57 is again chosen as the prediction target . Conclusions about the model 's stability and predictive ability are drawn from a comparison between this target and its prediction. Table 18 presents the results of estimating new coefficients for the full Model 6 . 95 · TABLE 18 PREDICTIVE ABILITY - FULL MODEL 6 {Fabrication Hours: F-4) Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Lot 57a Deviation 42 6,328 -0.221 -0.148 0.825 -7.84 40 6.166 -0.217 -0.150 0.830 -8,50 38 5.990 -0,212 -0.152 0,839 -9.67 36 5.971 -0.211 -0.155 0,840 -9.80 34 5.986 -0.209 -0.162 0.850 -11.11 32 5.991 -0.209 -0.164 0.849 -10.98 30 6.042 -0.212 -0.154 o.844 -10.33 aThe observed value of fabrication hours per pound at lot 57 (case 42) is 0.765. The full model appears to be quite stable when predicting fabrication hours per pound. This conclusion is indicated by the small range of the predictions produced by the seven sets of estimated coefficients. They vary from 0.825 to 0.850 hours per pound over spans from zero to 45 months. The accuracy of the seven predictions for lot 57 is not very good. The forecasts range from 7.84 to 11.11 per cent higher than the observed value for this particular point. These data suggest that a prediction of fabrication hours for lots beyond 57 would also be high. For such a forecast, logic would indicate a downward adjustment of the prediction to compensate for the error. The indicated predictive ability of the model is not good. According to the test plan, deviations larger . than five per cent are unacceptable. Therefore, hypothesis three is not accepted for Model 6. The predictive ability of the full Model 7 is also · tested. As compared to the tests run on Model 6, the results are not as good. The per cent deviation of each prediction is larger. Furthermore, the stability of the model is not as good with a range of O,OJ8 hours per pound for the seven estimates. Therefore, hypothesis three is also rejected for Model 7, COST MODELS 8 AND 9 F-4 ASSEMBLY HOURS PER POUND The hours required to assemble parts into an airframe are also assumed to be inversely affected by the production rate. An increase in rate would logically be implemented with higher loading per station and more stations, These management actions should lead to greater task speciali- zation per worker. This should be accompanied by a conservation of some time that would be expended when changing from one task to another at the lower rate. Furthermore, it is assumed that the sense of urgency that accompanies a higher rate of production motivates the 97 worker to produce faster. The opposite results are expected for a production rate decrease. Cost Models 8 and 9 examine assembly hours per pound as a function of cumulative production and production rate. The proxy for production rate in Model 8 is the program manufacturing rate while the program delivery rate represents the production rate in Model 9. Data for the models are also listed in Table 15, Regression analysis results are reported in Table 19 for Model 8 and Table 20 for Model 9, The statistics produced from Models 8 and 9 support the assumption that assembly hours are also inversely and importantly related to the production rate. For Model 8, with the manufacturing rate independent variable, the estimate of the B2 coefficient is negative. The F * incremental statistic is larger than the theoretical value of 4.09. Similarly, Mode l 9, with the delivery rate variable, also has a negative B2 estimate and an acce ptable F * incremental statistic of 10.96, Another measure of the contribution of the production rate to the model is the improvement in the coefficient of determi- nation from the reduced model to the full model. For Model 8, the improvement is 8,6 per cent while for Model 9 it is 7,5 per cent. Neither model with assembly hours fits the data as 98 TABLE 19 REGRESSION RESULTS - MODEL 8 Estimate BO = 9.016 R2 actual = 0.797 Estimate Bl = -0.279 t ratio B1 = 10.64 Estimate B2 = -0.112 t ratio B2 = 3.62 F * (full) = 56.59 F(2,39),0.05 = 3.24 F * (incremental) = 13. 10 F(l,39),0.05 = 4.09 MSE (full) = 0.0025 MSE (reduced) = 0.0033 R2 (full) = 0,744 R2 (reduced) = 0.658 TABLE 20 REGRESSION RESULTS - MODEL 9 Estimate BO = 10.400 R2 actual = 0,775 Estimate Bl = -0.278 t ratio B1 = 10.33 E;timate B2 = -0.097 t ratio B2 = J.Jl F (full) = 53.46 F(2,39),0.05 = J.24 ➔~ F (incremental) = 10.96 F(l,39),0.05 = 4,09 MSE (full) = 0.0026 MSE (reduced) = 0.0033 R2 (full) = 0.733 R2 (reduced) = 0.658 99 well as Models 4 and 5 with total hours or Models 6 and 7 2 with fabrication hours. For example, the R actual statistic for Model 8 is 0.797 and for Model 9 it is 0.775. The corresponding statistics for the total hours models are o.887 and o.854. Similarly, the corresponding statistics for the fabrication hours models are 0.919 and 0.889. Nevertheless, the statistical and subjective tests do not indicate model rejection and Models 8 and 9 are indicated as appropriate for the data set. Hypothesis two is accepted for both models. The results of the predictive ability test for Model 8 (Table 21) are good. Little change is observed when removing the first eight observations in steps of two. The forecast of the assembly hours required to produce lot 57 with a model developed from J4 cases spans about J4 months. Yet, the deviation of the prediction from the target is only 0.53 per cent. The prediction with 32 cases is the only one that exceeds the arbitrarily selected five per cent limit. The predictive ability of the model developed with different numbers of cases appears to be stable. The range of the seven forecasts is only 0,052 hours per pound. The predictive ability results for Model 9 (Table 22) are not quite as good. While the predictions center nicely 100 TABLE 21 PREDICTIVE ABILITY - FULL MODEL 8 (Assembly Hours: F-4) Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Lot 57a Deviation 42 9.016 -0.279 -0.112 0.750 -0.67 40 9,042 -0.279 -0.112 0.752 -0.94 JS 9,041 -0.279 -0.112 0.752 -0.01 36 9.111 -0.282 -0.105 0.745 o.oo 31.J. 9,060 -0.28J -0.097 0,741 0.53 32 8.812 -0.294 -0.041 0.700 6.04 JO 8.601 -0.283 -0.069 0.726 2.55 aThe observed value at lot 57 (case 42) is 0,745 assembly hours per pound. TABLE 22 PREDICTIVE ABILITY - FULL MODEL 9 (Assembly Hours: F-4) Cases Estimate Estimate Estimate Forecast Per Cent Bo Bl B2 Lot 57a Deviation 42 10,406 -0.278 -0,097 0,776 -4.16 40 10.402 -0.277 -0.098 0,780 -4.?0 38 10.365 -0.277 -0.099 0,775 -4.03 36 10.420 -0.281 -0.092 0.767 -2,95 34 10.142 -0.283 -0,081 0.755 -l.J4 J2 8,587 -0.294 -0.0lJ 0.692 7,11 JO 9.066 -0.28J -0.049 0.732 1,74 aThe observed value for lot 57 (case 42) is 0,745. 101 on the target of 0,745 assembly hours per pound, they range over 0,098 hours per pound. This is almost twice the range of the predictions from Model 8 and indicates that Model 9 is less stable than Model 8 . Tests for the predictive ability of reduced Models 8 and 9 are reported in Table 23. All of the predictions are low and the magnitude of the deviations is larger than those for full Models 8 or 9 . This reduced model is fairly stable with a prediction span of 0.056 hours per pound . TABLE 23 PREDICTIVE ABILITY - REDUCED MODELS 8 AND 9 (Assembly Hours: F-4) Cases Estimate Estimate Forecast Per Cent BO Bl Lot 57a Deviation 42 5.852 -0.245 0.737 1.07 40 5,912 -0.247 0.732 1.74 38 5,984 -0.249 0.729 2.15 36 6.430 -0.259 0.711 4 .56 34 6.996 -0.271 0.707 5.10 32 8.185 -0.294 0.6 81 8.59 30 7,931 -0.288 o.688 7.65 a The observed value for lot 57 (case 42 ) is 0,745 assembly hours per pound. The production rate variable helps in the accuracy of prediction in both models and hypothesis three is accepted. But i n comparing Models 8 and 9, Model 8 with the manu- 102 facturing rate representing the production rate is the better of the two. This correlates with the findings on all of the model pairs examined to this point. Another interesting fact is that the fit of the model to the data set improves with an increasing percentage of fabrication hours in the dependent variable. For example, Models 8 , 4 and 6 are identical except for the dependent variable. This variable is assembly, total and fabrication hours per pound respectively. The corresponding R2 actual statistics are 0,797, 0.887 and 0.919. One can estimate the fabricati on hours for Models 8, 4 and 6 as a percentage of total hours. This percentage is O, 33 and 100 respectively. This increasing percentage of fabrication hours with increasing R2 actual may indicate that prorating fabrication set up time is the most important of the reasons why the production rate is an explainer of variation in hours per pound. 103 THE F-102 PROGRAM The F-102 airframe production data are extracted from the "F-102 Program Cost History" which is prepared by the manufacturer to assist in pricing, estimating and cost evaluation. 44 This history is a comprehensive publication containing information on many facets of the F-102 program. One is left with the impression that great care has been taken to assign accurately the cost elements to the proper category or end item. For the purpose of this study, the data are limited in that lot release dates and hours for lower process levels by lot are not available. But total direct production hours by individual airframe are available. These data are matched with airframe delivery data to produce the variables needed to test the procedure in the cumulative production and production rate cost model. Deliveries of 1000 aircraft are made during the years 1953 through 1958. Of these 1000, 889 are F-102A and 111 are TF-102A aircraft. The monthly delivery rate of the 1000 is somewhat irregular as reflected in column 3 of Table 24. For the first 26 months, the maximum monthly delivery rate is three. Over these 26 months, 44 "F-102 Program Cost History ," CRA-1-7, a report prepared by the Cost Research Department, Fort Worth Division, General Dynamics Corporation, June 1965. 104 TABLE 24 VARIABLES FOR ANALYSIS OF F-102A TOTAL HOURS PER POUND Month Cumulative Monthly Total Hours Production Delivery Per Pound Plot Point Rate ( 1) (2) (J) (4) 1 1. 1 JJ.JJ J 1.5 1 Jl.lJ 7 J.O 2 21.72 8 4.5 1 20.72 9 5.5 1 20.38 10 7.0 2 18.81 11 8.5 1 17.44 12 9.5 1 17.28 15 10.5 1 26.92 16 11.5 1 23.57 17 12.5 1 22.32 18 13.5 1 21.74 21 14.5 1 20.35 22 16.0 2 18.27 23 17.5 J 15.22 24 21.0 2 13.44 27 26.0 6 lJ.JO 28 32.0 6 10.57 29 J8.0 6 9,66 JO 45.5 9 8 ,57 31 52.0 4 7.62 32 55.5 J 7,74 33 60.0 6 7.31 J4 68 .0 10 6.73 35 80.0 14 6.60 36 92.5 11 6.18 37 112.5 29 5.56 JS 148.5 43 5.19 39 185.5 31 4.69 40 221.0 40 4.79 41 257.5 33 4.14 42 293.5 39 4.01 43 337.0 48 3,84 44 384.5 47 3.72 45 435.0 54 3.38 1+6 482.0 40 3.17 47 526.5 49 J.26 105 TABLE 24-Continued Month Cumulative Monthly Total Hours Production Delivery Per Pound Plot Point Rate (1) (2) (J) (4) 48 567.5 JJ J.24 49 604.0 40 J.11 50 649.5 51 J.02 51 696.0 42 2.97 52 740.5 47 3.00 53 787.5 47 2.90 54 835.0 48 2.85 55 879.5 41 2.87 56 917.5 35 2.86 57 950.0 JO 2.85 58 975.0 20 2.92 59 990,0 10 J.02 60 997,5 5 J.42 SOURCE: The data for constructing the variables in this table are extracted from the "F-102 Program Cost History," numbered CRA-1-7 and dated June 1965. The summary is developed from the joint accounting records of the General Dynamics Corporation Convair and Fort Worth Divisions and published by the Fort Worth Division. The section for construction of the dependent variabl e is entitled "F-102A and TF-102A Final Tabulation of Charted Hours and Manhours per Pound". It includes direct labor hour and DCPR data for each airframe produced. The delivery information used for constructing the independent variables is taken from a section entitled "F-102 Deliver- ies". 106 there are ten during which no airframes are delivered. Then the rate increases gradually over the next 24 months to peak at 51. The last ten months are characterized by a generally decreasing rate. The cumulative production and production rate cost model, y = B B1 B2 0 • x1 • x 2 • 10 e , is once more used to examine the data. In this case, construction of the variables is easier because the hourly data are collected by individual airframe. The dependent variable is average direct production hours per pound for each month. It is constructed by calculating the arithmetic average of the hours required to build the F-102A airframes delivered in each month. This number is divided by the arithmetic average DCPR weight of those same airframes. Column 4 of Table 24 lists the dependent variable for the 50 months during which deliveries are made. The first independent variable is the cumulative production plot point. It is calculated on a monthly basis for the program total of 1000 airframes even though the dependent variable is constructed for the 889 F-102A air- frames. This is done because it is assumed that learning on all similar airframes contributes to the unit cost be- havior of a particular airframe model. The cumulative pro- duction plot point is simply the middle value for monthly 107 production plus the cumulative total production for the previous months. It is listed in column 2 of Table 24. The s econd i ndependent var i able is the program monthly delivery rate as determined by the date of customer acceptance of the airframe. It is listed in column 3 of Table 24 for those months in which some delivery is made. The ten months during which there are no deliveries are omi t ted as cases for the regression analysis. As previously indicated, the raw data needed to construct a manufacturing rate variable are not available. The F-102A data of Table 24 reflect the familiar toe-up phenomenon at the end of the program. For the last three months, the average hours per pound increase sharply due to the effects of program completion actions. This phenomenon is lacking in the F-4 data just examined because production of F-4 airframes continues through the most recent observations. For the F-102A part of the i nvestigation, two versions of t he cumulative production and production rate cost model are examined using the test procedures. They are Model 10 with all 889 F-102A airframes aligned in 50 monthly cases and Model 11 with the last 879 F-102A airframes delivered over 42 months. 108 COST MODEL 10 F-102A TOTAL HOURS - 50 CASES Table 25 lists t he results of regression analysis of the F-102A data in the cost model. The statistics reflect that the delivery rate is an important explanatory variable in the cost model. The F * incremental statistic of 38 . 39 is larger than the theoretical F value of 4.05. The R2 stat istic for the full model improves by 1.8 per cent over the same statis tic for the reduced model . Similarly, the MSE of the full model is about one-half that of the reduced model. The model also appears to fit the data well. The R2 statistic of 0.979 indicates that 97, 9 per cent of the variation in the logarithm of hours per pound is explained by the logarithm of the two independent variables. The R2 actual statistic of 0.933 indicates that some 93 per cent of the observed variation in hours per pound is explained by the cost model. The F * statistic of 1071.11 is much greater than the theoretical value of 3.20. This supports the theory of a relationship between the dependent variable and the set of independent variables. An examination of the residuals shows a sharp discontinuity between cases eight and nine. At this point the hours per pound dependent variable increases abruptly and begins what appears to be a new t rend. Perhaps there 109 TABLE 25 REGRESSION RESULTS - MODEL 10 (F-1 02A Total Hours - 50 Cases) Estimate 2 BO = 38.371 R actual = 0.933 Estimate Bl = -0.299 t ratio B1 = 14.95 Estimate B2 = -0.158 t ratio B2 = 6.08 F * (full) = 1070.11 F(2,47),0.05 = 3.20 F * (incremental) = 38.39 F(l ,47) ,0.05 = 4 .05 MSE (full) = 0.0029 MSE (reduced) = 0.0051 R2 (full) = 0.979 R2 (reduced) = 0.961 Case Observed Predicted Residual Per Cent Value Value Deviation 1 33.33 38 . 37 -5.04 -15.12 2 31 .1 3 33.99 - 2 . 86 -9.19 3 21.72 24.76 -3,04 -14.00 4 20.72 24.48 -J.75 -18.15 5 20.38 23.05 -2.68 -13.10 6 18.81 19.22 -0.41 -2.18 7 17.44 20.24 -2.80 -16.06 8 17.28 19.58 -2.30 -13.31 9 26.92 19.00 7.92 29.42 10 23 .57 18.49 5.08 21 .55 11 22.32 18.04 4 . 28 19 . 18 12 21 . 74 17.63 4.11 18.91 lJ 20.35 17.25 3 .10 15.23 14 18.27 15.01 3.25 17.84 15 15.22 13. 71 1.51 9.92 16 lJ.44 13.84 -0.40 -2.98 17 13.30 10.91 2.39 17.97 18 10.57 10.25 0.31 2.93 19 9,66 9.74 -0.08 -0. 83 20 8 .57 8.66 -0.09 -1.05 21 7,62 9,46 -1. 84 -24.15 22 7.74 9 ,71 -1. 96 -25.32 23 7.31 8 . 50 -1.19 - 6.28 24 6.73 7,55 -0.82 -12.18 25 6.60 6.82 -0.22 -3,33 26 6.18 6.78 -0.60 -9,71 110 TABLE 25-Continued Case Observed Predicted Residual Per Cent Value Value Deviation 27 5.56 5.49 0.07 1.26 28 5 .19 4.75 0.44 8.48 29 4.69 4.68 0.01 0,21 JO 4,79 4 . 26 0,53 11.06 Jl 4.14 4.20 -0.06 -1.45 32 4.01 J,93 0,08 2.00 33 3,84 3 ,65 0,19 4,95 34 3,72 3.52 0,20 5,38 35 3.38 3,32 0,06 1.78 36 3,17 3.38 -0.21 -6,62 37 3.26 3.18 0.08 2.45 38 3,24 3.31 -0.07 -2.16 39 3 .11 3.16 -0.05 -1. 61 40 3 .02 2,97 0,05 1.66 41 2.97 3.00 -0.0J -1,01 42 J.00 2.89 0.11 3,67 43 2 .90 2.84 0.06 2.07 44 2.85 2,78 0,07 2 . 46 45 2.87 2.81 0.06 2.09 46 2.86 2.84 0.02 0,70 47 2.85 2.88 -0.0J -1. 05 48 2,92 3.05 -0,lJ -4.45 49 3.02 3.39 -0.37 -12.25 50 3.42 3,78 -0,35 -10.23 111 is a process change or major design change introduced at this point. Except for the toe-up at the end, the rest of the residuals do not reflect abnormal distribution. Hypothesis one is therefore accepted for Model 10. The predictive ability of the full Model 10 is examined in the data of Table 26. Since the purpose of the model is to predict follow-on production, month 57 (case 47) prior to the toe-up effect is chosen as the prediction target. TABLE 26 PREDICTIVE ABILITY - FULL MODEL 10 Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Month 57a Deviation 50 38.371 -0.299 -0.158 2.88 -1.05 48 36.852 -0.273 -0.189 2.98 -4.56 46 36.171 -0.263 -0.200 3.02 -5.96 44 35.975 -0.260 -0.202 3.04 -6.67 42 35.933 -0.259 -0.202 3.06 -7.37 40 35.826 -0.258 -0.203 3.06 -7,37 38 35.655 -0.255 -0.205 3.09 -18.42 36 35,370 -0.252 -0.207 3.11 -9.21 J4 35.034 -0.247 -0.209 3.16 -10.88 aThe observed value for month 57 (case 47) is 2.85 total hours per pound. 112 The predictive ability of the procedure for this set of data deteriorates more rapidly than it did with similar F-4 data. As cases are removed from the original 50, the resulting models appear to systematically develop higher predictions. When predicting only 13 months into the future, the 34 case model predicts almost 11 per cent high. This exceeds the test limit and hypothesis three is rejected. The results of the predictive ability tests for a reduced Model 10 are presented in Table 27. Since the reduced model has a high R2 statistic of 0,961, one might expect the predictive ability of these reduced mo dels to be good. They are not. For a prediction of only one month from the model developed with 46 cases, the prediction is 8.77 per cent low. Longer predictions with coefficients develope·d from fewer cases are increasingly poor. When comparing the predictive results of the full model with those of the reduced model, one can conclude that the full model is better. For example, with 38 cases in the regression analysis , the full model predicts 8.4 per cent high while the reduced model predicts 11.6 per cent low. While neither prediction is particularly good for only a nine month span, the comparison shows that the delivery rate is an important additional variable in the model. 113 TABLE 27 PREDICTIVE ABILITY - REDUCED MODEL 10 Cases Estimat e Estimate Forecast Per Cent BO Bl Month 57 Deviation 50 44 .162 -0.408 2.69 5.61 48 44.883 -0.414 2.63 7,72 46 45.2 82 -0.417 2.60 8,77 44 45.623 -0.420 2.56 10.18 42 45,859 -0.521 2.55 10.53 40 46 .083 -o.423 2,53 11.23 38 46.178 -0.424 2.52 11.58 36 46.321 -0.425 2.51 11.93 34 45,966 -0. 422 2.55 10.53 The discontinuity in the dependent variable data between cases eight and nine suggests that a better model can be constructed from the last 42 cases of the 50 case F-102A data set. Model 11 is developed from these 42 cases. The results of the analysis are listed in Table 28. Once again the statistics indicate that the delivery rate is an important explainer of variation in hours per pound . In Model 11, the R2 statistic is increased by two per cent when adding the delivery rate to the reduced model. The F * incremental statistic of 36.37 is larger than the theoretical F value of 4 . 09 at the 0,05 signifi- cance level. 114 TABLE 28 REGRESSION RESULTS - MODEL 11 (F-102A Total Hours - 42 Cases) Estimate BO = 47 . 290 R 2 actual = 0,955 Estimate Bl = -0.344 t ratio B1 = 16.44 Estimate B2 = -0.144 t ratio B2 = 6 ,03 F * (full) = 909 , 46 F(2,39),0,05 = 3.24 ➔~ F (incremental) = 36,37 F(2,39),0.05 = 4.09 MSE (full) = 0.0022 MSE (reduced) = 0,0041 R2 (full) = 0,979 R2 (reduced) = 0.959 The model appears to be appropriate for the data set. The R2 actual statistic is 0,955 indicating that over 95 per cent of the variat ion in hours per pound is explained. The residuals indicate no important problems with the model, The F * statistic is much larger than the critical value. Therefore, hypothesis one is sustained for Model 11. Tests of the predictive ability of models estimated from the data set of Model 11 are also made, These pre- dictions of month 57 are low in every test with generally larger per cent deviations than those from the data set of Model 10, Hypothesis three is also rejected for Model 11. The stability of Model 11 is also not as good as that of Model 10, The range of nine forecasts using the data of Model 11 is 0,63 hours per pound. These forecasts span 16 115 months in steps of two. The corresponding forecasts for Model 10 range over only 0,27 hours per pound. As with the F- 4 program data, it appears that the early observations are important in both locating and stabilizing the re- gression plane. TABLE 29 PREDICTIVE ABILITY - FULL MODEL 11 Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Month 57a Deviation 42 47.290 -0.344 -0.144 2.75 3.51 40 45.994 -0.330 -0.158 2.80 1,75 38 46.017 -0.330 -0.158 2.80 1,75 36 47.263 -0.341 -0.149 2,74 3.51 34 51.254 -0.374 -0.125 2.58 9.47 32 48 , 961 -0.356 -0. 138 2.69 5.61 30 53.964 -0.394 -0.110 2,49 12.63 28 59.428 -0.433 -0.083 2.30 19.30 26 64.186 -0.462 -0.065 2.17 23.86 aThe observed value for month 57 ( case 39) is 2.85 hours per pound. 116 THE KC-lJSA PROGRAM The last program examined in this project is pro- duction of the KC-1J5A four engine tanker airframe. Deliveries from this program span the years 1957 through 1965. There are 820 aircraft in the C-135 family of which 732 are KC-1J5As. The balance of the aircraft are a mix of six other models produced during the last half of the program. The KC-1J5A model is selected for the analysis because it is produced·from the beginning to the end of the program and accounts for the great majority of the direct labor hours spent in production. Data for this analysis are derived from the historical records of the manufacturer. Two different data sets are constructed to facilitate the tests described in chapter four. One is based on 96 months of deliveries and the other is based on seven production lots. From these data, five models are developed and examined. COST MODEL 12 TOTAL HOURS: KC-lJSA-96 MONTHS , Table JO lists the variables needed to develop cost Model 12. The variables are constructed in the same manner as those for the F-102A. Deliveries per month are a count of all C-135 family airframes accepted in each month. The cumulative production plot point is the middle value of 117 TABLE 30 VARIABLES FOR ANALYSIS OF KC -135A TOTAL HOURS Month Cumulative Monthl y Total Hours Production Delivery Per Pound Plot Point Rate 1 1 . 1 9 , 82 2 1.5 1 7 , 59 3 2 . 5 1 7 , 31 4 4.0 2 7 , 36 5 6.5 3 7 , 62 6 10 . 0 4 6 . 05 7 14.0 4 5 . 69 8 16,5 1 5 , 52 9 19.0 4 5.29 10 23 . 5 5 5 . 22 11 28 . 0 4 5 .15 12 33.5 7 4.94 13 41.0 8 4 . 22 14 50.0 10 3 . 68 15 60 . 5 11 3 .16 16 72 . 0 12 2,75 17 84,5 13 2 . 46 18 98,5 14 2 . 34 19 112.5 15 2.15 20 127 , 5 15 2 . 04 21 142 . 5 15 2 . 01 22 157 . 5 15 1.99 23 172 . 5 15 1.96 24 187.5 15 1 . 91 25 202.5 15 1.86 26 217,5 15 1 . 82 27 232,5 15 1. 79 28 247.5 15 1. 75 29 262 . 5 15 1 . 60 30 277 . 5 15 1. 53 31 292 . 5 15 1 ,47 32 306.0 12 1 . 41 JJ 322.0 10 1 . 36 34 326,5 9 1 , 31 35 335,0 8 1 . 31 36 343 , 0 8 1 , 29 37 351 , 0 8 1.24 38 359 , 0 8 1 . 22 118 TABLE JO-Continued Month Cumulative Monthly Total Hours Production Delive ry Per Pound Plot Point Rate 39 366.5 7 1.18 40 373 , 5 7 1 .17 41 380 .5 7 1.16 42 J88.0 8 1.13 4J 395.5 7 1.12 44 402 . 5 7 1.11 45 409 . 5 7 1.09 46 417 . 0 8 1.06 47 424.5 7 1 . 06 48 4J2 .0 6 1 . 06 49 4J8 .0 6 1 . 05 50 444 .o 6 1 . 05 51 450,0 6 1.05 52 456.o 6 1.04 53 462 . 0 6 1 . 03 54 468.0 6 l . OJ .55 472 ,5 5 1.02 56 478,0 6 1 . 01 57 484 . 0 6 1 . 00 .58 490.0 6 1 . 00 59 497 ,0 8 1. 01 60 504.5 7 1 .01 61 511 .5 7 1.01 62 518 .5 7 1.00 63 527 .0 10 1 .06 64 537 ,0 10 1 . 09 65 548.0 12 1.11 66 560.0 12 1.10 67 569,5 9 1.09 68 579 , 0 . 8 1.06 69 587 .0 8 1. 04 70 596,5 11 l .O J 71 604 ,.5 5 1.04 72 611 .0 8 1.02 73 619.0 8 1.01 74 627.5 9 1. 01 75 635,5 7 1.04 76 54J.O 8 1 . 07 77 651 .0 8 1.00 78 6.59,0 8 1.00 79 667.0 8 1.00 80 675 . 0 8 0.98 119 TABLE JO-Continued Month Cumulative Monthly Total Hours Production Delivery Per Pound Plot Point Rate 81 683.0 8 0.98 82 691.0 8 0.99 8J 699~0 8 1,00 84 707 ,5 9 1.04 85 715,5 7 1.02 86 722.5 7 1.02 87 729,5 7 l,OJ 88 736.5 7 1.02 89 744.0 8 1.01 90 752 , 5 9 0,98 91 762.5 11 0,97 92 773,0 10 0,95 93 782.0 8 0,95 94 790,5 9 0.92 95 799,0 8 0,90 96 803,5 7 0.89 SOURCES: A summary of important dates for each air- frame is contained in a series of charts entitled "Military Program Actuals" prepared by the Program Planning Unit, Airplane Division, The Boeing Company. Each report in the series bears the chart number 1-135-25. The charts are dated 1 April 1960 (8 pages), JO June 1962 (2 pages), Jl March 1964 (2 pages) and Jl October 1966 (2 pages). These charts are the source of the monthly delivery rate and the cumulative production plot point. The total hours per pound per month are read and in some cases estimated from a summary chart entitled "Manhours Per Pound KC-135" and dated JO September 1965. This chart is numbered 4-135-7, consists of two pages and is prepared by the Manufacturing Department , Airplane Division, The Boeing Company. Although data for the first 20 and the last 13 airframes are reported individually, some of the intermediate airframes are reported in steps of five or ten. Therefore it is sometimes necessary to estimate the manhours per pound for the average airframe delivered in a particular month. 120 C-135 family acceptances in each month plus the cumulative total of prior acceptances . Total hours per pound are for the KC -1J5A airframe only. They represent the average total direct production labor hours divided by the average DCPR weight for the airframes accepted in each month. The data reflect an early peak production rate of 15 airframes pe r month , a decline to five per month followed by a second peak of 12 per month . After another decline to seven per month, there is a f inal peak of 11 per month. The data needed to develop fabrication and assembly hour variables t o match this total hour data set are not available. Also, a manufac turing rate variable cannot be produced from the available data. Therefore, only one model is examined with this data set. The results of the regression analysis are listed in Table 31, All of the statistics indicate that the model fits the data set well. The F * full statistic is much larger than the theoretical F value of J,10 indicating that the set of independent variables is related to the dependent variable. The R2 statistic (in logarithms) reflects that 97 per cent of the variance in the dependent variable is captured by the model. The R2 actual sta- tistic, showing explained variance after transformation back to the original variable form, indicates that 89 per cent of the variance in hours per pound is explained by 121 the model. TABLE 31 REGRESSION RESULTS: MO DEL 12 (KC -135A Total Hours-96 Cases) Estimate BO = 13.133 R2 actual = 0.890 Estimate Bl = -0.453 t ratio Bl = 47.60 Estimate B2 = o.16u t ratio B2 = 6.56 F * (full) = 1558,37 F(2,93),0.05 = 3 .10 F * (incremental) = 42,92 F(l ,93),0,05 = 3,95 MSE (full) = 0.0022 MSE (reduced) = 0.0032 R2 (full) = 0.971 R2 (reduced) = 0,958 The delivery rate is also shown to be an important explanatory variable. The F * incremental statistic of 42,92 is larger than the critical F value of 3,95 at the 0.05 significance level. The R2 statistic shows an improvement of 1,3 per cent after adding the rate variable to the reduced model. Similarly, the MSE statistic shows a one-third reduction due to inclusion of the rate variable in the model. The fact that the R2 statistic improves by only 1,3 per cent after adding the rate term to the reduced model raises doubt about the practicality of including the rate term in the KC-135A model. Further analysis of the variance eases these doubts. The SSE(x1 ) for the 96 cases is 0.301. It repres ents the unexplained variance (in 122 logarithms) when only cumulative production is in the model. SSE(x1_,x2 ) is 0,206 and r epresents the unexplained variance with both cumulative producti on and delivery rate in the model . The reduction in unexplained variance due to adding the rate variable when divided by SSE(x1 ) is the partial coefficient of determination. It is (O,JOl - 0,206)/0.JOl = 0.316. 45 From this analysis, one can conclude that about one-third of the unexplained variance from the reduced model is accounted f or by adding the rate variable. One unusual result of the KC-135A analysis is that the estimate of the B2 coefficient in the full model is positive. If one assumes that an increase in production rate causes a decrease in unit labor requirements, the expected sign of the B2 coe fficient is negative. The unexpected result can be explained as a twist in the regression plane brought about by the difference in relative strength of the independent variables and their collinearity. The reduced form of the full model provides a clue to this twisting effect cause. The estimate of the B1 coefficient with the x 2 variable removed is -0.415. The estimate of the B2 coeffici ent with the x1 variable removed 45Neter and Wasserman, Statistical Models , p. 411. 123 is -0.573. Yet when both the x1 and x 2 variables are in the full model, the B2 sign becomes positive. In their text on statistical analysis, Spurr and Bonini attribute this type of behavio r to collinearity among the independent vari. a bl es. 46 The x1 and x 2 variables are s omewhat correlated in this KC-135A case. The coefficient of correlation is 0.619. This statistic must be evaluated with some caution because as production rate declines, cumulative production continues to increase until the rate reaches zero. The x1 variable appears to be much stronger than the x 2 variable when explaining the variance of the dependent variable. For example, the R2 statistic of the reduced model with x 2 removed is 0.958. The R2 statistic of a model with x1 removed is only 0.265. This combination of collinearity and a lar ge differ- ence in relative strength of the independent variables causes the regression plane to tilt and the B2 coefficient to become positive. There does not appear to be any theo- retical significance to this change in sign and the model should be useful for predictive purposes. Subjective analysis of the model behavior and the 46William A. Spurr and Charles P. Bonini, Statistical Anal sis for Business Decisions (Homewood, Illinois: Richard D. Irwin, Inc., 1967 , p. 611. 124 residuals indicates no serious departures from the model assumptions. Accordingly, hypothesis one is accepted for the 96 case KC -135A data set. The predictive ability of the model is also examined. Results of the tests are listed in Tables 32 and 33, Parallel forecasts for the full and reduced models are made in steps of 12 months. Both model forms forecast fairly well. The full model form forecasts low in every case with a range of only 0.04 hours per pound over the five forecasts. The reduced model form predicts high in every case with a range of 0,05 hours per pound. One cannot say that adding the production rate improves the predictive ability of the full model over the reduced model very much . One can accept hypothesis three for data set of Model 12 . The data for Model 12 are not adaptable to evaluating the different levels of the KC - 1J5A manufacturing process. Therefore a different data set is constructed to evaluate hypothesis two for the KC-135A program . The 82 0 airframes in the C-135 family include seven different airframe models. They are 732 KC-135A, 17 EC-135C, 4 RC-1J5A, 10 RC-1J5B, 15 C-135A, JO C-1J5B and 12 C-135F aircraft. Labor requirement data are collected by the manufacturer at different process levels for these airframes by lot. These data are identified 125 TABLE 32 PREDICTIVE ABILITY - FULL MODEL 12 Cases Estimate Estimate Estimate Forecast Per Cent BO Bl B2 Month 96a Deviation 96 13.133 -0.453 0.164 0.87 2.2 84 lJ.244 -0.461 0.176 0.85 4.5 72 13.373 -0.468 0.186 0.84 5.6 60 13,421 -0.473 0.195 0.83 6.7 48 13.384 -0.468 0.186 o.84 5·.6 aThe observed value for month 96 is 0.89 hours per pound. TABLE 33 PREDICTIVE ABILITY - REDUCED MODEL 12 Cases Estimate Estimate Forecast Per Cent BO Bl Month 96a Deviation 96 14.782 -0.415 0.92 -3.3 84 14.891 -0.417 0.92 -3.3 72 14.945 -0.418 0.91 -2.2 60 14.917 -0.417 0.92 -3.3 48 14.225 -0.401 0.97 -9.0 aThe observed value for month 96 is o.89 hours per pound. 126 to a particular model through lot seven but are aggregated for all airframes for lots eight through ten, Therefore only seven lots of data are available for the KC-135A analysis. Table 34 lists the data needed to evaluate Models 13 through 16. The cumulative production plot point is constructed by adding one-half the lot size to the cumulative total number of airframes produced in prior lots. In accordance with learning curve technique, the plot value for the first lot is adjusted to compensate for the rapid change in the slope of the cumulative production curve for the first few units. A plot point value of 10.14 is read from the tables for a first lot size of 29 airframes with a slope of 78 per cent. 47 The lot average manufacturing rate is constructed by dividing the number of airframes in a lot by the time span in months required to manu facture those airframes. This span bridges the time the first batch of parts in the lot is released to fabrication until the last airframe in the lot is accepted. For this program, the first month that any fabrication hours are recorded for each lot is used as a proxy for the lot release date. The lot average delivery rate is constructed by 47 Boren and Campbell, Learning Curve Tables, p . 162. 127 TABLE 34 VARIABLES FOR ANALYSIS OF KC-135A FABRICATIO N AND ASSEMBLY HOURS Lot Cumulative Lot Lot Fabri- Assembly Production Ave rage Average cation Hours Plot Point Monthly Monthly Hours Per Manu- Delivery Per Pound facturing Rate Pound Rate 1 10.14 0 . 81 3,63 0 .509 2 ,933 2 63.00 2.52 9 ,71 0.235 1.65.5 3 156 .00 4.21 14.75 0.186 0.978 4 280.00 4.33 13.00 0.191 0.722 5 385,50 3.00 7.36 0,179 0.511 6 459.00 2.44 5,50 0.188 0.439 7 534,50 3 , 27 6.54 0.161 0,471 SOURCES: Lot composition data i s obtained from a report ent i tled "Ma jor Pro duc t ion Airplanes-Military" prepared by Industrial Engineering, Ai rplane Division, The Boeing Company and dated J J anuary 1966. The mont hs during which the first fabr ication hours are expended are extracted from a machine listing of the KC-135 Program data. The data bank is maintained by Production Planning , Industrial Engineering, 707/727/737 Divi sion of the Boeing Commercial Airplane Company. Delivery data needed to construct the two forms of the production rate are extracted from the "Military Program Actuals" series of reports more c·ompletely described in Table JO. Fabrication and assembly hours by lot are read from a three page report numbere d 132,3115 and entitled "Pro- duction Hours, Model KC-135, By ·v ork Order By Control Code." It is prepared by Program Planning, Industrial Engineering , Airplane Divi sion , The Boe ing Company and dated March 1968 . DCPR weights for constructing the dependent variables are estimated from a Boeing Company summary report entitled "Manufacturing On and Off-Site Dire ct Manhours by Lot." This one page report is da ted 1 March 1965. 128 dividing the number of airframes in a lot by the time span in months from the first to last acceptance, This span is ca lculated directly f rom the actual acceutance dates which are available in the data. The fabrication and assembly hours per pound variables are constructed by dividing the total hours required to fabricate or assemble the parts for each lot by the number of airframes in the lot and the lot DCPR weight. The assembly hours are the sum of two reporting codes used by the manufactu rer, minor and major assembly, The cumulative production and production rate cost model form is used t o examine four data sets lis ted in Table 34. The fabrication and assembly hours per pound are used as alternate dependent variables while the independent variables are the cumulative production plot point and one of the two forms of the production rate. It is unfortunate that the lots are so large and are released so infrequently (about once a year) . More obser - vations would permit a more precise evaluation of the rate effect on labor requirements. Nevertheless, after working with many different data sources for the KC-135A, one is left with the impression that the raw data for these seven cases are carefully collected and allocated. 129 COST MODELS 13 AND 14 KC-135A FABRICATIO N HOURS PER POUND Cos t Model 13 examines fabricati on hours per pound as a funct ion of cumulative production and .manufacturing rate. The r egression analysis results are presented in Table 35 . TABLE 35 REGRESSION RESULTS - MODEL 13 Estimate 2BO ::, o.674 R actual ::, 0 . 986 Est i mate Bl ::, -0.165 t ratio B1 = 4 . 47 Estimate B2 = - 0.305 t ratio B2 = 3.30 •,t- F (full) = 74 , 51 F(2 ,4 ),0,05 = 6 . 94 F * (incremental) ::, 10 . 88 F(l,4),0 .05 ::, 7 , 71 MSE (full) ::, 0 .0 011 MSE (reduced) = O.OOJ4 R2 ( fu ll) ::, 0 . 974 R2 (reduced ) ::, 0.903 Case Observed Predicted Residual Pe r Cent Value Value Deviation 1 0.509 0, 491 0 . 018 3,55 - 2 0 . 235 0,257 -0.022 - 9 . 40 3 0.1 86 0 .189 - 0 . 00J - 1 . 80 4 0.191 0 . 170 0,021 10,74 5 0 .179 0 . 181 - 0 , 002 - 1 . 04 6 0 .188 0.1 87 0 , 001 0,45 7 0.161 0.167 -0, 006 -3 . 69 Al though only seven observations are used in t he analysis , the statistics support the procedure and the theory . The F * statistic is large r t han its cri t ical 130 value indicating that the set of independent variables is related to the dependent variable. The F * incremental statistic is also larger than its critical value. This indicates that the manufacturing rate variable explains additional variation in the fabrication hours per pound variable in the presence of the cumulative production variable. Both of these conclusions are supportable at the 0,05 level of significance. The following observations indicate that the model is appropriate for the data. The residuals do not exhibit behavior that indicates violation of model assumptions. The R2 actual statistic of 0.986 indicates that almost 99 per cent of the variance in the fabrication hours per pound is explained by the model. Both the MSE and R2 regre ss ion statistics improve substantially upon addition of the manufac t uring rate variable to the reduced model. It is interesting to note that the sign of the B2 coefficient estimate does not become positive as it does in Model 12. Th i s fact supports the observation made in the F-4 analysis that fabrication hours are more sensitive to rat e than assembly .or total hours. Model 13 with the x1 variable excluded has a negative B2 coefficient estimate and an R2 s tatist ic of 0, 843 . With the x 2 variable excluded, the B1 coefficient is ne gative and the R 2 statistic is 0.903. The cumulative production variable is 131 not enough stronger than the rate variable t o cause a sign change as in Model 12. The analysis indicate s that hypothesis two is acceptable. Cost Mode l 13 is appropriate to explain variation in KC -135A fabrication hours per pound as a function of cumulative production and production rate for the first seven lots. The test of hypothesis three, predictive ability, is impractical in this and t he next thre e models because there are only seven observations. Removal of observations for further analysis leaves insufficient degrees of freedom to attach much importance to the results of a predictive ability test. Cost Model 14 examines KC-135A fabrication hours per pound as a function of cumulative production and delivery rate. The only difference between Models 13 and 14 is that 13 uses the manufacturing r a te and 14 uses the delivery rate form of the production rate. The regression results of Model 14 are listed in Table 36. As one would anticipate, the regression results are similar to those of Model 13 . The objective and subjective tests of the results do not indicate rejection and hypo- thesis two i s also accepted for Model 14. 132 TABLE 36 REGRESSION RESULTS - MODEL 14 Estimate BO = 1.1 23 R2 actual = 0.985 Est i mate Bl = -0. 2JJ t ratio B1 = 9 ,18 Estimate B2 = -0 .222 t ratio B2 = 3,03 F * (full) = 65.90 F(2 ,4),0,05 = 6 , 94 F * (incremental) = 9 . 20 F(l , 4) ,0, 05 = 7,71 MSE (full) = 0 , 0013 MSE (reduced) = O.OOJ4 R2 ( ful l) = 0, 971 R2 (reduced) = 0.903 Case Observed Predicted Residual Per Cent Value Value Deviation 1 0 ,509 0,491 0 . 018 3,57 2 0 , 235 0 ,257 -0,022 -9,57 3 0 .186 0 ,1 90 - 0 , 004 - 2 .09 4 0,191 0 ,1 70 0 . 021 10 . 80 5 0.179 0,179 -0, 000 -0. 25 6 0.188 0 .1 84 0, 004 2 . 22 7 0.161 0.171 -0. 010 - 6.03 As is the case with t he F- 4 data, comparison of the statistics of Mode l s 13 and 14 l eaves an indicat ion that the manufacturing rate form of the production r ate is better than the delivery r ate . For example, the F * incremental statistic is 10 , 88 for Model 13 as compared to 9 , 20 for Model 14 . Similarly, the MSE and R2 statistics are sl i ghtly bette r for Model 13. Again, there is insufficient evidence to conclude that one is clearly better than the other. lJJ COST MODELS 15 AND 16 KC-1J5A ASSEMBLY HOURS PER POUND Model 15 examines assembly hours per pound as a function of cumulative production and manufacturing rate. The results of regression analysis of the data in the cumulative production and nroduction rate cost model are presented in Table 37, TABLE 37 REGRESSION RESULTS-MODEL 15 Estimate BO = 13.338 R2 actual = 0.993 Estimate Bl = -0.608 t ratio Bl = 18.77 Estimate B2 = 0.361 t ratio B2 = 4.44 F * (full) = 327.85 F(2,4),0.05 = 6.94 F * (incremental) = 19,74 F(l,4),0.05 = 7,71 MSE (full) = 0.0009 MSE (reduced) = 0.0042 R2 (full) = 0. 994 R2 (reduced) = 0,964 Case Observed Predic ted Residual Per Cent Value Value Deviation 1 2 ,9 33 3,020 -0.087 -2.97 2 1.655 1.498 0.157 9 ,51 J 0,978 1.038 -0.060 -6.16 4 0.722 0,735 -0.013 -1.77 5 0.511 0,530 -0,019 -3.68 6 0.439 o.442 -0.003 -0.73 7 o.471 o.44 8 0.023 4 ,88 The stat istics show that the null assumption for the 134 three subhypotheses may be rejected. The F * full stat istic of 327,85 is larger than the theoretical F value of 6.94, The F * incremental statistic of 19.74 is larger than the theoretical F value of 7,71. A plot of the residuals against the test variables does not indicate that the model should be rejected. The subjective tests of the statistics also reflect a good fit and the importance of the rate variable. The R2 actual statistic of 0,993 shows that most of the variance is explained. The improvement in the R2 statistic from 0,964 to 0,994 due to add i ng the rate variable to the reduced model emphasizes the importance of the manufactur- ing rate. Similarly, adding the manufacturing rate variable to the reduced model decrease s the MSE statistic by more than 75 per cent. The behavior of the B2 coefficient estimate is similar to that demonstrated in the analysis of Model 12. The sign of the coefficient changes from negative to positive when cumulative production is added to a reduced model. The reduced model with x1 excluded has a B2 estimate of -0. 855 and an R2 statistic of 0.460. A reduced model with x 2 excluded has a B1 estimate of -0, 494 and an R2 statistic of 0,964. In the full model, the B2 estimate becomes positive. This again indicates the effect of the collinearity of x1 and x 2 and the large difference 135 in the relative explanatory strength of these two variables. There does not appear to be any theoretical significance to the change. The full model should be useful for prediction. None of the tests indicate that Model 15 is un- suitable for t he data set. Therefore hypothesis two is acce pted . Model 16 i s the last model examined in this study and is the companion to Model 15. It is used to evaluate assembly hours per pound as a function of cumulative production and delivery rate. The results of the regression analysis are listed in Table J8. TABLE 38 REGRESSION RES ULTS-MODEL 16 Estimate 7,303 2 BO = R actual = 0,992 Estimate Bl = -0,527 t ratio Bl = 22.28 Estimate B2 = 0.263 t ratio B2 = J.84 F * (full) = 259 ,14 F(2 , 4 ),0.05 = 6.94 F * (incremental) = 14.80 F(l,4),0 ,05 = 7,71 MSE (full) = 0.0011 MSE (re duced ) = 0,0042 R2 (full) = 0,992 R2 (reduced) = 0.964 These results are similar to those of Model 15. All the statistical tests indicate a good fit of the model to the data . Therefore hypothesis two is accepted for Model 16. 136 A compari s on of the statistics produced by Models 15 and 16 indicate s 15 t o be better . Since the only differ- ence in the models is the production rate proxy, once again one can conclude that manufacturing rate is the better of the two in this particular application. SUMMARY In this chapter sixteen di fferent sets of data are examined in the cumulative production and production rate cost model. There are nine from the F-4 program, two from the F-102 program and five from the KC-135 program . Wi th the F-4 program data, all three hypotheses are tested and accepted. The production rate is found to be an important explainer of variation in labor requirements at the three l eve ls of production when expressed i n the cost model . Furthermore, the rate variable in the presence of the cumulative production variable generally improves the ability of the model to predict as compared to the reduced model . It also appears to improve the stability of the model as predictions are made over longer spans of time. In the F-102 program, only the first and third hypotheses are evaluated as the labor requirement data for lower process levels are not available. As in the F-4 data, the production rate is found to be an important 137 explainer of total labor requirements. Again, the tests indicate that using the full model generally improves the capability to predict future requirements over that of the reduced model although the accuracy of the predictions are outside the test limits. The analyses of the KC -1J5A data also validate the model and the theory . The production rate variable generally improves the ability to explain variance in the labor requirement at the three levels of production. Tests of improvement in the predict ive ability are inconclusive for the model with total hours per pound. The tests for predictive ability are not made at lower process levels . Results of the F-4 and KC-135 tests indicate that the fabrication hours per pound variable is more sensitive to the production rate than are the total or assembly hours. However , the production rate variable improves the explanatory power of all three levels of the process that are investigated. Although no statistical proof exists, the manufactur- ing rate appears to be a better proxy for the production rate than the delivery rate. This observation is based on the relative strength of statistics produced from models with only the rate variable changed . 138 CHAPTER VI SUMMARY AND CONCLUSIONS There are many instances where the rate of production in an airframe manufacturing program is cllanged due to forces external to the production process. These changes are accompanied by a need for managers to know the effects on cost. The few methods that exist to estimate these effects are based on a limited number of cases and produce conflict ing results. The purpose of this research is to develop and test a procedure to consider the effect of a production rate change on the direct production labor requirements to produce additional airframes. The literature on the subject is inconclusive. Some writers believe that the change in production rate is an important predictor of variation in unit cost while others conclude that it is insignificant. Some writers indicate that an increase in production rate leads to an increase in unit cost and others believe the opposite effect occurs. This study shows through an empirical evaluation that the production rate can be an important predictor of variation in unit direct labor requirements. Furthermore, an increase in rate up to plant capacity can lead to a decrease in unit labor requirements. 139 One primary element in a cost estimate is the quantity of direct production labor hours. Therefore, a logical place to start the e s timates of the effects of a changed rate of production on airframe production cost is with these hours. Many estimators of direct labor hours use the learning curve cost model. But this model is driven only by the cumulative number of airframes produced, not the rate of production. Thus, even if the production rate is changed due to some requirement external to the process, the number of labor hours predicted by an unadjusted learning curve model to produce each unit is not changed. This result is contrary to logic. Up to some limit that is associated with plant capacity, an increase in production rate should result in a decrease in unit labor requirements . Greater spreading of fabrication set up charges, increased efficiencies resulting from labor specialization and a more highly motivated worker all contribute to this expected result. For the same reasons, a production rate decrease should result in a unit labor requirement increase. A cost model is proposed to capture this rate effect in the presence of the learning effect. It is y = Bo · x 1 B1 · x 2 B2 , lOe where: y represents the direct production labor hours per DCPR pound of airframe in each lot, 140 xa1 represents the cumulative production plot point sin the learning curve model and x 2 represents ~he production rate. The dependent variable is expressed as the lot average direct labor hours per DCPR pound of airframe. When the data permit, the effects are examined at three levels of aggregation, total hours, assembly hours and fabrication hours. The first independent variable is identical to that used in learning curve analysis. The cumulative production plot point is developed for all airframes of the same type produced in the plant. This procedure holds whether or not ·the dependent variable i s developed f rom data for a single model or aggregated from a number of models. Two variables are evaluated as representatives of the second independent variable. They are a manufacturing rate and a delivery rate. The manufacturing rate is the total number of airframes in a lot divided by the l ot manu- facturing time span. This span is bounded by the date the first airframe in a lot is released to fabrication until the last airframe in a lot is accepted. The delivery rate is the total number of airframes in a lot divided by the time span bounded by the first and last lot airframe acceptances. Sixteen data sets are constructed from data for three 141 airframe production programs. They include nine from the F-4 program, two from the F-102 program and five from the KC-135 program. The data sets are exami ned in the cumulative production and production rate cost model through regression analysis. The estimating procedure proposed here consists of three major steps. First, the data are analyzed to make sure the model is appropriate to the program. Then if the results of that analysis are favorable, the predictive ability and stability of the model are tested. Again assuming a favorable outcome, in a practical application a third step would be to forecast labor requirements for additional production using the cost model developed f r om the analysis. Each step is discussed in turn along with associated conclusions from the research. To linearize the model and facilitate the regression technique, each variable is transformed to logarithms so tha t: logy= log Bo+ Bl log xl + B2 log X2 + e. Then each data set is analyzed in the regression model producing estimated coefficients and statistics. These become the basis for answering questions about the process. Findings are summarized in paragraphs that follow. The cumulative production and production rate cost model fits the individual data sets well. In each of the sixteen sets of data the model is found to be appropriate. 142 Statistical tests to support this conclusion are conducted at the 0.05 level of significance. More specifically, one indicator of goodness of fit is the R2 statistic for the full model. As reflected in Table 39 which summarizes some regression results for all 16 models, the R2 full statistic is respectable in each case. The production rate variable contributes importantly to the explanatory ability of the model. A statistical test of this conclusion is performed and accepted at the 0.05 level of significance for each model. A more intui- tively appealing test is the improvement in the ' R2 sta- tistic when the production rate is added to a reduced model. Table 39 lists the R2 full and R2 reduced sta- tistics side by side for each model. In each case the R2 statistic is improved by an amount that indicates the production rate is a valuable contributor to the explana- tory ability of the model. Within some upper boundary related to plant capacity, production rate and unit labor requirements move in opposite directions with a cause and effect relationship. In every model examined, the rate variable is negatively correlated with unit direc t labor requirements. This negative correlation is also reflected in the negative sign for the B2 coefficient estimate in 13 of the 16 models (Table 39). In three KC-1J5A models, the combined effects TABLE 39 REGRESSION MODEL SUMMA RY Cost Airframe Cases Level Rate R2 R2 re- BO Bl B2 Model Model full duced 1 F-4A-F 57 Total Del 0.978 0.928 maske d -0.261 - 0.169 2 F-4B-F 55 Total Manu 0.973 0.904 " -0.246 -0. 183 3 F-4B-F 55 Total Del 0.966 0.904 " -0.257 -0.161 4 F-4B-F 42 Total Manu 0.853 0.585 " -0.230 -0.157 5 F-4B-F 42 Total Del 0.820 0,585 " -0.229 -0.136 6 F-4B-F 42 Fabri Manu o.889 0.618 6.328 - 0.221 - 0.148 7 F-4B-F 42 Fabri Del 0.851 0.618 7.601 - 0. 219 -0.127 8 F-4B-F 42 Assem Manu 0,744 0.658 9,016 -0.279 -0.112 9 F-4B-F 42 Assem De l 0,733 0,658 10.400 -0.278 -0.097 10 F-102A 50 To t al Del 0.979 0.961 38.371 -0.299 -0.158 11 F-102A 42 Total Del 0. 979 0.959 47.290 -0.344 -0.144 12 KC-135A 96 Total Del 0.958 0.971 13.133 -0.453 0,164 lJ KC-135A 7 Fabri Manu 0.974 0.903 o.674 -0.165 -0.305 14 KC-1J5A 7 Fabri Del 0.971 0.903 1.123 -0.2JJ -0.222 15 KC-1J5A 7 Ass em Manu 0,994 0,964 13,338 -0.608 0.361 16 KC-1J5A 7 Assem Del 0.992 0,964 7.303 -0.527 0.263 f--' ~ \...V 144 of collinearity and great relative strength of the cumulative production variable causes the· B2 estimate sign to change to positive in the full model. When comparing the effectiveness of a manufacturing or delivery rate representative of the production rate, the manufacturing rate gives better results in all six of the direct comparisons. But the difference is not great and either proxy is an important contributor to the explanatory power of the model. The model fits fabrication hours per pound data better than assembly or total data for the F-4 and KC-135 programs. This may indicate that the fabrication labor requirements are more sensitive to rate than the other two levels of the process examined. Fabrication and assembly hours are not evaluated for the F-102 program. This procedure does not appear to be suited to constructing general cost models with coefficients that are applicable to other programs. The wide variation in the coefficients listed in Table 39 for models with common production levels and rates suggests that any averaging of coefficients would lead to unreliable results. The procedure appears to be suited only for predicting ad- ditional production for cont inuing programs. When practical, the ability of a -model to predict 145 accurately beyond the data is tested. This is accomplished by estimating the model coefficients with successively smaller data sets formed by truncating the most recent observations. Omitted observations are predicted with the new model as if they are unknown. These forecasts are then compared with the known values and those forecast from a learning curve model to measure the predictive ability of the full cost model. The rate variable stabilizes and improves the pre- dictive ability of the cost model for the F-4 and F-102 program data. This improvement is particularly marked for the nine F-4 models analyzed. The predictive accuracy of the models developed from the F-4 data sets make them attractive alternatives for estimating labor requirements for additional production. Although the model fit for the F-102A data is excellent, it does not forecast the pre- diction target very well. Tests for predictive ability improvement are either inconclusive or impractical for the KC-135 program data. The procedure is well suited to forecasting the direct labor requirements for additional production lots. The steps for such a forecast are reviewed here. After the coefficients are estimated and the cost model is tailored to a particular program, one first must decide on the acceptability of the model using an approach similar to 146 that described in Chapter 5, If the model fits, its predictive ab i lity should be tested to gain some measure of confidence in a f ore cast . As s um i ng that t he results of this test are acciptable, one must calculate the cumulative production plot point and the delivery or manufacturing rate for the unknown production lot or lots. Then using the tailored cost mod el, the unknown direct labor requirement for the new lot can be predicted. The conclusions described above are necessarily limited to the data sets examined. But there is great temptation to generalize the results to other airframe production, In particular, the notion that a production rate variable correlates ne gatively and importantly with direct labor requirements has much application in the airframe industry if universally true. An obvious extension of this research effort is to duplicate the procedure on additional programs . I t would be of particular value to examine the procedure on new airframe production t o discover if changes in manufacturing technology are affecting the process. The behavior of other cost elements with respect to production rate changes is also an important research topic. APPENDIX A C THE CUMU LATI VE PROUU CTI C~ AND PRUOUC TI ON RA TE COS T MODEL C PROGRAM;\1EK IS LARR Y L. SMITh - MARCH 1S76 . C C TH I S PRUGKA~ I S TAI LOR ED TU EVA LUATE VAR I ATION IN UN lT UI~fC T LABOR C REQUIREMENTS AS A FUNC Tl CN UF CUMU LATI VE PRODU CTI ON ANO PR OCUC TIC N RATE . C THt: COST MODE L I S Y = BO'~ ( Xl ;'*t3 1) ,;, 1 xz,;,~'i32 l * {1 0 . ~'*t: ). C Y IS TH~ UN IT CR AVERA GE CI REC T MAN HOU~ S PER POUND . Xl lS THE C CUMU LATIV E PRCD UCTl CN PLC T PO INT AS I N THE LEARN I NG CURVE MODE L. C XZ IS THE PRO CLCTI CN RA TE PRCX Y SUCH AS DE L I VER I ES PEK ~UN TH. CE HEFRESEN TS THE ERR OR TERM . CAS~S ARE USUAL LY DI STI NGU I SHED BY C PR DOL CflCN LCl OR MLN Tr LY PRUO UCTl ON . EAC H TERM I N THE MOD EL lS C TRA ~S fGkMED TC LOG ARIT HMS TO FAC I LI TATE REGRESSION ANA LY S IS. C C THE FRCGRAM F I RS T TRANSfCRM S THE CATA TC COMMON LOGAR IT HMS . C REGHESS I ON RE .!:LLT S FOR A TWO VAR I Ad LE MCDE L Wl THO Ur THE PkOOUC TI ON RA TE C t RE CALC ULATED. TH IS ~ODE L IS EQ UIVAL ENT TO TH E UN IT LEARN I NG CURVE C MCDE L. THEN T~ E kES LLTS FOR A TWO VARlABL E MODE L WIT HOU T THE CU~ULATI VE C PKUC LCT IUl\i VA RI AB L E ARE CALCU LATED . THE STATI ST I CS AKE Or- TWO TY PES . C fl RS T, S TANCAR C RE GRESS I CN STATI ST I CS AkE CALCUL ATED . THE Y AR E I N C LOGAR IT H~ S Bf CALS[ UF T~E TRANSFORMED DAT A AND MODE L. I N AOO I T IC N, C ACI LA L RES I CU AL S BASED GN OBS~R VATI ONS I N TrlE IR OR I GI NA L fOR M ARE C CALCLL ATED . R** 2 ACTUA L ANC MSE ACTUA L STATI ST I CS ARE CALCULAT ED C BASE D ON ThESE RES I DUALS. FOR THE FU LL MODE L, THE OBSE~ VEU ANO C PRtC ICTEO V.A Ll.iES FDR t ACh CASE ARt Pk l NTEO OUT IN bG TH THE IR. CR IGI NAL C FOR~ AND l N LCGAR IT HMS . JES I DUALS A~0 PEK CE NT DE VI ATI ON FROM C THE CBSER VEO VALU E AR E ALS O PR I NTED . C C DA r A AR.t l l\jP U1 AS FO LL C~S : C T~ E Fl RS T C.A RC IS THE NUM~ER OF CASES RI GH T JUSTI F I ~D I N COLU MNS 1- 3 . C Tr ~ PROGR A~ l S NO~ SE T TO ACC EPT UNE CA~D FUR EACH CASE . THE VA R IA BLES C ARE I NP UT IN Th E Dk DEH Xl,X 2 ANO Y i,/ lTH 10 CO LUMN S FOR c AC H VA RIABLE C I~ f FORM AT. I □ ~ T EN ALT ER THE R~AD (l, 30 ) ANO 3J FORMAT STAT EME NTS C TC FI T TH ~ CATA CN THE CARO . T HE P~CGRAM IS SE T TU ACCiP T A MA XI MUM C Of sg CASES ~ LT CN E COULD ENLARGE THE MAT~ iC ES IN THE DI MENSION C STAlE MEN T TO DCCEPT MCR~ . C C Cl-MENS IC N PLOT(99 ), RATl:( 99 ) , HRS (9 9 ) ,Y{9 '-:!} , X( 2 , 99 } DA TA SUMHRS , SU MXl, SLMX2 , SU MY,SSXl , SSX2 , SU~X lY, SUMX2Y , SM XlX2,SSE R, 1 SSE ,SSTO,S SE RL,S STOL,S Scl , SSE KK L,SSERR/17*0./ C C I NP LT DATA A~U TKANSF CkM Th E VA R IA BLES ro LOGA RIT HMS. C fd ::A O (1,lOJf\CA S[S 1 0 F O RM A T ( I 3 l 08 50 1 = 1, NCASES R. EA C { 1 , 3 0 ) PLOT ( I ) , RAT [ ( l l , h R S ( I ) 3G FORMA T( 5X , 2F 5. 0 , 5X , F5.0l Xll,I) = ALOGlG{P LUT(I )) X {2 , Il = AL OG 1 0 ( k A TE (I) J Y(l ) - ALOGlO(HRS( I)l C C CALCUL ,~ TE lH E SUf'-'S OF Th e VA R.lAbLE PRCDUC l S . C. SL/v1HRS = SUMH RS -+ HRS ( I ) SLMXl = SUMX l +X( l,I) SUMX2 = SUMX 2 + X{2 ,I) sun = SUMY -+ y ( I ) SSX l = SSX l + X( 1,1 H '*2 S SX2 = SSX 2 + X(2,I) **2 SUMX lY = SUMX lY + X( l,IJ *Y (IJ SLMX2 Y = SUM X2 Y + X( 2 , I ) * Y { I ) SMX 1X2 = SMX 1X2 + X >1 ( s s x 2 - ( su ,; . x 2 * * 2 ; N c A s t. s l l dO = YEAR - B2*X2eAR AtW = 10.* *130 C C CALCULAT t. AND PRI ~T RESICUALS FOK THE ~E0UC D HRS V R~T~ MODE L. C CC 200 I = 1, NCA Sf:S YHATL = 80 + B2*X ( 2,I ) KES1CL = Y(I }- YhATL SSERRL = SS ERR L + KESIOL**2 YhAT = 10.** YH ATL iU: S l C = HRS ( l l - Yh AT SSERR = SSERK + RES I D**2 I-' \J1 2cc CLNTlNUE: 0 C C CALCULATE ANC PR l NT STATISTICS FOR THE KEDUCED HRS V RA TE MODEL . C R2L OG = (SSTCL - SS t: 1~1-{l)/SST OL FMSEL = SSERRL/(NCASES-2) FLCG = (SSlCL-SSERRL )/ FMSEL RSQI =lSSlO-SSE ~K J/ SSTO FMSER = SSE RR/(NCASES-2) C FRI ; T,' ' PRlNT,'RtSULTS OF A RECUCtO MODE L WITHOUT CUMULATIVE LEARNING .• FR l N 1, • * ~* ,::**,:::* ***~~** t~ ** ~t* **;'=*t; l{::** ****** **=-~***** *='.:~:** **** ** :(: * * , PRINT, I ' PRINT ,' ' PRINT,'YrlAT = ',A BC ,' •:< ( X2** ', 82,'}' PRINT,' I PKl NT ,' STA TIS TlC S I~ LOGA RITH MS ' PRINT,' I PRlNT ,'LOG R2 = ', R2LOG ,'L □ G MSE = ', FMSEL, 'L OG F = 1 , FLCG Pk l NT ,' ' PRI NT, 1 ST/lllSTICS OfvcLUPEO rRCM ORIGlNAL VARIA8LE FORM .' PR. INT, ' ' Pkl rn , ' 1--2 ACTU AL = ', KSGiR , 1 ~1 SE ACTUAL = ',F MSi::k C C CALCuLAT f 80 ,tl, A~C t2 f OR Tr. E FULL MCuEL . C C~Nl~ = ( (SSXl-X l e AR*SU~Xl J ~( SSX2-X28Ak*SUMX2 l -lSMX lX 2- XlJAR * t SLM>-2 ) ~:-:2) 81 = l { SSX2-X 2 t3AK*SU,iX2 ) *( SU MXlY- Xl3AfHSUMY ) - I ISMXLX2-XlBAR*SG ~ X2 l*(SUMX 2Y-XZ BAR*SUM Y))/O EN OM 82 = l ( SSXl-XlflAR,."SLMXl ) ,:, ( SUMX2Y-X 2BAk*SUMY ) - b (S MXlX2-XlBA R*SLMX2l*{ SUMX lY-XlBAR*SUMY)l/OENO~ e O = Yet K- e 1 * X 1 BAR- B 2 ,~ X 2 !J A ,~ ABO = l C. **GO I-' C V\ C ( ALCU LAl[ RcSlDUALS fOR T~ c FULL MODEL . I-' C Pf°. '** * *;(,:f,c**::<'** ,::: ,:, *~*Y.' ,~,:, ,:, I PR I I\: T' I I v.RIT E ( 3 ,21 0 ) 210 FORM Af( 1 U','CAS E OBSERVED PR ED ICTED RESIDUAL P ER CENT t 1 LC G LOG Cl fFERENCE 'l rd-UT i: ( 3 , 2 11) 2 l l F O RM AT ( ' ' , 6 X , -. V AL Li l VA L LJ E ' , 1 6 X , ' D ::: V I ..\ T I O N ') 3 S ER V 1:: D ' , 2 PKi::ClCTED') OU 250 I= 1, NCASf:S 'lt- AT L = BO+ 8 1>:' >-. (l,Il + B2"',X (2,I l RtS l DL = Ylll - Y~ATL SS~ L = S SEL • RESIDL**2 YhAl = 10.t*YhATL 1-U::S I D = HR S( I )-Y HA T SSt =SSE+ RtSIC**Z Pt:RCEN-= 10 0 . *RESlO / HR S( I} Wk l T [ ( 3 , 11 5 ) l , h R S ( I ) , YH A f , 1~ ES l D , PE RC t N , Y ( I ) , YH AT L , RE S I DL 11::i rC RM AT(' ',13,2 X ,3( F6 . 3 , 5 X),F6.2,5X,3(Fll. 5 )) 250 CCNT I NLf: C C CALCU LA TE SC~E STATI STIC S f□ R THE F ULL MODEL . C R2L UG = (SST OL- SS£:: ll/SST OL FMSL L = SSEL/( NCASE S-3 } ~INCL = (SS ER L-SSEL)/F MSE L FINCLZ= ( SSERR L- SSEL l/ FMSE L FLO - {SST CL-S SEL l/ 2./ FMS EL SSlST = SCRTl F~SE L) Stol = SQRT ( ( FMSl::L 0!: {SSX2-X2EAR*S UMX 2 ) )/ DENOM } S~ D2 ~ S~R l((F MSE L* (SS X1- x 1 eA 1 *SCMX lll/ DEN [MJ Tl::H = c i/SE 8 1 Tt)2 = b2 /SE 82 RSWK = {SST O-SSE U SSTO FMStA = S Sc/{ ~ CASES-3) C P R L\J T, ' I p R 1 f 4 T I I y HAT = I , A 6 0 , I * ( X l * ;~ 1 I • Q 1, ' ) ~: { X 2 t,: * I , 82 , I ) PR I NT, I I P KI ,\J 1,' S TATIST1 CS Ir\ LUGAK ITH t-1S ' P R 1 , l , ' ' PrUNT ,'LOG R2 = ', R2 LO G ,'LL1G MSE = ', FMSE L,'LO G F = ', F LOG P~ I NT,'LC G F 1NCHEME ~T A L F R CM ADDIN G X2 = ', FI NCL PRl N T,' I PRI 1n , 1 LCG F I NCREI-IE ~Tlll F RCM AD G l NG Xl = ', FINCLZ PK I i\J 1 , • ' Fkl ,\J T , 'T t l= ',T 8 1, 'T 22 =' ,T r.2 i-: ;~ l I ' l ' I ' PK l i'H, 1 STC ERf<- iJR Of f:ST IS ', SSE ST p R l :\I T , I I Prd /\ T,•ST.l)llSTIC S Dt VE LCPfJ FkCi~ CR l G l NAL VARIAB L E FOl{M .' Ph l N T,' I PK [ N T, • R2 AC TU !-\ L = ' , RSQR , 1 MSE ACTlJI\L = ', fMS EA wK ITU J ,5) S T tlP E-ND I-' \.J\ \.,J 154 SELECTED REFERENCES Alchian, Armen A. and Allen, William R. University Economics. Belmont, California: Wadsworth Publishing Company, Inc., 1964. Asher, Harold. Cost-Quantity Relationships in the Airframe Industry. Santa Monica, California: The Rand Corporation, 1956. Beals, Ralph E. Statistics for Economists. Chicago, Illinois: Rand McNally and Co., 1972. Cochran, E. B. Planning Production Cost: Using the Improvement Curve. San Francisco, California: Chandler Publishing Company, 1968. Dean, Joel, Managerial Ec onomics. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1961. Dunne , William E. "Microeconomic Theory Applied to Parametric Cost Estimation of Aircraft Airframes ." GOR/SM/75D-3, M.S. Thesis, Air Force Institute of Technology, Wright-Patters on Air Force Base, Ohio, 1975. Fazio, Peter F. and Russell, Stephen H. "An Analytical Approach To Optimizing Airframe Production Costs As A Function Of Production Rate." SLSR J0-74A, M.S . Thesis, Air Force Institute of Technology, Wright- Patterson Air Force Base, Ohio, 1974. Johnson, Gordon J. "The Analysis of Direct Labor Costs for Production Program Stretchouts." National Contract Management Journal (Spring 1969): 25-41. Kroeker, Herbert R. and Peterson, Robert. A Handbook of Learning Curve Techniques . Columbus, Ohio: The Ohio State University Research Foundation , 1961. Large, Joseph P.; Hoffmayer, Karl: and Kontrovich , Frank . Production Rate and Production Cost. R-1609-PA&E, Santa Monica, California: The Rand Corporation, December 1974. 155 Neter, John and Wasserman, William. Applied Linear Statistical Models . Homewood, Illinois: Richard D. Irwin, Inc., 1974, Noah, J. W. "Resource Input vs. Output Rate and Volume in the Airframe Industry." Draft Technical Report Number TR-204-USN, Contract Number N00014-73-C-0319, Washington, D.C.: J. Watson Noah Associates, Inc., December 1974. Noe, Charles G.; Smith, Larry L.; and Jacobs, Grady L. Defense Cost and Price Analysis. 2 Vols. Gunter Air Force Station, Alabama: Extension Course Institute, 1974. Orsini, Joseph A. "An Analysis of Theoretical and Empirical Advances in Learning Curve Concepts Since 1966." GSA/SM/72-12, M.S. Thesis, Air Force Institute of Technology, Wright-Patterson Air Force Base, Ohio, 1970. Spurr, William A. and Bonini, Charles P. Statistical Analysis for Business Decisions. Homewood, Illinois: Richard D. Irwin, Inc., 1967. U.S. Department of Defense, Armed Services Procurement Regulation, The 1973 Edition . Washington , D.C.: Government Printing Office, 1973, U.S. Department of Defense. Armed Services Procurement Regulation Manual for Contract Pricing . Washington, D.C.: Government Printing Office, February 1969, Wonnacott, Thomas H. and Wonnacott , Ronald J. Introductory Statistics. 2nd Ed. New York, New York: John Wiley and Sons, Inc., 1972. 156 Larry L. Smith