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Estimating learning curves for potential products
IndustrkrlMarketingManagement5 (1976) 147-154 ’ 0 ElsevierScientific PublishingCompany,Amsterdam- Printedin The Netherlands
147
ESTIMATING LEARNING CURVES FOR POTENTIAL PRODlJCTS
-Louis E. Yelle
.INTRODUCTION
The well-known phenomenon of the learning curve can be a useful tool with which to predict standard costs. The usual approach is to work with a learning curve which represents the expected cost behavior of the complete product. In the case of well-established products, sufficient data usually exist due to prior production experience on the actual product to determine the learning curve for that product. However, in the case of new potential products, sufficient data on the specific product in question are not generally available. Faced with this situation, the customary ap preach for proposed new products is to select the learning curve of a similar product to use as a predictive model. In this article, the point is made that many new products are manufactured on production lines which are aggregates of existing technology. It is further SUP gested that the learning curve approach to predicting standard costs on new products as described above can be refined by first determining the learning curve for each operation of the total manufacturing process. The author then
demonstrates that the individual learni for each operation of the manufacturi can be mathematically combined to yield the learning cuye for the proposed product. THE LEARNING CURVE PHENOIMEIUON
The learning curve phenomenon w covered in the 1920’s in the aircraft i was subsequently reported in 1936 [ 11 c ~irn~~y number of direc duce an individual unit de
hour to manufactu manufacture the fourth format which can
148 TABLE I Numericat vafucs for a 90% learning curve where the amount of time required TVmanufacture the frost unit is one direct labor aoto&e.,K=l> _~__ ~- ~~ c3mlillatlve Dht labor hours number of unit!s required to manufacture Inan. one unit (905S L-CJ 1 2 4 8 16 32 64 128 256 512 1024
1.0 O.!) Ml 0.73 0.66 0.59 0.53 0.48 0.43
0.39 0.35
Leaf&g cufves follow the’mrthematical fmction
where: Y = The number of direct labor hours required to produce the xth unit. K = The number of direct labor hours required to produce the first unit. X = ‘Xe cumulative unit number. log 4J n=loe: 0 = % learning curve divided by 100. Learning curves are therefore straight lines when plotted on log-log paper. Some typical learning curves (all requiring the same number of direct labor hours to manufacture the first uni” . . . i.e., K = 1 hour) having different leaming rat<:: are shown in Fig 1. One question which people often ask upon a cursory view of learning curves is, “Will the de crease in the amount of direct labor hours requrred to manufacture a unit ever cease?” The answer, for ail practical purposes, is yes. For example, the total decrease in time as a result of experience gainedmanufacturing the first 500 units (Table I) is (1.0 hours - 0.39 hours) approximately 0.6 hours. However, the total decrease achieved as a result of manufacturing the
second 500 units is only (0.39 hours 0.35 hours) 0.04 hours. It should be clear, there= fore, that a; the quantity of units manufactured increases the decrease in direct labor hours per unit achieved as a result of increased experience approaches zero for all practical purposes. PRESENT DETERMINATION LEARNING CURVES
AND USE OF
For established products or products for which a sufficient number of units have already been manufactured, the data usually exis’c:to determine the learning curve for that product. All that is generally required is a sufficient amount of production experience in order to allow the manufacturing process to stabilize. This stability generally leads to discernible trends in the number of direct labor hours required as the cumulative number of units produced increases. At this point, the data can be readily accumulated and plotted in order to determine if the process follows the learning curve phenomenon and, if so, what the rate of learning is. In the case where production experience does not exist or is insufficient, the usual approach has been to select the learning curve ‘of,a
Louis E. YelIe is an Associate Professor of Management in the College of Management Science at The University of Lowell, Lowell, Massachusetts. Prior positions were in tie semiconductor industry with major mar.Jfacturers. More recently he has been a consultant to the electronics industry.
149
1
10
100
1,000
CUMlJCATtVE UNIT NUMBER Fig. 1. TypicalIearningewes
alI requiringone directlaborhour to manufacturethe fhst tit.
similar product and use it as a predictive model, the underlying assumption being that similar products will generally follow the same learning curve. This approach can be refined [ 33. THE PROPOSED APPROACH
The thesis of this article is that similaritiesin manufacturing technology can be used to construct the learning curve for potential products whose cost behavior we wish to predict. A fundamental assumption being made is that many new manufacturing processes are, in many instances, nothing more than an aggregation of well=developedoperations which can be found in other established manufacturing processes.What I am suggestingis an approach to establishing new product learning curves which would utilize a data bank of known learning curves for individual production operations. Then, once the process design has been established, the learning curves stored in the data bank which are applicable to the new manuf&cturingprocess can be utilized to construct the learning curve for the proposed new pre
duct. In essence, the approach is somewhat similar to the well-known method of developi standard times catted the Predetermined Time Study Approach which utilizes tables of histol ically established values. AN EXAJVIPLE
In the early X960’sthe established semiconductor manufacturers possessedsufficient historical experience in semiconductor diode manufacturing ‘to construct a list of productio operations (i.e., a data bank of production operations) and their ass&ated iearning curve A representative list of some of these operatio is shown in Table II. During this period of timl interest in reducing the size of diodes develops due to the need to reduce the siz3 of electroni equipment in general. This move towards micr miniaturization was necessary for a variety of well-knownreasons. One specific example wai the need to reduce the size of the Silicon-Plan Diode packaged in what is commonly known i the DO-7 package. At that time, engineers be gan to look at a variety of production prom
TABLE II List of representative production processes and their approximate historical data Production operations
% Learning curve
Cumulative number of times this operation performed to date
fi.rect labor hours per unit (current)
Diffusion Photolithography Varnish Passivation Glass Passivation Evaporation Contact Plating Scribing Soldering Machine Sealing
85% 80% 65% 70% 75% 70% 89% 95% 90%
8,000,OOO 6,000,OOO 2,000,000 100,000 1,‘500,000 6,000,OOO 20,000,000 20,000,000 20,000,000
1.7 0.6 10.0 3.3 5.0 1.1 4.2 11.0 8.3
x do-4 x 10” x 1o-4 x 10’4 x 10” x 10’” x 10” Y 10” 3 1O-4
Source: The data in this table was gathered during interviews cond\:cted by this author in 1973. The approximations are rough and meant to be used for illustrative purposes only.
cesses whereby this could be accomplished. The Sealing” operation were an 85% learning curve end result was later called the Double Plugand a standard time of 20 x 10B4hours per unit. Diode. A point which should be made at this The total process is depicted in Table III, along time is that it was necessary to change many of with the associated learning curves and standard the individual operations in the overall manudirect labor data.. The mathematical expression facturing process to make this new diode. In for a typbzal operation’s learning curve is other words, in reconstructing the total manuderived in Appendix I. facturing process, some production operations HISTORICAL remained, others were discarded, and new ones LEARNING CURVE were added. With one exception, the new OPERATION AVAILABLE operations which were added had been developed previously on other diode manufacturing 1 Photollthc-raphy Yes lines. The initial process design decided upon by 2 Dlffuslon yes manufacturing engineers for the miniature diode is depicted in Fig. 2. With the sole excep 3 Contact Platlng Yes tion of the last operation entitled “Furnace Sealing,” the process was nothing more than an 4 Scribing yes aggregate of existing manufa.cturing operations developed on other diodes. And, historical learn5 Furnace Sealing no ing curve data existed for all but this last opera1 tion, When historical data do not exist for an END operation, the applicable learning curve along PRODUCT with the associated direct labor must be , estimated. When possible, small pilot runs Pig. 2. New diode process utilizing available manufacturing should be made as an aid in estimating. In this operations except for the last operation. spe~if’ic case, the estimates for the “Furnace
151 DETERMINING THE LEARNI CURVE FOR THE PROPOSEDPRODUCT The remaining task is to determine the learning curve for the potential product by mathe matically combining the individual learning curves for each operation comprising the manufacturing process. In our example, there are five separate manufacturing operations and, therefore, five separate learning curves. These leaming curves can be combined mathematically [4] to yield the learning curve which will describe the expected labor standard for the new product as a function of the cumulative number of units produced. The mathematical treatment is shown in Appendix II. The results yield the product learning curve for our example, which may be expressed as follows:
r, = 0.00157 Xp-o*0’99a The learning curve of the potential product is plotted in Fig. 3. SOME POTENTiAL USESOF THE PROPOSEDMETHQD
Some potential uses of the method of estimating the applicable learning curve presented in this article are: 1, To estimate standard costs on new product proposals. Predicting standard cost is
an essential step in estimating the future market potential of a new product. And, the future market potential will determine the advisability of taking the necessary steps to manufacture the product. 2. Accurate predictions of standard costs on proposed pr+cts are important for capital budgeting - particularly long-term projections. As the standard cost decreases, the product generally makes inroads into new marketing segments. Estimating how much of a capital investment will be needed in the future to manufacture the product in such a manner that the fm will remain competitive is important. A small firm may conclude that the required capital investment will exceed its capability to acquire capital. A large f^mnwould use the information to help allocate capital among alternative investment proposals. 3. Learning curves are useful when making pricing decisions. In the case of new products on which sufficient learning curve data do not exist, the cost of the initial contract or order is an important decision. Also, in situations where pricing for orders to be delivered far in the future is necessary, the learning curve can also serve as a useful predictive model. 4. Financial control is a necessary ingredient of all successful firms. The ability to predict standard costs in the future is an important step in making out pro-forma financial statements and budgets.
TABLEIII Double-Plug-Diodemanufacturingprocessstepsand associatedhistoricaldatagleanedfrom TableII Operation
96 Learning CUWd
1 2 3
Photolithography Diffusion Contact Plating
80% 85% 70%
4
Scribing
80%
5
FurnaceSealing
85%
Cumulativenumberof timesthis operation performedto date 6,000,OOO 8,000,000 6,000,OOO
Directlabor hoursper unit (current)
Learning curve equation*
0.6 x 10” 1.7 x 10”’ 1.1 x 10”’
0.009 12 x, =-@*rnll’@’ 0.00707 Xa”O”Mg* Y, m0.33800 Xglo*slM1
20,000,000
4.2 n lo-’
Y, - 0.09410 X,“*a”M
0
20.0 x 10”
*The learningcurvefor a typical operation is derivedin AppendixE.
Y, = 3.00200 &-@*~“4”
KP
= .00157
.
3.4%
learning
betweti:n doubled
Fig. 3. The learningcurveof the potential product.
5. In the case of new products, there is often a variety of alternative manufacturing line designs available from which to chocse. Being able to predict initial as well as potential future standard costs would provide information as to which manufacturing line design would be the most economical. A higher initial standard cost with a faster rate of learning may, in the fmal analysis, be the most economical way, particularly where significant capital expenditures are involved. CONCLIJDING COMMENTS
Estimating learning curves for new product proposals need not be pure guesswork In many casesdata already exist as a result of prior experience. It is demonstrated in this paper that a data bank of learning curves for individual operations can be a useful tool. Particularly in situations where the proposed manufacturing process will be, for the most part, an aggregation of manufacturing operations which have already been performed in other processes. The author draws the analogy between the proposed method suggested in this article and the well-known method of developing standard times from historical data known as the Predetermined Time Study Approach. Then, a mathematical approach for combining the individual operation learning curves into a single
quantities
CUMULATIVE
xp-•o4gg2
*
( $ = 96.6%
100,000 Uf’iT NUNBER
l
) 1,OOO,O~ -
learning curve which represents the expected learning behavior of the potential product is developed. Also, some of the more important questions which one usually attempts to answer in advance for most new product proposals are identified. Finally, the potential usefulness of the learning curve (derived from the method suggested in this article) as a predictive model to help answer these questions is explored. APPENDIX I Deriwation of a Typical Learning Curve Photolithography The generalexpressioofor the learningcurveis Y=KX” where: Y = The numberof directlabor hours requiredto producethe xth unit. K.= The numberof direct laborhours requiredto producethe fiist unit. X = The cumulativeunit number. log 9 n=log 2 @ = %learningcurvedivideJby 100. Substitutingthe appropriatevaluesfrom Table III into this relationshipyields y = KXlog 4log 2 Therefore Y = KX-0.31’91
(1)
153
s*&gfor K: logY=l~*gK+IIlo~X log 0.6 Y ’ 0” = log K i- (-0.32192) log 6,000,OOO
Yp - 0.00115
K = 0.0091 t
Substituting K irrto equation (1) yields the learning curve for the Photolithography operation listed in Table III. Y, = -0.00912 x,-“*s~‘#” The subscript (1) denoted the fact that the Photolithography operation is the first operation in our five stage manufacturing process.
APPENDIX
The general expression for the product learning curve is . Yp = Y, + Y, + . . . + Yn (1) The subscript p denotes product. In our example,there are 5 separateoperations.Therefore, equation (1) becomes +Y,+Y,+Y,
+Y,
yP 1,000
0.00115
10,000 100,000 1.000.000
0.00998 0.0~089 0.00081
Y - KXn, yields the least squares fit Yp = 0.00157 xp-o’e’m
Determination of the proposed Product’s learning Cu we
l.p=Y,
X
The next step in the mathernaticaI the least squares fit for the points d method provides us with voluta,for n andA: which, whemSW& stituted into the general expression for them aam which is
,
II
Similarly, substituting additional vahtes for x of 10#00, 109,000 and I,OOO,OOO into equation (4) will yield the loI= lowing:
(2)
t6)
Equation(6) is the learning curve for the potential new product‘ Tbir function is plotted in Fig. 3. The last step is to substitute two values of Xp into equation (6) In order to obtain specif%vahtes with which to plot the function. Substituting 1,000 and l,OOO,OOO yieids respectively Yp = 0.00111 for Xp = 1,000 Yp = 0.00078 for Xp = l,OOO,OOO
This expression can be further expanded to yield Yp = J&X,“& + K,X,“a + K,X,“s + K,X,“r + K,X,“r(3) Substitutingthe appropriatevaluesfrom TableIII into equation (3) yields Yp = 0.00912 ~o*s~‘*a + 0.00707 X1’O*~S”” +
0.338 ,~;0*5”6l + 0.0941 X,‘O*sll*st 0.002 x;o
34ss
(4)
In orderto solve for the productlearningcurve,the next requiredstep is to substituteseveraldifferentprojectedcumulative production quantitiesfor the new product into equation (4). Assuminga value of 1,000 units for our new product, the appropriate X values are X, - 6,000,OOO+ 1,000 - 6,001,OOO X, - 8,000,OOO+ 1,000 - 8,001,OOO X, - 6,000,OOO+ 1,000 - 6,001,OOO x, - 20,000,000 + 1,000 aW 20,001,000 x,=0+1,000=1,000
which upon solution yields
t, “FactorsAffectingthe Cost of Airplanes,,” Aeronautical Science, February, 1936, pp. 122-128. A 90% learning curve rn@msthat the doubled quantities is 10%.Whe eas an 80% learning curve indicates that 20% learning occurs between doubled quantities. See Yelle (1974). Alternatively, the individual learning curves may be plotted on loglog paper thereby yielding a simple framework for a graphical solution, This approach is well described in Fabrycky and To (1966,~~. 111-112).
Andress, FrankI. (1954). “The Learnin Tool,” Hmwd &&new Review, Vo February), pp. 87-97.
Equation (4) now becomes Yp = 0.00912 (6,001,000)“*s1’9a + 0.00707 (8,001,000)-“*‘s4ss+ 0.338 (6,001,000)-“*s416’+ 0.0941 (20,001,000)-“‘s1’9a+ 0.002 (l,ooo)-O’~“”
NOTES
(5)
Brumeek, Ronald (I 959). “Leaming Curve Techniques for More RotSable Contracts,“,? ‘AA. Bztlletin,Volume XL (July), pp. 59-69. Cole, Rena R (1958). “‘increasingUtilization of the CostQuantity Relationship in Manufacturing,” Jmmal of I#&ufH~ngi?le#i?Jg, vobme IX (May-June), pp. 173-177. Cc&y, P. (1978). “Experience Curves as a PlanningTool,”
JREESpacmrm, Vohrme VJI (June), pp. 63-68. Couway, R.W. and Andrew Schultz, Jr. (1959). “The ManProgressFunction,” Journal of Industrihl Bnghwrkg, Volume X (January-February), pp. 39-54.
Fabrycky, W.J. and Paul E. Torgersen (1966). Operations Econonty. New Jersey: Prentice-Hail, Inc. Hirschmann,W.B. (1964). “Profit from the Learning Curve,” HarvardBusiness&view, Volume XL11(January-February), pp. 125-139. Hoffmann, Thomas R (1968). “Effect of Prior Experience on Learning Curve Parameters,” Journal of Indusmizl Engineering, Volume XIX (August), pp. 412-413. J+udan,R.B. (1958). “Learning How to Use the LearningCurve.” N.A.A. BulleHn, Volume XXXIX(January), pp. 27-39. Yelle, Louis E. (1974). “Technulogical Forecasting: A Learning Curve Approach,” hdusrrial Managermar, Volume XVI (January), pp. 6-l 1.
147
ESTIMATING LEARNING CURVES FOR POTENTIAL PRODlJCTS
-Louis E. Yelle
.INTRODUCTION
The well-known phenomenon of the learning curve can be a useful tool with which to predict standard costs. The usual approach is to work with a learning curve which represents the expected cost behavior of the complete product. In the case of well-established products, sufficient data usually exist due to prior production experience on the actual product to determine the learning curve for that product. However, in the case of new potential products, sufficient data on the specific product in question are not generally available. Faced with this situation, the customary ap preach for proposed new products is to select the learning curve of a similar product to use as a predictive model. In this article, the point is made that many new products are manufactured on production lines which are aggregates of existing technology. It is further SUP gested that the learning curve approach to predicting standard costs on new products as described above can be refined by first determining the learning curve for each operation of the total manufacturing process. The author then
demonstrates that the individual learni for each operation of the manufacturi can be mathematically combined to yield the learning cuye for the proposed product. THE LEARNING CURVE PHENOIMEIUON
The learning curve phenomenon w covered in the 1920’s in the aircraft i was subsequently reported in 1936 [ 11 c ~irn~~y number of direc duce an individual unit de
hour to manufactu manufacture the fourth format which can
148 TABLE I Numericat vafucs for a 90% learning curve where the amount of time required TVmanufacture the frost unit is one direct labor aoto&e.,K=l> _~__ ~- ~~ c3mlillatlve Dht labor hours number of unit!s required to manufacture Inan. one unit (905S L-CJ 1 2 4 8 16 32 64 128 256 512 1024
1.0 O.!) Ml 0.73 0.66 0.59 0.53 0.48 0.43
0.39 0.35
Leaf&g cufves follow the’mrthematical fmction
where: Y = The number of direct labor hours required to produce the xth unit. K = The number of direct labor hours required to produce the first unit. X = ‘Xe cumulative unit number. log 4J n=loe: 0 = % learning curve divided by 100. Learning curves are therefore straight lines when plotted on log-log paper. Some typical learning curves (all requiring the same number of direct labor hours to manufacture the first uni” . . . i.e., K = 1 hour) having different leaming rat<:: are shown in Fig 1. One question which people often ask upon a cursory view of learning curves is, “Will the de crease in the amount of direct labor hours requrred to manufacture a unit ever cease?” The answer, for ail practical purposes, is yes. For example, the total decrease in time as a result of experience gainedmanufacturing the first 500 units (Table I) is (1.0 hours - 0.39 hours) approximately 0.6 hours. However, the total decrease achieved as a result of manufacturing the
second 500 units is only (0.39 hours 0.35 hours) 0.04 hours. It should be clear, there= fore, that a; the quantity of units manufactured increases the decrease in direct labor hours per unit achieved as a result of increased experience approaches zero for all practical purposes. PRESENT DETERMINATION LEARNING CURVES
AND USE OF
For established products or products for which a sufficient number of units have already been manufactured, the data usually exis’c:to determine the learning curve for that product. All that is generally required is a sufficient amount of production experience in order to allow the manufacturing process to stabilize. This stability generally leads to discernible trends in the number of direct labor hours required as the cumulative number of units produced increases. At this point, the data can be readily accumulated and plotted in order to determine if the process follows the learning curve phenomenon and, if so, what the rate of learning is. In the case where production experience does not exist or is insufficient, the usual approach has been to select the learning curve ‘of,a
Louis E. YelIe is an Associate Professor of Management in the College of Management Science at The University of Lowell, Lowell, Massachusetts. Prior positions were in tie semiconductor industry with major mar.Jfacturers. More recently he has been a consultant to the electronics industry.
149
1
10
100
1,000
CUMlJCATtVE UNIT NUMBER Fig. 1. TypicalIearningewes
alI requiringone directlaborhour to manufacturethe fhst tit.
similar product and use it as a predictive model, the underlying assumption being that similar products will generally follow the same learning curve. This approach can be refined [ 33. THE PROPOSED APPROACH
The thesis of this article is that similaritiesin manufacturing technology can be used to construct the learning curve for potential products whose cost behavior we wish to predict. A fundamental assumption being made is that many new manufacturing processes are, in many instances, nothing more than an aggregation of well=developedoperations which can be found in other established manufacturing processes.What I am suggestingis an approach to establishing new product learning curves which would utilize a data bank of known learning curves for individual production operations. Then, once the process design has been established, the learning curves stored in the data bank which are applicable to the new manuf&cturingprocess can be utilized to construct the learning curve for the proposed new pre
duct. In essence, the approach is somewhat similar to the well-known method of developi standard times catted the Predetermined Time Study Approach which utilizes tables of histol ically established values. AN EXAJVIPLE
In the early X960’sthe established semiconductor manufacturers possessedsufficient historical experience in semiconductor diode manufacturing ‘to construct a list of productio operations (i.e., a data bank of production operations) and their ass&ated iearning curve A representative list of some of these operatio is shown in Table II. During this period of timl interest in reducing the size of diodes develops due to the need to reduce the siz3 of electroni equipment in general. This move towards micr miniaturization was necessary for a variety of well-knownreasons. One specific example wai the need to reduce the size of the Silicon-Plan Diode packaged in what is commonly known i the DO-7 package. At that time, engineers be gan to look at a variety of production prom
TABLE II List of representative production processes and their approximate historical data Production operations
% Learning curve
Cumulative number of times this operation performed to date
fi.rect labor hours per unit (current)
Diffusion Photolithography Varnish Passivation Glass Passivation Evaporation Contact Plating Scribing Soldering Machine Sealing
85% 80% 65% 70% 75% 70% 89% 95% 90%
8,000,OOO 6,000,OOO 2,000,000 100,000 1,‘500,000 6,000,OOO 20,000,000 20,000,000 20,000,000
1.7 0.6 10.0 3.3 5.0 1.1 4.2 11.0 8.3
x do-4 x 10” x 1o-4 x 10’4 x 10” x 10’” x 10” Y 10” 3 1O-4
Source: The data in this table was gathered during interviews cond\:cted by this author in 1973. The approximations are rough and meant to be used for illustrative purposes only.
cesses whereby this could be accomplished. The Sealing” operation were an 85% learning curve end result was later called the Double Plugand a standard time of 20 x 10B4hours per unit. Diode. A point which should be made at this The total process is depicted in Table III, along time is that it was necessary to change many of with the associated learning curves and standard the individual operations in the overall manudirect labor data.. The mathematical expression facturing process to make this new diode. In for a typbzal operation’s learning curve is other words, in reconstructing the total manuderived in Appendix I. facturing process, some production operations HISTORICAL remained, others were discarded, and new ones LEARNING CURVE were added. With one exception, the new OPERATION AVAILABLE operations which were added had been developed previously on other diode manufacturing 1 Photollthc-raphy Yes lines. The initial process design decided upon by 2 Dlffuslon yes manufacturing engineers for the miniature diode is depicted in Fig. 2. With the sole excep 3 Contact Platlng Yes tion of the last operation entitled “Furnace Sealing,” the process was nothing more than an 4 Scribing yes aggregate of existing manufa.cturing operations developed on other diodes. And, historical learn5 Furnace Sealing no ing curve data existed for all but this last opera1 tion, When historical data do not exist for an END operation, the applicable learning curve along PRODUCT with the associated direct labor must be , estimated. When possible, small pilot runs Pig. 2. New diode process utilizing available manufacturing should be made as an aid in estimating. In this operations except for the last operation. spe~if’ic case, the estimates for the “Furnace
151 DETERMINING THE LEARNI CURVE FOR THE PROPOSEDPRODUCT The remaining task is to determine the learning curve for the potential product by mathe matically combining the individual learning curves for each operation comprising the manufacturing process. In our example, there are five separate manufacturing operations and, therefore, five separate learning curves. These leaming curves can be combined mathematically [4] to yield the learning curve which will describe the expected labor standard for the new product as a function of the cumulative number of units produced. The mathematical treatment is shown in Appendix II. The results yield the product learning curve for our example, which may be expressed as follows:
r, = 0.00157 Xp-o*0’99a The learning curve of the potential product is plotted in Fig. 3. SOME POTENTiAL USESOF THE PROPOSEDMETHQD
Some potential uses of the method of estimating the applicable learning curve presented in this article are: 1, To estimate standard costs on new product proposals. Predicting standard cost is
an essential step in estimating the future market potential of a new product. And, the future market potential will determine the advisability of taking the necessary steps to manufacture the product. 2. Accurate predictions of standard costs on proposed pr+cts are important for capital budgeting - particularly long-term projections. As the standard cost decreases, the product generally makes inroads into new marketing segments. Estimating how much of a capital investment will be needed in the future to manufacture the product in such a manner that the fm will remain competitive is important. A small firm may conclude that the required capital investment will exceed its capability to acquire capital. A large f^mnwould use the information to help allocate capital among alternative investment proposals. 3. Learning curves are useful when making pricing decisions. In the case of new products on which sufficient learning curve data do not exist, the cost of the initial contract or order is an important decision. Also, in situations where pricing for orders to be delivered far in the future is necessary, the learning curve can also serve as a useful predictive model. 4. Financial control is a necessary ingredient of all successful firms. The ability to predict standard costs in the future is an important step in making out pro-forma financial statements and budgets.
TABLEIII Double-Plug-Diodemanufacturingprocessstepsand associatedhistoricaldatagleanedfrom TableII Operation
96 Learning CUWd
1 2 3
Photolithography Diffusion Contact Plating
80% 85% 70%
4
Scribing
80%
5
FurnaceSealing
85%
Cumulativenumberof timesthis operation performedto date 6,000,OOO 8,000,000 6,000,OOO
Directlabor hoursper unit (current)
Learning curve equation*
0.6 x 10” 1.7 x 10”’ 1.1 x 10”’
0.009 12 x, =-@*rnll’@’ 0.00707 Xa”O”Mg* Y, m0.33800 Xglo*slM1
20,000,000
4.2 n lo-’
Y, - 0.09410 X,“*a”M
0
20.0 x 10”
*The learningcurvefor a typical operation is derivedin AppendixE.
Y, = 3.00200 &-@*~“4”
KP
= .00157
.
3.4%
learning
betweti:n doubled
Fig. 3. The learningcurveof the potential product.
5. In the case of new products, there is often a variety of alternative manufacturing line designs available from which to chocse. Being able to predict initial as well as potential future standard costs would provide information as to which manufacturing line design would be the most economical. A higher initial standard cost with a faster rate of learning may, in the fmal analysis, be the most economical way, particularly where significant capital expenditures are involved. CONCLIJDING COMMENTS
Estimating learning curves for new product proposals need not be pure guesswork In many casesdata already exist as a result of prior experience. It is demonstrated in this paper that a data bank of learning curves for individual operations can be a useful tool. Particularly in situations where the proposed manufacturing process will be, for the most part, an aggregation of manufacturing operations which have already been performed in other processes. The author draws the analogy between the proposed method suggested in this article and the well-known method of developing standard times from historical data known as the Predetermined Time Study Approach. Then, a mathematical approach for combining the individual operation learning curves into a single
quantities
CUMULATIVE
xp-•o4gg2
*
( $ = 96.6%
100,000 Uf’iT NUNBER
l
) 1,OOO,O~ -
learning curve which represents the expected learning behavior of the potential product is developed. Also, some of the more important questions which one usually attempts to answer in advance for most new product proposals are identified. Finally, the potential usefulness of the learning curve (derived from the method suggested in this article) as a predictive model to help answer these questions is explored. APPENDIX I Deriwation of a Typical Learning Curve Photolithography The generalexpressioofor the learningcurveis Y=KX” where: Y = The numberof directlabor hours requiredto producethe xth unit. K.= The numberof direct laborhours requiredto producethe fiist unit. X = The cumulativeunit number. log 9 n=log 2 @ = %learningcurvedivideJby 100. Substitutingthe appropriatevaluesfrom Table III into this relationshipyields y = KXlog 4log 2 Therefore Y = KX-0.31’91
(1)
153
s*&gfor K: logY=l~*gK+IIlo~X log 0.6 Y ’ 0” = log K i- (-0.32192) log 6,000,OOO
Yp - 0.00115
K = 0.0091 t
Substituting K irrto equation (1) yields the learning curve for the Photolithography operation listed in Table III. Y, = -0.00912 x,-“*s~‘#” The subscript (1) denoted the fact that the Photolithography operation is the first operation in our five stage manufacturing process.
APPENDIX
The general expression for the product learning curve is . Yp = Y, + Y, + . . . + Yn (1) The subscript p denotes product. In our example,there are 5 separateoperations.Therefore, equation (1) becomes +Y,+Y,+Y,
+Y,
yP 1,000
0.00115
10,000 100,000 1.000.000
0.00998 0.0~089 0.00081
Y - KXn, yields the least squares fit Yp = 0.00157 xp-o’e’m
Determination of the proposed Product’s learning Cu we
l.p=Y,
X
The next step in the mathernaticaI the least squares fit for the points d method provides us with voluta,for n andA: which, whemSW& stituted into the general expression for them aam which is
,
II
Similarly, substituting additional vahtes for x of 10#00, 109,000 and I,OOO,OOO into equation (4) will yield the loI= lowing:
(2)
t6)
Equation(6) is the learning curve for the potential new product‘ Tbir function is plotted in Fig. 3. The last step is to substitute two values of Xp into equation (6) In order to obtain specif%vahtes with which to plot the function. Substituting 1,000 and l,OOO,OOO yieids respectively Yp = 0.00111 for Xp = 1,000 Yp = 0.00078 for Xp = l,OOO,OOO
This expression can be further expanded to yield Yp = J&X,“& + K,X,“a + K,X,“s + K,X,“r + K,X,“r(3) Substitutingthe appropriatevaluesfrom TableIII into equation (3) yields Yp = 0.00912 ~o*s~‘*a + 0.00707 X1’O*~S”” +
0.338 ,~;0*5”6l + 0.0941 X,‘O*sll*st 0.002 x;o
34ss
(4)
In orderto solve for the productlearningcurve,the next requiredstep is to substituteseveraldifferentprojectedcumulative production quantitiesfor the new product into equation (4). Assuminga value of 1,000 units for our new product, the appropriate X values are X, - 6,000,OOO+ 1,000 - 6,001,OOO X, - 8,000,OOO+ 1,000 - 8,001,OOO X, - 6,000,OOO+ 1,000 - 6,001,OOO x, - 20,000,000 + 1,000 aW 20,001,000 x,=0+1,000=1,000
which upon solution yields
t, “FactorsAffectingthe Cost of Airplanes,,” Aeronautical Science, February, 1936, pp. 122-128. A 90% learning curve rn@msthat the doubled quantities is 10%.Whe eas an 80% learning curve indicates that 20% learning occurs between doubled quantities. See Yelle (1974). Alternatively, the individual learning curves may be plotted on loglog paper thereby yielding a simple framework for a graphical solution, This approach is well described in Fabrycky and To (1966,~~. 111-112).
Andress, FrankI. (1954). “The Learnin Tool,” Hmwd &&new Review, Vo February), pp. 87-97.
Equation (4) now becomes Yp = 0.00912 (6,001,000)“*s1’9a + 0.00707 (8,001,000)-“*‘s4ss+ 0.338 (6,001,000)-“*s416’+ 0.0941 (20,001,000)-“‘s1’9a+ 0.002 (l,ooo)-O’~“”
NOTES
(5)
Brumeek, Ronald (I 959). “Leaming Curve Techniques for More RotSable Contracts,“,? ‘AA. Bztlletin,Volume XL (July), pp. 59-69. Cole, Rena R (1958). “‘increasingUtilization of the CostQuantity Relationship in Manufacturing,” Jmmal of I#&ufH~ngi?le#i?Jg, vobme IX (May-June), pp. 173-177. Cc&y, P. (1978). “Experience Curves as a PlanningTool,”
JREESpacmrm, Vohrme VJI (June), pp. 63-68. Couway, R.W. and Andrew Schultz, Jr. (1959). “The ManProgressFunction,” Journal of Industrihl Bnghwrkg, Volume X (January-February), pp. 39-54.
Fabrycky, W.J. and Paul E. Torgersen (1966). Operations Econonty. New Jersey: Prentice-Hail, Inc. Hirschmann,W.B. (1964). “Profit from the Learning Curve,” HarvardBusiness&view, Volume XL11(January-February), pp. 125-139. Hoffmann, Thomas R (1968). “Effect of Prior Experience on Learning Curve Parameters,” Journal of Indusmizl Engineering, Volume XIX (August), pp. 412-413. J+udan,R.B. (1958). “Learning How to Use the LearningCurve.” N.A.A. BulleHn, Volume XXXIX(January), pp. 27-39. Yelle, Louis E. (1974). “Technulogical Forecasting: A Learning Curve Approach,” hdusrrial Managermar, Volume XVI (January), pp. 6-l 1.