176 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 39, NO. 2, MAY 1992

Computational Survey of Univariate and Multivariate Learning Curve Models

Adedeji B. Badiru

Abstract-This paper presents a computational survey of the various univariate and multivariate learning curve models that have evolved over the past several years. Discussions are pre- sented to show how the models might be used for cost analysis or productivity assessment in engineering management. A com- putational experiment comparing a univariate model to a bivari- ate model is presented. While the bivariate model provides only slightly better fit than the univariate model, it does provide more detailed information about factor interactions and better utiliza- tion of available data. The results of the computational experi- ment can be generalized for the appropriateness of multivariate models.

Keywords-Learning curve; univariate model; multivariate model; cost estimation; productivity assessment.

I. INTRODUCTION

EARNING, in the context of operations management, L refers to the improved efficiency obtained from repeti- tion of a production operation. Workers learn and improve by repeating operations. Learning is time-dependent and externally controllable. Several research studies have con- firmed that human performance improves with reinforcement or frequent repetitions [68], [56]. Several publications have documented the, historical aspects of the evolution of the learning curve [33], [48], [16], [68], [9]. Redubtions in operation processing times achieved through learning curve effects can directly translate to cost savings for manufacturers and improved morale for employees [5]. Learning curves, also known as progress function, cost-quantity relationship, cost curve, product acceleration curve, improvement curve, performance curve, experience curve, or efficiency curve, are essential for functions such as setting production goals [ S I , [52], [53], cost control [39], resource allocation [43], lot sizing [23], [41], and product pricing [19].

Typical learning curves present the relationship between production cost and cumulative output based on the effect of learning. An early study by Wright [64] disclosed the “80 percent learning effect” which indicates that a given opera- tion is subject to a 20% productivity improvement each time the production quantity doubles. Although extensive litera- ture is available on univariate and multivariate models of learning curves, there are no comprehensive surveys that

combine the univariate and multivariate literature. Such a combined survey is provided in this paper.

11. UNIVARIATE MODELS The conventional univariate learning curve expresses a

dependent variable (e.g., production cost) in terms of some independent variable (e.g., cumulative output). Since the first formal publication of learning curve theory by Wright [64], there have been numerous propositions concerning the repre- sentational geometry and functional forms [7], [37], [41], [54], [57], [65]. The most notable univariate models include:

1) The log-linear model [64]; 2) The S-curve [14]; 3) The Stanford-B model [4]; 4) DeJong’s learning formula [20]; 5) Levy’s adaptation function [42]; 6) Glover’s learning formula [25]; 7) Pegel’s exponential function [49]; 8) Knecht’s upturn model [39]; 9) Yelle’s product model [66];

10) Multiplicative Power Model [58].

A . The Log-Linear Model The log-linear model [64] is often referred to as the

conventional learning curve model. There are two basic forms of the log-linear model: the average cost function and the unit cost function.

I ) Average Cost Model: The average cost model is more popular than the unit cost model. It specifies the relationship between the cumulative average cost per unit and cumulative production. The relationship indicates that cumulative cost per unit will decre?se by a constant percentage as the cumula- tive production volume doubles. The model is expressed as:

c, = C , X b

where:

C, C , x = cumulative production count b

= cumulative average cost of producing x units = cost of the first unit

= the learning curve exponent (constant slope of the curve on log-log paper)

Manuscript received September 7, 1990.

Oklahoma, Norman, OK 73019. IEEE Log Number 9 10705 1.

When a linear graph paper is used, the log-linear learning curve is a hyperbola of the form shown in Fig. 1. On a log-log paper, the model is represented by the following

The author is with the School of Industrial Engineering, University of

0018-9391/92$03.00 0 1992 IEEE

BADIRU: SURVEY OF LEARNING CURVE MODELS 177

Cumulative Production

Fig. 1 . Log-linear learning curve model

straight line equation:

log C, = log C, + b log x

where b is the constant slope of the line. The log-log plot of the model is shown in Fig. 2. The expression for the learning rate, p , is derived by considering two production levels where one level is double the other. For example, given the two levels x1 and x2 (where x2 = 2x,) , we have the following expressions:

c x , = C , ( X d b

c,, = C,(2x,)b.

The percent productivity gain is then computed as:

Expression for Total Cost: Using the basic cumulative average cost function, the total cost of producing x units is computed as:

TCx = (x)C, = ( x ) C , X ~ = C , X ( ~ + ' ) .

Expression for Unit Cost: The unit cost of producing the xth unit is given by:

uc, = C,X'b+l)-C1(X - l )@+?

Expression for Marginal Cost: The marginal cost of pro- ducing the xth unit is given by:

MC, = ____ d[TCxl - - ( b + l)C,xb dx

An important application of learning curve analysis is the calculation of expected production time as illustrated by the following example. In a production run of a certain air craft component, it was observed that the cumulative hours re- quired to produce 100 units is 100OOO h with a learning curve effect of 85 % . For future project planning purposes, an analyst needs to calculate the number of hours spent in

1 10 1w loo0

Log of Cumulative Units

Log-linear model on log-log paper. Fig. 2.

producing the fiftieth unit. Following the notation used previ- ously, we have the following:

p = 0.85; X = 100 units;

C, = 100OOO h/100 units (i.e., 1 OOO h/unit).

Thus, 0.85 = 2', which yields b = -0.2345. Conse- quently, we have 1OOO = C,(lOO)b, which yields C, = 2,944.42 h. Since b and C, are now known, we can compute the cumulative average hours required to produce 49 units and 50 units, respectively:

C,, = C,(49)b = 1182.09 h

C,, = C,(50) = 1176.50 h .

Consequently, the number of hours spent in producing the fiftieth unit is given by: 50(1176.50 h) - 49(1182.09 h) = 902.59 h.

2) The Unit Cost Model: The unit cost model is ex- pressed in terms of the specific cost of producing the xth unit. The unit cost formula specifies that the individual cost per unit will decrease by a constant percentage as cumulative production doubles. The functional form of the unit cost model is the same as for the average cost model except that the interpretations of the terms are different. It is expressed as:

uc, = C , X b

where:

UC, = cost of producing the xth unit C, = cost of the first unit; X = cumulative production count; b = the learning curve exponent as discussed previ-

ously.

178 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 39, NO. 2, MAY 1992

From the unit cost formula, we can derive expressions for the other cost basis. For the discrete case, the total cost of producing x units is given by:

TC, = 9 I/ci = C, 5 ( i > b i = 1 i = 1

The cumulative average cost per unit is given by:

The marginal cost is found as follows:

d [ l + 2' + 3' + * e - + x b ] dx

= c, = C , b x b - ' .

For the continuous case, we have the following correspond- ing expressions:

The total cost for discrete production may be estimated by using integral approximation techniques [ 1 1 1 . The total cost for producing the first K units can be obtained from the learning curve equation as shown below:

K

Tc(b, K ) = '1 X b x= 1

which can be approximated by either of the following:

A

l + b = __ [ ( K + 0 . 5 ) 1 + b - 0 . 5 1 + b ] .

An analysis of the magnitude of the estimation error caused by the above approximations was reported by [ 1 1 1 .

B. The S-Curve Carr [14] proposed an S-shaped learning curve function

based on an assumption of a gradual start-up. The function has the shape of the cumulative normal distribution function for the start-up curve and the shape of an operating character- istics function for the learning curve. The gradual start-up is based on the fact that the early stages of production are

typically in a transient state with changes in tooling, meth- ods, materials, design, and even the workers. The basic form of the S-curve formula is

M C , = C , [ M + (1 - M ) ( x + B ) ~ ]

where

M B

= incompressibility factor (a constant); = equivalent experience units (a constant).

Assumptions about at least three out of the four parameters ( M , B, C,, and b ) are needed to solve for the fourth one. Alternately, the coefficients of the S-curve can be determined by fitting a cubic curve on a log-log plot. An example of such a cubic function is

log MC, = A + B(1og X ) + C(1og x') + D(1og x ' ) .

C. The Stanford-B Model An early study commissioned by the U.S. Defense Depart-

ment at the Stanford Research Institute [4] led to the develop- ment of the Stanford-B model, which was found to be more representative of World War II data. The model is repre- sented as:

Y, = C , ( X + B ) b

where

Y" = c, = b = B =

- -

- -

direct cost of producing the xth unit; cost of the first unit when B = 0; slope of the asymptote; constant (1 < B < lo); equivalent units of previous experience at the start of the process; equivalent to the number of units produced prior to first unit acceptance.

It is noted that when B = 0, the Stanford-B model reduces to Y, = C, x b , which is the conventional log-linear model. The Boeing Company found that the Stanford-B model was the best model for the manufacturing of the Boeing 707. Hoff- man [34] found that the inclusion of the B parameter resulted in smaller sums of squared deviations.

D. DeJong 's Learning Formula DeJong [20] presented a power function which incorpo-

rates parameters for the proportion of manual activity in a task. When operations are controlled by manual tasks, the time will be compressible as successive units are completed. If, by contrast, machine cycle times control operations, then the time will be less compressible as the number of units increases. DeJong 's formula introduces an incompressible factor, M , into the log-linear model to account for the man-machine ratio. The model is expressed as:

MC, = C , [ M + ( 1 - M ) x - b ]

where

M = incompressibility factor (constant)

BADIRU: SURVEY OF LEARNING CURVE MODELS 179

When M = 0, the model reduces to the log-linear model, which implies a completely manual operation. In completely machine-dominated operations, M = 1. In that case, the unit cost reduces to a constant equal to C,, which suggests that no cost improvement is possible in machine-controlled opera- tions. This represents a condition of high incompressibility. Regrettably, no significant published data is available on whether or not DeJong's model has been successfully used to account for the degree of automation in any given operation. With the increasing move towards automation in industry, this certainly is a topic for urgent research.

E. Levy's Adaptation Function Recognizing that the log-linear model does not account for

The total cost of producing x units is derived from the marginal cost as follows:

TC, = ( a a x - l + 6 ) dx J

where c is a constant to be derived after the other parameters are found. The constant can be found by letting the marginal cost, total cost, and average cost of the first unit to be all equal. That is, MC, = TC, = AC, . This yields

ff c = f f - -

In ( a ) , ,

leveling off of production rate and the factors that may influence learning, Levy [42] proposed the following learning cost function:

Pegel's exponential model assumes that the marginal cost of the first unit is known. Thus,

MC, = ff + p = y o . - 1

X b Pegels also presented another mathematical expression for the total labor cost in start-up curves [50]. He expressed the total cost as:

MC, = [; - [; - 4 k - q

a X 1 - b TC, = - 1 - b

where

/3 k

= production index for the first unit; = constant used to flatten the learning curve for large where

values of x. x = cumualtive number of units produced; a, b = empirically determined parameters.

The expressions for marginal cost, average cost, and unit cost can be derived as shown earlier for other models.

The flattening constant, k , forces the curve to reach a plateau instead of continuing to decrease or turning in the upward direction.

F. Glover's Learning Formula H. Knecht's Upturn Model

Glover [25] presented a model which incorporates a work commencement factor. The model is based on a bottom-up approach which uses individual worker learning results as the basis for plantwide learning curve standards. The functional form of the model is expressed as:

where

y i x i a = commencement factor; n m = model parameter.

= elapsed time or cumulative quantity; = cumulative quantity or elapsed time;

= index of the curve (usually 1 + b);

C. Pegel's Exponential Function

Knecht [39] presents a modification to the functional form of the learning curve to analytically express the observed divergence of actual costs from those predicted by learning curve theory when units produced exceed 200. This permits the consideration of nonconstant slopes for the learning curve model. If UC, is defined as the unit cost of the xth unit, then it approaches 0 asymptotically as x increases. To avoid a zero limit unit cost, the basic functional form is modified. In the continuous case, the formula for cumulative average costs is derived as:

C , X b + ' C, = l x C , z b d z = ~

( 1 + b) '

This cumulative cost also approaches zero as x goes to infinity. Knecht alters the expression for the cumulative curve to allow for an upturn in the learning curve at large cumula- tive production levels. He suggested the functional form below: Pegels [49] presented an alternate algebraic function for

the learning curve. His model, a form of an exponential function, is represented as:

C, = ClxbeCX

where c is a second constant. Differentiating the modified cumulative average cost expression gives the unit cost of the xth unit as shown below: MC, = cyax-I + /3

where a, 0, and a are parameters based on empirical data analysis.

d

dx UC, = - [ C , xbecx] = ClxbeCX

I80 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 39, NO. 2, MAY 1992

A model similar to Knecht’s model is the plateau model described by Baloff [7]. The plateau model assumes that production cost reaches a steady state at which point cost levels off.

I. Yelle’s Combined Product Learning Curve

Yelle [66] proposed a learning curve model for products by aggregating and extrapolating the individual learning curves of the operations making up a product on a log-linear plot. The model, which is similar to one of the cost estimat- ing relationships presented by Waller and Dwyer [%I, is expressed as shown below:

c, = k , X f ’ + k,X,b? + . . . +k,X,b,

where

cx n k ,x ,”~

= cost of producing the xth unit of the product; = number of operations making up the product; = learning curve for the ith operation.

Observations drawn from the mathematical expression above was then used to fit a least squares function of the log-linear form for the product linear curve. As pointed out by Howell [35], Yelle’s model contains several deficiencies. Some of the noted shortcomings are:

1) A learning curve formulated by aggregating several different learning curves (with different slopes) will not necessarily be a straight line on a linear plot.

2) A learning curve extrapolated from different learning curves will not necessarily be a straight line.

3) A product specific learning curve seems to be a more reasonable model than an integrated product curve.

For example, an aggregated learning curve with 96.6% learning rate obtained from individual learning curves with the respective learning rates of 80%, 70%, 8 5 % , 80%, and 85% does not appear to represent reality. If this type of composite improvement is possible, then one can always improve the learning rate for any operation by decomposing it into smaller integrated operations. Fig. 3 shows compara- tive plots of some of the models discussed above on a log-log scale.

111. MULTIVARIATE MODELS

Extensions and modifications of conventional learning curves are important for realistic analysis of productivity gain [57]. In many operations, several tangible and intangible, quantitative and qualitative factors intermingle to compound the productivity analysis problem [6]. There are numerous factors that can influence how fast, how far, and how well a worker learns within a given time horizon. Multivariate models have not received the attention they deserve in prac- tice [13]. This is perhaps due to the complexity of implement- ing the models for practical productivity assessment. Multi- collinearity is also a major problem in implementing multi- variate models [2], [ 1 11, [ 121. With the increasing prolifera- tion of numerical analysis tools, it can be expected that the models will begin to draw more appeal. A popular form of

T

1

I I I

10 loo 10W

Cumulative Production Units

Fig. 3 . Comparison of learning curve models on a log scale.

the multivariate learning curve model is shown below:

n

i = 1 c, = K n C;X,”l

where

C,

K x

x, n c, b,

For simplicity and ease of analysis, the above model is

= cumulative average cost per unit for a given set of

= model parameter (cost of first unit of the product); = vector of specific values of independent variables

= specific value of the ith factor; = number of factors in the model; = coefficient for the ith factor; = learning exponent for the ith factor.

often reduced to a bivariate model of the form below:

factor values;

(factors) ;

where y is a measure of cost and x , and x2 are independent variables of interest. The response surface for a generic bivariate model is shown in Fig. 4.

Conway and Schultz [17] first suggested the need for multivariate generalized progress functions. They point out that there are other factors that influence learning. For exam- ple, they present a hypothetical response surface relating cost to production rate and cumulative production volume. That idea and related investigations have since been studied by several authors including Alchian [2], Preston and Keachie [51], Graver and Boren [28], Goldberger [27], McIntyre [MI, Womer [61], Bemis [lo], Cox and Gander [18], Womer [62], Donath et al. [21], Gold [26], Waller and Dwyer [%I, Gulledge et al. [32], Gulledge and Khoshnevis [29], Camm, et al. [12], Camm et al. [11], and Dada and Srikanth [19].

BADIRU: SURVEY OF LEARNING CURVE MODELS 181

t xl

f

x2 \

Fig. 4. Generic response surface for bivariate learning curve

A. Alchian’s RAND Experiments A study conducted for the RAND Corporation [2] using

World War 11 data used manufacturing progress curves to estimate direct labor per pound of airframe needed to manu- facture the Nth airframe in a cumulative production of N planes. A study of the reliability of predictions made with the curve was also conducted. The curves were found to differ from one airframe type to another. However, the average error of prediction was found to range between 20 and 25%. Recognizing that lower direct labor costs occurred as the number of planes produced increased, the following ques-

were raised:

How long does the reduction in labor costs continue? Can the reduction be represented by a linear function on double log scale? Does the reduction fall at the same rate for all different airframe manufacturing facilities? Can reliable predictions of marginal and total labor requirements be obtained for a given facility based on an average progress curve for all airframe manufactur- ers? Can reliable predictions of labor requirements for a specific type of bomber be obtained from a curve fitted to the experience of all bomber production? How reliable is a single plant’s own experience for predicting later requirements for producing a particular type of airframe? What are the potential consequences of errors in the predictions obtained from the pregress functions?

It is not surprising that these are still the same questions faced by users of learning curves today. In addressing the second question, Alchian considered progress functions con- taining other variables in addition to cumulative production, N. This effort provided the first known report of experimen- tation with multivariate learning curves. Similar or related efforts were later reported by Preston and Keachie [51] and

Womer and Gulledge 1631. The alternate functions presented below were used by Alchian to describe the relationships between Direct Labor per pound of airframe ( m ) , Cumula- tive Production (N), Time ( T ) , and Rate of Production per month ( A N ) :

1) log m = a2 + & T ; 2) log m = a3 + @,T + p4 A N ; 3) log m = a4 + &(log T ) + &(log A N ) ; 4) log m = a5 + P7T + &(log A N ) ; 5 ) log m = cy6 + P,T + Pl0(log N ) ; 6 ) log m = a7 + Pll(log N ) + pI2(log A N ) .

Extensive experimentations with the above models failed to yield significantly better fits than the conventional univariate model. The major reason for the lack of significant improve- ments with the multivariate functions is the fact that there is a high degree of correlation among the independent variables, T , N , and AN. These discouraging results might have contributed to the initial decline of interest in multivariate learning models. The works of Preston and Keachie [51] and Graver and Boren [28] helped in reviving the interest. But, as noted by Camm et al. [ 12) and confirmed by a computational experiment later in this paper, the problem of multicollinear- ity (multiple correlation) is still a major concern in the use of multivariate learning curve models.

B. Cobb-Douglas Multiplicative Power Model The Cobb-Douglas function of the general form presented

below has been studied extensively in the literature [27], [lo], U81, [581, [291:

c = boX,blX:2.. . X F E

where

C = estimated cost; bo = model coefficient; x i b j E = error term.

= ith independent variable ( i = 1,2, . . . , n); = exponent of the ith variable;

In the above model, the disturbance term, E , is defined as:

E = e’ where U - N(0, a 2 ) and independent of XI, - , X , . Thus, the mean of the disturbance is given by E ( € ) = exp (a2/2) while its median is M ( E ) = 1. With these expressions, the conditional mean, E(C 1 X ) , and conditional median, M(C I X ) , were derived for the cost, C. Goldberger [27] pointed out the estimation problems associated with multi- plicative functions. The usual estimation approach is to use logarithmic transformations. While the approach works very well, it shifts the investigation focus from the conditional mean to the conditional median of the process under study. This shift may result in misleading interpretations of the fitted model. Goldberger [27] presents a modified procedure and numerical illustrations that provide minimum variance unbi- ased estimation of the conditional median or conditional mean. Details of the modified procedure are beyond the scope of this survey. Readers interested in the statistical

~

182 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 39, NO. 2, MAY 1992

implications of Cobb- Douglas functions should refer to the above reference and also Graver and Boren [28].

For parametric cost analysis, Waller and Dwyer 1581 pre- sent a variation of the Cobb-Douglas function and an addi- tive power model of the form:

c = C , X f ' + c * x p + . * * +c,x,bn + E

where c, ( i = 1 , 2 , . . . , n) is the coefficient of the ith inde- pendent variable. The multiplicative power model was fitted by using logarithmic transformation and standard multiple regression technique. The model was reported to have been fitted successfully for missile tooling and test equipment cost. Camm et al. [12] cautioned that the use of regression approach for the power model can be misleading because of the potential for collinearity and failure to appropriately represent actual behavior of the producer. This observation raises serious questions about the conventional approach to modeling multivariate learning curves. It is expected that future research efforts will be directed at careful investiga- tions of the assumptions and shortcomings of the regression approach to multivariate models.

A variation of the power model was used by Bemis [lo] to study an actual weapon system production. Cox and Gansler [18] also discuss the use of a multivariate model for the assessment of the costs and benefits of a single-source versus multiple-source production decision with variations in quan- tity and production rate in major DOD (Department of Defense) programs. A related study by Camm et al. [12] also uses the multiplicative power model to express program costs in terms of cumulative quantity and production rate in order to evaluate contractor behavior. A variation of the Cobb-Douglas function is used later in this paper to illustrate a computational analysis of a bivariate model involving train- ing time and cumualtive production.

C. Mclntyre's General Nonlinear Models McIntyre [44] introduced a nonlinear cost-volume-profit

model for learning curve analysis. The nonlinearity in the model is effected by incorporating a nonlinear cost function that expresses the effects of employee learning. McIntyre applied sensitity analysis to the nonlinear model to assess the impact of estimation errors in the learning rate and steady- state production time on estimated profit and break-even quantities. Several forms of the nonlinear model are pre- sented.

BASIC PROFIT FUNCTION The profit equation for the initial period of production for

a product subject to the usual learning function is expressed as :

P = px - c (axb+' ) - f where

P = profit; p = price per unit; x = cumulative production; c = labor cost per unit time; f = fixed cost per period; b = index of learning.

MUL TIPROCESS MODEL The profit function for the initial period of production with

n production processes operating simultaneously is given as:

p = p x - n c a ( X ) b+ 1 - f

where x is the number of units produced by n labor teams consisting of one or more employees each. Each team is assumed to produce x / n units. As McIntyre [44] observed, this model indicates that when additional production teams are included, more units are produced over a given time period. However, the average time for a given number of units increases because more employees are producing while learning. That is, more employees with low (but improving) productivity are engaged in production at the same time.

MUL TISKILL MODEL The preceding model can be extended to the case where

different skill levels of employees produce different learning parameters between production runs. Such an extension is

n n P = p C x i - CC aix!t+l - f

i = 1 i = 1

where a, and 6, denote the parameters applicable to the average skill level of the ith production run and x i repre- sents the output of the ith run in a given time period. This is a model that can find good use in modern manufacturing systems that call for simultaneous engineering.

D. Womer's Variable Production Rate Model Womer [61] presents a multivariate model that incorpo-

rates cumulative production, production rate, and program cost. Womer's approach presents a production function that relates output rate to a set of inputs with variable utilization rate. The production function is assumed to be:

q( t ) = AQ*( t ) XI''( t )

where

A = constant; q ( t ) = program output rate at time t ; Q ( T ) = cumulative production at time T ; 6 = learning parameter; h = returns to scale parameter; x ( t ) = rate of variable resource utilization at time t .

To optimize the discounted program cost, the cost function is defined as:

C = i ' x ( t ) e - p t dt

where p is the discount rate and T is the time horizon for the analysis. If V is defined as the planned cumulative produc- tion at time T (i.e., Q ( T ) = V ) , then the problem can be

BADIRU: SURVEY OF LEARNING CURVE MODELS 183

formulated as the minimization problem shown below:

min l T x ( t )ePP' dt

S.t. q ( t ) = A Q * ( ( t ) x ' / y ( t )

x ( t ) 2 0

Q(0) = 0

Q(T) = V

whose solution yields the estimated cost at time t , given V and T :

The optimized cost can be expressed in terms of general cumulative production, Q ( t ) , as shown below:

Along the same line, Gulledge et al. [32] present a dy- namic optimization model that considers the influence of production rate and learning on total cost. The model, which yields optimal production rates for contracted production, has been successfully applied to military production programs. Washburn [59] also addresses the effects of discounting on profit in the presence of learning.

Importance of Multivariate Models: As has been pointed out by several authors [61], [ 1 2 ] , many economic and pro- duction processes may require the inclusion of additional variables in learning curve models. In a simple bivariate model, it may be impossible to obtain accurate estimates of the effects of the two variables. Consequently, it may be necessary to consider more complicated models such as the one presented by Womer and Gulledge [63]. In addition to the more robust estimation potential of multivariate models, the models also provide a means for accounting for more of the available data.

IV. COMPUTATIONAL ANALYSIS This section presents a computational experiment with a

learning curve model containing two independent variables: cumulative production ( x , ) and cumulative training time ( x , ) and one dependent variable: expected production cost ( C ) . The analysis of the bivariate model compared to a univariate model reiterates the points made in the preceding section. Similar points can be made when going from a bivariate model to a model with more than two independent variables.

where

c, =

K =

The experimental model is given by:

c,,,* = Kc,xp~c2x,b2

cumulative average cost per unit for a given set of factor values; intrinsic constant;

x 1 x , ci b, The set of real test data used in the computational analysis

is shown in Table I. Two data replicates are used for each of the ten combinations of cost and time values. Observations are recorded for the number of units representing double production volumes. Data replicates were obtained by record- ing observations from identical setups of the process being studied. The model is represented in logarithmic scale to facilitate the curve fitting procedure as shown below:

log C, = [log K + log ( c , c ~ ) ] + b, log X , + b2 log X ,

= specific value of first factor; = specific value of second factor; = coefficient for the ith factor; = learning exponent for the ith factor.

= log a + b, log x , + b, log x2

where "a" represents the combined constant in the model such that:

a = (K)(CI) (CZ) . Using the multiple regression function in STATGRAPH-

ICs software, the following model was fitted for the data:

log C, = 5.70 - 0.21(10g x , ) - 0.13(10g x , )

which transforms into the multiplicative model given below:

C, = 2 9 8 . 8 8 ~ ; ~ . ~ ~ ~ , ~ . ' ~

where

a = 298.88 (i.e., log(a) = 5.70); C , = cumulative average cost per unit; x , = cumulative production units; x2 = cumulative training time in hours.

Since a = 298.88, we have (KC,.*) = 298.88. If two of the constants K , c , , or c2 are known, then the third can be computed. The constants may be determined empirically from the analysis of historical data.

Fig. 5 shows the response surface for the fitted model. A visual inspection of the response surface plot indicates that the cumulative average cost per unit is sensitive to changes in both independent variables. It does appear, however, that the sensitivity to cumulative units is greater than the sensitivity to training time. Diagnostic statistical analyses indicate that the model is a good fit for the data. The 95% confidence inter- vals for the parameters in the model are shown in Table 11. Fig. 6 shows a plot of the predicted and observed cost values with 95% confidence intervals for the predictions.

The result of Analysis of Variance for the full regression is presented in Table III. The P-value of O.oo00 in the table indicates that we have a highly significant regression fit. The R-squared value of 0.966962 indicates that most of the variabilities in cumulative average cost are explained by the terms in the model. Table IV shows the breakdown of the model component of the sum of squares. Based on the low P-values shown in the table, it is concluded that both units and training time contributed significantly to the multiple regression model. It is also noted, based on the sum of squares, that production units account for most of the fit in

184 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT. VOL. 39, NO. 2, MAY 1992

-- .-..-.- ......... ................. .L.X

........ ....... l5Oo 2500

Cum. Units

Fig. 5. Response surface for bivariate learning curve example

h c

v) 0 0 Y

5.6

5.1

4.6

4.1

3.6

3.1

2.6

- Fitted Observed 1 ~ " 1 ~ ~ ~ 1 ~ " 1 " ~ 1 ~ ~ ~ I 8 . I -.; ................ i. ............... 4 ................ ; ................ 4 ................ 1 ................ i.-

-.i ................ L ............... ................ : ................ L ................ : ................ :.- ~ I . I ~ . I I ~ . . , i , . , ~ , , , ~ , , , ~

TABLE 1 2.9 3.3 3.7 4.1 4.5 4.9 5.3 MULTIVARIATE LEARNING CURVE DATA

Predicted Cumulative Cumulative Cumulative

T~~~~~~~~ Observation A~~~~~~ Cost production ~ ~ ~ i ~ i ~ ~ ~i~~ Fig. 6. Predicted and observed cost with 95% C.I. for predictions. Number Number 6) (Units) (Hours)

1 1 120 10 11 2 140 10 8

2 3 95 20 54 4 125 20 25

3 5 80 40 100 6 75 40 80

4 7 65 80 220 8 50 80 150

5 9 55 160 410 10 40 160 500

TABLE III ANOVA FOR THE FULL REGRESSION OF THE LEARNING CURVE MODEL

Source Sum of Squares df Mean Square F-Ratio P-value

Model 7.41394 2 3.70697 248.778 O.oo00 Error 0.253312 17 0.0149007

Total 7.66725 19

R-squared = 0.966962 R-adjusted (adjusted for degrees of freedom) = 0.963075 Standard error of estimate = 0.122069

6 11 40 320 660 12 38 320 600 TABLE IV

14 36 640 750 Source Sum of Squares df Mean Square F-Ratio P-value

FURTHER ANOVA FOR THE VARIABLES IN THE MODEL FITTED 7 13 32 640 810

8 15 25 ._ 1280 ... 890 LOG (units) 7.28516187 1 7.2851619 488.91 O.OO0 16 25 1280 800 LOG (time) 0.12877864 1 0.1287786 8.64 0.0092

9 17 20 2560 990 Model 7.41394052 2 18 24 2560 900

10 19 19 5 120 1155 20 25 5120 loo0 TABLE V

CORRELATION MATRIX FOR COEFFICIENT ESTIMATES

TABLE I1 95 % CONFIDENCE INTERVAL FOR MODEL PARAMETERS

Parameter Estimate Lower Limit Upper Limit

1% (4 5.7024 5.4717 5.9331 -0.2093 -0.2826 - 0.1359 -0.1321 -0,2269 - 0.0373

b, b2

this particular bivariate learning curve model. This confirms the visual assessment obtained earlier from Fig. 5.

The correlation matrix for the estimates of the coefficients in the model is shown in Table V. It is seen that log of units and log of time are very negatively correlated and the con- stant is positively correlated with log of units while it is

Constant LOG (units) LOG (time)

Constant 1 .m 0.3654 -0.6895 LOG (units) 0.3654 1 .m -0.9189 LOG (time) - 0.6895 -0.9189 1 .m

negatively correlated with log of time. The strong negative correlation ( - 0.9189) between units and training time sug- gests that there is strong multicollinearity. Multicollinearity normally implies that one of the two correlated variables can be omitted without jeopardizing the fit of the model. The effect of dropping one of the independent variables is shown in the next section. Variables that are statistically independent will have an expected correlation of zero. As expected, Table V does not indicate any zero correlations. The source of

BADIRU: SURVEY OF LEARNING CURVE MODELS

0.28

0.18

0.08

-0.02

-0.12

-0 22

185

I ' " I ' ' ' I " ' I ' " I ~ ' ' , ~ ~ ~ , -i ...............................................................

-4 ............................... 1 ............... i ............... L.... ...... A .............. 2.- . ; e i

; . i

............................... ............... ............... .............. -A : : L L j . j : i i 0 ; ! O j

-4 ............... i ............ 9; ............... i ............... 4 .............. 4- ............. ;.- - j 0 i

- i o ! i O j

-+ ......... ...................................... ; ............... L .............. * .............. L- i p ;

. i 0 j i o i

............... i ............... i ............... <r ............. + .............. L .............. :..- . . -.; 1 . . . 1 . I . . . I . . , I . . . ~ . . . ~

v) m -

n [r W

strong correlations may be explained by the fact that it is difficult to separate the effects of training time from the effect of cumulative production. Obviously, the level of training will influence productivity, which will be reflected in the level of cumulative production within a given length of time.

Fig. VII shows the residual plot for the fitted model. The plot indicates a fairly random distribution of the residuals; thus suggesting a good fit. The following additional results are obtained for the residual analysis: Residual average = 4.66294E-16, Residual variance = 0.0149007, Coefficient of skewness = 0.140909, Coefficient of kurtosis = - 0.557022, and Durbin-Watson statistic = 1.21472. The normal proba- bility plot for the residuals is presented in Fig. VIII. The fairly straight line fit in the plot indicates that the residuals are approximately normally distributed-an essential condi- tion for a good regression model. A perfectly normal distri- bution would plot as a perfect straight line in a normal probability plot.

For a production level of 1750 units and a cumulative training time of 600 h, the fitted model indicates an estimated cumulative average cost per unit shown below:

C(1750,m) = (298.88)( 1750-0 .21) (600-0 . '3) = 27.12 .

Similarly, a production level of 3500 units and training time of 950 h yield the following cumulative average cost per unit:

C(3500,950) = (298.88)(3500-0.21)(950-0.13) = 22.08

For a numerical application of the fitted model, consider the following problem: Given that the standards department of a manufacturing plant has set a target average cost per unit of $12.75 to be achieved after loo0 h of training. Find the cumulative units that must be produced in order to achieve the target cost. Using the fitted model, the following relation-

99.9

99

95

8 80

.- $ 50

c c

e, 0.

c m 3 - E, 20 0

5

1

0.1

-0.22 -0.12 -0.02 0.08 0.18 0.28

Residuals

Fig. 8. Normal probability plot for the residuals.

ship is obtained:

$12.75 = (298.88)( X-0~21)(1000-0~'3)

which yields a required cumualtive production units of X = 46444 units.

Based on the large number of cumulative production units required to achieve the expected standard cost, the standards department may want to review the cost standard. The stan- dard of $12.75 may not be achievable if there is a limited market demand (i.e., demand much less than 46444 units) for the particular product being considered. The relatively flat surface of the learning curve model as units and training time increase implies that relatively more units will need to be produced in order to achieve any additional significant cost improvements. Thus, even though an average cost of $22.08 can be obtained at a cumulative production level of 3500 units, it takes several thousands of additional units to bring the average cost down to $12.75 per unit.

A . Comparison to Univariate Model The experimental bivariate model permits us to simultane-

ously analyze the effects of both production level and training time on the average cost per unit. If a univariate function was to be used with the conventional log-linear model, we would be limited to only a relationship between average cost and cumulative production. This might deprive us of the full range of information we could have obtained about the process productivity. However, the magnitude of additional information provided by a multivariate model over a univari- ate model is often a question of interest. This question can be addressed by considering only the third and fourth columns of the data in Table I for developing a univariate model. The univariate log-linear model is given by:

c, = C , X b

186 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 39, NO. 2, MAY 1992

5.3 I " ' -.i ........................ 1 ........................ d ........................ ! ......................... 1.-

2.3 4.3 6.3 8.3 10.3

250

200

150

LOG(units)

Fig. 9. Fitted values of univariate model with 95% C.1

I " " ' " I . " ' I " ' " " . . I -.i .................. .f ................... f ................... i ................... i ................... i.-

- ................... i ....................................... j ....................................... L-

-4 ................... i ................... i ................... i ................... i ................... i-

o

Cumulative Production

Fig. 10. Empirical univariate learning curve.

-+ ....................................... i ................... : ................... i ................... i.- ~ . . . . ~ . . . . i . . . . i , . . . i . . . . i

where x is cumulative units. The resulting univariate model is presented below:

with R 2 = 95.02%. The learning exponent, 6, is equal to -0.3032 and the corresponding percent learning, p , is 81.05%. This indicates that a productivity gain of 18.95% is expected whenever cumulative production doubles. This is consistent with what Wright (1936) observed in his pioneer learning curve experiments. The R 2 = 95.02% for the uni- variate model is not appreciably worse than the R 2 = 96.70% obtained for the bivariate model. Thus, if data collection and experimentation are expensive, time may be dropped from the model. However, if data is readily available and the modeling process is not time consuming and expensive, the little additional fit provided by the bivariate model may be worthwhile as shown by Bemis [ lo] . Bemis presented a bivariate model which yielded significant information about the relationship between unit cost, production rate, and cu- mulative quantity for defense systems. The model was suc- cessfully applied to generate reliable planning information for an actual weapon system. Fig. IX shows a plot of the fitted univariate model with 95 % confidence interval. The hyper- bolic form of the model is shown in Fig. X on a linear scale.

Even though we obtained a good fit for the univariate model, the information about cumulative training time is lost. For example, the univariate model indicates that the cumula- tive average cost would be $33.84 per unit when cumulative production reaches 640 units. By contrast, the bivariate model shows that the cumulative average cost is $32.22 per unit at that cumulative production level with a cumulative training time of 810 h. This lower cost is due to the additional effect of training time. Thus, the bivariate model provides a more detailed picture of the interactions between the factors associ- ated with the process. Projected productivity gains are more

C, = 240.03x(-0 .3032)

likely to be met if more of the factors influencing productiv- ity can be included in the analysis.

Carlson [13] showed that the validity of log-linear learn- ing curves may be suspect in many labor analysis problems. For manufacturing activities involving operations in different stations, several factors interact to determine the learning rate of workers. Multivariate curves can be of use in developing accurate labor standards in such cases.

V . CONCLUSION

The learning curve phenomenon has been of interest to researchers and practitioners for many years. The variety of situations to which learning curves are applicable has necessi- tated the development of various functional forms for the curves. This paper has presented a comprehensive survey of univariate and multivariate learning curves. Multivariate models are useful for detailed cost and productivity analysis in many economic and production processes. With a simple bivariate model, it may be impossible to obtain accurate estimates of the effects of the two variables involved. Conse- quently, it is often necessary to consider more complicated models. Multivariate models are more robust and help ac- count for more of the available data. A computational experi- ment comparing a univariate model to a bivariate model is presented. While the bivariate model provides only slightly better fit than the univariate model, it does provide more detailed information about factor interactions and better uti- lization of available data. The results of the computational experiment can be generalized to make a case for the appro- priateness of multivariate models in many learning curve analysis.

The multiplicative power model used in the computational experiment is just one of several possible models that can be investigated. Further research and detailed experimentations

BADIRU: SURVEY OF LEARNING CURVE MODELS 187

with alternate functional forms are necessary. The problem of multicollinearity is a concern that needs further investiga- tion. The paper points out the potential estimation errors associated with the regression approach to fitting multiplica- tive models. Error reduction techniques and enhanced model- ing approaches are germane research topics in multivariate learning curves.

Researchers and practitioners need to be aware of the utility of multivariate models. However, the benefit versus the cost of including additional variables in a learning curve model should be carefully assessed before implementing the model.

ACKNOWLEDGMENTS The author thanks the editor and two anonymous reviewers

for their excellent suggestions and helpful comments.

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Adedeji B. Badiru received the B.S. and M S degrees in industrial engineering and the M.S. degree in mathematics, all from Tennessee Techno logical University, Cookeville, TN. He received the Ph.D. degree in industrial engineering from the University of Central Florida, Orlando, FL.

He is an associate professor of industrial engi- neering at the University of Oklahoma and a regis- tered professional engineer in the state of Okla- homa. He is the director of the expert systems laboratory in the School of Industrial Engineering

at the University of Oklahoma. He has published numerous papers on project management, expert systems, and microcomputer applications. He is the author of several books including Project Management in Manufacturing and High Technology Operations (John Wiley & Sons, 1988), Computer Tools, Models, and Techniques for Project Management (TAB Profes- sional Reference Books, 1989), and Project Management Tools for Engi- neering and Management Professionals (IIE Press, 1991).

Dr. Badiru is a member of IIE, SME, TIMS, ORSA, PMI, and AAAI.