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A comparative study of learning curves with forgetting
A comparative forgetting Mohamad
study of learning curves with
Y. Jaber and Maurice Bonney
Department of Manufacturing Engineering and Operations Management, University of Nottingham, University Park, Nottingham, UK
Although there is almost unanimous agreement that the form of the learning curve is as presented by Wright,’ scientists and practitioners have not yet developed a full understanding of the behavior and factors affecting the forgetting process. Mathematical models of the forgetting process are reviewed, and then three models, the VRZF, VRVF, and LFCM models are compared and their differences and similarities are discussed. 0 1997 by ELseuier Science Inc. Keywords: learning,
1.
forgetting, minimum break
Background
Industrial learning curves have been studied for almost six decades, and the results have been comprehensively surveyed by Yelle’ and Belkaoui.3 However a full understanding of the behavior and factors affecting the forgetting process has not yet been developed. Hoffman4 and Adler and Nanda’ presented two refined mathematical techniques for incorporating the effects of production breaks into planning and control models. Elmaghraby6 presented the variable regression to invariant forgetting (VRIF) model in which he assumes that both learning and forgetting are functions of time.* The theory assumes that the longer the period of production, the more the productivity increases, whereas the longer the stoppage, the greater the forgetting. Elmaghraby hypothesised that there is a unique forgetting function that intercepts the axis representing the time to produce a unit at y (later called T’). The fixed value for y is either given a priori or is derived on the basis of the first lot produced and a “doubling factor,” F, which together define the forgetting function. This differs from the assumption made by the Carlson and Rowe model,’ referred to by Elmaghraby6 as the variable regression to variable forgetting (VRVF) model, which assumed that y varied with each new interruption. The Jaber and Bonney’ learn-forget curve model (LFCM) used a forgetting slope (rate) that is dependent on: (1) the minimum production break, t,, over which total forgetting is assumed to occur; (2) the learning slope (rate), 1; and (3) the amount of equivalent units accumulated, u, by the
Address reprint requests to Dr. M. Y. Jaber at the American University of Beirut, Faculty of Engineering and Architecture, 830 Third Avenue, 18th Floor, New York, NY 10022-6297. Appl. Math. Modelling 1997, 21523-531, August 0 1997 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
point of interruption. They showed that their model was able to predict the time estimates resulting from a laboratory study conducted by Globerson et al.” with less than 1% error. The Globerson et al9 experiment was performed in a computer-oriented manufacturing environment simulated in a microcomputer laboratory. The 120 subjects participated in the experiment in which they performed a data-entry task. Each subject participated in two sessions, separated by a break. In each session the participant had to key in 16 forms, containing information on 16 jobs. The break length varied from 1 to 82 days (a typical data set is presented in Figure I of Globerson et a1.9). In a later study Shtub et al.” used the forgetting model developed by Globerson et al.9 (simulated data-entry setting), as well as the data, to validate Bailey’s” (assembly/disassembly setting) hypothesis that forgetting is a function of the amount of learning prior to the interruption and the elapsed time of the interruption, and to identify a power forgetting model and estimate its parameters. Shtub et al.” used stepwise regression, i.e., the same method as Bailey,” to identify the most influential independent variables. Fourteen variables were used. Bailey’s hypothesis was confirmed, since the two variables, the amount of learning and the length of the break, entered the regression equation while the learning rate and initial performance time did not. The power forgetting model developed by Shtub et al.,” which was based on Wright’s power function, required the estimation of
*In this paper we will represent functions of the output quantity. Received
18 August
both
learning
and
1995; revised 7 May 1997; accepted
forgetting
as
23 May 1997
0307-904x/97/$17.00 PII S0307-904X(97f00055-3
Learning
curves
compared
with forgetting:
M. Y. Jaber
two parameters, the time to produce the first unit after an interruption and the learning slope, whereas the Globerson et a1.9 model required the estimation of one parameter, the time to produce the first unit after an interruption. Learning followed the same curve, that of Wright,’ as before the break. Badiru” presented an approach to developing multivariate (multifactor) models of learning curves that account for alternate periods of learning and forgetting. To demonstrate the usefulness of a multifactor learning curve Badiru ‘* added a production break variable into the univariate model given by Wright.’ The resulting bivariate model was of a similar form to that developed by Globerson et a1.9 Dar-El et a1.13 considered long-cycle tasks composed of a sequential set of short nonrepetitive cycle tasks. They addressed forgetting as a consequence of a specific sub-task, reappearing in the next cycle after a whole cycle time of other activities is completed. Dar-El et aLI used Wright’s power model to depict the learning phenomenon. In their authors experience Wright’s power model is the only model used in practice and possibly is the only learning model covered in textbooks on production and operations management. Dar-El et a1.13 concluded that the main factors involved in the learning behavior of long-cycle tasks are the task length, the learning slope, the time for executing the first cycle, and the length of the break over which forgetting is assumed to occur. The Jaber and Bonney’ assumptions are consistent with these. This paper examines the forgetting models by Elmaghraby6 (VRIF), Carlson and Rowe’ (VRVF), and Jaber and Bonney’ (LFCM), each of which uses the learning curve phenomenon reported by Wright.’ The rest of the paper is organised as follows. Section 2 provides an introduction to the theory of the learning and forgetting curves. Section 3 presents the mathematics required for the analysis of the VRIF, VRVF, and LFCM forgetting models. Section 4 illustrates numerically the differences and similarities among the three forgetting models presented in Section 3. Section 5 investigates the accuracy of the LFCM model. Section 6 offers a summary and conclusions.
and M. Bonney
of a power form:
9x= f,xf
as illustrated in Figure I where fX is the time for the xth unit of lost experience of the forget curve, x is the amount of output units that would accumulate if interruption did not occur, T, is the equivalent time for the first unit (intercept) of the forget curve, and f is the forgetting slope. Assume that an interruption occurs immediately after producing the qth unit and that, in intermittent production runs, the gap is of sufficient length that some of the learning accumulated when producing the previous lots is not retained. When forgetting is accounted for the effort required to produce the first unit after the interruption is higher than the effort required to produce the last unit in the previous cycle. The increase in time to produce the first unit in the next production run depends on the length of the interruption and the time to produce the 4 th unit. When the next production run starts, learning recommences. Three of the models mentioned above, namely, VRIF, VRVF, and LFCM, are now examined in greater detail. Elmaghraby6 in his VRIF model assumes that there is a unique forgetting function with a single intercept point. The rationale for this hypothesis is that the intercept, T,, and the forgetting slope, f, are system-dependent parameters similar to Tl and I of the learning function given in equation (1). The VRVF model assumes that the intercept of the forgetting function, T,, varies for each lot. The VRIF and the VRVF models each adopt a fixed forgetting slope. The Jaber and Bonney’ LFCM modifies the VRVF model by using a forgetting slope that is not fixed. The intercept of the forgetting function, as in Carlson and Rowe,’ also varies. A detailed discussion of each model is now presented. A study of both the VRVF and the VRIF models is discussed in Elmaghraby.6
3. The mathematics
The learning curve phenomenon reported by Wright’ implies that the unit production time decreases by a constant percentage (e.g., 90%, 80%, etc.) each time the cumulative quantity produced doubles. Wright’s learning curve expresses an exponential relationship between direct man-hour input and cumulative production in the form: 7; = T, j-’
(1)
where 5 is the time to produce the jth unit, j is the cumulative production count, Tl is the theoretical time to produce the first unit, and 1 is the learning slope. Similarly for each interruption some forgetting occurs. The forgetting curve relationship is commonly assumed to be
Appl.
of the forgetting
curves
This section presents the mathematics required for the VRIF, VRVF, and LFCM models to estimate the future performance of an intermittent production operation.
2. Introduction
524
(2)
Math.
Modelling,
1997,
Vol. 21, August
3.1 The KWF model
Elmaghraby’ hypothesised that there is a tmique forgetting function with a single origin point T, (previously called y), which either can be given a priori or can be derived on the basis of the first lot produced. This together with a “doubling factor” (forgetting slope), f, defines the forgetting function presented in equation (2). Consider now that there are several production runs. F&re 2 assumes that an interruption to manufacturing occurs at the point where a nominal qi units have been produced, where i is the number of the production run (i= 1,2,3,... ) and qi (qi = ui + ui) represents the equivalent units of continuous production experience accumu-
Learning
curves
compared
with forgetting:
M. Y. Jaber
and M. Bonney
Re-commencement after complete forgetting \ Time to produce a unit
(4)
(4 + R)
Units
k%Gi
Time
P-t----t~~ Forget
Learn Time to produce
Figure
1.
is the
minimum
(q),(q+R) number
time
are the of units
a unit versus
for total
number
produced
forgetting;
of units or which
units of output R is the
produced potentially
or
potential
which
additional
potentially
can be produced
lated by the point of interruption, where ui is the number of units remembered at the beginning of the cycle and ui is the quantity produced in cycle i. It is at this point that the forgetting function is defined by equating the time required to produce the qith unit on the learning curve to the time required on the forgetting curve. This @ done by equating equations (1) and (2) and solving for T,, which gives: ii, = T,qjP(‘+f)
where i = 1
Learn
for an interrupted
operation: quantity
(4)
that
can be produced in times
t,
and
is the time
would in times
t,ft,,
expressed
where
in production
be produced
tp
and
to produce
Q units; t,
if no interruption
r,+t,,
x Q g+R
and
respectively;
and
occurred; x is the
t, d ta.
as:
fl,Lfl = T&4j+,
+ 1>-’
Similarly for cycle i, fI,i is the time to produce the first unit in cycle i, after the break in cycle i - 1 and is given as:
fir = T,(u;
(3)
Assume that after making qi units the process is interrupted for a period of length tbi, during which, if there had been no interruption, an additional si units would have been produced. In any cycle i, si i R, when tbi G t,. Then, from equation (2), the time required to produce unit number qi + si by the end of the break period on the forget curve is expressed as:
t,
+ 1)~’
(6)
In the VRIF model the parameters constant for all cycles i.
f1 and f are held
3.2 The WWF model Carlson and Rowe7 assumed that the intercept of forget curve varies after each interruption based on number of units produced in that cycle. Following same method as in Section 3.1, equations (3), (4), and can be modified to, respectively:
As denoted in Fipre 2 u, + 1 is the amount of equivalent units remembered at the beginning of cycle i + 1 after interruption in cycle i. From equations (4) and (11, u~+~ is expressed as:
the the the (5)
(8)
(9)
(5)
Denote F, I+, as the time required to produce the first unit in cycle’ i + 1, after the break in cycle i, and it is
Elmaghraby6 argued that the value of fql+R does not converge to T1 when total forgetting is assuked in the VRVF model. Denote t, as the minimum interruption to
Appl.
Math.
Modelling,
1997, Vol. 21, August
525
Learning
curves
compared
with forgetting:
M. Y. Jaber
and M. Bonney
Tl
Forgetting curve
Time to produce r Unit
i Learning curve
I
L
%+I
hii)
si)
(4i +
Units Time
Figure 2. The learning-forgetting function at the end of cycle i: fPi is the time in production to produce 9i units in cycle i; tbi is the length of the interruption period in cycle i, tbi =gtB; si is the potential additional quantity that would be produced if no interruption occurred, si G Ri when rbi d ra. Note that t, is assumed to be fixed for all cycles i ii= 1,2,3,. . . ); (9i), (9;+s,) are the number of units produced or which potentially can be produced in times tpi and t,,St,;, respectively; and u,+, is the experience remembered at the beginning of production cycle i+ 1, knowing that the process was interrupted for a period of length tbi after accumulating 9i units in cycle i, i.e., the period of interruption t,; means that at the start of cycle iS 1 the experience remembered reverts to u;+,.
which total forgetting is assumed. If the production process is interrupted for t, units of time, then the time to produce the first unit after the interruption is T1 (i.e., tbi = t, and si ,= Ri). In the case of Elmaghraby,6 who assumed that Ti and f are constant for all cycles i, we have: $,+R , = f&ji
+ R,)f
ratio of t,, the minimum time for total forgetting, to t(qi), the amount of time required to produce qi units. t(qi) is determined by integrating equation (1) over the proper limits as: t(qi) = iq’TIj-‘dj
(10) Therefore
while in the case of Carlson and Rowe,’ who assumed fii is calculated for each cycle i while f is fixed for all cycles, we have: fq,+,, = ~,i(qi + Ri>f
3.3 The LFCM
expressed
1(1 - Olog qi fi
=
(11)
model
Jaber and Bonney’
log(C, + 1)
the forgetting
where i= 1,2,3,...
rate as:
Appl.
Math.
Modelling,
1997,
Ci is represented
&q;-l
ci = t, [
(12)
Vol. 21, August
as:
-1
I
At the point of interruption in cycle i in Fipre tion (1) is set equal to equation (2), T4, = Tql, intercept of the forget curve is determined from (14), which is derived by modifying equation (7), the changing value of the forgetting slope, to: fii = T1q;‘i+f~’
where fi, which varies in every cycle, is the forgetting slope after interruption in cycle i and Ci = tB/t(qi) is the
526
T,q/ -’ = (1 - 1)
(13) 2, equathen the equation to adopt
(14)
The coordinates (qi +~~,f~,+~,) on the forgetting curve have equivalent coordinates on the learning curve (ui+ 1, Tu,,,), where ui+ 1 is the number of units remembered at the beginning of cycle i + 1 and T, + , = Tp +s,. Equating (1) to (2) after substituting ui+ 1 = j kr equation
Learning
(1) and qi + si = x in equation (2) and solving gives (Jaber and Bonney’, equation [ll]):
for ui+,
compared
with forgetting:
= 0.2 x
(2) as si + R, (tbi + te), as
illustrated in Figures 1 and 2, the value of fq,+ R converges to T, when total forgetting is assumed, that is, when tbi = t,, and it is elaborated as:
ui+ , = 0 when
Similarly,
4. Numerical
= T,
si + Ri in equation
0
The forgetting (12) as:
fi =
.
2
x
1()()-".I52
1 - 0.152
slope is then
-1
I
= 300/l
determined
1.712
from equation
0.152 x (1 - 0.152) x log 100 log(1 + 300/11.712)
[
= 0.181
lb, +9,
1-I
T,
1 - 0.152
=
(15
example
300 x
1~~-(o.l"2+o.Ixl)
1
l/Cl-I)
~
(16)
This section illustrates the different solution methods of the three mathematical models. Suppose that there exists a production situation where 100 units are produced in each cycle. The elapsed time between ceasing production at unit Qi
=
f,, is determined
If the production run was not interrupted at unit number 100, but rather continued over the break period, a total of sr units would have been accumulated given by: l-l
+ R;)’
function,
0.0432 days
s,=
fq,+R, = &(qi
and M. Bonney
(15) =
By taking the limit of equation
M. Y. Jaber
The intercept of the forgetting from equation (3) as: f, = Tlqp”+f,’
+ S;) -f,/l
= qj’+fl)“(q,
ui+,
curves
x 10 +
-91
1~~l-".l52
- 100
0.2 = 107 units
I
Therefore the amount of equivalent units remembered at the beginning of cycle 2 for the LFCM is determined from equation (15) as: u2 I.FCM= =
~~~~(~l.lS2+ll
lXl)/l~.lS2
x
207-11
IXl/O.lSZ
42 units
The amount of equivalent units remembered at the beginning of cycle 2 for the VRIF is determined from equation (5) as:
=
o2
[
1
l/ll.lS2
0.0432 U 2,VRlF
x
207”.“’
.
= 42 units
Following the same procedure the value of u2 for the VRVF is 42 units. In the first cycle the learning and forgetting curves are the same for all models. Therefore the time required to produce the first unit in the second cycle after the first interruption (e.g., unit number 101) is given as: %,
= f~2,LFC%l = flZ,“RiF =
0.2 x (1 + 42)-“““*
=
%.VRVF
= 0.1129 days
Interruption occurs again at the end of the second cycle, i.e., after unit number 200 has been produced. At this stage the equivalent number of units of experience accumulated is 142 units. As before the forget slope, f2, and the number of potential units that would have been accumulated on the second break period, s2, are 0.213 and 111 units, respectively. The equivalent number of units of experience remembered at the beginning of cycle 3 for the VRIF, U3,VaIF, VRVF, U3,VRVFj and the LFCM, $LFCMj are 48 units, 70 units, and 63 units, respectively. Th_erefore the time required to produce unit_ number 201, _Tzol, for the three models, i.e., T,, VRIF, T12 VRVF, and T,, LFCM, are 0.1108 days, 0.1043 days, and 0.1663 days, respectively. Table I shows the results obtained from repeating the same procedure for all three models over five consecutive production cycles. The learning curve was plotted against the VRIF, VRVF, and LFCM forgetting curves in Figures 3, 4, and 5, respectively. Figure 3 shows that the learning
Appl. Math.
Modelling,
1997,
Vol. 21, August
527
Learning Table 1.
curves
compared
with forgetting:
Results
for the three
mathematical
M. Y. Jaber
models
VRIF
VRVF
(Elmaghraby’) Cycle i
ui
1
f,,
0
and M. Bonney
(Carlson
f,,
r,
“i
LFCM
and Rowe’)
f,,
(Jaber
?I,
f,
and Bonney’)
f,;
“,
71,
f,
0.2000
0.0432
0.181
0
0.2000
0.0432
0.181
0
0.2000
0.0432
0.181
2
42
0.1129
0.0432
0.181
42
0.1129
0.0384
0.181
42
0.1129
0.0327
0.213
3
48
0.1108
0.0432
0.181
71
0.1043
0.0361
0.181
63
0.1063
0.0289
0.228
4
48
0.1106
0.0432
0.181
93
0.1002
0.0347
0.181
74
0.1037
0.0272
0.235
5
48
0.1105
0.0432
0.181
111
0.0977
0.0337
0.181
80
0.1026
0.0263
0.238
curve and the VRIF forgetting curve did not conform to the hypothesised relationship that the curves intercept at the same point on all cycles. Unlike the VRIF curve the VRVF and LFCM curves did confirm the relationship presented in Figzue 2. This is due to the adjustment in both the forgetting rate and the intercept of the forgetting curve after each interruption. Assuming that the production break increases from 10 days to 300 days (tbi = t, = 300 days), then tota! forgetting occurs and si = Ri. To test the convergence (Tq,+R, + T,) of all three models, as given in equations (lo), (ll), and (16), the question is then, what if a break of 300 days is experienced in the first cycle? In the second cycle? In the third cy$e? . . . etc. Table 2 shows that the calculated values Of Tq +R LFCM converge to a value of 0.2 (T,) for all five cycle;, Ghile fq, + R,,vR,F converges to a value slightly higher than T, (0.2001 in cycle 1, then 0.2!07 in all remaining cycles). Conversely the value of Tq,+R,,VRVF diverged away from T, for each consecutive cycle (see column 5, Table 2). Figzue 6 illustrates the behavior of the three forget curves in the second cycle, where the LFCM started asymptotically with the VRVF and ends up asymptotically with the VRIF. Both the VRIF and the LFCM curves conform to the hypothesised relationship in Figure 1, which shows the convergence to a unique value of T,, while the VRVF did not. The above discussion shows that the LFCM was consistent with the hypothesised learningforgetting relationships of Figures 1 and 2, while the VRIF and the VRVF have some inconsistencies.
0.2
-
f
1
Learning curve
100
200
300
Potential Figure
3.
Wright’s tions
528
The
learning
occurring
Appl.
behavior curve
5. How accurate is the LFCM
of
400
the
VRIF
at 100,
200,
Math.
Modelling,
300,
.=
0.18
-
=
0.16
“,
0.14
Forgetting CUNI?
- - -
1
500
Learning curve
100
200
400,
300
Potential
units forgetting
over five production
and 500
1997,
curve
runs with
and
interrup-
potential
units.
Vol. 21, August
model?
As indicated earlier the Jaber and Bonneys learn-forget curve model (LFCM) presented in Section 3.3 produced results that are consistent with the results of the Globerson et aL9 experimentally derived learning-forgetting model. The Globerson et aL9 experiment was performed in a computer-oriented manufacturing environment simulated in a microcomputer laboratory. The 120 subjects divided into six groups participated in the experiment in which they performed a data-entry task. Each subject participated in two sessions, of 16 repetitions each, separated by a break. The average lengths of the break time between the two consecutive sessions were 1.7 (l-2 days), 6.1 (3-7 days), 11.5 (8-16 days), 19.1 (16-25 days), 40.8 (26-53 days), and 65.1 days (54-82 days) in Groups 1 through 6, respectively. Globerson et al.” use-d seven mathematical models to estimate the value of T,,,, the calculated performance time for the first unit after the break if interruption occurs, as a function of Tq+ ,, the
0.2
ForgettIng curve
- - -
In Table 1 the VRIF gav_e the highest values of ‘?,i for all cycles. The values of TIi obtained from the VRVF were slightly lower than those of the LFCM. The LFCM model combined the characteristics of the VRIF and the VRVF models by giving close estimates of the values of the VRVF and the convergence desired by the VRIF.
Figure
4.
Wright’s tions
The
learning
occurring
behavior curve at 100,
of
the
400
VRVF
forgetting
over five production 200,
300,
500
units
400.
curve
runs with
and 500
and
interrup-
potential
units.
Learning
.Z
= ; 2 B h 2 : i=
Forgetting curve
Cycle i
I
1
I,
I,
100
I,
1
200
The convergence
5.
The
learning
occurring
behavior curve at 100,
of
the
LFCM
forgetting
over five production
600
200,
300.
and 500
400,
curve
runs with
and
interrup-
potential
units.
calculated performance time for the q + 1 units after the break if no interruption occurs, and fb, the break time. The model that best describes the forgetting phenomenon was found to be:
t/+1= 1.87 x The model
9;fR;
and M. Bonney
of the time required
to produce
the
curve LFCM
VRVF Tq,+R,
9;fR;
fq,+~,
9;+ R;
f,,+ R,
1
4793
0.2001
4793
0.2000
4793
0.2000
2
4866
0.2007
4866
0.1784
4866
0.2000
3
4876
0.2007
4916
0.1680
4902
0.2000
4
4877
0.2007
4952
0.1615
5
4877
0.2007
4980
0.1571
4920 4930
0.2006 0.2000
,b
1
400
300
Potential units
tions
M. Y. Jaber
VRIF
04
Figure
with forgetting:
last unit on the forget
0.18 0.18 0.14 0.12 0.i 0.08 0.06 0.04 0.02
Wright’s
compared Table 2.
-
0.2
curves
T”.” q+l x to.“” I1
(18)
in equation (18) has an adjusted that equation (18) appropriately describes the forgetting phenomenon, Globerson et al.’ integrated it into the learning model to improve its predictability power. The learning curve model for the second session was of the form:
for 1
u* =
1.87 x 7-“.9 q+l x t”.“9 h
-l/I -1
Tl
For this study the average values of Wright’s model in equation (1) were T, = 544 seconds, 1 = 0.325. The time to produce the 17th repetition is determined from equation (1) as:
presented
T17 = 544 x 17P”.325 = 217 set
R* = 0.75 (R = 0.86). Assuming
iy = T,(j +
u2)-’
The calculated performance time for the first unit after an average break of 65.1 days is determined from equation (18): f,, = 1.87 X 217°.9 X 65.1°.09 = 344.50 z 345 set
(19)
A 0.2 -
0.180.16*g 0.14: 0.122 '0 9 0.1;
-vRlF LFCM _ _ _ VRVF
0.08-
E 0.06i= 0.040.02-
.
O,,,,,,,,,,,,,,,,,,,,~ 1
500
1008
1500
2000
2500
3000
'3500 4000
4500
Potential units Figure 6.
The convergence
of the three
forgetting
curves
in the second
cycle when
Appl.
Math.
total forgetting
Modelling,
is assumed
1997,
to occur (t,,=t,).
Vol. 21, August
529
Learning
curves
compared
with forgetting:
M. Y. Jaber
Similarly, for average breaks of 1.7, 6.1, 11.5, 19.1, and 40.8 days, the values of T,, are 248, 278, 295, 309, and 330 set, respectively. Following the same procedure as in Jaber and Bonney,s the level of units remembered at the beginning of the second run, c~rrc., is given from equation (15) and has values of 7.54, 5.59, 4.83, 4.30, 3.62, and 3.24 units for average breaks of 1.7, 6.1, 11.5, 19.1, 40.8, and 65.1 days, respectively. The time required to produce the first unit after an average break period of 65.1 days is:
~17.
LFCM
= 544 x (3.24 + 1 )P”‘325 = 340 set
which differs from the result obtained by - 1.45%, and it is calculated as:
%Dev =
f17,LFCM - ?I’ x
in Globerson
et a1.9
340 - 345 lOO=
f,, = - 1.45%
345
x 100
Similarly the values of f,, are 271, 295, 307, 316, and 331 set, respectively, and the percentage deviations are 12.1, 5.36, 4.07, 2.27, and 0.3%, respectively. The value of u2 is determined from equation (19). The value of u2 is then incorporated in equation (20) to determine the value of T,,. By treating u2 as an integer value, for average breaks of 1.7, 6.1, 11.5, 19.1, 40.8, and 65.1 days the values of T,, are 250, 289, 304, 322, 347, and 347 set, respectively. Also by treating the values of u2,LFCM as integers the values of ‘1, LFCM are 277, 304, 322, 322, 347, and 347 set, respectively and the percentage of deviations from the results of Globerson et a1.9 are 10.8, 5.2, 5.98, 0, 0, and O%, respectively. The results indicate that the LFCM mode1 presented in Section 3.3 produces results that are consistent with the results of the experimentally derived learning-forgetting curve by Globerson et aL9 for a break length between 16 and 82 days.
and M. Bonney
The VRIF, VRVF, and the LFCM models hypothesised two relationships in defining the learning-forgetting relationship. The first hypothesis indicates that when total forgetting occurs, the performance time on the forgetting curve reverts to a unique value equivalent to the time required to produce the first unit with no prior experience. The second hypothesis is that the performance time on the learning curve equals that on the forgetting curve at the point of interruption. The VRIF mode1 was consistent with the first hypothesis but was inconsistent with the second, whereas the VRVF mode1 was consistent with the second but was inconsistent with the first. The LFCM mode1 was consistent with both hypotheses. The accuracy of the LFCM mode1 was tested, and the results were shown to be consistent with those of the Globerson et a1.9 experimentally derived learning-forgetting mode1 with a negligible percentage of deviation for production breaks ranging between 16 and 82 days. Unlike earlier works5-’ it is believed that the work in this paper is the first that has used t, as a parameter. The primary assumption in the development of the learn-forget curve mode1 (LFCM) is that t, is of a fixed value. Intuitively using a fixed value does not seem unreasonable, provided one starts the production after a period of moderate training. This limiting property is characterised by the assumption of t, occurring over the interval [O,tB], disregarding any possibility of occurrence over the interval [tB, ~1. Another limitation to the work is that no field data was available to aid the authors in justifying the assumptions made to develop the learn-forget curve model (LFCM). Finally the above discussion suggests that the LFCM model should be tested with field data in several industrial settings. If the LFCM mode1 proves to be unsatisfactory to represent reality, a new modified mode1 should be developed to represent reality more faithfully.
Nomenclature 6. Summary
and conclusion
This paper compared three proposed mathematical models that describe the learning-forgetting relationship. The LFCM of Jaber and Bonney,’ calculates the value of the forgetting slope based on three factors. These factors were the equivalent number of units of continuous production accumulated by the point of interruption, the minimum break to which the manufacturer assumes total forgetting, and the learning slope. The LFCM estimates of the time required to produce the first unit after an interruption period were between the_ estimatesgiven by the VRIF and the VRVF (f,i,vRvF < T,i,LFCM G T,i,VR,F). The learn-forget curve mode1 (LFCM) allows the forgetting slope to be estimated, whereas the VRIF (Elmaghraby6) and VRVF (Carlson and Rowe’) models do not state the method by which they derive their forgetting slopes (rates), and so the LFCM mode1 was used to estimate these to give results consistent with the VRIF and VRVF equations. The equivalent time of qi + Ri on the VRVF curve, Tr;+R,.v~~~, failed to converge to a unique value of T, w en total forgetting was assumed.
530
Appl.
Math.
Modelling,
1997,
Vol. 21, August
L’i
u, 'i 9i X
Ri j tbi ci
fl,
break time to achieve total forgetting forgetting slope in cycle i learning slope number of actual units produced in cycle i number of equivalent units of experience with forgetting at the beginning of cycle i number of equivalent units of experience with no forgetting at the beginning of cycle i number of theoretical units of output accumulated at the end of cycle i number of units of output that would have been accumulated if interruption did not occur number of units that would have been produced during t, in cycle i production count/cumulative number of repetitions shorter break than t, experienced in cycle i the minimum value of the ratio of the break time to the production time in cycle i that will give total forgetting time for the first unit of the forgetting curve in cycle i
Learning
time for the xth unit of lost experience forgetting curve time to produce ui units in cycle i
curves
on the
time to produce the first unit in cycle i with forgetting after an interruption in production time to produce the first unit in cycle i with no forgetting after an interruption in production time to produce the jth unit on the learning curve
References Wright, T. Factors affecting the cost of airplanes. J. Aeronaur. Sci. 1936, 3, 122-128 Yelle, L. E. The learning curve: Historical review and comprehensive survey. Decision Sci. 1979, 10, 302-328 Belkaoui, A. The Learning Curve. Quorum Books, Westport, CT, 1986 Hoffman, Thomas R. Effect of prior experience on learning curve parameters. J. Indust. Eng. 1968, 19, 412-413
compared
with forgetting:
M. Y. Jaber
and M. Bonney
5. Adler, G. L. and Nanda, R. Effects of learning on optimal lot size determination-Single product case. AIZE Trans. 1974, 6, 14-20 6. Elmaghraby, S. E. Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF). Prod. Planning Confrol 1990, 1, 196-208 7. Carlson, J. G. and Rowe, R. G. How much does forgetting cost? Indust. Eng. 1976, 8, 40-47 8. Jaber, M. and Bonney, M. C. Production breaks and the learning curve: The forgetting phenomena. Appl. Math. Modefling 1996, 20, 162-169 9. Globerson, S., Levin, N. and Shtub, A. The impact of breaks on forgetting when performing a repetitive task. IIE Trans. 1989, 21, 376-381 10. Shtub, A., Levitt, N. and Globerson, S. Learning and forgetting industrial skills: Experimental model. Int. J. Hum. Factors Manufact. 1993, 3, 293-305 11. Bailey, C. D. Forgetting and the learning curve: A laboratory study. Manage. Sci. 1989, 35, 346-352 12. Badiru, A. B. Multivariate analysis of the effect of learning and forgetting on product quality. Inf. J. Prod. Rex 1995, 33, 777-794 13. Dar-El, E. M., Ayas, K. and Gilad, I. Predicting performance times for long cycle time tasks. IIE Trans. 1995, 27, 272-281
Appl.
Math.
Modelling,
1997,
Vol. 21, August
531
study of learning curves with
Y. Jaber and Maurice Bonney
Department of Manufacturing Engineering and Operations Management, University of Nottingham, University Park, Nottingham, UK
Although there is almost unanimous agreement that the form of the learning curve is as presented by Wright,’ scientists and practitioners have not yet developed a full understanding of the behavior and factors affecting the forgetting process. Mathematical models of the forgetting process are reviewed, and then three models, the VRZF, VRVF, and LFCM models are compared and their differences and similarities are discussed. 0 1997 by ELseuier Science Inc. Keywords: learning,
1.
forgetting, minimum break
Background
Industrial learning curves have been studied for almost six decades, and the results have been comprehensively surveyed by Yelle’ and Belkaoui.3 However a full understanding of the behavior and factors affecting the forgetting process has not yet been developed. Hoffman4 and Adler and Nanda’ presented two refined mathematical techniques for incorporating the effects of production breaks into planning and control models. Elmaghraby6 presented the variable regression to invariant forgetting (VRIF) model in which he assumes that both learning and forgetting are functions of time.* The theory assumes that the longer the period of production, the more the productivity increases, whereas the longer the stoppage, the greater the forgetting. Elmaghraby hypothesised that there is a unique forgetting function that intercepts the axis representing the time to produce a unit at y (later called T’). The fixed value for y is either given a priori or is derived on the basis of the first lot produced and a “doubling factor,” F, which together define the forgetting function. This differs from the assumption made by the Carlson and Rowe model,’ referred to by Elmaghraby6 as the variable regression to variable forgetting (VRVF) model, which assumed that y varied with each new interruption. The Jaber and Bonney’ learn-forget curve model (LFCM) used a forgetting slope (rate) that is dependent on: (1) the minimum production break, t,, over which total forgetting is assumed to occur; (2) the learning slope (rate), 1; and (3) the amount of equivalent units accumulated, u, by the
Address reprint requests to Dr. M. Y. Jaber at the American University of Beirut, Faculty of Engineering and Architecture, 830 Third Avenue, 18th Floor, New York, NY 10022-6297. Appl. Math. Modelling 1997, 21523-531, August 0 1997 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
point of interruption. They showed that their model was able to predict the time estimates resulting from a laboratory study conducted by Globerson et al.” with less than 1% error. The Globerson et al9 experiment was performed in a computer-oriented manufacturing environment simulated in a microcomputer laboratory. The 120 subjects participated in the experiment in which they performed a data-entry task. Each subject participated in two sessions, separated by a break. In each session the participant had to key in 16 forms, containing information on 16 jobs. The break length varied from 1 to 82 days (a typical data set is presented in Figure I of Globerson et a1.9). In a later study Shtub et al.” used the forgetting model developed by Globerson et al.9 (simulated data-entry setting), as well as the data, to validate Bailey’s” (assembly/disassembly setting) hypothesis that forgetting is a function of the amount of learning prior to the interruption and the elapsed time of the interruption, and to identify a power forgetting model and estimate its parameters. Shtub et al.” used stepwise regression, i.e., the same method as Bailey,” to identify the most influential independent variables. Fourteen variables were used. Bailey’s hypothesis was confirmed, since the two variables, the amount of learning and the length of the break, entered the regression equation while the learning rate and initial performance time did not. The power forgetting model developed by Shtub et al.,” which was based on Wright’s power function, required the estimation of
*In this paper we will represent functions of the output quantity. Received
18 August
both
learning
and
1995; revised 7 May 1997; accepted
forgetting
as
23 May 1997
0307-904x/97/$17.00 PII S0307-904X(97f00055-3
Learning
curves
compared
with forgetting:
M. Y. Jaber
two parameters, the time to produce the first unit after an interruption and the learning slope, whereas the Globerson et a1.9 model required the estimation of one parameter, the time to produce the first unit after an interruption. Learning followed the same curve, that of Wright,’ as before the break. Badiru” presented an approach to developing multivariate (multifactor) models of learning curves that account for alternate periods of learning and forgetting. To demonstrate the usefulness of a multifactor learning curve Badiru ‘* added a production break variable into the univariate model given by Wright.’ The resulting bivariate model was of a similar form to that developed by Globerson et a1.9 Dar-El et a1.13 considered long-cycle tasks composed of a sequential set of short nonrepetitive cycle tasks. They addressed forgetting as a consequence of a specific sub-task, reappearing in the next cycle after a whole cycle time of other activities is completed. Dar-El et aLI used Wright’s power model to depict the learning phenomenon. In their authors experience Wright’s power model is the only model used in practice and possibly is the only learning model covered in textbooks on production and operations management. Dar-El et a1.13 concluded that the main factors involved in the learning behavior of long-cycle tasks are the task length, the learning slope, the time for executing the first cycle, and the length of the break over which forgetting is assumed to occur. The Jaber and Bonney’ assumptions are consistent with these. This paper examines the forgetting models by Elmaghraby6 (VRIF), Carlson and Rowe’ (VRVF), and Jaber and Bonney’ (LFCM), each of which uses the learning curve phenomenon reported by Wright.’ The rest of the paper is organised as follows. Section 2 provides an introduction to the theory of the learning and forgetting curves. Section 3 presents the mathematics required for the analysis of the VRIF, VRVF, and LFCM forgetting models. Section 4 illustrates numerically the differences and similarities among the three forgetting models presented in Section 3. Section 5 investigates the accuracy of the LFCM model. Section 6 offers a summary and conclusions.
and M. Bonney
of a power form:
9x= f,xf
as illustrated in Figure I where fX is the time for the xth unit of lost experience of the forget curve, x is the amount of output units that would accumulate if interruption did not occur, T, is the equivalent time for the first unit (intercept) of the forget curve, and f is the forgetting slope. Assume that an interruption occurs immediately after producing the qth unit and that, in intermittent production runs, the gap is of sufficient length that some of the learning accumulated when producing the previous lots is not retained. When forgetting is accounted for the effort required to produce the first unit after the interruption is higher than the effort required to produce the last unit in the previous cycle. The increase in time to produce the first unit in the next production run depends on the length of the interruption and the time to produce the 4 th unit. When the next production run starts, learning recommences. Three of the models mentioned above, namely, VRIF, VRVF, and LFCM, are now examined in greater detail. Elmaghraby6 in his VRIF model assumes that there is a unique forgetting function with a single intercept point. The rationale for this hypothesis is that the intercept, T,, and the forgetting slope, f, are system-dependent parameters similar to Tl and I of the learning function given in equation (1). The VRVF model assumes that the intercept of the forgetting function, T,, varies for each lot. The VRIF and the VRVF models each adopt a fixed forgetting slope. The Jaber and Bonney’ LFCM modifies the VRVF model by using a forgetting slope that is not fixed. The intercept of the forgetting function, as in Carlson and Rowe,’ also varies. A detailed discussion of each model is now presented. A study of both the VRVF and the VRIF models is discussed in Elmaghraby.6
3. The mathematics
The learning curve phenomenon reported by Wright’ implies that the unit production time decreases by a constant percentage (e.g., 90%, 80%, etc.) each time the cumulative quantity produced doubles. Wright’s learning curve expresses an exponential relationship between direct man-hour input and cumulative production in the form: 7; = T, j-’
(1)
where 5 is the time to produce the jth unit, j is the cumulative production count, Tl is the theoretical time to produce the first unit, and 1 is the learning slope. Similarly for each interruption some forgetting occurs. The forgetting curve relationship is commonly assumed to be
Appl.
of the forgetting
curves
This section presents the mathematics required for the VRIF, VRVF, and LFCM models to estimate the future performance of an intermittent production operation.
2. Introduction
524
(2)
Math.
Modelling,
1997,
Vol. 21, August
3.1 The KWF model
Elmaghraby’ hypothesised that there is a tmique forgetting function with a single origin point T, (previously called y), which either can be given a priori or can be derived on the basis of the first lot produced. This together with a “doubling factor” (forgetting slope), f, defines the forgetting function presented in equation (2). Consider now that there are several production runs. F&re 2 assumes that an interruption to manufacturing occurs at the point where a nominal qi units have been produced, where i is the number of the production run (i= 1,2,3,... ) and qi (qi = ui + ui) represents the equivalent units of continuous production experience accumu-
Learning
curves
compared
with forgetting:
M. Y. Jaber
and M. Bonney
Re-commencement after complete forgetting \ Time to produce a unit
(4)
(4 + R)
Units
k%Gi
Time
P-t----t~~ Forget
Learn Time to produce
Figure
1.
is the
minimum
(q),(q+R) number
time
are the of units
a unit versus
for total
number
produced
forgetting;
of units or which
units of output R is the
produced potentially
or
potential
which
additional
potentially
can be produced
lated by the point of interruption, where ui is the number of units remembered at the beginning of the cycle and ui is the quantity produced in cycle i. It is at this point that the forgetting function is defined by equating the time required to produce the qith unit on the learning curve to the time required on the forgetting curve. This @ done by equating equations (1) and (2) and solving for T,, which gives: ii, = T,qjP(‘+f)
where i = 1
Learn
for an interrupted
operation: quantity
(4)
that
can be produced in times
t,
and
is the time
would in times
t,ft,,
expressed
where
in production
be produced
tp
and
to produce
Q units; t,
if no interruption
r,+t,,
x Q g+R
and
respectively;
and
occurred; x is the
t, d ta.
as:
fl,Lfl = T&4j+,
+ 1>-’
Similarly for cycle i, fI,i is the time to produce the first unit in cycle i, after the break in cycle i - 1 and is given as:
fir = T,(u;
(3)
Assume that after making qi units the process is interrupted for a period of length tbi, during which, if there had been no interruption, an additional si units would have been produced. In any cycle i, si i R, when tbi G t,. Then, from equation (2), the time required to produce unit number qi + si by the end of the break period on the forget curve is expressed as:
t,
+ 1)~’
(6)
In the VRIF model the parameters constant for all cycles i.
f1 and f are held
3.2 The WWF model Carlson and Rowe7 assumed that the intercept of forget curve varies after each interruption based on number of units produced in that cycle. Following same method as in Section 3.1, equations (3), (4), and can be modified to, respectively:
As denoted in Fipre 2 u, + 1 is the amount of equivalent units remembered at the beginning of cycle i + 1 after interruption in cycle i. From equations (4) and (11, u~+~ is expressed as:
the the the (5)
(8)
(9)
(5)
Denote F, I+, as the time required to produce the first unit in cycle’ i + 1, after the break in cycle i, and it is
Elmaghraby6 argued that the value of fql+R does not converge to T1 when total forgetting is assuked in the VRVF model. Denote t, as the minimum interruption to
Appl.
Math.
Modelling,
1997, Vol. 21, August
525
Learning
curves
compared
with forgetting:
M. Y. Jaber
and M. Bonney
Tl
Forgetting curve
Time to produce r Unit
i Learning curve
I
L
%+I
hii)
si)
(4i +
Units Time
Figure 2. The learning-forgetting function at the end of cycle i: fPi is the time in production to produce 9i units in cycle i; tbi is the length of the interruption period in cycle i, tbi =gtB; si is the potential additional quantity that would be produced if no interruption occurred, si G Ri when rbi d ra. Note that t, is assumed to be fixed for all cycles i ii= 1,2,3,. . . ); (9i), (9;+s,) are the number of units produced or which potentially can be produced in times tpi and t,,St,;, respectively; and u,+, is the experience remembered at the beginning of production cycle i+ 1, knowing that the process was interrupted for a period of length tbi after accumulating 9i units in cycle i, i.e., the period of interruption t,; means that at the start of cycle iS 1 the experience remembered reverts to u;+,.
which total forgetting is assumed. If the production process is interrupted for t, units of time, then the time to produce the first unit after the interruption is T1 (i.e., tbi = t, and si ,= Ri). In the case of Elmaghraby,6 who assumed that Ti and f are constant for all cycles i, we have: $,+R , = f&ji
+ R,)f
ratio of t,, the minimum time for total forgetting, to t(qi), the amount of time required to produce qi units. t(qi) is determined by integrating equation (1) over the proper limits as: t(qi) = iq’TIj-‘dj
(10) Therefore
while in the case of Carlson and Rowe,’ who assumed fii is calculated for each cycle i while f is fixed for all cycles, we have: fq,+,, = ~,i(qi + Ri>f
3.3 The LFCM
expressed
1(1 - Olog qi fi
=
(11)
model
Jaber and Bonney’
log(C, + 1)
the forgetting
where i= 1,2,3,...
rate as:
Appl.
Math.
Modelling,
1997,
Ci is represented
&q;-l
ci = t, [
(12)
Vol. 21, August
as:
-1
I
At the point of interruption in cycle i in Fipre tion (1) is set equal to equation (2), T4, = Tql, intercept of the forget curve is determined from (14), which is derived by modifying equation (7), the changing value of the forgetting slope, to: fii = T1q;‘i+f~’
where fi, which varies in every cycle, is the forgetting slope after interruption in cycle i and Ci = tB/t(qi) is the
526
T,q/ -’ = (1 - 1)
(13) 2, equathen the equation to adopt
(14)
The coordinates (qi +~~,f~,+~,) on the forgetting curve have equivalent coordinates on the learning curve (ui+ 1, Tu,,,), where ui+ 1 is the number of units remembered at the beginning of cycle i + 1 and T, + , = Tp +s,. Equating (1) to (2) after substituting ui+ 1 = j kr equation
Learning
(1) and qi + si = x in equation (2) and solving gives (Jaber and Bonney’, equation [ll]):
for ui+,
compared
with forgetting:
= 0.2 x
(2) as si + R, (tbi + te), as
illustrated in Figures 1 and 2, the value of fq,+ R converges to T, when total forgetting is assumed, that is, when tbi = t,, and it is elaborated as:
ui+ , = 0 when
Similarly,
4. Numerical
= T,
si + Ri in equation
0
The forgetting (12) as:
fi =
.
2
x
1()()-".I52
1 - 0.152
slope is then
-1
I
= 300/l
determined
1.712
from equation
0.152 x (1 - 0.152) x log 100 log(1 + 300/11.712)
[
= 0.181
lb, +9,
1-I
T,
1 - 0.152
=
(15
example
300 x
1~~-(o.l"2+o.Ixl)
1
l/Cl-I)
~
(16)
This section illustrates the different solution methods of the three mathematical models. Suppose that there exists a production situation where 100 units are produced in each cycle. The elapsed time between ceasing production at unit Qi
=
f,, is determined
If the production run was not interrupted at unit number 100, but rather continued over the break period, a total of sr units would have been accumulated given by: l-l
+ R;)’
function,
0.0432 days
s,=
fq,+R, = &(qi
and M. Bonney
(15) =
By taking the limit of equation
M. Y. Jaber
The intercept of the forgetting from equation (3) as: f, = Tlqp”+f,’
+ S;) -f,/l
= qj’+fl)“(q,
ui+,
curves
x 10 +
-91
1~~l-".l52
- 100
0.2 = 107 units
I
Therefore the amount of equivalent units remembered at the beginning of cycle 2 for the LFCM is determined from equation (15) as: u2 I.FCM= =
~~~~(~l.lS2+ll
lXl)/l~.lS2
x
207-11
IXl/O.lSZ
42 units
The amount of equivalent units remembered at the beginning of cycle 2 for the VRIF is determined from equation (5) as:
=
o2
[
1
l/ll.lS2
0.0432 U 2,VRlF
x
207”.“’
.
= 42 units
Following the same procedure the value of u2 for the VRVF is 42 units. In the first cycle the learning and forgetting curves are the same for all models. Therefore the time required to produce the first unit in the second cycle after the first interruption (e.g., unit number 101) is given as: %,
= f~2,LFC%l = flZ,“RiF =
0.2 x (1 + 42)-“““*
=
%.VRVF
= 0.1129 days
Interruption occurs again at the end of the second cycle, i.e., after unit number 200 has been produced. At this stage the equivalent number of units of experience accumulated is 142 units. As before the forget slope, f2, and the number of potential units that would have been accumulated on the second break period, s2, are 0.213 and 111 units, respectively. The equivalent number of units of experience remembered at the beginning of cycle 3 for the VRIF, U3,VaIF, VRVF, U3,VRVFj and the LFCM, $LFCMj are 48 units, 70 units, and 63 units, respectively. Th_erefore the time required to produce unit_ number 201, _Tzol, for the three models, i.e., T,, VRIF, T12 VRVF, and T,, LFCM, are 0.1108 days, 0.1043 days, and 0.1663 days, respectively. Table I shows the results obtained from repeating the same procedure for all three models over five consecutive production cycles. The learning curve was plotted against the VRIF, VRVF, and LFCM forgetting curves in Figures 3, 4, and 5, respectively. Figure 3 shows that the learning
Appl. Math.
Modelling,
1997,
Vol. 21, August
527
Learning Table 1.
curves
compared
with forgetting:
Results
for the three
mathematical
M. Y. Jaber
models
VRIF
VRVF
(Elmaghraby’) Cycle i
ui
1
f,,
0
and M. Bonney
(Carlson
f,,
r,
“i
LFCM
and Rowe’)
f,,
(Jaber
?I,
f,
and Bonney’)
f,;
“,
71,
f,
0.2000
0.0432
0.181
0
0.2000
0.0432
0.181
0
0.2000
0.0432
0.181
2
42
0.1129
0.0432
0.181
42
0.1129
0.0384
0.181
42
0.1129
0.0327
0.213
3
48
0.1108
0.0432
0.181
71
0.1043
0.0361
0.181
63
0.1063
0.0289
0.228
4
48
0.1106
0.0432
0.181
93
0.1002
0.0347
0.181
74
0.1037
0.0272
0.235
5
48
0.1105
0.0432
0.181
111
0.0977
0.0337
0.181
80
0.1026
0.0263
0.238
curve and the VRIF forgetting curve did not conform to the hypothesised relationship that the curves intercept at the same point on all cycles. Unlike the VRIF curve the VRVF and LFCM curves did confirm the relationship presented in Figzue 2. This is due to the adjustment in both the forgetting rate and the intercept of the forgetting curve after each interruption. Assuming that the production break increases from 10 days to 300 days (tbi = t, = 300 days), then tota! forgetting occurs and si = Ri. To test the convergence (Tq,+R, + T,) of all three models, as given in equations (lo), (ll), and (16), the question is then, what if a break of 300 days is experienced in the first cycle? In the second cycle? In the third cy$e? . . . etc. Table 2 shows that the calculated values Of Tq +R LFCM converge to a value of 0.2 (T,) for all five cycle;, Ghile fq, + R,,vR,F converges to a value slightly higher than T, (0.2001 in cycle 1, then 0.2!07 in all remaining cycles). Conversely the value of Tq,+R,,VRVF diverged away from T, for each consecutive cycle (see column 5, Table 2). Figzue 6 illustrates the behavior of the three forget curves in the second cycle, where the LFCM started asymptotically with the VRVF and ends up asymptotically with the VRIF. Both the VRIF and the LFCM curves conform to the hypothesised relationship in Figure 1, which shows the convergence to a unique value of T,, while the VRVF did not. The above discussion shows that the LFCM was consistent with the hypothesised learningforgetting relationships of Figures 1 and 2, while the VRIF and the VRVF have some inconsistencies.
0.2
-
f
1
Learning curve
100
200
300
Potential Figure
3.
Wright’s tions
528
The
learning
occurring
Appl.
behavior curve
5. How accurate is the LFCM
of
400
the
VRIF
at 100,
200,
Math.
Modelling,
300,
.=
0.18
-
=
0.16
“,
0.14
Forgetting CUNI?
- - -
1
500
Learning curve
100
200
400,
300
Potential
units forgetting
over five production
and 500
1997,
curve
runs with
and
interrup-
potential
units.
Vol. 21, August
model?
As indicated earlier the Jaber and Bonneys learn-forget curve model (LFCM) presented in Section 3.3 produced results that are consistent with the results of the Globerson et aL9 experimentally derived learning-forgetting model. The Globerson et aL9 experiment was performed in a computer-oriented manufacturing environment simulated in a microcomputer laboratory. The 120 subjects divided into six groups participated in the experiment in which they performed a data-entry task. Each subject participated in two sessions, of 16 repetitions each, separated by a break. The average lengths of the break time between the two consecutive sessions were 1.7 (l-2 days), 6.1 (3-7 days), 11.5 (8-16 days), 19.1 (16-25 days), 40.8 (26-53 days), and 65.1 days (54-82 days) in Groups 1 through 6, respectively. Globerson et al.” use-d seven mathematical models to estimate the value of T,,,, the calculated performance time for the first unit after the break if interruption occurs, as a function of Tq+ ,, the
0.2
ForgettIng curve
- - -
In Table 1 the VRIF gav_e the highest values of ‘?,i for all cycles. The values of TIi obtained from the VRVF were slightly lower than those of the LFCM. The LFCM model combined the characteristics of the VRIF and the VRVF models by giving close estimates of the values of the VRVF and the convergence desired by the VRIF.
Figure
4.
Wright’s tions
The
learning
occurring
behavior curve at 100,
of
the
400
VRVF
forgetting
over five production 200,
300,
500
units
400.
curve
runs with
and 500
and
interrup-
potential
units.
Learning
.Z
= ; 2 B h 2 : i=
Forgetting curve
Cycle i
I
1
I,
I,
100
I,
1
200
The convergence
5.
The
learning
occurring
behavior curve at 100,
of
the
LFCM
forgetting
over five production
600
200,
300.
and 500
400,
curve
runs with
and
interrup-
potential
units.
calculated performance time for the q + 1 units after the break if no interruption occurs, and fb, the break time. The model that best describes the forgetting phenomenon was found to be:
t/+1= 1.87 x The model
9;fR;
and M. Bonney
of the time required
to produce
the
curve LFCM
VRVF Tq,+R,
9;fR;
fq,+~,
9;+ R;
f,,+ R,
1
4793
0.2001
4793
0.2000
4793
0.2000
2
4866
0.2007
4866
0.1784
4866
0.2000
3
4876
0.2007
4916
0.1680
4902
0.2000
4
4877
0.2007
4952
0.1615
5
4877
0.2007
4980
0.1571
4920 4930
0.2006 0.2000
,b
1
400
300
Potential units
tions
M. Y. Jaber
VRIF
04
Figure
with forgetting:
last unit on the forget
0.18 0.18 0.14 0.12 0.i 0.08 0.06 0.04 0.02
Wright’s
compared Table 2.
-
0.2
curves
T”.” q+l x to.“” I1
(18)
in equation (18) has an adjusted that equation (18) appropriately describes the forgetting phenomenon, Globerson et al.’ integrated it into the learning model to improve its predictability power. The learning curve model for the second session was of the form:
for 1
u* =
1.87 x 7-“.9 q+l x t”.“9 h
-l/I -1
Tl
For this study the average values of Wright’s model in equation (1) were T, = 544 seconds, 1 = 0.325. The time to produce the 17th repetition is determined from equation (1) as:
presented
T17 = 544 x 17P”.325 = 217 set
R* = 0.75 (R = 0.86). Assuming
iy = T,(j +
u2)-’
The calculated performance time for the first unit after an average break of 65.1 days is determined from equation (18): f,, = 1.87 X 217°.9 X 65.1°.09 = 344.50 z 345 set
(19)
A 0.2 -
0.180.16*g 0.14: 0.122 '0 9 0.1;
-vRlF LFCM _ _ _ VRVF
0.08-
E 0.06i= 0.040.02-
.
O,,,,,,,,,,,,,,,,,,,,~ 1
500
1008
1500
2000
2500
3000
'3500 4000
4500
Potential units Figure 6.
The convergence
of the three
forgetting
curves
in the second
cycle when
Appl.
Math.
total forgetting
Modelling,
is assumed
1997,
to occur (t,,=t,).
Vol. 21, August
529
Learning
curves
compared
with forgetting:
M. Y. Jaber
Similarly, for average breaks of 1.7, 6.1, 11.5, 19.1, and 40.8 days, the values of T,, are 248, 278, 295, 309, and 330 set, respectively. Following the same procedure as in Jaber and Bonney,s the level of units remembered at the beginning of the second run, c~rrc., is given from equation (15) and has values of 7.54, 5.59, 4.83, 4.30, 3.62, and 3.24 units for average breaks of 1.7, 6.1, 11.5, 19.1, 40.8, and 65.1 days, respectively. The time required to produce the first unit after an average break period of 65.1 days is:
~17.
LFCM
= 544 x (3.24 + 1 )P”‘325 = 340 set
which differs from the result obtained by - 1.45%, and it is calculated as:
%Dev =
f17,LFCM - ?I’ x
in Globerson
et a1.9
340 - 345 lOO=
f,, = - 1.45%
345
x 100
Similarly the values of f,, are 271, 295, 307, 316, and 331 set, respectively, and the percentage deviations are 12.1, 5.36, 4.07, 2.27, and 0.3%, respectively. The value of u2 is determined from equation (19). The value of u2 is then incorporated in equation (20) to determine the value of T,,. By treating u2 as an integer value, for average breaks of 1.7, 6.1, 11.5, 19.1, 40.8, and 65.1 days the values of T,, are 250, 289, 304, 322, 347, and 347 set, respectively. Also by treating the values of u2,LFCM as integers the values of ‘1, LFCM are 277, 304, 322, 322, 347, and 347 set, respectively and the percentage of deviations from the results of Globerson et a1.9 are 10.8, 5.2, 5.98, 0, 0, and O%, respectively. The results indicate that the LFCM mode1 presented in Section 3.3 produces results that are consistent with the results of the experimentally derived learning-forgetting curve by Globerson et aL9 for a break length between 16 and 82 days.
and M. Bonney
The VRIF, VRVF, and the LFCM models hypothesised two relationships in defining the learning-forgetting relationship. The first hypothesis indicates that when total forgetting occurs, the performance time on the forgetting curve reverts to a unique value equivalent to the time required to produce the first unit with no prior experience. The second hypothesis is that the performance time on the learning curve equals that on the forgetting curve at the point of interruption. The VRIF mode1 was consistent with the first hypothesis but was inconsistent with the second, whereas the VRVF mode1 was consistent with the second but was inconsistent with the first. The LFCM mode1 was consistent with both hypotheses. The accuracy of the LFCM mode1 was tested, and the results were shown to be consistent with those of the Globerson et a1.9 experimentally derived learning-forgetting mode1 with a negligible percentage of deviation for production breaks ranging between 16 and 82 days. Unlike earlier works5-’ it is believed that the work in this paper is the first that has used t, as a parameter. The primary assumption in the development of the learn-forget curve mode1 (LFCM) is that t, is of a fixed value. Intuitively using a fixed value does not seem unreasonable, provided one starts the production after a period of moderate training. This limiting property is characterised by the assumption of t, occurring over the interval [O,tB], disregarding any possibility of occurrence over the interval [tB, ~1. Another limitation to the work is that no field data was available to aid the authors in justifying the assumptions made to develop the learn-forget curve model (LFCM). Finally the above discussion suggests that the LFCM model should be tested with field data in several industrial settings. If the LFCM mode1 proves to be unsatisfactory to represent reality, a new modified mode1 should be developed to represent reality more faithfully.
Nomenclature 6. Summary
and conclusion
This paper compared three proposed mathematical models that describe the learning-forgetting relationship. The LFCM of Jaber and Bonney,’ calculates the value of the forgetting slope based on three factors. These factors were the equivalent number of units of continuous production accumulated by the point of interruption, the minimum break to which the manufacturer assumes total forgetting, and the learning slope. The LFCM estimates of the time required to produce the first unit after an interruption period were between the_ estimatesgiven by the VRIF and the VRVF (f,i,vRvF < T,i,LFCM G T,i,VR,F). The learn-forget curve mode1 (LFCM) allows the forgetting slope to be estimated, whereas the VRIF (Elmaghraby6) and VRVF (Carlson and Rowe’) models do not state the method by which they derive their forgetting slopes (rates), and so the LFCM mode1 was used to estimate these to give results consistent with the VRIF and VRVF equations. The equivalent time of qi + Ri on the VRVF curve, Tr;+R,.v~~~, failed to converge to a unique value of T, w en total forgetting was assumed.
530
Appl.
Math.
Modelling,
1997,
Vol. 21, August
L’i
u, 'i 9i X
Ri j tbi ci
fl,
break time to achieve total forgetting forgetting slope in cycle i learning slope number of actual units produced in cycle i number of equivalent units of experience with forgetting at the beginning of cycle i number of equivalent units of experience with no forgetting at the beginning of cycle i number of theoretical units of output accumulated at the end of cycle i number of units of output that would have been accumulated if interruption did not occur number of units that would have been produced during t, in cycle i production count/cumulative number of repetitions shorter break than t, experienced in cycle i the minimum value of the ratio of the break time to the production time in cycle i that will give total forgetting time for the first unit of the forgetting curve in cycle i
Learning
time for the xth unit of lost experience forgetting curve time to produce ui units in cycle i
curves
on the
time to produce the first unit in cycle i with forgetting after an interruption in production time to produce the first unit in cycle i with no forgetting after an interruption in production time to produce the jth unit on the learning curve
References Wright, T. Factors affecting the cost of airplanes. J. Aeronaur. Sci. 1936, 3, 122-128 Yelle, L. E. The learning curve: Historical review and comprehensive survey. Decision Sci. 1979, 10, 302-328 Belkaoui, A. The Learning Curve. Quorum Books, Westport, CT, 1986 Hoffman, Thomas R. Effect of prior experience on learning curve parameters. J. Indust. Eng. 1968, 19, 412-413
compared
with forgetting:
M. Y. Jaber
and M. Bonney
5. Adler, G. L. and Nanda, R. Effects of learning on optimal lot size determination-Single product case. AIZE Trans. 1974, 6, 14-20 6. Elmaghraby, S. E. Economic manufacturing quantities under conditions of learning and forgetting (EMQ/LaF). Prod. Planning Confrol 1990, 1, 196-208 7. Carlson, J. G. and Rowe, R. G. How much does forgetting cost? Indust. Eng. 1976, 8, 40-47 8. Jaber, M. and Bonney, M. C. Production breaks and the learning curve: The forgetting phenomena. Appl. Math. Modefling 1996, 20, 162-169 9. Globerson, S., Levin, N. and Shtub, A. The impact of breaks on forgetting when performing a repetitive task. IIE Trans. 1989, 21, 376-381 10. Shtub, A., Levitt, N. and Globerson, S. Learning and forgetting industrial skills: Experimental model. Int. J. Hum. Factors Manufact. 1993, 3, 293-305 11. Bailey, C. D. Forgetting and the learning curve: A laboratory study. Manage. Sci. 1989, 35, 346-352 12. Badiru, A. B. Multivariate analysis of the effect of learning and forgetting on product quality. Inf. J. Prod. Rex 1995, 33, 777-794 13. Dar-El, E. M., Ayas, K. and Gilad, I. Predicting performance times for long cycle time tasks. IIE Trans. 1995, 27, 272-281
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Math.
Modelling,
1997,
Vol. 21, August
531