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\title{Interval Approach to Learning Curves}
\author{Mario Beruvides and Vladik Kreinovich}

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ }

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\auffil{The author is with the 
$^1$Department of Industrial Engineering,
College of Engineering,
Texas Tech University,
Lubbock, TX 79409, 
phone (806) 742-3543 (Dept.),
email mberuvides@coe3.coe.ttu.edu, and $^2$Department of Computer
Science, Univeristy of Texas at El Paso, El Paso, TX 79968, email
vladik@cs.utep.edu.}

\section{What is a Learning Curve}
In order to make reasonable manufacturing decisions, we must 
be able to predict the workers' productivity in producing different
objects. 

Of course, we can ask several workers to perform the desired task and
thus measure their current productivity. However, this measured
productivity is a very good estimate for the future productivity: 
the workers'
performance improves as they produce more and more units; in other
words, they {\it learn}. To predict the performance after a certain
amount of units, we must measure the performance at different moments
of time, and then extrapolate the resulting dependency $t(n)$
of the time $t$ that is required to produce a single unit on the
total (cumulative) amount $n$ of the units produced. The functions
that are used to model this dependency are called {\it learning
curves} (see, e.g., \cite{Niebel 1976,Martin 1991} and reference
therein). 

Usually, the {\it hyperbolic} dependency is used: $t(n)=a\cdot n^\alpha$
with unknown $a$ and $\alpha$. This dependency can be further
simplified if we move to a logarithmic scale: $T=\alpha N+A$, where
$T=\ln(t)$, $N=\ln(n)$, and $a=\ln(A)$. So, to 
estimate $A$ and $\alpha$ from the
experimental data $(n_i,t_i)$, $1\le i\le K$, we can:
\begin{itemize}
\item compute the
logarithms $T_i=\ln(t_i)$ and $N_i=\ln(n_i)$;
\item find the $\alpha$ and $A$ from the equations $T_i=\alpha N_i+A$;
and
\item compute $a=\exp(A)$.
\end{itemize}

\section{Problems with Learning Curves}
The dependency represented by a learning curve is usually only
approximate. 

Traditionally, statistical methods are used to describe the approximate
character of this curve. However, it is well known (see, e.g.,
\cite{Fein}) that the difference between the model and the actual
productivity is not of purely stochastic nature. It is more of a bias
than of a random component. A more adequate description of this
difference is that it belongs to a certain {\it interval}
$[-\Delta,\Delta]$, and that we know nothing about the probabilities of
different values from this interval.

\section{Our Idea}
In view of this description, it is very natural to apply {\it interval
computations} to learning curves. 
Namely, if we have the measurement results $t_i$ and $n_i$, and the
accuracy $\Delta$, then we must find $a$ and $\alpha$ from the
conditions that $t_i-\Delta\le a\cdot n_i^\alpha\le t_i+\Delta$. In
logarithmic scale, these conditions take the form:
$$\ln(t_i-\Delta)\le \alpha N_i+A\le \ln(t_i+\Delta).$$ 
Using linear programming, we can find the smallest and the largest
possible values of $A$ and $\alpha$ that satisfy these inequalities. 
In other words, we will have the {\it intervals} ${\bf A}=[A^-,A^+]$ 
and $[\alpha^-,\alpha^+]$ that are guaranteed to contain
the actual values of these parameters.
From the interval for $A$, we can find an interval for $a$: 
${\bf a}=[a^-,a^+]$, where $a^\pm=\exp(A^\pm)$. 

Now, if we are interested in the value of $t(n)$ by the time when
we have already produced $n$ units, we get an {\it interval} of
possible values: 
$$t(n)\in [a^-\cdot n^{\alpha^-}, a^+\cdot n^{\alpha^+}].$$ This interval must
be used for decision making: The cautious
estimate should be based on the largest possible time, the most risky
ones can use the fastest possible time. 

\begin{thebibliography}{99}

\bibitem{Fein}
M. Fein, ``How `reliability', `precision', and `accuracy' refer to use
of work measurement data'', {\it Industrial Engineering}, July 1981;
reprinted in: R. E. Shell (ed.), {\bf Work Measurement: Principles and
Practice}, IIE Press, Atlanta, GA, 1986, pp. 30--37.   

\bibitem{Martin 1991}
J. C. Martin, {\bf Labor productivity control. New approaches for
industrial engineers and managers}, Praeger, NY, 1991.

\bibitem{Niebel 1976}
B. W. Niebel, {\bf Motion and time study}, R. D. Irwin, Homewood, IL,
1976.

    
\end{thebibliography}


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