1
Some s
ingle
machine scheduling
problems
with a learning effect
Dar

Li Yang
(
楊達立
)
Department of Information Management
(
資管系
)
National
Formosa
University
(
虎尾科技大學
)
Yun

Lin, Taiwan 632, R.O.C.
E

mail:
dlyang@nfu.edu.tw
Abstract
In classical scheduling prob
lem
s
, the processing times of jobs are assumed to be
constant values. However, this assumption is not
appropriate
for the modeling of
many modern industrial processes where very often a job, executed in the same or
almost the same conditions, has a vari
ed
processing time. For example, a worker has
to assemble a large number of similar products. The processing time for the worker to
assemble one product depends on his knowledge, skills, organization of the working
place, and other
aspects
.
With repetition t
h
e worker learns how to
produce
the product
more efficiently
.
Consequently, h
e is better skilled, his knowledge has increased, and
the working place is better organized.
As a result
, the processing time for the worker
to assemble one product decreases. This
phenomenon is known as the “learning
effect” in
the
literature.
T
he learning effect was first applied to
the
industry more than
60 years ago (Wright [1]).
Since t
hen it has been studied extensively in a number of
areas (Nadler and Smith [2], Yelle [3]). H
owever, the learning effect has not been
studied in the context of scheduling until recently.
Biskup [4] was the first to investigate the learning effect in scheduling problems.
He assumed a learning process that reflects a decrease in the process time as
a
function of the number of repetitions i.e. as a function of the job position in the
sequence. Later, Mosheiov [5

6
] continued to study Biskup’s model
and
considered
the scheduling problem of minimizing flow time on parallel identical machines. He
showed
that the problem has a solution which is polynomial in the number of jobs.
T
he assumption of these papers is that the learning effect is
jo
b

independent
, i.e.
a
common
learning curve for all the jobs is assumed. However, in many realistic
situations, learn
ing of some jobs may be better than th
at
of others. Therefore,
M
osheiov and Sidney [
7
] further considered learning in the production process of
some jobs to be faster than th
at
of others, i.e. the learning
is
job

dependent
. They
showed that
the scheduling
problems of
makespan and total flow time minimization
on a single

machine, a due

date assignment problem, and total flow time
2
minimization on unrelated parallel machines remain polynomial
ly
solvable.
However, it still can not clearly describe
time

dependen
t
learning situations. That
is, the more time you practice, the better learning performance you will obtain. In this
discussion [8

9]
, we will introduce a time

dependent learning effect and incorporate it
into
some
single
machine scheduling problems i.e.
a
makespan minimization problem
,
and
a
total completion time minimization problem.
Keywords:
Scheduling, Time

dependent, Learning effect
References
[1] Wright TP. Factors affecting the cost of airplanes.
Journal of Aeronautical
Sciences
1936;
3:
122

128
.
[2]
Nadler
G,
Smith
WD.
Manufacturing progress functions for types of processes
.
International Journal of Production Research
1963;
2
:
115

135.
[3]
Yelle
LE.
The learning curve: historical review and comprehensive survey.
Decision Science
1979;
10
:
302

328
.
[4] Biskup D. Single

machine scheduling with learning considerations.
European
Journal of Operational Research
1999;
115:
173

178.
[5] Mosheiov G.
Scheduling problems with learning effect.
European Journal of
Operational Research
2001a;132:
687

693.
[6] Mosheiov G.
Parallel machine scheduling with learning effect.
Journal of the
Operational Research Society
2001b;
52:
1

5.
[7]
Mosheiov G, Sidney
JB. Scheduling with general job

dependent learning curves.
European Journal of Operational Research
2003;
1
47 :
665

670.
[8]
Kuo
WH,
Yang
DL.
Single

machine group scheduling with a time

dependent
learning effect.
Computers & Operations Research
2
005a;
(i
n press)
.
[9]
Kuo
WH,
Yang
DL.
Minimizing the
total completion time
in a single machine
scheduling problem wi
th a time

dependent learning effect.
European Journal of
Operational Research
2005b;
(under
the 2
nd
revision)
.
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