JOURNAL
OF
OPTIMIZATION THEORY
AND
APPLICATIONS: Vol.
97, No. 1, pp.
7192,
APRIL
1998
Nonlinear
Programming
Analysis
to
Estimate
Implicit
Inventory
Backorder
Costs
1
S.
CETINKAYA
2
AND M.
PARLAR
3
Communicated
by D. G.
Luenberger
Abstract.
In
this paper,
we use
nonlinear programming
to
provide
an
alternative
treatment
of the
economic order quantity problem with
planned
backorders. Many businesses, such
as
capitalgoods
firms
that
deal
with
expensive products
and
some service industries that cannot
store their services, operate with substantial backlogs.
In
practical prob
lems,
it is
usually very
difficult
to
estimate accurately
the
values
of the
two
types
of
backorder
costs,
i.e.,
the
timedependent unit backorder
cost
and the
unit backorder
cost.
We
redefine
the
original problem
without
including these backorder
costs
and
construct
a
nonlinear pro
gramming problem with
two
service measure constraints which
may be
easier
to
specify
than
the
backorder
costs.
We find
that, with this
differ
ent
formulation
of our new
problem,
we
obtain results which give
implicit
estimates
of the
backorder
costs.
The
alternative formulation
provides
an
easiertouse model
and
managerially
meaningful
results.
Next,
we
show that,
for a
wide range
of
parameter values,
it
usually
suffices
to
consider only
one
type
of
backorder
cost,
or
equivalently,
only
one
type
of
service measure constraint. Finally,
we
develop expres
sions which bracket
the
optimal values
of the
decision variables
in a
narrow range
and
provide
a
simple method
for
computing
the
optimal
solution.
In the
most complicated case, this method requires
finding the
unique
root
of a
polynomial.
Key
Words.
Nonlinear programming, inventory theory, backorders
costs.
1.
Introduction
One
of the
wellknown generalizations
of the
economic order quantity
(EOQ) model
of
inventory theory
is the
case where backorders
are
allowed
This research
was
conducted while
Dr.
Cetinkaya
was PhD
Student
in
Management
Science/
Systems
at
McMaster University, Hamilton, Ontario, Canada.
2
Assistant
Professor, Department
of
Industrial Engineering, Texas
A&M
University, College
Station,
Texas.
3
Professor.
DeGroote
School
of
Business, McMaster University, Hamilton,
Ontario,
Canada.
71
00223239/98/0400001IJ15.00/0
©
1998
Plenum
Publishing
Corporation
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(Refs.
17).
The
optimal solution
is
determined
by
minimizing
the
average
annual cost function, which
is the sum of
three terms:
(i)
ordering cost, (ii)
holding cost,
and
(iii)
two
types
of
backorder costs.
It is a
long established
fact
(Hadley
and
Whitin, Ref.
2, pp.
1819)
that
in
practice
the
backorder
costs
are
very
difficult
to
specify,
since they
usually
represent intangible
factors
such
as the
loss
of
goodwill resulting
from
a
delay
in
meeting
demands. However,
for
those cases where backorder costs
can be
accurately
estimated, Hadley
and
Whitin (Ref.
2) use a
nondecreasing function
of
time
n(t)
to
represent
the
cost
of
each unit backordered
for
which
the
backorder
remained on the books. They specialize the function n(t) to the linear form
;r(0
=
nt + n,
where
n
is the
timedependent
per
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