A
Model for Resource Constrained Production and Inventory Management
Kurt
zyxwvuts
retthauer, Bala Shetty, Siddhartha
zyxw
yam,
nd Susan White
zyx
eparment
of
Business Analysis and Research,
Texas
A&M
University, College Station,
zyxwvu
X
77843
ABSTRACT
We present a general model for multiitem production and inventory management problems that include
a
resource restriction. The decision variables in the model can take on a variety of interpretations, but
zyxwvu
ill
typically represent cycle times, production batch sizes, number of production runs,
or
order quantities for each item. Weconsider environments
where
item
demand rates
are
approximately constant
and
performing an activity such
as
producing
a
batch
of
a
product
or
placing an order results
in
the consumption of
a
scarce resource
that
is shared among the items. Some examples of shared resources include limited machine capacity,
a
restriction on the amount of money that can be tied up in stock,
or
limited storage capacity.
We
focus
on
the
case
where
the decision variables must be integer valued
or
selected from
a
discrete set of choices, such as when
an
integer number
of
production runs is desired for each item,
or
in order quantity problems where the items come in pack sizes containing more
than
one unit and, therefore, the order quantities must
be
an
integer multiple of the pack
sizes.
Wedevelop
a
heuristic and
a
branch
and
bound
algorithm for solving the problem.
The
branch
and
bound algorithm includes reoptimization procedures and the heuristic to improve its performance. Computational testing indicates that the algorithms are effective for solving the general model.
Subject Areas: Mathematical Programming and ProductiodOperations Management.
INTRODUCTION
We present a general model for multiitem production and inventory management problems that include
a
resource restriction and decision variables that must be integer valued or selected from
a
discrete set of choices. The decision variables will typically represent cycle times, production batch sizes, number of production runs, or order quantities for each item. We consider environments where performing any of these activities results in the consumption of a scarce resource that is shared among the items. For example, there may be limited machine capacity when items
are
manufactured
z
6]
[16] [18], a restriction on the amount of money that can be tied up in stock
[8] [9] [12]
[13] [17], or limited storage capacity for the items
[9] [Ill [12] [16].
These situations all require including a single constraint in the model to enforce the resource restriction. The objective function of the model includes reciprocal, linear, and constant terms which often arise from setup (or ordering) costs, inventory holding costs, and purchase costs. The model also includes lower and upper bounds on the decision variables. This allows for additional nonshared restrictions that affect each item individually, such as requiring a minimum and/or maximum number of production runs
or
a supplier specifying a minimum allowable order quantity.
zyx
561
Decision Sciences
Volume
25
Number
4
Printed in the
U.S.A.
562
Resource Constrained Production
z
Specific examples of
our
integer model have received
zyx
ome
ttention in the literature. Consider, for instance, a setting such
as
the one described in Sundararaghavan and Ahmed
zyxwvut
zyxw
81
where
zyxwv
different products
zyxw
re
produced in a common facility. One of the problems addressed involves determining the minimum cost integer number of batches to produce for each item with a restriction on the number of days the facility is available for production. Sundararaghavan and Ahmed assume processing times
are
independent
of
the batch
sizes
and the products
are
usable only after a complete batch has been produced. Their model
is
appropriate for producing many chemical products like resins, paints,
inks,
pharmaceuticals, and dyes [18]. As another example, consider “basic period” approaches to the economic lot scheduling problem (ELSP) where the integer variables represent integer multiples of a basic period cycle time. Elmaghraby
z
7]
presents a formulation of the ELSP that includes a single feasibility constraint and can be stated in the form of our general model. In a problem unrelated to production and inventory management, optimal allocation in stratified sampling
[4] [51 [lo] [ll]
is a special case of our general model where the integer variables represent sample sizes. In specific hierarchical production planning models with continuous variables
[
13
[2],
the cost function has the same structure as our general model and the additional constraint becomes necessary to ensure consistency between the aggregate production plan and a more detailed production plan. Other integer variable production and inventory management problems with a single resource constraint representing a restriction on inventory investment include Goyal
[8],
Lawrence and Schaefer
[13],
and Smith and Schaefer
[17].
Goyal considers situations where a product must be packaged into a number of container sizes after manufacture. Lawrence and Schaefer and Smith and Schaefer develop models to determine the
size
of the spare parts inventory for the maintenance center of a repairable system.
The
resource constrained production and inventory management model proposed in this paper is formulated as a nonlinear integer program. To the best of our knowledge, there is no method in the literature for solving
this
problem to optimality. Its continuous relaxation, however,
zyxwv
an
be
solved easily via manipulations of the KuhnTucker conditions when the bounds on the decision variables are ignored.
Hax
and Candea
[9],
Ibaraki
and Katoh
[ll],
Johnson and Montgomery
[12],
Maloney and Klein
[14],
Ventura and Klein
[19],
and Ziegler
[20]
all present continuous variable models and solution techniques that are based on satisfying the KuhnTucker conditions. None of
these
studies address the integer problem and, therefore, we develop a heuristic that provides good integer solutions to the general model and a branch and bound algorithm that provides optimal integer solutions. Our solution methodology for the continuous variable subproblem at each node of the branch and bound tree generalizes the above mentioned approaches to handle lower and upper bounds on the variables. A branch and bound algorithm for determining the optimal integer solution is desirable because the heuristic solution can, in some cases, yield much higher costs than the optimal solution (see, for example, Table
zyxwvu
).
The branch and bound algorithm seeks integer variable values that minimize average total cost per year while satisfying the resource constraint and the lower and upper bounds on the variables. We assume item demand rates are approximately constant. The form of the total cost function and other assumptions will
be
discussed further.
Bretthauer, Shetty,
zyxwvu
yam,
and
White
zyxwv
63
z
THE GENERAL RESOURCE CONSTRAINED MODEL
zy
he general resource constrained production and inventory management model will be formulated
as
subject to where
zyxwvut
Xi
n
(P)
Min
C(ai
+
bpi
+
zyx
i
xi ,
i=
zyxwvut
li
s
xi
I
zyxwv
i,
i=
1,
...,
n,
(3)
xi
a discrete variable,
i
=
1,
,
n,
4)
the decision variable,
i=l,
...,
n
(for now, its interpretation will be left general), number
of
items, cost coefficients for item
i
ai,
b,
c120),
total amount of the limited resource available
(f>O),
consumption of the limited resource per unit of item
i
(di>o),
proportion
of
xi
consuming the limited resource
(OceiSl),
lower bound on decision variable
xi (li>o,
li=O
will be discussed), and upper bound on decision variable
zyx
i
(ui>li,
ui==
will
be
discussed). The interpretation of the decision variables, the cost terms
in
the objective func tion, and the resource constraint will depend on the particular environment being modeled. We assume item demand rates are approximately constant and the cost pa rameters
ai,
b,
and
ci
are timeinvariant and independent
of
xi
for all
i.
Any additional assumptions will depend on the application being considered. Of course, the product of the two parameters
di
nd
ei
n the resource constraint could be replaced by a single parameter.
Also,
the solution techniques presented in later sections can
be
easily modified to solve problem
(P)
hen the resource constraint is an equality instead of an inequality. Next we show how the general model can
be
specialized for four different applications.
THE NUMBER OF BATCHES PROBLEM WITH LIMITED MACHINE CAPACITY
As
discussed in a previous section, Sundararaghavan and Ahmed
[
181
study a problem that requires determining the minimum
cost
number
of
batches to produce for each item when there is limited machine capacity. The decision variable
xi
represents the integer number of batches of item
i
to produce and the general model can be
written
as:
564
zyxwvusrq
esource Constrained Production
z
subject to
liIxiIui,
zyx
=
zyxwvuts
,
...,
n,
xi
integer,
i
=
zyx
,
,
n,
where
xi
=
the number of batches of item
i
to
be
produced,
n
=
number of items being produced,
Si
=
setup cost per batch
of
item
i,
Hi
=
annual inventory carrying charge per unit of item
i,
Di
units demanded per year for item
i
Ti
=
machine time consumed per batch
of
zyx
tem
i,
and
Tt
=
total machine time available. In the objective function of problem
(NB),
zyxw
y lSpi
represents yearly setup costs and
Zy .lHiDi/(2xi)
epresents average yearly inventory holding costs. Note that cost parameter
ui
from problem
(P)
is zero. Also, because every batch of product
i
is
assumed
to
consume
Ti
units
of
whine time, we set
el=l
for
i=l,
,
n.
The production batch size for item
i,
qi,
is
easily computed given
xi
via
qi=Di
xi,
i=
1,
.
n.
Order Quantities with an Inventory Investment Limit
Various authors have addressed production and inventory management problems with
an
inventory investment constraint and integer variables
[8]
[13]
[17].
We consider an environment where items
are
ordered and there is a restriction on inventory investment.
As
discussed by Silver and Peterson
[16],
some products are sold in pack sizes that contain more than one unit and, therefore, order quantities must be selected from an integer multiple of the pack sizes. The general model presented earlier can
be
written
as
n
(OQ
Min
c(PiDi
+
HPpi
/2
+
AiDi
xi),
i=l
subject to
n
i=
1
Bretthuuer, Shetty,
Syam,
and
White
li
5
xi
zyxwvu
i,
zyxwvu
=
zyxwvut
,
zyx
..,
n,
565
xi
a discrete variable,
i
=
1,
..
,
n,
where
xi
=
order quantity of item
i,
n
=
number of items,
Pi
=
unit variable purchase cost of item
i
Di
=
units demanded per year for item
i,
H
=
annual inventory carrying charge
zyx
s
a percentage
of
unit purchase cost
P,
Aj
=
fixed
cost of ordering item
i,
ei
=
a parameter satisfying O<eill, and
F
=
upper limit on inventory investment. In the objective hnction of problem
OQ),
CY=,PiDi
represents yearly item purchase costs,
Zn ,HPpi
/
2
represents average yearly inventory carrying costs, and
Cn=lAiDi
xi
represents yearly fixed ordering costs. We assume instantaneous replenishment and no backorders. Different values of the parameter
ei
yield different interpretations of the resource constraint. Setting
el=l
for all
i
imposes an upper limit on maximum inventory investment.
This
upper limit may
be
overly restrictive because it implicitly assumes that orders for all
n
items
are
received simultaneously. If we use e,=S instead of
z
l=l
for all
i,
then the resource constraint can
be
interpreted
as
imposing an upper limit on average inventory investment instead of maximum inventory investment.
Basic Period Approaches to the Economic Lot Scheduling Problem
The economic lot scheduling problem
(ELSP)
nvolves scheduling the production
of
multiple items in a common facility on a repetitive basis. Basic period approaches to the
ELSP
limit the choice
of
cycle times for the items to integer multiples
of
a basic period which is long enough to accommodate the production of all items. Elmaghraby
[7]
presents the following formulation of the basic period
ELSP.
n
(ELSP)
Min
X(Si
(
Wxi)
+
HiDi(
1

Di
Ei)>Wxi
2),
i=
1
subject to
n
C(Gi
+
DiWxi
/Ei)
5
W,
zyx
=
1
1
<xi
5
zyxwv
i
i
=
1,
...,
n,
xi
integer,
i
=
1,
,.
,
n,