European Journal of Operational Research 214 (2011) 179–198
Contents lists available at ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Invited Review
A survey of deterministic models for the EOQ and EPQ with partial backordering
David W. Pentico a,⇑, Matthew J. Drake b
a
b
Management Science, Palumbo-Donahue School of Business, Duquesne University, Pittsburgh, PA 15282-0180, USA
Supply Chain Management, Palumbo-Donahue School of Business, Duquesne University, Pittsburgh, PA 15282-0180, USA
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 23 January 2010
Accepted 26 January 2011
Available online 1 February 2011
Models for the basic deterministic EOQ or EPQ problem with partial backordering or backlogging make all
the assumptions of the classic EOQ or EPQ model with full backordering except that only a fraction of the
demand during the stockout period is backordered. In this survey we review deterministic models that
have been developed over the past 40 years that address the basic models and extensions that add other
considerations, such as pricing, perishable or deteriorating inventory, time-varying or stock-dependent
demand, quantity discounts, or multiple-warehouses.
Ó 2011 Elsevier B.V. All rights reserved.
Keywords:
Inventory
Partial backordering
Lot sizing
1. Introduction
The first, and by far the best known, inventory model is the classic square-root economic order quantity (EOQ) model developed
by Harris [43]. While this model has been criticized for its unrealistic assumptions, it has been widely and successfully used in practice. Possibly more important, it forms the basis for many other
models that relax one or more of those assumptions.
The earliest paper we could find that relaxed one of these basic
assumptions was Taft [106], who used a finite production rate,
leading to the basic economic production quantity (EPQ) model,
also known as the economic manufacturing quantity (EMQ), economic lot size (ELS), or production lot size (PLS) model.
Relaxation of the basic EOQ and EPQ models’ assumption that
stockouts are not permitted led to the development of models for
the two basic cases for stockouts – backorders and lost sales. Zipkin
[151] discusses these two basic model types for the EOQ, which are
easily generalized to the EPQ.
What took longer to develop were models that recognized that,
while some customers are willing to wait for delivery, others are
not. Either these customers will cancel their orders, or the supplier
will have to fill them within the normal delivery time by using
more expensive alternative supply methods. The first models for
this variation on the basic EOQ model – partial backordering or
backlogging – appeared in 1967 [38] and 1970 [7], although neither author showed how to solve his model. The first paper that
developed a model for the basic EOQ with partial backordering
(EOQ–PBO) and a solution procedure for it appeared in 1973
⇑ Corresponding author. Tel.: +1 412 396 6252; fax: +1 412 396 4764.
E-mail addresses:
(M.J. Drake).
[email protected]
(D.W.
Pentico),
[email protected]
0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2011.01.048
[60], with others taking somewhat different approaches appearing
up to the present time. These models are summarized in Section
3.1. Comparable models for the basic EPQ with partial backordering (EPQ–PBO), the first of which appeared in 1987 [58], are summarized in Section 5.1.
Over the 40-plus years since the first basic EOQ–PBO model
appeared, many authors have developed models that have relaxed
additional assumptions of the basic EOQ–PBO and EPQ–PBO
models, including such things as time- or backlog-dependent backordering probabilities, inventory deterioration, time- or inventorydependent demand functions, and quantity discounts. Other
authors have examined scenarios that included making pricing
decisions or multiple items. Approximately 90% of this work has
appeared since 1990 and about 75% since 2000. In this paper we
provide brief summaries of most of the deterministic models in
this area, focusing on the assumptions behind the models and
the approaches taken to developing and solving them. Stochastic
models will be covered in a second survey.
We will not attempt to describe the models in detail. However,
it will be necessary to use symbols for some of the models’ characteristics. Unfortunately, as in most areas of quantitative analysis,
there is no consistency in how symbols are used in the literature
in this area. For example, b has been used to mean the maximum
backorder level, the maximum stockout level, the unit cost of a
backorder, the probability a unit of demand will be backordered,
and the probability a unit of demand will not be backordered. At
the risk of making it a little more difficult to translate directly from
our summary to the original paper’s discussion, we will use, for the
most part, a single set of notation, given in Table 1, in order to
make it easier to see the similarities and differences among the
papers. (Note, however, that an author may use one or more of
the symbols defined in this table to mean something completely
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D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
other exception is Mak [57], who assumed that b is a random variable with a constant mean. Both of these exceptions are discussed
in Section 3.
Table 1
Summary of symbols used and their meanings.
Symbol
Meaning
b
Fraction of demand backordered during a stockout, which may
depend on the time until the next replenishment order is received
(b(s))
Maximum value of b(s) over a stockout interval
Maximum backorder level
Demand per period, which may depend on time (D(t)), price (D(p)),
both (D(p, t)), or something else, such as the inventory level
Fraction of demand filled from stock, fill rate
Unit selling price, which may depend on time (p(t))
Production rate, which may depend on time (P(t))
Order quantity
Maximum stockout level
Length of an inventory cycle
Length of time during an inventory cycle for which there is
inventory
Length of time during an inventory cycle for which there is a
stockout
Length of time the start of the deterioration of inventory is delayed
Time remaining until the next replenishment order is received
Maximum amount of time any customer will wait
Total demand during an inventory cycle
Maximum inventory level
Fictitious demand rate = V/T
bM
B
D
F
p
P
Q
S
T
T1
T2
td
s
sM
U
V
X
different that what it means in this survey.) Where a paper uses
variables or coefficients that do not appear in other papers, we
use the original paper’s notation for them, defining these symbols
as they are used.
The structure of this paper is as follows: After a brief discussion
of various models for determining the percentage of demand backordered at any given time during the stockout interval (Section 2),
we will review models for the basic EOQ–PBO and no additional
complicating factors (Section 3). In Section 4 we will cover models
that include, in addition to partial backordering, additional features, such as deteriorating inventory, non-constant demand patterns, an uncertain replenishment quantity, pricing, and multiple
warehouses. In Section 5 we will review models with a finite production rate, beginning with the basic EPQ–PBO and moving onto
models that include additional features.
2. Models for determining the backordering percentage
One of the variations on the basic EOQ–PBO and EPQ–PBO models is the inclusion of time- or backlog-dependent backordering
probabilities. Since this is a significant way in which the models
to be discussed differ from one another, we will briefly discuss
the approaches to determining the backordering percentage that
have appeared in the literature.
With two exceptions, all of the models prior to 1996 assumed
that the percentage of demand backordered is constant over the
length of the stockout interval. One exception is Montgomery
et al. [60], in which the primary development was for a constant
b, but which also discussed what we call the Linear 1 model for
time-dependent b(s). They noted that they tried other, unidentified, forms forb(s), but they were more difficult to analyze. The
2.1. Backordering percentage based on the time to replenishment
A number of authors, most notably Abad [1–5], have noted that
it is more realistic to assume that more customers are willing to
wait if the waiting time is short. That is, they assume that b is a
function of s, the time remaining until replenishment. We provide
brief descriptions of the models proposed for b(s), discussing them
in the order in which they first appeared. Table 2 summarizes the
equations for b(s) for each model type. (Note that, as we stated in
Section 1 with reference to Table 1, where the notation used in this
paper is defined, the notation we use in discussing the equations in
Table 2 is not necessarily the same as that used by the authors of
the papers using these models for b(s).)
1. Linear 1: The first model for a non-constant b, which we call
Linear 1, was covered in Montgomery et al. [60], along with
their constant-b model. b(s) has an initial value, b0, at the time
the stockout begins (i.e., when s has its maximum value) and
increases linearly until it reaches its maximum value, bM, when
the replenishment order arrives (when s = 0). Linear 1 was also
analyzed by San José et al. [82].
In 1996 Abad [1] was the first to introduce both the Exponential
and the Rational models for b(s). Although he provided a brief
rationale there (‘‘Since in general, consumers do not like to wait,
we assume that b(s) is a decreasing function of s’’.), Abad [3] gives
a more extensive rationale for using a b(s) function like these. After
noting that ‘‘the conventional approach to modeling the backlogging phenomenon requires the use of the backorder cost and the
lost sale cost’’, he states that ‘‘these costs. . . are difficult to estimate
in practice [, so a] new approach in which customers are considered impatient. . . and the fraction of demand that gets backlogged
at a given point in time is a decreasing function of waiting time’’
will be used instead. While not justifying either the exponential
or the rational form explicitly, Abad [1] used both types. In both,
b(s) increases toward its maximum value, bM, which Abad calls
the backordering intensity, at an increasing rate as s decreases
toward 0, the actual rate being determined by a, the backordering
resistance. He also noted that as a ? 1 (i.e., customers are highly
impatient), b(s) ? 0.
2. Exponential: b(s) is bM multiplied by a negative exponential
function of s. Abad [1] introduced this model originally, but
Papachristos and Skouri [67] were the first to refer to it as
‘‘exponential’’. San José et al. [80,81] analyzed this form for
the basic EOQ with partial backordering, while Abad’s models
also included deteriorating inventory and pricing decisions
and Papachristos and Skouri included deterioration.
3. Rational: b(s) is bM divided by a positive linear function of s. In
addition to Abad, who used it first, San José et al. [80] and Sicilia
et al. [98] used this form and were the first to use this name for it.
The exponential and rational forms of b(s) are the only forms that
have been used extensively by other authors. As we discuss the
Table 2
Equations for functional forms of b(s).
Form of b(s)
Equation
Range for s
Linear 1
Exponential
Rational
Linear 2
Step
Mixed exponential
b(s) = bM (bM b0)(s/T2)
b(s) = bMeas, a > 0
b(s) = bM/(1 + as), a > 0
b(s) = bM (bM/sM)s
b(s) = 1
bðs1 ; s2 Þ ¼ b1 ea1 s1 þ b2 ea2 s2 ; b1 ; b2 P 0; b1 þ b2 6 1; a1 ; a2 > 0
0 6 s 6 T2
06s
06s
0 6 s 6 sM
0 6 s 6 sM
0 6 s1, s2
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
EOQ–PBO and EPQ–PBO models that include other features in Sections 4 and 5, we will identify which form of b(s) the authors used.
The remaining forms for b(s) to be discussed have been analyzed by
only one set of authors: San José, Sicilia, and García-Laguna.
4. Linear 2: Although the linear 1 form is relatively easy to solve
and does provide for an increase in the percentage of customers
who are willing to backorder as the time to replenishment
decreases, it assumes that the supplier would use the same b0
without knowing how long the stockout interval will be. While,
as Montgomery et al. [60] suggested, linear 1 is probably a satisfactory approximation in many cases, the linear 2 form could
be better in other cases since it allows the initial value of b to be
determined by the length of the stockout period. In linear 2,
b(s) = 0 until a time sM before the time at which the replenishment order will arrive and then increases linearly until it
reaches its maximum value, bM, at s = 0. All demands occurring
more than sM periods before the replenishment order is
received will be lost sales if there is no inventory. San José
et al. [79] introduced and analyzed this form.
5. Step: This form is like linear 2 in that b(s) = 0 until a time sM
before the time at which the replenishment order will arrive,
but it differs in that the step form then immediately increases
to b(s) = 1 until the replenishment order is received rather than
gradually increasing as linear 2 does. San José et al. [80] proposed and analyzed this form.
The final b(s) form to be identified differs significantly from the
preceding ones in that, rather thanb(s) being a non-decreasing
function of s, it is U- or dish-shaped.
6. Mixed exponential: The basic assumption behind the mixed
exponential form, analyzed only in Sicilia et al. [99], is that
there are two types of customers. For some of them the reluctance to backorder decreases as the replenishment time
approaches. However, there are others who are the opposite
of this: they are more likely to backorder if the wait is longer.
Sicilia et al. suggested that this might be the case if the customer believes that this will ensure that he or she will get a
higher quality item. (Note: It seems to the authors of this survey
that the likelihood of this type of reasoning applying for the
types of items that would be managed by a basic EOQ model
or one of its variants is extremely low, so that the applicability
of this form would be rare.) The combination of these two types
of customers results in a backordering-likelihood function that
is the sum of two exponential forms. One, with a maximum
value of b1, is based on s (referred to as s1), and the other, with
a maximum value of b2, is based on T2 s (referred to as s2), the
amount of time from when the stockout begins until the unit of
demand that will wait for s1 periods arrives. Adding the two
exponential forms gives a dish-shaped function.
181
is the case very soon after the stockout interval begins, the probability a new demand will be backordered is high and when the
existing backlog is large, which would be the case as the stockout
interval nears its end, the probability a new demand will be
backordered is low. Papers [64,24,33] will be reviewed in
Section 3.3, [65] will be reviewed in Section 4.2.3, and [61] in
Section 4.2.6.
Comment: All the forms for b(s) considered here, except the
constant-b model, the step form, and the mixed exponential form,
assumed that the probability of backordering will be higher (lower) if the replenishment time is closer (further away). One implication of that assumption, not only for linear 1 and 2, the exponential
and the rational, but also for the step form and the constant-b
models, and even for the mixed exponential form as s approaches
0, is that when the backlog is high, the probability that the next demand will be backordered does not decrease; if anything, it increases. The reason for this is clear: The backlog is increasing
because it is getting closer to the time of replenishment and, since
replenishment is instantaneous in the EOQ model, all the backorders will be filled soon and simultaneously. For the models in
[64,65,61,24,33], the reverse is true. As noted above, these models
assume that the probability of backordering will be higher when
the stockout period has just begun and will become lower as the
stockout continues and the replenishment time gets closer, so that
the customer will wait for a shorter time. It is possible that this
might be true for ‘‘fashionable commodities’’, as suggested by Ouyang et al. [61], although they provide no justification for that statement. Even then, those are not the type of items that would
generally be controlled by EOQ-type models since their lifetime
tends to be short and demand is more stochastic. The relationship
between the longer line and lower probability of waiting makes
sense for a queuing system using a first come-first served queue
discipline. A customer arriving when the queue is long may balk
because she knows that her wait will be longer due to all the customers to be served before her. It does not make a lot of sense for
an EOQ system, where a longer line implies that service will be
sooner, not later.
3. Deterministic EOQ–PBO models
In Section 3.3 we will discuss each of these b(s) forms somewhat more extensively. We note that, in addition to the six forms
for b(s) identified, some authors have used more general forms
for which no equation is given, but general properties are stated.
The deterministic EOQ–PBO models satisfy all, or at least most,
of the basic assumptions of the classic EOQ model with full backordering except that only a percentage of the demand when the supplier is out of stock will be backordered. In Section 3.1 we briefly
summarize papers that developed models for the basic deterministic EOQ–PBO model with a constant b. In Section 3.2 we consider
several papers that addressed the issues of the sensitivity of those
models or that identified potential problems with their formulation or solution. In Section 3.3 we examine basic models that included one of the time-based backordering percentage forms
identified in Section 2. In Section 3.4 we consider models that used
a backlog-based probability of backordering.
2.2. Backordering percentage based on the size of the backlog
3.1. Basic deterministic EOQ–PBO models with constant b
All of the forms for b(s) discussed in Section 2.1 began with the
assumption that the percentage of demand during the stockout
period will be backordered is a function of s, the time remaining
to the receipt of the replenishment order. In this section we discuss
modeling backordering or backlogging as a function of the size of
the existing backlog.
Padmanabhan and Vrat [64,65], Ouyang et al. [61], Chu et al.
[24], and Dye et al. [33] assumed that the probability of backordering is negatively related to the size of the existing backlog when a
demand arrives. That is, when the existing backlog is small, which
All the models reviewed in this section are single-item models
that assumed that all parameters are known and constant over
an infinite time horizon, that replenishment is instantaneous with
a known lead time, and that the cost to place and receive an order
is a constant, independent of the size of the order. Where they differ is in their choice of variables, their assumptions about the backordering rate, and their cost structures. All but one assumed that b
is known and constant. Most made the usual EOQ model assumptions about costs: (1) the holding cost is an amount per unit per
year, (2) the backorder cost is an amount per unit per year plus,
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D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
possibly, a fixed cost per unit, and (3) the lost sale cost is an
amount per unit. These papers will be reviewed in Section 3.1.1.
Papers that used alternative cost structures will be reviewed in
Section 3.1.2.
3.1.1. Models using the classical cost structure
The earliest papers that specifically modeled the basic deterministic EOQ–PBO were Fabrycky and Banks [38] and Ali [7]. Both
developed cost functions to be minimized, but neither provided a
solution procedure.
Montgomery et al. [60] included a fixed cost per unit short,
which is the same whether that unit is backordered or not, and a
cost per year per unit backordered. Their decision variables were
Q (the order quantity) and S (the maximum stockout level), which
they replaced by two new variables: U (the total demand during an
inventory cycle = Q + (1 b)S) and V (the maximum inventory
level = Q bS). Their solution procedure found V⁄(U) and then optimized U, although they recognized that, because the cost function
is not convex, their solution could not be guaranteed to be optimal.
They determined a critical value for b, above which partial backordering is optimal.
Leung [52] showed that an algebraic approach called the complete squares method could be used to determine the optimal values
of U and S for Montgomery et al.’s [60] model, even though that
model’s quadratic cost function is not convex. As a by-product of
using this method, Leung determined the same critical value for
b as in Montgomery et al. and was able to determine the conditions
under which the optimal strategy is to partially backorder all demand or to not allow backordering at all. In addition, by choosing
appropriate values for different parameters, he was able to develop
formulas equivalent to those in Mak [57] (to be reviewed later in
this section) and for variations on the basic EOQ scenario previously developed by other authors (see [52] for references to other
authors who have used the complete squares method for models
without partial backordering).
Rosenberg [76] used the same cost structure as Montgomery
et al. and also used Q and S as the initial decision variables, which
he replaced by T (the cycle length) and X (a fictitious demand rate,
X = V/T). He used a two-stage solution procedure, first finding X⁄(T)
and then T⁄. Rosenberg also determined a critical value for b, above
which partial backordering is optimal.
Except for slight changes in notation, Park presented the same
model in [70,71]. He dropped the fixed cost for a backordered unit
and used U (R in [71]) and S as the decision variables. He proved
that his cost function is convex as long as U P S P 0 and developed
formulas for S⁄ and U⁄(S). Park also determined a critical value for
b, above which partial backordering is optimal.
Wee [121], with the same cost structure as Park [70,71], used
T and T2 (the length of the stockout phase of an inventory cycle)
as the decision variables. He proved his objective function is convex and determined equations for T 2 and T⁄ (T2). Wee also determined a critical value for b, above which partial backordering is
optimal.
Although he never referred to partial backordering or any of the
other terms typically used, Urban [115] developed a basic EOQ–
PBO model by assuming the demand rate is a constant D during
the in-stock period of an inventory cycle and changes to kD during
the stockout period. While his model allows for the possibility that
k > 1, k = 1 means full backordering, and having 0 < k < 1 corresponds to partial backordering with k equal to a constant b. The
objective is profit maximization and his decision variables are
the maximum (initial) inventory level, which he calls S (not the
same as S in Table 1), and the most-negative shortage (backorder)
level or reorder point, which he calls s (B in Table 1). A difference
from the other models is that he does not include a ‘‘goodwill’’ cost
for lost sales, recognizing (indirectly) only the lost profit component of a lost sale cost.
Pentico and Drake [72] used T and F (the fraction of demand
filled from stock or the fill rate) as the decision variables. They used
the same cost structure as Park [70,71] and Wee [121], but included a fixed cost per backordered unit in an Appendix. They
proved that their solution is optimal for T > 0 and 0 < F 6 1 and
derived equations for T⁄ and F⁄(T). They also determined a critical
value for b, above which partial backordering is optimal and below
which the optimal solution is to either use the basic EOQ model
without backordering or not stock the item at all, whichever costs
less.
San José et al. [80] developed a general approach to analyzing
deterministic EOQ–PBO models with the cost structure used by
Montgomery et al. [60] and Rosenberg [76] when the value of
b(s) is a non-decreasing function of s, the time until the replenishment order is available to fill the backorders (see Section 2.1). This
will be discussed in Section 3.3, but is mentioned here because it
applies when b is a constant.
Mak [57] considered the basic problem without a fixed cost per
unit backordered, but with the added complication that, while the
mean percentage of demand backordered during the stockout period is given by the constant b, the actual amount backordered is a
random variable, with its mean given by bS and a standard deviation. He considered two cases: (1) the standard deviation of B is
independent of the cumulative shortage S (rB is a constant), and
(2) the standard deviation of B is proportional to the cumulative
shortage (rBjS = aS). As Rosenberg [76] did, Mak used Q and S as
the initial decision variables, but then changed to T and X. His
two-stage solution procedure was to find X⁄(T) and then optimize
over T. Mak’s equations are identical to Rosenberg’s except they
do not include the fixed cost per unit backordered and they do include terms to reflect the variance of the amount backordered. Mak
developed the same condition on the minimum value of b for
which partial backordering is optimal as in Park [70,71], Wee
[121], and Pentico and Drake [72]. If rB or rBjS = 0 (i.e., there is
no uncertainty about the amount backordered), then Mak’s equation for T⁄ is identical to the one in Pentico and Drake [72] and
his equation for X⁄(T) is identical to their D F.
3.1.2. Models using alternative cost structures
In a series of papers, San José et al. considered three variations
on the problem with constant b in which the cost structure
differed from the classical cost structures used in the papers in
Section 3.1.1. San José et al. [84] included both fixed and timedependent unit costs for backordering (as in [60,76]) and for lost
sales, a cost structure that is not considered by any other authors.
Their decision variables were U and B (the maximum backorder
quantity). In San José et al. [83], the holding cost per unit is a constant per unit time, lost sales have a fixed cost per unit and the
cost of a backorder is an increasing quadratic function of the
waiting time. Their decision variables were T1 (the length of the
in-stock phase during an inventory cycle) and T2. In San José
et al. [78] they used a constant unit holding cost per period,
but they assumed that the backordering cost per unit has a fixed
component plus a time-dependent component that is nondecreasing, continuous and positive. As in many of their other
models, the decision variables were T1 and T2, but the objective
was profit maximization rather than cost minimization. In
[78,83,84], they used the combinations of two control parameters
to determine: (1) whether the optimal policy was to not stock the
item at all, to stock but not allow backorders, or to allow partial
backordering, and (2) how to find the optimal values of the decision variables.
Hu et al. [46] considered the constant-b problem when the unit
backorder cost increases linearly with the duration of the shortage.
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
Their decision variables were T and T1. They established a critical
value for the holding cost per unit, at or below which the optimal
solution is to allow no backordering and above which partial backordering is optimal. The optimal solution is found by solving a
complicated equation for T1 and then using that value of T1 in
the quadratic formula to solve for T2, and thus for T. They noted
that an explicit closed form solution for the optimal policy may
not be obtained, but it can be solved for by a numerical search.
3.2. Sensitivity of the basic model and related issues
While some of the authors discussed in Section 3.1 performed
some basic sensitivity analysis on their models, we review here papers that only address the sensitivity issue. But first, we consider a
paper that raised issues about whether the specification of the conditions under which a model gives optimum results is complete.
Whitin [128], commenting on papers by Montgomery et al. [60],
Rosenberg [76], and Park [71], noted that none of those authors
mentioned a simple but necessary economic constraint on the
optimality of the proposed solution: ‘‘[a]n item will be bought or
sold only if positive profits result’’. Using the example in [76], he
showed that ignoring this constraint meant that the example solution in [76], which met the condition for the optimality of partial
backordering given in the paper, actually resulted in a negative
profit, in which case the optimal solution would be to not stock
the item at all. He also showed that, in the example in [71],
whether the presumably optimal solution met this constraint or
not depended on how the cost of a lost sale is divided between
the lost profit and the rest of the cost of a stockout.
We note that the formal sensitivity analysis studies found considered only the effects of changing b, not the effects of changing
any of the other parameters in the models.
As noted above, most of the authors who developed models
with a constant b also determined a condition that can be used
to determine whether partial backordering is optimal or not. These
conditions, whether expressed in these terms or not, can all be
translated into a condition on the minimum value of b. For Park
[70,71], Wee [121], Pentico and Drake [72] and Mak [57], none of
whom included a fixed cost per unit backordered in their models,
this minimum value of b for which partial backordering is optimal
is given by
average cost per period of using the basic EOQ model
:
b ¼1
average cost per period of all lost sales
For Montgomery et al. [60] and Rosenberg [76], who included a
fixed cost per unit backordered, the formula for b⁄ is more complicated due to that cost. b⁄ is important in the following.
Chu and Chung [23] stated that sensitivity analyses for models
of inventory systems that analyze the results of solving those models for a variety of numerical cases are ‘‘questionable since different conclusions may be made if different sets of numerical
examples are analyzed’’. They illustrated this concern by considering the results of the experiment-based sensitivity analysis performed by Park [70]. Limiting their attention to cases in which
b⁄ P 0, which both Park and they argued is the only situation that
needs to be considered, Chu and Chung conducted a formal mathematical analysis that concluded that Park was correct in finding
that Q⁄(b) increases and the optimal cost decreases as b increases,
but they found that S⁄(b) increases with b only if another condition
is met. This condition is an upper bound for the ratio of the unit
holding cost per period to the unit backorder cost per period that
is based on a more complicated expression involving b⁄.
Yang [137] extended Chu and Chung’s [23] work to consider
what happens if b⁄ < 0. While this means that the cost of all lost
sales is less than the cost of using the basic EOQ with no backorder-
183
ing, which was Park’s [70] and Chu and Chung’s justification for
limiting their analyses to b⁄ P 0, it does not necessarily mean that
having all lost sales is better than using the EOQ–PBO. Yang addressed two basic issues. First, he extended the work by Park and
Chu and Chung to more completely justify the result that, when
b 6 b⁄ and b⁄ P 0, the optimal solution is to use the basic EOQ
without backordering. Second, he showed that when b⁄ < 0, then
the upper bound for the ratio of the unit holding cost per period
to the unit backorder cost per period determined by Chu and
Chung [23] is decisive in determining the optimal inventory policy.
While Chu and Chung [23] and Yang [137] both based their sensitivity analyses on Park’s [70] model, Leung [53] based his on
Montgomery et al. [60]. Other than the choice of decision variables,
the significant difference between the models in Park and Montgomery et al. is that the latter included a fixed cost per unit backordered, whereas Park did not. However, Leung’s model differed
from Montgomery et al.’s in that his fixed penalty cost per unit
backordered is less than the fixed penalty cost (exclusive of the lost
profit) of a lost sale. Leung derived the conditions for b under
which the optimal solution is to use the equations for partial backordering. He then conducted a sensitivity analysis on the decision
variables and the optimal cost, determining the range of b within
which each would be monotonically increasing or decreasing.
3.3. Basic deterministic EOQ–PBO models with b a function of the time
to replenishment
In Section 3.1 we reviewed models in which a constant fraction
b of the demand during the stockout period will be backordered,
with the remainder being lost sales. In this section we review models in which b is a function of the time remaining until a replenishment order is received. We also review some papers that model the
backordering rate as a function of the size of the backlog.
As discussed in Section 2.1, there are a number of ways in which
authors have assumed that b will change with the time remaining
until a replenishment order is received. Brief descriptions of the
functional forms ofb(s), where s is the time remaining until a
replenishment order is received, and of the solution procedures
used follow. Table 2 summarizes the equations for b(s) for each
case.
3.3.1. Linear 1
Montgomery et al. [60] noted that, while ‘‘many different functional forms for the manner in which the mixture of backorders
and lost sales will occur’’ are possible and ‘‘the identification of
the proper functional form may be difficult in practice. . . the constant or linear ratio (linear 1, in our categorization) models are
probably satisfactory approximations’’. Certainly, the linear 1 form
is the easiest to analyze other than the constant-b model reviewed
in Section 3.1. Montgomery et al. did not show their analysis, but
did state that the results are basically the same as for the constant-b model with the substitution of two new parameters in their
¼ ðb þ 1Þ=2, the average value of
equations, both based on using b
0
b over the time the stockout exists, instead of b. (Note: They assumed, as most authors have, that bM = 1.)
San José et al. [82] expressed the concept of the linear 1 form
somewhat differently than Montgomery et al. [60] did, saying that
they ‘‘assume that the fraction of the customers who are not willing to wait is proportional to the ratio between the waiting time
(i.e., s) and the length of the shortage cycle (i.e., T2)’’. However,
expressing the decrease in 1 – b(s) in this way is the same as
expressing the increase in b(s) as the linear 1 form does. As in most
of their other work, San José et al. defined two control parameters
as functions of the basic parameters in the model and then identified, for the different combinations of those control parameters,
when the basic EOQ model without backordering is optimal, when
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all sales should be lost, and when partial backordering is optimal.
They also gave, for the latter case, the equations for the optimal
values of T and T2, which were their decision variables.
3.3.2. Exponential
b(s) is bM multiplied by a positive exponential function of s; i.e.,
b(s) = bMexp(as), with with s P 0, a > 0, and 0 6 bM 6 1. This is
one of the four forms considered by San José et al. in [80]. Their
decision variables were T2 and T. While the authors described the
general procedure for solving for the optimal values of T2 and T
for all four of the forms of b(s) that they identified in [80], they
did not specifically address the solution for the exponential form.
San José et al. [81] did address the exponential form in detail,
using T1 and T2, rather than T2 and T, as the decision variables. They
defined four control parameters and identified, for their various
combinations, what the form of an optimal solution is. Since the
equation to be solved when partial backordering is optimal includes exponential terms, determining T 2 for those cases requires
the use of a numerical method.
3.3.3. Rational
b(s) is bM divided by a positive linear function of s; i.e.,
b(s) = bM/(1 + as), with s P 0, a > 0, and 0 6 bM 6 1. This is another
of the four forms considered by San José et al. in [80]. Their decision variables were T2 and T. For the rational form of b(s), finding
the optimal value of T2 for one of the combinations of the two control parameters they defined involves solving an equation that includes ln(1 + aT2), which requires using a search procedure rather
than evaluating a closed-form expression.
Sicilia et al. [98] combined the rational form of b(s) with a backorder cost function that is a quadratic and increasing function of s.
As in their other papers discussed here as San José et al. [79,80,83],
their decision variables were T2 and T. Due to the more complicated backordering cost function, they used four control parameters. For most combinations of the control parameters, the
optimal solution is to either use the basic EOQ without backordering or not stock the item at all. For the remaining cases, finding the
optimal value of T2 involves solving an equation that includes
ln(1 + aT2), which, as noted, requires using a search procedure
rather than substituting the problem’s parameters into a closedform expression.
3.3.4. Linear 2
Unlike the linear 1 form, in which b(s) has an initial value of b0
when the stockout interval begins, in the linear 2 form b(s) = 0 until sM periods before the end of the inventory cycle, after which it
increases in a linear fashion until it reaches its maximum value,
bM, when the replenishment order arrives. The only paper that analyzed the basic EOQ model with partial backordering andb(s) modeled by the linear 2 form is San José et al. [79], in which T1 and T2
were the decision variables. Given the need to concern themselves
with the issue of whether T2 is greater than sM, in which case there
is an interval at the start of the stockout period during which all
sales will be lost, the cost function to be minimized is more complicated than the one they used for the linear 1 form. As a result,
their characterization of the nature of an optimal solution is also
more complicated, using combinations of three control parameters, rather than two, as they did in [82]. For most combinations
the optimal solution is to either use the basic EOQ model without
backordering or to not stock the item. If partial backordering is
optimal, determining the optimal value of T2 may involve solving
for the roots of a fourth-power polynomial.
3.3.5. Step
Like the linear 2 form, the step form includes specifying a maximum value for s, sM, beyond which no customers will be willing to
wait. The major difference between the linear 2 and step forms is
that in the step form all customers will wait (i.e., b(s) = 1) as long
as s is less than sM, whereas in the linear 2 form b(s) increases linearly after s < sM. San José et al. [80] addressed the step form as one
of the four they considered using a general solution approach for
problems in which b(s) is a non-decreasing function of s. Their
decision variables were T2 and T. The solution procedure begins
by ignoring sM and solving the problem as if there were full backordering. If T2 P sM, then the problem is solved. If not, a more
complicated solution procedure is used.
3.3.6. Mixed exponential
All of the forms for b(s) discussed so far, except the constant-b
models in Section 3.1 and the step form in Section 3.3.5, assumed
that a customer who would have to wait longer would be less
likely to backorder. As discussed in Section 2.1, the basic assumption behind the mixed exponential form analyzed in Sicilia et al.
[99] is that there are two types of customers, those whose reluctance to backorder decreases as the replenishment time approaches and others who are more likely to backorder if the wait
is longer. The combination of these two types of customers results
in a backordering-likelihood or customer-impatience function that
is the sum of two exponential forms, resulting in a curve that is
dish-shaped. How high the two lips of the dish are depends on
the relative sizes of b1 and b2, but since their sum must not exceed
1, the maximum probability that a customer will wait for any value
of s is no more than 1. Sicilia et al. [99] used T1 and T2 as the decision variables. As they did in their other work [98,78–84], they defined control parameters as functions of the basic model
parameters, and identified, for each combination of them, the nature of the optimal solution. Since the equation to be solved when
partial backordering is optimal includes exponential terms, determining T 2 for those cases requires the use of a numerical method.
3.4. Backordering likelihood based on the size of the backlog
All of the forms for b(s) discussed so far began with the assumption that the probability that a demand during the stockout period
will be backordered is a function of the time remaining to the receipt of the replenishment order. In Section 2.2 we discussed the
concept behind basing the backordering likelihood on the size of
the existing backlog. Here we discuss three papers that used that
approach.
Padmanabhan and Vrat [64] assumed that the probability of
backordering is a negatively related to the size of the existing backlog when a demand arrives. The cost function is convex, so the
optimum values for T and T1, the decision variables, can be found
by simultaneously solving a pair of equations, both of which include exponential terms, which they do numerically.
Chu et al. [24] revisited the work by Padmanabhan and Vrat
[64] in order to analyze and improve on their solution method.
They determined that if the setup cost is less than a critical value,
then T 1 and T ðT 1 Þ can be obtained from two straightforward equations rather than having to solve a pair of non-linear equations
simultaneously; otherwise, the best policy is to prolong the shortage period as long as possible.
Dye et al. [33] noted that neither Padmanabhan and Vrat [64]
nor Chu et al. [24] recognized either the cost of lost sales due to
shortages or the purchase cost of backordered units in their cost
function. However, in addition to including these two costs, Dye
et al. made other changes to the model. They used a finite time
horizon, assumed a positive log-concave demand function rather
than a constant demand rate, and assumed inventory deterioration
at a constant percentage. As result, their model, other than making
the assumption that the backordering rate decreases as the number of backorders increases, is like the models considered in
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
Section 4.2.2.2 and the solution procedure is like the one described
there.
4. Deterministic EOQ–PBO models with additional
considerations
All the models reviewed in Section 3 except Dye et al. [33] have
the form of a basic deterministic EOQ–PBO model. Although there
are differences with respect to the cost structure or whether b is a
constant or a function of either the time remaining until the
replenishment order will be received or the size of the existing
backlog, they all have a constant demand rate and instantaneous
replenishment and a constant cost structure over an infinite time
horizon. In this section we review papers that introduce additional
considerations, such as deteriorating or perishable inventory, demand that varies with time or the stock level, quantity discounts,
inflation, uncertain receipt quantity, pricing decisions, or multiple
stocking locations. We begin with models that introduce one additional consideration and then review models that have two or
more added considerations. (Note: Unless defined where used,
symbol definitions are given in Table 1. Descriptions of the timebased backordering functions b(s) are in Section 2.1 and Table 2.)
4.1. Models with a single additional consideration
4.1.1. Delayed backordering
Abboud and Sfairy [6] developed a model for the basic EOQ in
which partial backordering with a constant b is delayed for m periods after the stockout phase begins. Prior to that time all customers
will backorder. The objective is to minimize the average cost per
period; the decision variables are Q and t, the length of the partial
backordering phase exceeding m. By examining the roots of a quadratic function for t⁄, they determine whether (1) partial backordering is optimal and Q⁄ is determined from a simple linear
function of t⁄, (2) t⁄ = 0 and Q⁄ is determined from a square root formula that adjusts the basic EOQ formula for m, or (3) the item
should not be stocked.
4.1.2. Deteriorating or perishable inventory
Although deteriorating or perishable inventory is by far the
most popular additional consideration in the development of
EOQ–PBO models, almost all the papers that include this also include other additional features. Only two papers were found that
add only deteriorating inventory to the basic partial backordering
model. Wee and Mercan [126] used a constant b and inventory
deteriorating at a constant percentage rate. Their decision variables
were T1 and T. They proved that their cost function is convex if b
exceeds a specified lower bound and the optimal solution is found
by simultaneously solving a pair of non-linear equations. Shah and
Shukla’s [86] scenario is the same as that in Wee and Mercan except that they assumed that b(s) has a rational form. They also
used T1 and T2 and found a solution by solving a pair of non-linear
equations, although they did not prove that the solution is optimal.
4.1.3. Demand pattern
Urban [115] extended his discussion of a basic EOQ model with
a constant demand rate that changes to a different constant demand rate when the inventory level drops to 0 (Section 3.1), to
consider the two cases in which the demand rate during the instock period is either (a) a constant that is a function of the initial
stock level or (b) given by a power function of the instantaneous
stock level until that reaches a specified level, at which point D becomes constant. In both cases he recognizes partial backordering
by having the demand rate change to kD (which corresponds to
partial backordering at a constant rate b if 0 < k < 1) when the
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inventory level reaches 0. As was the case with his basic EOQ–
PBO model in Section 3.1, the objective was profit maximization
and he did not include a ‘‘goodwill’’ cost for lost sales.
Hsieh et al. [45] examined Urban’s [115] case (b) with two
important changes: (1) b(s) is not constant but has a rational form,
and (2) there is a goodwill cost per unit of lost sales. They proved
that there is a unique optimal replenishment policy and gave a relatively simple procedure for finding it.
Zhou et al. [150] used a finite time horizon. They assumed that
the demand rate is an increasing function of time, although they
also proved some results for a decreasing function. They used both
the rational and exponential forms for b(s). The decision variables
were (1) n, the number of complete cycles during the horizon, (2)
the starting time for each cycle, which is the start of its stockout
phase, and (3) each cycle’s replenishment time, which is the start
of its in-stock phase. They recognized that, since the demand rate
changes over time, the optimal lengths of the n cycles and the relative times of the replenishments within each cycle are not necessarily the same. Their solution procedure finds solutions for n = 1,
2, etc. and identifies the value of n with the lowest cost. They could
not prove their cost function is convex, but they proved results that
narrow the search ranges for n and the optimal times for each
cycle.
4.1.4. Uncertain replenishment quantity
Most inventory control models assume that the quantity received is the same as the quantity ordered, Q. However, it is
possible that the quantity received, Y, is a random variable, with
E (YjQ) = aQ, where the bias factor, a, may exceed 1.0.
Kalro and Gohil [50] extended Silver [100] on adjusting the
basic EOQ model without backordering for an uncertain replenishment quantity to consider both full and constant-b partial backordering. They considered Silver’s two cases with respect to the
variability of YjQ: (1) rYjQ = r, independent of Q; (2) rYjQ = r1Q,
where r1 = rYjQ=1. Kalro and Gohil used the same cost structure
and solution approach as Rosenberg [76] did for the basic EOQ–
PBO (Section 3.1), adapted for the randomness of Y. They replaced
their original decision variables Q and S with: (1) the cycle length,
T = (aQ + (1 b)S)/D, and (2) a fictitious demand rate, X = (aQ bS)/T. Their solution procedure was to find an equation for X⁄(T)
and use that to find an equation for T⁄. They established conditions
similar to Rosenberg’s under which partial backordering is optimal.
As in Mak [57], who included uncertainty of the quantity backordered (Section 3.1), Kalro and Gohil’s results were similar to Rosenberg’s, adjusted for including rYjQ.
4.1.5. Pricing
The vast majority of papers about partial backordering assume
that the vendor’s price is given. If it is considered at all, it is because
the objective is profit maximization or, if the objective is cost minimization, the price becomes part of the penalty for a lost sale. In
addition to papers that also include deteriorating or perishable
inventory, which will be covered later, there are a few papers that
include price as a decision variable, recognizing that by giving a
customer a discount when out of stock, the vendor may increase
backordering and reduce lost sales, thus increasing the overall
profit.
Assuming that the price charged during the in-stock phase of a
cycle is given, Drake and Pentico [29] modified their basic constant-b EOQ model with both fixed and time-dependent costs per
unit backordered (Appendix B of [72]), which used T and F as the
decision variables, to include d, the discount offered during the
stockout phase, as a decision variable. They assumed that b is a linear function of d. They determined the range for d within which
offering a discount to increase b is feasible by determining when
b(d) is greater than b⁄(d), the minimum value of b(d) for which
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partial backordering is optimal if d is included as part of the fixed
cost per unit backordered. Their equations for T⁄ and F⁄(T) from
[72] can then be used to search over the feasible range of d to
determine the optimal discount.
Bhargava et al. [11] used T1 and T2, the lengths of the in-stock
and stockout phases of a cycle, and p1 and p2, the prices to be
charged during those two intervals, as their decision variables.
They assumed a ‘‘customer valuation’’ function that describes the
demand distribution as a function of the price. Given standard
assumptions about demand elasticity, their primary results are
that the ratio of the optimal values of T1 and T2 is the ratio of the
unit backordering cost per unit time to the unit holding cost per
unit time and that the vendor will set ‘‘equal effective prices’’ for
the two phases, meaning that the prices during the two phases
are set so that the effective demand rates (sales plus backorders)
during the two phases are the same. They proved additional results
about the characteristics of an optimal solution under different
conditions and distributions for the customer valuation function.
They discussed the possible effects of customers deliberately waiting to purchase during the stockout phase to secure the lower
price. They also compared the optimal policy with three alternatives: (1) a no-stockout policy, (2) a stockless-operation policy,
and (3) a backorder policy that offers no discount.
You and Wu [144] analyzed a system in which a finite horizon L
is divided into n equal-length periods. Each period consists of an advance sales period of length k with price p1, during which customers
place orders which will be delivered at the end of that period, followed by a spot sales period with price p2 > p1, during which customer orders are filled immediately from stock. Advance orders
may be canceled with a partial refund. You and Wu assumed that
the cancelation rate is a percentage of the current backlog or advance-sales amount. They divided the analysis into two parts. First
they assumed that k is fixed, so the decision variables are n, p1, and
p2. They proved results about the nature of the solution under reasonable conditions for the demand function and developed a solution procedure based on determining the optimal prices, order
quantity and profit for n = 1, 2, etc. Then they allowed k to be a
fourth decision variable. For this case the theoretical results are less
definite and solving for k may require an iterative search procedure.
You [143] examined a variation on the system in You and Wu
[144] in which the time horizon is not predetermined. As in You
and Wu, n(6 nmax) time intervals, not necessarily of equal length,
are each divided into two parts, an advance sales period with partial backordering at b(s) with a rational form, followed by a spot
sales period with no stockouts. During interval j the sales price is
pj = p (j 1)r and the demand is given by d(j) = a jk bpj. During each interval there is a replenishment of size Q when the total
amount backordered is s. The objective is to determine the values
of n, Q, and the starting and replenishment times for the n intervals
that will maximize the average profit per unit time. To simplify the
computations, You redefined the time variables in terms of n and
z = Q s, the maximum inventory level. His solution procedure
finds z⁄ for n = 1, 2, . . . , nmax and identifies the n with the highest
profit per period.
Ding et al. [28] considered ‘‘the problem of purchasing and
allocating inventory to several classes of customers. . . when
price-based partial backlogging of unfilled demand is possible.
The classes are distinguished by the prices they pay’’. Unlike the
papers by You and Wu [144] and You [143], which deal with two
demand classes that occur in sequence, the multiple demand classes in Ding et al. occur simultaneously. As in Drake and Pentico [29],
Ding et al. assumed that the fraction of demand from a given class
that would be backordered is a linear function of the discount size.
Drake and Pentico, however, maintained the same discount
through the entire stockout phase, while Ding et al. allowed the size
of the discount to vary over time. Also, since there are multiple de-
mand classes that have different prices and, therefore, different
lost-sale costs, discounts may be offered to one class of customer
while a different class is still having its demand filled. The objective
was to maximize the average profit per period. The decision variables were T, the ‘‘run-out times’’ for each demand class (i.e., the
times at which the demand classes no longer have their orders
filled but are offered a discount to backorder instead), and the discount to be offered to each class at each point in time from its runout time to the end of the cycle. They developed equations for the
optimal discount at any time for each demand class and the optimal
run-out time for each demand class and an equation to be solved
for T⁄.
Finally, we note that Abad [1,3] considered pricing as a way of
influencing demand, not backordering specifically, as a special
non-deterioration case within his more general work on EOQ models with partial backordering, pricing, and deterioration. These papers are reviewed in Section 4.2.1.
4.1.6. Multiple items
Zhang et al. [147] considered the scenario in which there are
two items – a major item that has the characteristics of a basic
EOQ with partial backordering at a constant rate b and a minor
item, a fraction of the demand for which is derived directly from
the sales of the major item. The authors assume the minor item
is to be stocked with no backordering and that any demands for
the minor item that are directly tied to the sale of the major item
will be lost if the major item is not available. Using, as in Pentico
and Drake [72], T and F for the major item as their primary decision
variables, Zhang et al. develop models for two basic cases: (1) the
major and minor items are ordered at the same time, and (2) the
ordering frequency for the minor item is T/k, where k is a positive
integer. For each model they determined a critical value for b and
described and illustrated by example a procedure for determining:
(a) whether the major item should be stocked or not, and (b) if so,
what the optimal values for T and F are.
4.2. Models with multiple additional considerations
Most of the models that add two or more additional considerations to the basic partial-backordering EOQ include deteriorating
or perishable inventory as one of them. Thus we start with models
that add deterioration and one other feature, such as pricing or
time-dependent demand, and then review models that add deterioration and two or more other features.
4.2.1. Deterioration and pricing
Abad [1] added partial backordering to a model by Rajan et al.
[75] that included deterioration and pricing. Abad noted that
‘‘[w]hen customers are willing to wait to obtain fresh stock, resellers may use backlogging as a strategy to control cost especially
when the good is highly perishable’’ and that ‘‘the reseller may
vary price within the inventory cycle to take into account the age
of the product and the value drop associated with the product’’.
Abad used T1, T2, and p(t), the price at time t, as his decision variables. He allowed the deterioration to vary with time and modeled
demand as a function of both t and p(t), decreasing with respect to
increases in both, and analyzed both the exponential and rational
forms of b(s). Unlike the approaches taken by Drake and Pentico
[29], Bhargava et al. [11] or Ding et al. [28] (Section 4.1.5), he did
not include using the price during the stockout phase as a way of
increasing the backordering rate. Abad also did not include any
costs of stockouts, whether backorders or lost sales, on the basis
that ‘‘these costs are not easy to estimate in practice’’ [3]. Instead,
in both [1,3] he used a time-based b(s) as a way to reflect the idea
that longer backorders are less desirable and, therefore, less likely
to be backordered (see Table 2 and Section 2.1).
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Abad [1] used a non-linear programming model and, applying
the Kuhn–Tucker conditions, established conditions under which
partial backordering is optimal and developed a procedure for finding the optimal solution. However, ignoring stockout costs tends to
bias the results toward higher stockouts.
Abad continued his analysis of the combination of backordering,
deterioration, and pricing in [3]. As in [1], his objective was profit
maximization, he considered both the exponential and rational
forms of b(s), and he did not include costs for either backordering
or lost sales. In [3] he used a fixed price. He again used a non-linear
programming model, but in this case the profit function cannot be
proved to be pseudo-convex, so there might be multiple local maxima. His solution procedure was to fix the price and find the optimal values of T1 and T2, then determine the optimal price for that
T1 and T2, solve for T1 and T2 for the new price, repeating this sequence until a local maximum is found. To allow for the possibility
of multiple local maxima, he suggested changing the starting price
and repeating the process.
Abad [5] extended his model in [3] to include both backordering
and lost sales costs, using T1, T2, and a fixed price as the decision
variables. He used a non-linear programming model and the same
solution procedure as in [3].
Dye [32] and Dye et al. [34] also extended Abad [3] by including
backordering and lost sales costs. Both used T1, T2, and a fixed price
as the decision variables. Both assumed that the demand rate is
any non-negative, continuous, convex, decreasing function of the
selling price and that deterioration is at a varying rate. Where these
two papers differ is b(s): Dye [32] used a rational form and Dye
et al. [34] used an exponential form. Both proved the optimality
of their solution approach if the price exceeds a specific lower
bound and used an iterative procedure that begins with setting
the price at that lower bound, determining T1 and T2 from that
price, determining a new price from that T1 and T2 and repeating
until the price converges.
Yang et al. [135] considered basically the same scenario as Dye
[32] except that they allowed for a delay of td periods before the
start of deterioration, which is at a constant percentage rate. Their
objective was maximization of profit, with T1, T, and p as the decision variables. Their optimization procedure iterates between
determining the optimal values of T1 and T for a given p and determining the optimal p for that pair T1 and T. The set of equations
used for finding T1jp and Tjp are based on a criterion function of
p, which determines whether T1 is less than, equal to, or greater
than td.
4.2.2. Deterioration and time-varying demand
Because of the time-varying demand assumption, most of the
papers reviewed in this section used a finite planning horizon of
length H. (Note: In these papers, H is assumed to be given. In reality, the length of a planning horizon is somewhat arbitrary since in
most cases only the first few cycles’ decisions will be implemented
before replanning takes place. None of the authors of these papers
considered the possibility of changing H to see whether that would
reduce (increase) the average cost (profit) per period.)
4.2.2.1. Equal length intervals within H. Except for Yang and Zhou
[142], all the papers in this section assumed that the horizon H is
to be divided into n equal-length intervals and that the second
decision variable is r, the fraction of each inventory cycle during
which there will be stock, which will be the same for all the cycles,
except, possibly, for the last interval. Note that this is not the same
as F, as defined in Table 1, unless the demand rate is constant since
a changing demand rate means that the demand may differ over
the same length of time in different equal-length cycles.
Wee [124] assumed a constant deterioration percentage, exponentially decreasing demand, and that r = 1 for interval n. His mod-
187
el development was done for full backordering, but he showed how
to convert the cost function to include constant-b partial backordering. He developed necessary conditions for the optimality of
n⁄ and r⁄ and used a search procedure that tries n = 1, 2, etc. until
the condition on n is met, using the necessary condition for the
optimality of r given n to find r⁄(n).
Chung and Tsai [26] considered the exact same problem as Wee
[124]. They noted that with Wee’s formulation, using Newton’s
method to find r⁄(n), as was done by Wee, will not necessarily converge with an arbitrary starting point for r. By reconfiguring the
first partial of the objective function with respect to r, Chung and
Tsai are able to develop an alternate solution procedure for which
Newton’s method will converge and can establish an upper bound
on r, which they use as the starting point.
Chu and Chen [22] considered the same problem as Wee [124]
and Chung and Tsai [26]. They noted that there may still be problems getting an optimal solution with Chung and Tsai’s approach.
They determined an approximately optimal solution for r⁄(n) to
use as a starting point for the search and suggested that using that
starting point as a heuristic solution for r(n) would give a high
quality solution.
Chang and Dye [16] analyzed the same scenario as Wee [124]
except that b(s) has a rational form, using the same decision variables as Wee. Their solution procedure is to find r⁄ for n = 1, 2, etc.,
stopping when the cost increases for an increase in n.
Deng et al. [27] reanalyzed the scenario in Chang and Dye [16].
By reformulating the objective function, they were able to prove
that there is a unique optimal value for r⁄(n). They used the same
search procedure over n and, since, like Chang and Dye, they did
not prove convexity in n, they cannot guarantee an overall
optimum.
Lin et al. [54] generalized Wee’s [124] model by using a general
time-varying demand function and a deterioration percentage that
is linear in time. Their other assumptions were the same as Wee’s.
Their solution procedure first derives an upper bound on n (nmax)
which guarantees that there will be a solution for r within the
interval [0, 1]. They then use an approach that, starting with
n = 1, finds r⁄(n), if it exists. It then increments n and repeats the
process, stopping when nmax is reached, and identifies the best n.
Lin and Lin [55] included inflation (and discounting) at a constant rate, a general continuous time-varying demand function,
deterioration at a linearly increasing rate, and an exponential form
for b(s). They developed an expression for a minimum value of n,
above which there will be a value of r between 0 and 1 for which
the first partial with respect to r of the total discounted cost will
equal 0. (Note: Their proposition does not say that this cannot happen if n is less than this criterion value. Nor does it say that this value of r will be unique.) Their solution procedure is to start with the
smallest integer value of n above this criterion value and try successively larger values of n until the total discounted cost increases. Given the caveats raised and the fact that they do not
prove convexity with respect to n, this may not guarantee an optimal solution.
Yang and Zhou [142] included inflation in a scenario with
replenishments at both the beginning and end of the planning
horizon. They assumed a general time-varying demand function
and that b(s) is any non-increasing function. The decision variables
were n, the number of complete inventory cycles of equal length
within the horizon, and the times at which each of the in-stock
and stockout phases begin. All costs were subject to inflation at
the same constant rate and the objective was the maximization
of discounted cash flow over the horizon. For both a simplified scenario in which b(s) is a constant and the scenario in which b(s) is
more general, they proved that the percentage of time there is
inventory is the same in each cycle, and for each scenario they
developed a procedure for finding an optimal solution.
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4.2.2.2. Unequal interval lengths within H. The papers in this section
did not assume that the cycles are of equal length, as those in Section 4.2.2.1 did. The decision variables were (1) n, the number of
complete cycles during the horizon, (2) the starting time for each
cycle’s in-stock phase (ti), and (3) the starting time for each cycle’s
stockout phase (si). The solution procedures were very similar:
Find the values of {ti} and {si} for different values of n and identify
which n gives the best overall cost or profit, depending on which
was the objective. All of these authors proved that the sets {si}
and {ti} were optimal for a given n. Most proved, in addition, that
the cost (profit) function is convex (concave) in n, so that their
search procedure guarantees an overall optimal solution. However,
several [15,41,39] did not, so they cannot guarantee an overall
optimal solution.
Chang and Dye [15] allowed shortages in all cycles but did not
assume the service level would be the same in all cycles. They assumed a constant deterioration percentage, a demand function
that has D(t)/D0 (t) non-decreasing in t, and a rational form for b(s).
Goyal and Giri [41] identified several typographical errors in
Chang and Dye [15] and proposed a simpler procedure than the
one Chang and Dye used.
Wang [117] suggested a change in the equation for the rational
form of b(s) used by Chang and Dye [15] to make it unit-free. (Note:
This change would apply to any authors’ use of the rational b(s),
not just Chang and Dye’s. However, this really only has the effect
of rescaling the multiplier for s, and thus has no significant implications.) Wang also provided a different argument for the monotonicity of the replenishment quantity with respect to the demand.
Teng et al. [113] also revisited Chang and Dye’s [15] model.
They noted that Chang and Dye did not include the purchase cost
for a non-constant order quantity in their cost function and did
not define the opportunity cost due to lost sales clearly. They used
a positive log-concave demand function. After changing the opportunity cost of a lost sale to be at least as large as the unit purchase
cost and adding the total purchase cost into the cost function, they
showed that the optimal replenishment schedule is unique and the
total cost is a convex function of n. They provided an intuitively
good starting point for the search over n.
Teng and Yang [112] addressed basically the same problem as
Teng et al. [113] with the following differences: the demand function and unit cost change in any patterns over the horizon and b(s)
has a fairly general form. Like Teng et al. [113], they provided a
good starting point for the search over n and proved the optimality
of their solution.
Chern et al. [21] addressed basically the same problem as Teng
and Yang [112] with the addition of inflation at possibly different
internal and external rates, a constant unit cost except for inflation,
and minimization of the total discounted cost as the objective.
Wang [118] considered the same basic problem as Chang and
Dye [15] except that D(t) is any continuous positive function. He
only included the purchase cost for deteriorated items in the cost
function and clearly defined the opportunity cost due to lost
sales.
Ghosh and Chaudhuri [39] addressed the same basic problem as
Chang and Dye [15]. They assumed the demand function is a linear
function of time with a non-zero slope, b(s) has a rational form,
and deterioration has a two-parameter Weibull distribution.
Wang [119] assumed a general time-varying demand function,
a constant deterioration percentage, and that b(s) has a rational
form. In addition to proving his approach gives an optimal solution,
he gave a good heuristic for determining the value of t1, the starting point for finding an optimal solution for a given n.
Papachristos and Skouri [67] assumed a log-concave demand
function, deterioration at a constant percentage, and an exponential form for b(s). Unlike Teng et al. [113], they did not include lost
sales costs or the cost of units bought except those that deterio-
rated. They proved that their cost function is convex with respect
to n and that there is a unique optimal set of times for a given n.
Teng et al. [110] added the costs missing in Papachristos and
Skouri [67] to the objective function, used a log-concave demand
function, and allowed any b(s) that satisifies b(s) + Hb0 (s) P 0.
They proved that the total cost is convex in n and gave a good starting value and search pattern for n.
Skouri and Papchristos [103] considered the same basic problem as Teng et al. [110] except that b(s) has an exponential form
and they only considered minimizing the cost.
Papachristos and Skouri [68] included the time value of money
in a scenario with the both the beginning and ending inventory
levels equal to zero. They assumed a log-concave demand function,
deterioration at a constant percentage, and an exponential form for
b(s). Their objective was minimization of discounted total cost, but
they only included a discount rate and not inflation rates in the
model. They did prove convexity in n and overall optimality.
Yang’s [138] objective was to determine which of four sequences of in-stock and stockout phases leads to the highest profit
over a finite horizon. He assumed the demand function is positive
and fluctuating with time, deterioration is at a constant percentage, and b(s) satisfies b(s) + Hb0 (s) P 0. He identified four different
sequences of in-stock and stockout phases that have appeared in
the literature:
Model 1: Start at the beginning of an in-stock phase and end at
the end of an in-stock phase. (Note the extra in-stock phase.)
Model 2: Start at the beginning of an in-stock phase and end at
the end of a stockout phase.
Model 3: Start at the beginning of a stockout phase and end at
the end of an in-stock phase.
Model 4: Start at the beginning of a stockout phase and end at
the end of a stockout phase. (Note the extra stockout phase.)
Yang developed cost and profit models for all four models. He
proved that, for any given n, (1) Model 4 has the highest and Model
1 has the lowest total profit over the planning horizon, with Models 2 and 3 between, in no specified order, and (2) Model 4 has the
lowest and Model 1 has the highest total cost over the planning
horizon, with Models 2 and 3 between, in no specified order. (Note
that he did not prove that the same n is best for all four models, so
he did not prove that the two statements are necessarily true without the qualifier ‘‘for any given n’’.)
Skouri and Papachristos [104] analyzed the same basic problem
as Yang [138] with the following changes: (1) the demand function
is log-concave, (2) b(s) has an exponential form, and (3) the objective is to minimize cost. They proved there is a unique optimal
solution for each of the four model types and established the same
result about the ordering of the costs as Yang did.
Chern et al. [20] carried the analysis in Yang [138] further and
developed an algorithm for finding the most profitable of the four
models. They found that if the net revenue per unit is at least a critical value, then Model 3 has the highest profit and, if it not, then
Model 4 does.
4.2.2.3. Models with one inventory cycle. With the exception of Shah
[85], all the models in this section used a single inventory cycle of
specified length T, which might be repeated over an infinite horizon, and had a single decision variable, T1, the length of the instock interval.
Dye [31] extended a model by Lee and Wu [51] that included a
power function for demand and two-parameter-Weibull distributed deterioration by replacing Lee and Wu’s assumption of full
backordering by partial backordering with b(s) having a fairly general distribution. Dye proved that there is a unique optimal solution if the first derivative condition based on T1 is satisfied.
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
Wu [129] assumed that b(s) has a rational form, deterioration
follows a two-parameter Weibull distribution, and demand follows
a general time-varying pattern that is continuous and positive. He
proved that the solution to the first derivative necessary condition
for the optimal average cost per period, which must be found using
software, makes the second partial positive and is, therefore,
optimal.
Singh et al. [101] assumed that b(s) has a rational form, demand
is given by a power function of t, and deterioration has the following pattern: From t = 0 to td (l in the paper) there is no deterioration; from t = td to T1, deterioration is at a rate of ht times the
inventory level. Finding T 1 requires solving a cubic equation.
Skouri et al. [102] assumed that b(s) satisfies b(s) + Tb0 (s) P 0
and that deterioration has a two-parameter Weibull distribution.
They assumed that demand follows a ramp-type function with
parameter l; i.e., the demand distribution is D(t) for t < l, and
D(l) for t P l, where D(t) is a positive, continuous function of t.
Their analysis is divided into two cases: (1) the cycle starts at the
beginning of the in-stock phase and ends at the end of the stockout
phase, and (2) the phases are reversed. For both cases they showed
how to determine the optimal length of the initial phase of the cycle and, from that, the optimal order quantity and cost.
Hung [47], examining the same basic scenario as Skouri et al.
[102] with a general deterioration function, proved that the optimal solution is independent of the type of demand function, which
led to a simpler solution procedure.
Unlike the previous papers in this section, Shah [85] assumed
that T, the length of the inventory cycle, and T1 are both decision
variables. She further assumed that b(s) has a rational form and demand is time-dependent, decreasing exponentially during the instock interval for 0 6 t 6 T1 and then remaining constant for
T1 6 t 6 T. She gives the first partial derivative equations to be
solved using software to find an optimal solution, but does not
prove that the cost function is convex.
4.2.3. Deterioration and stock-level-based demand
Padmanabhan and Vrat [65] developed models for no backlogging, full backlogging and partial backlogging. They assumed an
infinite planning horizon, demand determined by an increasing linear function of the inventory level at time t, and a constant deterioration percentage. For their partial backordering model, they
assumed, as they did in [64], that the backlogging rate decreases
with the size of the backlog. (Note: See our comment in Section
2.2 for a critique of this assumption.) Their objective was maximization of average profit per unit time. They developed a cost
expression for the opportunity cost of lost sales, but did not include
it in the profit expression and, therefore, ignored it in determining
the solution. The decision variables were T and T1, with the optimal
values being determined by numerically solving an equation that
includes exponential functions.
Dye et al. [37] noted that, while Chung et al. [25] developed necessary and sufficient conditions for the existence and uniqueness
of the optimal solutions for the no- and full-backlogging cases in
Padmanabhan and Vrat [65], they did not address the partial backlogging case. Retaining the rest of Padmanabhan and Vrat’s model,
they adjusted the profit function in [65] by including the lost sales
cost and proved there is a unique optimal solution if and only if a
specific condition based on the model parameters is satisfied.
Dye and Ouyang [35] also noted that Padmanabhan and Vrat
[65] had not included the opportunity cost of lost sales in their
model and criticized the approach used for determining the
amount backordered. They modified Padmanabhan and Vrat’s
model by including the lost sales costs and by using a rational form
for b(s) instead of using Padmanabhan and Vrat’s approach. They
proved that there is no optimal solution if an expression which
can be interpreted as the value of building inventory is non-nega-
189
tive and that there is a unique optimal solution if it is negative.
They developed an equation to be solved for T 1 , which must be
done numerically, and an equation for T⁄(T1), which has a closed
form.
Chang et al. [13] complemented Dye and Ouyang [35] by creating a model in which building up inventory is profitable and developed an algorithm for finding an optimal solution. Their model is
identical except: (1) they included an upper limit on the level of
inventory, and (2) their decision variables were T1 and T2 instead
of T1 and T. They established a closed form expression for T 1 if
building up inventory is profitable and, given that, necessary and
sufficient conditions for partial backordering to be optimal (i.e.,
for T 2 > 0) and an equation to be solved numerically for T 2 .
Wu et al.’s [131] model is identical to Dye and Ouyang’s [35] except that they assumed, like Singh et al. [101] (Section 4.2.2.3), that
there is a period (0 6 t 6 td) at the beginning of the in-stock phase
during which there is no deterioration. They proved that if a specific condition is met, then there is an equation which must be
solved numerically for T 1 , and if not, then T 1 ¼ t d . The same closed
form equation for T⁄(T1) is used in either case.
Chang et al. [14] extended Wu et al.’s [131] model by changing
the objective to maximizing the total profit and including a maximum inventory level. As in Chang et al. [13], they used T1 and T2 as
the decision variables and used a similar optimization procedure.
Yang et al. [140] modified Teng et al.’s [110] model (Section
4.2.2.2) by using stock-level dependent demand – instead of
time-varying demand – and inflation and using discounted total
profit as their objective. They allowed any b(s) with b0 (s) < 0, but
used the exponential form in their analysis. They proved there is
a unique optimal solution and provided a good estimate for the value of n.
Like Wu et al. [131] and Chang et al. [14], Jain et al. [48] assumed that there is a period (0 6 t 6 td, which they called l) at
the beginning of the in-stock phase during which there is no deterioration, after which deterioration is at a constant percentage rate.
In addition, they assumed that demand decreases linearly with the
inventory level up to time td, but then decreases at a faster linear
rate until all the inventory is gone. Their objective is to maximize
profit by determining values for T1 and T, which they do by using
the two partial derivative equations to create equations for T1
and T(T1). However, they do not prove concavity of their profit
function and, possibly more significant, they do not appear to recognize, as was done in Yang et al. [135] (Section 4.2.1) that the
profit function will differ depending on whether T1 is less than or
greater than td.
Uthayakumar and Geetha [116] also assumed that there is a
period of length td at the beginning of the in-stock phase during
which there is no deterioration, after which deterioration is at a
constant percentage rate. Unlike all the other papers in this section,
they assumed that there is a finite planning horizon of length H,
which is to be divided into m equal-length intervals, each beginning with the receipt of an order of size Q and ending with a backlog level of size Ib, which must be eliminated at the end of the
horizon. Demand during each interval is given by D(t) = a + bI(t),
with a > 0 and 0 6 b 6 1. The objective is to minimize the present
value of the total cost over the horizon by determining the optimal
values of m and T1, which they call k. Their solution procedure is to
try successively larger values of m, in each case finding the value of
T1 by solving the partial derivative of the cost with respect to T1,
stopping when increasing the value of m increases the cost. However, they do not prove that the cost is convex in m. Nor, as was the
case with Jain et al. [48], do they recognize that the cost function
will differ depending on whether T1 is less than or greater than td.
Yang et al. [136] assumed an infinite horizon with inventory
deterioration at a constant percentage rate and D(t) = a + bI(t), with
a and b both positive. They also assumed limited storage space, so
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the maximum inventory at the start of an inventory cycle is U (not
the same U as in Table 1). The relaxed terminal condition is that the
inventory at the end of a cycle does not have to be zero or negative
(backorders), but may be positive since the higher level of inventory during the cycle may result in sufficient extra demand and
sales to more than compensate for the higher carrying costs and
deterioration costs. The objective is to find the length of a cycle,
T, that will maximize the average profit per period when the starting inventory level is U (given) and the ending inventory level is L,
which is determined by the values of T and t1, the length of time it
would take all the inventory to be used up given U. They develop
an equation for t1 and divide the analysis into two cases: T 6 t1
(i.e., L > 0 and there is no backordering), and T P t1 (i.e., L < 0 and
there is backordering at a constant b), each case having its own
profit function. They prove, for each case, that there is a unique
optimal solution and show how to determine what T⁄ is for that
case.
4.2.4. Deterioration, a constant demand rate, and one other
consideration
Kalro and Gohil [50] (Section 4.1.4) extended the basic constant-b partial backordering model by including uncertainty about
the amount received. Warrier and Shah [120] extended this model
further by including deterioration of inventory at a constant percentage. Their decision variables were Q and S. They combined
the two receipt-uncertainty cases considered by Kalro and Gohil
by assuming that r2YjQ ¼ r20 þ r21 Q 2 , where Y is the amount actually
received. Q⁄ and S⁄ are found by simultaneously solving two cubic
equations. Although they said that the resulting Q⁄ and S⁄ are optimal, they did not prove it. They noted that the resulting model is
the same as the one in Kalro and Gohil if there is no deterioration.
Chang and Dye [17] combined deterioration with a twoparameter Weibull distribution and partial backordering with a
rational form of b(s) with a delay of up to M periods for the buyer
to pay for a delivery. Any revenues received up to t = M earn interest and delaying payment until after t = M incurs interest charges
at a higher rate than that earned. The decision variables were T1
and T, and models were developed for two cases: T1 > M and
T1 6 M. By ignoring the second-order and higher terms of the scale
parameter in the deterioration function, they developed a pair of
non-linear equations to be solved for T 1 and T⁄ for each case and
compared to determine the overall optimal solution.
Ouyang et al. [62] considered a variation on Chang and Dye’s
problem [17]. They assumed deterioration is at a constant percentage and b(s) may be any differential function for which b0 (s) < 0.
Ouyang et al. considered four cases, combining the two cases relating the length of T1 to M in Chang and Dye with the issue of how
the buyer pays for the order: (1) paying only for the goods sold
at the deadline, paying for the rest when they are all sold, or (2)
paying for the entire order, whether all sold or not, at the deadline.
Ouyang et al. used T2 and T as the decision variables and modeled
the time process starting with the beginning of a stockout phase of
length T2, followed by an in-stock phase of length T1, rather than
the reverse, as in Chang and Dye. Ouyang et al. proved that the cost
function for three of their four cases is pseudo-convex and is in the
fourth case if the Hessian condition is met. Their solutions for all
four cases involve search procedures.
4.2.5. Warehouse-based models
Dye et al. [36] combined deterioration and a rational form for
b(s) with two warehouses, one owned, with capacity W, and the
other rented, with no capacity limit. Each warehouse has its own
constant deterioration percentage. They assumed that the warehouses’ costs are such that the customer delivery policy is to ship
from the rented warehouse first. The decision variables were Tw,
the time the rental warehouse is empty, and T2, the length of the
shortage period at the end of an inventory cycle. They determined
a critical value W⁄ for the owned warehouse’s capacity and determined how to find the values of T w ; T 1 (the time at which all the
inventory is depleted) and T 2 if W P W⁄, in which case T w ¼ 0,
and if W < W⁄.
Wee et al. [127] added inflation to Dye et al.’s problem [36].
They assumed a constant b and deterioration with a two-parameter Weibull distribution. The decision variables were T1 = the time
with positive inventory at the rented warehouse (Note: there is an
error in the notation list, where this is designated as T2.), T2 = the
additional time with positive inventory at the owned warehouse,
and T3 = the length of the stockout phase. They approximated the
cost function, which included discounted cash flows, by using Taylor series expansions and dropping terms above the second order.
The solution is found by simultaneously solving the three partial
derivative equations, but they did not prove that the solution will
meet the sufficient condition for optimality.
Yang [139] addressed basically the same scenario as Wee et al.
[127] except that he assumed different constant deterioration rates
for the two warehouses and a general b(s) for partial backordering.
His objective was to compare two timing models, Model 1 starting
with at the beginning of an in-stock interval and ending with at the
end of a stockout interval and Model 2 the opposite. As in Dye et al.
[36] and Wee et al. [127], he assumed that the cost structure was
such that the inventory in the rented warehouse would be used
up before starting to use the inventory in the owned warehouse.
For both models the decision variables are the time at which the
rented warehouse becomes empty and the length of the inventory
cycle. For both models he proves that there is a uniquely determined optimal solution that is found by solving a pair of partial
derivative equations. His basic conclusion is that Model 2 is less
expensive to operate than Model 1 if the net discount rate is
positive.
Zhou [148] assumed there are several warehouses, each with a
capacity limit and its own holding cost per unit. One is owned and
has the lowest holding cost per unit. The others are rented and
each has a per-unit cost to ship to the owned warehouse. He assumed that the demand rate is a continuous function of time and
increases at a decreasing rate over an infinite time horizon. Shortages are partially backordered with a constant b. His approach was
to ignore the future cycles and determine the optimal solution for
one inventory cycle with length T and replenishment quantity Q,
determining how many and which of the rented warehouses to
use in addition to the owned warehouse to minimize the average
cost per period. The decision variables were T and T1. Zhou developed equations that satisfy the necessary and sufficient conditions
for an optimum solution and a procedure for finding it. After the
first cycle is completed, the problem is solved for a second cycle
with a new demand level, determining new values for T, T1, and
Q. This is repeated as far into the future as desired.
4.2.6. Other multiple-consideration models
Pal et al.’s [66] model optimized the average profit per period
over an infinite horizon for a problem with deterioration at a constant percentage, a rational form for b(s), limited storage space
(W), and incoming transportation cost that has a pseudo-step function form. The demand rate depends on the displayed stock level
(DSL) between specified limits S0 and S1, p, and the number of
advertisements per period. The decision variables were T1, T2,
the selling price, and the number of full vehicles needed to transport the order quantity Q. Pal et al. identified three basic scenarios
based on the relative sizes of W, S0, and S1. They divided these basic
scenarios into six sub-cases based on the size of the inventory
immediately after order delivery. After ruling out three of the
sub-cases due to the demand rate never depending on the DSL,
they proved the profit functions for the other three sub-cases are
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
pseudo-convex and developed a solution procedure that finds and
compares the optimal solutions for the sub-cases to determine the
overall optimum solution.
Wee [123] assumed deterioration at a constant percentage, a
constant b, and a constant price, p, over a finite horizon, which is divided into n equal-length intervals, all with the same value for r, the
percentage of time there is stock (r is equivalent to F in
[29,30,72,73] since they assumed that demand is at a constant rate).
The initial demand level is given by a decreasing linear function of p,
and demand decreases exponentially over time. The objective was
maximization of the total net profit, with n, r, and p the decision
variables. The solution procedure has the basic form used by most
of the papers discussed above that used a finite horizon.
Ouyang et al. [61] also developed a model with a finite horizon
divided into n equal-length intervals, each having the same value
of r, except for the last interval, for which r = 1. They assumed deterioration at a constant percentage, a demand rate that depends linearly on the level of the inventory, and, as in Padmanabhan and
Vrat [64,65], Chu et al. [24] and Dye et al. [33], a backordering rate
that decreases with the amount already backordered (see our comment on this assumption in Section 2.2). They assumed inflation at
constant internal and external rates. The decision variables are n
and r and the objective is minimization of the net present value
of cost. The solution procedure is to determine r⁄ for n = 1, 2, etc.,
until the cost for n + 1 exceeds the cost for n, thus determining
the presumed n⁄. They note, however, that the cost function is
too complex to prove convexity analytically, although an numerical evaluation can be done.
Wee [125] assumed an infinite horizon with demand at a constant rate that depends linearly on the price, p, a unit cost with
quantity discounts, deterioration according to a two-parameter
Weibull distribution, and backordering at a constant b. The objective was maximization of the average net profit per period, with p,
T, and T1 the decision variables. While p and T1 are both continuous, T is discrete. The solution procedure is similar to that in
Wee [123], but iterating over T: try a value for T, finding p⁄ and
T 1 for that T and each possible unit cost by simultaneously solving
a pair of non-linear equations. Increment T by 1 and repeat the process, stopping when the optimal profit decreases. While he claims
that this results in an optimal solution, Wee does not prove concavity with respect to T.
Papachristos and Skouri [69] generalized the model in Wee
[125] by allowing the demand rate to be determined by any nonnegative, continuous, convex, decreasing function of the price,
p 6 pmax, and using a rational form for b(s). As in Wee, the objective
was maximization of average net profit per period and the decision
variables were T, p, and T1. They proved limited results about the
optimality of the solution found by simultaneously solving the partial derivative equations and also discussed solving the problem by
using the Kuhn–Tucker conditions.
Chang et al. [18] added the pricing decision to the problem treated in Teng et al. [110] (Section 4.2.2.2). Demand, which is initially
a function of the price, decreases over a finite planning horizon
according to a log-concave function. The decision variables were
the same as in Teng et al. plus the price. Their solution procedure
is the same as it was for Teng et al. but it iterates over the price
as well as over the number of periods. This gives an overall optimal
solution if a stated condition on p is met.
Hsieh and Dye [44] further generalized the problem in Chang
et al. [18] by assuming a constant deterioration rate, a general form
for b(s), and demand that is a function of both time and price,
which is allowed to change at the start of each of the n inventory
cycles within the planning horizon H. As in most of the models
in Section 4.2.2.2, the decision variables were n, {ti} (the starting
times for the in-stock periods), and {si} (the starting times for the
stockout periods), but they also included {pi}, the prices for each
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of the inventory cycles. Their objective was to maximize the total
profit over H, which they achieve by using the Nelder–Mead algorithm, a direct search method for optimizing a function with multiple variables, iterating over the values of n. To avoid a brute force
enumeration over n, they developed an intuitively good starting
value based on the one in Teng et al. [110].
Ouyang et al. [63] started with the basic assumptions of the
model in Wu et al. [131] (Section 4.2.3) and added (1) quantity discounts and (2) a probability distribution for b, the ‘‘constant’’ backordering fraction, instead of b(s) having a rational form. The second
addition had no real impact since they assumed that E(b) is a constant and only used that mean in the objective function, which was
the expected cost. As in Wu et al., the decision variables were T and
T1. They developed theoretical results and a solution procedure
that are basically the same as the ones in Wu et al. except that they
recognize the different possible prices and minimum order quantities in the quantity discount statement.
Tsao [114] assumed a finite planning horizon divided into n
equal-length intervals, each having the same value for K (the percentage of time there is stock), except for the last interval, for
which K = 1 (K is the same as r in [123,61]). Deterioration follows
an exponential distribution. Demand, which is a decreasing function of time (as is generally the case with fashion goods), is also
a function of the different prices p1 and p2 < p1 during the in-stock
and stockout phases of the inventory cycles. Tsao also includes the
effects on revenue and profit of the extension by the supplier of
trade credit, the period of time after delivery to the seller before
payment to the supplier must be made, recognizing that prior to
the payment due date the seller can earn interest on sales and after
that date must pay interest on the money borrowed until the sales
are made. The objective is the maximization of profit by determining values for n, K, p1 and p2. Tsao breaks the analysis into three
cases, recognizing the possibilities that tc (the length of the trade
credit) is less than KT, between KT and T, or greater than T. His solution procedure is to determine, for successively larger values of n,
the optimal values of K, p1, and p2, if they exist, for each the three
cases relating tc to KT and T and then compare the optimal profits
for the three cases to determine the overall optimal solution. Tsao
also discusses the possibility of having a different value of K for
each of the n intervals, but notes that this is a much more complex
problem. (Note: One thing Tsao does not consider is the possibility
that, in response to the generally decreasing level of demand over
time, the set of prices could vary over the n intervals within the
horizon, whether K remains a constant or not.)
5. Deterministic EPQ–PBO models
The deterministic basic EPQ–PBO models satisfy all, or at least
most, of the basic assumptions of the basic EPQ model with full
backordering except that only a percentage of the demand when
the supplier is out of stock will be backordered. In Section 5.1 we
briefly summarize papers that address models for the basic deterministic EPQ–PBO model with constant b. In Section 5.2 we review
papers that introduce additional considerations, such as deteriorating or perishable inventory, demand that varies either with time or
the stock level, pricing decisions, or multiple items. (Note: Unless
defined where used, symbol definitions are given in Table 1.)
All papers reviewed in Section 5 assumed partial backordering
with a constant b unless stated otherwise. Descriptions of timebased forms forb(s) are in Section 2.1 and Table 2.
5.1. Basic deterministic EPQ–PBO models
All the models reviewed in this section are single-item models
that assumed all parameters are known and constant over an
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infinite time horizon and replenishment is at a finite rate P. All except Sharma and Sadiwala [97] made the usual EPQ model assumptions about costs.
Where the models differ is in their choice of variables and their
assumptions about backordering after production starts. Pentico
et al. [73] distinguished between what they called LIFO (last-in/
first-out) and FIFO (first-in/first-out) policies for filling demands
at the beginning of a production interval. Under a LIFO policy,
the incoming orders are filled first, with any excess production
(P D units per period) being used to fill the backorders. Thus, if
the maximum backorder level is B, the time required to eliminate
all the backorders is B/(P D). Under a FIFO policy, the new production is used to eliminate the existing backorders first and, since
the incoming orders will not be filled immediately, a fraction b of
them will be backordered and the time required to eliminate all
the backorders is B/(P bD).
The first paper to address the basic EPQ–PBO was Mak [58],
which used T and T2 as the decision variables and followed a LIFO
policy on filling demands. He proved that his cost function is convex and developed equations for T⁄ and T 2 along with an inequality
condition for b to establish that partial backordering is optimal.
Zeng [145] also assumed a LIFO policy on filling demands. She
initially used Q and S as the decision variables, but then changed
to T and S. (Note: Zeng used b instead of S and defined it as ‘‘the
number of backorders during a replenishment cycle’’, but it is clear
from her use of b in the model that it is really the maximum number of stockouts, including both backorders and lost sales.) Zeng
established a condition for b under which partial backordering is
feasible and showed that if the condition is met, then the cost function is convex and equations for T⁄ and S⁄(T⁄) can be developed.
Pentico et al. [73] assumed a FIFO policy on filling demands and
used, as they did in Pentico and Drake’s [72] model for the basic
EOQ–PBO (Section 3.1), T and F, the percentage of demand filled
from stock, as the decision variables. They proved the optimality
of their solution and developed equations for T⁄ and F⁄(T⁄) that
are identical to the equations in [72] if the unit holding and backordering costs are adjusted for continuous rather than instantaneous replenishment and the use of a FIFO order-filling policy.
They also developed a simple statement for the condition on b under which partial backordering is optimal.
Pentico et al. [74] extended their model in [73] to allow for the
possibility that the backordering rate might increase when the
backorders reach their maximum level and a new production run
starts, a structure that includes their original model and the models by Mak [58] and Zeng [145] as special cases. They showed that,
with appropriate redefinition of the unit holding and backordering
costs, the objective function for this problem has the same form as
the objective function for their original model in [73] and, therefore, the optimal solution has the same basic form.
Sharma and Sadiwala [97] considered a more extensive scenario; they included yield losses, transportation, and inspection
costs. Although they did recognize lost sales, they did not include
their cost in the objective function. They used a LIFO order-filling
policy. Their objective was to minimize average cost per period
using Q and S (which they called J) as the decision variables.
Obtaining a solution required solving a fourth-order equation for
Q⁄, after which S⁄ could be found from a simple expression. By
dropping the lost sales term in the objective function, which they
stated would have little effect on the cost, the value of Q⁄ could
be found from a square root expression.
The last paper in this section addressed an issue that was considered in Section 3.2 for the basic EOQ with partial backordering:
the danger in ignoring the alternative of not stocking an item at all.
Zhang [146], commenting specifically on Pentico et al. [73], but
with applicability also to both Mak [58] and Zeng [145], pointed
out by example that meeting the condition that shows that using
the EPQ–PBO is preferable to using the basic EPQ with no backordering does not guarantee that it is preferable to not stocking and
having all lost sales. He developed a two-part condition that guarantees that partial backordering is optimal.
5.2. Deterministic EPQ–PBO models with additional considerations
Several authors have extended the basic EPQ–PBO model to include one or more additional considerations, such as deteriorating
or perishable inventory, pricing, demand that varies with either
time or the stock level, or multiple items.
5.2.1. Deteriorating or perishable inventory
Sachan’s [77] model was a basic EPQ–PBO with deterioration at
a constant percentage and a LIFO policy on filling demands. His
decision variables were T and the maximum inventory level. His
objective was to maximize the average profit per period, which
was given by a simplified objective function that used approximations of the exponential terms. By examining the signs of the pieces
of a reconfiguration of his objective function, Sachan concluded
that shortages should not be allowed for 0 < b < 1. By examining
what happens to the objective function coefficients as the production rate approaches infinity, Sachan came to the same conclusion
for the EOQ–PBO with deterioration. (Note: A possible explanation
for this result, which is contrary to what was found by other
researchers, is that Sachan’s objective function does not account
for either the lost revenue on deteriorated inventory or the revenue
on backorders that are filled after production begins again.)
Wee [122] addressed the same problem as Sachan [77] except
that he used a policy on filling demands during the backorder elimination phase that is neither LIFO nor FIFO. While the rate at which
the backorders are eliminated is P D in LIFO and P bD in FIFO,
in Wee’s model it was b(P D). The demand filling policy that
would result in this form was not explained. Wee’s objective was
to minimize the average cost per period by determining optimal
values for T and the length of the cycle phase from the time that
production stops until the inventory level reaches 0. (Note: Wee
called this T2, but it is not the same as the definition of T2 from
Table 1 or as used in other models discussed here.) Recognizing
that his objective function may not be convex, Wee used an iterative approach in which, starting with b = 1, he checked for convexity and, if that condition is met, determined the optimal values of
his decision variables. By reducing b, he determined the minimum
value for b for which partial backordering is feasible and, within
that range, determined the optimal solution.
Abad [2] used the same assumptions as Sachan [77]. His objective was to minimize the average cost per period. Using T and T2 as
the decision variables, he proved that his objective function, which,
unlike those in Sachan [77] and Wee [122], did not use approximations of the exponential terms, is pseudo-convex. Thus there is a
unique optimizing solution, which can be found using standard
non-linear programming software.
5.2.2. Deterioration and time-varying demand
Yan and Cheng [132] presented a model in which the demand
rate, the production rate, and the deterioration rate are all functions of time, b is a constant, and a LIFO policy on filling demands
is followed. Although the overall problem has an infinite time horizon, they said it would be solved as a series of single-cycle problems. To further complicate their problem description, they
stated that each ‘‘cycle starts at a given inventory level (which
can be zero, positive, or negative), which is the ending inventory
of the last cycle; denote this by B; and B = 0 in the first cycle’’.
The objective was to determine the optimal stopping and restarting times for the production phases of a cycle that will minimize
the cost. Although Yan and Cheng developed some theoretical
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results about a possible solution for the problem, they acknowledged that even in the simplest case, in which the demand, production, and deterioration rates are all constant, the function to
be minimized is not convex and there may not be a solution.
Yan and Cheng’s lack of specificity about certain critical characteristics of their problem led to a dialog between Balkhi [8,9], Yan
and Cheng [133,134], and Balkhi et al. [10] about the correctness of
Yan and Cheng’s [132] model, particularly about whether the setup
cost should or should not be included in the objective function
(Balkhi – yes; Yan and Cheng – no) and whether Yan and Cheng left
part of the backordering cost out of their objective function, with
Balkhi [8] presenting his own version of the objective function
and a solution procedure. Balkhi et al. [10] noted a mistake in
one of Balkhi’s earlier comments, suggested two possible ways of
resolving some of the issues, and developed an optimization
approach.
Goyal and Giri [42] developed a model for the same scenario as
in Yan and Cheng [132] that corrected the problems that Balki [8]
identified. They developed models using two approaches that had
been proposed by other researchers for the same basic problem
with full backordering. Both approaches had cost minimization
as the objective, but they differed with respect to when a typical
cycle begins (one starts when the inventory first becomes positive
and ends when the backorder is completely eliminated; the second
starts when the backorder first begins and ends when the inventory is completely used up). Both used the same two decision variables (the times when production starts and ends) and
simultaneously solved the two partial derivative equations to obtain an optimal solution. Goyal and Giri then extended both modeling approaches to the case of a finite planning horizon and
developed near-opimal solution procedures for both models.
Skouri and Papachristos [105] commented on some of the problems Balki [8] identified in Yan and Cheng’s [132] model and developed their own model that assumed a rational form forb(s) rather
than a constant b. They noted that a condition that Yan and Cheng
had stated as sufficient for a minimum cost does not ensure this.
They developed their own conditions under which a minimizing
solution can be found.
Chen et al. [19] assumed a ramp-type demand function with a
linearly increasing rate until time l, production at a constant multiple of demand, deterioration at a rate linearly increasing with
time, a rational form for b(s), and a LIFO policy on filling demand
after production begins. They developed models for both the nostockout and partial backordering cases. The decision variables
for the PBO case were t1, the time at which production ends (which
they assume is greater than l), and t3, the time the backlog reaches
its maximum and the next production phase begins. The cost-minimizing solution for the PBO case is found by using software to
solve the two partial derivative equations, but they do not prove
that this meets the sufficient condition for an optimum, as they
do for the no-stockout model.
Zhou et al. [149] considered a system with a finite planning
horizon H, demand that varies over time in an unspecified fashion,
a constant deterioration percentage, and lost sales at a constant
percentage of the size of the backlog. (Note: This is the same
assumption about the probabilities of backordering and lost sales
that was used by Padmanabhan and Vrat [64,65], Chu et al. [24],
Dye et al. [33], and Ouyang et al. [61], discussed in Section 2.2.
The comment there about this approach to determining lost sales
is equally applicable during the initial part of the stockout phase
(i.e., before production starts) in Zhou et al. It is irrelevant after
production starts in Zhou et al. since they assumed a LIFO policy
for filling demands.)
Zhou et al.’s [149] objective was maximizing total profit, which
required determining n, the number of complete productioninventory cycles, and four times for each cycle: (1) the time back-
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orders reach their maximum and production starts, (2) the time
the backorders are eliminated and inventory begins accumulating,
(3) the time inventory reaches its maximum, and (4) the time the
cycle ends. Their approach was to find the optimum values of the
four times for a given value of n, searching over n, beginning with
what they identify as a good starting point. Their stopping rule assumes that the profit is concave in n, but they do not prove that.
5.2.3. Deterioration and stock-level-based demand
Wu and Liu [130] assumed the demand is a linear function of
the inventory level, deterioration at a constant percentage, and a
FIFO policy on filling demands. They assumed that the lost sales
cost is an amount/unit/unit time, rather than the usual assumption
of an amount/unit. The objective was to minimize the average cost
per period. The decision variables were the maximum inventory
and shortage levels. (Note: Wu and Liu stated that this was the
maximum shortage or backorder level, but it is clear that they
meant the total shortage.) Their solution approach was to simultaneously solve the two partial derivative equations, which will give
an optimal solution if the Hessian is positive definite at that point.
They do not discuss optimality if that condition is not met.
Jolai et al. [49] used the same demand rate and backordering
assumptions as Wu and Liu [130], but assumed deterioration
according to a Weibull distribution, a LIFO policy on filling demands, and inflation at a constant rate. Their objective was to minimize the discounted total costs over a finite planning horizon
divided into m identical inventory cycles. The decision variables
were m and the length of the cycle subinterval during which the
inventory decreases from its maximum level to zero and the stockout phase begins (which they called T2, although it is not the same
as the T2 defined in Table 1). After approximating all the exponentials in the objective function, they solved the problem by using the
first partial derivative to find T 2 for each value of m and identifying
the best of those solutions,
checking numerically to see that the
second partial at m ; T 2 is non-negative since they could not do
it analytically.
5.2.4. Deterioration and pricing
Abad’s [4] model for the EPQ with pricing, deterioration, and
partial backordering has, other than a finite replenishment rate,
most of the characteristics of his model for the EOQ in [3] (Section
4.2.1). As in [3,1], Abad did not include either backordering or lost
sales costs on the basis that ‘‘these costs are not easy to estimate in
practice’’. As noted in the discussion of [3], not including these
costs tends to bias the results toward higher stockouts. Instead of
using a constant b, he used a time-dependent b(s) to reflect the
idea that longer backorders are less desirable.
As in [3], Abad [4] maximized the average profit per period by
determining optimal values for p, T1, and T2. Also as in [3], he allowed demand to be determined by any fairly general function of
p. Where the two papers differed is in the deterioration function
and the allowable form of b(s). In [4], the deterioration percentage
can be almost any function of time, instead of a constant, as in [3],
and b(s) can be either exponential or rational, rather than only
exponential. The theoretical results and suggested solution procedure in [4] are identical to those in [3].
Teng et al. [111] did an analytic comparison of the net profit per
period between models based on those in Abad [4] and Goyal and
Giri [42] (Section 5.2.2). We say ‘‘based on’’ because the assumptions in those two papers are not identical, so Teng et al. made
the changes necessary to create models that were identical in all
respects except one: the sequence of the in-stock and stockout
phases, with the model based on Abad [4] having the in-stock
phase first and the model based on Goyal and Giri [42] having
the stockout phase first. Teng et al.’s objective was to determine
whether that makes any difference in the average profit per period.
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Their basic conclusions were: (1) If all the costs are constant, then
the maximum profits are the same for both sequences. (2) If one of
the costs is non-decreasing or non-increasing over time and all the
rest are constant, then one of the models gives a maximum profit
which is at least as large as the maximum profit from the other
model; which model is better depends on which cost is time
dependent and whether that cost is non-decreasing or nonincreasing.
5.2.5. Other single-item models
Yang et al. [141] developed a model for an integrated vendor–
buyer system in which the vendor produces on a continuous basis
for a period of time long enough to provide n shipments of size Q to
the single buyer. The vendor satisfies all of the buyer’s demand
with no backordering. Thus the vendor’s inventory position looks
similar to that for an EPQ model except that the demands on it
are in lots of size Q rather than continuous. For the buyer, the
inventory system is a basic EOQ with partial backordering at a constant b. The objective was to minimize the average combined costs
per period for the vendor and the buyer by determining values for
n, Q, and S, the maximum stockout level for the buyer (called B in
the paper). The solution procedure was to find, for successively larger values of n, the optimal values of Q and S for that n, stopping
when the cost increases if n is increased, which they state, but
do not prove, will give an overall optimal solution.
Lo et al. [56] also modeled an integrated vendor–buyer system,
but included additional features in the scenario: (1) The vendor’s
production process is imperfect, with the process starting in control and producing perfect output, but shifting to an out-of-control
state in which a fixed percentage of the output is defective, which
is repaired at a cost. (2) The raw materials and the finished goods
deteriorate over time according to Weibull distributions. (3) All
costs are subject to inflation at a constant rate. As in Yang et al.
[141], the vendor produces continuously for a long enough period
to make k (n in Yang et al.) shipments to the buyer, so the vendor’s
finished goods inventory position is similar to an EPQ with Weibull
deterioration and no stockouts. The vendor’s raw material inventory system is a basic EOQ with Weibull deterioration. For the
buyer, the inventory system is a basic EOQ with Weibull deterioration and partial backordering at a constant b. The objective was to
minimize the discounted sum of the total costs per period for the
vendor and the buyer. The decision variables were k, the length
of the non-production period per cycle for the vendor, and the
length of time the buyer is out of stock during each of its inventory
cycles. The solution procedure tries k = 1, 2, etc., determining the
optimal values for the two time variables, stopping when the discounted cost increases with an increase in k. They claim optimality,
but do not prove convexity of the cost function with respect to k.
Giri et al.’s [40] model included increasing demand over time
with the added feature that the production level could be reset
at the start of each cycle, the cost of change being a linear function
of the size of the increase. They assumed a LIFO policy on filling demands. Although they assumed an infinite horizon for their primary model, their solution procedure, as in Yan and Cheng [132],
was to solve a series of single-cycle problems. After the first cycle,
which starts with an inventory level of zero and production taking
place, cycle i begins at time Ti with the backlog at its maximum level. Production at a rate of Pi starts at that point and continues until
(1) the backlog is eliminated at time ti, (2) the inventory reaches its
maximum level, and (3) all the inventory is used up at time si.
(Note: This is unlike a standard EPQ model in which production
stops when the inventory reaches its maximum level.) At that
point backordering begins and the cycle finally ends when the
backlog again reaches its maximum level and cycle i + 1 begins.
The decision variables for cycle i were Pi, Ti, ti, and si, and the objective was to minimize the average cost per period for the cycle. Giri
et al. did not show how to solve the problem for a general increasing demand function, but they showed how to find an optimal
solution if demand increases linearly. They also showed how the
solution procedure could be adapted for a specified finite horizon.
Mirzazadeh [59] developed a model for the basic EPQ with b(s)
having an exponential form, deterioration at a constant percentage
rate, and a demand rate that is a function of the inflation rate. The
inflation rate is stochastic, and its probability distribution can
change after a specified number of periods during the planning
horizon of length H. Thus it can have one rate, determined by
one distribution, during the first few periods and a different rate,
determined by a possibly different distribution, during the last
few periods. The decision variables were n, the number of equal
length periods into which H is divided, and k and k1, which are
the lengths of time during the inventory cycles for which the
inventory level is positive prior to and following the change in
the distribution of the inflation rate. The solution procedure was
to use the two partial derivative functions to determine the values
of k⁄ and k1 for a given n, increment n, continuing until the minimum total expected discounted cost is found, although Mirzazadeh
does not state how that minimum is to be identified. (Note: The paper is difficult to read due to several typographical errors, including
using notation that differs from what is defined.)
A major area of interest currently is planning recoverable inventory, i.e., items that can be recovered from customers and processed for reuse. Teng et al. [109] developed a model for
planning the ‘‘new’’ purchase and subsequent reprocessing of such
items. They assumed the demand, recovery, and reprocessing rates
are constant over time and that demands during periods when
there is no new or reprocessed inventory available are partially
backordered at a constant b. They developed cost expressions to
be minimized for three basic cases: (1) the reprocessing rate
P > the demand rate D, (2) P < D, and (3) P = D. In all three cases,
they assumed that the recovery rate is less than D. Their decision
variables were R, the inventory level of recoverable items at which
the reprocessing starts, B, (which they designated by J), and m, the
number of recycling runs during the cycle time for a lot (Q) of new
items. Their solution procedure finds the optimal values of R and B
that minimize the average cost per period for m = 1, 2, etc., stopping when the cost for m + 1 is greater than the cost for m. Once
the values of m⁄, R⁄, and B⁄ are determined, T, the length of the purchase/reprocessing cycle, and Q⁄, the optimal amount of new items
to purchase, can be determined from simple equations. (Note:
Although they assumed that a new unit costs more than the cost
of reprocessing a recovered unit, they did not include either of
these costs in their objective function. Nor did they consider the
possibility of multiple purchases of new items during each recycling run, rather than the reverse.)
Sharma [93] developed models for the situations where (1) a
buyer faces an imminent price increase for an item or (2) has the
opportunity to take advantage of a temporary price reduction for
an item. He assumed: (1) the time frame within which the price
reduction applies or the time until the price increase takes effect
is short enough that only one special order can be placed, (2)
although the item is purchased, it is delivered at a finite rate rather
than all at once, (3) a fraction of the incoming units are unusable,
and (4) the same shortage cost per unit is incurred whether the
item is backordered or a lost sale. He developed expressions for
the potential cost savings during the period to be covered by the
special order and, using basic calculus, determined equations for
the optimal size of a special order for the two situations.
Sharma wrote a series of papers dealing with flexibility in EPQ
systems, some on single-item systems, which we review here,
and others on multiple-item systems, reviewed in Section 5.2.6.
Recognizing that the production rate in an EPQ system may be
flexible, Sharma [89] examined the effect of reducing P in order to
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reduce the average cost per period of its operation. By comparing
the average cost per year for an EPQ system with production rate
P against the cost with a lower production rate P1, he established
lower and upper bounds for P1 within which the cost would be
reduced. In [90], Sharma recognized that if the production rate P
is reduced, then that portion of the unit cost attributable to the
production process may also change. If so, the holding cost should
also be adjusted. To recognize this he modified his analysis in [89]
to include the change in the holding cost per unit. Sharma [92]
examined the same basic issue as Sharma [89] except that he considered the effects of increasing or decreasing the demand rate D
instead of the production rate P in an EPQ system. While his discussion recognized that a change in the demand rate will result in
changes in revenue as well as costs, he did not specifically incorporate that effect into his analysis but, instead, again focused on the
change in cost from increasing or decreasing D. In [89,90,92], Sharma’s primary analysis was for systems without backordering, but
he extended it to systems with partial backordering and FIFO order
filling.
Bhunia et al.’s [12] model assumed that all the costs except the
unit production cost are constant, inventory deteriorates at a rate
that changes linearly with time, and the b(s) has a rational form.
All other aspects of the model are decision variables (DV) or are
based on those variables. Demand is a function of A, the marketing
cost per unit (a DV), and the selling price, which is a function of the
mark-up rate (a DV) and the unit production cost, which is a
function of A and the production rate (a DV). The other decision
variables are T, the cycle length, the length of the production run
after the accumulated backorders are eliminated, and the maximum backorder level. They used a FIFO policy on filling backorders,
and the objective was maximization of profit. Since the profit function is neither concave nor pseudo-concave, they used a heuristic
search procedure that two of the authors had previously developed
called a tournament genetic algorithm.
5.2.6. Multi-item models
All the models discussed up to this point have considered only
one item. In this section we review models that simultaneously
determine policies for multiple items.
Drake et al. [30] developed a model for a two-level system that
has one final product with multiple components. The final product
is planned by an EPQ–PBO model; each component is planned by a
basic EPQ or EOQ model with no backordering. The objective was
to minimize the average cost per period and, as in their other work
[29,72,73], they used T and F for the final product as two of the
decision variables and assumed a FIFO policy on filling backorders.
The other decision variables were the number of times (Ni, an integer) that each component should be produced during each cycle of
the final product. They proved that their solution is optimal if the
integer requirement is ignored and, using a test set that varied six
scenario factors, showed that a procedure for integerizing the Ni
values gave solutions that were very close to the lower bounds
found by using the continuous Nis.
Sharma [96] developed a model for the same basic scenario as
Drake et al. [30], but with the following differences: (1) he allowed
yield losses in producing the final product; (2) he assumed a LIFO,
rather than a FIFO policy on filling backorders (see Section 5.1 for a
discussion of the difference between the LIFO and FIFO policies);
(3) he used the same cost per unit per unit time for both backorders and lost sales; (4) he did not recognize the timing implications
of the difference between backorders and lost sales when determining the equation for the fraction of time during a cycle for
which shortages exist; and (5) he did not require Ni (Ki in his notation) to be an integer.
If the same production facility (machine or line) is used to produce a set of products and can only produce one product at a time,
195
then the planning problem is to determine an optimal cycle time
(and, possibly, sequence) during which all the products will be produced. (Note: It is also possible that not every product will be produced in every cycle and that some products may be produced
more than once during a cycle. None of the papers reviewed here
addressed these variations on the basic problem.) A number of
authors have considered the cycle-determination problem without
backordering or with full backordering.
Taleizadeh et al. [107] developed a model for scheduling the
production of n items in a fixed cycle on one machine. In addition
to making the usual EPQ assumptions, they assumed a LIFO policy
on filling backorders, a stochastic scrap rate with no repair of the
defective units, and an overall service level constraint. The n + 1
decision variables were T, the length of the production cycle, and
Sj, j = 1, . . . , n, the maximum total shortage level for each item
(which they called Bj). The objective was the minimization of total
expected cost per period, which they proved is a convex function,
subject to two constraints: (1) the maximum capacity of the machine is not exceeded (i.e., the setups for and production of all
the items can be completed within the cycle time), and (2) the
minimum combined service level is met. Assuming that the resulting values for T and the Sjs meet the constraints, the optimal solution is obtained by simultaneously solving the n + 1 equations
found by setting the partial derivatives equal to 0, which gives a
square root formula for T⁄ and linear equations of T⁄ for the Sj s.
If the constraints are not met, they provide a procedure for adjusting the solution values to meet the constraints.
Taleizadeh et al. [108] extended the model in [107] to include
repairing the scrapped units from the initial production phase.
The repair phase uses the same machine and is subject to its
own failure rate. As in [107] they prove convexity of the cost function and provide a solution procedure for determining the optimal
values of T and the Sjs if the capacity and combined service level
constraints are met and an alternate solution procedure if either
of those constraints is not met.
One alternative to assuming that an item will be usable for an
unlimited time is to assume that units deteriorate over time. Several single-item models that made this assumption are included in
the preceding subsections, as well as in models in Section 4. A different approach is to assume that units all have the same limited
shelf-life and that the unit is unusable when that limit is reached.
Sharma [87] developed a model for the multi-product cycle
time determination problem with two added features: (1) partial
backordering, and (2) limited shelf-lives. Drawing on previous
researchers’ work, Sharma considered three options for determining an optimal (minimum cost) cycle: (1) reduce the production
rate for some or all of the products, (2) reduce the cycle time, or
(3) reduce both. Sharma made the usual cost assumptions except
that he assumed the same cost/period/unit for both backorders
and lost sales. Although Sharma stated that inventory is consumed
on a FIFO basis, this really related to his including a shelf-life constraint. For the time to eliminate backorders, his model implied a
LIFO order-filling policy. Sharma developed an expression for the
average cost per period and an equation for T⁄ if the shelf-life
restrictions are ignored. He then showed how to modify the cost
expression and solution for each of the three options to satisfy
the shelf-life constraints.
In [90] Sharma recognized that if the production rate P is reduced, then the portion of the holding cost that depends on the
unit’s production time should also be adjusted. His primary analysis was for single-item systems, but he also included a section on
incorporating this change into models for multiple-item systems
with no, full, or partial backordering. In [94] Sharma reformulated
his models from [87] to include the change in the cost to hold a
unit in inventory if the production rate is decreased and redid
the analysis.
196
D.W. Pentico, M.J. Drake / European Journal of Operational Research 214 (2011) 179–198
In [91] Sharma discussed the basic concepts of his ‘‘theory of exchange’’, which suggests that when several items are produced
sequentially in a common cycle time on a given facility, the systems’ productivity may be improved or its operating cost reduced
by exchanging, for a pair of items, a parameter value of one for the
comparable parameter value of another.
Sharma [88] first explored this idea by exchanging the production rates of two items. He first developed a model for full backordering, using T, the common cycle time, and the maximum
stockout levels of the items, which he called Jis. For this model T⁄
is found using a square root formula similar to the one for the basic
EPQ with full backordering and the J i s are found as simple multiples of T⁄. His heuristic for determining which production rates to
switch was to try each pair and see which reduces the overall cost
the most. He then showed how to adjust the cost function and
solution for partial backordering. Sharma [91] extended this idea
to consider the possibility of switching the values of any one of a
set of parameters: production rates, demand rates, inventory carrying costs, or shortage costs, using the same heuristic as he did in
[88]. Sharma [95] extended the concept of pair-wise switching of
the values of one parameter discussed in [91] to the idea of simultaneously switching the values of two or three of the parameters
for an item, using a simple extension of the heuristic in [91,88]:
try all combinations of changes to see which reduces the cost the
most. In [88,91,95], Sharma first developed the model for full backordering and then showed how to adjust the cost function and
solution for partial backordering.
6. Conclusion
Deterministic models for the basic EOQ or EPQ with partial
backordering make all or most of the assumptions of the basic
EOQ or EPQ with full backordering except that only a fraction of
the demand during the stockout interval is backordered. We have
summarized most of the papers on deterministic models for the
basic problems and for more complex scenarios that add additional
considerations, such as pricing, perishable or deteriorating
inventory, time-varying or stock-dependent demand, quantity discounts, or multiple-warehouses. Stochastic models will be covered
in a second survey.
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