A SURVEY ON INVENTORY MODELS UNDER TRADE CREDIT
Mingfang Li
Guanzhuang Campus, University of Science and Technology Beijing, 100024, Beijing, China
Keywords: Inventory models, Trade credit, Permissible delay in payments, EOQ, EPQ.
Abstract: This paper presents a review of the advances of inventory literature under permissible delay in payments
since 1990s. The models available in the relevant literature have been suitable classified by the objective
function that focuses only on the supplier’s or buyer’s performance or an integrated vendor-buyer inventory
system. The motivations, extensions and generalizations of various models have been discussed. Finally, we
conclude with suggestions for future research.
1 INTRODUCTION
The traditional economic order quantity (EOQ)
model assumes that the retailer’s capital are
unrestricting and must be paid for the items as soon
as the items were received. This is not always true in
the actual business world. In practice, the supplier
will offer the retailer a delay period, that is the trade
credit period, in paying for the amount of purchasing
cost. The supplier often makes use of this policy to
promote his sale and attract new purchasers who
consider it to be a type of price reduction. Before the
end of trade credit period, the retailer can sell the
goods and accumulate revenue and earn interest. A
higher interest is charged if the payment is not
settled by the end of the trade credit period.
Inventory problems for permissible delay in
payments have been studied extensively by many
researchers from time to time. Inventory problems
for permissible delay in payments can be broadly
classified into two categories: (1) The models focus
only on the supplier’s or buyer’s performance under
trade credit. (2) The models focus on the integrated
vendor-buyer inventory system under trade credit.
2 THE STUDIES FOCUSED ONLY
ON THE SUPPLIER’S OR
BUYER’S PERFORMANCE
UNDER TRADE CREDIT
The inventory models available in the relevant
literature can be classified into the following two
categories according to the replenishment rate that is
infinite or not.
(1) Optimal retailer’s replenishment decisions in the
EOQ model under trade credit.
(2) Optimal retailer’s replenishment decisions in the
EPQ model under trade credit.
2.1 The EOQ Model under Trade
Credit
Goyal (1985) was the first to establish an EOQ
model with a constant demand rate under the
condition of permissible delay in payments. Then a
number of researchers variously extended the model.
Chu et al. (1998) extended Goyal (1985) to the case
of deterioration. Aggarwal et al. (1995) considered
the inventory model with an exponential
deterioration rate under the condition of permissible
delay in payments. Jamal et al. (1997) and Chang et
al. (2001) extended this issue with allowable
shortage. Sarker et al. (2000) addressed the optimal
payment time under permissible delay in payment
with deterioration. Hwang et al. (1997)developed the
optimal pricing and lot sizing for the retailer under
the permissible delay in payments for an order of a
product whose demand rate is represented by a
constant price elasticity function. Jamal et al. (2000)
developed a retailer’s model for optimal cycle and
payment times under permitted delay in payment by
the wholesaler. Teng et al. (2005) then amended
Goyal (1985) by considering the difference between
unit price and unit cost, and found that it makes
economics sense for a well-established retailer to
order less quantity and take the benefits of payment
511
Li M..
A SURVEY ON INVENTORY MODELS UNDER TRADE CREDIT.
DOI: 10.5220/0003581905110516
In Proceedings of the 13th International Conference on Enterprise Information Systems (DMLSC-2011), pages 511-516
ISBN: 978-989-8425-55-3
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

delay more frequently.
Then many scholars extended the EOQ models
under permissible delay in payments which can be
classified broadly into the following three
categories.
2.1.1 Inventory Models under Trade Credit
and Cash-discount Policy
The supplier offers a specified period without
interest charge to the buyer that is to be paid off
within a permissible delay period. As a result, with
on incentive for making early payments, and earning
interest through the accumulated revenue received
during the credit period, the buyer postpones
payment up to the last moment of the permissible
period allowed by the supplier. Therefore, from the
supplier’s perspective, offering trade credit leads to
delayed cash in flow and increase the risk of cash
flow shortage and bad debt. To accelerate cash
inflow and reduce the risk of a cash crisis and bad
debt, the supplier may provide a cash discount to
encourage the buyer to pay for goods quickly.
Many studies develop the inventory models
under the supplier’s trade credit policy and cash-
discount policy. Huang et al. (2005)
investigated the
buyer’s optimal cycle time and optimal payment
time under the supplier’s trade credit policy and
cash-discount policy. Ouyang et al. (2005)
developed an inventory model with non-
instantaneous receipt under trade credit, in which the
supplier provided not only a permissible delay
payment but also a cash discount to the retailer.
Recently, Chang (2002), Ouyang
(2002) and Huang
(2005) developed inventory models in which the
supplier provides a permissible delay payment and a
cash discount for early payment.
2.1.2 Inventory Models under Trade Credit
Linked to Order Quantity
All previous models implicitly assumed that credit
terms are independent of the order quantity. In order
to stimulate the buyer’s demand, the suppliers offer
the retailer the trade credit period depending on the
ordering quantity.
Shinn et al. (2003) determined the retailer’s
optimal price and order size simultaneously under
the condition of order-size-dependent delay in
payments. They assumed that the length of the credit
period is a function of the retailer’s order size, and
also the demand rate is a function of the selling
price. Chang et al. (2003) then established an EOQ
model for deteriorating items under supplier trade
credits linked to order quantity. Chung et al. (2004)
dealt with the problem of determining the economic
order quantity for exponentially deteriorating items
under permissible delay in payments depending on
the ordering quantity and developed an efficient
solution-finding procedure to determine the retailer’s
optimal ordering policy. Chang (2004)
extended
Chung et al. (2004) by taking into account inflation
and finite time horizon. Ouyang et al.
(2008)
presented an integrated inventory model with
variable production rate and price-sensitive demand
rate.
However, all above published papers dealing
with economic order quantity in the presence of
permissible delay in payments assumed that the
supplier only offers the retailer fully permissible
delay in payment if the retailer ordered a sufficient
quantity. Otherwise, permissible delay in payments
would not be permitted. We know that this policy of
the supplier to stimulate the demands from the
retailer is very practical. But this is just an extreme
case. That is, the retailer would obtain 100%
permissible delay in payment if the retailer ordered a
large enough quantity. Otherwise, 0% permissible
delay in payments would happy.
In reality, the supplier can relax this extreme case
to offer the retailer partially permissible delay in
payments rather than 0% permissible delay in
payments when the other quantity is smaller than a
predetermined quantity. That is, the retailer must
make a partial payment to the supplier when the
order is received to enjoy some portion of the trade
credit. Then, the retailer must pay off the remaining
balances at the end of the permissible delay period.
Huang (2007) established an EOQ model in
which the supplier offers a partially permissible
delay in payments when the order quantity is smaller
than the predetermined quantity. Ouyang et al.
(2009)
extended Huang (2007) by taking into
account the deteriorating items.
In all the above models, however, the effects of
the optimal replenishment decisions under two
levels of trade credit policy linked to ordering
quantity are not taken into consideration. Kreng et
al. (2010) established an inventory model to
determine the optimal replenishment decisions under
two levels of trade credit policy if the purchaser’s
order quantity is greater than or equal to a
predetermined quantity. The major assumptions used
in the above research articles are summarized in
Table 1.
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512

Table 1: Major characteristics of inventory models on selected researches.
References
Allowing for
deterioration
Payment linked
to order
quantity
Allowing for
partial
payments
Constant
demand rate
Two levels of
trade credits
Taking into
account
inflation
Shinn et al. (2003) No Yes No No No No
Chang et al. (2003) Yes Yes No No No No
Chung et al. (2004) Yes Yes No Yes No No
Chang (2004) Yes Yes No Yes No Yes
Ouyang et al. (2008) No Yes No No No No
Huang (2007) No Yes Yes Yes No No
Ouyang et al. (2009) Yes Yes Yes Yes No No
Kreng et al. (2010) No Yes No Yes Yes No
2.1.3 Inventory Models under Two-level
Trade Credit
Almost all of the above-models under trade credit
assumed that the supplier would offer the retailer
trade credit but the retailer would not offer the trade
credit to his/her customer. That is one level of trade
credit. Huang (2003) modified the assumption to
assume that the retailer will adopt the trade credit
policy to stimulate his/her customer demand to
develop the retailer’s replenishment model. That is
two levels of trade credit. This new viewpoint is
more matched real-life situations in the supply chain
model. Several researchers have studied in this area.
Huang (2006)
incorporated Huang (2003) and Teng
(2002) by two levels of trade credit. Chung et al.
(2007) modified Huang (2003) by developing a two-
warehouse inventory model for deteriorating items
under permissible delay in payments.
Liao (2008)
developed an EOQ model with non-
instantaneous receipt and exponentially deteriorating
items under two-level trade credit. Chang et al.
(2010) developed Liao (2008)
by relaxing some
dispensable assumptions that the retailer received
the supplier trade credit and provided the customer
trade credit. Huang et al. (2008) extended EOQ
model to study the situation under which the retailer
has the powerful decision-making right. That is, the
retailer can obtain the full trade credit offered by the
supplier and the retailer just offers the partial trade
credit to his/her customer. There are several
interesting and relevant papers related to two-level
trade credit such as Teng (2007), Huang (2007),
Teng (2009), Thangam (2009) and Min (2010).
2.2 The EPQ Model under Trade
Credit
Almost all of the above-mentioned models assume
that replenishment is done instantaneously. In real
life cases, however, the ideal case is not quite
applicable. Inventories are often replenished
periodically at certain production rate which is
seldom infinite.
Chung et al. (2003) extended Goyal (1985) to the
case that all items are replenished at a finite rate
under permissible delay in payment. When the
replenishment rate approaches to infinite, Goyal
(1985) will be a special case of Chung et al.
(2003).
But the model in Chung et al. (2003) assumed that
the unit selling price and the unit purchasing price
are equal. Huang (2004) assumed that the selling
price was not equal to the purchasing price to
modify Chung et al. (2003). Liao (2008) extended
Huang (2004)
to the case of deterioration. Hu et al.
(2009) extended Chung (2009) to allow for
shortages. Huang (2007) investigated the optimal
retailer’s replenishment decisions under two levels
of trade credit policy within the economic
production quantity(EPQ) framework. In Teng et al.
(2009) the authors studied the inventory model in
which the permissible delay period offered by the
retailer is independent of the permissible delay
period offered by the supplier to the retailer. Hence,
it extended the EPQ model to complement the
shortcoming of Huang (2007). There are several
interesting and relevant papers related to two-level
trade credit such as Huang et al. (2008)
and Ma et al.
(2009).
3 COLLABORATIVE
INVENTORY SYSTEM WITH
PERMISSIBLE DELAY IN
PAYMENTS
Almost all of the above-mentioned models studied
the optimal policy for the buyer or the vendor only.
However, these one-sided optimal inventory models
A SURVEY ON INVENTORY MODELS UNDER TRADE CREDIT
513

neglected the complicated interaction and
cooperation opportunity between the buyer and the
vendor. In practice, many companies learn that
actions taken by one member of the chain can
influence the success of all others in the same value
chain. Recognizing this principle, the vendor and
buyer may consider how to relieve the conflict
relationship and attempt to become partners to create
a win–win strategy.
Goyal (1976)
first developed a single vendor–
single buyer integrated inventory model.
Subsequently, Banerjee (1986) extended Goyal
(1976) and assumed that the vendor followed a lot-
for-lot shipment policy with respect to a buyer.
Goyal (1988) relaxed the lot-for-lot policy and
illustrated that the inventory cost can be reduced
significantly if the vendor’s economic production
quantity is an integer multiple of the buyer’s
purchase quantity.
Lately, some researchers discussed the impact of
delay payment strategy on the integrated inventory
models. Abad and Jaggi (2003) provided a seller–
buyer integrated inventory model under trade credit
and followed a lot-for-lot shipment policy. Jaber et
al. (2006) proposed a supplier–retailer supply chain
model in which the permissible delay in payments is
considered as a decision variable. Yang et al. (2006)
developed a vendor–buyer integrated inventory
model for deteriorating items with permissible delay
in payment. Ho et al. (2008) investigated the
production and ordering policy under a two-part
trade credit in an integrated supplier–buyer
inventory model. Chang et al. (2010) presented a
stylized model to determine the optimal strategy for
an integrated vendor-buyer inventory system under
the condition of trade credit linked to the order
quantity, where the demand rate is considered to be
a decreasing function of the retail price. Ouyang et
al. (2008) gave a best policy that aimed maximizing
the joint total profit while the trade credit and freight
rate are simultaneously linked to the order quantity.
There are several interesting and relevant papers
related to two-level trade credit such as Chen et al.
(2007), Luo et al. (2007), Yang et al. (2006), Sheen
et al. (2007), Sarmaha et al. (2008), Huang (2010)
and Chen et al. (2010) and so on.
4 CONCLUSIONS
In this paper we have provided an up-to-date review
inventory literature under permissible delay in
payments. A future study will incorporate more
realistic assumptions in the proposed models. The
proposed models can be extended in several ways.
First, demand can be extended to a more generalized
demand pattern that fluctuates with time, price or
stock-dependent demand rate. Second, models can
be extended to corporate some more realistic
features, such as quantity discount, the inventory
holding cost and others are also fluctuating with
time. Third, integrated models can be extended to
more general supply chain networks, for example,
multi-echelon or assembly supply chains.
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