real-life situations in the supply chain model. For example, the Toyota Company can require his
supplier offered the trade credit to him and he must also offer the trade credit to his dealership.
Then, the Toyota Company can offer shorter delay period to his dealership than his supplier offered
to him. In this transaction, the Toyota Company can obtain maximum advantages. But
Huang [21] implicitly assumed that the unit selling price and the purchasing price per unit of
the retailer are equal. However, as we know, the unit selling price for the retailer is usually significantly
higher than the purchasing price per unit in order to obtain profit. Consequently, the viewpoint
of Huang [21] can be extended.
Such trade credit policy is one kind of encouragement for the retailer to order large quantities
because a delay of payments indirectly reduces inventory cost. Hence, the retailer may purchase
more goods than that can be stored in his/her own warehouse (OW). These excess quantities are
stored in a rented warehouse (RW). The proposed model is applicable for the business of small
and medium sized retailers since their storage capacity are small and limited. Especially, Taiwan
has traditionally relied on its small and medium sized firms to compete in international markets
since the 1970s. Therefore, this proposed model is more applicable for the special industrial environment
in Taiwan. In general, the inventory holding charges in RW are higher than those in OW.
When the demand occurs, it first is replenished from the RW which storages those exceeding
items. This is done to reduce the inventory costs. It is further assumed that the transportation
costs between warehouses are negligible. Several researchers have studied in this area such as
Benkherouf [22], Bhunia and Maiti [23], Goswami and Chaudhuri [24], Pakkala and Achary
[25], Sarma [26] and Wu [27].
Therefore, this paper tries to incorporate both Huangs model [21] and Tengs model [1] by
considering the retailers storage space limited. That is, we want to investigate the retailers
inventory policy under two levels of trade credit and limited storage space. Furthermore, we
adopt Tengs viewpoint [1] that the retailers unit selling price and the purchasing price per unit
are not necessarily equal. In addition, we try to use the more easily algebraic approach to find
the optimal solution in this paper. In recent papers, Ca
´
rdenas-Barro
´
n [28] and Grubbstro
¨
m
and Erdem [29] showed that the formulae for the EOQ and EPQ with backlogging could be
derived without differential calculus. Yang and Wee [30] developed algebraically the optimal
replenishment policy of the integrated vendor–buyer inventory system without using differential
calculus. Wu and Ouyang [31] modify Yang and Wee [30] to allow shortages using algebraic
method.
Consequently, this paper deals with the retailers inventory replenishment problem under
two levels of trade credit and limited storage space derived without derivatives. In addition, we
develop the easy-to-use procedures to efficiently find the optimal cycle time for the retailer under
minimizing annual total relevant cost. Finally, a numerical example is given to illustrate these
results and managerial insights are drawn.
2. Model formulation
In this section, we want to develop the retailers inventory model under two levels of trade credit
and limited storage space. For convenience, most notation and assumptions similar to Huang [21]
will be used in this paper.
real-life situations in the supply chain model. For example, the Toyota Company can require hissupplier offered the trade credit to him and he must also offer the trade credit to his dealership.Then, the Toyota Company can offer shorter delay period to his dealership than his supplier offeredto him. In this transaction, the Toyota Company can obtain maximum advantages. ButHuang [21] implicitly assumed that the unit selling price and the purchasing price per unit ofthe retailer are equal. However, as we know, the unit selling price for the retailer is usually significantlyhigher than the purchasing price per unit in order to obtain profit. Consequently, the viewpointof Huang [21] can be extended.Such trade credit policy is one kind of encouragement for the retailer to order large quantitiesbecause a delay of payments indirectly reduces inventory cost. Hence, the retailer may purchasemore goods than that can be stored in his/her own warehouse (OW). These excess quantities arestored in a rented warehouse (RW). The proposed model is applicable for the business of smalland medium sized retailers since their storage capacity are small and limited. Especially, Taiwanhas traditionally relied on its small and medium sized firms to compete in international marketssince the 1970s. Therefore, this proposed model is more applicable for the special industrial environmentin Taiwan. In general, the inventory holding charges in RW are higher than those in OW.When the demand occurs, it first is replenished from the RW which storages those exceedingitems. This is done to reduce the inventory costs. It is further assumed that the transportationcosts between warehouses are negligible. Several researchers have studied in this area such asBenkherouf [22], Bhunia and Maiti [23], Goswami and Chaudhuri [24], Pakkala and Achary[25], Sarma [26] and Wu [27].Therefore, this paper tries to incorporate both Huangs model [21] and Tengs model [1] byconsidering the retailers storage space limited. That is, we want to investigate the retailersinventory policy under two levels of trade credit and limited storage space. Furthermore, weadopt Tengs viewpoint [1] that the retailers unit selling price and the purchasing price per unitare not necessarily equal. In addition, we try to use the more easily algebraic approach to findthe optimal solution in this paper. In recent papers, Ca´rdenas-Barro´n [28] and Grubbstro¨mand Erdem [29] showed that the formulae for the EOQ and EPQ with backlogging could bederived without differential calculus. Yang and Wee [30] developed algebraically the optimalreplenishment policy of the integrated vendor–buyer inventory system without using differentialcalculus. Wu and Ouyang [31] modify Yang and Wee [30] to allow shortages using algebraicmethod.Consequently, this paper deals with the retailers inventory replenishment problem undertwo levels of trade credit and limited storage space derived without derivatives. In addition, wedevelop the easy-to-use procedures to efficiently find the optimal cycle time for the retailer underminimizing annual total relevant cost. Finally, a numerical example is given to illustrate theseresults and managerial insights are drawn.2. Model formulationIn this section, we want to develop the retailers inventory model under two levels of trade creditand limited storage space. For convenience, most notation and assumptions similar to Huang [21]will be used in this paper.
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