Optimal Economic Ordering Policy with Trade Credit and Discount Cash-Flow Approach

In this paper, an inventory model for deteriorating items under two levels of trade credit will be established. The trade credit policy depends on the retailer’s order quantity. When the retailer’s order quantity is greater than or equal to a predetermined quantity, both of the supplier and the retailer are taking trade credit policy; otherwise, the delay in payments is not permitted. Since the same cash amount has different values at different points of time, the discount cash-flow (DCF) is used to analysis the inventory model. The purpose of this paper is to find an optimal ordering policy to minimizing the present value of all future cash-flows cost by using DCF approach. The method to determine the optimal ordering policy efficiently is presented. Some numerical examples are provided to demonstrate the model and sensitivity of some important parameters are illustrated the optimal solutions.


Introduction
In the traditional economic order quantity models assumed the purchaser must pay for the items as soon as the items received.However, in real markets, to stimulate retailer's ordering qualities the supplier allows a certain fixed permissible delay in payment to settle the amount.Similarly, a retailer may offer his/her customers a permissible delay period to settle the outstanding balance when he/she received a trade credit by the supplier, which is a two-level trade credit.Huang [1] was the first to explore an EOQ model under the two-level trade credit.Kreng and Tan [2] and Ouyang et al [3] proposed to determine the optimal replenishment decisions if the purchasers order quantity is greater than or equal to a predetermined quantity.Teng et al [4] extended the constant demand to a linear non-decreasing demand function of time and incorporate supplier offers a permissible delay linked to order quantity under two levels of trade credit.Teng et al [5] established an EOQ with trade credit financing for a linear nondecreasing demand function of time.Paulus [6] shed light on how search strategy can be used to gain the maximum benefit of information search activities.Feng et al [7] investigated the retailer's optimal cycle time and optimal payment time under the supplier's cash discount and trade credit policy within the EPQ framework.Wang et al [8] established an economic order quantity model for deteriorating items with maximum lifetime and credit period increasing demand and default risk.Liao [9] developed an inventory model by considering two levels of trade credit, limited storage capacity.Wu et al [10] discussed an economic order quantity model under two levels trade credit, and assumed deteriorating items have their expiration dates.Enda et al [11] presented a generic solution to the sensitive issue of PCI Compliance.Teng et al [12] proposed an EPQ model from the seller's prospective to determine his/her optimal trade credit period, and in his paper production cost declined and obeyed a learning curve phenomenon.
However, the above inventory models did not consider the effects of the time value of money.In fact, as the value of money changes with time, it is necessary to take the effect of the time value of money on the inventory policy into consideration.Chang et al [13] investigated the DCF approach to establish an inventory model for deteriorating items with trade credit based on the order quantity.Chung and Liao [14] adopted the DCF approach to discuss the effect of trade credit depending on the ordering quantity.Liao and Huang [15] extended the inventory model to consider the factors of two levels of trade credit, deterioration and time discounting.
In this paper, we develop an inventory system for deteriorating items.Firstly, the items start deteriorating from the moment they are put into inventory.Secondly, if the retailer's order quantity is greater than or equal to a predetermined quantity, both of the supplier and the retailer are taking trade credit policy; otherwise, the delay in payments is not permitted.Thirdly, the present value of all future cash-flows cost instead of the average cost.The theorems are developed to efficiently determine the optimal cycle time and the present value of the total cost for the retailer.Finally, numerical examples and sensitive analysis of major parameters are given to illustrate the theoretical result obtain some managerial insight.

Notations
The following notations are used throughout this paper.
A the ordering cost one order; ( ) PV T  the present value of all future cash-flow cost.

Assumptions
The assumptions in this paper are as follows: (1) Time horizon is infinite, and the lead time is negligible; replenishment are instantaneous, and shortage is not allowed; (2) A constant  ( 0 1    ) fraction of the on-hand inventory deteriorates per unit of time and there is no repair or replacement of the deteriorated inventory; (3) If Q W  , both the fixed trade credit period M offered by the supplier and the trade credit N offered by the retailer are permitted.Otherwise, the delay in payments is not permitted.The retailer can accumulate revenue and earn interest after his/her customer pays for the amount of purchasing cost until the end of the trade credit period offered by the supplier.That is to say, the retailer can accumulate revenue and earn interest during the period N to M with rate e I under the condition of trade credit; When T M  , the account is settled at and the retailer would pay for the interest charges on items in stock with rate p I over the interval   , M T ; when T M  , the account is also settled at T M  and the retailer does not need to pay any interest charge of items in stock during the whole cycle; The fixed credit period offered by the supplier to the retailer is no less to his/her customers, i.e. 0 N M   .

Mathematical Model
Based on above assumptions, depletion due to demand and deterioration will occur simultaneously.The inventory level of the system can be described by the following differential equation ' ( ) The solution to the above equation is     1 So the retailer's order size per cycle is If Q W  , we get d T : .
The present value of all future cash-flow cost

 
PV T  consists of the following elements: (1) The present value of order cost:   (3) The present value of purchasing cost: (4) The present values of interest charged and earned are addressed as follows: when , there is no interest earned, that is , there is no interest charged, that is The present value of interest earned is Therefore, the present value of all future cash-flow cost, Consequently, based on the values of d T , N , M ,three possible cases: (1)

D h Dr e e D PV T
A ce e pI T e e e r r r .

Theoretical Results
The objective in this paper is to find the replenishment time * T to minimize the present value of all future cash-flow cost of the retailer.
To simply the proof process of this model, the following lemma is given. .We have the following results.

Case 1
Taking derivative of  

1
PV T with respect to T , we obtain From the above analysis and lemma 1(1-2), we have the following results.

Case 2
Taking derivative of  

2
PV T with respect to T , we obtain 1 , Similarly, taking derivative of with respect to , we know that       (1) when   0 , where a r g m in ,

T T 
where (3) when   0 arg min , (1) when   0 From lemma 1, the results are proofed. ( , the equation of   0 g T  has a unique root (say # 2 T ).In this situation, As follows, we will discuss the property of   T N  .
(A).In this case we discuss the property of   From the above analysis, we know that    , and have the following results.
(B) In this case we discuss the property of   From (A) and (B), we have the following results a r g m in , T N  .For the convenience of that problem, we denote 0 2 T is the unique solution of    (3) when   0 g N  , we have   0 g T  .Thus,   , there exists a unique solution 3   2 arg min , From the above analysis, we know when arg min ,

Case 3
Taking derivative of   3 PV T with respect to T , we obtain 1 .
Thus, we know that From the above analysis and lemma 1, we obtain the lemma 4.

Case 4
Taking derivative of  

4
PV T with respect to T , we obtain 1 , Therefor, we know that   From the above analysis and lemma 1(1-2), we obtain the lemma 5. From the lemmas of 2, 3and 6, we have the follow theorem.Results are summarized in Table 1.(3) For fixed other parameters, when Q W  , the larger the value of M , the smaller the values of   T when the parameter of the discount rate r is changed from (0,1).T is not impact to the value of r .

Conclusions
In this paper, we develop an inventory system for deteriorating items under permissible delay in payments.The primary difference of this paper as compared to previous studies is that we introduce a generalized inventory model by relaxing the traditional EOQ model in the following three ways: (1)   greater than or equal to a predetermined quantity, then both of the supplier and the retailer are taking trade credit police; otherwise, the delay in payments is not permitted; (3) the present value of all future cash-flows cost instead of the average cost.The proposed of the paper is minimizing the present value of all future cash-flow cost of the retailer.In addition, the optimal solutions to the model have been discussed in detail under all possible situations.Three easyto-use theorems are developed to find the optimal ordering policies for the considered problem, and these theoretical results are illustrated by some numerical examples.
In regards to future research, one could consider incorporating more realistic assumptions into the model, such as the demand dependents the selling price, quantity discounts, supply chain coordination, etc.

Lemma 1 .
Let * x denotes the minimum point of the function of   F x on interval   , a b .Suppose   f x is continuous function and increasing on   , a b , and     Policy with Trade Credit and Discount Cash-Flow … (Hao Jiaqin)

2 fT
is decreasing on   ,

Lemma 5 .* 4 T 4 PV , where 0 4 TTheorem 1 .
Let is the minimum point of   is the unique solution of   From lemmas 2-5, the following theorem is obtained.The optimal cycle time * T and the present value of all future cash-flow cost

 , where 0 6 T
Policy with Trade Credit and Discount Cash-Flow … (Hao Jiaqin) the optimal cycle time * T and the present value of all future cashflow cost is the unique solution of   , the optimal cycle time * T and the present value of all future cash-flow cost examples To illustrate the results obtained in this paper, we provide the following numerical examples.

( 1 )
For fixed other parameters, the larger the value of A , the larger the values of , the larger the value of W , the larger the values of the value of N , the larger the values of the values of M and N , the larger the value of * T ; when * d T T , the value of * T is keeping a constant when the values of M and N .

Figure 1 .Figure 2 .
Figure 1.The impact of change of r on 1494

Table 1 .
The impact of change of A andW on *PV T

Table 2 .
The following inferences can be made based on table 1 and table2:

Table 2 .
The impact of change of M and N on the items deteriorate continuously; (2) if the retailer's order quantity is Optimal Economic Ordering Policy with Trade Credit and Discount Cash-Flow … (Hao Jiaqin)