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This paper investigates the optimal replenishment policy for the retailer with the ramp type demand and demand dependent production rate involving the trade credit financing, which is not reported in the literatures. First, the two inventory models are developed under the above situation. Second, the algorithms are given to optimize the replenishment cycle time and the order quantity for the retailer. Finally, the numerical examples are carried out to illustrate the optimal solutions and the sensitivity analysis is performed. The results show that if the value of production rate is small, the retailer will lower the frequency of putting the orders to cut down the order cost; if the production rate is high, the demand dependent production rate has no effect on the optimal decisions. When the trade credit is less than the growth stage time, the retailer will shorten the replenishment cycle; when it is larger than the breakpoint of the demand, within the maturity stage of the products, the trade credit has no effect on the optimal order cycle and the optimal order quantity.

Along with the globalization of the market and increasing competition, the enterprises always take trade credit financing policy to promote sales, increase the market share, and reduce the current inventory levels. As a result, the trade credit financing play an important role in business as a source of funds after Banks or other financial institutions. In the traditional inventory economic order quantity (EOQ) model, it is assumed that the retailer must pay for the products when receiving them. In practice the suppliers often provide the delayed payment time for the payment of the amount owed. Usually, there is no interest charged for the retailer if the outstanding amount is paid in the allowable delay. However, if the payment is unpaid in full by the end of the permissible delay period, interest is charged on the outstanding amount.

The optimal inventory policy is also influenced by the market demand and the production rate with the trade credit. For the market demand, it is always changing fast and influenced by many factors such as the price [

The optimal inventory policy is not only related to the market demand but also influenced by the production rate. As a result, in-depth research is required on the inventory replenishment decisions. Therefore, this study extends the EOQ models in the several ways as follows.First, trade credit financing is introduced to the traditional EOQ models. The retailer is offered by the supplier with a delayed payment time. Second, the ramp type demand is introduced to the EOQ models with the trade credit financing. The demand is nearly a constant in the maturity stage. During the growth time, the demand of the products is increasing with time. Furthermore, the production rate dependent on demand is introduced to the EOQ models with trade credit financing considering the ramp type demand for the first time.

The trade credit is studied by many scholars from the aspects of finance, accounting, and operations management. Finance and accounting research mainly focused on the study of trade credit enterprise's cash flow, discussing the nature of the trade credit and its impact. In the area of operation management research, mainly from the angle of cash flow and logistics coordination, based on the traditional economic order quantity model framework, weigh the cost of capital cost elements such as fixed ordering cost, storage cost, and profit (or other elements), to analyze the supply chain inventory control and coordination problems. Our research is mainly felt in the operation management flow.

The inventory replenishment policies under trade credit financing have been studied intensively. Most papers discussed the EOQ or EPQ inventory models under trade credit financing all based on the assumptions that the demand rate is a constant and the infinite production rate. The EOQ model with the trade credit financing is put forward for the first time by Goyal [

A number of papers in published literature have extensively studied inventory problem by assuming this ramp type demand rate. Under the assumption, Hwang and Shinn [

However, the problems of payment delay linked to the ramp type demand rate have not received much attention. The most related literatures to our study are as follows. Mishra and Singh [

For the trade credit, the researchers attempting to solve the problems mostly assume that the production rate is infinite or a constant value. However, the infinite or a constant replenishment rate of the inventory models is inconsistent with the actual industrial practices. When the market demand is better, the supplier will provide a higher production rate; if the market demand is shrinking, the supplier will reduce the production rate. In the traditional EOQ or EPQ models without the trade credit financing, Darzanou and Skouri [

Therefore, based on the literatures above, we find that none of the above models explore the optimal replenishment policies of the retailer under trade credit financing with the ramp type demand and demand dependent production rate.

Given the analysis above, this paper developed inventory models under trade credit financing with ramp type demand and demand dependent production rate. In this inventory system, they might attribute to intricate correlations among the period to maturity stage

To build the mathematical models, the following notations and assumptions are adopted in this paper.

The following assumptions are used throughout in this paper.

There is a single supplier and single retailer and they deal with a single product.

The lead time is zero. The planning horizon is infinite. The shortage is not allowed.

The initial and final inventory levels are both zero.

The market demand for the item is assumed to be a ramp type function of time. At the first stage, the demand is increasing with the time, such as the introduction stage of the new products; but as the time increases, the demand will keep a constant, such as the maturity stage of the new product. That is,

For the production rate, when the market demand is high, the production will be improved; but when the market demand is low, the production rate will be down. Therefore, the production rate is related to the demand closely. In this paper, the production rate is assumed

The retailer would settle the account at

The break point of the demand function

For the inventory system, we analyze it in the Section

Given the above, it is possible to build the mathematical inventory EOQ model with the trade credit financing.

Based on the above notations and assumptions, the inventory system can be considered in the following. At the beginning, the stock level is zero. The production starts with zero stock level at time

The inventory model for

Hence, the variation of the inventory level

During the growth stage in the interval

Similarly, during the maturity stage

Finally, at the maturity stage

Solving (

In addition, using the boundary condition

Solving (

The inventory system for

The inventory model for

During the growth stage in the interval

Similarly, during the maturity stage

Finally, at the maturity stage

Solving (

In addition, using the boundary condition

Solving (

The annual total relevant cost consists of the following elements: ordering cost, holding cost, interest payable, and interest earned. The components are evaluated as in the following.

Annual ordering cost:

Annual stock holding cost (excluding the interest charges).

There are eight cases to occur in interest earned and interest charged for the items kept in stock per year.

From the value of

Interest charged and interest earned for

Interest charged for

Interest earned for

During the time

Interest charged and interest earned for

Interest charged for

Interest earned for

During the time

Interest charged and interest earned for

Interest charged for

Interest earned for

For the interval

The results in previous Section

If

If

The problem is

(1) If

(2) If

For

It is similar to analyze the optimal solutions for other branches as Theorem

The annual interest earned during the trade credit period is shown in the following:

The annual interest earned during the trade credit period is shown in the following:

The annual interest earned during the trade credit period is shown in the following:

The annual interest earned during the trade credit period is shown in the following:

The annual interest earned during the trade credit period is shown in the following:

Combine the inventory models of Cases 2 and 3, the results in previous subsections lead to the following total cost function.

In order to obtain the optimal replenishment decisions, we should study each branch of the cost functions.

(1) If

(2) If

It is similar to analyze the optimal solutions for other branches as Theorem

In this section, we carry out some numerical examples to illustrate the algorithms obtained in the previous sections. Additionally, we also provide a sensitivity analysis of the values of most important parameters on the optimal decisions of the retailer and the total cost.

The input parameters are

Using Algorithm

The input parameters are

Using Algorithm

The input parameters are

Using Algorithm

The input parameters are

Using Algorithm

In the model, there are many parameters influencing the decisions of the members. But according to the past research, the effects of some parameters on the decisions are analyzed in many literatures such as the ordering cost per order. Therefore, we choose three parameters, which are directly related to our innovations, to conduct the sensitivity analysis for obtaining interesting management insights. In this subsection, we present the sensitivity analysis of the models mentioned above with respect to the three parameters of

Basic parameters’ values for the case

The impact of the changes of the parameter

The optimal order cycle

The optimal order quantity

The total cost for the retailer

Basic parameters’ values for the case

The impact of the changes of the parameter

The optimal order cycle

The optimal order quantity

The total cost for the retailer

Based on Figures

For the total cost, as the increase of the parameter

Basic parameters’ values are

Based on Figure

The impact of the changes of the parameter

The optimal order cycle

The optimal order quantity

The total cost for the retailer

When

However, the total cost for the retailer is decreasing with the delayed payment time for the product life cycle. Therefore, the retailer hopes the supplier can offer them the delayed payment time as long as possible to obtain more profits.

Basic parameters’ values are

Based on Figure

The impact of the changes of the parameter

The optimal order cycle

The optimal order quantity

The total cost for the retailer

The total cost for the retailer is increasing with the demand breakpoint. Hence, for the products, if the growth stage of the product life cycle is longer, the cost of the retailer will be higher. Therefore, the retailer should make some measures, such as advertisement, to induce the products’ demand moving to the maturity stage as soon as possible.

Most of the existing inventory models under trade credit financing are assumed that the demand is a constant and the replenishment rate is infinite or a constant. However, in practice, the demand rate is the ramp type function of time for some cases, such as the new products and the holiday related products. On the other hand, the production rate is related to the market demand. When the market is better, the production rate will be improved. When the retailer makes an order, it can be met faster. That is, the replenishment of the retailer is related to the market demand.

Therefore, in this paper, we developed an EOQ model under trade credit financing with ramp type demand and the demand dependent production rate. Subsequently, the algorithms are proposed to decide the optimal replenishment cycle and the optimal order quantity for the retailer. Finally, the numerical analysis is demonstrated to illustrate the models and the sensitively analysis is carried out to give some management insights.

Based on the study, we mainly found the following.

If the value of production rate is small, the retailer will lower the frequency of putting the orders to cut down the order cost. If the demand dependent rate is higher, the demand dependent production rate has no effect on the optimal order cycle and the order quantity. For the limit case, when the production rate is infinite, as most papers presented, the production rate has no influence on the members’ decisions.

When the delayed payment time is less than the growth stage time of the new products’ introduction, the retailer will shorten the replenishment cycle to take advantage of the trade credit more frequently for accumulating the interest; on the other hand, the short order cycle can make the retailer adjust its order decisions more quickly for meeting the changeable demand within the growth stage of the products. When the delayed payment time is larger than the breakpoint of the demand, within the maturity stage of the products, the delayed payment time has no effect on the optimal order cycle and the optimal order quantity. The total cost for the retailer is decreasing with the delayed payment time. Therefore, the retailer hopes the supplier can offer them the delayed payment time as long as possible.

As the demand breakpoint is increasing, the optimal order cycle and the optimal order quantity are decreasing first and then increasing. The total cost for the retailer is increasing with the demand breakpoint. Hence, for the products, if the growth stage of the product life cycle is longer, the cost of the retailer will be higher. Therefore, the retailer should make some measures, such as advertisement, to induce the products’ demand moving to the maturity stage as soon as possible.

The research presented in this paper can be extended in several ways. For example, other demand functions can be further discussed considering the demand dependent production rate in the EOQ inventory model with the trade credit financing. Additionally, the models can be generalized to consider the shortage or the partial backlogging. Furthermore, the influence of the poor quality products on the EOQ model can be discussed to obtain some management insights.

The first derivative for a minimum of

Based on (

The second derivatives for

We can know that

Let us set

If

If

Based on the analysis above, it is easy to obtain Theorem

Calculate the first derivatives of the

Therefore, we have

The second derivatives for

We know that

Let us set

If

If

Based on the analysis above, it is easy to obtain Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The paper is funded by The National Natural Science Foundation of China (71302115, 71172018), The Ministry of education of Humanities, and Social Sciences Project (13YJC630121).