Optimal replenishment and credit policy in supply chain inventory model under two levels of trade credit with time and creditsensitive demand involving default risk
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Abstract
Traditional supply chain inventory modes with trade credit usually only assumed that the upstream suppliers offered the downstream retailers a fixed credit period. However, in practice the retailers will also provide a credit period to customers to promote the market competition. In this paper, we formulate an optimal supply chain inventory model under two levels of trade credit policy with default risk consideration. Here, the demand is assumed to be creditsensitive and increasing function of time. The major objective is to determine the retailer’s optimal credit period and cycle time such that the total profit per unit time is maximized. The existence and uniqueness of the optimal solution to the presented model are examined, and an easy method is also shown to find the optimal inventory policies of the considered problem. Finally, numerical examples and sensitive analysis are presented to illustrate the developed model and to provide some managerial insights.
Keywords
Supply chain management Trade credit Inventory Time and credit period sensitive demand Default riskIntroduction
The managing of inventories is one of the most significant tasks that every manager must do efficiently and effectively in any organization so that their organization can grow. Now, all the organizations are involved in a global competitive market and then these organizations are taking seriously the activities related to manage inventories. Thus, recently the practitioners and researchers have been increasing their interest in optimizing the inventory decisions in a holistic way. Thus, the manager tries to find out different ways through which the cost associated with inventory can reduce and the profit of the organization can increase. Recently, the business operations such as share marketing and trade credit financing have been a powerful tool to increase sales and profits. In practice, vendors allow a fixed period to settle the payment without penalty for their customers to increase sales and reduce onhand inventory. In fact, a permissible delay in payment decreases the cost of inventory holding because this action decreases the amount of capital invested in inventory for the duration of the permissible period. Furthermore, for the period of the delay in payment, the retailer can accumulate revenue on sales and earn interest on that revenue via share marketing investment or banking business. This type of inventory problem is known as inventory problem with permissible delay in payments. In today’s severely competitive business environment, business men are increasingly using trade credit to stimulate the demand of the products and attract more customers. As a result, the trade credit financing plays an important role in modern business operations as a source of funds aside from banks and other financial institutions (Yang and Birge 2013). In India, the nonstateowned enterprises often obtain limited support from banks. Therefore, the trade credit policy is adopted as a very important shortterm financing method. On the other hand, the policy of granting trade credit adds not only an additional cost but also an additional dimension of default risk (i.e., the event in which the buyer will be unable to payoff its debt obligations) to the retailer (Teng et al. 2005).
During the past few years, many articles dealing with a range of inventory models under trade credit have appeared in various journals. At the earliest, Goyal (1985) established a singleitem inventory model under permissible delay in payments when selling price equals the purchase cost. Aggarwal and Jaggi (1995) extended Goyal (1985) model to consider the deteriorating items. Jamal et al. (1997) further generalized the model to allow for shortages. Teng (2002) amended Goyal (1985) model by considering the difference between selling price and purchasing cost, and found that it makes economic sense for a wellestablished retailer to order less quantity and take the benefits of payment delay more frequently. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Huang (2003) extended Goyal (1985) model to develop an EOQ model with upstream and downstream trade credits in which the length of downstream trade credit period is less than or equal to the length of the upstream trade credit period. Teng and Goyal (2007) complemented the shortcoming of Huang (2003) model and proposed a generalized formulation. Mahata and Goswami (2007) developed an inventory model to determine an optimal ordering policy for deteriorating items under twolevel trade credit policy in the fuzzy sense. Huang (2007) incorporated Huang (2003) model to investigate the twolevel trade credit policy in the EPQ frame work. Recently, Mahata (2012) proposed an EPQ model for deteriorating items with upstream full trade credit and downstream partial trade credit with a constant demand. Annadurai and Uthayakumar (2012) analyzed partial trade credit financing in a supply chain by EOQbased model for decaying items with shortages. Taleizadeh et al. (2013) considered an EOQ problem under partial delayed payment and partial backordering. Chen et al. (2014) discussed the EPQ model under retailer partial trade policy. Taleizadeh and Nematollahi (2014) investigated the effects of time value of money and inflation on the optimal ordering policy in an inventory control system to manage a perishable item over the finite horizon planning under backordering and delayed payment. Ouyang et al. (2015) discussed the integrated inventory model with the ordersize dependent trade credit and a constant demand. Lashgari et al. (2015) proposed a lotsizing model with partial upstream advanced payment and partial downstream delayed payment in a threelevel supply chain. Recently, Zia and Taleizadeh (2015) presented a lotsizing model with backordering under hybrid linkedtoorder multiple advance payments and delayed payment. Many related articles can be found in Chang and Teng (2004), Chung (2008), Chung and Liao (2004), Goyal et al. (2007), Huang and Hsu (2008), Mahata and Mahata (2011, 2014a, b), Lashgari et al. (2016) and their references.
All the Above researchers established their EOQ or EPQ inventory models under trade credit financing based on the assumption that the demand rate is constant over time. However, in practice, the market demand is always changing rapidly and is affected by several factors such as price, time, inventory level, and delayed payment period. From a product life cycle perspective, it is only in the maturity stage that demand is near constant. During the growth stage of a product life cycle (especially for hightech products), the demand function increases with time. Moreover, the marginal influence of the credit period on sales is associated with the unrealized potential market demand. To obtain robust and generalized results, we extend the constant demand to a creditsensitive and linear nondecreasing function of time.
Related to this context, few papers explore the optimal replenishment policies of the retailer under trade credit financing by considering demands that are sensitive to either time stock and/or credit period. Sarkar (2012), Teng et al. (2012) and Mahata and Mahata (2014b) build the economic order quantity model with trade credit financing for timedependent demand. Mahata (2015a) discuss the inventory replenishment problems with upstream full trade credit and downstream partial trade credit financing by considering a pricesensitive demand. Soni (2013) developed the optimal replenishment policies under trade credit financing by considering price and stocksensitive demand. Some studies focus on demand dependence on delayed payment time. The inventory model with the creditlinked demand are discussed by Jaggi et al. (2008), Jaggi et al. (2012). Thangam and Uthayakumar (2009) discuss trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period. Lou and Wang (2012) study optimal trade credit and order quantity by considering trade credit with a positive correlation of market sales, but are negatively correlated with credit risk. Giri and Maiti (2013) discuss the supply chain model with price and trade creditsensitive demand with trade credit by considering the fact that a retailer shares a fraction of the profit earned during the credit period. Wu (2014) explore optimal credit period and lot size by considering demand dependence on delayed payment time with default risk for deteriorating items with expiration dates. Annadurai and Uthayakumar (2015) develop the decaying inventory model with stockdependent demand and shortages under twolevel trade credit. Dye and Yang (2015) discuss the sustainable trade credit and replenishment policy with creditlinked demand and credit risk considering the carbon emission constraints. Mahata (2015b) propose an EOQ model for the retailer to obtain his/her optimal credit period and cycle time under twolevel trade credit by considering demand dependence on delayed payment time with default risk for deteriorating items.
In traditional supplierretailerbuyer supply chain inventory models, the supplier frequently offers the retailer a trade credit of M periods, and the retailer in turn provides a trade credit of N periods to her/his buyer to stimulate sales and reduce inventory. From the seller’s perspective, granting trade credit increases sales and revenue but also increases opportunity cost (i.e., the capital opportunity loss during credit period) and default risk. Also, from a product life cycle perspective, it is only in the maturity stage that demand is near constant. During the growth stage of a product life cycle (especially for hightech products), the demand function increases with time. To obtain robust and generalized results, we extend the constant demand to a linear nondecreasing demand function of time. As a result, the fundamental theoretical results obtained here are suitable for both the growth and maturity stages of a product life cycle. However, none of the paper discusses the optimal trade credit and order policy by considering demand to a creditsensitive and linear nondecreasing function of time involving default risk. Therefore, indepth research is required on the inventory policy that considers demand sensitive to time and credit period involving default risk to extend the traditional EOQ model. The main aspects of this paper is to develop an economic order quantity model for the retailer where: (1) the supplier provides an upstream trade credit and the retailer also offers a downstream trade credit, (2) the retailer’s downstream trade credit to the buyer not only increases sales and revenue but also opportunity cost and default risk, and (3) the demand rate of which varies simultaneously with time and the length of credit period that is offered to the customers. We model the retailer’s inventory system under these conditions as a profit maximization problem. We then show that the retailer’s optimal credit period and cycle time not only exist but also are unique. Furthermore, we run some numerical examples to illustrate the problem and provide managerial insights.
Problem description and formulation

A ordering cost per order for the retailer

h unit stock holding cost per item per year (excluding interest charges) for the retailer

c unit purchasing cost of the retailer

s unit selling price with \(s>c\)

\(I_{\rm{e}}\) interest earned per \( \$ \) per year by the retailer

\(I_{\rm{p}}\) interest charged per \( \$ \) in stocks per year to the retailer

M the retailer’s trade credit period offered by supplier in years

N the customer’s trade credit period offered by retailer in years

T replenishment time interval for the retailer, where \(T\ge 0\)

Q the order quantity for the retailer

I(t) the inventory level in units at time t

D(t, N) the market annual demand rate in units, which is a function of time t and the trade credit period.

\(T^*\) the optimal replenishment period for the retailer

\(N^*\) the optimal trade credit period offered to the customers by the retailer

\(\Pi (N,T)\) the profit per year for the retailer
 1.
Time horizon is infinite.
 2.
Replenishment is instantaneous; shortage is not allowed.
 3.
The supplier provides the trade credit period M to the retailer, and the retailer offers its customers trade credit period N.
 4.
Demand rate D(t, N) is a function of time t and the credit period N. For simplicity, we assume that \(D(t,N)=a+bt+\lambda (N)\), where a and b are nonnegative constants, and t is within a positive time frame, governing the linear trend of the demand with respect to time t as the demand rate in today’s hightech products increases significantly during the growth stage; \(\lambda '(N)>0\) reflects the demand increasing with the credit period as trade credit attracts new buyers who consider it a type of price reduction.
 5.
When the credit period is longer, the default risk to the retailer is higher. For simplicity, the rate of the default risk given the credit period offered by the retailer is \(F(N)=1{\rm{e}}^{kN}\), where \(k>0\). This default risk pattern is used in some studies (Lou and Wang 2012; Zhang et al. 2014).
Model formulation
 1.
Annual ordering cost per cycle is \(\frac{A}{T}\).
 2.
Annual purchasing cost per cycle is \(\frac{cQ}{T}=\frac{c}{T}\Big (\rho T+\frac{1}{2}bT^2\Big )\).
 3.Annual holding cost excluding interest charges per cycle is$$\begin{aligned} \frac{h}{T}\int _0^TI(t){\rm{d}}t=\frac{h}{T}\Big (\frac{1}{2}\rho T^2 +\frac{1}{3}bT^3\Big ). \end{aligned}$$
 4.The sales revenue considering default risk is$$\begin{aligned} \frac{se^{kN}}{T}\int _0^TD(t){\rm{d}}t=\frac{se^{kN}}{T}\Big (\rho T+\frac{1}{2}bT^2\Big ). \end{aligned}$$
 5.
The interest earned and interest charged is shown in Case 1 and 2 below.
Case 1
\(N\le M\) In this case, we have two possible subcases: (1) \(T+N\le M\) and (2) \(T+N \ge T+N\). Now, let us discuss the detailed formulation in each subcase.
Subcase 1.1
Subcase 1.2
Case 2
Theoretical results and optimal solutions
The main purpose of this paper is to determine the optimal replenishment and trade credit policies that correspond to maximizing the total profit per unit time. To achieve our purpose, in all cases, we first establish conditions where for any given N, the optimal solution of T not only exists but also is unique. Then, for any given value of T, there exists a unique N that maximizes the total profit per unit time.
Optimal solution for the case of \(N\le M\)
Theorem 1
 (i)
Equation (8) has only one positive solution.
 (ii)
If the only positive solution \(T_1\) to Eq. (8) is less than or equal to \(MN\), then \(T_1\) is the only optimal solution to \(\Pi _1(TN)\) in Eq. (4).
 (iii)
If \(\Delta <0\), then \(\Pi _1(TN)\) has the unique optimal solution \(T_1\), which is less than \(MN\). Otherwise, if \(\Delta \ge 0\), the optimal solution is \(T_1=MN\).
Proof
Theorem 2
 (1)
\(\Pi _1(NT)\) is strictly concave function in N, hence exists a unique maximum solution.
 (2)
If \(\Delta _{N_1}\le 0\), then \(\Pi _1(NT)\) is maximizedat \(N_1^*=0\).
 (3)
If \(\Delta _{N_1}>0\), then there exists a unique \(\widehat{N}_1>0\) such that \(\Pi _1(NT)\) is maximized at \(N_1^*=\widehat{N}_1>0\).
Theorem 3
 (i)
Equation (15) has only one positive solution.
 (ii)
If the only positive solution \(T_2\) to Eq. (15) is greater than or equal to \(MN\), then \(T_2\) is the only optimal solution to \(\Pi _2(TN)\) in Eq. (5). Otherwise, the only optimal solution to \(\Pi _2(TN)\) is \(T_2=MN\).
 (iii)
If \(\Delta >0\), then \(\Pi _2(TN)\) has the unique optimal solution \(T_2\), which is greater than \(MN\). Otherwise, if \(\Delta \le 0\), the optimal solution is \(T_2=MN\).
Proof
See Appendix B.
Based upon the above two Theorems 1 and 3, the following main theorem can be derived to determine the optimal solution \(T^*\) to be \(T_1\) or \(T_2\). \(\square\)
Theorem 4
 (1)
If \(\Delta <0\), then \(\Pi (T^*N) =\Pi _1(T_1N)\). Hence \(T^*=T_1\), and \(T_1<MN\).
 (2)
If \(\Delta =0\), then \(\Pi (T^*N) =\Pi _1(MNN)=\Pi _2(MNN)\). Hence \(T^*=MN\).
 (3)
If \(\Delta >0\), then \(\Pi (T^*N) =\Pi _2(T_2N)\). Hence \(T^*=T_2\), and \(T_2>MN\).
Proof
Theorem 5
 (1)
\(\Pi _2(NT)\) is strictly concave function in N, hence exists a unique maximum solution.
 (2)
If \(\Delta _{N_2}\le 0\), then \(\Pi _2(NT)\) is maximized at \(N_2^*=0\).
 (3)
If \(\Delta _{N_2}>0\), then there exists a unique \(\widehat{N}_2>0\) such that \(\Pi _2(NT)\) is maximized at \(N_2^*=\widehat{N}_2>0\).
Optimal solution for the case of \(N>M\)
Theorem 6
Theorem 7
 (1)
\(\Pi _3(NT)\) is strictly concave function in N, hence exists a unique maximum solution.
 (2)
If \(\Delta _{N_3}\le 0\), then \(\Pi _3(NT)\) is maximized at \(N_1^*=M\).
 (3)
If \(\Delta _{N_3}>0\), then there exists a unique \(\widehat{N}_1>M\) such that \(\Pi _3(NT)\) is maximized at \(N_1^*=\widehat{N}_3>M\).
Numerical analysis
In this section, we assume that \(\lambda (N)=de^{uN}\) (Teng et al. 2014; Wu 2014) to conduct the numerical analysis for illustrating the theoretical results and obtaining the optimal solutions using the software Mathematica 7.
Numerical examples
Example 1
Example 2
Sensitivity analysis and real usage of the model
Sensitivity analysis of the parameters
Parameter  \(N^*\)  \(T^*\)  \(\Pi ^*(N,T)\)  

A  10  1.5486  0.0256  24867.26 
14  1.5444  0.0279  24573.32  
18  1.5434  0.0297  24321.57  
22  1.5424  0.0312  24115.45  
a  100  1.5486  0.0256  24867.26 
150  1.5448  0.0256  24947.58  
200  1.5445  0.0257  25031.84  
250  1.5439  0.0258  25112.57  
s  30  1.5486  0.0256  24867.26 
35  1.7989  0.0198  89352.75  
40  2.0194  0.0168  283278.72  
45  1.9909  0.0155  813827.72  
c  10  1.5486  0.0256  24867.26 
11  1.3879  0.0315  10719.69  
12  1.0196  0.0389  6972.56  
13  0.8813  0.0483  3853.75  
h  5  1.5486  0.0256  24867.26 
7  1.5447  0.0242  24659.75  
9  1.5449  0.0232  24473.46  
11  1.5436  0.0225  24303.52  
k  0.5  1.5486  0.0256  24867.26 
0.6  1.2781  0.0375  8348.57  
0.7  1.0676  0.0538  3387.75  
0.8  0.8857  0.0751  1796.78  
d  5  1.5486  0.0256  24867.26 
7  1.5468  0.0238  36342.87  
9  1.5479  0.0223  47861.14  
11  1.5484  0.0214  59404.67  
b  0.2  1.5486  0.0256  24867.26 
0.3  1.5456  0.0257  24885.31  
0.4  1.3234  0.0258  24904.72  
0.5  1.5454  0.0260  24923.88  
u  5  1.5486  0.0256  24867.26 
6  1.5868  0.0179  120907.56  
7  1.6139  0.0151  800056.97  
8  1.6334  0.0137  4331072.12 
 1.
If the value of A increases, then the optimal order cycle \(T^*\) increases while the values of \(N^*\) and the optimal profit \(\Pi ^*(N,T)\) decrease. When the ordering cost is higher, the retailer orders the products in the longer replenishment period to reduce the order frequency. Thus, he pays less ordering cost.
 2.
If the value of a increases, then the values of \(N^*\) and the optimal order cycle \(T^*\) decrease while the value of the optimal profit \(\Pi ^*(N,T)\) increases. When the initial market demand is greater, the retailer can make more profit.
 3.
If the value of b increases, then the values of \(N^*\) decrease, but the optimal order cycle \(T^*\) and the value of the optimal profit \(\Pi ^*(N,T)\) increase. When the market demand is more sensitive to time, the retailer provides shorter delayed payment time and longer order cycle to make more profit.
 4.
If the value of h increases, then the values of \(N^*\), the optimal order cycle \(T^*\), and the value of the optimal profit \(\Pi ^*(N,T)\) decreases. The retailer can adopt some measurements to reduce the holding cost to make more profit.
 5.
If the value of c or k increases, then the values of \(N^*\) and the optimal profit \(\Pi ^*(N,T)\) decrease while the value of the order cycle \(T^*\) increases. When the unit purchasing price of the retailer is higher, the retailer makes less profit. When the default risk of the customers is higher, the retailer should offer a shorter delayed payment time to his customers. The retailer can take some measurements to reduce the default risk of the customer, such as by adopting the partial delayed payment policy, to make more profit.
 6.
If the value of s, d, or u increases, then the values of \(N^*\) and the value of the optimal profit \(\Pi ^*(N,T)\) increase while the optimal order cycle \(T^*\) decreases. When the sales price is higher, the retailer offers a shorter delayed payment time and longer order cycle for his customers to make more profit. When the market demand is more sensitive to the trade credit, the retailer should provide a longer delayed payment time to make more profit.
Conclusions
Most of the existing inventory models under trade credit financing are assumed that the demand rate remains constant. However, in practice the market demand is always changing rapidly and is affected by several factors such as price, time, inventory level, and delayed payment period. In today’s hightech products demand rate increases significantly during the growth stage. Moreover, the marginal influence of the credit period on sales is associated with the unrealized potential market demand. In this paper, we have developed an EOQ model under twolevel trade credit financing involving default risk by considering demand to a creditsensitive and linear nondecreasing function of time. The objective is to find optimal replenishment and trade credit policies while maximizing profit per unit time. For any given credit period, we first prove that the optimal replenishment policy not only exists, but is also unique in some conditions. Second, we show how the optimal credit period for any given replenishment cycle can be decided. Furthermore, we use the Mathematica 7.0 to obtain the optimal inventory policies for the proposed model. Sensitivity analysis is conducted to provide some managerial insights. To the authors’ knowledge, this type of model has not yet been considered by any of the researchers/scientists in inventory literature. Therefore, this model has a new managerial insight that helps a manufacturing system/industry to gain maximum profit. The model can be extended in several ways, for example, we may consider the deteriorating items with a constant deterioration rate. Also, we can extend the model to allow for shortages and partially backlogging. Finally, the effect of inflation rates on the economic order quantity can also be considered.
Notes
Acknowledgements
The authors are grateful to the Editorinchief, Associate editors and the anonymous reviewers for their valuable and constructive comments which have led to a significant improvement of this manuscript.
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