 Research
 Open Access
Two retailer–supplier supply chain models with default risk under trade credit policy
 Chengfeng Wu^{1}Email author and
 Qiuhong Zhao^{2}
 Received: 21 October 2015
 Accepted: 21 September 2016
 Published: 6 October 2016
Abstract
The purpose of the paper is to formulate two uncooperative replenishment models with demand and default risk which are the functions of the trade credit period, i.e., a Nash equilibrium model and a supplierStackelberg model. Firstly, we present the optimal results of decentralized decision and centralized decision without trade credit. Secondly, we derive the existence and uniqueness conditions of the optimal solutions under the two games, respectively. Moreover, we present a set of theorems and corollary to determine the optimal solutions. Finally, we provide an example and sensitivity analysis to illustrate the proposed strategy and optimal solutions. Sensitivity analysis reveals that the total profits of supply chain under the two games both are better than the results under the centralized decision only if the optimal trade credit period isn’t too short. It also reveals that the size of trade credit period, demand, retailer’s profit and supplier’s profit have strong relationship with the increasing demand coefficient, wholesale price, default risk coefficient and production cost. The major contribution of the paper is that we comprehensively compare between the results of decentralized decision and centralized decision without trade credit, Nash equilibrium and supplierStackelberg models with trade credit, and obtain some interesting managerial insights and practical implications.
Keywords
 Supply chain
 Trade credit
 Default risk
 SupplierStackelberg game
 Nash game
Background
In the past 100 years, a huge of extensions of the traditional Economic Order Quantity (EOQ) model has been proposed by lots of researchers. Recently, the International Journal of Production Economics published a special issue “Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris”. Among them, Andriolo et al. (2014) and Glock et al. (2014) respectively adopted different methodologies to analysis the evolution and main streams of these research emerged from Harris’ seminal lot size during 100 years of history and proposed a new research opportunities for future research, such as, sustainability issue and cash flows. Latest works include those by Battini et al. (2014), and Marchi et al. (2016), among others.
One of most important extensions is that incorporating trade credit into the EOQ. It assumes that supplier offers retailer/buyer a permissible delay in payments (trade credit period). The account is not settled during trade credit period and there is no interest charge. In fact, as shortterm financing tool, trade credit is widely implemented in fierce competitive circumstance and has important influence on inventory holding cost (Azzi et al. 2014).
Goyal (1985) first fully analysis the impact of fixed permissible delay in payments on the retailer’s ordering decision. Since then, the effect of fixed trade credit on the replenishment policy has been studied in extensive literatures. For instance, Aggarwal and Jaggi (1995) extended Goyal’s model (1985) to consider exponentially deteriorating items. Teng (2002) further established an easy analytical closedform solution about Goyal’s model (1985) by considering the difference between the purchase cost and the retail price. Furthermore, Huang (2003) proposed a brand new inventory model under two levels of trade credit where the manufacturer offers trade credit to the retailer, and the retailer also offers his or her customer partial trade credit.
Teng et al. (2012a, b) extended the constant demand to a linear increasing demand under trade credit. Wu and Zhao (2015a) recently established an EOQ model with a constant deterioration rate, a current inventorydependent and linearly increasing timevarying demand under trade credit and presented some fundamental theoretical results. Lots of related articles can be seen in Khouja and Mehrez (1996), Chu et al. (1998), Chang and Dye (2001), Teng and Chang (2009), Jain and Aggarwal (2012), Chung (2013), Zhou et al. (2013), Chen and Teng (2014), Ouyang et al. (2014), among others. However, most of these models assumed that a trade credit period is fixed parameter and the retailer sets up its own strategy only from its individual perspective. Elaborating on this subject, Chang et al. (2008), Seifert et al. (2013), and Molamohamadi et al. (2014) conducted comprehensive literature reviews of different model settings.
On the other hand, like quantity discount, price discount, etc., as a profit transfer means, trade credit has been deeply studied in supply chain coordination. For instance, Yang and Wee (2006) presented a collaborative model for deteriorating items with pricesensitive demand and finite replenishment rate under trade credit and proposed a negotiation factor to share the additional profit between the vendor and buyer. Sarmah et al. (2008) considered the issue of coordination with trade credit term in a single supplier and multiple heterogeneous retailers at same replenishment cycle time. Wu and Zhao (2014a) recently established a collaborative model under trade credit for inventorydependent and timevarying demand during the finite planning horizon. Other related articles can be seen in Jaber and Osman (2006), Chen and Kang (2007), Huang et al. (2010), Chan et al. (2010), Krichen et al. (2011), Teng et al. (2012a, b), Hsu and Hsu (2013), Wu and Zhao (2014b), Glock and Kim (2015), and Marchi et al. (2016), among others. These papers assumed that trade credit is a decision variable and coordination mechanism, and discussed the effect of trade credit in coordinating supply chain for different settings.
However, how to determinate an optimal trade credit period for the supplier has been received limited attention for a long time. Although, Kim et al. (1995) first proposed a strategy to determine the optimal trade credit period for supplier and the optimal pricing for the retailer in supplierStackelberg game. And then, Abad and Jaggi (2003) extended the model of Kim et al. (1995). The two literatures did not arouse scholars’ attention at that time. The question can be really solved until Teng and Lou (2012) proposed the demand rate is an increasing function of the trade credit period (a decision variable). Although, Jaggi et al. (2008) first assumed that trade credit period has a positive impact on demand and set up a polynomial function, where trade credit period is still given parameter. Of course, other researcher also proposed inventory model with demand rate is dependent on the trade credit period, such as, Ho (2011), Giri and Maiti (2013), the trade credit period is constant.
At present, determining optimal trade credit period is being more and more attention from researchers. There are two main research views, one is trade credit provider perspective, and other is game perspective. From trade credit provider perspective, for instance, Lou and Wang (2012) extended Teng and Lou’s model (2012) to establish an EOQ model to derive optimal trade credit period and lot size simultaneously. But, in their model, they didn’t concern the retailer’s benefit and an additional capital opportunity cost the supplier burdens for offering trade credit. Recently, Teng et al. (2014) extended Lou and Wang’s model (2012) to consider learning curve phenomenon and the loss of capital opportunity during the delay payment period. Dye and Yang (2015) further extended Lou and Wang’s model (2012) to include cases with partial backorder and the supplier’s opportunity cost and two carbon emission constraints. Chen and Teng (2015) recently extended Teng and Lou’s model (2012) to consider timevarying deteriorating items and default risk rates under two levels of trade credit by discounted cash flow analysis. Other prominent and latest works include those by Wang et al. (2014), Wu et al. (2014), and Shah and CárdenasBarrón (2015), among others.
Determining optimal trade credit period from the game perspective is becoming concerned. Only a few corresponding articles may be found in latest literatures. Zhou et al. (2012) established an uncooperative inventory model for items with stockdependent demand where the retailer has limited displayedshelf space, and optimized the trade credit period in a twoechelon supply chain. Zhou and Zhou (2013) investigated two trade credit scenarios, i.e., unconditional and conditional trade credit, and discussed how the supplier sets up trade credit period to minimize his or her cost under supplierStackelberg game in a twoechelon supply chain. Based on the models of Zhou and Zhou (2013) and Teng et al. (2014), Wu and Zhao (2015b) established an uncooperative replenishment model with timevarying demand and timevarying price and learning curve phenomenon under finite planning horizon and supplierStackelberg game. However, these cited references do not consider the effect of trade credit period on market demand and default risk.
Additionally, based on the achievements of Teng and Lou (2012) and Lou and Wang (2012), Chern et al. (2013) recently established a vendor–buyer Stackelberg equilibrium model with compounded interest rate and relaxing lotfor lot replenishment policy, and derived the vendor’s optimal ordering policy and trade credit period. Chern et al. (2014) extended the model of Chern et al. (2013) to establish a vendor–buyer supply chain model in Nash game. But the two references ignored the results of decentralized decision and centralized decision without trade credit, and didn’t compare with the results of Nash equilibrium and supplierStackelberg models with trade credit in detail.
In this paper, we discuss about two retailer–supplier uncooperative replenishment models with trade credit where the demand and default risk are liked to trade credit period, i.e., a Nash equilibrium model and a supplierStackelberg model. We comprehensively compare between the results of decentralized decision and centralized decision without trade credit, and Nash game and supplierStackelberg models with trade credit. We distinguish the impact of trade credit period on the demand and default risk to observe two parties’ profit and behavior.
The remainder of the paper is organized as follows. In “Assumptions and notation” section, assumptions and notation are presented. In “Mathematical formulation of the model without trade credit” section, we present the decentralized and centralized inventory models without trade credit. In “Mathematical formulation of the models with trade credit” section, we derive uncooperative supply chain inventory models with trade credit in Nash game and supplierStackelberg game, respectively. In “Numerical examples and analysis” section, we present a numerical example and sensitivity analysis, and propose important conclusions on managerial phenomena. The last section summarizes the paper’s findings and suggests areas for future research.
Assumptions and notation
The following assumptions and notation are used throughout the paper. Some assumptions and notation will be presented later when they are needed.
Assumptions
 (i)Permissible delay in payments or trade credit attracts new buyers who consider it to be a type of price reduction. According to the previous literatures, such as that by Teng and Lou (2012), Lou and Wang (2012), Chern et al. (2013), and Teng et al. (2014), among other authors, demand rate is assumed to be a polynomial or exponential function of the trade credit period. For convenience, the demand rate D(t) may be given bywhere, \(K > 0\), \(a{ \ge }0\).$$D(M) = Ke^{aM} ,$$(1)
 (ii)The longer the trade credit period is to the retailer, the higher the default risk is to the supplier. The default risk function with respect to trade credit period is given bywhere, \(b{ \ge }0\).$$F(M) = 1  e^{  bM} ,$$(2)
 (iii)
Shortages are not permitted and lead time is zero.
 (v)
The replenishment is instantaneous and the production rate is finite. Furthermore, the demand for the product does not exceed the production rate in model.
 (vi)
The supplier follows a lotforlot replenishment policy.
 (vii)
To simplify the problem and obtain main conclusions, we further assume that the retailer’s capital opportunity cost equal to its opportunity gain.
Notation
 A :

the production rate per year for the supplier.
 b :

the default risk coefficient.
 Ke ^{ aM } :

the demand rate per year, \(A \ge Ke^{aM}\), \(M_{\rm max} = {{\ln \left( {{A \mathord{\left/ {\vphantom {A K}} \right. \kern0pt} K}} \right)} \mathord{\left/ {\vphantom {{\ln \left( {{A \mathord{\left/ {\vphantom {A K}} \right. \kern0pt} K}} \right)} a}} \right. \kern0pt} a}\), where, \(K\) the basic demand rate, \(a\) the increasing demand coefficient.
 S _{ i } :

the setup cost, $/order, \(i = s,r\).
 C :

the production cost per unit, $/unit.
 W :

the wholesale price per unit, $/unit.
 P :

the retail price per unit, $/unit, with \(P > W > C\).
 h _{ i } :

the inventory holding cost, $/unit/year, \(i = s,r\).
 I _{ i } :

the interest charged per dollar per year, $/year, \(i = s,r\).
 \(\Pi_{i}^{j}\) :

total annual profit. \(i = s,r,sc\), \(j = 0\), 1, 2, 3. \(j = 0\) decentralized decision; \(j = 1\) centralized decision; \(j = 2\) the Nash game; \(j = 3\) the supplierStackelberg game.
 \(M^{j}\) :

the length of the trade credit period offered by the supplier in years, decision variable, \(j = 2\), 3.
 \(Q^{j}\) :

the order quantity, decision variable, \(j = 0\), 1, 2, 3.
Mathematical formulation of the model without trade credit
In this section, we first propose two inventory models without trade credit, i.e., decentralized decision and centralized decision. The corresponding results of the two scenarios will be used as comparison benchmarks when the supplier permits delay in payments to the retailer for supply chain coordination.
Mathematical formulation of the models with trade credit
Two parties’ decision making in Nash game
In this subsection, we assume that the supplier and the retailer have the same bargaining power, i.e., neither side has the monopoly strength. Under this background, the optimal equilibrium solution is Nash equilibrium.
It includes a single decision variable \(M\).
Theorem 1
The supplier’s optimal trade credit period is zero (i.e., \(M^{2*} = 0\) ) if (i) \(a \le b\) , or (ii) \((a  b)W \le aC\) , or (iii) \((a  b)W \le aC + CI_{s}\).
Proof
From Eq. (15), if \(a \le b\), \(\frac{{d\Pi_{s}^{2} (M)}}{dM} < 0\). Consequently, the optimal trade credit period is zero, i.e., \(M^{2*} = 0\). Similarly, if \((a  b)W \le aC\), or \((a  b)W \le aC + CI_{s}\), we have the same results \(\frac{{d\Pi_{s}^{2} (M)}}{dM} < 0\), and \(M^{2*} = 0\). This completes the proof.
That is to say, the two parties don’t achieve any coordination or improvement in Theorem 1.
Note that the left and right sides of Eq. (19) are functions of \(M\). Due to the complexity of the problem, it seems difficult to derive a closedform expression. Additionally, in Eq. (19), we obviously observe that the left side increases and the right side decreases as \(M\) increases. There is only one intersection point when the two sides of Eq. (19) intersect, i.e., unique optimal positive solution \(M^{{2\bar{*}}}\).
Theorem 2
When \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\) , (i) if \(M^{{2\bar{*}}} < M_{\rm max}\) , the final optimal trade credit period is \(M^{2*} = M^{{2\bar{*}}}\) ; (ii) if \(M^{{2\bar{*}}} \ge M_{\rm max}\) , the final optimal trade credit period is \(M^{2*} = M_{\rm max}\).
Proof
Next, we have two alternative cases: (i) \((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} \le 0\) and (ii) \((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} > 0\).
Case 1
\((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} \le 0\).
In this case, we know that \((a  b)^{2} We^{  bM}  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} Ke^{aM} }}{{2h_{r} }}}  2aCI_{s} < 0\), further, \(\frac{{d^{2} \Pi_{s}^{2} (M)}}{{dM^{2} }} < 0\). Therefore, \(\Pi_{s}^{2} (M)\) is a strictly concave function in \(\left[ {0,\infty } \right)\). Therefore, combining with Eq. (20) and Eq. (21), we know that there exists a unique positive optimal solution such that \(\frac{{d\Pi_{s}^{2} (M)}}{dM} = 0\), denoted as \(M^{{2\bar{*}}}\).
Case 2
\((a  b)^{2} W  a^{2} C  \frac{{3a^{2} h_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}  2aCI_{s} > 0\)
In this case, we know that the value of \(\frac{{d^{2} \Pi_{s}^{2} (M)}}{{dM^{2} }}\) moves from positive to negative as \(M\) increases, that is to say, \(\Pi_{s}^{2} (M)\) is a convexconcave function of \(M\). Therefore, combining with Eq. (20) and Eq. (21), we know that \(\Pi_{s}^{2} (M)\) is a unimodal function in \(\left[ {0,\infty } \right)\). There also exists a unique positive optimal solution such that \(\frac{{d\Pi_{s}^{2} (M)}}{dM} = 0\), denoted as \(M^{{2\bar{*}}}\).
In a word, if \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\), the solution of Eq. (19) is a unique optimal positive solution \(M^{{2\bar{*}}}\) for \(\Pi_{s}^{2} (M)\). Then, we consider the upper bound of \(M\), i.e., \(M_{\rm max}\). If \(M^{{2\bar{*}}} < M_{\rm max}\), the final optimal trade credit period is \(M^{2*} = M^{{2\bar{*}}}\). If \(M^{{2\bar{*}}} \ge M_{\rm max}\), the final optimal trade credit period is \(M^{2*} = M_{\rm max}\). This completes the proof.
From Eq. (19) and Theorem 2, we obtain the following results.
Corollary 1
(i) A higher value of \(a\), \(W\), \(A\) and a lower value of \(b\), \(C\), \(S_{s}\), \(h_{s}\), \(I_{s}\) cause a higher value of \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}\), and \(M^{{2\bar{*}}}\).
(ii) The change of \(P\) and \(I_{r}\), i.e., the retailer’s profit parameters, do not affect the supplier as to whether to offer trade credit to the retailer.
Proof
The above is apparent from \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\), Eq. (19) and Theorem 2.
A simple economic interpretation is as follows. A higher value of \(a\) (i.e., increasing demand coefficient) leads to a higher demand, and higher values of \(W\) and \(A\) lead to higher revenue. Hence, the supplier is willing to offer a longer trade credit period. On the other hand, lower values of \(b\) (i.e., default risk coefficient) and \(C\) lead to a higher expected revenue for supplier, and lower values of \(S_{s}\), \(h_{s}\), and \(I_{s}\) lead to a lower ordering and inventory cost. Hence, the supplier willing to offer a longer trade credit period to the retailer.
Furthermore, according to Theorem 2, Theorem 1 can be modified to Theorem 3.
Theorem 3
The supplier’s optimal trade credit period is zero (i.e., \(M^{2*} = 0\) ) if (i) \(a \le b\) , or (ii) \((a  b)W \le aC\) , or (iii) \((a  b)W \le aC + CI_{s}\) , or (iv) \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} \le 0\).
Corollary 2
The supplier’s optimal trade credit period is zero (i.e., \(M^{2*} = 0\) ) if \(\left( {a  b} \right)W  aC  CI_{s}  a\sqrt {\frac{{S_{s} h_{s} }}{A}} \le 0\).
Proof
We use the theorem that the arithmetic mean is not always less than the geometric mean. It is omitted.
Note that \(\Pi_{r}^{2} (M^{2*} )\) is an increasing function of \(M^{2*}\) only if \(\Pi_{r}^{0} \ge 0\), which is a reasonable assumption. That is to say, as long as the trade credit period is offered by the supplier, \(\Pi_{r}^{2} (M^{2*} )\) is greater than \(\Pi_{r}^{0}\), i.e., \(\Pi_{r}^{2} (M^{2*} ) \ge \Pi_{r}^{0}\). Additionally, it is obvious that \(\Pi_{s}^{2} (M^{2*} ) \ge \Pi_{s}^{0}\). Proof is omitted.
Two parties’ decision making in a supplierStackelberg game

(i) The retailer’s optimal response

(ii) The supplier’s optimization
After observing the optimal response of the retailer (given by Eq. (25)), the supplier selects optimal \(M\) so that his or her total annual profit is maximized.
Theorem 4
The supplier’s optimal trade credit period is zero (i.e., \(M^{3*} = 0\) ) if (i) \(a \le b\) , or (ii) \((a  b)W \le aC\) , or (iii) \((a  b)W \le aC + CI_{s}\).
Proof
We omit the proof of Theorem 4 since it mimics that of Theorem 1.
That is to say, the two parties don’t achieve any coordination or improvement in Theorem 4.
Theorem 5
When \(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} > 0\) , (i) if \(M^{{3\bar{*}}} < M_{\rm max}\) , the final optimal trade credit period is \(M^{3*} = M^{{3\bar{*}}}\) ; (ii) if \(M^{{3\bar{*}}} \ge M_{\rm max}\) , the final optimal trade credit period is \(M^{3*} = M_{\rm max}\).
Proof
We omit the proof of Theorem 5 since it mimics that of Theorem 2.
From Eq. (31) and Theorem 5, we can obtain the following results.
Corollary 3

(i) A higher value of \(a\), \(W\), \(A\) and a lower value of \(b\), \(C\), \(S_{s}\), \(h_{s}\), \(I_{s}\) cause a higher value of \(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}}\) and \(M^{{3\bar{*}}}\).

(ii) The change of \(P\) and \(I_{r}\), i.e., the retailer’s profit parameters, do not affect the supplier as to whether to offer trade credit to the retailer.
Proof
It is omitted.
Likewise, according to Theorem 5, Theorem 4 can be modified to Theorem 6.
Theorem 6
The supplier’s optimal trade credit period is zero (i.e., \(\Pi_{r}^{3} (M^{3*} ) \ge \Pi_{r}^{0}\) ) if (i) \(a \le b\) , or (ii) \((a  b)W \le aC\) , or (iii) \((a  b)W \le aC + CI_{s}\) , or (iv) \(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} \le 0\).
Corollary 4
The supplier’s optimal trade credit period is zero (i.e., \(M^{3*} = 0\) ) if \(\left( {a  b} \right)W  aC  CI_{s}  a\sqrt {\frac{{3S_{s} h_{s} }}{4A}} \le 0\).
Proof
It is omitted.
Likewize, \(\Pi_{r}^{3} (M^{3*} )\) is an increasing function of \(M^{3*}\) only if \(\Pi_{r}^{0} \ge 0\). Therefore, as long as the trade credit period is offered by the supplier, \(\Pi_{r}^{3} (M^{3*} )\) is greater than \(\Pi_{r}^{0}\), i.e., \(\Pi_{r}^{3} (M^{3*} ) \ge \Pi_{r}^{0}\). Additionally, it is obvious that \(\Pi_{s}^{3} (M^{3*} ) \ge \Pi_{s}^{0}\). Proof is omitted.
Numerical examples and analysis
According to the analysis and arguments in Sect. 4, when the final optimal trade credit period is zero, the retailer’s and the supplier’s total annual profits will not be improved. Therefore, the following numerical example is proposed to illustrate the improvement process for the two games.
Example
Given \(A = 10000\) units/year, \(K = 3600\) units/year, \(a = 1\), \(b = 0.3\), \(P = \$ 35\)/unit, \(W = \$ 23\)/unit, \(C = \$ 12\)/unit, \(S_{r}\) = $200/order, \(h_{r} = \$ 5\)/unit/year, \(I_{r} = 0.12\)/year, \(S_{s}\) = $300/setup, \(h_{s} = \$ 4.5\)/unit/year, \(I_{s} = 0.1\)/year, respectively.
By applying the corresponding expressions, the results are obtained as follow.
In the decentralized decision, the economic order quantity \(Q^{0*} = 537\) units, \(\Pi_{r}^{0} = \$ 40517\)/year, \(\Pi_{s}^{0} = \$ 37153\)/year, the annual profit of the supply chain \(\Pi_{sc}^{0} = \$ 77670\)/year.
In the centralized decision, the optimal joint order quantity \(Q^{1*} = 737\) units. The optimal annual profit of the supply chain \(\Pi_{sc}^{1} = \$ 77918\)/year.
In Nash game, we obtain \(\left( {a  b} \right)W  aC  CI_{s}  aS_{s} \sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{ah_{s} }}{A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} = 2.22 > 0\). Consequently, \(M^{{2\bar{*}}} = 0.3989\) years \(= 145\) days, and \(M_{\rm max} = 1.0217\) years. According to Theorem 2, for \(M_{\rm max} > M^{{2\bar{*}}}\), the final optimal trade credit period \(M^{2*} = M^{{2\bar{*}}} = 0.3989\) years. Consequently, \(Q^{2*} = 655\) units; \(D^{2} = 5365\) units/year, an increase of 49.03 % (\(\frac{5365  3600}{3600}*100\,\% = 49.03\,\%\)); \(\Pi_{\text{r}}^{2} (M^{2*} ) = \$ 67010\)/year, an increase of 65.39 % (\(\frac{67010  40517}{40517}*100\,\% = 65.39\,\%\)) from the decentralized decision; \(\Pi_{\text{s}}^{2} (M^{2*} ) = \$ 39279\)/year, an increase of 5.72 % (\(\frac{39279  37153}{37153}*100\,\% = 5.72\,\%\)) from the decentralized decision; \(\Pi_{\text{sc}}^{2} (M^{2*} ) = \$ 106289\)/year, an increase of 36.41 % (\(\frac{106289  77918}{77918}*100\,\% = 36.41\,\%\)) from the centralized decision. However, we notice that the default risk is \({\text{F}}^{ 2} = 11.3\,\%\).
In a supplierStackelberg game, we obtain \(\left( {a  b} \right)W  aC  CI_{s}  \frac{{aS_{s} }}{2}\sqrt {\frac{{h_{r} }}{{2S_{r} K}}}  \frac{{3ah_{s} }}{2A}\sqrt {\frac{{S_{r} K}}{{2h_{r} }}} = 2.44 > 0\). Consequently, \(M^{{3\bar{*}}} = 0.4273\) years \(= 156\) days. According to Theorem 5, for \(M_{\rm max} > M^{{3\bar{*}}}\), the final optimal trade credit period \(M^{3*} = M^{{3\bar{*}}} = 0.4273\) years. Consequently, \(Q^{3*} = 664\) units/order; \(D^{3} = 5519\) units/year, an increase of 53.31 %; \(\Pi_{\text{r}}^{3} (M^{3*} ) = \$ 69420\)/year, an increase of 71.34 % from the decentralized decision; \(\Pi_{\text{s}}^{3} (M^{3*} ) = \$ 39291\)/year, an increase of 5.75 % from the decentralized decision; \(\Pi_{\text{sc}}^{3} (M^{3*} ) = \$ 108711\)/year, an increase of 39.97 % from the centralized decision. However, default risk is \({\text{F}}^{ 3} = 12\, \%\).
The sensitivity analysis on parameters (Note that \(M^{j*}\) is in days, \(F^{j}\) is in percentage)
Parameters  Nash game  Decision without trade credit  SupplierStackelberg game  \(\frac{{\Pi_{r}^{3} }}{{\Pi_{r}^{0} }}\)  \(\frac{{\Pi_{s}^{3} }}{{\Pi_{s}^{0} }}\)  \(\frac{{\Pi_{sc}^{3} }}{{\Pi_{sc}^{0} }}\)  

\(M^{2*}\)  \(F^{2}\)  \(D^{2}\)  \(\Pi_{r}^{2}\)  \(\Pi_{s}^{2}\)  \(\Pi_{sc}^{2}\)  \(\Pi_{r}^{0}\)  \(\Pi_{s}^{0}\)  \(\Pi_{sc}^{0}\)  \(\Pi_{sc}^{1}\)  \(M^{3*}\)  \(F^{3}\)  \(D^{3}\)  \(\Pi_{r}^{3}\)  \(\Pi_{s}^{3}\)  \(\Pi_{sc}^{3}\)  
a  
1.4  266  19.7  10,001  135,670  50,654  186,324  40,517  37,153  77,670  77,918  266  19.7  10,001  135,670  50,654  186,324  3.349  1.363  2.400 
1.2  228  17.1  7606  100,451  43,880  144,331  40,517  37,153  77,670  77,918  234  17.5  7780  103,210  43,889  147,099  2.547  1.181  1.894 
1  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
0.8  13  1.1  3707  42,141  37,187  79,328  40,517  37,153  77,670  77,918  28  2.3  3827  43,961  37,199  81,160  1.085  1.001  1.045 
0.6  0  0  3600  40,517  37,153  77,670  40,517  37,153  77,670  77,918  0  0  3600  40,517  37,153  77,670  1.000  1.000  1.000 
0.4  0  0  3600  40,517  37,153  77,670  40,517  37,153  77,670  77,918  0  0  3600  40,517  37,153  77,670  1.000  1.000  1.000 
A  
16,000  149  11.6  5421  67,885  39,588  107,473  40,517  37,316  77,833  78,148  162  12.4  5606  70,778  39,605  110,382  1.747  1.061  1.418 
14,000  149  11.5  5408  67,674  39,514  107,188  40,517  37,277  77,794  78,092  160  12.3  5585  70,450  39,529  109,979  1.739  1.060  1.414 
12,000  147  11.4  5390  67,395  39,416  106,811  40,517  37,225  77,742  78,019  158  12.2  5558  70,017  39,430  109,447  1.728  1.059  1.408 
10,000  147  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
8000  143  11.1  5328  66,436  39,077  105,514  40,517  37,044  77,561  77,771  152  11.8  5463  68,543  39,087  107,629  1.692  1.055  1.388 
6000  139  10.8  5268  65,503  38,747  104,250  40,517  36,863  77,380  77,535  146  11.3  5372  67,127  38,752  105,879  1.657  1.051  1.368 
W  
29  372  26.3  9962  90,595  75,818  166,412  18,917  58,753  77,670  77,918  373  26.4  10,000  91,081  75,820  166,901  4.815  1.291  2.149 
27  302  22  8235  83,902  61,923  145,825  26,117  51,553  77,670  77,918  308  22.4  8372  85,782  61,930  147,711  3.285  1.201  1.902 
25  227  17  6706  75,911  49,779  125,690  33,317  44,353  77,670  77,918  235  17.6  6855  78,092  49,788  127,881  2.344  1.123  1.647 
23  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  155  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
21  56  4.5  4202  57,570  30,316  87,886  47,717  29,953  77,670  77,918  70  5.6  4357  60,137  30,331  90,468  1.260  1.013  1.165 
19  0  0  3600  54,917  22,753  77,670  54,917  22,753  77,670  77,918  0  0  3600  54,917  22,753  77,670  1.000  1.000  1.000 
b  
0.4  0  0  3600  40,517  37,153  77,670  40,517  37,153  77,670  77,918  0  0  3600  40,517  37,153  77,670  1.000  1.000  1.000 
0.35  64  6  4294  50,689  37,652  88,341  40,517  37,153  77,670  77,918  76  7  4431  52,732  37,665  90,397  1.302  1.014  1.164 
0.3  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
0.25  246  15.5  7061  94,095  42,779  136,873  40,517  37,153  77,670  77,918  254  16  7230  96,865  42,788  139,653  2.391  1.152  1.798 
0.2  373  18.5  10,000  143,730  49,868  193,598  40,517  37,153  77,670  77,918  373  18.5  10,000  143,730  49,868  193,598  3.547  1.342  2.493 
C  
16  0  0  3600  40,517  22,753  63,270  40,517  22,753  63,270  63,518  0  0  3600  40,517  22,753  63,270  1.000  1.000  1.000 
14  1  0.1  3612  40,691  29,956  70,647  40,517  29,953  70,470  70,718  14  1.2  3743  42,579  29,970  72,549  1.051  1.001  1.030 
12  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
10  316  22.9  8562  119,074  54,116  173,191  40,517  44,353  84,870  85,118  323  23.3  8722  121,782  54,123  175,905  3.006  1.220  2.073 
8  373  26.4  10,000  143,730  75,745  219,475  40,517  51,553  92,070  92,318  373  26.4  10,000  143,730  75,745  219,475  3.547  1.469  2.384 
S _{ s }  
400  135  10.5  5212  64,632  38,436  103,069  40,517  36,482  76,999  77,452  151  11.7  5443  68,231  38,464  106,695  1.684  1.054  1.386 
350  140  10.9  5288  65,817  38,858  104,675  40,517  36,817  77,334  77,680  153  11.8  5481  68,825  38,877  107,702  1.699  1.056  1.393 
300  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
250  150  11.7  5442  68,207  39,701  107,908  40,517  37,488  78,005  78,169  158  12.2  5558  70,017  39,707  109,724  1.728  1.059  1.407 
200  156  12  5519  69,412  40,122  109,534  40,517  37,824  78,340  78,434  161  12.4  5596  70,615  40,125  110,740  1.743  1.061  1.414 
150  161  12.4  5596  70,624  40,543  111,167  40,517  38,159  78,676  78,716  163  12.6  5634  71,216  40,544  111,759  1.758  1.063  1.421 
h _{ s }  
5.5  143  11.1  5332  66,499  39,100  105,599  40,517  37,056  77,573  77,787  153  11.8  5469  68,639  39,109  107,748  1.694  1.055  1.389 
5  144  11.2  5348  66,753  39,189  105,942  40,517  37,105  77,621  77,852  154  11.9  5494  69,027  39,200  108,227  1.704  1.057  1.394 
4.5  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
4  147  11.4  5381  67,266  39,370  106,636  40,517  37,201  77,718  77,985  158  12.2  5545  69,817  39,383  109,200  1.723  1.059  1.405 
3.5  148  11.4  5398  67,525  39,461  106,987  40,517  37,249  77,766  78,053  159  12.3  5570  70,218  39,476  109,694  1.733  1.060  1.411 
3  149  11.5  5415  67,786  39,553  107,340  40,517  37,298  77,814  78,122  161  12.4  5596  70,624  39,569  110,194  1.743  1.061  1.416 
I _{ s }  
0.14  104  8.2  4785  58,091  38,356  96,447  40,517  37,153  77,670  77,918  114  9  4923  60,196  38,368  98,565  1.486  1.033  1.269 
0.12  124  9.7  5053  62,183  38,769  100,952  40,517  37,153  77,670  77,918  134  10.4  5199  64,433  38,781  103,214  1.590  1.044  1.329 
0.1  146  11.3  5365  67,010  39,279  106,289  40,517  37,153  77,670  77,918  156  12  5519  69,420  39,291  108,711  1.713  1.058  1.400 
0.08  170  13  5733  72,778  39,910  112,688  40,517  37,153  77,670  77,918  180  13.8  5898  75,370  39,921  115,291  1.860  1.075  1.484 
0.06  197  14.9  6176  79,790  40,688  120,478  40,517  37,153  77,670  77,918  207  15.7  6350  82,581  40,700  123,280  2.038  1.096  1.587 
0.04  228  17.1  6715  88,476  41,655  130,131  40,517  37,153  77,670  77,918  237  17.7  6901  91,484  41,667  133,151  2.258  1.122  1.714 
The sensitivity analysis reveals the following.
(i) \(M^{2*}\), \(F^{2}\), \(D^{2}\), \(\Pi_{r}^{2}\), \(\Pi_{s}^{2}\), \(M^{3*}\), \(F^{3}\), \(D^{3}\), \(\Pi_{r}^{3}\), \(\Pi_{s}^{3}\) increase as a, \(A\) and \(W\) increase, and decrease as \(b\), \(C\), \(S_{s}\), \(h_{s}\) and \(I_{s}\) increase. This coincides with the Corollary 1 and Corollary3, and the purpose of two games. Note that \(M^{j*} = 0\) or \(M^{j*} = M{}_{\rm max}\)(\(j = 2\), 3), the conclusion is invalid.
(ii) \(M^{2*}\), \(F^{2}\), \(D^{2}\), \(\Pi_{r}^{2}\), \(\Pi_{s}^{2}\), \(M^{3*}\), \(F^{3}\), \(D^{3}\), \(\Pi_{r}^{3}\), \(\Pi_{s}^{3}\) are high sensitive to a, \(W\), \(b\) and \(C\), moderate sensitive to \(S_{s}\) and \(I_{s}\), low sensitive or insensitive to \(A\) and \(h_{s}\). On the one hand, the result shows that the change of trade credit period is greatly influenced by increasing demand coefficient \(a\), default risk coefficient \(b\), wholesale price \(W\), and production cost \(C\). Hence, the two parties should make joint promotional effort to improve the value of \(a\), and to reduce the value of \(b\), such that lead to higher the demand and higher the two parties revenue. Meanwhile, the supplier should strive to reduce the production costs \(C\) through a variety of efficient measures, or raise the wholesale price \(W\). Both of strategies can incentive the supplier to willing to offer a longer trade credit period to raise the profit of the retailer and the supplier. On the other hand, the result also imply that some errors in estimating \(A\) and \(h_{s}\) may result in little deviation from the optimal results. Hence, in practice, the supplier does not have too high surplus production capacity, and does not need to accurately estimate on the inventory holding cost \(h_{s}\).
(iii) For the same conditions, the profits of the retailer, the supplier, and supply chain in a supplierStackelberg game are better than the results in Nash game, i.e., \(\Pi_{r}^{3} > \Pi_{r}^{2}\), \(\Pi_{s}^{3} > \Pi_{s}^{2}\), and \(\Pi_{sc}^{3} > \Pi_{sc}^{2}\) in the sensitivity analysis. This is because the optimal trade credit period \(M^{3*}\) is greater than \(M^{2*}\), such that the market demand from supplierStackelberg game is greater than from Nash game, i.e., \(D^{3} > D^{2}\). However, we find that the supplier will burden higher default risk in a supplierStackelberg game, i.e., \(F^{3} > F^{2}\). Likewise, if \(M^{j*} = 0\) or \(M^{j*} = M{}_{\rm max}\)(\(j = 2\), 3), the conclusion is invalid.
(iv) Under the same conditions, we have \({{\Pi_{r}^{3} } \mathord{\left/ {\vphantom {{\Pi_{r}^{3} } {\Pi_{r}^{0} }}} \right. \kern0pt} {\Pi_{r}^{0} }} > {{\Pi_{s}^{3} } \mathord{\left/ {\vphantom {{\Pi_{s}^{3} } {\Pi_{s}^{0} }}} \right. \kern0pt} {\Pi_{s}^{0} }}\) when \(M^{3*} > 0\). There are two major reasons can explain the phenomenon. First, the supplier burdens an additional capital opportunity cost, i.e., \(CKe^{aM} I_{s} M\). Second, the supplier burdens the default risk of trade credit from the retailer reduces his or her expected net revenue. Moreover, as shown in Table 1, we find that if \(0 < M^{3*} < M_{\rm max}\), \({{\Pi_{r}^{3} } \mathord{\left/ {\vphantom {{\Pi_{r}^{3} } {\Pi_{r}^{0} }}} \right. \kern0pt} {\Pi_{r}^{0} }}\), \({{\Pi_{s}^{3} } \mathord{\left/ {\vphantom {{\Pi_{s}^{3} } {\Pi_{s}^{0} }}} \right. \kern0pt} {\Pi_{s}^{0} }}\) and \({{\Pi_{sc}^{3} } \mathord{\left/ {\vphantom {{\Pi_{sc}^{3} } {\Pi_{sc}^{0} }}} \right. \kern0pt} {\Pi_{sc}^{0} }}\) also increase as a, \(A\) and \(W\) increase, and decrease as \(b\), \(C\), \(S_{s}\), \(h_{s}\) and \(I_{s}\) increase. Additionally, \({{\Pi_{r}^{3} } \mathord{\left/ {\vphantom {{\Pi_{r}^{3} } {\Pi_{r}^{0} }}} \right. \kern0pt} {\Pi_{r}^{0} }}\), \({{\Pi_{s}^{3} } \mathord{\left/ {\vphantom {{\Pi_{s}^{3} } {\Pi_{s}^{0} }}} \right. \kern0pt} {\Pi_{s}^{0} }}\) and \({{\Pi_{sc}^{3} } \mathord{\left/ {\vphantom {{\Pi_{sc}^{3} } {\Pi_{sc}^{0} }}} \right. \kern0pt} {\Pi_{sc}^{0} }}\) is highly sensitive to a, \(W\), \(b\) and \(C\), moderately sensitive to \(I_{s}\), and has a low sensitive or insensitive to \(S_{s}\), \(A\) and \(h_{s}\).
(v) In most situations, \(\Pi_{sc}^{3} > \Pi_{sc}^{1}\), \(\Pi_{sc}^{2} > \Pi_{sc}^{1}\), i.e., the total profits of supply chain under the two games are better than the results under centralized decision, only if the optimal trade credit period should not be too short. That is to say, the supply chain’s total profits with longer trade credit period under the two games are both greater than the profit of centralized decision. It indicates that trade credit can be used as coordination parameter.
Conclusions
How to determinate an optimal trade credit period? This question is gaining more and more attention from researchers. In this paper, we discuss about two retailer–supplier uncooperative replenishment models with default risk under trade credit policy, i.e., a Nash equilibrium model and a supplierStackelberg model.
Generally, the main trait of this paper compared to most existing uncooperative inventory model is that the developed model includes the following aspects: (i) Nash equilibrium game and supplierStackelberg game; (ii) the results of decentralized decision and centralized decision without trade credit as comparison benchmarks; (iii) the retailer’s capital opportunity cost equal to its opportunity gain; (iv) trade credit period is a decision variable; (v) the demand and default risk both are exponential functions of trade credit period; (vi) lotforlot policy; and (vii) the production rate is finite but the replenishment is instantaneous. The major contribution of the paper is that we fully compare between the results of decentralized and centralized decision without trade credit, Nash equilibrium and a supplierStackelberg model with trade credit in detail, and obtain some interesting managerial insights and practical implications.
In this paper, we first derive the existence and uniqueness conditions of the optimal solutions for the retailer and the supplier under noncollaborative replenishment policies, i.e., Nash equilibrium game and supplierStackelberg game. Moreover, we develop a set of theorems and corollaries to determine the optimal solution and obtain some managerial insights. For instance, a higher value of \(a\), \(W\), \(A\) and a lower value of \(b\), \(C\), \(S_{s}\), \(h_{s}\), \(I_{s}\) cause a higher value of \(M^{{2\bar{*}}}\) and \(M^{{3\bar{*}}}\). Finally, we provide an example and sensitivity analysis to illustrate the proposed strategy. Sensitivity analysis reveals that the total profits of supply chain under the two games are better than the results under centralized decision when the optimal trade credit period isn’t too short, also suggests that the size of \(M^{j*}\), \(F^{j}\), \(D^{j}\), \(\Pi_{r}^{j}\), and \(\Pi_{s}^{j}\) (\(j = 2,3\)) have a strong relationship with a, \(W\), \(b\) and \(C\). In addition, we present other main managerial insights.
The previous results have some practical implications. On the one hand, a supplier may offer a retailer trade credit to expand the demand under certain conditions, especially for the growth phase or launch phase of a product life cycle, even to avoid lasting price competition from competitors. We usually observe that sales volume increases with trade credit period, but production cost decreases with time during the two stages of product life cycle. On the other hand, trade credit is an important financing tool for retailers, especially, the small and micro or startingup retailer having lack of capital. However, for trade credit of default risk from retailers, the supplier should carefully select good retailers. Moreover, the retailer should set up a fine credit record in the markets, or a longterm relationship with the supplier.
For further research, we may extend the model to allow for other demand functions, such as quadratic trade credit period demand, varying demand both with trade credit period and time, etc. In addition, we may further consider deteriorating items, shortages, environmental impact, warehouse capacity constraint and single supplier/multiretailer noncoordination and others. Therefore, the effects of all of these additional scenarios may be incorporated in future research.
Declarations
Authors’ contributions
CFW drafted the manuscript and analyzed the data. QHZ designed the study. All authors read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (71471006, 91224007), Ministry of education of Humanities and Social Science Youth Fund Project (16YJC630135), Shandong Province Soft Science Research Plan Project (2014RKB01289).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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