Skip to main content

Advertisement

Log in

A game-decision-theoretic approach to optimize the dynamic credit terms in supply chain finance

  • Original Research
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

An optimal credit term decision in supply chain finance often needs to be made in a dynamic way considering the varying market demand among other factors. We study the dynamic credit term optimization problem (DCTOP), where a supplier determines the credit term in conjunction with its production and inventory decision, while anticipating a buyer’s order quantity in a leader-follower game setting. The DCTOP is first approached to using a continuous time optimal control model, with analytical results characterizing the structural properties of the optimal solution. To complement the structural properties, we then develop a discrete time bilevel programming model to provide computationally tractable and implementable numerical solutions. A comprehensive computational study shows significant advantage of our optimal solutions over the heuristic credit term rules in practice, and provides managerial insights regarding the impacts of key problem parameters on the optimal solutions and coordination scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Abad, P. L., & Jaggi, C. (2003). A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive. International Journal of Production Economics, 83(2), 115–122.

    Article  Google Scholar 

  • Brennan, M.J., Maksimovic, V., & Zechner, J. (1988). Vendor financing. The Journal of Finance43(5).

  • Dong, G., Liang, L., Wei, L., Xie, J., & Yang, G. (2021). Optimization model of trade credit and asset-based securitization financing in carbon emission reduction supply chain. Annals of Operations Research, 1–50.

  • Du, R., Banerjee, A., & Kim, S.-L. (2013). Coordination of two-echelon supply chains using wholesale price discount and credit option. International Journal of Production Economics, 143(2), 327–334.

    Article  Google Scholar 

  • Emery, G. W. (1984). A pure financial explanation for trade credit. Journal of Financial and Quantitative Analysis, 19(3), 271–285.

    Article  Google Scholar 

  • Ferrando, A., & Mulier, K. (2013). Do firms use the trade credit channel to manage growth? Journal of Banking & Finance, 37(8), 3035–3046.

    Article  Google Scholar 

  • Ferrara, M., Khademi, M., Salimi, M., & Sharifi, S., et al. (2017). A dynamic stackelberg game of supply chain for a corporate social responsibility. Discrete Dynamics in Nature and Society.

  • García-Teruel, P. J., & Martínez-Solano, P. (2010). A dynamic perspective on the determinants of accounts payable. Review of Quantitative Finance and Accounting, 34, 439–457.

    Article  Google Scholar 

  • Guariglia, A., & Mateut, S. (2006). Credit channel, trade credit channel, and inventory investment: Evidence from a panel of uk firms. Journal of Banking & Finance, 30(10), 2835–2856.

    Article  Google Scholar 

  • Huyghebaert, N. (2006). On the determinants and dynamics of trade credit use: Empirical evidence from business start-ups. Journal of Business Finance & Accounting, 33(1–2), 305–328.

    Article  Google Scholar 

  • Jaber, M. Y., & Osman, I. H. (2006). Coordinating a two-level supply chain with delay in payments and profit sharing. Computers & Industrial Engineering, 50(4), 385–400.

    Article  Google Scholar 

  • Kim, J., Hwang, H., & Shinn, S. (1995). An optimal credit policy to increase supplier’s profits with price-dependent demand functions. Production Planning & Control, 6(1), 45–50.

    Article  Google Scholar 

  • Li, H., Mai, L., Zhang, W., & Tian, X. (2019). Optimizing the credit term decisions in supply chain finance. Journal of Purchasing and Supply Management, 25(2), 146–156.

    Article  Google Scholar 

  • Love, I., Preve, L. A., & Sarria-Allende, V. (2007). Trade credit and bank credit: Evidence from recent financial crises. Journal of Financial Economics, 83(2), 453–469.

    Article  Google Scholar 

  • Norden, L., & Kampen, S., et al. (2015). The dynamics of trade credit and bank debt in sme finance: International evidence| conference–2015

  • Pei, Q., Chan, H.K., Zhang, T., & Li, Y. (2022). Benefits of the implementation of supply chain financez, 1. Annals of Operations Research, 1–33.

  • Petersen, M. A., & Rajan, R. G. (1997). Trade credit: Theories and evidence. The Review of Financial Studies, 10(3), 661–691.

    Article  Google Scholar 

  • Rogers, D., Leuschner, R., & Choi, T. (2016). The rise of fintech in supply chains. Harvard Business Review.

  • Sarmah, S. P., Acharya, D., & Goyal, S. (2007). Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option. European Journal of Operational Research, 182(3), 1469–1478.

    Article  Google Scholar 

  • Schiff, M., & Lieber, Z. (1974). A model for the integration of credit and inventory management. The Journal of Finance, 29(1), 133–140.

    Article  Google Scholar 

  • Smith, J. K. (1987). Trade credit and informational asymmetry. The Journal of Finance, 42(4), 863–872.

    Article  Google Scholar 

  • Sun, J., Yuan, P., & Hua, L. (2022). Pricing and financing strategies of a dual-channel supply chain with a capital-constrained manufacturer. Annals of Operations Research, 1–21.

  • Tang, W., Li, H., Cai, K. (2020). Optimising the credit term decisions in a dual-channel supply chain. International Journal of Production Research, 1–18.

  • Tawarmalani, M., & Sahinidis, N. V. (2005). A polyhedral branch-and-cut approach to global optimization. Mathematical programming, 103(2), 225–249.

    Article  Google Scholar 

  • Tirole, J. (2010). The theory of corporate finance. New Jersey: Princeton University Press.

    Google Scholar 

  • Wilson, N., & Summers, B. (2002). Trade credit terms offered by small firms: Survey evidence and empirical analysis. Journal of Business Finance & Accounting, 29(3–4), 317–351.

    Article  Google Scholar 

  • Wilson, N., & Summers, B. (2002). Trade credit terms offered by small firms: Survey evidence and empirical analysis. Journal of Business Finance & Accounting, 29(3–4), 317–351.

    Article  Google Scholar 

  • Wilson, N., & Summers, B. (2002). Trade credit terms offered by small firms: Survey evidence and empirical analysis. Journal of Business Finance & Accounting, 29(3–4), 317–351.

    Article  Google Scholar 

  • Yang, P.-C., & Wee, H.-M. (2006). A collaborative inventory system with permissible delay in payment for deteriorating items. Mathematical and Computer Modelling, 43(3–4), 209–221.

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful for the support from the University Transportation Center (UTC) program sponsored by the U.S. Department of Transportation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Haitao Li.

Ethics declarations

Conflict of interest

Dr. Haitao Li declares that he has no Conflict of interest. Dr. Wenguang Tang declares that he has no Conflict of interest. Dr. Liuqing Mai declares that she has no Conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: an illustration of DCTOP

As an illustration of DCTOP, Fig. 10 shows an example with T time periods. Suppose that the supplier sets the credit term \(\delta _1=2\) in Period 1. Then the buyer orders \(y_1\) units of products meeting the demand of \(D_1\) with some revenue gain (indicated by the dashed inflow arrow) and carrying an inventory of \(O_1\) to Period 2. The supplier produces \(x_1\) units to fulfill the supplier’s order of \(y_1\) units while incurring its production cost (indicated by the dashed outflow arrow) and carrying an inventory of \(I_1\) to Period 2. Since \(\delta _1=2\), the supplier will not receive the payment from the buyer until Period 3 (indicated by the dashed inflow arrow). In Period 2, suppose that the supplier’s credit term \(\delta _2\) is set to be 2, then the buyer orders \(y_2\) units meeting the demand of \(D_2\) with certain revenue and carrying an inventory of \(O_2\) to Period 3; and the supplier produces \(x_2\) units and carries \(I_2\) units of inventory to Period 3. Since \(\delta _2=2\), the supplier will not receive the payment from the buyer until Period 4.

Fig. 10
figure 10

An illustration of DCTOP

Appendix B: proofs

Proof of Theorem 1

To obtain the necessary conditions, we first solve the buyer’s (the follower’s) problem, taking the supplier’s control variables \(x_t\) and \(\delta _t\) as given. We set

$$\begin{aligned} \frac{\partial L^B}{\partial y_t}=-(1-r\delta _t)f'(y_t)+\lambda ^B_t+\mu ^B_t=0 \end{aligned}$$
(40)

where

$$\begin{aligned} {\dot{\lambda }}^B_t=-\frac{\partial L^B}{\partial O}=h-\eta ^B_t,\\ \lambda ^B_T=-\alpha ,\alpha \ge 0,\alpha (\Gamma -O_T)=0. \end{aligned}$$

Because \(f'(y_t)=aby_t^{b-1}\), we obtain

$$\begin{aligned} y_t^*=\left( \frac{ab(1-r\delta _t)}{\lambda ^B_t+\mu ^B_t}\right) ^{\frac{1}{1-b}}. \end{aligned}$$
(41)

Next, we solve the supplier’s (the leader’s) problem by plugging in the buyer’s decision variable \(y_t^*\). Setting

$$\begin{aligned} \frac{\partial L^S}{\partial x_t}=-g'(x_t)+\lambda ^S_t+\mu ^S_{1t}-\mu ^S_{2t}=0 \end{aligned}$$
(42)

and

$$\begin{aligned} \frac{\partial L^S}{\partial \delta _t}=-rf(y^*_t)+[(1-r\delta _t)f'(y_t^*)-\lambda ^S_t]\times \frac{\textrm{d} y^*_t}{\textrm{d}\delta _t}+\mu ^S_{3t}-\mu ^S_{4t}=0. \end{aligned}$$
(43)

Because \(g'(x_t)=cdx_t^{d-1}\), from Eq. (42) we get

$$\begin{aligned} x_t^*=\left( \frac{cd}{\lambda ^S+\mu ^S_{1t}-\mu ^S_{2t}}\right) ^{\frac{1}{1-d}}. \end{aligned}$$

From 41, we get

$$\begin{aligned} \frac{\textrm{d}y_t^*}{\textrm{d}\delta _t}=\frac{r}{b-1} (\frac{\lambda ^B_t+\mu ^B_t}{ab})^{\frac{1}{b-1}}(1-r\delta _t)^{\frac{b}{1-b}}. \end{aligned}$$
(44)

Substituting \(f'(y_t)=aby_t^{b-1}\) and (44) into (43), we get

$$\begin{aligned} \frac{ra}{b-1}\left( \frac{ab(1-r\delta ^*_t)}{\lambda ^B_t+\mu ^B_t}\right) ^{\frac{b}{1-b}}\frac{\lambda ^B_t+\mu ^B_t-b\lambda ^S_t}{\lambda ^B_t+\mu ^B_t}+\mu ^S_{3t}-\mu ^S_{4t}=0, \end{aligned}$$
(45)

further we get

$$\begin{aligned} \delta ^*_t=\frac{1}{r}\left( 1-\left( \frac{(1-b)(\mu ^S_{3t}-\mu ^S_{4t})(\lambda ^B_t+\mu ^B_t)}{ra(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t)}\right) ^{\frac{1-b}{b}}\frac{\lambda ^B_t+\mu ^B_t}{ab}\right) . \end{aligned}$$
(46)

Substituting (46) into (41), we get

$$\begin{aligned} y_t^*=\left( \frac{(1-b)(\mu ^S_{3t}-\mu ^S_{4t})(\lambda ^B_t+\mu ^B_t)}{ra(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t)}\right) ^{\frac{1}{b}}. \end{aligned}$$

\(\square \)

Proof of Lemma 1

From the third and fourth inequalities of (2), we have: if \(\mu ^S_{3t}>0, \delta _t=0\) or \(\mu ^S_{3t}=0\); and if \(\mu ^S_{4t}>0, \delta _t={\bar{\delta }}\) or \(\mu ^S_{4t}=0\). Clearly, it is not possible for both \(\mu ^S_{3t}\) and \(\mu ^S_{4t}\) to be positive. \(\square \)

Proof of Proposition 1

From Eq. (40), since \(f'(y_t)>0\) and \(1-r\delta _t>0\), we have

$$\begin{aligned} \lambda ^B_t+\mu ^B_t>0. \end{aligned}$$
(47)

Since \(0<a<1\), \(0<b<1\), \(1-r\delta _t^*>0\), and from (47) and 45, we can get

  1. (1)

    If \(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t<0\), \(\mu ^S_{3t}-\mu ^S_{4t}<0\);

  2. (2)

    If \(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t>0\), \(\mu ^S_{3t}-\mu ^S_{4t}>0\);

  3. (3)

    If \(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t=0\), \(\mu ^S_{3t}-\mu ^S_{4t}=0\).

If \(\mu ^S_{3t}-\mu ^S_{4t}<0\), according to Lemma 1, we have \(\mu ^S_{3t}=0\), and \(\mu ^S_{4t}>0\). Thus from (5), we get \(\delta ^*_t={\bar{\delta }}.\)

If \(\mu ^S_{3t}-\mu ^S_{4t}>0\), according to Lemma 1, we have \(\mu ^S_{3t}>0\), and \(\mu ^S_{4t}=0\). Thus from (5), we get \(\delta ^*_t=0.\)

If \(\mu ^S_{3t}-\mu ^S_{4t}=0\), according to Lemma 1, we have \(\mu ^S_{3t}=0\), and \(\mu ^S_{4t}=0\). Thus from (5), we get \(0<\delta ^*_t<{\bar{\delta }}.\) \(\square \)

Proof of Proposition 2

Taking derivative of Eq. (45) with respect to r on both sides, we obtain

$$\begin{aligned} \delta ^*_t+r\frac{\textrm{d}\delta _t^*}{\textrm{d}r}=\frac{(1-b)(1-r\delta ^*_t)}{br}, \end{aligned}$$
(48)

thus

$$\begin{aligned} \frac{\textrm{d}\delta _t^*}{\textrm{d}r}=\frac{1-r\delta ^*_t-b}{br^2}, \end{aligned}$$

Therefore

  1. (1)

    If \(1-r\delta ^*_t-b>0\), \(\frac{\textrm{d}\delta _t^*}{\textrm{d}r}>0\);

  2. (2)

    If \(1-r\delta ^*_t-b<0\), \(\frac{\textrm{d}\delta _t^*}{\textrm{d}r}<0\);

\(\square \)

Proof of Proposition 3

Taking derivative of Eq. (40) with respect to r on both sides, we obtain \((1-r\delta ^*_t)f''(y^*_t)\frac{\textrm{d}y_t^*}{\textrm{d}r}=(\delta ^*_t+r\frac{\textrm{d}\delta _t^*}{\textrm{d}r})f'(y^*_t)\) Since \((1-r\delta ^*_t)>0\), \(f'(y^*_t)>0\), \(f''(y^*_t)<0\), and from (48) we have \(\delta ^*_t+r\frac{\textrm{d}\delta _t^*}{\textrm{d}r}>0\), thus \(\frac{\textrm{d}y_t^*}{\textrm{d}r}<0\). \(\square \)

Appendix C: a case study

ABC Inc. builds a special cogwheel as a part in mowers for its customer Evergreen Inc., a distributor of equipment and hardware in the region. Due to the seasonality of mower demand, the demand of the cogwheel varies over a year. The marketing department at ABC Inc. is able to provide historical data to characterize the seasonal demand of the cogwheel sold to Evergreen Mowing. Using the accounting data, ABC estimates its production cost function to be \(g(x_t)=80x_t^{0.9}\) with moderate economies of scale, with a maximum production capacity of 150 units of cogwheel per week. ABC Inc. offers quantity discount for Evergreen Inc. such that the purchasing cost is a function of the buyer’s order quantity \(y_t\), i.e., \(f(y_t)=120y_t^{0.9}\). The market sales price p of the cogwheel is $500 per unit. The inventory holding cost h is assumed to be $0.5 per unit during the planning horizon at both ABC Inc. and Evergreen Inc. ABC Inc. would not want to hold more than \(\Gamma =500\) units of inventory at the end of the planning horizon. The marginal increase of demand is conservatively estimated to be 0.1 units per day of delayed payment. The maximal credit term offered by ABC Inc. to Evergreen Inc. cannot exceed 150 days.

ABC Inc. has been implementing the most commonly used net 30 payment term in industry, meaning that the buyer has a 30-day length of time to pay the total amount of the invoice. Occasionally, ABC Inc. adjusts the payment terms to be longer for long-existing customers, or to be shorter for new or small-size customers. The company soon realizes that a drawback of such heuristic rule is that it does not exploit the incentive or disincentive of purchasing due to the length of payment term, nor does it capture the potential impact on production lot size and inventory level. Thus ABC Inc. would like to explore the potential adoption of a flexible and dynamic credit term solution in a multi-period setting.

Working with a team of Supply Chain Analytics experts, a bilevel programming model is built for the addressed DCTOP. Solving it using the exact method based on the KKT conditions in Sect. 4.2 yields the following optimal solution to the base scenario as shown in Fig. 11. The optimal payment term prescribes 100 days in Period 3 and 42 days in Period 9, with all other periods have zero days of delayed payment. This corroborates Proposition 1 that the optimal payment term may vary between 0 and the upper bound of the length of delayed payment. It is optimal for ABC Inc. (the supplier) to produce at its maximum capacity in certain time periods, while having no production in others and keeping varying levels of inventory in multiple periods. For Evergreen Inc. (the buyer), it is optimal to order the exact demand quantity with no inventory. Note that in Periods 3 and 9, the buyer’s order quantities are higher than the nominal exogenous demand due to the offer of delayed payment.

Fig. 11
figure 11

Optimal solution to the base scenario

To see how the optimization solution compares to the status quo, the base scenario is also solved using three heuristic rules by fixing the length of delayed payment to be 10, 30 and 60, respectively. Figure 12 compares the supplier’s total profit of the four solutions. The optimal solution generates 2.9% higher total profit than the mostly used 30-day payment term, and 0.7% higher than the better performed 10-day payment term.

Fig. 12
figure 12

The supplier’s maximum profit of four solutions

While the case study shows how the optimal DCTOP solution functions and its benefit over the heuristic payment rules, it also raises the following questions: (i) How would the optimal payment term behave when some key input parameters vary, e.g., would it tend to increase or decrease?; (ii) In what problem space characterized by the input parameters would the optimal DCTOP solution be more beneficial over the heuristic payment rules? To address these questions, we performed a comprehensive computational experiment for a sensitivity analysis to be presented next.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, H., Tang, W. & Mai, L. A game-decision-theoretic approach to optimize the dynamic credit terms in supply chain finance. Ann Oper Res 340, 913–941 (2024). https://doi.org/10.1007/s10479-024-06178-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-024-06178-z

Keywords

Navigation