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.
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The authors are grateful for the support from the University Transportation Center (UTC) program sponsored by the U.S. Department of Transportation.
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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.
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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.
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
where
Because \(f'(y_t)=aby_t^{b-1}\), we obtain
Next, we solve the supplier’s (the leader’s) problem by plugging in the buyer’s decision variable \(y_t^*\). Setting
and
Because \(g'(x_t)=cdx_t^{d-1}\), from Eq. (42) we get
From 41, we get
Substituting \(f'(y_t)=aby_t^{b-1}\) and (44) into (43), we get
further we get
Substituting (46) into (41), we get
\(\square \)
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
Since \(0<a<1\), \(0<b<1\), \(1-r\delta _t^*>0\), and from (47) and 45, we can get
-
(1)
If \(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t<0\), \(\mu ^S_{3t}-\mu ^S_{4t}<0\);
-
(2)
If \(\lambda ^B_t+\mu ^B_t-b\lambda ^S_t>0\), \(\mu ^S_{3t}-\mu ^S_{4t}>0\);
-
(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
thus
Therefore
-
(1)
If \(1-r\delta ^*_t-b>0\), \(\frac{\textrm{d}\delta _t^*}{\textrm{d}r}>0\);
-
(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.
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.
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.
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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
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DOI: https://doi.org/10.1007/s10479-024-06178-z