DESIGNING A SUPPLY CHAIN NETWORK UNDER A DYNAMIC DISCOUNTING-BASED CREDIT PAYMENT PROGRAM

This study examines the effects of dynamic discounting based credit payment on a supply chain network design problem. Dynamic discounting based credit payment is a supply chain finance policy wherein the supplier provides a credit period to a distribution center (DC) with a discount applied if the DC pays the supplier before the end of the credit period. This study also considers the time value of money and applies discounted cash flows to formulate a model that determines the DC’s optimal replenishment cycle, selling price, and influence area while maximizing the present value of the total profit. The continuous approximation approach is applied to formulate a mathematical model of the problems, and an algorithm based on non-linear optimization is established to solve the problem. A numerical example and a sensitivity analysis are provided to present the proposed model and solution approach and to illustrate the effect of each cost on the decisions and profit. Mathematics Subject Classification. 90B05. Received September 11, 2020. Accepted July 21, 2021.

are increasingly turning to dynamic discounting to solve the growing problem of late payments. Procurement software provider, Wax Digital, asked 100 UK procurement professionals about dynamic discounting, and 27% said they are already using it. Another 30% said that they are planning on implementing the approach in the next 12 months, and 20% said it is a long-term objective. In total, 77% of businesses (more than three quarters) is using or tend to use dynamic discounting in the future [6]. Specially, in the coronavirus era, many businesses have lost significant revenue, many see early pay via dynamic discounting as a win-win -help suppliers get cash, and buyers manage cash at much better yields than alternatives [13]. Therefore, more researches of dynamic discounting is necessary to see how to apply it into the real case. Gelsomino et al. [9] examined dynamic discounting in a consumer goods retailer in Italy and found that the amount of daily discount has a linear effect on profit. Meanwhile, no studies considered dynamic discounting in supply chain network design problems. To fill this gap, our study considers the dynamic discounting-based credit payment in designing a supply chain network.
There are two kind of models (discrete model and continuous model) in supply chain network design literature. The discrete model is a common method used in formulating the mathematical model [2,3,8,28]. Generally, a discrete model can obtain an optimal solution, but it needs a high level of data and intensive computations, especially if the study tries to illustrate real operational networks [24,27]. Furthermore, a high level of data leads to a low level of data reliability and model accuracy [24]. Thus, to overcome these disadvantages, some studies use a continuous approximation (CA) approach, which provides a near-closed form solution and reduces the complexity of the problem [5]. The basic idea of the CA approach is to reduce the complexity of the problem by using a continuous function when defining the decision variables [24]. For example, the location of facilities under a CA approach is defined as an influence area, not as an exact location. Dasci and Verter [5] used a CA approach instead of a discrete approach for designing a production-distribution system. Pujari et al. [21] presented a CA approach for determining the optimal number and size of shipment in an inventory distribution model. Tsao et al. [24] proposed a CA approach for designing a distributor center network. Furthermore, Wang and Ouyang [27] developed a CA approach in a competitive facility location problem that also considered facility disruption risk. Other studies that also used the CA approach for model formulation are Tsao [22], Tsao and Linh [23], and Tsao et al. [25].
This study also applies the time value of money concept or discounted cash flow analysis (DCF). Theoretically, DCF assumes that the current price is equal to the cumulative discounted future cash flows at a constant rate [26]. This theory is supported by the fact that both outflow and inflow in a supply chain happen at different times; thus, they have different values. For example, if r is the annual compound interest rate per dollar per year, then $1 today is equal to $e r one year later, and $1 one year from now is equal to $e −r today [16]. Therefore, it is important to consider the time value of money and apply discounted cash flow analysis to illustrate real-life situation. Many studies, such as Li et al. [16] and Tsao et al. [25], have applied DCF analysis.
In this study, dynamic discounting is used as the payment scheme in a supply chain network design problem. To the best of the author's knowledge, this is the first study that examines the effects of a dynamic discounting scheme on a supply chain network problem. In such a scheme, the supplier provides a credit period to the distribution center (DC) and a discount is applied if the DC pays the supplier before the end of the credit period. This study also considers the time value of money and conducts a DCF analysis. Under a dynamic discounting scheme, the DC determines the replenishment cycle, selling price, and influence area to maximize the present value of the total profit. The problems are formulated using the CA approach, which requires fewer data and computational time. In using this approach, the derived theoretical results, as well as the closed-form solutions, are provided directly. Moreover, this study develops an algorithm based on non-linear optimization to solve the problem. The proposed algorithm looks for the optimum solutions that maximize the present value of the total profit. Lastly, this study provides numerical and sensitivity analyses that provide managerial insights.
The rest of the paper is organized as follows. Section 2 describes the problem, including assumptions and notations. Section 3 discusses the model formulation. Section 4 introduces the theoretical results and the proposed solution approach, as well as the algorithm. Section 5 provides a numerical analysis to gain managerial insights. Section 6 presents the conclusions and avenues for future studies.

Problem definition
In this study, a network consists of a single supplier, multiple DCs, and multiple retailers, as illustrated in Figure 1. In practice, this kind of network is common, Target, Frito-Lay, and Wal-Mart are examples [22,24]. Dynamic discounting is provided by the supplier to the DC. Supplier provides a credit period (µ) to the DC and a discount is applied if the DC pays before the end of the credit period. The sooner the payment is settled, the higher the discount is. If the DC does not pay before the credit period, he must pay in the end of credit period. This study develops the model from the viewpoint of the DC. The credit bring benefits for both supplier and distribution centers (DCs). First, the supplier who provides the dynamic discounting based credit payment policy may increase his sale volume because the DCs want to pay before the end of credit time, they need to quickly sell out products (reduce the replenishment cycle time) and order more products to meet customer demand (increase the order quantity). Second, DCs do not need to pay at the ordering time, so they could invest this amount and earn some interests. Because, they can order more than ususal in a replenishment cycle time and do not need to pay immediately. The model formulation is carried out using the CA approach.
The supplier serves a given large service area which is divided into N smaller clusters i, (i = 1, 2, . . . , N ). The area of each cluster is C i . DCs are responsible for holding, packaging, and delivering to retailer. The number and location of retailers are known and fixed. Let A dci be the service area of each DC in the same cluster C i . Then, the number of DCs should be opened in a cluster, C i , is Ci A dc i . The objective of this paper is to determine the service area of each DC, C i , to know which retailer should be served by which DC. Assume that the DC's service area is roughly circular in shape [24]; irregularly shaped service areas could be roughly circular, hexagonal, or square-shaped, but they have little effect on the optimal solution [5]. Instead of finding the exact location of each DC, we assume that in any case, each DC is located at the center of its service area. Let F t be the constant that depend on the shape of DC's service area, the outbound distance from a DC to retailers in the same cluster i is F t A dci [5].
The following assumptions are used in this study.
(1) The demand for each retailer in cluster i is similar, D ri . The demand function for each retailer in cluster i is a price-dependent demand model with a linear model, as shown in equation (2.1): The demand for each DC in cluster i is the total of the demand from retailers within its influence area, as shown in equation (2.2): The total demand for DCs in cluster i is shown in equation (2.3): The supplier offers a dynamic discounting program to DC. This study uses the average discount, as shown in equation (2.5): The buyer is cash-rich and is more trustworthy than the supplier; therefore, r b < r s [10]. (7) The invoice is approved after a few days, and the buyer can pay within the credit period [9]. (8) Since this study considers the time value of money, there is an exponential function e −rt [16,17].
The following notations are used in this study.

Model formulation
The objective of the mathematical model is to maximize the present value of the total profit derived from the following elements per replenishment cycle.
(a) Revenue is obtained by multiplying the total demand in the DCs in cluster i with the DCs' selling price, as shown in equation (3.1): Facility cost is the total cost of renting and operating the DC per unit of time [24], and it is derived by multiplying the facility cost for each DC with the number of DCs in cluster i, as shown in equation (3.2): (c) Inbound transportation cost is the cost of sending shipments from the supplier to the DC [24], and it is derived by multiplying the transportation cost in a single inbound shipment, C f +C V t Q dci , with the number of shipments, as shown in equation (3.3): Outbound transportation cost is the total cost of shipping goods to retailers located within the DC's influence areas [24], and it is derived by multiplying the delivery cost with the average distance between the DC and the retailer, F t A dci , with the total demand in the DCs in cluster i, as shown in equation (3.4): Ordering cost is the total cost every time an order is placed, assuming the ordering time is T i [24]. It is derived by multiplying the cost per order with the number of DCs and the frequency of orders 1 / T i as shown in equation (3.5): Purchasing cost is the total cost of purchasing products from the supplier, and it is derived by multiplying the unit cost C V with the total demand in the DC in cluster i with the percentage of discount as an effect of early payment as shown in equation (3.6): (g) Inventory holding cost is the total cost for storing inventories at the DC [24], and it is derived by multiplying the holding cost with the average amount of inventories as shown in equation (3.7): (h) Interest charged and earned : the DCs can generate revenues by selling products before paying the supplier, thereby earning interest. However, if the DCs have inventories, they need to pay interest charged to the supplier until all inventories are sold. There are three scenarios as follows: Interest earned is shown in equation (3.8) and illustrated in Figure 2: Interest earned and interest charged in this scenario are shown in equations (3.9) and (3.10) and further illustrated in Figure 3: Interest charged and interest earned in this scenario are shown in equations (3.11)-(3.13) and further illustrated in Figure 4: Therefore, the profit function, PTP, for each case is as follows:

Theoretical results
To prove Theorem 4.1, the Hessian Matrix is used for each case as follows: Based on the sufficient condition of concavity, if L < 0, M < 0, and LM − K 2 > 0, then the Hessian Matrix (H j ) associated with PTP ij T ij , p ij , A dcij is negative definite, and PTP ij T ij , p ij , A dcij is concave in p ij and A dcij . First, the second-order derivative of PTP ij T ij , p ij , A dcij with respect to p ij is carried out. Then, the second-order derivative of PTP ij T ij , p ij , A dcij with respect to A dcij is carried out. Last, the second partial derivative of PTP ij T ij , p ij , A dcij with respect to p ij and A dcij is carried out. The second derivative for each case is presented in Appendix A. According to the result of the second derivation, the sufficient condition of concavity is satisfied for each case. Therefore, for any given T ij , PTP ij T ij , p ij , A dcij is concave in p ij and A dcij . Hence, there is an optimal solution or closed form for p ij and A dcij that is obtained from the first-order derivative of PTP ij T ij , p ij , A dcij with respect to p ij and A dcij . The closed forms of p * ij and A * dcij are shown in Appendix A.
Theorem 4.2. For any given A dcij and p ij , P T P T ij , p ij , A dcij is strictly pseudo-concave in T ij .
To prove Theorem 4.2, first, we set PTP T ij , p ij , A dcij = fj (Tij ) gj (Tij ) where g j (T ij ) = T ij . Then the second-order derivative of f j (T ij ) with respect to T ij is carried out. The result is presented in Appendix B. According to the result of the second derivation, the sufficient condition of concavity is satisfied for each case. Therefore, for any given p ij and A dcij , PTP ij T ij , p ij , A dcij is strictly pseudo-concave in T ij . Hence, there is an optimal solution or closed form for T ij that maximizes PTP ij T ij , p ij , A dcij . However, due to the high level of complexity, the closed form of T * ij is not found.

Solution approach
To solve this problem, a non-linear optimal solution approach is applied. We first find the optimal closed-form of all decision variables based on the theoretical results in Section 4.1. Two closed forms of p * ij and A * dcij for all the cases (j = 1, 2, 3) are found. However, both closed forms are dependent on each other. Therefore, we Then, we substitute p ij (T ij ) and A dcij (T ij ) into PTP ij T ij , p ij , A dcij to reduce the number of decision variables as N * 3 decision variables as follows: Due to the complicated derivative, the closed form of T * ij cannot be found from the first derivation of PTP ij (T ij ) with respect to T ij . Thus, a searching algorithm is developed to find the DC's optimal replenishment cycle, T * ij , as follow

Numerical example
This section is to show if the proposal model and solution approach could be applicable in the real data case. All used parameters are referred from company data and Indonesia market according to model's required  Figure 5. Graphic of optimal DC's selling price in cluster 1.
parameters. The Indonesian batik business is Batik Butimo, located in Yogyakarta, Indonesia is considered. Batik Butimo produces batik tulis with natural colors and sells it to several retailers. Each retailer's demand is represented by the function D ri = 405 − 2p i . Batik Butimo plans to expand its business by building DCs in Central Java. Based on historical data, there are three clusters of demand in Central Java, represented as follows: C 1 = 11 200 km 2 ; C 2 = 9220 km 2 ; and C 3 = 12 380 km 2 . Batik Butimo buys its materials from a single supplier, which uses a dynamic discounting program. Payment can be made on the same day the invoice is approved until the end of the credit period. Generally, it takes five days or 0.014 year to approve an invoice. In this example, Batik Butimo agrees to pay the supplier on the same day the invoice is approved, that is, at t p = 0.014 year. Suppose that the following parameters are used: C f = $5; C t = $8; C v = $26.32/unit; C vt = $1/unit; F dc = $8194; F t = $0.01; h dc = $0.8; I c = 0.05; I e = 0.04; O c = $25/order; r = 0.07; r b = 0.02; r s = 0.1; µ = 0.1 year; δ 1 = 0.0047; δ 2 = 0.0063; and δ 3 = 0.0069. First, a case with dynamic discounting is considered. The optimum solution for this problem is Case 2, as shown in Table 1. The optimum solution for each decision variable is illustrated in Figures 5-13. Second, Table 2 presents the results of the case without dynamic discounting. Based on the results of Table 2, we can see that the replenishment cycle time for three clusters in the case with dynamic accounting is shorter that in the case without dynamic accounting. The reason is that the DCs want to reduce the cycle time to get the benefits from early payment when dynamic discounting scheme is applied. The optimal price does not change so much when applying dynamic discounting. That means the dynamic discounting scheme affects the price slightly. However, the service areas of DCs become smaller in the case of a non-dynamic discounting application. The reason is that the DCs tend to reduce the service area when they cannot get the benefits from dynamic discounting. Finally, according to the results in Tables 1 and 2, dynamic discounting can help to increase the total profit by 0.15%. Thus, dynamic discounting could be considered in the supply chain network to increase profit.

Sensitivity analysis
A sensitivity analysis is carried out to analyze the effect of changing the cost parameter from -40%, -20%, +20%, and +40% to the present value of the total profit, as well as the optimal decision variables. These calculations are conducted to help the decision makers make the decisions when the market condition changes. The results of the sensitivity analysis are presented in Table 3.
The sensitivity analysis reveals the following.   (1) As the price effect parameter, b, increases, the DC's replenishment cycle and influence area increase, but the selling price and present value of the total profit decrease. This is understandable because a decrease in the present value of the total profit implies an increase in the present value of the total cost. Therefore, the DC's influence area and replenishment cycle are increased to reduce the number of DCs that operate and reduce the frequency of orders and the volume of inventories. (2) As the fixed cost per inbound shipment, C f , increases, the selling price and the DC's replenishment cycle and influence area increase, but the present value of the total profit decreases. The selling price is increased to maintain the present value of the total profit, and the demand decreases as an effect of increasing the   selling price. A decrease in demand means that there are not many customers to serve. Therefore the DC's influence area is increased to reduce the number of DCs that operate. Besides, as the inbound shipment cost increases, the DC's replenishment cycle is increased to reduce the frequency of product deliveries from the supplier. (3) As the delivery cost, C t , increases, the DC's selling price and replenishment cycle increase as well, but the DC's influence area and the present value of the total profit decrease. The selling price is increased to maintain the present value of the total profit and the replenishment cycle is increased to reduce the number of products for delivery. Moreover, the DC's influence area is decreased to increase the number of DCs that operate and to reduce the distance covered for delivery. (4) As the unit cost, C v , increases, the DC's influence area and selling price increase, but the DC's replenishment cycle and the present value of the total profit decrease. The selling price is increased to maintain the present value of the total profit, and the demand decreases as a result of increasing the selling price. A decrease in demand means that there are not many customers to serve. Therefore, the DC's influence area is increased to reduce the number of DCs that operate. Moreover, as the unit cost increases, the ordered quantities also decrease; therefore, the DC's replenishment cycle is decreased. (5) As the variable cost per inbound shipment, C vt , increases, the DC's influence area and selling price increase, but the DC's replenishment cycle and the present value of the total profit decrease. The selling price is increased to maintain the present value of the total profit, and the demand decreases as a result of increasing the selling price. A decrease in demand means that there are not many customers to serve. Therefore, the DC's influence area is increased to reduce the number of DCs that operate. Moreover, as the variable cost per unit of shipment increases, the ordered quantities decrease; therefore, the DC's replenishment cycle is decreased. (6) As the facility cost, F dc , increases, the DC's selling price and influence area increase, but the DC's replenishment cycle and the present value of the total profit decrease. The selling price is increased to maintain the present value of the total profit, and the DC's influence area is increased to reduce the number of DCs that operate. (7) As the holding cost, h dc , increases, the present value of the total profit and the DC's replenishment cycle decrease. However, the selling price is slightly increased to maintain the present value of the total profit, and the DC's influence area is increased to reduce the number of DCs that operate. Meanwhile, as the holding cost increases, the DC's replenishment cycle is decreased to reduce the ordered quantities and thereby reduce the inventory cost. (8) As the interest charged, I c , increases, the selling price and the DC's influence area increase, but the present value of the total profit and the DC's replenishment cycle decrease. The selling price is increased to maintain the present value of the total profit, and the demand decreases as a result of increasing the selling price. A decrease in demand means that there are not many customers to serve. Therefore, the DC's influence area is increased to reduce the number of DCs that operate. Furthermore, when the interest charged increases, it is reasonable to decrease the DC's replenishment cycle to reduce the number of inventories so that the interest charged can be reduced. (9) As the interest earned, I e , increases, the selling price and the DC's replenishment cycle decrease, but the DC's influence area and the present value of the total profit increase. The selling price is decreased to stimulate and increase the demand when the interest earned is high, thereby increasing the present value of the total profit.
(10) As the ordering cost, O c , increases, the DC's selling price, influence area, and replenishment cycle increase, but the present value of the total profit decreases. The selling price is increased to maintain the present value of the total profit, and the demand decreases as the selling price increases. A decrease in demand means that there are not many customers to serve. Therefore, the DC's influence area is increased to reduce the number of DCs that operate. Meanwhile, when the ordering cost increases, it is reasonable to increase the DC's replenishment cycle to reduce the replenishment frequency so that the ordering cost can be reduced. (11) As the annual interest rate, r, increases, the DC's selling price and influence area increase, but the DC's replenishment cycle and the present value of the total profit decrease. This is reasonable because the higher the annual interest rate, the smaller the present value of money. Moreover, when the annual interest rate increases, the selling price is also increased to maintain the profit, and the demand decreases as a result of increasing the selling price. A decrease in demand and the present value of the total profit means that there are not so many customers to serve and that the total cost has increased. Therefore, the DC's influence area is increased to reduce the number of DCs that operate and to reduce the number of ordered quantities and the volume of inventories.
As the cost of debt of the buyer, r b , or the supplier, r s , increases, the DC's influence area and selling price decrease, but the DC's replenishment cycle and the present value of the total profit increase. Here, it is reasonable to decrease the selling price because the buyer's cost of debt in a dynamic discounting scheme is used as a basis to determine the amount of discount. Therefore, the total cost that should be covered by revenues decreases because of the higher discount. Moreover, the demand increases as the selling price decreases, and when the demand increases, the DC's influence area is decreased to increase the number of DCs that operate. Moreover, the DC's replenishment cycle is increased to increase the ordered quantities so that the purchasing cost can be reduced while optimizing the discount.
Aside from the results discussed above, we have several key findings as follows.
(1) The cost elements that have the most significant effect on the present value of the total profit are the price effect parameter and the unit cost. Therefore, under a dynamic discounting scheme, the manager should pay more attention to both of these cost elements before agreeing to pay at a certain time and to use dynamic discounting as the payment scheme so as to maximize the total profit. (2) The cost element that has the least significant effect on the present value of the total profit is the holding cost. Thus, under a dynamic discounting scheme, the manager can prioritize other costs in the decisionmaking process.

Conclusions
This study presented a supply chain network design that consists of a single supplier, multiple DCs, and multiple retailers. It also considered the time value of money by conducting a DCF analysis. Specifically, the analysis involved a supplier that provides a dynamic discounting program for a DC. A mathematical model was built under three cases with three decision variables, namely, the DC's replenishment cycle, selling price, and influence area so that the present value of the total profit is maximized. To solve the problem, an algorithm based on non-linear optimization is developed. A numerical example was provided to illustrate the proposed model and solution approaches. The numerical example illustrated that the proposed model can be applied to a real case problem to obtain the optimal solution. A sensitivity analysis was also conducted to demonstrate the effect of changing the cost parameter in relation to the total profit, as well as the optimal decision variables. The sensitivity analysis revealed that the price effect parameter and the unit cost have the most significant effect on the present value of the total profit. Meanwhile, the holding cost has the least significant effect on the present value of the total profit. Future studies can use a network that consists of multiple suppliers, where all suppliers provide dynamic discounting and a trade credit payment is provided to the downstream of the network (i.e., from the DC to the retailers). Furthermore, future studies can consider a perishable product and a shortage of products because both scenarios are common in real life.
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