Lot‐sizing policies for deterioration items under two‐level trade credit with partial trade credit to credit‐risk retailer and limited storage capacity

The main purpose of this article is to investigate the optimal wholesaler's replenishment decisions for deterioration items under two levels of the trade credit policy and two storage facilities in order to reflect the supply chain management situation within the economic order quantity framework. In this study, each of the following assumptions have been made: (1) The own warehouse with limited capacity always is not sufficient to store the order quantity, so that a rented warehouse is needed to store the excess units over the capacity of the own warehouse; (2) The wholesaler always obtains the partial trade credit, which is independent of the order quantity offered by the supplier, but the wholesaler offers the full trade credit to the retailer; (3) The wholesaler must take a loan to pay his or her supplier the partial payment immediately when the order is received and then pay off the loan with the entire revenue. Under these three conditions, the wholesaler can obtain the least costs. Furthermore, this study models the wholesaler's optimal replenishment decisions under the aforementioned conditions in the supply chain management. Two theorems are developed to efficiently determine the optimal replenishment decisions for the wholesaler. Finally, numerical examples are given to illustrate the theorems that are proven in this study, and the sensitivity analysis with respect to the major parameters in this study is performed. Copyright © 2016 John Wiley & Sons, Ltd.     

Seller’s optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits

In practice, a supplier often offers its retailers a permissible delay period M to settle their unpaid accounts. Likewise, a retailer in turn offers another trade credit period N to its customers. The benefits of trade credit are not only to attract new buyers who consider it a type of price reduction, but also to provide a competitive strategy other than introduce permanent price reductions. On the other hand, the policy of granting credit terms adds an additional cost to the seller as well as an additional dimension of default risk. In this paper, we first incorporate the fact that trade credit has a positive impact on demand but negative impacts on costs and default risks to establish an economic order quantity model for the seller in a supply chain with up-stream and down-stream trade credits. Then we derive the necessary and sufficient conditions to obtain the optimal replenishment time and credit period for the seller. Finally, we use some numerical examples to illustrate the theoretical results.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. Furthermore, we assume that the retailer's trade credit period offered by supplier M is not shorter than the customer's trade credit period offered by retailer N(M?N). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.     

Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit

This study models a finite horizon inventory problem for deteriorating and fashion goods under trade credit and partial backlogging conditions. Demand may vary with price or time. The supplier can extend credit to the retailer. As a result, the retailer does not have to pay for goods immediately upon acquiring them, and can instead earn interest on the retail price of the goods between the time they are sold and the end of the credit period. The proposed model considers two-phase pricing and inventory decisions. In other words, it determines both the optimal prices and the lengths of the in-stock and stock-out period. This paper is the first to consider different price decisions for in-stock and stock-out periods under trade credit. We develop an algorithm to determine the optimal pricing and replenishment strategy while still maximizing the total profit. Further, this study shows that the proposed two-phase pricing strategy is superior to a one-phase pricing strategy in terms of profit maximization. Computational analysis illustrates the solution procedures and the impacts of the related parameters on decisions and profits. The results of this study can serve as references for business managers or administrators.     

Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing

In a supplier-retailer-buyer supply chain, the supplier frequently offers the retailer a trade credit of S periods, and the retailer in turn provides a trade credit of R periods to her/his buyer to stimulate sales and reduce inventory. From the seller’s perspective, granting trade credit increases sales and revenue but also increases opportunity cost (i.e., the capital opportunity loss during credit period) and default risk (i.e., the percentage that the buyer will not be able to pay off her/his debt obligations). Hence, how to determine credit period is increasingly recognized as an important strategy to increase seller’s profitability. Also, many products such as fruits, vegetables, high-tech products, pharmaceuticals, and volatile liquids not only deteriorate continuously due to evaporation, obsolescence and spoilage but also have their expiration dates. However, only a few researchers take the expiration date of a deteriorating item into consideration. This paper proposes an economic order quantity model for the retailer where: (a) the supplier provides an up-stream trade credit and the retailer also offers a down-stream trade credit, (b) the retailer’s down-stream trade credit to the buyer not only increases sales and revenue but also opportunity cost and default risk, and (c) deteriorating items not only deteriorate continuously but also have their expiration dates. We then show that the retailer’s optimal credit period and cycle time not only exist but also are unique. Furthermore, we discuss several special cases including for non-deteriorating items. Finally, we run some numerical examples to illustrate the problem and provide managerial insights.     

基于二层信用策略的两货栈库存模型

假设供应商向零售商提供信用支付期的同时,零售商也向顾客提供信用支付期,研究了两货栈的变质物品库存模型,并讨论了模型最优解的唯一性,最后给出了最优订购策略的算法步骤与数值例子.     

部分延期付款下易腐品联合经济订货批量模型

针对易腐品供应链的联合库存决策问题展开研究.假设供应链内存在唯一的供应商和零售商,供应商提供商业信用期给零售商,但零售商需要在收到订货后,立即交付部分货款,且零售阶段由于条件限制,产品存在常数腐败率,而联合决策模型的目标是确定供应商的订货量乘数n和零售商的订货周期使得供应链的总成本最低.通过建立该问题的数学模型,证明了目标函数的性质,说明当给定n时,目标函数在每种情况下都存在唯一最优解.以此为基础,给出了相应的求解算法对该联合批量决策模型进行了求解.最后,结合运作管理实践,并通过数值算例说明了模型的有效性.     

Channel coordination under two-level trade credits and demand uncertainty

This study considers a decentralized supply chain where a retailer has an opportunity to order a product from a supplier prior to the sales season to satisfy uncertain demand. The retailer provides trade credit to end customers and makes credit period and order quantity decisions to maximize profits. The end demand is both random and credit period-dependent. On the basis of the newsvendor model, this paper focuses on channel coordination when a retailer provides trade credit to end customers. When the supplier also provides trade credit to the retailer, we show that the traditional trade credit contract cannot coordinate the channel. Four composite contracts based on trade credit (trade credit cost sharing with buy back or quantity flexibility; modified trade credit with buy back or quantity flexibility) are provided to induce the retailer to make decisions while optimizing the channel profit. This paper shows that the retailer provides a longer credit period to its customers and orders a larger quantity from the supplier under the composite contracts. With these contracts, the profit sharing between both parties depends on the wholesale price (Pareto improvement) for the fixed retail price and the purchasing cost.     

Determination of the optimal trade credit policy: a supplier-Stackelberg model

This paper considers a two-echelon supply chain, where one supplier sells through a retailer a product with a stable market demand. We focus on how the supplier induces the retailer through trade credit to order more to reduce his/her own inventory-related cost. Under a ‘supplier-Stackelberg’ setting, we provide the supplier with the method of determining two trade credit scenarios: unconditional and conditional trade credit. We show that the unconditional trade credit scenario is always beneficial to the retailer but harmful to the supplier in most situations, while the conditional trade credit scenario is always beneficial to both parties. In addition, we specify the conditions under which the provision of unconditional trade credit is beneficial to the supplier. The three insights obtained in this paper are the following: (i) When the retailer’s per-unit opportunity cost is less than his/her per-unit opportunity gain, unconditional trade credit can induce the retailer to order less instead of more. (ii) If the supplier offers the retailer unconditional trade credit, the length of trade credit offered will have an upper bound. (iii) A well-designed conditional trade credit policy can realize a win-win outcome but also enables the supplier to occupy all the savings in the channel's cost incurred by trade credit, but any unconditional trade credit policy does not.     

Some modified mathematical analytic derivations of the annual total relevant cost of the inventory model with two levels of trade credit in the supply chain system

Several recent studies in supply chain system and related areas explored various economic order quantity (EOQ) models for noninstantaneous deteriorating items with imperfect quality and trade credit financing. In particular, in the year 2007, Teng et al investigated an EOQ model in which the supplier offers the retailer the permissible delay period M and the retailer, in turn, provides the trade credit period N (with ) to his/her customers. The main purpose of this article is twofold: (a) It modifies the annual total relevant cost TVC(T) in the study of Teng et al and presents the correct derivations of TVC(T) by applying mathematical analytic tools and techniques. (b) It exposes some logical and mathematical problems in the proof of Theorem 1 in Teng et al. It also corrects and overcomes all of the errors and shortcomings by systematically presenting the complete and mathematical solution procedures in order to locate all optimal solutions for the model in Teng et al.     

随机需求下供应链商业信用契约协调

研究一个供应商和一个零售商构成的两级供应链面对随机需求时的商业信用契约协调问题。通过分析商业信用对供应商与零售商各自目标利润及供应链的总利润的影响,建立了商业信用契约协调模型,推导了最优信用期和最优订货量,指出通过信用策略可以使双方实现对利润的合理分配与协调。最后通过数值算例说明了所得结论。     

An inventory model for increasing demand under two levels of trade credit linked to order quantity

During the growth stage of a product life cycle especially for high-tech products, the demand function increases with time. In this paper, we extend the constant demand to a linear non-decreasing demand function of time and incorporate a permissible delay in payment under two levels of trade credit into the model. The supplier offers a permissible delay linked to order quantity, and the retailer also provides a downstream trade credit period to its customers. The objective is to find the optimal replenishment cycle that minimizes the retailer’s annual total relevant cost per unit time. The condition for an optimal solution to the generalized model is presented and some fundamental theoretical results are established. Finally, numerical examples to illustrate the proposed model are provided. Sensitivity analysis is performed and some relevant managerial insights are obtained.     

Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit

In real life situation, it is observed that demand of an item depends on the length of the credit period offered by the retailer to his customers which has a positive impact on demand of an item. But the impact of credit period on demand has received a very little attention by researchers. Furthermore, by allowing shortages as backlogging, the impact on the cost from the decay of the products can be balanced out. A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the permissible delay in payments. Recently, Jaggi et al. (Eur J Oper Res 190:130–135, 2008) have investigated the impact of credit linked demand on the retailer’s optimal replenishment policy. The objective of this study is to extend Jaggi et al. (Eur J Oper Res 190:130–135, 2008) model by incorporating deterioration and backlogging. That is, we formulate a two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under two-level trade credit financing and determine the retailer’s optimal replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. Furthermore, we establish some useful theorems to characterize the optimal solution and provide an easy and useful computational algorithm with the help of computer code using the software Matlab 7.0 to determine the optimal shortage point, cycle length, ordering quantity and credit period. A numerical example is included to illustrate the solution procedure for the mathematical model developed. Finally, we implement sensitivity analysis of the optimal solution with respect to the major parameters of the system and obtain some important managerial insights.     

An uncooperative order model for items with trade credit,inventory-dependent demand and limited displayed-shelf space

This paper considers a two-echelon supply chain where a supplier sells a single product through a retailer, who faces an inventory-dependent demand. The supplier hopes to incentive the retailer to order more items by offering trade credit. The retailer places the ordered items on the display shelf (DS) with limited space and stocks the remaining items (if any) that exceed the shelf capacity in his/her backroom/warehouse (BW). From the supplier’s perspective, we focus mainly on under which conditions the supplier should offer trade credit and how he/she should design such trade credit policy and corresponding ordering policy to obtain much more benefits. From the retailer’s perspective, we discuss whether the retailer needs BW and exactly how many items need to be stocked in BW when the supplier offers trade credit. We formulate a “supplier-Stackelberg” game model, from which we obtain the conditions under which the presented simple trade credit policy not only increases the overall chain profit but also each member’s profit. We also show that the trade credit policy is always more beneficial to the retailer than to the supplier if it is offered.     

The optimal order and payment policies for deteriorating items in discount cash flows analysis under the alternatives of conditionally permissible delay in payments and cash discount

This study investigates a deteriorating inventory problem in which the supplier simultaneously offers the retailer either a conditionally permissible delay in payments or a cash discount. In the case of a conditionally permissible delay, if the retailer orders more than a predetermined quantity, then he/she has a grace period to make the full payment. Otherwise, he/she must pay the payment for goods of certain proportion first while receiving the goods and has a grace period to pay off the rest. As to a cash discount, if the retailer pays for the entire amount of the order within a certain short period, then he/she will receive a cash discount from the supplier. In additions, from a financial standpoint, all cash outflows related to the inventory control that occur at different points of time have different values. Hence, it is necessary to take account of the factor of time value of monetary when drafting the replenishment policy. In a word, this paper uses an alternate approach-discount cash flow (DCF) analysis to establish an inventory problem for deteriorating items in which the supplier provides the retailer either a conditionally permissible delay or a cash discount. We then study the necessary and sufficient conditions for finding the optimal solution. Furthermore, we establish several theoretical results to obtain the solution that provides the smallest present value of all future cash flows. Finally, several numerical examples are given to illustrate the results and obtain some managerial insights.     

基于收益共享契约的零售商押金补货策略研究

押金补货是指零售商在第一阶段向供应商交纳一定的押金,在接下来的阶段中不需要再交纳额外的补货费用的补货方式。对于中小型零售商,它的补货策略往往受到自有资金的约束,当自有资金不足以订到报童模型的最优补货量时,不仅自身收益受到影响,还会影响到整条供应链的收益,本文研究了在随机市场需求,且零售商能够随时向银行贷款的情况下,基于收益共享契约的零售商的押金融资订货策略。本文从两阶段出发,分析零售商的最优订货策略,研究表明,零售商的补货策略受初始资产和收益共享系数的影响。最后,文章将其扩展到多阶段问题进行讨论。     

信用期下时滞变质品供应链库存和定价的Stackelberg均衡策略

在允许缺货和考虑资金机会成本情况下,根据时滞变质品的基本库存模型,分别构建了信用期下供应商为领导者(SL)和零售商为领导者(RL)的Stackelberg博弈模型。通过分析SL和RL下的Stackelberg博弈模型唯一均衡解,得到两个模型中均衡解的解析表达式。最后,根据数值算例分析得出:(1)在SL供应链中信用期并不总使整个供应链协调,然而在RL供应链中信用期的协调效果较好;(2)延长信用期或增加零售价格均能刺激零售商多订货;(3)在两个模型中,零售价格均随变质时刻递增,且整条供应链达到Pareto改进;(4)SL中供应商变动信用期与RL中零售商变动价格相比,SL供应链收益更高;而当信用期和价格固定,其他参数变动时,RL供应链收益更大。     

Retailer’s optimal ordering policy for deteriorating items with maximum lifetime under supplier’s trade credit financing

Many products such as fruits, vegetables, pharmaceuticals, volatile liquids, and others not only deteriorate continuously due to evaporation, obsolescence, spoilage, etc. but also have their expiration dates (i.e., a deteriorating item has its maximum lifetime). Although numerous researchers have studied economic order quantity (EOQ) models for deteriorating items, few of them have taken the maximum lifetime of a deteriorating item into consideration. In addition, a supplier frequently offers her/his retailers a permissible delay in payments in order to stimulate sales and reduce inventory. There is no interest charge to a retailer if the purchasing amount is paid to a supplier within the credit period, and vice versa. In this paper, we propose an EOQ model for a retailer when: (1) her/his product deteriorates continuously, and has a maximum lifetime, and (2) her/his supplier offers a permissible delay in payments. We then characterize the retailer’s optimal replenishment cycle time. Furthermore, we discuss a special case for non-deteriorating items. Finally, we run several numerical examples to illustrate the problem and provide some managerial insights.     

Consignment contracting: Who should control inventory in the supply chain?

Consignment is a popular form of business arrangement where supplier retains ownership of the inventory and gets paid from the retailer based on actual units sold. The popularity of such an arrangement has come with some continued debates on who should control the supply chain inventory, the supplier or retailer. This paper aims at shedding light on these debated issues. We consider a single period supply chain model where a supplier contracts with a retailer. Market demand for the product is price-sensitive and uncertain. The supplier decides his consignment price charged to the retailer for each unit sold, and the retailer then chooses her retail price for selling the product. We study and compare two different consignment arrangements: The first allows the retailer to choose the supply chain inventory, together with her retail price, and is labeled as a Retailer Managed Consignment Inventory (RMCI) program; and the second calls for the supplier to decide the inventory, together with his consignment price, and is labeled as a Vendor Managed Consignment Inventory (VMCI) program. We show that with an RMCI program, the supply chain loses at least 26.4% of its first-best (expected) profit, while with VMCI, it loses just or no more than 26.4% of the first-best profit. Second, we demonstrate that both programs lead to an equal split of the corresponding channel profit between the supplier and the retailer. These results indicate that it is beneficial both to the supplier and to the retailer when delegating the inventory decision to the supplier rather than to the retailer in the channel.