A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit

In this paper, we proposed a generalized, integrated, supplier–retailer inventory model using a trade credit policy. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides customers a permissible delay period N. Cases where M > N and M ? N are explored thoroughly. In addition, the demand rate is assumed to be a function of both retail price and the customers’ credit period. Consequently, this paper deals with the problem of determining the optimal retail price, economic order quantity, and the number of shipments from the supplier to the retailer in one production run for an integrated inventory system under both two-level trade credit and price-and-credit-linked demand rate. Algorithms are developed in order to determine the joint optimal policies. Numerical examples are presented to illustrate the proposed models, as well as sensitivity analysis of key parameters.     

Two-period supply chain with flexible trade credit contract

This paper studies a two-period supply chain that consists of a retailer and a supplier. A newsvendor-like retailer is capital constrained and orders products using a supplier's trade credits to satisfy uncertain market demand. Most existing studies show that the retailer always postpones payment until the due date. To recall the loans earlier, we present a case in which the supplier, as a Stackelberg leader, offers an incentive of a discounted wholesale price in the second order to entice the retailer to choose flexible early payment. The proposed incentive is related to the retailer's early payment time in the first period. In the presence of bankruptcy risks for both the retailer and supplier, we propose a continuous newsvendor model of a two-period supply chain to analyze the decisions involved in the flexible trade credit contract. The analytic forms confirm that such an incentive can improve the decentralized supply chain efficiency and decreases the supplier's trade credit risk. The retailer always prefers early payment to payment around the due date to increase revenues. Furthermore, the action of paying early might help the retailer adjust cash flow between the two periods. We also find that a revenue sharing contract significantly affects the retailer's payment behavior and supplier's wholesale price. The numerical simulations support our results.     

Optimal manufacturer's pricing and lot-sizing policies under trade credit financing

In this paper, we extend Goyal's economic order quantity (EOQ) model to allow for the following four important facts: (1) the manufacturer's selling price per unit is necessarily higher than its unit cost, (2) the interest rate charged by a bank is not necessarily higher than the manufacturer's investment return rate, (3) the demand rate is a downward‐sloping function of the price, and (4) an economic production quantity (EPQ) model is a generalized EOQ model. We then establish an appropriate EPQ model accordingly, in which the manufacturer receives the supplier trade credit and provides the customer trade credit simultaneously. As a result, the proposed model is in a general framework that includes numerous previous models as special cases. Furthermore, we provide an easy‐to‐use closed‐form optimal solution to the problem for any given price. Finally, we develop an algorithm for the manufacturer to determine its optimal price and lot size simultaneously.     

Supply chain model with price- and trade credit-sensitive demand under two-level permissible delay in payments

This article develops a single-manufacturer and single-retailer supply chain model under two-level permissible delay in payments when the manufacturer follows a lot-for-lot policy in response to the retailer's demand. The manufacturer offers a trade credit period to the retailer with the contract that the retailer must share a fraction of the profit earned during the trade credit period. On the other hand, the retailer provides his customer a partial trade credit which is less than that of the manufacturer. The demand at the retailer is assumed to be dependent on the selling price and the trade credit period offered to the customers. The average net profit of the supply chain is derived and an algorithm for finding the optimal solution is developed. Numerical examples are given to demonstrate the coordination policy of the supply chain and examine the sensitivity of key model-parameters.     

Optimal transfer,ordering and payment policies for joint supplier–buyer inventory model with price-sensitive trapezoidal demand and net credit

This study aims at formulating an integrated supplier–buyer inventory model when market demand is variable price-sensitive trapezoidal and the supplier offers a choice between discount in unit price and permissible delay period for settling the accounts due against the purchases made. This type of trade credit is termed as ‘net credit’. In this policy, if the buyer pays within offered time M₁, then the buyer is entitled for a cash discount; otherwise the full account must be settled by the time M₂; where M₂ > M₁ ? 0. The goal is to determine the optimal selling price, procurement quantity, number of transfers from the supplier to the buyer and payment time to maximise the joint profit per unit time. An algorithm is worked out to obtain the optimal solution. A numerical example is given to validate the proposed model. The managerial insights based on sensitivity analysis are deduced.     

Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period

A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

Optimal payment time for retailer's inventory system

A retailer's inventory control system for the optimal delay in payment time for initial stock-dependent consumption rate when a wholesaler permits delay in payment is developed. Shortages are not allowed in the inventory system. The effect of inflation rate, deterioration rate, initial stock-dependent consumption rate and a wholesaler's permissible delay in payment is discussed. A mathematical model is derived when a wholesaler permits that the credit period is less than or equal to the retailer's optimal payment time, and that the retailer's optimal payment time is less than or equal to the cycle time for settling the account. Besides, an expression for a retailer's inventory system's total cost derived for this case and five special cases will be discussed. Moreover, a computational procedure is proposed to obtain the optimal cycle time, retailer's payment time and order size. The results could help retailer's managers to determine the optimal total cost and strive for the wholesaler's permission to delay the payment period. Thus, a retailer can attempt to gain two very big advantages: (1) that the retailer will have more money to run in that period because of the extension of the optimal payment time and (2) that extra interest will be earned in that period. Finally, a numerical example demonstrates the applicability of the proposed model and a sensitivity analysis is also discussed.     

Pricing strategies for dual‐channel supply chains under a trade credit policy

This paper discusses the impact of a trade credit policy on alleviating conflicts arising on a dual‐channel supply chain that includes one manufacturer and one value‐added retailer. We use the Stackelberg game to model the problem and characterize optimal pricing strategies for each supply chain partner, examining different circumstances in terms of retail price and trade credit contracts. When a consistent price strategy is applied in the dual channels under conditions of an exogenous credit period, trade credit can help both partners to achieve win‐win situations in the following circumstances: (1) when the retail channel's market share is small and the retailer's interest rate is high; or (2) when the retail channel's market share is large and the retailer's interest rate is lower than the manufacturer's. The study also concludes that when an inconsistent price strategy is applied, a trade credit contract can alleviate channel conflicts when the retailer's interest rate is higher than the manufacturer's. Otherwise, the partners may terminate cooperation. However, when the manufacturer has the power to determine and set the credit period, trade credit cannot alleviate channel conflicts under consistent price and inconsistent price scenarios.     

An integrated inventory model with capacity constraint and order-size dependent trade credit

Trade credit has many forms in today’s business practice. The most common form of trade credit policy that is used to encourage retailers to buy larger quantities is order-size dependent. When the number of ordered units exceeds the capacity of the own warehouse, an additional rented warehouse is required to store the excess units. Therefore, to incorporate the concept of order-size dependent trade credit and limited storage capacity, we proposed an integrated inventory model with capacity constraint and a permissible delay payment period that is order-size dependent. In addition, the unit production cost, which is a function of the production rate, is considered. Three theorems and an algorithm are developed to determine the optimal production and replenishment policies for both the supplier and the retailer. Finally, numerical examples are presented to illustrate the solution procedure and the sensitivity analyses of some key parameters are provided to demonstrate the proposed model.     

不对称信息下供应链赊销交易中的激励契约

赊销交易中,若零售商的 销售能力是私人信息,则供应商的赊销风险将增加.为给供应商提供甄别零售商销售能力的方法和 依据以控制赊销风险,运用激励理论构建激励模型并求解得到激励契约,将其与对称信息下赊销交易 和不对称信息下现金交易中的激励契约进行比较.结果表明,该激励契约能够甄别零售商的能力,不对称 信息下的产品交易量均向下扭曲,赊销交易能够减轻产品交易量的扭曲程度和降低甄别成本.最后 通过算例分析验证了主要结论.     

基于价格折扣的有条件延期支付策略

在延期支付及价格折扣策略下,建立以供货商为主导的主从斯坦伯格模型,从供货商的角度研究基于价格折扣的有条件延期支付策略设置问题.通过对模型的理论分析和模型求解,给出了各情形下供货商的延期支付策略、零售商的最佳订单量和货款的最佳支付时间.数值结果表明,此策略在激励零售商加大订单量的同时,还能吸引其尽早交付货款,从而加快供货商的资金周转,实现双赢.     

The supplier's optimal guarantee policy in newsvendor finance

The retailer is a capital‐constrained newsvendor and can borrow money from the bank if necessary. To help the retailer get a bank loan at a lower interest rate, the supplier provides guarantee for the retailer's loan up to a prespecified amount. In a Stackelberg game, the supplier decides the wholesale price and the guarantee amount as a leader, and then the retailer determines the order quantity and the amount of the loan as a follower. The supplier is risk‐neutral while the retailer's risk preference is reflected by a spectral risk measure (risk‐neutral, risk‐averse, or risk‐seeking). For a given wholesale price and guarantee amount, the retailer's objective function is quasi‐concave in the order quantity. The optimal solutions for the supplier and the retailer are derived. The supplier's expected profit with optimized wholesale price increases with the guarantee amount, and thus the supplier's optimal policy is to provide a full guarantee for the retailer's loan. When the supplier can limit his guarantee responsibility by a proportion of the outstanding loan obligation, the supplier's optimal policy is also to provide a full guarantee. Even if the retailer incurs bankruptcy costs in the event of repayment default, the supplier's optimal guarantee policy remains the same in these two different forms of limited guarantee. However, when the wholesale price is exogenous, that is, not a decision variable, the full guarantee is not necessarily optimal for supplier.     

Retailer's optimal policy under inflation in fuzzy environment with trade credit

Trade credit plays an important role in financing many industries. In the classical inventory model it is assumed that the buyer must pay for the items as soon as the items are received. In this problem, it is considered that the retailer can pay the supplier either at the end of the credit period or later pay interest on the unpaid amount for the overdue period. Here, the retailer's inventory model for the optimal cycle time and payment time for a retailer is developed. The effects of the inflation rate, deterioration rate and delay in payment have been discussed. The whole study is performed in a fuzzy environment by taking the opportunity cost, interest earned and interest paid rate as a triangular fuzzy number. Fuzzy profit functions, which involve fuzzy arithmetic operation, are defined using the function principle. We use the signed distance method to defuzzify the fuzzy profit function. Moreover, numerical and sensitivity analysis is performed to validate the proposed model.     

考虑消费者缺货容忍的变质品订购与信用期策略研究

考虑消费者存在缺货容忍行为,研究变质品的联合订购和信用期决策问题.假设延迟订购消费者面对缺货等待时存在一个容忍期限(即在该容忍期限内零售商无需支付缺货成本),同时考虑市场需求受商业信用期的影响,库存系统允许缺货且短缺量部分延迟订购.以零售商的平均利润最大化为目标,分两种情形构建变质品的订购和信用期决策模型.从理论上证明最优解的存在性和唯一性,给出相关定理结论,并在此基础上设计一个寻找最优解的两阶段迭代算法.最后通过数值算例展示了模型和算法的应用,并完成主要参数的灵敏度分析.研究结果表明:消费者的缺货容忍行为可以有效增加零售商利润,减低产品变质损失,同时还可以激励零售商提供更长的商业信用期,进而实现买卖双方的共赢.     

Coordinating a two-level supply chain with delay in payments and profit sharing

Achieving effective coordination among suppliers and retailers has become a pertinent research issue in supply chain management. Channel coordination is a joint decision policy achieved by a supplier(s) and a retailer(s) characterized by an agreement on the order quantity and the trade credit scenario (e.g., quantity discounts, delay in payments). This paper proposes a centralized model where players in a two-level (supplier–retailer) supply chain coordinate their orders to minimize their local costs and that of the chain. In the proposed supply chain model the permissible delay in payments is considered as a decision variable and it is adopted as a trade credit scenario to coordinate the order quantity between the two-levels. Computational results indicate that with coordination, the retailer orders in larger quantities than its economic order quantity, with savings to either both players, or to one in the supply chain. Moreover, a profit-sharing scenario for the distribution of generated net savings among the players in the supply chain is presented. Analytical and experimental results are presented and discussed to demonstrate the effectiveness of the proposed model.     

信用担保与下侧风险控制下零售商资金约束供应链订货与定价策略

针对供应商、零售商和商业银行组成的供应链系统,考虑当零售商面临资金约束时,通过供应商信用担保贷款模式缓解零售商资金约束对供应链总体利润的影响.构建零售商和供应商期望利润模型,分别讨论银行风险中性及下侧风险控制下的零售商订货与供应商定价决策,并利用斯坦克尔伯格博弈求解.最后,对供应商担保系数、银行风险容忍系数进行灵敏度分析.研究表明:1)给定担保系数,只有当风险容忍系数小于一定的阈值时,银行下侧风险控制才会影响零售商以及供应商的最优决策;2)给定风险容忍系数,只有当担保系数小于一定的阈值时,供应商担保系数才会影响零售商以及供应商的最优决策;3)银行贷款限额是担保系数、风险容忍系数的严格递增函数.     

Trade credit insurance in a capital‐constrained supply chain

We examine the role of trade credit insurance in a capital‐constrained supply chain with one (or two) loss‐neutral retailer(s) and one loss‐averse manufacturer. We model the interplay between these supply chain participants as a Stackelberg game and analyze their operating and financing decisions. In one capital‐constrained retailer case, we find that either the manufacturer's high loss aversion level or the retailer's low initial capital motivates the manufacturer to adopt insurance. Insurance drives more credit financing with more attractive financing terms (a lower wholesale price), which promotes the manufacturer to collect better product sales and performance. Although the retailer enjoys improved profit from insurance, its default risk increases. In contrast, in one capital‐constrained retailer and one well‐funded retailer scenario, numerically, when the manufacturer's loss aversion level is high or the weak retailer's initial capital is low, insurance is also adopted but is not always preferred by the capital‐constrained retailer due to competition. In addition, as the demand substitution rate increases, the manufacturer is more likely to prefer insurance due to better performance.     

基于现金折扣和延期支付的零售商补货和付款策略

在考虑供应商给予零售商延期支付、现金折扣的基础上,加入了非瞬时补货的条件,建立了相应的零售商库存决策模型.通过模型的分析求解,可以得出零售商在上述情况下的最优订货周期及最优付款时间的简单判定方法.最后,通过算例验证了模型的可行性,得出了与实际相符的结论.