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.     

Retailer's optimal ordering policies with trade credit financing

In this article, we extended Goyal's model to develop an Economic Order Quantity (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 N?≤?M) to his/her customers. In addition, we assume that (1) the retailer's selling price per unit is necessarily higher than its unit cost, and (2) the interest rate charged by a supplier or a bank is not necessarily higher than the retailer's investment return rate. We then establish an appropriate EOQ model with trade credit financing, and provide an easy-to-use closed-form solution to the problem. Furthermore, we find it is possible that a well-established buyer may order a lower quantity and take the benefit of the permissible delay more frequently, which contradicts to the result by the previous researchers. Finally, we perform some sensitivity analyses to illustrate the theoretical results and obtain some managerial results.     

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.     

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.     

How to make the replenishment and payment strategy under flexible two-part trade credit

In the existing inventory models concerning the two-part trade credit, a common assumption is that the retailer either pays for all the ordered items within a short permissible delay period and receives a cash discount or pays for all the ordered items within a long permissible delay period at the regular price. In this paper, this unrealistic assumption is relaxed. We assume that the retailer may pay any fraction of the purchase cost within the short permissible delay period and receives a cash discount and then the rest is paid within the long permissible delay period. A decision model is proposed for a retailer to determine the optimal ordering policy and payment plan. The closed-form optimal solution to the model is developed and analyzed. Numerical studies show that a retailer can obtain more benefits from the proposed payment plan than from the extreme payment plan in the existing literature.     

Two-echelon competitive integrated supply chain model with price and credit period dependent demand

This study considers a two-echelon competitive supply chain consisting of two rivaling retailers and one common supplier with trade credit policy. The retailers hope that they can enhance their market demand by offering a credit period to the customers and the supplier also offers a credit period to the retailers. We assume that the market demand of the products of one retailer depends not only on their own market price and offering a credit period to the customers, but also on the market price and offering a credit period of the other retailer. The supplier supplies the product with a common wholesale price and offers the same credit period to the retailers. We study the model under a centralised (integrated) case and a decentralised (Vertical Nash) case and compare them numerically. Finally, we investigate the model by the collected numerical data.     

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.     

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.     

Dynamic pricing,promotion and replenishment policies for a deteriorating item under permissible delay in payments

We study the problem of dynamic pricing, promotion and replenishment for a deteriorating item subject to the supplier's trade credit and retailer's promotional effort. In this paper we adopt a price- and time-dependent demand function to model the finite time horizon inventory for deteriorating items. The objective of this paper is to determine the optimal retail price, the promotional effort and the replenishment quantity so that the net profit is maximized. We discuss the properties and develop an algorithm for solving the problem described. The numerical analyses show that an appropriate promotion policy can benefit the retailer and that the promotion policy is important, especially for deteriorating items. Furthermore dynamic decision-making is shown to be superior to fixed decision-making in terms of profit maximization. Some special cases, such as with no credit period and for non-deteriorating items, are discussed as is the influence of the time-varying demand, the rate of deterioration and the credit period on the retailer behavior.     

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.     

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.     

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.     

Inventory policy of a deteriorating item with variable demand under trade credit period

In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailer’s profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.     

A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and trade credit financing

Pricing is a major strategy for a retailer to obtain its maximum profit. Furthermore, under most market behaviors, one can easily find that a vendor provides a credit period (for example 30 days) for buyers to stimulate the demand, boost market share or decrease inventories of certain items. Therefore, in this paper, we establish a deterministic economic order quantity model for a retailer to determine its optimal selling price, replenishment number and replenishment schedule with fluctuating demand under two levels of trade credit policy. A particle swarm optimization is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived in this paper. Some numerical examples are used to illustrate the features of the proposed model.     

The effect of inflation‐induced demand and trade credit on ordering policy of exponentially deteriorating and imperfect quality items

This paper develops an inventory model for exponentially deteriorating and imperfect quality items when a trade credit is offered by the supplier. As it has been seen that the changing rates of inflation do not just have an effect on the ordered quantity but they also stimulate the demand for products, so this paper considers the combined effect of inflation, deterioration, imperfect quality and ‘’ trade credit policy on the economic ordering policy using a discounted cash flow (DCF) approach. In the study, mathematical models are derived under two different circumstances, i.e., Case I: When the discount offered by the supplier with a condition of making the payment by a specified period, M₁ is accepted; and Case II: Discount is not taken and instead the payment is made by the end of the full credit period M (M > M₁). Finally, a numerical example demonstrating the applicability of the proposed model has been included.     

Three stage trade credit policy in a three-layer supply chain–a production-inventory model

The main purpose of this paper is to investigate the optimal replenishment lot size of supplier and optimal production rate of manufacturer under three levels of trade credit policy for supplier–manufacturer–retailer supply chain. The supplier provides a fixed credit period to settle the accounts to the manufacturer, while the manufacturer gives a fixed credit period to settle the account to the retailer and the retailer, in turn, also offers a credit period to each of its customers to settle the accounts. We assume that the supplier supplies the raw material to the manufacturer and sends back the defective raw materials to the outside supplier after completion of inspection at one lot with a sales price. The system always produces good items in the model. Also, we consider the idle times of supplier and manufacturer. Finally, numerical examples are provided to illustrate the behaviour and application of the model with graphical simulation.     

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.     

Determining optimal retail price and lot size under day-terms supplier credit

This paper deals with the problem of determining the retailer's optimal price and lot size simultaneously under the condition of permissible delay in payments. It is assumed that the customer's demand rate is represented by a constant price elasticity function which is a decreasing function of retail price. Investigation of the properties of an optimal solution allows us to develop an algorithm whose validity is illustrated using an example problem. Also, the effect of credit period on the retailer's pricing and lot sizing policy is examined through sensitivity analysis.     

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.