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.     

延期支付条件下零售商的最优售价和订货策略

在供应商给予零售商延期支付和现金折扣的优惠政策下,进一步假设产品的年需求量依赖于零售商产品售价的基础上,建立了由一个零售商和一个供应商所构成的库存决策模型,扩展了经典的经济生产批量(EPQ)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、最优售价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。     

两货栈及延期支付和现金折扣情形下变质产品的最优订货策略

已有的两货栈库存模型通常不考虑将延期支付和现金折扣相结合的情形,但实际上,供应商在给予销售商延期支付政策的同时,也会实施现金折扣策略以激励销售商尽快付款,加快资金周转,减少坏账损失。为此,本文建立了延期支付和现金折扣情形下变质产品的两货栈库存模型,并对模型的最优解进行理论分析,给出了最优解的求解步骤。最后通过数值算例对模型的可行性进行了验证,并分析了模型参数变化对最优订货策略和最优付款时间的影响。     

两货栈及延期支付和现金折扣情形下变质产品的最优订货策略

已有的两货栈库存模型通常不考虑将延期支付和现金折扣相结合的情形,但实际上,供应商在给予销售商延期支付政策的同时,也会实施现金折扣策略以激励销售商尽快付款,加快资金周转,减少坏账损失。为此,本文建立了延期支付和现金折扣情形下变质产品的两货栈库存模型,并对模型的最优解进行理论分析,给出了最优解的求解步骤。最后通过数值算例对模型的可行性进行了验证,并分析了模型参数变化对最优订货策略和最优付款时间的影响。     

An EOQ model for deteriorating items under trade credits

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the supplier is paid for the items immediately after the items are received. However, in practices, the supplier may simultaneously offer the customer: (1) a permissible delay in payments to attract new customers and increase sales, and (2) a cash discount to motivate faster payment and reduce credit expenses. In this paper, we provide the optimal policy for the customer to obtain its minimum cost when the supplier offers not only a permissible delay but also a cash discount. We first establish a proper model, and then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Furthermore, we also compare the optimal order quantity under supplier credits to the classical economic order quantity. Finally, several numerical examples are given to illustrate the theoretical results.     

Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy

In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution.     

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.     

A comprehensive note on: An inventory model under two levels of trade credit and limited storage space derived without derivatives

In 2006, Huang proposed an inventory model with two warehouses when the supplier offers the retailer a permissible delay of M periods, and the retailer also provides its customers a permissible delay of N periods. He then solved it without derivatives. In this note, we extend his model to complement the shortcomings of his model. In contrast to the complicated and tedious quadratic–algebraic method suggested by Huang, we propose a simple arithmetic–geometric method to solve the inventory problem. Finally, we run computer programs for several numerical examples to illustrate the proposed model and obtain some managerial implications.     

A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation

In the business transactions, the supplier usually offers a permissible delay in payment to his retailer to attract more sales. In addition, a permissible delay in payment may be applied as an alternative to price discount. Based on the above phenomena, we incorporate a permissible delay in payment into the model of Yang [1] and develop a two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation. The objective of this study is to derive the retailer’s optimal replenishment policy that maximizes the net present value of the profit per unit time. The necessary and sufficient conditions for an optimal solution are characterized. An algorithm is developed to find the optimal solution. Finally, numerical examples are provided to illustrate the proposed model. Sensitivity analysis is made and some managerial implications are presented.     

The EOQ with defective items and partially permissible delay in payments linked to order quantity derived algebraically

Most researchers established their inventory lot-size models under trade credit financing by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, in the real business environment, it often can be observed that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the predetermined quantity Q _d . In addition, an arriving order lot usually contains some defective items due to imperfect production processes or other factors. To capture this reality, the paper extends Huang (2007) economic order quantity (EOQ) model with partially permissible delay in payments to consider defective items. We formulate the proposed problem as a profit maximization EOQ model in which the replenishment cycle time is the decision variable. Then we use the arithmetic-geometric mean inequality approach to determine the optimal solution under various situations. An algorithm to obtain the optimal solution is also provided. Finally, the numerical examples and sensitivity analysis are given to illustrate the results.     

Managing a retail-competition distribution channel with incentive policies

In reality, the multi-retailer channel of distribution has occurred in many industries, such as electronics, appliances, and apparel. The practice of the supplier providing incentives to retailers to promote sales is also quite prevalent in today’s business. This paper considers a two-retailer supply chain in which the supplier provides a cash discount to a specific retailer (a senior company owning more market share and having less pressure on the utilization of capital investment) and a credit period to another (a junior company owning less market share and having more pressure on the utilization of capital investment). The end demand for each channel has a substitute effect in terms of retail prices. The research problem originates from operations of dual-retailer channels in the automotive parts and accessories industry. The objective of the supplier is to determine the cash discount and credit period to maximize his profit. Both retailers determine pricing and ordering policies to maximize profits. This paper applies a Supplier Stackelberg framework to obtain equilibrium solutions for every entity. Numerical analysis is conducted to discuss the influence retailer competition has on the decision behaviors of the supplier and retailers.     

Analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain

This paper investigates the economic order quantity (EOQ) — based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier yet the retailer just offers a partial trade credit to customers. The demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers. Under these conditions, the retailer can obtain the most benefits. Study also investigates the retailer’s inventory policy for deteriorating items in a supply chain management situation as a cost minimization problem in the fuzzy sense. The annual total variable cost for the retailer in fuzzy sense is defuzzified using Graded Mean Integration Representation method. Then the present study shows that the defuzzified annual total variable cost for the retailer is convex, that is, a unique solution exists. Mathematical theorems and algorithms are developed to efficiently determine the optimal inventory policy for the retailer. Numerical examples are given to illustrate the theorems and the algorithms. Finally, the results in this paper generalize some already published results in the crisp sense.     

A deterministic EOQ model with delays in payments and price-discount offers

It is the purpose of this paper to model the retailer’s profit-maximizing strategy when confronted with supplier’s trade offer of credit and price-discount on the purchase of merchandise. Generally, retailers have to face many types of demands for different kinds of goods. In real situation, retailers have to correlate between the selling price and supplier’s trade offer, keeping in mind profit-maximization strategy. In the proposed model, all increasing deterministic demands are discussed analytically, numerically and graphically in the environment of permissible delay in payment and discount offer to the retailer.     

基于提前订货折扣和延期支付契约的双渠道供应链运作策略研究

在随机需求环境下, 构建了四种不同情形下零售商和双渠道制造商利用契约机制进行竞争的供应链决策模型, 并给出了各情形下供应链成员最优决策和利润。研究发现, 相对于供应链无任何契约情形, 双渠道供应链中各主体均提供契约来增加自身需求和利润的策略并非始终是有效的, 一定条件下供应链各主体利润均受损。当供应链中仅零售商为下游顾客提供提前订货折扣契约, 而制造商不提供任何契约时, 供应链各主体利润均达到最大。因而, 对双渠道制造商来说, 当零售商采用一定契约策略增大其渠道需求并降低制造商网络渠道需求时, 制造商最优的策略并非是采用“敌对”的契约策略来进一步增加自身网络渠道需求, 而是采用“搭便车”策略, 不为下游顾客提供任何契约优惠, 而从零售商渠道获得更多批发收益, 并最终实现自身总收益的最大化。     

Economic order quantity under conditionally permissible delay in payments

Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.     

顾客部分延期付款下两级商业信用易腐品订货策略

针对易腐品的经济订货批量问题展开研究。在供应商给零售商提供延迟还款的同时,零售商也给顾客提供部分延期还款条件。分五种情况分别讨论零售商的成本构成,并由此建立数学模型以求解最优订货周期使得零售商单位时间总成本最小化。通过数学证明得到了目标函数的解析性质,结果显示每种情况下在可行域范围内至多存在一个极小值点。以此为基础给出了相应的命题以有效地确定零售商的最优决策。最后通过数值算例说明了模型的有效性。     

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.     

基于信用支付和现金折扣的变质物品库存模型

本文在供应商提供给零售商定期信用支付和现金折扣情况下,研究了零售商的变质物品最优库存问题。基于信用支付和现金折扣的两种支付条件下,分四种情况建立库存模型,并给出了寻求变质物品最优订购周期和最优付款时间的有效算法。最后,给出算例以及最优解,以说明本模型及求解过程。     

An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity

To attract more sales suppliers frequently offer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity W. In this paper, we generalize [Goyal, S.K., 1985. EOQ under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338] economic order quantity (EOQ) model with permissible delay in payment to reflect the following real-world situations: (1) the retailer’s selling price per unit is significantly higher than unit purchase price, (2) the interest rate charged by a bank is not necessarily higher than the retailer’s investment return rate, (3) many items such as fruits and vegetables deteriorate continuously, and (4) the supplier may offer a partial permissible delay in payments even if the order quantity is less than W. We then establish the proper mathematical model, and derive several theoretical results to determine the optimal solution under various situations and use two approaches to solve this complex inventory problem. Finally, a numerical example is given to illustrate the theoretical results.