EOQ under Date-terms Supplier Credit: A Near-optimal Solution

This paper shows that under date-terms supplier credit, making explicit the separate effects of carrying cost, the financing and other marginal holding costs, does not invalidate Kingsman's original result that the optimal order quantity is given by an integer multiple of monthly demands, provided the capital investment component of the inventory holding costs is equal to or greater than 30% of the component due to the physical holding of inventory. The analysis is extended to the case when orders of less than a month's demand are optimal. Here it is shown that the order quantity should be an integer fraction of a month's demand, provided that the capital investment component of the inventory holding charge is equal to or greater than one quarter of the component due to the physical holding of inventory. It is argued that these conditions are likely to be satisfied for most if not all practical inventory situations. Combining these results with those of Carlson and Rousseau leads to a simple formula for the general optimal policy. The EOQ can still be expressed as a simple formula, so for practical situations generally there is no need to use the numerical search procedure these authors propose.     

Economic Order Quantity and Quantity Discounts under Date-terms Supplier Credit: A Discounted Cash Flow Approach

This paper examines economic order quantity and optimal order quantity under both all-units and incremental-quantity discounts when purchase cost, order cost, and carrying cost are all incurred on date-terms supplier credit. Payment dates for the three cost components need not be the same. The traditionally-used periodic-cost minimization methodology which is insensitive to differences in the timing of various within-period cash flows is replaced with a discounted cash flow methodology. Differences in the characteristics of day-terms and date-terms solutions to the quantity discount case are high-lighted.     

Note: An application of the EOQ model with nonlinear holding cost to inventory management of perishables

In this note, we consider a variation of the economic order quantity (EOQ) model where cumulative holding cost is a nonlinear function of time. This problem has been studied by Weiss [Weiss, H., 1982. Economic order quantity models with nonlinear holding costs. European Journal of Operational Research 9, 56–60], and we here show how it is an approximation of the optimal order quantity for perishable goods, such as milk, and produce, sold in small to medium size grocery stores where there are delivery surcharges due to infrequent ordering, and managers frequently utilize markdowns to stabilize demand as the product’s expiration date nears. We show how the holding cost curve parameters can be estimated via a regression approach from the product’s usual holding cost (storage plus capital costs), lifetime, and markdown policy. We show in a numerical study that the model provides significant improvement in cost vis-à-vis the classic EOQ model, with a median improvement of 40%. This improvement is more significant for higher daily demand rate, lower holding cost, shorter lifetime, and a markdown policy with steeper discounts.     

The impact of dynamic pricing on the economic order decision

This paper analyzes the impact of dynamic pricing on the single product economic order decision of a monopolist retailer. Items are procured from an external supplier according to the economic order quantity (EOQ) model and are sold to customers on a single market without competition following the simple monopolist pricing problem. Coordinated decision making of optimal pricing and ordering is influenced by operating costs – including ordering and inventory holding costs – and the demand rate obtained from a price response function. The retailer is allowed to vary the selling price, either in a fixed number of discrete points in time or continuously. While constant and continuous pricing have received much attention in the literature, problems with a limited number of price changes are rather rare. This paper illustrates the benefit of dynamically changing prices to achieve operational efficiency in the EOQ model, that is to trigger high demand rates when inventories are high. We provide structural properties of the optimal time instants when the price should be changed. Taking into account costs for changes in price, it provides numerical guidance on number, timing, and size of price changes during an order cycle. Numerical examples show that the benefits of dynamic pricing in an EOQ framework can be achieved with only a few price changes and that products being unprofitable under static pricing may become profitable under dynamic pricing.     

基于数量折扣合同和缺货损失的EOQ模型的改进

经济订购批量模型假定需求率、单位持有成本、订货成本为常数下得到总成本最低的订购批量,这些参数常数化的假设在现实中通常难以满足.假定需求和订货费为不确定的、库存成本包括年固定成本(与订货量无关)和年可变成本(与订货量有关),用三角模糊数表示年需求和订货费,通过引入数量折扣合同来量化单位产品进价,分别在不允许缺货和考虑缺货损失两种情况下得到最佳订货量.最后的算例表明了模型的合理性.     

损耗性物品信用交易下的斯坦博格库存模型

目前众多的信用交易模型是在供应商给定的信用交易期限条件下,零售商确定最优订购数量或订购周期,而很少考虑供应商信用交易策略的制定问题。本文针对损耗性物品,在最终需求为价格的线性函数条件下,利用斯坦博格博弈模型给出了信用交易下供应商信用交易策略的制定和零售商的最优订购决策,最后通过算例对模型进行了验证。     

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.     

A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts

Lin [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165] recently proposed an EOQ model with imperfect quality and quantity discounts, where the lot-splitting shipments policy is adopted. In this note we first rectify the holding cost terms showed in Lin to obtain a new objective function, then resolve the problem and develop an easy to implement algorithm to find the overall optimal solutions for the model. Besides, we present a new model for items with imperfect quality, where lot-splitting shipments and different holding costs for good and defective items are considered. The closed-form formulas for determining the optimal ordering and shipping policies are derived. Also, the results are examined analytically and numerically to gain more insights of the solutions.     

The EOQ model with defective items and partially permissible delay in payments linked to order quantity derived analytically in the supply chain management

A lot of researchers develop their inventory models under trade credit by assuming that the supplier offers the retailer fully permissible delay in payments and the products received are all non-defective. However, from the viewpoint of practice, it can often be found that the supplier offers the retailer a fully permissible delay in payments only when the order quantity is greater than or equal to the specific quantity. Furthermore, the products received usually contain some defective items. This paper establishes the EOQ model with defective items and partially permissible delay in payments linked to order quantity. It also uses the rigorous method of mathematics to derive the solution procedure to locate the optimal solution. Finally, numerical examples are given to illustrate all theoretical results in this paper.     

An incentive-compatible solution for trade credit term incorporating default risk

This paper studies the optimal trade credit term decision in an extended economic ordering quantity (EOQ) framework that incorporates a default risk component. A principal-agent bilevel programming model with costs minimization objectives is set up to derive the incentive-compatible credit term. The supplier determines the credit term as the leader in the first level programming, by balancing her/his financing capacity with the retailer’s default risk, order behavior and cost shifting. At the second level, the retailer makes decisions on ordering and payment time by reacting on the term offered by the supplier. A first order condition solution procedure is derived for the bilevel programming when credit term is confined within the practically feasible interval. Two key results are obtained – the condition to derive incentive-compatible credit term, and an equation system to derive threshold default risk criterion filtering retailers suitable for credit granting. Numerical experiments show that the capital cost of the supplier is the most important factor determining the credit term. Default risk acts like a filtering criterion for selecting retailers suitable for credit granting. Empirical evidence supporting our theoretical considerations is obtained by estimating three panel econometric models, using a dataset from China’s listed companies.     

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.     

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.     

The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach

This paper discusses the optimum order quantity of the EOQ model that is not only dependent on the inventory policy but also on firm’ credit policy. Here, the conditions of using a discounted cash-flows (DCF) approach and trade credit depending on the quantity ordered are discussed. We consider that if the order quantity is less than at which the delay in payments is permitted, the payment for the item must be made immediately. Otherwise, the fixed trade credit period is permitted.     

A complement to “A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts”

Chang [1] [H.-C. Chang, A comprehensive note on: an economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 35 (10) (2011) 5208-5216] corrects a flaw in Lin’s inventory model [T.Y. Lin, An economic order quantity with imperfect quality and quantity discounts, Appl. Math. Model. 34 (10) (2010) 3158–3165]. Then, he develops an algorithm to find the optimal solution for the corrected Lin’s inventory model and furthermore derives close form expressions to determining the optimal solution to an EOQ inventory model considering items with imperfect quality with different holding costs for good and defective items. In both models there is a discrete variable and he presents some inequalities in order to find the integer value. This paper provides some simple formulas to obtain, in an easy way, the integral value for the discrete variable.     

An EOQ Model with Substitutions Between Products

In this paper, we present an economic order quantity (EOQ) model when two products are required, and one can be substituted for the other, if necessary, at a given unit cost. We consider three cases: (i) when there is no substitution between the products, (ii) when there is full substitution between the products, and (iii) when there is partial substitution between the products. In a deterministic setting with proportional substitution costs, we would expect to find full substitution or no substitution being optimal, depending on the cost parameters. However, we observe that full substitution is never optimal; only partial substitution or no substitution may be optimal. This result can best be explained due to the non-linearity of the decision variables in the total cost expression. Finally, we present an algorithm to compute the optimal order quantities.     

A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity

This paper explores the inventory replenishment policy for deteriorating items in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. As a matter of fact, the inventory system discussed by this paper is the same as that of Chang et al. [C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credit credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996]. However, their approach in solving the problems needs further analysis. This article deals with an alternative approach to present a simple procedure in order to determine the optimal ordering policy when the supplier provides a permissible delay in payments linked to order quantity. Numerical examples reveal that the solution algorithm described in this paper is accurate and rapid.     

关于信用期内含缺陷产品EOQ模型

考虑到在实际中供应链上游供应商提供给下游零售商的信用支付期通常为一个订货周期,建立了缺陷率服从一定分布的缺陷产品在信用支付策略下的最优订货批量模型.模型中允许缺货发生并且以最大期望利润为目标函数,通过分析得到模型最优解.最后给出仿真实验,并且分析了模型参数变化对最优解的影响.     

On explicit solutions of a two-echelon supply chain coordination game

A contracting game under asymmetric information specific to two-echelon supply chain coordination between a retailer of unknown type and a supplier is studied. When the parameter which is private information to the retailer (holding cost) is known up to an interval of uncertainty, a uniform discrete approximation for retailer types leads to closed-form solutions where the joint (coordinated) optimal order quantity for a modified holding cost plays a major role. Furthermore, the closed-form solutions result in increasing information rent for higher types under easy-to-verify conditions involving strict lower limits on the total holding costs of retailer and supplier and the difference between uncoordinated optimal costs of consecutive retailer types.     

An Experimental Comparison of MRP Purchase Discount Methods

This paper addresses the managerial issue of how best to order purchased materials in MRP environments when discounts are available from vendors. The least unit cost, least period cost, McLaren's order moment, revised part-period balancing, incremental part-period balancing, traditional discount order quantity, and an optimal algorithm are experimentally investigated under a variety of simulated scenarios. Other experimental factors include the coefficient of variation in demand, forecast uncertainty beyond the current period, the average time between orders, the ratio of the discount quantity to the EOQ, the attractiveness of the discount, the length of the planning horizon, inventory holding costs, and the autocorrelation of demand. All factors tested in this comprehensive experiment significantly affected the performance of the discount ordering procedures. Furthermore, the results from this study suggest that the least unit cost, McLaren's order moment, the traditional discount order quantity, and the optimal procedures significantly out-perform the others. A further choice among these alternative methods was found to be a function of the operating environment and limitations that may exist on available computing time.     

A generalized discount structure and some dominance rules for selecting the price-break EOQ

This paper introduces a generalized discount structure that combines the features of incremental and all-units quantity discount policies. General properties of the EOQ model under this discount structure are studied, and dominance rules for comparing order quantity intervals under either type of discount policy are established. In addition, procedures for developing an iso-cost function and the minimal feasible set for optimal order quantity are proposed.