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.     

A Note on Fixed and Minimum Order Quantity Stock Systems

It is often believed that the square root formula called the EOQ only applies to situations where customers demand small quantities at a fairly constant rate. This note shows that essentially the same formula can be derived for "lumpy" customer orders occurring at a variable rate. Also, as the EOQ is derived assuming that a fixed quantity is always ordered for replenishment it does not allow for the over-shoot when customer orders exceed the available stock. This note shows that when the EOQ is large relative to the average customer order the overshoot problem can be ignored. However, when the EOQ is small some tentative results involving exponentially distributed customer order quantities indicate that substantial savings can be made using a "minimum order quantity" policy.     

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.     

Optimal inventory policies for an economic order quantity model with decreasing cost functions

In this paper, three total cost minimization EOQ based inventory problems are modeled and analyzed using geometric programming (GP) techniques. Through GP, optimal solutions for these models are found and sensitivity analysis is performed to investigate the effects of percentage changes in the primal objective function coefficients. The effects on the changes in the optimal order quantity and total cost when different parameters of the problems are changed is also investigated. In addition, a comparative analysis between the total cost minimization models and the basic EOQ model is conducted. By investigating the error in the optimal order quantity and total cost of these models, several interesting economic implications and managerial insights can be observed.     

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.     

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.     

An EOQ model for deteriorating items under supplier credits linked to ordering quantity

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the purchaser must pay for the items received immediately. However, in practices, the supplier usually is willing to provide the purchaser a permissible delay of payments if the purchaser orders a large quantity. As a result, in this paper, we establish an EOQ model 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. We then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Finally, several numerical examples are given to illustrate the theoretical results.     

Entropic order quantity (EnOQ) model for deteriorating items

Jaber et al. [M.Y. Jaber, R.Y. Nuwayhid, M.A. Rosen, Price-driven economic order systems from a thermodynamic point of view, Int. J. Prod. Res. 42 (24) (2004) 5167–5184] suggested that it might be possible to improve production systems performance by applying the first and second laws of thermodynamics to reduce system entropy (or disorder). They then used these laws to modify the economic order quantity (EOQ) model to derive an equivalent entropic order quantity (EnOQ). The results suggested that larger quantities should be ordered than is suggested by the classical EOQ model.     

Economic order quantity model and Taguchi’s cost of poor quality

In this paper we consider defective products and Taguchi’s cost of poor quality in the economic order quantity (EOQ) model. We assume that the product quality performs a normal distribution function, and the Taguchi’s poor quality cost has been involved. From our analysis, it has been found that the annual profit will be decreased if the poor quality of product and Taguchi’s quality cost are involved in the model. It has also been found that economic order quantity in our model is larger than that in a traditional EOQ model.     

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.     

Robust economic order quantity models

In this paper, we study the classical economic order quantity (EOQ) model under significant. In particular, the problem under consideration is the economic order quantity model with the input data of the demand rate, the order cost, and the holding cost rate being uncertain. A robustness approach based on scenario characterization of the input data is adopted to describe the uncertainties. The aim of the approach is to find an inventory policy that performs well under all realizable input data scenarios. An efficient linear time algorithm is devised to find the robust decisions. Analytical results are obtained for the case where input data are defined in continuous intervals. Comparisons on performances between the robust decisions and the stochastic optimization decisions are conducted. The results demonstrate the advantages of robustness approach.     

EOQ under Date-terms Supplier Credit

This paper examines EOQ under date-terms supplier credit, making explicit the separate effects on inventory policy of the two components of carrying cost-namely, financing cost and other variable holding costs. When a distinction between these types of holding costs is made, EOQ can no longer be expressed as a simple formula. Rather, optimal order quantity must be determined by search over a well-defined range of order quantities which encompasses the classical EOQ. The conclusion currently contained in the literature that the optimal order quantity under date terms is always given by an integer multiple of monthly demands no longer applies. In particular, a unique feature of date-terms credit is the possible existence of multiple EOQs.     

Optimal Ordering Policy for Deteriorating Items with Partial Backlogging under Permissible Delay in Payments

In 1985, Goyal developed an Economic order quantity (EOQ) model under conditions of permissible delay in payments. Jamal et al. then generalized Goyal’s model for deteriorating items with completely backlogging. However, they only ran several simulations to indicate that the total relevant cost may be convex. Recently, Teng amended Goyal’s model by considering the difference between unit price and unit cost, and provided an alternative conclusion that it makes economic sense for some retailers to order less quantity and take the benefits of the permissible delay more frequently. However, he did not consider deteriorating items and partial backlogging. In this paper, we establish a general EOQ model for deteriorating items when the supplier offers a permissible delay in payments. For generality, our model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. Consequently, the proposed model includes numerous previous models as special cases. In addition, we mathematically prove that the total relevant cost is strictly pseudo-convex so that the optimal solution exists and is unique. Finally, our computational results reveal six managerial phenomena.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

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.     

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.     

An inventory model under two levels of trade credit and limited storage space derived without derivatives

This paper tries to incorporate both Huang’s model [Y.F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, J. Oper. Res. Soc. 54 (2003) 1011–1015] and Teng’s model [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] by considering the retailer’s storage space limited to reflect the real-life situations. That is, we want to investigate the retailer’s inventory policy under two levels of trade credit and limited storage space. Furthermore, we adopt Teng’s viewpoint [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] that the retailer’s unit selling price and the purchasing price per unit are not necessarily equal. Then, an algebraic approach is provided and three easy-to-use theorems are developed to efficiently determine the optimal cycle time. Some previously published results of other researchers can be deduced as special cases. Finally, a numerical example is given to illustrate these theorems and managerial insights are drawn.     

Dynamic demand lot-sizing rules for incremental quantity discounts

Numerous studies have developed and compared lot-sizing procedures for finite-horizon dynamic demand, material requirements planning (MRP), environments when either no purchase discounts exist or for the case of all-units quantity discounts. This paper examines lot-sizing rules when product price schedules follow incremental quantity discounts. The optimal (non-discount) procedure and some traditional heuristic procedures are modified to incorporate incremental quantity discounts. We further modify two heuristics with a ‘look-ahead enhancement’ that performs very well under experimentation. Numerical tests revealed the overall best-performing heuristic in this study to be a modified ‘least-unit cost’ method with a look-ahead enhancement. That procedure produced an average cost penalty vs optimal of 0.26%.     

An economic order quantity model with partial backordering and a special sale price

A constant unit purchase cost is one of the main assumptions in the classic economic order quantity model. In practice, suppliers sometimes offer special sale prices to stimulate sales or decrease inventories of certain items. In this paper we develop an EOQ model with a special sale price and partial backordering. We prove the convexity of the cost-reduction function if a special order is placed at the special sale price. A solution method is proposed and numerical examples are presented.