A note on an economic order quantity model and Taguchi’s cost of poor quality

Tsou [Appl. Math. Model. 31 (2) (2007) 283–291] previously proposed an economic order quantity model with Taguchi’s cost of poor quality. In this note, we correct an error appearing in their numerical results and provide some simple formulas that can be applied to numerical studies with the specific objective of calculating the defective rate and the total profit of his model.     

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 EOQ model with process reliability considerations

The classical economic order quantity (EOQ) model assumes that items produced are of perfect quality and that the unit cost of production is independent of demand. However, in realistic situations, product quality is never perfect, but is directly affected by the reliability of the production process. In this paper, we consider an EOQ model with imperfect production process and the unit production cost is directly related to process reliability and inversely related to the demand rate. In addition, a numerical example is given to illustrate the developed model. Sensitivity analysis is also performed and discussed.     

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.     

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.     

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.     

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.     

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.     

Some comments on “A simple method to compute economic order quantities”

In this note, we emphasize that the arithmetic–geometric-mean-inequality approach proposed by Teng [Teng, J.T., 2008. A simple method to compute economic order quantities. European Journal of Operational Research. doi:10.1016/j.ejor.2008.05.019] is not a general solution method. Teng’s approach happens to work and give the correct results when the two terms in an objective function are any functions such that their product is a constant. The classical EOQ model works fine since the product of the two terms is indeed a constant! When the product is not a constant, Teng’s approach is of little use. This is exemplified in Comment 1 via solving the EOQ model with complete backorders (where the model is regarded as having two decision variables). Comment 2 is generally valid for an algebraic method when it is used to solve an objective function with two decision variables.     

Random fuzzy EOQ model with imperfect quality items

This paper investigates an economic order quantity (EOQ) problem with imperfect quality items, where the percentage of imperfect quality items in each lot is characterized as a random fuzzy variable while the setup cost per lot, the holding cost of each unit item per day, and the inspection cost of each unit item are characterized as fuzzy variables, respectively. In order to maximize the expected long-run average profit, a random fuzzy EOQ model is constructed. Since it is almost impossible to find an analytic method to solve the proposed model, a particle swarm optimization (PSO) algorithm based on the random fuzzy simulation is designed. Finally, the effectiveness of the designed algorithm is illustrated by a numerical example.     

Learning-by-doing may not be enough to sustain competitiveness in a market

The economic order quantity model and its variants are the oldest reported scientific inventory models in the literature. They are mathematically simple to use and understand. This characteristic contributed to their popularity. However, they have limitations that restrict their application in practice. Researchers have extended those models by relaxing some of their restrictive assumptions. One of the main limitations of the EOQ model is the estimation of its cost parameters, such as setup and holding costs, which include some hidden (or difficult to estimate) components. Emerging research proposes using the second law of thermodynamics when to measure the hidden (entropy) costs of inventory systems. Another research stream focuses on learning-by-doing, which reduces the unit cost of a product, thus reducing its price and enhancing a firm's competitiveness. This paper assumes that a buyer receives a shipment from its vendor following the EOQ model. The buyer invests in the vendor's process to accelerate learning (mainly workers training, among other things). The buyer's unit purchase cost reduces because of learning at the vendor's side. The buyer's competitors also learn, increasing competitiveness. For this purpose, we develop an EOQ model where price-dependent demand is analogous to a heat transfer equation, which is used to determine the entropy cost. It accounts for improvement by learning as a result of the buyer's investment to compete in a market. The implementation and management of improvement programs are not usually smooth, resulting in disorder quantified by an entropy cost function. The developed EOQ model is investigated with and without entropy costs. Numerical results representing different scenarios are provided to illustrate the behavior of the models and identify the factors affecting the decision variables and, subsequently, the buyer's profitability. The results showed that in a competitive market, the buyer has to maintain a learning rate faster than those of its competitors to ensure a prolonged, but indefinite, profitability. The results also showed that investment in learning is meaningful when the competition is fierce.     

A note on fuzzy inventory model with storage space and budget constraints

In 1997, Roy and Maiti developed a fuzzy EOQ model with fuzzy budget and storage capacity constraints where demand is influenced by the unit price and the setup cost varies with the quantity purchased [T.K. Roy, M. Maiti, A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity, Eur. J. Oper. Res. 99 (1997) 425–432]. However, their procedure has some questionable points and their numerical examples contain rather peculiar results. The purpose of this paper is threefold. First, for the same inventory model with fuzzy constraints, based on the max–min operator, we proposed an improved solution procedure. Second, we review the solution procedure by Roy and Maiti that is based on Kuhn–Tucker approach to point out their questionable results. Third, we compare Roy and Maiti’s approach with ours to explain why our approach can solve the problem and theirs cannot. Numerical examples provided by them also support our findings.     

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.     

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.     

Weibull变质率和短缺量拖后的易变质物品EOQ模型

基于需求和采购价格均为时变的EOQ模型,考虑物品的变质率呈更符合现实情况的三参数Weibull分布,同时考虑短缺量拖后和资金时值对易变质物品库存管理的影响,构建了相应的EOQ模型.应用数学软件Matlab对该库存模型进行仿真计算和主要影响参数的灵敏度分析.结果表明,该模型存在最优解,且各主要影响参数对最优库存控制各有不同程度的影响,资金时值对库存总成本净现值的影响程度要甚于短缺量拖后的影响,故在制定科学的库存策略时资金时值需要更加关注.     

EOQ holds under stochastic demand,a technical note

We consider a variant of the economic order quantity (EOQ) model. Mainly, we assume that demand occurs at random, one unit at a time, and is characterized by independent and identically distributed times between two demand epochs. We also assume that the ordering policy is characterized by ordering the same amount whenever the inventory level drops to zero, and a demand occurs. Surprisingly, we show that the optimal order quantity that minimizes the expected inventory cost follows the familiar EOQ formula.     

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.     

Optimal investment in setup reduction in manufacturing systems with WIP inventories

A number of models have been proposed to predict optimal setup times, or optimal investment in setup reduction, in manufacturing cells. These have been based on the economic order quantity (EOQ) or economic production quantity (EPQ) model formulation, and have a common limitation in that they neglect work-in-process (WIP) inventories, which can be substantial in manufacturing systems. In this paper a new model is developed that predicts optimal production batch sizes and investments in setup reduction. This model is based on queuing theory, which permits it to estimate WIP levels as a function of the decisions variables, batch size and setup time. Optimal values for batch size and setup time are found analytically, even though the total cost model was shown to be strictly non-convex.     

Optimal inventory policies under decreasing cost functions via geometric programming

In this paper, we establish and analyze two economic order quantity (EOQ) based inventory models under total cost minimization and profit maximization via geometric programming (GP) techniques. Through GP, optimal solutions for both models are found and managerial implications on the optimal policy are determined through bounding and sensitivity analysis. We investigate the effects on the changes in the optimal order quantity and the demand per unit time according to varied parameters by studying optimality conditions. In addition, a comparative analysis between the total cost minimization model and the profit maximization model is conducted. By investigating the error in the optimal order quantity of these two models, several interesting economic implications and managerial insights can be observed.     

Calculating safety stocks for assembly systems with random component procurement lead times: A branch and bound algorithm

In this paper, a discrete single-level multi-component inventory control model for assembly systems with random component procurement lead times is considered. The economic order quantity (EOQ) policy is used for a type of finished product. The requirements of the components are constant and cyclic (periodic), and their values per period are deduced from the EOQ for the finished product. The paper focuses on the components safety stock calculation. The objective is to minimise the average holding cost of the components while keeping the desired service level for the finished product. For this, an upper bound, two lower bounds, two dominance properties and an efficient branch and bound algorithm are suggested. Several tests are executed and conclusions are drawn. The proposed model provides a substantial saving for assembly systems with a large number and unreliable delivery of components as in semi-conductor and automotive industries.