Classical Economic Order Quantity Model Research Articles

An EOQ model with fuzzy demand and learning in fuzziness

This paper develops an economic order quantity (EOQ) model with fuzzy demand that may vary between upper and lower limits. The imprecision in demand is assumed to reduce with time because of learning. The results from the developed model are compared to those of an EOQ model with fuzzy demand and no learning. It is shown that learning in fuzziness improves the information base for future orders by reducing uncertainty, which favours delivering demand in smaller lots which are delivered more frequently. As the learning rate increases and fuzziness in demand reduces, the results were shown to converge to those of the classical EOQ model.

Inventory models for imperfect quality items with shortages and learning in inspection

The classical economic order quantity model, although well known and useful; assumes that all items received conform to quality characteristics. However, in practice, items may be damaged due to transportation and/or production conditions. This requires a buyer to screen each lot it receives from its vendor to separate the good from the nonconforming (due to imperfect quality) items. While screening is usually a manual task performed by inspectors, it may improve with learning. Besides, it was observed in some studies that coordinating activities (e.g., quality) between a buyer and a vendor may be subject to learning effects and results in improving the quality of each lot (as it contains less nonconforming items) delivered or produced.This paper investigates an EOQ model with imperfect quality items and shortages, where the fraction of imperfect quality in each shipment reduces because of learning. Mathematical models are developed for the cases of infinite and finite planning horizons. Numerical results, which have several interesting managerial insights and implications, are presented and discussed. This paper may be useful to managers who wish to benefit from the learning opportunities in their logistics and inventory systems, especially, when comparing different production strategies and making choices regarding the selection of suppliers or the processes that need improvements.

Deterministic ordering policy with price- and stock-dependent demand under fluctuating cost and limited capacity

In this paper, we extend the classical economic order quantity model to allow for the demand rate influenced by both selling price and displayed stock level. A finite period system is considered under inflation for deteriorating items. In addition, we impose a ceiling on the number of on-display stocks. We also allow for shortages and general partial backlogging. The objective is to maximize the discounted total profit by finding the optimal replenishment number, replenishment schedule and selling price changed periodically. An effective search procedure is provided to find the optimal solution by employing the properties derived in this paper and particle swarm optimization algorithm. Several numerical examples are used to illustrate the features of the proposed model.

Economic order quantity model with demand influenced by dynamic innovation effect

Inventory control policies for new-product items are highly perceptive to different marketing policies especially for innovation effects at the earlier stage of the product life cycle but unfortunately classical economic order quantity (EOQ) model do not recognises the innovation driven demand model. In this paper, a time dependent innovation driven demand model has been introduced in the basic EOQ model to calculate the different optimal policies. The proposed model acknowledged relationship between the innovation coefficient and the optimal policies. Four hypotheses were framed in this paper based on the numerical exercise that could explain the impact of dynamic pattern of the innovation coefficient on different optimal policies.

An Algebraic Approach for Determining the Optimal Lot Size for EPQ Model with Rework Process

This paper presents a simple algebraic approach for deriving the optimal lot size for economic production quantity (EPQ) model with rework process. Conventional methods for solving lot size problems are by using differential calculus on the long-run average production-inventory cost function with the need to prove optimality first. A few recent articles proposed the algebraic approach to the solution of classic economic order quantity (EOQ) and EPQ model without reference to the use of derivatives. This paper extends it to an EPQ model with reworking of defective items. We demonstrate that optimal lot size and optimal production-inventory cost for such an imperfect EPQ model can be derived without derivatives. As a result, it may enable the practitioners or students who with little knowledge of calculus to understand or handle with ease the realistic production systems.

A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and unit purchasing cost

In this paper, we extend the classical economic order quantity model to allow for not only a function of price-dependent and time-varying demand but also fluctuating unit purchasing cost. The joint replenishment problem is subject to continuous decay and a general partial backlogging rate. The objective is to find the optimal replenishment number, time scheduling and periodic selling price to maximize the discounted total profit. An effective search procedure is provided to find the optimal solution by employing the properties derived in this paper and particle swarm optimization algorithm. Several numerical examples are used to illustrate the features of the proposed model.

Backorder penalty cost coefficient “b”: What could it be?

The classical economic order quantity (EOQ) model with planned penalized backorders (PB) relies on postulating a value for the backorder penalty cost coefficient, b, which is supposed to reflect the intangible adverse effect of the future loss of customer goodwill following a stockout. Recognizing that the effect of the future loss of customer goodwill should be not a direct penalty cost but a change in future demand, Schwartz [1966. A new approach to stockout penalties. Management Science 12(12), B538–B544] modified the classical EOQ-PB model by eliminating the backorder penalty cost term from the objective function and assuming that the long-run demand rate is a decreasing, strictly convex function of the customer's “disappointment factor” (defined as the complement of the demand fill rate) following a stockout, which in turn is an increasing, strictly convex function of the demand fill rate. He called the new model a perturbed demand (PD) model. Schwartz provided convincing justification for his PD model and presented several variations of it in a follow-up paper, but he did not solve any of these models. In this paper, we solve Schwartz's original PD model and its variations, and we discuss the implications of their solutions, thus filling a gap in the literature left by Schwartz. Moreover, having been convinced that Schwartz's approach is more valid than the classical approach for representing the effect of the loss of customer goodwill following a stockout, but also recognizing that the classical approach is far more popular than the PD approach, because of its simplicity and because of tradition, we use the solution of the PD model to infer the value of b in the classical model, thus providing one possible answer to the question, what could b be? A noteworthy implication of the solution of Schwartz's original PD model is that the optimal fill rate is always 0 or 1, rendering the inferred value of b in the classical model 0 or ∞, respectively. Suspecting that the property of the PD function which is most likely responsible for producing this “bang-bang” type of result is strict convexity, we show that for the case where the PD function is proportional to an integer power, say n, of the fill rate, the optimal fill rate is always 0 or 1, if and only if n>1, in which case the PD function is strictly convex in the fill rate.

An economic order quantity model for an imperfect production process with entropy cost

Among the assumptions of the classical economic order quantity (EOQ) model is that all units that are purchased (or produced) are of perfect quality. However, this is frequently unrealistic since production processes deteriorate resulting in the production of defective products requiring rework. Some recent studies suggest that production systems performance might be improved by applying the first and second laws of thermodynamics to reduce system entropy (or disorder). This paper applies the concept of entropy cost to extend the classical EOQ model under the assumptions of perfect and imperfect quality. Mathematical models are developed and numerical examples illustrating the solution procedure are provided. Accounting for entropy cost suggests that order quantities should be larger than the figures derived from the classical EOQ model.

AN EOQ MODEL WITH CONTROLLABLE SELLING RATE

According to the marketing principle, a decision maker may control demand rate through selling price and the unit facility cost of promoting transaction. In fact, the upper bound of willing-to-pay price and the transaction cost probably depend upon the subjective judgment of individual consumer in purchasing merchandise. This study therefore attempts to construct a bivariate distribution function to simultaneously incorporate the willing-to-pay price and the transaction cost into the classical economic order quantity (EOQ) model. Through the manipulation of the constructed bivariate distribution function, the demand function faced by the supplier can be expressed as a concrete form. The proposed mathematical model mainly concerns how to determine the initial inventory level for each business cycle, so that the profit per unit time is maximized by means of the selling price and the unit-transaction cost to control the selling rate. Furthermore, the sensitivity analysis of optimal solution is performed and the implication of this extended inventory model is also discussed.

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. This paper extends the work of Jaber et al. (2004) to consider items which are subject to deterioration. Such a situation increases entropy, and so intuitively one would expect deterioration to reduce the order quantity because smaller order quantities will be used up more quickly and are less likely to deteriorate. A mathematical model is developed that determines batch sizes for deteriorating items while minimising entropy. The numerical results obtained by using the model are then discussed.

Economic order quantity for items with imperfect quality: Revisited

We rectify a flaw in an economic order quantity (EOQ) model with unreliable supply, characterized by a random fraction of imperfect quality items and a screening process, developed by Salameh and Jaber [2000. Economic production quantity model for items with imperfect quality. International Journal of Production Economics 64(1–3), 59–64]. This leads to simple expressions of the optimal order quantity and expected profit per unit time. We then analyze the effect of screening speed and variability of the supply process on the order quantity, and show that the order quantity in our model is larger than that of the classical EOQ model when the variability of the yield rate is reasonably low. In addition, we extend the model by allowing for several batches of imperfect quality items to be consolidated and shipped in one lot. This is likely to be useful when there are economies of scale in shipping of imperfect quality items. Finally, we present numerical results to illustrate the application of the model and draw insights.

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 application of cost-effective fuzzy inventory controller to counteract demand fluctuation caused by bullwhip effect

This paper develops a fuzzy inventory model to counteract the demand fluctuation in supply demand networks, which combines fuzzy logic controller with (s, S) policy based on economic order quantity (EOQ) model. Following a literature review and a discussion of counteractions to the bullwhip effect and the obstruction of general counteraction in supply demand networks, a multi-echelon fuzzy inventory model in supply demand networks is proposed. A simulation model with one- and two-echelon supply demand network is built and tested for (s, S) policy based on the classical EOQ model and the proposed fuzzy inventory model. Based on the simulation, results of the relevance performance are presented and discussed, which show that the proposed multi-echelon fuzzy inventory model provides not only a cost-effective management of inventory (e.g. lower inventory levels and cost) in market uncertainty, but also another effective alternative for counteracting demand fluctuation. In particular, the proposed multi-echelon fuzzy inventory model shows benefit in counteracting demand fluctuation in multi-echelon supply demand networks. Finally, some conclusions and suggestions for further research works are presented.

A Model for Comparative Analysis of Inventory Costs of JIT and EOQ Purchasing in the Ready Mixed Concrete Industry

A model for comparing the inventory costs of purchasing under the economic order quantity (EOQ) system and the just-in-time (JIT) order purchasing system in existing literature concluded that JIT purchasing was virtually always the preferable inventory ordering system especially at high level of annual demand. By expanding the classical EOQ model, this paper shows that it is possible for the EOQ system to be more cost effective than the JIT system once the inventory demand approaches the EOQ-JIT cost indifference point. The case study conducted in the ready-mixed concrete industry in Singapore supported this proposition.

An Analytical Model of E-Recruiting Investment Decision: An Economic Employment Approach

The online recruiting market is one of the most rapidly growing e-commerce areas. Since the mid-1990s, a number of e-recruiting methods such as job boards, corporate career Web sites, and e-recruiting consortia have been introduced into the labor market. Among them, e-recruiting with the use of a corporate career Web site has been touted as the most efficient and cost-effective recruiting method. While the corporate career Web sites have experienced a phenomenal growth, no formal economic models have been developed for the analysis and assessment of investment decisions in various e-recruiting technologies. In this paper, a mathematics-based economic decision model for e-recruiting technology investment is introduced to evaluate the economic impact of various e-recruiting technologies. A classical economic order quantity (EOQ) model was extended to include costs uniquely associated with recruiting activities such as staffing and coordination costs. We derive a minimum employment level for an optimal investment in e-recruiting technologies. The model suggests that the optimal investment cost increases rapidly near the minimum number of employees to be recruited, but the growth of the optimal investment cost slows down as the total number of employees becomes greater than the minimum number of employees to be recruited.

Note on “Effects of Random Defective Rate and Imperfect Rework Process on Economic Production Quantity Model”

Conventional approaches for deriving optimal production lot size are by using the differential calculus on the production-inventory cost function with the need to prove optimality first. Recent articles proposed the algebraic approach to the solution of classic economic order quantity and economic production quantity (EPQ) model without reference to the use of derivatives. This note extends it to an EPQ model taking the random defective rate and imperfect rework process into consideration. We demonstrate that the optimal lot size can be solved algebraically and the expected inventory cost can be derived immediately as well.

Deterministic inventory lot-size models with shortages for fluctuating demand and unit purchase cost

The classical economic order quantity (EOQ) model assumes not only a constant demand rate but also a fixed unit purchasing cost. In today's time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle while its demand increases. Therefore, using the classical EOQ formulation for a high-tech product will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial. Consequently, adding the purchasing strategy into the EOQ model is vital. In this paper, we assume that not only the demand function but also the unit purchase cost is fluctuating with time. We then provide an easy-to-use algorithm to find the optimal replenishment number and schedule. In a numerical example, we show that the total cost obtained by our proposed model is 32.4% less expensive than that obtained by the classical EOQ model.

Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time

In today's time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand increases. In this paper, we extend the classical economic order quantity model to allow for not only time-varying demand but also fluctuating unit cost. In addition, we also allow for shortages and partial backlogging. We then prove that the optimal replenishment schedule not only exists but also is unique. In addition, we also show that the total cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Moreover, we further simplify the search process by providing an intuitively good starting search point.

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.

An EOQ model for deteriorating items with time varying demand and partial backlogging

In the classical economic order quantity model, it is often assumed that the shortages are either completely backlogged or completely lost. However, in some inventory systems, it is more reasonable to assume that the backlogging rate is dependent on the length of the waiting time for the next replenishment. The longer the waiting time is, the smaller the backlogging rate would be. In this paper, we focus on the effect of the backlogging rate on the economic order quantity decision. Numerical examples are presented to illustrate the model.