Classical Economic Order Quantity Research Articles

A Variant Deterministic Model of Classical EOQ Formula

Inventory management and transportation have been the principal areas of focus in industrial engineering and management for a long time. Inventory management attracts considerable attention in logistics and supply chain management today because new supply chain models have become more integrative and complex. New market forces have introduced many complex elements which affect the performance of the supply chain in general and inventory level in particular. Inventory decisions are high risk and high impact for supply chain management. Hence, this paper compiles all the derivations of classical deterministic lot size economic order quantity models and proposes a new method to verify the formula. Keyword: Inventory Management, Supply Chain Management (SCM), Economic Order Quantity (EOQ) I. INTRODUCTION At the very basic level any firm faces two main decisions concerning the management of inventory: When should new stock be ordered and in what quantities? With regard to the order quantity, that minimizes inventory related costs. The classical EOQ (economic order quantity) model remains the basic inventory model even when it is not applicable in real life business situations in most cases. In inventory related literature, the answer to the question of when to order is given with reference to the ROP (reorder point), and the point at which the replenishment order should be initiated so that the facility receives the inventory in time to maintain its target level of service. In the static and deterministic model, the ROP is the simple multiplication of the number of lead days and the daily demand. It means that every time the inventory falls to the ROP level, an order must be initiated. And the order quantity is given by the EOQ model which is based on cost minimization. Figure-1:~ A simple inventory model based on fixed demand and fixed lead time (1). The EOQ is the balance between order and holding costs attached with the inventory. The order cost is made up of fixed and variable costs, whereas the holding cost consist of costs of maintenance. The formula is: Q = √ (2CoD/Cc) Q is the order quantity per order, D is the demand per year Co is the fixed cost which the warehouse incurs every time it places an order Cc is the inventory carrying or holding cost per unit per year, and Notice that it highlights two important insights regarding the EOQ model. These are: 1. Optimum order size is a balance between the holding cost and the fixed order cost. 2. Total inventory cost is related with order size, but the relationship is not significant.

Optimal Inventory Policy of Production Management: A Present Value Framework

The classical economic production quantity (EPQ) assumes that the replenishments are instantaneous. As a manager of a factory, there is a problem must be taken into consideration. If the establishment buys all of the raw materials at the beginning, the stock-holding cost for the raw materials should be counted into the relevant costs. So the main purpose of this paper will add the raw materials stock-holding cost to the EPQ model and take the time value of money into consideration. Therefore, we will calculate the present value and compare the difference between take and does not take the time value of money into consideration. From these procedures of calculating, we found some interesting results: 1) the present value of total stock-holding cost of raw materials plus products from the beginning to time t is the same as the stock-holding cost of classical economic order quantity (EOQ) model; 2) the present value of total relevant cost is independent of the production rate (if the production rate is greater than the demand rate); 3) the optimal cycle time of total relevant cost not taking the time value into consideration is the same the optimal cycle time of classical EOQ model; 4) the purchasing cost per unit time is irrelevant to time.

Analysis of purchasing activity with discounted cash flow inventory models

The classical economic order quantity model has two types of costs: ordering and inventory holding costs. In this paper we try to investigate the effect of purchasing activity on cash flow of a firm. In the examinations we use a cash flow identity similar to that of in inventory modeling. In our approach we analyze the purchasing and ordering process with discounted costs. The cost function of the model consists of linear cash holding, linear opportunity cost of spending cash, and linear interest costs. We show the optimal solution of the proposed model. The optimal solutions will be presented by numerical examples.

Optimal Lot Sizing with NPV Approach for Inventory Systems with Constant Demand

This paper presents two new economic order quantity (EOQ) formulations supplemented with net present value (NPV) approach. While NPV approach is broadly accepted as the most realistic framework in modelling inventory systems, the literature possesses rather contradicting results on the application. In this paper, we introduce a NPV based EOQ model for an inventory system with constant demand, under fixed ordering cost and linear holding and penalty cost assumptions. For this particular problem, Gurnani (1983) is the first to provide some results on NVP approach. His study is criticized by Kim and Chung (1985) afterwards. In this study, we show that the earlier works of Gurnani, and Kim and Chung are both flawed, and we provide the optimal formulations for both cost minimization and profit maximization objectives. Through an extensive numerical study, we demonstrate that classical EOQ model may lead to sub-optimal solutions without an embedded cash flow analysis. Meanwhile, we also show how EOQ models proposed by Gurnani and Kim and Chung deviate from optimality.

Robust Optimization in Simulation: Taguchi and Krige Combined

Optimization of simulated systems is the goal of many methods, but most methods assume known environments. We, however, develop a “robust” methodology that accounts for uncertain environments. Our methodology uses Taguchi's view of the uncertain world but replaces his statistical techniques by design and analysis of simulation experiments based on Kriging (Gaussian process model); moreover, we use bootstrapping to quantify the variability in the estimated Kriging metamodels. In addition, we combine Kriging with nonlinear programming, and we estimate the Pareto frontier. We illustrate the resulting methodology through economic order quantity (EOQ) inventory models. Our results suggest that robust optimization requires order quantities that differ from the classic EOQ. We also compare our results with results we previously obtained using response surface methodology instead of Kriging.

An economic order quantity model with multiple partial prepayments and partial backordering

The classic Economic Order Quantity model assumes that an order is paid for at the time of its receipt. In practice, suppliers may require purchasers to pay a fraction of the order’s cost in advance and sometimes allow them to divide the prepayment into multiple equal-sized parts to be paid during a fixed lead time. In this paper, an EOQ model with multiple prepayments under three different conditions: (a) no shortage, (b) full backordering and (c) partial backordering, are developed. Numerical examples illustrate the proposed models and solution method.

Including sustainability criteria into inventory models

Research on sustainability performance has considerably enriched operations management literature in recent years. However, work with quantitative models is still scarce. This paper contributes by revisiting classical inventory methods taking sustainability concerns into account. We believe that reducing all aspects of sustainable development to a single objective is not desirable. We thus reformulate the classical economic order quantity model as a multiobjective problem. We refer to this model as the sustainable order quantity model. Then, a multi-echelon extension of the sustainable order quantity model is studied. For both models, the set of efficient solutions (Pareto optimal solutions) is analytically characterized. These results are used to provide some insights about the effectiveness of different regulatory policies to control carbon emissions. We also use an interactive procedure that allows the decision maker to quickly identify the best option among these solutions. The proposed interactive procedure is a new combination of multi-criteria decision analysis techniques.

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.

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.

Economic Order Quantity for Items with Two Types of Imperfect Quality

The classical economic order quantity considers that the ordered items are of perfect quality. In this research, a model for the economic order quantity of imperfect quality items is developed, where the incoming lot has fractions of scrap and reworkable items. These fractions are considered to be random variables with known probability density functions. The demand is satisfied from perfect items and reworked items; whereas the scrap items are sold in a single batch at the end of the cycle with a salvage cost. A numerical analysis is provided to illustrate the sensitivity of the model to the fractions of scrap and reworked items.

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.

Analysis of a two-party supply chain with random disruptions

We study a continuous-review inventory problem of a two-echelon supply chain with random disruptions, identify properties of the optimal cost function, compare the optimal order quantity with the classical economic order quantity, analyze the sensitivity of the optimal solution, and explore the conditions under which zero-inventory ordering policy is preferred.

Alternative approach for solving replenishment lot size problem with discontinuous issuing policy and rework

Conventional approach for solving the replenishment lot size problem is by using differential calculus on the long-run average production cost function with the need to prove optimality first. Recent studies proposed an algebraic approach to the solution of classic economic order quantity (EOQ) and the economic production quantity (EPQ) models without reference to the use of derivatives. This paper extends it to the solution of a specific EPQ model as was examined by Chiu et al. [Chiu, S. W, Chen, K. -K, Lin, H. -D. Numerical method for determination of the optimal lot size for amanufacturing system with discontinuous issuing policy and rework. International Journal for Numerical Methods in Biomedical Engineering. doi: 10.1002/cnm.1369. (in Press; Published online March-10-2010).]. As a result, optimal replenishment lot size and a simplified optimal production-inventory cost formula for such a particular EPQ model can be derived without derivatives. This alternative approach may enable practitioners who with little knowledge of calculus to understand the realistic production systems with ease.

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.

Analysis of economic order quantity under fuzzy environments

In the economic order quantity (EOQ) model, the decision maker has vague information about holding cost, ordering cost and market demand. With these uncertainties characterized as fuzzy variables, a new formula is developed by analyzing the fuzzy total cost. By comparing with other four EOQ formulas, i.e., using the crisp numbers with the highest membership values in classic EOQ formula, using the expected values of fuzzy parameters in classic EOQ formula, using the fuzzy variables in classic EOQ formula and then calculating the expected value, and calculating EOQ by hybrid intelligent algorithm simulation, the effectiveness of this formula is illustrated.

The integrality of the lot size in the basic EOQ and EPQ models: Applications to other production-inventory models

In this paper we present a method to obtain the solution of the classic economic order quantity (EOQ) and economic production quantity (EPQ) models when the lot size must be an integer quantity. This approach is operatively very simple and allows obtaining a rule to discriminate between the situation in which the optimal solution is unique and when there are two optimal solutions. Also, this method is applicable to the resolution of other production-inventory models. We expose some of them and illustrate the use of the method with numerical examples.

Robust optimization in simulation: Taguchi and Response Surface Methodology

Optimization of simulated systems is tackled by many methods, but most methods assume known environments. This article, however, develops a ‘robust’ methodology for uncertain environments. This methodology uses Taguchi's view of the uncertain world, but replaces his statistical techniques by Response Surface Methodology (RSM). George Box originated RSM, and Douglas Montgomery recently extended RSM to robust optimization of real (non-simulated) systems. We combine Taguchi's view with RSM for simulated systems. We illustrate the resulting methodology through classic Economic Order Quantity (EOQ) inventory models, which demonstrate that robust optimization may require order quantities that differ from the classic EOQ.