Classical Economic Order Quantity Model Research Articles

Optimal ordering policies in response to a disposable coupon

This study investigates a retailer's ordering strategy under the framework of the economic order quantity (EOQ) model. A supplier offers a retailer a disposable coupon and allows it to place a special order at any time in a promotion period. The promotion period is not necessary short and shortages are allowed throughout the time horizon. In addition to the special order time and the special order quantity, the retailer needs to decide whether to place some regular orders in the promotion period before placing the special order for the purpose of making better use of this coupon. We show that the coupon should be used to the retailer's first order in the promotion period regardless of the duration of the promotion period. Moreover, the retailer's maximum inventory level is higher than that in the classical EOQ model. We find that a longer promotion period can benefit the retailer by endowing it with more flexibility in its decision-making. Therefore, the supplier can improve the cash flow and reduce the overstock by integrating a disposable coupon with a longer promotion period. Numerous managerial insights are obtained from sensitivity analysis and numerical experiments.

EOQ Models for Imperfect Items under Time Varying Demand Rate

In the classical Economic Order Quantity (EOQ) model, the common unrealistic assumptions are that all the purchased items are of perfect quality and the demand is constant. However, in a real-world environment, a portion of the purchased items might be damaged due to mishandling or an accident during the shipment process, and the demand rate may increase or decrease over time. Many companies are torn between repairing or replacing the imperfect items with new ones. The right decision on that options is crucial in order to guarantee that there is no shortage of stocks while at the same time not jeopardising the items’ quality and maximising the company’s profit. This paper investigates two EOQ models for imperfect quality items by assuming the demand rate varies with time. Under Policy 1, imperfect items are sent for repairs at an additional cost to the makeup margin; under Policy 2, imperfect items are replaced with equivalent quality items from a local supplier at a higher price. Two mathematical models are developed, and numerical examples along with sensitivity analyses are provided to illustrate these models. Our results reveal that Policy 1 is preferable to Policy 2 most of the time. However, Policy 2 outperforms Policy 1 if there is no minimum threshold on the purchased stock quantity. This research allows a company to discover solutions to previously identified inventory problems and make the inventory-patching process more controlled.

Inventory policies under fuzzy and cloud fuzzy environment

This article is an attempt to extend the classical economic order quantity (EOQ) model for deteriorating items under fuzzy and cloud fuzzy environment. Inventory parameters such as holding cost, purchase cost, ordering cost, demand rate and deterioration rate are considered as triangular fuzzy numbers as well as cloud triangular fuzzy numbers to develop fuzzy and cloud fuzzy models respectively. Yager's ranking method and De and Beg's ranking methods are used for defuzzification. A comparative study reveals the superiority of cloud fuzzy model over crisp and fuzzy model. Further, numerical example, graphical illustrations and sensitivity analysis are carried out for better understanding of the application of cloud fuzzy approach to inventory optimisation problem.

The average-cost formulation of lot sizing models and inventory carrying charges: a technical note

It is generally recognised that the present-value criterion should be preferred to the average-cost formulation in developing lot sizing models. Despite the advantages of the present-value measure, average-cost lot sizing models are far more widely applied. Because of the nature of the average-cost formulation, inventory carrying costs are evaluated according to a look-back approach, relying on historical values. In this regard, a general misconception in the inventory management literature concerned with average-cost models is that the unit stockholding cost rate should be established considering fixed warehouse costs, which are costs that are, in the short term, independent of the inventory level. This paper develops arguments supporting our belief that inventory carrying charges used in lot sizing models should take into account only those costs varying with the inventory level in the warehouse, and that considering fixed warehouse costs leads to pitfalls when making inventory replenishment decisions. To this aim, we first present an analytical treatment based on the classical Economic Order Quantity (EOQ) model, as its full analytical tractability permits us to better discuss the problem we are interested in. Finally, we present numerical experiments to assess the effect of the correct procedure to establish the unit stockholding cost rate on inventory management decisions. These experiments are performed considering warehouse costs taken from some industrial case studies presented in the literature.

Optimal integrated production-inventory system considering shortages and discrete delivery orders

This paper proposes an optimal integrated production-inventory model with multiple discrete delivery and shortage to manage the step-wise delivery of the products to the retailers. The main objective of the model is to determine the optimal order and backorder quantities to obtain the minimum total cost. A novel smoothing strategy and an original heuristic algorithm have been introduced to cope with the non-linearity of the model and solve it. The encouraging and quite simple results obtained for practical purposes show the superiority of the proposed framework compared to the classical Economic Order Quantity (EOQ) model. These empirical findings lead to some managerial implications that enable the system to select the best modeling approach based on different amounts of cost.

Optimal inventory policies for short life cycle successive generations’ technology products

In this paper, a new Economic Order Quantity (EOQ) model for a successive generation of technology products has been discussed. The classical EOQ model is based on the assumption that the demand rate is constant. Hence it cannot be used for technology products where competition-substitution among products is a usual phenomenon. To address this problem, the EOQ model proposed in this article is considered a demand model for a technology product that follows the innovation-diffusion process. A numerical example has been illustrated and a comprehensive sensitivity analysis is conducted to understand the path of the optimal planning horizon and optimal costs under varied innovation and imitation effect. The sensitivity analysis of the introduction timing of the second generation has been performed to know the applicability of the model in actual circumstances. The behavior of the model has been discussed in detail in the numerical illustration section.

Economic Order Quantity Model for Growing Items with Incremental Quantity Discounts, Capacitated Storage Facility, and Limited Budget

The development of the inventory model started when Harris introduced the classic inventory model. It was firstly published by Wilson using the optimization method. He derived a mathematical equation model to obtain economic order quantities. Later, this model is known as the classic Economic Order Quantity (EOQ) or Wilson Model. The classic inventory EOQ model has some limitations. The model assumed that order items do not have physical changes during a planning period. This assumption becomes the weakness of the classical EOQ inventory model. Many items have material changes during a planning period, such as amelioration, deterioration, and growth. This research proposed a new mathematical model. The model relaxes three implicit assumptions of the classical EOQ: (1) the ordered items do not grow; (2) unlimited capacity; and (3) unlimited budget. A solution procedure to solve the model was developed and illustrated with a numerical example. A numerical example was performed to compare the result between the reference model and the new model. The number of ordered items per cycle time increased by 7%, and cycle time increased by 28%. It increased because the proposed model tends to choose large purchased quantities to get a cheap price. It caused the number of ordered items per cycle time to be larger and the cycle time to be smaller than the reference model. This research also provided sensitivity analysis. It showed the response of decision variables to some changes in input parameters.

Effects of discount policies on economic order quantity and total cost for perishables: a case study

The classical economic order quantity (EOQ) model assumes that demand stays constant over time. This study examined the effect of changes in the economic order quantities resulting from discount po...

The EOQ model with items of imperfect quality and replenishment from different suppliers

The classical economic order quantity model is extended to the case where the items may be acquired from various suppliers. It is assumed that any lot size received from a supplier contains perfect and imperfect quality items. The percentage of perfect quality items in a lot size is a random variable having a known probability distribution. The imperfect quality items are detected through a 100% screening process conducted at the start of the inventory cycle. When the screening process is concluded, the imperfect quality items are sold in one batch at a discounted price. A mathematical model is developed to determine the total profit function. The optimal order quantity and proportions of the order acquired from the suppliers are obtained by maximising the total profit function. Iterative numerical algorithms that determine the optimal solution are proposed. Numerical examples are presented to illustrate the calculations in the case when the percentages of imperfect quality items follow the uniform distribution.

On the Economic Order Quantity Model with Compounding

The classical Economic Order Quantity (EOQ) model assumes simple interest to represent the opportunity cost of capital tied up in the inventory. Recently, the classical model has been extended to incorporate compounding, and an intuitive closed-form solution has been proposed. The compounding based model is more realistic than the original EOQ model because compound interest is the standard practice in finance and banking. The resulting closed-form solution proposed for this recent model is based on an approximation of the annual compound interest-based opportunity cost. However, the derivation of the approximation model is rather long and complicated, involving the use of the L’Hôpital’s rule several times. Here, we show an easier way to approximate the annual compound interest-based opportunity cost. Our derivation is shorter and it does not require the use of the L’Hôpital’s rule. We also demonstrate that the approximation is remarkably close to the exact model, and it results in the same intuitive closed-form solution as the earlier one.

Environmental sustainability EOQ model for closed-loop supply chain under market uncertainty: A case study of printer remanufacturing

The traditional economic order quantity (EOQ) model determines the optimal ordering strategy based on the premise that the market demand is constant, which leads to obvious deviations under uncertain circumstances. This work relaxes the limitations of the classical EOQ model and extends the application to closed-loop supply chain (CLSC) systems. First, through stochastic analysis, this revised model quantifies the impacts of market uncertainty on remanufacturing and operations processes. Then, by converting all the production indexes into carbon equivalent emissions, it conducts a comparative analysis of environmental efficiency under different acquisition scenarios. More importantly, based on the quantitative relationship between market uncertainty and production indexes, we devise an optimal ordering strategy for the maximization of environmental benefits under an arbitrary distribution of demand from the perspective of the entire CLSC. Finally, validated by numerical experiments and grounded with the data of printer remanufacturing, we demonstrate that the extended EOQ method can greatly reduce carbon equivalent emissions and improve environmental efficiency. This work provides a theoretical foundation for remanufacturers to formulate optimal ordering strategies under market uncertainty conditions while being more environmentally efficient without significantly increasing production input.

An Optimization Approach to the Low-Frequency Entire Train Formation at the Loading Area

It is well known that the shift of transporting bulk cargo from roads to railways is an important measure to reduce carbon emissions of the overall transportation systems. In order to increase the attractiveness of railway transport, companies usually provide some discounts to the customers with great transport demand so that entire trains can be operated. Since the operation of entire trains can reduce the reclassification times of shipments, the expenses of railway operations can be reduced. However, when the volume of shipment is not sufficient, the door-to-door direct transportation (in the railway industry specifically, “door-to-door” means running trains from supplier’s warehouse to customer’s warehouse) of the entire train often leads to a decrease in the frequency of delivery, which increases the average stock of users, thus increasing the inventory cost of users. Therefore, how to balance the pros and cons of the two is exactly the problem to be studied. In this paper, the optimal operation plan is obtained by minimizing the total cost of the stockholding of suppliers and customers, as well as the transportation costs of an entire train and non-direct train. Based on the classic economic order quantity (EOQ) model, a 0-1 integer programming model with the constraint of the maximum stock level is proposed to solve this problem. And an innovative approach is used to calculate the actual average stock of the customer. Finally, the model is validated and its effectiveness is confirmed using a real-world case, which is carried out using data from the China rail system.

A comprehensive study of an economic order quantity model under fuzzy monsoon demand

This article deals with a classical economic order quantity (EOQ) model under monsoon type fuzzy demand rate. It is nothing but the generalization of cloudy fuzzy model. We split the EOQ model into three parts according to the real-time fuzzy components of the demand rate. To defuzzify the model we develop an algorithm and the solution is obtained with the help of a nonlinear optimization technique that requires maximum aspiration level of the fuzzy membership of the objective function. Moreover, for comparative study we take numerical results of the crisp, general fuzzy and cloudy fuzzy model also. By this study we have shown that the decision maker might have to choose the monsoon type fuzzy environment all the time to control the proposed inventory. Moreover, sensitivity analysis and graphical illustrations are made to justify the new fuzzy approach.

Carbon-Regulated EOQ Models with Consumers’ Low-Carbon Awareness

In the context of a low-carbon economy, firms must make positive responses in their operation management, including inventory management. Carbon-emission regulation policies have marked their influence on the optimization of low-carbon inventory systems. In addition to regulation policies, consumers’ low-carbon awareness can also influence inventory systems by affecting demand. This study investigates the influence of regulation policies and consumers’ low-carbon awareness on optimal order size, emission levels, and total costs. Two widely used regulation policies, i.e., the carbon-tax mechanism and cap-and-trade mechanism, are incorporated into the classical Economic Order Quantity (EOQ) model. Analytical conclusions were obtained by optimization methods to indicate the influences of regulation policies and consumers’ low-carbon awareness. Our study implies that inventory systems under different regulation policies perform similarly except with regard to total cost. Numerical examples provide more support for these analytical conclusions. Some managerial insights can be derived from the analytical conclusions and numerical examples.

Production-inventory and emission reduction investment decision under carbon cap-and-trade policy

The increasing amount of carbon emissions has caused global warming and challenged the sustainable development of environment. Governments around the world have implemented carbon policies including carbon cap-and-trade policy. In this paper, we focus on how a two-echelon supply chain manages its carbon footprints in production and inventory under carbon cap-and-trade policy. We extend the classical EOQ (economic order quantity) model and study decisions on production-inventory, carbon trading and emission reduction investment in the decentralized and centralized situations. The results show that emission permit sharing can effectively reduce the total cost and total carbon emissions of the supply chain. Moreover, the manufacturer’s emission reduction effort rises with the increase of the buying and selling prices of emission permits under centralized decision-making. In addition, a compensation mechanism is proposed for the centralized supply chain with emission permit sharing. It is observed that the buying and selling prices of emission permits have a positive influence on the permit sharing price in the compensation mechanism. Meanwhile, the retailer pays less for using the emission permits if it has a higher carbon cap, while the manufacturer with a higher carbon cap is more capable to provide a high compensation for the retailer.

EES-EOQ model with uncertain acquisition quantity and market demand in dedicated or combined remanufacturing systems

• Build up a mathematical model analyzing environmental benefits of remanufacture over manufacture. • Overcome the limitations of the classical EOQ model and assume acquisition and demand are both uncertain. • Prove the existence and uniqueness of optimal strategy mathematically. • Close-form expression of optimal solutions that maximize the benefits and minimize costs. • Optimal strategies for dedicated and combined remanufacturing models. The classical economic order quantity (EOQ) model assumes that market demand is deterministic and would be invalid without this premise. Thus far, available literature on the EOQ model do not consider any situation where both market demand and acquisition quantity are uncertain. Therefore, this work develops an evolved environmental sustainability EOQ (EES-EOQ) model to overcome the limitations of the classical EOQ model. Employing the remanufacture-up-to ratio to control the production quantity indirectly, we deduce three optimal strategies: (I) minimize the inventory cost, (II) maximize the overall environmental benefit in a dedicated remanufacturing model, and (III) coordinate forward logistics and reverse logistics to achieve environment optimization in the combined remanufacturing model. With stochastic analysis, we mathematically prove the existence and uniqueness of the optimal remanufacture-up-to ratio in each strategy. Validated by numerical experiments and sensitivity analysis, we demonstrate that the replenishment of new products in the combined model can effectively eliminate the risks of being out of stock and strike an optimal balance between uncertain supply and demand. This study provides the theoretical foundation for remanufacturers to develop optimal remanufacturing strategies for mitigating environmental issues caused by excessive carbon emissions.

A carbon-constrained EOQ model with uncertain demand for remanufactured products

The classical economic order quantity (EOQ) model is simple and practicable, but it would be invalid without the rigorous assumption that demand is constant. This work breaks through the limitation and develops an extended EOQ model under uncertain demand, in order to derive different optimal acquisition strategies. Under the carbon cap and trade mechanism, the optimal quantities in environmental and economic optimization are usually different, while we deduce the optimal acquisition strategies (III and IIII) to solve multi-objective optimization problems. We rigorously prove the uniqueness of optimal order quantity in each strategy and provide close-form expressions for possible holding and opportunity costs caused by the uncertain supply-demand relationship. Other than conducting an optimization analysis from the perspective of firms, we also analyze the effects of taxation policy on the optimal strategies from the perspective of the government. An interesting phenomenon is that increasing carbon tax will not only reduce profit margins, but will also change the optimal acquisition decisions. Consequently, remanufacturing enterprises can be guided in a sustainable direction, by adopting more energy-efficient strategies to enter the “profitable region”.

A lot sizing model with advance payment and planned backordering

One of the assumptions in the classic economic order quantity model is that the buyer pays the purchasing cost for an order immediately after receiving the goods. In the real world, the vendors sometimes offer the buyers to pay all or a proportion of the purchasing cost after receiving the items to encourage them to increase their orders (i.e., delayed payment). Furthermore, in some cases, the powerful vendors may ask the buyers to prepay the entire, or a fraction of the purchasing cost before the delivery to mitigate the risk of cancellation or procuring the primary parts (i.e., advance payment). Advance payment is widely used by firms, but its effects on customer’s inventory decisions are seldom discussed. In this paper, we investigate the customer’s inventory policy by considering two different conditions: (a) full prepayment with shortage, (b) partial prepayment–partial delayed payment with shortage. We discuss the effect of parameters such as price discount linked to prepayment and length of prepayment on optimal periods of inventory and shortage quantity. The conclusions shows that the length of the advance payment period does not influence the inventory cycle, but that parameters such as discount factor linked to advance payment and delayed payment period affect the optimal inventory cycle in all cases. Numerical examples and a sensitivity analysis are presented to demonstrate the performance of the model and the results.

Bertrand-nash Equilibrium in the Retail Duopoly Model under Asymmetric Costs

In this paper, the Bertrand's price competition in the retail duopoly with asymmetric costs is analyzed. Retailers sell substitute products in the framework of the classical economic order quantity (EOQ) model with linear demand function. The market potential and competitor price are considered to be the bifurcation parameters of retailers. Levels of the barriers to market penetration depending on the bifurcation parameters are analyzed. The conditions of Bertrand-Nash equilibrium in parametric and trigonometric forms are found.

An EOQ model for decaying item with full advanced payment and conditional discount

The classic economic order quantity model assumes that purchasing cost should be paid immediately after the delivery time. In practice, sometimes the vendors ask the buyers to prepay the entire or a percentage of the purchasing cost before delivery time. In this paper the buyer’s inventory control system for a decaying item under full prepayment scheme based on various conditions consisting of (1) no shortage, (2) full backordering shortage is allowed and (3) partial lost sale is permitted, are developed. Numerical analysis is provided to show the performance of the model and some managerial insights are presented based on the proposed solution technique and sensitivity analysis.