BUS300: Operations Management

Abstract

Inventory control and management is an important task for every organization, wherein they need to determine the order quantity and order period to maintain enough inventory to meet the demand. The original Economic Order Quantity (EOQ) model was developed by Ford W. Harris to address this problem and determine the optimal order quantity that helps in optimizing the cost incurred by the organization. Even though this model gives a very good approximation on these parameters, there were many inherent restrictions to the model that were based on unrealistic expectations. In over a century since its original publication, many researchers have made extensions to this model to address the limitations and align it with real world scenarios. In this review paper, we intend to provide a representation of the original EOQ model, its limitations and the various extensions made to the model to address these limitations.

Keywords: EOQ, Inventory cost, Holding cost, Perishable items, Delayed payments, Lead time, Replenishment, Price discount, Backorder, Backlog

Introduction

The material cost contributes a significant portion to the total cost of a manufactured product, and any additional inventory held by the company locks up its capital from being put to better use. Given these, it is very essential that the Inventory Management and control of an organization should be a top priority for the management. This planning involves things such as how much quantity of raw material should be ordered and at what time or frequency for optimal cost structures. Economic order quantity (EOQ) model, which was originally published by Ford W. Harris and extensively studied and applied by R.H. Wilson gives an optimal quantity to order which minimizes the total cost to the organization. This classical EOQ model has served as a basis for multiple extensions with time to address varying conditions and specific requirements.

Classical EOQ model

There are three types of cost associated with inventory management in general-

1. Ordering Cost (Co) – New orders need to be placed with the supplier based on the requirements, and there are multiple activities like supplier selection, price negotiation, purchase order generation, receipt of material etc. Each of these activities have an associated cost which makes up the ordering cost. For large quantity orders, we need fewer orders and hence lower Co and vice versa.
2. Holding Cost (Ch) – This includes the cost of storing the inventory, insurance cost, obsolescence cost, pilferage and wastage etc. These costs make up the Holding cost. The Holding costs usually increase linearly with the inventory level.
3. Shortage Cost – This cost results from shortage in inventory, which can result in lost sales, backorder, customer dissatisfaction etc

The Classic EOQ model was made with the following assumptions –

1. A deterministic and constant annual demand of D
2. A constant and known Lead time – L
3. The ordering cost is constant and does not depend on order size – Co
4. No limit on order size either for the supplier or the firm
5. Constant cost of holding a unit – Ch (per unit).
6. Fixed unit cost of the item, does not change with time or order quantity – p
7. No stockouts are allowed and all the demand is met.
   

With the above assumptions, the total cost of the plan (TC) is given by,

TC(Q) = Total Holding Cost + Total Ordering cost, i.e.,

TC(Q) = ((Q/2)*Ch) + ((D/Q)*Co))

Here Q is the quantity per order and Q/2 is the average holding per cycle. The optimal or EOQ (Q*) is determined by finding the minimum of the Total cost, which is given by

Q* = √((2Co*D)/Ch)

and the optimal number of orders is given as (D/Q*).

A graphical representation of the cost and the EOQ is as given in Fig. 1

Fig. 1 Economic Order Quantity and Costs

Limitations of the Classic EOQ model

The assumptions made for the classic EOQ model result in many limitations for practical applications. Some of these limitations are –

1. Fixed unit cost
   This assumption is reasonable when there is a long-term contract for supply of item and the per unit cost can be fixed. But in reality, we can have situations where there are price discounts based on quantity, items subject to inflation etc. which are not addressed. Delayed payment terms provided by suppliers can also help in reducing the total cost for the firm.
   
2. No restrictions on order quantity
   This assumption does not account for situations where perishable items are being handled. The inventory needs to closely match the demand to avoid wastage. Also, there could be warehouse / storage space restrictions that could put a cap on the order size.
   
3. Stockouts are not allowed
   This assumes that all the demand can be met always. But there can be cases where it is beneficial to allow backordering if the backordering costs are substantially low. Also, this situation happens when there is variable demand with time or any unexpected delays in supply etc.
4. All items are perfect, no wastage
   This assumption does not address the case when there are imperfect items produced due to various production constraints, perishable items remaining in inventory beyond their shelf- life etc.
   

Extensions to the Classic EOQ model

In the earlier section, we had looked at some of the limitations that were present in the classic EOQ model because of the assumptions. Many researchers have worked on extensions to the original model to address these limitations as needed. We will now look at how a few of these model extensions were done.

Fixed Unit cost limitation

Suppliers at times provide a price discount or delayed payments to the retailer so that it gives them an incentive to buy more. Biswajit Sarkar presented a model that provides for optimal replenishment quantity under delayed payment terms along with stock dependent demand and in the presence of imperfect items. The author was able to establish a model that helped in optimizing cost based on the terms for delayed payment segmented into 3 distinct intervals. This kind of model is specifically applicable to goods like sugar, clothes and spices etc. that have stock dependent demand. Ata Allah Taleizadeh et. al. developed an extension model that would address the case of a known price increase in the presence of partial backordering. In case of a permanent increase or temporary reduction in price of an item, the buyer can place a special order that is bigger in size than normal to take advantage. This model covers this aspect and helps in calculating the optimal order quantity for such a situation when the price change happens under inventory available or under shortage conditions. Sheng-Chin Chen et. al. developed a simple arithmetic- geometric method instead of the differential calculus model to be applied to a retailer with permissible delayed payments from the supplier. They noted that the fraction of permissible delay payment was inversely proportional to the optimal reorder time, and the delay payment threshold was directly proportional to the reorder time

No restrictions on order quantity

Even though the classic EOQ model placed no constraints on the order quantity, there are restrictions in terms of how much storage space a firm can afford. Many studies have been done in this respect to address these limitations. T.K. Roy et. al. developed a fuzzy model that introduced fuzziness in both the storage space available as well as the unit price which was not constant, but varies inversely with the current demand. With their model, they were able to show better results than the usual crisp model.

Stockouts are not allowed

Stockouts happen when there is customer demand, but the firm does not have enough items in inventory to meet the demand. This can lead to two possibilities – lost sales for those customers who do not want to wait, and backorders where the customers are willing to wait for stock replenishment, but potentially adding cost to the firm. Even though stockouts are costly and not a desirable condition, it is very much a reality and firms need to address them in their inventory management through partial or full backorders or accounting for lost sales.

All items are perfect, no wastage

Even with the constantly improving quality control measures in firms, it is unrealistic to assume that all produced items would be perfect and no wastage or rework needs to be done. So, it is essential to incorporate the quality aspect into the classic EOQ model to mimic situations that are closer to real world.

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Source: Uma Raju, https://iopscience.iop.org/article/10.1088/1742-6596/2332/1/012019/pdf
This work is licensed under a Creative Commons Attribution 3.0 License.