Economic Order Quantity
Read this paper on EOQ modeling, which is fairly technical but should help you understand the fundamental function of this equation. EOQ is important because it helps minimize the total holding and ordering costs related to inventory. Pay close attention to when this applies in the production process.
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.
In this paper, we will provide a basic representation of the classical EOQ model, the limitations of
the classical model and subsequently discuss a few extensions to this model that were developed
for specific requirements and practical scenarios.
Classical EOQ model
There are three types of cost associated with inventory management in general-
- 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.
- 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.
- Shortage Cost – This cost results from shortage in inventory, which can result in lost sales, backorder, customer dissatisfaction etc
- A deterministic and constant annual demand of D
- A constant and known Lead time – L
- The ordering cost is constant and does not depend on order size – Co
- No limit on order size either for the supplier or the firm
- Constant cost of holding a unit – Ch (per unit).
- Fixed unit cost of the item, does not change with time or order quantity – p
- 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 –
- 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. - 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. - 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. - 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.
In case of perishable items such as food or certain electronic items or chemicals, they usually
have a shelf life after which they cannot be used for the original intended purpose. The problem
could be due to various reasons such as decay, evaporation, loss of quality etc. Richard P.
Covert et. al. developed an extension that modelled the deterioration of the item over
time using a Weibull distribution with assumptions of constant demand, no delay in delivery
and no shortage. Their results were converging to other models that accounted for deterioration
and also with the original model when the deterioration was negligible. Gede Agus Widyadana
et. al. worked on a model for the deteriorating items with and without backorder using
a simplified approach. They used a modified cost-difference comparison for the same and the
results were comparable to other models, and hence provides for a mechanism that is easier to
solve. Cinzia Muriana presented a mathematical model that had stochastic demand
for perishable items with fixed shelf life. The varying demand was modelled using a normal
distribution, and the model was used to determine the probability of having a perished item in
inventory and based on this determine the optimized cost conditions. This helped in providing
a better retailer strategy based on the parameter
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.
David W. Pentico et. al. modelled a deterministic EOQ that accounted for partial backordering. The model for a partial backorder is more difficult to approach than a full backorder or lost sales case and, in this respect, their model provided an easy approach to get the EOQ under such conditions. Their results also matched with those of full backordering and the basic EOQ model under similar conditions. N. Kazemi et. al. developed a fuzzy model with backorders using triangular and trapezoidal fuzzy numbers for the input parameters and decision variables. This approach provided a better method than using a deterministic one. Kuhn-Tucker conditions were used to determine the optimal policy in this method. They established a directly correlated relationship between the decision variables and the differences in cost between the crisp and their fuzzy model.
Sujit Kumar De et. al. approached the problem of backordering by considering
promotional efforts such as price discounts, free gifts, delay in payments or better services etc.
under stockout conditions. They developed a model with pentagonal strong / weak fuzzy
variables applied to the classical backordered EOQ model with shortage time dependent
demand rate and in presence of promotional efforts.
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.
M.K. Salameh et. al. provided a framework for the EOQ model that would account
for imperfect items. Their model used a 100% screening to identify the items that were
imperfect and these were separated and subsequently sold at a discounted price for alternate
use. So, screening cost became a significant cost in their model. But full inspection is very
costly and may not be a reasonable approach. Building up on this, Jafar Rezaei worked on a EOQ model that used sampling inspections for identification of imperfect items.
Based on pre-determined thresholds, there were 3 options based on sampling – no inspection
of the lot if the number of imperfect items is lesser than a lower threshold and these can be
addressed using replacements eventually, full inspection if the sample value is between the
lower and upper thresholds and finally reject the lot if the sample value is higher than the upper
threshold. With this approach, the firm gets a better optimized cost and EOQ as it accounts for
imperfect items.
Amir Hosein Nobil et. al. extended the imperfect item study to determine the reorder
point, which was missing from earlier such studies. Their research helped in incorporating
certain rates of imperfect items, system costs and lead time to get maximum profit and
efficiency
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.