If large quantities of material are ordered at any one time, rather than small quantities, then the inventory ordering costs per unit, and the probability of stock out will be low. However, inventory carrying costs will be high.

Thus, there is an inventory cost component that tends to decrease - ordering costs and stockout costs - when large quantities are ordered and one other that tends to increase – the carrying costs - when large quantities are held in stock.

As seen more in details in the article **what is the Inventory cost structure** - the total inventory cost is the sum of the carrying costs, stockout costs and ordering costs – and the economic ordering cost is when this total cost has a minimum value.

We present here 4 EOQ models depending on usage parameters from constant to variable demand and lead-time.

# EOQ Model 1: basic model

The basic model for determining the optimal economic order quantity (EOQ) was first proposed by Harris (1913).

This basic model assumes the following parameters are constant and known with certainty:

- Demand rate per unit time : consumption is linear
- Replenishment lead time: no delays
- Per-unit purchase cost: no discounts or price breaks
- Per-unit inventory-holding cost
- Per-unit ordering cost

ln addition, the model assumes that all demand must be met and the complete order is delivered instantaneously. Hence, the basic EOQ model does not consider purchase quantity discounts, back orders, or inventory shortage costs.

Moreover, there is no safety stock considered in this model.

* *

*Terms*

**D**is the annual demand by the client for inventory in units/year. However, the time period could be weeks, months, or some other period.**Q**is the quantity of material ordered each time a requisition is made and is measured in units/ purchase order.**C**is the cost of carrying one unit in inventory for one year, $/unit/year when one year is the time period in consideration. This value of C may be a given financial amount such as $10/unit/year or may be expressed as a percentage of the unit purchase price of the material.**S**is the average cost of placing a purchase order, $/order.**TSC**is the total annual inventory cost, $/year. The assumptions of the model are that the values of D, C, and S can be determined precisely, and remain**constant**.

Since the assumption is that the entire inventory is used up before the next order quantity arrives then the minimum value of inventory is 0, and the maximum is Q.

Therefore:

- Average inventory: One-half the maximum and minimum inventory = Q/2.
**Annual carrying cost**: Average inventory * Unit stocking cost = (Q/2)*C.- Number of orders made per year is the annual consumption/order quantity = D/Q.
**Annual ordering cost**is the Number of orders * Ordering cost = (D/Q)*S.- Total costs
**TSC**associated with inventory handling is sum of the carrying and ordering costs – assuming no stockout cost:

Reorganizing and making Q the subject leads to **EOQ:**

This is the economic order quantity or the quantity to purchase to make inventory-related costs a minimum:

The expected order cycle length, or the time between orders (TBO), is **TBO = EOQ/D **in years.

An order is placed when inventory position reaches the reorder point **R = d*L**, where L is the length of the replenishment lead time and d is the expected demand per unit of time.

Consider now the following:

• Demand increase: If demand in the marketplace doubles, the optimal order quantity, and hence the average inventory level, increases only by the square root of 2 or by 41% over the original amount.

• Warehouse consolidation: The total average inventory required to maintain identical regional warehouses is n(Q*/2). However, consolidating the regional warehouses into a central facility results in system order cycle inventory of only square root(n)* Q/2. Hence, a regional system required square root(n) times as much inventory as a central system.

• Reduction in ordering costs: Suppose an investment in an automated electronic ordering system (EDI…) reduces ordering costs by 75% of their former level. The optimal order cycle inventory reduces by 50% reduction in inventory over the previous level of ordering cost.

# EOQ Model 2: Simultaneous replenishment and consumption

The basic model assumes the replenishment order arrives instantaneously at the end of the order cycle. However, in aproduction environment, inventory may be replenished gradually over time.

This type of model might be applied, for example, when:

1. ln distribution, inventory items of finished goods are being delivered to a distribution centre and placed in the storage area. Simultaneously during this delivery, items are being withdrawn from this storage area, called 'picking', for customer needs.

2. ln production, upstream Work post N° 1 is producing units destined for the adjacent downstream Work post N° 2. Work post N° 2 is using the inventory items at the same time they are being delivered

**
**

*Maximum inventory*

ln contrast to the basic model, the inventory level in Model 2 for either distribution or a production operation never reaches the maximum value of Q but at a smaller level given by:

- Qmax =
**(p-d)*Q/p**where p is the delivery rate and d is the usage rate.

Thus:

- Inventory growth rate = p – d (per unit per bucket)
- Delivery time up to Qmax = Q / p
- Qmax = (p-d)*Q/p

* *

*Distribution*

Here inventory is being delivered and used at the same time and the profile for this model as shown above. The shaded area is when there is both delivery and usage, and the non-shaded area for usage only. The figure also shows the profile if there was immediate delivery and now usage or EOQ model I.

The assumptions for Model II are essentially the same as for Model 1except that terms supply rate, p, and demand rate, d, appear:

**D**is the annual demand for material in units/year.**Q**is the quantity of material ordered in units/purchase order.**C**is the cost of carrying one unit in inventory for one year, $/unit/year.**S**is the average cost of placing a purchase order, $/order.**d**is the usage rate of the material in units/hour (or other time period) and is considered as**constant**(see EOQ model 4 for variable usage rate)**p**is the delivery rate of the material in units/hour (or other time period) and considered as**constant****TSC**is the total annual inventory stocking cost, $/year. Values of D, C, p, d, are known and are constant.

The inventory level never reaches the maximum value of Q, the quantity delivered, but at a maximum value given by the relationship [(p - d)/p]Q. ln the two extreme cases:

- The value of d = 0 (the inventory is not being used), then the maximum value is Q, or the same as for Model I.
- The value of d = p (supply is the equal to demand), then there is zero inventory or a completely balanced system

As before, the minimum inventory level is 0:

- Average inventory is one-haIf the maximum and minimum inventory, or [(p - d)/(2p)]Q.
- Annual carrying cost is Average inventory* Unit stocking cost, or [(p - d)/(2p)]QC.
- Number of orders made per year is the annual consumption/order quantity, or D/Q.
- Annual ordering cost is the Number of orders * Ordering cost, or (D/Q) S.

Total costs **TSC** associated with inventory handling are the sum of ordering and carrying costs:

Thus, total costs are a minimum when the annual ordering costs are equal to the annual carrying costs. Reorganizing and making Q the subject leads to **EOQ _{2} or here to POQ:**

* *

*Economic production run size (POQ)*

ln a situation where an upstream work centre is producing units used by a downstream centre, the EOQ analysis can be used to identify an optimum run size.

There are three possible situations:

1. If usage (demand) and production rate are equal there will be no build-up of inventory and the question of run size is irrelevant.

2. If the demand rate (usage rate) d, exceeds supply rate, p, then there will be a stockout,

3. If production, p, exceeds demand, d, then the EOQ analysis can be applied.

- As long as production occurs, inventory continues to build.

- It will be at a maximum when production ceases.

- Demand occurs over the entire cycle. When inventory is exhausted, then production will be again.

- Since the firm makes the product itself the ordering or purchasing costs are now the **machine setup costs** and other associated preparation activities.

- Setup costs are considered independent of lot size

The difference is that one is for ordering outside, or a purchase cost, and the other is for ordering inside, or setup and other preparation costs. The cycle time or the time between starting one production run and the start of the next is a function of the run (lot) size, and the demand or usage rate:

Cycle time = **EOQ/d**

* *

*Balancing the run size*

This EOQ formula for a production run can be rewritten as:

This can be interpreted as saying the closer that the demand rate, d, is to the production rate, p, the greater the value of the lot size, Q. ln other words, the greater the production run. This would be the case of an assembly line operation where the production of the upstream work post is equal to the usage (demand) of the downstream work post. Both posts are in balance. ln an assembly operation when p is greater than d, inventory starts building up. When d becomes greater than p, stockouts occur.

# EOQ model 3: quantity discounts

ln inventory situations, quantity discounts may be possible the greater the quantity of material ordered:

- ln purchasing products from an outside supplier, quantity discounts are offered because there is an economy in transportation, and order preparation. For example, in purchasing a quantity of goods a supplier may offer price breaks according to volume.
- Similarly, in a production operation, the greater the lot size, or production run, the lower the unit cost because the number of setups would be fewer.

* *

*Model development*

**D**is the annual demand**Q**is the quantity of material ordered each time a requisition is made**C**is the cost of carrying one unit in inventory for one year**S**is the average cost of placing a purchase order**TSC**is the total annual inventory stocking cost

Now two others terms are added:

**P**is the price per unit of product (inventory) in $/unit**TPC**is the total product cost and includes the total price paid for the inventory, the total carrying costs, and the total ordering costs.

The total price paid for the material is the price times annual demand or P * D, where P is now a variable.

Thus for either Model I or Model II:

**TPC = TSC + DP**

The economic order quantity is now when the total product costs given by the sum of the total inventory stocking costs plus total price paid is a minimum.

*Model A: Immediate delivery*

Here the delivery is immediate and so: TPC = (Q/2)*C + (D/Q)*S + D*P

*Model B: Continuous replenishment and consumption *

Here there is continuous supply and usage and so:

TPC = (Q/2)*[(p-d)/p]*C + (D/Q)*S + D*P

*Calculation procedure*

The model EOQ is computed using each of the sales prices using either Model I or Model II depending on whether the consideration is instantaneous delivery or supply and usage at the same rime. The value of C, the carrying cost, is usually a function of sales price, or C = f(P). Thus, for example, if the carrying cost is 20% of sales price then the EOQ will change as P changes.

The feasible EOO from step 1 is determined. That is that value of the EOO that lies in the quantity range for the given price level. ln some cases, it may not make sense to purchase the quantity, 0, calculated at the unit price given. ln other cases, it may not be possible, according to the supplier's criteria, to purchase the quantity 0 at the given unit price.

The total annual product cost, TPC, is computed for each feasible EOO. ln addition, values of TPC are calculated for purchase quantities, which may not arrive at from the EOO relationship, but at which level a new lower unit price is possible according to the supplier's criteria. There will be different TPC curves for each price level

The order quantity with the lowest total annual product cost, TPC, is now the economic order quantity; this value may have no relationship to the EOQ calculated from the models.

## EOQ model 4: Variable demand and replenishment lead-time

Variable demand and replenishment lead-time assumptions do not change the optimal order quantity, but it does alter the reorder point, ROP, indicating when the order should be placed.

**d**is the usage rate of the material in units/hour (or other time period) and is variable over time.**L**is the replenishment lead-time in hour or day (or any other time period) and is variable over-time.

Under the EOQ basic model 1, the reorder point is at ROP = d*L (i.e., the expected demand during replenishment lead time) assuming that d and L are constants.

Now we can express **d = d _{avg} + **

**Delta**

**where d**

_{d }_{avg }is the average demand rate and ?

_{d }is the demand variability from the average demand rate.

Same with the replenishment lead-time (or delivery lead-time) which can be express as **L =****L _{avg} + **

**Delta**

_{L}If the usage rate or demand rate is not constant over replenishment lead-time we have then a demand variability that can lead to over consumption than usual and as a consequence to a stock out. Same for replenishment lead-time when the supplier (internal or external) does not meet its standard lead-time, we have then a replenishment variability or in other words, delivery variability.

The demand or replenishment variability can lead to a stock out or an over stocking depending on the actual deviation from the constant model or EOQ basic model 1.

In order to tackle the demand or replenishment variability, the solution is to size an adequate buffer stock according to the desired service level to avoid any stock out during the replenishment lead-time.

The ROP formula is the following one: ROP = **d _{avg }* L_{avg }+ Z*EDDLT**

Where:

**d**is the average usage rate during the replenishment lead-time_{avg }* L_{avg }**Z**is the safety factor depending on the desired confidence level - 95% or 98% risk coverage - given by the normal distribution formula**EDDLT**is the expected demand during lead-time**variability**equal to:

**d**is the usage rate during replenishment and**L**the replenishment lead-time- is the standard deviation for the usage variability
- is the standard deviation for the replenishment variability

** **

**Z * EDDLT** express the safety stock according to the desire service level such that the ROP or reorder point is the sum of the safety stock plus the average demand during replenishment lead-time.

Please find **here in the toolbox area** the complete spreadsheet to calculate EOQ and other inventory threshold.