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EOQ Model with Purchases

Home | What is Economic Order Quantity? | Its Cost Components | EOQ Model with Purchases | Assumptions of the Model | Optimizing Economic Order Quantity | Graphical Solutions | Sample Problems | Answers



 How much should we order?

 There are different approaches to how much (and when to order) - Just in Time and Economic Order Quantity vary in the approach to order size and timing.

Economic Order Quantity (EOQ) Model


*       Demand is constant and known

*       Supply is certain, and replenishment lead time is constant and known

*       Items are ordered in lots (batches), and complete orders are received

*       Ordering decisions for one item can be made independently of other items

*       Stockouts and shortages can be completely avoided


To determine the quantity to order which minimizes the total annual inventory management cost.

Total annual inventory management cost = annual inventory holding costs + annual ordering/setup costs.

The formula for the economic order quantity (EOQ) is derived by developing an expression for total annual inventory management cost, then taking its derivative with respect to quantity Q to find its minimum point.

Total annual inventory management cost = S (D / Q*) + H (Q* / 2)


*       Q* = optimal order quantity

*       D = demand per time period

*       S = cost of placing an order

*       H = cost of holding one unit in inventory for one time period

Optimal order quantity Q* = square root of (2 D S/ H )

Number of orders placed per year = D / Q*

Average cycle inventory = Q* / 2

Time between orders = Q*/ D (expressed in years)

The optimal order quantity Q* occurs at the point where total "fixed costs" (eg., ordering costs) equal total "variable costs" (eg., inventory holding costs).

Example 1.

Assumption – no lead time and constant demand

Annual demand = 20,000 units per year

Cost/unit = $30

Order cost = $100/order

Holding cost = 12% of unit cost/per year

H = .12(30) = $3.60/unit/year – holding costs and demand must be in the same time frame


EOQ Model with Purchase Quantity Discounts (Price Breaks; All-Quantity Discounts)

Differences from the basic EOQ model:

Because the per-unit price of the items purchased changes as the quantity changes, the purchase price must be included in the calculation of total annual inventory management cost. As the purchase price changes, the inventory-holding cost also may change since the investment in inventory is different. Because each discount category may represent a different inventory holding cost, we must calculate the EOQ for each discount category. The optimal order quantity may or may not occur at the quantity where total ordering cost equals total holding cost.

The optimal quantity will occur either:

i) at one of the EOQs; or

ii) at the minimum quantity of one of the discount categories that is larger than the feasible EOQ with the lowest price finding the optimal order quantity involves calculating the total annual cost (including purchase cost) for each of these quantities, and picking the quantity that gives the lowest total cost.


Step 1: Calculate the EOQ using the lowest price. If this EOQ is feasible, this is the best order quantity, so stop.

Step 2: Solve the EOQ for the next higher price. If this EOQ is feasible, go to Step 4.

Step 3: If the EOQ found in Step 2. is not feasible, repeat Step 2. for the next higher price until a feasible EOQ is found.

Step 4: Calculate the total annual inventory management cost for the (first) feasible EOQ (found in Step 2.) and for the minimum quantity in all discount categories that are larger than the feasible EOQ.

Select the order quantity with the lowest total annual inv. management cost.

Example 2: Example 1 with quantity discounts



Continuous review system: review the on-hand quantity of an item each time an inventory withdrawal occurs, and decide whether a replenishment order should be placed at that time order quantity is fixed but the time between orders ("order cycle") varies.

Reorder point system (fixed order quantity system): reorder a fixed quantity Q whenever the inventory position falls to or below a predetermined reorder point R.

Inventory position: the ability of inventory to satisfy future demand for an item.

IP = OH + SR - BO where:

*       IP = inventory position of the item (in units)

*       OH = number of units on-hand

*       SR = number of units scheduled to be received ("open order")

*       BO = number of units back-ordered or allocated


Reorder point R = amount of inventory required to meet expected demand during the lead time plus amount of safety stock held to meet unanticipated demand.

R = L + B


*       R = reorder point

*       L = expected demand during lead time

*       B = amount of safety stock

Calculating the level of safety stock is the challenging part of this process. It involves considering the level of service which an organization wishes to provide to its customers. This will be discussed in class.

Example 3

  Q = 1000

  Lead time between placing an order and receiving it is 1 week – L = 1

If weekly demand is normally distributed with mean 100 and st. dev. 20.  When should we place the order?

 We want to be 95% sure that we don’t run out of inventory in any cycle. Cycle service level = 0.95



 Inventory Levels

  Average Inv = Q/2 + Safety Stock


  Measuring relative inventory levels

Days in inventory = Average inventory

                                      Average daily sales

Inventory Turnover =  Annual Cost of Goods Sold

                                    Average Inventory

Or by unit - Inv. Turnover = Annual unit sales

                                                            Average inventory (in units)


Example 4 – Reorder Point Example 2


Q = 500 found by EOQ

L = 2

Cycle service level = 95%

Weekly demand d = N(200, 40)

We must convert weekly demand to demand during lead time and must also convert sigma from one week to two.

R = dL + z sigmaL 

R = 500(2) + 1.645 (40) square root (2)

R = 1000 + 93.05 = 1094 (round up)

Assumptions of the Model

1.      Demand is known and is deterministic, ie. constant.

2.      The lead time, ie. the time between the placement of the order and the receipt of the order is known and constant.

3.      The receipt of inventory is instantaneous. In other words the inventory from an order arrives in one batch at one point in time.

4.      Quantity discounts are not possible, in other words it does not make any difference how much we order, the price of the product will still be the same. (for the Basic EOQ-Model)

5.      That the only costs pertinent to the inventory model are the cost of placing an order and the cost of holding or storing inventory over time

Important Note: When calculating the Economic Order Quantity, be aware of the assumptions mentioned above!

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