**
** 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

**Assumptions **

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

**Objective:**

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) **

where:

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.

**EOQ MODEL WITH PURCHASE QUANTITY DISCOUNTS**

**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 INVENTORY SYSTEMS (Q
SYSTEMS)**

**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

**CONTINUOUS REVIEW INVENTORY SYSTEMS (Q
SYSTEMS): SELECTING THE ****REORDER**** ****POINT**** ****R**

**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**

**where: **

**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**

####

#### Average

**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!**