INVENTORY MANAGEMENT - ECONOMICAL ORDER QUANTITY



Inventory Management

§  Introduction

§  Economic Order Quantity (EOQ) Model

§  Economic Production Quantity Model

§  Quantity Discounts Model

§  Reorder Point (Q System)

§  Shortages and Service Levels

§  Single Period Model

Definition of Independent Demand Inventory

Independent demand inventory consists of items for which demand is influenced by market conditions and is not related to production decisions for any other item held in stock.

Contrast this with dependent demand inventory, consisting of items required as components or inputs to a product or service.  We will talk about managing dependent demand inventory in manufacturing using a material requirements planning (MRP) system.

Types of Inventory

·      Cycle inventory

·      Safety stock

·      Anticipation inventory

·      Pipeline inventory

Managing Independent Demand Inventory

Managing independent demand inventory involves answering two questions:

·      How much to order?

·      When to order?

Five Assumptions of EOQ

·      Demand is known and constant

·      Whole lots

·      Only two relevant costs

·      Item independence

·      Certainty in lead time and supply

Total Annual Relevant Cost

a.  Annual holding cost

       Annual holding cost =

b.  Annual ordering cost

       Annual ordering cost =

c.  Total annual relevant cost:

Derivation of Economic Order Quantity (EOQ) and Time Between Orders (TBO)

Total annual relevant cost:  C =

Take the first derivative of cost with respect to quality: 

Setting  and solving for Q: 

Time between orders: 

Overland Motors Example

Overland Motors uses 25,000 gear assemblies each year (i.e. 52 weeks) and purchases them at $3.40 per unit.  It costs $50 to process and receive each order, and it costs $1.10 to hold one unit in inventory for a whole year.  Assume demand is constant.

Ralph U. Reddie has been ordering 1,000 gear assemblies at a time, but can adjust his order quantity if it will lower costs.

a.     What is the annual cost of the current policy of using a 1,000-unit lot size?

b.    What is the order quantity that minimizes cost?

c.     What is the time between orders for the quantity in part b?

d.    If the lead time is two weeks, what is the reorder point, R?

Economic Production Quantity

1.  Maximum Cycle Inventory

  

2.  Total cost = Annual holding cost + Annual ordering or setup cost

  

3.  Economic Production Lot Size (ELS)

      

Economic Production Quantity Example

A domestic automobile manufacturer schedules 12 two-person teams to assemble 4.6 liter DOHC V-8 engines per work day.  Each team can assemble five engines per day.  The automobile final assembly line creates an annual demand for the DOHC engine at 10,080 units per year.  The engine and automobile assembly plants operate six days per week, 48 weeks per year.  The engine assembly line also produces SOHC V-8 engines.  The cost to switch the production line from one type of engine to the other is $100,000.  It costs $2,000 to store one DOHC V-8 for one year.

a.    What is the economic lot size?

b.   How long is the production run?

c.    What is the average quantity in inventory?

d.   What are the total annual relevant costs?

Quantity Discounts

In the case of quantity discounts (price incentives to purchase large quantities), the price, P, is relevant to the calculation of total annual cost (since the price is no longer fixed).

Total cost = Annual holding cost + Annual ordering cost + Annual cost of materials

Quantity Discounts

Two-Step Procedure

Step 1:	Beginning with lowest price, calculate the EOQ for each price level until a feasible EOQ is found.  It is feasible if it lies in the range corresponding to its price.
Step 2:	If the first feasible EOQ found is for the lowest price level, this quantity is best.  Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level.  The quantity with the lowest total cost is optimal.

Quantity Discounts Example

Order Quantity	Price Per Unit
0-99	$50
100 or more	$45

If the ordering cost is $16 per order, annual holding cost is 20 percent of the per unit purchase price, and annual demand is 1,800 items, what is the best order quantity?

Step 1.                =

                           =

Step 2.                  =

                           =

Reorder Point (Q System)

A continuous review (Q) system tracks the remaining inventory of an item each time a withdrawal is made, to determine if it is time to reorder.

Decision rule:  Whenever a withdrawal brings the inventory down to the reorder point (R), place an order for Q (fixed) units.

Reorder Point

Demand pattern	Lead time for ordering	ROP
Known and constant	None	ROP = 0
Known and constant	Known and constant	ROP =
Variable, normally distributed,  known	Known and constant	ROP =
Variable, normally distributed,  known	Known and constant	ROP =
Known and constant	Variable, normally distributed,  known	ROP =
Uncertain, discrete probability distribution	unknown	Determine ROP for a given service level based on the cumulative probabilities of demand during lead time.

Shortage and Service Levels

Expected Shortage per order cycle:

E (n) = E (z)

E(z) = standardization parameter obtained from Table 11.3.

 = standard deviation of lead time demand

Expected shortage per year:

E (N) = E (n)

Annual Service Level:

Q System Example

You are reviewing the company’s current inventory policies for its continuous review system (Q system), and began by checking out the current policies for a sample of items.  The characteristics of one item are:

·       Average demand = 10 units/wk (assume 52 weeks per year)

·       Ordering and setup cost (S) = $45/order

·       Holding cost (H) = $12/unit/year

·       Average lead time (L) = 3 weeks

·       Standard deviation of demand during lead time = 17 units

·       Service-level = 70%

a)     What is the EOQ for this item?

b)    What is the desired safety stock?

c)     What is the desired reorder point R?

d)    What is the decision rule for replenishing inventory?

e)  What is the expected shortage per year?

If instead of the above situation, suppose the lead time is known and constant at 3 weeks and the standard deviation of demand during lead time is unknown.  However, we do know the standard deviation of weekly demand to be 8 units.  How do your answers change?

Cycle-Service Level with Discrete Distribution

·       Set R so that the probabilities of demand at or below its level total the desired cycle-service level.

·       To find safety stock, subtract expected demand during lead time from R.

Application:

The demand during lead time distribution is shown below, along with possible R values and their corresponding cycle-service levels.

Demand Level	Probability	R	Cycle-Service Level (%)
0	0.30	0
50	0.20	50
100	0.20	100
150	0.15	150
200	0.10	200
250	0.05	250

a.     What reorder point R would result in a 95% cycle-service level?

b.    How much safety stock is provided with this policy?

Single Period (Newsvendor) Model

Used to handle ordering of perishables and items that have a limited useful life.

shortage costs = unrealized profit per unit

excess costs = the unit cost less the salvage value

1.  Calculate the shortage and excess costs:

2.  Calculate the service level (SL), which is the probability that demand will not exceed the stocking level:

SL =

3.  Determine the optimal stocking level, , using the service level and demand distribution information.

 = Mean demand + z_SL*s_demand