Lecture 31

Independent Demand and Ordering System (Revision)

Books• Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming

College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M. Clive, P.E., CFPIM, Fleming College

• Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005, N.Y.: McGraw-Hill/Irwin.

• Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of Business, Rollins College, Prentice Hall

Objectives

• What is Inventory• Cost of Inventory• Benefits of Inventory• Multi period model• Optimal quantity to order• Safety stock• Periodic review system• Single period Inventory model

What is inventory?

Inventory is the raw materials, component parts, work-in-process, or finished products

that are held at a location in the supply chain.

Why do we care?

At the macro level:

Investment in inventory is currently over $1.25 Trillion (U.S. Department of Commerce).

This figure accounts for almost 25% of GNP.

Enormous potential for efficiency increase by controlling inventories

Inventory is one of the biggest corporate assets ($).

– Sales growth: right inventory at the right place at the right time

– Cost reduction: less money tied up in inventory, inventory management, obsolescence

Higher profit

Why do we care?

At the firm level:

Why do we care?Each of Solectron’s big customers, which include Cisco, Ericsson, and Lucent was expecting explosive growth for wireless phones and networking gear….when the bottom finally fell out, it was too late for Solectron to halt orders from all of its 4,000 suppliers. Now, Solectron has $4.7 billion in inventory. (BW, March 19, 2001)

“When Palm formally reported its quarterly numbers in June, the damage was gruesome. Its loss totaled $392 million, a big chunk of which was attributable to writing down excess inventory - piles of unsold devices.” (The Industry Standard, June 16, 2001)

“Liz Claiborne said its unexpected earnings decline is the consequence of higher than anticipated excess inventories”. (WSJ, August 1993)

How do you manage your inventory?How much do you buy? When?

• Soda• Milk • Toilet paper• Gas• Cereal• Cash

What Do you Consider?

• Cost of not having it. • Cost of going to the grocery or gas station (time,

money), cost of drawing money.• Cost of holding and storing, lost interest.• Price discounts.• How much you consume.• Some safety against uncertainty.

Costs of Inventory

• Physical holding costs:– out of pocket expenses for storing inventory (insurance,

security, warehouse rental, cooling)– All costs that may be entailed before you sell it

(obsolescence, spoilage, rework...)• Opportunity cost of inventory: foregone return on

the funds invested.• Operational costs:

– Delay in detection of quality problems.– Delay the introduction of new products.– Increase throughput times.

• Hedge against uncertain demand

• Hedge against uncertain supply

• Economize on ordering costs

• Smoothing

Benefits of Inventory

To summarize, we build and keep inventory in order to match supply and demand in the most cost effective way.

Modeling Inventory in a Supply Chain…

WarehouseRetail

Supplier

Home Depot

• “Our inventory consists of up to 35,000 different kinds of building materials, home improvement supplies, and lawn and garden products.”

• “We currently offer thousands of products in our online store.”

• “We offer approximately 250,000 more products through our special order services.”

Different types of inventory models

1. Multi-period model

• Repeat business, multiple orders

2. Single period models

• Single selling season, single order

Multiperiod model

• Key questions:– How often to review?– When to place an order?– How much to order?– How much stock to keep?

orders

Supply

On-handinventory

• Ordering costs

• Holding costs

Multiperiod model – The Economic Order Quantity

• Demand is known and deterministic: D units/year

• We have a known ordering cost, S, and immediate replenishment

• Annual holding cost of average inventory is H per unit

• Purchasing cost C per unit

Supplier DemandRetailer

What is the optimal quantity to order?

Total Cost = Purchasing Cost + Ordering Cost + Inventory Cost

Purchasing Cost = (total units) x (cost per unit)

Ordering Cost = (number of orders) x (cost per order)

Inventory Cost = (average inventory) x (holding cost)

Finding the optimal quantity to order…

Let’s say we decide to order in batches of Q…

Number of periods will be

D

Q

Time

Total Time

Period over which demand for Q has occurred

Q

Inventory position

The average inventory for each period is…

Q

2

Finding the optimal quantity to order…

Purchasing cost = D x C

Inventory cost =

Ordering cost = D

Qx S

Q

2x H

So what is the total cost?

TC = D C + +

In order now to find the optimal quantity we need to optimize the total cost with respect to the decision

variable (the variable we control)

Which one is the decision

variable?

D

QS

Q

2 H

What is the main insight from EOQ?

There is a tradeoff between holding costs and ordering costs

Order Quantity (Q*)

Cost

Total cost

Holding costs

Ordering costs

Economic Order Quantity - EOQ

Q* = 2SD

H

Example:

Assume a car dealer that faces demand for 5,000 cars per year, and that it costs $15,000 to have the cars shipped to the dealership. Holding cost is estimated at $500 per car per year. How many times should the dealer order, and what should be the order size?

548500

)000,5)(000,15(2* Q

Receive order

Time

Inve

nto

ry

OrderQuantity

Q

Placeorder

Lead Time

If delivery is not instantaneous, but there is a lead time L:When to order? How much to order?

ROP = LxD

Receive order

Time

Inve

nto

ry

OrderQuantity

Q

Placeorder

Lead Time

ReorderPoint(ROP)

If demand is known exactly, place an order when inventory equals demand during lead time.

D: demand per periodL: Lead time in periods

Q: When shall we order? A: When inventory = ROPQ: How much shall we order? A: Q = EOQ

Example (continued)…

What if the lead time to receive cars is 10 days? (when should you place your order?)

10

365D = R =

10

3655000 = 137

So, when the number of cars on the lot reaches 137, order 548 more cars.

Since D is given in years, first convert: 10 days = 10/365yrs

Time

Inventory Level

OrderQuantity

But demand is rarely predictable!

Demand???

Receive order

Placeorder Lead Time

ROP = ???

XX

Inventory at time of receipt

Receive Receive orderorder

TimeTime

Inventory Inventory LevelLevel

OrderOrderQuantityQuantity

PlacePlaceorderorder

Lead TimeLead Time

Actual Demand < Expected Demand

ROP

Lead Time Demand

StockoutPoint

Unfilled demand

Receive Receive orderorder

TimeInve

nto

ry

OrderOrderQuantityQuantity

PlacePlaceorderorder

Lead TimeLead Time

If Actual Demand > Expected, we Stock Out

ROP = Expected Demand

Average

TimeTime

Inventory Inventory LevelLevel

OrderOrderQuantityQuantity

If ROP = expected demand, service level is 50%. Inventory left 50% of the time, stock

outs 50% of the time.

Uncertain Demand

To reduce stockouts we add safety stock

Receive Receive orderorder

TimeTime

PlacePlaceorderorder

Lead TimeLead Time

Expected

Lead-timeDemand

InventoryLevel

ROP =Safety Stock +

Expected LT

Demand

Order QuantityQ = EOQ

ExpectedLT Demand

Safety Stock

Service level

Safety Stock

Probabilityof stock-out

Decide what Service Level you want to provide (Service level = probability of NOT stocking out)

Service level

Safety Stock

Probabilityof stock-out

Safety stock =(safety factor z)(std deviation in LT demand)

Read z from Normal table for a given service level

Caution: Std deviation in LT demand

Variance over multiple periods = the sum of

the variances of each period (assuming

independence)

Standard deviation over multiple periods is

the square root of the sum of the variances,

not the sum of the standard deviations!!!

Average Inventory = (Order Qty)/2 + Safety Stock

Receive Receive orderorder

TimeTime

PlacePlaceorderorder

Lead TimeLead Time

InventoryLevel

Order Quantity

Safety Stock (SS)

EOQ/2Average

Inventory

How to find ROP & Q

1. Order quantity Q =2. To find ROP, determine the service level (i.e., the

probability of NOT stocking out.) Find the safety factor from a z-table or from the graph. Find std deviation in LT demand: square root law.

Safety stock is given by:

SS = (safety factor)(std dev in LT demand) Reorder point is: ROP = Expected LT demand + SS

3. Average Inventory is: SS + EOQ/2

2SDEOQ

H

( )

LT D

std dev in LT demand std dev in daily demand days in LT

LT

Example (continued)…

Back to the car lot… recall that the lead time is 10 days and the expected yearly demand is 5000. You estimate the standard deviation of daily demand demand to be d = 6. When should you re-order if you want to be 95% sure you don’t run out of cars?

168)36(1065.1137 ROP

Since the expected yearly demand is 5000, the expected demand over the lead time is 5000(10/365) = 137. The z-value corresponding to a service level of 0.95 is 1.65. So

Order 548 cars when the inventory level drops to 168.

Why Companies Don’t Always Use Optimal Order Quantity

It is not unusual for companies to order less or more than the EOQ for several reasons:

• They may not have a known uniform demand;• Some suppliers have minimum order quantity that are

beyond the demand.

Justifying Smaller Order Quantities

JIT or “Lean Systems” would recommend reducing order quantities to the lowest practical levels

• Benefits from reducing Q’s:– Improved customer responsiveness (inventory = Lead time)– Reduced Cycle Inventory– Reduced raw materials and purchased components

• Justifying smaller EOQ’s:

• Reduce Q’s by reducing setup time (S). “Setup reduction” is a well documented, structured approach to reducing S

H

2DSQ

Determining Safety Stock and Service Levels

• If demand or lead time is uncertain, safety stock can be added to improve order-cycle service levels– R = dL +SS

– Where SS =zσdL, and Z is the number of standard deviations and

σdL is standard deviation of the demand during lead time

• Order-cycle service level– The probability that demand during

lead time will not exceed on-hand inventory

– A 95% service level (stockout risk of 5%) has a Z=1.645

Periodic Review Systems

• Orders are placed at specified, fixed-time intervals (e.g. every Friday), for a order size (Q) to bring on-hand inventory (OH) up to the target inventory (TI), similar to the min-max system.

• Advantages are:– No need for a system to continuously monitor item– Items ordered from the same supplier can be reviewed on the

same day saving purchase order costs• Disadvantages:

– Replenishment quantities (Q) vary – Order quantities may not quality for quantity discounts– On the average, inventory levels will be higher than Q

systems-more stockroom space needed

Periodic Review Systems: Calculations for TI

• Targeted Inventory level:

TI = d(RP + L) + SS

d = average period demand

RP = review period (days, wks)

L = lead time (days, wks)

SS = zσRP+L

• Replenishment Quantity (Q)=TI-OH

P System: an auto parts store calculated the EOQ for Drive Belts at 236 units and wants to compare the Total Inventory Costs for a Q vs. a P Review

System. Annual demand (D) is 2704, avg. weekly demand is 52, weekly σ is 1.77 belts, and lead time is 3 weeks. The annual TC for the Q system is $229;

H=$97, S=$10.

• Review Period • Target Inventory for 95% Service Level

• Average On-Hand

OHavg= TI-dL=424-(52belts)(3wks) = 268 belts

• Annual Total Cost (P System)

5wksx522704

23652weeksx

D

QRP

belts 4248416351.771.64535units 52TI

zσL)d(RPSSL)d(RPTI LRP

$16$229$245DifferenceCost Annual

$245130115$0.972

268$10

5

52TCp

Single Period Inventory Model

The SPI model is designed for products that share the following characteristics:– Sold at their regular price only during a single-time period– Demand is highly variable but follows a known probability distribution– Salvage value is less than its original cost so money is lost when these

products are sold for their salvage value

• Objective is to balance the gross profit of the sale of a unit with the cost incurred when a unit is sold after its primary selling period

SPI Model Example: T-shirts are purchase in multiples of 10 for a charity event for $8 each. When sold during the event the selling price is $20. After the event their salvage

value is just $2. From past events the organizers know the probability of selling different quantities of t-shirts within a range from 80 to 120

Payoff TableProb. Of Occurrence .20 .25 .30 .15 .10Customer Demand 80 90 100 110 120# of Shirts Ordered Profit

80 $960 $960 $960 $960 $960 $96090 $900 $1080 $1080 $1080 $1080 $1040

Buy 100 $840 $1020 $1200 $1200 $1200 $1083 110 $780 $ 960 $1140 $1320 $1320 $1068 120 $720 $ 900 $1080 $1260 $1440 $1026

Sample calculations:Payoff (Buy 110)= sell 100($20-$8) –((110-100) x ($8-$2))= $1140Expected Profit (Buy 100)= ($840 X .20)+($1020 x .25)+($1200 x .30) + ($1200 x .15)+($1200 x .10) = $1083

End of Lecture 31