Chapter 10

Inventory Management

Independent vs. Dependent Demand Items

Independent demand inventory items– demand cannot be computed, it is random (uncertain)– items such as finished goods or end items

Dependent demand inventory items– demand is directly related to that of another item– items like raw materials or subcomponents (related to

end item)

Big Question When and how much to order?every day |-----------------------------------------------------| once a year

2 Basic Approaches to Ordering Inventory

Fixed order quantity– always order same quantity– order whenever inventory level gets low (order point)

Fixed order period– always order every n days– order a different quantity each time

Economic Order Quantity (EOQ) model is most common fixed order quantity approach

Inventory Costs

Ordering Costs– clerical costs, postage, material handling costs, etc.– setup or changeover cost

# orders/year = D/Q

Annual ordering cost = S(D/Q)

where D = annual units demanded (forecast)

Q = quantity of one order

S = average cost of processing one order

Carrying Costs

-- costs incurred to keep items in storage

average inventory level = Q/2

(for basic EOQ model)

Annual carrying cost = C(Q/2)

where C = carrying cost rate ($ per unit per year)

Acquisition Costs– cost to purchase or produce the items

Annual acquisition cost = D(ac)

where ac = cost to purchase or produce one unit of the item

Stockout Costs– estimated per unit cost of a stockout (running

out of items)– extra paperwork, lost sales, late fees, lost

goodwill, etc.

Cost of Capital

Interest paid to bank to borrow money to finance inventory

Example:

8% annual interest rate

60,000 units annual demand

$20 per unit—cost

order size = 5000 units

average inventory on hand = 2500 units

How much interest should they expect to pay next year?

Balancing Carrying Costs Against Ordering Costs

0

100

200

300

400

500

600

0 200 400 600 800 1000 1200

Q = Quantity of Material per Order (Units)

An

nu

al C

ost

of S

tock

ing

a

Ma

teri

al (

$)

x

xEOQ

Annual Ordering Costs

Annual Carrying Costs

Total Annual Stocking Costs

MinimumTotal Annual

Stocking Costs

Time Graph of Inventory

Q

OP

LT LTLT

Average Inventory Level = Maximum Inventory + Minimum Inventory 2 = (Q + 0)/2 = Q/2

Time

InventoryLevels

EDDLT

LTLT0

Time Graph of Inventory

Q

OP

LT LTLT

Average Inventory Level = Maximum Inventory + Minimum Inventory 2 = [(Q+SS) + SS]/2 = Q/2 + SS

Time

InventoryLevels

EDDLT

LTLT

SS0

EOQ Assumptions

– Demand, ordering cost rate, carrying cost rate, unit cost, and lead time are known constants

– An order arrives all at once– No stockouts occur– No safety stock is carried

Total annual inventory cost = ordering + carrying + acquisition costs

TC = S(D/Q) + C(Q/2) + D(ac)

We want to find the order quantity that results in the minimum total annual inventory cost.

At the minimum total cost, the slope of the total cost curve is zero, so the derivative of TC with respect to Q is zero.

TC = S(D/Q) + C(Q/2) + D(ac)

Solve for Q to get:

002

C

Q

SD

d{Q}

d{TC}2

C

2DSEOQ Q

C

2DSQ *2

Basic EOQ ExampleAnnual demand = 6000 units

Ordering cost rate = $100 per order

Acquisition cost = $24 per unit

Carrying cost rate = 25% of unit value per year

250 work days per year

What quantity should be ordered to minimize total annual inventory cost?

EOQ = C

2DS

What is the total annual inventory cost with this order quantity?

TC = S(D/Q) + C(Q/2) + D(ac)TC =TC =

On average, how many orders per year should be expected?

On average, how many work days should one order last?

What is the expected minimum, maximum, and average inventory level?

EOQ with Quantity Discounts

Step 1: Calculate EOQ for each price

Step 2: For feasible EOQs, calculate total annual

cost

Step 3: Calculate total annual cost at the lowest allowable quantity for each lower price

Step 4: Pick quantity with lowest total annual cost

Graph of EOQs and price break quantities

Example: An office supplies wholesaler sells copier paper by the ream. Ordering cost is $20/order. Carrying cost rate is 30% of the dollar value per year. Annual demand is 1000 reams.

#Reams Cost/Ream 1-49 3.90 50-199 3.75200-499 3.65500+ 3.60

EOQ3.90 =

EOQ3.75 =

EOQ3.65 =

EOQ3.60 =

Calculate TC for: 189 reams @ $3.75

200 reams @ $3.65

500 reams @ $3.60

TC = SD/Q + CQ/2 + D(ac)

TC3.75 =

TC3.65 =

TC3.60 =

Order Point

Perpetual inventory accounting – inventory records are updated anytime inventory levels change (typically used with fixed order quantity inventory systems)

Order point – the inventory level that triggers an order

Lead time – lead time is the amount of time between when a replenishment order is placed until it is received

Stockout – inventory level drops to zero

For most fixed order quantity systems, stockouts can only occur during the lead time

If demand is constant, set the order point equal to the expected demand during the lead time

OP = EDDLT

Stockouts and Safety Stock

2 main reasons for a stockout:

-- demand during lead time is greater than expected

-- lead time is longer than expected

Safety stock is extra inventory held during the lead time (beyond EDDLT amount) and is the most common approach to reducing stockouts

OP = EDDLT + SS

What are the disadvantages of:

-- too little safety stock? -- too much safety stock?

Setting Order Points

Problem: What inv. level should order point be set at?

2 common approaches

-- set OP to achieve a desired customer service level

-- set OP to minimize costs of to much or too little inv.

There are many ways to measure customer service.

We will define customer service level as:

-- the % of DDLT filled with stock on hand

(What is safety lead time?)

Order Point Example

Annual demand = 8000 units

Lead time = 4 working days

260 working days per year

Safety stock = 200 units

What inv. level should the order point be at?

Two-Bin System (for inventory control)

4 Examples of Setting the OP

• Achieve a desired service level Example#– discrete demand (small numbers) 1 & 2– continuous demand (normal distr.) 3

• Minimize stockout and carrying costs– payoff table 4

1. Sue’s Jewelry orders 20 men’s Rolex watches (style #41B) each time the inventory level of this item gets low. There is a two week lead time once the order is placed with the supplier. Sue’s records show that for the past 20 times an order has been placed, the demand during the 2-week lead time has always been 3, 4, 5, 6, or 7 watches. The number of occurrences of these demands has been 4, 7, 6, 2, or 1, respectively (a total of 20 DDLT observations). Since the carrying cost for Rolex watches is quite high, Sue wants to determine what order point to use so that there are enough watches on hand during the lead time to sell to 80% of the customers who request one.

Sue’s Jewelry

DDLT Frequency Prob. Service Level

3 4

4 7

5 6

6 2

7 1

20

Find OP for an 80% service level

2. So that it can get a volume discount, Kendall Ford orders 20 F-150 extended cab pickup trucks each time it places an order from the manufacturer. The lead time to receive the trucks is 22 days. The frequency of different demands during the lead time has been 3, 4, 7, 8, 9, 12, and 5 occurrences for demands of 9, 10, 11, 12, 13, 14, and 15 trucks, respectively. Due to the cost of having extra trucks on hand, management has decided it is not cost effective to try to avoid all stockouts during the lead time. They would like to set the order point for the F-150 so that it is out of stock for no more than 30% of the customers who would buy this truck. Kendall Ford should place a new order when how many trucks are left on the lot? How many trucks should they expect to sell during the new lead time?

Kendall Ford Trucks

DDLT Occurrences Prob. Service Level9 3

10 4 11 7 12 8 13 9 14 12 15 5

48

3. A distributor of aircraft jet fuel orders 180,000 gallons each time its supply gets low. The lead time is 3 days. The average daily demand for jet fuel is 18,500 gallons. Past records show that the standard deviation of demand during the lead time is 12,500 gallons. Because of stiff competition from another distributor, it is desired to have enough fuel on hand so that a stockout occurs no more than 5% of the times that customers place orders during the lead time. What should the level of safety stock be? How many gallons should be on hand when an order is placed?

Safety Stock and Order Point

OPEDDLT

SS = zσDDLT

Probabilityof Stockout

ActualDDLT

Payoff Table:Long cost – the cost of one unit left over on hand when an

order arrivesShort cost – the cost of being one unit short during the lead

time (stockout cost)

4. Each year the Payless Drug Store on Coburg road places orders for cases of natural Christmas wreaths and pays $20 for a case of ten wreaths. The sales price is $5 per wreath. Records for the past 20 orders show that demand during the lead time has been 6 cases on 2 occasions, 7 cases on 6 occasions, 8 cases on 10 occasions, and 9 cases on 2 occasions. Any wreath left on hand when a new order arrives will be all dried out and must be thrown away. What is the long cost and short cost? What should the order point be? What service level would this order point provide?

Payoff Table

Long cost = Short cost =

DDLT Freq. Prob.

6 2

7 6

8 10

9 2

20

Next, fill in payoff table and compute expected costs (EC)

6 7 8 9 EC

6

7

8

9

Prob:

OP

actual DDLT(Long cost or shortcost in table)

Reducing Lot SizesCutting setup costs is key to reducing production lot sizes.Setup reduction examples

Summary of Benefits of Reduced Lot Sizes• shorter lead times• less inventory investment• defectives are caught quicker – less scrap, rework, & future

errors• need less floor space – employees closer together – better

communication• processes more closely linked – encourages joint problem

solving• simplified inventory management• lower material handling costs• avoid lumpy workloads

Processing Schedule

Alternative 1: process batch size = 100 units; transfer batch size = 100 units

batch 1

batch 1

batch 1

batch 1

batch 1

batch 1

Machine

1

2

3

4

5

0 500 1000 1500 2000 2500 3000

Elapsed Time (minutes)

3000

6

Processing Schedule

Machine

1

2

3

4

5

0 500 1000 1500 2000 2500 3000

Alternative 2: process batch size = 100 units; transfer batch size = 50 units

Elapsed Time (minutes)

6

batch1

batch2

batch1

batch2

batch1

batch2

batch1

batch2

batch1

batch2

batch1

batch2

1750