Inventory- a stock or store of goods

• Dependent demand items- components or sub-assemblies (In a Roland piano, the bench, for example). Forecast is based on # of related finished goods

• Independent demand items- finished goods that have their own demand curve (subject to randomness we discussed during forecasting section

Types of inventories- piano example

• Raw materials & parts (e.g. piano keys)

• Work In Process (keyboard assembly)

• Finished Goods (keyboard, stand and bench)

• Replacement items (keyboard cover handle)

• In-transit inventory

Why keep inventory if it costs so much?

• There are times in which the cost of keeping inventory is less than the benefits derived:

• smooth production requirements as seen in Agg. Planning examples

• decouple operations A distribution company wants to keep distributing even if the ship carrying the next shipment is late!

• To meet our stockout goals. Software is quick-decision purchase- many companies have 0% stockout strategies as a result (I.e. opportunity cost = 100%; inventory cost may equal 50%)

• To capitalize on opportunities. If we have excess warehouse and staff capacity, we may save by buying a lot at a great price.

Why keep inventory if it costs so much?

Ordering: quantity & timing Realities in the real world

• Your order quantity may have to be done for political reasons (new product the president is behind- Edirol example)

• We may not be able to affect the timing of orders. Distribution companies usually have to place 3 or even 6 month orders for highly technological products to smooth production planning. So fixed interval models are developed.

Counting Inventory

• Periodic systems count physically at regular intervals and re-order when necessary. Your accounting audit will require this.

• Perpetual systems (that count inventory as it changes in real time and re-ordering when we hit a reorder point) are almost universally used as the cost of computing has decreased.

• Most companies combine use of both.

Adding math models to your tool kit

• What is the lead time of your order (time between submission & receipt)

• What is your holding cost (includes interest, insurance obsolescence, theft, wear, warehousing, etc.)

• What is your ordering cost (including the cost of the transaction and receipt

• What is your Shortage cost (opportunity)

What inventory do we evaluate?

• Pareto principle tells us that 20% of our items will account for 80% of our orders/ supply requests

• So, use the ABC system to classify value Item demand Unit Cost Annual $ value Class

1 10 50 5002 100 1 1003 75 300 225004 5 2500 125005 50 35 17506 130 50 6500

More on ABC System

• Can be used to determine number of re-counts in physical counts (e.g. A’s get 3; B’s get 2; C’s get 1)

• Can also be used to determine who does counts (A’s counted by controller, staff & warehouse; B’s by staff & warehouse; C’s by warehouse only)

The Inventory Cycle

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Time

So we’ve evaluated the right inventory. Now let’s order.

• EO Q Model minimizes the sum of holding and ordering costs by finding the optimal order quantity.

• Assumptions: 1) one product at a time; 2) we’re confident in our annual demand forecast; 3) demand is even; 4) lead time is constant (management issue); 5) orders received in one delivery; 6) no qty discounts

Getting to EOQ: we’re balancing...

• ANNUAL CARRYING COST = (Q/2)*H (Q= order quantity units; H- carrying cost/unit)

• ANNUAL ORDERING COST = (D/Q)*S (D= annual unit demand; Q= order

size; and S= ordering cost

• calculus then gives us EO Q, the optimal order quantity

Total cost = annual carrying cost + annual order cost

The Total-Cost Curve is U-Shaped

Ordering Costs

QO Order Quantity (Q)

An

nu

al C

os

t

(optimal order quantity)

TCQ

HD

QS

2Carrying

Costs

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

• Given that demand = 405/month

• Carrying cost = $30/yr/unit Order Cost = $4/order

• 1) EOQ= SQR(2*(405*12)*4)/30)= 36

• 2)What is average # of bags on hand? Q/2= 18

• 3) # of orders per year= (405*12)/36 =135

• 4) Carrying cost = (36/2)*30=540; ordering cost= (4860/36)*4=540 total cost = 540 +540 =1080

• **We need figures represented as annual costs.

Determining the economic run quantity of production

• When company is producer and user, determines optimum production run size (since production usually happens faster than usage)

• When we’re producing our own goods, assumes setup costs are the same as order costs in formula

• so total cost = carrying cost +setup cost• TC = (Max. Inventory/2)*H + (D/Q)*S• Economic Run quantity = SQRRT(2DS/H)* SQRRT(p/(p-u))

where p=prod. Rate u=usage rate• cycle time =Q/u run time = Q/p

Quantity discounts if carrying costs are constant

• Goal: minimize total cost, where TC =(Q/2)*H + (D/Q)*S + PD

where P= unit price

• Step 1: compute the common EOQ (if carrying cost is a constant $ figure, it won’t vary)

• Step 2: compute total cost at EOQ and price breaks and compare

Assume: D5000,/yr h= $2/unit/yr s=$48

Units Price

1-399 $10

400-599 $9

600+ $8

Quantity discounts if carrying costs are constant

• STEP 1: compute the common EOQ= SQR ((2DS)/H) =SQR((2*5000*48)/2)= 489.90

• STEP 2: compute the TC @ EOQ (490) = (Q/2)*H + (D/Q)*S + PD = (490/2)*2 + 5000/490)*48 + (9*5000) = $45980 (with rounding)

• STEP 3: compare with TC at discount levels TC = (Q/2)*H + (D/Q)*S + PD TC @600 = (600/2)*2 + (5000/600)*48 + (8*5000)= 41000

• 600 is the optimum order quantity account for discounts

We know how much to order… now, when do we reorder?

• ROP: predetermined inventory level of an item at which a reorder is placed.

• Demand (d) and Lead TIME (LT)

• ROP= d*LT

• Example: Monthly demand is 400. Lead Time is two weeks (.50 months). ROP= 400 *.50 =200

• Reorder when inventory level reaches 200.

• This model assumes static d and LT

What if demand or lead time is variable?

• Then we add a safety stock to help us satisfy orders if demand is higher than expected.

• Company policy: What is our service level? It is the number: 1- stock-out risk. “Our service level goal is 95%. In other words, there’s a 95% probability we won’t stock out.

Handling variability, 2

• We assume the variability is characterized by the normal distribution.

• Turn to page 889. The shaded area under the curve represents the probability of us having inventory, given the variability in the average demand or average lead time.

• So let’s say we have a service level goal of 95%. What is the Z score that characterizes 95% of the area under the normal curve?

• About 1.645

When lead time is variable:

• First example: LEAD TIME variable.

• When lead time is variable, ROP= d* avgLT + z*d(LT) where d= demand rate; LT= lead time; LT=std. Dev. Of lead time

• Get the z score (based on your service level goal) from the table as we saw on the last slide based on company’s stockout policy..

• Given: demand during lead time =400/day

• Lead Time = 5 days, =2acceptable stockout risk= 5%

• STEP 1: get your Z score1-.05 = .95 z (.95) =1.65

• STEP 2: plug in400*5 + 1.65*400*2= 3320

• Reorder when inventory = 3320

ROP= d* avgLT + z*d(LT)

If demand rate is variable:

• ROP= avgd* LT + z* sqr.root of LT * (d)

• assume: avg d =1000; d= 14; LT=4; company stockout policy = 10% risk.

• Z score for .90 = 1.28

• 1000*4 + 1.28* 2 * 14= 4000+ 35.84= 4036

• in real world, d is derived by managers keeping careful records to determine it.

For next time

• PROBLEMS (not questions) Ch 12 #s 1,6,13,19,

• Page 587- know models 1,2,3, and 4a,b,c

Problem 1Item Usage Unit Cost Value Class

4021 90 1400 126000 A

9402 300 12 3600 C

4066 30 700 21000 B

6500 150 20 3000 C

9280 10 1020 10200 C

4050 80 140 11200 C

6850 2000 10 20000 B

3010 400 20 8000 C

4400 5000 5 25000 B

Problem 6• D=800/MO @ $10/UNIT S=$28

H= 35% OF UNIT COST/YR

• D- 9600/YR H= $3.50/UNIT/YR

• CURRENT TC = (q/2)*H + (D/Q)*S

• CURRENT TC= (800/2)*3.50 + (9600/800)*28 =1736

• EOQ= square root of ((2DS)/H)

• EOQ = SQR ((2*9600*28)/3.50)=SQR 153600 =391.91= 392

• TC at 392= (392/2 )*3.5 + (9600/392)*28=1371.71

• Cost savings =1736-1372=364

Problem 13: carrying costs are constant

• D=18000 H= $0.60/yr S=$96

• STEP 1: Common EOQ= SQR ((2DS)/H) =SQR((2*18000*96)/.6)= 2400

• STEP 2: TC @2400 = (Q/2)*H + (D/Q)*S + PD =(2400/2)*0.60 + (18000/2400)*96 + (1.20*18000)=$23040

• TC @5000 = (5000/2)*0.60 + (18000/5000)*96 + (18000*1.15)=$22545.60

• TC@10000= (10000/2)*.60 + (18000/10000)*96 + 18000*1.10=22972.80

• 5000 is the optimum order quantity account for discounts

Problem 19, page 597• see page 573, equation 12-12. The estimate of standard deviation of lead time demand

is available, so you can use this simpler equation

• Expected demand during LT = 300 Std dev of LT demand = 30

• a) Step 1 z=2.33• a) step 2 300+(2.33*30)=69.9=370• b) from a)--> 70 units• c)less safety stock is required because we’d be

carrying an amount of inventory causing us to stock out more often.

Problem 23• Hint: plot the information you do have under the equation, then solve for

what you don’t have.• When the book says “the delivery time is normal” that means we’ve got a

variable lead time problem.

• When lead time is variable, ROP= d* avgLT + z*d(LT) where d= demand rate; LT= lead time; LT=std. Dev. Of lead time

• 625= 85*6 + z*85*1.10• solving for z, z=1.22• from table on p. 883, that shows an 89% probability of

supply, implying an 11% probability the supply will be exhausted.