3/7: Inventory Planning & Control• Roll call / collect homework / hand back hmwk

• Go over homework (?)

• Understanding inventory issues

• The basic numbers involved– Holding cost, ordering cost, demand

• Basic EOQ model

• EOQ model with allowed shortages

• Assign homework

• Have a good Spring Break

Holding Cost• Made of many things:

– Cost of capital (money tied up in inventory)• Expressed as a % of amount invested (% of purch. price)

– Insurance– Breakage– Pilferage (Theft)– Overhead, etc.

• Also expressed as a % of amount invested

Holding Cost• Made of many things:

– Cost of capital (money tied up in inventory)• Expressed as a % of amount invested (% of purch. price)

– Other holding costs• Also expressed as a % of amount invested

• EX: I have $10,000 worth of 100 comic books. I estimate my cost of capital to be 11% and other holding costs to be 4%. What is my total holding cost in dollars? What is my holding cost per unit?

Ordering Cost• A fixed cost

– Doesn’t change with quantity ordered– Is charged each time a order is placed

• Made of many things:– Salaries of purchasing department– Cost of preparation of the order documents– Cost of processing the order, etc.

Demand• How much is required of the product

• In the Economic Order Quantity (EOQ) model, we assume that demand is CONSTANT.

• EX: Each month, I sell 200 comic books.

EOQ: The Dilemma• We seek a balance while satisfying demand:

• Ordering costs must be kept as low as possible (over time), and

• Holding costs must be kept as low as possible.

• So how much do we order each time to minimize overall inventory costs?

EOQ: The Dilemma• So how much do we order each time to

minimize overall inventory costs?In

vent

ory

( Q

)

timeCycle time (T)Lead time ( L, l )

Reorder Point (r)

Daily demand(slope) (d)

The Basic EOQ Model• EOQ: Economic Order Quantity

• Assumptions of EOQ models:– Demand is constant (unvarying ), expressed as

annual demand (units per year (or other time unit) ).– 2 variable costs: setup cost and holding cost.– Lead time is constant & known.– Models use continuous review, not periodic review.– Quantity discounts are not possible.

EOQ: Symbols & Assumptions• Q: Maximum in inventory, as well as order

quantity. What is the average inventory?

timeCycle time (T)Lead time ( L, l )

Reorder Point (r)

Inve

ntor

y (

Q )

Daily demand(slope) (d)

EOQ: Symbols & Assumptions• Lead time (L or l ) is constant & known, so we

can order replenishment to be received when inventory hits zero. Measured in time units.

timeCycle time (T)Lead time ( L, l )

Reorder Point (r)

Inve

ntor

y (

Q )

Daily demand(slope) (d)

EOQ: Symbols & Assumptions• Reorder point (r) is the level of inventory at

which a replenishment order will be triggered. Measured in units of inventory (portion of Q).

timeCycle time (T)Lead time ( L, l )

Reorder Point (r)

Inve

ntor

y (

Q )

Daily demand(slope) (d)

EOQ: Symbols & Assumptions• Cycle time (T) is the length of time it takes to

use up the inventory (Q).

timeCycle time (T)Lead time ( L, l )

Reorder Point (r)

Inve

ntor

y (

Q )

Daily demand(slope) (d)

EOQ: Calculating it• EOQ = Annual holding cost of average inventory

+ Annual ordering cost

• EOQ = AHC + AOC

EOQ: Annual Holding Cost• I = annual holding cost rate (note: RATE, %)

• C = unit cost of the inventory item

• Ch = annual cost of holding one unit in inventoryCh = I * C

• Annual holding cost of the average inventory is:avg. inventory level * ann. holding cost per unit

hC2

QAHC

EOQ: Annual Ordering Cost• D = annual demand for item (measured in units

of item)

• D / Q = number of orders per year

• Co = Cost per order

oCQ

DAOC

EOQ: Total Annual Cost

oh CQ

DC

2

QCostTotal

AOCAHCCostTotal

Example: Magazine Distributor• Annual demand: 150,000 copies of Vogue

• Cost of ordering (Co) is $10

• Cost per magazine is $1.50

• Annual holding cost rate is 10%

Q

000,500,1Q075.010

Q

000,150)50.110.0(

2

QTC

CQ

DC

2

QCostTotal oh

Ex: Magazine Distributor• Annual demand: 150,000 copies of Vogue

• Cost of ordering (Co) is $10

• Cost per magazine is $1.50

• Annual holding cost rate is 10%

• We still need to know what the order quantity is.

Q

000,500,1Q075.010

Q

000,150)50.110.0(

2

QTC

Total Cost vs. Order QuantityA

nnua

l Cos

t

Order Quantity

Holding cost curve

Setup cost curve

Combined curve:holding & setup.

Minimumannual

cost

Optimal order quantity

We’ll findan equationfor this amount

So How Much Should We Order?• The best order quantity will be found where

AOC = AHC.

Holding cost curveMinimum

annual cost

Ann

ual C

ost

Order Quantity

Setup cost curve

Combined curve:holding + setup.

Optimal order quantity

Where AOC = AHC• We replace AOC & AHC

with their respective equations and then solve for Q.

• This value of Q is the Economic Order Quantity. We use Q* as its symbol.

h

o

h

o2

oh

C

CD2*Q

C

CD2Q

CQ

DC

2

Q

Back to EX: Magazine Distributor• Annual demand: 150,000 copies of Vogue

• Co is $10, cost per magazine is $1.50

• Annual holding cost rate is 10%

• EOQ = 4472 magazines per order

447210.050.1

10000,1502

C

CD2*Q

h

o

But When Should We Order It?• The reorder point (r) is expressed in units of

inventory.

• Related to the lead time (m) (time it takes for an order to be fulfilled) by looking at the demand per day (d).

• Days per year is not necessarily 365 – it’s working days per year.

yearperdays

demandAnnualdayperDemand

dmr

And How Long Will the Order Last?• Since we know how many orders will be placed

per year ( D / Q* ), we can calculate the cycle time in days.

D

*QyearperdaysworkingTor

*QD

yearperdaysworking)T(TimeCycle

Go to Excel setup

New Situation: Planned Shortages• Allows for backordering

• Q: amount of order, S: greatest shortage

• ThereforeQ – S isamountof greatestinventory

timeCycle time (T)Lead time ( L, l )

r

Inve

ntor

yQ - S

S

Daily demand(slope) (d)

Shortages: Cycle Time Sections• T is divided into two distinct phases: t1 & t2

• t1 is timewithpositiveinventory.

• t2 is timewith ashortage.

timeCycle time (T)Lead time ( L, l )

r

Inve

ntor

yQ - S

S

t1 t2

Daily demand(slope) (d)

Shortages: Average Inventory Cost• Calculating the average inventory:

• Q – S is greatest inventory, and S is greatest shortage, but you can’t go lower than zero.

• We need a weighted average of:– The average inventory in t1 and 0 in t2.

T

t)SQ(21

tt

t0t)2

SQ(

inventoryavg1

21

21

Shortages: Average Inventory Cost• Calculating the average inventory:

• Since we know that t1 = (Q–S) / d , & T = Q/d,

Q2

)SQ(

Q2

)SQ(

dQ

d2)SQ(

dQ

d)SQ(

2)SQ(

T

t)2

SQ(

invavg

2

22

1

Shortages: Average Backorder Level

• And since and

T

t)2S

(

T

t)2S

(t0levelbackorder.avg

221

d

St 2

d

QT

Shortages: Average Backorder Level• We can calculate the average backorder level as:

Q2

S

dQ

dS

2S

levelbackorder.avg2

Total Inventory Cost for Shortages• The total cost of the inventory system that allows

for backorders is= AHC + AOC + annual cost of backordering

• Where Ch = cost to inventory 1 unit for 1 year Cb = cost to backorder 1 unit for 1 year Co = cost per order

b

2

oh

2

CQ2

SC

Q

DC

Q2

)SQ(TC

So the EOQ for Shortages is…• (Trust me…)

)C

CC(

C

DC2*Q

b

bh

h

o

)CC

C(*Q*S

bh

h

Homework due 3/21• Ch. 11 #1 a-d (note: “Total Annual Cost” of the

Inventory System) (do by hand)

• Ch. 11 #4 a-d (do with Excel)

• Ch. 11 #6 a-d (do with Excel)

• Ch. 11 #15 a-e (do by hand)

• Ch. 11 #17 (do with Excel)