Inventory Control Model

Kusdhianto Setiawan

Gadjah Mada University

Inventory Planning & Control

Planning on whatInventory to stock And how to acquire

it

Planning on whatInventory to stock And how to acquire

it

ForecastingParts/Product

Demand

ForecastingParts/Product

Demand

ControllingInventory

Levels

ControllingInventory

Levels

Feedback MeasurementsTo revise plans and forecasts

Feedback MeasurementsTo revise plans and forecasts

Importance of Inventory Control

• The Decoupling Function….. Inventory as a buffer

• Storing Resources…. Where JIT is not possible

• Irregular Supply and Demand

• Quantity Discount

• Avoiding stockouts and shortages

Inventory Decision

• How much to order• When to order

With respect to (constraint) inventory cost:• Cost of the items• Cost of ordering• Cost of carrying/holding• Cost of safey stock• Cost of stockouts

Inventory Cost FactorsOrdering Cost Factors Carrying Cost Factor

Developing & Sending Purchase Order (PO)

Cost of Capital

Processing & inspecting incoming iventory

Taxes

Bill Paying Insurance

Inventory Inquiries Spoilage

Utilities, phone bills, etc for the purchasing dept.

Theft

Salaries & wages for purchasing dept. employees

Obselescence

Supplies e.g: paper, toner printer, billing form, etc. for the purchasing dept.

Salaries & wages for warehouse employees

Utilities & building cost for the warehouse

Supplies such as forms & papers for the warehouse.

Economic Order Quantity (EOQ)

• Objective: Determining how much to order• Assumptions:

– Demand is known and constant– Lead time, the time between the placement of the

order and the receipt of the order, is known and constant

– The receipt of inventory is instantaneous– Quantity discount are not possible– Variable costs: ordering cost and holding/carrying

cost– If orders are placed at the right time,

stockouts/shortages can be avoided completely

EOQ Continued….

Order Quantity = Q = Maximum inventory level

Inve

ntor

y Le

vel

0

Minimum Inventory Level

Time

EOQ Continued….Cost

Minimum Total Cost

Carrying Cost C

urve

Ordering Cost Curve

Optimal Order Quantity

Order Quantity

Computing Average Inventory

DayInventory Level

Beginning Ending Average

1 (order received) 10 8 9

2 8 6 7

3 6 4 5

4 4 2 3

5 2 0 1

Demand: Constant, 2 units/day

Ending Inventory is assumed to be always zero

Maximum level = 10 units

Total of Daily average = 9 + 7 + 5 + 3 + 1 = 25

Number of days = 5

Average inventory level = 25/5 = 5 …… Q/2

Finding the EOQ

• Expression:Q = number of pieces per order

Q* = optimal number of pieces per order

D = annual demand in units for the inventory items

C0 = ordering cost for each order

Ch = holding cost per unit per year

Finding the EOQ1. Annual Ordering Cost

= no. of order placed per year x order cost per order

2. Annual Holding or Carrying Cost= Average inventory level x carrying cost per unit per year= (Q/2) Ch

3. Optimal Order Quantityordering cost = carrying cost(D/Q)Co = (Q/2)Ch

4.

)()(Q

D

order)per cost (ordereach in units of no.

demand annual

oo CQ

DCx

orderx

IP

DC

C

DCQ o

h

o 22*

IP

DC

C

DCQ o

h

o 22*

Finding Reorder Point (ROP)

• ROP = (demand/day) x (lead time for a

new order in days)

• ROP = d x L

Inve

ntor

y Le

vel (

Uni

ts)Q* Slope = units/day = d

Lead Time = L

Time(days)

ROP(units)

EOQ Without The Instantaneous Receipt Assumption

Inventory Level

Maximum Inventory

Part of inventory cycle during whichProduction is taking placeThere is no productionDuring this part of the inventory cycle

t

Production Run Model

time

Annual Carrying Cost• New terms:

t = length of the production run (days)p = daily production rate

1. Annual inventory holding/carrying cost= average inventory level x carrying cost/unit/year= average inventory level x Ch

2. Average inventory level = ½ Maximum inventory level3. Maximum inventory level

= (total produced during the production run)- (total used during the production run)Q = pt t = Q/pMax Inv. Level = p(Q/p) – d(Q/p) = Q – (d/p)Q = Q(1-d/p)

4. Annual Inventory carrying cost= ½ (max. inv. Level) x Ch

= ½ Q(1-d/p)Ch

Annual Setup/Ordering Cost

1. Annual setup cost= (no. of setup/year) x (setup cost/setup)

= (D/Qp)Cs

where: D = annual demand in units

Qp = Quantity produced in one batch

Cs = setup cost per setup2. Annual Ordering Cost

= (D/Q)Co

Optimal Order Quantityfor Production Run Model

• Ordering Cost = Carrying Cost• (D/Q)Co = ½ ChQ(1-d/p)• Optimal Order Quantity

pd

C

DCQ

pd

C

DCQ

h

o

h

o

1

2*

1

22

Optimal Production Quantity, Q*p

pd

C

DCQ

h

s

1

2*

Quantity Discount Model

Discount Number

Discount Quantity

Discount (%)Discount Cost ($)

1 0-999 05.00 (normal

cost)

2 1,000 – 1,999 4 4.80

3 2,000 – over 5 4.75

Quantity Discount Schedule

Total Cost = material cost + ordering cost + carrying cost = DC + (D/Q)Co + ½ QCh

Total Cost CurveTC for Disc. 1

TC for Disc. 2

TC for Disc. 3

Q* for Disc. 2

1,000 2,0000

Total Cost

Order Quantity

Use of Safety Stock

• Safety stock: additional stock that is kept on hand

• It is used only when demand is uncertain• Main purpose: to avoid stockouts when the

demand is higher than expected• ROP = d x L (normal condition)• ROP = d x L + SS (demand is uncertain)• Because it is dealing with decision under risk,

knowing the probability of demand is necessary.

Safety Stock with Known Stockout Costs

Case of ABCO• ROP = 50 units (= d x L)• Ch = $5 (per unit per year)• Cso = $40/unit (stockout cost)• Optimal number of orders per year is 6• Objective: to find the reorder point, including

safety stock, that will minimize total expected cost

• Total expected cost is the sum of expected stockout cost plus expected additional carrying cost

Probability of Demand for ABCO

Number of Units Probability

30 0.2

40 0.2

50 (ROP) 0.3

60 0.2

70 0.1

Total 1.0

Annual Expected Stockout Cost

• When the ROP < demand over lead time

Total Cost = Stockout Cost

= no. of units short x stockout cost/unit x

no. of orders per year

• When the ROP > demand over lead time

Total Cost = total additional carrying cost

= no. of surplus units x carrying cost

ABCO’s Stockout CostsProbability 0.20 0.20 0.30 0.20 0.10

State of Nature

Alternative

30 40 50 60 70 EMV

30 0 2,400 4,800 7,200 9,600 4,320

40 50 0 2,400 4,800 7,200 2,410

50 100 50 0 2,400 4,800 990

60 150 100 50 0 2,400 305

70 200 150 100 50 0 110

Safety Stock with Unknown Stockout Cost

• There are many situation when stockout cost are unknown or extremely difficult to determine, i.e: major stockout cost is the loss of goodwill, how to measure it?

• Alternative approach: using service level

• Service level = 1 – probability of a stockout or

• Probability of a stockout = 1 – service level

Hinsdale Company Example

• Average demand = 350 units

• Standard Deviation = 10

• Hinsdale wants to follow a policy that result in stockout occuring only 5% of the time.

• How much safety stock should be maintained?

Safety Stock & Normal Distribution

X=?

SS

μ=350

σ=10X = mean + safety stockSS = safety stock = X – μZ = (X – μ) / σ = SS/ σ

Z value for an area under the normal curve of 0.95 (=1-0.05) is 1,65 (see appendix A)

SS = 1.65 (10) = 16.5 units or 17 units

SS = Z σ