1

Chapter 10: Inventory

• Types of Inventory and Demand

• Availability

• Cost vs. Service Tradeoff

• Pull vs. Push

• Reorder Point System

• Periodic Review System

• Joint Ordering

• Number of Stocking Points

• Investment Limit

• Just-In-Time

2

Chapter 10: Inventory

• Skip the following:– Single-Order Quantity: pp. 322-323– Lumpy Demand: pp. 344-345, – Box 10.23 Application: pp. 347-348, – Poisson Distribution: pp. 356-357)

3

Inventory

• Inventory includes:– Raw materials, Supplies, Components, Work-in-progress,

Finished goods.

• Located in:– Warehouses, Production facility, Vehicles, Store shelves.

• Cost is usually 20-40% of the item value per year!

4

Why Keep Inventories?

• Positive effects:– Economies of scale in production & transportation.

– Coordinate supply and demand.

– Customer service.

– Part of production.

• Negative Effects:– Money tied up could be better spent elsewhere.

– Inventories often hide quality problems.

– Encourages local, not system-wide view.

5

Types of Inventories

• Regular (cycle) stock: to meet expected demand between orders.

• Safety stock: to protect against unexpected demand.– Due to larger than expected demand or longer than expected lead

time.– Lead time=time between placing and receiving order.

• Pipeline inventory: inventory in transit.

• Speculation inventory: precious metals, oil, etc.

• Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.

6

Types of Demand

• Perpetual (continual): – Mean and standard deviation (or variance) of demand are known (or can be

calculated).– Use repetitive ordering.

• Seasonal or Spike: – Order once (or a few time) per season.

• Lumpy: hard to predict.– Often standard deviation > mean.

• Terminating: – Demand will end at known time.

• Derived (dependent): – Depends on demand for another item.

7

Performance Measures

• Turnover ratio:

• Availability:– Service Level = SL– Fill Rate = FR– Weighted Average Fill Rate = WAFR

Annual demand

Average inventoryTurnover ratio =

8

Measuring Availability: SL

• Want product available in the right amount, in the right place, at the right time.

• For 1 item: SLi = Service Level for item iSLi = Probability that item i is in stock.

= 1 - Probability that item i is out-of-stock.

Expected number of units out of stock/year for item iAnnual demand for item i

SLi = 1 -

9

Measuring Availability: FR and WAFR

• For 1 order of several items: FRj = Fill Rate for order jFRj = Product of service levels for items ordered.

• For all orders: WAFR (Weighted Average Fill Rate)– Sum over all orders of (FRj) x (frequency of order j).

FRj = SL1 x SL2 x SL3 x ...

10

WAFR Example

• Example: 3 items – I1 (SL=0.98); I2 (SL = 0.90); I3 (SL = 0.95)

Order Frequency FR Freq.xFRI1 0.4 0.98 0.392I1,I2,I2 0.1 0.98x0.90x0.90=0.7938 0.07938I1,I3 0.2 0.98x0.95=0.931 0.1862I1,I2,I3 0.3 0.98x0.90x0.95=0.8379 0.25137

WAFR = 0.90895

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Fundamental Tradeoff

• Level of Service vs. Cost

Level of Service

$

Cost

Revenue

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Fundamental Tradeoff

• Level of Service (availability) vs. Cost

• Higher service levels -> More inventory.-> Higher cost.

• Higher service levels -> Better availability.-> Fewer stockouts.-> Higher revenue.

13

Inventory Costs

• Procurement (order) cost: – To prepare, process, transmit, handle order.

• Carrying or Holding cost: – Proportional to amount (average value) of inventory.– Capital costs - for $ tied up (80%). – Space costs - for space used.– Service and risk costs - insurance, taxes, theft, spoilage, obsolecence, etc.

• Out-of-stock costs (if order can not be filled from stock).– Lost sales cost - current and future orders.– Backorder cost - for extra processing, handling, transportation, etc.

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Fundamental Cost Tradeoff

Inventory carrying cost vs. Order & Stockout cost

• Larger inventory -> Higher carrying costs.

• Larger inventory -> Fewer larger orders.-> Lower order costs.

• Larger inventory -> Better availability. -> Few stockouts.

-> Lower stockout costs.

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Retail Stockouts

On average 8-12% of items are not available!

• Causes:– Inadequate store orders.

– Not knowing store is out-of-stock.

– Poor promotion forecasting.

– Not enough shelf space.

– Backroom inventory not restocked.

– Replenishment warehouse did not have enough • True for only 3% of stockouts.

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Pull vs. Push Systems

• Pull:– Treat each stocking point independent of others. – Each orders independently and “pulls” items in.– Common in retail.

• Push: – Set inventory levels collectively.– Allows purchasing, production and transportation economies

of scale.– May be required if large amounts are acquired at one time.

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Push Inventory Control

• Acquire a large amount.

• Allocate amount among stocking points (warehouses) based on:– Forecasted demand and standard deviation.– Current stock on hand.– Service levels.

• Locations with larger demand or higher service levels are allocated more.

• Locations with more inventory on hand are allocated less.

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Push Inventory Control

TRi = Total requirements for warehouse i

NRi = Net requirements at i

Total excess = Amount available - NR for all warehouses

Demand % = (Forecast demand at i)/(Total forecast demand)

Allocation for i = NRi + (Total excess) x (Demand %)

= Forecast demand at i + Safety stock at i

= Forecast demand at i + z x Forecast error at i

= TRi - Current inventory at i z is from Appendix A

19

Push Inventory Control Example

Allocate 60,000 cases of product among two warehouses based on the following data.

Current Forecast ForecastWarehouse Inventory Demand Error SL 1 10,000 20,000 5,000 0.90 2 5,000 15,000 3,000 0.98

35,000

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Push Inventory Control Example

Current Forecast Forecast DemandWarehouse Inventory Demand Error SL % 1 10,000 20,000 5,000 0.90 0.5714

2 5,000 15,000 3,000 0.98 0.4286 35,000

TR1 = 20,000 + 1.28 x 5,000 = 26,400TR2 = 15,000 + 2.05 x 3,000 = 21,150

NR1 = 26,400 - 10,000 = 16,400NR2 = 21,150 - 5,000 = 16,150

Total Excess = 60,000 - 16,400 - 16,150 = 27,450

Allocation for 1 = 16,400 + 27,450 x (0.5714) = 32,086 casesAllocation for 2 = 16,150 + 27,450 x (0.4286) = 27,914 cases

21

Pull Inventory Control - Repetitive Ordering

• For perpetual (continual) demand.

• Treat each stocking point independently.

• Consider 1 product at 1 location.

Determine:

How much to order:

When to (re)order:

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Pull Inventory Control - Repetitive Ordering

• For perpetual (continual) demand.

• Treat each stocking point independently.

• Consider 1 product art 1 location.

Reorder Periodic

Determine: Point System Review System

How much to order: Q M-qi

When to (re)order: ROP T

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Reorder Point System

Order amount Q when inventory falls to level ROP.• Constant order amount (Q).• Variable order interval.

0

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0 10 20 30 40 50 60 70 80 90

Time (day)

INV

EN

TO

RY

ROP Inventory Level

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0

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INV

EN

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ROP Inventory Level

Reorder Point System

Receive 1st order

Place 1st order

Receive 2nd order

Place 2nd order Receive

3rd order

Place 3rd order

LT1 LT2 LT3

Each increase in inventory is size Q.

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Time (day)

INV

EN

TO

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ROP Inventory Level

Reorder Point System

Receive 1st order

Place 1st order

Receive 2nd order

Place 2nd order Receive

3rd order

Place 3rd order

LT1 LT2 LT3

Time between 1st & 2nd order

Time between 2nd & 3rd order

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Periodic Review System

Order amount M-qi every T time units.

• Constant order interval (T=20 below).• Variable order amount.

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Time (days)

INV

EN

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Inventory Level

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Time (days)

INV

EN

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Inventory Level

Periodic Review System - T=20 days

Receive 1st order

Place 1st order

Receive 2nd order

Place 2nd order

Receive 3rd order

Place 3rd order

LT1 LT2 LT3

Each increase in inventory is size M-amount on hand.(M=90 in this example.)

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0

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INV

EN

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Inventory Level

Periodic Review System - T=20 days

Receive 1st order

Place 1st order

Receive 2nd order

Place 2nd order

Receive 3rd order

Place 3rd order

LT1 LT2 LT3

Time between 1st & 2nd order

(20 days)

Time between 2nd & 3rd order

(20 days)

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Optimal Inventory Control

• For perpetual (continual) demand.

• Treat each stocking point independently.

• Consider 1 product art 1 location.

Reorder Periodic

Determine: Point System Review System

How much to order: Q M-qi

When to (re)order: ROP T

Find optimal values for: Q & ROP or for M & T.

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D = demand (usually annual) d = demand rateS = order cost ($/order) LT = (average) lead timeI = carrying cost k = stockout cost

(% of value/unit time) P = probability of being inC = item value ($/item) stock during lead time

sd = std. deviation of demandsLT = std. deviation of lead times’d = std. deviation of demand during lead time

Q = order quantity N = number of orders/yearTC = total cost (usually annual)ROP = reorder pointT = time between orders

Inventory Variables

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No variability in demand and lead time (sd = 0, sLT = 0).Will never have a stock out.

Simplest Case - Constant demand and lead time

Invento

ry

Time

ROPQ

Suppose: d = 4/day and LT = 3 days

Then ROP = 12 (ROP = d x LT)

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Constant demand and lead timeIn

vento

ry

Time

ROPQ

TC = Order cost + Inventory carrying cost

Order cost = N x S = (D/Q) x S

Carrying cost = Average inventory level x C x I

= (Q/2) x C x I

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Economic Order Quantity (EOQ)In

vento

ry

Time

ROPQ

Select Q to minimize total cost.

Set derivative of TC with respect to Q equal to zero.

TC = QD S + IC 2

Q

0 = -Q2

D S +2IC

Q =IC

2DS

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Optimal OrderingIn

vento

ry

Time

ROPQ

Economic order quantity:

Optimal number of orders/year:

Optimal time between orders:

Optimal cost:

TC = Q*D S + IC 2

Q*

2DSQ* =

IC

Q*D

DQ*

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Example

D = 10,000/yearS = $61.25/orderI = 20%/yearC = $50/item

TC = Q*D S + IC 2

Q*

2DSQ* =

IC = 2(10,000)(61.25)(0.2)(50)

= 350 units/order

=10,000

350(61.25) + (0.2)(50)

3502

= 1750 + 1750 = $3500/year

N = 10,000350

= 28.57 orders/year

T = 35010,000

= 0.035 years = 1.82 weeks

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Example - continuedQ* = 350 units/orderN = 28.57 orders/yearT = 1.82 weeks

This is not a very convenient schedule for ordering!

Suppose you order every 2 weeks:T = 2 weeks, so N = 26 orders/year

TC = QD S + IC 2

Q =10,000384.6

(61.25) + (0.2)(50)384.6

2

= 1592.56 + 1923.00 = $3515.56/year

Q =DN

= 384.6 units/order (10% over EOQ)10,000

26=

Q = 384.6 is 9.9% over EOQ, but TC is only 0.4% over optimal cost!!!

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0

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2000

3000

4000

5000

6000

100 200 300 400 500 600 700

Q

Cost

Total Cost

Carrying Cost

Order Cost

Model is Robust

Q* = 350 TC = $3500/year

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Model is Robust

Changing Q by 20% increases cost by a few percent.

0

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2000

3000

4000

5000

6000

100 200 300 400 500 600 700

Q

Cost

Carrying Cost

Order Cost

Total Cost

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Model is Robust

• A small change in Q (or N or T) causes very little increase in the total cost.– Changing Q by 10% increases cost < 1%.– Changing Q changes N=D/Q, T=Q/D and TC.– Changing N or T changes Q!

• A near optimal order plan, will have a very near optimal cost.

• You can adjust values to fit business operations.– Order every other week vs. every 1.82 weeks.– Order in multiples of 100 if required rather than Q*.

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Non-instantaneous Resupply

• Produce several products on same equipment.

• Consider one product.

p = production rate (for example, units/day)

d = demand rate (for example, units/day)

• Inventory increases slowly while it is produced.

• Inventory decreases once production stops.

• Stop producing this product when inventory is “large enough”.

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Inventory Level

Suppose: p = 10/day (while producing this product).

d = 3/day (for this product).

Put p-d = 7 in inventory every day while producing.

Remove d = 3 from inventory every day while not producing this product.

Invento

ry

Time

Produce Q Do not produce

Slope=7Slope=-3

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D = demand (usually annual) d = demand rateS = setup cost ($/setup) p = production rateI = carrying cost

(% of value/unit time)C = item value ($/item)

Assume d and p are constant (no variability).

Q = production quantity (in each production run) N = number of production runs (setups)/yearTC = total cost (usually annual)

Also want: Length of a production run (for example, in days)Length of time between runs (cycle time)

Variables

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Inventory LevelIn

vento

ry

Time

Maximuminventory

Inventory pattern repeats:Produce Q units of product of interest.Then produce other products.

Every production run of Q units requires 1 setup.Find Q to minimize total cost.

Produce Q Do not produce

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Inventory LevelIn

vento

ry

Time

Maximuminventory

TC = Setup cost + Inventory carrying cost

Setup cost = N x S = (D/Q) x S

Carrying cost = Average inventory level x C x I

= (Max. inventory/2) x C x I

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Maximum Inventory LevelIn

vento

ry

Time

Maximuminventory

Length of a production run = Q/p (days)

Max. inventory = (p-d) x Q/p = Q

Carrying cost = IC

p-dp

p-dp

Q2

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Optimum Production Run Size: QIn

vento

ry

Time

Maximuminventory

TC = QD S + IC 2

Q

Q =IC

2DS

Select Q to minimize total cost.

Set derivative of TC with respect to Q equal to zero.

p-dp

p-dp

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Non-instantaneous Resupply Equations

TC = QD S + IC 2

Q p-dp

Q =IC

2DSp-dp

N = D/Q

Length of a production run = Q/p

Length of time between runs = Q/d

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Non-instantaneous Resupply Example

TC = 63,246 + 63,246 = $126,492/year

Every 7.91 days begin a 2.64 day production run.

Q =0.2x6000

2x5000x2000

60-2060

Q/p = 158.11/60 = 2.64 days

Q/d = 158.11/20 = 7.91 days

D=5000/year assume 250 days/yearI = 20%/yearS = $2000/setupC = $6000/unit p=60/day

First, calculate d=5000/250 = 20/day

= 158.11 units

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Adjust Values to Fit Business Cycles

Change cycle length to 8 days -> Q/d = 8 days

Then: Q = 160 unitsQ/p = 2.67 daysTC = 62,500 + 64,000 = $126,500/year

8 16 240

Production runs

Produce other products

2.7 10.7 18.7

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Cost is Insensitive to Small Changes

Change cycle length to 10 days=2 weeks (+26%)

Then: Q/d = 10 daysQ = 200 units Q/p = 3.33 daysTC = 50,000 + 80,000 = $130,000/year

TC is only 2.8% over minimum TC!

10 200

Production runs

Produce other products

51

Scheduling Multiple Products

Suppose 3 products are produced on the same equipment.Optimal values are:

P1: Q/d = 7.9 1 Q/p = 2.64

P2: Q/d = 13.4 Q/p = 4.8

P2: Q/d = 25.8 Q/p = 5.9

Adjust cycle lengths to a common value or multiple.

For example 8 days

P1: Q/d = 8 -> Q/p = 2.7

P2: Q/d = 12 -> Q/p = 4.3

P2: Q/d = 24 -> Q/p = 5.5

Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days.

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Scheduling Multiple Products - continued

P1: Q/d = 8 -> Q/p = 2.7

P2: Q/d = 12 -> Q/p = 4.3

P2: Q/d = 24 -> Q/p = 5.5

Now schedule 3 runs of P1, 2 runs of P2 and 1 run of P3 every 24 days.

P1

P2

P3

Idle

2.70 7 9.7 15.2 17.9 22.2 24

P1 P1 P1P2 P2P3

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Reorder Point System - Variability

Order amount Q when inventory falls to level ROP.

If demand or lead time are larger than expected -> stockout

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ROP Inventory Level

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Variability

Variability in demand and lead time may cause stockouts.

d = mean demandsd = std. deviation of demand

LT = mean lead timesLT = std. deviation of lead time

s’d = std. deviation of demand during lead time

LT x sd2 + d2 x sLT

2s’d =

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Safety Stock

Use safety stock to protect against stockouts when demand or lead time is not constant.

Safety stock = z x s’d

z is from Standard Normal Distribution Table and is based on P = Probability of being in-stock during lead time.

ROP = expected demand during lead time + safety stock

= d x LT + z x s’d

Average Inventory Level (AIL) = regular stock + safety stockAIL =

2Q

+ z x s’d

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Special Cases

1. Constant lead time, variable demand: sLT = 0

2. Constant demand, variable lead time: sd = 0

3. Constant demand, constant lead time: sd = 0, sLT = 0

LT x sd2 s’d = = sd LT

d2 x sLT2s’d = = dsLT

s’d = 0

57

Total Cost

TC = Order cost + Regular stock carrying cost

+ Safety stock carrying cost + Stockout cost

TC = QD S + IC 2

Q + ICz s’d +

QD k s’d E(z)

k = out-of-stock cost per unit short

s’d E(z) = expected number of units out-of-stock in one order cycle

E(z) = unit Normal loss integral

P -> z (from Appendix A) -> E(z) (from Appendix B)

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3 Cases

1. Stockout cost k is known; P is not known.

-> Calculate optimal P by repeating (1) and (2) until z does not change.

2. Stock cost k is not known; P is known.-> Can not use last term in TC.

3. Stockout cost k is known; P is known.-> Could use k to calculate optimal P.

P = 1 - Dk

QIC(1)

Q = IC2D[s + ks’dE(z)

(2)

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Reorder Point Example

D = 5000 units/year d = 96.15 units/weekS = $10/order sd = 10 units/week

C = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0

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Reorder Point Example - Case 1

D = 5000 units/year d = 96.15 units/weekS = $10/order sd = 10 units/week

C = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0

• k = $2/unit; P is not given

• Iterate to find optimal P.

Q =0.2x5

2x5000x10 = 316.23 units

s’d = sd LT = 14.14= 10 2

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Case 1 (continued) - Find best P

P = 1 - 5000(2)

316.23(0.2)5

Q = 0.2(5)2(5000)[10 + 2(14.14)0.0123

= 0.9684

z = 1.86 E(z) = 0.0123

= 321.68

P = 1 - 5000(2)

321.68(0.2)5

Q = 0.2(5)2(5000)[10 + 2(14.14)0.0126

= 0.9678

z = 1.85 E(z) = 0.0126

= 321.81

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Case 1 (continued)

P = 1 - 5000(2)

321.81(0.2)5= 0.9678

z = 1.85 E(z) = 0.0126

• z does not change, so STOP

Solution: Q = 322 z = 1.85 E(z) = 0.0126

TC = 155.28 + 161.00 + 26.16 + 5.53 = $347.97/year

ROP = d x LT + z x s’d = 96.15(2) + 1.85(14.14) = 218.46

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Reorder Point Example - Case 2

D = 5000 units/year d = 96.15 units/weekS = $10/order sd = 10 units/week

C = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0

• k is not known; P =90%

Q =0.2x5

2x5000x10 = 316.23 units s’d = 14.14 (as in Case 1)

Solution: z = 1.28

TC = 158.23 + 158.00 + 18.10 = $334.33/year

ROP = d x LT + z x s’d = 96.15(2) + 1.28(14.14) = 210.40

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Reorder Point Example - Case 3

D = 5000 units/year d = 96.15 units/weekS = $10/order sd = 10 units/week

C = $5/unitI = 20% per yearLT = 2 weeks (constant) sLT = 0

• k =$2/unit; P =90%

Q =0.2x5

2x5000x10 = 316.23 units s’d = 14.14 (as in Case 2)

Solution: z = 1.28

TC = 158.23 + 158.00 + 18.10 + 21.25 = $355.58/year

ROP = d x LT + z x s’d = 96.15(2) + 1.28(14.14) = 210.40

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Reorder Point Example - Case 3

• k =$2/unit; P =90%

Q = 316.23Solution:

TC = $355.58/year

ROP = 210.40

• Could use k=$2/unit to find optimal P

• It would be P = 96.78% as in Case 1!

• Order size would be slightly larger (322 vs. 316).

• Cost would be slightly less ($347.97 vs. $355.58).

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Reorder Point Example - Case 4

• Suppose we keep no safety stock

Q =0.2x5

2x5000x10 = 316.23 units

Solution:

TC = 158.23 + 158.00 + 0 + 178.50 = $494.73/year

ROP = d x LT = 96.15(2) = 192.30

• With no safety stock there is a stockout whenever demand during lead time exceeds expected amount (dxLT).

• Therefore: P = 0.5

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Reorder Point Example - Summary

Case k P Q ROP TC($/year)

1 2 .9678 322 218 347.97

2 - .90 316 210 334.33

3 2 .90 316 210 355.58

4 2 .50 316 192 494.73

• A small amount of safety stock can save a large amount!– Case 4 vs Case 3

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P and SL

• Suppose that on average:– There are 10 orders/year.– Each order is for 100 items (Q=100).– We are out-of-stock 2 items per year on one order.

P = probability of being in stock during lead time. = 1 - probability of being our-of-stock during lead time.

= 1 - 1/10 = 0.90

SL= Service level = % of items in-stock = 1 - % of items out-of-stock = 1 - 2/1000 = 0.998

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Expected number of units out-of-stock/year

Service Level - Reorder Point

Annual demand

SL= 1 - % of items out-of-stock

= 1 -

(D/Q) x s’d x E(z)

D= 1 -

s’d E(z)

Q= 1 -

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Service Levels for Cases 1-4

Case 1: SL = 1 -

Case 2: SL = 1 -

Case 3: SL = 1 -

Case 4: SL = 1 -

14.14(.0126)322

= 0.9994

14.14(.0475)316

= 0.9979

14.14(.0475)316

= 0.9979

14.14(.3989)316

= 0.9822

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Reorder Point Example - Summary

Case k P Q ROP TC($/yr) SL

1 2 .9678 322 218 347.97 .9994

2 - .90 316 210 334.33 .9979

3 2 .90 316 210 355.58 .9979

4 2 .50 316 192 494.73 .9822

• Note difference between P and SL!

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Out-of-Stock for Cases 1-4

Case 1: Out-of-stock: 3 items per year and 0.5 orders/year

SL = 0.9994 -> (1-.9994)x5000 = 3 items/yearP = 0.9678 -> (1-.9678)x5000/322 = 0.5 orders/year

Case 2 & 3: Out-of-stock: 10.5 items per year and 1.58 orders/year

SL = 0.9979 -> (1-.9979)x5000 = 10.5 items/yearP = 0.90 -> (1-.90)x5000/316 = 1.58 orders/year

Case 4: Out-of-stock: 89 items per year and 7.9 orders/year

SL = 0.98229 -> (1-.9822)x5000 = 89 items/yearP = 0.50 -> (1-.50)x5000/316 = 7.9 orders/year

73

Lead Time Variability in Example

D = 5000 units/year d = 96.15 units/weekS = $10/order sd = 10 units/week

C = $5/unitI = 20% per yearLT = 2 weeks (constant)

Suppose sLT = 1.2 (not 0 as before)

Now:

For constant lead time (sLT = 0) s’d =14.14

Additional safety stock due to lead time variability = z(116.24-14.14)

LT x sd2 + d2 x sLT

2s’d = = 116.24

74

Optimal Inventory Control

• For perpetual (continual) demand.

• Treat each stocking point independently.

• Consider 1 product art 1 location.

Reorder

Determine: Point System

How much to order: Q

When to (re)order: ROP