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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

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)

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!

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.

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- Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.

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.

Performance Measures

- Turnover ratio:
- Availability:
- Service Level = SL
- Fill Rate = FR
- Weighted Average Fill Rate = WAFR

Annual demand

Turnover ratio=

Average inventory

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

SLi = 1 -

Annual demand for item i

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 i
SLi = Probability that item i is in stock.

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

Measuring Availability: FR and WAFR

- For 1 order of several items: FRj = Fill Rate for order j
FRj = 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 ...

WAFR Example

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

Order Frequency FR Freq.xFR

I1 0.4 0.98 0.392

I1,I2,I2 0.1 0.98x0.90x0.90=0.7938 0.07938

I1,I3 0.2 0.98x0.95=0.931 0.1862

I1,I2,I3 0.3 0.98x0.90x0.95=0.8379 0.25137

WAFR = 0.90895

Fundamental Tradeoff

- Level of Service (availability) vs. Cost
- Higher service levels -> More inventory.
-> Higher cost.

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

-> Higher revenue.

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.

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.

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.

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.

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.

Push Inventory Control

= Forecast demand at i + Safety stock at i

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

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 %)

= TRi - Current inventory at i

z is from Appendix A

Push Inventory Control Example

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

Current Forecast Forecast

Warehouse Inventory Demand Error SL

1 10,000 20,000 5,000 0.90

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

Push Inventory Control Example

Current Forecast Forecast Demand

Warehouse InventoryDemand 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 = 26,400

TR2 = 15,000 + 2.05 x 3,000 = 21,150

NR1 = 26,400 - 10,000 = 16,400

NR2 = 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 cases

Allocation for 2 = 16,150 + 27,450 x (0.4286) = 27,914 cases

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:

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

Reorder Point System

Order amount Q when inventory falls to level ROP.

- Constant order amount (Q).
- Variable order interval.

Reorder Point System

Place 1st

order

LT1

LT2

LT3

Place 2nd

order

Place 3rd

order

Receive

3rd order

Receive

1st order

Receive

2nd order

Each increase in inventory is size Q.

Reorder Point System

Place 1st

order

LT1

LT2

LT3

Place 2nd

order

Place 3rd

order

Receive

3rd order

Receive

1st order

Receive

2nd order

Time between

1st & 2nd order

Time between

2nd & 3rd order

Periodic Review System

Order amount M-qi every T time units.

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

Periodic Review System - T=20 days

Place 1st

order

LT3

LT1

LT2

Place 3rd

order

Receive

3rd order

Place 2nd

order

Receive

1st order

Receive

2nd order

Each increase in inventory is size M-amount on hand.

(M=90 in this example.)

Periodic Review System - T=20 days

Place 1st

order

LT3

LT1

LT2

Place 3rd

order

Receive

3rd order

Place 2nd

order

Receive

1st order

Receive

2nd order

Time between

1st & 2nd order

(20 days)

Time between

2nd & 3rd order

(20 days)

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.

Inventory Variables

D = demand (usually annual) d = demand rate

S = order cost ($/order) LT = (average) lead time

I = carrying cost k = stockout cost

(% of value/unit time) P = probability of being in

C = item value ($/item) stock during lead time

sd =std. deviation of demand

sLT =std. deviation of lead time

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

Q = order quantity

N = number of orders/year

TC = total cost (usually annual)

ROP = reorder point

T = time between orders

ROP

Time

Simplest Case - Constant demand and lead timeNo variability in demand and lead time (sd =0, sLT =0).

Will never have a stock out.

Q

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

Then ROP = 12 (ROP = d x LT)

ROP

Time

Constant demand and lead timeQ

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

Q

D

S + IC

TC =

2

Q

ROP

Time

IC

D

S +

0 = -

2

Q2

Economic Order Quantity (EOQ)Q

Select Q to minimize total cost.

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

2DS

Q =

IC

Q* =

IC

ROP

Time

Q*

D

S + IC

TC =

2

Q*

Optimal OrderingQ

2DS

Economic order quantity:

Optimal number of orders/year:

Optimal time between orders:

Optimal cost:

D

Q*

Q*

D

IC

Q*

D

S + IC

TC =

2

Q*

ExampleD = 10,000/year

S = $61.25/order

I = 20%/year

C = $50/item

2DS

2(10,000)(61.25)

=

= 350 units/order

(0.2)(50)

350

10,000

(61.25) + (0.2)(50)

=

2

350

= 1750 + 1750 = $3500/year

10,000

N =

= 28.57 orders/year

350

350

= 0.035 years = 1.82 weeks

T =

10,000

Example - continued

Q* = 350 units/order

N = 28.57 orders/year

T = 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

10,000

D

Q =

= 384.6 units/order (10% over EOQ)

=

26

N

384.6

10,000

Q

D

(61.25) + (0.2)(50)

=

S + IC

TC =

2

2

Q

384.6

= 1592.56 + 1923.00 = $3515.56/year

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

Carrying Cost

Order Cost

Model is RobustChanging Q by 20% increases cost by a few percent.

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*.

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”.

Inventory

Slope=-3

Time

Produce Q

Do not produce

Inventory LevelSuppose: 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.

Variables

D = demand (usually annual) d = demand rate

S = setup cost ($/setup) p = production rate

I = 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)/year

TC = total cost (usually annual)

Also want:

Length of a production run (for example, in days)

Length of time between runs (cycle time)

Inventory Level

Inventory

Maximum

inventory

Time

Do not produce

Produce Q

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.

Maximum

inventory

Time

Inventory LevelTC = 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

Maximum

inventory

Time

Maximum Inventory LevelLength of a production run = Q/p (days)

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

Carrying cost = IC

p-d

p

Q

p-d

2

p

inventory

Time

Q

D

S + IC

TC =

2

Q

Optimum Production Run Size: QInventory

p-d

p

Select Q to minimize total cost.

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

p

2DS

Q =

IC

p-d

2DS

Q =

IC

p-d

Q

D

S + IC

TC =

2

Q

Non-instantaneous Resupply EquationsN = D/Q

p-d

p

Length of a production run = Q/p

Length of time between runs = Q/d

Non-instantaneous Resupply Example

D=5000/year assume 250 days/year

I = 20%/year

S = $2000/setup

C = $6000/unit

p=60/day

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

2x5000x2000

60

Q =

= 158.11 units

0.2x6000

60-20

Q/p = 158.11/60 = 2.64 days

Q/d = 158.11/20 = 7.91 days

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

Every 7.91 days begin a 2.64 day production run.

Adjust Values to Fit Business Cycles

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

Then: Q = 160 units

Q/p = 2.67 days

TC = 62,500 + 64,000 = $126,500/year

8

10.7

0

16

18.7

24

2.7

Production runs

Produce other products

Cost is Insensitive to Small Changes

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

Then: Q/d = 10 days

Q = 200 units

Q/p = 3.33 days

TC = 50,000 + 80,000 = $130,000/year

TC is only 2.8% over minimum TC!

10

20

0

Production runs

Produce other products

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.

P1

P3

P2

P2

P1

0

2.7

22.2

17.9

24

9.7

15.2

7

P1

P2

P3

Idle

Scheduling Multiple Products - continuedP1: 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.

Reorder Point System - Variability

Order amount Q when inventory falls to level ROP.

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

Variability

Variability in demand and lead time may cause stockouts.

d = mean demand

sd =std. deviation of demand

LT = mean lead time

sLT =std. deviation of lead time

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

s’d =

LT x sd2 + d2 x sLT2

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 stock

Q

AIL =

+ z x s’d

2

Special Cases

s’d =

LT x sd2

= sd

LT

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

s’d =

d2 x sLT2

= dsLT

s’d = 0

D

D

S + IC

k s’d E(z)

+ ICz s’d +

TC =

2

Q

Q

Total CostTC = Order cost + Regular stock carrying cost

+ Safety stock carrying cost + Stockout cost

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)

Dk

3 Cases1. 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 -

(1)

2D[s + ks’dE(z)

(2)

Q =

IC

Reorder Point Example

D = 5000 units/year d = 96.15 units/week

S = $10/order sd = 10 units/week

C = $5/unit

I = 20% per year

LT = 2 weeks (constant) sLT = 0

Reorder Point Example - Case 1

D = 5000 units/year d = 96.15 units/week

S = $10/order sd = 10 units/week

C = $5/unit

I = 20% per year

LT = 2 weeks (constant) sLT = 0

- k = $2/unit; P is not given
- Iterate to find optimal P.

2x5000x10

= 316.23 units

Q =

0.2x5

s’d =

= 14.14

sd

2

= 10

LT

Case 1 (continued) - Find best P

316.23(0.2)5

P = 1 -

= 0.9684

5000(2)

z = 1.86 E(z) = 0.0123

2(5000)[10 + 2(14.14)0.0123

= 321.68

Q =

0.2(5)

321.68(0.2)5

P = 1 -

= 0.9678

5000(2)

z = 1.85 E(z) = 0.0126

2(5000)[10 + 2(14.14)0.0126

= 321.81

Q =

0.2(5)

Case 1 (continued)

321.81(0.2)5

P = 1 -

= 0.9678

5000(2)

- z does not change, so STOP
Solution:Q = 322 z = 1.85 E(z)= 0.0126

z = 1.85 E(z) = 0.0126

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

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

Reorder Point Example - Case 2

D = 5000 units/year d = 96.15 units/week

S = $10/order sd = 10 units/week

C = $5/unit

I = 20% per year

LT = 2 weeks (constant) sLT = 0

- k is not known; P =90%

Solution:z = 1.28

2x5000x10

= 316.23 units

s’d = 14.14 (as in Case 1)

Q =

0.2x5

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

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

Reorder Point Example - Case 3

D = 5000 units/year d = 96.15 units/week

S = $10/order sd = 10 units/week

C = $5/unit

I = 20% per year

LT = 2 weeks (constant) sLT = 0

- k =$2/unit; P =90%

Solution:z = 1.28

2x5000x10

= 316.23 units

s’d = 14.14 (as in Case 2)

Q =

0.2x5

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

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

Reorder Point Example - Case 3

Solution:

- k =$2/unit; P =90%

Q = 316.23

ROP = 210.40

TC = $355.58/year

- 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).

Reorder Point Example - Case 4

Solution:

- Suppose we keep no safety stock

2x5000x10

= 316.23 units

Q =

0.2x5

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

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

- With no safety stock there is a stockout whenever demand during lead time exceeds expected amount (dxLT).
- Therefore: P = 0.5

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

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

Service Level - Reorder Point

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

Expected number of units out-of-stock/year

= 1 -

Annual demand

(D/Q) x s’d x E(z)

= 1 -

D

s’d E(z)

= 1 -

Q

Service Levels for Cases 1-4

14.14(.0126)

= 0.9994

Case 1: SL = 1 -

Case 2: SL = 1 -

Case 3: SL = 1 -

Case 4: SL = 1 -

322

14.14(.0475)

= 0.9979

316

14.14(.0475)

= 0.9979

316

14.14(.3989)

= 0.9822

316

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!

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/year

P = 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/year

P = 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/year

P = 0.50 -> (1-.50)x5000/316 = 7.9 orders/year

Lead Time Variability in Example

D = 5000 units/year d = 96.15 units/week

S = $10/order sd = 10 units/week

C = $5/unit

I = 20% per year

LT = 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)

s’d =

LT x sd2 + d2 x sLT2

= 116.24

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

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