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Chapter 10: Inventory

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- 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

- 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 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!

- 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.

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- Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.

- 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.

- 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

- 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.

- 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 ...

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

OrderFrequencyFRFreq.xFR

I1 0.40.98 0.392

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

I1,I3 0.20.98x0.95=0.931 0.1862

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

WAFR = 0.90895

Revenue

$

Cost

Level of Service

- Level of Service vs. Cost

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

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

-> Higher revenue.

- 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.

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.

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:
- 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.

- 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.

= 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

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

CurrentForecast Forecast

WarehouseInventoryDemandErrorSL

1 10,000 20,0005,000 0.90

2 5,000 15,0003,000 0.98 35,000

Current Forecast Forecast Demand

WarehouseInventoryDemandErrorSL%

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

- For perpetual (continual) demand.
- Treat each stocking point independently.
- Consider 1 product at 1 location.
Determine:

How much to order:

When to (re)order:

- 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

Order amount Q when inventory falls to level ROP.

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

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.

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

Order amount M-qi every T time units.

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

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.)

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)

- 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.

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

Inventory

ROP

Time

No 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)

Inventory

ROP

Time

Q

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

Inventory

Q

D

S + IC

TC =

2

Q

ROP

Time

IC

D

S +

0 = -

2

Q2

Q

Select Q to minimize total cost.

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

2DS

Q =

IC

Inventory

Q* =

IC

ROP

Time

Q*

D

S + IC

TC =

2

Q*

Q

2DS

Economic order quantity:

Optimal number of orders/year:

Optimal time between orders:

Optimal cost:

D

Q*

Q*

D

Q* =

IC

Q*

D

S + IC

TC =

2

Q*

D = 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

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!!!

Total Cost

Carrying Cost

Order Cost

Q* = 350TC = $3500/year

Total Cost

Carrying Cost

Order Cost

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

- 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*.

- 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”.

Slope=7

Inventory

Slope=-3

Time

Produce Q

Do not produce

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.

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

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.

Inventory

Maximum

inventory

Time

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

Inventory

Maximum

inventory

Time

Length 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

Maximum

inventory

Time

Q

D

S + IC

TC =

2

Q

Inventory

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

p

2DS

Q =

IC

p-d

Q

D

S + IC

TC =

2

Q

N = D/Q

p-d

p

Length of a production run = Q/p

Length of time between runs = Q/d

D=5000/yearassume 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.

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

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

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

P1: Q/d = 7.91Q/p = 2.64

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

P2: Q/d = 25.8Q/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

P1

P3

P2

P2

P1

0

2.7

22.2

17.9

24

9.7

15.2

7

P1

P2

P3

Idle

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.

Order amount Q when inventory falls to level ROP.

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

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

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

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

Q

D

D

S + IC

k s’d E(z)

+ ICz s’d +

TC =

2

Q

Q

TC = 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)

QIC

Dk

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 -

(1)

2D[s + ks’dE(z)

(2)

Q =

IC

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

S = $10/ordersd = 10 units/week

C = $5/unit

I = 20% per year

LT = 2 weeks (constant) sLT = 0

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

S = $10/ordersd = 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

316.23(0.2)5

P = 1 -

= 0.9684

5000(2)

z = 1.86E(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 = 1.85E(z) = 0.0126

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

= 321.81

Q =

0.2(5)

321.81(0.2)5

P = 1 -

= 0.9678

5000(2)

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

z = 1.85E(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

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

S = $10/ordersd = 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

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

S = $10/ordersd = 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

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).

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

Casek P Q ROP TC($/year)

12.9678 322 218 347.97

2-.90 316 210 334.33

32.90 316 210 355.58

42.50 316 192 494.73

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

- 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

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

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

Casek P Q ROP TC($/yr) SL

12 .9678 322 218 347.97.9994

2- .90 316 210 334.33.9979

32 .90 316 210 355.58.9979

42 .50 316 192 494.73.9822

- Note difference between P and SL!

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

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

S = $10/ordersd = 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

- 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