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
Lumpy Demand: pp ,
Box Application: pp ,
Poisson Distribution: pp )

3. Inventory Inventory includes: Located in:
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: Negative 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Obsolete/Shrinkage stock: out-of-date, lost, stolen, etc.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 Terminating: Derived (dependent):
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: Annual demand
Service Level = SL
Fill Rate = FR
Weighted Average Fill Rate = WAFR
Annual demand
Turnover ratio =
Average inventory

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 i
SLi = 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 i
Annual demand for item i
SLi = 1 -

9. 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 ...

10. WAFR Example Example: 3 items
I1 (SL=0.98); I2 (SL = 0.90); I3 (SL = 0.95)
Order Frequency FR Freq.xFR I
I1,I2,I x0.90x0.90=
I1,I x0.95=
I1,I2,I x0.90x0.95=
WAFR =

11. Fundamental Tradeoff $ Level of Service Level of Service vs. Cost
Revenue

12. 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: Carrying or Holding 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.

14. 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.

15. 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.

16. Pull vs. Push Systems Pull: Push:
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.

17. 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.

18. 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 Forecast
Warehouse Inventory Demand Error SL
, ,000 5,
, ,000 3, ,000

20. Push Inventory Control Example
Current Forecast Forecast Demand
Warehouse Inventory Demand Error SL %
, , ,
, , , ,000
TR1 = 20, x 5,000 = 26,400
TR2 = 15, x 3,000 = 21,150
NR1 = 26, ,000 = 16,400
NR2 = 21, ,000 = 16,150
Total Excess = 60, , ,150 = 27,450
Allocation for 1 = 16, ,450 x (0.5714) = 32,086 cases
Allocation for 2 = 16, ,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:

22. 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

23. Reorder Point System Order amount Q when inventory falls to level ROP.
Constant order amount (Q).
Variable order interval.

24. Reorder Point System Each increase in inventory is size Q. LT1 LT2 LT3
Place 1st
order
Place 2nd
order
Place 3rd
order
Receive
3rd order
Receive
1st order
Receive
2nd order
Each increase in inventory is size Q.

25. Reorder Point System LT1 LT2 LT3 Place 1st order 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

26. Periodic Review System
Order amount M-qi every T time units.
Constant order interval (T=20 below).
Variable order amount.

27. Periodic Review System - T=20 days
LT1
LT2
LT3
Place 3rd
order
Place 2nd
order
Receive
3rd order
Place 1st
order
Receive
1st order
Receive
2nd order
Each increase in inventory is size M-amount on hand.
(M=90 in this example.)

28. Periodic Review System - T=20 days
LT1
LT2
LT3
Place 3rd
order
Place 2nd
order
Receive
3rd order
Place 1st
order
Receive
1st order
Receive
2nd order
Time between
1st & 2nd order
(20 days)
Time between
2nd & 3rd order
(20 days)

29. 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.

30. 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

31. Simplest Case - Constant demand and lead time
No variability in demand and lead time (sd = 0, sLT = 0).
Will never have a stock out.
Inventory
Time
ROP Q
Suppose: d = 4/day and LT = 3 days
Then ROP = 12 (ROP = d x LT)

32. Constant demand and lead time
Inventory
Time
ROP 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

33. Economic Order Quantity (EOQ)
Inventory
Time
ROP Q
TC = Q D
S + IC 2
Select Q to minimize total cost.
Set derivative of TC with respect to Q equal to zero.
0 = - Q2 D
S + 2 IC
2DS
Q = IC

34. Optimal Ordering 2DS Economic order quantity: Q* = IC
Inventory
Time
ROP Q
Q* = IC
2DS
Economic order quantity:
Optimal number of orders/year:
Optimal time between orders:
Optimal cost: D Q* Q* D
TC = Q* D
S + IC 2

35. Example D = 10,000/year S = $61.25/order I = 20%/year C = $50/item 2DS
Q* = IC
2DS
2(10,000)(61.25) =
= 350 units/order
(0.2)(50)
TC = Q* D
S + IC 2
10,000
350 =
(61.25) + (0.2)(50)
350 2
= = $3500/year
10,000
N =
= orders/year
350
350
T =
= years = 1.82 weeks
10,000

36. 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 D
10,000
Q = =
= units/order (10% over EOQ) N 26 D Q
10,000
384.6
S + IC =
(61.25) + (0.2)(50)
TC = Q 2
384.6 2
= = $ /year
Q = is 9.9% over EOQ, but TC is only 0.4% over optimal cost!!!

37. Model is Robust Q* = 350 TC = $3500/year Total Cost Carrying Cost
Order Cost

38. Model is Robust Changing Q by 20% increases cost by a few percent.
Carrying Cost
Order Cost
Total Cost

39. 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*.

40. 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”.

41. Inventory Level Suppose: p = 10/day (while producing this product).
Time
Produce Q
Do not produce
Slope=7
Slope=-3
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.

42. 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)

43. Inventory Level Inventory pattern repeats:
Maximum
inventory
Time
Produce Q
Do not produce
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.

44. Inventory Level TC = Setup cost + Inventory carrying cost
Time
Maximum
inventory
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

45. Maximum Inventory Level
Time
Maximum
inventory
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

46. Optimum Production Run Size: Q
Time
Maximum
inventory
Inventory
TC = Q D
S + IC 2
p-d p
Select Q to minimize total cost.
Set derivative of TC with respect to Q equal to zero.
2DS p
Q = IC
p-d

47. Non-instantaneous Resupply EquationsIC
2DS
p-d p
N = D/Q
TC = Q D
S + IC 2
p-d p
Length of a production run = Q/p
Length of time between runs = Q/d

48. 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 =
= units
0.2x6000
60-20
Q/p = /60 = 2.64 days
Q/d = /20 = 7.91 days
TC = 63, ,246 = $126,492/year
Every 7.91 days begin a 2.64 day production run.

49. 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, ,000 = $126,500/year
2.7 8
10.7 16
18.7 24
Production runs
Produce other products

50. 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 = $130,000/year
TC is only 2.8% over minimum TC! 10 20
Production runs
Produce other products

51. Scheduling Multiple Products
Suppose 3 products are produced on the same equipment. Optimal values are:
P1: Q/d = 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.

52. 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.
2.7 7
9.7
15.2
17.9
22.2 24 P1 P2 P3 P1 P2 P3
Idle

53. Reorder Point System - Variability
Order amount Q when inventory falls to level ROP.
If demand or lead time are larger than expected -> stockout

54. 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

55. 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

56. 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
s’d =
LT x sd2
= sd LT
s’d =
d2 x sLT2
= dsLT
s’d = 0

57. Total Cost TC = Order cost + Regular stock carrying cost
+ Safety stock carrying cost + Stockout cost
TC = Q D
S + IC 2
+ ICz s’d +
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)

58. 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. Dk
QIC
P = 1 -
(1)
2D[s + ks’dE(z)
Q =
(2) IC

59. 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

60. Reorder Point Example - Case 1
D = 5000 units/year  d = 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
Q =
= units
0.2x5
s’d = sd
= 14.14 LT
= 10 2

61. Case 1 (continued) - Find best P
316.23(0.2)5
P = 1 - =
5000(2)
z = 1.86 E(z) =
2(5000)[10 + 2(14.14)0.0123
Q = =
0.2(5)
321.68(0.2)5
P = 1 - =
5000(2)
z = 1.85 E(z) =
2(5000)[10 + 2(14.14)0.0126
Q = =
0.2(5)

62. Case 1 (continued) 321.81(0.2)5 P = 1 - = 0.9678 5000(2)
z = 1.85 E(z) =
z does not change, so STOP
Solution: Q = 322 z = E(z) =
ROP = d x LT + z x s’d = 96.15(2) (14.14) =
TC = = $347.97/year

63. Reorder Point Example - Case 2
D = 5000 units/year  d = 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
Q =
= units
s’d = (as in Case 1)
0.2x5
ROP = d x LT + z x s’d = 96.15(2) (14.14) =
TC = = $334.33/year

64. Reorder Point Example - Case 3
D = 5000 units/year  d = 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
Q =
= units
s’d = (as in Case 2)
0.2x5
ROP = d x LT + z x s’d = 96.15(2) (14.14) =
TC = = $355.58/year

65. Reorder Point Example - Case 3
k =$2/unit; P =90%
Solution:
Q =
ROP =
TC = $355.58/year
Could use k=$2/unit to find optimal P
It would be P = 96.78% as in Case 1!
Order size would be slightly larger (322 vs. 316).
Cost would be slightly less ($ vs. $355.58).

66. Reorder Point Example - Case 4
Suppose we keep no safety stock
Solution:
2x5000x10
Q =
= units
0.2x5
ROP = d x LT = 96.15(2) =
TC = = $494.73/year
With no safety stock there is a stockout whenever demand during lead time exceeds expected amount (dxLT).
Therefore: P = 0.5

67. Reorder Point Example - Summary
Case k P Q ROP TC($/year)
A small amount of safety stock can save a large amount!
Case 4 vs Case 3

68. 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

69. 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

70. Service Levels for Cases 1-4
14.14(.0126)
Case 1: SL = 1 -
Case 2: SL = 1 -
Case 3: SL = 1 -
Case 4: SL = 1 - =
322
14.14(.0475) =
316
14.14(.0475) =
316
14.14(.3989) =
316

71. Reorder Point Example - Summary
Case k P Q ROP TC($/yr) SL
Note difference between P and SL!

72. Out-of-Stock for Cases 1-4
Out-of-stock: 3 items per year and 0.5 orders/year
SL = > ( )x5000 = 3 items/year
P = > ( )x5000/322 = 0.5 orders/year
Case 2 & 3:
Out-of-stock: items per year and 1.58 orders/year
SL = > ( )x5000 = 10.5 items/year
P = > (1-.90)x5000/316 = 1.58 orders/year
Case 4:
Out-of-stock: 89 items per year and 7.9 orders/year
SL = > ( )x5000 = 89 items/year
P = > (1-.50)x5000/316 = 7.9 orders/year

73. Lead Time Variability in Example
D = 5000 units/year  d = 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( )
s’d =
LT x sd2 + d2 x sLT2 =

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 