Distribution Inventory
Systems
Dr. Everette S. Gardner, Jr.
Competing interests in inventory
management
Controller:
Inventory investment
Marketing manager:
Customer service
Operations manager:
Stock replenishment
workload
Inventory
2
Average inventory behavior with
stable demand
12
11
On hand
10
Stock
9
8
7
6
Avg. inv.
5
4
3
2
ROP
1
0
0
4
8
12
16
Day
Inventory
20
24
28
3
Average inventory behavior with stable
demand (cont.)
Demand = 1 unit per day
Leadtime = 2 days
Leadtime demand (LTD) = 2 units
Reorder point (ROP) = LTD = 2 units
Order quantity (Q) = 10 units
Maximum inventory = Q = 10 units
Avg. investment = Q/2 = 5 units
Inventory
4
The economic order quantity
Objective
Minimize total variable costs (TVC)
Ordering costs (OC)
OC = Cost per order x Nbr. of orders
Nbr. of orders = Demand/Order qty.
Holding costs (HC)
HC = Holding cost per unit per year x Avg. inv. balance
Avg. inv. balance = Order qty./2
Total variable costs
TVC = OC + HC
Inventory
5
The economic order quantity (cont.)
$
Total costs
Ordering costs
EOQ
Inventory
6
Economic order quantity (cont.)
EOQ in units of stock
QU = ((2 * demand in units * cost per order) / holding cost per unit per year)1/2
EOQ in dollars of stock
Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2
where the holding rate is a fraction of inventory value
Inventory
EOQ.xls
7
Economic order quantity (cont.)
Example
annual demand
cost per order
holding cost per unit
unit price
holding rate
=
=
=
=
=
100 units
$10
$5
$25
.20
QU = ((2 * 100 * 10) / 5)1/2
= 20 units
Q$ = ((2 * 25 * 100 * 10) / .20)1/2
= $500
Inventory
8
EOQ in perspective
• Ordering and holding costs should be marginal
(out of pocket) costs.
• Accounting systems generate average costs.
• In reality, ordering costs are semifixed.
Inventory
9
EOQ in perspective (cont.)
Assumption:
Reality:
Total
ordering
costs
Nbr. of orders
Nbr. of orders
• In reality, holding costs depend on executive judgments on
the cost of capital.
Inventory
10
Using the EOQ when costs are
unknown
Costs can be taken out of EOQ formulas and used as
policy variables to achieve management goals for
workload and average cycle stock investment.
Formula for EOQ in dollars
Q$ = ((2 * demand in dollars * cost per order) / holding rate)1/2
Inventory
11
Using the EOQ when costs are
unknown (cont.)
Remove all constants and unknowns
K = ((2 * cost per order) / holding rate)1/2
K is called the EOQ constant
Simplified EOQ formula
Q$ = K (demand in $)1/2
Inventory
12
Calculations with the EOQ cost
constant
Let unit price = $10, annual demand = 100 units
Low investment, high workload policy
K=1
Q = K (demand in $)1/2
= 1 ($10 * 100)1/2 = $31.62
Avg. investment = order qty./2
= $31.62/2 = $15.81
Workload
= demand/order qty.
= $1000/$31.62 = 31.62 orders
Inventory
13
Calculations with the EOQ cost
constant (cont.)
High investment, low workload policy
K=6
Q = K (demand in $)1/2
= 6 ($10 * 100)1/2 = $189.74
Avg. investment = order qty./2
= $189.74 / 2 = $94.87
Workload
= demand/order qty.
= $1000 / $189.74 = 5.3 orders
Inventory
14
Tradeoffs between investment and
workload
K
1
2
3
4
5
6
6.32
8
10
20
Avg. invest. =
order qty./2
$15.81
31.62
47.43
63.24
79.04
94.86
100.00
126.48
158.10
316.20
Inventory
Workload =
demand/order qty.
31.62 orders
15.8
10.5
7.9
6.3
5.3
5.0
4.0
3.5
1.6
15
Tradeoffs between investment and
workload (cont.)
$
Avg. investment
300
The optimal
policy curve
250
200
150
100
50
0
5
10
15
20
25
30
Workload
Inventory
16
Optimal policies for multi-item
inventories
Given only the sum of square roots of demand in $, you can
compute aggregate workload and investment.
Read Σ as “the sum of”:
Investment formula
Single-item
Multi-item
Q$ = K (demand in $)1/2
Σ Q$ = Σ K (demand in $)1/2
Σ Q$ = K Σ (demand in $)1/2
Σ Q$ / 2 = K/2 Σ (demand in $)1/2
Q$ / 2 = (K/2) * (demand in $)1/2
Inventory
17
Optimal policies for multi-item
inventories (cont.)
Workload (F) formulas
Single-item
Multi-item
F = (demand in $) / Q$
Σ F = Σ (demand in $) / Q$
F = (1/K) * (demand in $)1/2
Σ F = 1/K Σ (demand in $)1/2
Inventory
18
Multi-item example
5,000 line-item inventory
Σ (demand in dollars)1/2 = $250,000
K
1
2
5
10
K/2
0.5
1.0
2.5
5.0
Investment
$125,000
250,000
625,000
1,250,000
1/K
1.0
0.5
0.2
0.1
Inventory
Workload
250,000 orders
125,000
50,000
25,000
19
Multi-item example (cont.)
For K = 5:
avg. investment = Σ Q$/2
Σ Q$/2
workload
= (K/2) Σ (demand in $)1/2
= 2.5 * 250,000
= 625,000
= 1/K Σ (demand in $)1/2
= 0.2 * $250,000
= 50,000 orders
Inventory
20
Achieving management goals for
investment
Goal = Σ Q$/2
Goal = (K/2) * Σ (demand in $)1/2
Solving for K yields:
K = (2 * goal) / Σ (demand in $)1/2
This value of K meets the investment goal exactly.
The workload for that K is:
Σ F = (1/K) * Σ (demand in $)1/2
= (Σ (demand in $)1/2)2 / (2 * goal)
Inventory
21
Average inventory behavior with
uncertain demand
On hand
12
11
10
Stock
9
8
Avg. inv.
7
6
5
4
ROP
3
2
SS
1
0
0
4
8
12
16
Day
Inventory
20
24
28
22
Average inventory behavior with
uncertain demand (cont.)
Demand = 1 unit per day
Leadtime = 2 days
Leadtime demand (LTD) = 2 units
Safety stock (SS) = 2 units
Reorder point (ROP) = LTD + SS = 4 units
Order quantity (Q) = 10 units
Maximum inventory = Q + SS = 12 units
Avg. investment = Q/2 + SS = 7 units
Inventory
23
Reorder point with uncertain
demand
Assumption
Length of leadtime is constant
Concepts
Reorder point = mean demand
during leadtime
+
safety
stock
standard deviation
Safety stock = safety factor * of forecast errors
during leadtime
Inventory
24
Reorder point with uncertain
demand (cont.)
The standard deviation is a measure of variability of the
forecast errors.
With a perfect forecast, the standard deviation is zero.
As forecast accuracy gets worse, the standard deviation gets
larger.
The larger the safety factor, the smaller the risk of running out
of stock.
Inventory
25
Safety factor and probability of shortage
z
P(z)
0.00
0.50000
0.10
0.46017
0.20
0.42074
0.30
0.38209
0.40
0.34458
0.50
0.30854
0.60
0.27425
0.700.24196
0.80
0.21186
0.90
0.18406
1.00
0.15866
1.10
0.13567
1.20
0.11507
1.30
0.09680
1.40
0.08076
1.50
0.06681
1.60
0.05480
1.700.04457
1.80
0.03593
1.90
0.02872
2.00
0.02275
2.10
0.01786
2.20
0.01390
z
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
4.00
4.10
4.20
4.30
4.40
4.50
P(z)
0.01072
0.00820
0.00621
0.00466
0.00347
0.00256
0.00187
0.00135
0.00097
0.00069
0.00048
0.00034
0.00023
0.00016
0.00011
0.00007
0.00005
0.00003
0.00002
0.00001
0.00001
0.00001
0.00000
Inventory
26
Probability of shortage on one order cycle
P(Z) = Probability demand will exceed Z standard deviations
of safety stock on one order cycle
Example:
Mean demand during LT = 100 units
Std. dev. = 20 units
Safety factor (Z) = 1.5
Reorder point = mean demand + Z (std. dev.)
during LT
= 100 + 1.5 (20)
= 130 units
From table, P(Z) = .06681
Inventory
ROP.xls
27
Probability of shortage on one order
cycle (cont.)
From table, P(Z) = .06681
.50
.43319
Z
0
1.5
100
130
X
Inventory
28
Number of annual shortage
occurrences (SO)
The probability of shortage on one order cycle is misleading
since the ordering rate can vary widely across the
inventory. A better measure of customer service is the
number of annual shortage occurrences.
SO
=
Probability of
shortage on one
order cycle
*
Inventory
Number of
annual
order cycles
29
Number of annual shortage
occurrences (SO) (cont.)
Example:
Safety factor = 1.5
Probability = .06681
Annual demand = 1000 units
Order quantity = 50 units
Number of annual order cycles = 1000/50 = 20
SO = .06681 * 20 = 1.34
Inventory
30
Units or dollars backordered as a
service measure
E(Z)
=
E(Z)σ =
Expected units backordered for a distribution
with mean = 0 and standard deviation = 1 on
one order cycle
Expected units backordered for a distribution
with mean = X and standard deviation = σ
on one order cycle
Inventory
31
Units or dollars backordered as a
service measure (cont.)
Example:
Annual demand = 1000 units
Order quantity = 50 units
X = 25 units
σ=5
Z = 1.2
From table, E(Z) = .05610
Reorder point = 25 + 5 (1.2) = 31
Units short per cycle = .05610 (5) = .2805
Annual order cycles = 1000/50 = 20
Units short per year = 20 (.2805) = 5.61
Inventory
32
Quiz #1: Computing customer
service measures
Given:
Annual demand = $2,000
Order quantity = $100
Mean demand during leadtime = $80
Standard deviation = $60
Suppose we want the probability of shortage on one order cycle to be
.09680. Compute the following:
Safety stock
Reorder point
Number of annual shortage occurrences
Dollars backordered during one order cycle
Dollars backordered per year
Average cycle stock investment
Average inventory investment
Inventory
33
Quiz #2: Computing customer
service measures
For the same data as the previous problem, what reorder
point will yield:
a. 3 shortage occurrences per year?
b. 5% of annual sales backordered?
Inventory
34
Inventory tradeoff curves
A variety of different workload and investment combinations
yield exactly the same customer service.
To develop a tradeoff curve for dollars backordered:
1. Compute and plot the optimal policy curve showing tradeoffs
between cycle stock investment and workload.
2. Select a percentage goal for dollars backordered.
3. For each workload, compute the safety stock needed to meet the
goal.
4. Add cycle stock to safety stock to obtain total investment.
Inventory
35
Inventory tradeoff curves (cont.)
“Isoservice curve” -- each
point yields the same
dollars backordered
Investment
$
Safety
stock
Optimal
policy or
cycle stock curve
Workload
Inventory
36
U.S. Navy application of tradeoff
curves
Inventory system
8 Naval supply centers
Average inventory statistics at each center
80,000 line items
$25 million investment
Budget constraint
Average investment limited to 2.5 months of
stock
Inventory
37
U.S. Navy application of tradeoff
curves (cont.)
Investment allocation strategies
Safety stock
months
Cycle stock
months
Total
Old
1.5 months
New
1.0
1.0 months
1.5
2.5
2.5
Results of new allocation
Reordering workload cut from 840,000 to 670,000 per year
$2 million annual savings in manpower
Inventory
38
Workload/service tradeoffs
(Total investment fixed at 2.5 months)
90%
Customer service
New policy
Previous policy
85%
80%
75%
0.0
0.5
1.0
1.5
2.0
Safety stock (months)
112
121
184
240
289
Workload (thousands of orders)
Inventory
39
Strategic problems in managing
distribution inventories
1. Controlling inventory growth as sales increase
2. Controlling inventory growth as new locations are
added
3. Push vs. pull decision rules
4. Continuous review of stock balances vs. periodic review
5. Choosing a customer service measure
Inventory
40
Inventory growth
Inventories should grow at slower rate than sales.
Why? Order quantities are proportional to the square root of sales.
Example:
One inventory item
K=1
Q$ = K (demand in $)1/2
Sales
Sales
Growth
Q$
Average
Investment
100
200
--100%
10
14
10/2 = 5
14/2 = 7
Inventory
Investment
Growth
--40%
41
Effects of adding inventory locations
Inventories must increase as new locations are added.
One reason is that forecasting is easier when customer demands are
consolidated. Thus forecast errors are smaller and less safety stock
is required.
Another reason stems from the EOQ:
One inventory item
Sales of $100
K=1
With one location:
Q$ = K (demand in $)1/2
Q$ = 1 (100)1/2 = 10
Average investment = 10/2 = 5
Inventory
42
Effects of adding inventory locations
(cont.)
With two locations:
Location 1: Q$ = 1 (50)1/2 = 7.07
Location 2: Q$ = 1 (50)1/2 = 7.07
Average investment = (7.07 + 7.07) / 2 = 7.07
Investment increase = 7.05 – 5 = 2.05
Percentage increase = 2.05 / 5 = 41%
Inventory
43
Continuous review vs. periodic review
systems
Continuous review
Check stock balance after each transaction
If stock on hand below reorder point, place new order in a fixed
amount
Periodic review
Check stock balance on a periodic schedule
If stock on hand below reorder point, place new order:
in a fixed amount, or
in a variable amount (maximum level – on hand)
Investment requirements
Periodic review always requires more investment than continuous
review to meet any customer service goal.
Inventory
44
Push vs. pull control systems
Push or centralized system
Central authority forecasts demand, sets stock levels,
and pushes stock to each location.
Pull or decentralized system
Each location forecasts its own demand and sets its own
stock levels.
Investment requirements
A pull system always requires more investment than a
push system to meet any customer service goal.
Inventory
45
Comparison of shortage values
Inventory statistics
5,790 line items
$45 million annual sales
Shortage values at investment constraint of $5 million
Shortage value
minimized
Shortage
occurrences
Dollars
backordered
shortage occurrences
dollars backordered
1,120
2,812
$3.46 million
1.48
Inventory
46