# DISTRIBUTION INVENTORY SYSTEMS

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```					Distribution Inventory
Systems
Dr. Everette S. Gardner, Jr.
Competing interests in inventory
management
Controller:
Inventory investment

Marketing manager:                Operations manager:
Customer service                 Stock replenishment
Inventory                         2
Average inventory behavior with
stable demand
12
11
10                                     On hand
9
8
Stock

7
6
5                                                Avg. inv.
4
3
2                                                     ROP
1
0
0   4   8   12      16       20       24    28
Day
Inventory                              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              =   100 units
cost per order             =   \$10
holding cost per unit      =   \$5
unit price                 =   \$25
holding rate               =   .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
Avg. invest. =          Workload =
K       order qty./2            demand/order qty.
1       \$15.81                  31.62 orders
2        31.62                  15.8
3        47.43                  10.5
4        63.24                   7.9
5        79.04                   6.3
6        94.86                   5.3
6.32    100.00                   5.0
8       126.48                   4.0
10      158.10                   3.5
20      316.20                   1.6
Inventory                       15
Tradeoffs between investment and
\$

300

250           The optimal
Avg. investment

policy curve
200

150

100

50

0   5   10    15       20      25   30
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
Q\$ / 2 = (K/2) * (demand in \$)1/2      Σ Q\$ / 2 = K/2 Σ (demand in \$)1/2

Inventory                                  17
Optimal policies for multi-item
inventories (cont.)

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     K/2     Investment                1/K    Workload
1     0.5        \$125,000                1.0   250,000 orders
2     1.0         250,000                0.5   125,000
5     2.5         625,000                0.2    50,000
10    5.0       1,250,000                0.1    25,000

Inventory                           19
Multi-item example (cont.)
For K = 5:

avg. investment = Σ Q\$/2

Σ Q\$/2         = (K/2) Σ (demand in \$)1/2
= 2.5 * 250,000
= 625,000

workload       = 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
12                                     On hand
11
10
9
8
Stock

7                                                     Avg. inv.
6
5
4                                                     ROP
3
2                                                     SS
1
0
0   4   8   12      16       20      24     28
Day
Inventory                                   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         +   safety

standard deviation
Safety stock = safety factor * of forecast errors

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)       z          P(z)
0.00   0.50000     2.30      0.01072
0.10   0.46017     2.40      0.00820
0.20   0.42074     2.50      0.00621
0.30   0.38209     2.60      0.00466
0.40   0.34458     2.70      0.00347
0.50   0.30854     2.80      0.00256
0.60   0.27425     2.90      0.00187
0.70   0.24196     3.00      0.00135
0.80   0.21186     3.10      0.00097
0.90   0.18406     3.20      0.00069
1.00   0.15866     3.30      0.00048
1.10   0.13567     3.40      0.00034
1.20   0.11507     3.50      0.00023
1.30   0.09680     3.60      0.00016
1.40   0.08076     3.70      0.00011
1.50   0.06681     3.80      0.00007
1.60   0.05480     3.90      0.00005
1.70   0.04457     4.00      0.00003
1.80   0.03593     4.10      0.00002
1.90   0.02872     4.20      0.00001
2.00   0.02275     4.30      0.00001
2.10   0.01786     4.40      0.00001
2.20   0.01390     4.50      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
X
100            130

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.

Probability of                 Number of
SO    =   shortage on one    *           annual
order cycle                    order cycles

Inventory                         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)   =   Expected units backordered for a distribution
with mean = 0 and standard deviation = 1 on
one order cycle

E(Z)σ =    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
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

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
Old                    New
Safety stock                   1.5 months             1.0
months
Cycle stock                    1.0 months             1.5
months
Total                          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
(Total investment fixed at 2.5 months)
90%
New policy         Previous policy
Customer service

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

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                      Average      Investment
Sales           Growth            Q\$       Investment   Growth

100               ---             10       10/2 = 5        ---
200             100%              14       14/2 = 7       40%

Inventory                              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                    Shortage       Dollars
minimized                         occurrences    backordered

shortage occurrences              1,120          \$3.46 million
dollars backordered               2,812           1.48

Inventory                      46

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