4 Distribution Inventories

8/13/2019 4 Distribution Inventories

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Inventory 13

Calculations with the EOQ costconstant

Let unit price = $10, annual demand = 100 units

Low investment, high workload policyK = 1Q = 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


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Inventory 14

Calculations with the EOQ costconstant (cont.)

High investment, low workload policyK = 6Q = 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


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Inventory 15

Tradeoffs between investment andworkload

Avg. invest. = Workload =K order qty./2 demand/order qty.1 $15.81 31.62 orders2 31.62 15.8

3 47.43 10.54 63.24 7.95 79.04 6.36 94.86 5.3

6.32 100.00 5.08 126.48 4.010 158.10 3.520 316.20 1.6


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Inventory 16

Tradeoffs between investment andworkload (cont.)

$

300

250 The optimalpolicy curve

200

150

100

50

0 5 10 15 20 25 30

Workload

A v g .

i n v e s

t m e n

t


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Inventory 17

Optimal policies for multi-iteminventories

Given only the sum of square roots of demand in $, you cancompute aggregate workload and investment.

Read as the sum of:

Investment formula

Single-item Multi-item

Q$ = K (demand in $) 1/2 Q$ = K (demand in $) 1/2 Q$ = K (demand in $) 1/2

Q$ / 2 = (K/2) * (demand in $) 1/2 Q$ / 2 = K/2 (demand in $) 1/2


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Inventory 18

Optimal policies for multi-iteminventories (cont.)

Workload (F) formulas

Single-item Multi-item

F = (demand in $) / Q $ F = (demand in $) / Q $

F = (1/K) * (demand in $) 1/2 F = 1/K (demand in $) 1/2


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Inventory 19

Multi-item example

5,000 line-item inventory

(demand in dollars) 1/2 = $250,000

K K/2 Investment 1/K Workload1 0.5 $125,000 1.0 250,000 orders2 1.0 250,000 0.5 125,0005 2.5 625,000 0.2 50,00010 5.0 1,250,000 0.1 25,000


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Inventory 20

Multi-item example (cont.)

For K = 5:

avg. investment = Q$ /2

Q$ /2 = (K/2)

(demand in $)

1/2

= 2.5 * 250,000= 625,000

workload = 1/K (demand in $) 1/2

= 0.2 * $250,000= 50,000 orders


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Inventory 21

Achieving management goals forinvestment

Goal = Q $ /2

Goal = (K/2) * (demand in $) 1/2

Solving for K yields:

K = (2 * goal) / (demand in $) 1/2

This value of K meets the investment goal exactly.

The workload for that K is: F = (1/K) * (demand in $) 1/2

= ( (demand in $) 1/2 )2 / (2 * goal)


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Inventory 22

Average inventory behavior withuncertain demand

S t o c k

Day12 160 4 8 20 24 28

On hand

Avg. inv.

ROP

SS

01

23

45

6

789

10

11

12


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Inventory 23

Average inventory behavior withuncertain demand (cont.)Demand = 1 unit per dayLeadtime = 2 daysLeadtime demand (LTD) = 2 units

Safety stock (SS) = 2 unitsReorder point (ROP) = LTD + SS = 4 unitsOrder quantity (Q) = 10 unitsMaximum inventory = Q + SS = 12 units

Avg. investment = Q/2 + SS = 7 units


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Inventory 24

Reorder point with uncertaindemand

AssumptionLength of leadtime is constant

ConceptsReorder point = mean demand + safety

during leadtime stock

standard deviation

Safety stock = safety factor * of forecast errorsduring leadtime


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Inventory 25

Reorder point with uncertaindemand (cont.)

The standard deviation is a measure of variability of theforecast errors.

With a perfect forecast, the standard deviation is zero.

As forecast accuracy gets worse, the standard deviation getslarger.

The larger the safety factor, the smaller the risk of running outof stock.


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Inventory 26

Safety factor and probability of shortagez P(z) z P(z)

0.00 0.50000 2.30 0.010720.10 0.46017 2.40 0.008200.20 0.42074 2.50 0.006210.30 0.38209 2.60 0.004660.40 0.34458 2.70 0.003470.50 0.30854 2.80 0.002560.60 0.27425 2.90 0.001870.70 0.24196 3.00 0.001350.80 0.21186 3.10 0.000970.90 0.18406 3.20 0.000691.00 0.15866 3.30 0.000481.10 0.13567 3.40 0.000341.20 0.11507 3.50 0.000231.30 0.09680 3.60 0.000161.40 0.08076 3.70 0.000111.50 0.06681 3.80 0.000071.60 0.05480 3.90 0.00005

1.70 0.04457 4.00 0.000031.80 0.03593 4.10 0.000021.90 0.02872 4.20 0.000012.00 0.02275 4.30 0.000012.10 0.01786 4.40 0.000012.20 0.01390 4.50 0.00000


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Inventory 27

Probability of shortage on one order cycle

P(Z) = Probability demand will exceed Z standard deviationsof safety stock on one order cycle

Example:

Mean demand during LT = 100 unitsStd. dev. = 20 unitsSafety factor (Z) = 1.5Reorder point = mean demand + Z (std. dev.)

during LT= 100 + 1.5 (20)= 130 units

From table, P(Z) = .06681ROP.xls


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Inventory 28

Probability of shortage on one ordercycle (cont.)

From table, P(Z) = .06681

.50 .43319

Z0 1.5

X100 130


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Inventory 29

Number of annual shortageoccurrences (SO)

The probability of shortage on one order cycle is misleadingsince the ordering rate can vary widely across theinventory. A better measure of customer service is thenumber of annual shortage occurrences.

Probability of Number ofSO = shortage on one * annual

order cycle order cycles


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Inventory 30

Number of annual shortageoccurrences (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


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Inventory 31

Units or dollars backordered as aservice measure

E(Z) = Expected units backordered for a distributionwith mean = 0 and standard deviation = 1 onone order cycle

E(Z) = Expected units backordered for a distributionwith mean = X and standard deviation = on one order cycle


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Inventory 32

Units or dollars backordered as aservice measure (cont.)

Example:

Annual demand = 1000 unitsOrder quantity = 50 unitsX = 25 units = 5Z = 1.2From 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


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Inventory 33

Quiz #1: Computing customerservice measures

Given: Annual demand = $2,000Order quantity = $100Mean demand during leadtime = $80Standard deviation = $60

Suppose we want the probability of shortage on one order cycle to be.09680. Compute the following:

Safety stockReorder point

Number of annual shortage occurrencesDollars backordered during one order cycleDollars backordered per year

Average cycle stock investment Average inventory investment


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Inventory 34

Quiz #2: Computing customerservice measures

For the same data as the previous problem, what reorderpoint will yield:

a. 3 shortage occurrences per year?

b. 5% of annual sales backordered?


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Inventory 35

Inventory tradeoff curves

A variety of different workload and investment combinationsyield exactly the same customer service.

To develop a tradeoff curve for dollars backordered:

1. Compute and plot the optimal policy curve showing tradeoffsbetween cycle stock investment and workload.

2. Select a percentage goal for dollars backordered.

3. For each workload, compute the safety stock needed to meet thegoal.

4. Add cycle stock to safety stock to obtain total investment.


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Inventory 36

Inventory tradeoff curves (cont.)

$ Isoservice curve -- eachpoint yields the same dollars backordered

Safetystock

Optimalpolicy orcycle stock curve

Workload

I n v e s t m e n

t


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Inventory 37

U.S. Navy application of tradeoffcurves

Inventory system8 Naval supply centers

Average inventory statistics at each center80,000 line items$25 million investment

Budget constraint Average investment limited to 2.5 months ofstock


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Inventory 38

U.S. Navy application of tradeoffcurves (cont.)

Investment allocation strategies

Old NewSafety stock 1.5 months 1.0

monthsCycle stock 1.0 months 1.5months

Total 2.5 2.5

Results of new allocationReordering workload cut from 840,000 to 670,000 per year

$2 million annual savings in manpower


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Inventory 39

Workload/service tradeoffs

C u s t o m e r s e r v

i c e

New policy Previous policy

90%

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)

(Total investment fixed at 2.5 months)


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Inventory 40

Strategic problems in managingdistribution inventories

1. Controlling inventory growth as sales increase

2. Controlling inventory growth as new locations areadded

3. Push vs. pull decision rules

4. Continuous review of stock balances vs. periodic review

5. Choosing a customer service measure


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Inventory 41

Inventory growthInventories should grow at slower rate than sales.

Why? Order quantities are proportional to the square root of sales.

Example:One inventory itemK = 1Q$ = K (demand in $) 1/2

Sales Average Investment

Sales Growth Q$ Investment Growth100 --- 10 10/2 = 5 ---200 100% 14 14/2 = 7 40%


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Inventory 42

Effects of adding inventory locationsInventories must increase as new locations are added.

One reason is that forecasting is easier when customer demands areconsolidated. Thus forecast errors are smaller and less safety stockis required.

Another reason stems from the EOQ:One inventory itemSales of $100K = 1

With one location:Q$ = K (demand in $) 1/2 Q$ = 1 (100) 1/2 = 10

Average investment = 10/2 = 5


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Inventory 43

Effects of adding inventory locations(cont.)

With two locations:Location 1: Q $ = 1 (50) 1/2 = 7.07Location 2: Q $ = 1 (50) 1/2 = 7.07

Average investment = (7.07 + 7.07) / 2 = 7.07

Investment increase = 7.05 5 = 2.05Percentage increase = 2.05 / 5 = 41%


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Inventory 44

Continuous review vs. periodic reviewsystemsContinuous review

Check stock balance after each transaction

If stock on hand below reorder point, place new order in a fixedamount

Periodic reviewCheck 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 requirementsPeriodic review always requires more investment than continuousreview to meet any customer service goal.


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Inventory 45

Push vs. pull control systems

Push or centralized systemCentral authority forecasts demand, sets stock levels,and pushes stock to each location.

Pull or decentralized systemEach location forecasts its own demand and sets its ownstock levels.

Investment requirements A pull system always requires more investment than apush system to meet any customer service goal.


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Inventory 46

Comparison of shortage values

Inventory statistics5,790 line items$45 million annual sales

Shortage values at investment constraint of $5 millionShortage value Shortage Dollarsminimized occurrences backordered

shortage occurrences 1,120 $3.46 milliondollars backordered 2,812 1.48

4 Distribution Inventories

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