Estimating And Auditing Aggregate Inventory Levels ... - Science Direct

Estimating And Auditing Aggregate Inventory Levels At Multiple Stocking Points Ronald H. Ballou* ABSTRACT

~,~~ati~?ely little attention has been given to methods for estimating and auditing distribution net\l.ork inventories in the aggregate. This article shou,s that a simple polynomial equation whose cotgficients can be determined by means of regression unajysis t~~ch~~i~~esis a good, basic tool for e~~t~mati~go~~erallin~~entory Iel’els and can be rrsed to provide insight into the effectiveness of inllcutory control policies at different stocking poirlts. It also serljes as the basis for uuditing and co,ltroliing current imtentory policies on an ongoirlg basis. Tests of the Rolodexin a ~lurnber of actrwl cases revealed a predict~~ble dissociation betu*een the inventory policy being used and the coefficient l’alues in the polynomial equation. When multiple product invento~es are to be positioned in a distribution channel, it is necessary to estimate how much stock should be held at each stocking point. This estimate is useful in the determination of system-wide inventory costs where that may be of interest or as part of a broad analysis where inventory costs are balanced against other costs such as production, transportation, and order processing to determine the optimal number, location, and size of the stocking points. This paper describes a polynomial model that has been useful in planning, analyzing, and auditing the inventory component of logistics systems, as well as in appraising the performance of in-place inventory stocking policies on a company-wide basis. It illustrates the insights to be gciined from a “top down” or aggregate representation of a firm’s execution of its inventory policy which serves as a basis for estimating the impact of inventory policy change as well as monitoring the effect of that change on distribu-

-YYa\e Western Reserve University. -The author wishes to acknowledge the assistance of Mr. Henry Blain. Chief Executive Officer. United Management Consultants in the preparation of this article. Journal of Operations

Management

tion network costs. It also shows the range of the relationship experienced in practice that serves as a guide for auditing as well as estimating inventory levels when little or no data are available. NATURE OF INVENTORY EFFECTS

CONSOLIDATION

Estimating aggregate inventory levels for products held at multiple sites is complicated by the fact that the amount of stock needed to support a particular level of demand and a given service level is usually not proportional to the number of stocking points. For example, if sales are routed through 112 the number of existing stocking points, the total inventory required may drop by 113of its previous level. This disproportionality is referred to as the “inventory consolidation effect” [I]. This effect encourages management to seek the smallest number of stocking points possible. A countervailing economic force to the consolidation effect is transportation cost. The relatively lower inbound rates to a stocking point, as compared to outbound rates, encourages the selection of a large number of stocking points close to major concentrations of demand. The decision concerning the number of stocking points and how much inventory should be placed at each location is a result of optimally balancing these costs. Determining transportation costs is relatively simple and accurate since rates are published for common carriers or, in the case of private carriage, they can be determined through straightforward accounting practice. The inventory consolidation effect is much less accurately determined since it is a result of established inventory replenishment rules, and the execution of these rules is highly individualized by company and by stockkeeping person. Therefore, inventory levels and costs must be determined on a company-to-company basis. A simple method for estimating overall inventory levels would be useful for planning and auditing purposes since it replaces the aIternative item-by-item estimating procedures that are impractical when there are hundreds of stockkeeping items held at many stocking points. 143

A SIMPLE RELATIONSHIP

N

Inventory level by stocking point, and for the distribution system as a whole, is a result of the execution of a stocking policy. Surveys have shown that over 90 percent of the U.S. firms now use computerized procedures for controlling inventory stocking levels, especially for finished goods [2]. Where demand is reasonably random, as is usually the case when it is generated from a large number of customers acting independently of each other and no one customer’s order dominates, statistical inventory control procedures apply. Most computer models such as the popular IBM IMPACT program are based on such procedures. With these procedures, it is possible to determine system-wide inventory levels through an item-by-item analysis. However, such an approach when there may be thousands of stock items located in many stocking points becomes impractical to execute. As an alternative, an estimating equation that is a derivative of basic inventory theory can be developed. As shown in the Appendix, such a relationship can have the following general form: I-, = $ (w+mD,+aDF) ,=I

(1)

Where: IT = System-wide inventory, units or $. Di = Annual warehouse throughput, units or $. N = Number of stocking points at which the products are held. w,m, a,b = Constants to be determined from company data usually by curve fitting procedures. The terms in the equation have the following general meaning: The average amount of promotional, speculative, obsolete, or production overrun stock at a stocking point. mDi= The amount of safety stock at stocking point i. aDib = The amount of regular stock at stocking point i.

W

titularly useful in practice since the broad-form model becomes a little trickier to curve tit and all terms usually are not statistically significant. Consider the nature of one of these reduced forms. T .eoretically, when inventory control is based on statistical inventory theory, the total inventory throughout a system of multiple warehouses can be determined from the following expression.

Regular stock Where: Di = SD= K= c= s= z=

No N m IT

= = = =

Safety stock

Period demand throughput at warehouse i. Standard deviation of demand. Carrying cost in percent. Average product value. Order processing cost. Number of standard deviations for a given service level. Initial number of stocking points. Revised number of warehouses. Average replenishment lead time. Total system inventory.

Therefore, the system inventory level is sensitive to the number of stocking points in the system and to the demand throughput at each stocking point. More simply, Equation 2 reduces to the form N

I, =

_

a c d/D, ,=I

=

Although any number of stock control policies may be encountered in practice, the model is sufficiently robust to accurately represent a wide variety ofinventory policies. Yet, by combining terms, it can be made quite simple with only a minor loss in accuracy. In fact, reducing the broad-form model to a form such 144

N

as I, = c aDp or 1, = c (w + mD,) has proved par,=I ,=I

when there is equal demand on each warehouse and all other factors remain constant. Thus, theoretically, we would expect total inventory to be a function of the number of stocking points, or the throughput levels at the stocking points, and be related to them in a square root fashion, assuming the same inventory policy for all items in the product line. For a number of reasons, we will find that the square root relationship does not strictly hold in practice, but it still is useful in showing the maximum consolidation effect that theoretically can be achieved. American

Production

and Inventory Control Society

OBSERVING

THE MODEL IN PRACTICE

The benefit that the simple model has is that it can be structured from existing data that are readily available within most firms. Inventory level information is typically maintained on a periodic basis, usually monthly, for each stocking point. The data may be aggregated as a total dollar value or on a product-by-product basis as given in stock status reports. Which data to use depends on how different product groups vary as to the way in which order quantities are determined and the service level set on each, and on the level of estimating accuracy expected from the model. An example of the use of aggregated data is shown in Figures 1 through 3. Individual item data appear later in Figure 4, but of course with somewhat increased variability compared with the aggregated data. Figure 1 shows the average inventory level for all products as a function of the total annual

Inventory

Consolidation

throughput for 23 warehouses out of approximately 100 in operation by a firm that produced and distributed industrial cleaning compounds for industrial and institutional use. If a power function is fitted to the data, the estimated average inventory level (Ii) in a warehouse having a throughput of Di is given by Ii = 2.986 Di0.63’. The system inventory is an extension of this formula and is given by

I, = i I, = 2.986 ,=I

i D,““7s 1=1

(4)

This line fits the data with a coefficient of determination of R2 = 0.82. Looking at the data in Figure 1, one might ask why a linear representation of the data (the second simplitied model that was previously suggested) might not offerreasonableaccuracy. Alinearrelationshipofthe

FIGURE 1 Curve for a Producer of industrial

Cleaning

Compounds

500 -

_

0

R2 = 0882

200

400

600

800

1000

1230

1400

1600

ANNUALWAREHOUSETHROUGHPUT,000s LB, Journal of Operations

Management

145

form IT = 5 (w+mDJ fits the data with acoefficient I=1 of determination of R2 = 0.85, about the same as the power function. However, if we believe the concave nature of the inventory function to be correct due to theory-based inventory policies and a predominance of cycle stocks, as was the case in this company, reducing 23 warehouses down to 1 will result in at least a 250 percent overstatement of the network inventory levels with the use of the linear expression. A second example involves a manufacturer of water treatment chemicals. In this case there were 10 stocking points at both plants and warehouses. The plot of the data is shown in Figure 2. The computed curve was I.1 = 44 * 4 ~.053* (3 I

Inventory-Demand

with a coefficient of determination of R* = 0.48, which was not particularly good. This was due mainly to inconsistent inventory policies at several plant stocking points. A third example, as shown in Figure 3, is for a producer and distributor of dry grocery products. In this case the curvilinear relationship of Ii = 0.311 Di”.68

offered the best tit with an R* = 0.83. The constant term was nearly zero and has been elimiHowever, the nated from the relationship. straightline relationship of Ii = 0.596 + 0.091 Di

500

(7)

with an R2 = 0.76 was selected. The reason was that the constant term in the equation better rep-

FIGURE 2 Data for a Water Treatment

- DALLAS ' PLANT

(61

Chemical

Producer.

0 - PHOENIX PLANT

1000

1500

ANNUALWAREHOUSETHROUGHPUT,000s LB, 146

American

Production

and Inventory Control Society

Two Inventory-Throughput

Relationships

FIGURE 3 for a Producer

and Distributor

of Dry Grocery

Products

O-1977 DATA n-1978 DATA

0

10

20

30

40

ANNUALTHROUGHPUT($000,000~) resented the high level of promotional merchandise and production overrun stocks typical in these food finished goods inventories than did the curvilinear relationship. That is, the merchandise was allocated to warehouses based on a long term estimate of national demand, warehouse space available, and customer service requirements such that inventory levels held little relationship to demand levels in the short run. Inventory policq was not tied closely to demand as demand actually materialized. Journal of Operations Management

Another example involved a steel products distributor operating eight warehouses. Forty-two (42) products were sampled within a common product group from the company’s stock status report. The results are shown in Figure 4. The curve in this case was 1..I zz 7.10 L>.o.“O .I

(f9

where Dj represents the demand for a particular item within a product group. The lR2 = 0.64 and 147

By-Item

Inventory-Demand

FIGURE 4 Data for a Steel Products

Distributor

l

MONTHLYITEM WAREHOUSETHROUGHPUT, 000s LB.

total inventory is the sum across all items in all warehouses. Other similar analyses, which are summarized in Table 1, have shown the exponent on demand for the curvilinear relationship usually to range between 0.65 and 0.75 with a mean of about 0.7. Thus, depending on the nature of the inventory policy used, we now have a guideline with which to evaluate a given company’s inventory control against its stated policy. That is, differences between the model fitted to actual inventorydemand data compared with the model as suggested by a company’s stated policy may raise questions about how effective inventories are being managed. Also, why the exponent on demand is approximately 0.7 rather than the 0.5 as theory suggests requires further discussion in the next section. 148

DEPARTURES

FROM Di0.j

The reasons for the exponent on demand being greater than the 0.5, as would theoretically be expected when cycle stocks are dominant in the inventory, are important to note as the model provides insights into the operation of the inventory control system. Since observed exponents are generally greater than 0.5, somewhat more inventory is being held after consolidation of stocking points than would be anticipated based on theory alone. There are several practical reasons for the exponent to be greater than 0.5 1. A number of the items in the product group may be joint ordered causing increased inventory levels so as to realize the benefits of volume buying or shipping. American

Production and Inventory Control Society

TABLE 1 Selected Examples of Aggregate Inventory Relationships and Range of Exponent “b” for a Variety of Companies Fit to

Product description

Estimated relationship”

Inventory policy description

data,

Over the counter drugs

I-r=c

Stock to 6 weeks demand

0.99

1

Mixed pushpull strategy

NA

0.60

0.96

1

(0.53+0.1lD,)

Vacuum ).r x45.6

cleaners Dry groceries

D,“.“”

c

(12.8+0.09D,)

Ir= c

Varies by stocking point and by inventory controller IT =5.5 1

D,“.“’

1.r=21 1

D,“.“X

Outboard beat

motors”

Outboard boat motors”

1.r= 2

Marine parts & accessories

l.r ~20

Medical gloves

Medical needles syringes

I.r ~0.6

(85,230+0.3[3,)

1

D,“-7’

1

& l.l_ = 16.9

Small appliance service parts

l_r ~37.8

Shoes

,.,=0.5x

l-J,“-!”

c

D,“.”

1

D,” 37

D,

Recreational I., =O.l

1

High,

but

only 4 points in data base

0.84

MIN-MAX control with some pushing of production overruns into field warehousing

0.50

0.68

MIN-MAX control with substantial volume buying pushed into field warehousing

0.95

1

Pull strategy with MIN-MAX control

0.91

0.74

Predominantly push strategy due to large production runs.

0.64

0.91

Pull strategy based on weeks of demand with some push overrides

0.88

0.67

NA

NA

0.57

Fit to policy

1

Stock Stock

goods

Exponent “b”

Order quantities set to 1 week forecast plus a fixed safety stock.

Replacement parts for large equipment

R2

to demand to demand

D,

Fit to policy

1

parts

1.rT26.9 1

D,“.“”

Mixed push-pull strategy

A

0.60

Candy products

Ir =98.9x

D,“.‘”

Unknown

NA

0.78

I, =67.7x

D,““’

Unknown

NA

0.65

Pull strategy based on IBM’s IMPACT program.

0.96

0.62

Replacement

Drugs & sundries

I, =0.7x

D,““’

“A relationship expressing the execution of the firm’s Inventory policy. “Competing divisions of same parent company except second division ’ Statement of the firm’s actual inventory policy. Source, Various distribution planning studies.

Journal of Operations

Management

makes

substantial

buys from

foreign

manufacturers

149

2. Forward buying, or promotional stocking, or dead stocking or over production to current requirements may be occurring in anticipation of future market needs. 3. Some modified control policy different from an optimal “pull” policy for single items may be in effect. 4. A “push” distribution system strategy may be used where production considerations override replenishment stocking policies. 5. There may be multiple product types with different service policies represented within the aggregated product group. 6. There may be departures from the assumptions on which the theory is based such as equal item service levels and costs. 7. An inventory policy may be followed that relates inventory levels directly to demand. There may be poor inventory management. 8. USES OF THE I., = 2: (w + mDi +aDr) I I

MODEL

There are two primary uses that the model has in the strategic planning of distribution systems. First, it permits the current inventory policies to be audited. Second, it provides a method by which overall inventory levels may be estimated for a varying number of stocking points. As an Audit Tool

A common question raised by management during an audit of operations is whether inventory levels are being controlled as effectively as they might be. Although item sampling could be conducted and relevant cost data collected to estimate inventory levels by means of economic order quantity formulations, this is expensive and time consuming when only initial impressions are desired. Thus, the power function model can provide the necessary insights that can lead to more in-depth analysis, if appropriate. First, the model provides information about the nature of the control process. The value of the exponent on demand of the power term in the model is the key. If cycle stocks are scientifically managed using “pull” inventory theory, we would expect the exponent to approach 0.5. However, as the exponent approaches 1.O, inventory stocking rules that tie inventory levels directly to the demand level would be expected. Exponents outside the range of 0.5 to 1.O usually indicate inventory management that is out of control. For example, Figure 1 shows the nature of the model to be expected from reasonably man150

aged inventory with little promotional stock or excessive safety stock. Approximately 750 product line items located in about 100 public warehouses were controlled by means of a computerized MIN-MAX inventory program. Even though product items were aggregated into one group, the exponent is respectably close to 0.5 and the R*=0.82 is reasonably high. Contrast this with the situation in Figure 4. The exponent of 0.90 departs significantly from 0.5. The level of the exponent should raise questions about the nature of the inventory control policy. In fact, the policy was one of maintaining inventories at a level equal to four-months’ demand. Since the exponent is close to 1.0, the model is reflecting the linear nature of this policy. Second, the scatter of the data about the estimating line provides important insights as to the effectiveness of the firm’s inventory control policies. Again, the companies represented in Figures 1 and 3 can serve as benchmarks. We expect something less than a perfect fit of the model to the data due to normal variability in the execution of inventory policies. However, the case does show the level of the accuracy that can be achieved in reflecting actual inventory policy. A sharp contrast is seen between the fit of the line in Figure 2 compared with Figures 1 and 3. Only 48 percent of the variation in the inventory level in Figure 2 is explained by the model. This is not particularly impressive. When management was asked about those stocking points that departed significantly from the line, the response was that the points represented inventories maintained at the plant sites. In those cases, the plant managers controlled their own inventory levels. Differences in attitudes about producing to order or producing to inventory explained the high variability. The company is in the process of centralizing and automating the inventory control throughout its production-distribution system. The potential inventory reduction, if the variability in inventory control practice between stocking points can be eliminated, is estimated to be 17 percent of the current average inventory level. As an Estimating

Tool

The use of the model as a tool for estimating the overall inventory level is important to strategic planning. Because of the substantial investment often required in inventories and annual carrying costs typically ranging from 20 to 40 percent of the inventory value [3], top management must continually be concerned with how average inventory levels change with changes in the number American

Production and Inventory Control Society

of stocking points and the demand assigned to them. Once the model has been established for the existing inventory policy and number of stocking points, it is a relatively straightforward matter to estimate inventory levels at individual stocking points or for the entire network. It is necessary to forecast or otherwise determine the throughput at each stocking point. The substitution of the appropriate Di value for each stocking point i into the formula gives the estimated inventory level at that point. Accounting for all demand in this manner gives the overall inventory level. The number of stocking points and, therefore, the demand on each point may be changed and the inventory levels estimated again in this manner to accomplish a sensitivity analysis. The usefulness of the model does not end here. In addition, management often wishes to know the minimum inventory levels that could be achieved under a well-managed control system. After adjusting the exponent downward to the theoretically lower bound value of0.5 and assuming the power function reduced form of the polynomial model, to make an estimate of the inventory levels at all stocking points, it is necessary toapproximate the constant in the formula, i.e., the “a” value in I, = a c D,“’ for each item sampled. A number of “a” values will be found which can then be averaged. To illustrate, consider a sampling of the points presentedinFigure4sincetheyareforanumberofitems. First, we know that S = $.Yorder, K = 20 percent, N,, ==8, N = 8, LT = 1.0 month, and Z = 1.65 for normally distributed demand at all stocking points with a 95 percent service level. There is an insignificant amount of promotional stocking, over buying, or production overruns. Next, the inventory level for the item is determined theoretically when a given number of stocking points is assumed. Table 2 shows the calculations. The estimating formula for the theoretically optimum policy would be IT = 12.8

i

,=~,

D,‘ji.

Thus, for an item or group of items having a monthly demand of 1000 lb. in each of 5 warehouses, the total inventory would be 2023 lb. According to the formula in Figure 4, the firm is estimated to be now stocking 17,800 lb. of steel products in this particular product group. a Control Tool In addition, the model can be used as an evaluation tool on a continuing basis to monitor invenAs

Journal of Operations

Management

tory control performance. Actual or idealized aggregate inventory relationships may be established to represent desired inventory policy. Since most companies report inventory status by stocking location on a regular basis, it is a simple matter to track the execution of the inventory policy. Should the actual and the desired inventory relationships diverge beyond acceptable limits, corrective action would be indicated.

SUMMARY

The model fills a need in inventory theory that traditionally has focused on item-by-item models rather than on aggregate models for overall network inventory analysis. Distribution and production planning often require an assessment of the impact that changes in warehouse configuration and the number of Iocations have on inventory levels. Also, changes in inventory stocking policies need to be evaluated broadly when considered in light of their impact on transportation mode choices, production schedules, and the like. It has been the intent of this article to show that a polynomial equation of the form IT = i (w+ mDi+aDi) I_ I can be an effective tool for estimating the level of inventories on a by-product group basis for a varying number of stocking points in a distribution system. Perhaps more importantly, the research has shown that for companies using a pull-type inventory control system with control procedures based on statistical inventory control theory the formula will be a power function with an exponent ranging between 0.65 and 0.75. Deviations from this are indications of poorly executed policies, of policies tied more directly to demand such that economies of scale are not present, or of a push strategy, volume buying, or production overruns that dominate inventory management. Also, we would expect this exponent to be no lower than 0.5 nor higher than 1.0. Thus, knowing the nature of a company’s inventory policy, an estimating equation can be fashioned. In addition, this simple formulation serves as a basis for auditing and controlling inventories on an aggregate level. The data for the model are readily obtained from a company’s normal operating data such as from periodic stock status reports. Only information on inventory levels and stocking point throughput (demand) is required, and a knowledge of the inventory policy being implemented. 151

Approximating

Product

C,

the Constant

TABLE 2 in the Inventory

SC,

1

.lO

25

2

.12

186

Level Estimating

Formula

(1)

(2)

(3)=(2)+(l)

I

6

a

,,,"."

D, 2312:1

578.83"

48.08

12.0

16586

128.79

1621.32

12.6

3

.09

13

1192

34.52

428.34

12.4

4

.08

44

2271

47.66

668.28

14.0

5

.lO

111

12435

1429.90

111.51

Simple

12.8

average

a = 12.8

bimonthlydemand

+ 1.65(25)&

llExample:l,=

= 578.83 from Equation 2

The model form is quite simple but surprisingly effective. It is sure to be incorporated into many future auditing and strategic planning analyses. It already has had repeated application in warehouse location studies. REFERENCES I. Baumol.

William J. and Philip Wolfe. “A Warehouse-Location Problem,” Opermtions Re.seard. Vol. 6 (March-April, 1958). pp. 252-63. and Robert G. Brown. Dec,ision Rules fhr In~~rntoq

Derivation

Monrrgr,?r~nf (New York: Holt. Rinehart and Winston, 1967). p. 353. House, Robert G. and George C. Jackson. “Trends in Computer Applications in Transportation Distribution Management.” Intermtionul Joumcrl of Physicd Disfrihrrfion. Vol. 7, No, 3 ( 1977). pp. l76- 187. LaLonde, Bernard J. and Douglas M. Lambert. “A Methodology for Determining Inventory Carrying Costs: Two Case Studies” (Supplement in Proceedings of the Fifth Annual Transportation and Logistics Educators’ Conference. Edited by James F. Robeson and John R. Grabner. Chicago. IL. October 12. 1975).

APPENDIX of an Aggregate Inventory

The aggregate, polynomial inventory estimating equation has its theoretical underpinnings in inventory theory. Consider how it is developed.

Estimating

the product class, the total items in a warehouse is:

Regular stock is that amount of inventory to meet average demand over the period of time from one stock replenishment to the next. Based on the Wilson EOQ formulation, regular stock in a warehouse for the jth item is, on the average

(RS), =

Cj =

for n

(A.3)

(A. 1)

Average regular stock for item j, units. Annual demand for item j, units/year. Procurement cost, $/order. Inventory carrying cost, 5% of item value per year. Item value for item j, $/unit.

Assuming all items in a warehouse have the same K and S, and C represents the average value for 152

stock

If Di is the total demand (throughput) on warehouse i and if there is not a great deal of difference in the levels of dj as can be the case for products of a particular product class,’ then

where:

dj = S= K=

regular

2 (RS), =

Regular Stock

(RS)j =

Equation

=J

Sn __

2KC

is a reasonable all warehouses

-

‘v/D,

(A.3)

approximation to Equation A.2. If contain roughly the same number

‘A product class might be all high volume items or conversely all low volume items. The number of warehouses would likely be different for each. American

Production and Inventory Control Society

of items for a particular product class, the total regular stock in a warehouse can be approximated as (A.4j

IRsi = a fi

Since various inventory policies can prevail besides the Wilson EOQ and the noted assumptions may not strictly hold in specific circumstances, the above equation might more generally be represented as IRsi = aDih

(A.5)

More simply, by

the total safety

stock is represented

I 5s = mD,

(A. 10)

Note that if g is less than 1, the effect of safety stock will become incorporated in the regular stock term. Other stocks Additions may be made to inventories in the form of speculative stock, warehouse start-up stock, and production overruns. These stocks can be represented by

Safely Stock under a reorder Safety stock determination, system, requires point type of inventory control _ ._ . . estimating the standard devlatlon of the demand during lead time distribution. This has been shown by Brown” to be (A.6) when both lead time and demand distributed and independent; where S Il1’1l.T

Standard units. Average Standard years. Standard

=

_1-T = SLT = S,, =

deviation

are normally

lead time,

lead time, years. deviation of lead

time,

deviation

units.

of demand,

= ,-$ Z, [m(hdfY

2 (SS), = Z

to Appendix warehouse is the sum of stocks, and other stocks. Ii in warehouse i is

+ regular stock

(A. 12)

(mh

(A.7)

+ dfSt,]“5

+ S:.,)‘)s D,

G.

Brown,

Smoothing.

F~orecustinl:

Time

.Seric,\

(Englewood

Cliff\.

19631 pp. 366-367.

Management

NJ:

of warehouses

N, the total

has been rea-

(A.8)

(A.9)

--

Journal of Operations

Conclusion The total stock in a regular stocks, safety Hence, the total stock

For the entire network system inventory is

Further, if g = 1 and Zj is the same for all productrs in the product class, and factoring out dj,

‘Robert

stock for item j in a ware-

= w + mD, + aDh,

where g has been observed to range from 0.6 to 1.0. If Zi is the number of standard deviations on a normal distribution curve representing the service level for item j, the safety stock for all items in a warehouse is

Di.tc r’fc

where Ti is the added house.

= w + Is,, + IRS,

S, = hdf

(SS),

(A. 11)

[=I

I, = other stock + safety stock

during

The standard deviation of demand sonably approximated by

2

w=iT,

cd

Predic,tion

Prentice-Hall.

of

Inc..

I, = i I, 171

(A. 13)

The constants w, m, a, and b can be determined empirically from warehouse throughput and inThe exact form of the ventory level data. simplified equation for the inventory in a single warehouse accounts for the fact that the Wilson EOQ model may not be used to set regular stock levels, the lead time distribution may not be strictly normally distributed, there may not be an identical number of items in each warehouse, the service level may not be exactly the same for all items in a warehouse even within the same product class, empirical determination of the constants may result in one or more of the terms being combined for the best estimation. The assumptions made in this derivation will not strictly hold in some specific situations. However, the theory does show that the polynomial equation is a reasonable general model to apply to empirical data. 153