Naval Research Logistics (NRL) Volume 43 Issue 6 1996 [Doi 10.1002%2F%28sici%291520-6750%28199609%2943%3A6-839%3A%3Aaid-Nav4-3.0.Co%3B2-5] Narendra Agrawal; Stephen a. Smith -- Estimating

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Estimating Negative Binomial Demand for Retail Inventory

Management with Unobservable Lost Sa le s

Narendra Agrawal and Stephen A . Smith

Decision

and Information Sciences, Santa Clara University, San ta C lara,

Calijbrnia

95053

The importance of effective inventory management has greatly increased for many major

retailers because of more intense competition. Retail inventory management methods often

use assumptions and demand distributions that were developed for application areas other

than retailing. For example, it is often assumed that u n m e t demand is backordered and that

demand

is

Poisson or normally distributed. In retailing, unmet demand is often lost and

unobserved. Using sales data from a major retailing chain, our analysis found that the neg-

ative binomial

fit

significantly better than the Poisson or the normal distribution. A param-

eter estimation methodology that compensates for unobserved lost sales is developed for the

negative binomial distribution. The method's effectiveness is demonstrated by comparing

parameter estimates from the complete data set to estimates obtained by artificially trun-

cating the data to simulate lost sales.

996

John Wiley Sons. Inc.

1.

INTRODUCTION

Achieving

a

high level of customer service is one of the most important objectives for

firms that sell both durable and nondurable goods. Increased competition has made the

retail inventory management system a key strategic weapon for large retailers such as su-

permarkets and depar tment stores (see [ 6 ] ) . For high-priced durable goods such as furni-

ture or appliances, unmet demand is typically backordered and thus can be observed; how-

ever, for low-cost, nondurable merchandise, sales are typically lost and not reported when

the items are out of stock. Most inventory management methods d o not explicitly account

for lost sales in updating demand forecasts. This can lead to systematic understocking of

items that are in high demand. Industry surveys [141have revealed that for certain popular

items, retail in-stock positions are actually no more than 85 , despite the fact that the

retailers' targets are typically 95-9970. Fisher, Hammond, Obermeyer, and Raman [61 also

note the strategic importance of estimating lost sales.

Demand forecasting based on a parametric distribution requires statistical estimation of

its mean and variance o r other parameters. In the case of unobserved lost sales, the pa-

rameter estimates must be adjusted appropriately to account for the unobserved compo-

nent of demand. Most inventory methods use either the Poisson or the normal, which are

analytically convenient distributions for modeling the demand per period. In general, the

normal is preferred when the demand per cycle is relatively large, while the Poisson is

better for low-demand items because it is discrete. When all demand is observed, the actual

frequency distribution of historical data might also be used. However, when demand ob-

' A n

exception is catalog sales, for which unmet demands can be recorded.

Nuval Rcseurch

Logislics,Vol. 43, pp. 839-861

( 1

996)

Copyright 996 by John Wiley &Sons, Inc.

CCC 0894-069X/96/060839-23



servations are consistently truncated at the same value, nonparametric estimation methods

have no basis for estimating the demand distribution beyond the truncation point. This is

a crucial shortcoming of nonparametric methods because inventory stocking criteria rely

on the tail of the demand distribution.

The negative binomial distribution fits our data significantly better than e ither the Pois-

son or the normal, and is only slightly less manageable analytically than the Poisson. The

negative binomial also has the advantage of providing a single discrete distribution for all

SKUs (stock keeping units) with a wide range of demand rates, removing the need for two

separate distributions. Additionally, from a practical standpoint, the negative binomial is

an efficient (and analytically tractable) distribution to represent the high variability in de-

mand that occurs

i n

retailing environments due to weather. competitors' promotions, and

other random fluctuations. Consequently, using the negative binomial distribution for in-

ventory management decisions will lead to more reliable levels of customer service and

lower costs.

This article presents both theoretical and empirical support for the contention that the

negative binomial is an appropriate demand distribution for retail inventory management

applications and develops a parameter estimation methodology that compensates for the

effects of unobservable lost sales. The method's effectiveness is demonstrated by artificially

truncating sales data from a major retailing chain and comparing the resulting parameter

estimates to those obtained from t h e full data set. Sufficient conditions are provided for

convergence of this method.

The remainder of the article is organized as follows. The reasons for selecting the negative

binomial distribution are described in Section

2.

Section 3 contains a brief review of the

relevant literature. Analysis of the data used for this study is described

i n

Section 4. The

estimation methodology is presented in Section 5 . and theempirical validation of the mcth-

odology is i n Section 6. The details of the statistical analysis are in Appendix I and Iength-

ier proofs are deferred to Appendix

2 .

Conclusions are summarized in Section

7.

2. PROBABILITY DISTRIBUI'IONS FOR D E M A N D

The following notation will be used for our analysis:

I

p

cr

random demand per week at the SKU-store level

true mean of the demand distribution

true standard deviation

of

the demand distribution

Thc Poisson distribution ( 1 ) is often used to describe retail demand. because i t arises

from the assumption of independent random arrivals at a uniform rate:

with mean

=

variance = A.

When demand per period is large, the normal distribution ( 2

is

often used because it

approximates the Poisson well for large mean demand and because

it

allows independent

selection

of

t h e mean and variance. For the discrete case. the probability distribution of

demand can be approximated as follows:



Agrawal and Smith: Estimating Negative Binomial Demand

84 1

P{Demand = k }

x

@ ( k +

I y a)

@ ( k

- ;Iy,

a),

k =

0, 1 , 2 , . . .

( 2 )

where @ X I y,

a

= normal cumulative distribution with mean y and variance a*.

The normal distribution may fit low-demand items poorly, however, because it assigns

probability to negative values and because it must be symmetric about its mean. For the

retailing data analyzed in this article, the Poisson distribution did not fit low-demand data

well either, because its fixed variance to mean ratio of one is too small. There are some

practical reasons why actual demand may be more variable than the Poisson. Random

variations may occur in the underlying Poisson arrival rate due to the weather, competitors’

promotions, or special events that are not captured by the inventory system’s forecask2

Second, customers whose purchases are Poisson arrivals may introduce additional vari-

ability by purchasing multiple items of the same type.

The negative binomial distribution is capable of capturing either of these increased vari-

ability effects. The negative binomial distribution with parameters

N

and p has the follow-

ing discrete probability function:

fYD

=

k l N , P )

= h N , p )=

(”,’”; ) P ” ( l k

O < p < l ,

N > O ,

where the cumulative probability distribution is

N S j - 1

F , , ( N , P )=

c

N - ) P ” . ( l

- P Y .

J = O

The mean and variance are

. = I - l ) ,

k = 0 , I , . . . , (3)

( 4 )

The probability distribution function described in

( 3 )

can be interpreted as the proba-

bility of having exactly k failures before the Nth success with independent events where the

probability of success is p . Notice that the ratio of the variance to mean is 1 / p , which is

greater than

I

and can be arbitrarily large. This makes the negative binomial distribution

attractive for retailing applications, which tend to have high variability.

There are other ways to describe the genesis of the negative binomial distribution that

make it intuitively appealing to model the demand process in retailing applications. For

example,

(3)

is equivalent to the probability of

k

arrivals when customer arrivals occur

Several corporate membersof the SantaClara Universi ty Retail Workbench

w h o hold such

promo-

tions reported to us that forecasting slow sellers was a particular problem

for

their inventory man-

agement systems because of larger

than

expected variability in demand.



according

to a

Poisson process with a random arrival rate

A,

which is sampled from a

gamma distribution of the form [8, p. 1241

where,

E [ ] = N / x ,

Var[

A ]

= N i x 2 ,

and

p = x / ( l + x )

Alternatively, if customer purchases occur according to a Poisson distribution, and the

quantity of each purchase is described by a logarithmic series distribution, the resulting

demand distribution is also negative binomial [ 8 , p. 1251. Multiple items per purchase

could be expected to occur in a wide variety of retail merchandise. including apparel,

housewares, and grocery items. Retail data captured currently generally do not allow the

items per customer to

be

determined. However, the logarithmic distribution appears to be

at least a reasonable candidate distribution for purchase quantity. Boswell and Patil

[21

give

I 5

different derivations

of

the negative binomial distribution.

T hus ,

it can be seen

that this distribution has a

good

intuitive basis as a model for the demand process

in

re-

tailing. Finally, the negative binomial is superior from a practical standpoint because it

accurately describes both low-demand and high-demand items, eliminating the need for

two distributions.

3.

LITERATURE REVIEW

In this section, we briefly review some of the literature

on

distributions used to model

the demand process for inventory systems. We also highlight the approaches that have

been

used to estimate the parameters of these distributions.

Most inventory research

has

focused on systems where unmet demand is backordered,

in

part because the resulting models are simpler

to

analyze, and because many inventory

models have evolved from durable goods and military applications, where demand is back-

ordered.

A

survey of research on single- and multiechelon inventory systems that consider

lost sales can be found in Nahmias and Smith

[

12. 131. Estimation methods that include

lost sales are reviewed in Nahmias

[

1 I]. Most commonly used methods for unobserved

lost sales assume that the demand process follows either a Poisson or a normal distribution.

For example, Conrad [4 ] considers a single-period newsboy model

in

which the excess

demand is lost. Under the assumption of i.i.d. Poisson demand, a maximum-likelihood

estimator

for

the mean

of

the demand distribution

is

derived by observing actual

sales

in

a

certain number of time periods. Hill [7] develops procedures to derive the moments of

customer demand based on the data obtainable from point-of-sale scanning systems. The

customer arrival rate is assumed to be Poisson and the model was not tested empirically.

Wecker

[ 18)

considers the effect of stockouts on forecasting bias. Demand

in

any period

is assumed to be generated by an autoregressive process, where the error term has a normal



Agrawal and Smith: Est imat ing Negative Binomial Dem and

843

distribution with zero mean and a given standard deviation. An unbiased estimator for the

forecast is determined under the assumption that ( i ) exactly one stockout has occurred

i n

the most recent sales period, and (ii) exactly one stockout has occurred in some prior

period. When more than one stockout has occurred, this approach leads to cumbersome

algebra.

Sarhan and Greenberg

[

15, 161 develop estimators for the normal distribution for the

case in which the samples may be doubly censored, that is, when the r , smallest and r,

largest observations are unobserved. Braden and Freimer

[

31 characterize the class of dis-

tributions for which sufficient statistics exist when observations are truncated from the top.

These distributions are termed newsboy distributions because of their ease of application

in inventory systems, but the negative binomial does not have this property. Nahmias [ 1

11

develops an estimator for truncated demand data

in

the normal-demand case, which is a

simple algebraic function of the sample data. He also shows how the estimator can be

incorporated into sequential updating routines. A nonparametric estimation method for

truncated data was developed by Kaplan and Meier [

93,

and maximum-likelihood esti-

mates are obtained. However, when sales data are always truncated at the same threshold

(the base stock level), this method provides no information regarding the shape of the tail

of the distribution, which is needed for analyzing inventory stocking policies. Anraku and

Yanagimoto [ 11and Van De Ven [171 describe estimation methodologies for the parame-

ters

of

the negative binomial distribution, but do not consider truncated data.

In

the statistical literature, an iterative estimation algorithm, known as the EM method,

has been developed for obtaining maximum-likelihood estimates from data that has been

censored in any known manner [ 5 ] It has been noted, however, that this method can

become computationally cumbersome unless the maximum-likelihood estimates for the

uncensored case are easily computed

[101.

In the case of the negative binomial, maximum-

likelihood estimates in the uncensored case require an iterative solution of an infinite series

relationship [8, p. 1321. Johnson and Kotz

[

81 present alternative simple formulas for the

uncensored case, which we use as the basis for our method.

The approach in this article differs from the existing literature in several respects. It first

compares the effectiveness of three common distributions: normal, Poisson, and negative

binomial,

in

fitting actual sales data and finds that the negative binomial

is

clearly superior

for this application. A parameter estimation method is developed for the negative binomial

with unobserved lost sales, and is empirically validated with the use of the actual data. In

most cases, this method requires a single one-dimensional search. Because our method has

minimal data requirements and is relatively simple to solve, it is attractive for use in in-

ventory replenishment applications.

4. A N A LY Z IN G T H E S A M P L E D A T A

A

comparison of the negative binomial, the Poisson, and the normal distributions was

made using sales and inventory data from a major retailer. The data contained 5 2 weeks of

unit sales at each store for a particular type of men’s slacks at a major retailing chain.

Based on the various combinations of sizes and styles, there were 4 1 different SKU s for this

product (with average weekly sales per store ranging from 0.03 to 1.27) and 24 different

stores (with average weekly sales per SKU ranging from 0.03 to 2.04). These items were

stocked at very high levels relative to sales in that an ending stock level of zero occurred in

less than

0.0

1 of the samples. This made it possible to assume that reported sales are equal

to actual demand. This product was considered a basic item; that is, it is sold throughout



844

Naval R m w r h Logistics,

Vol. 43 ( 1996)

8

7

0

the year. Replenishment was done once per week. This retailer uses media advertising to

boost its overall sales level, but does not use periodic price promotions-thus the product

price was constant throughout the year.

In order to compare the goodness of fit for alternative frequency distributions, it is nec-

essary to obtain sample sizes large enough to estimate the probabilities in the tail of the

demand distribution. For most retail sales data this is not a straightforward task. Replen-

ishment cycle times of one week or less are common for many department-store chains

and apparel merchants. Average sales per week at the store-SKU level tend to be low, that

is, usually in the range of 1-5, and less than

0.1

for slow-moving SKUs (see Figure 1

).

Demand per period needs to be forecasted at the store and SKU level, because that is the

level at which base stock levels must be determined. Sales rates

for

each class of item tend

to fluctuate by season in predictable ways. However, this means that the sales data must

either be deseasonalized

or

partitioned into peak and off-peak seasonal times.

Although data for any one store-SKU combination are quite limited, there are abundant

data for similar SKUs and stores. That is, department-store chains have hundreds of stores

carrying the same merchandise, thus offering potential groupings of stores that have similar

sales volumes and seasonal patterns. Also, retailers have tens

or

hundreds

of

thousands of

different SKUs, providing many sets of SKUs that have similar demand patterns. Because

the data analyzed for this article had these characteristics, it was decided

to

group sales data

in order to obtain sufficient sample sizes for goodness-of-fit comparisons.

We partitioned the

5

1,168 combinations of stores, SKUs, and weeks of the season into

eight categories and then treated all samples in each category as multiple observations of

the same demand distribution. For inventory management, all SKU-store combinations

in the same category would use the same base stock level. The grouping methodology was

subjective, based on a graphical analysis of the data along the following two dimensions:




845

1.80

1.60

40

1.20

00

0

v

-

m

0.80

0.60

0.40

0.20

0.00

Pea..

Weeks rl

- m = ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ f i k F

u * u m

Week

of

Season

Figure 2. Weekly sales for average SKU-Store.

1 .

Various SKU-store combinat ions were ranked into the top 30, the next

90,

the

next

270,

and the lowest

594

by weekly sales averages (Figure

1 ) .

2.

The weeks of the year were separated into two categories: Peak ( top

1

I

weeks)

or off-peak (next 4 1 weeks) based on total sales volume per week (Figure 2 ) .

Because of the highly skewed distribution of demand per week, the four categories based

on weekly sales volume in Figure 1 were spaced logarithmically. That is, the number of

SKU-store combinations in each pair of adjacent groups has an approximate ratio ofthree,

which is the best integer choice for achieving four logarithmically spaced groups. This spac-

ing has the effect of dividing the SKU-store categories in such a way that each category has

approximately an equal fraction o fthe total weekly demand. This means that each category

would have approximately equal financial significance.

We found that the

1

I

peak weeks of the year, which were the back-to-school and Christ-

mas seasons, had a significantly higher mean than the remaining weeks of the year. These

two groups were selected as the peak and off-peak groupings. The peak weeks fell into two

contiguous periods, one in August and a second in November and December (see Figure

2 ) .

For convenience, the same SKU-store groupings were used for both the peak and off-

peak weeks.

The characteristics of the eight categories are summarized in Table

1.

Th e sample size is

determined by multiplying the number of SKU-store combinations in each category by the

number ofweeks. These categories passed a goodness-of-fit test using the negative binomial

distribution, but failed when the normal and Poisson distributions were used. A detailed

discussion of the goodness-of-fit testing is presented in Appendix I .

Figure 3 compares the estimated distributions obtained with the negative binomial, Pois-



846

Naval

Rcscarc~lr

Logislicx. Vol. 43 ( 1996)

Table I

Data groupings based on SKU-store and peak/off-peak combinations.

Peak Off Peak

No.

of

No. of

30 Mean

=

5.7 330 Mean

=

2.4 1.230

90 Mean = 2.5 990 Mean

=

1.19

3,690

2

70 Mean = 1.16

2,970 Mean

=

0.51 1 1,070

594 Mean = 0.28 6,534 Mean

=

0. I2 24,354

SKU-store Statistics observations Statistics observations

Var = 13.4

Var = 4.8

Var = 1.60

Var

=

0.35

Var = 4.6

Var = 1.61

Var = 0.60

Var

=

0.13

son, and normal demand assumptions to the actual data for the high demand, peak case.

The figure illustrates why the fit obtained with the negative binomial distribution is sub-

stantially better-the Poisson tends to understate the demand in the right tail of the distri-

bution, and the mode of the normal is shifted to the right.

The nature of these categories allows the implementation of peak and off-peak stock

levels for each given store and S K U combination because there are only two transitions

between peak and off-peak periods. This is consistent with practices observed at this re-

tailer. as well as at a vendor of men's slacks with whom we have discussed this research.

0.18 -

I .

-

0.16

0.14

0.1

2

0

,s 0.08

-

d

0 06

0.04

0 02

~

, egat ive B inomia l

-Data

Po i sson

- _

0

0

2 4

6 8

10 12

14

16

18 20

Sales Per

Week

Figure 3. Comparing distribution models to data (high, peak, mean

=

5.7, Var = 13.4).



Agrawal

and

Smith: Estimating

Negative Binomial

Demand

847

Both use two separate inventory plans, one for the peak period beginning in August and

extending through Christmas, and the second for the remaining off-peak period.

For retailers that are promotional, for example, one week out of each month has lower

prices and increased advertising, the peak and off-peak weeks alternate. This leads to a

more complex inventory decision problem, because higher inventory stocked in prepara-

tion for a peak week may remain unsold during the off-peak weeks.

To

our knowledge,

no specific optimization methodologies (other than dynamic programming) have been

developed for solving the alternating peak and off-peak inventory problem.

5. THE PARAMETER ESTIMATION METHODOLOGY

The following additional notation will be used for our analysis:

mean of the distribution with observed demand truncated at

s

sample mean

sample mean with demand truncated at

s

observed frequency of demand k ,

k =

0,

1 , . . .

s

1

observed frequency that demand is less than or equal to L

estimator for

N

estimator for p

base stock level

When all demands are observed, a common method for deriving the estimators Na n d b

is to match the sample mean Yto

p

and the sample frequency of

0

demand tofo(N,

p ) .

(See Johnson and Kotz [

8,

pp. 131- 1351). This gives the following two equations to solve

for

k

and i

:

Y = N i - 1

( 7 )

When the base stock level is s, all demands greater than swill be reported ass . However,

the observed frequencies for demand values smaller than s will be unbiased estimates of

the true probabilities, and the observed frequency that demand is less than or equal to s is

an unbiased estimate of

1 F , N ,p ) .

The mean

p , in

the truncated case can be expressed

as follows:

which involves o n l y frequency values less than

s

This last formula allows us

to

calculate

the observed sample mean for the truncated distribution with the use of o n l y the observed

frequencies, { , fyh ’ ) k = 0,

1,

. . . s I , which are unbiased estimators of the true fre-

quencies:



848

Nuvul

Rescurch

Logi.sIic.s.

Vol.

43 (

1996)

\ -

I

x = k s f p +

s

h=O

We obtain one relationship for the est imators6 and Nby matching the truncated sample

mean X o the mean p, ( 8 ,6) from (

9

) :

The other relationship for the estimators is obtained by matching the observed frequency

F:~hsor some L < s to the formula for Fl, N , p ) ; hat is,

Solving the Esiimuior fiqirutions.

The two relationships

(

12) and

(

13) for N and I;

cannot be solved in closed form. However, the following properties of the relationships

lead to a solution procedure that is quite efficient compared to a general two-dimensional

search. We first note that for each N , 1 3 ) has a unique solution p N ) defined by

where

p

N s increasing in

N .

This fact follows from Lemma

I .

LEMMA

1:

For every value of

N ,

and

L

<

s , there is

a

unique p =

p N )

defined by

F,

N , N ) )=

Fyha.

PROOF:

Recall that F h ( N ,p ) can be interpreted as the probability that k or fewer

failures are required to obtain N successes. Therefore, it follows that FA

, p )

s strictly

increasing in p and strictly decreasing in N . (Noninteger values for N are allowed as well.)

lf we take the total derivative of

(

14), it follows that

thus establishing monotonicity. QED

Using the relationship in

( 14) ,

we let functions with a single argument denote the sub-

stitution of

p N

or

p .

That is,

and

FA

N )

=

FA

N ,

?(N ) ) .




849

For any N , the pair

{

N ,

p ( N )

can be substituted into

(

1 2 ) to compute a value p s N )

defined by

The solution then follows from a one-dimensional search on N to find the value such

that p , ( N ) closely matches the observed value X, . or L = 0, the uniqueness of the solution

to p5 N )

= X,

is guaranteed by Theorem I which is proved in Appendix 2.

THEOREM 1:

b y F I ~

N )= FFhs .

For L = 0, p,\ N )

s

monotone decreasing in N , where p (

N )

s defined

For

L

=

0,

a necessary and sufficient condition for the existence of a solution to p.\ N )

=

X,

is given by Theorem 2.

THEOREM

2:

When L = 0, the algorithm will converge to a solution that satisfies ( 12)

and ( 1 3 ) if and only

if

the observed sample mean

h;)

satisfies the following condition:

e-yyh

-

I

h = O k

h- 7

x >

c ( k - s ) - + s = lim p l ( N ) ,

and

where y = -In

f g h r .

The above procedure is a generalization of an iterative procedure proposed by Johnson

and Kotz [8, p. 13 ] .

I t

makes use of only the truncated mean, and matches the cumulative

probability F'y.hs,rather than only

J:hr

When

L

=

0

is used, our procedure matches./"(N)

to the observed frequency f : .However, when

f z h s

is too small, this estimator tends to

be very noisy, because it is based on too few observations. Consequently, the resulting

estimators for Na nd p also become noisy. T o overcome this difficulty, we sometimes used

larger values of L to obtain more stable and accurate estimators during our empirical vali-

dation. In six of eight cases tested (see Table

2 ) ,

our data met the conditions necessary to

guarantee convergence of our method.

The following procedure was successful in determining an appropriate value of L > 0.

First, ifJ':h5 < 0. L could be increased until FYb5 0.1. Second, we performed agoodness-

of-fit test to compare the frequencies estimated using a given value of L to the frequencies

observed in the truncated sample. If the result ingp value was not satisfactory, the value of

L was increased further. This is illustrated in Section 6, where the empirical validation of

our methodology is discussed. If no value of

L

< s can be found that provides a successful

fit, it indicates that the negative binomial is not appropriate for the given set of data or that

s is too small. It should also be noted that if the variance in the sample is smaller than the

mean. then the use of the negative binomial distribution is not appropria te. However, for

the retail sales data we have observed to date. the variance has been greater than the mean.



850

Naval Rcstw rch Lo& ic.s, Vol. 43 ( 1996)

Table 2. Results ofgoodness-of-fittests for all SKU groupings.

Chi- Estimated Estimated

SKUgrouping

s L

(fitted) dof value meana variance Mean Var

s o 9

square p distribution distribution Sampleb Sampleb

High, Peak

High,

Off

Peak

MedHi. Peak

MedHi. Off Peak

MedLo. Peak

McdLo.

Off

Peak

Low,

Peak

Low.

Off Peak

10 2

7 3

2 0

3 0

I 0

1 0

I 0

1 0

20.29

26.93

17.49

11.36

5.41

2.38

2.87

4.38

20 0.44 5.77

17

0.06 2.42

16 0.35 2.62

9 0.25 1.19

10 0.86 1.17

6 0.88 0.5

5 0.72 0.29

4 0.36 0.12

15.12

5.60

5.27

I

.60

1.71

0.59

0.37

0. I4

5.70 13.40

10

2.40 4.60

5

2.50 4.80 5

1.19 1.61

3

1.16 1.60 3

0.5 0.60

I

0.28 0.35 1

0.12 0.13

I

a

These statistics are

for

the original, untruncated data set.

These estimates

used

only the truncated data.

6.

EMPIRICAL VALIDATION

OF

THE ESTIMATION METHODOLOGY

As described previously, the base-stock levels in the data are high enough that essentially

all demands were observed. The effect of

lost

sales for different choices of s was simulated

by modifying the data as follows:

modified demand = max { actual demand, s 3 .

The modified data were used to determine values of Y-,nd FY.hs,which serve as inputs to

the algorithm. The algorithm then determines the parameter estimates for

a

negative bino-

mial fit.

The fitted distributions were compared to the observed frequencies using the p value of

a chi-square statistic. The following expression was used to compute the chi-square statistic

where

.fi = predicted frequency of demand k from the negative binomial model

z = total number of periods for which data were observed

Y

= number of entries k withji

0.000

I

dof

=

degrees of freedom for the

xz

est

=

Y

1 .

The correspondingp values are then computed to determine the likelihood that data could

have been generated by the fitted model.

Our analysis was performed for the eight data categories described earlier. and is shown

in Table 2. For each category, the smallest value of s (the second column in Table 2 ) was

determined that resulted in a chi-squarep value of at least 0.10.The high-demand, off-peak

case was an exception. The best p value we could obtain in this case was 0.06, which is

smaller than the other cases, but is still considered acceptable.

The sample mean and variance refer to the statistics for the entire sample (before

truncation). The estimated mean and variance, calculated from the truncated sample, are

referred to as the estimated distribution mean and variance. The last column shows the



Agrawal and Smith: Estimating Negative B inomial Demand

85

I

Table 3.

Results ofgoodness-of-fit tests for high-peak grouping.

Estimated distribu tion

vs

Estimated distribution vs

observed truncated observed complete

distribution distribution

stimated Estimated

distribution distribution

L s

meana variance” Chi-square dof

p

value Chi-square

dof p

value

0 10 5.34

9.44 25.03

10 0 62.00 20 0

1 10 5.77

15.12 6.99 10

0.73

20.29 20

0.44

2

10

5.77 15.12

6.99

10 0.73

20.29 20 0.44

~

a These estimates used only the truncated da ta.

stock level

(so.u)

needed

to

provide a 90 service level. T he values for so.y n the las t colum n

are greater tha n

or

equal to the required truncation levels in Co lum n 2 in all cases but on e.

Th us, it app ear s tha t the typical base stock levels used in practice would provide sufficient

da ta for param eter estimation in the case of unobserved lost sales.

In

six of the eight cases in Table

2,

L

=

0

yielded a

p

value

greater than

0.1.

In the high-deman d, peak case, a marginally acce ptable p value of 0.025

was obtained for L

= 0.

T he algorithm was then tried for

L =

1 and

L = 2,

as shown in

Table 3. Th e frequencies from th e estimated distribution were comp ared t o the observed

frequencies from the trun cated sample, as well as from th e complete (u ntr un ca ted ) sample.

Th e results of the goodness-of-fit tests are sho wn in T ab le 3. For

s = 10, L = 1

and

2

yielded

identical estimated parameters and p values. Th e p values indicate th at t he fits were good.

For the high-off-peak case, a similar procedure was used to select the best L, which resulted

in

L = 3,

which had the best

p

value of

0.06.

Som e of the com mo n inventory mana gemen t metr ics were also

calculated

for

the fi ts obtain ed from th e trunca ted data. In particular, the me an, variance,

an d the e xpected lost sales

(E[

Lost Sales]) for the fitted distribution were co m pu ted for a

range of trunc atio n levels ob tain ed by assu m ing different values of stock levels. T he

fol-

lowing expression was used fo r

E [

Lost Sales]

:

Obtaining

valiies,fi,r

L.

Culculation

o f E r r o r s .

T he calculated values were compa red to the values tha t would be obtain ed withou t trun-

cation, tha t is,

s

= co, n Figures 4 and 5. Th e horizontal axes in these figures correspond

to the trunca tion level,

or

base stock level, expressed a s

a

multiple of th e stock level needed

to provide 90 service (see last colum n in Ta ble 2 ) . Because errors will only m atter if

dem an d exceeds the stock level, this seem s appropriate.

A

value of 1 on the horizontal axis

corresponds

to

the base stock

level

tha t yields 90 service, an d higher values correspond

to

higher base stock levels. T h e vertical axes me asure t he e rrors, again expressed as multiples

of the stock level needed fo r 90 service.

Figure 4 shows the error in the m ean e st imate due to t runcat ion. T he range of the per-

centage error is from to

2.4%.

W hen th e stock level equals that needed for

90%

cus-

tom er service, the ab solute errors are less tha n

2 .

Fo r stock levels tha t would yield cus-

tom er service levels higher than 90 , the ab solute errors are less tha n 1 . Figure 5 shows

that th e range of errors in E [Lost Sales]

is

from

-0.6% to 2.3%.

Fo r stock levels tha t yield

90 cus tom er service, the absolute errors are less tha n

1 ,

an d for stock levels that would



852

8

A

0

8

m

0

0

0

A

A

~ _ _ _ _ _ _

m

High-Peak

~ o High-Off Peak

MedHigh-Peak

o MedHigh-Off Peak

MedLow-Peak

MedLow-Off Peak

Low-Peak

o Low -Off Peak

.

A

0

0

0

0

0

A

0

0

0

A

- -

0.5 1 5 2 2.5 3

Truncation Level

I

(Stock Level

for

90

Service)

Figure4. Error in computed mean versus t runca t ion level.

yield custo me r service levels higher tha n 90%, he absolute errors are less than 0.5%.Sim-

ilarly, for the st and ard deviation, the errors are relatively sm all, and range fro m

.3

t o

2.9 . As expected,

i n

each case,

as

the stock level increases, smaller errors result, because

mo re de m and information is available.

7 CONCLUSION

In

retailing, increased competitive pressure has forced m an y com pan ies to improve the

efficiency of their inventory m anag em ent systems. For basic merchandise, the large num -

ber

of

low-cost suppliers has made supply chain management

a

focal point for

gaining

competitive advantage through providing the highest level

of

customer service with the

least

inventory investment. Accurate methods for estimation of customer demand and

cus tom er service level achieved are therefore crucial to m ajor retailers. Th is article identi-

fies an d addresses two significant practical sho rtcom ings of most estimation me thods used

for retail inventory m anag em ent. First, the negative binomial de ma nd d istribution

was



Agrawal

and Smith: Estimating Negative Binomial Demand

853

2 -

8

0

A

A

8

High-Peak

High-Off Peak

+ MedHigh-Peak

o

MedHigh-Off Peak

A MedLow-Peak

A

MedLow-Off Peak

Low-Peak

o

Low-Off Peak

B

A

A

d

-0.5

-

~~~ ~~~ ~

~ ~ _ _ ~

.. .

____

-1

0 0.5 1

1.5 2 2.5 3

Truncation Level /(Stock Level for

90

Service)

Figure

5.

Error

in

computed

E [

ostsale]

versus truncation level.

found to be significantly better than either the Poisson

or

the normal, which are the pre-

dominant distribution choices for inventory models. Second, this article develops parame-

ter estimation methods for the case of unobservable lost sales, which

are

prevalent in re-

tailing, but are omitted from most demand models.

The choice of negative binomial distribution is supported in two ways. First, behavioral

assumptions for the underlying demand process suggest the use

of

the negative binomial.

Second, using the

p

value

of

the chi-square statistic to compare the fits of the three distri-

butions to the actual frequency distribution of the data found that with the negative bino-

mial all groups of data but one had p values substantially greater than 0.1, whereas for the

Poisson and the normal, all groups had

p

values less than 0.005 except one. These obser-

vations indicate that the negative binomial is far superior to the other distribution choices

for these data. Consequently, the normal and Poisson models can lead to stocking levels

that are significantly in error for most commonly used customer service levels in retailing.

However, when the required service levels are low ( e g , 70 or 8 0 ) , he stock levels tend

to be the same for all three distributions.

Estimation methods were developed for the parameters of the negative binomial distri-

bution, using demand data truncated at the base stock level. It was shown, using base stock

levels sufficient to provide at least a

90%

customer service level, that these methods yielded

satisfactory parameter estimates, based on the chi-square

p

values obtained from compar-



854 Naval Rrseurch Logistics, Vol. 43 ( 1996)

ing the model to the actual frequency distribution of the data. Furthermore, these param-

eter estimates led to small errors in estimates

for

the mean and variance of demand and

expected lost sales.

We have worked with the members of Santa Clara University’s Retail Workbench in

obtaining, analyzing, and interpreting the store-level data to test

our

methodology and

verify our conclusions. We believe that these research results can lead to more effective

applications of inventory management methods for retailers based on improved demand

models. Second, we believe that the demand models and estimation methods developed

here can provide other researchers with a more appropriate representation of customer

demand for the development of new inventory management methods for retailing ap-

plications.

One area seems particularly important for future research. Cyclical demand fluctuations

due to periodic retailer promotions commonly occur. Because excess inventory from peak

periods carries forward, a multiperiod optimal inventory policy is required for these appli-

cations. Optimization methods that determine multiple base stock levels or perhaps

multiple replenishment frequencies for cyclical random demand patterns with an underly-

ing seasonal variation would be particularly valuable.

APPENDIX 1 . S A M P L E DATA

A N A L Y S I S

A variety of ranking methods were explored to identify groupings for the data. S K Us

were ranked based

o n

average weekly sales, and stores were ranked by sales volume. Store

and SKU combinations were ranked together as in Figure 1 . Ultimately, the SKU-store

rankings in Figure 1 were separated into the top 30, the next 90, the next

270,

and the lowest

594, by selecting breakpoints on the graph of ranked sales averages. These breakpoints are

spaced approximately logarithmically. Larger numbers of logarithmically spaced points

were also tested, but the goodness-of-fit results were not as good. Next, the average weekly

sales rankings

in

Figure

2

were used to separate the weeks into two classifications of weeks,

peak and off peak, corresponding to the top 1 1 and the bottom

41

weeks, respectively.

All eight of the resulting clusters yielded chi-square p values of at least 0.10 when com-

pared to the frequency values that would be generated by a negative binomial distribution

with the same mean and variance. Several aggregations of sales by S K U and store were

tested without separating weeks of the season into separate categories. All of these gave p

values

of

less than

0.

.

Because

of

the simplicity of using just eight groups of data, subse-

quent analyses were performed using the four groups of SKU-store classifications together

with the peak and off-peak weekly classifications.

Table 4 compares the

p

values for the three distributions using the eight groupings of

data in Table

1 .

When compared to the actual frequency distribution of the data, the neg-

ative binomial resulted in chi-squarep values greater than 0.1 for all eight groups. Both the

Poisson and normal resulted in p values significantly less than those obtained with the

negative binomial. Indeed, all

p

values but two for these two alternative distributions were

less than 0.005.

Note that the degrees of freedom in this table differ from those in Table 2 because of

the difference in estimation methodologies for the frequencies

{

,h

1 . In each case, data

corresponding to,fi

<

0.000 were dropped for calculating the chi-square value in ( 18). In



Agrawal and Sm ith: Estimating Negative Binomial Dem and

855

Table

4. Chi-square errors and p values for Poisson. normal, and negative binomial distribution

fits.

x2 p alue) x2

p

alue) x 2 p alue)

SK U Cluster Poisson Normal N.B in dof Mean Var

High , peak

High.

off

peak

Med Hi, peak

MedHi, off peak

MedL o, peak

MedLo.

off

peak

Low . peak

Low,

off

Deak

35 (0.02)

569

(0.00)

266 (0.00)

I35

(0.00)

204 (0.00)

143

(0.00)

192(0.00)

100(0.00)

24 (0.24)

293

(0.00)

I59

(0.00)

753

(0.00)

1004(0.00)

262 I

(0.00)

I237

(0.00)

960

(0.00)

22.2 (0.33)

27.2

(0.13)

16.5 (0.42)

1 I .9 (0.22)

4.82 (0.90)

4.78 (0.57)

1.06 (0.96)

1.98 (0.85)

20 5.7

20 2.4

16 2.5

9 1.19

10

1.16

6

0.5 1

5

0.28

5

0.12

13.4

4.60

4.80

1.61

1.60

0.60

0.35

0.13

Table 4, he

{h .}

ere obtained by matching the mean and variance to the actual data,

whereas in Table 2, they were obtained by using our estimation methodology.

Because the intended application for these fitted distributions is inventory management,

it is natural to ask what impact the choice of distribution has on the selection of base stock

levels. The recommended inventory levels associated with each of the three distributions

are shown for

a

variety ofsample cases in Table 5. These are compared to the best inventory

policy based on the frequency distribution of the actual sales data. Service levels of 99%,

95%, and 90% were considered, because these are typical target levels used by retailers.

That is, the stock level

s

must be chosen so that P { Demand } is greater than a target

service level, for the appropriate demand distribution.

It is clear that the negative binomial is much closer to the stock level that would have

been selected using the actual demand distribution. The normal and Poisson stock levels

are consistently low, when in error. The mean absolute percentage errors (MAPEs) for all

stock levels considered in Table 5 are 18.2% for the Poisson, 19.2% for the normal, and

0.3%

for the negative binomial. The results show that the normal and Poisson models can

lead to stock levels that are significantly in error in some cases, and are inferior to the

negative binomial on average.

Table

5.

Optimal base stock levels

for

various dem and distributions. Order of

the figures in table: actual distribution. negative binomial. normal. Poisson.

Target Service Level

S K U

Cluster

99% 95% 90

High, peak

High,

off

peak

MedHi, peak

MedHi, off peak

MedLo, peak

MedLo, off peak

Low, peak

Low,

off

peak



APPENDIX 2. PROOFS O F THEOREMS

1

AND 2

Th e proof of Theorem

1

requires the following key result, which is stated and proved in

Lem ma 2.

LEMM A 2: For any

j >

L ,

we have

(

N

+

1

)

>

F,

) ,

or

all

N .

PROOF: Define

w h e r e a ( N ) = [ l

- p N +

l ) ] / [ l - p ( N ) ] < l , s i n c e p ( N + l ) > p ( N ) ( f ro m L e m m a 1 ) .

Using (2 0 ) and the fact that

==,J i N ) =

1, for each N , w e have

Similarly, since 1

F , ( N )

= 1 F 1 - ( N + 1 ), it follows that

It can be further shown that ( 2 0 ) - ( 23 lead to the following two conditions:

i )

rh(N ) s unimod al in k , and ,

i i ) there exists k2(N ) > L such that

r h ( N )

1,

for 0 < k

I

, ( N )

and

r A ( N )

<

1 ,

f o r k

>

k , ( N ) .

( 2 4 )

In order t o establish ( i ) , we show that the derivative of r h ( N ) in ( 2 0 ) changes s ign at

mo st once from positive to negative.

If

we hold N fixed an d tre at

k

as a c ontin uou s variable

for simplicity, it can be seen that

d r k ( N ) / d k= r k ( N ) [ l n ( a ( N ) )+ l / ( N + k ) ] .

( 2 5 )

Because In(a

N ) )

< 0 a n d

1

/ (N

+

k ) > 0 an d is decreasing in k , there is at m ost on e value

o f k , denoted by k , ( N ) , uch that

d r k ( N ) / d k 2 0 , f o r k s k , (N )



Agrawal

an d Smith: Estimating Negative Binomial Dem and

857

and

wh ere, if it exists, is unique ly defined by

k , ( N )

=

- N - l / l n ( a ( N ) ) . ( 2 6 )

This establishes condition ( i ) .

In order to establish ( ii ) , consider first the case when

L =

0, so that

Yo

N )

= 1

for all N .

From

the preceding discussion, there m ust exist a finite positive v alue k , N ) ; therw ise all

values

of rh N )

1. Because r,: N )

s

unimodal , YO

N ) =

1 , a n d

(

2

1 )

holds, there mu st be

values of

k

> k , N ) uch tha t rA N )

<

1. Thus , because r,: N ) s unimodal, k 2 (N ) xists for

L = 0.

For

L

>

0, we first show tha t rl.(

N ) > 1.

Suppose

Y, ( N ) 1 . If

k l

N )

, hen

r k ( N )

<

I

f o r a l l k >

L,contradicting(23).lfkl(N)>

L , t h e n r k ( N ) < f o r a l l k s L ,c on tra dic tin g

(

22 ). Therefore

rI N ) 1

mu st hold.

Given that

r12(

)

>

1,

(

23

)

implies tha t

r k

N )<

1

for som e

k

> L . Because r h ( N )

s

also

un im oda l, this establishes the existence of k 2 (N ) or L > 0 as well. Figure 6 illustrates the

tw o cases graphically.

Now, the lem m a is proved by considering the following two cases: I f L < j <

k 2 ( N ) ,

we

have

Th e first sum in

( 2 7 )

is zero in view

of

(2 2 ) , whereas

all

terms in the second sum must be

positive bec ause rh ( N )>

I

for L

+ I

k <

k 2 (

N ) . T h u s F, N

+ I ) >

F,( N ) n this case. If

.j 2 k2(N ) , using ( 2 ) to rewrite the sum s, we have

rr w

F , ( N +

1 )

F , ( N )

= c [ . / i N . L N + 1 ) 1 = z: . / i (N)[ l - k ( N I 1 ( 2 8 )

h = j + l h = J i I

Because

r h ( N )<

1 when k k 2 ( N ) , t again follows tha t

F, N

+

1 ) > Fl

N ) .

QED

T H E O R E M

1:

For

L =

0, p5

N )

s m ono ton e decreasing in N , w h e r e p ( N )

is

defined by

F L ( N )

=

F i b s .

PROOF:

Using summation by parts, it can be verified that

p s ( N )

an be written as

follows:

When = 0, I

F o ( N )=

1 - f , ( N ) =

1

’gh’, which

is

a constant .

For k

> 0, all other

terms in the sum in ( 2 9 ) are decreasing in N by L emm a 2. Thus, for L

=

0, the result is

proved.

QED

For L

>

0,

the following sufficient con ditio n can be stated for the theorem:



858

1 2

1 1

1

rh(N)

0 9

0 8

0 7

0 6

0 5

0 4

L=2

=O

Figure

6 Ratio of probability densities for ( N

+

1 )

and

N

If some value K

2

L exists such that

C o

[ 1 FA N ) ] s decreasing in N , then

p,JN ) s decreasing in N for all s

2

K .

The proof of this condition again follows from Lemma

2,

because

all

F A

N )

are increas-

ing inNfork> L .

T H E O R E M 2:

When L = 0, the algorithm will converge

to

a solution that satisfies ( 12)

and ( 13) if and only

if

the observed sample mean

X,)

atisfies the following condition:

e-7yl’

-

I

h = O k h a

x >

c ( k - s ) -

+ s = lirn p , ( N )

and

where = -In f b s




859

PROOF: For ease of exposition, we will drop the superscript from f Z h s n the remain-

When

L =

0, from Eq. ( 13

),

we know that for a given value of N , p N ) can be calculated

der of this proof. We will first establish the first part of the condition specified in ( 3 0 ) .

as follows:

Using this relation and the definition ofa negative binomial distribution function, we have

Multiplying and dividing this expression by

Nl ' ,

we get

The term N ( 1

N ) )

can now be written as follows, with the use of the relationship forp

given above:

Using L'H6pital's rule, the limit of N ( 1

N ) )

an be found as

N

tends to infinity:

lim

N (

1

N ) )

=

-Info.

K-. a

Thus, the limit of./; ( N ) s N tends to infinity is calculated as follows:

From Eq. ( 1 2 ) , we now have

3-

1

lim p 5 N ) C

( k

) lim j i ( N ) + s

N+

a

h = O

K+

Substituting the result from ( 3 2 ) in

( 3 3 ) ,

we get

( 3 3 )

Because p s N ) converges monotonically down to this limit by Theorem 1 it follows that if

X, is strictly greater than the lower limit, finite convergence must occur. This proves the

first part of the condition specified in ( 3 0 ) .

In order to prove the second part of the condition specified in 3 0 ) ,we begin by noting

that the expression for p 5 N ) iven in ( 1 2 ) can be rewritten as



860

Nuvul Rescwrcli

Logi.slic:v.

Vol.

43 ( 1996)

From (3 ), we can obtain the following expressions:

and

In

3 ),

taking the limit ofp( N) as

N

tends to 0, we get

lim p N ) =

0.

N-0

Also, in( 37 ), takingthelimit of.f; (N)asNtends oo , weget

l im

1;

N ) = 0.

fi-0

Now, from ( 3 6 ) and (39), the limit ofh.(N ) can be obtained as follows:

limfi.(N)

= 0.

h-0

With the use of the result in

(40) ,

he limit of

p , ( N ) [

from 3 5 ) ] s

N

tends to

0

is

lim p , ( N )

=

s( 1 -1; .

N-0

(37)

( 3 8 )

( 3 9 )

Because monotonicity of p,,

N )

with respect to

N

has already been established in Theo-

rem

I Eq. (4 )

provides an upper bound for the value of

p , N )

calculated from the fitted

distribution. Therefore,

ifx,

is strictly less than this upper bound, finite convergence must

occur, thus establishing the second part of the condition in

(30).

QED

ACKNOWLEDGMENTS

The authors thank the corporate sponsors

of

the Santa Clara University Retail

Work-

bench for providing the data for this analysis and for their valuable insights and financial

support for this research. The opinions expressed in this article are those of the authors

and d o not necessarily reflect those of the sponsors. We also thank Dale Achabal, Shelby

McIntyre, and Steven Nahmias

for

their valuable comments during the course of this re-

search. Comments by two anonymous referees and the Associate Editor have led to sig-

nificant improvements

in

this article.

A n y

remaining errors are the authors’ responsibility.

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A c c e p t e d March 6, I996

Documents

Naval Research Logistics (NRL) Volume 43 Issue 6 1996 [Doi 10.1002%2F%28sici%291520-6750%28199609%2943%3A6-839%3A%3Aaid-Nav4-3.0.Co%3B2-5] Narendra Agrawal; Stephen a. Smith -- Estimating