use of preview exercises to forecast demand for new lines in mail order

Use of Preview Exercises to Forecast Demand for New Lines in Mail OrderAuthor(s): M. L. Chambers and R. W. EgleseSource: The Journal of the Operational Research Society, Vol. 37, No. 3 (Mar., 1986), pp. 267-273Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/2582206 .

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J. OpI Res. Soc. Vol. 37. No. 3. pp. 267-273, 1986 Printed in Great Britain. All rights reserved

0 1 60-5682/86 $3.00 + 0.00 Copyright ((". 1986 Operational Research Society Ltd

Use of Preview Exercises to Forecast Demand for New Lines in Mail Order

M. L. CHAMBERS and R. W. EGLESE University of Lancaster

In a mail-order preview exercise, a sample of customers is given the opportunity to place orders before the main catalogue season begins. The orders received are then scaled up to forecast demand for new lines in the coming season. The accuracy of such estimates is examined, and it is shown that mail-order previews can provide useful information on demand, particularly for high sellers. For the approach to be both effective and economic, however, care needs to be exercized in analysing the preview data and in choosing the preview sample size. Methods for doing both are described.

INTRODUCTION

Forecasting demand for new lines is never easy, and is especially difficult for seasonal fashion goods with a finite selling period. Nevertheless, forecasts have to be made, even before new lines are launched, so that rational decisions can be taken on the amounts of stock required to cover the early-season demands.

A company will, of course, form some initial views on the likely demands for new lines before deciding to include them in its product range. The estimates made at this stage are, however, usually subject to a large amount of error. Preview exercises are therefore undertaken in a number of industries, in an effort to obtain more reliable estimates before normal selling begins.

Thus, before launching a new catalogue, a mail-order company may send out copies of the catalogue to a sample of customers, and invite them to place orders before the season proper starts. The preview orders received are then scaled up to provide estimates of the total season's demand for each new line. In other industries (e.g. shoe manufacturing), new lines are exhibited at trade fairs, and orders taken during the fair are used to estimate total demands.

This paper is based on the authors' experience of mail order, and explores the usefulness of preview exercises in this case. Many of the ideas discussed will be relevant in other industries, but the detailed methods and results will differ, in particular because of the different statistical properties of the preview demand data.

The paper begins by describing how preview exercises can be organized and analysed in mail order. By examining the accuracy of the demand estimates obtained, it is shown that preview exercises can provide useful information on demand, particularly for high sellers. The paper then goes on to show how preview exercises can be made more efficient and economic by careful grouping of lines in the preview analysis and by appropriate choice of the preview sample-size.

Forecasting and inventory management for seasonal fashion goods have been studied by a number of authors. Peterson and Silver' provide a useful source of references to the general problem at the end of Chapter 10 of their book. However, these references are mostly concerned with forecasting and inventory management once the season is under way, rather than at the preview stage.

PREVIEWING IN MAIL ORDER

Various, mainly subjective, methods have been proposed for making initial estimates of demand for mail-order lines (see, for example, Green and Harrison2). These, however, tend to be very inaccurate, with errors of +50% being typical rather than exceptional, which makes the determination of initial order sizes hazardous. Once the season has started and actual demand data becomes available, it is, however, possible to obtain much better estimates, using trend profile (or other) methods (see, for example, Miller,3 Chang and Fyffe,4 Murray and Silver5).

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Journal of the Operational Research Society Vol. 37, No. 3

The purpose of preview is to bridge this gap by providing more reliable estimates of demand before the season begins. Initial orders can then be kept lower, to reduce the dangers of creating end-of-season surpluses, and then topped up after preview, to overcome the risks of early-season stock-outs. There is usually no possibility of cancelling or reducing an initial order if the preview estimate indicates it was too large.

Preview exercises are usually started several weeks before the proper catalogue season begins, by mailing catalogues to a sample comprising several thousand regular customers. The preview catalogue will not necessarily include the full range of merchandise offered in the main catalogue, but may be restricted to those merchandise areas, mainly fashion, that contain most of the new lines. Customers in the sample will be encouraged to place orders before the season begins by being given a discount on orders received by a specified date.

The ratio of final to preview demand naturally varies appreciably with the type of merchandise. When scaling up the observed preview demands to obtain estimates of total demand by line, the different departments into which merchandise is typically divided (e.g. dresses, coats, woollens) are therefore considered separately. The ratio of final to preview demand can also vary significantly from year to year (even for the same department), because of changes in such factors as the number and type of customers and their spending power. Preview exercises on their own cannot therefore provide valid estimates of the total departmental demands. These can, however, be readily estimated using other methods, and preview data can then be used to split the overall departmental estimates between lines.

The simplest way of doing this is to estimate the final demands for individual lines as

d=-s (Ia)

where S and D denote the estimated final demands for the line and department respectively, and s and d denote the corresponding observed preview demands. Applying the same scaling factor to all lines in a department may, however, be too coarse. For example, when a winter catalogue is previewed in late summer, the ratio of final demand to preview demand will normally be higher for fur-lined boots than ordinary shoes. It may, therefore, be better to divide the lines in a department into product groups, and to write

I0- - -

g (G0/Do)D (Ib)

where Go/go and Do/do denote the ratios of final to preview demand for the appropriate group and department in previous years.

Both equations (la) and (lb) involve estimating a department's 'line splits' as

S= is, (2)

and it is in this form that we shall consider the estimation procedure below.

ACCURACY OF PREVIEW LINE-SPLIT ESTIMATES

When the line-split estimates are determined according to the formula in (2), there are two distinct sources of error in the estimate:

(i) random error in the observed value of the preview demand, s; (ii) errors in the value used for the multiplying factor, m.

For a given line, s will be a Poisson variable with mean ps, say. The multiplying factor, k, that is strictly appropriate to this line will then be such that

S/D = kys,(3

where S and D denote the actual line and departmental demands respectively. In any product group for which the same multiplying factor is to be applied, there will be some variation in the values of k for the lines in the group. If /1k is the mean value of k, then /1k is the multiplying factor that

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M. L. Chambers and R. W. Eglese Preview Exercises in Mail Order

should be used for the group, and the accuracy of the line-split estimates will depend on the variance of k, say ad = K 242, where K is the coefficient of variation of k.

If r is the ratio of estimated to actual line-split, then

r = mrs/ky,,.

Taking expectations over the Poisson variability in s, we have

E(r) = m/k

and

E(r2) = Mr2(l/jP + 1)/k2.

If m = Pk, then next taking expectations over the variability in k (given S, D and km, = S/D), it may be shown that to second-order terms in K

E(r - 1)2 = kD (1 + K 2)/S + K 2.

Letting p = S/Dik denote the expected preview demand for the line, we therefore have

E(r - 1)2 = I/p + K2(l + lIp), (4)

which is a measure of the mean squared proportional error in estimating the line split. The relationship described in equation (4) is illustrated in Figure 1. This shows that when the

expected preview demand for a line is small, then the first component of error (random error in the observed preview demand) dominates, and poor accuracy is obtained whatever the value of K for the product groups used. When the expected preview demand for a line is high, it is the second component of error (errors for assuming the same multiplying factor for the whole group) that is most important, and in this case a root mean squared error of, say, 25% is obtainable, if K can be kept below 20%.

The value of K can be reduced by good grouping of lines, which is discussed in the next section. The accuracy of the forecasts then depends on the value of p, which is influenced by the sample size adopted. The determination of preview sample-size is examined in the following section.

SPLITTING LINES INTO GROUPS

In order to achieve good accuracy, lines need to be grouped in such a way that the lines in any one group have similar values for the preview multiplying factor k. By equation (3), k =S(Dys); estimates of k for lines offered in a previous season can therefore be obtained by setting k = S/(Ds).

80

N- 60 -

K = 40%

40-

K =30 %

K _20 %

1W0 20 30

FIG. 1. Accuracy of preview estimates.

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However, the preview demand for a line, s, can often be in single figures and so may differ markedly in percentage terms from its expected value, ys. This means that the values of k can be estimated only very inaccurately. It is not therefore possible to determine what sorts of lines should be grouped together using a purely statistical approach.

A more realistic method is to first propose provisional groups on a priori grounds, and then to check whether these groups are reasonable by calculating their K' values. A natural way of defining the groups is in terms of merchandise type, e.g. the lines in the footwear department might be grouped according to such criteria as men's/women's, shoes/sandals/boots, into men's shoes, women's shoes, men's sandals, etc. The groups must obviously be described in such a way that it will be clear to which group a new line should be assigned. They should not therefore be too small or specialized; this would in any case make it difficult to obtain a good estimate of the group multiplying factor [kut

The following calculations can then be used at the end of a season to check on the suitability of the groups used. The calculations are based on the classical chi-squared criterion for testing the hypothesis that all lines in a group have the same k values. The purpose of the calculations is, however, not really to determine whether there is a statistically significant variability in the values of k (there almost invariably is!), but rather to provide an estimate of U 2(which measures the extent of the variability).

Suppose, then, a group comprises n lines, and let si and Si denote the observed preview and final demands for the ith line. Then if the same multiplying factor m were appropriate for all lines, the expected preview demand for the ith line would be pi = Si/(Dm). The chi-squared criterion for testing whether all lines have the same multiplier is therefore

1=1

It is shown in the Appendix that if m = Pk, then

E(X2)/n = 1 + K2(1 +ft), (6)

where n

= Zpi/n.

Thus K' may be estimated as

2 X2/n-I 1 +i

Experience will show what constitutes an acceptable value for K2 in a particular context. Guidelines of the following type may then be developed:

k2 a-a suitable group;

a < k2 fl-reductions in K2 may be possible by splitting or rearranging groups;

/l > Ki2 an unsuitable group, should be modified;

where a and ft are appropriate constants. (Alternatively, depending on how the cost of forecasting errors varies with the level of demand, a and ft could be made functions of the estimated mean demand per line in a group.)

The above calculations can be readily incorporated into a post-season preview analysis programme. As well as checking the suitability of the groups used, this programme can also provide guidance on how unsuitable groups should be revised, by indicating which lines have high values of (s, -Pi-

Our experience suggests that the average value of K can be reduced by about one third by splitting the lines in a department into smaller groups rather than treating them all as a single group. Equation (4) shows that the resultant reduction in error in the preview estimates will be rather

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M. L. Chambers. and R. W. Eglese Preview Exercises in Mail Order

less than a third, but for high sellers this can reduce the root mean squared error from, say, 35 to 25%.

The above discussion implicitly assumes that the group multiplying factors, Pk, can be accurately estimated from previous seasons' data. Where this is not so, the multipliers obviously have to be estimated from the current season's data before the groups can be checked. The Appendix shows how the formula for estimating K 2 then needs to be modified.

DETERMINATION OF PREVIEW SAMPLE-SIZE

The value of the expected preview demand for a line, p, will be directly proportional to the sample size employed. Therefore, once values of K2 for the product groups have been estimated by the method described in the previous section, equation (4) can be used to determine how the accuracy of the preview estimates depends on sample size.

For given stock-control procedures, the preview sample-size adopted should imply a level of accuracy of preview estimates which provides a satisfactory level of service to customers. The sample size should then be chosen to minimize the associated costs subject to this customer service-constraint. The first of these costs is the cost of over-ordering, which results from the need to dispose of surplus stock at the end of the season. This will depend on how such stock is cleared; in the mail order case, the cost may be small if, for example, the stock can be offered in a subsequent catalogue, but the cost may be relatively high if it must be disposed of in job lots to other traders. Data should normally be available to enable the average cost of clearing a surplus item to be estimated. The expected cost of over-ordering will decrease as the preview sample-size is increased.

There are two other costs which depend on the preview sample-size. These increase as the preview sample-size is increased. The first of these is the cost of printing and mailing the preview catalogues to the sample of customers. The second is the cost to the mail-order company of offering a discount in the price of preview orders to encourage preview demand. This cost is difficult to estimate with precision since it depends on what items customers in the preview sample would have purchased, if they had not been given the opportunity to order at the preview stage. They may have ordered the same items at full price or only ordered items to the same total value as they spent on their preview orders. The discount may have generated extra demand for some lines. However, as will be explained later, the inaccuracy in estimating this and other costs is not likely to be crucial in determining the preview sample-size.

The effect of sample size on over- and under-ordering might be attempted analytically, but there are likely to be difficulties in doing this in the mail-order situation, particularly in gauging the extent of under-ordering. This depends in a complex way on the patterns of demand received during the catalogue season and how stock orders are placed.

An alternative approach, used by the authors, is to write a computer program to simulate the estimation of demands, the placing and receipt of stock orders, and the receipt of customer demand. The simulation incorporates the current stock-control procedures, and outputs information on the levels of overstocking at the end of a season and the extent of stock-outs. Past data on estimated and actual customer demand for a sample of lines is used to drive the simulation. The accuracy of the preview estimates used may be adjusted according to equation (4), and so estimates can be made of how the extent of over-ordering and level of customer service depend on preview sample-size. The simulation may also be used to investigate other factors, e.g. the stock-ordering procedures used.

In the case investigated by the authors, the results from the simulation showed that the customer service-level (as measured by the length of stock-outs and the number of customers affected) changed little for large variations in preview sample-size with the current stock-control procedures. There was, however, a significant effect on over-stocking. Calculation of total expected costs for various preview sample-sizes revealed a relationship described by curve B in Figure 2. The sensitivity of the total expected costs to the cost of clearing surplus stock and to the cost of offering a discount at preview was examined. The combination of cost estimates which implied the highest optimum sample size is illustrated by curve A in Figure 2. Curve C illustrates the relationship implying the lowest optimum sample size. The sample size giving the minimum total expected cost was little affected by the range of cost estimates thought to be reasonable. The curves illustrated

271




A

B

0 o ~~~~~~~~~~C 0

Sample size

FIG. 2. Graph of total costs against preview sample-size.

in Figure 2 were shallow in the region of their minima, and so it was possible to select a suitable value of the preview sample-size which would give total expected costs close to the minimum for all cost estimates investigated.

CONCLUSIONS

Preview exercises of the type described in this paper form a useful part of the forecasting process in mail order. They are, in particular, useful for detecting high selling-lines and so enabling orders to be placed which will prevent stock-outs.

The way lines are split into groups when making the preview estimates has a significant effect on the accuracy of these estimates. The grouping test described here is an important aid in checking the suitability of groups which are formed.

Economies may be made by adjusting the sample size adopted at preview. Though costs are not too sensitive to changes in sample size around the optimum, intuitive estimates of the required sample size may be a long way out.

APPENDIX

Estimation of K2

From equation (5),

X 2= (Si _pi)21pi.

Let si be distributed as a Poisson variable with mean pi, then taking an expectation over the variability in s, we have

E (X2) = EE {- 2si + Pi} i=1 Pi

=E{iH_ pi + Pi}

From equation (3), ,i= Si/Dki, and pi= Si/Dm. So

tlm Sim 2Si Si E(X2) = _E _

E(X4)= E + Dk- Dk, Dad}

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M. L. Chambers and R. W. Eglese Preview Exercises in Mail Order

For lines within this product group, take ki to be distributed with mean 11k and variance K 4llX .

Then

(X2) E {1 2+ K (1 + 3K2 -2(1 + K2) + 1)

(to second order in K).

When m =Pk,' then pi= Si/Dik, so that

E(X2)/n = I + K2(l +p),

where

= Z pin,

i.e.

E(X2)/- 2 - E( I)n - I 1 +K

The above analysis assumes m = Ilk. If 1Pk cannot be accurately estimated from previous years' data, then 11k can be estimated from the most recent year's data by setting

m =( Si /D Esi)

so that

Pi= Si/Dm = SZSi ( Si).

This modifies the analysis to obtain E(X2), but, after taking expectations over the variability in si and ki, it may be shown that to second-order terms in K,

n n

ESi ~i2

E (X2) = (n- 1) + i x 2{ ( n

(Those who require details of the proof should contact the authors.) This expression may then be used to estimate K.

REFERENCES 1. R. PETERSON and E. A. SILVER (1979) Decision Systems for Inventory Management and Production Planning. Wiley,

Chichester. 2. M. GREEN and P. J. HARRISON (1973) Fashion forecasting for a mail order company using a Bayesian approach. Opl

Res. Q. 24, 193-205. 3. D. W. MILLER (1956) Operations research in the allocation of resources. In Operations Research, p. 50. American

Management Association. 4. S. H. CHANG and D. E. FYFFE (1971) Estimation of forecast errors for seasonal-style-goods sales. Mgmi Sci. 18,

B89-B96. 5. G. R. MURRAY JR and E. A. SILVER (1966) A Bayesian analysis of the style goods inventory problem. Mgmt Sci. 12,

785-797.

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use of preview exercises to forecast demand for new lines in mail order

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