Forecasting Systems for Production and Inventory ...

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Received April 1991 Revised September 1991

Forecasting Systems for Production and Inventory Control •^

Robert Fildes Lancaster University, UK, and

_

Glasgow Business School, UK

, '

Charles Beard Introduction The most common area of application of quantitative forecasting techniques is as part of production and inventory systems. While standard texts in the area such as Silver and Petersen|l] and Vollmann e(fl/.[2] discuss basic forecasting concepts, the recent production and mventory literature has paid little attention to specifying the requirements of a complete forecasting system designed to support core production activities, Brown[3l being the major exception. Instead, authors such as Plossl[4] and Kenworthy[5j have contented themselves with exhortations to improve forecasting while failing to offer specific remedies. Indeed, Goddardf6] has even gone so far as to write "Let's scrap forecasting", although his recommendations turn out to be less controversial — we should, he says, "reduce lead time". Such inattention by academic and practitioner alike leaves serious gaps in the knowledge necessary to design an effective forecasting system. This article places new research in forecasting within the framework of planning for production and inventory control. From the practitioner's view, the accuracy of any particular forecasting technique can only play a secondary role to the broader issues that place the forecasting system in the context of the functioning organization. The article begins by demonstrating how the inventory-control forecasting problem differs from other forecasting applications in its use of information. Any discussion of effective forecasting technique must start with an examination of the data to which it is intended to be applied. The next section therefore discusses the characteristics of inventory-type data, arguing that different product types require different forecasting methods, but that conventional attempts to simplify the problem are flawed. Having defined the context in which forecasting for production and inventory control occurs, we go on to evaluate a variety of potentially suitable forecasting methods. The usefijlness of any particular method is best assessed through continuous measurement of its performance over time — the possibility of which

International Jouiiul of Operations

Press, 0144-3577

Thisresearchwas carried out while both authors were at the Manchester Business School. A preliminary vcrsion of this paper was given at the TIMS/ORSA Meeting in \\^shington. DC, 1988.

is a distinctive feature of production and inventory forecasting. This section therefore also includes a discussion of monitoring schemes that are used in practice. The information available to the forecaster working within a functioning organization extends beyond that contained within the time-series history. Other sources of information relating to the activities of the organization, such as orders or marketing, can improve forecast accuracy, and these are discussed in the next section on extending the information base beyond the time-series history. Finally, the "ideal" system for production and inventory-control (P&I) forecasting is discussed. The ideal system is designed with reference to commercially available forecasting systems for inventory control applications. It is these packages that determine the effectiveness of most organizations' P&I systems since they are adopted wholesale into manufacturing and distribution without being appraised as to how well they ' 'fit" the particular user's needs. The paper conclude s with a critique of these influencial software packages and makes recommendations that should be employed to improve current practice.

The Context of Production/Inventory Forecasting Forecasts are needed to support production and inventory decisions both in the short and the long term. In the long term they are needed to plan capacity changes. However, these forecasts are rarely the province of the production department. Rather it is their job to examine the consequences of such a^regate market forecasts to establish any capacity constraints in the production and distribution facilities (including staffing). For the remainder of the article we will concentrate on manufacturing planning rather than distribution, noting any major differences where appropriate. In contrast, shorter-term forecasts, generated often from within production {or operations), are produced regularly (perhaps monthly or weekly, even daily) for a small number of periods ahead and drive the manufacturing planning and control system. While not all manufacturing activity requires such short-lead-time forecasts, for example those that "make to order" (to adopt Vollmann et a/.'s|2| terminology), MRP, JIT and flow systems are all dependent on these short-term forecasts. Increasingly, Vollmann et al. add, tbe goal of manufacturing, planning and control systems is to lessen the time between distinct manufactured units and decrease the number of sub-parts, and this therefore increases the reliance on forecasting. A number of studies have examined the importance of forecasting for capacity planning and tbe master production schedule. For example, in the longer term. Price and Sharpf7l analysed the consequences of forecasting error for the electricity supply industry during the period 1959-77 and showed that the forecasts adopted by the industry had cost the taxpayer substantial sums of money (several £billion more than would have been necessary if the best forecasting model had been used). Evidence on short-term effects has been presented by Adsbead and Price[8] for a make-to-order manufacturing unit, Gardnerl9l in a distribution system, and Lee etal.\lO\ who simulated an MRP system. All authors concluded that the accuracy of the chosen forecasting method was important to the overall cost (or service level) of the system being modelled.

Forecasting for Inventory Control

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The many repetitions and short lead times, when forecasting for P&I, permit the forecaster and the forecasting system to collect immediate feedback on forecast error, which has the potential for being used to improve the forecasting process. There are typically many simultaneous applications; a tentative estimate by Mentzer and Cox[ll| in a survey of US forecasters gave the average number of products forecast as 2 ^ 0 although the distribution was badly skewed to the few organizations whose forecasters are responsible for 10,000 or more. Similarly, experience with the Bell Telephone system showed that each of the company's systems had to be designed for over 1,000 items, with some systems over 100,000. Recent analysis of a UK manufacturing comparQ' showed six product ranges, each of wliich has perhaps 40 features with four or five alternatives within each feature. In short, forecasting production and inventory (FPD systems have to cope with well over 100 time series with numbers over 10,000 being quite common. The individual forecaster using such a system also hasresponsibilityfor a large number of these products. Fortunately each individual forecast will have small-scale consequences for the overall performance of the system. In Figure 1 we show a hierarchical view of how the demand for the company's product is represented. Here the overall activities in the economy (extended if necessary to include other countries) lead to consumer demand. A part of that demand is supplied direct by the company's products and this is supplemented by the company supplying other businesses, which in turn are meeting that consumer demand. Except in the case of a monopoly, both parts of the company's business will be affected by competition. The aggregate company demand can be broken down into demand for various product classes (or ranges), and divided again by region. In turn the product Mac'oeconomic variables e.g. growth

Consumer demand e.g. cars Business demand Aggregate company demand Competitors' demand

r .

Figure 1. Hierarchical Forecasting

class is the aggregate of individual products. The demand for these products generates a dependent demand for their components and the labour and machines needed to complete their manufecture. This dependent demand can (and should) be estimated from the forecasts made for the products themselves. There may also be an independent demand for the same components, generated, for example, by a malfunction in a product already sold. Interrelationships hetween the products may lead to components in common that can be capitalized on when generating forecasts and deciding safety stocks[12l. An FPI system is required to produce medium-term forecasts by product class, which inevitably are required in money terms and short-term forecasts by product and component volumes. Similarly, in a distribution system a natural hierarchy exists with product groups (e.g. washing powder) broken down first into the various product types (e.g. biological) and then further subdivided by pack size, region and store. In summary, information at the upper levels of the hierarchy will affect demand at lower levels. Certain structural features of the market can be used to eliminate unnecessary complexity, permitting the forecaster to identify the important influences in the hierarchy more easily. But, as Schwarzkopf et al.[\3\ make clear, the disaggregate forecasts themselves may contain information helpful to the aggregate forecast. Any FPI system must provide forecasts at a detailed level while being compatible with more aggregate market forecasts and allowing a two-way comparison between levels. An ideal FPI system cannot be designed in the abstract, away from its intended user. Survey information has begun to accumulate on forecasting practices within organizations, for example, in the work of Mentzer and Cox|ll], Dalrymple|14|, and Fildes and Hastings|15|, although these researchers have mostly concentrated on the marketing/sales activity rather than its counterpart in production and distribution. This suggests that any system should be designed to be equally useful both to production and to marketing as the link between the two has strengthened with the advent of modem manufacturing systems such as MRPII and their extensions into distribution. Unfortunately, perhaps, neither production nor marketing are likely to employ forecasters well versed in statistical methods, as Fildes and Hastings'|15| survey carried out in a UK multinational makes clear. Statistical support to resolve technical queries is usually unavailable. Instead, users rely on the programs and the manuals to lead them towards sensible forecasts. Anecdotal evidence repeatedly suggests that programs that have flaws in them are used from year to year without the flaws being detected, and programs which make much play of the users' selecting parameters, usually run in default mode. Any forecasting system should take into account these user characteristics.

Characterizing Inventory Data When attempting to define suitable forecasting procedures to deal with inventorystyle data, it is obviously important to understand what such data typically look like. It might then be possible to link the known characteristics of the data to different forecasting methods. Variables such as the number of data points available for analysis, a ratio of signal to noise in the series and the correlation


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between the current data point and previous values of the series (the autocorrelation) might prove useful in choosing between methods. In this section we examine these data characteristics in an attempt to understand the type of forecasting model tbat should be considered when forecasting for P&I. Simple exponential smoothing has performed well in empirical comparative testing. Some authors(3,16| have suggested a smootbitig parameter lying between 0.1 and 0.3. This implies first-order autocorrelation in the stationary series of about -0.5, and a number of software packages (e.g. American) encourage the user to look within this range. Figure 2 shows tbe distribution of the first-onder autocorrelation coefficient of the differenced data for 795 series from telecommunications. The data are notable for the fact tbat -0.5 is close to the 99 per cent value, the mean is close to zero and tbe bistogram is positively skewed. Optimal smoothing parameters are far from these recommended values. A second, more limited set of data from an establisbed UK manufacturing company is sbown on tbe same Box plot where the typical series is stationary witb positive autocorrelation of about 0.4 on average. The contrast between the two classes of series is extreme and we can conclude that any automatic cboice of method (e.g. setting the trend to zero or a positive value) and associated parameters (constraining the smoothing parameter to lie between 0.1 and 0.3) without reference to tbe data is likely to produce highly inaccurate forecasts. An alternative way of characterizing inventory data is to examine tbe errors produced from some base-line forecasting model such as simple exponential smoothing. If a variant of exponential smoothing[17j is applied to a subset of the 795 series, tbe forecast error distribution is approximately normal but with fat tails, in tbe sense that too many extreme observations are observed to correspond strictly with tbe normality assumption. 1

06

- "''.. 0.2

1! 1 1

Figitre 2. Multiple Box-andWhisker Data Plot: Telecommunications and Manufacturing

Telecommunications

Manufacturing

The average absolute percentage error observed for each of these telecommunications series is 1.8 per cent Oead 1), 5.5 per cent Qead 6) and 8.2 per cent flead 12) as measured by the median of the mean absolute percentage errors (MAPE): (L) is for the different lead times: MAPE (L) = _


\et(L)\

9 where e^(L) is the Z^step ahead error made in forecasting Y^^^ at time /. The median was used to overcome the problem of extreme observations. Estimates based on stirvey evidence are shown in Table I and similar figures were obtained from respondents in US telephone operating companies. The actual errors found in a UK manufecturing company were also comparable, as are those given by VoDmann, et al. |2|. Both the survey figures and the actual errors observed in the two case studies suggest that improvements in forecasting accuracy should have a substantial impact on inventory costs. Up to 3 months

Lead time 3 nK)nths-2 years

Product group 10 00) Product line 11 (12) Product 16 (16) S(mrce:\\\\. Figures in parentheses are from a UK

15 (10) 16 (12) 21 (20) multinational.

Over 2 years 20 20 26

(15) (19) (27)

Model Stability The previous section demonstrates that inventory data often have little structure and arelativelyh i ^ level of randomness. A fiorther characteristic is the low stability of forecasting performance across time. This can be tested by usitig two simple forecasting methods and examining their relative perfonnance from one sub-sample to another. In an ideal world the entries on the 2 x 2 table would be diagonal, that is to say, if Method A performed best over the first data set it would perform best on the second and vice-versa. In fact, there are likely to be major cross-over effects between the two methods, caused by sampling variability on the one hand and lack of stability in perfonnance on the other (see Fildes[17] for an example). Two conclusions follow for forecasting series such as these. The first is that there is little point in attempting to establish the "best forecasting model" for a particular series without considerable analysis of its stability over time. Even then such an approach might well prove unprofitable. The second is that, when the focus is on aggregate forecasting performance across the whole population of data series as part of a P&I system, a method which is robust to curious features of the data is likely to outperform any methodfine-tunedto a particular stnjcturell8j. Simplifying and Segmenting Because the FPI problem is concerned with a large number of series, early forecasters such as Brown argued infavourof the adoption of A-B-C classification

Table I. Estimated Forecast Error (Mean Absolute Percentage Error) b>' Level of Product Aggregation and Lead Ttme

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schemes by which the high-volume data series are considered carefully while those with a C classification, typically including about 50 per cent of the series but only 10 per cent in value terms, are forecast by default. Methods such as exponential smoothing are reserved for the B classification while class A series are seen as requiring further individual attention. Such an approach has the apparent advantage of minimizing computer usage -jQ and storage, while concentrating effort on the high-total-value series. However, ^ ^ _ ^ ^ ^ ^ ^ _ ^ ^ such arguments had greater force in days where storage and computer time were much more limited than they are today. A second issue is that inventory investment is proportional not only to unit value but also to forecast error as measured by the standard error of the forecast. In order, first, to avoid estimating the error variance for each product and, second, to overcome the disadvantages of basing the estimates on only a limited data set, authors such as Stevens[19] have discussed various models linking the mean level of the series M to the error standard deviation a^ through relationships such as: o^ = C(M)" where a < 1 and C an appropriate constant of proportionality. (This relationship can be used for items with a limited history or when no error history is availablefl9, pp. 143-4|.) Unfortunately such an aggregate relationship neglects those series where a^ is large relative to the mean. Figure 3 shows a scattergram of the relationship while Figure 4 shows Box plots of Og for the telecommunications data, broken down into the three A-B-C groups according to mean demand. The total sales in each group are also shown on the Box plot, underlying the approximately 80-20 split. A rough regression model offers some support for the above relationship (Stevens[19| points out some of the econometric difficulties of estimating such a model). However the Box plot shows the danger of such simplifications. The variability within each group is high and, with o^ large compared to M, inventory costs are not proportionate to the series' classification. With computer storage becoming cheaper and more easily accessed this suggests that simple forecasting schemes should be adopted for all series, while type A series should be more broadly defined to include those series which are moving up the list in importance (but would be naively classified as in B) or alternatively whose associated forecast errors are unusually large. The AB-C scheme can be extended to include additional criteria such as the product's operational characteristics[20, 21j. Most FPI systems assume that the same forecasting technique is suitable for all the data series under consideration. Such an assumption may weU be invalid. The benefits of matching a forecasting method to a homogeneous subset of the data series may yield substantial accuracy benefits. Consequences of Forecast Error and Measures of Forecast Accuracy Ideally a forecasting method should be chosen in order to minimize unnecessary costs such as stock holding. This can be done through simulating the system as Gardner[9| showed when he modelled a complex distribution system with 50,000 parts. He estimated that the work cost $150,000 but saved $30 million. The actual consequences of error depend on such items as the nature of the


11

100

1.000

10,000

100,000

Figure 3. Plot of Standard Error versus Series Mean

1 [

-

1 -

1 11

1

-

-

1 1

-2

Figure 4. Standard deviat'on =

Mean = 724 Mea Standard deviaiion = 414 Standard deviation = 4,519

Multiple Box-andWhisker Plot: A-B-C Classification

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production process, the valuation of the inventory, both finished and semifinished, as well as the inventory and lot-sizing rules used. If only the inventory is valued, the value of the stock is LCjko^ where ka^ is the volume of safety stock held for the ith product and Q is its unit value. However few companies have gone down the simulation route chosen by Gardner for the US Navy. Instead they have used standard statistical accuracy measures as a surrogate. They ^ ^ "*^^ ^^ ^" ^^^^ match but it is hard to argue in favour of the proposition that, so long as you have chosen a suitable master production schedule (MPS), higher levels of uncertainty lead to improved MPS performance. The measurement of forecast accuracy is not straightforward. Textbooks concentrate on mean absolute deviation (MAD)|2|, while standard safety-stock formulae for independent demand depend on the standard deviation of the forecast errors. Even when considering dependent demand, lead time can harxlly be regarded as fixed and again any safety margins built into the master production schedule (MPS) should depend on tbe standard deviation. Brown[3] has pointed out tbat there is no need to use MAD and that its utility arose because of lack of computing power[22l. Thus most computer programs calculate both these error measures as well as MAPE. These error measures are, however, usually calculated using the within-sample fit of tbe chosen method to the data; i.e. the exponential smoothing parameters can be chosen to minimize the sum of squared errors and the mean square error (MSE) or calculated from the same data as have been used to select the parameters. These summary statistics for the model's fit are typically used to derive the inventory policy parameters. Unfortunately this is not appropriate, as MakridaMs and W^mkler(23l show. There is a limited relationship between fit and forecast accuracy and it is the latter concept that is appropriate in determining inventory policy. To estimate forecasting accuracy, a forecasting method must first be selected and the corresponding parameters estimated. The chosen model must then be used on line by adding an extra data point to the estimation database and calculating new forecasts and new errors, thereby generating "true" forecast errors. P&I texts suggest that the MSE (or MAD) is updated as new data (and errors) become available through a simple smoothing equation with initial values derived from the fitted MSE. This is unlikely to be adequate unless the data series is long. The problem is further complicated by the need to calculate the error distributions across various lead times, because as Yar and Chatfieldf24] point out, the approximations found in the literature are ill-founded[21. Silver and Petersen[l] are careful in their analysis and argue that the incorrect (and often inadequate) approximation of V^â^ for the forecast standard error over the replenishment lead time L should be established empirically. It is doubtful whether any commercially available software has this facility. Currently the only solution is to calculate the empirical standard error across all relevant lead time as well as the lead 1 standard error. Even when based on true forecast errors (rather than within-sampie errors) the accuracy measures that are typically used to compare methods can be distorted by heterogeneity in the data series. Thus MSE for lead L, defined as

where e,-/L'' is the L-step ahead error made in forecasting Ya+i at time / and Nis the number of series, staffers from two weaknesses: it is not scale invariant, so that a change of scale for one of the data series leads to a change in the measure, and it is particularly sensitive to extreme error values[25]. If the costs of forecast error can be regarded as proportional to the scale of the data as in the previous formula, then it is only the latter problem that is important. However, such a situation is unlikely in that forecasting should usually be based on volume measures and not distorted by price effect. Some computer programs permit the user to examine the data in value terms as well and, with this an option, would overcome the problem of scale. MAPE^(L), while being scale invariant, is sensitive to errors in near-zero series (location dependent). Nor does it have any direct role to play in deciding inventory levels. Forecasting researchers have argued over how best to deal with this probleml26l, typically using a wide range of error statistics aimed at capturing diversity in the population of data series. Certainly any measure used shotild be robust to outliers and "rogue" series. One straightforward way of eliminating part of the problem is to examine the performance of one forecasting method relative to a {usually simple) altemative[18l. (The use of a relative geometric mean-squared error overcomes the problem of scale. Alternatively the relative median APE has similar performance characteristics.) Bias, the average error (or percentage error) is also an important error statistic, in that methods that are consistently biased have the potential for improvement; for example, the historical bias may just be subtracted from the forecast. Authors such as Lee and Adam|27] have argued that bias in a forecast may improve MRP performance (depending on lot-sizing rules) but that is just to confiise the process of forecasting (which looks to minimum error and absences of bias) with the use to which the forecast is put. Once the appropriate forecasting method has been selected, the forecast errors are usually assumed to be normal and independent {or Poisson for slow moving items — special forecasting methods have been proposed for slow moving items) in the calculation of the appropriate inventory policies. Typically the effect of non-normality is small|l,28]. However, for leads greater than 1 the errors will typically be autocorrelated and this issue has received very limited attention] 29|. In summary, measuring forecast error over production/delivery lead time requires careful attention in any forecasting system. Standard methods are inadequate (and typically lead to gross underestimates of error). Except in those rare cases where the appropriate loss function has been established, based on the value of the inventory, a variety of error meastires shotild be considered including relative error measures that measure the performance of the method under consideration compared to some simple base-line alternative.

Forecasting Methods and Monitoring Schemes The nature of the inventory-control forecasting problem is that, of necessity, any forecasting methods must be, broadly speaking, automatic. Two choices


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are therefore possible, an automatic method, together with a monitoring scheme for the errors to ensure that the method remains ' 'in control", or alternatively an adaptive forecasting method such as that proposed by Trigg and Leach|30| or Harrison and Stevens [31]. In recent years a number of new forecasting procedures have been proposed. Tables II and III provide a checklist of forecasting methods, together with non-technical references, that in various forms have been considered suitable for inventory control, with comments on their accuracy and data/computer reqtiirements.

Model

Comments on application

(1) Decomposition: moving averages\32, chapter 6]

Data intensive Computer intensive in ^-11 form Designed to adjust for outliers and trading days Designed for batch processing of many series Readily Interpretable Limited empirical testing which is not supportive

where 7, is the trend and is forecast by a trend curve (usually linear), C, is a cyclical factor of period perhaps four years or more wbicb is often Incorporated in tbe trend term, F, is a seasonal factor incorporating all witbinyear regular fluctuations and e, is the random error term (2) Exponential smoothing and HoltWinters\l, 33J The ft-step ahead forecast is Y k = (S + kTJ»F where S, is a smootbed estimate of the level of the series defined by the equation a being a smootbing parameter, and 7 being a second smoothing parameter, with a similar equation for updating tbe seasonal factors |F^j. Tbe equations for updating tbe smoothed level and trend are based on tbe notion of taking the previous smoothed and modifying it according to the forecasting error just made

Table II. Autoregressive Forecasting Methods

Limited data required Optimal smootbing parameters needed. As argued earlier, many computer programmes do not help tbe user to select smootbing parameters but differing characteristics amongst time series mean tbat tbe user must match the parameters to tbe series. Searcb for optima] parameters is computer intensive Initialization of So and To controversial Limited data storage and processing required once initial conditions and parameters are set Various specifications including additive or multiplicative seasonality, or no trend. Gardner and McKenzie|37| discuss methods of identifying the appropriate model Readily interpretable Strong performance in tbe major study of forecasting accuracy, tbe Af-Competition |341

(3) Damped trend\32, pp. 88-91; 36] where is the dampening parameter, S, = S,_i + "iiT,.! + ae, 7, = «i»7,_i + Q!-i«,. The dampening parameter < 1 has tbe effect of damping tbe trend from the linear trend line towards tbe bodzontal so tbat for 0 = 0 tbe model excludes trend while for ^ = 1 it is equivalent to Holt's method, It can be modified to take into account seasonality (4) ARIMA or Box-Jenkins models\32, pp. 126-48; 35) Z, = 6 + L^f_k - '^Qkî^k

+ h

where Z, is a transform of the variable being forecast, y, and the ;0is and )9!s are unknown parameters to be estimated from the data. The model can include seasonality, non-linearity, outliers, and "interventions" that interrupt tbe regular development of the data series. Exponential smootbing is a special case of the above. Has well-defined statistical features and associated "optimal" estimation. Z, is chosen to be stationary (constant mean and variance, as well as autocorrelation structure) and tbis often necessitates using differenced data so

Strong empirical performance on tbe A/-Competition data[36| — as tbis is an extension of exponential smoothing, tbe other comments apply equally Recently included in some FPI software


15

High data requirement particularly with seasonal models Computer intensive Model and output not readily interpretable and expertise needed Automatic programs available to belp to overcome need for expertise Problems fitting tbe model to some types of data series Not easily modified for FPI applications Disappointing performance in the A/-Competition Few FPI applications available because of computational demands, but see Goodrich [38]

that

Table II. Continued

Forecastir^ Methods Makridakis et al.'s book[34] describes many of these methods and offers an evaluation of their accuracy in the context of a forecasting competition that examined their relative performance on 1,001 data series. Fildesf44] describes a wider range of methods and gives an update and interpretation of the empirical results on relative accuracy. The results suggest strong if not conclusive evidence in favour of simple robust methods such as is apparently offered by exponential smoothing and its extensions. In contrast, Fildes[18] shows how, for a specific population of time series, methods designed (or the data substantially outperform smoothing methods. However, the PSrzen approach, an extension in the ARIMA class|34, pp. 267-87] or Gardner and McKenzie's[36] "damped trend model", both of which attempt to model trend in unusual ways, are apparently more effective than, for example, the Box-Jenkins ARIMA approach to dealing with trend in the data by transforming the series to stationarity through differencing.

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Model

• Commeqts on application

(1) Adaptive smoothing\l3\

16

Smoothing parameters a and (sometimes) 7 in the above exponential smoothing model are made to depend on recent errors

-...

(2) Slate-space or Bayesian mo(ifi/5J3I,39| Y

1

= a +V

t "/ / the observation or measurement equation, and 8 &i = 0t-l

Table III. \toable- E^rameter Models

+ ^2t'

the transition equations. This is a specific example of the very general class of state space models 2,42]. The observation equation describes how the variable to be forecast varies around the (unobserved) level of the series, while the transition equations show how the level changes over time, depending on trend, and how the trend changes over time. Changing seasonality can also be included in a similar fashion

• Has the flexibility of exponential smoothing, supplemented by the theoretical ability to adapt to (in)stability in the data by modifying smoothing parameter. However performance is apparently poor|33| although researchers still attempt to 1 ' capitalize on its attractive features[40,41 • Often the basis for FPI forecasting systems. • Parameters can be estimated but with high computational requirements|42| or pre-set|43| with likely loss in accuracy • Flexible in that few data are required • Few FPI applications but still in operation in ICI Paints for which it was initially developed • Hard to understand technical details • General model not used for FPI

1

Adaptive schemes such as that proposed by Trigg and Leacb|30l or Bayesian procedures of Harrison and Stevensf31] seem to be hyper-responsive to changes in the series behaviour, and this in turn leads to ineffective forecasting performance[33,39l. In contrast, Williams[41] has suggested a modification of Trigg and Leach which has tbe potential for overcoming this problem. Two conclusions emerge from this brief summary of forecasting accuracy of common inventory-style forecasting methods: (1) Conventional and well-established forecasting methods such as exponential smoothing perform adequately if operationalized correctly. (2) Major improvements are achievable in principle by adopting the appropriate forecasting method to match the problem to hand. However, the newer methods are not ideally suited to large FPI systems without modification because of their computational complexity.

In view of tbe above, is it possible to establish which method is appropriate for a particular segment of the population of time series under study? Fildes|17, 18] bas concluded that method selection should be based on the analysis of a large sample (or preferably the full population) of tbe time series of interest over time. More conventional evaluation techniques rely on taking a snapshot of comparative performance, but this approach is undermined by sampling variability. Equally important for applied forecasing is the conclusion that careful method selection leads to major improvements in forecasting accuracy. Gains of 20 per cent are practical and, given tbe investment in inventory even in medium-sized systems, selection improvements in forecasting accuracy would be well worth establishing. Monitoring Schemes\32, chapter 14\ The forecast errors made in routine forecasts of Class B items contain key information on whether or not the forecasting procedure is operating within its expected level of accuracy. If not, tbe forecaster will wish to know of the problem as soon as possible and make adjustments to, perhaps, the parameters of the method in use, tbe current forecasts, and tbe estimated standard deviation of the error. A number of proposals have been made as to how best to monitor the errors to provide speedy feedback as to when the forecast errors are unacceptably high. Gardner[45j provides a survey and an evaluation. Using Gardner's notion, let • j. . i

where F^ is one step ahead forecast of Y^, = e^ •¥ Si_i

(l-a)*MD,_i SUM/MAD then the two standard monitoring schemes that Gardner considers are the simple cusum Cf and the smoothed error tracking signal, Tf. On balance, Gardner argues for the simple cusum while McClain[46] comes to the opposite conclusion. The simulations supporting either conclusion are not wholly convincing and would be strengthened by a wider choice of data-generation process/forecasting method. Both agree that a should be small and should not necessarily be chosen as equal to the corresponding smoothing parameter if the forecasting scheme is exponential smoothing. However, as Gardner points out, the discrepancy does not bave to be resolved as both signals are easily computerized witbout excessive storage costs and, together with a third measure, the autocorrelation signal, should all be included.


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Extending the Information Base ' • Demand in many organizations is too volatile to be accurately forecast using only tbe corresponding historical data alone. There is consequently the need to extend the information base beyond its past history. Volatility within the organization and its markets can arise from many sources, which tbe forecasting literature has generally dealt with under four information categories; orders, changes in the marketing mix, aggregate information (such as economic indicators), and management information. These extra sources of information will be of varying importance from organization to organization, leaving the decision to include all or any of them in the forecasting system to the experience of the forecaster. Orders Many organizations receive orders over tbe period of time before a sale is finalized. The importance of tbese orders depends on the nature of the manufacturing operation[2, pp. 408-9]. These orders vary in their relationship to total sales: on tbe one hand they may be firm orders from key customers that say little about the normal ordering pattern; alternatively, they may contain lead information on realized demand for the period. A simple model linking demand to orders is: where Ot(k) are the orders placed in period t~k for delivery in the rth period. Two issues dominate whether order data are particularly helpful in forecasting demand: tbe reliability of the data, {often quite low due to cancellations and over-ordering), and the predictability of the unknown orders over the forecast lead time. For example, in the case of the manufacturing company under consideration, there was little if any relationship between initial orders and realized demand. As a consequence, one priority for improving tbe forecasting system lay in amending the ordering system so that the salesforce wbo had responsibility for completing the order forms specified the order in such a way as to add information to tbe forecasting system. However, cancellations and alterations to tbe initial order still remained a problem. Kelle and Silver|47) demonstrated the usefulness of additional information in the context of returns of reusable containers. The equation above can be estimated directly using regression or, if the parameters \0[ are thought to vary, simple extrapolative forecasting models can be used to update these estimates[48,49j. Changes in the Marketing Mix • ' -' " Tbe fast-moving consumer non-durables market is particularly affected by changes in tbe marketing mix, for example, advertising campaigns, price and pack-size promotions, and in-store displays. Durables and industrial products and services are affected much less. FPI systems incorporate such effects directly, by extending the models, or indirectly, by allowing the product manager to intervene and modify the automatic forecast.

Direct methods include multivariate ARIMA and extensions of regression analysis, the latter being the preferred approach on grounds of comprehensibility, accessible computer programs and theoretical rigour (see Fildes|50| for a description). However, product promotions which, for example, include packbased advertising and a change in product size, are not readily interpretable as data for use in a regression model. Instead, the problem may be solved by modelling the observations as: where Bf are the base-line sales unaffected by promotions, P/ are the additional sales owing to a particular promotion and e, is random error. Bt and Pi cannot be observed directly and any objective technique must allocate sales to the base line or to the promotion to minimize the residual variation. Abraham and Lodish|511 offer an ad hoc but persuasive method for estimating the base line. Indirect methods have more regularly been used in FPI systems where the user is permitted to identify a number of periods as abnormal and override the base-line forecast with a subjective assessment of the overall sales as they are affected by the promotion. Aggregate Information Over periods of three months or more, product group sales can be expected to be affected by the economy as a whole, as well as by competitive activity. FPI systems which do not include this information are_ likely to incur unnecessary errors. Heuts and Bronckers|52|, for example, linked an index of constructionindustry activity to fluctuations in the truck market. As noted above, regression analysis is best suited to carrying out this analysis. While establishing market links to macroeconomic information has proved straightforward, relationships between the market and price charged and market share and relative price, advertising and promotions variables have proved more difficult to identify as the survey by Brodie and de Kluyver|53| made clear. An appraisal|54| based on scanner data suggests that the problems can be resolved if sufficiently disaggregate data are used, and such short-term disaggregate forecasts could link directly to the P&I system. Management Information Product sales are regularly affected by extraordinary events, ranging from a strike by competitors to an interruption in demand from a particular customer due to weather. The forecasting methods we have described are unable to include these effects. Subjective intervention on the part of the knowledgeable product/sales manager liaising with the FPI forecaster is the only route likely to prevent what might prove to be important forecasting errors. (Changes in the marketing mix may also be taken into account through managerial intervention.) Two approaches are available: (1) Combining subjective and objective forecasts: the forecaster makes an independent forecast of sales {Y^f, say) and this is then combined with


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the method-based forecast (K^p using a weighting scheme to produce a revised forecast, K^? = aiYg^ -f- a2Âff- Clemen's recent survey and bibliography [55] summarizes the issues. Few authors have argued that "combining" is easily applied, despite its impressive record in improving forecasting acctu:acy. The problem stems from a natural unwillingness on management's part to concede the part which an objective forecast iTiust play. Dalrymple's survey evidence[14] found that 38 per cent of sales forecasters combined forecasting sources, but most are likely to do this sequentially as described below in (2). (2) An alternative approach is where the objective forecast is adjusted by the manager. The revised forecast is than anchored to the original objective forecast. Various papers by Mathews and Diamantopoulos|56| have suggested that this leads to improved forecasting. It is a common presumption that subjective forecasts are more accurate than their objective alternative [5]. The usual reasons given for using objective methods are to minimize cost and the use of managerial time. However, substantial research has been carried out which examines the biases and resulting inaccuracy when forecasts are based solely on judgement!57, 58|. Management interventions can be improved by: (1) carefully prepared graphs and tables of the sales history[59]; (2) substantive additional information available; (3) feedback on its effectiveness being made available to the managerf60l.

Forecasting Systems The nature of the data, the relative accuracy of the different forecasting methods now available, and the additional sources of information that can be incorporated all have implications for the design of effective software. The user's skills and training must also be taken into account. In describing the context of FPI we noted that the typical user of the system would not be well trained, nor would there be much expert help on hand to overcome any problems encountered. As a consequence, such FPI systems should run effectively in "default" mode, encotiraging the user towards good practice at every choice that he or she faces. A number of computer software producers have developed forecasting systems for inventory control applications. In the United States these include American Software, Brown's forecasting module in his "Comprehensive Manufacturing Control" suite of programs, the modules of IBM's MAPICS and COPICS (or INFOREM for distribution systems). Focus Forecasting[611 and in the UK, Datasolve's MURCO system. In this section we will' 'design*' an ideal forecasting system, referring to these existing products for comparison. The Data (1) History. All available data should be available for analysis. A number of the products (American Software and Datasolve) have poorly chosen defaults limiting the data history readily available. Products should be permitted different start

dates. Random sampling of the data series should be carried out easily for use in an experimental module (described below). (2) Data smoothing. Data series should be adjusted easily for exceptional observations. Standard adjustments to take into account trading days should be available. Data series should be identifiable by date and value to remove problems arising from unusual series, e.g. new products and "dead" products. (3) Series identifiers. Series can be identified as "new product" (included in most software), "intermittent/lumpy" by which is meant a series which contains many zeros, "declining" (Datasolve) implying a negative trend. Different forecasting procedures can then be applied to these product categories. The system should allow the user to select by identifier and to change identification. (4) Forecasting dimensionality. As argued above, forecasts are required at both individual item level and in aggregate. In a distribution system the hierarchy may extend across a number of levels and contain subunits of analysis within

a level. For example, American Software models the hierarchy as a pyramid

and aggregation can take place across levels. However, certain products are more naturally categorized by a number of dimensions and forecasting carded out by totalling sales across the unwanted dimensions (Brown's LOGOL). In contrast, MAPICS does not contain the necessary options, thereby limiting the flow of information in the hierarchy. Forecasting Method Selection and On-line Application As this article has argued, the range of forecasting methods available should be at the heart of any FPI system. The choice of method requires the user to carry out an initial analysis of the data series which may be done when the system is implemented or is an option within the system that the user would activate at infrequent but regular intervals (the forecasting review). (1) Initial analysis. Summary statistics to describe the data series should be readily calculated. For example, to carry out the analysis presented earlier in this article, means, medians, standard deviations, autocorrelation coefficients, etc., were calculated for all data series under consideration and these statistics summarized further through aggregate statistics and graphs. None of the programs under review has these facilities. What is worse is that they cannot carry out the analysis on a single series. (2) Forecasting methods. Because of the characteristics of inventory data a number of forecasting methods should be considered before a single method is chosen for application to all series (until the next forecasting review). Such a requirement implies that the forecaster should have available an experimental module in which analysis can be carried out easily and an operational module in which the method(s) selected through the experiment are applied online. This is not available in any of the products. So far, attempts to automate method choice through an ' 'expert system'' type of approach have proved inadequate.


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^^ general, the programs under consideration here offer few alternative methods and are unclear as to the algorithms they choose to employ. This is important because the basic statistical principles underlying a method may be ' incorrect or poorly implemented. Focus Forecastmg|61|, for example, a method given prominence by APICS and BPICS, is apparently unable to identify some simple deterministic patterns in the data. Datasolve in its basic form rejects the possibility of a negative trend(62|. American Software's approach damps ^^ the trend when forecasting more than two periods ahead. IBM's MAPICS offers ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ "experiential roughing" (sic) which, despite its claims, has not been subject to the public scrutiny of Brown's exponential smoothing (described in detail by Brown|3| and used in his LOGOL programs), to which it undoubtedly owes some allegiance. In short, these programs usually offer a single unvalidated method, a variant of exponential smoothing, which the typical user will adopt in default mode. The effect can only be unnecessarily poor forecasts. Many of these software products (e.g. Focus) are poorly documented technically; they may have been shown to have limited performance characteristics in empirical tests (see Flores[63| on Focus or Gardner[33| on Trigg and Leach) (implemented in INFOREM); or they may not have been the subject of rigorous, impartial review at all. Therefore users must be particularly wary in that the algorithms employed in software systems may be subject to error, both statistical and computational.

]

(3) Method selection. In the experimental module, individual series or a group of series need to be analyzed online. The flexibility of a microcomputer would be helpful here although not completely necessary. The database should be easily split into three segments: initialization, estimation and validation. To use exponential smoothing as an example, the initial smoothed values need to be specified within the program. With the initial values calculated over the initialization period, the smoothing parameters need to be selected. As Gardner[33] points out, this must necessarily be done by evaluating forecast accuracy over the ' 'estimation'' data set to calculate best fit. A priori selection of the parameters is inadequate. Here again the reference programs are at best unclear as to how the parameters are selected and some seem to rely on the users or on a limited search routine. While computer limitations would previously have necessitated some compromise, there is little reason now to accept it. The validation data set is kept aside for further testing of the selected models (and parameters). As was argued earlier, the fact that the "optimal" parameters have been chosen on the "estimation" data has limited implications for performance in forecasting proper. Summary statistics of the forecasting accuracy of the different methods under review need to be provided. The standard deviation over the forecast lead time is also needed for inventory decisions and again the reference programs make no comment on its calculation, despite it being a crucial system parameter. (4) On-line application. With the method chosen, all the programs are on more solid ground. Key issues arise in the format of the management reports and the ability of users to examine and modify the forecasts of particular series.

Typical designflawsare to overload the user with information, however relevant. Use of on-screen highlighting and help screens, for example, could overcome the confusion. Filtering of outliers or exceptions must take place as well as identification of changes in the demand pattern. Fortunately our review has shown that the choice of monitoring techniques seems to be unimportant. The summary statistics included in the reference programs' item and product group reports are generally ill thought out with, for example, too much reliance being placed on MAD (Brown's LOGOL being the exception). ; The Extended Information Base (1) Forecasting hierarchy. Information on demand is often available from other parts of the hierarchy described above. In the initial data analysis and model building, the system should permit the user to identify market relationships in the hierarchy through regression analysis. Typically, the user will only wish to examine a limited number of relationships so it is quite practical for the analysis to be kept separate from the FPI system. However the aggregate forecasts are established and transmitted to the FPI system, the user may wish to constrain the aggregate of demand at the lower level to the upper-level forecast. Equally, bottom-up forecasting, whereby aggregate demand is best forecast as the sum of the individual product forecasts, may prove more effective. The initial analysis should be able to evaluate these constraints. What is at issue is not just the accuracy of the forecasting system but its plausibility — without the constraints imposed the system could all too easily produce inconsistent forecasts. • -. ^. Of the reference systems, American concerns itself with this issue and much of the database design is premissed on constrained forecasting. (2) Orders and the effects of the marketing mix. The reference programs pay little attention to the information available to many firms through orders, although Brown's system incorporates firm orders into thefinalforecast. The importance of this omission in the "ideal" system can only be judged by the frequency of occurrence. Promotional profiles, whereby the user estimates the intensity and the time phasing of the promotion, are considered, for example, by American. The demand data can then be normalized by removing the effect of the promotion. In the terminology adopted earlier Pf is estimated subjectively and used to calculate the base-line sales, B^. No attempt to evaluate this subjective approach is available. (3) Managerial information. Our earlier discussion showed that managerial information is a mixed blessing. While designing a system to permit managerial overrides is both necessary (to respond to out-of-control signalsfix)mmonitoring) and desirable in that it increases user acceptability, the danger is that bias may overwhelm the benefits from the increased range of information. First the user must be able to select the series needed for further analysis. The screen-based


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selection might be based on product name or product group, A-B-C analysis, 0^ ^''foi' analysis. Base-line forecasts from a range of methods including, of course, that already in use should then be available. Any interventions should then be added easily back into the forecast file. Improvements in judgement only come from feedback. The ideal system should offer feedback on the accuracy of the management interventions compared with the base system (as does American). Conclusions Forecasting for production and inventory control has been a neglected area of the forecasting and OR literature. Brown and Gardner being the exceptions. This has left to the software engineers the specification of the computer programs designed for FPI and the links to the various new manufacturing systems. This orientation is reflected in the reference programs considered above, where the manuals often concentrate on the intricacies of the dataprocessing and reporting systems without any serious attempt to justify the fundamentals of the forecasting approach and the stock-control parameters that should lie at the program's heart. As this article has demonstrated, recent forecasting research has shown that new methods offer the potential for substantial improvements in accuracy, that methods must be chosen to match the data characteristics, and that evaluation has to be carried out with considerable care. Achievable improvements in accuracy lead directly to worthwhile savings. Existing FPI systems do not yet include these new ideas. Users are therefore experiencing the results of unnecessarily large forecast errors — high stock levels or poor customer service. References 1. Silver, E.A. and Petersen, R., Decision Systems for Inventory Management and Production Planning, 2nd edition, Wiley, New York, 1985. 2. Vollmann, T.E., Berry, W.L. and Whybark, D.C., Manufacturing Planning and Control Systems, 2nd edition, Irwin, Homewood, IL, 1988. 3. Brown, R.G., Materials Management Systems: A Modular Library. Wiley, New York, 1977. 4. Plossl. G.W., Getting the Most from Forecasts, American Production and Inventory Control Society. Falls Church, VA, 1979. 5. Kenworthy, J.G., "Some Practical Hints on Sales Forecasting", Control. \bl. 15 No. 2, 1989, pp. 25-9. 6. Goddard, W.E., "Let's Scrap Forecasting". Modem Materials Handling. Vol. 39, September 1989. p. 39. 7. Price, D. and Sharp. J.. "A Comparison of the Performance of Different Univariate Forecasting Methods in a Model of Capacity Acquisition in UK Electricity Supply", International Journal of Forecasting. \o\ 2. 1986. pp. 333-48. 8. Adshead, N.S. and Price, D.H.R.. "Demand Forecasting and Cost Performance in a Model of a Real Manufacturing Unit'', International Journal of Production Research, Vol. 25,1987, pp. 1251-66. 9. Gardner. E.S. Jr, "Evaluating Forecast Performance in an Inventory Control System", Management Science, W. 36, 1990, pp. 490-9. 10. Lee, T.S., Adam. E.E. and Ebert, RJ.. 'An Evaluation of Forecast Error in Master Production Scheduling for Material Requirements Planning", Decision Sciences. \bl. 18, 1987. pp. 292-307.

11. Ment2er, J.T and Cox, J.E. Jr, "Familiarity, Application and Performance of Sales Forecasting Techniques", foumal of Forecasting, Vol. 3, 1984. pp. 27-36. 12. Baker. K.R., "Safety Stocks and Component Commonality", Journal of Operations Management, Vol. 6, 1985. pp. 13-22. 13. Schwarzkopf, A.B., Tersine. R.J. and Morris, J.S., "Top-down versus Bottom-up Forecasting", International Journal of Production Research, Vol. 26, 1988, pp. 183343. 14. Dalrymple, DJ., "Sales Forecasting Practices; Results from a United States Survey", International Journal of Forecasting, Vol. 3. 1987, pp. 379-91. 15. Fildes, R. and Hastings, R., The Organization and Improvement of Forecasting Procedutvs, Wbrking Paper, University of Lancaster, 1990. 16. Coutie. G.A.. Short-term Forecasting: ICI Monograph No. 2, O^vev and Bayd, Edinburgh. 1963. 17. Fildes, R.. "Evaluation of Aggregate and Individual Forecast Method Selection Rules", Management Science, Vol. 35, 1989, pp. 1056-65. 18. Fildes, R., "The Evaluation of Extrapolative Forecasting Models", Internationat Journal of Forecasting, Vol. 8, 1992 (forthcoming). 19. Stevens, C.F, "On the Variability of Demand for Families of Items", Operational Research Quarterly, Vol. 25, 1974. pp. 411-19. 20. Flores, B.E. and Whybark, D.C.. "Implementing Multiple Criteria ABC Analysis", yowraa/ of Operations Management, Vol. 7. 1987, pp. 79-85. 21. Cohen. M.A. and Emst. R., "Multi-item Classification and Generic Inventory Stock Control Policies", Production and Inventory Management Journal, Vol. 3. 1988, pp. 6-8. 22. Silver and Petersen(l] in contrast to Brownl3| continue to argue, somewhat implausibly, in favour of using the MAD approximation to the standard error on ground of ease of interpretation. A case could be made out for using MAD in that it is less sensitive to extreme observations and these should not be included in the calculation of the safety stock (Bretschneider, S., "Estimating Forecast Variance with Exponential Smoothing: Some New Results", International Journal of Forecasting, Vol. 2, 1986. pp. 349-55). 23. Makridakis, S. and Winkler. R.L., "Sampling Distributions of Post-sample Forecasting Errors", Applied Statistics, \bl. 38, 1989, pp. 331-42. 24. Yar, M. and Chatfield. C . "Prediction Intervals for the Holt-Winters Forecasting Procedure". International Journal of Forecasting, Vol. 6, 1990. pp. 127-37. 25. Chatfield. C , "Apples, Oranges and Mean Square Error", International Journal of Forecasting. Vol. 4. 1988, pp. 515-8. 26. Fildes, R. and Makridakis, S.. "Loss Functions and Forecasting", International Journal of Forecasting, Vol. 4, 1988, pp. 545-50. 27 Lee. T.S. and Adam. E.E. Jr, "Forecasting Error Evaluation in Material Requirements Planning (MRP) Production-inventory Systems". Management Science, y}o\.Z2,1986, pp. 1186-205. 28. Fortuin, L., "Five Popular Probability Density Functions; A Comparison in the Field of Stock Control Models", 7o«ma/ ofthe Operational Research Society, \bl. 31,1980, pp. 937-42. 29. Williams. W.W.. Peters, M.H. and Raiszadeh, M.E., "Time-dependent Demand in Requirements Planning: An Exploratory Assessment of the Effects of Serially Correlated Demand Sequences on Lot-Sizing Performance"/ourwa/o/O/)^rafto«5 A/an