To start with, we have plotted the data on the number of dealers and the number of consumers versus the number of the month (Fig.1). To show the correlation between the number of dealers and the number of consumers we have plotted both on the same graph using the two different appropriately scaled Y axes. The well defined seasonal variation of all curves can be observed. The number of consumers is approximately ten times larger than the number of dealers throughout the observation. The averaged over 12 months number of consumers showed a 34% growth from year 1 to year 2 while the number of dealers has increased by only 11%. The average number of dealers outside has decreased by 2% in year 2 and the number of dealers inside has increased approximately 29%. It is worth noticing that in the year 1 the market was the most active from May to November while in year 2 this period spanned April to November.
One objection to the data collection should be made here. The data has been averaged over a month while one month may contain 4 or 5 full weekends when the market activity is expected to increase. Thus it was more natural to use the weekly averages.
The correlation coefficients between the studied numbers are shown in Table 1. All values are strongly correlated; that is usually a sign of some common background reason for them to vary. In our case it may be the seasonal variation in the demand and the weather conditions.
The linear trend fitted to the number of consumers is shown by the yellow line in Fig.1. The trend predicts that the number of consumers will exceed 20,000 in month 41. Obviously this is not the case because the linear trend does not account for the seasonal variations. The multiplicative model (yellow curve in Fig.1) predicts that the number of consumers will exceed 20,000 in month 29 which is May of year 3.
The average monthly seasonal effects for outside dealers are shown in Table 2. The multiplicative model (purple curve in Fig.1) predicts that the number of dealers outside will reach 800 in month 29 (May, year 3), 900 in month 41 (May, year 4), and 100 in month 53 (May, year 5).
The multiplicative model predicts that the number of dealers inside (blue curve in Fig.1) will exceed 600 in month 29 (May, year 3), so that it is worth expanding the inside stalls during the quiet time between high seasons of year 2 and year 3.
Presented forecasts are useful for the Management Committee of the fleamarket to estimate the number of staff required to serve the fleamarket and to make the plans for the market expansion. It has been suggested to use the multiplicative model as a forecast tool, which is obviously more relevant than using the linear trend because the strong seasonal variations are observed. At the same time, the multiplicative model is not so obviously better than the additive model as shown below.
The software products MS Excel and MS Word have been used to prepare this report. The two products are well matched, which allows tables and graphs to be copied and pasted without making extra efforts on the picture or table editing. MS Excel allowed building of the multiplicative models keeping all calculations in the spreadsheet that makes it easy to spot and correct arising errors. Plotting graphs is straightforward in MS Excel which allows assessing the data graphically. Simple data organisation allowed building the additive model promptly by introducing the minor corrections to the multiplicative model.
In this work, the data collected during the two years has been used to make the forecast for the three years ahead. The linear trend and seasonal variations have been calculated as described by L. Swift (1997). Obviously, predicted numbers will not coincide with the numbers that will be actually observed and it is important to estimate the possible difference between the forecast and the reality. Knowing the error will make it possible to build decision trees as described by T. Lucey (1996). For example, the decision node can be building the new inside stalls after the high season of year 2 or delaying it for a year. Estimating of the probability that the number of dealers inside will exceed 600 in year 3 will help to decide whether it is less harmful to loose dealers who will not fit to inside stalls if the number of dealers inside will exceed 600 or to maintain the empty extra stalls for an extra year if the number of dealers inside will remain below 600. Although the estimate of the error is out of the scope of the current work, we note that the predicted number of dealers inside in year 3 is marginally above 600 so that the correct decision is not obvious. A number of decisions should be made on the number of cleaning staff and the amount of car parking space necessary for dealers and consumers.
As mentioned above, the choice has been arbitrarily made to the favour of the multiplicative model instead of the additive model. The first one is better when the variation increases with the trend and the second is better when the variations are constant. In the case with the fleamarket, the growth can be restricted by the number of habitats in the hosting town (city) which is not considered at all.
Also, the two year observation does not look enough at making predictions about further development. Indeed, the increase of the number of dealers and consumers in Year 2 may be a random variation or may be due to the better weather conditions (observe that the period of activity of the fleamarket increased by one month in year 2). Thus it is not obvious that the numbers in year 3 will increase at all. In all the textbooks where the time series modelling is described, the cases with several periods are considered but not the case with the data available on the two years only.
The final illustration of the reliability of our forecast is the prediction of the time instant for the dealers outside to reach 800, 900, and 1,000 using the multiplicative or the additive model. The multiplicative model (Fig.1) gives months 29, 41 and 53 correspondingly while the additive model (Fig.2) gives months 30, 56 and somewhere beyond 60 correspondingly.
As mentioned above, the available data is insufficient to reliably forecast the number of dealers and the number of consumers in the fleamarket.
Figure 1. Numbers of dealers and consumers. Data and the multiplicative model.
Figure 2. Numbers of dealers and consumers. Data and the additive model.