Workload Assessment (Forecasting)
3.3 Forecasting Methods for Time-Series Analysis
3.3.4 Forecasting with a Seasonal Cycle
If sudden change occurs, then the response will be sluggish irrespective of whether change is genuine or noise is present when α is small. On the other hand, if α is close to 1, there will be substantial adjustment for any error. With a large value of α, the bona fide changes will be reflected immediately in the new average.
It is difficult to say beforehand which value of α will serve the purpose best.
Different values of α starting with a small value may be chosen to make a forecast.
The actual observed data and the forecast using different values of α may be plot- ted on a graph and by examination that value of α which tracks the actual demand best is chosen. Alternatively, for different values of α, the errors—mean absolute deviation (MAD) and mean squared error—may be calculated and that value of α which gives a lower error may be chosen. See appropriate details in the section ahead on forecast errors.
As in the case of MAs, if there is a fundamental systems change, then prior his- tory of patterns is not a help, and ES should be applied as if to a new startup system for the first time.
ES has been found to be very effective in a variety of situations. Many forecast- ing and control systems employ ES because it works better than the older methods of MAs and WMAs. It has proved effective for diverse applications. Fighter aircraft use ES to aim their guns at moving targets. In effect, they forecast the location of enemy jets during flying missions. This application shows how fast the ES method can track a consistent, but dynamically changing pattern. Manufacturers use ES to forecast demand levels, which experience the same kind of nonrandom but volatile shifts from time to time. In such cases, the recent past has the most information about the near future. ES can catch these shifts and make rapid adjustments to inventory levels. Other manufacturers and service organizations use it because it requires less computational work and is readily understood.
or monthly basis. Table 3.4 shows the historical forecast technique applied to data that exhibit a seasonal cycle. No modification has been made for an overall change in annual sales, which is the basis of the historical forecast.
When cycles are stable, they can provide insights that are very important for P/OM tactical planning. When they work, historical cycles allow P/OM to excel at capacity planning and production scheduling for mature manufactured products and services. The historical forecast is not appropriate for a new product introduction unless there is similarity to some other product that is already on the market.
Historical seasonal cycles can become apparent by studying the calendar. In the case of P/OM, there are also other cyclical regularities and patterns that lend themselves to capacity planning, production scheduling, and purchasing decisions for both manufacturers and service systems.
If the time-series pattern remains fixed, but the demand level has increased overall, then a base series modification can be used. Assume that in 2013 the quar- terly demands were 10, 30, 20, and 40. This gives a yearly demand of 100 units.
Further, assume that in 2014 the yearly demand is expected to increase to 120 units. Then the quarterly forecasts would be adjusted:
Table 3.4 An Example of Seasonal Cycles (Historical Forecast with Seasonal Data)
Month Actual Sales
Last Year Forecast of Sales Next Year
January 1500 1500
February 1600 1600
March 1800 1800
April 2000 2000
May 2300 2300
June 2500 2500
July 2350 2350
August 2100 2100
September 1850 1850
October 1650 1650
November 1550 1550
December 1400 1400
Forecast for 2014
Quarter 1: 120 (10/100) = 12 Quarter 2: 120 (30/100) = 36 Quarter 3: 120 (20/100) = 24 Quarter 4: 120 (40/100) = 48
The adjusted quarterly demands total 120 units. The cyclical patterns are matched. In the year 2015, assuming the pattern continues and the base level increases to 150, the time series would be 15, 45, 30, and 60 with the sum of 150.
The above reasoning can be extended to those situations where the demand data exist for several years with similar demand patterns.
Consider the data in Table 3.5 that gives the quarterly demand for last 4 years.
There is a clear pattern of demand variation among the four quarters. Based on these data, we can forecast the demand for the four quarters of year 5. Assume that the annual demand is expected to be, 2800, in year 5. This demand has to be esti- mated using other techniques. The model discussed in this section will divide the yearly demand into quarterly demands based on the past quarterly demand pattern.
The following five-step process will be used to obtain the quarterly demand.
Step 1: Find average quarterly demand for each year.
The total demand for each year is divided by four (number of quarters in each year) to obtain the average demand in each year. Table 3.5 gives the average demand for each quarter.
Step 2: Compute seasonal index (SI) for each quarter for each year.
The seasonal index for a quarter for a given year is obtained by dividing the demand in that quarter by the average quarterly demand for that year. For Table 3.5 Demand
Quarter Year 1 Year 2 Year 3 Year 4
Fall 2530 2690 2790 2860
Winter 2300 2420 2410 2600
Spring 1900 2000 2105 2175
Summer 1510 1775 1875 1945
Average 2060 2221 2295 2395
Calculation = (1510 + 1900 + 2300 + 2530)/4)
= (1775 + 2000 + 2420 + 2690)/4)
= (1875 + 2105 + 2410 + 2790)/4)
= (1945 + 2175 + 2600 + 2860)/4)
example, the seasonal index for the winter quarter of year 2 is 1.090 which is obtained by dividing 2420 (demand for the winter quarter in year 2) by 2221 (average quarterly demand for year 2). The seasonal indices for each quarter of each year are given in Table 3.6. The formulas to calculate the seasonal indices are also given in this table.
Step 3: Calculate the average SI for each quarter.
The next step is to find the average seasonal index for each quarter which is simply the average of the seasonal indices calculated in step 2. Table 3.7 gives the average seasonal index for each quarter. For example,
Average SI for the summer quarter =0 790. = 0 733 0 799. + . +0.817+0 812.
44 .
Step 4: Calculate the average quarterly demand for next year.
In this step, we find the average quarterly demand for the next year. The yearly demand estimate for next year is 2800. Therefore, the average quarterly demand = 2800/4 = 700. This step is shown in Table 3.8.
Step 5: Forecast demand for quarters of next year.
The quarterly demand for each quarter of next year is obtained by multiply- ing the average demand by the SI for each quarter. Table 3.9 gives the forecast Table 3.6 Step 2: Compute Seasonal Index (SI) for Each Quarter
for Each Year
Quarter Year
1
Calculation for SI Year 1
Year 2
Calculation for SI Year 2
Year 3
Calculation for SI Year 3
Year 4
Calculation for SI Year 4
Fall 1.228 = 2530/2060 1.211 = 2690/2221 1.216 = 2790/2295 1.194 = 2860/2395 Winter 1.117 = 2300/2060 1.090 = 2420/2221 1.050 = 2410/2295 1.086 = 2600/2395 Spring 0.922 = 1900/2060 0.900 = 2000/2221 0.917 = 2105/2295 0.908 = 2175/2395 Summer 0.733 = 1510/2060 0.799 = 1775/2221 0.817 = 1875/2295 0.812 = 1945/2395
Table 3.7 Step 3: Calculate the Average SI for Each Quarter
Quarter Average SI Calculation
Fall 1.212 = (1.228 + 1.211 +1.216 + 1.194) /4 Winter 1.086 = (1.117 + 1.09 +1.05 + 1.086) /4 Spring 0.912 = (0.922 + 0.9 +0.917 + 0.908) /4 Summer 0.790 = (0.733 + 0.799 +0.817 + 0.812) /4
of the quarterly demand of next year. For example, forecast for the Spring quar- ter = 638 = 700 * 0.912, where 700 is the average quarterly demand and 0.912 is the average SI for the Spring quarter.
In the above example, we have used year as the time period in which the seasons are represented by quarters. The seasons could have been represented by months if the time series shows variations by month. In some instances, the week could be the time period and the days could be the “seasons” if the demand fluctuates on a daily basis. Restaurants are a good example of where the demand fluctuates on a daily basis. Even hour of the day can be the “season” if the demand fluctuates hourly in a given day, for example, the power generation requirement in a power utility or the number of calls in a telephone company vary on an hourly basis.