BUS 2100 Module 4 Control Charts and a Seasonal Forecast Example

Heavy Loaves and Holiday Peaks: Control Charts and a Seasonal Forecast for a Bakery

Student Name

Bachelor of Science in Business Administration, Thomas Edison State University

BUS-2100: Quantitative Skills for Business

Mentor Name

Month Day, Year

Heavy Loaves and Holiday Peaks: Control Charts and a Seasonal Forecast for a Bakery

The Situation

A composite regional bakery in Wisconsin packs whole-wheat loaves labeled 800 grams. Underweight loaves risk complaints and regulatory trouble, while overweight loaves give away dough and raise costs. The operations manager also needs a quarterly sales forecast to plan flour purchases and staffing for the coming year. Wisniewski and Shafti (2020) treat quality control and forecasting together because both use past data to manage future operations, and both are covered in this module.

Part 1: Control Charts for Loaf Weight

Eight samples of five loaves were weighed over two days. For each sample, the mean and the range, the heaviest minus the lightest loaf, were calculated.

The grand mean of the sample means is 801.6 grams and the average range is 6.25 grams. For samples of five, the standard control chart factors are A2 = 0.577 for the means chart and D3 = 0 and D4 = 2.114 for the range chart (Montgomery, 2019). The means chart limits are therefore 801.6 plus or minus 0.577 x 6.25, or 798.0 to 805.2 grams, and the range chart's upper limit is 2.114 x 6.25, about 13.2 grams.

Every range falls within its limit, so the variation within samples is stable. Sample 6, however, has a mean of 810.8 grams, well above the upper limit. Investigation found that the divider's scale had been recalibrated incorrectly after cleaning that morning. Because the cause was found and corrected, sample 6 is removed and the limits recalculated from the remaining seven samples: a center line of 800.3 grams, limits of 796.6 to 803.9 grams, and a range limit of about 13.3 grams. The revised chart becomes the standard for routine monitoring, with a sample taken every two hours. Shewhart's idea of separating ordinary variation from variation with an assignable cause is exactly what the chart did here: it flagged a real problem that a glance at individual loaves might have missed.

Part 2: A Seasonal Forecast

Quarterly sales over three years, in thousands of loaves, show growth and a clear seasonal pattern, with the fourth quarter highest because of holiday demand.

Each seasonal index is the average, across the three years, of that quarter's sales divided by the year's quarterly mean, adjusted so the four indices sum to four. Dividing each quarter's sales by its index removes the seasonal pattern; a straight line fitted to the twelve deseasonalized values gives about 440.6 plus 4.64 per quarter, in thousands. Multiplying the trend projection for each quarter of year four by its seasonal index gives the forecast: about 456,000 loaves in the first quarter, 511,000 in the second, 487,000 in the third and 579,000 in the fourth, a total of about 2.03 million loaves, 3.4 percent above year three.

Applied to the three years of history, the model's fitted values have a mean absolute deviation of about 4,700 loaves and a mean absolute percentage error of about 1.0 percent. That fit is optimistic, since the model was estimated from the same data, so the bakery should compare the forecast with actual results each quarter. Hyndman and Athanasopoulos (2021) recommend judging forecasting methods on data not used to fit them for this reason.

Decisions

The bakery should adopt the revised control chart for loaf weight, add a scale check to the cleaning procedure and train line staff to stop and investigate whenever a sample mean falls outside the limits. For planning, it should order flour and schedule seasonal staff to the quarterly forecast, with the fourth quarter about 12 percent above an average quarter, and update the indices and trend each year.

Using the Tools Together

The two tools support each other. If loaf weights drift upward and go unnoticed, the bakery gives away dough on every loaf, and at the fourth-quarter forecast of about 579,000 loaves, 11 extra grams per loaf would amount to more than six metric tons of dough in a single quarter. Monitoring weight with the control chart therefore matters most in exactly the quarters the forecast says will be busiest.

Conclusion

The control chart caught an assignable cause that was giving away about 11 grams of dough per loaf, and the seasonal model gives a forecast precise enough for purchasing and staffing. Both tools turn routine data into decisions, which is the purpose of this module.

References

Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.

Montgomery, D. C. (2019). Introduction to statistical quality control (8th ed.). Wiley.

Wisniewski, M., & Shafti, F. (2020). Quantitative analysis for decision makers (7th ed.). Pearson.