Modeling Complex Seasonality Patterns

1. Modeling Complex Seasonality Patterns Paul J. Fields and Phillip Witt Brigham Young University Utah, USA

2. The Challenge Diagnose a Noisy Time Series Like This One …

3. Noisy Time Series: Seasonality?

4. The Problem • Potentially Multiple Underlying Processes • Fluctuating Processes Can Augment or Cancel Each Other → • Underlying Processes Can Be Disguised

5. Purposes • Identify the Underlying Processes • Forecast How Much to Produce to Maximize Profit Potential (Not Forecast of Demand) • Find Opportunities to Intervene to Change Demand Pattern Advantageously

6. Questions to Answer • What Processes Are Going On? • What Are Their Relative Contributions? • How to Use the Patterns?

7. Usefulness of Models “All Models Are Wrong, But Some Are Useful” George Box Usefulness is to Aid in Making Decisions with Desirable Results

8. Context • Daily Demand • Profit Maximizing Objective • Direct Costs = 1/3 Unit Price • Perishable Goods • No Carry-Over • No Salvage Value • No Lost Goodwill from Stock-Outs • No Shrinkage

9. Daily Weekly Bi-Weekly Monthly Bi-Monthly Quarterly Trimester Semi-Annual Annual Complex Seasonality Potential Seasonal Components

10. Poly-Trigonometric Model y = b0 + b1 t + b2 SIN θt + b3 COS θt Level Trend Seasonal ∑ b I SIN θ kt + ∑ b J COS θ kt Complex Seasonality

11. Objective Function • Maximize Operating Income = Revenue – Direct Costs • Demand > Prediction (Sell All Produced) Profit = Prediction - 1/3 Prediction • Demand < Prediction (Sell What Demanded) Profit = Demand – 1/3 Prediction

12. Diagnostic Modeling • Estimate Coefficients with Non-Linear Optimization • Calculate Marginal Contribution to Operating Income from Each Component • Identify ‘Useful’ Terms via Pareto Principle – 80-20 Rule • Re-optimize Coefficients

13. Contributions of Seasonal Components Weekly Trimester Bi-Weekly Bi-Monthly

14. Useful Seasonal Components

15. Final Model

16. Starting with Trimester Seasonality

17. Adding Bi-Monthly Seasonality

18. Adding Bi-Weekly Seasonality

19. Adding Weekly Seasonality

20. Effect Sizes • Weekly: 80% Tue to Sat • Trimester: 12% Apr, Aug, Dec • Bi-Weekly: 5% “Pay Day” • Bi-Monthly: 3% Shifts Tri Peaks

21. Forecasting Model • In the Absence of Marketing Interventions … • Add Smoothing Term for Highest Contributing Seasonal Component • y Adj = y CS + αε 7α Opt = .28

22. With Smoothing of Weekly Errors

23. Compared to ‘Crystal Ball Perfect’ Profit Potential • Diagnostic Model: 92.7% • Forecasting Model: 93.3% • Smoothing Effect was Second Largest Contribution: 38% Weekly Effect and 2.5x Trimester Effect

24. Results • Operating Decisions to Maximize Operating Income: Daily Production Batch Sizes • Marketing Decisions to Minimize Fluctuations: Specials at Weekly Trough on Tuesday Advertising Campaigns Starting at Trimester Peaks April 16, August 5 and December 10

25. Conclusions • Effective for Diagnosing Complex Seasonality • Identify Underlying Seasonal Processes Not Clearly Seen Otherwise • Intuitively Understandable and Easy to Implement

26. Conclusions • With Asymmetric Fluctuations – Higher Order Seasonal Terms Could Be Included and the ‘Useful’ Terms Identified Similarly • Could Calculate Approximate Effectiveness of Marketing Interventions

27. Conclusions • Useful for Operating Decisions for Production and Inventory and Managing the Present • Useful for Marketing Decisions for Intervening in the Process and Making the Future