Modeling Complex Modeling Complex Seasonality Seasonality

PatternsPatterns

Paul J. Fields and Phillip Paul J. Fields and Phillip WittWitt

Brigham Young UniversityBrigham Young University

Utah, USAUtah, USA

The ChallengeThe Challenge

Diagnose a Diagnose a Noisy Time SeriesNoisy Time Series Like This One …Like This One …

Noisy Time Series: Noisy Time Series: Seasonality?Seasonality?

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The ProblemThe Problem Potentially Multiple Underlying Potentially Multiple Underlying

ProcessesProcesses

Fluctuating Processes Can Fluctuating Processes Can Augment or Cancel Each Other Augment or Cancel Each Other →→

Underlying Processes Can Be Underlying Processes Can Be DisguisedDisguised

PurposesPurposes

Identify the Underlying ProcessesIdentify the Underlying Processes

Forecast How Much to Produce to Forecast How Much to Produce to Maximize Profit Potential Maximize Profit Potential (Not Forecast of (Not Forecast of

Demand)Demand)

Find Opportunities to Intervene to Find Opportunities to Intervene to Change Demand Pattern Change Demand Pattern AdvantageouslyAdvantageously

Questions to AnswerQuestions to Answer

What Processes Are Going On?What Processes Are Going On?

What Are Their Relative What Are Their Relative Contributions?Contributions?

How to Use the Patterns? How to Use the Patterns?

Usefulness of ModelsUsefulness of Models

““All Models Are Wrong, But Some All Models Are Wrong, But Some Are Useful”Are Useful”

George BoxGeorge Box

Usefulness is to Aid in Making Usefulness is to Aid in Making Decisions with Desirable ResultsDecisions with Desirable Results

ContextContext Daily DemandDaily Demand

Profit Maximizing ObjectiveProfit Maximizing Objective Direct Costs = 1/3 Unit PriceDirect Costs = 1/3 Unit Price

Perishable GoodsPerishable Goods No Carry-OverNo Carry-Over No Salvage ValueNo Salvage Value No Lost Goodwill from Stock-OutsNo Lost Goodwill from Stock-Outs No ShrinkageNo Shrinkage

Potential Seasonal Potential Seasonal ComponentsComponents

DailyDaily WeeklyWeekly Bi-WeeklyBi-Weekly MonthlyMonthly Bi-MonthlyBi-Monthly

QuarterlyQuarterly TrimesterTrimester Semi-AnnualSemi-Annual AnnualAnnual Complex Complex

SeasonalitySeasonality

Poly-Trigonometric ModelPoly-Trigonometric Model

y = by = b00 + b + b11 t + b t + b22 SIN SIN θθ t + bt + b33 COS COS θθ tt

Level TrendLevel Trend SeasonalSeasonal

∑∑ b b II SIN SIN θθ k k t + t + ∑∑ b b J J COS COS θθ k k tt

Complex SeasonalityComplex Seasonality

Objective FunctionObjective Function

Maximize Operating Income = Revenue – Maximize Operating Income = Revenue – Direct CostsDirect Costs

Demand > Prediction (Sell All Produced)Demand > Prediction (Sell All Produced)

Profit = Prediction - 1/3 PredictionProfit = Prediction - 1/3 Prediction

Demand < Prediction (Sell What Demanded)Demand < Prediction (Sell What Demanded)

Profit = Demand – 1/3 Prediction Profit = Demand – 1/3 Prediction

Diagnostic Diagnostic ModelingModeling

Estimate Coefficients with Non-Linear Estimate Coefficients with Non-Linear OptimizationOptimization

Calculate Marginal Contribution to Operating Calculate Marginal Contribution to Operating Income from Each Component Income from Each Component

Identify ‘Useful’ Terms via Pareto Principle –Identify ‘Useful’ Terms via Pareto Principle –80-20 Rule80-20 Rule

Re-optimize CoefficientsRe-optimize Coefficients

Contributions of Seasonal Contributions of Seasonal ComponentsComponents

0.0000

0.5000

1.0000

1.5000

2.0000

2.5000

β4 β14 β7 β10 β9 β15 β5 β12 β1 β6 β13 β11 β8 β16 β17 β18 β19 β2 β3

Weekly

Trimester

Bi-Monthly

Bi-Weekly

Useful Seasonal Useful Seasonal ComponentsComponents

Weekly Sin Trimester Sin Bi-Weekly Cos Bi-Monthly Sin

β4 β14 β7 β10 β9

1.9849 0.3278 0.1627 0.0973 0.0791

X X X X

69.2% 11.4% 5.7% 3.4% 2.8%

69.2% 80.6% 86.3% 89.6% 92.4%

Final ModelFinal Model

Weekly Sin Trimester Sin Bi-Weekly Cos Bi-Monthly Sin

β4 β14 β7 β10 β1

2.0729 0.3124 0.1348 0.0744 0.0015

X X X X

79.8% 12.0% 5.2% 2.9% 0.1%

79.8% 91.8% 97.0% 99.9% 99.9%

Starting with Trimester Starting with Trimester SeasonalitySeasonality

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Adding Bi-Monthly Adding Bi-Monthly SeasonalitySeasonality

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Adding Bi-Weekly Adding Bi-Weekly SeasonalitySeasonality

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Adding Weekly SeasonalityAdding Weekly Seasonality

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Effect SizesEffect Sizes

Weekly:Weekly: 80%80% Tue to SatTue to Sat

Trimester:Trimester: 12%12% Apr, Aug, DecApr, Aug, Dec

Bi-Weekly:Bi-Weekly: 5%5% “Pay Day”“Pay Day”

Bi-Monthly: Bi-Monthly: 3%3% Shifts Tri Peaks Shifts Tri Peaks

Forecasting ModelForecasting Model

In the Absence of Marketing In the Absence of Marketing Interventions …Interventions …

Add Smoothing Term for Highest Add Smoothing Term for Highest Contributing Seasonal ComponentContributing Seasonal Component

yy Adj Adj = y = y CS CS + + αα εε 7 7 αα Opt Opt = .28 = .28

With Smoothing of Weekly With Smoothing of Weekly ErrorsErrors

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Compared to ‘Crystal Ball Compared to ‘Crystal Ball Perfect’Perfect’

Profit PotentialProfit Potential Diagnostic Model:Diagnostic Model: 92.7%92.7% Forecasting Model:Forecasting Model: 93.3%93.3%

Smoothing Effect was Second Largest Smoothing Effect was Second Largest Contribution:Contribution:

38% Weekly Effect and38% Weekly Effect and

2.5x Trimester Effect 2.5x Trimester Effect

ResultsResults Operating Decisions to Maximize Operating Operating Decisions to Maximize Operating

Income:Income:

Daily Production Batch SizesDaily Production Batch Sizes

Marketing Decisions to Minimize Fluctuations:Marketing Decisions to Minimize Fluctuations:

Specials at Weekly Trough on TuesdaySpecials at Weekly Trough on Tuesday

Advertising Campaigns Starting at Trimester Advertising Campaigns Starting at Trimester Peaks April 16, August 5 and December 10Peaks April 16, August 5 and December 10

ConclusionsConclusions

Effective for Diagnosing Complex Effective for Diagnosing Complex SeasonalitySeasonality

Identify Underlying Seasonal Identify Underlying Seasonal Processes Not Clearly Seen Processes Not Clearly Seen OtherwiseOtherwise

Intuitively Understandable and Easy Intuitively Understandable and Easy to Implementto Implement

ConclusionsConclusions

With Asymmetric Fluctuations –With Asymmetric Fluctuations –

Higher Order Seasonal Terms Could Higher Order Seasonal Terms Could Be Included and the ‘Useful’ Terms Be Included and the ‘Useful’ Terms Identified SimilarlyIdentified Similarly

Could Calculate Approximate Could Calculate Approximate Effectiveness of Marketing Effectiveness of Marketing InterventionsInterventions

ConclusionsConclusions

Useful for Operating Decisions Useful for Operating Decisions for Production and Inventory and for Production and Inventory and

Managing the PresentManaging the Present

Useful for Marketing Decisions Useful for Marketing Decisions for Intervening in the Process for Intervening in the Process and and Making the FutureMaking the Future